ABSTRACT
Title of dissertation: INVESTIGATION OF IRON-BASED AND
TOPOLOGICAL SUPERCONDUCTORS
VIA POINT-CONTACT SPECTROSCOPY
Steven Ziemak, Doctor of Philosophy, 2015
Dissertation directed by: Professor Johnpierre Paglione
Department of Physics
I report results of point-contact spectroscopy (PCS) measurements preformed
on a variety of superconductors which are predicted to exhibit unconventional Cooper
pairing mechanisms. Point-contact spectra of the iron pnictide BaFe1−xPtxAs2 are
consistent with a two-gap isotropic s-wave model. This conclusion is supported by
previously published results from thermal conductivity, angle-resolved photoemission
spectroscopy (ARPES), and Raman spectroscopy, which confirm a lack of nodes in
the order parameter and the presence of a gap of magnitude 3 meV. Conductivity
spectra were also measured for the half-Heusler materials YPtBi and LuPdBi using
the soft point contact method. I argue that the repeated observation of a single peak
in dI/dV at zero bias is not consistent with a conventional s-wave model. Based on
attempts to fit my data to the Blonder-Tinkham-Klapwijk theory and comparison
to previous experimental and theoretical work, I conclude that my results are most
consistent with a model of triplet Cooper pairs and an order parameter with an
odd-parity component.
INVESTIGATION OF IRON-BASED AND TOPOLOGICAL
SUPERCONDUCTORS VIA POINT-CONTACT SPECTROSCOPY
by
Steven Ziemak
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
2015
Advisory Committee:
Professor Johnpierre Paglione, Chair/Advisor
Professor Richard Greene
Professor Ichiro Takeuchi
Professor Frederick Wellstood
Professor Steven Anlage
c©Copyright by
Steven Ziemak
2015
Foreword
The results of my work have contributed to the following published works,
with the publications that have portions represented in this thesis indicated with
an asterisk.
*Steven Ziemak, K. Kirshenbaum, S. R. Saha, R. Hu, J.-Ph. Reid, R. Gor-
don, L. Taillefer, D. Evtushinsky, S. Thirupathaiah, B. Bu¨chner, S. V. Borisenko, A.
Ignatov, D. Kolchmeyer, G. Blumberg, and J. Paglione, Isotropic multi-gap super-
conductivity in BaFe1.9Pt0.1As2 from thermal transport and spectroscopic
measurements. Superconductor Science and Technology 28, 014004 (2014).
Xiaohang Zhang, N. P. Butch, P. Syers, S. Ziemak, Richard L. Greene, and
Johnpierre Paglione, Hybridization, Inter-Ion Correlation, and Surface States
in the Kondo Insulator SmB6. Physical Review X 3, 011011 (2013).
Kevin Kirshenbaum, S. R. Saha, S. Ziemak, T. Drye, and J. Paglione, Univer-
sal pair-breaking in transition-metal-substituted iron-pnictide supercon-
ductors. Physical Review B 86, 140505 (2012).
S. R. Saha, N. P. Butch, T. Drye, J. Magill, S. Ziemak, K. Kirshenbaum, P. Y.
Zavalij, J. W. Lynn, and J. Paglione, Structural collapse and superconductivity
in rare-earth-doped CaFe2As2. Physical Review B 85, 024525 (2012).
The results presented in Chapter 5 are currently in the pre-publication stage
and will likely be submitted for publication soon pending other experimental results.
ii
Dedication
To JP for showing me the way, and to Cait for her endless support, patience,
and encouragement.
iii
Acknowledgments
This thesis is the result of a massive collaboration that has taken place over the
course of the last five-plus years. It was completed with the assistance of dozens of
colleagues not just at the University of Maryland, but at universities and institutions
around the world. I would like to take a moment to thank everyone who helped make
this work a reality.
First, I’d like to thank my advisor Johnpierre Paglione, who has guided me
through the whole process. As a fellow graduate of an Engineering Physics program,
JP sought me out and invited me to work with his group before I’d even finished my
undergraduate degree. I am grateful to him for giving me the opportunity to work in
such an interesting field and for generally being a great advisor to work for.
Next, I’d like to acknowledge all of the current and former postdoctoral re-
searchers and research scientists who have helped me throughout the years. I would
especially like to thank Xiaohang Zhang for teaching me almost everything I know
about the point-contact spectroscopy technique. From preparing samples to per-
forming measurements and analyzing data, he helped me out every step of the way.
Xiaohang was also responsible, along with Dr. Richard Greene and the rest of his
research group, for building the needle-anvil point-contact probes and Labview pro-
grams that I have used in my research. Their hard work truly laid the foundation
for my work, and these contributions are greatly appreciated. I’d also like to thank
Shanta Saha, Kui Jin, Rongwei Hu, Xiangfeng Wang, and Yeping Jiang for their
assistance with my experiments.
iv
I would also like to acknowledge the contributions of Professors Richard Greene
and Ichiro Takeuchi for their valuable discussions on my results, as well as the
CNAM technical staff, Brian Straughn and Doug Benson, for keeping our labs running
smoothly.
I have also had the pleasure of working with some very talented graduate stu-
dents who have helped me along the way: Kevin Kirshenbaum, Tyler Drye, Paul
Syers, Richard Suchoski, and more recently, Renxiong Wang and Connor Roncaioli.
I’d like to thank all of them for their support and assistance in the lab, to congratu-
late those of them who have already graduated, and to wish the best of luck to those
who haven’t yet.
In addition to my colleagues in College Park, several other collaborators helped
make this work possible. Many thanks to the research groups of Louis Taillefer,
Sergey Borisenko, and Girsh Blumberg for their contributions to my first and only
first author paper. I’d also like to thank Dr. Laura Greene and her group at the
University of Illinois for hosting me for another collaborative research project. It
was a great opportunity to work with some of the world’s foremost experts on the
technique my thesis is based on.
Finally, I’d like to thank the people closest to me who have kept me sane
throughout the past five and a half years. Thanks to my parents for always being
there to commiserate on the ups and downs of the Ph.D. process and to my brother
Chris for supporting me as he made his own way through college. Last, but certainly
not least, I’d like to thank my wife Caitlin. I will be forever grateful to her for her
unwavering support and encouragement while I finished my education. Perhaps more
v
than anyone else, all of this would not have been possible without her.
vi
Table of Contents
List of Figures ix
1 Iron-Based Superconductors 1
1.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Gap Structure and Pairing Symmetry . . . . . . . . . . . . . . . . . . 3
1.3 122 Iron Pnictides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.1 Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.2 Chemical Substitution and Superconductivity . . . . . . . . . 9
1.3.3 Magnetic Properties . . . . . . . . . . . . . . . . . . . . . . . 16
1.3.4 Fermi Surfaces and Pairing Mechanisms . . . . . . . . . . . . 18
1.4 Experimental Work on 122 Gap Structure . . . . . . . . . . . . . . . 22
2 Topological Superconductors 29
2.1 Topological Insulators . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 Noncentrosymmetric Superconductors . . . . . . . . . . . . . . . . . . 32
2.3 Half-Heusler Compounds . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4 Experimental Work on Topological and Noncentrosymmetric Super-
conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3 Experimental Methods 48
3.1 Sample Preparation and Initial Characterization . . . . . . . . . . . . 48
3.1.1 Crystal Growth . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.1.2 Resistivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.1.3 Magnetic Properties . . . . . . . . . . . . . . . . . . . . . . . 56
3.2 Point-Contact Spectroscopy: Theory . . . . . . . . . . . . . . . . . . 57
3.3 Point-Contact Spectroscopy: Experimental Setup . . . . . . . . . . . 68
3.3.1 Needle-Anvil Method . . . . . . . . . . . . . . . . . . . . . . . 72
3.3.2 Soft Point Contact Method . . . . . . . . . . . . . . . . . . . 75
3.4 Point-Contact Spectroscopy: Data Collection and Analysis . . . . . . 77
vii
4 Multi-experimental Study of BaFe1.9Pt0.1As2 Gap Structure 78
4.1 Initial Characterization of Samples . . . . . . . . . . . . . . . . . . . 78
4.2 Point-Contact Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 82
4.2.1 Measured dI/dV Spectra from Multiple Samples . . . . . . . . 82
4.2.2 Fitting to BTK Model . . . . . . . . . . . . . . . . . . . . . . 83
4.2.3 Discussion of Fit Parameters . . . . . . . . . . . . . . . . . . . 87
4.3 Other Thermodynamic and Spectroscopic Measurements . . . . . . . 92
4.3.1 Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . 92
4.3.2 Angle-Resolved Photoemission Spectroscopy . . . . . . . . . . 97
4.3.3 Raman Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 99
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5 Point-Contact Spectroscopy in Half Heusler Compounds 109
5.1 Initial Characterization of Superconducting Materials . . . . . . . . . 109
5.2 Soft Point-Contact Measurements on YPtBi . . . . . . . . . . . . . . 111
5.2.1 Preparation of Junctions and the 3He Probe . . . . . . . . . . 111
5.2.2 Measured dI/dV Curves . . . . . . . . . . . . . . . . . . . . . 112
5.2.3 Comparison to Previous Studies . . . . . . . . . . . . . . . . . 114
5.2.4 Multiple Samples and >100% Zero Bias Enhancement . . . . . 118
5.3 Attempts at LuPdBi Tunnel Junctions . . . . . . . . . . . . . . . . . 120
5.3.1 Preparation of Junctions . . . . . . . . . . . . . . . . . . . . . 122
5.3.2 Behavior of dI/dV with Increasing Temperature and Field . . 123
5.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6 Conclusions 131
Bibliography 135
viii
List of Figures
1.1 Comparison of order parameter pairing symmetries . . . . . . . . . . 5
1.2 Proposed pairing symmetries for 122 iron pnictides . . . . . . . . . . 6
1.3 Unit cells of iron pnictide compounds . . . . . . . . . . . . . . . . . . 8
1.4 Resistivity vs. temperature curves and phase diagram for
SrFe2−xPtxAs2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5 Comparison of superconducting domes for transition metal-substituted
SrFe2As2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.6 Phase diagrams of BaFe2As2 with chemical substitution or applied hy-
drostatic pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.7 Magnetic stripe ordering in pnictide superconductors . . . . . . . . . 16
1.8 Magnetic susceptibility data and phase diagram for SrFe2−xNixAs2 . . 18
1.9 Calculated Fermi surface of SrFe2−xCoxAs2 from density functional
theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.10 Nesting vectors in the 122 iron pnictide Fermi surface . . . . . . . . . 21
1.11 Point-contact spectroscopy data for K- and Co- substituted BaFe2As2 24
1.12 Thermal conductivity vs. temperature curves for BaFe2−xCoxAs2 . . . 26
1.13 Energy distribution curves from ARPES measurement of
BaFe2−xCoxAs2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.1 Energy band diagrams illustrating band inversion . . . . . . . . . . . 30
2.2 Spin-momentum locking in a two-dimensional topological insulator . . 31
2.3 Singlet and triplet Cooper pairs and associated gap symmetries . . . 34
2.4 Unit cells of half-Heusler and Heusler crystal structures . . . . . . . . 36
2.5 Resistivity and Hall resistance vs. temperature in YPtBi . . . . . . . 39
2.6 Calculated band inversion strength in a variety of half-Heusler com-
pounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.7 Predicted dI/dV spectra for a superconductor with triplet Cooper pairing 43
3.1 Single crystals of BaFe1.9Pt0.1As2 and LuPdBi . . . . . . . . . . . . . 51
3.2 Differential interference contrast image of LuPdBi crystal showing Bi
inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3 Schematic of four-point probe resistivity measurement . . . . . . . . . 54
3.4 Diagram of a superconducting quantum interface device (SQUID) . . 57
ix
3.5 Density of states at a normal metal-superconductor junction and dia-
gram of Andreev reflection . . . . . . . . . . . . . . . . . . . . . . . . 59
3.6 dI/dV vs. bias voltage curves in the Andreev reflection and tunneling
regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.7 dI/dV spectra showing effects of inelastic scattering . . . . . . . . . . 64
3.8 dI/dV spectra over a range of temperatures . . . . . . . . . . . . . . 66
3.9 dI/dV spectra for anisotropic s-wave and d-wave superconductors . . 67
3.10 Sharvin regimes based on point-contact junction size . . . . . . . . . 69
3.11 Diagrams of needle-anvil and soft point contact junctions . . . . . . . 71
3.12 Image of low temperature needle-anvil probe . . . . . . . . . . . . . . 73
3.13 Circuit diagram for point-contact spectroscopy measurement . . . . . 74
3.14 Soft point contact junction on a BaFe1.9Pt0.1As2 sample . . . . . . . . 76
4.1 Resistivity and magnetic susceptibility vs. temperature for
BaFe1.9Pt0.1As2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.2 Resistivity vs. temperature of BaFe1.9Pt0.1As2 in magnetic field . . . . 81
4.3 Raw dI/dV curves for multiple BaFe1.9Pt0.1As2 samples . . . . . . . . 84
4.4 Normalized BaFe1.9Pt0.1As2 dI/dV spectra over a range of temperatures 85
4.5 BTK fit to 4.2 K dI/dV spectrum of BaFe1.9Pt0.1As2 . . . . . . . . . 88
4.6 dI/dV Spectra and ∆ vs. T data for Ba1−xKxFe2As2 . . . . . . . . . 91
4.7 Thermal conductivity vs. temperature data for BaFe1.9Pt0.1As2 . . . . 93
4.8 Residual thermal conductivity of BaFe1.9Pt0.1As2 compared to other
superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.9 Angle-resolved photoemission spectroscopy Fermi surface and energy
density curves for BaFe1.9Pt0.1As2 . . . . . . . . . . . . . . . . . . . . 98
4.10 Raw Raman spectroscopy data for BaFe1.9Pt0.1As2 . . . . . . . . . . . 101
4.11 Normalized BaFe1.9Pt0.1As2 Raman spectra in multiple symmetry chan-
nels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.1 Resistivity vs. temperature in YPtBi . . . . . . . . . . . . . . . . . . 110
5.2 Resistivity ρ vs. magnetic field H in YPtBi at 0.4 K . . . . . . . . . . 111
5.3 Image of YPtBi crystal showing soft point contact setup . . . . . . . 113
5.4 dI/dV vs. bias voltage curves for YPtBi over a range of temperatures 115
5.5 dI/dV vs. bias voltage curves for YPtBi over a range of applied mag-
netic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.6 Comparison of YPtBi dV/dI spectrum to p-wave Sr2RuO4 . . . . . . 119
5.7 dI/dV spectra from multiple YPtBi samples . . . . . . . . . . . . . . 121
5.8 Schematic of LuPdBi tunnel junction deposition process . . . . . . . 124
5.9 LuPdBi dI/dV vs. temperature and dI/dV vs. bias voltage over a range
of temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.10 LuPdBi dI/dV vs. magnetic field and dI/dV vs. bias voltage over a
range of field values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.11 dI/dV spectra for three LuPdBi samples . . . . . . . . . . . . . . . . 129
x
Chapter 1: Iron-Based Superconductors
1.1 History
Superconductivity was first observed in mercury in 1911 by Dutch physicist
Heike Kamerlingh Onnes [1]. Not coincidentally, Onnes was also the first to produce
liquid helium, which was necessary at the time to achieve the extremely low temper-
atures required to enter the superconducting state [1]. In the following four decades
higher transition temperatures were found in various metallic elements and alloys, but
the physical mechanism that was causing the phenomenon was not well understood.
The situation changed in 1957 when John Bardeen, Leon Cooper, and John
Schrieffer published what has come to be known as the BCS theory [2–4]. The princi-
ple insight of the theory is that superconductivity results when electrons in a material
form quasiparticles known as Cooper pairs. Whereas electrons are fermions, which
are subject to the Pauli exclusion principle, Cooper pairs are bosons, meaning they
can collect in the same low energy ground state that is the superconducting state.
The theory posits that electrons that would normally repel each other due to the
Coulomb interaction can form attractive pairs with the assistance of phonons, i.e.,
quantized lattice vibrations. The distortions that phonons generate in a crystal lattice
result in centers of positive charge, which can attract two electrons and cause them
1
to pair. BCS theory has proven quite successful in describing and predicting physi-
cal, electronic, and magnetic properties observed in many superconducting materials.
However, in other materials, especially high-temperature superconductors, the theory
falls short.
For the first two decades of the BCS theory’s existence superconductivity was
believed by many to have a finite limit with transition temperatures above about 30
K forbidden [5]. This view was proven to be wrong in 1986 when a copper oxide-based
ceramic material was discovered with a Tc of 35 K [6]. Further study of these new
cuprate superconductors led to even higher transition temperatures, and the current
record for a material at atmospheric pressure stands at 133 K for HgBa2Ca2Cu3Ox [7],
though higher transition temperatures up to 150 K have been measured under hy-
drostatic pressure [8]. Recently other classes of unconventional superconductors (so
called because they do not seem to obey the conventional BCS theory) have been
discovered [9]. One of the more notable groups of compounds is known as iron pnic-
tides, with pnictides referring to group 15 on the periodic table primarily phosphorus,
arsenic, and antimony. These materials are notable because they are able to supercon-
duct despite containing elements with large magnetic moments. It was widely believed
that high-temperature superconductors would not contain magnetic elements such as
iron because magnetic fields are known to suppress superconductivity. These mate-
rials are one of the main topics of my thesis, and I discuss their electronic structure
and magnetic properties in greater detail in later sections.
2
1.2 Gap Structure and Pairing Symmetry
An important property associated with a superconductor’s pairing mechanism
is the superconducting order parameter, denoted as ∆k because it is a function of
the electronic wave vector k. ∆k is a complex quantity with units of energy whose
magnitude represents the strength of the energy gap as a function of direction in
momentum space, the energy gap being the difference in energy between the ground
state, i.e. the condensate of Cooper pairs, and the excited state of quasiparticles. The
order parameter describes how strongly two electrons are bound in a Cooper pair due
to interaction with a pairing boson field. For BCS superconductors that boson is
a phonon. For unconventional superconductors like iron pnictides it has yet to be
definitively proven what the pairing boson is. More generally, the order parameter
can be said to represent the macroscopic wave function of the Cooper pairs [10].
While the magnitude of the order parameter is important, its phase is critical
in determining the fundamental nature of a materials electronic structure. The sign
of ∆ on each of the conducting bands of a structure with multiple bands and the
potential presence of a sign change in ∆ within a single band lead to different types
of symmetries in the order parameter, also known as pairing symmetries.
Determining the symmetry of the order parameter is potentially useful for un-
derstanding the mechanism of superconductivity in these materials, which is one of
the goals of my research. The physical process by which electrons in a BCS super-
conductor interact with phonons to form Cooper pairs is well understood [2–4]. Since
an analogous description does not yet exist for the iron pnictides, more information
3
about the gap structure could prove useful in determining the pairing mechanism [11].
Several theories of pairing mechanisms have been proposed, and these theories predict
different pairing symmetries and conditions on the order parameter. Thus, investigat-
ing this aspect of electronic structure provides a way of verifying theories of Cooper
pair formation and represents another step towards understanding the fundamental
nature of the superconducting state in these materials.
As I discuss in Chapter 2, the iron pnictides are layered materials with most
electron transport occurring in two dimensions. Thus, I only consider gap structures
for two-dimensional materials. For the simplest case of a single-band superconductor
in two dimensions with a perfectly circular Fermi surface, there are two primary
options for pairing symmetries. The simplest is s-wave symmetry, pictured in Fig.
1.1a. This isotropic order parameter represents the highest possible symmetry, with
the pairing symmetry matching that of the underlying Fermi surface. It has been
proven that s-wave symmetry is required for BCS superconductors, which typically
have a single band in their electronic structure [5]. Another single-band possibility is
known as d-wave symmetry (Figure 1.1b). In this case, the phase of ∆ has a sinusoidal
dependence on angle in the kx-ky plane. Due to this anisotropy, d-wave pairing has
a lower rotational symmetry than s-wave. Because of this sinusoidally varying phase
there are directions in momentum space where ∆ is positive, directions where it is
negative, and specific directions where ∆ = 0, that is, the gap disappears. These
points are known as nodes in the order parameter.
When additional bands are considered, which is necessary given the multi-band
structure that has been observed for pnictides, more complex pairing symmetries
4
Figure 1.1: Plots of ∆ vs. kx and ky for various two-dimensional pairing
symmetries. a. Single band s-wave. b. Single-band d-wave. c. Two-band
s++-wave. d. Two-band s±-wave. (Adapted from Ref. [12].)
become possible. For two bands, each with s-wave pairing, there are two possibilities
depending on the relative signs of the s-waves on each band. If ∆ has the same sign
on each band the result is known as s++ (Fig. 1.1c). If ∆ has a sign change between
bands the pairing symmetry is called an s+− (Fig. 1.1d). Note that these two-band
models have four-fold rotational symmetry. This is a consequence of the pnictides
tetragonal crystal structure, which will be detailed in the next section. A general rule
is that a materials Fermi surface tends to reflect the symmetry of its crystal structure.
