ABSTRACT
Title of dissertation: Electronic and Magnetic Properties of
MnP-Type Binary Compounds
Daniel James Campbell
Doctor of Philosophy, 2019
Dissertation directed by: Professor Johnpierre Paglione
Department of Physics
The interactions between electrons, and the resulting impact on physical prop-
erties, are at the heart of present-day materials science. This thesis looks at this
idea through the lens of several compounds from a single family: the MnP-type
transition metal pnictides. FeAs and FeP show long range magnetic order with
some similarities to the high temperature, unconventional iron-based superconduc-
tors. CoAs lies on the border of magnetism, with strong fluctuations but no stable
ordered state. CoP, in contrast, shows no strong magnetic fluctuations but serves as
a useful baseline in determining the origin (from composition, structure, or magnetic
order) of behavior in the other materials.
For this work, single crystals were grown with two different techniques: solvent
flux and chemical vapor transport. In the case of FeAs the flux method resulted in
the highest quality crystals yet produced. Extensive work was then performed on
these samples at the University of Maryland and the National High Magnetic Field
Laboratory. Quantum oscillations observed in high magnetic fields, in combination
with density functional theory calculations, give insight into the Fermi surfaces of
these materials. Large magnetoresistance in the phosphides, but not the arsenides,
demonstrates differences in the choice of pnictogen atom that cannot be simply a
product of electron count. Angle-dependent linear magnetoresistance in FeP is a
sign of a possible Dirac dispersion and topological physics, as has been hinted it
in other MnP-type materials. Ultimately, it is possible to examine results for all
four compounds and draw conclusions on the role of each of the two elements in the
formula, which can be extended to other members of this family.
ELECTRONIC AND MAGNETIC PROPERTIES OF MNP-TYPE
BINARY COMPOUNDS
by
Daniel James Campbell
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
2019
Advisory Committee:
Professor Johnpierre Paglione, Chair/Advisor
Professor Nicholas P. Butch
Professor Richard L. Greene
Professor Efrain E. Rodriguez, Dean?s Representative
Professor James R. Williams
?c Copyright by
Daniel James Campbell
2019
Foreword
The following is a list of all the papers that I have contributed to during my
time as a graduate student, either published or intended for publication, starting
with the most recent. Those whose results are covered extensively in this thesis are
marked with an asterisk.
*D.J. Campbell, J. Collini, K. Wang, B. Wilfong, Y.S. Eo, P. Neves, D.
Graf, E.E. Rodriguez, N.P. Butch, and J. Paglione, ?Large, Anisotropic, Linear
Magnetoresistance in FeP?, in preparation.
D.J. Campbell, Z.E. Brubaker, C. Roncaioli, P. Saraf, Y. Xiao, P. Chow,
C. Kenney-Benson, D. Popov, R.J. Zieve, J.R. Jeffries, and J. Paglione, ?Pressure-
Driven Valence Increase and Metallization in Kondo Insulator Ce3Bi4Pt3?,
arXiv:1907.09017. [1]
K. Wang, R. Mori, Z. Wang, L. Wang, J.H.S. Ma, D.W. Latzke, D. Graf,
J.D. Denlinger, D.J. Campbell, B.A. Bernevig, A. Lanzara, and J. Paglione,
?Crystalline-Symmetry-Protected non-Trivial Topology in Prototype Compound
BaAl4?, under review.
Y. Nakajima, T. Metz, C. Eckberg, K. Kirshenbaum, A. Hughes, R. Wang,
L. Wang, S.R. Saha, I.-L. Liu, N.P. Butch, D.J. Campbell, Y.S. Eo, D. Graf, Z.
Liu, S.V. Borisenko, P.Y. Zavalij, and J. Paglione, ?Planckian dissipation and scale
invariance in a quantum-critical disordered pnictide?, arXiv:1902.01034. [2]
C. Eckberg, D.J. Campbell, T. Metz, J. Collini, H. Hodovanets, T. Drye,
P. Zavalij, M.H. Christensen, R.M. Fernandes, S. Lee, P. Abbamonte, J. Lynn,
and J. Paglione, ?Sixfold enhancement of superconductivity in a tunable electronic
nematic system?, arXiv:1903.00986 -accepted, Nat. Phys., October 2019. [3]
S. Ran, I.-L. Liu, Y.S. Eo, D.J. Campbell, P. Neves, W.T. Fuhrman, S.R.
Saha, C. Eckberg, H. Kim, J. Paglione, D. Graf, J. Singleton, and N.P. Butch,
?Extreme Magnetic Field-Boosted Superconductivity? Nat. Phys., 2019. [4]
R.L. Stillwell, X. Wang, L. Wang, D.J. Campbell, J. Paglione, S.T.
