ABSTRACT
Title of dissertation: THE LOW TEMPERATURE
MAGNETOCONDUCTANCE OF
DELTA-DOPED SILICON
Dan Francis Sullivan, Doctor of Philosophy, 2005
Dissertation directed by: Dr. Bruce E. Kane
Department of Physics
The low temperature electronic transport properties of delta-doped silicon are
established here, in an e?ort to determine the technological limitations facing the
construction of semiconductor wafers with monolayer doping profiles. Both
phosphorous (Si:P) and boron (Si:B) delta-doped silicon wafers prepared by
molecular beam epitaxy have been investigated. Devices fabricated from samples
of these wafers were cooled to approximately 130 mK in a dilution refrigerator,
where four wire resistance measurements were performed in the presence of
magnetic fields both perpendicular and parallel to the plane of the devices. The
data obtained was interpreted using the theory of weak localization.
For Si:P samples, the e?ective thickness of the delta-doped region can be
inferred by comparing the magnetoconductance signal in parallel magnetic fields to
that of perpendicular fields. Using this technique on several di?erent samples
which were annealed at 850
?
C, an estimate of the di?usivity of phosphorus in
silicon at this temperature is established. In addition, it is shown that the primary
mechanism for the loss of quantum mechanical phase coherence in these samples at
low temperatures is the electron-electron interaction, and provide evidence for
lattice defects enhancing this dephasing. For Si:B samples, the spin-orbit
interaction is shown to dominate the dephasing, and the temperature dependence
of this interaction is found to be substantially di?erent from that observed in Si:P
devices.
Based on these results, I conclude that specific epitaxial growth techniques
must be followed if monolayer doping profiles are ever to be achieved in silicon. In
addition, because this weak localization technique for measuring the thickness of
delta-doped regions is highly sensitive and non-destructive, it may ultimately
prove to be the most e?ective method yet established for probing ultra-thin doping
profiles. Finally, several theoretical desiderata are established, which would enable
a more accurate interpretation of experimental data.
THE LOW TEMPERATURE MAGNETOCONDUCTANCE OF DELTA-DOPED
SILICON
by
Dan Francis Sullivan
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2005
Advisory Committee:
Professor Frederick C. Wellstood, Chair
Doctor Bruce E. Kane
Professor Chia-Hung Yang
Professor James R. Anderson
Adjunct Professor Keith C. Schwab
Professor Thomas D. Cohen
c?Copyright by
Dan Francis Sullivan
2005
To Amanda and Rosalee.
ii
Acknowledgments
It is unfortunately impossible to acknowledge every individual who has
contributed in one way or another to this dissertation. Accordingly the people who
are mentioned below are those who have figured most prominently in these e?orts.
I would first like to thank my dissertation advisor, Dr. Bruce Kane, for
supporting me financially through the five plus years I worked for him. I also
appreciate him giving me the opportunity to work on this project essentially
independently, and for the many conversations we had which helped to deepen my
understanding of delta-doping in silicon.
Dr. Phillip E. Thompson was a great help in preparing the epitaxially grown
delta-doped wafers which were investigated in this dissertation. This research
would not have been possible without his assistance, and I am grateful for the
service he provided.
Professor Richard Webb, one of the pioneers in low temperature experiments
involving weak localization, graciously provided extensive experimental
consultations. His insights and recommendations have been incorporated
throughout this work. Although he was originally slated to be the committee chair
for this dissertation, an unfortunate set of circumstances prevented him from being
able to do so. His presence is sorely missed, and I wish him the best of luck in his
new post at the University of South Carolina.
I was blessed to have an outstanding group of coworkers at the Laboratory for
iii
Physical Sciences (LPS), where this research was carried out. These people include
Carlos ?el chingon? Sanchez (whose electrical engineering background proved to be
an invaluable resource), the ?nanomechanics boys? Matt LaHaye and Olivier
?Buu-yah? Buu (for numerous political conversations and excursions to the Town
Hall), Kenton Brown (for helium loans and the use of his L
A
T
E
Xbook in writing
this document), Keith Schwab (for numerous conversations), and Luyan Sun. I
especially thank Carlos, Matt, and Olivier for making the trek out to Utah for
Amanda?s and my wedding.
I would also like to acknowledge several members of the sta? and management
at LPS for their aid. The skilled machining and design aid provided by J.B.
Dottellis helped me to fix a broken dilution refrigerator and rewire it so that my
experiments could be performed more e?ciently. Greg Latini and Mark Thornton
were always available to help with computer and software related issues, not to
mention interesting lunchtime conversations. Geo? Burdge, the director of LPS
when I arrived, provided many an adventure on Stampede, and introduced me to
Bob Putnam and Better Mousetrap, both much-needed diversions from the
graduate school grind. Also assisting me with escape from certain ?imperial
entanglements? were Ron Andre, Bernadette Preston, and Marc Manheimer. I
appreciate their common sense, loyalty, and unity.
Finally, I would like to thank those members of my family who have been a
constant source of support to me throughout this process. My Mom deserves
special praise, for everything she has done for me in my life, especially teaching me
how to work. I also thank my Dad, who introduced me to science through
iv
mathematics, and who will no doubt be pleased with the continuing evolution of
the ?new and improved version? he so often cites. My wife Amanda has truly been
phenomenal with her patience and support throughout the long and arduous
struggle that is graduate school. It?s finally over, honey. Lastly, and perhaps most
of all, I thank my beautiful daughter Rosalee, for being so impossibly perfect. You
are the light of my life.
v
Contents
1 Introduction 1
1.1 Background ................................ 1
1.2 Definition ................................. 2
1.3 ThicknessDetermination......................... 3
1.4 Motivation................................. 7
1.5 Prospectus................................. 9
2Theory 11
2.1 BandStructure .............................. 11
2.1.1 GeneralConsiderationsforSi .................. 12
2.1.2 DensityFunctionalTheory.................... 18
2.1.3 ?-LayerBandStructure...................... 20
2.1.4 Perspective ............................ 36
2.2 WeakLocalization ............................ 40
2.2.1 HistoricalDevelopment...................... 40
2.2.2 Quantum Conductivity . . . . . . . . . . . . . . . . . . . . . . 42
2.2.3 Calculation of ??(H)....................... 47
vi
2.2.4 T Phenomenology ........................ 52
2.3 BerryPhase................................ 55
2.3.1 2DEGBerryPhase........................ 56
3 Fabrication 63
3.1 ?-layerRequirements........................... 63
3.1.1 GeneralConsiderations...................... 64
3.1.2 PreviousWork .......................... 65
3.2 ?-dopedWaferPreparation........................ 72
3.3 DeviceFabrication ............................ 79
3.4 DeviceTesting............................... 87
4Measurement 89
4.1 Background ................................ 89
4.2 The Two-Dimensional Conductivity Tensor . . . . . . . . . . . . . . . 93
4.2.1 HallBarDevices ......................... 95
4.2.2 vanderPauwDevices ...................... 97
4.3 Measurement ............................... 99
4.3.1 Instrumentation..........................101
4.3.2 Grounding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.3.3 ExperimentalProcedure .....................109
4.4 MeasurementErrors ...........................120
4.4.1 E ..................................121
4.4.2 H .................................126
vii
4.4.3 J ..................................130
4.4.4 T..................................132
4.4.5 ? ..................................133
5 Results 135
5.1 Si:P ?-Layers ...............................135
5.1.1 Di?usivityMeasurements.....................136
5.1.2 DensityDependence .......................177
5.1.3 FinalRemarksandSummary ..................185
5.2 Si:B ?-layers................................192
5.2.1 SamplePreparation........................193
5.2.2 Theory...............................195
5.2.3 Results...............................196
6 Conclusion 203
6.1 FutureDirections.............................203
6.2 Summary .................................205
6.3 Conclusion.................................206
A Elementary Transport Theory 208
A.1 Classical Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . 208
A.2 HallE?ect.................................210
A.3 MeanFreePath..............................212
A.4 Mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
viii
A.5 DensityofStates .............................215
A.5.1 IsotropicMass...........................216
A.5.2 AnisotropicMass .........................218
B Data Analysis Routines 221
B.1 ??(H
?
) ..................................221
B.1.1 WL1p3...............................222
B.1.2 deltaGWL1p3...........................223
B.1.3 perpfitWL1p3...........................224
B.1.4 deltaLp ..............................225
B.2 vanderPauwTheorem..........................226
B.3 SIMS T ..................................228
C Dilution Refrigerator Modifications 231
C.1 Wiring...................................232
C.2 Repair...................................236
ix
Chapter 1
Introduction
This introductory chapter provides a comprehensive framework for
understanding the data and theoretical analysis presented throughout this
dissertation. A brief outline of the material I will cover is included at end of this
chapter.
1.1 Background
The importance of semiconductor physics in everyday life, and that of silicon,
in particular, can hardly be overstated. Since the invention of the transistor by
Bardeen, Brattain, and Shockley in 1948, semiconductor technology has steadily
advanced, with transistors simultaneously becoming both faster and smaller.
These devices have found applications in an endless list of areas, including basic
scientific research, personal computers, and all aspects of communication and
information technology. Indeed, the myriad of roles the internet now plays in our
1
lives would be impossible without the computational power provided by
microprocessors, with billions of transistors integrated onto a single silicon wafer.
State-of-the-art transistors now (2005) have dimensions of less than 100 nm, and
the trend towards ever smaller devices, with a concurrent doubling of the
transistor density on a single wafer every two years (?Moore?s law?), is expected to
continue for at least another decade.
There are, however, physical limits to this scaling. As the dimensions of these
devices become smaller and smaller, the atomic nature of matter can no longer be
ignored, and quantum mechanical phenomena, such as charging energies and
tunneling events, become manifest. Ultimately, the ability to control doping
profiles in semiconductors with atomic precision, will determine how small
functional devices, such as transistors, can be built. The technique known as
delta-doping of semiconductors represents the frontier in this e?ort [1], and is the
subject of research in this dissertation.
1.2 Definition
A single delta-doped region in a semiconductor is referred to as a delta-layer
(?-layer), and is simply an extremely thin layer of impurity atoms in an otherwise
pure crystal. Ideally, the dopants occupy a single atomic layer (monolayer) in the
lattice. A schematic representation of this situation is shown in Figure 1.1. Figure
1.1 (a) shows that the impurity atoms (dopants) are confined to a narrow, planar
region of the crystal, ?sandwiched? between two layers nominally devoid of
2
dopants. Figure 1.1 (b) shows a plot of the doping density, n(z), as a function of
vertical position in the crystal. We see that n(z) = 0 everywhere except at the
position z
0
, hence the name ?-layer implies a ?Dirac delta? doping distribution, ?(z
-z
0
). Mathematically, this ideal delta-doping profile n(z) is represented by
n(z) = n
2D
?(z?z
0
) (1.1)
where n
2D
is the two-dimensional density of atoms in the ?-layer. The relevant
consideration for the work in this thesis (?-layers in Si) is that an actual physical
?-layer, composed of real atoms, must possess a finite width, or have a thickness,
T , associated with it. Although the physical limit of T is a monolayer, all the
samples investigated in this work, and thus far reported in the literature, far
exceed this lower bound. Nonetheless, very thin (T?5nm)?-layers have been
produced in a variety of semiconductors, and the ultimate limits in preparing these
structures are still being explored.
1.3 Thickness Determination
One of the central questions I will address in this thesis is to quantify precisely
how two-dimensional a given ?-doped sample is (that is, measuring T ). The way I
approach this problem is through a weak localization analysis of low temperature,
four-wire conductance measurements (?), performed on ?-doped samples in the
presence of a magnetic field (H); hence the term magnetoconductance, ?(H).
Both perpendicular (?(H
?
)) and parallel (?(H
bardbl
)) magnetic field configurations
play an important role in this T measurement, as does as the temperature, T. The
3
(a) (b)
Figure 1.1: (a) Schematic representation of a ?-doped wafer, and (b), a plot of the
doping density versus vertical position in the crystal for an ideal ?-layer. The doping
density is zero everywhere except at the position z
0
.
theory of weak localization will be discussed in Chapter 2, but for now it su?ces
to say that such a measurement of T amounts to a comparison of the e?ective
cross sectional areas subtended by the ?-layer in H
?
and H
bardbl
. To illustrate this
idea, consider the right circular cylinder of radius R and height T ,sketchedin
Figure 1.2.
When viewed from ?above? (corresponding to the H
?
configuration), the
observed cross sectional area is simply that of a circle: A
?
= ?R
2
. If the cylinder
is rotated and viewed from the side (corresponding to H
bardbl
), the cross section
becomes rectangular, with area A
bardbl
=2RT . Solving these two equations for T
gives T =A
bardbl
radicalbig
?/4A
?
. Clearly, an accurate estimate of T requires a similar
accuracy in determining the areas A
?
and A
bardbl
. The discussion of weak localization
4
(a) (b)
Figure 1.2: A weak localization measurement of the ?-layer thickness,T , corresponds
to a comparison of two e?ective cross sectional areas in perpendicular and parallel
magnetic fields. In (a) (corresponding to H
?
), the appropriate e?ective area is A
?
= ?R
2
. In (b) (corresponding to H
bardbl
), the e?ective area is A
bardbl
=2RT .Fromthese
relations one obtains T =A
bardbl
radicalbig
?/4A
?
.
in Chapter 2 will make clear how this is done, as well as taking into account some
of the details neglected in this simplified example.
To see how these considerations are involved in ?(H) measurements, typical
weak localization spectra for a phosphorous-doped silicon (Si:P) ?-layer sample are
shown in Figure 1.3 (these measurements were taken at the base temperature of
the dilution refrigerator used to cool the samples, T similarequal 130 mK). The upper curve
corresponds to H
?
data, the lower to H
bardbl
data. The di?erence in the response of
the conductance to H between the two configurations is obviously well-resolved. In
general, ?(H
?
)isalwaysgreaterthan?(H
bardbl
), and the more of a contrast that
exists between the two configurations, the thinner is the ?-layer in the device.
Again, by exploiting the asymmetry between ?(H
?
)and?(H
bardbl
) and applying the
5
Figure 1.3: Typical weak localization data (points), with theoretical fits (solid lines).
The upper curve corresponds to ?(H
?
) data, the lower to ?(H
bardbl
) data. The greater
the di?erence between ?(H
?
)and?(H
bardbl
), the thinner is T .
appropriate theory to estimate the relevant cross sectional areas, a measurement of
T is obtained.
Now that the reader has some feel for what will be presented in this
dissertation, some justification for carrying out this work will be discussed.
6
1.4 Motivation
This dissertation research was originally motivated by an interest in quantum
computing (QC), specifically the proposal of my advisor, Bruce Kane, [4]. The
idea was to use individual P donors in crystalline Si (c-Si) as quantum bits
(?qubits?). The requirements and implications of QC have been elaborated upon
extensively elsewhere, and will not be discussed in any detail here, except to say
that such an achievement would render certain formidable computations (e.g.
factoring large numbers [5]) tractable.
Before QC can be achieved, however, many fundamental and practical
questions must be addressed. One is how accurately can P donors can be placed in
a Si crystal? This issue turns out to be of high importance for QC [6]. Various
groups have made e?orts to investigate the issue of precise P donor placement in
c-Si through techniques such as scanning tunneling microscope positioning [7] and
ion implantation [9], but perhaps the most readily available system for study are
?-layers grown by molecular beam epitaxy (MBE), as used in this thesis. Although
other techniques for constructing ?-layers have been demonstrated (such as
chemical vapor deposition (CVD) [8]), MBE o?ers many advantages, which will be
presented in Chapter 3.
QC is far from the only application for ?-layer structures, however. They are
widely used in charge coupled devices (CCDs) as highly e?cient detectors for
photons and low-energy fluxes in particle physics [10]. Another application is in
resonant interband tunnel diodes (RITDs), which dissipate little power and are
7
therefore desirable for use in integrated circuits [12]. A possible future use of
?-layers is in the construction of single electron transistors (SETs), which have a
host of potential applications, including use as a current standard in metrology
[13] and ultra-sensitive charge detection [14].
Ialsonotethat?-layers are of interest from a purely theoretical perspective.
The electronic system associated with a ?-distribution is inherently
two-dimensional, and some of the most striking discoveries in solid state research
over the past 20 years stem from the physics of associated with two-dimensional
electron gases (2DEGs). One of the most prominent is von Klitzing?s discovery in
1980 [15] of the quantum Hall e?ect (QHE) in an Si MOSFET (Metal Oxide
Semiconductor Field E?ect Transistor). This e?ect now forms the basis for a
resistance standard [16]. Another important discovery was of that the fractional
quantum Hall e?ect (FQHE), first observed by St?ormer and Tsui in 1984 [17] and
explained theoretically by Laughlin in 1985 [18]. In the FQHE, charge carriers
with a fractional electronic charge (e.g. 1/3, 2/3, 1/5, etc.) are observed in very
high mobility (typically GaAs) heterostructures at low temperatures and high
magnetic fields. ?-layers are a member of this exotic 2DEG family, but display
carrier mobilities many orders of magnitude lower than their ballistic transport,
high mobility MOSFET and GaAs cousins, and will consequently display rather
di?erent physical characteristics.
8
1.5 Prospectus
Chapter 2 will present those theoretical considerations most pertinent to this
work. Specifically, the band structure of ?-layers in Si will be discussed, along with
the impact this structure has on magnetoconductance measurements. Since this
thesis hinges on interpreting such data using the theory of weak localization, a
systematic treatment of this phenomenon is covered. Lastly, the subtle e?ect of a
Berry phase and its influence on interpreting T using weak localization will be
explored.
Chapter 3 is concerned with the samples used in my experiments. I include a
brief discussion of previous e?orts in preparing Si:P ?-layers, the MBE conditions
necessary for constructing a ?-doped wafer, and the appropriate methods used in
device fabrication. The presentation of the device fabrication process is rather
?hands-on?, and is meant to serve as a reference recipe to be followed in order to
produce high quality devices for measurement.
Chapter 4 is perhaps the most important of the entire dissertation, and deals
with the measurement techniques and equipment which were used to obtain my
weak localization data. Since this is an experimental thesis, Chapter 4 will be
somewhat more detailed in its exposition than the other chapters, and explains the
precise conditions under which my magnetoconductance measurements were
performed.
In Chapter 5 I present the results and analysis of my experiments, along with
a discussion of the various systematic errors involved in the measurements. One
9
set of measurements is concerned with annealing several di?erent devices
constructed from the same Si:P ?-doped wafer, extracting T from each device as
described in Chapter 2, and using this information to determine the di?usion
coe?cient of Si:P at T = 850
?
C. Other experiments involve weak localization T
measurements performed on devices constructed from different wafers with
varying carrier densities, n. A brief account of measurements on a device
constructed from a boron-doped silicon (Si:B) ?-layer is also presented, showing
the dramatic influence of spin-orbit scattering has on ?(H).
Chapter 6 summarizes my results and suggests new directions for future
investigation.
The appendices cover ancillary points relevant my work, including a discussion
of elementary transport theory, the programming code I used in data analysis, and
the modifications and improvements I made to the dilution refrigerator I used.
The works of the numerous authors who helped to lay the foundations for this
research are listed in the bibliography.
10
Chapter 2
Theory
This chapter presents three theoretical aspects crucial to understanding the
physics of delta-doped silicon. The first section is on the electronic band structure
of a ?-layer system in Si, and how this structure influences the interpretation of
magnetoconductance data. The second is an overview of the theory of weak
localization and its use in probing ?-layer structures. Lastly, the phenomenon of a
Berry or geometric phase is discussed, and I describe how this phase manifested
itself in the magnetoconductance measurements I performed.
2.1 Band Structure
This section will describe the energy characteristics of electrons in Si,
culminating with a theory particular to Si:P ?-layers.
11
2.1.1 General Considerations for Si
I will outline the main results for the band structure of Si here to establish
specific numbers and notions pertinent to my work. These results are reasonably
well-known, and are taken primarily from [19, 20, 21].
The term ?band structure? refers to specifying the energy of an electron (or
hole) in a given crystalline potential as a function of either wavevector k, position
r,orboth.TheE(k) representation is often referred to as a dispersion relation,
and in essence quantifies how the kinetic energy of an electron in a crystal depends
on its direction of propagation. Since thorough discussions of this topic are in
standard texts on solid state physics [2, 3], I merely summarize here.
Qualitatively, when in the presence of a periodic crystalline potential,
solutions of the Schr?odinger equation (H? =E?) have a range of energies over
which no solutions exist. This is the origin of the electrical characteristics of solids,
and, for the case of interest here, produces the energy (or band) gap, E
g
,of
crystalline Si (c-Si). To appreciate how this comes about, consider an isolated Si
atom, with electronic configuration 1s
2
2s
2
2p
6
3s
2
3p
2
(i.e. four valence electrons
outside a closed Ne shell). Combining many such Si atoms into their lowest energy
state produces a covalently bonded crystal, identical in structure to that of C
atoms in diamond. In this crystal, each atom occupies the center of a tetrahedron,
with nearest neighbor atoms positioned at the vertices of the tetrahedron, as
depicted in Figure 2.1 (a). This is equivalent to a lattice constructed from two
face-centered cubic (fcc) lattices (Figure 2.1 (b)), one copy of which is displaced
12
(a) (b)
Figure 2.1: Crystal structure of Si. (a) tetrahedral structure, where each Si atom
occupies the center of a tetrahedron. (b) fcc lattice. The c-Si lattice consists of two
fcc lattices: one at the at the origin, the other 1/4 of the distance along the main
body diagonal.
from the other 1/4 of the distance along the main cubic diagonal. To provide the
appropriate length scale, x-ray crystallography has determined that the lattice
constant for c-Si is similarequal 0.543 nm.
A generic semiconducting energy band diagram (in position space) is shown
schematically in Figure 2.2. The states below the gap comprise the valence band
(E
v
), those above the gap the conduction band (E
c
), and states with energies
forbidden by the Schr?odinger equation lie in E
g
. In order for an electron (hole) to
be transported through the crystal under the application of an arbitrarily small
electric field E, it must have an energy greater (less) than E
c
(E
v
). Of course, c-Si
is an indirect gap semiconductor, meaning that the maximum of E
v
and the
minimum of E
c
occur at di?erent points of k-space. Therefore the actual E(k)
13
Figure 2.2: One-dimensional semiconductor energy band diagram (energy vs. posi-
tion, E(x)). States with energies forbidden by the Schr?odinger equation lie in the
energy gap, which for Si at T = 0 is E
g
=1.17eV.Thek-space diagram [20] is
anisotropic,andE
g
varies with k.
band diagram for Si is much more complicated than that of Figure 2.2. Numerous
measurements have determined that the maximum of E
v
occurs at the center of
the Brillouin zone (the ??? point), whereas the minimum of E
c
occurs along the
< 100 > directions of k-space, approximately 3/4 of the distance from the center
of the Brillouin zone to its boundary (the ??? points).
Due to the symmetry of the lattice, there is a six?fold degeneracy associated
with the minimum of E
c
in Si, as depicted in Figure 2.3. The shape of these six
conduction band valleys (or pockets) can be approximated by an ellipsoid whose
eccentricity is specified by the e?ective mass tensor of the electrons. Cyclotron
resonance measurements have established [22] that two e?ective masses are
14
Figure 2.3: Si energy ellipsoids showing the six-fold conduction band degeneracy.
The curvature near the ends of the ellipsoids corresponds to the transverse e?ective
mass m
?
= 0.190m
e
, while the curvature near the middle of the ellipsoids corre-
sponds to the longitudinal e?ective mass, m
bardbl
= 0.916m
e
appropriate for characterizing these Si ellipsoids. Taking the center of a single
pocket to be the k-space origin, E(k) takes the form
E(k)=
planckover2pi1
2
k
2
x
2m
bardbl
+
planckover2pi1
2
(k
2
y
+ k
2
z
)
2m
?
(2.1)
with analogous expressions holding for the other symmetry-related pockets of
Figure 2.3. Since E ? (1/m), a lighter mass corresponds to a more sharply curved
portion of the energy ellipsoid. The results of [22] are m
bardbl
= 0.916m
e
and m
?
=
0.190m
e
,wherem
e
is the free electron mass (m
e
similarequal 9.11x10
?31
kg). As the labels
suggest, m
bardbl
corresponds to a k parallel to the major (long) axis of the ellipsoid,
while m
?
corresponds to k perpendicular to the major axis.
Pure Si, being an intrinsic semiconductor, becomes an insulator at su?ciently
low temperatures, with E
g
= 1.17 eV at T = 0. In order to obtain a Si sample
15
which conducts even at T = 0, the crystal must be doped to a level greater than
the metal-insulator transition (MIT) for the particular impurity element involved,
defining the critical density, n
c
. This in turn establishes a limit between the
two-dimensional dopant density n in a ?-layer and its thickness T .Ifn< n
c
T ,the
sample will not conduct at low temperatures, and cannot be investigated using
weak localization techniques. The two Si dopants studied here, P and B, have n
c
=
3.7x10
18
/cm
3
and 4.0x10
18
/cm
3
, respectively, while their crystalline ionization
energies are 0.045 eV below E
c
for Si:P, and 0.045 eV above E
v
for Si:B (since B is
a hole donor). Because these energies are near E
c
and E
v
respectively, and small
compared with most other elements, P and B are known as ?shallow donors? in Si.
I note here an interesting relationship which exists between n
c
and the Si:P
Bohr radius, a
B
, which I shall refer to again in Chapter 5. First, at low
temperatures it is legitimate to consider an isolated P atom in Si as an ?e?ective
Hydrogen atom? [23], since P has one more valence electron than Si, and the
remainder of the crystal is insulating. Furthermore, the other Si atoms in the
crystal act as a renormalized vacuum with dielectric constant ?.ForSi,? = 11.9
epsilon1
0
,whereepsilon1
0
is the permittivity of vacuum (in SI units epsilon1
0
= 8.85x10
?12
C
2
/N?m
2
).
Relating the kinetic energy (KE) to the potential energy (PE) of this system
through the virial theorem (i.e. KE = PE/2) [24], the following relationship is
obtained for a
B
:
a
B
=
4??planckover2pi1
2
me
2
. (2.2)
Using m
?
for m, I obtain a
B
similarequal 3.23 nm, quite large compared to the Si lattice
16
Figure 2.4: Sketch of the relationship between the conductivity of a sample and
doping density. If the doping level is such that the mean donor d spacing is less
than 2a
B
, the ?Bohr diameter?, the electron clouds (open circles) have considerable
overlap, and the sample will conduct at T = 0 (i.e. it is metallic). If d > 2a
B
,the
sample will insulate, and when d = 2a
B
, the MIT occurs.
spacing of 0.543 nm. The use of m
?
is only an approximation, and obviously
equation (2.2) implies that a lighter electron will on average lie a greater distance
from the nucleus.
The mean donor spacing at the MIT is given by taking the cube root of n
c
for
Si:P, for which I obtain d similarequal 6.47 nm. Therefore the relationship d = 2a
B
is nearly
satisfied, to which a simple physical interpretation can be ascribed. If each P
donor is thought of as a ?conducting ball? of diameter 2a
B
, then Si:P samples will
become insulating when these balls no longer touch. Thus, from an atomic point of
view, the better a sample conducts, the greater is the electronic overlap bewtween
donors. A depiction of this situation is shown in Figure 2.4. All of the samples
studied in this work were metallic.
17
2.1.2 Density Functional Theory
The task of calculating the band structure for an arbitrarily doped,
three-dimensional Si sample from first principles is, in general, intractable.
Nevertheless, accurate numerical techniques have been developed to overcome this
problem. For crystalline samples the most important result is Bloch?s theorem,
which states that the appropriate electronic wave functions can be expressed as
the product of a plane wave and a periodic function, whose spatial period matches
that of the crystal: ?
k
(r)=e
ik?r
?
n
(k,r)(?
n
(k,r) is the lattice periodic function).
The label n is the band index, and simply arranges the bands in order of increasing
energy. Thus determining the band structure of a given crystal reduces to finding
the functions ?
n
(k,r). Many techniques have been developed which are in
excellent agreement with experimental results for the energy gap, magnitude of
carrier e?ective masses, etc., including Hartree-Fock calculations, density
functional theory (DFT), and Monte Carlo and Green?s function methods. I
present a brief outline of DFT here. The other methods are discussed in [20] and
the refernces therein.
DFT, first introduced by Hohenburg and Kohn in 1964 [25], asserts that the
ground state energy of a many-body (N-electron) wave function can be
transformed into N one ?electron equations in which the electrons are subjected
to an e?ective local potential, V
eff
(r). Since the properties of a homogeneous
electron gas are well-known over a wide range of densities [26], characteristics of
inhomogeneous systems can be calculated by ?integrating over? a given
18
distribution, using the V
eff
(r) appropriate for a homogeneous gas. This DFT
approach is known as the ?local density approximation? (LDA), and requires that
the density of electrons per unit volume, n(r), be a su?ciently slowly-varying
function of position. Precisely what constitutes ?su?ciently slowly-varying? is a
problem-specific question, which is addressed in [25].
Once the potential of interest has been specified, DFT requires the
self ?consistent solution of two coupled partial di?erential equations. These are
the Schr?odinger equation (H?
i
=E
i
?
i
) for the subband wave functions and
energies, and Poisson?s equation (?
2
V(r)=-?(r)/?), relating the electrostatic
potential V(r) to the charge density ? and dielectric constant ?.
To proceed, I first note Hohenberg and Kohn?s result [25] for the ground-state
energy of an inhomogeneous, interacting electron gas in a static potential, v(r):
E = e
integraldisplay
v(r)n(r)dr +
parenleftbigg
e
2
8??
parenrightbiggintegraldisplayintegraldisplay
n(r)n(r
prime
)drdr
prime
|r?r
prime
|
+ G[n]. (2.3)
In this equation G[n] is a universal functional of the density, G[n] = T[n] + E
xc
[n],
where T[n] is the kinetic energy and E
xc
[n] the exchange-correlation energy for a
system of non?interacting electrons. In the LDA, E
xc
[n]=
integraltext
n(r)epsilon1
xc
(n(r))dr,
where epsilon1
xc
is the exchange-correlation energy per electron,givenn(r).
In this LDA limit, the ith electron?s Schr?odinger equation can be cast in the
form [27]
{
?planckover2pi1
2
?
2
2m
+[?(r)+?
xc
(n(r))]}?
i
(r)=E
i
?
i
(r). (2.4)
Thus we have for the e?ective potential V
eff
(r)=?(r)+?
xc
(n(r)) (i.e. an
19
?e?ective? Poisson equation). The two contributions to V
eff
(r)aregivenby
?(r)=ev(r)+
parenleftbigg
e
2
4??
parenrightbiggintegraldisplay
n(r
prime
)dr
prime
|r?r
prime
|
(2.5)
and
?
xc
=
d
dn
(nepsilon1
xc
(n(r))). (2.6)
Finally, n(r) is given by the sum of the contributions to the density from the
individual electrons:
n(r)=
N
summationdisplay
i=1
|?
i
(r)|
2
. (2.7)
The pieces are now in place to self-consistently calculate a crystal?s band
structure. The typical procedure is that some initial n(r) is assumed, and ? and
?
xc
are calculated according to equations (2.5) and (2.6). Placing these quantities
in the Schr?odinger equation (2.4), the ?
i
?s are determined. These wave functions
are then used in (2.7) to obtain a new estimate for n(r). By iterating this
procedure many times, a solution is (hopefully) converged to rapidly.