The order parameter may have the same symmetry as the Fermi surface (as seen in
an s-wave) or a lower symmetry (as with d-waves and other possible symmetries).
Currently the s+−wave is considered the leading candidate for the pairing sym-
metry of the 122 iron pnictides [11]. Unfortunately, based on the evidence available
this has not been proven definitively. One possible complication, the potential pres-
5
Figure 1.2: Most likely pairing symmetries for 122 iron pnictides: isotropic
s±-wave (left), anisotropic s±-wave, and d-wave (right). (From Ref. [11].)
ence of accidental nodes, is illustrated in Fig. 1.2.
Figure 1.2 shows three possible pairing symmetries superimposed on a two-
dimensional cross-section of the Fermi surface. Though they appear different, the
isotropic and anisotropic s+−-wave (i.e., s+− without and with nodes) are fundamen-
tally the same symmetry because they both have four-fold rotational symmetry in
the kx-ky plane. Similar nodes are also seen in the d-wave case, which has a lower
two-fold symmetry. If the true pairing symmetry were an anisotropic s+−-wave, it
would be difficult to discern from a d-wave based on their similarities. Experimen-
tal techniques exist that detect the presence of nodes or a sign change in the order
parameter; however, it would still be challenging to distinguish anisotropic s-wave
symmetry from d-wave. The approach I used to differentiate anisotropic s+−- from
d-wave is point-contact spectroscopy, which is discussed in Chapter 4.
6
1.3 122 Iron Pnictides
1.3.1 Crystal Structure
The crystal structure of the iron pnictide superconductors has been well char-
acterized by x-ray diffraction and neutron diffraction. Such analysis has revealed
several common characteristics of compounds in the pnictide family. Each compound
features a tetragonal unit cell with a distinct layered structure, which explains the
materials quasi two-dimensional electronic properties [11]. Common to all of the iron
pnictides are Fe-As (or Fe-P, etc.) layers consisting of a square lattice of Fe atoms with
As atoms forming separate square lattices above and below the Fe plane and forming
bonds such that each Fe atom is tetrahedrally coordinated. For all iron pnictides
except the binary compound FeAs, these Fe-As layers are separated by spacing layers
containing atoms such as oxygen, fluorine, alkaline earth metals, and rare earth ele-
ments. Unit cells illustrating five of the best-characterized iron pnictide compounds
are shown in Fig. 1.3.
The first of the group to be discovered [9] was the LaFeAsO structure (fourth
from the left in Figure 1.3a). In these materials the spacer layer between Fe-As
layers is similarly structured, with a square lattice of oxygen and/or fluorine atoms
tetrahedrally coordinated to square lattices of a metallic element (usually rare earth
elements such as La or alkaline earth elements like Sr) above and below the plane.
This structure is noted for exhibiting the highest transition temperature observed for
iron pnictides, 55 K [13].
7
Figure 1.3: a. Unit cells of five of the compounds in the iron pnictide
family. b. Closer view of the common Fe-As layers showing the plane of
Fe atoms (red) with As atoms (yellow) above and below. (From Ref. [11].)
Of particular interest for my thesis is the SrFe2As2 structure, the third from
the left in Figure 1.3a. This compound and its derivatives are known collectively
as 122s (based on the subscripts in their chemical formulas). The 122 unit cell is a
body-centered tetragonal lattice with space group I4/mmm. Coordination numbers
for each atom present are 8 for Sr (bonds to 8 As atoms), 4 for Fe (bonds to 4 As
atoms), and 8 for As (bonds to 4 Fe atoms and 4 Sr atoms). Based on the elements
present and simple electron counting the valences of each element should be Sr2+,
Fe2+, and As3−.
One set of quantities that is frequently used to characterize 122 compounds is
their lattice parameters, i.e., the dimensions of their unit cells. For pure SrFe2As2
these values are approximately a = 3.9 A˚ and c = 12.2 A˚ as determined by x-ray
diffraction [14, 15]. In many experiments different elements are substituted for Sr,
8
Fe, and/or As. For most of these cases the measured unit cell dimensions behave
according to Vegards Law, the principle that lattice parameters should change linearly
with the atomic percentage of a substituted element in a solid solution [14–18].
1.3.2 Chemical Substitution and Superconductivity
One of the most notable features of the 122 iron-based superconductors is that
superconductivity is not observed in the parent compounds (e.g. SrFe2As2) under
normal conditions. Rather, superconductivity must be induced by either chemical
substitution or the application of hydrostatic pressure. As the atomic percentage of
a substituted element or the applied pressure is increased from zero, the electronic
and magnetic properties of the material change dramatically. These effects can be
summarized in a phase diagram, with temperature as a function of either the amount
of the doping element or applied pressure. The diagram reveals the regions where
different phases are observed and the temperatures at which physical and/or electronic
phase transitions occur.
As an example, consider the platinum-substituted system SrFe2−xPtxAs2, where
x is used to denote the amount of substituted platinum (x = 0 means pure SrFe2As2,
x = 0.1 means 5% of the Fe has been replaced with Pt, etc.) [15]. Single crystals
were grown for compounds with Pt concentrations ranging from x = 0 to x = 0.36.
Wavelength dispersive x-ray spectroscopy (WDS) was used to measure the precise
amounts of Pt in each sample, as well as to verify the expected stoichiometry.
Electrical resistance was measured as a function of temperature for crystals of
9
each Pt concentration. This was accomplished by attaching gold wires to the crystals
with silver paint in the standard four-point probe layout and applying AC current
over a range of temperatures in a Quantum Design Physical Property Measurement
System (PPMS). The resistances measured were then converted to resistivities using
the known physical dimensions of each crystal. The results of these measurements
can be seen in Fig. 1.4 (top left) with the resistivity curves for each concentration
offset for clarity.
The most striking feature of these curves is the abrupt drop in resistivity at
low temperatures for x = 0.09 to x = 0.27. This is the characteristic zero resistance
indicating superconductivity. This drop is not observed in the lower concentrations
(x = 0 and 0.035) as well as the highest concentration x = 0.36. The resistivity
curves also indicate the position of two other transitions. The first is a structural
transition from a tetragonal lattice to an orthorhombic lattice, indicated by a kink in
ρ(T ) for the x = 0 and 0.035 or a minimum in ρ(T ) for x = 0.09 to 0.13. It has been
shown that the 122 materials exhibit this relaxing of bonds in one of the tetragonal
a-directions in order to reduce the magnetic frustration associated with magnetic
ordering, and that this structural transition manifests itself as a minimum in ρ(T ).
The final noteworthy feature of the resistivity curves is the points of inflection seen
in the curves up to x = 0.13. Such an inflection point has been shown occur at the
Ne´el temperature, the point where the iron sublattice shifts from a paramagnetic
state to an ordered antiferromagnetic state. The minima and points of inflection in
the resistivity curves were determined precisely by plotting dρ(T )/dT , shown in Fig.
1.4 (bottom left), and locating the points where the derivative was zero or reached a
10
maximum. Note that the width of these dρ(T )/dT peaks was estimated as the error
in the Ne´el temperatures.
The temperatures at which these transitions occurred were used to create the
phase diagram seen on the right side of Fig. 1.4. Phase diagrams of this form have
been observed for substitution of several different transition metal elements at the
Fe site (e.g., BaFe2−xPtxAs2) [19], as well as for substitution of alkali metals or rare-
earth elements at the alkaline earth metal site (e.g., Ba1−xKxFe2As2) [20] and for
substitution at the pnictide site (e.g., BaFe2As2−xPx) [22]. All of these diagrams
share two common phase regions. One is the superconducting dome, the finite range
of concentrations of the substituted element where Tc rises, reaches a peak value, and
then falls off. The other is a region of magnetic ordering (either antiferromagnetic
or spin density wave, two similar states which will be discussed later), which also
coincides with a structural transition from a tetragonal lattice to an orthorhombic
lattice. These two transitions tend to occur at or near the same temperature.
The generally accepted explanation for this behavior is that the magnetic order-
ing in the Fe sublattice causes magnetic frustration [11]. That is, it is energetically
favorable for each Fe up spin (up meaning a spin pointing in the [110] direction) to be
surrounded by neighboring down spins, but this is forbidden by the geometry of the
lattice. Thus, the lattice expands slightly in one direction to relieve this frustration
by separating neighboring spins that point in the same direction.
Several attempts have been made to compare the effects of different substituted
elements in order to draw more general conclusions about how these impurities change
the electronic structure. Figure 1.5 shows the superconducting dome portions of the
11
Figure 1.4: Restivity curves for several samples of SrFe2−xPtxAs2 for a
range of x values (top left), the smoothed derivative curves -dρ/dT for
selected values of x to determine minima and points of inflection in ρ(T )
curves (bottom left) and the phase diagram that was generated from these
curves (right). In the phase diagram key, Tc(ρ) and Tc(χ) are the transi-
tion temperatures as determined from resistivity data and magnetic sus-
ceptibility data, respectively; T0 is the temperature of the tetragonal to
orthorhombic structural transition; and TN is the Ne´el temperature, where
the magnetic transition was observed. (From Ref. [15].)
12
phase diagrams for five crystal systems of the form SrFe2−xMxAs2, where M is a
transition metal in column 9 or 10 of the periodic table [15, 16, 23, 24]. Based on the
SC domes for Co, Ni, Rh, and Pd one might conclude that the maximum Tc and the
concentration at which it will occur can be predicted using a simple electron counting
picture. Co and Rh both have one more 3d electron than Fe, and their domes peak
at approximately x = 0.3 and Tc = 20 K. Ni and Pd have two more 3d electrons than
Fe, and their domes peak around x = 0.15 and Tc = 10 K. By adding twice as many
valence electrons per atom, only half the amount of the doping element is needed
to achieve the same effect, and all substituted elements in the same column of the
periodic table should behave in the same way.
This trend was complicated by the addition of SrFe2−xPtxAs2, the most recent
system out of the five to be characterized, to the plot (also shown in Figure 1.5).
The Pt dome is noticeably wider than that of Ni and Pd, and it has a maximum at
around 18 K, much higher than Ni and Pd, though at around the same Pt concentra-
tion (about x = 0.15) [15]. This anomaly illustrates one of the main difficulties with
understanding the effects of chemical doping and other factors: accurately predict-
ing transition temperatures for new superconductors is nearly impossible, especially
considering that the exact mechanism for Cooper pair formation is still unknown for
this class of materials. It is widely assumed that scattering of conduction electrons
(within bands and between bands in the Fermi surface) plays a major role [11], but
a comprehensive model for predicting Tc remains elusive.
The two methods of inducing superconductivity, chemical substitution and hy-
drostatic pressure, though quite different experimentally, both serve to apply pressure
13
Figure 1.5: Phase diagram showing detail of the superconducting domes
for single crystals of SrFe2As2 with different transition metal substitutions
at the Fe site.
to the crystal lattice. With hydrostatic pressure, physical pressure compresses the
lattice from all sides and pushes atoms together reducing bond lengths and changing
the size of the unit cell. In the case of chemical substitution, systematically replacing
one type of atom in the lattice for another with a different atomic radius also applies
pressure and has a similar effect on bond lengths and unit cell dimensions. Experi-
mental evidence for a wide range of compounds has shown that chemical substitution
and hydrostatic pressure have very similar effects on the electronic properties of 122
materials [11]. Figure 1.6 compares several phase diagrams for BaFe2As2 with the
x-axis representing either chemical substitution or hydrostatic pressure.
The similarities between the phase diagrams in Figure 1.6 suggest that the same
phenomenon is inducing magnetic order and superconductivity. Clearly, increasing
14
Figure 1.6: Comparison of phase diagrams for BaFe2As2 with different
chemical substitutions (left) and different types of hydrostatic pressure
cells (right). The chemical substitution and pressure axes have been nor-
malized so that the antiferromagnetic transitions overlap. (From Ref.
[11].)
the concentration of a substituted atom causes an increase in chemical pressure in the
same way that a pressure cell applies physical pressure. Furthermore, this suggests
that charge doping may not play a major role since hole doping (Ba1−xKxFe2As2),
electron doping (BaFe2−xCoxAs2), and isovalent substitution (BaFe2As2−xPx) seem to
have similar effects. If this is the case, the primary effect of substituted elements may
be to simply change the lattice parameters and bonding structure, which in turn could
change the band structure and Fermi surface in such a way that superconductivity
is energetically favorable. The true challenge, then, is to determine exactly how
chemical substitution affects the electronic structure and how it can be optimized to
create better superconductors with higher transition temperatures.
15
Figure 1.7: Stripe ordering in the Fe-As layer of a pnictide superconductor
with one stripe highlighted. Fe atoms are in red; As atoms above and
below the Fe plane are in yellow. Dotted line shows redrawing of primitive
unit cell when AFM ordering is taken into account. (From Ref. [11].)
1.3.3 Magnetic Properties
As I mentioned previously, some form of magnetic ordering occurs in all Fe-based
122 materials at low temperatures. This is usually in the form of antiferromagnetic
(AFM) or spin density wave (SDW) ordering, which are fundamentally very similar.
Both entail a spatial modulation of the direction of spins of Fe atoms within the Fe-As
planes. The form of AFM ordering seen in many 122s, known as stripe ordering, is
shown in Fig. 1.7.
As the name suggests, lines of adjacent atoms with the same spin are present
(ordering along the [11¯0] direction), with atoms on neighboring stripes (separated by
a translation along the [110] direction) having spins in the opposite direction. That
16
is, the lattice exhibits ferromagnetic ordering in one direction and antiferromagnetic
ordering in another. This lattice of atomic spins changes the size of the primitive unit
cell to the configuration shown in Figure 1.7 with two iron atoms per layer instead of
one. This change in lattice size results in a folding of the Fermi surface resulting in
the positioning of the electron pockets at (±pi, ±pi) in the Brillouin zone.
SDW ordering in these materials looks very similar, but the spatial period be-
tween neighboring up spins in the [110] direction is greater. If one imagines the angle
of the Fe spins as sinusoidally varying in the [110] direction, the period of this sinu-
soid is exactly two interatomic distances for AFM. For SDW the period is different,
and it is not necessarily an integer number of interatomic distances - this case is
known as an incommensurate SDW. Experimentally, neutron diffraction can be used
to differentiate between AFM, SDW, and other forms of magnetic ordering [25–29].
One of the most fundamental properties of a material in the superconduct-
ing state is perfect diamagnetism. As a superconductor is cooled below its critical
temperature, it expels all external magnetic fields such that its volume-normalized
magnetic susceptibility drops to a value of 4piχ = -1 [5]. This property is quite useful
in both verifying Tc values and for determining the superconducting volume fraction
of a sample. For a sample that is composed of only a single fully superconducting
phase, one would expect the volume-normalized susceptibility to approach -1 at low
temperatures and low applied fields. If, however, the entire sample did not enter the
SC state or it contains non-superconducting impurity phases, the low temperature
susceptibility will reflect this. As a simple example, for a superconducting sample
with 50% of its volume composed of a paramagnetic impurity phase 4piχ will ap-
17
Figure 1.8: Magnetic susceptibility vs. temperature for SrFe2−xNixAs2
samples with the labelled values of Ni concentration x (left) and the phase
diagram for the system (right). (From Ref. [16].)
proach -0.5 at low temperatures, a reflection of the average susceptibilities of all of
the phases in the sample.
Examples of such susceptibility vs. temperature curves are shown in Fig. 1.8
for samples in the SrFe2−xNixAs2 system with several different concentrations of Ni
[16]. The phase diagram for the system is also shown for comparison. Note that
concentrations at the middle of the superconducting dome (x = 0.15, 0.16, 0.18)
show full diamagnetic screening at low temperatures, while concentrations closer to
the edge of the dome have SC volume fractions of less than 100%. As expected, Ni
concentrations outside the dome show no diamagnetism.
1.3.4 Fermi Surfaces and Pairing Mechanisms
The highly layered crystal structure of the 122s iron pnictides suggests that con-
duction should be more favorable in the ab plane of the crystal lattice. Thus, these
materials ought to have a quasi two-dimensional Fermi surface. This has been con-
18
Figure 1.9: Fermi surface of SrFe2−xCoxAs2 mapped out using density
functional theory calculations. Different colors correspond to separate
sheets in the FS as a result of multiple bands crossing the Fermi energy.
(From Ref. [30].)
firmed by angle-resolved photoemission spectroscopy (ARPES) measurements that
reveal the Fermi surface to have roughly cylindrical surfaces [31–34]. Note that for
a three-dimensional free electron gas the Fermi surface is spherical, reflecting the
isotropy of the system. Likewise, for a two-dimensional conducting surface the Fermi
surface is circular in the plane of conduction. The Fermi surfaces measured for 122
materials are not completely cylindrical, as shown in the model in Figure 1.9. This
suggests some interlayer conduction, but for my purposes the assumption of a 2D
Fermi surface is a good one. Another notable feature of the Fermi surface is the
presence of multiple bands. The ARPES data confirm the presence of a hole-like
pocket centered at the Γ point in the Brillouin zone and an electron-like pocket cen-
tered at the M point, and other experimental results are consistent with a two-band
material [31–34].
Based on analysis of Fermi surfaces and the measured effects of elemental sub-
19
stitution, one can extract information about the possible mechanisms of Cooper pair
formation in the iron pnictides. Several studies have shown that superconductivity
in these materials does not appear to be purely phonon-mediated. Calculations of
electron-phonon coupling strength have failed to account for the high Tc values that
have been observed [35]. Studies of the isotope effect in some 1111 and 122 pnic-
tides have shown some electron-phonon interaction - replacing elements in a BCS
superconductor with heavier isotopes should decrease Tc by altering the frequency of
lattice vibrations [36] - but the effect are small, suggesting that other factors influence
pairing.
It has been suggested that magnetic fluctuations could be driving Cooper pair-
ing in the pnictides, a theory that is supported by several observations. One is the
presence of nesting vectors in the Fermi surface. In theoretical 122 Fermi surfaces,
such as Figure 1.10, as well as in experimentally observed ones the hole and elec-
tron pockets have matching shapes. Thus, they can be connected by a scattering
momentum vector, known as a nesting vector, as shown in Figure 1.10.
The direction of this nesting vector in momentum space is identical to that of the
ordering vector (in real space) of the spin density wave state. The strong connection
between magnetic ordering and Fermi surface structure suggests there could also be
a link between magnetism and superconductivity. That is, Fermi surface nesting may
lead to multiple ordered states: the magnetically ordered spin density wave and the
superconducting state [11].
Several experiments have supported the connections between Fermi surface
structure, magnetic ordering, and superconductivity. According to density functional
20
MX
Γ
Q
SDW
Figure 1.10: Cross-section of 122 Fermi surface showing the nesting vector
QSDW connecting electron and hole pockets. This scattering vector in
momentum space matches the ordering vector of the spin density wave
state in real space and suggests a connection between magnetic order and
superconducting order.
21
theory calculations, doping pnictides with holes or electrons can change the size of
hole and electron pockets on the Fermi surface and possibly disrupt nesting [37]. Elec-
tronic structure is also significantly influenced by small changes in crystal structure,
including lattice parameters and bond angles [38,39]. These factors all underline the
effects of chemical substitution, which provides charge doping (for most substituted
elements) and changes lattice parameters and bond angles. An additional link be-
tween interband scattering and superconductivity has been established for optimally
doped 122 pnictides suggesting that transition temperature is tied directly to a scat-
tering rate determined using Abrikosov-Gorkov theory. [48] Finally, as I discussed
above, superconductivity emerges in the 122 pnictides upon suppression of magnetic
order. The combination of these factors strengthens the case for a model of Cooper
pairing that is at least partially mediated by magnetic fluctuations.
1.4 Experimental Work on 122 Gap Structure
This section provides a brief summary of some of the experimental work that
has been done since the discovery of the 122 iron pnictides to determine their gap
structure and pairing symmetry. I provide a description of the experimental details
of these techniques is provided in Chapter 3, while Chapter 4 summarizes the results
of our experiments using these four methods.
Point-contact spectroscopy (PCS), also known as quasiparticle scattering spec-
troscopy, is a versatile technique that can reveal information about the electronic
structure of a a superconductor. Measurements of electrical conductance vs. DC bias
22
voltage across a normal metal/superconductor junction provide an indirect probe of
density of states. In principle, by fitting such data to the Blonder-Tinkham-Klapwijk
(BTK) theory [40] one can determine the of gap size(s) as well as pairing symme-
try. [41]
PCS measurements on various doped FeSC systems have resulted in widely
varying conclusions about their pairing symmetries, at times contradicting the results
from other measurement techniques. Point-contact measurements in the Andreev
reflection regime have shown evidence for a two-gap structure in Ba1−xKxFe2As2
(Refs. [42, 43]), while BaFe2−xCoxAs2 spectra have been fit to both one- and two-
gap s-wave models depending on the study and even the fitted features in a single
spectrum [41, 42] (i.e. inclusion or exclusion of zero-bias or higher bias features).