Weir, Y.K. Vohra, and J.R. Jeffries, ?Observation of two collapsed phases in
CaRbFe4As4?, Phys. Rev. B 100 045152 (2019). [5]
D.J. Campbell, C. Eckberg, P.Y. Zavalij, H.-H. Kung, E. Razzoli, M.
Michiardi, C. Jozwiak, A. Bostwick, E. Rotenberg, A. Damascelli, and J. Paglione,
ii
?Intrinsic Insulating Ground State in Transition Metal Dichalcogenide TiSe2?,
Phys. Rev. Mater. 3 053402 (2019). [6]
S. Lee, G.A. de la Pea, S.X.-L. Sun, M. Mitrano, Y. Fang, H. Jang, J.-S.
Lee, C. Eckberg, D.J. Campbell, J. Collini, J. Paglione, F.M.F. de Groot,
and P. Abbamonte, ?Discovery of a charge density wave instability in Co-doped
BaNi2As2?, Phys. Rev. Lett. 121 147601 (2019). [7]
Z.E. Brubaker, R.L. Stillwell, P. Chow, Y. Xiao, C. Kenney-Benson, R. Ferry,
D. Popov, S.B. Donald, P. So?derlind, D.J. Campbell, J. Paglione, K. Huang,
R. E. Baumbach, R. J. Zieve, and J. R. Jeffries, ?Pressure dependent intermedi-
ate valence behavior in YbNiGa4 and YbNiIn4?, Phys. Rev. B 98 214115 (2018). [8]
H. Hodovanets, C.J. Eckberg, P.Y. Zavalij, H. Kim, W.-C. Lin, M. Zic, D.J.
Campbell, J.S. Higgins, and J. Paglione, ?Single crystal investigation of proposed
type-II Weyl semimetal CeAlGe?, Phys. Rev. B 98 25132 (2018). [9]
C. Eckberg, L. Wang, H. Hodovanets, H. Kim, D.J. Campbell, P. Zavalij,
P. Piccoli, and J. Paglione, ?Evolution of structure and superconductivity in
Ba(Ni1?xCox)2As2?, Phys. Rev. B 97 224505 (2018). [10]
*D.J. Campbell, L. Wang, C. Eckberg, D. Graf, H. Hodovanets, and J.
Paglione, ?CoAs: The line of 3d demarcation?, Phys. Rev. B 97 174410 (2018). [11]
B. Wilfong, X. Zhou, H. Vivanco, D.J. Campbell, K. Wang, D. Graf, J.
Paglione, and E.E. Rodriguez, ?Frustrated magnetism in the tetragonal CoSe
analog of FeSe?, Phys. Rev. B 97 104408 (2018). [12]
*D.J. Campbell, C. Eckberg, K. Wang, L. Wang, H. Hodovanets, D. Graf,
D. Parker, and J. Paglione, ?Quantum oscillations in the anomalous spin density
wave state of FeAs?, Phys. Rev. B 96 075120 (2017). [13]
C. K. H. Borg, X. Zhou, C. Eckberg, D.J. Campbell, S.R. Saha, J. Paglione,
and E.E. Rodriguez, ?Strong anisotropy in nearly ideal tetrahedral superconducting
FeS single crystals?, Phys. Rev. B 93 094522 (2016). [14]
iii
Acknowledgments
The first person to thank is my advisor, Johnpierre Paglione, whose support, in
terms of funding, advice, and encouragement, made all of this work possible. I came
in having never taken a condensed matter class and at most a passing knowledge
of what a superconductor was. It says a lot that he was willing to take the time to
mentor someone who needed to build up his knowledge from quite a low baseline.
Many people from the Paglione group have also been essential to helping me
learn how to be a scientist. Some were directly involved in these projects, including
accompanying me on various trips to the High Magnetic Field Laboratory in Tal-
lahassee: former members Chris Eckberg (who gave me my initial training when I
joined the group), Kefeng Wang, and Limin Wang (who is responsible for all the
density functional theory calculations in this dissertation), current members John
Collini, Yun Suk Eo, Halyna Hodovanets, Shanta Saha, and Brandon Wilfong. Of
course, everyone else that has passed through or is in the group has either given ad-
vice or been a key part of other projects: former members I-Lin Liu, Yasuyuki Naka-
jima, Renxiong Wang, and Xiangfeng Wang, current members Ian Hayes, Hyunsoo
Kim, Wen-Chen Lin, Tristin Metz, Connor Roncaioli, Prathum Saraf, and Mark Zic.
Others at UMD also helped a great deal, especially Josh Higgins who was always
helpful with cryostat operation and general experimental knowledge.