2.1.3 ?-Layer Band Structure
I now present a calculation for the T = 0 band structure of ?-doped Si:P. The
theory associated with this problem is discussed in Chapters 1 and 2 of [1]. A
more general numerical calculation for Si:P ?-layers which I shall summarize
subsequently has been performed in [31].
The approach I take in this calculation is to assume a planar doping
distribution, and find the associated electronic potential energy. I then substitute
this potential into the Schr?odinger equation, and solve it for the energy eigenvalues
20
Figure 2.5: Planar charge distribution for modeling delta-doped Si:P. ? is the charge
density per unit area.
E
i
and subband wave functions ?
i
. Finally, I determine the density of states, ?(E),
of the system, and use ?(E) to find the Fermi energy, E
F
, as a function of doping
density by demanding charge neutrality.
Energy Eigenstates
As discussed in [28], the wavefunction for a general 2DEG can be
approximated by the product of a plane wave (e
ik?x
)lyingintheplanez=0,and
a subband wavefunction, ?
i
(z), perpendicular to the plane. This section
establishes the nature of the ?
i
.
Referring to Figure 2.5, I assume that a planar charge distribution, with
charge density per unit area ?, lies in the plane z = 0. Such a distribution acts as
21
a ?quantum well?, providing a confinement potential for the donor electrons. I
further assume that ? is proportional to the doping density n, and take ? = en for
the remainder of this calculation (e is the electron charge, e = 1.602x10
?19
C). A
more general approach would be to take ? = ?en, and examine the dependence of
the results on ?.
The electric field E produced by ? is determined by applying Gauss?s law in
the standard ?pillbox? form to this system, which yields a constant field in the
+/-?z direction:
E =
?
2?
?z. (2.8)
Such a constant field leads to an electrostatic potential which is linear in z.
Therefore the potential energy, V(z), of an electron a distance +/- z from the
planez=0is
V(z) = eE|z| =
e?
2?
|z|. (2.9)
The use of the absolute value of z in equation (2.9) reflects the fact that the energy
of an electron in this potential is a symmetric function of z: E(z) = E(-z).
Given the potential energy defined by (2.9), the appropriate one-dimensional
Schr?odinger equation for subband wave function ?
i
(z) with energy eigenvalue E
i
becomes
parenleftbigg
?planckover2pi1
2
2m
z
parenrightbigg
d
2
?
i
dz
2
+
parenleftbigg
e?
2?
parenrightbigg
|z|?
i
= E
i
?
i
. (2.10)
To solve equation (2.10), I note that the di?erential operator d
2
/dz
2
- |z| has been
studied since the 19th century, and the solutions to
d
2
f
dz
2
?|z|f = ??f (2.11)
22
are the so-called Airy functions, Ai(z) and Bi(z), with eigenvalues ? [32]. An
obvious physical boundary condition which the ?
i
(z) must satisfy is that ?
i
(z) ?
0as|z|??, which excludes the Bi(z) as possible wave function candidates. A
second boundary condition is imposed on the z = 0 derivative of the ?
i
(z) by the
triangular symmetry of the potential (2.9). If the eigenvalues of (2.11) are indexed
in increasing order, then the even index eigenvalues (0,2,4,...) are given by the
negatives of the zeros of the derivative of Ai(z), while the odd index eigenvalues
(1,3,5,...) correspond to the zeros of the Ai(z) itself [32]. Similarly, the even or odd
parity of a particular ?
i
(z) is determined by the parity of its index, with the even
valued eigenfunctions satisfying d?
i
/dz = 0 at z = 0.
To cast equation (2.10) in the form of (2.11), I rescale z according to x = Az,
and substitute back into (2.10) to obtain:
d
2
?
i
dx
2
?
parenleftbigg
em
z
?
?planckover2pi1
2
A
3
parenrightbigg
|x|?
i
= ?
parenleftbigg
2m
z
E
i
A
2
planckover2pi1
2
parenrightbigg
?
i
. (2.12)
Choosing A = (em
z
?/?planckover2pi1
2
)
1/3
, the left hand side of equation (2.12) is identical in
form to that of (2.11), and therefore the right hand sides also equal each other.
Thus the energy eigenvalues for the subband wave functions are given by
E
i
= ?
i
parenleftbigg
e?planckover2pi1
?
?
8m
z
parenrightbigg
2/3
= ?
i
parenleftbigg
e
2
planckover2pi1n
?
?
8m
z
parenrightbigg
2/3
. (2.13)
Since electrons in Si have two possible values for m
z
(m
z
=m
?
and m
z
=m
bardbl
), a
discrete spectrum of states exist for each of these masses. The complete energy
spectrum for this system is shown plotted in Figure 2.6. In Table 2.1, the 10
lowest eigenvalues of equation (2.11) are listed [29], as well as the associated
(rescaled) subband energies for either mass. Since E ? n
2/3
, all of the energies in
23
Figure 2.6: Energy spectrum of states corresponding to m
?
= 0.19m
e
and m
bardbl
=
0.916m
e
masses for Si. The energy of the various states have been rescaled such
that E
0
=1,whereE
0
is the lowest available energy of the system (corresponding
to ? = 1.0187 and m = m
bardbl
in equation (2.13)).
Figure 2.6 and Table 2.1 have been scaled relative to E
0
, the lowest energy
subband (i.e. E
0
? 1). From equation (2.13), it is clear that E
0
has m
z
=m
bardbl
, ? =
1.0187, while the next lowest subband has m
z
=m
?
, ? = 1.0187, the next lowest
m
z
=m
bardbl
, ? = 2.3381, etc.
A plot of the probability density |?
i
(z)|
2
for i = 0,1,2,3 is shown in Figure 2.7.
Each ?
i
(z) satisfies
integraltext
?
??
|?
2
i
|(z)dz = 1, and for each increment in the index, an
additional ?lobe? appears in the wave function profile. The boundary conditions
require that the origin be shifted for each ?
i
(z), i.e. ?
0
(z) ? Ai(z-?
0
), ?
1
(z) ?
Ai(z-?
1
), ?
2
(z) ? Ai(z-?
2
), etc [32].
24
Table 2.1: Table showing eigenvalues of equation (2.11) and corresponding (scaled)
energy states for Si masses.
index ? E(m
bardbl
) E(m
?
)
0 1.0187 1.00 1.68
1 2.3381 2.30 3.86
2 3.2482 3.19 5.36
3 4.0879 4.01 6.75
4 4.8201 4.73 7.95
5 5.5206 5.42 9.11
6 6.1633 6.05 10.17
7 6.7867 6.66 11.20
8 7.3722 7.24 12.16
9 7.9441 7.80 13.11
25
Figure 2.7: Probability densities |?
i
(z)|
2
of the subband wavefunctions in ?-doped
Si:P. As the band index increases, the wave functions become more extended. The
?
i
have been o?set from each other vertically.
26
Density of States
In this section I calculate the density of states, ?(E), for an Si:P ?-doped
system. The motivation here is to understand the interplay between the
three-dimensional Si energy ellipsoids of Figure 2.3, and the two-dimensional
doping plane of Figure 2.5. Appendix A discusses the physical meaning of ?(E)
(which has dimensions of 1/(energy)(area)), and calculates ?(E) for isotropic
(circular dispersion relation) and anisotropic (elliptical dispersion relation)
two-dimensional electron systems.
As shown in Appendix A, ?
1
(E) for an isotropic, two-dimensional, single
subband electron system, with e?ective mass m
?
, is a constant, given by
?
1
(E)=
m
?
?planckover2pi1
2
. (2.14)
For an elliptical dispersion relation in two dimensions, two masses, m
1
and m
2
,are
required to specify ?
2
(E):
?
2
(E)=
?
m
1
m
2
?planckover2pi1
2
. (2.15)
The two-fold spin degeneracy of the electrons is included in both (2.14) and (2.15).
Figure 2.8 shows why the two relations (2.14) and (2.15) are necessary for
modeling Si:P band structure. This figure is essentially the result of projecting the
ellipsoids of Figure 2.3 onto the doping plane of Figure 2.5. The cross section of
the ellipsoids perpendicular to the plane is circular, corresponding to ?
1
(E), while
the cross section of the in-plane ellipsoids is elliptical, corresponding to ?
2
(E).
Combining these two contributons, and ignoring, for the moment, the quantized
27
Figure 2.8: The two-dimensional density of states for Si (100). The energy ellipsoids
of Figure 2.3 have been projected into the k
z
= 0 plane, superimposing Si band
structure on the doping distribution of Figure 2.5. Since equation (2.10) is concerned
with motion perpendicular to this plane, the m
bardbl
subbands correspond to the circular
area(s), the m
?
subbands to the ellipses.
28
nature of the energy states in this system, ?(E) for ?-doped Si:P is given by:
?(E)=
2m
?
?planckover2pi1
2
+
4
?
m
?
m
bardbl
?planckover2pi1
2
= ?
bardbl
+ ?
?
. (2.16)
As suggested by the second equality, the first term in (2.16) corresponds to
electrons with an m
bardbl
mass, the second to the m
?
mass. The prefactors of each
reflect the degeneracies of the ellipses in Figure 2.8. I remark here that a
counterintuitive distinction exists between the transport mass (m
t
)andthedensity
of states mass (m
?
). For m
t
=m
bardbl
,m
?
=m
?
, while for m
t
=m
?
,m
?
=
?
m
?
m
bardbl
.
To arrive at the final expression I shall use for ?(E) in Si:P ?-layers, I now take
into account the discrete nature of the energy states calculated in the previous
section. By plotting the energies in Table 2.1 in ascending order on a line (i.e.
turning Figure 2.6 ?on it?s side?), it is (hopefully) clear that a given subband
becomes occupied only if the most energetic electrons in the system (i.e. those at
the Fermi energy, E
F
) exceed the subband?s energy eigenvalue. The lowest energy
subband (E
0
in Table 2.1) corresponds to m = m
bardbl
, the second lowest (E
1
= 1.68)
to m = m
?
, the next two lowest (E
2
=2.30andE
3
= 3.19) to m = m
bardbl
,etc.Thus
for E
0
? E
F
? E
1
, only a single subband is occupied, and ?(E) = ?
bardbl
.ForE
1
? E
F
? E
2
, two subbands are occupied, one for each mass, and ?(E) = ?
bardbl
+ ?
?
.ForE
2
? E
F
? E
3
, three subbands are occupied (two for ?
bardbl
,onefor?
?
)and?(E) = 2?
bardbl
+ ?
?
. Continuing in this way, one can build up the ?staircase? of the density of
states for Si:P ?-layers, shown in Figure 2.9 (the vertical axis is scaled such that ?
bardbl
= 1). Comparing the two terms in equation (2.16), we see that for ?
bardbl
=1,?
?
=
4.39, and consequently it is easy to determine which steps in Figure 2.9 correspond
29
Figure 2.9: Staircase density of states ?(E) for ?-doped Si:P. The short vertical steps
correspond to states with m = m
bardbl
, large vertical steps correspond to states with m
=m
?
.
30
to m
?
or m
bardbl
states. The states with short (unit) steps correspond to m
bardbl
,those
with large steps to m
?
states.
With the step-like nature of ?(E) in Si:P ?-layers established, the final form I
shall use in this dissertation is:
?(E)=
summationdisplay
i
?
i
?(E
F
?E
i
). (2.17)
Here ?(x) is the Heaviside step function, defined as ?(x) = 0 for x < 0and?(x)
=1forx? 0. ?
i
is either ?
bardbl
or ?
?
, depending on whether E
i
corresponds to an m
bardbl
or an m
?
state.
Fermi Energy
The final aspect of this calculation will be to determine the Fermi energy, E
F
,
of an Si:P ?-layer for arbitrary doping density, n.
For a single subband system, the following relation holds between n and ?(E)
[2, 3]:
n =
integraldisplay
E
max
E
min
?(E)f(E)dE. (2.18)
f(E) is the Fermi-Dirac distribution function. The integral (2.18) gives the number
of electrons per unit area with energies between E
min
(the lowest allowed energy in
the band) and E
max
. Since I am considering T = 0 in this computation, f(E) = 1
for E ? E
F
, and f(E) = 0 for E > E
F
. Furthermore, having seen that ?(E) is
constant for a 2DEG with a single subband, (2.18) becomes
n = ?(E
max
?E
min
)=?(E
F
?E
min
) (2.19)
since the integral is cut-o? for E
max
> E
F
.
31
I now use (2.19) to determine the electron density at which the first subband
is fully occupied. By setting ? = ?
bardbl
,E
min
=E
0
and E
F
=E
1
in (2.19), and using
equation (2.13) for the energies, I find that the first subband is filled when n =
3.39x10
12
/cm
2
. Thus for densities greater than this, at least two subbands will be
occupied. All of the samples I investigated in this work had densities greater than
this value.
An analogous computation can be carried out for the second subband (i.e. for
what n are the first two subbands filled?). In this case, n is given by:
n = ?
bardbl
(E
2
?E
0
)+?
?
(E
2
?E
1
). (2.20)
Substituting in the appropriate quantities I obtain n = 6.46x10
14
/cm
2
,which
exceeds the highest density sample I investigated (n similarequal 1.5x10
14
/cm
2
) by roughly a
factor of 4. Thus I conclude that all of the samples I measured in this work had
two subbands occupied: one with m
z
=m
bardbl
, the other with m
z
=m
?
.
A relevant consideration here are the physical limits imposed on n. Obviously,
n can be made arbitrarily small (zero doping limit), but in order to investigate a
sample using weak localization it must have n ? n
c
= 2.39x10
12
/cm
2
(corresponding to the two-dimensional MIT for Si:P). There is also an upper limit
to n: it can be no greater a monolayer Si (100) surface (1 ML = 6.78x10
14
/cm
2
).
In other words, for n > 1 ML, the doping distribution is no longer ??-like? since
the impurity atoms must begin to inhabit other lattice planes. Therefore, for truly
planar Si:P doping distributions, a maximum of three subbands are occupied at T
=0.
32
The results of this calculation for 10
11
/cm
2
? n ? 10
15
/cm
2
are displayed in
Figure 2.10. Plotted are the two lowest energy subbands for the m
bardbl
and m
?
masses, E
F
, and the densities of the samples I have measured (labeled A, B, C, D).
E
F
was found by specifying n, calculating the necessary E
i
from (2.13),
substituting these into (2.19) or (2.20), and then solving the resulting expression
for E
F
. This ensures that the system obeys charge neutrality. Also labeled are the
densities corresponding to the MIT and a ML doping level. According to this plot,
all of the samples I measured had two subbands occupied.
Calculational Limitations
To illustrate some theoretical limitations of my calculational approach, I now
compare my results for the band structure of ?-doped Si:P to those reported by
Qian et al. in [31].
First, it should be clear that the potential energy approximation I used (2.9)
cannot be correct, since V(z) increases indefinitely with |z|. In reality, if one
assumes charge neutrality, the electric field of the donor nuclei must eventually be
?screened? by the conduction electrons. In other words, su?ciently far from the
?-layer, V(z) should be flat, corresponding to zero electric field.
Accounting for this ?band-bending? (and other factors) is done in [31]. These
authors use a more general Schr?odinger equation, approximating the ?-doped
system by
[H
0
+ V
dop
+ V
H
+ V
xc
]?(r)=E?(r) (2.21)
Here H
0
is the potential associated with a 1000 ML Si lattice, V
dop
is the potential
33
Figure 2.10: Results of calculation. E
0
and E
2
correspond to the two lowest energy
subbands with with m
z
=m
bardbl
,E
1
and E
3
to the lowest energy subbands with m
z
=m
?
. The dashed line represents the Fermi energy, E
F
.TheE
1
subband starts
to become occupied for a density n > 3.39x10
12
/cm
2
,theE
2
subband for n >
6.64x10
14
/cm
2
. For a 1 ML doping level (n = 6.78x10
14
/cm
2
), three subbands are
occupied (two with m
z
=m
bardbl
,onewithm
z
=m
?
). The approximate MIT density
for ?-doped Si:P is shown, as well as the densities of the four samples measured in
this work (A,B,C,D), discussed in Chapter 5. All of the samples measured in this
work had two subbands populated: one with m
z
=m
?
, the other with m
z
=m
bardbl
.
34
due to the P dopants, V
H
is the Hartree potential due to the electron-electron
interaction, and V
xc
the exchange-correlation potential of the electron gas. They
assume a planar distribution of donors, in which the P atoms lie in a c(4x2) Si
(100) surface reconstruction (see Chapter 5 of [20]). They ignore both spin-orbit
scattering (which is weak in Si:P [30]), and ?inter-pocket? scattering between the
various energy ellipsoids in Figure 2.3. The band structure for this system is solved
by by expanding ?(r) in a planar Wannier orbital (PWO) basis, which transforms
the di?erential equation (2.21) into a finite-dimensional eigenvalue problem,
enabling the appropriate eigenvalues and eigenvectors to be found. Since a PWO is
the product of a two-dimensional plane wave and a one-dimensional Gaussian
function, products of such PWOs have convenient computational properties.
Two classes of subbands result, labeled the ??? and ??? bands. Two ? bands
are found two exist, lying approximately 20 meV apart near the ? point of the
Brillouin zone, which correspond to the circles of Figure 2.8. The four equivalent
? bands correspond to the in-plane ellipsoids of Figure 2.8. The dispersion
relations for these subbands are
E
i
(k)=E
(?)
i
+
k
2
m
?
?
(2.22)
and
E
i
(k)=E
(?)
i
+
(k
x
?k
?
)
2
m
?
bardbl
+
(k
y
)
2
m
?
?
. (2.23)
The e?ective masses are determined to be m
?
?
= 0.211m
e
and m
?
bardbl
= 0.95m
e
,close
to the values of m
?
and m
bardbl
for ?normal? Si. Should the need arise, I will use these
masses in subsequent calculations, and I define m
?
t
,thetransport e?ective mass
35
[34], by m
?
t
=2m
?
?
m
?
bardbl
/(m
?
?
+m
?
bardbl
) similarequal 3.15x10
?31
kg.
Although [31] provides a more general calculation than I have performed, the
results of both approaches are compatible. As an experimentalist, this is
heartening, since it suggests a strong understanding of the fundamental physics
governing ?-doped Si:P.
2.1.4 Perspective
The significance of subband structure in the context of 4-wire measurements
can be appreciated by the fact that such a measurement actually sums the
contribution of each subband to the total conductance signal. For example, when
measuring the Hall resistance to obtain the carrier density n (see Appendix A),
this quantity can be considered as the sum of the carrier densities n
i
in each
individual subband. Therefore, if there are M independent subbands populated in
a given device, n is given by
n=
M
summationdisplay
i=1
n
i
. (2.24)
Similar considerations apply to the measured conductance ?, assuming
independent (decoupled) subbands. Labeling the ith subband?s contribution by ?
i
,
and using the Drude expression for each component (see Appendix A), we arrive at
? =
M
summationdisplay
i=1
?
i
=
M
summationdisplay
i=1
parenleftbigg
n
i
e
2
?
i
m
i
parenrightbigg
. (2.25)
Since I used ?(H) to probe the thickness T of ?-layers (see Chapter 5), it is
important to understand the contribution of each ?
i
to T . In the calculation I
performed above, a perfectly planar distribution was assumed. However, Figure
36
2.7 clearly shows that even for a perfectly flat ?-layer, the spatial extent of the
electronic distribution depends on which subband is considered. To establish a
consistent definition for T , I first define the thickness of the ith subband, T
i
,in
terms of its mean-square variance in the ?z direction:
T
i
=2
radicalBig
< z
2
i
> ? < z
i
>
2
(2.26)
where <f(z) > =
integraltext
?
??
f(z)?
?
i
(z)?
i
(z)dz. Assuming only the i = 0 (lowest energy)
subband is occupied, this reduces to [1]
T =2
radicalbigg
7
5
parenleftbigg
4planckover2pi1
2
18emE
parenrightbigg
1/3
, (2.27)
and evaluating this with ?prototypical? device parameters (n = 10
14
/cm
2
,m=m
?
t
= 3.15x10
?31
kg), I obtain Tsimilarequal1.1 nm. Since E?n, as in (2.27), in general for
this approximation T?n
?1/3
. Thus even for a single subband and a perfectly
planar distribution, di?erent 2DEG thicknesses are obtained depending on the
doping density n.
I note that although it is completely reasonable from a physical perspective
that T of a 2DEG should vary with n, the assumed thickness of the donor
distribution does not change with doping density. In fact, the donors in my
calculation as well as [31] have been assumed to possess zero spatial extent; only
the electrons are ?spread out?. I will neglect this e?ect in what follows, and assume
equality between the thickness of the 2DEG, as inferred from ?(H) measurements,
and the donor distribution itself. This is reasonable if the donors are distributed in
a non-planar fashion, which is the case for all of the samples I examined.
37
From equations (2.24) and (2.25), one might be tempted to define T as
T =
summationtext
T
i
. However, given this definition, higher energy subbands, although
generally less populated, would be weighted equally in the sum as the lower energy
bands, whence T would be artificially enhanced. A more appropriate definition
should take into account the fact that the T
i
do not contribute uniformly to T :
they should be weighted by the subband population. Given this consideration, we
are finally in a position to give a reasonable definition of T . Using the fact that
each subband contributes to n through (2.24), ? through (2.25), and T through
(2.26) I define the ?total 2DEG thickness signal? T by
T =
M
summationdisplay
i=1
parenleftBig
n
i
n
parenrightBig
T
i
. (2.28)
Since this is an experimental dissertation, it would have been preferable to
determine the subband populations n
i
through an actual measurement, instead of
resorting to approximate theoretical calculations. Unfortunately, being able to
resolve the contribution of each subband to the total conductance ? requires
capabilities beyond those discussed in this thesis. For example, in order to infer
subband occupations due to Shubnikov de-Haas oscillations in the
magnetoresistance [2, 3], one needs to apply a field |H| of su?cient strength to
render the cyclotron frequency ?
c
at least as large as the elastic scattering rate
1/?
e
.Since?
c
=e|H|/m and 1/?
e
=v
F
/lscript
e
, this puts a lower bound on |H| of |H|
? planckover2pi1k
F
/elscript
e
. For our prototypical device n similarequal 10
14
/cm
2
and lscript
e
similarequal 25 nm, which
yields |H|similarequal65 T. This field is over 5 times stronger than the maximum H (12 T)
the cryomagnetic system used in these measurements could attain at 4.2 K.
38
Accordingly the subband occupations could not be experimentally determined.
To conclude this section, I now clarify the model for the subband structure I
will use for the remainder of this dissertation. What has been omitted from the
work [1, 31] I have discussed are any physical e?ects which could lead to a
broadening of the ?-layer band structure. This is significant, since if the energy
levels of two subbands were broadened to the extent that their distributions
overlaped considerably, they would then behave as as one effective subband.
Some of the physical processes which could lead to subband broadening include
inter-subband scattering, non-zero temperature, defects, a finite thickness of the
?-layer, fluctuations of T in a given sample, impurties, and surface e?ects. All of
these play a role in ?randomizing? the band structure of the ?-layer system. The
issue of inter-subband scattering has been discussed theoretically in [33, 56], and
experimentally in [34] (the measurements of [34] were on Si:Sb doped ?-layers).
The results of [34] were that analysis with single subband models proved su?cient
for modeling their data. Since my samples and those of [34] were both low
mobility, ?-doped Si samples (? similarequal 100 cm
2
/V?s), it seems reasonable to conclude
that a similar approach applies here. In fact, single subband models were used for
the data which will be dicussed in Chapter 5, and very good theoretical fits to the
data were obtained, which experimentally justifies this approach. Thus I shall
henceforth assume that a single, e?ective subband is responsible for the transport
characteristics observed in this work.
With this single subband model in mind, I now discuss the theory of weak
localization, and the application of this theory to ?-layer systems.
39
Figure 2.11: Longitudinal resistance of a ?-doped sample at T similarequal 130 mK with
a perpendicular magnetic field H applied. The sample displays a negative
magnetoresistance, characteristic of the phenomenon of weak localization.
2.2 Weak Localization
2.2.1 Historical Development
Before stating any theoretical results, let us first examine magnetoresistance
data from an Si:P ?-doped device taken in the presence of a perpendicular
magnetic field H at T similarequal 130 mK, shown in Figure 2.11.
We see that the resistance of the sample displays a maximum at H =0,and
40
decreases monotonically for |H| negationslash= 0. Such negative magnetoresistance had been
observed in two-dimensional systems (such as depletion layers in MOSFETs) long
before it was given a solid theoretical interpretation in the early 1980s. The
questions I will address here are what causes this negative magnetoresistance to
occur,andhowcanitbeusedtoprobea?-layer? Before doing so, however, a brief
digression into the historical evolution of weak localization is appropriate.
The modern theory of weak localization starts with Anderson?s observation
[35] that a conductor which is su?ciently disordered will in fact insulate at T = 0.
Subsequent diagrammatic calculations carried out by Langer and Neal [36] showed
that the so-called ?maximally crossed? diagrams which contribute to the T = 0
conductivity are divergent, and as we shall see, these are precsiely the diagrams
used for carrying out computations of the weak localization correction to the
conductivity. The scaling theory of localization, first espoused by Thouless [37],
furthered our understanding of localization behavior by subdividing a conductor
into phase coherent units, and investigating the e?ect boundary conditions
between these units had on the macroscopic conductance (analogous to the overlap
viewpoint of Figure 2.4). This led to the theoretical discoverybyAbrahamset al.
in 1979 that all two-dimensional systems insulate at T = 0 [38]. The importance of
the electron-electron interaction in understanding the behavior of disordered
conductors was elaborated upon by many workers in the early 1980?s, including
seminal theoretical contributions by Altshuler [39] and Lee [40], among others,
while experiments validating these theories have been demonstrated in numerous
experiments, beginning with Dolan and Osherho? [41]. Thorough presentations of
41
the theory of weak localization with references to the original literature can be
found in the review articles by Bergmann [43] and Chakravarty and Schmid [44],
as well as in [39, 40]. A more recent treatment can be found in the paper by
Aleiner, et al. [45].
2.2.2 Quantum Conductivity
Often referred to as the ?quantum correction? to the classical conductivity,
the weak localization data shown in Figure 2.11 can be expressed as
?(H)=?
0
+ ??(H) (2.29)
where ?
0
is the Drude (H = 0) conductivity and ??(H) is the weak localization
(H negationslash= 0) contribution to the magnetoconductance. Here ?(H) = 1/R
a50
(H), and R
a50
is the ohms/square resistivity of the device (see Chapter 4). This section will
explore the nature of ??(H), including its quantum mechanical origin, magnitude
and sign, etc.
As it turns out, the data of Figure 2.11, and therefore ??(H), can be
understood by invoking a particular length scale into the problem. In addition to
the mean free path, lscript
e
(Appendix A), the other length scale which can be inferred
from such data is the dephasing length, lscript
?
. The physics behind lscript
?
is in some sense
much more involved than that of lscript
e
, but is nevertheless conceptually easy to
understand. Whereas lscript
e
is the average distance an electron can travel in a solid
before undergoing an elastic (energy-conserving) collision, lscript
?
is the corresponding
distance for an inelastic (nonconservative) collison. Since the time evolution of a
42
quantum state with energy E is proportional to the phase e
?iEt/planckover2pi1
, any change in
the state?s energy will alter the evolution of the phase of the wave function. Given
the fact that quantum mechanical probability distributions are extremely sensitive
to phase relationships between state kets (i.e. matrix elements), measureable
quantities including magnetoconductance are strongly influenced by these same
phase correlations. The phase-breaking mechanisms in solids are the
electron-electron, electron-phonon and spin-flip interactions [44]. As we shall see in
Chapter 5, the electron-electron interaction is the most important for Si:P ?-layer
physics.
To proceed with our explanation of Figure 2.11, consider Figure 2.12 (a),
which shows a particle (electron) located at the initial position r
A
at time t = 0. I
now ask, what is the probability that it will be found at position r
B
at some later
time t? If we label the probability amplitude of an arbitrary path satisfying the
specified conditions by A
?
, elementary quantum mechanics dictates that the
desired probability, P
A?B
, is given by the following expression [47]:
P
A?B
=
parenleftBigg
summationdisplay
?
A
?
?
parenrightBiggparenleftBigg
summationdisplay
?
A
?
parenrightBigg
(2.30)
i.e. the squared-modulus of the sum over all appropriate amplitudes A
?
. Carrying
out the prescribed computation, I obtain
P
A?B
=
summationdisplay
?
|A
?
|
2
+
summationdisplay
?negationslash=?
A
?
?
A
?
. (2.31)
The first sum represents the classical probability of propagating from r
A
to r
B
along the various paths ?. The second sum represents the quantum mechanical
interference between arbitrary pairs of paths, labeled by ? and ?.Formostpairs
43
(a) (b)
Figure 2.12: Paths from A to B. (a) The probability of propagating from A to B
is given by summing the quantum mechanical amplitudes (equation (2.30)) for all
appropriate trajectories. (b) When A = B, there is an enhanced probability for a
particle to return to A (remain localized), and the resistance is maximal at H =
0 (?coherent backscattering?). For H negationslash= 0, a phase shift proportional to the flux
through the loop is induced between oppositley directed paths, and the constructive
interference is reduced, making the sample less resistive.
44
of paths, the phase accumulated along the separate trajectories is incoherent
(oscillating near a given point r
B
on the order of the Fermi wavelength, ?
F
); thus
the interference terms typically average to zero. However, if r
A
= r
B
,thatis,we
consider the closed trajectories of Figure 2.12 (b), the situation changes
dramatically. Now each closed loop can be paired with a ?time-reversed?
counterpart, and whether a given loop is traversed clockwise (?forward in time?)
or counterclockwise (?backward in time?) has no influence on the phase
accumulation around the path. To explicitly illustrate this, let A
1
and A
2
represent the amplitudes for propagation around a given loop, corresponding
respectively to clockwise and counterclockwise trajectories. Denoting the
contribution of these two paths to P
A?B
by p,wehave
p = |A
1
+ A
2
|
2
=(A
1
+ A
2
)(A
1
+ A
2
)
?
=2|A
1
|
2
+2Re(A
1
A
2
)=4|A
1
|
2
. (2.32)
Ihaveused|A
1
|
2
= |A
2
|
2
in (2.32) due to the fact these trajectories are identical
except for the direction of traversal. Note that if the contribution of the
interference terms A
1
A
?
2
and A
?
1
A
2
to p had been neglected, we would just have the
classical probability p =2|A
1
|
2
, one-half of that predicted by (2.32). Thus we see
that these time-reversed paths are coherent (in phase), interfere constructively,
and contribute to a higher probability of returning to r
A
after a time t. These
closed loop paths, collectively referred to as ?Cooperons?, are the source of the
weak localization correction to the conductivity.
Suppose now that a uniform magnetic field H is ?turned on? in the vicinity of
these paths, and identify A as the vector potential corresponding to H in the
45
usual way [46] (i.e. H = ??A). The presence of A breaks the time-reversal
symmetry of the problem, and causes the wavefunction of a charged particle to
acquire an additional phase ?? as it propagates around a loop, given by [47]
?? =
parenleftBig
e
planckover2pi1
parenrightBig
contintegraldisplay
A?dl (2.33)
where dl is an infinitesimal length element tangent to the path. Since we are
considering here closed paths, Stokes? theorem (
contintegraltext
A?dl =
contintegraltext
(??A)?da =
contintegraltext
H?da)
allows us to rewrite this expression for ?? as
?? =
parenleftBig
e
planckover2pi1
parenrightBig
? (2.34)
where ? is the magnetic flux enclosed by the path. This path-dependent phase
accumulated by a charged particle in a magnetic field shows that, ultimately, the
phenomenon of weak localization is a particular manifestation of the
Aharonov-Bohm e?ect [48]. Accordingly, whether the charge traverses the loop
clockwise or counterclockwise a?ects the sign of the phase acquired, and the
amplitudes A
i
in (2.30) must be transformed according to A
i
? e
i??