Fig. 1.11 shows an example of point-contact spectra measured on BaFe2As2 crystals
with either K or Co substitution. For each curve the data have been fit to the BTK
theory, which will be discussed in detail in Section 3.2, to determine the gap size(s).
Finally, PCS measurements of BaFe2−xNixAs2 have been used to argue for a two-gap
s-wave structure at low Ni concentrations with nodes or deep minima emergent above
optimal doping. [44]
Thermal conductivity measurements provide a powerful probe of electronic ex-
citations in the zero temperature limit, when phonons have been frozen out. In a
conducting solid at low temperatures, the temperature dependence of thermal con-
ductivity κ can be modeled as κ(T ) = aT + bTα, where the T -linear term aT repre-
sents the electronic contribution to heat conduction, while the Tα term represents the
phonon contribution, where 2 ≤ α ≤ 3. By measuring κ/T in the zero temperature
23
Figure 1.11: Conductance vs. bias voltage spectra for different substi-
tuted 122 iron pnictides. (a) Measured dI/dV vs. Vbias spectra for
Ba1−xKxFe2As2. (b) The same curves from (a) after normalization to
remove background conductance. Red lines represent fit curves, and the
inset shows the evolution of the size of the two observed SC gaps as a
function of temperature up to Tc. (c) Similar dI/dV spectra for multi-
ple BaFe2−xCoxAs2 crystals with different fit curves representing either a
one-gap fit (blue lines) or two-gap fit (red lines). (d) Conductance spec-
tra for a single sample measured over a wide range of temperatures. The
inset shows the two gap sizes as extracted from fit curves as a function of
temperature. (From Ref. [41].)
24
limit the electron contribution can be isolated, given by a ≡ κ0/T . It has been shown
that the presence or absence of this residual thermal conductivity at zero magnetic
field and its evolution in field can reveal the presence or absence of nodes or zeroes in
the SC order parameter and provide additional information about gap structure. [45]
For example, in the d-wave cuprate superconductors κ0/T is nonzero at zero mag-
netic field. [46] The presence of this same residual thermal conductivity has also been
used as evidence for nodal s-wave or d-wave pairing in FeSCs such as BaFe2−xCoxAs2
citeReid064501 and KFe2As2 [48]. Figure 1.12 shows an example of measured κ/T
vs. T 2 data for BaFe2−xCoxAs2 crystals with a range of Co concentrations.
Conversely, the absence of this residual term has been taken as evidence for
a nodeless s-wave gap structure in the 18 K superconductor LiFeAs (Ref. [49]). As
a function of magnetic field, the evolution of κ0/T can provide further information
about energy scale anisotropies or the presence of multiple gaps with different energy
scales, and has been used to differentiate between s-wave superconductors with a
single gap (or gaps of equal magnitude on multiple Fermi surface (FS) pockets) and
those with multiple gaps of significantly different sizes, e.g. differentiating doubly-
gapped NbSe2 from single-gap V3Si [50].
Angle-resolved photoemission spectroscopy (ARPES) is another technique that
can provide information about the magnitude of the gap in different directions. In the
case of 122 FeSCs, previous work has found two nodeless and nearly isotropic gaps of
different sizes on various FS pockets in Ba1−xKxFe2As2 [31]. ARPES measurements
on BaFe2−xCoxAs2 have shown evidence for hole pockets disappearing upon increasing
Co substitution [32], while a separate study focusing on gap structure [51] observed
25
Figure 1.12: Thermal conductivity vs. temperature curves for
BaFe2−xCoxAs2 samples with a wide range of x values and measured at
applied magnetic fields of up to 15 T. The extrapolated value of κ/T at
T = 0 approaches roughly zero for some samples at low field, but at high
field an increasing residual thermal conductivity is observed. The presence
of a nonzero residual value in the absence of applied field is evidence of
nodes in the SC order parameter, and the evolution of the residual term’s
value in increasing magnetic field reveals further information about the
gap structure. (From Ref. [47].)
26
the presence of two isotropic gaps and suggested a connection between FS nesting
and Cooper pairing based on comparison with the hole-doped Ba1−xKxFe2As2 case. A
sample of ARPES data used to extract gap structure information is shown in Figure
1.13 for the case of a BaFe2−xCoxAs2 sample. Finally, to date less extensive studies of
Raman spectroscopy have been completed in the 122 FeSC materials [41, 52], where
one study has shown band anisotropy in BaFe2−xCoxAs2 supporting a nodal s-wave
model [53].
27
Figure 1.13: Symmetrized energy distribution curves (EDCs) obtained
from ARPES showing the density of states at several points on the Fermi
surface on the hole-like (A) and electron-like (B) Fermi surface pockets of
BaFe2−xCoxAs2. The specific locations in momentum space where these
EDCs were measured are indicated in (C), while (D) shows the magnitude
of the two SC gaps as a function of angle in momentum space, demon-
strating that the two gaps are isotropic within error. (From Ref. [51].)
28
Chapter 2: Topological Superconductors
2.1 Topological Insulators
Many material properties can be characterized by symmetries and the phase
transitions that break them. For example, the transition from a liquid to a solid phase
reduces the spatial symmetry of a substance, while the application of a magnetic field
breaks time reversal symmetry of moving electrons. Recently, topological order has
emerged as a new classification of matter that does not break symmetries.
Topological order is manifested in materials known as topological insulators
(TIs), a class of materials that show great promise both as a source of new and in-
teresting physics but also for potential technological applications. The hallmark of
a topological insulator is a phenomenon known as band inversion, in which energy
bands with fundamentally different spatial symmetries reverse order. To compare
this phenomenon to mathematical topology, the bands of a topological insulator cant
be adiabatically transformed into the structure of a conventional insulator or semi-
conductor, much in the same way that a sphere cannot be adiabatically transformed
into a torus. This fundamental rearrangement of bands results in an accidental cross-
ing of energy states at the Fermi level, which turns a nominally insulating material
into a conductor at the surface. Fig. 2.1 contrasts the calculated band structures of
29
Figure 2.1: Band diagrams for two sets of materials illustrating the effects
of band inversion as determined by density functional theory calculations.
CdTe and ScPtSb are topologically trivial semiconductors and thus have
a gap in the density of states. HgTe and ScPtBi (same crystal structures
as CdTe and ScPtSb, respectively) experience an inversion in the order of
the bands with Γ8 symmetry (red) and Γ6 symmetry (blue) resulting in a
crossing at the Fermi energy. (From Ref. [54].)
two predicted topological insulators with their topologically trivial counterparts. As
illustrated in Fig. 2.2, these surface states are also spin-momentum locked, meaning
that electrons moving along the surface have their momentum locked at a 90-degree
angle to their spin.
This spin-momentum locking means that TIs could be useful for spintronic
devices. Another interesting behavior has been predicted when superconductivity
is induced in a topological insulator by the proximity effect. It has been predicted
30
Figure 2.2: Illustration of spin-momentum locking in a topologically insu-
lating surface. Note that electrons with opposite spins move in opposite
directions along the edge of the surface. (From Ref. [55].)
that when a TIs surface state becomes superconducting it can support Majorana
fermions (MFs). The Majorana fermion is an elusive quasiparticle interaction that is
its own antiparticle. It has been proposed that a Majorana bound state, an anyon
that obeys non-Abelian statistics, can be formed when a MF is bound to a defect [56].
Because of the unique entanglement expected to occur between these bound states,
they could form the basis of a topological quantum computer. This would represent
a groundbreaking technological opportunity, but detecting and manipulating MFs is
quite difficult. For now, there is still much insight to be gained by studying the unique
properties of topological materials themselves.
While some TIs simply have a conductive surface state, others enter a super-
conducting state at very low temperatures. These topological superconductors are
especially promising for technological applications, but they are also interesting purely
31
because of their differences from other superconductors. Proving that a superconduc-
tor is topological or that experimental results are consistent with the presence of MFs
can be difficult, but there are some proposed signatures. Section 2.3 will discuss this
issue in greater detail as it pertains to a specific class of topological superconductors
known as half-Heuslers.
2.2 Noncentrosymmetric Superconductors
It is generally assumed that the superconducting half-Heuslers are unconven-
tional superconductors, but the nature of their superconducting states is still unclear.
However, some predictions can be made based on their noncentrosymmetric (NCS)
crystal structure. The electronic structure and pairing symmetry of NCS supercon-
ductors has been thoroughly explored from a theoretical perspective [57, 58]. Other
SCs lacking inversion symmetry have also been detailed, many of which have been
shown to be heavy fermion materials.
The lack of inversion symmetry in the crystal lattice affects in how electrons
interact with the crystal field via spin-orbit (SO) coupling. This noncentrosymmetry
lifts the spin degeneracy of the Bloch bands, thus changing the spin texture of the
band in k-space. Perhaps most interestingly, NCS superconductors are predicted to
support triplet pairing states (S = 1) and chiral or mixed-parity pairing states [58,59],
the implications of which will be discussed in this section. The effects of spin-orbit
coupling on band structure in NCS materials [57] and the behavior of topological SCs
in general [56, 60–62] have been explored in great detail in the literature, but this
32
work will only focus on the general conclusions from these works.
In a conventional superconductor, the two electrons in each Cooper pair are in
a spin singlet (S = 0) quantum state. Furthermore, the gap symmetry is generally
an isotropic s-wave. This combination of features is not a coincidence. Because a
Cooper pair is an interaction between two fermions its wavefunction must obey the
Pauli exclusion principle, imposes the exchange symmetry condition ∆αβ(k) = -∆αβ(-
k) on the order parameter. As a consequence, a spin singlet state (antisymmetric with
respect to particle exchange) is associated with spatial wavefunctions that are even
under applications under the parity operator such as s- and d-wave. Likewise, a spin
triplet state should have an odd parity spatial component such as p-wave [63]. While
this decoupling of the spin and spatial components of the wavefunction is complicated
by spin orbit coupling, the association between triplet pairing and p-wave symmetry
is still very strong. These combinations of pairing state and gap symmetry can be
seen in Figure 2.3.
The noncentrosymmetric (NCS) crystal structure of materials like the half-
Heuslers has led to speculation that they should be unconventional superconductors
with a topological nature. In a material whose crystal lattice lacks inversion symme-
try the spin degeneracy of the bands is removed and an electronic state that breaks
time reversal symmetry becomes possible. The absence of inversion symmetry also
introduces a non-trivial topology into the band wavefunctions.
The first NCS superconductor to be discovered was CePt3Si (Ref. [64]). Many
others have been discovered since then, [65–70] and while evidence has been found for
unconventional superconductivity the exact nature of the gap structure and pairing
33
+ +
+
+
-
-
-
Singlet
Cooper pair
Triplet
Cooper pair
s-wave d-wave
p-wave
Figure 2.3: Singlet Cooper pairing states are associated with pairing sym-
metries that are symmetric under application of the spatial parity operator
such as s- and d-wave. Likewise, triplet pairing is associated with chiral
pairing symmetries such as p-wave.
34
symmetry has not been settled in most cases. Although it has a centrosymmet-
ric crystal structure, the perovskite Sr2RuO4 provides a useful point of comparison.
Sr2RuO4 is the only material that is widely accepted as a triplet-pairing supercon-
ductor with a p-wave gap symmetry. [71] Previous experimental results on Sr2RuO4
will be discussed at length in Section 2.4.
2.3 Half-Heusler Compounds
Several materials have been proposed as topological materials and studied in
detail, including SmB6 [72–74] and binary telluride compounds such as HgTe and
Sn1−xInxTe [75, 76]. Another system of note is a system of materials known as half-
Heuslers. This family of materials that has been known for decades, but only recently
characterized as superconductors with a potentially topological nature.
The crystal structure of half-Heuslers consists of a face-centered cubic lattice
in the F4¯3m space group, which is also known as the MgAgAs-type structure. The
typical chemical formula is expressed as XYZ where X is typically Sc, Y, or a rare
earth element; Y is a transition metal; and Z is a pnictide (or occasionally a Group
14 element), most often Bi or Sb. The half-Heusler structure, shown in Figure 2.4, is
characterized by a face-centered cubic (FCC) sublattice of the X atoms, with the Y
and Z atoms also arranged in offset FCC sublattices which are located at the Miller
indices (1/4, 1/4, 1/4) and (3/4, 3/4, 3/4), respectively. Another way of visualizing
this structure is as an overlap of a zinc-blende lattice with the formula XY overlapping
with a rock salt (NaCl) structure lattice with formula XZ overlapping at the X sites.
35
Figure 2.4: a. Half-Heusler and b. Heusler crystal structures with the
chemical formulae XYZ and X2YZ. The non-centrosymmetric half-Heusler
lattice consists of face-centered cubic sublattices of X, Y, and Z atoms.
Application of the parity operator at the center of the unit cell leads to
the red X atoms occupying different sites on the imaginary cube drawn
with a dotted line. The centrosymmetric Heusler structure is shown for
comparison. (From Ref. [77].)
Perhaps the most striking feature of this crystal structure is that it lacks a cen-
ter of inversion. It is a well-established principle of solid state physics that crystal
structure has a direct effect on electronic structure [78]. In this case, the noncen-
trosymmetric (NCS) nature of the half-Heusler structure is predicted to affect the
superconducting properties of these materials, as was discussed in Section 2.2.
As a note on nomenclature, the half-Heusler structure is a modification of an-
other FCC structure known simply as Heusler compounds. The similar (but cen-
trosymmetric) Heusler structure, also shown in Figure 2.4, is seen in compounds with
the stoichiometric formula X2YZ and is identical to the half-Heusler structure, but
with an additional FCC sublattice of X atoms offset by (1/2, 0, 0).
Among superconductors that are predicted to be topological, a wide range of
electronic states has been observed at high temperature. For the half-Heusler SCs
36
some significant differences in electron transport properties have been measured de-
pending on the elements present in the structure. For example, YPtBi (which will
be discussed at length in Chapter 5) is insulating at high temperature, but as it
is cooled down below roughly 150 K a crossover to metallic behavior is observed
and resistivity decreases down to Tc. In the chemically similar material YPdBi, no
transition is observed and resistivity increases monotonically down to Tc. Making a
different elemental substitution to LuPtBi results in similar semimetallic behavior in
ρ vs. T [79], but LuPtBi and YPdBi behave significantly differently when comparing
magnetic susceptibility vs. temperature. [cite] Significant differences have also been
observed in the residual resistivity ratio (RRR), expressed as ρ(300 K)/ρ(0 K), where
ρ(0 K) is extrapolated from low temperature resistivity, ignoring the zero resistiv-
ity SC transition if present. These differences in resistivity highlight the complex
effects produced by elemental substitution and the difficulty of making theoretical
predictions.
Several studies have investigated the electron transport properties of different
half-Heusler compounds, but here I will focus on YPtBi. YPtBi was first charac-
terized by Canfield et al. [80], who found that the resistance of YPtBi increases by
a factor of roughly 25-30% as temperature is decreased from 300 K to 150 K. Su-
perconductivity in the compound was first characterized by Butch et al. [81], who
observed a SC transition temperature of Tc = 0.77 K and a critical field of Hc2 =
1.5 T when Hc2(T ) was extrapolated to zero temperature. Above Tc, semimetallic
behavior was observed, with ρ(T ) increasing with decreasing temperature, and then
leveling off below roughly 100 K. At high magnetic fields, Shubnikov-de Haas os-
37
cillations were observed with a frequency corresponding to a hole-majority carrier
density of 1.7 ×1018 cm−3. Further analysis of these quantum oscillations revealed
a node that introduced a pi phase shift and could be consistent with two spin-orbit
split Fermi surfaces of similar size. Differences in magnetoresistance behavior were
also observed after mechanically roughening the surface, which may be evidence of
the predicted surface states. Fig. 2.5 shows electron transport data from Butch et al.
over a range of temperatures and magnetic fields demonstrating the superconducting
transition and quantum oscillations [81].
More recent studies of YPtBi have used other experimental techniques to probe
the electromagnetic properties and gap structure. Bay et al. [82] observed an unex-
pected metallic behavior in ρ(T ), with resistivity decreasing monotonically down to
Tc. Carrier concentration was also much more temperature independent compared
to the previous study, though the precise reason for this difference was not clear.
Application of hydrostatic pressure caused a consistent increase in Tc up to p =
2.51 GPa. Analysis of normalized upper critical field vs. temperature found that Hc2
was approximately linear in temperature for all applied pressures. Comparison to
the Werthamer-Helfand-Hohenberg (WHH) model and a polar-state model for this
data showed a better fit to a p-wave model than s-wave. This unusual Hc2 behavior,
coupled with a large calculated mean free path, was cited as evidence for a dominant
odd-parity component in the order parameter.
Further work by Bay et al. [83] showed that YPtBi is a bulk superconductor
with a volume fraction of approximately 70%. They also found a very small lower
critical field, which was argued to be consistent with a non-unitary Cooper pair state.
38
Figure 2.5: Resistivity (ρ, green) and Hall resistance (RH , purple) vs.
temperature (left) in YPtBi with inset showing detail of resistive transi-
tion and magnetic susceptibility around Tc and magnetoresistance data
(right) with background normalized quantum oscillations shown in the in-
set. At high temperatures resistivity behavior is consistent with a single-
band semimetal, and RH follows a similar temperature dependence. A
sharp transition to zero resistance is observed at Tc = 0.77 K, and dia-
magnetic screening is also seen at this same transition temperature. At
3 K relatively linear magnetoresistance was measured for fields applied
both parallel and perpendicular to the direction of current flow and quan-
tum oscillations are seen at high magnetic fields. Similar oscillations were
observed at 0.1 K, seen in the lower inset with the background resistance
subtracted. The oscillations have a period of 0.02 T−1, which corresponds
to an effective mass of 0.15 me, and a node appears due to beating result-
ing in a pi phase shift. (From Ref. [81].)
39
Finally, Shi et al. [84] used nuclear magnetic resonance (NMR) measurements to
detect enhanced orbital diamagnetism in YPtBi and YPdBi, induced by spin-orbit
coupling.
Most published experimental work on half-Heuslers has focused on a single mea-
surement on one or two compounds. Fewer systematic studies examining a range of
similar materials have been published. One example is a study by Gofryk et al. [85] of
polycrystalline RPdBi (where R represents six different rare earth compounds), which
included measurements of magnetic susceptibility, resistivity, thermopower, and Hall
resistance. It was found that most of the compounds undergo an antiferromagnetic
transition at low temperature but that YPdBi is diamagnetic. Electrical resistivity
and thermopower measurements were consistent with semimetals with hole-majority
carriers. Overall, results were considered to be consistent with theoretical predictions
for topological insulators, especially in the case of YPdBi, which has been predicted
to have a very small band inversion strength.
Multiple theorists have analyzed the half-Heusler crystal structure using density
functional theory (DFT) [54, 86–88]. Calculations of the band structure are able to
predict band inversion and the resultant surface states in some half-Heusler materials,
but others are predicted to be normal band insulators or semiconductors. As shown
previously in Fig. 2.1, significant differences are predicted for various compounds at
the Fermi level. For example, ScPtSb is predicted by Chadov et al. to be a trivial
insulator, while ScPtBi is predicted to undergo inversion of two bands with Γ6 and
Γ8 symmetries, experience a crossing at the Fermi level, and behave as a topological
insulator.
40
The aggregate of these results can be displayed qualitatively, as shown in Fig.
2.6. These DFT calculations yield a quantity, ∆E representing the difference in energy
between the Γ6 and Γ8 bands, which represents effective band inversion strength.
When ∆E is plotted against the lattice constant or atomic number, general trends
emerge. For example, half-Heusler compounds with heavier elements are more likely
to be calculated as topologically nontrivial. This is consistent with the role that spin
orbit coupling plays in NCS materials, as will be discussed in the next section.
2.4 Experimental Work on Topological and Noncentrosymmetric Su-
perconductors
Experimentally proving that a superconducting state is topological can be chal-
lenging. However, finding evidence of a p-wave or mixed parity gap or triplet pairing
can be done with standard experimental techniques. Most notably for this work, I
will use point-contact spectroscopy to distinguish between different gap structures
and pairing symmetries, as discussed in detail in Chapter 3.