External collaborators were also essential, most of all Dave Graf from the
NHMFL, who dealt with naive or confusing demands over the course of many ex-
periments. There is also David Parker from Oak Ridge National Laboratory, and on
iv
other projects Andrea Damascelli at the University of British Columbia and Sean
Kung, Matteo Michiardi, and Elia Razzoli from his group. Something must be said
also for Zach Brubaker, Ryan Stillwell, and Jason Jeffries from Lawrence Livermore
National Laboratory, who allowed me to work there for a six month sojourn and
teach me the basics of diamond anvil cell operation, despite it taking a lot of time
away from their own work.
Lastly, I need to thank my family: Dad, Mum, Denis, and Theresa. All of
them have experienced the stress of graduate school through me, and are probably
just as happy in their own way that I will be finishing as I am myself.
v
Table of Contents
Foreword ii
Acknowledgements iv
Table of Contents vi
List of Tables ix
List of Figures x
List of Abbreviations xii
1 Introduction 1
1.1 Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The MnP-Type Structure . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 This Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Theoretical Background 10
2.1 Magnetic Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.1 Antiferromagnetism: Helimagnetism and Spin Density Waves . 14
2.1.2 Near Ferromagnetism and Magnetic Fluctuations . . . . . . . 17
2.2 Magnetoresistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.1 The Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.2 Large Magnetoresistance . . . . . . . . . . . . . . . . . . . . . 27
2.2.3 Linear Magnetoresistance . . . . . . . . . . . . . . . . . . . . 32
2.2.4 The Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3 Quantum Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3.1 Origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3.2 Detection and Analysis . . . . . . . . . . . . . . . . . . . . . . 41
2.3.3 Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3 Experimental Methods 46
3.1 Single Crystal Growth . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.1.1 Liquid Solvent Flux . . . . . . . . . . . . . . . . . . . . . . . . 49
3.1.2 Chemical Vapor Transport . . . . . . . . . . . . . . . . . . . . 53
3.1.3 Growth Details . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2 Measurements at the University of Maryland . . . . . . . . . . . . . . 59
vi
3.2.1 Structural and Compositional Characterization . . . . . . . . 59
3.2.2 Electrical Transport . . . . . . . . . . . . . . . . . . . . . . . 60
3.2.3 Heat Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.2.4 Magnetic Susceptibility . . . . . . . . . . . . . . . . . . . . . . 66
3.3 High Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.3.1 Electrical Transport . . . . . . . . . . . . . . . . . . . . . . . 71
3.3.2 Magnetic Torque . . . . . . . . . . . . . . . . . . . . . . . . . 71
4 Crystal Quality and Quantum Oscillations in FeAs 75
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2 Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3 Transport Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.4 Quantum Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.4.1 Angular Dependence . . . . . . . . . . . . . . . . . . . . . . . 83
4.4.2 Fermi Surface Shape . . . . . . . . . . . . . . . . . . . . . . . 84
4.4.3 Temperature Dependence . . . . . . . . . . . . . . . . . . . . 87
4.4.4 Dingle Temperature and Scattering . . . . . . . . . . . . . . . 90
4.5 FeAs: Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5 A Magnetoresistance Study of FeP 95
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.2 Methods and Characterization . . . . . . . . . . . . . . . . . . . . . . 96
5.3 Magnetoresistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.3.1 Field Dependence . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.3.2 Angular Dependence . . . . . . . . . . . . . . . . . . . . . . . 103
5.3.3 Temperature Dependence . . . . . . . . . . . . . . . . . . . . 104
5.4 Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.6 FeP: Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6 Near Ferromagnetism in CoAs 115
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.2 Crystal Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.3 Physical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.4 Theoretical Calculations . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.5 Quantum Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.5.1 Fermi Surface Geometry . . . . . . . . . . . . . . . . . . . . . 131
6.5.2 Effective Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.5.3 Dingle Temperature . . . . . . . . . . . . . . . . . . . . . . . . 136
6.6 CoAs: Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7 CoP: A Nonmagnetic Contrast 140
7.1 Crystal Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.2 Physical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7.3 Quantum Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
vii
7.3.1 Angular Dependence . . . . . . . . . . . . . . . . . . . . . . . 148
7.3.2 Temperature and Field Dependence . . . . . . . . . . . . . . . 150
7.4 CoP: Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
8 Co-doped FePn 155