A
i
,as
required by gauge invariance. Taking this into account an re-computing p,Iobtain
p =2|A
1
|
2
(1 + cos(2??)). (2.35)
Thus, ?H negationslash=0, the constructive interference which contributed to the ?enhanced
backscattering? at H = 0 is lost, and it becomes less probable for the charge to
return to its initial position. Hence the application of H renders the electrons in
the sample ?less localized?, which makes the device less resistive, and we are
finally able to appreciate the physical origin of the negative magnetoresistance
46
signal shown in Figure 2.11. In addition, the symmetry of the signal about H =0
can be understood by the ? cos(2??) dependence of p on H.
2.2.3 Calculation of ??(H)
Although a start-to-finish calculation of ??(H) is too lengthy to be shown
here, a brief sketch of the standard computational procedure is nonetheless
appropriate, in order to appreciate the origin of the expressions which will be used
for fitting data. The references cited earlier can be consulted for details.
As was shown in the previous section, Cooperons are responsible for the
resistance maximum at H =0.Thustocalculate?(H) for a particular system,
one must merely add up (integrate) the contribution of each Cooperon to obtain
the total conductance signal. Several equivalent approaches include calculating
?(H) using the Kubo-Greenwood formula [49], as done in [43], employing
Abrikosov?s result [50] for the residual resistance of a normal metal, as in [39], or
through the use of linear response theory [51]. Since weak localization stems from
the interference of electron waves as they propagate through an impurity
distribution, it should come as no surprise that all of these methods employ
impurity averaged Green?s functions [50].
The interference between the time reversed paths which comprise the
Cooperon can be conveniently represented using Feynman diagrams (a pedagogical
introduction to the use of Feynman diagrams in many-body problems can be found
in [52]). A representation of the appropriate diagrams for use in weak localization
47
calculations in k-space are shown in Figure 2.13 (a). Here a solid line corresponds
to a single-particle propagator (Green?s function), a wavy line to the amplitude for
scattering o? of an impurity, and the impurity itself by an open circle. Note that
the order of impurity scattering between the upper propagator line (?forward in
time?) and lower line (?backwards in time?) is reversed, as one would expect for
the previously explained origin for the interference manifested by the Cooperon.
Although only two scattering events are depicted here, the term ?maximally
crossed? diagrams is apparent from the figure, and simply refers to switching the
order of the scattering sequence. If the lower propagator line is reversed, taking
care to preserve the scattering sequence, the diagram is ?uncrossed?, and the
interference between the two paths can be expressed by the ?ladder diagrams? of
Figure 2.13 (b). Taking into account all possible scattering sequences by summing
the obvious geometric series suggested by Figure 2.13 (b), an integral equation is
obtained for the Cooperon, commonly referred to as the Dyson equation.The
diagrammatic representation for Dyson?s equation is shown in Figure 2.13 (c).
A few final comments about calculations of ??(H) will now be made.
First, it is important to note that not all closed paths in the sample contribute
to the weak localization signal: there is a cut-o? parameter, namely lscript
?
,which
defines the average size of the phase coherent region in the sample. Furthermore,
lscript
?
is temperature dependent. In general, this dependence is logarithmic on the
temperature, raised to a power p: lscript
?
? ln(T
?p
) [40]. In other words, lscript
?
should
diverge as T ? 0. This can be understood by the fact that the phase-breaking
rate corresponding to the electron-electron interaction varies as 1/?
e?e
similarequal T
2
/E
F
,
48
Figure 2.13: Feynman diagrams in k-space for calculating ??(H). In (a), the inter-
ference between ?forward? in time electrons (upper lines), and ?backward? in time
electrons (lower lines) is accounted for by reversing the scattering sequence under
consideration, hence the term ?maximally crossed? diagrams. This is equivalent to
the ?ladder diagram? in (b), where the interference is between two forward in time
paths, but with k vectors switched. Summing all such diagrams leads to the Dyson
equation, represented by (c). See [52] for a discussion of Feynman diagrams in the
many-body problem.
49
while that of the electron-phonon interaction varies as 1/?
e?ph
similarequal T
3
/?
2
D
[45] (E
F
is the Fermi energy, ?
D
the Debye frequency). For c-Si, ?
D
similarequal 1 THz [20], whence
it is clear that 1/?
e?e
will dominate at low temperatures.
Second, since the phase accumulated around a given path is extremely
sensitive to magnetic fields, the device under consideration must be free of
magnetic impurities. In addition, if any spin-orbit e?ects are present in the
sample, this will also cause a path-dependent phase shift. In fact, the spin-orbit
interaction is quite strong in Si:B ?-doped samples, and as we shall see in Chapter
5, produces very di?erent magnetoresistance behavior than that of Figure 2.11.
Finally, the e?ective dimensionality of the sample is important, and di?erent
corrections apply depending on the geometric properties assumed in the
calculation. All of the samples in this work are considered to be e?ectively
two-dimensional, but one-dimensional weak localization has also been extensively
studied experimentally.
Having said all this, it should be apparent that a great deal of theory has been
developed to produce useful expressions for fitting experimental data obtained
from 2DEGs. Here I simply state that when placed in a perpendicular magnetic
field (H
?
), the prediction for the measured change in conductance of a 2DEG
system is given by [53]:
??(H
?
)=(
?e
2
2?
2
planckover2pi1
)
bracketleftbigg
?
parenleftbigg
1
2
+
H
1
H
?
parenrightbigg
?
parenleftbigg
3
2
parenrightbigg
?
parenleftbigg
1
2
+
H
2
H
?
parenrightbigg
?
parenleftbigg
1
2
parenrightbigg
?
parenleftbigg
1
2
+
H
3
H
?
parenrightbiggbracketrightbigg
(2.36)
where ?(x) is the digamma function, the ?logarithmic derivative? of the gamma
50
function. ?(x)=
d
dx
ln(?(x)), and asymptotically approaches the natural logarithm
ln(x) for large x. The characteristic fields H
1,2,3
are associated with the transport
relaxation mechanisms of impurity potential scattering (H
0
), inelastic scattering
(H
i
, including electron-electron and electron-phonon interactions), spin scattering
(H
s
), and spin-orbit scattering (H
so
), as follows [43, 53]:
H
1
= H
0
+ H
so
+ H
s
(2.37)
H
2
=
4
3
H
0
+
2
3
H
s
+ H
i
H
3
=2H
s
+ H
i
.
In turn, a relaxation time, ?
i
, can be associated with the H
i
through
H
i
?
i
=
planckover2pi1
4eD
(2.38)
where D is the electronic di?usivity. In two dimensions, D = v
2
F
?
e
/2. Although in
principle knowledge of the carrier e?ective mass would allow one to estimate D
fairly accurately, the best way of independently determining D is through
measurements at high frequencies [57]. Since this was not done here, I will
henceforth equate an e?ective length with H
i
and ?
i
via
lscript
i
=
radicalbig
D?
i
. (2.39)
Fortunately, equation (2.36) can be simplified considerably in the case of Si:P
?-layers, as previous work has shown that both spin and spin-orbit scattering in
Si:P are weak [58]. Therefore, the relaxation times associated with these processes
are very large, and can be neglected when compared with the electron-electron and
51
electron-phonon interactions. Taking this into account, I arrive at the final
expression I will use for modeling ??(H
?
) in Si:P ?-layers:
??(H
?
)=(
e
2
2?
2
planckover2pi1
)[?(
1
2
+
planckover2pi1
4eH
?
lscript
?
2
)??(
1
2
+
planckover2pi1
2eH
?
lscript
e
2
)+ln(
2lscript
?
2
lscripte
2
)] (2.40)
where lscript
?
is the previously defined dephasing length (lscript
?
=
radicalbig
D?
?
), and lscript
e
the
elastic mean free path.
If instead the magnetic field is applied parallel to the 2DEG (H
bardbl
), the
appropriate expression takes the form [54, 55, 56]
??(H
bardbl
)=(
e
2
2?
2
planckover2pi1
)ln(1+?H
2
bardbl
). (2.41)
The physical interpretation of the parameters lscript
?
and ? in the context of measuring
T will become apparent presently.
As discussed in Chapter 1, using weak localization to measure T corresponds
to a cross section comparison of the 2DEG in H
?
and H
bardbl
configurations. Now
that I have discussed the physical origin of the negative magnetoresistance in
Figure 2.11 (namely magnetic field dependent quantum interference e?ects), I
present a phenomenological approach for determining T .
2.2.4 T Phenomenology
This section uses straightforward dimensional analysis to find an expression
for T in a ?-layer using weak localization data. Although reasonably simple, this
analysis reveals certain features of this approach to measuring T which all models
have in common.
52
To proceed, consider the data in Figure 1.3, which shows both ?(H
?
)
(corresponding to the upper curve in the figure) and ?(H
bardbl
)(lowercurve).Fora
particular value of magnetic field, say H
0
, the graphs show that the maximum
value of ?(H
0
) is attained when H
0
= H
?
, while the minimum occurs for H
0
=
H
bardbl
. Furthermore, both curves are symmetric functions of H (i.e. ?(H)=?(-H)),
which suggests that ?(H) can be represented by a Taylor series expansion with
only even powers of H:
?(H
?,bardbl
)=
?
summationdisplay
n=0
?
2n
H
2n
?,bardbl
. (2.42)
Then to lowest non-vanishing order in H,wecanwrite
??(H
?
)=?
?
H
2
?
(2.43)
and
??(H
bardbl
)=?
bardbl
H
2
bardbl
(this is the lowest order Taylor expansion of equation (2.41)). From our discussion
of weak localization and the data of Figure 1.3, it is clear that a greater magnetic
flux penetrates the 2DEG when H = H
?
(since the conductance is higher), and
therefore that we can associate an e?ective area, A
eff
(or cross section), with ?
?
and ?
bardbl
, as follows.
First, I adopt the convention that ??(H) is dimensionless, measured in
?Cooperon units? of e
2
/2?
2
planckover2pi1 =e
2
/?h. Then in order for both sides of equation(s)
(2.43) to be dimensionless, ? must have units of 1/T
2
. Because weak localization is
due to interference between pairs of paths, the natural units of magnetic flux in
53
this problem are planckover2pi1/2e, and this combination of physical constants has dimensions
of T?m
2
. Hence the desired relationship between ? and A
eff
in this approximation
is given by
? =
parenleftbigg
2eA
eff
planckover2pi1
parenrightbigg
2
. (2.44)
The ?cross section? argument presented in Chapter 1 is shown again in Figure
2.14. When H = H
?
, all directions in the 2DEG should be treated symmetrically,
which suggests that a circle is the appropriate cross sectional area to be associated
with A
?
. Furthermore, the characteristic length scale of the 2DEG in this
configuration is lscript
?
, and using this (subjectively!) as the diameter of the circle, we
have
?
?
=
parenleftbigg
e?lscript
2
?
2planckover2pi1
parenrightbigg
2
. (2.45)
On the other hand, when H = H
bardbl
, all directions in the device are not
equivalent, since, due to the confinement potential, the wavefunction is coherent in
the direction perpendicular to the doping plane. In fact, if the sample is to be
considered two-dimensional from a quantum mechanical perspective, the condition
T <lscript
?
must hold [54]. Assuming this is the case, the cross sectional area for H =
H
bardbl
is rectangular, with length lscript
?
and height T .Thatis,
?
bardbl
=
parenleftbigg
2eT lscript
?
planckover2pi1
parenrightbigg
2
. (2.46)
Solving equation (2.46) for T yields
T =
planckover2pi1
?
?
bardbl
2elscript
?
. (2.47)
Note that this expression relates T of the ?-layer to the perpendicular field
54
(a) (b)
Figure 2.14: E?ective cross sectional areas for H
?
(a) and H
bardbl
(b), and the corre-
sponding weak localization length scales for defining these areas.
parameter lscript
?
(derived from (2.45)) and the parallel field parameter ? (from (2.46))
as T?
?
?/lscript
?
. Because ?(H) is symmetric with respect to H but highly
anisotropic, all weak localization models of T will have this form.
The rectangular model used in this analysis is of course too simple to
accurately describe real ?-layer systems. This ties in with the principal di?culty
one is faced with when using parallel field weak localization techniques to measure
T : proper modeling of ? in equation (2.41). I discuss this further in the following
section on Berry phase.
2.3 Berry Phase
The notion of a Berry phase, also known as a geometric phase, is ubiquitous in
physics. Although the modern form of Berry?s theorem was not formulated until
1984 [59], the importance of path dependent phenomena in various physical
55
contexts has been appreciated for centuries. A familiar classical example of a
geometric phase is the precession of a (Foucault) pendulum due to the rotation of
the earth. The observed angular deflection of the pendulum from it?s initial
trajectory corresponds to a geometric phase, and depends on parameters such as
how long the pendulum has been in motion and it?s latitudinal position on earth.
The phenomenon of weak localization itself is an example of a Berry phase, since it
is based on the interference between oppositely directed Aharonov-Bohm paths, as
described in the previous section. I now discuss the particular considerations of
Berry phase pertinent to this work. A comprehensive discussion of Berry phase,
with numerous physical examples, is contained in the book
Geometric Phases in Physics, by Wilczek and Shapere [60].
2.3.1 2DEG Berry Phase
The importance of Berry?s phase in the context of 2DEGs subjected to parallel
magnetic fields has been appreciated and elaborated upon previously [61, 62].
Since in essence weak localization measures a magnetic field induced electronic
phase di?erence, it is important to distinguish whether this phase is due to
Aharonov-Bohm interference, or results from the intrinsic geometry of the system.
An artificial example of this e?ect is illustrated in Figure 2.15. The top panel (a)
shows a perfectly flat 2DEG with thickness T , the middle panel (b) a 2DEG also
of thickness T , but with a periodic ?ripple? superimposed upon it. If an electron is
confined to follow the boundary of these samples, say through the application of
56
(a)
(b)
(c)
Figure 2.15: Three di?erent models for the thickness of a 2DEG. The bottom fig-
ure (c) is the most realistic description for a ?-layer 2DEG. Geometric phase is
acquired by electrons in these systems when a H
bardbl
field is applied (perpendicular to
the thickness direction).
57
an electric field, then in the ?flat? sample, no additional phase is acquired, whereas
in the ?rippled? sample a nonzero phase is accumulated. If we take the parallel
magnetic field H = H
0
to be in the ?z direction, and work in the gauge A = H
0
x?y,
this accumulated phase ?? will be proportional to (e/planckover2pi1)
integraltext
A?dl ? (eH
0
T
2
/planckover2pi1).
Since ?? is due entirely to the geometry of the system, the term geometric phase
is indeed appropriate.
Of course, an actual ?-layer system will be quite di?erent from these idealized
examples. Rather than having a single thickness or spatial frequency which
characterizes a given sample, a more realistic description requires a distribution of
thicknesses and frequencies to be adequately specified. This is sketched in Figure
2.15 (c), where T is a fluctuating quantity, i.e. T depends on where in the sample
plane it is measured. Representing this dependence using Cartesian coordinates,
we have T = T (x,y). Since I have suggested that the rms value of T is the quantity
of interest in a given device, we see that in this situation an expression such as
< T >=
parenleftbigg
1
A
parenrightbiggintegraldisplay
T (x,y)dxdy (2.48)
should be used, where A is the total area of the device.
To account for how the T (x,y) distribution influences ??(H), I introduce an
extremely useful construct which captures the essential statistical information. I
define the ?thickness autocorrelation function?, C
T
(r) as follows:
C
T
(r)=< T (0)T (r) >. (2.49)
The bracket < ? > implies ?averaging over all values of the origin?, and r
represents the distance from the origin to the point of interest. In other words,
58
equation (2.49) quantifies the average correlation between the thickness at an
arbitrary point in the sample (taken to be the origin) with all other points a
planar distance |r| away. Using the Wiener-Khintchine theorem,thiscanbeshown
to be equivalent to the following expression:
C
T
(r)=
integraldisplay
?
??
T
2
(r)e
ik?r
dk (2.50)
i.e. C
T
(r) is the Fourier transform of the absolute square of T .Atomicforce
microscope (AFM) images of surfaces have established that C
T
(r)takesthe
following form for random Gaussian surfaces:
C
T
(r)=?
2
?
e
?(r/lscript
c
)
2H
(2.51)
Where ?
?
is the rms surface roughness, lscript
c
the correlation length,andH the
so-called ?scaling exponent? (usually Hsimilarequal1). Although such a surface model is
probably of limited utility in ?-layer physics, we can nonetheless identify lscript
c
as the
appropriate statistical quantity for taking into account how the ?roughness? of the
?-layer influences the magnetoresistance.
A precise estimate of the ?-layer thickness will depend on the relationship
between lscript
e
, lscript
?
,andlscript
c
. For example, as discussed in [62], if the inequality lscript
c
lessmuch lscript
e
lessmuch lscript
?
holds, the situation is analogous to ?homogeneous broadening? in nuclear
magnetic resonance, and the weak localization line shape (?(H)) is unaltered, as
sketched in Figure 2.16 (a). In the ?inhomogeneous broadening? limit (Figure 2.16
(b)), lscript
e
lessmuch lscript
?
lessmuch lscript
c
, and the line shape is altered, showing a change of sign from
the ideal line shape (zero roughness limit) as a function of H
?
. I apologize for my
lack of artistry in crafting this figure.
59
(a)
(b)
Figure 2.16: NMR analogy for parallel field weak localization measurements based
on the length scales (lscript
c
, lscript
e
, lscript
?
) characterizing the ?-layer. The dashed curves rep-
resent ?(H)forzero parallel field, while the solid curves represent a finite (and
constant) parallel field. The ?homogeneous broadening? limit corresponds to (a),
applicable when lscript
c
lessmuch lscript
e
lessmuch lscript
?
. The ?inhomogeneous broadening? limit corresponds
to (b), and is applicable when lscript
e
lessmuch lscript
?
lessmuch lscript
c
. Previous measurements, not reported
in this thesis, suggest that the samples studied here are homogeneously broadened.
60
I will consider only two models for lscript
c
in this thesis. In the first case, I
completely ignore any e?ects of lscript
c
on ?(H), and estimate an ?uncorrelated?
thickness, T :
T =
planckover2pi1
?
?
2elscript
?
. (2.52)
This corresponds to the cross section argument presented earlier.
In the second case, I will assume homogeneous broadening of the line shape
(lscript
c
lessmuch lscript
e
lessmuch lscript
?
), and estimate a ?correlated? thickness, T
c
:
T
c
=
parenleftbigg
1
4?
parenrightbigg
1/4
bracketleftBigg
parenleftbigg
planckover2pi1
elscript
?
parenrightbigg
2
parenleftbigg
lscript
e
?
lscript
c
parenrightbigg
bracketrightBigg
1/2
(2.53)
Since independently measuring lscript
c
is presently impossible (due to the fact that the
?-layer lies beneath the device surface), all estimates of T in this dissertation will
use the uncorrelated estimate, equation (2.52). The lone exception to this will be
when I discuss Si:P di?usivity measurements in Chapter 5, and model the data
using (2.53), to be consistent with the published results.
Regardless of the theoretical limitations facing a given thickness model, it
seems safe to assume that the devices measured in this work were homogeneously
broadened. This is suggested by previous measurements, not reported in this
thesis, which more closely mimicked the homogeneous broadening case of Figure
2.16 (a). In addition, for our prototypical device lscript
e
similarequal 25 nm and lscript
?
similarequal 100 nm,
while AFM images on device surfaces have shown lscript
c
similarequal 5 nm. Although lscript
c
of the
surface and that of the ?-layer are undoubtedly distinct, the likelihood that lscript
c
of
the ?-layer exceeds 100 nm seems remote.
A natural question to ask is what actually causes the fluctuations in T ?It
61
seems likely that this disorder stems from two sources. The first is the intrinsic
roughness of Si surfaces. A perfectly flat atomic surface does not exist, and for Si
in particular the quality of the surface depends heavily on the temperature of the
substrate during growth. Because of the technological importance of Si, such
surface features have been extensively studied and characterized. According to the
experts who actually grow ?-doped samples, this source of disorder is likely to be
the limiting factor to achieving monolayer ?-doped samples, once other, less
intrinsic issues, are overcome (assuming they can be).
The second source of fluctuations in T is from the phenomenon of segregation.
This will be discussed more fully in Chapter 3, but for now it su?ces to define
segregation as the tendency of donors to migrate towards the sample surface during
epitaxial growth. In other words, the donors do not remain on the surface where
they were deposited, which leads to a broadening of the ?-layer. This tendency is
much more pronounced in Si:P than in Si:B, and has recently been a subject of
considerable interest. At the time of this writing, it is safe to say that segregation
is the dominant contribution to a non-uniform T in Si:P ?-doped samples.
62
Chapter 3
Fabrication
The details of how a ?-doped wafer is constructed, and the processing methods
used to fabricate devices for experimental investigation, are presented in this
chapter. A comparison is drawn with the approaches implemented by other
research groups, and the use of SIMS analysis in characterizing the quality of the
samples is discussed. The wafers investigated in this dissertation were prepared by
Dr. Philip E. Thompson, head of the Si MBE section at NRL. This work would
not have been possible without his expertise, and his contribution is greatly
appreciated.
3.1 ?-layer Requirements
Referring to Figure 1.1, it should be obvious that ?-layers do not occur in
Nature: they must somehow be built. As mentioned in the Introduction, several
methods of ?-layer construction have been demonstrated, including ion
63
implantation [9], chemical vapor deposition (CVD) [8], and molecular beam
epitaxy (MBE), as in this and many other works. I shall first outline the physical
characteristics all ?-layers should have.
3.1.1 General Considerations
Without question, MBE is by far the most powerful technique for growing
?-doped samples. What makes MBE so elegant is that, under suitable conditions,
the lattice planes are grown atom by atom.Thistypeofcontrolispreciselywhat
is sought in a ?-layer sample, where ideally donor placement in the c-Si lattice lies
in a single lattice plane (i.e. in a monolayer). The basic idea behind MBE is
straightforward. A substrate is placed in the MBE chamber to serve as the ?base?
for the heterostructure to be grown. Various ?e?usion cells? (specialized ovens) are
distributed throughout the chamber, which act as sources for the di?erent
materials to be grown into the sample. The growth rate of a given material
depends on the temperature of its cell, as well as the temperature of the substrate.
Thus, in a loose sense, MBE is similar to the common technique of metallic
evaporation. The MBE chamber is held at ultra-high vacuum (UHV), and
therefore contamination by unwanted impurities is kept to a minimum.
Regardless of whether MBE or another approach is used, there are essentially
only two factors which characterize the quality of a ?-doped sample. One, of
course, is the thickness, T , and the smaller is T , the better. The other aspect is
multifaceted, namely the defect density inherent in a particular method of
64
preparation. Obviously a sample with a lower defect density is ?better?, and there
are many ways of quantifying this. Some examples include an asymmetry of the
?-layer (which, due to segregation, tends to be much worse in Si:P than Si:B),
lattice defects (such as vacancies and interstitials), and impurities which ?pollute?
the ?-layer in a fashion other than intended. While quantifying T is relatively
straightforward in that it reduces to a length measurement, the matter of defect
specification is much more di?cult. For instance, in the case of MBE grown layers,
the temperature sequence incorporated during the various stages of device
processing has a profound impact on the electrical characteristics a device will
display. The details of this will be discussed in Chapter 5 with respect to how the
resistivity of a device is influenced by a high temperature (850
?
C) anneal.
At present, eliminating dopant segregation is a major area of research in the
production of Si:P ?-layers. This phenomenon was described in Chapter 2 as the
tendency of the donors to migrate toward the surface of the sample during growth,
leading to a broader ?-layer. To see how other workers have addressed this issue,
and to contrast their approaches with those used in the course of this research, I
briefly digress to discuss the published results.
3.1.2 Previous Work
Three papers [64, 65, 66] on Si:P ?-layers are summarized here. The basic idea
behind the MBE growth techniques used by all workers who have reported results
on Si:P ?-layers, is illustrated by the ?naive? MBE growth sequence shown in
65
Figure 3.1: A naive approach to MBE ?-layer growth. A minimum of three steps
are necessary to isolate the ?-doped region from the (100) Si substrate: 1) bu?er
growth (or annealing), to prepare a clean surface, 2) ?-doping, in which the donors
are deposited, and 3) ?cap? growth, which isolates the ?-layer and keeps unwanted
impurities out of sample. Layer widths are not to scale.
Figure 3.1. In this approximation, there are only three growth steps to perform. A
clean substrate of high-resistivity p-type (100) Si is used as the ?base? for the rest
of the heterostructure. The reason for using a p-type (100) wafer is to better
isolate the n-type ?-layer from the substrate. The first step is to prepare a pristine
surface for deposition of the P atoms, either by growing a so-called ?bu?er? layer
of pure Si, or by annealing the sample at high temperatures. In the second step,
?-doping takes place by depositing P atoms (in one form or another) for a specified
amount of time. This controls the doping density, n. The last step in the process is
to encapsulate the ?-layer by growing pure Si (the ?cap? layer) on top of the
?-doped region. This serves to isolate the P donors from the environment.
66
T.-C. Shen, et al., at Utah State University, were the first to report results for
a Si:P ?-doped wafer [64]. This work focused on the surface morphology of the
sample during the various stages of growth, rather than end-result electrical
characteristics. The preparation sequence employed by this group was more
elaborate than the ?naive? approach described above, and consisted of the
following steps:
1. Start with a high-resistivity (0.1 ??cm) Si(100) substrate.
2. ?Flash? anneal the sample at T = 1500 K to prepare the surface for P
deposition.
3. Expose the substrate to a ?dose? of phosphine (PH
3
) gas at room
temperature (RT). The P density in the ?-layer is controlled by the pressure
and duration of the exposure.
4. Anneal at T = 510 K to dissociate the PH
3
and adsorb the P atoms into the
substrate.
5. Deposit 0.4 nm of Si at T = 428 K.
6. Anneal at T = 773 K.
7. Deposit the cap layer, an additional 6.3 nm of Si at T = 468 K.
8. Anneal at T = 673 K.
Presumably, the annealing steps were employed to reduce the defect density in
the sample. This is born out by the STM (scanning tunneling microscope) analysis
67
they performed of the growth stages, which showed low surface roughness and
c(2x2) structures, characteristic of high-quality Si surfaces. For the two samples
discussed in [64], transport measurements yielded for one sample n =
1.3x10
14
/cm
2
, with a mobility of ? =13cm
2
/V?s, while for the other sample they
found n = 8.8x10
13
/cm
2
,and? =38cm
2
/V?s. No measure of T in these samples
was reported, nor were values for lscript
?
, although they were able to observe weak
localization in devices fabricated from these wafers.
A similar (and independent) approach to Si:P ?-layer construction is reported
in [65], obtained by L. Oberbeck, et al., at the University of New South Wales
(UNSW) in Australia. For comparison with [64], I also enumerate their
preparation sequence:
1. Start with a high-resistivity (? 1??cm) Si(100) substrate.
2. ?Flash? anneal the sample at T = 1150
?
Cfor? 30 seconds to prepare the
surface.
3. Expose the substrate to PH
3
at RT.
4. Anneal at T = 550
?
Cfor5minutes.
5. Deposit 24 nm of Si at T = 250
?
C.
These workers performed a slightly more thorough characterization of their
samples by performing a SIMS (Secondary Ion Mass Spectrometry) analysis, in
addition to STM and weak localization measurements. This led to identification of
the annealing step after the PH
3
dose as the key element to limiting segregation.
68
Their explanation of this e?ect is that as the P atoms are adsorbed into the Si
crystal, they become bound by three covalent bonds, instead of just one as when
the P atom resides on the surface. Transport measurements yielded n = (2.0 +/-
0.4)x10
14
/cm
2
,and? similarequal 140 cm
2
/V?s. They also observed weak localization, but
did not report values for lscript
?
.
The final work I shall discuss was performed by K.E.J. Goh, et al.,alsoat
UNSW [66]. This is by far the most complete study on segregation in Si:P ?-doped
samples reported to date. Here a systematic study of the substrate temperature on
P segregation into the cap layer was performed, with di?erent samples prepared at
RT, 250, 400, and 600
?
C. This paper does report their weak localization results,
which serves as an important piece of the puzzle for optimizing ?-layers grown
using phosphine. The largest value of lscript
?
(and therefore the ?best? from a quantum
transport perspective) was obtained for the sample whose cap layer was grown at
T = 250
?
C, and for this sample the corresponding device parameters were
determined to be ? ? 61 cm
2
/V?s, n = 1.64x10
14
/cm
2
,andlscript
?
=72nm.Their
transport measurements were performed at T = 4.2 K.
Since the research in [65, 66] relied heavily on the use of SIMS to assess the
quality of their process, a brief discussion of that technique is in order.
SIMS
SIMS is a very powerful technique for acquiring profiles of elemental
concentrations versus depth in a given material sample. An example of a SIMS
profile from an Si:P ?-doped sample is shown in Figure 3.2. The way such a profile
69
is obtained is to bombard the sample with a high-energy (? keV) primary ion
beam, which sputters matter away and leaves a ?crater? behind. The ejected
material consists of both neutral atoms and secondary ions. The secondary ions
are separated from the other matter electrostatically, then directed into a mass
spectrometer tuned to detect the particular elemental mass of interest. The
primary beam is rastered in order to create a flat-bottomed crater, but the ions are
acquired only from the center of this crater, which ensures that each data point
comes from a unique depth. The intensity of the secondary ion signal is
proportional to the ion concentration at a given depth, whose scale is quantified by
the use of reference samples and surface profilometer measurements. By
correlating the crater depth with the secondary ion intensity, the desired depth
profile is constructed. It is also worth noting that SIMS can distinguish the signals
between di?erent isotopes of a given element, which has been used to explain the
large surface signal in Figure 3.2. According to [65, 66], this signal is due not to P
(with atomic number Z = 15), but to ?false counts? induced by the simultaneous
detection of Si (Z = 14) and H (Z =1).
The detailed procedures incorporated in a given SIMS analysis are rather
extensive, and must be optimized for the particular application involved. The
doping profile shown in Figure 3.2 was performed by the analytical group at
Charles Evans and Associates, and used a quadrupole SIMS instrument with Cs as
the primary ions. The incident ion energy was 1 keV, and the angle of incidence
was 60
?
. The P detection limits for this analysis were 4x10
15
/cm
3
. The ?straggle?
of the doping distribution toward the surface is a clear manifestation of
70
Figure 3.2: SIMS data for a Si:P ?-doped sample, with a large amount of segregation
apparent (the region between 10 and 50 nm). The signal peak at the surface of the
sample is likely due to Si and H (total protons = 14 + 1 = 15) ?masquerading? as
P (15 protons) [65, 66].
71
segregation.
Although this SIMS analysis has yielded a detailed cross section of the doping
profile in the sample, several limitations of this technique are worth pointing out.