The basic idea was modeled by Honerkamp and Sigrist [89], who showed that
a triplet pairing state should produce a zero bias peak in the conductivity spectrum
as shown in Fig. 2.7. The zero bias peak generally corresponds to an approximate
doubling of the high bias conductance, but with a peak morphology that is slightly
different from the s-wave shapes seen in, for example Fig. 1.11, and which will be
discussed in greater detail in Chapter 3. Furthermore, for all but the largest values
of the Z parameter, shallow dips in conductance are predicted at a bias voltage
41
Figure 2.6: Band inversion strength plotted against different parameters
for several half-Heusler compounds, as calculated from density functional
theory. Note that in the upper figure (from Ref. [54]) band inversion is
identified by a negative value of EΓ6-EΓ8 while in the lower figure (from
Ref. [86]) topological materials are identified by a positive value of band
inversion strength.
42
Figure 2.7: Predicted dI/dV spectra for a triplet superconductor for a
range of Z values. (From Ref. [89].)
43
corresponding to ±∆. This feature is not uncommon for predictions of dI/dV curves
for unconventional superconductors (it is also seen in a d-wave model under certain
conditions, for example). However, the specific shape of the peaks and dips still set
the triplet model apart, and other experimental methods could also be used to confirm
the triplet pairing state.
Laube et al. performed point contact measurements on the proposed spin-triplet
p-wave SC Sr2RuO4 [90] to test this model. Using the needle-anvil technique in a
dilution refrigerator they observed two distinct types of conductivity spectra. Some
junctions showed a double peak structure at zero bias, which is often a characteristic
of s-wave SCs, but with very small enhancement in conductivity at zero bias (about
3% compared to high-bias values). Other junctions had dI/dV curves with a single
zero bias peak (10-15% enhancement) and small dips at higher bias. Both of these
morphologies were fit to a triplet pairing model allowing a tunable transparency, and
the temperature dependence of the gaps was consistent with that expected for an
triplet pairing.
Recent work by Sasaki et al. sheds more light on how topological superconduc-
tivity manifests itself in point-contact measurements with studies of CuxBi2Se3 (Ref. [91])
and Sn1−xInxTe (Ref. [92]). In both cases a clear zero bias conductance peak was ob-
served in dI/dV spectra accompanied by dips on either side of the peaks. Based on
the shape of these features and their evolution in magnetic field it was argued that the
peaks are caused by an Andreev bound state (ABS) rather than Andreev reflection,
reflectionless tunneling, or magnetic scattering. Based on analysis of the symmetry
of the Hamiltonian, it was argued that in each case the ABS must be caused by an
44
unconventional, odd-parity bulk SC state. Furthermore, in the case of Sn1−xInxTe,
the zero bias enhancement observed was greater than 100% compared to the high
bias value [92], which is impossible for typical Andreev reflection. This was cited as
further evidence for the ABS. It should be noted, however, that more recent work on
CuxBi2Se3 using scanning tunneling spectroscopy shows a full gap in the density of
states with no zero bias features [93], which puts this work into question.
Another technique that could potentially distinguish triplet pairing is nuclear
magnetic resonance (NMR) measurements of the Knight shift. In NMR spectroscopy
the nuclear spins in the target material are aligned using an external field, then a
radio frequency pulse is applied and the samples relaxation response is measured.
The Knight shift is a characteristic shift in the frequency of nuclei due to the spins
of conduction electrons. For a conventional superconductor with spin-singlet Cooper
pairs the Knight shift should detect a sharp change in spin susceptibility as a function
of temperature below Tc. In the presence of a small magnetic field, singlet Cooper
pairs will not be broken, meaning the SC state has a low net magnetization compared
to that produced by the unpaired electrons in the normal state. In a triplet SC
however, the low-field spin susceptibility should not change below Tc because the
conducting electrons transition from a collection of single electrons with aligned spins
to a collection of Cooper pairs with similarly aligned spins.
This phenomenon has provided direct evidence for triplet pairing in Sr2RuO4,
one of the only materials in which the presence of a triplet pairing state has been
convincingly established. As detailed in a review by Mackenzie and Maeno [71],
spin susceptibility χs is the result of Zeeman splitting of electron states into spin-up
45
and spin-down Fermi surfaces. In a SC with a singlet pairing state, this splitting
suppresses Cooper pair formation, as the two electron states necessary to form a pair
(k↑ and k↓) no longer exist at the Fermi surface. In the superconducting state, the
free energy gain from condensation into the SC state dominates and χs is reduced
to zero as T → 0. For a spin triplet SC, however, this spin splitting does not affect
Cooper pair formation. Thus, for a triplet superconductor in a low applied field, spin
susceptibility is strong both above and below Tc, as both individual electrons or spin-
aligned Cooper pairs will align in the direction of the field. This persistence of χs
below Tc in Sr2RuO4 was confirmed experimentally by Ishida et al. [94] in an applied
field of 0.65 T. The stark contrast between expected behavior for singlet and triplet
superconductors makes Knight shift measurements a prime candidate for unearthing
the nature of the pairing state in YPtBi and other half-Heusler materials.
Evidence for unconventional superconductivity could also be found from mea-
surements of the penetration depth and muon spin relaxation. The London pene-
tration depth λL of a SC can be measured by several methods, including magnetic
force microscopy, SQUID microscopy, and microwave cavity perturbation. For a con-
ventional s-wave superconductor, the quantity ∆λL = λL(T ) λL(0) should have an
exponential T -dependence at low temperature. Deviations from this expected behav-
ior, for example a linear λL(T ) in the cuprates, can be evidence of nodes [95]. In
Sr2RuO4, ∆λL vs. T is quadratic at low temperatures as measured by Bonalde et
al. [96]. This may be evidence in favor of non-s-wave superconductivity, but isnt com-
pletely conclusive due to complicating factors that may have caused this quadratic
behavior.
46
Finally, muon spin relaxation (µSR) is a useful probe for measuring the local
distribution of magnetic field in a solid. In this technique, a beam of spin-polarized
muons are directed at a sample so that they come to rest inside it before decaying
into a positron and two neutrinos. The positrons will be emitted in a direction
that correlates with the local magnetic field at the site of the decay, and by observing
positron polarization as a function of time one can extract information about magnetic
fields inside the sample. In the case of Sr2RuO4, the muon spin relaxation rate has
been shown to increase below Tc, which is likely due to the internal magnetic field
caused by spin triplet Cooper pairs [97]. Enhancement of this relaxation rate is
believed to be an indicator of broken time reversal symmetry. But like the case of
non-exponential penetration depth, unconventional superconductivity is not the only
possible explanation, although it appears to be the most plausible.
47
Chapter 3: Experimental Methods
3.1 Sample Preparation and Initial Characterization
3.1.1 Crystal Growth
All the experiments I carried out for this thesis involved measurements of single
crystal superconducting samples. Working with single crystals is especially impor-
tant to maintain consistency between research groups and even materials made by
the same individuals. The introduction of microstructure and polycrystallinity adds
another potential complication, as it can be difficult to ascertain how much the grain
size and shape or presence of grain boundaries is affecting an electronic or magnetic
measurement. This section details the crystal growth process with a particular em-
phasis on the growth of 122 materials.
One of the reasons the 122 iron pnictides are so attractive to study is that large
single crystals can be grown relatively easily with the flux method. In flux-aided
crystal growth the component elements are melted together along with an additional
non-reacting component that helps the other components dissolve and mix together.
The flux also provides a liquid medium in which single crystals of the desired solid
material can grow [16,98].
48
The flux method requires a few extra steps when dealing with arsenic, which
is toxic and sublimes at 614 ◦C. Sublimation can be avoided by pre-reacting arsenic
powder to form a compound that melts, usually the binary compound FeAs. This is
achieved by sintering (i.e., heating a powder below its melting point) stoichiometric
amounts of powdered Fe and As together, which results in the formation of the FeAs
phase.
The FeAs powder is then combined with a stoichiometric amount of Ca, Sr, or
Ba metal pieces as well as the correct amount of any doping element in an alumina
crucible along with a large amount of flux, either extra FeAs or Sn. The crucible is
then sealed under argon gas in a quartz tube. This is necessary to prevent the for-
mation of oxides at the high temperatures required to melt the component materials.
The quartz tube is heated and cooled in a box furnace according to estab-
lished heating schedules that usually last approximately one week. These temper-
ature vs. time profiles generally require holding the mixture at multiple high tem-
peratures to allow each component of the mixture to melt for a long enough time
for all components to combine via diffusion. The mixture is then cooled slowly, al-
lowing large single crystals to grow. For these materials, large means sizes on the
millimeter scale. Typical crystals used for resistivity measurements are on the order
of 1 mm × 2 mm × 0.05 mm, though under optimal conditions the size of the crystals
is limited only by the size of the container usually about 1 cm.
When excess FeAs flux is used, the mixture is allowed to cool by simply pro-
gramming the furnace to shut off once the heating schedule is completed. Once it
reaches room temperature, the alumina crucible is broken and shiny, platelet-shaped
49
single crystals can be extracted from the solid, dull gray flux with tweezers and a
razor blade. For Sn flux, the crystals cannot be extracted so easily. In this case, the
quartz tube must be removed from the furnace at a temperature that is below the
melting point of the 122 crystals but above the melting point of the Sn. The hot
tube is then spun in a centrifuge to separate out most of the liquid flux, leaving the
crystals behind. Usually these crystals still have some Sn flux on the surface. This
can be removed by etching in HCl.
Fig. 3.1 shows an example of 122 crystals grown using the FeAs flux method.
Note that the platelet shape observed in these crystals is caused by the relative
strength of bonds within the lattice. The Fe-As bonds are stronger than Sr-As bonds,
thus growth occurs preferably in the ab-plane, which reflects the layered crystal struc-
ture.
In the case of the half-Heuslers the same flux method is used, but with excess Bi
as the liquid medium. This is possible because Bi has the lowest melting point of the
component elements in this family of compounds. Much like Sn flux growths of 122
iron pnictides, Bi flux must be separated from the crystals at high temperature. This
is done either by the centrifuge method or by decanting, that is, inverting the quartz
tube containing the crucible immediately after removal from the furnace so that the
molten Bi can flow downward into an empty crucible, thus leaving the crystals behind.
The Bi-flux method is currently the only known effective technique for growing
half-Heusler materials and it typically yields fairly large crystals as shown in Fig. 3.1.
Although not as large as optimal 122 crystals, the largest half-Heusler crystals are
typically a few mm in length, which is large enough for the purposes of most of the
50
Figure 3.1: Single crystals of BaFe1.9Pt0.1As2 grown in FeAs flux (above)
and crystals of the half-Heusler LuPdBi grown in Bi flux (below).
51
measurements. Also of note is the shape of the crystals. Because of their cubic crystal
structure, half-Heusler single crystals do not grow as platelets. As a result, I had to
sand the crystals flat to expose a large face for the application of electrical contacts.
One final complication of working with half-Heusler crystals is the presence of
residual Bi flux inside the material. In nearly all of my samples I typically observed
under a microscope, small discolored areas have been observed on the surface. El-
emental analysis using energy dispersive x-ray spectrometry in a scanning electron
microscope reveals that they represent Bi inclusions. Figure 3.2 shows a close-up of
these inclusions on a polished LuPdBi crystal as observed with a differential inter-
ference contrast microscope. Because the inclusions are small and fairly sparse it is
generally assumed that they do not have a significant effect on experimental results.
3.1.2 Resistivity
The resistance of large samples can be determined by a simple two-point mea-
surement with an ohmmeter. Such a measurement would be difficult for the small and
highly conductive crystals used in my experiments, as the resistance of the contacts
would not be negligible compared to the sample resistance. This problem can be
remedied by using a four-point probe method, which utilizes four electrical contacts
on the sample. In this method a known current I is applied to two outer contacts and
the voltage drop V caused by that current is measured between two inner contacts
(see Fig. 3.3). Since very little current passes through the wires on the voltage con-
tacts, the voltage dropped across them is very small, and the voltage drop measured
52
Figure 3.2: Image of polished LuPdBi crystal captured with a differential
interference contrast microscope. The small spots scattered across the
bottom of the crystal are inclusions of Bi flux.
53
Figure 3.3: Schematic of a resistivity measurement using the four-point
probe method. Gold wires are applied across the width of the sample
using silver paste or Pb-Sn solder.
can be considered to be entirely across the sample. Using Ohm’s Law, the measured
current and voltage are then used to calculate the resistance R = V/I between the
two voltage leads. I converted the resistance R to resistivity ρ = RA/l by measuring
the length l between the voltage leads and the cross-sectional area A of the sample.
When making a four-point resistivity measurement on a 122 crystal I attached,
four parallel gold wires, usually 50 µm thick, to the sample with conductive silver
paste or Pb-Sn solder. The sample was then mounted on a puck specially designed for
use with a Quantum Design Physical Property Measurement System (PPMS). Once
inserted into the PPMS, the sample was cooled to as low as 1.7 K and I measured
resistivity as a function of temperature. The system was also capable of applying
54
magnetic field and measuring other physical properties such as specific heat.
While the PPMS was suitable for many measurements, in some cases lower
temperatures than 1.7 K were required. Such low temperatures can be attained
with a dilution refrigerator, which achieves cooling by taking advantage of a unique
property of liquid helium. Helium has two stable isotopes, 3He and 4He. Because
the two isotopes are immiscible below a critical temperature, when a mixture of 3He
and 4He is cooled below about 1 K they separate in an endothermic process. This
process can be then repeated by pumping out the dilute 4He phase and properly
adding it back into the remaining 3He phase so that continuous cooling power can be
achieved. [57] Theoretically, this could be repeated ad infinitum so that the system
exponentially approaches 0 K. Practically, the minimum temperature obtainable is
limited by radiative and conductive heating in the sample chamber and is typically in
the 10-20 mK range. Resistivity, magnetoresistance, and specific heat measurements
can all be obtained with a properly configured dilution refrigerator. The advantage of
a dilution refrigerator is the extremely low base temperature, which is necessary for
detecting superconducting transitions in the 20 mK to 1 K range and observing the
behavior of transport properties (such as resistivity and specific heat) as T approaches
zero.
A useful intermediate option is a 3He probe produced by Quantum Design for
use with a PPMS. This specialized insert uses a small amount of liquid 3He which is
continuously circulated throughout the probe. Given the limited cooling power of a
smaller probe with less 3He than a dilution refrigerator, the minimum temperature for
the system is only around 400 mK. This is still less than the Tc of many half-Heusler
55
materials, and the fact that it offers all of the functions of the PPMS and much faster
cooling than a dilution refrigerator made it a viable option for many measurements.
3.1.3 Magnetic Properties
The primary tool used for determining the magnetic structures of samples is neu-
tron diffraction, which yields the same crystal structure information as x-ray diffrac-
tion while also giving information about magnetic ordering. This is possible because,
unlike x-rays, neutrons have a net magnetic moment. Thus, they can be diffracted
by an array of periodic magnetic moments (( e.g.), a magnetically ordered lattice).
In analyzing neutron diffraction data, one can use Rietveld refinement to separate
the crystal lattice peaks (the same peaks one would observe from x-ray diffraction)
from additional magnetic peaks. Based on the Bragg angles and peak heights, one
can determine the spatial orientation of spins of the atoms in the crystal lattice (e.g.,
AFM, SDW) through further refinement [25–29].
Magnetic susceptibility can be measured using a superconducting quantum in-
terface device (SQUID), which is capable of detecting very small changes in magnetic
field. A SQUID is essentially two Josephson junctions [58, 59] connected to form a
loop (see Fig. 3.4). It is known that the magnetic flux through such a loop is quan-
tized in units of the flux quantum Φ0 = h/2e ≈ 2.07 × 10−15 T*m2 [99, 100]. When
properly current biased, small changes in magnetic field applied to the loop cause the
voltage across the loop to vary periodically in the flux. In order to measure magnetic
susceptibility, a superconducting sample is placed in an inductance coil that is cou-
56
VMagnetic field
Applied current
Josephson junction
Figure 3.4: Diagram of a superconducting quantum interface device
(SQUID), a loop of superconducting material with two Josephson junc-
tions.
pled to the SQUID while a magnetic field is applied. The sample is then physically
oscillated in order to induce changes in the field, which in turn causes changes in the
voltage across the SQUID. This voltage is a measure of the change in magnetic field,
from which one can calculate the net magnetization of the material causing the change
and thus the magnetic susceptibility of the sample. The Quantum Design Magnetic
Property Measurement System (MPMS), is capable of measuring susceptibility as a
function of temperature and applied magnetic field.
3.2 Point-Contact Spectroscopy: Theory
While resistivity and magnetization measurements can reveal a great deal of
information about the SC state, they fall short when it comes to understanding the
nature of the superconducting order parameter and its symmetries. In contrast,
57
phase-sensitive experiments are designed to reveal information about gap structure
and pairing symmetry. Point-contact spectroscopy (PCS) is a particularly versatile
tool that can be used to extract gap sizes and provide some information about phase.
The basis of point-contact spectroscopy is a small junction created between a
normal metal tip (N) and a superconducting sample (S) below its transition temper-
ature. Consider the density of states of both materials at the interface. The states in
the normal metal are continuous up to the Fermi surface, while the superconductor
has a gap between the lower occupied band of Cooper pairs and the unoccupied higher
band of excited states. When the Fermi energy of the normal metal falls within this
gap, an electron travelling at the Fermi momentum in the normal metal cannot pass
through the gap. Since the electron doesnt have sufficient energy to reach an excited
quasiparticle state in the superconductor, it must either form a Cooper pair in or-
der to pass into the superconductor or simply be reflected back. This single-electron
transmission across the junction is forbidden [5]. In the case where the electron does
pass through the barrier and form a Cooper pair in the superconductor, a hole with
spin opposite that of the incident electron must be reflected back into the normal
metal (this is known as retroreflection) in order to conserve charge, spin, and mo-
mentum. This process is known as Andreev reflection and is illustrated in Figure
3.5.
The extent of Andreev reflection can be measured as a function of bias voltage
across the N/S junction, which moves the Fermi level of the normal metal up and
down relative to the gap. When the bias voltage is such that the Fermi level of the
normal metal is above the gap of the superconductor, single electrons can pass through
58
Δ-Δ
N(E)
E
N S
N
S
Figure 3.5: Density of states of normal metal (N) and superconductor
(S) (top) with the Fermi energy of the normal metal within the SC gap;
illustration of Andreev reflection (bottom) with an incident electron at
the junction creating a Cooper pair in S and a retroreflected hole in N
with opposite spin.
59
the junction and the conductivity of the junction reaches a fixed value. When the
Fermi energy is within the gap (at least in the simplest case), each incident electron
generates a Cooper pair and a hole moving in the opposite direction. Since the amount
of charge moving has effectively doubled compared to the high bias value, one would
expect the conductivity of the junction to double as well.
In practice, the scattering process, and the behavior of the conductivity, is more
complex. One very typical complication is the presence of a potential barrier at the
junction, often in the form of a thin oxide layer at the surface of the normal metal
or superconductor. Similar effects can result from a mismatch in Fermi velocities
between the normal metal and the SC [5]. As the effective magnitude of this barrier
increases, more electrons are reflected back at the junction rather than transmitted
via Andreev reflection. In the extreme case where the barrier is infinitely high, 100%
of all incident electrons are reflected and no current passes through, meaning the
conductivity drops to zero for all bias voltages within the gap [5]. The height of this
barrier is quantified by a unitless fitting parameter labeled Z, which can vary from
zero (no barrier, conductivity should double inside the gap) to infinity (strong barrier,
no conduction inside the gap).
Blonder, Tinkham, and Klapwijk have quantified the effect of a potential barrier
and other factors on conductivity in what has become known as the BTK theory [40].
In modeling the behavior of electrons approaching a N/S junction, the barrier is
treated as a delta function potential with height Z. This model [41] assumes that
the densities of states for quasiparticles and Cooper pairs in the superconductor are
given as
60
Nq(E) =
E + iΓ
√
(E + iΓ)2 −∆(E)2
(3.1)
Np(E) =
∆(E)
√
(E + iΓ)2 −∆(E)2
(3.2)
where E is the energy that is controlled by adjusting bias voltage, ∆(E) is the
energy gap, and Γ is a term with units of energy that accounts for inelastic scattering.
This is the density of states function illustrated in Figure 3.5. For the BTK model
of the barrier as a delta function of magnitude Z, the transmission coefficient for
tunneling through the barrier is
τN(θ) =
cos2(θ)
cos2(θ) + Z2
(3.3)
where θ is the angle between the momentum of the particle incident on the
barrier and the normal direction to the barrier n (e.g., θ = 0 for normal incidence).