8.1 Crystal Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
8.2 Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
8.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
9 Chapter 9: Conclusion 162
9.1 Summary of Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
9.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
A Additional MnP-Type Compounds 167
A.1 RhPn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
A.1.1 Crystal Growth . . . . . . . . . . . . . . . . . . . . . . . . . . 168
A.1.2 Physical Properties . . . . . . . . . . . . . . . . . . . . . . . . 169
A.1.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
A.2 WP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
A.2.1 Crystal Growth and Characterization . . . . . . . . . . . . . . 173
A.2.2 Normal State Properties . . . . . . . . . . . . . . . . . . . . . 174
A.2.3 Superconducting Properties . . . . . . . . . . . . . . . . . . . 180
A.2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
Bibliography 184
viii
List of Tables
1.1 Lattice parameters and information about ordered magnetic states of
MnP-type compounds . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 Quantities characterizing the double helical magnetic state in MnP-
type compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.1 Parameters extracted from FeAs quantum oscillations data . . . . . . 93
6.1 Parameters extracted from CoAs quantum oscillations data . . . . . . 138
7.1 Parameters extracted from CoP quantum oscillations data . . . . . . 153
ix
List of Figures
1.1 The MnP-type (B31) unit cell . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Illustration of different kinds of magnetic order . . . . . . . . . . . . . 13
2.2 Depiction of candidate and actual magnetic arrangements of FeAs . . 16
2.3 An example of an Arrott plot used to find the Curie temperature . . 21
2.4 The difference between open and closed Fermi surface orbits . . . . . 30
2.5 An illustration of Landau level quantization in a metal . . . . . . . . 40
3.1 Single crystals grown by various techniques . . . . . . . . . . . . . . . 47
3.2 Binary phase diagrams of the As-Fe and As-Bi systems . . . . . . . . 50
3.3 A furnace, sealed quartz ampule, and centrifuge used for solvent flux
growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.4 A diagram depicting the formation of FeP single crystals with I2
chemical vapor transport . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.5 Chemical vapor transport ampules before and after growth . . . . . . 55
3.6 Illustrations and pictures of wiring for electrical transport measure-
ments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.7 Instruments used at the National High Magnetic Field Laboratory
DC Field Facility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.8 Pictures of the setup for high magnetic field measurements of mag-
netic torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.1 Temperature and magnetic-field dependent resistivity of a Bi flux-
grown FeAs crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2 High field magnetic torque data at different orientations for FeAs . . 80
4.3 Fast Fourier transforms of FeAs torque data, and angular dependence
of quantum oscillation frequencies . . . . . . . . . . . . . . . . . . . . 82
4.4 DFT calculated Fermi surfaces of FeAs in an antiferromagnetic state 88
4.5 Temperature dependence of quantum oscillation amplitude of differ-
ent bands for FeAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.1 Zero-field resistivity as a function of temperature for FeP . . . . . . . 99
5.2 Magnetoresistance as a function of field at various angles for FeP . . 100
x
5.3 Angle-dependent magnetoresistance of FeP at constant field . . . . . 102
5.4 Temperature-dependent resistivity of in fields up to 14 T near the
maximum angle of magnetoresistance . . . . . . . . . . . . . . . . . . 105
5.5 Hall effect data for FeP crystals with field applied along three different
crystallographic directions . . . . . . . . . . . . . . . . . . . . . . . . 108
6.1 Comparison of the morphology of CoAs crystals grown by different
methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.2 Resistivity of CoAs single crystals grown by different methods, and
heat capacity in different fields . . . . . . . . . . . . . . . . . . . . . . 120
6.3 Magnetic susceptibility of CoAs . . . . . . . . . . . . . . . . . . . . . 122
6.4 An Arrott plot for CoAs at various temperatures . . . . . . . . . . . 124
6.5 DFT-calculated Fermi surfaces of CoAs in a paramagnetic state . . . 127
6.6 Comparison between CoAs and FeAs paramagnetic band structures
and Fermi surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.7 High field magnetic torque data for CoAs . . . . . . . . . . . . . . . . 132
6.8 Temperature dependence of quantum oscillation amplitude in CoAs . 137
7.1 Electrical resistivity and low temperature specific heat of CoP . . . . 142
7.2 High field magnetoresistance of CoP . . . . . . . . . . . . . . . . . . 144
7.3 Temperature-dependent magnetic susceptibility of CoP . . . . . . . . 146
7.4 High field magnetic torque of CoP and FFT of the oscillatory signal
at various angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
7.5 Angular dependence of CoP quantum oscillation frequencies . . . . . 149