First, T at only one point in the crystal is sampled, and examining many areas
would require a substantial amount of time (and money). Second, the technique is
destructive. By boring a crater in the sample through ionic bombardment, the P
atoms in the ?-layer are to some extent forced further into the crystal. This often
manifests itself in the profile appearing steeper near the surface of the sample than
on the ?backside?, and thus leads to an erroneous estimate of T . Third, little to
no information in regard to electrical characteristics or defect densities can be
inferred from SIMS. Fourth, and perhaps most importantly, the spatial resolution
of SIMS is limited. Determining the precise resolution limits of SIMS for Si:P
?-layers is presently an unresolved question, but it is safe to say that
sub-monolayer resolution is not possible using SIMS. In contrast, weak localization
measurements for determining T su?er from none of these limitations. Chapter 5
will explore this more fully.
3.2 ?-doped Wafer Preparation
With the necessary background in place, I now discuss the preparation
sequence for the ?-doped wafers used in this work. I shall first comment on some
particular aspects of our approach.
First, the primary di?erence between the samples used here versus those of
72
Figure 3.3: ?Old? ?-doping technique. In an e?ort to minimize P segregation, a
sequence of bu?er layers are incorporated in the growth as the substrate temperature
is reduced. Bu?er 1 corresponds to 170 nm of Si grown at T = 650
?
C, bu?er 2 to
18 nm of Si grown as T was reduced to 400
?
C,andbu?er3to12nmofSiwithT
held at 400
?
C. The ?-layer and cap layer (50 nm of Si) were grown with T = 400
?
C. No trap layer was implemented, and the layer thicknesses are not to scale.
73
[64, 65, 66] is that the P was obtained through thermal decomposition (?burning?)
of GaP. The Ga was separated from the P by passing the GaP ?fumes? through a
B filter, apparently producing a very pure P source. Second, the substrates were
always high-resistivity p-type Si(100) wafers. As mentioned earlier, a p-type
substrate more e?ectively isolates the n-type ?-layer from the rest of the sample,
but more importantly, a ?-layer in Si can only be grown along the (100) lattice
direction [63]. Finally, typical vacuum levels during growth were ? 5x10
?9
Torr,
and the wafers were rotated at ?30 rpm during growth to promote a uniform
doping density across the wafer.
An actual growth sequence for a ?-doped sample is depicted in Figure 3.3. The
major di?erence between this recipe and the ?naive? one described earlier is the
temperature sequence the substrate is taken through in the course of growth. After
placing the substrate in the MBE chamber, the wafer?s temperature is raised to T
= 650
?
C and a 170 nm bu?er layer of pure Si is grown. Next, the substrate
temperature was reduced to T = 400
?
C over the course of 3 minutes, then held at
T = 400
?
C for the remainder of the growth process. The total Si growth during
the time the sample has its temperature reduced and stabilized is ? 30 nm (a
constant Si growth rate of ? 0.1 (nm/s) is assumed). At this point the P ?-layer
itself is grown by opening the shutter to the P source for 200 seconds, which
produces a doping density of ?10
14
Patoms/cm
2
. The heterostructure is
completed by growing a 50 nm cap layer with the substrate temperature held at T
= 400
?
C. The SIMS analysis for the ?-layer produced by this process was shown
in Figure 3.2. The most obvious feature of this plot is that the ?-layer itself is not
74
Figure 3.4: Growth sequence of ?-doped sample incorporating both substrate cooling
and an Si trap layer. This technique produces the sharpest ?-profiles observed in
this work. Bu?er 1 corresponds to 150 nm of Si grown at T = 650
?
C, and bu?er 2
to 18 nm of Si grown as T was reduced to 320
?
C. The ?-layerand4nm?traplayer?
were grown with T = 320
?
C, while the 50 nm cap layer was grown at T = 500
?
C.
All magnetoconductance data presented in this thesis were taken on samples using
this type of growth sequence. Layer thicknesses are again not to scale.
very sharp, but rather displays a skewness or ?straggle? towards the surface. This
is segregation, and is especially pronounced in Si:P. The low growth temperature
used (400
?
C) was intended to minimize this segregation, but clearly was not
entirely e?ective.
The best growth technique used in this work is shown in Figure 3.4. Two
bu?er layers are grown to isolate the ?-layer from the substrate, the first being a
layer 150 nm thick grown at T = 650
?
C, the second a 18 nm layer grown as the
75
substrate temperature was reduced from 650
?
C to 320
?
C over the course of three
minutes. The ?-layer was then grown at 320
?
C by opening the P shutter for 200
seconds. The next step is the most crucial part of the entire process: 4 nm of
undoped Si were grown on top of the delta-layer at 320
?
C in an e?ort to ?trap?
the P donors to a narrower region than had been obtained by the methods of
Figure 3.3. Incorporating this step was motivated by [65]. All growth (Si and P)
was then stopped, and the substrate temperature was raised from 320
?
C to 500
?
C
over a 3 minute interval. Finally, a 50 nm cap layer was grown to complete the
process. A SIMS analysis from a sample of this wafer is shown in Figure 3.5.
Clearly something rather dramatic has occurred. Whereas when using the
?old? growth method the sample showed a great deal of segregation, the ?new?
sample shows almost none. In addition, the doping profile is much more symmetric
about its maximum doping density level, and in general appears to be much more
??-like?. These are two very significant improvements to the quality of the
?-doping profile, and can be traced back to the implementation of the trap layer in
the growth sequence. A heuristic explanation for this e?ect goes somewhat as
follows. Since segregation is more pronounced at higher temperatures, it is
desirable to grow the ?-layer at as a low a temperature as possible. Unfortunately,
the growth of high?quality c-Si has just the opposite temperature dependence:
the higher the Si growth temperature, the fewer intrinsic lattice defects (vacancies
and interstitials) are produced [20]. It is also well-known that di?usion is enhanced
by the presence of such defects in the crystal [85]. Thus we see that growing Si at
a higher temperature will reduce the intrinsic di?usivity of P in Si, and
76
Figure 3.5: SIMS analysis of a ?-doped wafer grown with the 4 nm thick trap layer.
A much narrower and more symmetric ??-like? profile is obtained, showing the
importance of the trap layer for Si:P ?-layer growth.
77
consequently will help to produce a narrower ?-layer because the P di?uses less in
the c-Si lattice. These two opposing requirements are balanced by the trap layer.
By growing a narrow (? 4 nm) Si trap layer on top of the ?-doped region and then
heating the substrate, the P donors do segregate to the surface, but no farther:
they are bound to this region. Thus the name ?trap layer? is highly suggestive of
its function in that it limits the P segregation to a well-defined portion of the
crystal. The cap layer can then be grown at a higher temperature (T = 500
?
C)
without being overly concerned with P segregation. It should also be mentioned
that a sample with a lower defect density generally has a higher electronic
mobility, which is often desirable in nanostructure devices.
This insight presents interested researchers with a wide range of potential
avenues for investigation. For example, does the ?-layer width correlate strongly
with the trap layer width? That is, if the trap layer were grown to be only 1, 2 or
3 nm thick, would this correspondingly produce a narrower ?-profile? Another
interesting question is precisely what temperatures and growth rates optimize the
process? In other words, can true Dirac-delta (monolayer)dopingprofilesbe
achieved in Si:P using MBE? The available parameter space to be explored is
enormous, and will not be discussed further here. Instead, we shall content
ourselves to simply note that very sharp, ?-like doping distributions can indeed be
achieved in MBE grown Si:P ?-layers, provided that a trap layer is incorporated in
the growth process. For reference, a step-by-step algorithm to grow a high-quality
?-layer is included below.
78
1. Clean p-type Si(100) substrate (includes NRL clean and 2.5% HF dip to
remove native oxide) and place in MBE chamber.
2. Increase substrate temperature to T = 650
?
C for bu?er layer growth.
3. Bu?er layer growth: ? 150 nm grown at T = 650
?
C. Si growth rate is ? 0.1
nm/s.
4. Reduce substrate temperature to T = 320
?
Coverthreeminutesfor?-layer
growth. Total Si growth in this interval is ? 18 nm.
5. Grow ?-layer with substrate temperature at T = 320
?
C. A 200 second
exposure to the P source produces a doping density of ? 10
14
/cm
2
.
6. Trap layer growth: 4 nm of Si grown at T = 320
?
C.
7. No growth of either Si or P as substrate temperature is increased from T =
320
?
C to T = 500
?
Coverthreeminutes.
8. Cap layer growth: 50 nm of Si grown with T = 500
?
C.
Once a ?-doped wafer has been prepared as described, the next step in the
experimental investigation is to prepare mesoscopic devices for measurement. This
fabrication process will now be described.
3.3 Device Fabrication
One of the primary goals of the fabrication process is to keep the sample as
free from contaminants as possible. Therefore, a substantial number of the
79
processing steps are devoted to ?cleaning? the device. In addition, the samples are
inspected optically throughout the process to be sure that nothing obvious has
gone awry. Even if things ?look? fine with the device (i.e. no defects are
apparent), the only meaningful check is actual electronic testing. Such testing will
be discussed at the end of this chapter.
Before any processing is done to a sample of the ?-doped wafer, it is advisable
to simply check whether a piece of it conducts at low (liquid helium) temperatures.
This is done by ?scribing? a small piece o? of the wafer, placing 4 In solder
?blobs? on the edges of the chip, connecting Au wires to the blobs, attaching the
chip to a DIP (dual inline pin) header, and dipping the sample in liquid helium. If
thesampledisplaysalowtwo-terminal resistance (R ? 10 k?) between any two
contacts, then the enumerated procedure described below can be followed to make
an actual device. If this test does not work, i.e. the two-terminal resistances are
very high (R ? 1 M?), then obtaining a new sample is in order, rather than going
through the laborious process of fabrication only to find, in the end, that the
device does not work!
Here, then, is a step-by-step algorithm for producing a Hall bar device suitable
for measurement.
1. Soak wafer in concentrated HCl to remove any residual ?scum? from
previous handling (? 2 hours).
2. Dice wafer into approximately 5 mm x 5 mm pieces for use in actual device
fabrication. The DIP header on which completed devices were mounted has
80
awidthof?7 mm, therefore the dies used must be smaller than this.
3. Non-chlorinated (NC) solvent clean. In order, the chip is sprayed with
acetone, methanol, and isopropanol alcohols, then ?blow-dried? with N
2
gas.
4. Spin on resist. Typically Futurrex NR7 1500 P negative resist was used. The
resist was spun at 5000 rpm for 1 minute.
5. Bake sample in oven at T = 120
?
C for 1 minute to perform initial ?cure? of
resist.
6. Perform electron beam lithography using the scanning electron microscope
(SEM) on the resist to define a device pattern. The reason e-beam
lithography was used is twofold. For one, it is very easy to make changes to a
given pattern or design entirely new devices by using the NPGS software
which controls the SEM. Photomasks lack this flexibility. Second, by using
the SEM, annoying issues associated with photomasks are avoided, such as
proper contact between the sample and mask or the sample sticking to the
mask. In a nutshell, the use of e-beam lithography makes the fabrication
process more reliable. For reference, a typical exposure dose was ? 10
?C/cm
2
.
7. Post-bake sample in oven at T = 120
?
C for 1 minute for final ?cure? of
resist.
8. Develop resist in appropriate solvent (RD6 for Futurrex negative resists).
9. Dip in de-ionized (DI) water to remove solvent from exposed sample surface.
81
10. Etch device pattern into ?-doped sample using RIE (Reactive Ion Etch). An
SF
6
plasma was used to etch the Si 20-30 nm below the ?-layer in order to
expose an edge of the doping plane for subsequent electrical contact. A
?mesa? structure results from this etch (see Figures 3.6 and 3.7).
11. Clean resist from sample by soaking in a ?piranha? bath ?30 minutes. The
piranha bath consists of a mixture of H
2
S0
4
and H
2
O
2
in a 3:1 ratio.
12. Measure the mesa profile to ascertain how much Si the SF
6
etch removed.
The ?-layer should nominally lie somewhere near the middle of the ?wall?
profile of the mesa. For example, if the chip were from the same wafer as the
SIMS analysis of Figure 3.5, the wall should have a height of at least 60-75
nm.
13. Perform 0
2
plasma clean (?descum?) in RIE to further remove residual resist
(and possibly other) contaminants.
14. Dip sample in concentrated HF solution to remove the SiO
2
produced by the
O
2
plasma of the previous step.
15. Clean ?polymer scum? (CF
x
, Si-C, Si-F, etc.) o? of sample surface left as a
result of the SF
6
etch. This is done by placing the sample in a bath of the
?barracuda? clean, a mixture of NH
4
OH, H
2
0
2
,andH
2
0 in the ratio 1:1:5 for
? 30 minutes [67]. This barracuda cleaning step is crucial, since the polymer
scum otherwise causes contacts to the ?-layer to become insulating at low
temperatures, making electrical measurements impossible.
82
16. Dip sample in DI water, and perform NC solvent clean.
17. Spin and bake resist for second layer of lithography as in steps 4 & 5.
18. Perform optical lithography to define ??-contact? pads (see Figure 3.7). The
resist exposure is done with a photomask and ultraviolet light on a contact
aligner.
19. Post-bake and develop resist as in steps 7 & 8.
20. Dip sample in HF:H
2
O (1:100) solution to remove any oxide which has
accumulated at the surface of the ?-layer.
21. Perform first metalization. In a thermal evaporator, deposit ? 2.5 nm of Cr,
then enough of the Au-Sb (99:1) alloy to exceed the mesa wall height by
10-20%. The reason Au-Sb is used is to lower the Schottky barrier
associated with the metal-semiconductor contact.
22. ?Lift-o?? resist and unused Au-Sb in RR2 solvent. For this step, the RR2
has T = 100
?
C, and the sample is allowed to soak for ?20 minutes.
23. If the RR2 lift-o? still has not removed all of the unused Au-Sb, place the
sample in a beaker of DI water and subject the beaker to ultrasonic
treatment for 1 minute intervals until the lift-o? is complete.
24. Anneal the sample in a rapid thermal annealer (RTA) for 5 minutes at T =
450
?
C. The ambient atmosphere in the RTA is a combination of N
2
and H
2
gas. The purpose of the annealing is to di?use the Au-Sb alloy into the
83
?-doped region of the sample and further lower the Schottky barrier. This
step is important for establishing an Ohmic electrical connection with the
?-layer (i.e. a connection which displays a linear I-V curve).
25. NC solvent clean.
26. Spin and bake resist for third and final layer of lithography.
27. Perform optical lithography (including development) to define ?bonding
pads? which overlap with the ?-contact pads (see Figure 3.7).
28. Perform second metalization. In a thermal evaporator, deposit ? 2.5 nm Cr
and ? 100 nm Au.
29. Lift o? using RR2 as in step 22.
30. NC solvent clean.
31. Connect Au wires to bonding pads with a wire bonder.
32. Fasten sample to DIP header using either GE varnish or vacuum grease.
33. Use silver paint to finally connect Au wires to DIP header contacts. The
reason for using Ag paint is that the header pins cannot be directly wire
bonded to, and the use of a solder such as In generally causes magnetic
shielding due to the Meissner e?ect (to be discussed more fully in Chapter 4).
Generally speaking, this process yields high quality devices for
measurement. Figures 3.6 and 3.7 show the evolution of the sample from
84
Figure 3.6: A ?horizontal profile? of the device fabrication process flow. In a),
the device?s dimensions are defined by electron beam lithography. In b) the device
pattern is etched into the ?-doped Si with an SF
6
etch. Electrical contacts are made
via 2 metalizations as in c), and finally wires bonded are to the metal in d). At this
point the device is ready to be mounted on a header and tested at T = 4.2 K.
85
(a)
(b)
(c)
Figure 3.7: Approximate device evolution following process flow sequence of Figure
3.6. First the ?mesa? is defined with electron beam lithography and a dry SF
6
etch,
as in (a). Two metalizations follow to make electrical contacts with the ?-layer. The
first evaporation is Au-Sb (b), for the actual ??-contacts?, the second (c) is pure
Au, which overlaps the Au-Sb contacts and provides a larger surface area (? 250
?m x 250 ?m) for wire bonding.
86
horizontal and vertical perspectives, to give the reader a visual feel for the
major fabrication steps required.
3.4 Device Testing
Assuming the ?-doped wafer is suitable for use in device construction
(i.e. conducts current at low temperatures, etc.), at this stage the sample
could be mounted on the dilution refrigerator and cooled to base
temperature for measurement. However, it is prudent to perform some final
diagnostic tests to ensure that the device will behave as desired and be
straightforward to measure.
The first such test is a simple ?continuity? check at room temperature to
verify that all probes on the sample are in fact electrically connected with
one another. This is quickly and easily done with a hand-held DVM.
The second check is to ?dip? the sample in a liquid helium dewar and
then re-measure the contact resistances. If there are any open circuits at 4.2
K but not at room temperature, then it is necessary to evaluate what may
have gone wrong in the processing to cause this. Depending on the intrinsic
resistivity of the sample, the voltage probes should display a low resistance
between each other, and the resistance measured between adjacent contacts
around the sample should be very nearly equal.
The third test is to establish whether or not any conduction through the
87
substrate is taking place at 4.2 K. This is accomplished by the use of an
ancillary contact, connected to the ?-doped device only through the
substrate. The resistance between this ?floating? contact and any probe on
the sample should be infinite, or at least much higher than the resistance
between any two probes on the device. Otherwise, the current in the sample
is not confined to the lithographically defined device region, and resistance
measurements become meaningless.
Finally, the sheet resistance (resistivity) can be checked with a four-wire
measurement to get a rough idea of what the signals will be like at base
temperature of the dilution refrigerator. This merely serves as a benchmark
for the more accurate magnetoresistance measurements to be performed,
which will be discussed in the next chapter.
88
Chapter 4
Measurement
In this chapter I will discuss how four-wire conductance measurements
on ?-doped devices were performed. The necessary background is developed
by discussing the conductivity tensor for two-dimensional systems, the
instrumentation used in making the measurements, the particular techniques
employed for obtaining data, and the systematic errors involved in these
experiments.
4.1 Background
A proper 4-wire measurement [68] of a resistance R
s
is depicted in
Figure 4.1 (a). The device is biased with a known current I using one pair of
wires, and the voltage V induced by the current is measured with a second
pair of wires. R
s
is given by Ohm?s law, i.e. by forming the ratio R
s
=V/I.
89
Typically the current source is realized by placing a large, known resistance
R
L
in front of a known voltage source V
0
. The much smaller resistance of R
s
is then placed in series with this arrangement, as shown in Figure 4.1 (b).
Since R
L
dominates the circuit impedance (R
L
greatermuch R
s
), the current I is
e?ectively given by I = V
0
/R
L
. Voltage measurements across R
s
are
generally made using the combination of a di?erential preamplifier and a
digital voltage meter (DVM). This high input impedance arrangement limits
?shunting? of current from the device, which would otherwise corrupt the
measurement. Obviously, the more accurately V and I are known, the better
the measurement.
As straightforward as this 4-wire technique is, it is also incredibly
powerful. Since the resistance of a device depends on numerous parameters
of physical interest, such measurements can be used to extract useful
information about a particular electronic system. For example, the
conductance measurements on ?-doped devices discussed in this thesis
depend on ?external? variables such as the sample dimensions, temperature
T, magnetic field H, current density J, and applied electric field E,aswellas
the frequency of these fields, ?
i
. Quantities ?internal? to the ?-layer, such as
the carrier density n, mean free path lscript
e
, dephasing length lscript
?
, e?ective mass
m
?
, and, of course, the ?-layer thickness T , can be deduced through the
application of appropriate theoretical models to properly carried out
resistance measurements. It should also be noted that resistivity (?) refers
90
Figure 4.1: 4-wire measurement schematic. In (a), a current bias, I, is passed
through the device, and a voltage V is measured. (b) shows a practical realization
of this, which incorporates a voltage source V
0
in series with a large resistance
R
L
to serve as the current source. The resistance of the sample, R
s
, is deduced
by measuring the voltage V
s
across the sample with a di?erential preamplifier and
DVM, then forming the ratio R
s
=V
s
/I.
91
to the intrinsic electrical characteristics of the material system under study.
Given ? and the geometry of a given device, one should be able to predict
what resistances will be measured under a given set of conditions. I shall
discuss this in more detail later.
An elaborate theory of conductance measurements and their symmetry
properties has been developed over time. We cannot go into this in any
detail here, but merely cite a few results pertinent to this thesis. Since this
work concerns magnetoconductance measurements of c-Si, Onsager?s result
[69] that the local resistivity tensor of a cubic crystal obeys the symmetry
relation ?
ij
(H)=?
ji
(-H) suggests that an analogous (though not identical)
expression should hold here. Landauer?s many contributions to the modern
understanding of conduction processes include quantifying how local
variations of E in the conductor due to scatterers influence transport [70],
modeling conduction using the basic quantum mechanical notions of
reflection (R) and transmission (T) coe?cients, and generalizing these
notions to real, multichannel conductors [71]. van der Pauw?s result [72],
showing that the resistivity of an arbitrarily shaped conductor could be
found through a conformal mapping, further established the power of 4-wire
measurements, and will be used extensively in Chapter 5. Finally, B?uttiker?s
reciprocity theorem [73] regarding 4-wire measurements in the presence of a
magnetic field H leads to the global symmetry relation R
ij,kl
(H)=
R
kl,ij
(-H). In this expression, the first pair of indices refers to the current
92
leads, the second to the voltage leads. Thus a perfect conduction symmetry
is observed in a device under magnetic field reversal, if, in addition, the
current and voltage leads are also exchanged with one another. I shall
discuss this further in Chapter 5 as well.
4.2 The Two-Dimensional Conductivity
Tensor
Since the electron system associated with a ?-layer is inherently
two-dimensional, in general a tensorial relationship will exist between the
electric field E (related to the measured voltage V by V = -
integraltext
E?dl)andthe
current density J in such a device. It is immediately apparent that the units
of J in two dimensions are (amperes/meter) instead of the usual
(amperes/meter
2
), as it is for three-dimensional conductors. I shall elaborate
on this shortly.
In this work, R = V/I is a function of both magnetic field H and
temperature T:
R(H,T)=
V (H,T)
I(H,T)
. (4.1)
Such an expression makes the use of the term magnetoresistance obvious,
and is really the ?scalar? version of the aforementioned tensorial relation. In
order for this expression to be meaningful, a linear response must be
observed between E and J, and therefore a more precise definition would be
93
R = dV/dI. This linear dependence was observed in all of the samples
discussed here.
Depending on the causal point of view one wishes to take, one can consider
either the resistivity ? or the conductivity ?. In the first case the application
of E ?causes? J to flow in the device, and Ohm?s law takes the form
J = ?E. (4.2)
One can also take the ?inverse? approach, and write
E = ?J. (4.3)
Accordingly, ?? = 1,where1 is the 2x2 identity matrix.
For a Cartesian two-dimensional system, equation (4.3) is expressed in
matrix notation as:
?
?
?
?
E
x
E
y
?
?
?
?
=
?
?
?
?
?
xx
(H,T) ?
xy
(H,T)
?
yx
(H,T) ?
yy
(H,T)
?
?
?
?
?
?
?
?
J
x
J
y
?
?
?
?
. (4.4)
This can be further simplified for the devices investigated here, since not all
of the components ?
ij
were found to be independent. The diagonal elements,
?
xx
and ?
yy
equal one another: ?
xx
= ?
yy
? ?. This is easily understood due
to the fact that the samples were fabricated from (100) Si, and all <100>
directions in Si are equivalent [20]. Furthermore, the o?-diagonal
components were found to be antisymmetric: ?
xy
=-?
yx
.Thisisalsoeasy
to interpret, by considering how the Lorentz force (F =qv ? H)actsona
charge q moving with velocity v. If we assume q > 0 and motion along the
94
+?x direction, the Lorentz force deflects q towards -?y, in e?ect experiencing
an e?ective electric field in that direction. On the other hand, for motion
along +?y, the deflection (and corresponding e?ective electric field) is in the
+?x direction. Thus I arrive at the final form of the resistivity tensor for
?-doped silicon:
?
?
?
?
E
x
E
y
?
?
?
?
=
?
?
?
?
?
xx
(H,T) ?
xy
(H,T)
??
xy
(H,T) ?
xx
(H,T)
?
?
?
?
?
?
?
?
J
x
J
y
?
?
?
?
. (4.5)
Thus there are only two independent components of the resistivity
(conductivity) tensor for ?-layers in Si.
4.2.1 Hall Bar Devices
To see how these notions apply to actual device geometries, consider
Figure 4.2, which shows a schematic of a typical Hall bar device constructed
for use in ?-layer transport studies. The ?current channel? has a width W,
and the voltage contacts are spaced a distance L apart. There are essentially
only two independent resistance measurements one can perform on such a
device. One is the ?longitudinal? resistance, when the induced voltage V
WL
(for ?weak localization voltage?) is measured between two adjacent contacts
on the same side of the device (i.e. parallel to the current bias). The other is
the ?transverse? resistance where the induced voltage V
H
(?Hall voltage?) is
measured between contacts directly across the sample from one another
(perpendicular to the current bias). Other resistances on the sample will be
95
Figure 4.2: Hall bar measurement schematic. A current bias, I, is passed through
the device, and two voltages are measured, as discussed in the text. The weak local-
ization voltage, V
WL
,ismeasuredparallel to I, the Hall voltage, V
H
, perpendicular
to I. When the voltage contact spacing, L, is equal to the sample?s width, W, the
measured longitudinal resistance is the device?s intrinsic resistivity in units of ?/a50.
linear combinations of these ?basis? measurements.
It should be clear from the figure that V
WL
scales like V
WL
? (L/W)I,
whereas V
H
is independent of L and W (see Appendix A for a derivation).
Thus the significant resistances of the device given a rectangular geometry
are R
WL
= (W/L)(V
WL
/I), and R
H
=V
H
/I. If the current channel width W
equals the voltage contact spacing L, W/L = 1, and the measured
longitudinal resistance is by definition the two-dimensional resistivity: R
WL
96
= ?. This two-dimensional ? is often referred to as the sheet resistivity,
symbolized by R
a50
, and has dimensions of ?/a50 (?ohms per square?), as
opposed to ??cm for three dimensional samples. The set of equalities R
WL
=
? =R
a50
will be used interchangeably throughout the remainder of this
dissertation. Given this two-dimensional definition of resistivity, we now see
that in order to be dimensionally consistent, the current density J must have
units of (amperes/meter), as mentioned earlier.
4.2.2 van der Pauw Devices
Not all of the samples fabricated in the course of this work used the
rectangular geometry of Figure 4.2. Such devices, with arbitrary geometries,
are collectively referred to as van der Pauw devices. A general van der Pauw
sample is shown in Figure 4.3, and is simply a device with 4 contacts (labeled
A, B, C, D) placed on its perimeter. Given a device with such an arbitrary
geometry, it is clear that if a current is passed between any two electrodes,
the current density will be non-uniform throughout the sample. This is very
di?erent from a Hall bar, and given such non-uniformity, one might at first
assume that being able to determine the samples conductivity tensor would
be a di?cult task. Fortunately, this is not the case, and a conformal mapping
exists which translates the resistances measured on these arbitrarily shaped
samples to that of ?
xx
and ?
xy
as measured on the standard Hall bar device
of Figure 4.2. This result, mentioned earlier, was established by L.J. van der
97
Figure 4.3: Arbitrarily shaped van der Pauw device. The electrodes A, B, C, D
must be placed on the devices perimeter to apply van der Pauw?s theorem.
Pauw in 1958 [72], and has proved extremely useful in this work.
To extract ? from such a sample such as the one shown in Figure 4.3 one
must measure two appropriate resistances, labeled R
AB,CD
and R
AD,BC
,then
numerically solve the following equation for ?:
exp(
??R
AB,CD
?
)+exp(
??R
AD,BC
?
)=1. (4.6)
The notation here is such that R
AB,CD
refers to a resistance measurement
where current is sourced and drained in leads A and B, while voltage is
measured between contacts C and D (similarly for R
AD,BC
). The key point is
that if one is interested in measuring ?
xx
, the current contacts in both
R
AB,CD
and R
AD,BC
must be adjacent to one another. To perform a Hall
measurement, the current contacts lie ?across? from one another, and the
usual relationship between magnetic field H and carrier density n is
obtained, i.e. ?
xy
=R
AC,BD
= IH/ne [74].
98
One set of experiments I will discuss in Chapter 5 used square van der
Pauw samples, with the contacts placed at the vertices of the square. The
appropriate measurement configurations are shown in Figure 4.4. Obviously,
such a perfectly symmetrical device should yield very nearly the same
resistance for all ?
xx
measurements, and the same slope for ?
xy
measurements. Letting R
12,34
=R
13,24
= R and substituting into
equation (4.6) yields ? = ?R/ln(2) similarequal 4.532R.
Now that the basic notions of 2DEG transport measurements have been
established, I shall discuss the systematic details of a typical
magnetoconductance investigation.
4.3 Measurement
Ultimately, the results of this thesis can be reduced to establishing the
interrelationships between four physical quantities: an electric field, E,a
magnetic field, H, a current density, J, and a temperature, T. In abstract
terms these quantities define the appropriate ?phase-space? for modeling
electronic transport in ?-layers, and this work, in a loose sense, establishes
the appropriate ?metric? on this (E,H,J,T) space. In what follows I shall
frequently refer to this idea. First, I will discuss the various instruments and
apparatus which were used in my experiments.
99
(a)
(b)
Figure 4.4: Square van der Pauw device and appropriate configurations for mea-
surement of the weak localization (a) and Hall (b) signals. In a perfectly symmetric
device, all the measured weak localization signals will have the same magnitude, and
the resistivity is related to these measured resistances (R = V/I) by the relation ?
= ?R/ln(2) similarequal 4.532R.
100
4.3.1 Instrumentation
Resistance measurements were performed in an Oxford Kelvinox 25
dilution refrigerator which cooled the devices to the low temperatures
necessary to observe weak localization. After extensive repairs and
modifications (discussed in Appendix C), the lowest achievable temperature
of this instrument was T similarequal 130 mK. The temperatures of the 1 Kelvin pot,
mixing chamber and still were determined via 4-wire measurements on
ruthenium oxide (RuO
2
) thermometers (produced by Lakeshore cryotronics).
These measurements were performed continuously by a Neocera LTC-21
temperature controller, using the calibration curve (resistance vs.
temperature) supplied by Lakeshore. Although no definitive measure of the
instrument?s cooling power at base temperature was made, it seems
reasonable to assume it was at least as large as the 50 ?W specified by
Oxford for operation at T = 50 mK.
The devices were mounted on a tilting stage in the refrigerator whose
orientation was controlled mechanically via a hermetically sealed
feedthrough. By rotating this stage, the angle ? between J and H could be
placed at an arbitrary angle between 0
?
and 90
?
, as depicted in Figure 4.5.
Theprecisevalueof? was determined using the Hall e?ect and the relation
for the Hall voltage, V
H
=IHsin(?)/ne, derived in Appendix A.
Electrical connections to the samples were made with coaxial cables via a
?breakout box? at room temperature equipped with 20 BNC connectors.
101
Figure 4.5: Tilting stage diagram. The current density J (proportional to the bias
current I) lies in the plane of the sample, and the magnetic field H is fixed along the
z-axis. The stage orientation is controlled mechanically via a hermetically sealed
feedthrough, and the angle ? is determined through a measurement of the Hall
resistance: R
H
=Hsin(?)/ne.