The relative transparency of the barrier, σS, can be written as:
σS(E, θ) =
1 + τN(θ)|γ(E)|2 + (τN(θ)− 1)|γ(E)2|2
|1 + (τN(θ)− 1)γ(E)2|2
(3.4)
where γ is given by
γ(E) =
Nq(E)− 1
Np(E)
(3.5)
This quantity can then be integrated over all possible incident angles in the
two-dimensional plane to arrive at the expected normalized conductance:
61
G2D(E) =
∫ pi/2
−pi/2 σS(E, θ)τN(θ) cos(θ) dθ
∫ pi/2
−pi/2 τN(θ) cos(θ) dθ
(3.6)
Equations 3.1-3.6 assume that the energy gap has no dependence on angle,
that is, an isotropic s-wave superconductor. The BTK model can be easily extended
to include angular dependence to the gap, as will be demonstrated later. For the
simplest case of an isotropic s-wave with no inelastic scattering (Γ = 0) and at zero
temperature, the model produces conductance vs. bias voltage curves (with bias
voltage expressed in units of the energy gap divided by electron charge, ∆/e) as
shown in Fig. 3.6.
I note in particular the doubling in conductivity within the gap (-∆/e ≤ Vbias ≤ ∆/e)
for the zero barrier case and the reduction of conductivity to zero in the gap for
Z  1. The large barrier case also yields high conductivity close to bias voltages
of V = ± ∆/e, due to the high density of states in the superconductor at those
energies [41]. One can include inelastic scattering using nonzero values of Γ, resulting
in curves such as those in Figure 3.7.
Generally speaking, increasing the magnitude of the inelastic scattering term
tends to change the shape of the curve by causing more reflection inside the gap
for shorter barriers and less tunneling close to the gap for a higher barrier. Other
changes occur when accounting for the effect of non-zero temperature. At nonzero
temperatures thermal motion causes the density of states to smear out. This is
reflected in Figure 3.8 for a low-Z case. The most obvious effect is that the height of
the conductance peaks decreases with increasing temperature; this is consistent for
62
00.5
1
1.5
2
2.5
dI/
dV
(no
rm
aliz
ed
)
Vbias
--2 0  2
Z = 0
Z = 1
Z >> 1
Figure 3.6: Expected conductivity vs. bias voltage curves for Andreev re-
flection and tunneling with potential barriers of different sizes. Computed
from BTK theory with T = 4.2 K, ∆ = 5 meV, and γ = 0.
63
11.2
1.4
1.6
1.8
2
 = 0.0
 = 0.5
 = 1.0
dI/d
V(
nor
ma
lize
d)
Vbias
--2 0  2
Z = 0.1
0
0.5
1
1.5
2
2.5
 = 0.0
 = 0.5
 = 1.0
dI/d
V(
nor
ma
lize
d)
Vbias
--2 0  2
Z = 5.0
Figure 3.7: Examples of conductance curves showing the effect of increas-
ing Γ for small Z (Andreev limit, above) and large Z (tunneling limit,
below) at T = 4.2 K with ∆ = 5 meV. Tc can be assumed to be large
compared to 4.2 K. Temperature only affects the conductance via the
Fermi-Dirac term f(E) = (eE/kT+1)−1 in the BTK theory.
64
all values of Z and Γ. Also, the magnitude of the gap ∆ decreases too as T increases
to Tc [5]. Qualitatively this makes sense - the energy gap should disappear when
the superconductor undergoes a transition to the normal state. This shrinking of the
gap is predicted by the BCS theory, and similar behavior is observed for non-BCS
superconductors, and is approximated in equation (3.7) [5] and plotted in the inset
of Figure 3.8.
∆(T )
∆(T = 0)
≈ 1.74(1−
T
TC
)1/2 (3.7)
As T is increased beyond Tc, the gap disappears and the conductivity becomes
that of the normal state. This background conductivity can be used to normalize
lower temperature conductivity curves so that they may be properly fit to BTK
curves predicted by equation (3.6).
As I noted above, one can extend the BTK theory to account for pairing sym-
metries in which ∆(E) is not constant as a function of angle. The anisotropic s-wave
and d-wave symmetries (see Fig. 3.9) can be modeled as follows:
∆s,anis(E, θ) = ∆(E) cos
4[2(θ − α)] (3.8)
∆d(E, θ) = ∆(E) cos[2(θ − α)] (3.9)
The parameter α introduced in Equations (3.8) and (3.9) represents the dif-
ference in angle between the a-direction of the crystal lattice of the superconductor
and the vector n that is normal to the N/S interface [41]. In the case of my iron
65
Figure 3.8: Effect of increasing temperature on expected PCS conductivity
curves and gap size (inset). From Ref. [41].
66
Figure 3.9: Effect of inelastic scattering, barrier height, and tunneling
direction on conductance curves with angular dependence taken into ac-
count for anisotropic s-wave (α = pi/4, left) and d-wave (α = pi/4, center;
and α = pi/8, right). Note that (a) and (b) correspond to current injection
in the direction of the nodes in the gap. (From Ref. [41].)
pnictide samples the interface was always parallel the ab-plane (so the normal to the
surface was the c-direction) meaning α = pi/2. Figure 3.9 shows a few examples of
the effect of including the angular dependence for in anisotropic s-wave or d-wave
superconductor.
It should also be noted that Andreev reflection can only be observed when the
radius of the N-S junction a is small in comparison to the mean free path l of electrons.
Based on the ratio of these two lengths, the behavior of a point-contact junction can
be divided into three general regimes.
67
The ideal case in which al is known as the ballistic regime because electrons
will be accelerated through the contact area without scattering and gain a kinetic
energy of eV , where V is the applied voltage. Contacts in which the junction radius
is greater than the elastic scattering length, but still less than the diffusion length,
are in the diffusive regime. In this regime, elastic scattering occurs inside the junction
area, but energy resolution is still possible. The other extreme where al is known
as the thermal regime. In this case, elastic scattering causes energy loss and local
heating in the vicinity of the point contact. Figure 3.10 illustrates the differences
between these three regimes and the types of scattering observed in each.
For an N-S junction, the energy resolution needed for a successful spectroscopic
measurement is possible in both the ballistic and diffusive regimes. Ballistic contacts
are preferred, as anomalies in point-contact spectra have been observed for contacts
in the diffusive regime. Thus, in general it is desirable to obtain the smallest junction
possible for a PCS measurement.
3.3 Point-Contact Spectroscopy: Experimental Setup
There are several variations of the point-contact technique that can be used to
create a normal metal-superconductor junction. I used two different methods, known
as needle-anvil (NA) and soft point contact (SPC), and each has advantages and
disadvantages. With the NA method a sharp normal metal tip is pressed into the
sample. The SPC technique uses a small spot of conductive paint or paste on the
surface of the SC as the normal metal contact, as described by Daghero et al. [101].
68
Figure 3.10: Diagrams of point contacts in the (a) ballistic regime, (b)
diffusive regime, and (c) thermal regime. In a ballistic contact there is
no scattering at the junction, while for diffusive and thermal contacts one
observes inelastic and elastic scattering, respectively. The diagrams on
the right side indicate how the Fermi surface changes upon application of
an electric field for a normal metal-normal metal heterojunction. (From
Ref. [101].)
69
Diagrams of both types of junctions are shown in Figure 3.11.
The needle-anvil technique has the advantage of being adjustable, making it
possible to explore a wide range of junction resistances in the same measurement run.
On the other hand, the mechanical apparatus required for these measurements can be
sensitive to vibrations, and there is evidence that pushing a sharp tip into a sample
can create pressure that may affect results (see, for example, Ref. [102]).
The soft point contact method produces junctions that are very mechanically
stable and offers greater flexibility in the variety of measurement probes it can be used
with. However, soft point contact junctions cannot be adjusted after fabrication, and
depending on the fabrication technique the junction size can be larger than those
obtained using electrochemically sharpened tips.
The choice of method I used was based on the materials’ properties and available
equipment. I used the NA technique for the 122 iron pnictide samples because of
the adjustability, which allowed for up to five different junction resistances to be
measured during each measurement run. Based on the specific probe I used, the
lower temperature limit for the NA method is either 4.2 K or 1.7 K, both of which
were too high to reach the very low transition temperatures of the superconducting
half-Heuslers. Therefore, I used the SPC method for my half-Heusler samples, and
cooled them in the 3He insert designed for the PPMS.
70
Figure 3.11: a. Needle-anvil point contact junction with a sharp normal
metal tip pressed into a superconducting sample. b. Soft point contact
junction with a small spot of silver paste forming the normal metal con-
tact. (From Ref. [101].)
71
3.3.1 Needle-Anvil Method
Needle-anvil point-contact measurements were performed using a specialized
probe that allowed for precise control of experimental parameters. This probe was
used in conjunction with external voltage and current sources and a cryogenic tem-
perature controller. To create the junction, I first prepared a normal metal tip by
using a razor blade to shave either a piece of thick gold wire into a sharp tip or a piece
of lead shot into a sharp pyramid. Two copper wires were attached to the tip with
DuPont silver paint, and the tip was mounted on a movable stage at one end of the
probe. I also soldered wires onto the superconducting sample, which were connected
to a breakout box at the room temperature end of the probe. This gave a total of
four leads that were connected to the junction (V+ and I+ on the tip, V− and I− on
the sample). A closeup view of the stage end of the probe is shown in Figure 3.12.
A vacuum can was secured over the end of the probe containing the tip and
sample and sealed with indium wire. The probe was then inserted into a cryostat and
the other end connected to an instrument panel. The system is cooled to 4.2 K by
flushing the cryostat with liquid nitrogen, then pushing out the N2 and filling with
liquid He. The stage containing the tip was carefully moved using a system of gears
until contact is made with the sample and the junction formed. While moving the
tip the voltage between the tip and sample was closely monitored, as a sharp change
indicated that contact had been made.
I used a Labview script to collect conductivity data. Fig. 3.13 shows a diagram
of the circuit used to adjust the bias voltage and measure conductivity. Two voltage
72
Figure 3.12: Needle-anvil probe used for experiments in Chapter 4 (above)
with a close up image of the stage end of the probe showing the gear
mechanism that pushes the tip into the sample (below).
73
Figure 3.13: Wiring diagram for measuring PCS conductance curves. Rac
and Rdc are adjustable, and for typical values RacRdcRJ . The junc-
tion is represented by the “X” shape labeled as RJ , with the I− and V−
corresponding to the leads on the tip and I+ and V+ to the leads on the
superconducting sample.
sources (DC and AC) are connected in parallel with the junction. The DC voltage
is used to sweep the bias point, generally from about +50 mV to -50 mV. A lock-
in amplifier is then used to superimpose a small AC signal on top of the bias and
measure the output, which is proportional to the sample resistance.
A typical measurement run consisted of measuring resistance as a function of
bias voltage at a fixed temperature, then precisely increasing the temperature (usually
by 1 or 2 K) using a Lakeshore cryogenic temperature controller and repeating the
resistance measurement until the temperature exceeded Tc. I then moved the stage
to push the tip further into the sample and repeated the measurements. Moving the
tip changed both the junction resistance and the effective barrier height, which was
useful for verifying that the apparent gap size ∆ did not change with increasing Z or
RJ .
74
3.3.2 Soft Point Contact Method
I made soft point contact junctions by first attaching 50 µm gold wires (the
V− and I− “bottom” electrodes) to the edges of a sample using DuPont 4929N silver
conductor paste diluted with paint solvent. Next, I painted a clear epoxy (Stycast
1266 two-part epoxy) over the surface of the crystal, leaving small openings on the
order of 0.1 mm in diameter. This was done to prevent the diluted silver paste used
for the top wire from wetting the surface so that the contact area between the tip and
sample will be as small as possible. Using silver paste in this way requires somewhat
delicate control. With this method, the electrically resistive epoxy allows one to use
enough paste to ensure that the wire stays in place without all of it coming in contact
with the sample
Once the epoxy set, I attached the remaining two electrodes (V+ and I+) by
making a sharp bend in a length of 25 µm gold wire and adhering it to the exposed
sample area with a dilute silver paste solution. The sample was then mounted on a
specialized puck for use with the 3He probe designed for a Quantum Design Physi-
cal Property Measurement System (PPMS). This particular setup allowed me to use
most of the measurement capabilities of the PPMS while also achieving the low tem-
peratures necessary to go below the transition temperatures of the half-Heuslers. Fig.
3.14 shows a soft point contact junction on a sample of BaFe1.9Pt0.1As2 before and
after the “top” electrodes were added.
I cooled the samples to temperatures as low as 400 mK with the PPMS 3He
option. As with the NA method, I measured conductivity spectra by sweeping a DC
75
Figure 3.14: Soft point contact junction on a BaFe1.9Pt0.1As2 sample be-
fore (above) and after (below) application of top electrodes. The sample
was cleaved and the bottom electrodes were soldered onto the edges of the
crystal. Epoxy was applied across the surface with a small hole left in the
center to ensure a small junction area. Finally, the top wires were applied
with silver paint to complete the junction.
76
bias voltage and using a lock-in amplifier to apply a small AC signal and measure the
response at several points. This process was repeated over a range of temperatures
and applied magnetic fields up to and above Tc and Hc2.
3.4 Point-Contact Spectroscopy: Data Collection and Analysis
Once a measurement was complete, I converted the current vs. voltage curves
to normalized conductivity vs. bias voltage. This required the removal of a nontriv-
ial “background” conductivity, which usually persists above Tc (but did not change
significantly when temperature was increased further). Since the exact Vbias values
differed for the dI/dV curves measured at different temperatures, I could not simply
subtract the high temperature curve from the low temperature curves. Instead, I fit
the background conductivity to a polynomial function. The low temperature conduc-
tivity values were divided by their corresponding points on the polynomial fit curve,
which removed the curvature resulting from the “background” and yielded normal-
ized conductivity data that approached a value of 1.0 away from zero bias. Once the
curves were properly normalized, I calculated the expected conductivity curves such
as those in Figures 3.6 through 3.9. I then manually varied the parameters to gener-
ate the best fit estimates for the gap size(s), barrier height, and inelastic scattering
energy, as well as an assessment of the pairing symmetry of the superconducting order
parameter.
77
Chapter 4: Multi-experimental Study of BaFe1.9Pt0.1As2 Gap Struc-
ture
Much of this chapter is based on the published paper by Ziemak et al., Super-
conductor Science and Technology 28, 014004 (2014).
4.1 Initial Characterization of Samples
Single crystals of BaFe1.9Pt0.1As2 were grown from prereacted FeAs and PtAs
powders and elemental Ba using the FeAs self-flux method [103]. This yielded crystals
with typical dimensions 0.1 × 1 × 2 mm3 (Ref. [14]). Single crystal x-ray diffraction
and Rietveld refinement determined the Fe:Pt ratio and the precise composition of
BaFe1.906(8)Pt0.094(8)As2. Previous x-ray measurements had shown that Pt substitu-
tion reduces the c-axis length and c/a ratio while increasing the a-axis length and
unit cell volume compared to pure BaFe2As2 (Ref. [14]). Characterization by energy-
and wavelength-dispersive x-ray spectroscopy verified the same substitutional con-
centrations across several specimens.
As shown in Fig. 4.1, resistivity measurements exhibited the expected metallic
behavior and a sharp transition to the superconducting state with an onset of Tc=23 K
and zero resistance achieved by 21.5 K for a transition width of ∆Tc < 1.5 K. No kink
78
was observable at higher temperatures, which would have indicated a magnetically
ordered phase, as seen in the parent compound and underdoped samples. [21, 104]
Figure 4.2 shows a series of resistivity vs. temperature curves for BaFe1.9Pt0.1As2
in a magnetic field of up to 14 T. As field strength is increased, the zero resistivity
transition broadens slightly and Tc is significantly decreased, but superconductivity
is not fully suppressed. The inset plots the extracted upper critical field as a function
of temperature. Magnetic field suppressed Tc to approximately 16 K at 14 T as
shown in Fig. 4.2. This is consistent with an upper critical field of approximately
45 T as determined by fits to the Werthamer-Helfand-Hohenberg approximation. [14]
However, Hc2(0) may be as high as 65 T based on a linear fit to Hc2(T ), which would
be similar to the value found in materials such as SrFe2−xNixAs2 (Ref. [105]).
DC magnetic susceptibitity was measured with field applied along the ab-plane,
under zero-field-cooled conditions and field cooled with a 1 mT applied field. As shown
in the inset in Fig. 4.1, a sharp transition is observed at Tc=23 K into the diamagnetic
state with a full-volume shielding fraction observed, as indicated by 4piχ = −1 by
17 K. Previous specific heat measurements [14] have confirmed the bulk nature of the
transition, with a jump in Cp(T ) that occurs slightly below Tc. The size of the jump
was estimated as ∆Cp/Tc ≈ 20 mJ mol−1K−1. Assuming the BCS weak-coupling
ratio of ∆Cp/γTc=1.43 applies, this yields γ = 16(2) mJ mol−1K−2. [14].
79
050
100
150
200
0 50 100 150 200 250 300
r
(m
W
cm
)
Temperature (K)
-1.2
-0.8
-0.4
0
0 10 20 30
4
p
c
ZFC
FC
BaFe Pt As1.9 0.1 2
Figure 4.1: Characteristic properties of the superconducting transition in
single-crystal samples of BaFe1.9Pt0.1As2 with resistivity (main panel) and
magnetic susceptibility (inset) measurements exhibiting sharp transition
features consistent with a transition temperature at Tc=23 K, as observed
in other bulk measurements. [14] Magnetic susceptibility measurements
performed in zero-field-cooled (ZFC) and field-cooled (FC) conditions with
a magnetic field of 1 mT indicate a 100% superconducting volume fraction
(inset).
80
Figure 4.2: Normalized in-plane resistivity of BaFe1.9Pt0.1As2 in magnetic
fields (0, 0.01, 0.02, 0.03, 0.04, 0.05, 0.1, 0.2, 0.5, 1, 2, 4, 6, 10, and 14 T in
order as indicated by the arrow) applied along the c-direction. Resistivity
values are normalized to the 25 K normal state value for clarity. The inset
shows the upper critical field Hc2 as a function of temperature based on
the 50% positions of the resistive transitions for each field value from the
main figure. (From Ref. [14]).
81
4.2 Point-Contact Spectroscopy
4.2.1 Measured dI/dV Spectra from Multiple Samples
I performed needle-anvil point-contact measurements on several single crystal
BaFe1.9Pt0.1As2 samples using the techniques described in Chapter 3. Electrical con-
tacts consisted of thin Cu wires that I soldered onto the platelet-shaped crystals or
attached to the sharpened Pb or Au tips with silver paste.
Conductance spectra were obtained on several crystals from the same synthesis
batch. It was necessary to measure multiple samples to ensure that features observed
in the spectra were repeatable and not irreproducible artifacts.
For each junction, the normal metal tip was slowly brought closer to the crystal
using a gear mechanism until a sharp decrease in resistance indicated that electrical
contact had been made between the tip and sample. Next, dI/dV spectra were
measured for bias voltages in the ±40-50 mV range to ensure that potential gap
features could be detected. The temperature was then increased, and a similar bias
voltage sweep was performed at 1-2 K intervals over temperatures ranging from 4 K
to roughly 30 K (well above Tc). After recooling to 4 K, the tip was pushed further
into the crystal to decrease the junction resistance and the process was repeated. In
each individual measurement run, I took data for at least three different tip positions
and three different junction resistances. This was necessary to ensure that features
in the spectrum could be observed consistently over a range of RJ values.
A sampling of my measured dI/dV curves are shown in Fig. 4.3, presented as
82
raw data without background normalization but with normalizion to their individual
high-bias values (i.e. at 40 meV for each curve) in order to account for differences
in junction resistance. Disregarding the slight differences in the shape of the curves,
which results from differing background or scattering (γ) contributions, each of the
dI/dV curves in Fig. 4.3 features a relatively sharp low-bias peak with a width of
approximately 2-3 meV, consistent with Andreev reflection in the superconducting
state. Furthermore, all curves exhibit depressions or enhancements in conductance
at higher energies (i.e. 5-6 meV). This overlap is most easily seen in the inset to
Fig. 4.3, which presents the same data without vertical offsets.
Figure 4.4 shows the temperature evolution of the PCS conductance for sample
S2, with dI/dV normalized to the normal state spectra to remove the background
contribution to conductance (c.f. Fig. 4.3). This was done by dividing out a polyno-
mial fit to the dI/dV data measured at 18 K from each temperature data set. Upon
cooling below Tc [106], the features described above emerge and evolve with decreasing
temperature to reveal a sharp conductance enhancement and a depression at higher
bias, suggesting features with at least two energy scales in the order parameter.