7.6 Temperature and magnetic field dependence of CoP quantum oscil-
lation frequencies with field along the b-axis . . . . . . . . . . . . . . 152
8.1 Temperature-dependent resistivity and TN suppression with doping
in Fe1?xCoxP and Fe1?xCoxAs . . . . . . . . . . . . . . . . . . . . . . 160
A.1 Resistivity and specific heat of RhAs and RhSb . . . . . . . . . . . . 170
A.2 Different product compounds from a vapor transport growth of WO3
and P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
A.3 Normal state resistivity and specific heat of WP . . . . . . . . . . . . 177
A.4 High field magnetoresistance of WP . . . . . . . . . . . . . . . . . . . 179
A.5 Superconducting properties of WP . . . . . . . . . . . . . . . . . . . 182
xi
List of Abbreviations
AC Alternating current
ADR Adiabatic demagnetization refrigerator
AFM Antiferromagnet(-ic, -ism)
AMRO Angular magnetoresistance oscillations
BZ Brillouin zone
CVT Chemical vapor transport
DC Direct current
DFT Density functional theory
dHvA de Haas-van Alphen
EDS Energy-dispersive x-ray spectroscopy
FFT Fast Fourier transform
FM Ferromagnet(-ic, -ism)
LK Lifshitz-Kosevich
LL Landau level
MPMS Quantum Design Magnetic Properties Measurement System
(L)MR (Linear) magnetoresistance
NHMFL National High Magnetic Field Laboratory
PM Paramagnet(-ic, -ism)
PPMS Quantum Design Physical Properties Measurement System
RRR Residual resistivity ratio
QC(P) Quantum critical(-ity, point)
QO Quantum oscillation
SdH Shubnikov-de Haas
SDW Spin density wave
SKEAF Supercell k -Space Extremal Area Finder
SQUID Superconducting quantum interference device
UMD University of Maryland
VSM Vibrating Sample Magnetometer
XRD X-ray diffraction
xii
Chapter 1: Introduction
1.1 Foundations
The discovery and application of new materials is fundamental to society, so
much so that ancient Western civilization is traditionally divided into epochs named
after the primary material of the era: the Stone Age, followed by the Bronze Age,
followed by the Iron Age. Skipping ahead a few millenia, the realization of the power
of semiconductors, specifically silicon, has led to a technological revolution within
the past century that has fundamentally altered the nature of human existence. It
is the investigation of new compounds, or of old compounds with new methods, that
drives societal advancement.
Basic materials research focuses in depth of the small changes that can have
a significant impact on physical properties. This thesis will cover the study of four
materials: FeAs, FeP, CoAs, CoP, as well as mixed Fe1?xCoxP and Fe1?xCoxAs
substitution series. All of these have the same crystal structure, with changes at
either the transition metal or pnictogen1 site.
With this similarities in atomic arrangement and composition come, naturally,
1The pnictogens are the elements in Group 15 of the periodic table (the fourth from the right).
The ones relevant to this thesis are P, As, Sb, and Bi; N is generally very chemically different in
compounds from the others and Mc is a synthetic, extremely radioactive element produced only
in very small quantities.
1
common threads in behavior. The interactions between electrons, and specifically
a tendency toward magnetic order, are relevant to describing all of them. For the
iron compounds, this means that at low temperatures the electron spins are slightly
rotated relative to their closest neighbors, leading to a spiral pattern along a specific
direction. The cobalt-based materials do not have a long range magnetic structure,
but as will be shown there is clear evidence that in CoAs electrons still have magnetic
fluctuations with those those nearby, hinting at a proximity to a phase transition to
ordered magnetism. Compared to the others, CoP does not show the same degree
of interesting magnetic behavior. But that makes it useful as a nonmagnetic basis
for comparison, helping to determine whether the interesting properties in the other
compounds come from the emergence of magnetic states, or are related to their
structural and elemental makeup.
The interest in magnetic order, from the perspective of this thesis, comes
from the states that can emerge as that order is on the verge of disappearing. The
first member of what would come to be the iron-based superconductor family was
discovered a little over a decade ago [15], and many others have since followed.
Structure and composition vary, but common to all is a unit cell of several dis-
tinct layers, one of which is made up of bonded Fe and As. Furthermore, most are
derived from nonsuperconducting parent compounds with a low temperature mag-
netic state similar to some of the binaries. When magnetic order is suppressed by
applied pressure or chemical doping, an unconventional high temperature supercon-
ducting state arises around the quantum critical point (QCP) of the phase diagram
where the transition goes to zero temperature [16]. ?Unconventional? means that
2
the electron-electron pairing mechanism that opens the superconducting gap is not
the phonon-mediated seen in the vast majority of materials and described by the
Bardeen-Cooper-Schrieffer theory. Instead, evidence has accumulated that magnetic
interactions are critical to opening a superconducting gap.