Each of these 20 measurement lines was filtered in two stages, as depicted in
Figure 4.6. At room temperature, a ??? section filter with a cuto? frequency
of 1 MHz was used. Following this, RC filters with a cuto? frequency of 10
kHz were implemented at the 1 Kelvin pot.
Contained in the same dewar as the dilution refrigerator was an Oxford
cryomagnetic system. The maximum field this magnet was capable of
reaching when immersed in liquid helium (T = 4.2 K) was H =12T.The
homogeneity of the field (as specified by Oxford) was 1 part in 1000 over a 1
cm
3
spherical volume, at the center of the magnet bore. The power supply
controlling the applied field H was equipped with an analog output voltage
102
Figure 4.6: Filtering schematic. Two stages of electronic filtering were incorporated
to isolate the devices from high frequency noise. At room temperature (T similarequal 300 K)
alow-pass??? section filter was used, with a cuto? frequency of 1 MHz. At T similarequal 1
Kelvin, a low-pass RC filter was used, with a cuto? frequency of 10 KHz. Twenty
such lines comprise the total number of leads available for device measurement at
base temperature, T similarequal 130 mK.
103
connection, whose output signal was proportional to H.AsH was varied
(usually ?swept? at a constant rate
?
H), this output voltage was recorded
with a DVM to determine the field applied to the sample.
In order to determine a suitable (low-noise) test frequency and investigate
whether any anomalous signals were present in the experiment, a Stanford
SR785 dynamic signal analyzer was used to measure the spectral density of
voltage noise in the measurement configuration. Depending on the voltage
signal levels induced in the devices due to current bias and magnetic field
configuration (recall that ?(H
bardbl
)isalwayslessthan?(H
?
) in a given device),
the signals were often preamplified and further filtered using battery-powered
Stanford SR560 di?erential preamplifiers, with their bandwidth limited to ?
300 Hz. Figure 4.7 shows a typical noise spectrum, obtained by connecting
two weak localization voltage contacts to the A and B inputs of the preamp,
and feeding the ouput signal into the spectrum analyzer. No appreciable
di?erence was observed between this spectrum and one taken with the A and
B inputs shorted together. Consequently the noise shown in Figure 4.7 is
actually the input referred voltage noise of the preamp itself.
The voltages V
WL
and V
H
, as well as the bias current I, were measured
using Perkin-Elmer digital signal processing (DSP) 7280 lock-in amplifiers.
The principal advantage of using lock-in detection for conductance
measurements lies in the ability to reduce the noise bandwidth of the
measurement, as well rejecting out of band noise, by increasing the
104
Figure 4.7: Noise spectrum from a ?-doped sample. The spectrum shows the voltage
noise V
N
( in units of V/
?
Hz) as a function of frequency present in the measurement
configuration. A typical test frequency used was f = 2?eHz? 17.08 Hz, which
was su?ciently far from the low-frequency (1/f) portion of the spectrum to allow
measurements with good signal-to-noise ratio (SNR).
105
measurement time constant, ?, and thus increasing the signal-to-noise ratio
(SNR) in the experiment. Also equipped with an analog output voltage
which was proportional to the signal level at the input of the lock-in, the
data from these instruments were also recorded using a DVM. The coaxial
cables connecting the device (BNC) contacts to the measurement
instruments were configured using the standard ?twisted pair? configuration
to minimize noise pickup due to time-varying magnetic fluxes, especially at
60 Hz. The DVMs used were Agilent 34401A 6-1/2 digit multimeters.
To record a typical magnetoconductance spectrum, automated data
acquisition was performed using a personal computer (PC) with LabView
(version 5.1) software. The particular ?VI? (?virtual instrument?) used for
recording data was rather simple in that it merely queried the DVMs and
temperature controller for their readings at a user-specified frequency. I
typically used an acquisition rate of 1 data point every 5 seconds. At a
minimum, each data point consisted of four numbers, corresponding to the
(E,H,J, T) parameters discussed earlier. The PC was connected by a GPIB
cable to an IOtech 488/F bus extender, which converted the GPIB
commands (voltage pulse sequences) into photonic signals (light pulses).
These photonic commands were then transmitted via fiber optic cable to a
second IOtech extender, which was in turn connected to the measurement
DVMs via GPIB cables. A schematic of this situation is shown in Figure 4.8.
106
Figure 4.8: Data acquisition schematic. The lock-in amplifier measures the voltage
across a resistance, and the analog output of the lock-in is measured by a DVM.
GPIB commands from the PC, which controlled the data acquisition, were converted
to light pulse signals with the IOtech extender, transmitted via fiber optic cable
to another extender, and converted back to GPIB commands to read the DVMs.
This arrangement eliminated all electrical connections between the computer and
measurement instruments.
107
4.3.2 Grounding
Given the electrical noise generated by the PC, the fiber optic cable was
used to break all electrical connections with this instrument, eliminating
high-frequency noise, cross-talk, ground loops, etc. In addition, several other
steps were taken to further isolate the measurement system from unwanted
noise sources.
Each pumping line connected to the dilution refrigerator was clamped
with non-metallic (insulating) centering rings and clamps, which eliminated
electrical connections with the pumps associated with the 1 Kelvin pot and
the
3
He:
4
He mixture. All instruments were powered through a single ?mains?
120 V AC power line, which was first fed through a Topaz ultra-isolation
transformer. The ?earth ground? connection of this transformer was
removed, and it?s case ?hard wired? to the earth ground of the magnet power
supply. This served as the single point ground for the entire experiment,
including the cryostat dewar. To prevent ground loop currents from flowing
through the BNC shells of the lock-in input BNCs, the coupling inputs were
operated in ?float? mode, which connected the shells of the BNCs to the
chassis ground of the lock-ins via 1 k? resistors, establishing a fairly high
impedance path to ground. This high impedance would help to minimize any
residual issues with ground loops not dealt with by the previous methods.
While measurements were being recorded, every e?ort was made to power
down all unnecessary instruments (level meters, magnet thermometers,
108
high-powered lasers in the neighboring lab, etc.) in order to operate in as
quiet an environment as possible. Although these e?orts were largely
successful, I am certain that the system could be further optimized.
I now turn to the experimental procedures followed in acquiring
magnetoconductance data.
4.3.3 Experimental Procedure
The following steps comprise a fairly typical sequence of measurements I
performed on ?-doped devices in my work.
After the sample was mounted on the tilt stage in the dilution refrigerator
and cooled to base temperature, a noise spectrum was recorded to determine
a suitable measurement frequency. A typical example of such a measurement
is shown in Figure 4.7. As mentioned previously, the noise shown in Figure
4.7 is actually the input referred noise of the amplifier itself (we shall see
later that the noise produced by the samples was generally less than this).
For whimsical reasons, the test frequency was often chosen to be 2?eHz(similarequal
17.08 Hz). This frequency is su?ciently low that the measurements in this
thesis can in essence be considered as DC resistance measurements.
After this, the bias current was varied (from approximately 10 nA to 1
?A) to determine whether or not the V
WL
vs. I curve was linear (i.e.
whether or not the device was ohmic). If the device displayed ohmic
behavior, a linear fit to the data was performed, and from the slope of this
109
fit, the H = 0 resistivity of the device was determined. An example of this is
shown in figure 4.9. If the curve was nonlinear, the sample generally was not
investigated further, due to the di?culty of interpreting such data using the
theory of weak localization. Assuming a linear response was observed, the
bias current was chosen to be as large as possible in order to maximize the
SNR, but still low enough such that no heating was apparent (as observed on
the mixing chamber thermometer). Oftentimes I = 1 ?A was found to be a
suitable bias current, with heating usually becoming evident in the
neighborhood of I similarequal 5 ?A.
Next, a large magnetic field (|H|similarequal5 T) was applied to the sample in
order to establish the H = H
?
configuration. Since the Hall voltage V
H
in a
field H is given by V
H
=IHsin(?)/ne, it is clear from this expression that
V
H
attains it?s maximum value when ? =90
?
, i.e. when I and H are
perpendicular. By gradually rotating the tilt stage and tracking the
dependence of V
H
on the stage position, H = H
?
was easily established.
At this point meaningful magnetoconductance data for H = H
?
could be
acquired. The usual way this was done was to first set H = -1 T, sweep the
field down to H = 0 while recording data, then set H = +1 T and sweep to
0 again. Thus positive field data and negative field data were recorded as
separate files. The reason for ?partitioning? the field in this way was due to
the unfortunate fact that the magnet power supply was really a ?unipolar?
current source. In other words, H could not be continuously swept through
110
Figure 4.9: Data showing ohmic behavior in an Si:B ?-doped sample. The slope of
the line through these data points gives the resistivity of the device, in this case R
a50
similarequal 2.5 k?/a50.
111
Figure 4.10: Typical magnetoconductance weak localization data (points) with the-
oretical fits (lines) for Si:P ?-layers. The upper curve corresponds to H
?
measure-
ments, the lower to H
bardbl
.
112
zero. If one attempted to do so, when changing field polarities an audible
?clunk? could be heard as the power supply mechanically switched the
polarity of the magnet leads. This wreaked havoc with the measurements,
producing unseemly spikes in the data. This probelm was eliminated by
taking the two sets of data separately, and the full magnetoconductance
spectra shown throughout this thesis were constructed by ?sewing together?
the data for H ? 0, and H ? 0.
The sweep rate of the magnetic field (
?
H) was varied to find a rate slow
enough to render any heating e?ects (due Faraday induced eddy currents)
negligible, but fast enough to allow a quick turnaround in data acquisition.
In addition, depending on the time constant (?)ofthelock-inusedinagiven
measurement, the sweep rate needed to be su?ciently slow to allow the
lock-in to ?track? the voltage signals. Good data was obtained using
?
H=
0.025 Tesla/minute, ? = 5 seconds, and was found to be compatible with
results for measurements taken with smaller values of
?
H. The noise levels in
the system were su?ciently low that values of ? greater than 5 seconds were
seldom necessary. Typical data for ?(H
?
)and?(H
bardbl
)with|H|?[-1,+1] Tesla
are shown in Figure 4.10.
After the H
?
data had been recorded, the tilt stage was gradually turned
until the H = H
bardbl
configuration was obtained. Again the Hall e?ect (V
H
=
IHsin(?)/ne) made this determination possible. Now, instead of maximizing
V
H
,theslope of the R
H
vs. H was used to find the angle ? between I and H.
113
Explicitly, we have
R
H
=
Hsin(?)
ne
. (4.7)
Taking the derivative of this expression with respect to H we obtain
dR
H
dH
=
sin(?)
ne
(4.8)
which is (by definition) the slope of the R
H
(H)curve.ForH = H
?
,sin(?)=
1, and I denote the slope in this configuration by s
0
(i.e. s
0
= 1/ne).
Obviously, we learn the value of the carrier density, n, from this
measurement. Upon rotating to a non-perpendicular orientation, the slope of
the Hall curve will be reduced because sin(?) is now less than 1. Let this new
valueoftheslopebes
1
. It is clear that the two slopes are related to one
another by the equality s
1
=s
0
sin(?). Dividing through by s
0
and taking the
inverse sine of the ratio s
1
/s
0
, we obtain the desired expression for the stage
orientation ?:
? = sin
?1
parenleftbigg
s
1
s
0
parenrightbigg
. (4.9)
This analysis should make it clear that when the sample is in the H = H
bardbl
configuration, the slope of R
H
(H)iszero. Figure 4.11 shows several Hall
traces taken as the stage was turned to H = H
bardbl
, along with the appropriate
? extracted from (4.9) for each trace. It is worth noting here that no
evidence for the stage ?drifting? over time was observed, so that once it was
rotated to a particular position, it stayed there.
?(H
bardbl
) data was taken in precisely the same fashion as ?(H
?
), as
previously described. A complete data set for both configurations was shown
114
Figure 4.11: Hall data showing the angle ? between J and H, as determined by
equation (4.9). For ? = 0, a horizontal line is produced.
115
in Figure 4.10.
Once the ?(H
?
)and?(H
bardbl
) data sets had been recorded, some final
temperature-dependent measurements were performed to investigate the
influence of the electron-electron interaction in the samples. By applying the
maximum amount of power to both the mixing chamber (20 mW) and the
still (50 mW), the mixture was removed from the dilution refrigerator and
the unit ?warmed up? to slightly above T = 4.2 K. At this point, the powers
were reduced to their typical base temperature operating levels (0 for the
mixing chamber, ? 1.5 mW for the still), and a fairly low value of ? set on
the lock-ins (? ? 1-2 seconds). Data acquisition was begun with T ? 4.2 K,
and the refrigerator was cooled in the usual fashion to base temperature.
Once base temperature was reached, data taking was stopped, and the
process was repeated if additional measurements were desired. An example
of such temperature-dependent resistance data is shown in Figure 4.12. The
(logarithmic) resistance increase as the temperature is lowered is another
characteristic of weak localization, as it is for the electron-electron
interaction. Taking such data in the presence of a magnetic field (also shown
in the figure) enables one to distinguish between these two phenomena, as I
discuss in Chapter 5.
One final detail in regard to the resistance measurements discussed in this
dissertation will now be made. Although the configurations shown in
Figure 4.1 were used to perform measurements, at times the limitations of
116
Figure 4.12: Temperature-dependent resistance data, taken in both zero and non-
zero magnetic fields. Analysis of such curves quantifies the magnitude of the
electron-electron interaction in the device.
117
Figure 4.13: Resistance bridge schematic. A room temperature potentiometer, R
p
was adjusted to balance the magnetoresistance of the sample R
s
at H =0. The
application of an H negationslash= 0 throws the bridge o? balance, inducing a voltage ?V in
the voltage lock-in. A second lock-in measures the current I through the bridge.
the instruments became apparent. In particular, since a DSP lock-in
amplifier was used, the finite resolution of the digital-to-analog converters
(DACs) on these instruments would occasionally reveal the need for greater
measurement sensitivity. (An example of this in a current measurement is
shown in Figure 4.16.) The finite resolution of the DACs was especially
apparent in samples with low values of R
a50
in parallel fields, when the
induced change in the resistance was exceedingly small. To overcome this
deficiency, a custom-made resistance bridge was designed and constructed to
improve the measurements.
A schematic of this bridge is shown in Figure 4.13. The current source for
118
the bridge was supplied by driving an AC voltage from the ?current lock-in?
through a 500 k? resistor, which was then split along the two branches of
the bridge before being recombined and ?drained? (measured) at the lock-in
input. At the top of each branch was a 1 % precision 1 M? resistor. Along
one branch (in the cryostat) was the sample resistance R
s
, and along the
other was the balancing element of the bridge, a variable resistance in the
form of a room temperature potentiometer, R
p
. The idea behind this
arrangement was that the 1 M? resistors at the top of the bridge would
balance the current through each branch to almost exactly equal I/2. The
voltages across R
s
and R
p
(V
s
and V
p
, respectively) were preamplified and
measured with the ?voltage lock-in?, which was operated in differential
mode. The choice of lock-in inputs (A and B) was chosen such that the
voltage di?erence between the two signals, ?V, was given by ?V = V
p
-V
s
= (I/2)(R
p
-R
s
). At H =0,R
p
was adjusted until ?V = 0 was obtained,
i.e., until the bridge was balanced. Turning on a magnetic field of course
causes R
s
to decrease (due to the negative magnetoresistance associated with
weak localization), throwing the bridge out of balance. Since R
p
is
independent of H,forH negationslash= 0, the signal di?erence always satisfied ?V ? 0.
Although only changes in the magnetoresistance are relevant for fitting weak
localization data, for reference the total magnetoresistance of R
s
in a field H
is given by R
s
(H)=R
p
-2(?V/I).
After a particular experimental run had been concluded, the mixture was
119
again extracted from the refrigerator, and the dilution unit was allowed to
warm up to room temperature. The next step in the experimental
investigation involves data reduction and error analysis. Before presenting
these results, however, a brief discussion of the errors in the measurements
themselves is warranted.
4.4 Measurement Errors
This section is motivated by the question, ?What could systematically
go wrong with these measurements??. It is important to distinguish this
concern from other types of errors a?ecting these experiments, including
random errors, such someone (Kenton) bumping the cables during
measurements, device-specific anomalies (some of which will be discussed
here), or theoretical errors in interpreting the data. Two broad ways of
discussing this question are in terms of measurement accuracy (i.e. how close
to the ?true? value was the recorded data?), and precision (how reproducible
was a given result?). Since each recorded data point corresponds to the
specification of a point in the (E,H,J, T) phase space, errors associated
with quantifying these physical parameters shall be considered separately.
120
4.4.1 E
To begin with E, it is perhaps more convenient to discuss the
corresponding voltage, defined by V = -
integraltext
E?dl, since this is what was
actually measured. For the Hall bar device in Figure 4.2, V
WL
= |E
bardbl
|L, and
V
H
= |E
?
|W, where E
bardbl
and E
?
are the components of E parallel and
perpendicular to the current bias. L and W were lithographically defined
according to the relation L/W = 1, and measurements of these quantities
with tools such scanning electron microscopes (SEM) and atomic force
microscopes (AFM) revealed that the feature sizes were generally within 5%
of their designed dimensions. Consequently, geometric issues associated with
a given sample are not a source of error in a voltage measurement. In
addition, even if L/W varied substantially from 1 between di?erent devices,
for measurements on a particular device this ratio would be a constant.
Thus by determining the actual value of L/W with an SEM or AFM, the
data could accordingly be rescaled.
To further discuss voltage measurement errors, I shall consider the
recorded signal V to be composed of two parts: the signal of interest, V
S
,
and an error signal (?noise?) ?V. Mathematically, V = V
S
+?V,where?V
can be negative or positive. The SNR is defined as
SNR =
vextendsingle
vextendsingle
vextendsingle
vextendsingle
V
S
?V
vextendsingle
vextendsingle
vextendsingle
vextendsingle
2
(4.10)
whence it is clear that the higher the SNR, the ?better? the measurement.
Since we are concerned with resistance measurements, V
S
can be controlled
121
by simply adjusting the device?s current bias. ?V is not as easily
manipulated, and several noise sources contribute to its magnitude. These
include the Johnson (Nyquist) noise of the device itself, amplifier noise
(including digitization errors), o?set signals due to ground loops or a poor
common mode rejection ratio (CMRR), instrumentation drifts induced by
gradual changes in the room temperature or humidity, etc. The remainder of
this section will demonstrate that the SNR in these experiments was so large
as to be essentially irrelevant to the conclusions which were drawn from this
work. As an experimentalist, this is a nice situation to be in.
Consider first the voltage error ?V
T
due to the thermal noise of the
device itself. The largest values of R
a50
measured in the course of this work
were R
a50
similarequal 3k?/a50. The spectral density of the noise produced by such a
resistance is given by the well-known Nyquist expression
?
4k
B
TR
a50
,and
assuming V
T
was measured with ? = 1 second at T = 4.2 K, we have ?V
T
similarequal 1 nV. This would of course be lower at the base temperature of the
refrigerator, and for lower resistance samples.
Next, I consider Figure 4.7 to estimate the voltage error, ?V
A
, due to the
preamplifier. From Figure 4.7, I estimate the spectral density of the amplifier
noise to be similarequal 3nV/
?
Hz for the typical test frequencies I used of 10 - 20 Hz.
Again using ? = 1 second for the lock-in time constant, I find ?V
A
=3nV.
The final voltage error I consider is that due to the finite CMRR of the
amplifier, ?V
C
.Toestimate?V
C
, I use the specified value of CMRR = 100
122
dB for the 7280 lock-in amplifier. Assuming a 1 ?A current bias through a
total common mode resistance of 2k? (including the RC filter resistance of
Figure 4.6), the voltage signal applied to both inputs of the lock-in is V
0
=2
mV. Since the common mode error signal is defined [75] by CMRR =
20log
10
(V
0
/?V
C
) = 100, we obtain V
C
=20nV.
The total voltage error in these measurements, ?V, is given by summing
the separate contributions in quadrature, since they are indendent of one
another. That is,
?V =
radicalbig
(?V
T
)
2
+(?V
A
)
2
+(?V
C
)
2
(4.11)
which yields ?V similarequal 21 nV. Thus we see that the finite value of the amplifer?s
CMRR dominates the volatge errors in these measurements. Nevertheless,
this noise level is extremely small when compared with actual signals. Even
for the lowest resistance device measured in this work, R
a50
similarequal 250 ?/a50,anda
modest bias current of I = 100 nA, I obtain a SNR = (25 ?V/21 nV)
2
similarequal
1.4x10
6
! Thus, the limitations of these experiments lie in interpreting the
results, and not with the voltage measurements themselves.
Although this analysis is rather convincing, a variety of other cross-checks
were performed to ensure the quality of these measurements. One check
included examining the dependence of the measured resistance on the test
frequency, f. By varying f from DC (f = 0) to f = 1 kHz, it became obvious
that current leakage to ground was a real issue at higher frequencies,
presumably due to the reduced impedance of the capacitance in the coaxial
123
cables and RC filters, which varies like 1/fC. This implies that low frequency
measurements are easier to interpret, since there is only one signal channel to
consider at DC, while there are two for f negationslash= 0 (the ?in-phase? and
?quadrature? signal components). The usual way of establishing the
appropriate signal phase was to ?autophase? the lock-in at H =0tobein
phase with the current through the device. Nonetheless, even at the low
measurement frequencies used in these experiments, the quadrature signal
showed non-zero response to the varying magnetic field. No e?ort was made
to analyze this component of the signal.
Another step which was taken to enhance the quality of these
measurements was to judiciously allocate the gain of the DSP lock-in to
eliminate digitization errors. The 7280 had two stages of gain: AC gain
(which amplified the signal before DSP), and demodulator gain (which
amplified the signal using DSP). The largest possible value of AC gain was
used to minimize such digitization errors, and it is worth mentioning that
when using the bridge of Figure 4.13, no evidence of digitization was
observed. With regard to the accuracy of the lock-in?s gain, Perkin-Elmer
claims the 7280 gain is accurate to 0.6%.
Finally, in order to investigate whether temperature or humidity drifts in
the laboratory were an issue in these experiemnts, a long (17 hour) trace was
taken of a sample with H = 0, shown in Figure 4.14. This shows that,
although some drifts were apparent, the variation of the resistance over this
124
Figure 4.14: Measurement drift over time. Although the measured resistance does
vary over time, and shows a slightly increasing trend, the variation is quite low.
The maximum resistance over the 17 hour measurement period is R
max
= 55.868 ?,
the minimum R
min
= 55.859 ?, and are therefore 99.98% consistent. Typical weak
localization sweeps occurred on much shorter timescales (? 1 hour) than this trace,
and produced much larger signal variations, therefore such drifts are inconsequential
to data analysis.
125
interval was extremely small (approximately 2 parts in 10
4
), and, when
compared with the changes induced by application of a magnetic field and
the typical sweep times (? 1 hour), are negligible.
4.4.2 H
In reality, a direct measurement of H was not performed. The reason for
this is that the power supply would source a given current, and I assumed
that the field-to-current ratio was accurate. For this system, the conversion
factor was 0.1258 (Tesla/ampere). However, because of hysteresis e?ects
inherent to all superconducting magnets [76], the field experienced by the
sample was in general di?erent from the intended value. To illustrate this
point, consider the R(H) data shown in Figure 4.15. The abrupt
discontinuity of R near H similarequal 275 gauss can be traced to the presence of the
superconductor indium (In) near the sample, which was used as a solder on
this device. One of the basic properties of type I superconductors such as In
is the existence of a critical magnetic field H
c
,belowwhichthemetalis
superconducting, above which the metal is ?normal? (non-superconducting).
For In, H
c
= 293 gauss [2], very close to the discontinuity in Figure 4.15.
When H < H
c
, type I superconductors completely expel magnetic flux from
their interior (the Meissner effect). On the other hand, weak localization
measurements are extremely sensitive to the vectorial nature of the applied
H.
126
Figure 4.15: Impact of the Meissner e?ect on the magentoresistance due the use of
In solder. The transition from the superconducting to normal state in In occurs at
H
c
= 293 gauss. This distorts the local magnetic field applied to the device, and
consequently the measured resistance.
127
Given these two considerations, the explanation of the data in Figure 4.15
is that instead of the device experiencing a gradual change in flux bias, when
sweeping through H
c
the flux bias changes discontinuously. In e?ect, the
magnetoresistance of the device acts as an independent magnetometer of the
field H. Clearly whether H < H
c
or H > H
c
has a profound impact on the
local H applied to the sample, and consequently the measured resistance.
Assuming that the In near the sample really did have H
c
= 293 gauss, then
the onset of the transition region in Figure 4.15 gives an estimate of the
actual H applied to the sample. Examining this figure shows that the
discontinutiy occurs at roughly 275 gauss, and is therefore reasonably close
to the expected value of 293 gauss. This di?erence is quite small when
compared with the typical sweep ranges (+/- 1 Tesla). Incidentally, this
analysis suggests that a convenient and highly accurate method of
calibrating both H and T in a given low temperature apparatus is through
4-wire measurements on type I superconductors near their critical
temperatures and magnetic fields.
The homogeneity of the field is another concern in my measurements.
Because the magnetoresistance is highly anisotropic, stray perpendicular
fields would lead to a larger response than for H strictly parallel to the
2DEG plane. Given that ??B =0always, magnetic field lines are never
truly straight: they always have a slight curvature. As mentioned earlier,
Oxford claims that the homogeneity at the center of the magnet bore was
128
good to 1 in 1000 over a 1 cm
3
spherical volume. Therefore one way of
estimating the influence of this e?ect is to ?scale down? this figure to a
typical device size of 50 ?mx50?m. Doing this yields a field homogeneity
of (10
?3
)
2/3
= 0.01 per cm
2
, and since the device area was (50?m)
2
=
2.5x10
?5
cm
2
, this suggests that the field was homogeneous to better than 3
parts in 10
7
across each square of the device! This analysis is probably ?too
good to be true?, and thus I take the following approach.
The superconducting solenoid which was used in these experiments had a
length L of approximately 35 cm, and a bore diameter of 52 mm, hence a
radius a of 2.6 cm. Given the field/current ratio of 0.1258 T/A of this
solenoid, we can find the approximate number of turns per unit length, N,
from the equation [78]
N =
H
?
0
I
=
0.1258(
T
A
)
?
0
. (4.12)
Since ?
0
=4?x10
?7
N/A
2
, this yields N similarequal 10
5
turns/meter.
The issue of the field inhomogeneity near the center of a solenoid is
treated in chapter 5 of [46], where the following result is obtained for the
radial component H
r
of H, an on-axis distance z and radial distance r from
the geometric center of the solenoid:
H
r
=
24?
0
NIa
2
rz
L
4
. (4.13)
Inserting the values above, and a current corresponding to 1 T (7.949 A), I
find H
r
similarequal 2 gauss. Since the radial field is proportional to the magnet
current I, H
r
will generally be even less than this. When compared with the
129
axial field (1 T), we see that field inhomogeneities were not a concern in
these measurements.
4.4.3 J
The current density J was determined by the lithographically defined
width of the samples, W, and measurement of the current bias, I, with the
7280 lock-in operated as a current amplifier. As mentioned earlier, the
dimensions of the sample are relatively unimportant, since their precise
values can be independently determined with SEMs or AFMs. Therefore I
was primarily concerned with determining I.
A measurement of I for |H|?[0,+1] T is shown in Figure 4.16. The
resolution limitations of the DACs are strikingly apparent, and the discrete
nature of the recorded data points is obviously unphysical. This
measurement was performed in a manner analogous to that shown in
Figure 4.1, with a 1 M? resistor placed in front of the 1 Volt AC source of
the lock-in serving as the current source. The behavior displayed by this
graph is easily understood by considering the fact that the entire sample,
including the contact wires, etc., display weakly localized behavior. The
entire sample has a resistance maximum at H =0,andasthefieldis
increased, the entire sample becomes less resistive, therefore a larger current
flows through the device. To overcome this, one might be tempted to
construct a ?current bridge?, analogous to the resistance bridge of
130
Figure 4.16: Finite resolution of DAC in a bias current measurement. Despite the
limitations of this measurement, the current over the full magnetic field range is still
constant to 2 parts in 10
4
.
Figure 4.13, but such an e?ort is unnecessary, since I is constant to roughly 2
parts in 10
4
. Thus we see that determination of the bias current was not a
significant limitation of these measurements.
131
4.4.4 T
The most uncertain quantity in these experiments was T, the
temperature of the device. Several studies (see [77] and references therein)
have demonstrated that, in general, the electronic temperature T
e
and the
phonon (lattice) temperature T
p
are unequal. In the experiments discussed
here, the sample and thermometer were physically separated by ? 2cm.
Therefore assuming that T
e
was equal to the phonon (mixing chamber)
temperature T
p
is inaccurate. However, this is insignificant for most of the
considerations discussed in this thesis. The reason for this is that, for the
most part, experiments were performed only at constant temperatures (T =
4.2 K and the base temperature of the refrigerator, T similarequal 130 mK). The only
time the sample temperature was absolutely certain was at T = 4.2 K, when
the inner vacuum chamber of the dilution refrigerator was filled with
3
He
exchange gas and the cryostat was filled with liquid
4
He. T
e
was certainly
colder at base temperature, and looking closely at the data in Figure 4.12, it
is apparent that the resistance saturates in Si:P devices below T similarequal 200 mK.
I also observed such saturation in Si:B devices, but in these samples the
onset occurred at much higher temperatures: T similarequal 800 mK. The physical
origin of this behavior will be discussed in Chapter 5, but Figure 4.12
suggests that despite T
e
being unknown at base temperature, the observed
saturation makes it plausible that T
e
was constant during the
measurements. Therefore, imprecise knowledge of T was not a significant
132
source of additional (propagated) error in these experiments: the samples
were pretty cold regardless.
4.4.5 ?
Although I have now discussed the errors associated with the physical
variables (E,H,J, T), an additional parameter which is important to the
conclusions drawn from these measurements is the angle ? between H and J.
At this point it is reasonable to ask how sensitive the stage orientation
technique was, or what angular resolution, ??, was achievable? The answer
to this is straightforward.
Since ? is determined through a measurement of V
H
,avoltage signal,the
voltage noise in the measurement system, V
N
, will ultimately limit the
determination of ? = 0. A lock-in amplifier operated with a time constant ?
will, on average, lead to an uncertainty ?V in a voltage measurement given
by ?V=V
N
/
?
?. Intuitively, this makes sense, since one would expect that
by averaging a given signal for a longer time a more precise measurement
would be produced. Once again, I use V
H
=IHsin(?)/ne to derive an
expression for ?? given ?V.
Since in parallel fields we are concerned with very small angles between I
and H (? lessmuch 1), we can expand sin(?) to lowest order in a Taylor series and
use the approximation sin(?) similarequal ?. I equate this small angle ? with the
resolution limit ??.WhenIandH are as parallel as possible, the ability to
133
distinguish V
H
from ?V is lost, or the equality
?V =
IH??
ne
(4.14)
holds. Solving this equation for ??, I obtain the desired expression for the
angular resolution limit:
?? =
ne?V
IH
=
neV
N
IH
?
?
. (4.15)
What is interesting here is that the numerator consists of quantities over
which the experimentalist has no (or at best, little) control, whereas the
denominator is under complete control. Thus, in order to improve (reduce)
??, one should work with a large bias current I in a large magnetic field H,
and average the Hall signal for a long time ?.Using?V=5nV,n=
10
14
/cm
2
,I=1?A, and H = 1 T, leads to ?? =8x10
?4
radians (similarequal 0.05
?
).