4.2.2 Fitting to BTK Model
To better understand the features observed in the conductance spectra, I at-
tempted fitting the Blonder-Tinkham-Klapwijk (BTK) model to my data. The BTK
model includes several parameters which I varied to fit my data. [40] As discussed pre-
viously, the I − V characteristics of a sufficiently small normal metal-superconductor
83
BaFe1.9Pt0.1As2
1
1.2
1.4
1.6
-40 -30 -20 -10 0 10 20 30 40
dI/d
V(
nor
ma
lize
d,
offs
et)
Vbias (mV)
S1
S2
S3
S4
S5
1
1.1
1.2
1.3
-10 -5 0 5 10
dI/d
V(
nor
ma
lize
d)
4 K
Needle-Anvil Method
Figure 4.3: Conductance curves obtained from point contact spectroscopy
measurements performed on five BaFe1.9Pt0.1As2 samples from the same
synthesis batch, offset vertically for clarity. Each has a strong zero bias
enhancement of similar width and either a depression or secondary en-
hancement at higher bias. These curves represent raw data without back-
ground subtraction, and each curve is normalized to their high bias values
but includes an offset. The inset presents the conductance curves for all
five samples plotted without an offset in order to show the overlap of the
gross gap features. The green curve (S2) corresponds to the data analyzed
in Figs. 4.4 and 4.5.
84
11.1
1.2
1.3
1.4
1.5
-20 -15 -10 -5 0 5 10 15 20
dI/d
V(
nor
ma
lize
d)
Vbias (mV)
18 K
16 K
14 K
12 K
10 K
8 K
6 K
4.2 K
BaFe1.9Pt0.1As2
Needle-Anvil
Method
Figure 4.4: Normalized dI/dV spectra over a range of temperatures for
single crystal BaFe1.9Pt0.1As2 with high temperature background conduc-
tance divided out. Curves are offset for clarity. Dashed curves serve as
a guide to the eye to indicate two possible superconducting gap features,
the low-bias conductance enhancement and higher bias wells, decrease as
expected up to Tc (see text for details).
85
(N-S) junction are determined by the gap size ∆, a unitless barrier strength Z, the
temperature T , and an inelastic scattering energy γ. The value of Z determines the
strength of tunneling across the junction: Z = 0 indicates complete Andreev reflec-
tion, while Z →∞ indicates the weak tunneling limit. In the low-Z case, the dI/dV
vs. Vbias curve contains a peak at zero bias extending out to bias voltages correspond-
ing to approximately +∆ and -∆, whereas for large Z, dI/dV is suppressed at zero
bias with enhancements at ±∆ due to quasiparticle tunneling. The γ term affects
the shape of the dI/dV curves and is critical in obtaining a good fit to measured
data. In the case of a two-gap BTK model, the fit parameters include two gap sizes,
∆1 and ∆2, two barrier strengths, Z1 and Z2, two scattering energies, γ1 and γ2, and
an additional weight factor w, which indicates the proportion of electrons incident
on the barrier that interact with either gap. The composite two-gap BTK curve is
simply a weighted average of two single-gap curves.
It should be noted that the fits I obtained were not true least-squares statistical
fits. Rather, the fitting process was done by manually adjusting the parameters
until the agreement appeared to be good to my eye. Due to the large number of
parameters involved, particularly for a two-gap fit, a proper least-squares fit would
be quite difficult to converge because of the time required to generate even a single
BTK curve.
Figure 4.5 shows the best fit that I obtained for the normalized dI/dV curved
labelled as sample S2 in Figure 4.3. The black points represent my data, while the
blue curve corresponds to the two-gap isotropic s-wave BTK fit. I made several
attempts to obtain a single-gap s-wave fit. While a single-gap model can be matched
86
relatively well to the central zero bias peak, in each case it failed to account for the
sharp dips closer to 5 meV. In contrast, the double gap fit was much more successful
at reproducing the measured data, matching both the wells and zero bias peak in
amplitude and energy scale.
I conclude, therefore, that the two-gap nodeless s-wave model more accurately
describes the conductance observed via PCS. The fit assumed the existence of a
nodeless gap with energy ∆1 = 2.5 meV, while the high bias depression was well
modeled by including a second gap of magnitude ∆2 = 7.0 meV. These features are
strikingly similar to those observed in PCS measurements performed on the two-gap
superconductor MgB2, which also exhibit peak-dip features that were understood in
terms of a two-gap scenario with similar gap energies (but of course a much higher
Tc of 40 K). [107]
4.2.3 Discussion of Fit Parameters
Interestingly, I found that the best fit values for Z and γ depended on the gap.
For instance, the parameter set for the smaller gap spectrum (∆1= 2.5 meV, Z1 =
0, γ1 = 1.0 meV) indicated complete Andreev reflection with little broadening. In
contrast, the larger gap spectrum (∆2 = 7.0 meV , Z2 = 7.0, γ2 = 7.0 meV) appeared
closer to the tunneling limit with a larger Z and much greater broadening. The weight
factor of w=0.6 indicated comparable amounts of Andreev reflection and tunneling
into each gap with more conduction for the smaller gap.
While it is not clear why such significant differences were observed in the mod-
87
0.95
1
1.05
1.1
1.15
1.2
-15 -10 -5 0 5 10 15
Two-gap fit
Measured dI/dV
dI/d
V(
nor
ma
lize
d)
Vbias (mV)
BaFe1.9Pt0.1As2
4 K
Needle-Anvil Method
Figure 4.5: Point contact conductance spectrum of BaFe1.9Pt0.1As2 sample
S2, at 4.2 K, normalized to remove the normal state background conduc-
tance (black points). The blue curve shows the fit to an isotropic s-wave
two-gap Blonder-Tinkham-Klapwijk model, yielding ∆1= 2.5 meV, Z1 =
0, γ1 = 1.0 meV for the smaller gap and ∆2 = 7.0 meV , Z2 = 7.0, γ2 =
7.0 meV for the larger gap, with a weighting factor of w=0.6.
88
eled barrier heights and scattering strengths of the two gaps, such differences are not
unprecedented. For example, a two-gap model whose larger gap also had a larger Z
was observed in Ba1−xKxFe2As2 [42]. The unusually large scattering strength deduced
from the fit to BTK for the larger gap, with γ2 ' ∆2, is also unusual. However, a
high quasiparticle scattering rate has also been deduced from pair-breaking experi-
ments. [49,108] For example, Kirshenbaum et al. investigated pair-breaking scattering
in a range of 122 iron pnictide materials with transition metal substitutions at the Fe
site. [108] From their observations one can estimate the transport scattering rate of
Γ ' 2.5 × 1013 s−1 for BaFe1.9Pt0.1As2 based on a Tc value of 23 K. This translates
to an energy scale of h¯Γ ≈ 16 meV, which is actually higher but of the same order of
magnitude as γ2, suggesting strong scattering in the FeSC materials [108].
Alternatively, the contrast between Z1 and Z2 may be due to a relatively large
“impedance mismatch” between the tip and the large-gap band than for the small-gap
band. Thermal conductivity measurements on LiFeAs [49] also indicate the possibility
of a mismatch in Fermi velocities, so it is possible that my observation of this large
contrast in Z values are valid.
Finally, the appearance of spectral features at energy scales much larger than
the predicted weak-coupling BCS gap value for a 23 K superconductor, ∆BCS =
1.76kBTc = 3.5 meV, is puzzling but also not unprecedented in tunneling and PCS
studies. For normal metal-superconductor junctions outside the ballistic limit, it has
been shown that sharp dips can occur at higher bias voltage. [109] A conservative
estimate gives an electron mean free path on the order of nanometers, [110] which
would place my micron-size junctions (assuming a perfect and full contact) in the
89
diffusive rather than ballistic regime, and outside the Sharvin limit. However, as
evidenced in Fig. 4.3 my spectra exhibit features at higher bias voltages, including
both gap-like dips and Andreev-like shoulders.
I note that many other PCS experiments have observed dominant spectral
features in a lower-energy range consistent with our prominent features near ∆1.
For example, BaFe1.86Co0.14As2 [42] and several 122’s exhibit 2∆/kBTc ' 3.1 [111].
Data from the latter study are presented in Fig. 4.6 for multiple concentrations of
Ba1−xKxFe2As2. These dI/dV curves were fitted to a single gap BTK fit, which
provides a useful contrast to the two-gap spectra observed for BaFe1.9Pt0.1As2. In
the case of LiFeAs, observations of a small gap of 1.6 meV (2∆/kBTc = 2.2) have
been reported from PCS measurements using Pb and Au tips [112]. They do not
observe features associated with any larger gap, such as seen in ARPES experi-
ments. [113, 114] However, other studies have revealed prominent two-gap features,
such as in Ba0.55K0.45Fe2As2 [42,43] and BaFe2−xNixAs2 up to x = 0.10 [44]. Further-
more, a wide range of scanning tunneling spectroscopy experiments have observed
superconducting gap magnitudes indicative of strong coupling, with 2∆/kBTc ratios
far above the BCS weak-coupling expectation of 3.5 [115]. For example, optimally
doped BaFe2−xCoxAs2 shows a superconducting tunneling gap with coherence peaks
corresponding to a much larger (average) single-gap value of ∆ = 6.25 meV, corre-
sponding to 2∆/kBTc = 5.73 [116].
90
Figure 4.6: Needle-anvil point-contact spectra for Ba0.51K0.49Fe2As2 (top
left) and Ba0.71K0.29Fe2As2 (top right) showing normalized dI/dV spectra
overlaid with BTK fits. These single-gap spectra are clearly different in
form than the two-gap spectra I observed for BaFe1.9Pt0.1As2. The lower
figure shows the normalized gap extracted from BTK fits plotted against
temperature (bottom left) and ∆ vs. Tc for four different compounds (bot-
tom right) demonstrating 2∆/kBTc ' 3.1 for these iron pnictide materials.
From Ref. [111]
91
4.3 Other Thermodynamic and Spectroscopic Measurements
4.3.1 Thermal Conductivity
All thermal conductivity data I discuss in this section was measured at Univer-
site´ de Sherbrooke in Quebec, Canada by Kevin Kirshenbaum and the group of Dr.
Louis Taillefer.
Thermal conductivity was measured using a one-heater, two-thermometer steady
state technique, as a function of temperature, to 60 mK in a dilution refrigerator.
Temperature sweeps were repeated for fixed magnetic fields from 0 T to 15 T, in
both parallel (H ‖ ab) and perpendicular (H ‖ c) orientations with respect to the
crystallographic basal plane. Measured κ/T vs. T are shown in Fig. 4.6, along with
fits to κ/T = a+ bTα−1. Extrapolating to the T → 0 limit, one obtaines the residual
electronic term a = κ0/T .
The residual electronic contribution a = κ0/T clearly approaches a value of zero
as T → 0, indicating no zero energy quasiparticles in the superconducting state. The
fitted value of κ0/T is actually slightly negative, but is within experimental error of
zero. For comparison, in the T → 0 limit one can calculate the electronic contribution
to thermal conductivity from the Wiedemann-Franz law, κ0/T = L0/ρ0, where L0 =
(pi2/3)(kB/e)2=2.44×10−8 W Ω K−2. Using a fictitious residual zero-temperature
resistivity value of ρ0=125 µΩ cm based on extrapolating the zero-field data (c.f.
Fig. 4.1) yields an approximate value of L0/ρ0=0.195 mW K−2 cm−1, shown in Fig. 4.7
as blue diamonds on the T=0 axis. As evident in Fig. 4.6, this estimated normal state
92
00.5
1
1.5
k 0
/T
(m
W
/K
2 c
m
)
0
0.1
0.2
0.3
0.4
0.5
0 0.1 0.2 0.3
0
0.5
1
1.5
0 0.2 0.4 0.6 0.8 1
k 0
/T
(m
W
/K
2 c
m
)
Temperature (K)
0
0.1
0.2
0.3
0.4
0.5
0 0.1 0.2 0.3
b) H || c-axis
a) H || ab-plane
0 T
15 T
L /r (0 T)0 0
0 T
15 T
L /r (0 T)0 0
BaFe Pt As1.9 0.1 2
Figure 4.7: (a) Low temperature thermal conductivity of a crystal of
BaFe1.9Pt0.1As2 measured in zero (black, closed symbols) and 15 T (red,
open symbols) applied magnetic fields, oriented along the crystallographic
ab basal plane (circles) and (b) the c-axis (squares). Zero field measure-
ments were repeated in each orientation using the same crystal for direct
comparison. Insets show a detailed view of the low-temperature data,
with solid lines representing power law fits to κ/T = a+ bTα−1 below 150
mK. Blue diamonds in all panels represent the normal state electronic
thermal conductivity estimated using the Wiedemann-Franz law.
93
limit is many orders of magnitude larger than the maximum possible extrapolated
value of κ0/T allowed by error, suggesting a fully gapped superconducting order
parameter with no nodes or deep minima. This is to be contrasted with the finite κ0/T
expected for superconductors with either symmetry-imposed or accidental nodes in
the gap, such as observed in the d-wave superconductor Tl2Ba2CuO6+δ (Tl-2201) [46]
or in c-axis transport measurements of BaFe2−xCoxAs2 away from optimal doping [47],
respectively.
The absence of low-energy excitations is also evident in magnetic field measure-
ments up to 15 T, or approximately 25% of Hc0. Figure 4.7(a) and (b) show κ/T vs.
T for both H ‖ ab and H ‖ c orientations, respectively. Figure 4.8 shows κ0/T as a
function of field for several materials. Single-gap isotropic s-wave superconductors at
T = 0 such as Nb [117] do not possess quasiparticles until the magnetic field becomes
so large that vortex core bound states begin to delocalize. This process proceeds
exponentially with the ratio of intervortex spacing to coherence length (or essentially
the magnitude of magnetic field) up to the uppper critical field. Multiband s-wave
superconductors such as NbSe2 also have κ0/T = 0 in zero field due to lack of nodes
in the gap structure. However, vortex core delocalization can be accelerated in a
multi-band superconductor due to a reduced gap magnitude on one or more bands.
This can produce an onset of low-energy excitations and a finite and increasing value
of κ0/T at fields much smaller than Hc2. [50].
Figure 4.8 (a) shows the evolution of κ0/T with magnetic field in BaFe1.9Pt0.1As2
and other superconductors. Figure 4.8 (b) shows the behavior observed in the FeSC
family. The lack of absence of a residual term in BaFe1.9Pt0.1As2 is clear: the evolu-
94
00.2
0.4
0.6
0.8
1
k 0
/k N
Tl-2201
NbSe2
Nb
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
k 0
/k N
H/Hc2
KFe2As2
LiFeAs
H || ab
H || c
a)
b)
H || ab
H || c
Ba0.6K0.4Fe2As2
BaFe1.85Co0.15As2 (H || c)
(H || a) BaFe1.85Co0.15As2
BaFe Pt As1.9 0.1 2
BaFe Pt As1.9 0.1 2
Figure 4.8: (a) Residual thermal conductivity κ0/T of BaFe1.9Pt0.1As2
extracted from temperature sweep measurements and normalized to the
normal state conductivity κN/T , plotted as a function of reduced mag-
netic field and compared to several characteristic superconductors. The
lack of increase in κ0/κN in BaFe1.9Pt0.1As2 is comparable to the behav-
ior of isotropic, single-gapped s-wave superconductors such as elemen-
tal Nb [117] that exhibit an exponentially slow activated increase with
field, and is in contrast to the cases for multi-band s-wave superconductor
NbSe2 [50] and d-wave nodal superconductor Tl2Ba2CuO6+δ [46], which
exhibit a much faster rise with field due to nodal or small-gap low energy
quasiparticle excitations. (b) Comparison of κ0/T in BaFe1.9Pt0.1As2 to
that in a range of iron-based superconductors including KFe2As2 [118],
Ba0.6K0.4Fe2As2 [118], BaFe1.85Co0.15As2 [47], and LiFeAs [49].
95
tion in BaFe1.9Pt0.1As2 is comparable to that of elemental Nb [117]. This is in stark
contrast to that of the d-wave and multiband s-wave superconductors Tl-2201 and
NbSe2, respectively. Within the FeSC family, such a flat response is not unprece-
dented. In fact it it what one expects for a fully isotropic s-wave gap, as deduced for
both Ba0.6K0.4Fe2As2 [118] and LiFeAs [49].
In a multi-band scenario, which is the case for BaFe1.9Pt0.1As2 as measured by
ARPES (see Sect. 4.3.2), the suppression of the maximum energy gap on portions
of the band structure can lead to an enhancement of tunneling between vortex core
bound states, resulting in an effectively reduced critical field for those portions. In
the case of NbSe2, this field scale is identified with a shoulder in κ0/κN(H) near
H∗ ∼ Hc2/9. This is consistent with an energy gap ratio ∆min/∆max ' 1/3 since
Hc2 ∝ ∆2 [50]. In contrast, in BaFe1.9Pt0.1As2 the lack of a rise in κ0/κN(H) up to
Hc2/4 suggests little if any anisotropy in the gap. However, as with LiFeAs [49], the
lack of a rise in κ0/κN(H) does not rule out the existence of a smaller gap in the
band structure: it may be that the conductivity contribution of the small-gap Fermi
surface is much smaller that that of the large-gap component, or if coherence lengths
are comparable due to scaling of Fermi velocities (i.e. since ξ0 ∼ vF/∆).
LiFeAs is relevant to BaFe1.9Pt0.1As2 because recent ARPES experiments on
LiFeAs show two gap magnitudes of approximately 2-3 meV and 5-6 meV [113, 114,
119]. The observed anisotropies are weak, with gap minima not reaching below
∼2 meV. BaFe1.9Pt0.1As2 may be similar, or have similar coherence lengths across
bands, making it difficult to determine the gap anisotropy via thermal conductivity.
96
4.3.2 Angle-Resolved Photoemission Spectroscopy
ARPES allows the measurement of electronic structure of materials and charac-
terization of the order parameter of the superconductor. If suitably clean surfaces can
be prepared, ARPES can be particularly useful for identifying and characterizing the
momentum-resolved energy scales of the gap function ∆(k) and elucidating the gap
structure of multi-band superconductors. The ARPES data I discuss in this section
was measured at the Leibniz Institute for Solid State and Materials Research (IFW)
in Dredsen, Germany by the group of Dr. Sergey Borisenko.
ARPES was performed on clean BaFe1.9Pt0.1As2 sample surfaces, which were
cleaved in high vacuum at low temperatures, exposing mirror-like surfaces. Figure
4.9(a) shows the resulting Fermi surface map of BaFe1.9Pt0.1As2. There is a large
electron pocket at the X point and a smaller hole pocket at the Γ point. This is
similar to other electron-doped 122 FeSCs [120]. Energy-momentum cuts presented
in Figs. 4.9(b) and (c) show occupied bands above and below Tc. Figures 4.9(d)-(g)
show the corresponding energy distribution curves (EDCs). The EDC curves show
a clear peak, consistent with a gap opening below approximately 17 K in both the
electron and hole pockets. For example, upon cooling below Tc a sharp coherence
peak and gap at the Fermi energy are clearly seen in the EDC curve in Fig. 4.9(d).
The width of these features corresponds to approximately a 3 meV gap, consistent
with the small gap size ∆1 that I extracted from fits of the point contact spectroscopy
data (see Sect. 4.2).
Integrated energy distribution curves (IEDCs) were also recorded at different
97
Electron pocket
Kinetic energy (eV)
22.64 22.68 22.72
12
10
8
6
4
2
In
te
ns
ity
 (
ar
b.
 u
.)
4
3
2
1
Γ pocket
22.68 22.72
In
te
ns
ity
 (
ar
b.
 u
.)
10
8
6
4
2
In
te
ns
ity
 (
ar
b.
 u
.)
22.70 22.72
Momentum
M
om
en
tu
m
110
cut 1
2
3
4
5
6
7
8
9
10
Γ
X
Momentum
M
om
en
tu
m
Momentum
Momentum
K
in
. e
n.
 (
eV
)
22.66
22.70
K
in
. e
n.
 (
eV
)
22.66
22.70 single EDC
 1K
 30K
Kinetic energy (eV)
22.60 22.70 22.80
12
8
4I
nt
en
si
ty
 (
ar
b.
 u
.)
(a) (b)
(c)
(d)
(e) (f) (g)
Kinetic energy (eV) Kinetic energy (eV)
BaFe Pt As1.9 0.1 2
Figure 4.9: ARPES measurements of BaFe1.9Pt0.1As2. (a) Fermi surface
map, revealing large electron-like pockets around Brillouin zone corner
and small hole-like pockets at center. (b) Energy-momentum cut, passing
through the electron pocket recorded below Tc. (c) Same cut, recorded
above Tc. (d) Single energy distribution curve (EDC) recorded above and
below Tc shows appearance of a sharp peak in the superconducting state.
(e) Temperature dependence of the integrated EDC (IEDC) recorded from
electron pocket for heating up to 30 K and cooling down to 1 K. (f)
Temperature dependence of the IEDC for hole pocket. (g) IEDC recorded
from different parts of the electron pocket. Positions of cuts are indicated
in the mini-map in inset.