Transition temperatures in the iron superconductors can exceed 50 K, while
typical transition temperatures for conventional materials are below 20 K. Another
family of unconventional superconductors is the heavy fermion materials, so called
because electron-electron interactions lead to effective electron masses orders of mag-
nitude higher than the rest mass. The phase diagram looks similar; when magnetic
order is suppressed, unconventional superconductivity emerges and increases in tem-
perature, though in this case with transition temperatures below 5 K. The case of
the other high temperature superconducting family, the cuprates, first discovered
in the 80s, also follows this pattern. The difference is that it is charge order or
the pseudogap phase that are often suppressed, though magnetism plays a role in
these materials as well [17], though there are indications of magnetic fluctuations or
order in the cuprates as well. It seems in all these materials that there is a remnant
energy waiting to be applied to correlated electron effects. If it is not taken up by
magnetic order, then some other interesting state ought to emerge. This knowledge
also motivated the Fe-Co substitution work to be presented. Knowing that the Fe
end members are magnetically ordered and the Co ones are not, there must be a
point at which the system loses long range order.
The MnP-type compounds offer the chance to isolate the transition metal-
pnictogen layer that is crucial to the iron-based superconductors. It has also been
3
known for decades that many of the binaries take on a similar magnetic ordering
to that seen in the parent compounds to the iron superconductors. For that reason
researchers quickly jumped on them after the importance of the iron pnictide su-
perconductors began to be understood. In two cases, a similar effect was observed.
CrAs and MnP, both helimagnetic at low temperature, became superconductors un-
der pressure when magnetism disappeared [18?21]. While transition temperatures
for both were found to be less than 3 K, more work with other MnP-type compounds,
especially those with strong magnetic interactions, was clearly of importance.
1.2 The MnP-Type Structure
The structure that the four compounds to be explored in this work take on is
an orthorhombic2, Pnma structure, known as the MnP or B31 type. A depiction
is in Fig. 1.1, and the full list of materials with this structure is given in Table 1.1
along with lattice parameters and, in the case of long range magnetic order, transi-
tion temperatures. The structure is a distortion of the hexagonal NiAs structure in
which both the transition metal and pnictogen atom are displaced and the overall
symmetry lowered [22]. Several more of the possible transition metal-pnictogen com-
pounds take the original hexagonal structure, and there are some materials (CrAs,
CoAs, and from work in the Appendix possibly RhAs) that transition between the
two at temperatures of roughly 1000 ?C [23].
The basic properties and some crystal growth information for these compounds
were well described starting in the 1960s, primarily by the group at the University
2Meaning that the unit cell is a rectangular box, with all three axes having a different length.
4
c
a b
Figure 1.1: Depiction of the MnP (B31) type structure (Pnma, space group no. 62),
showing octahedral coordination of the pnictogen (blue) atoms around the transition
metal (red). There are four formula units per unit cell. Taken from Ref. [32].
5
of Oslo [24?28]. Further work was done to examine the magnetic ordering present in
FeAs, FeP, CrAs, and MnP [29?31]. After a lull, these materials received renewed
interest with the discovery of the high-temperature iron pnictide superconductors
in 2008 [15] for the reasons outlined earlier in this chapter.
While similarities between the MnP-type binaries and the Fe-based supercon-
ductors have been highlighted so far, there are some differences. Arsenic atoms are
tetrahedrally coordinated to Fe in the high-Tc pnictides, but form octahedra around
the transition metals in the MnP structure. The lower symmetry of the crystal
structure also results in different Fe-As bond lengths Additionally, only three of the
orthorhombic compounds have been found to superconduct, and none at high tem-
peratures: WP at 0.7 K [33], CrAs at 2.2 K and 1 GPa of applied pressure [18,34],
and MnP at about 1 K and 8 GPa [20]. Despite Tc values more than an order of
magnitude lower than the multicomponent Fe-based materials, the results of high
pressure measurements are intriguing. Beyond inducing superconductivity, applying
pressure to CrAs and MnP also monotonically decreases the onset temperature of
magnetic order, which ultimately disappears completely in a manner similar to the
quantum critical physics seen in other material families [21]. When this happens for
the iron-based, heavy fermion, or cuprate superconductors, it is an indicator that
Cooper pairing is driven by magnetic fluctuations or correlated electron physics [35].
This would explain why Tc typically reaches a maximum value just after suppression
of SDW order; fluctuations are still strong, but are working to stabilize supercon-
ductivity rather than magnetism.
Another reason for interest in MnP-type materials is that the crystal structure
6
Formula [Ref.] a (A?) b (A?) c (A?) Magnetic Order?