Now that the measurement details of this research have been established,
I turn to Chapter 5, and a discussion of my experimental results.
134
Chapter 5
Results
The results of my low temperature magnetoconductance measurements
on ?-doped Si are presented in this chapter. Although the primary focus will
be upon Si:P ?-layers, a limited study of Si:B ?-layers is also presented. The
theory of weak localization developed in Chapter 2 will serve as my primary
tool for interpreting the data. In addition, I critically examine the various
experimental and theoretical approaches incorporated throughout this work
for their advantages and limitations.
5.1 Si:P ?-Layers
The principal motivation for a detailed study of Si:P ?-layers lies in the
potential these structures have in device applications. Some particular
examples were mentioned in the Introduction (Chapter 1), and will not be
135
discussed further here. Instead, I shall concentrate on two particular aspects
of Si:P ?-layer physics. One study examines the impact a high temperature
anneal has on the electronic characteristics of the samples. The other
investigates the dependence of the magnetoconductance signals on n, the
sample carrier density. I also establish several limitations the present theory
of weak localization has with respect to ?-doped Si:P systems.
5.1.1 Di?usivity Measurements
The desire to establish weak localization as a tool for measuring the
thickness of ?-layers can be traced back to a paper of Mensz and Wheeler
[80]. By performing weak localization measurements on Si MOSFETs in
perpendicular and parallel magnetic fields, these workers estimated that the
rms value of the 2DEG thickness in their samples was similarequal 0.21 nm, i.e. less
than a Si monolayer. Being able to achieve this type of resolution in ?-doped
samples would far exceed the limitations of other established methods for
measuring T , including capacitance-voltage (CV) profiling (chapter 9 of [1]),
x-ray di?ractometry [81], transmission electron microscopy (TEM) [34], and
SIMS (Secondary Ion Mass Spectrometry) analysis. Each of these techniques
has its own advantages and disadvantages, and those of SIMS, the most
common approach, were discussed in Chapter 3. I now turn to my
experiments using weak localization for measuring T , reported in [79].
136
Experimental Procedure
For my measurements, four distinct 50 ?mx50?m van der Pauw
devices were fabricated, as described in Chapter 3. The wafer used for these
devices corresponds to the point labeled ?A? in Figure 2.10 (n similarequal 10
14
/cm
2
),
and a SIMS analysis from a sample of this wafer is shown in Figure 5.1 (this
is the same plot as Figure 3.5). The reason for going through the somewhat
elaborate fitting procedure associated with van der Pauw devices is simple.
For measurement of a Hall bar, 6 wires are needed (source, drain, and four
voltage contacts), whereas with a van der Pauw device it is the bare
minimum of 4. In addition, one lead per device was necessary as an ancillary
contact to check for conduction through the substrate (Chapter 3), making 7
contacts necessary for a Hall bar device. As discussed in Chapter 4, the
dilution refrigerator used in these experiments had 20 total experimental
leads. Thus using Hall bars, realistically only two devices could be measured
in a given experimental run (cooldown). In contrast, van der Pauw devices
allowed four di?erent samples to be measured in a single cooldown, in e?ect
doubling the amount of data obtained. Since the work required to cycle a
dilution refrigerator from room temperature to base temperature and back is
time-consuming and non-trivial, the use of van der Pauw devices was an
obvious choice.
After defining the device ?mesa? and cleaning o? the resist mask as
described in the device processing recipe of Chapter 3 (steps 10 through 15),
137
Figure 5.1: SIMS data from a sample of wafer A, with n similarequal 10
14
/cm
2
.Alofthe
devices used in the di?usivity study were fabricated from this wafer. This SIMS
profile corresponds to the cook 0 (unannealed) device, and is the same as shown in
Figure 3.5.
138
three of the four samples were subjected to an 850
?
C anneal in a nitrogen
atmosphere, performed in a Barnstead Thermaline 79300 tube furnace (not a
rapid thermal annealer). The annealing times were for 5, 10 and 15 minutes,
and henceforth these devices will be referred to by the time each was
annealed as cook 0, cook 5, cook 10 and cook 15. Two considerations led to
this choice of annealing times. First, prior experience had shown that if the
devices were annealed for too long (40+ minutes), T in these devices would
?saturate? and become indistinguishable from one another. This leads to the
second rationale for the chosen annealing times, which was to explore
precisely how sensitive this weak localization approach is for measuring T .
As will be shown, with an appropriate (and as yet undeveloped) theoretical
footing, this technique should prove to be the most precise method of
measuring ?-layer thicknesses yet demonstrated.
Following the anneals, the fabrication process was completed in a parallel
fashion for all of the devices (resuming at step 16, Chapter 3), the only
di?erence being I used In solder instead of wire bonds (step 31) to connect
wires to the Au contact pads. Thus, aside from statistical variations in the
wafer itself, the only di?erence between the devices was annealing time. The
samples were then mounted in the dilution refrigerator and measured as
described in Chapter 4.
139
Results
Before fitting data to the appropriate expressions, a standard
symmetrization procedure was followed with weak localization data. This
consisted of transforming the data according to:
R
WL
(+/?H)=
R
WL
(+H)+R
WL
(?H)
2
. (5.1)
The e?ect of this was to remove any of the residual asymmetries which are
present in all 4-wire magnetoresistance measurements [82] , and fit data only
to the symmetric component of the conductivity tensor. This technique is
both experimentally and theoretically well-founded, and has been discussed
extensively elsewhere [68]. Figure 5.2 shows a plot of typical
magnetoresistance data in both ?raw? (unsymmetrized) and symmetrized
forms. Clearly, the data is highly symmetric to begin with, and this
procedure has a negligible e?ect on the interpretation. Henceforth, all weak
localization data presented in this thesis will be in symmetrized form. The
reader should thus bear in mind that the plots I present are somewhat
redundant, since by symmetrizing I am displaying how the data depends on
the magnitude of H, not it?s sign.
For completeness, I re-state the equations cited in Chapter 2 which I used
for fitting data and calculating T . First, in perpendicular fields, the change
in the magnetoconductance, ??(H
?
), is expected to take the form
??(H
?
)=(
e
2
2?
2
planckover2pi1
)[?(
1
2
+
planckover2pi1
4eH
?
lscript
?
2
)??(
1
2
+
planckover2pi1
2eH
?
lscript
e
2
)+ln(2(
lscript
?
lscript
e
)
2
)] (5.2)
140
Figure 5.2: Raw (unsymmetrized) weak localization data (solid points) and sym-
metrized data (open squares). The measurements are highly symmetric before ap-
plying equation (5.1), and hence this procedure has a negligible e?ect on data anal-
ysis.
141
where?isthedigammafunction,lscript
e
the mean free path, and lscript
?
the
dephasing length.
In parallel fields, ??(H
bardbl
)isgivenby
??(H
bardbl
)=(
e
2
2?
2
planckover2pi1
)ln(1+?H
2
bardbl
) (5.3)
requiring the single parameter ? for fitting.
The model for T used in [79] assumes that the inequality lscript
c
lessmuch lscript
e
lessmuch lscript
?
was valid in all of the devices measured, which corresponds to the
?homogeneous broadening? limit described in Chapter 2 and [62]:
T =
parenleftbigg
1
4?
parenrightbigg
1/4
bracketleftBigg
parenleftbigg
planckover2pi1
elscript
?
parenrightbigg
2
parenleftbigg
?lscript
e
lscript
c
parenrightbigg
bracketrightBigg
1/2
. (5.4)
A ?ball park? estimate of lscript
c
=1/
?
n was used in [79]. I return to this point
later when I discuss the systematic errors associated with my work.
The data and theoretical fits for my four devices are shown in Figures 5.3
through 5.6. The di?erence between the ?(H
?
) (upper curves) and ?(H
bardbl
)
(lower curves) configurations is easily understood using the
phenomenological ?cross section? argument presented in Chapter 2. When H
= H
bardbl
, the e?ective cross section of the 2DEG is significantly smaller than for
H
?
. Thus less flux penetrates the electron system, and less of a change in
conductance is induced. Furthermore, we see that for |H| = 0.5 T, the signal
?contrast?, ??, decreases with annealing time (?? ? ?(H
?
)-?(H
bardbl
)).
Equivalently, the greater ??, the smaller is T of the 2DEG in the device,
independent of the model used to interpret the data. I now turn to an
142
Figure 5.3: Cook 0 weak localization data. The best-fit parameters to equations
(5.2) and (5.3) are lscript
e
=30nm,lscript
?
= 101 nm, and ? = 0.49/T
?2
. The signal contrast
at |H| = +/- 0.5 T is ?? similarequal 1.4.
analysis of these results. Before doing so, however, I note that the fitting
lines seem to systematically deviate from the data near H =0.Myguessis
that this comes from a combination of e?ects I discussed in Chapter 3,
including some amount of In shielding (these devices did have In contacts!),
being unable to sweep the field continuously through H = 0, and field
inhomogeneities across the DIP header e?ecting each device di?erently. In
any case, my conclusions are independent of these discrepancies.
143
Figure 5.4: Cook 5 weak localization data. The best-fit parameters are lscript
e
=33nm,
lscript
?
= 117 nm, and ? = 5.31/T
?2
. The signal contrast ?? similarequal 1.1.
144
Figure 5.5: Cook 10 weak localization data. The best-fit parameters are lscript
e
=37
nm, lscript
?
= 124 nm, and ? = 7.54/T
?2
. The signal contrast ?? similarequal 0.8.
145
Figure 5.6: Cook 15 weak localization data. The best-fit parameters are lscript
e
=38
nm, lscript
?
= 127 nm, and ? = 11.25/T
?2
. The signal contrast ?? similarequal 0.7.
146
The best-fit parameters extracted from cook 0 through cook 15, along
with their associated uncertainties, are shown in Table 5.1. Only the errors
in lscript
e
and lscript
?
(as determined by the corresponding fitting routines) are
included, since these uncertainties dominate the corresponding error in T .
ThemeansquareerrorinT was determined by ?propagating? these errors in
the usual way [83], given by
(?T )
2
=
parenleftBigg
C
2
4lscript
e
lscript
2
?
parenrightBigg
2
(?lscript
e
)
2
+
parenleftBigg
C
2
lscript
e
lscript
4
?
parenrightBigg
2
(?lscript
?
)
2
(5.5)
where C =(1/4?)
1/4
(planckover2pi1/e)
radicalbig
?/lscript
c
.
Examination of Table 5.1 shows several trends, which I shall interpret by
considering the e?ect the anneal had on the samples. I will primarily discuss
lattice-type defects (e.g. vacancies and interstitials), since it is well-known
that the quality of the Si crystal improves with annealing time, ?t, and
therefore that these defects are reduced, or ?cooked out?, with annealing [20].
First, I note that ? monotonically decreases with ?t. Since this number is
formed by the ratio of two directly measurable physical quantities (? ? V/I),
it does not hinge on any particular model for interpretation. The fact that ?
decreases with ?t is easily understood, since the scattering of electrons o?
defects is the origin of resistivity in the first place, and annealing removes
defects. Similar considerations will be invoked when discussing the mean free
path, lscript
e
.
No such monotonic trend is observed with n. However, if one arbitrarily
(for the time being) ignores the cook 5 data point, n does become a
147
monotonically increasing function of ?t. The reason for pointing this out is
that previous measurements, not presented in this thesis, demonstrated this
e?ect very clearly, and it is in fact quite common; see [84], for example. Such
behavior has to do with the notion of electrical activation, which essentially
asserts that even if a P donor is placed in the crystal, it does not necessarily
contribute an electron to the conduction band. This phenomenon may come
about due to donors ?clustering? about one another during epitaxial growth,
with some type of molecular bonding acting to confine the electrons, or
perhaps another cause related to defects ?trapping? the electrons. Whatever
the origin may be, these e?ects are small (< 10%), which supports the
assumption that the wafer was quite homogeneous.
Themeanfreepath,lscript
e
, steadily increases with ?t. Essentially the same
argument as was used for ? applies here. As the density of defects is reduced,
so are the number of scattering sites which disrupt an electronic trajectory as
it propagates through the crystal. The average distance traveled by an
electron between scattering events is lscript
e
, so again this result is intuitively
appealing.
Perhaps the most interesting result in Table 5.1 is the trend the of the
dephasing length, lscript
?
, to increase with ?t. Recall that lscript
?
is the mean
distance an electron can travel through a sample before its phase coherence
is destroyed, such as occurs during an inelastic scattering event. The
previous arguments related to the defect density require a ?renormalization?,
148
Table 5.1: Experimental results for ?-layer devices annealed (?cooked?) at 850
?
C
forintervalsof0to15minutes.Theparameterslscript
e
and lscript
?
were extracted from fits
to equation (5.2), ? from fits to equation (5.3). The mean-square thickness of a
given ?-layer was estimated using equation (5.4), and is proportional to the parallel
field parameter ?. The dominant contribution to the error in T comes from lscript
e
.
sample resistivity electron mean dephasing ? rms
? density free length thickness
n path lscript
?
T
lscript
e
?/a50 10
14
/cm
2
nm nm Tesla
?2
nm
30 101 14
cook 0 428.6 1.40 +/- +/- 0.49 +/-
13 3 3
33 117 42
cook 5 300.9 1.39 +/- +/- 5.31 +/-
14 11 10
37 124 51
cook 10 267.9 1.42 +/- +/- 7.54 +/-
13 3 9
38 127 61
cook 15 251.6 1.49 +/- +/- 11.25 +/-
14 9 12
149
so to speak, since the static lattice defects I have been discussing do not alter
the electron?s energy in the course of a scattering event, only its momentum.
Therefore understanding this result requires more elaborate considerations.
As mentioned in Chapter 2, the electron-electron interaction is expected
to be the dominant dephasing mechanism in Si:P ?-doped samples (data
supporting this assertion will be presented shortly). Theoretical work by
various authors [39, 40] has shown that the electron-electron interaction is
enhanced by disorder. In brief, the explanation for this enhancement is that
defects disrupt the translational symmetry of the lattice, resulting in a
reduction of conserved momentum. This in turn increases the rate of
inelastic scattering, and thus the increase of lscript
?
with ?t can be interpreted as
a manifestation of this disorder enhanced e?ect at work. This observation
could lead to an interesting series of experiments exploring such behavior,
possibly incorporating crystallographic x-ray data to quantify the defect
density.
The parallel field parameter, ?, also steadily increases with ?t. Given the
relation between ? and the expression for T in equation (5.4), i.e. that T?
?
?, such behavior is anticipated. This leads us to consider the final column
in Table 5.1, T itself.
As expected, T is a monotonically increasing function of ?t. This result
is consistent with thermal di?usion of the ?-layer induced by the anneals. To
check this, I employ a simple model of the data to estimate the di?usion
150
coe?cient of Si:P at T = 850
?
C. These are standard results, see Shewmon
[85], for example.
Di?usion Model
From an atomistic point of view, as an Si:P sample is heated, each
particle in the crystal acquires an rms thermal energy of order k
B
T
(Boltzmann?s constant k
B
= 1.38x10
?23
J/T). If the temperature is raised
su?ciently high, the atoms become mobile, and perform three-dimensional
?random walks? throughout the lattice. By considering the system?s entropy
it is clear that if di?usion is allowed to proceed indefinitely, the steady-state
solution will be one of complete homogeneity. That is, if there are N
P
phosphorous atoms in a sample of crystal with total volume V, then, after
di?using for a su?ciently long time, locating a small volume with a density
appreciably di?erent from N
P
/V ?almost never? happens (in the sense of
Poincar?e cycles and ergodic theory). So the ?-layer gets wider after
annealing, with probability 1. Assuming isotropic di?usion (which is the case
for P in Si [20]), these three-dimensional random walks can be factored into
the product of three, one-dimensional random walks. Therefore, I now show
how one-dimensional random walks can be used to model the results for
T (?t) in Table 5.1.
I start by considering a single P atom located at the origin (z = 0), and
assume that it undergoes a one-dimensional random walk for a time ?t.
151
Then the probability that after this time the atom lies in the infinitesimal
interval (z, z + dz) is given by P(z, ?t)dz, where P(z, ?t) is the Gaussian
expression [86]
P(z,?t)=
exp(
?z
2
4D?t
)
?
4?D?t
. (5.6)
Here D is the diffusion coefficient characterizing the random walk. Note
that P(z, ?t) is a normalized probability density (
integraltext
?
??
P(z, ?t)dz = 1), and
satisfies the one-dimensional di?usion equation [87]
?P(z,?t)
?t
= D
?
2
P(z,?t)
?z
2
(5.7)
both of which are essentially mathematical statements that the dopant
atoms are conserved during the process of di?usion.
The mean square displacement, <z
2
>, an atom undergoes from its
original (z = 0) position is given by
<z
2
>=
integraldisplay
?
??
z
2
P(z,?t)dz (5.8)
which yields <z
2
> = 2D?t. This relationship will be used subsequently to
define the di?usion length, L
D
. If I assume a Gaussian-distributed ?-layer
with initial square thickness T
2
0
, the e?ect of annealing is to simply add
<z
2
> to the distribution (since these are two independent Gaussians [88]).
Thus a new Gaussian ?-layer is produced, with square thickness T
2
related
to T
0
,Dand?tas
T
2
= T
2
0
+2D?t. (5.9)
Accordingto(5.9),ifT
2
is plotted versus ?t, a linear relation should be
152
observed, with slope 2D and intercept T
2
0
. Such a plot is shown in Figure
5.7, with the cook 0 data point omitted from the linear fit. The reason for
the omission is that the initial rapid increase in T observed between cook 0
and cook 5 likely corresponds to an enhanced di?usion due to the defects, as
mentioned earlier. As ?t is increased, the defects are cooked out, the curve
?settles down? from this initial e?ect, and an accurate estimate of D can be
obtained. Fitting a line to the annealed data points, I obtain a slope of 186
nm
2
/min, which converts to D = 3.1x10
?14
cm
2
/s in conventional di?usivity
units.
A legitimate question at this point is to ask how well this data agrees with
other measurements of D in Si:P? This is somewhat di?cult to quantify,
since di?usivity is heavily dependent on the techniques incorporated during
sample preparation, and the interpretation of these weak localization results
is of course model-dependent. A fairly recent treatment using SIMS analysis
[89] claims D = 4.14x10
?16
cm
2
/s, while values of D over two orders of
magnitude higher than this are cited in [90]. Given this spread in values, it is
clear that when quoting a measured value for D, more is required than
simply specifying the temperature. Therefore I conclude that my work [79] is
in accord with other published results, and that weak localization in parallel
fields can be used for measurement of D.
Before I can unambiguously assert that this technique is better than SIMS
for measuring T in ?-layers, however, a discussion of the systematic errors
153
Figure 5.7: Di?usion coe?cient data. The slope of the line yields D = 3.1x10
?14
cm
2
/s. Note that the intercept of the line lies well above the cook 0 data point,
which has been omitted from the fit. This is consistent with transient enhanced
di?usion in the annealed devices, as discussed in the text.
154
associated with this approach is necessary. The two categories I will explore
are errors associated with the experimental techniques themselves, and those
with the theoretical approach I have taken to interpreting the data.
Experimental Systematic Errors
Several issues with regard to the experimental procedures described
previously require a more in-depth analysis to establish that this approach is
on a sound footing. Since the systematic errors associated with the
resistance measurements were discussed in Chapter 4, I will primarily
concentrate on issues pertinent to the samples themselves. These include
variances intrinsic to the wafer, non-uniformities in the devices due to
unintended processing di?erences, and artifacts associated with the
annealing procedure. Although this list is far from exhaustive, it seems likely
that these issues dominate the systematic errors involved in this di?usivity
study. At this point it should also be mentioned that the techniques
associated with magnetic field application cited in [80] are probably better
than those used in this work. The authors of [80] were able to apply H
?
and
H
bardbl
independently and simultaneously, while I was only able to rotate the tilt
stage to some fixed angle ? and then vary |H|. Although the two approaches
should be equivalent, the flexibilty of controlling two fields independently
would have been preferable.
A thorough investigation into the uniformity of the wafer used for these
155
measurements was not carried out, but there are several reasons to discount
this as a significant source of error. First, the pieces of the wafer used to
construct the devices were taken from the same part of the wafer, near the
center. As discussed in Chapter 3, the wafer is rotated during epitaxial
growth in an e?ort to eliminate the type of non-uniformities I am considering
here. Thus, one would expect that variance in the doping density would
appear as a radial e?ect. Since the center of the wafer was essentially at rest
during growth, this area should be the most uniform on the entire sample.
Second, as shown in Table 5.1, even after the anneals, the density was found
to vary by less than 10% from sample to sample. As discussed earlier, given
the psuedo-trend of n increasing with ?t and the phenomenon of electrical
activation, it seems quite likely that the variance in n was due to the anneals,
not intrinsic anomalies in the wafer. Third, given that T increased from cook
0 through cook 15 precisely as expected, there is little reason for believing
that this e?ect was from some bizarre coincidence in the wafer precisely
corresponding to the annealing time of the device. In short, the data shown
in Table 5.1, taken as a whole, essentially eliminates the possibilty that the
observations presented here were due to non-uniformities in the wafer.
Undoubtedly, some part of the observed results were due to unequal
processing techniques among the devices. Small di?erences in device features
would lead to a corresponding change in the associated 4-wire measurements.
A trivial example is that the farther apart two voltage contacts are spaced,
156
the larger is the resistance between them. Using the same device for the
entire series of experiements would eliminate such issues, but was not the
method followed here, for two reasons. One was prior experience with this
?recycling? approach. To subject the same device to multiple anneals
requires the electrical contacts to be removed and re-established each time.
The way this was accomplished originally was by using In solder to make
contact directly to the ?-layer, and removing the In by soaking the device in
a room-temperature HCl bath. However, as discussed in Chapter 4, In, being
a superconductor, expels magnetic flux from its interior when immersed in a
field |H| < H
c
.ForIn,H
c
= 293 gauss. This unwanted superconducting
behavior would manifest itself by shielding portions of the device from the
appliedfieldfor|H| < H
c
, an example of which was shown in Figure 4.15. In
light of this, one of the main experimental lessons of this work is that the use
of superconductors should be assiduously avoided when performing weak
localization measurements. The small amounts of In I used for wiring
apparently had little to no e?ect.
No such ?discontinuities? were observed when samples using Au-Sb
contacts were measured. However, these contacts were ?annealed in? (step
24, Chapter 3), and therefore essentially permanent. Furthermore, at T =
850
?
C, it is a bad idea to anneal Au in Si, given gold?s large di?usivity in Si
and the associated trap states in the middle of the Si energy gap [91], which
wreak havoc with proper measurements. The second reason for measuring
157
separate devices was mentioned earlier: working with a dilution refrigerator
takes time, and instead of doing 4 cooldowns to obtain the data in [79], one
(long) cooldown using van der Pauw devices was su?cient.
My own belief is that such device-dependent e?ects are irrelevant to the
conclusions which I have drawn from these measurements. The reasons for
this are twofold: the use of the symmetrization procedure and the van der
Pauw theorem. As discussed in [68], all 4-wire measurements are a mixture
of the symmetric and antisymmetric components of the conductivity tensor.
By symmetrizing the data, one automatically obtains the symmetric
component of the magnetoresistance, which is precisely what the weak
localization correction to the conductivity, equation (5.2), is calculated for.
In addition, no matter how convoluted a devices geometry, van der Pauw?s
theorem allows one to calculate the associated resistivity, again the pertinent
quantity for the weak localization correction. That is, the theorem relates
the resistance measured in a given configuration to the sample?s intrinsic
resistivity by a single number, appropriate to that particular device. Once
the two appropriate, lead-switching van der Pauw measurements have been
performed and ? determined via equation (4.6), any other measured
resistance is simply renormalized to equal ?. Thus, for arbitrarily shaped
samples the appropriate correction factor is easily calculated (as described in
Chapter 4). Finally, since the data will ultimately be used with a
least-squares fitting routine, this factor can be further optimized by making
158
it into a fitting parameter and finding the best fit. Since a single number
multiplies all magnetoresistance data, the legitimacy of the procedure should
be clear.
An issue I have glossed over in the above argument is the role the voltage
probes play in the measurements. Probes will always be invasive, to some
degree influencing the data. My guess is that as long as the contacts are
su?ciently ?point-like?, the above arguments should hold. Point-like
essentially means that the spatial extent of the probes should be small
compared to the perimeter of the device (the e?ect of a contact with finite
extent was discussed by van der Pauw [72]). Although potentially interesting
and useful, a study of the e?ect of probe size on weak localization data was
not carried out in this work. In any case, inspection of the lithographically
defined devices throughout the fabrication process indicated samples which
were highly uniform. Thus I assert that device-dependent di?erences are
insignificant to the results in Table 5.1 when compared with the e?ect of
annealing.
The systematics of the annealing procedure will be the final experimental
errors I consider. Both the temperature and cooking times introduce errors
into the results, and this is most likely the dominant error contribution to
the estimate of the di?usivity.
For Si:P, the well-established relationship between the di?usion coe?cient
159
and temperature takes the Arrhenius form [85, 89]
D(T)=D
0
exp
parenleftbigg
??
k
B
T
parenrightbigg
, (5.10)
where D
0
is a constant (with di?usivity units of cm
2
/s) and ? is the
activation energy (also constant). This exponential dependence on
temperature is the primary culprit for the experimental systematic errors in
my measurements. To see this, consider the plot in Figure 5.8, which depicts
an idealized annealing process. After being loaded in the furnace, the sample
temperature is ?ramped up? linearly to the desired maximum temperature
T
M
(850
?
C here), ?dwells? at T
M
for the desired time ?t, and is then
symmetrically ramped back down to room temperature. However, di?usion
happens not only during the cooking step proper (the rectangular area below
T
M
in Figure 5.8) but also as the temperature is ramped up and down
(triangular areas). The obvious geometric picture is that the total (square)
di?usion length induced by the annealing process is the sum of these three
areas (really, a trapezoid), and therefore that the actual di?usion in the
cooked devices is larger than quoted in [79], which neglected the
contribution of the ramps. This dependence on the ramps will be represented
by writing the total induced di?usion length as
L
2
D
= L
2
a50
+2L
2
triangle
, (5.11)
where L
2
a50
corresponds to the di?usion during the dwell time ?t, and L
2
triangle
to
the contribution of the ramps. Since the temperature profile is assumed to
be symmetric, 2L
2
triangle
must be included in L
2
D
.
160
Figure 5.8: Model of annealing procedure. Although the di?usion was only intended
to occur for the cooking time ?t at temperature T
M
, di?usion actually occurs
throughout the ramping process while T ? T
C
. The magnitude of this error is
calculated in the text.
161
To estimate the magnitude of this accounting error, I will assume that
whenever the sample?s temperature is above a critical temperature T
C
(arbitrarily defined), D obeys equation (5.10), while below T
C
,Dis
identically 0. In order to calculate the relevant quantities of interest,
Hadara?s result that D = (1.71x10
?3
)exp(-2.81 eV/k
B
T) cm
2
/s will be used
[89], although it should be stressed that sample preparation methods are
often as important as the annealing times and temperatures themselves, and
the use of this D is only meant to give an estimate of the error. In fact, this
probably sets a lower bound on the error, given the disparity between the
estimates of D reported in [89] and [79]. Once again being somewhat
arbitrary, I ignore values of D for which the di?usivity is less than 1/100 of D
at 850
?
C, which leads to T
C
similarequal 700
?
C.
For a variable D, the di?usion length L
2
triangle
must be weighted by an integral
over the product of the di?usion coe?cient and the infinitesimal time dt [85]:
L
2
triangle
=2
integraldisplay
?
0
D(T)dt =2D
0
integraldisplay
?
0
exp
parenleftbigg
??
k
B
T
parenrightbigg
dt. (5.12)
? is the time interval over which D varies. The exponential dependence on
1/T in this integral requires invoking the incomplete gamma function to
obtain a closed form solution, and since this is expressed yet again in terms
of an integral, provides little physical insight to the problem. Instead, I have
calculated the integral numerically by partitioning the domain of integration
into a finite number of subintervals, calculating the corresponding di?usion
length for each of these elements, and summing the results to approximate
162
L
2
triangle
. Carrying out this computation with a 180 element partition (one for
each second of the ramp between 700 and 850
?
C), I obtain L
2
triangle
=3.7nm
2
.
Therefore, assuming a symmetric ramp profile, the total contribution of the
ramps to L
2
a50
is 2L
2
triangle
,or7.4nm
2
.
Since all of the devices studied used the same ramp profile but di?erent
dwell times, the error introduced by L
2
triangle
is larger for shorter anneals. This is
clear by forming the ratio L
2
triangle
/L
2
a50
for each device, for which one obtains 0.297
for cook 5, 0.149 for cook 10, and 0.099 for cook 15. Thus, the L
2
triangle
error is
clearly a non-trivial contribution, especially for cook 5, where the error is
nearly 30 % of L
2
a50
. To combat this e?ect, fast ramp profiles should be used,
which would make the ?annealing trapezoid? I have considered in this
analysis more rectangular, and therefore closer to ideal. This also suggests
that the resolution limits of the weak localization technique used here could
be probed by subjecting the samples to very brief ?heat pulses? (for
example, in a rapid thermal annealer), whose duration is systematically
reduced until the ?-layer broadening can no longer be resolved.
To consider the final aspects of the annealing process experimental errors,
I take a much less quantitative approach than for the ramping e?ects. One
issue is the fact that, in reality, the true temperature of the sample is not
precisely known as it is cooked. No thermometer was near the sample to
monitor the temperature, so e?ects such as ?overshooting? T
M
and
fluctuating about T
M
cannot be ruled out. The anneals were performed in a
163
Barnstead Thermaline 79300 tube furnace, whose total length was
approximately 1 meter. Obviously there are substantial temperature
gradients in such a processing tool, with the center presumably being hotter
than the ends. Although every e?ort was made to place the samples in the
exact center of the tube, this procedure was approximate at best, and thus
one sample may have actually been annealed at a temperature slightly higher
than 850
?
C, another slightly below, etc. In addition, the ramp was not even
symmetric, let alone trapezoidal in shape, since the ramp down from T
M
always took longer than the ramp up. To stress the point one final time,
since the di?usivity depends exponentially on 1/T, these phenomena could
have played a noticeable e?ect in the reported results, but a quantitative,
systematic analysis of these points was not carried out.
Despite the considerations discussed here, the experimental results
produced by these measurements, shown in Table 5.1 and discussed in detail
subsequently, are quite convincing. In particular, the weak localization data
in Figures 5.3 through 5.6, showing the signal contrast ?? in these samples
monotonically decreasing with ?t, provides strong evidence that
experimental systematic errors were not the primary limitation facing the
interpretation of this data. The next section of this chapter is devoted to
exploring the theoretical interpretation of this data, which we shall see is the
limiting factor of this weak localization analysis.
164
Theoretical Systematic Errors
Although the weak localization data shown in Figures 5.3 through 5.6
was well fit by equations (5.2) and (5.3), alternatives to this are possible. For
example, another common expression for ??(H
?
) which also produces good
fits is [53]
??(H
?
)=(
e
2
2?
2
planckover2pi1
)[?(
1
2
+
planckover2pi1
4eH
?
lscript
?