98
positions along the electron pocket, as shown in Fig. 4.9(g). Each IEDC has a peak
of approximately the same width, confirming that the observed electron pocket gap
is isotropic. This behavior is similar to several other 122 FeSCs including optimally
doped Ba1−xKxFe2PtAs2 [121], BaFe2−xCoxAs2 [51] and LiFeAs [119], which all show
multiple gap sizes with little or no anisotropy.
4.3.3 Raman Spectroscopy
All Raman spectroscopy data for this work was obtained by the group of Dr.
Girsh Blumberg at Rutgers University.
Raman spectra were obtained using a Kr+ laser with 647 and 476 nm wave-
lengths 1.2 to 12 mW of incident laser power was focused to a spot of 50×100 µm2
on the freshly cleaved ab-plane crystal surface. The backscattered light was collected
close to the backscattered geometry was focused onto 100×240 µm2 entrance slits of
a custom triple-stage spectrometer equipped with 1800 lines/mm gratings. To obtain
symmetry-resolved Raman spectra of BaFe1.9Pt0.1As2, the Rutgers group employed
both linearly- and circularly-polarized light. [53] Data were collected from 14 total
samples at 3 and 30 K. The estimated local heating in the laser spot did not exceed
4 K for laser power less than 2 mW.
Figure 4.10 shows examples of the raw Raman data, dσ(ω)/dω, normalized
to the power and laser frequency in the B2g channel. The plot illustrates the im-
portance of measuring the normal state response and highlights the contribution
of the major superconducting phase to the Raman response. I also note that the
99
residual flux gives rise to a related, large background of laser-induced luminescence.
The magnitude of this background luminescence (bottom arrow) sets the minimum
polarization contribution for each measured polarization. This background contri-
bution is not related to Raman scattering and must be subracted before dσ(ω)/dω
can be converted into the Raman response. The final spectra are calculated as χ′′ =
(dσ(ω)/dω−BG)(n(ω, T )+1), where n(ω) is the Bose factor. The resulting response
is further decomposed into contributions from the major superconducting phase and
impurity phase(s). These potential impurity phases may not be superconducting or
may simply have a much lower Tc. Based on the flattened section of the experimental
data (from the cutoff near 10 cm−1 to roughly 25 cm−1) at 3 K and general flattened
response at 30 K, we assumed that contributions from any impurity phase(s) were
well approximated by a constant, much like the luminescence term. To summarize,
the total background (BG) gapping B2g was removed and dσ(ω)/dω is converted to
the Raman response, χ′′major = (dσ(ω)/dω −BG)(n(ω, T ) + 1).
The normal state response (red curve in Fig. 410) was essentially flat down
to the cutoff value of the spectrometer. The superconducting response (blue curve)
exhibited a broad peak around 80-90 cm−1. This feature was clearly seen in previous
Raman studies [53, 122, 123] as well as a threshold near 45 cm−1, marked by a black
dashed line in Fig. 4.10. The response flattens out below approximately 25 cm−1.
The flattened background (uppermost black dashed line) is likely a result of sample
impurity content. All 14 samples showed non-vanishing backgrounds with amplitude
depending on spot position within the cleave and laser excitation wavelength.
Here we focus on properties of the major superconducting phase and attempt
100
00.2
0.4
0.6
0.8
1
1.2
0 50 100 150 200 250 300
 d
σ
(ω
)/
d
ω
 (
a
rb
it
ra
ry
 u
n
it
s
)
Raman shift (cm  )
-1
2Δ
BG gapping B
2g
Luminescence
Raman from impurity phase(s)
30 K
3 K
BaFe Pt As1.9 0.1 2
Figure 4.10: Raw measured data, containing contributions from both Ra-
man and luminescence channels, for sample S3, obtained with laser exci-
tation of 467 nm. Blue diamonds represent data at 3 K, while red circles
represent 30 K data. The superconducting gap is indicated by the vertical
dashed line labeled 2∆. Note that the gray area indicates energies below
the cutoff range of the spectrometer.
101
to remove the contributions from impurity phases. The measured electronic Raman
response χ′′(ω) in the B2g and A1g channels is shown in Figs. 4.11(a) and (b), re-
spectively. The data in the SC state exhibit a threshold around 45-50 cm−1 (vertical
black dashed line, labelled as 2∆), clearly seen in both B2g and A1g polarizations.
This is the gap, also seen in ARPES and point-contact spectroscopy. The value
∆ = 3± 0.3 meV was confirmed for multiple BaFe1.9Pt0.1As2 samples.
4.4 Discussion
To understand all of these results it is instructive to review the conclusions
drawn from similar experiments on Co-substituted BaFe2As2. Results from multiple
studies on optimally doped BaFe2−xCoxAs2 tend to confirm a similar nodeless gap
structure. Comprehensive analysis of the phase diagram by Reid et al. [47] looked at
the evolution of the gap structure with increasing Co concentration based on thermal
conductivity. Based on the presence or absence of residual thermal conductivity
κ0/T and its evolution in field, it was argued that line nodes are present in under-
and overdoped BaFe2−xCoxAs2 but that optimally doped x = 0.148 samples were
nodeless.
Daghero et al. measured several PCS spectra for overdoped samples with x =
0.2 [41]. Current was injected in the ab-plane and parallel to the c-direction using
the soft point-contact method. Fitting to BTK theory showed better agreement for
a two-gap isotropic s-wave model than for a single nodeless gap. The estimated gap
values were ∆1 = 3.8-5.2 meV and ∆2 = 8.2-10.9 meV. The authors also reported
102
00.2
0.4
0.6
0.8
1
0 50 100 150 200
χʺ
 -
 B
G
 (
a
rb
itr
a
ry
 u
n
its
)
Raman shift (cm  )-1
2Δ
(b)
A
1g
0
0.2
0.4
0.6
0 50 100 150 200 250 300
χʺ
 -
 B
G
 (
a
rb
itr
a
ry
 u
n
its
)
Raman shift (cm  )-1
B
2g
2Δ
(a)
30 K
3 K
30 K
3 K
Figure 4.11: Electronic Raman response (background removed) of sample
S3, obtained with 476 nm laser excitation in (a) B2g and (b) A1g channels.
Blue (red) dots show superconducting (normal state) response averaged
over four (two) experimental scans. Both thick lines represent 5-point
smoothed data. Black dashed lines mark the energies corresponding to
the superconducting gap observed, 2∆. The gray area again indicates
areas below the cutoff energy of the spectrometer.
103
features in the dI/dV curves at higher bias voltage that deviated from the BTK fit
which they attributed to strong electron-boson coupling.
PCS measurements were also performed by Samuely et al. with sharpened Pt
tips pressed into freshly cleaved optimally doped (x = 0.14) crystals. [42] BTK fits
to the measured dI/dV curves showed only a single isotropic gap of magnitude ∆
= 5-6 meV, suggesting that if multiple gaps were present they were close together
in magnitude. Given the significant differences in gap structure observed through
thermal conductivity across the BaFe2−xCoxAs2 phase diagram, it is possible that
the difference in these results may be due to different Co concentrations.
Terashima et al. examined the gap structure of x = 0.15 samples using ARPES
and found strong evidence for two isotropic gaps. [51] Energy distribution curves
showed two gaps with ∆1 = 5 ± 1 meV on the electron pocket and ∆2 = 7 ± 1 meV
on the hole pocket. EDCs measured over a range of angles shows that both gaps are
isotropic within error.
Muschler et al. performed Raman spectroscopy measurements on BaFe2−xCoxAs2
samples with optimal doping (x = 0.122) and slight overdoping (x = 0.17). [53] Based
on the strong low-temperature shift in the B2g spectrum which varies as
√
Ω the au-
thors propose an s-wave state with accidental nodes in the optimally doped sample.
Slight overdoping resulted in a B2g peak with greatly diminished amplitude suggest-
ing that the gap is strongly effected by doping and disorder and potentially sample
quality.
The most commonly agreed upon conclusion regarding the gap structure of
BaFe2−xCoxAs2 seems to be that there is an s-wave gap which is has no nodes at
104
optimal doping, but nodes appear at higher and lower Co concentrations. The nodes
do not appear to be imposed by symmetry and are present away from the ab-plane.
However, some results appear to contradict each other. For example, PCS studies
have consistently supported an isotropic gap while Raman spectra at multiple dopings
suggest the presence of nodes. Some have suggested reconstruction of the gap at the
surface as a possible explanation for the difference, but this alone does not reconcile
disagreements between thermal conductivity, PCS, and Raman spectroscopy, all of
which are bulk probes. Another possible explanation is differences in crystal quality,
and thus impurity scattering, as has been proposed by Muschler and others.
Compared to the Co-substituted system, our results on BaFe1.9Pt0.1As2 offer
a more comprehensive conclusion of an isotropic gap structure. All of the samples
were made in the same batch and therefore are expected to have similar impurity
concentrations and defect structures. Furthermore, our results all support the con-
clusion of at least one gap with a consistent magnitude of approximately 3 meV, and
all experiments point to an isotropic s-wave gap symmetry.
Comparing the Co- and Pt-doping results, both optimally doped materials lack a
residual thermal conductivity at zero field, indicating a fully-gapped superconducting
order parameter. For the Co compound, a nonzero κ0/T emerges at small magnetic
fields, while for the Pt compound it remains zero at all fields observed. While the
thermal conductivity of each material could only be observed up to a small fraction
of Hc2, this may suggest subtle differences in gap morphology. BTK fits to PCS
measurements on both yielded good fits to nodeless s-wave models. Results from our
study with optimal Pt doping seem to compare more directly with those of overdoped
105
BaFe2−xCoxAs2 rather than optimally doped. Both point to a two-gap structure with
features present in the spectrum that do not perfectly match the BTK fit. This may
indicate strong electron-boson coupling in both materials. ARPES measurements
on optimally doped samples in both systems showed no variation in gap magnitude
as a function of angle. [51] In the Co-doped compound, two gaps were consistently
seen, while only one was observed in our study of BaFe1.9Pt0.1As2. Finally, Raman
spectra on both compounds feature low temperature enhancements in the A1g and
B2g channels at positions which agreed with other reported values of gap size.
While we found no evidence of nodes for optimally doped BaFe1.9Pt0.1As2,
under- and overdoped compounds in the BaFe2−xPtxAs2 system have yet to be ex-
plored as far as gap structure is concerned. It has yet to be seen whether similar nodes
would appear or if the Pt-substituted system would exhibit significant differences from
the Co-substituted system. Future investigation of the entire BaFe2−xPtxAs2 phase
diagram is certainly warranted.
Our results on BaFe1.9Pt0.1As2 can also be used to draw interesting parallels
with the multiband superconductor LiFeAs. Both compounds present no evidence
of quasiparticle excitations in thermal conductivity measurements at low magnetic
fields, and ARPES shows minimal gap anisotropy. Both also exhibit comparable
small and large gap magnitudes as extracted from PCS measurements and ARPES
measurements (in the case of LiFeAs), with 2∆/kBTc ' 2 for the smaller gap in each
system. The multi-gap nature of both materials would seem to be in disagreement
with the thermal conductivity observations, but can be reconciled if one assumes
the coherence length is similar between the two gaps. In our PCS observations of
106
BaFe1.9Pt0.1As2, the large discontinuity between Z values for the two gaps is consistent
with large differences between Fermi velocities, which means that vF/∆ could be
comparable for the two gaps as suggested for LiFeAs. [49]
4.5 Conclusions
In conclusion, I examined results from four measurement techniques that probed
the superconducting gap in single-crystal samples of BaFe1.9Pt0.1As2, finding reliable
evidence for an isotropic one- or possibly two-gap s-wave model. Thermal conduc-
tivity measurements showed no sign of low energy quasiparticle excitations even at
relatively high magnetic fields, with behavior comparable to other isotropic, single-
gapped s-wave superconductors. While there was no indication of a reduced energy
gap, the presence of a second gap cannot be completely excluded because of the small
relative field range that was studied.
The conductivity spectra that I measured by point-contact Andreev reflection
spectroscopy exhibit sharp enhancements and notable suppression of dI/dV at lower
and higher bias, respectively, which suggests the presence of two gaps. Fitting to
an isotropic two-gap Blonder-Tinkham-Klapwijk model results in gap size estimates
of 2.5 meV and 7.0 meV, corresponding to features in the spectra that have been
replicated in several crystals from the same batch.
Angle-resolved photoemission spectroscopy measurements show an isotropic gap
on both electron and hole bands with a magnitude of approximately 3 meV. Finally,
Raman spectroscopy revealed excitations in the superconducting state in both A1g
107
and B2g channels whose energy scales correspond with the 3 meV gap magnitude
observed in other measurements.
Overall, I can conclude that the data on the optimally doped iron-based super-
conductor BaFe1.9Pt0.1As2 support an isotropic, fully gapped superconducting order
parameter with no nodes or deep minima, and possibly with band-dependent energy
scales of order 3 meV and 7 meV. Combining results from different experimental
probes allowed me to rule out several extrinsic parameters, allowing for better eluci-
dation of the pairing mechanism of this iron-based superconductor.
108
Chapter 5: Point-Contact Spectroscopy in Half Heusler Compounds
5.1 Initial Characterization of Superconducting Materials
Single crystal samples of several half Heusler materials including YPtBi and
LuPdBi have been synthesized using the Bi flux method described in Chapter 3. A
detailed characterization of YPtBi was first described by Butch et al. (Ref. [81]) and
revealed a lattice parameter of 6.6522(10) A˚ and the expected F 4¯3m space group.
Figure 5.1 shows my measurement of resistivity as a function of temperature.
Note that as YPtBi is cooled below 300 K, the resistivity first increases as expected
for a band insulator. Below approximately 130 K however, ρ(T ) decreases before an
abrupt zero resistance transition ending at Tc0 = 0.7 K. This behavior is consistent
with a semimetallic normal state. It should be noted that resistivity in this sample
experiences a maximum at 133 K, while ρ(1 K) is nearly equal to ρ(300 K). This
contrasts with the samples measured by Butch et al., where resistivity leveled off below
approximately 100 K rather than decreasing, and as a result ρ(1 K) was approximately
twice ρ(300 K). The reason for this difference is not immediately clear, but it may be
related to differences in crystal quality that affected the transition.
I also measured the low-temperature resistivity as a function of magnetic field,
as shown in Fig. 5.2 for the same sample at 0.4 K. The observed R vs. H behavior
109
00.1
0.2
0.3
0.4
0.5
0.6
0.7
0 50 100 150 200 250 300

(m

cm
)
Temperature (K)
0
0.1
0.2
0.3
0.4
0.5
0.6
0 0.5 1 1.5 2
YPtBi
Sample R5
Figure 5.1: Resistivity vs. temperature between 0.4 K and 300 K showing
semimetallic behavior. The inset shows a close-up of the data below 2 K
and reveal Tc ' 0.7 K.
110
00.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 2 4 6 8 10 12 14

(m

cm
)
Field (T)
YPtBi
Sample R5
Figure 5.2: Resistivity ρ vs. magnetic field H in YPtBi at 0.4 K
Zero resistivity is only observed below 0.2 T, but based on the midpoint
of the resistive transition, the true Hc2 is closer to 1 T.
indicates a critical field of approximately Hc2 = 1 T, although the resistivity continues
to increase above the fairly broad transition.
5.2 Soft Point-Contact Measurements on YPtBi
5.2.1 Preparation of Junctions and the 3He Probe
I measured point-contact spectra for these YPtBi samples using the soft point
contact method with silver paste as the normal metal contact (see Chapter 3). Only
111
the soft point contact method was feasible because the samples had to low of a Tc to
be accessible in our needle-anvil probe.
I prepared junctions by applying two “bottom” wires to the edges of a YPtBi
sample that had sanded to reveal a large, flat face. I then coated the surface of the
crystal with a clear epoxy, leaving a small contact area for the counter-electrode. This
counter-electrode “tip” was applied by adhering a bent wire to the exposed area of
the crystal with diluted silver paste. Fig. 5.3 shows a typical junction created in this
fashion. In this case I left with three holes in the epoxy layer so that several junctions
could be made from the same sample.
The four electrodes were attached to a measurement puck to prepare them for
insertion into a PPMS using the specialized 3He probe. Compared to the needle-anvil
probe, this system offers a few advantages. In addition to a lower base temperature
and faster cooling, it also allows for the application of magnetic fields of up to 14 T,
which enables a wider range of data to be measured.
5.2.2 Measured dI/dV Curves
Samples were cooled to temperatures as low as 400 mK with the PPMS 3He
option. Conductivity spectra were measured by slowly sweeping the DC bias voltage
Vbias while using a lock-in amplifier to apply a small AC current signal I and measure
the response V . This process was repeated over a range of temperatures and applied
magnetic fields up to and above Tc and Hc2, respectively.
I − V curves were converted to dI/dV vs. Vbias and were divided by their high
112
Figure 5.3: YPtBi single crystal sample R5 used for soft point contact
measurements as shown in Fig. 3.11. Gold wires on far left and right rep-
resent the V− and I− ”bottom” electrodes. Clear epoxy has been painted
over most of the surface except for three spots, each approximately 100 µm
in diameter. The uppermostmost spot is covered with silver paste, as it
was used in a previous measurement for which the tip wire has since been
removed. The leftmost spot contains an intact junction with a thinner
gold wire used as the V+ and I+ “top” electrodes. The rightmost spot
shows the still-exposed crystal surface through the slightly opaque epoxy.
113
bias values in order to normalize them. Fig. 5.4 shows dI/dV data taken in zero
applied field over a range of temperatures, while Fig. 5.5 shows data over a range of
applied magnetic fields at a constant temperature of 0.4 K. For all samples I observed
a relatively flat background conductance on the order of 0.1-1 Ω in the normal state.
The sample denoted as R5 used for point-contact measurements in Figs. 5.4
and 5.5 is the same that was used for the R vs. T and R vs. H measurements in
Figs. 5.1 and 5.2, but not during the same measurement run. After the resistivity
measurements I removed the sample from the cryostat, cleaned it in acetone to remove
the silver paste from the resistivity contacts, and prepared the soft point contact
junction as described previously. Although these data sets were not taken in situ, the
sample was exposed to air for a short period of time (less than an hour) and showed
no obvious signs of oxidation.
5.2.3 Comparison to Previous Studies
The most obvious feature of the dI/dV spectra shown in Figs. 5.4 and 5.5 is the
strong enhancement in conductivity at zero bias. This zero bias conductance peak
(ZBCP) is characteristic of point-contact spectra in the Andreev reflection regime.
The ZBCP diminishes in height and disappears above both Tc and Hc2, which is a
strong indication that this feature represents a superconducting effect rather than
some type of anomalous scattering. A less obvious aspect of the data is that dI/dV
increases monotonically up to Vbias = 0. Typically for soft point-contact spectra one
observes a “split peak” morphology, which is characterized by a small dip in the
114
0.9
1
1.1
1.2
1.3
1.4
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
0.4 K
0.5 K
0.7 K
0.85 K
1.0 K
1.4 K
1.8 K
dI/
dV
(n
or
ma
liz
ed
)
Vbias
YPtBi
Sample R5
Soft Point Contact
Method
Figure 5.4: Normalized dI/dV vs. Vbias for sample R5 for selected temper-
atures. An enhancement of roughly 40% is observed at zero bias compared
to the high bias dI/dV value. The peak is not split, as would be expected
for a conventional superconductor. This peak persisted above 1.0 K, which
is unusual considering the resistive transition temperature of Tc = 0.7 K.
115
0.8
0.9
1
1.1
1.2
1.3
1.4
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
0 T
0.1T
0.2T
0.5T
1.0T
2.0T
5.0T
dI/
dV
(n
or
ma
liz
ed
)
Vbias
YPtBi
Sample R5
Soft Point Contact
Method
Figure 5.5: Normalized dI/dV vs. Vbias for selected applied magnetic fields
measured at a temperature of 0.4 K. The zero bias peak decreases in
magnitude before disappearing at roughly 2 T. This is also unexpected, as
the upper critical field appears to be closer to 1 T as shown in Fig. 5.2. The
considerable amount of noise in the 5.0 T spectrum and part of the 2.0 T
spectrum is likely due to deterioration of the junction due to repeated
application of current. This change in the junction occurred during the
2.0 T sweep and persisted during the 5.0 T sweep (which were performed
in that sequential order at the end of the measurement). Despite the noise,
those sections of the dI/dV spectrum still look as expected on average,
with no strong changes in the average conductance for both Vbias < −∆
at 2.0 T and the entire 5.0 T sweep, for which H > Hc2.
116
dI/dV peak at zero bias. This dip is usually small, both in terms of the relative
decrease in conductivity and the range of bias voltages over which it is measured, but
it has been consistently observed for soft point contact measurements for a variety
of samples close to T = 0 [41]. While the single peak structure is predicted by the
BTK theory for Z = 0 case of complete Andreev reflection, [40] this is generally not
observed in soft point contact measurements. In the case of YPtBi, this lack of a
split in the ZBCP is a possible indication of unconventional superconductivity, as I
discuss below.