CrP [36] 5.362 3.113 6.018 No
CrAs* [37] 5.649 3.463 6.212 AFM, 264 K
CoP [This work] 5.08 3.28 5.59 No
CoAs* [This work] 5.28 3.49 5.87 No
FeP [This work] 5.10 3.10 5.79 AFM, 120 K
FeAs [This work] 5.44 3.37 6.02 AFM, 70 K
MnP [20] 5.26 3.17 5.92 FM, 291 K and AFM 50 K
MnAsa [38] 5.72 3.676 6.379 Nob
MoAs [39] 5.97 3.36 6.41 No
RhAs* [This work] 5.65 3.60 6.06 No
RhSb [This work] 5.97 3.87 6.34 No
RuP [36] 5.520 3.168 6.120 No
RuAs [40] 5.717 3.338 6.313 No
RuSb [41] 5.961 3.702 6.580 No
VAs [40] 5.850 3.366 6.289 No
WP [33] 5.722 3.243 6.211 No
Table 1.1: Lattice parameters and information about magnetic ordering for MnP-
type compounds. Those marked with an asterisk transition at high temperatures to
the NiAs structure (for RhAs this is not known definitively, but suspected-see the
Appendix). AFM stands for ?antiferromagnetic? and FM for ?ferromagnetic?.
aOnly from 40-120 ?C, hexagonal NiAs structure at room temperature.
bFM in the NiAs structure at room temperature and below.
is nonsymmorphic. A symmorphic structure is one in which all symmetry operations,
barring translational ones, leave a single point fixed. In contrast, nonsymmorphic
symmetry operations do not leave any part of the unit cell fixed. This leads to
further degeneracies in the Brillouin zone (BZ) [32]. These degeneracies, and the
protected crossings between different bands that can result from nonsymmorphic
symmetry, can produce topological phases. Topology has received significant recent
attention in the condensed matter community, because topologically states (of any
origin) are more robust to perturbations. For that reason materials with topological
properties are one potential route to qubits to be used in quantum computers, as
any topological qubits could be much more robust to outside perturbations. Any
7
potential interplay between topological properties and superconductivity is also of
interest.
Work in magnetic fields has led to further signs of topological physics
in the B31 group. Linear magnetoresistance near the ordered magnetism-
superconductivity crossover in CrAs was linked to a ?semi-Dirac? dispersion near
the Fermi energy EF . A typical Dirac k -space dispersion is linear, resulting in mass-
less quasiparticles that are described by different physics than typical electrons or
holes with a quadratic dispersion. Linear dispersions are commonly seen in materi-
als with small Fermi surface pockets termed Dirac or Weyl (in the case of additional
broken symmetries) semimetals. A semi-Dirac dispersion is linear in one direction
but parabolic in the other [19]. The existence of a semi-Dirac point seems to be
common to all materials in this class. It was also reported in the same paper that
CrP had a semi-Dirac point at the same point in k -space located further from EF ,
and density functional theory (DFT) calculations for CoAs done for work related to
Ch. 6 have found the same for that material.
1.3 This Work
This chapter has laid out the basic concepts motivating this work. Chapter 2
will cover the theoretical grounding needed to understand the results to be presented.
Chapter 3 will go over sample synthesis and the types of experiments subsequently
performed on samples. Each of Chapters 4-7 will focus on a specific material: FeAs,
FeP, CoAs, and CoP, in that order. Chapter 8 will briefly look at the effect of
8
chemical substitution with Fe1?xCoxP and Fe1?xCoxAs. Different growth methods
will be compared, as well as basic physical properties that had not been previously
reported. The behavior of these compounds in high magnetic fields will be treated in
depth, especially quantum oscillatory phenomena in the arsenides and CoP, and the
large, anisotropic magnetoresistance of both of the phosphides. While the materials
are all related, the work done and discoveries made about each are not all exactly the
same. In the Conclusion, the main findings and unifying themes will be recapitulated
and ideas for future experiments suggested. The Appendix will round up related
work, on the Rh-based pnictides and WP, that was motivated by the main results
but not pursued to a sufficient degree to merit a standalone chapter.
9
Chapter 2: Theoretical Background
The behavior described the main section of this thesis, Chapters 4-8, will
cover a broad scope of physics of magnetism and magnetic fields. This chapter
will outline those concepts in advance, before more detail is added when discussing
specific materials. The first thing to cover is the role of magnetic fluctuations and the
type of magnetic order seen in these materials. This will be followed by a discussion
of magnetoresistance, a term referring to change of a material?s resistance in field.
Related to magnetoresistance (and, more broadly, the effects of magnetic fields) is
the phenomenon of quantum oscillations, the inverse field-periodic behavior seen in
many quantities (such as resistance) that has its origin in the quantization of energy
levels by an applied field. The techniques to measure the quantities to be described
here will be covered in the next chapter.