2
)?ln(
planckover2pi1
2eH
?
lscript
2
e
)] (5.13)
which can be obtained from equation (5.2) by replacing ?(1/2 + planckover2pi1/2eHlscript
2
e
)
with its asymptotic form, the natural logarithm. More generally, one might
want to take into account all types of scattering (electron-electron,
electron-phonon, and electron spin interactions) and fit data using the full
expression for the weak localization correction, equation (2.36), mentioned in
Chapter 2. The point is that although there will be certain features common
to all two-dimensional electron systems (such as the magnetoresistance being
anisotropic), there will also be important di?erences particular to each, and a
theory specific to weak localization in Si:P ?-layers including band structure
e?ects, density dependence, etc., has yet to be developed. Because of this,
the results shown in Table 5.1 have theoretical systematic errors associated
with them, stemming from the use of imperfect modeling, and it is my
assertion that these modeling limitations dominate the errors in this work.
I have identified several desiderata for a theoretical model of the weak
localization correction ??(H) in Si:P ?-layers, which I shall write as ??(H,?)
to emphasize the vectorial nature of the measurement (again, ? is the angle
165
between H and the current density J; see Figure 4.5). In addition to being
as specific as possible to the system under study (i.e. accounting for subband
e?ects, etc.), these criteria include that ??(H,?) should be a continuous
function of ?, have well-defined parameter regimes over which the theory
applies, predict crossover behaviors, be consistent with previous results, and
incorporate lscript
c
throughout the analysis, not merely for parallel fields. I shall
touch on these points only briefly.
For purely physical reasons, if measurements are performed over a finite
range of magnetic field (say |H|?[-1,+1] T), ??(H,?) should be a continuous
function of ?.GivenafixedH = H
0
, the magnetoconductance is maximal
for ? = ?/2, minimal for ? = 0, and should monotonically decrease as ? is
reduced from ?/2 to 0. This behavior is shown in Figures 5.3 through 5.6.
Thus far theoretical papers, especially [62], have addressed this point by
decomposing H into components perpendicular (H
?
= |H|sin(?)) and
parallel (H
bardbl
= |H|cos(?)) to the 2DEG, and phenomenologically introducing
an additional dephasing rate due to the parallel field: 1/?
bardbl
? 1/H
2
bardbl
.
Although phenomenology is acceptable and ?on the right track?, a
first-principles theory would be preferable.
That well-defined parameter regimes and predictable crossover behaviors
are lacking in the present work is demonstrated by the data shown in Figures
5.9 through 5.13 and the fitting results in Table 5.2. Figure 5.9 captures the
essence of the point of view I advocate here. Although quite simple
166
Figure 5.9: Large field Hall data for the cook 0 device. The slope yields a carrier
density of n = 1.4x10
14
/cm
2
.
167
(Appendix A), the Hall e?ect is an extremely accurate theory; the measured
transverse magnetoresistance is highly linear over the entire accessible
magnetic field range in these experiments. The same cannot be said of using
equation (5.2) to fit the data shown in Figure 5.10. The longitudinal
magnetoconductance shows an almost ?resonant? behavior near H = 0, but
instead of ??(H
?
) asymptotically approaching (2e
2
/?h)ln(2lscript
?
/lscript
e
)as
predicted by (5.2), the derivative d?/dH instead changes sign near H similarequal 2T,
and progressively decreases as the field magnitude is increased. This e?ect is
likely due to the classical magnetoresistance (CM) contribution to the
conductivity, a well-known phenomenon [2] wherein the resistance of a doped
semiconductor increases like |H|
2
for large |H|. Such considerations have
been omitted from the majority of weak localization analyses, and being able
to predict when the CM begins to dominate based on lscript
e
, lscript
?
,etc.wouldbe
extremely valuable for interpreting data.
With regard to a consistent theory of weak localization in Si:P ?-layers,
Figures 5.11 and 5.13 immediately show several features that are necessary,
namely characteristic magnetic field and temperature dependences. The
graph in Figure 5.11 shows two measurements of the longitudinal resistance
of a device as the temperature is reduced to the dilution refrigerator base
temperature. The upper curve corresponds to zero applied magnetic field,
the lower curve to a perpendicular field of 2 T. Except for an approximately
constant o?set, these two curves are nearly identical. Such behavior is
168
Figure 5.10: Large field weak localization data. The fact that ?(H) is not a mono-
tonically increasing function of |H| shows the limited validity of equation (5.2). The
reduction in ?(H)for|H|?2 T is likely due to classical magnetoresistance.
169
expected in a system where dephasing is dominated by the electron-electron
interaction [40, 34] (a fit to the predicted logarithmic dependence on T is
shown). Thus a theory of localization in these systems should from the
beginning consider this interaction as the primary mechanism responsible for
the magnetoconductance.
Turning to Figure 5.13, the measured change in resistance versus H
?
is
plotted for a sample at two fixed temperatures, T = 4.2 K and T = 130 mK
(a complement of the data in Figure 5.11, where the temperature was varied
and the field held constant). Since the width of this line shape gives a
measure of the dephasing time (the sharper the curve, the longer the
dephasing time), it is reasonable that the higher the temperature data
displays a broader line shape (presumably in part due to phonon-induced
dephasing). The canonical interpretation of these results is to argue that the
dephasing time becomes infinite as T ? 0 according to a power law, i.e. ?
?
? 1/T
p
. However, if one looks closely at the data corresponding to the
lowest temperatures of Figure 5.11, it is evident that the resistance saturates
(ceases to change) for T ? 200 mK, implying a concurrent saturation of ?
?
.
To make this saturation more apparent, the H = 2 T data is plotted on a
logarithmic temperature scale in Figure 5.12. Such saturation behavior has
been observed in all experimental weak localization studies thus far reported,
and a possible explanation in terms of so-called intrinsic decoherence was
advanced in [92]. The theoretical fervor which this prompted is still
170
Figure 5.11: Temperature dependent data (constant field). The upper curve (solid
points) was recorded with|H|= 0, the lower curve (open squares) with H = 2 T. The
fact that the form of curve does not change with the large field applied demonstrates
that the electron-electron interaction is responsible for the dephasing in Si:P ?-doped
samples. The solid line is a fit to the equation R(T) = R
0
+ Aln(BT), with values
R
0
= 88.1 ?, A = -0.36 ?, and B = 0.86/K. The jumble of data points for |H| =0
near T = 1 K is due to the refrigerator ?hanging up? (ceasing to cool continuously)
at this point as mixture was circulated.
171
Figure 5.12: Data of Figure 5.11 plotted on a logarithmic temperature scale, high-
lighting the saturation of R(T) for T ? 200 mK.
172
unfolding at the time of this writing, and remains unresolved.
The final theoretical point I consider is the appropriate incorporation of
the correlation length, lscript
c
, throughout the magnetoconductance analysis.
Although the influence of lscript
c
on ??(H
bardbl
) has been considered at length in
terms of estimating T , the e?ect of lscript
c
on ??(H
?
) has been neglected.
Obviously, since weak localization is ultimately reducible to the
Aharanov-Bohm e?ect, and this is itself a manifestation of a geometric
phase, the discussion of Chapter 2 makes it clear that the inclusion of lscript
c
in a
calculation of ??(H
?
) is necessary for a complete theory. Furthermore,
calculations for systems with multiple subbands, intersubband scattering,
etc., should be undertaken for a better understanding of magnetotransport in
?-layers.
Without resorting to any particular model of the ?-layer thickness, a
reasonable estimate of T in the cook 0 device can be obtained by considering
the SIMS data of Figure 5.1 and the Hall data of Figure 5.9 (which yields n
=1.4x10
14
/cm
2
). Since the SIMS data gives a three-dimensional P density
versus depth in the sample, while the Hall data gives a two-dimensional
electron density, integration of the SIMS profile produces a two-dimensional
density. By starting at the peak of the SIMS profile and progressively
stepping away from this point (symmetrically in either direction), the
number of two-dimensional electrons in each infinitesimal interval is summed
until the resultant two-dimensional density equals the measured Hall density,
173
Figure 5.13: Temperature dependent weak localization data. The width of the line
shape is narrower at 130 mK, due to the logarithmic dependence of the dephasing
length on temperature: lscript
?
? ln(T).
174
n. The MATLAB routine which carries out this computation is shown in
Appendix B. The result, which I denote by T
S
(for ?SIMS?), is T
S
=7.5nm.
This shall serve as an important benchmark for the theoretical models of T I
consider below.
To obtain an order of magnitude estimate for how theoretical limitations
influence the data analysis, consider the results displayed in Table 5.2.
Shown are the best-fit results to equations (5.2) and (5.3) for data obtained
from the cook 0 device as the fitting range of |H| is reduced from +/- 1 T to
+/- 0.1 T. Also shown for each range are estimates for the ?-layer thickness
in this device based on the ?uncorrelated? (T ) and ?correlated? (T
C
)models
discussed in Chapter 2. Several features are evident. First, the best-fit values
of lscript
?
steadily decrease as the range of |H| is reduced, and the same trend
occurs for lscript
e
, although the variation in this case is much greater. Second, the
best-fit values of ? are by contrast rather uniform, suggesting that equation
(5.3) is a su?ciently accurate model for parallel field data. Lastly, the T and
T
C
estimates are actually quite consistent (T = 2.3 +/- 0.06 nm, T
C
= 15.8
+/- 0.6 nm), provided one only considers ranges of |H|?0.4 T. For smaller
ranges of |H|, the parallel field magnetoconductance changes very little, and
the fact that a negative value of ? was found for |H|?0.1 T is a
consequence of the data being more noise than signal. Therefore, thickness
estimates were excluded in this range.
Obviously, the average thickness of the donors in the sample is
175
Table 5.2: Dependence of the parameters lscript
?
, lscript
e
, ?, and consequentlyT , on magnetic
field range used in fitting routine.
H lscript
?
lscript
e
? T T
c
limits
T nm nm T
?2
nm nm
+/- 1.0 103.2 41.1 0.455 2.2 15.9
+/- 0.9 102.2 39.9 0.467 2.2 16.1
+/- 0.8 101.2 38.6 0.483 2.3 16.2
+/- 0.7 100.2 37.0 0.492 2.3 16.2
+/- 0.6 99.3 35.0 0.488 2.3 15.8
+/- 0.5 98.6 33.2 0.493 2.3 15.6
+/- 0.4 97.8 29.9 0.47 2.3 14.6
+/- 0.3 96.7 3.3 0.37 2.1 4.3
+/- 0.2 96.7 3.0 0.02 0.5 1.0
+/- 0.1 92.7 4.1 -0.8 - -
176
independent of the range of |H|, so the less the fitting parameters vary with
the |H| interval, the better. Given the enormous variance between the
best-fit values of lscript
e
, it is clear that equation (5.2) does not provide a reliable
estimate of this parameter. Ignoring this for the time being and using the
result T
S
= 7.5 nm, it seems that the ?uncorrelated? thickness model (T =
planckover2pi1
?
?/2elscript
?
) underestimates the ?-layer thickness by roughly a factor of 3.5,
while the model corresponding to T
C
(equation (5.4)) overestimates the
thickness by approximately a factor of 2. Even using the estimate for lscript
e
obtained from n and ? (Appendix A), T
C
= 11.7 nm, which is still nearly
60% larger than T
S
.
What this analysis shows is that although the parallel field weak
localization technique is in principle very powerful, and should ultimately
prove to be more sensitive than SIMS, the theory necessary for interpretation
of the data is still far from adequate for this purpose. Nevertheless, this work
has provided a good starting point for further study.
5.1.2 Density Dependence
This subsection of my Si:P ?-layer study is concerned with how the
considerations I have discussed depend on the carrier density, n. Originally
my hope was to perform the perpendicular and parallel field weak
localization measurements on Si:P samples with progressively larger sheet
resistivities, apply the analysis previously described, and establish the
177
critical density (n
c
) for the metal-insulator transition in two-dimensions.
However, I was not able to carry this study to its conclusion, due to my
inability to obtain appropriate samples.
SIMS data for one successfully measured sample is shown in Figure 5.14.
This sample came from wafer ?B? in Figure 2.10. The corresponding Hall
data is shown in Figure 5.15 (the linear fit yields n = 6.5x10
13
/cm
2
)andthe
base temperature, zero-field resistivity was ? = 625 ?/a50. The corresponding
data for perpendicular and parallel field weak localization measurements is
shown in Figure 5.16. The fitting parameters established for this device via
equations (5.2) and (5.3) are lscript
?
=94nm,lscript
e
=28nm,and? = 1.12/T
?2
.As
discussed earlier, a better estimate of lscript
e
= 20.4 nm can be obtained by
combining n and ? as described in Appendix A. Using the uncorrelated
estimate of the thickness, I obtain T = 4.3 nm in this sample, while
integration of the SIMS plot yields T
S
= 20.8 nm. I o?er no explanation for
this rather large discrepancy.
SIMS data from wafer ?C? of Figure 2.10 is shown in Figure 5.17. Hall
data for this sample is shown in Figure 5.18. The density in this device was
n = 3.7x10
13
/cm
2
, the zero-field resistivity ? = 1063 ?/a50, and combining
these parameters yields lscript
e
= 15.9 nm. The best-fit parameters for the
corresponding weak localization data, shown in Figure 5.19, are lscript
?
=81nm,
? = 1.6/T
2
. Putting everything together, T = 5.1 nm for this device, while
integration of the SIMS profile yields T
S
= 11.6 nm.
178
Figure 5.14: SIMS data from a sample of wafer B (see Figure 2.10).
179
Figure 5.15: Hall data for a device fabricated from wafer B. The linear fit yields n
=6.5x10
13
/cm
2
.
180
Figure 5.16: Weak localization data from the same device as the Hall data of Figure
5.15 (wafer B). The best-fit weak localization parameters for this device were lscript
?
=
94 nm, ? = 1.12/T
2
.
181
Figure 5.17: SIMS data from a sample of wafer C (see Figure 2.10)
182
Figure 5.18: Hall data for a device fabricated from wafer C. The linear fit yields n
=3.7x10
13
/cm
2
in this device.
183
Figure 5.19: Weak localization data from the same device as the Hall data of Figure
5.18 (wafer C). The best-fit weak localization parameters for this device were lscript
?
=
81 nm, ? = 1.6/T
?2
.
184
One additional sample was measured, but only at T = 4.2 K. The reason
for the abbreviated investigation of this device is that after measuring the
density (n = 2.7x10
13
/cm
2
) and resistivity (? = 2670 ?/a50), it was clear that
this sample was still far from the desired parameter range of R
a50
similarequal h/e
2
.
Although a cursory perpendicular field measurement showed the expected
negative magnetoresistance, instead of performing the standard weak
localization measurements and analysis, the dilution refrigerator was cycled
to room temperature, and additional samples were sought. As mentioned
earlier, despite four di?erent attempts, I was unable to obtain adequate
samples, and was forced to conclude this aspect of my investigation.
The results of my work with respect to the two-dimensional MIT in Si:P
are graphed in Figure 5.20. We see the expected behavior in this figure that
as n is reduced, R
a50
increases. Furthermore, weak localization was clearly
observed in all of these samples, although a thorough investigation was only
performed on the three highest density devices. Completing this graph with
progressively lower density samples would be my first research goal if I were
to continue studying Si:P ?-layers.
5.1.3 Final Remarks and Summary
This section concludes my discussion of Si:P ?-layers. The material I
present here is somewhat ?sprawling?, and follows no particular pattern. I
include it merely to document my own thoughts and perspectives on the
185
Figure 5.20: Metal-insulator transition data. As expected, the samples become more
resistive as the carrier (doping) density n is decreased.
186
work I have performed, and to clarify why my density dependence work was
truncated.
The reason for the di?usivity study was hinted at in Chapter 3, namely to
determine how temperature influences T . Being able to precisely quantify
both T and how T varies with temperature is significant since this
information sets thresholds on the temperatures and times which can be
incorporated in various fabrication scenarios. Given a particular ?-doped
wafer, it seems reasonable to assume that T throughout the sample is more
or less uniform. It is also clear that heating the wafer to a su?ciently high
temperature should cause the dopants to di?use from their original positions
in the crystal and increase T . Given these considerations it is obvious that if
oneweretomeasureT of two devices from the same wafer, the device which
was annealed at the same temperature but for a longer time should display a
larger value of T , and indeed this is shown by my data, as reported in [79].
The density dependence aspect of this work was concerned with
establishing how a weak localization measurement of T depends on n, as well
as establishing n
c
, the crtical density for the metal-insulator transition in
two-dimensions. Particularly interesting for study would have been devices
with R
a50
? h/e
2
(h/e
2
similarequal 25.8 k?), since this condition is equivalent to k
F
lscript
e
? 1 (appendix A), and the usual assumptions made when calculating ??(H)
are invalid. Such an investigation would have allowed the crossover behavior
of localization (from weak to strong) to be experimentally established, a
187
useful result for future theoretical work in this area.
Four separate attempts were made to obtain appropriately doped wafers,
but each time the measured resistivity was either too low or too high to be
meaningful. Lightly doped samples were tested, but all showed nonlinear I-V
characteristics, and even measuring the Hall coe?cient on these samples
proved impossible (thus n could not be determined). The probable
explanation for this behavior is that n in these samples was actually below
n
c
, the observed conduction was due to transport through an impurity band,
and was consequently non-ohmic. As discussed in Chapter 3, n is controlled
by varying the wafer?s exposure time to the P source during growth (lower
exposure times correspond to lower doping densities, and therefore higher
resistivities). The non-ohmic samples were grown with exposure times of less
than 40 seconds, while the low resistivity samples were exposed to the P
source for 180 seconds or longer. In a seemingly foolproof e?ort to obtain
highly resistive but ohmic wafers, I requested a series of samples with shutter
times of 60, 80, 100, 120, 140, 160, and 180 seconds. Unfortunately, this
attempt also failed. After roughly one week of trying to understand these
results by performing a battery of processing tests, the e?ort proved hopeless
and was abandoned.
Perhaps the failure to obtain proper samples can be explained by
considering the SIMS data shown in Figures 5.1, 5.14, and 5.17. Several
features are common to all of these plots. First, all show that the
188
background P level is much higher after opening the shutter (similarequal 3x10
18
/cm
3
),
than before ?-layer growth (similarequal 5x10
17
/cm
3
). This is obviously undesirable.
Second, the ?bumps? in the spectra before the doping peak (i.e. deeper in
the sample) likely correspond to the changes in substrate temperature during
growth. These features are clearly much worse in samples B and C, which
are decidedly ?un-?-like?. Third, although the large surface signal common
to all of these spectra is supposedly due to Si:H (see Chapter 3), I found that
the surfaces did conduct T = 4.2 K, which is also unwanted. My guess is
that the degradation in sample quality I observed and my inability to obtain
wafers with specified doping levels had to do with some repairs that Phil
made to his MBE system.
An interesting idea for future work would be to convolve a ?SIMS
potential? with my calculation for band structure. Since a higher doping
density corresponds to a deeper potential well, ?inverting? a SIMS spectrum
would essentially define the potential energy environment experienced by the
electrons in the sample. In other words, I expect that the eigen-energies
would be given by an expression roughly of the form
E =
integraldisplay
n(z)E(n(z))dz (5.14)
where n(z) is specified by the SIMS data, and E(n(z)) is found from my
calculation, given n. This is analogous to density functional theory, discussed
in Chapter 2.
To summarize my results, the data shown in Figures 5.3 through 5.6, and
189
the corresponding parameters in Table 5.1, demonstrate that the e?ective
conduction thickness, T , of the 2DEG associated with a ?-doped layer can be
measured using weak localization. Furthermore, all models for T are related
to the perpendicular and parallel field weak localization parameters as
T?
?
?
lscript
?
. (5.15)
By determining T in several di?erent samples annealed at T = 850
?
C, an
estimate of the di?usivity of Si:P has been established at this temperature
[79].
Second, the ability to measure T was shown in samples with carrier
densities ranging from approximately 10
13
/cm
2
to 10
14
/cm
2
,andthisweak
localization technique works best in very thin ?-layers, since the signal
contrast implied by (5.15) is higher in this situation. However, because lscript
?
decreases with doping density n, for su?ciently low values of n this technique
will ultimately fail, presumably when R
a50
similarequal h/e
2
. The influence of the Si:P
Bohr radius on this limit has yet to be determined, although since the
present theory of weak localization works best for samples with low values of
R
a50
, this implies a large overlap between the donor wave functions.
Finally, temperature and magnetic field dependent resistance
measurements suggest that the phase coherence length, lscript
?
, is limited by the
electron-electron interaction, and my annealing experiments provide evidence
for an enhancement of this dephasing due to lattice defects. Although a
great deal of experimental knowledge has been gained about Si:P ?-layers in
190
Table 5.3: Results for all measured Si:P delta-doped samples discussed in this work,
excluding the ?cooked? devices of Table 5.1. The carrier density n was obtained
from a linear fit to Hall data, and R
a50
is the zero-field resistivity. lscript
e
was estimated
using equation (A.16), while lscript
?
and ? correspond to the best fit values of the data
to equations (5.2) and (5.3), respectively. T was estimated using T =(planckover2pi1/2e)
?
?/lscript
?
.
Weak localization data was not obtained for the sample with n = 2.7x10
13
e
?
/cm
2
.
wafer n R
a50
lscript
e
lscript
?
? T T
S
e
?
/cm
2
?/a50 nm nm Tesla
?2
nm nm
A 1.4x10
14
429 20.3 101 0.49 2.3 7.5
B 6.5x10
13
625 20.4 94 1.12 3.7 20.8
C 3.7x10
13
1063 15.9 81 1.6 5.1 11.6
D 2.7x10
13
2670 7.4 - - - -
191
the course of these investigations, it is also clear that more theoretical work
is needed to more accurately model the data, in particular with respect to
the influence the correlation length lscript
c
has on the magnetoresistance.
These observations are contained in the data presented in Table 5.3 for all
of the Si:P samples measured here.
5.2 Si:B ?-layers
Although the majority of the experimental work I performed was on Si:P
?-layers, the other shallow donor in Si, namely Boron (B), was investigated
using the same techniques. Being a dopant from column III of the periodic
table, B is an acceptor impurity in Si, and therefore the low temperature
conduction characteristics displayed in Si:B samples are due to holes,not
electrons. This makes these samples interesting to study in their own right,
but having the two sets of data side-by-side is rather striking, demonstrating
several di?erences that cannot be explained simply by changing the sign of
the charge carriers. An accounting of the MBE growth and device
fabrication methods, theoretical background, and results of these
experiments now follows. The measurement techniques are exactly the same
as for Si:P samples (see Chapter 4).
192
5.2.1 Sample Preparation
Epitaxially grown Si:B ?-doped wafers have, to this point, been studied
much more intensively than Si:P samples; see, for example [93]. Segregation
in these samples is not an issue, as evidenced by the SIMS analysis shown in
Figure 5.21 (performed on a piece of the Si:B wafer used in these
measurements). Unlike Si:P, the backside of the doping region in this sample
appears to be more extended than the front, which is possibly an artifact of
the SIMS measurement itself. For this profile, O
2
primary ions were used to
bombard the sample, with an incident energy of 3 keV at angle of 52
?
.
A standard 50 ?mx50?m Hall bar was constructed for measurement as
described in Chapter 3. In order to make ohmic contacts to the B ?-layer,
aluminum was evaporated and annealed, instead of the Au-Sb alloy used for
Si:P devices. The reason for this choice is purely one of suitable interface
chemistry, as trying to use Au-Sb to contact the ?-layer produced insulating
samples at low temperatures. This behavior is easily understood when one
considers that the Schottky barrier energy is raised when an n-type metallic
alloy is brought into contact with a p-type semiconductor [19]. Aside from
this contact detail, this sample was lithographically identical to the Si:P
devices of the previous section and shown in Figure 4.2.
193
Figure 5.21: SIMS data for an Si:B ?-doped sample. The density of data points was
su?ciently small that a line has been drawn through them to guide the eye.
194
5.2.2 Theory
Since we are considering hole transport in this section, several theoretical
considerations come into play. One is the that the wave function of a hole
has a p?wave character, instead of the s-wave type P donors exhibit. Thus
the symmetries obeyed by these wave functions will be di?erent, and in turn
so will the transport properties of devices fabricated from an Si:B ?-doped
wafer. The primary manifestation of this asymmetry is a strong spin-orbit
coupling, which leads to a positive magnetoresistance. The appropriate
equation for modeling the magnetoconductance for H = H
?
is given by [53]
??(H
?
)=?(
e
2
4?
2
planckover2pi1
)[?(
1
2
+
planckover2pi1
4eH
?
D?
?
)+ln(
4eH
?
D?
?
planckover2pi1
)]. (5.16)
Note that this produces the desired positive magnetoresistance (??(H
?
) < 0
? H), and the prefactor is (1/2) of that shown in (5.2). As before, instead of
working with ?
?
Iconsiderlscript
?
through the relation lscript
?
=
radicalbig
D?
?
.
In parallel fields, I will use a slightly modified version of (5.3):
??(H
bardbl
)=?(
e
2
2?
2
planckover2pi1
)ln(1+?H
2
bardbl
) (5.17)
where ? is an additional fitting parameter which I have introduced ?by
hand? (post hoc). In other words, I have no physical argument justifying the
use of ?, except that it?s introduction allows the data to be fit better!
195
Figure 5.22: Hall data for the Si:B ?-doped device. The slope yields a hole density
of n = 1.43x10
14
/cm
2
.
5.2.3 Results
First, I note that the Hall data in Figure 5.22 yields n = 1.43x10
14
/cm
2
in this sample, a density comparable to the cooked Si:P devices of [79]. The
zero-field sheet resistivity, however, is substantially higher: R
a50
= 2450 ?/a50.
Combining these quantities, I estimate the mean free path lscript
e
= 3.5 nm in
this device. The density of data points in Figure 5.21 was too small to
integrate and thus estimate the conduction thickness.
196
The weak localization data shown in Figure 5.23 yields the following
results for the best-fit parameters: lscript
?
= 44.3 nm, ? = 0.09, and ? = 5.35/T
2
.
The fact that ? is significantly di?erent from 1 strongly suggests the need for
a better parallel field fitting equation than (5.17). Putting this question
aside for future work and plowing ahead with the canonical analysis, I obtain
an uncorrelated estimate of T = 17.2 nm for this device.
Carrying out the analogous temperature and field dependent
measurements on this sample produced several interesting results. First, by
measuring the longitudinal resistance as the sample was cooled from T = 4.2
K to base temperature, a very strong saturation of the resistance was
observed, setting in near T similarequal 800 mK; see Figure 5.24. Although the
physical reason for this behavior is unknown, similar results have been
reported previously [94, 95]. Second, performing weak localization magnetic
field sweeps at constant temperatures (Figure 5.25) showed that the width of
the line shape decreases with T, as one would expect for a
temperature-dependent dephasing time. I also note that the longitudinal
magnetoconductance is well-fit by equation (5.16) all the way out to |H| =
+/- 10 T. The best-fit value of lscript
?
for this data is lscript
?
= 47.7 nm, within 10%
of the value obtained for sweeps between +/- 1 T. Thus another di?erence
between Si:B and Si:P samples has been established: evidence of classical
magnetoresistance e?ects is harder to discern in Si:B devices.
To summarize my results for this limited Si:B ?-layer study, I first note
197
Figure 5.23: Weak localization data for the Si:B sample. Note the
negative magnetoconductance, as compared to the positive magnetoconductance of
Si:P samples. The best-fit parameters to equations (5.16) and (5.17) are lscript
?
= 44.3
nm, ? = 0.09, and ? = 5.35/T
2
.Thefactthat? is substantially di?erent from
unity suggests that (5.17) is an inadequate model for the parallel field magnetocon-
ductance.
198
Figure 5.24: Si:B temperature dependent data in zero magnetic field. The saturation
of this data occurs at a much higher temperature (T similarequal 800 mK) than in Si:P samples
(T similarequal 200 mK). The origin of this behavior is unknown.
199
Figure 5.25: Si:B temperature-dependent data. As with Si:P samples, the lower the
temperature, the narrower the line shape, due to the logarithmic dependence of lscript
?
on temperature.
200
Figure 5.26: Large field sweep for Si:B sample. This data is to be contrasted with
that of an Si:P sample (Figure 5.10), where non-monotonic behavior is observed.
The line fit to this data yields lscript
?
= 47.7 nm, close to that obtained when the field
limits were only +/- 1 T (lscript
?
= 44.3 nm).
201
that a positive magnetoresistance is observed for all fields and orientations,
presumably due to the strong spin-orbit scattering e?ects associated with the
p-wave B acceptor impurity. The device was substantially more resistive (?
6x) than its Si:P counterparts of comparable density. The asymmetry
between the H
?
and H
bardbl
data suggests that one should be able to measure T
using weak localization in these samples, and from the best-fit parameters I
estimate T = 17.2 nm. However, the anomalously low value of ? in equation
(5.17) suggests that this area of the theory needs further development.
Perhaps some interesting results relating to the spin-orbit e?ect in parallel
fields are involved. Finally, I note that the R(T) curve saturates at a much
higher temperature than in Si:P devices.
202
Chapter 6
Conclusion
A brief look at possible directions of future research and a summary of
the results I obtained in the course of this work comprise the final chapter of
this dissertation.
6.1 Future Directions
Several avenues for further work would be worthwhile pursuing. For one,
determining the two-dimensional metal-insulator transition densities for both
Si:P and Si:B would provide answers to a variety of interesting questions. Is
this transition of the Anderson or Mott type? For what value of the carrier
density does the canonical theory of weak localization, used throughout this
dissertation, collapse? How do holes behave di?erently from electrons in this
limit? Can aspects of the two-dimensional scaling theory of localization be
203
observed in ?-doped devices? Can x-ray di?raction studies be used to
quantify lattice disorder and therefore probe the electron-electron interaction
theory? Do Hall measurements yield additional information on the
electron-electron dephasing time, ?
e?e
?Howdoesthenoise of these samples
change when in the presence of a magnetic field, including perpendicular and
parallel to the plane of the ?-layer? The list is extensive, and no doubt will
be eventually undertaken.
Another aspect of these samples worth exploring is in terms of
one ?dimensional transport physics. This includes phenomena such as
universal conductance fluctuations, di?erent weak localization
characteristics, etc [68]. Simply by reducing the width of the Hall bar to less
than the phase coherence length (W lessmuch lscript
?
), the device becomes e?ectively
one-dimensional, literally a nanowire. Such an approach would be fairly easy
to address from a fabrication standpoint using electron beam lithography,
and a wealth of experimental possibilities also exist in this context.
In terms of actual device fabrication, constructing silicon single electron
transistors (SETs), using either Schottky barriers or electrostatic gates,
would be useful from a variety of perspectives. In immediate terms, having
an alternative SET design would produce more knowledge on the charge
o?set phenomenon, which presently prevents SETs from being used in a
current standard [13]. From a long-term perspective, investigating these
devices would clarify whether or not their use as controllable charge
204
detectors is feasible.
Finally, detailed theoretical calculations of ?-doped silicon need to be
realistically addressed. Most importantly for this dissertation, models of
magnetoconductance phenomena incorporating the correlation length lscript
c
suitably defined for ?-layers need to be performed, in order that a competent
experimentalist has the modeling necessary for a quantitative comparison
with measurements.
6.2 Summary
To summarize this dissertation, ?-doping in silicon is technologically
important, and Chapter 1 motivated my experiments. Chapter 2 discussed
the three theoretical aspects of ?-layers crucial for interpreting the
measurements presented in this work, including band structure, weak
localization, and Berry phase. In Chapter 3, I discussed the epitaxial growth
methods and device fabrication techniques incorporated in these
experiments, along with the research of other workers. The measurement
conditions and data acquisition approaches were the topics of Chapter 4.