In addition to the shape of the curves, other aspects of the data are unusual
compared to what is expected for conventional superconductivity. First, the zero
bias enhancement shown in Figs. 5.4 and 5.5 persists at values of temperature and
magnetic field that are above the values of Tc and Hc2 that were determined from
the resistivity measurements in Figs. 5.1 and 5.2. The reason for the persistence of
these gap features is unknown, but it may be suggestive of unconventional super-
conductivity. Second, the size of these gap features is also quite large compared to
the expected weak-coupling BCS value of ∆BCS = 1.76kBTc = 0.1 meV for a Tc of
0.7 K. In fact, based on the peak width the gap size appears to be at least twice the
BCS value. This large deviation from weak-coupling provides some evidence against
a conventional weak-coupling s-wave model.
It is interesting to compare these point-contact spectroscopy results, with those
on the triplet p-wave SC Sr2RuO4 (see Chapter 2). Laube et al. measured soft
point contact spectra on Sr2RuO4 and observed conductance curves with two distinct
morphologies [90]. dV/dI resistance curves are compared with our YPtBi data at
117
0.4 K and 0 T in Fig. 5.6.
In the case of Sr2RuO4, some SPC junctions yielded a typical split-peak shape
(which manifests as a split well for dV/dI data). However, other junctions showed an
unsplit zero bias conductivity enhancement, similar to what I observed. This behavior
in Sr2RuO4 was coupled with enhancements in dV/dI at higher bias that very closely
mirror the higher bias dips in my conductivity data so that the shape of the dV/dI
curves is quite similar between the two materials.
Laube et al. concluded that the presence of two types of zero bias features
was consistent with a p-wave superconducting state and triplet pairing. Fits to the
measured data showed that the dI/dV curves with a zero bias “double-minimum”
are consistent with a high transparency barrier between a normal metal and a triplet
superconductor, while the curves with the single-minimum zero bias dip corresponded
to a low transparency condition. It is possible that our results are also in the low
transparency limit for a normal metal-triplet superconductor junction. Of course,
it is also possible that some other effect is producing the single minimum, but the
positive comparison to Sr2RuO4 is encouraging.
5.2.4 Multiple Samples and >100% Zero Bias Enhancement
I repeated conductance measurements on different YPtBi samples in order to
verify that the zero bias feature was repeatable and not an anomaly. Figure 5.7 shows
dI/dV curves for three samples, including the one from Figs. 5.4 and 5.5, at 0.4 K in
zero magnetic field. The zero bias unsplit peak is a consistent feature, but the height
118
0.9
1
1.1
1.2
1.3
1.4
1.5
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
dV/
dI(
nor
mal
ized
)
Vbias
YPtBi
Figure 5.6: Comparison between dV/dI curves for YPtBi (top) and
Sr2RuO4 (bottom). YPtBi data is the inverse of the 0.4 K, 0 T curve
from Figs. 5.4 and 5.5. Sr2RuO4 data is from Ref. [90]. Laube et al.
observed two different forms of dV/dI spectra using the same soft point
contact technique on p-wave Sr2RuO4. Curves 3 and 4 strongly resemble
our YPtBi data, with a smooth zero bias suppression of dV/dI and small
enhancements at higher bias.
119
of the ZBCP is not constant. Indeed, for one sample the zero bias conductance was
over twice the high bias value. I note that Sasaki et al. [92] have claimed a zero bias
enhancement of over 100% is evidence of topological superconductivity in Sn1−xInxTe.
Because I did not consistently observe this high level of enhancement, I do not claim
that this material is topological or even that this large zero bias enhancement is
repeatable. It is also notable that the energy scale of these gap features is fairly
consistent, if one considers the higher bias “dip” observed for two of the three samples.
This consistency between samples with a variety of junction resistances (with a range
of 0.3 Ω – 8 Ω at high bias) supports the assumption that the ZBCP represents the
SC gap.
5.3 Attempts at LuPdBi Tunnel Junctions
In order to confirm the unconventional nature of the superconducting gap in the
half-Heuslers I attempted to artificially alter the impedance of the junctions to force
them into the tunneling regime. Recent work on Fe nanowires suggests that a zero bias
enhancement in an STM tunneling spectrum can be due to Majorana fermions [124].
Furthermore, measuring dI/dV curves over a range of junction resistances and Z
values should strengthen the argument that the previously observed zero bias peaks
represent a true superconducting gap.
The work described in this section was performed at the University of Illinois
at Urbana-Champaign in conjunction with the group of Prof. Laura Greene. Dr.
Greene’s group, especially Dr. Wan Kyu Park, provided me with experimental assis-
120
0.5
1
1.5
2
2.5
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
Sample R1
Sample R3
Sample R5
dI/
dV
(n
or
ma
liz
ed
)
Vbias
YPtBi
0.4 K
Soft Point
Contact Method
Figure 5.7: dI/dV vs. Vbias at T = 0.4 K and H = 0 T for YPtBi samples
R1, R3, and R5. The shape of the curves and height of the zero bias
enhancement changes significantly between samples, but the consistent
width of the overall curves possibly suggests the same energy scale for the
gap features. Although sample R1 has a much broader peak, it lacks the
higher bias dips in dI/dV present in the other two samples.
121
tance. Dr. Park taught me this method of sample preparation and how to do the
SiO2 sputtering and Ag evaporation steps described below.
5.3.1 Preparation of Junctions
To artificially increase the junction resistance, a thin film layer of Al2O3 was
deposited onto the surface of LuPdBi crystals. Although my previous measurements
focused on YPtBi, I chose LuPdBi in this case because larger crystals were available
and these were better suited for the tunnel junction preparation process. LuPdBi
is also predicted to be a topologically non-trivial material according to DFT. Fur-
thermore, with a Tc of 1.7 K, a cryostat with a base temperature of 0.4 K would be
capable of reaching well below Tc so that temperature-dependent trends would be
more apparent.
Prior to thin film deposition, I polished the samples to create as smooth a
surface as possible; if the insulating SiO2 layer is deposited on a rough surface, it
may result in cracks or other defects and compromise the oxide layer′s insulating
benefits. Samples were embedded in Stycast 1266 two-part epoxy in order to hold
them flat for polishing and make the millimeter-scale crystals easier to manipulate.
They were then polished with increasingly fine lapping films containing either Al2O3
or diamond particles to achieve smoothness on the scale of less than 1 µm. In between
polishing steps, samples were inspected using a differential interference contrast (DIC)
microscope to assess the state of the surface and make sure that any scratches had
been removed.
122
Once the surface was sufficiently smooth, I masked each sample with thin strips
of aluminum foil so that the edges of the crystal would remain exposed. This was
necessary so that the bottom wires could be applied directly to the LuPdBi sample
after film deposition. Samples were then loaded into a sputtering chamber and a
70 A˚ Si film was deposited onto the surface. The Si film was then oxidized, either by
exposing the sample to O plasma, or simply by filling the chamber with O2 gas and
heating the sample. For some samples, an additional step was taken of masking the
sample again and depositing strips of Ag to serve as the top wires in the junction.
Figure 5.8 illustrates the general appearance of the sample at different points in the
deposition process.
5.3.2 Behavior of dI/dV with Increasing Temperature and Field
Samples were then mounted on measurements pucks and cooled in a PPMS
Dynacool cryostat using the external 3He probe. I obtanined conductivity vs. bias
voltage data for a for several samples over a range of temperatures and applied mag-
netic fields.
Figure 5.9 presents my conductivity data for one LuPdBi sample over a range
of temperatures. The top figure shows the zero bias conductance as the sample was
cooled from 300 K to 0.4 K. As expected, the conductivity changes abruptly at Tc
= 1.7 K. This change was an increase in dI/dV suggesting that the junction is in
the Andreev reflection regime. This is confirmed by the dI/dV vs. Vbias curves in the
bottom figure, in which a zero bias enhancement is observed consistently up to Tc.
123
VI
I
V
(a) (b) (c)
(d) (e) (f)
Figure 5.8: Schematic of steps in the process of fabricating LuPdBi tunnel
junctions. (a) Sample (gray) is embedded in a disk of Stycast 1266 two-
part epoxy (black) and polished to achieve as flat a surface as possible.
(b) The sample is masked by covering the edges with strips of Al foil and
a 70 A˚ film of Si is sputtered on the surface and oxidized. When the
mask is removed the center of the crystal is covered with SiO2 while the
edges are less oxidized. (c) For a quicker measurement, gold wires are
simply attached to the sample with silver paint for a soft point contact
measurement. (d) Alternatively, the sample is prepared for an additional
deposition of Ag strips to create a more stable electrical contact. To
make sure the junction is as small as possible, most of the sample is
covered with transparent, electrically insulating Duco cement. (e) Next,
the sample is masked with Al strips once again and coated with a layer
of Ag using an evaporator. (f) Gold wires are applied to the deposited
Ag strips to complete the junction. Areas highlighted in red represent
the points of contact with the crystal. The two bottom wires connect
directly to the crystal through the silver contacts. The junction itself is
made at the small section of the middle silver strips in between the spots
of Duco cement. With proper masking before Ag deposition and careful
application of cement, this junction can be quite small (on the order of
0.1 × 0.1 mm2).
124
A few conclusions can be drawn from my data. First, it is quite apparent
that the device was not a tunnel junction. In fact, these dI/dV curves look quite
similar to those of YPtBi discussed previously. Each higher temperature dI/dV
spectrum has the same single-peak structure, and a slight decrease in conductivity at
higher bias, though this dip is not as pronounced as the ones I observed for YPtBi.
This suggests possible similarities between YPtBi and LuPdBi and possibly that
LuPdBi is a topological superconductor. Another point worth noting is that the
dI/dV vs. T plot is roughly linear close to Tc. This is somewhat unexpected as the zero
bias conductance should behave like an order parameter in increasing temperature,
however the significance of this behavior is unclear.
Similar measurements were performed with applied magnetic fields up to 3 T
with the results shown in Fig. 5.10. The upper figure shows that the dI/dV enhance-
ment disappears above Hc2 = 2 T, but with a similarly unusual linear morphology that
does not resemble the expected behavior for an order parameter. The enhancement in
dI/dV vs. bias voltage, in the lower figure, decreases up to Hc2 as expected, though
the high-field background conductance is not quite as flat as its high-temperature
counterpart. These results confirm that the zero bias enhancement represents a true
superconducting gap, although the persistence of a zero bias peak height that de-
creases linearly as the superconducting state is suppressed remains puzzling.
125
0.14
0.15
0.16
0.17
0.18
0.19
0.2
0 50 100 150 200 250 300
dI/d
V(1
/
Temperature (K)
0.14
0.15
0.16
0.17
0.18
0.19
0.2
0.5 1 1.5 2 2.5
LuPdBi +70 Å SiO
Sample J5
2
Soft PointContactMethod withevaporatedAg contacts
0.95
1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
-3 -2 -1 0 1 2 3
0.4 K0.6 K0.8 K1.0 K1.2 K1.4 K1.6 K1.8 K2.0 K
dI/d
V(n
orm
aliz
ed)
Vbias (mV)
LuPdBi +70 Å SiO
Sample J5
2
Soft PointContactMethod withevaporatedAg contacts
Figure 5.9: Conductivity vs. temperature of a LuPdBi point contact junc-
tion. The top figure shows that conductivity is roughly constant down to
Tc = 1.7 K, at which point it increases abruptly. The inset shows a de-
tailed view of the data below 2.5 K. The bottom plot shows a series of
dI/dV vs. Vbias curves up to and above Tc. This junction is in the Andreev
limit rather than tunneling, and exhibits a strong single-peaked enhance-
ment in dI/dV as well as very small dips at higher bias (not visible at
this scale).
126
0.14
0.15
0.16
0.17
0.18
0.19
0 0.5 1 1.5 2 2.5 3
dI/d
V(1
/
Field (T)
LuPdBi + 70 Å SiO
Sample J50.435 K
2
Soft Point ContactMethod with evaporatedAg contacts
0.95
1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
-3 -2 -1 0 1 2 3
0 T0.2 T0.4 T0.6 K0.8 T1.0 T1.25 T1.5 T2.0 T3.0 T
dI/d
V(n
orm
aliz
ed)
Vbias (mV)
LuPdBi + 70 ÅSiO
Sample J50.435 K
2
Soft Point ContactMethod withevaporated Agcontacts
Figure 5.10: Zero bias dI/dV of LuPdBi as a function of magnetic field
(above) and dI/dV vs. Vbias for a range of applied field values (below).
Conductivity decreases and levels off above Hc2 = 2 T, but the roughly
linear decrease is unexpected. The zero bias enhancement retains the
single-peak shape up to Hc2. Note that these measurements were obtained
at at T = 0.4 K.
127
5.3.3 Discussion
My efforts to create well-defined point contact tunnel junctions with LuPdBi
ultimately led to mixed results. On the positive side, the process of polishing and
oxide deposition were successful in producing junction with somewhat higher resis-
tance. Junctions made without an SiO2 layer typically had resistances on the order
of 0.1 Ω. The addition of the oxide layer increased the resistance by two orders of
magnitude - for example, the junction used in the measurements from Figs. 5.9 and
5.10 had a resistance of about 7 Ω.
My results also compare favorably to earlier measurements taken at the Uni-
versity of Maryland. For example, Fig. 5.11 shows the low temperature normalized
dI/dV spectrum from Figs. 5.9 and 5.10 compared to similar spectra measured on
LuPdBi samples that were not polished and did not have sputtered SiO2. Both the
junction measured at UIUC (J5, in red) and the junctions measured at UMD (R9
and R13, in blue) have similar unsplit peak structures (although the R9 data is too
noisy to claim that definitively). Furthermore, the size of the gaps seems to be similar
between the three junctions. Although it is not as evident from Fig. 5.11, the dI/dV
curve for sample J5 does have a slight dip and a minimum around 0.7 mV, which is
clearer in Figs. 5.9 and 5.10. This demonstrates that similar results can be obtained
for junctions with a wider range of resistances.
Despite being unable to produce LuPdBi junctions in the tunneling regime, my
junctions still yielded some interesting results. The presence of a consistent unsplit
zero bias peak in the dI/dV vs. Vbias curves is similar to behavior seen in YPtBi,
128
LuPdBi
0.4-0.435 K
Soft Point Contact
Method
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
Sample J5
Sample R9
Sample R13
dI/
dV
(n
or
ma
liz
ed
)
Vbias (mV)
Figure 5.11: Low temperature, zero magnetic field dI/dV spectra for three
LuPdBi soft point contact junctions. Sample J5 (red) was polished and
sputtered with SiO2 and was prepared at UIUC. Samples R9 and R13
(blue) did not undergo this polishing and deposition process and were
prepared at UMD. The high-bias junction resistances of these samples
were 7 Ω, 1.8 Ω, and 1.6 Ω for J5, R9, and R13, respectively.
129
suggesting LuPdBi might possibly be a topological superconductor. I also note that
I did make junctions with even higher junction resistances, one as high as RJ =
1 kΩ. Unfortunately, these high resistance junctions resulted in dI/dV data that was
extremely noisy to the extent that no discernible features could be extracted. This
is still a promising development, as it shows that point contact tunnel junctions with
these materials may not be too far off.
5.4 Conclusions
In conclusion, I have performed soft point-contact spectroscopy measurements
on the half-Heusler superconductors YPtBi and LuPdBi and observed a consistent
single-peak enhancement in dI/dV at zero bias. This result suggests unconventional
superconductivity as expected for a noncentrosymmetric and possibly topological su-
perconductor. The unsplit zero bias peak combined with high-bias dips was consistent
with theoretical predictions for a superconductor with a triplet pairing state. This
behavior was also consistent with previous experimental work on the unconventional
triplet p-wave material Sr2RuO4. Future work would be needed to verify the pos-
sibility of triplet pairing and to positively identify a p-wave or mixed parity order
parameter
130
Chapter 6: Conclusions
In conclusion, I have used point-contact spectroscopy for measurement of the
electronic structure of superconductors. My work, in collaboration with other re-
searchers, has expanded knowledge of a variety of unconventional superconductors.
My work has yielded positive results, opened more questions about these materials,
and suggests many possible future experiments.
I used the needle-anvil point contact method to measure AC conductivity vs.
DC bias voltage in the iron pnictide superconductor BaFe1.9Pt0.1As2. dI/dV curves
were obtained for several single crystal samples down to 4 K using a probe with an
adjustable mechanical stage that was used to create junctions between sharpened
Pb or Au tips and the ab-plane of the samples. After background subtraction and
normalization, the dI/dV data showed a sharp peak at zero bias and sharp decreases
in conductivity at higher bias voltages. I fit my results to a two-gap isotropic s-wave
model using the Blonder- Tinkham-Klapwijk theory and found reasonable agreement.
Analysis of these fit parameters revealed a high-transparency gap of magnitude
3 meV and a 7 meV gap that was closer to the tunneling limit with a high degree
of inelastic scattering. This apparent discrepancy in transparency was possibly due
to the high scattering rate in the 122 iron pnictides and/or an impedance mismatch
131
between the normal metal tip and superconductor. My measurements on multiple
samples confirmed the consistent presence of two gap-like features at roughly 3 and
7 meV.
I also discussed three other experimental techniques that were applied to crys-
tals from the same growth batch to check the interpretation of my point-contact spec-
troscopy results. Thermal conductivity revealed no quasiparticle excitations at low
temperature, consistent with a nodeless gap structure. Angle-resolved photoemission
spectroscopy and Raman spectroscopy confirmed the presence of the smaller 3 meV
gap but did not show convincing evidence for the larger 7 meV gap. The ARPES
result also showed no evidence for nodes or deep minima in the order parameter.
Overall, this represents consistent evidence of an isotropic 3 meV gap in BaFe1.9Pt0.1As2.
This stands in contrast to much other work on similar materials, in which different
groups reached different conclusions on gap size and symmetry for seemingly identical
materials. The success of our approach, which used crystals from the same growth
batch, highlights the importance of crystal quality in studies of superconducting gap
structure. Differences in impurity content and/or defect structures seem to be the
most likely cause of these inconsistencies.
Finally, I applied the soft point contact technique to single crystal samples of
the noncentrosymmetric half-Heusler superconductor YPtBi. Samples were sanded
down to reveal flat faces for the application of electrical contacts. I attached gold
wire electrodes with conductive silver paste, and used electrically insulating epoxy to
reduce contact between the paste and crystal surface and ensure a small contact area
for the junction. I measured conductivity vs. bias voltage in a 3He probe capable of
132
reaching temperatures below 400 mK with applied magnetic fields up to 14 T.
dI/dV measurements on YPtBi samples revealed a consistent peak at zero bias
and shallow dips at higher bias. A single peak is inconsistent with the results pre-
dicted for soft point contact measurements of an s-wave material, in which case a
double-maximum “split peak” structure is typically observed. Conductivity curves
measured at higher temperatures and magnetic fields confirmed that the observed
peak is from the superconducting state rather than a temperature effect or from
anomalous scattering. Repeating the measurement on other samples confirmed the
consistency of this single peak structure. Comparison to previous work showed the
dI/dV curves resemble those measured for the triplet-pairing p-wave superconductor
Sr2RuO4 as well as theoretical predictions for a generic noncentrosymmetric super-
conductor. This suggests that YPtBi may be a triplet p-wave superconductor, or at
least has a mixed-parity pairing state with a p-wave component.
My work suggests some possible directions for future studies. I found that a
range of experiments performed on iron-based superconductors from the same batch
of crystals can yield highly consistent results. Similar studies on different pnictide ma-
terials would be interesting. This could help shed light on compounds that have been
previously studied but found to give contradictory results, such as BaFe2−xCoxAs2. It
could also be useful for investigating new superconductors or materials for which the
gap symmetry has not been decisively settled, such as Ba1−xKxFe2As2 compounds on
the K-rich end of the phase diagram.
In the realm of topological superconductors, point-contact spectroscopy could
be useful for investigating the electronic structure. All of the superconducting half-
133
Heuslers share the same noncentrosymmetric crystal lattice, but it has yet to be
shown whether they also share the p-wave characteristics predicted for chiral materi-
als. Further complicating matters are theoretical models indicating that some half-
Heuslers should be topologically trivial while others should exhibit band inversion.
The fact that both sides of the topological/non-topological divide feature known su-
perconductors suggests the possibility of using point-contact spectroscopy to separate
superconductors with different topologies. Finally, several of the half-Heuslers con-
tain highly magnetic rare earth elements and as a consequence exhibit magnetically
ordered states at low temperature. Here, PCS and other spectroscopic techniques
could be used to detect changes in gap structure above and below the magnetic tran-
sition temperatures to determine what effect, if any, these magnetic interactions have
on the topological superconducting state.
134
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