The results of my magnetoconductance experiments were presented in
Chapter 5, including a study of the di?usivity of Si:P at T = 850
?
C, testing
the parallel field weak localization technique in Si:P devices with di?ering
carrier densities, and an analogous set of measurements on Si:B ?-layers.
Many di?erences were apparent in these samples, most notably that Si:P
205
?-layers exhibit a negative magnetoresistance, while a positive
magnetoresistance is observed in Si:B ?-doped devices, as expected.
6.3 Conclusion
Several noteworthy results were achieved in the course of this work.
First, by adapting a previously demonstrated experimental technique
employed on Si MOSFETs, I have established that a comparison of weak
localization signals in perpendicular and parallel magnetic fields enables a
precise measurement of the e?ective conduction thickness, T ,ina?-doped Si
device. Second, by measuring T in several devices subjected to T = 850
?
C
anneals, an estimate of the di?usivity in these epitaxially grown Si:P ?-layers
has been produced. Third, by investigating the temperature and magnetic
field dependence of these measurements, I have established that the
electron-electron interaction is the primary mechanism limiting quantum
mechanical phase coherence in Si:P ?-layers, and shown evidence for the
enhancement of this e?ect due to lattice disorder, inherent in MBE growth
techniques. Fourth, these characteristic weak localization phenomena were
observed in numerous samples (four were discussed here), with carrier
densities ranging from approximately 10
13
to 10
14
cm
?2
, showing a
reasonably broad range of applicability. Fifth, a brief study of Si:B ?-layers
suggests that a complementary localization investigation in these structures
would be informative and straightforward to carry out. Sixth and finally,
206
several theoretical issues with regard to the interpretation of these
measurements have been suggested. In all, this work has been broad in
scope, informative in its results, and, on the whole, enjoyable to carry out.
These results establish weak localization as a powerful tool in the
materials analysis of ultra-thin conducting samples. Although weak
localization has long been appreciated from rather esoteric theoretical and
experimental perspectives, there are indeed practical applications of this
phenomenon, as is evident by the work I have presented here.
207
Appendix A
Elementary Transport Theory
In this appendix I will derive several useful results concerning 2DEG
transport. Although these are standard relations and straightforward to
derive, having them all in one appendix is beneficial from a completeness
standpoint, as well as establishing equivalent ways of interpreting my
measurements. The quantities I consider include the conductivity, ?;the
carrier density, n; the mean free path lscript
e
; the carrier mobility, ?;andthe
density of states, ?(E).
A.1 Classical Conductivity
Recall that the classical conductivity of a 2DEG, ?, is related to its
resistivity ? (or, equivalently, R
a50
)by? =1/?. An expression for ? using
Newton?s second law, combined with Ohm?s law, can be obtained as follows.
208
First, the application of an electric field E (in the form of a voltage bias)
accelerates the electrons of a conducting sample in the direction opposite to
E. The electrons (with e?ective mass m
?
), on average, accelerate for a time
?
e
before experiencing a scattering event in which their momentum is
randomized. Therefore, they acquire an average drift velocity, v
d
,before
scattering. For the time-independent, steady state situation, the acceleration
due to E and the momentum relaxation balance each other, i.e., we have the
equality
F = m
?
a = eE?
m
?
v
d
?
e
=0. (A.1)
From this we obtain
v
d
=
parenleftBig
e?
e
m
?
parenrightBig
E. (A.2)
On the other hand, Ohm?s law relates the current density J to the applied
field E via
J = ?E. (A.3)
If we assume that the sample has n electrons per unit area, J can also be
expressed as
J = nev
d
. (A.4)
Substituting (A.2) into (A.4), and equating this with (A.3), we obtain the
well-known Drude result for the classical conductivity
? =
ne
2
?
e
m
?
. (A.5)
I now discuss how n can be determined with Hall e?ect measurements.
Before proceeding, however, I note that even with ? and n experimentally
209
determined (and the electron charge e obviously well-known), we still have
incomplete information about the electrons in the sample, since the
parameters ?
e
and m
?
are unknown. Other measurements, beyond the scope
of this thesis (such as Shubnikov de-Haas oscillations), must be performed to
determine these quantities.
A.2 Hall E?ect
Since the results of this thesis are very sensitive to the precise angle ?
between the magnetic field H and the current density J, I now derive the
relationship between these quantities and the Hall resistance, R
H
.
To proceed, I consider the square Hall bar geometry of Figure 4.2, and
assume, for the moment, that the electric current is carried by particles with
positive charge q (I will later take q = e, the electron charge). When q
moves with a velocity v in a magnetic field H, it experiences a Lorentz force
F which is orthogonal to both these vectors, given by
F = q(v?H). (A.6)
Since this force is proportional to the cross product of v and H,the
magnitude of F depends on the angle ? between these vectors:
|F| = q|v||H|sin(?). (A.7)
In words, if v and H are parallel (? = 0), no force acts on q, whereas if they
are perpendicular (? = ?/2), F is maximal. For perpendicular fields, q is
210
deflected towards one side of the device, determined by applying the
?right-hand rule? to (A.6).
Next, I take v to be the drift velocity v
d
of the electric current in a
sample with n charges per unit area:
J = nqv
d
. (A.8)
From (A.7), I define an e?ective electric field, E
H
, associated with the Hall
e?ect, given by
|E
H
| = |v
d
||H|sin(?). (A.9)
Solving (A.8) for |v
d
| and substituting this into (A.9) I obtain
|E
H
| =
|J||H|sin(?)
nq
. (A.10)
This ?deflection? field pushes the electrons towards one side of the device,
thereby inducing a charge gradient across the sample, and consequently a
transverse Hall voltage,V
H
.SinceV
H
=-
integraltext
E
H
?dl, and the line integral is
simply the transverse distance W across the width of the sample,
V
H
=
|J||H|Wsin(?)
nq
. (A.11)
However, as mentioned in Chapter 4, for a 2DEG, |J| = I/W, where I is the
current bias through the device. Incorporating this relation into (A.11) and
taking q = e, I arrive at our final expression for the Hall voltage induced in a
2DEG:
V
H
=
I |H|sin(?)
ne
. (A.12)
211
It is worth noting that V
H
is independent of the device width W, and that
V
H
can naturally be associated with a Hall resistance,R
H
, by dividing
through by the bias current I:
R
H
=
|H|sin(?)
ne
. (A.13)
The sin(?) dependence was crucial in this research, since the Hall e?ect was
used to determine the angle between the J and H.
As implied by (A.13), plotting R
H
versus |H| in perpendicular fields
(sin(?) = 1), produces a line whose slope determines the carrier density, n.
The importance of n can be appreciated by noting the frequency with which
it appears in the following derivations.
A.3 Mean Free Path
An extremely useful length scale which can be inferred from
magnetoconductance measurements is the mean free path, lscript
e
. The obvious
relationship which holds on a microscopic scale is
lscript
e
= v
F
?
e
(A.14)
since the electrons are ?whizzing? around the lattice at the Fermi velocity v
F
and are scattered, on average, every ?
e
seconds. Substituting this for ?
e
in
the Drude expression (A.5) for the resistivity yields
1
R
a50
=
ne
2
lscript
e
mv
F
=
ne
2
lscript
e
p
F
. (A.15)
212
The advantage of doing this is that now lscript
e
canbeexpressedintermsoftwo
measurable quantities: the resistivity R
a50
and the density n (via Hall
measurements). Substituting p
F
= planckover2pi1k
F
= planckover2pi1
?
2?n into (A.15) and solving for
lscript
e
, I find
lscript
e
=
parenleftbigg
h
R
a50
e
2
?
2?n
parenrightbigg
=
parenleftbigg
h
e
2
parenrightbiggparenleftbigg
1
R
a50
k
F
parenrightbigg
. (A.16)
Several features about this expression are noteworthy. For one, the prefactor
of the resistance quantum, (h/e
2
) highlights the quantum mechanical nature
of electronic transport. Second, the fact that lscript
e
scales like (1/R
a50
)is
physically appealing, since a device with a low value of R
a50
should have a
long mean free path. Similarly, given that the donor atoms become ionized
scattering centers for the electrons to bounce o? of, the fewer of these
scatterers are present, the longer lscript
e
should be, as suggested by the (1/
?
n)
dependence.
Most relevant for this work, however, is the fact that lscript
e
? k
?1
F
.As
mentioned in Chapter 2, calculations of the weak localization correction to
the Drude conductivity, ??(H), are predicated on the so-called ?disorder
parameter?, k
F
lscript
e
, being large compared to unity: k
F
lscript
e
greatermuch 1. What is
immediately apparent from (A.16) is that this condition is equivalent to the
resistivity being small compared to the resistance quantum:
k
F
lscript
e
greatermuch 1 ? R
a50
lessmuch
h
e
2
. (A.17)
This inequality was satisfied for all of the samples investigated in the course
of my research, with the smallest observed value of k
F
lscript
e
being approximately
213
8. One of the original goals of this work was to determine precisely for what
values of k
F
lscript
e
did the ?weakly localized? regime break down, which,
according to (A.17) should occur for R
a50
similarequal (h/e
2
) (approximately 25.8 k?).
Unfortunately (as mentioned in Chapter 5), I was unable to obtain suitable
samples for investigation, and consequently could not carry this program to
fruition.
A.4 Mobility
An alternative measure of ?how well a sample conducts? is the electronic
mobility, ?, defined as the proportionality constant between the drift
velocity, v
d
, and the applied electric field E:
v
d
= ?E. (A.18)
Of course, we can also express the current density as in (A.3) and (A.4):
J = ?E = nev
d
. (A.19)
Substituting (A.18) into (A.19), I find that
ne? = ? =
1
?
. (A.20)
Therefore our final expression for ? becomes
? =
1
ne?
. (A.21)
214
That nothing new has been introduced by considering ? can be seen by
solving (A.21) for ? and equating the result with that of (A.5). The result is
? =
e?
e
m
?
. (A.22)
We are still left with the di?culty of disentangling the elastic scattering time
?
e
with the e?ective mass m
?
, as mentioned in section A.1.
A.5 Density of States
I present here two calculations for the density of states, ?(E), of a
two-dimensional electron gas. The first calculation assumes an isotropic mass
(circular dispersion relation), while the second assumes an anisotropic mass
(elliptical dispersion). Both of these results are used in my band structure
calculation described in Chapter 2.
The physical significance of ?(E) is that it describes the number of
electrons per unit area in a 2DEG as a function of energy. That is, if there
are n
0
electrons in a given subband of a 2DEG with energies between E
1
and
E
2
,then?(E) is defined by
n
0
=
integraldisplay
E
1
E
0
?(E)dE. (A.23)
Equivalently, ?(E) can also be defined by di?erentiating both sides of
equation (A.23) with respect to E:
?(E)=
dn
dE
. (A.24)
215
Before carrying out the calculations, I briefly digress to make an important
point regarding low temperature electronic transport.
Because electrons possess an intrinsic ?spin? angular momentum equal to
planckover2pi1/2, they are classified as spin-1/2 fermionic particles. Consequently, they
are subject to the Pauli exclusion principle, and obey Fermi-Dirac statistics.
This is significant for my work since at low temperatures electronic
conduction is entirely due to electrons within roughly k
B
ToftheFermi
energy, E
F
. This is apparent by considering the low temperature behavior of
the Fermi-Dirac distribution function, which gives the probability that an
electronic state with energy E is occupied [87]:
f(E)=
1
e
?(E?E
F
)
+1
(A.25)
where ? = 1/k
B
T. For T ? 0, all states with E < E
F
are occupied, while all
states with E > E
F
are unoccupied. Given the low temperatures associated
with the measurements in my experiments (T similarequal 130 mK), the assumption
that the transport physics is dominated by electrons very close to E
F
is
clearly a good approximation.
A.5.1 Isotropic Mass
To derive ?(E) for a 2DEG with an isotropic mass m
?
, I consider a
system of N such electrons confined to a square of side L at T = 0.
According to the Pauli principle, the lowest single-electron energy levels are
216
doubly occupied up to the highest energy in the system, E
F
. I now establish
how E
F
depends on N and L.
In k-space, the electrons will occupy all states out to some radius k
F
,the
Fermi wavevector, and states with |k| > k
F
will remain unoccupied. The
area of this ?Fermi circle? is ?k
2
F
. Due to the boundary conditions that the
wavefunction must be zero on the perimeter of the square, the smallest
allowed wavevector, k
min
will have a magnitude of ?/L, and all other allowed
wavevectors in the system will be integral multiples of k
min
. For spin-zero
particles, N is simply the ratio of the total volume occupied in k-space (?k
2
F
)
divided by the volume occupied by the standing wave mode corresponding to
k
min
((2?/L)
2
) [78]. For spin-1/2 electrons this ratio must be multiplied by 2
to take account of the spin degeneracy, and we have
N =
k
2
F
L
2
2?
. (A.26)
Clearly the density of electrons n = N/L
2
, and thus I obtain the following
relation (used previously) between k
F
and n:
k
F
=
?
2?n. (A.27)
Since I am assuming an isotropic dispersion relation (i.e. E = planckover2pi1
2
k
2
/2m
?
), the
Fermi energy is given by
E
F
=
?planckover2pi1
2
n
m
?
. (A.28)
Therefore E
F
in a 2DEG is directly proportional to n. Furthermore, since E
F
is purely kinetic in nature, we can also identify E
F
=m
?
v
2
F
/2 = p
2
F
/2m
?
,
217
where v
F
is the Fermi velocity and p
F
the Fermi momentum of the
conduction electrons (p
F
= planckover2pi1k
F
).
Given the discussion in Chapter 2 on band structure, it should be noted
that this derivation neglects e?ects associated with interactions between the
electrons, or band degeneracies due to the lattice symmetry, which reduce
the value of k
F
from that of (A.27) [28]. ?(E) is now easily found by solving
equation (A.28) for n and then di?erentiating with respect to E
F
.Theresult
is
?(E)=
m
?
?planckover2pi1
2
(A.29)
A.5.2 Anisotropic Mass
Inowcalculate?(E) for a 2DEG with an elliptical dispersion relation. In
other words, I consider a system (such as Si) in which the energy of an
electron depends not only on the magnitude of k, but also it?s direction.
Representing this dependence in Cartesian coordinates
E(k
x
,k
y
)=
planckover2pi1
2
k
2
x
2m
x
+
planckover2pi1
2
k
2
y
2m
y
. (A.30)
Generalizing my calculation for the isotropic mass, I assume that N electrons
occupy a spatial region of dimensions L
x
by L
y
at T = 0. In k-space, the
length of the Fermi wave vector will now be directionally dependent, i.e. k
F
=(k
F
x
,k
F
y
) (I use a superscript to keep the notation uncluttered). Thus the
occupied area is given by an ellipse with semi-major and semi-minor axes
218
lengths of k
F
x
and k
F
y
. Again equating N with the ratio of occupied k-space
to the minimum allowed based on L
x
and L
y
, I find
N =
k
F
x
k
F
y
L
x
L
y
2?
(A.31)
or
n =
k
F
x
k
F
y
2?
(A.32)
since the sample area A = L
x
L
y
.
E
F
can be attained either for k
x
=0,k
y
=k
F
y
or k
x
=k
F
x
,k
y
= 0 (i.e. k
F
x
=
radicalbig
2m
x
E
F
/planckover2pi1
2
,k
F
y
=
radicalbig
2m
y
E
F
/planckover2pi1
2
). Substituting k
F
x
and k
F
y
into (A.32), I
obtain
n =
E
F
?
m
x
m
y
?planckover2pi1
2
. (A.33)
Finally, by di?erentiating (A.33) with respect to E
F
, I arrive at my final
expression for ?(E) in an anisotropic 2DEG:
?(E)=
?
m
x
m
y
?planckover2pi1
2
. (A.34)
Both of the above results for ?(E) shed light on why quantum Hall e?ect
physics is of little concern for 2DEGs in ?-layers. The first quantum Hall
plateau occurs when the ratio of the magnetic field to the carrier density
equals h/e: |H|/n = h/e. Inserting our canonical density of 10
14
/cm
2
, I find
that this requires a field of |H|similarequal4000 Tesla! It is highly improbable that
such fields will ever be achieved on earth, and even samples with densities
near the metal-insulator transition limit (n similarequal 5x10
12
/cm
2
)require|H|similarequal200
T; still impractical fields. Thus, it seems unlikely that the quantum Hall
219
e?ect will ever be observed in ?-doped Si samples, even if one considers
higher index plateaus (since the spacing between these plateaus decreases as
the Landau level index increases).
220
Appendix B
Data Analysis Routines
This appendix contains the MATLAB (version 6.5) code which was used
in various aspects of this work. Included are algorithms for fitting data to
the perpendicular field correction to the conductivity, ??(H
?
), determining
the resistivity of a device using the van der Pauw theorem, and estimating
the ?-layer thickness based on SIMS data and Hall measurements. Linear fits
to Hall data and logarithmic fits to ??(H
bardbl
) data were performed using
pre-packaged routines in the Origin (version 6.1) library.
B.1 ??(H
?
)
The following three sets of code were used to fit perpendicular field weak
localization data. The algorithm used by MATLAB was a nonlinear
least-squares routine, which, according to MATLAB, incorporated a
221
?subspace trust region method based on the interior-reflective Newton
method? (whatever that means). There were three subroutines to the which
comprised the complete fitting algorithm: WL1p3, deltaGWL1p3, and
perpfitWL1p3. The reason MATLAB was used for this task is that the
digamma function, ?(x), (?psi(x)? in MATLAB) is a built-in function in
MATLAB, but not in Origin. Similar sets of code were written for fitting
Si:B data, but have been omitted from this section.
B.1.1 WL1p3
The first part of the fitting routine, WL1p3 (for ?weak localization 1, 3
parameters?) is simply the MATLAB version 6.5 code for equation (5.2),
with fitting parameters p(1), for the o?set conductance (in units of (e
2
/?h)),
p(2) for the dephasing length lscript
?
(in nm), and p(3) for the mean free path lscript
e
(in nm)). H obviously stands for the magnetic field value.
function Y = WL1p3(H,p)
if H==0
Y=p(1);
else
222
Y = p(1) + (psi(0.5 + (164.57)/(abs(H)*p(2)
2
)) - psi(0.5 +
(329.14)/(abs(H)*p(3)
2
)) + log(2*(p(2)/p(3))
2
));
end
B.1.2 deltaGWL1p3
The second part of the fitting routine, deltaGWL1p3 (for ?di?erence
(delta) between weak localization theory and measured data?), ?loaded? the
measured data (gm) into the routine, calculated the theoretically expected
conductance (gc) using WL1p3 based on the values of the fitting parameters
p(i) defined above, and compared the di?erence between the two.
function G = deltaGWL1p3(p)
gm = load(?C:\ data, etc\ perpSiPdata\ filename.dat?);
[M,N]=size(gm);
k=1:M;
gc=zeros(M,N);
gc(k,1)=gm(k,1);
223
for i=1:M
gc(i,2)=WL1p3(gc(i,1),p);
end
for i=1:M
x(i)=gm(i,1);
G(i)=gm(i,2)-gc(i,2);
end
B.1.3 perpfitWL1p3
The last part of the routine, perpfitWL1p3 (for ?fitting perpendicular
field data to WL1p3?) iterated the fitting procedure until the sum
summationtext
i
(gc(i)-gm(i))
2
was minimized, which is essentially the same as minimizing
?
2
. The fit was started with the initial parameters p0 and was constrained to
remain within the physically reasonable bounds defined by lb and ub. This
was carried out using the MATLAB?s built-in non-linear least-squares fitting
routine, ?lsqnonlin?.
p0 = [100 100 10];
lb = [0,10,1];
224
ub = [1e5,1e4,1e4];
[p,resnorm] = lsqnonlin(@deltaGp3,p0,lb,ub)
B.1.4 deltaLp
Unfortunately, the lsqnonlin routine in MATLAB 6.5 only produces the
best-fit values for the parameters p(i), and not the uncertainties in these
parameters. To overcome this deficiency, separate routines were written to
perform this task, and here we show deltaLp (for ?uncertainty in lscript
?
?). The
way the uncertainties were determined was to find the best-fit values of the
parameters using perpfitWL1p3, gradually increment lscript
?
away from its
minimum while holding all other values fixed at their best-fit values,
calculating a new value of ?
2
for each step, and iterating this procedure until
?
2
was doubled from its minimum value: ?
2
? 2?
2
. According to [83], this
procedure probably overestimates the errors in the fitting parameters, but
when compared with the theoretical uncertainties facing the interpretation of
these measurements, is relatively unimportant. The routine for finding the
error in lscript
e
is virtually identical to that of lscript
?
, and will not be shown here.
function f = deltaLp(gm,p)
[m,n]=size(gm);
225
for i=1:m
H(i)=gm(i,1);
gc(i)=WL1p3(H(i),p);
end
r0=chi2(gm,gc);
r=r0;
while (r <= 2*r0)
p(3)=p(3) + 0.001;
gc(i)=WL1p3(H(i),p);
r=chi2(gm,gc); end
r0
r
f=p(3);
B.2 van der Pauw Theorem
This routine takes two properly measured resistances from a van der
Pauw device, R1 and R2, and applies the van der Pauw theorem to
226
determine the resistivity ? (here denoted by R).
R1 = input(?Enter the first resistance:?);
R2 = input(?Enter the second resistance:?);
R=(R1 + R2)/2;
x=exp(-pi*R1/R) + exp(-pi*R2/R);
if x < 1
while (1-x)>.0001
R= R + .001;
x = exp(-pi*R1/R) + exp(-pi*R2/R);
end
else
while (x-1)>.0001
R = R - .001;
x = exp(-pi*R1/R) + exp(-pi*R2/R);
end
end
227
R
B.3 SIMS T
As discussed in Chapter 5, an estimate of the conduction thickness T can
be obtained once given a SIMS profile and a Hall density, n. The way this is
accomplished is by integrating the three-dimensional density of the SIMS
plot until the resulting two-dimensional density equals that obtained by the
Hall e?ect. The integration is started at the maximum value of the density
in the SIMS plot, then symmetrically stepping away from the maximum and
summing the contribution from each interval. The density of each interval is
estimated by averaging the values of the endpoints, while the length of the
interval is given by the di?erence in depths. The algorithm is shown below,
where the parameter nH is the input Hall density, n2D is the ?running tally?
of the integrated SIMS density, and T the corresponding thickness estimate.
function F = SIMS(nS,nH)
nS=load(?C:\data, etc\perpSiPdata\filename.dat?);
nH=1.43e14;
[m,n]=size(nS);
228
for i=5:m
n(i-4,1)=1e-8*nS(i,1);
n(i-4,2)=nS(i,2);
end
[k,l]=size(n);
M=max(n);
for i=1:k
if (n(i,2)?=M)
continue
else
dM=n(i,1);
j=i;
end
end
T=(n(j+1,1)-n(j-1,1))*1e7;
229
n2D=((n(j-1,2)+n(j,2))/2)*(n(j,1)-n(j-1,1)) +
((n(j,2)+n(j+1,2))/2)*(n(j+1,1)-n(j,1));
p=0;
while (n2D<nH)
p=p+1;
T =(n(j+p+1,1)-n(j-p-1,1))*1e7;
n2D = n2D + ((n(j-p-1,2)+n(j-p,2))/2)*(n(j-p,1)-n(j-p-1,1)) +
((n(j+p+1,2)+n(j+p,2))/2)*(n(j+p+1,1)-n(j+p,1));
end
j
p
T
n2D
230
Appendix C
Dilution Refrigerator
Modifications
This appendix is primarily devoted to the wiring modifications which
were performed on the dilution refrigerator used in these experiments.
However, twice in the course of this work, the flow of the mixture became
blocked at liquid nitrogen temperatures (presumably by pump oil), which
necessitated the disassembly and cleaning of the refrigerator. Therefore,
should the same fate befall some future graduate student, the procedures
which were followed to remove the blockage will be briefly outlined.
231
C.1 Wiring
Originally, the dilution unit had ten wires available for measuring
samples on it. Consequently, a maximum of one Hall bar or two van der
Pauw samples could be measured in a single cooldown. After the second
blockage occurred, it became clear that the time required to rewire the unit
would be well spent by enabling much more data to be obtained in a given
cooldown. Given the spatial constraints of the magnet bore (chapter 4), it
was decided that twenty sample wires would be the maximum number which
could be conveniently housed in the inner vacuum can (IVC). The various
control and monitoring elements of the refrigerator, including the
RuO
2
thermometers of the mixing chamber (MC), 1 Kelvin pot (1KP), and
still, along with the MC and still heaters, also required additional wires.
It is probably easier to understand the wiring through the use of diagrams.
To this end, figure C.1 shows the room temperature ?breakout box? which
serves as the electrical ?point of entry? to the dilution unit. The twenty BNC
connectors leading to the sample wires were each equipped with a toggle
switch, allowing a direct connection to the corresponding sample wire at the
MC when in the ?up? position, and to ground when in the ?down? position.
One detail which deserves mention is in regard to the thermometers. As
mentioned in chapter 4, the thermometry was achieved via 4-wire
measurements on calibrated RuO
2
resistors. Although 4 individual wires
were used for measuring the MC thermometer, the 1KP and still
232
Figure C.1: Breakout box schematic used in these experiments. In addition to the
20 sample wires, this box also enabled the thermometers of the mixing chamber
(MC), 1 Kelvin pot (1KP) and still to be monitored, as well as allowing power to be
applied to the MC and still via the heaters. The 1KP and still thermometers were
connected in series, and the toggle switch, when in the ?up? position, measures the
voltage across the 1KP thermometer. When toggled to the ?down? position, the
voltage across the still is measured.
233
Figure C.2: Wiring diagram for the dilution refrigerator. As labeled, the outermost
column corresponds to the wiring of the 40 pin connector at the 1KP. The next
column in on either side corresponds to the wires associated with the 37 pin cable
connector, which connects the breakout box to the refrigerator. The middle columns
specify the function of each contact. An ?x? denotes a pin which is either open or
shorted to ground. Note that 1KP contacts 20 and 21 are available for use, should
the need arise.
234
thermometers were connected in series, with the bias current first flowing
through the 1KP resistor, then the still resistor. Thus the ?bottom? (V-, low
voltage end) of the 1KP and the ?top? (V+, high voltage end) of the still
were at the same potential. Because the Neocera temperature controller
could only read two thermometers at a time, and one of these was dedicated
to the MC, a toggle switch was implemented allowing one to read the 1KP
when in the ?up? position, and the still when in the ?down? position.
The wires from the breakout box were connected to the dilution unit via a
37 pin cable connector, which before entering the unit were filtered through 1
MHz ? filters, as discussed in chapter 4. The wires were then thermally sunk
at 4.2 Kelvin, and then at the 1KP. At this stage, the wires were soldered to
two, 20 pin DIP connectors, and from this point, the wires were distributed
throughout the dilution unit to perform their respective functions. The 20
device measurement lines were further filtered by passing them through 10
kHz RC filters, and were thermally sunk at the still and the MC before being
soldered to the sample mount on the tilt stage.
The correspondence between the breakout box wires, the 37 pin cable
connector, and the 40 pin 1KP stage is shown in figure C.2. This figure
assumes one is viewing the 1KP wires from the ?front?, that is, on the side
opposite to where the wires enter the IVC, are thermally sunk on a 4.2 K
post, then at the 1KP. The outer most columns represent the 40 contacts at
the 1KP stage. Moving in from either side, the next set of columns
235
correspond to the wires associated with the 37 pin cable connector. The
middle of the figure simply labels what function was performed by each pin.
For example, the device wires correspond to pins 9-18 and 22-31 at the 1KP,
and pins 11-20 and 2-10, 23, respectively on the cable connector, while the
mixing chamber is measured with the current source wire I+ (1KP pin 7,
cable pin 34) , current drain wire I-, (1KP pin 34, cable pin 37), and voltage
contacts V+ (1KP pin 8, cable pin 35), and V- (1KP pin 33, cable pin 36).
Those pins marked with an ?x? are dysfunctional for one reason or another;
either because they are open circuits, or shorted to ground. Note that there
remain two unused wires (1KP pins 20 and 21), which could be serve as
replacements for one of the components should the need arise.
Having said all of this, if a problem does arise (as it inevitably will at
some point - this is experimental low temperature physics after all!),
straightforward continuity tests with a hand held DVM go a long way
towards understanding the source of the problems, and the diagram in figure
C.2 should help. If you are reading this appendix because some electrical
aspect of the apparatus isn?t functioning properly, all I can say is good luck!
You?ll need it.
C.2 Repair
The last item which shall be discussed here are the steps to take should
the dilution unit become blocked. A new rotary pump was purchased to
236
circulate the mixture, which hopefully will make this concern a non-issue.
Unfortunately, if the unit does become blocked again, there are no quick-fix
solutions. My experience shows that the blockage occurs in the condenser
line of the refrigerator, which shouldn?t be too surprising considering this is
the smallest impedance in the unit. The only way to fix the problem is to
disassemble the refrigerator, which is straightforward to do.
First, disconnect all wires below the still, so that the MC can be easily
removed without being attached to the rest of the nit via wires. Next,
carefully remove the mixing chamber from the rest of the refrigerator, by
disconnecting both the MC and still, which requires nothing more than
turning screws and bolts. Be especially careful with the condenser line,
which is the most fragile part of the unit. After this, remove the ?tail? from
the mixing chamber (again by turning screws), which basically ?splits open?
the MC. Upon looking inside, you can see a small orifice (the condenser,
where the mixture enters the MC) and a larger tube (the still line) which
extends approximately 1 cm into the MC. At this point, you?re ready to roll.
Support the MC so that it is in the upright position, and flush the
condenser line with TCE (trichloroethylene). Be careful with this stu?,
because it is carcinogenic. I had a fan blowing while I was dispensing it, as
well as wearing gloves and a mask. J.B. Dotellis made a special apparatus to
do this, so see if it?s still around. It?s an unmistakable contraption, since the
two attachments are of just the right size to be attached to the still and
237
condenser lines. If you want, you can pressurize the condenser line with
either nitrogen or helium gas to help force the TCE through. The TCE
should start to drip fairly soon, but the ?drip rate? is very slow, again due to
the high impedance of the condenser line. The best thing to do is probably
keep the condenser line pressurized with gas for about a week to be sure that
all of the TCE has evaporated.
After the week has elapsed, the block should be cleared. You can now
proceed to reassemble the refrigerator, and carry on with your research. My
experience has been that by doing a ?throughput? test with helium gas at
liquid nitrogen temperatures, you?ll know whether or not your e?orts have
been successful. A throughput test is nothing more than a mechanical
continuity test to see if gas flows through the unit, and by pressurizing the
condenser side of the unit to roughly 0.5 atmospheres with helium gas, then
opening a valve to let it flow through the unit, do you see the pressure rise
on the still side? If yes, the block is gone, if not...sorry.
I apologize if this explanation seems a bit terse and cryptic, but it really is
this simple to fix a blocked dilution refrigerator. Rather than relying on the
few paragraphs I have written here, I would also highly recommend calling
Oxford instruments and speaking to one of their technicians about the
problem. The people who talked me through this process for the first time
were Mark Jackson and Nick Dent, both of whom were very helpful,
especially ?Jacko?. Again, if you find yourself reading this because the unit
238
has become blocked, all I can say is good luck. I hope this appendix helps.
239
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[4] Kane, B.E., Nature,vol.393, p. 133 (1998).
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