2.1 Magnetic Ordering
In a sense, magnetism was the first quantum effect harnessed by humans, with
the discovery of ferromagnetic minerals and their use for navigation in the form of
compasses guided by the Earth?s magnetic field. Of course, it took a few thousand
years for the accompanying theoretical explanation of magnetism. That being said,
10
the gap between the discovery and mathematical formulation of gravity is even
larger. Humans are not only experimentalists to take advantage of magnetism; birds,
bats, bacteria, and other organisms can sense the Earth?s magnetic field internally
and use it for orientation.
A thorough treatment of magnetism is found in Ref. [42], and a very condensed
version will be given here. Magnetic order is the coherent alignment of spins within
a lattice. In paramagnetic or diamagnetic materials, there is no spontaneous order.
They can only respond to an external field by either aligning or antialigning with
it, respectively. However, in the absence of field spins will be randomly oriented,
with interactions between spin sites too weak to generate a consistent structure. In
contrast, ferromagnetic, ferrimagnetic, and antiferromagnetic materials are charac-
terized by long range ordering that results from interactions between neighbors [43],
though often it takes an external field to be able to measure them1.
Ferromagnetic materials have a spontaneous magnetization below the Curie
temperature TC , coming from the positive alignment of all the spins [Fig. 2.1(a)].
Ferrimagnetism is similar, except that instead of a single magnetic structure there
are two, intertwined with each other. Moments on the two lattices point in opposing
directions, but with different magnitudes, and so there is still a nonzero magneti-
zation [Fig. 2.1(b)]. The oldest known magnetic mineral, Fe3O4 or magnetite, is
actually a ferrimagnet. The distinction between the two can only be seen with
probes like neutron scattering that can probe the magnetic structure on each site,
1In spin glasses, moments ?freeze? into locked positions at low temperatures. However, the
alignments are random, hence the term glass, and so they are better described as disordered
magnets.
11
rather than a bulk measurement such as magnetic susceptibility.
Antiferromagnets can in one sense be thought of as a specific case of ferri-
magnets. In AFMs, the net magnetization approaches zero below the transition
temperature, as the spins combine to cancel each other out [Fig. 2.1(c)]. The transi-
tion temperature is referred to as the Ne?el temperature, TN . In the simplest picture,
spins alternate between being up and down at each neighboring site, though in mul-
tiple dimensions a material may have an AFM arrangement in one direction, but a
full alignment of moments in another. More complicated structures that still result
in zero net magnetization, helimagnets and spin density waves [Fig. 2.1(d) and (e)],
will be treated in more depth in the forthcoming section.
Magnetic order and a tendency toward it are critical to describing the behavior
the MnP-type pnictides. This should comes as no surprise: only five elements
are magnetically ordered at room temperature, and four of them (AFM Cr and
FM Fe, Co, and Ni) are in the middle of the 3d block2. As mentioned in the
introduction, the competition between ordered magnetism and superconductivity
under pressure in MnP and CrAs has lead to speculation about potential quantum
criticality or an unconventional superconducting pairing mechanism [21, 44]. MnP
can order magnetically in several different ways depending on temperature, pressure,
and applied field. The materials to be described here-CoAs, CoP, FeAs, and FeP-
all have some relation to magnetism. The first two are paramagnetic but sits on
the border of ferromagnetism, while the latter two have similar antiferromagnetic
2The other is Gd, which with TC = 292 K may technically not quite count as ?room tempera-
ture? magnetism.
12
Figure 2.1: Illustrations of different kinds of magnetic structures. (a) Ferromag-
netism, where all spins point in the same direction. (b) Antiferromagnetism, where
spins point in opposite directions in a specific pattern, resulting in zero net mag-
netization. (c) A spin glass, where spins lock into place below the transition tem-
perature, but with an overall random orientation. (d) Spiral magnetism, where
the moment rotates with some canting into the propagation direction (e) Helical
magnetism (or helimagnetism), where the moment rotates in a plane perpendicular
to the propagation direction. Note that various arrangements within each of these
categories are also possible. Taken from Ref. [42].
13
orderings to CrAs and MnP.
2.1.1 Antiferromagnetism: Helimagnetism and Spin Density Waves
Four transition metal-pnictogen binaries are antiferromagnetically ordered at
low temperatures [29]: CrAs (TN = 264 K) [18], FeAs (TN = 70 K) [24], FeP
(TN = 120 K) [31], and MnP (TN = 50 K) [45]. In the last of these, the story
is more complicated, as it actually first enters a ferromagnetic state at 291 K and
below about 100 K can have one of five different magnetic orderings, with first or
second order phase transitions between them which occur at different combinations
of temperature with applied field direction and magnitude. But at base temperature
and low field, it takes on a ?screw? structure similar to helimagnetic state of CrAs
and FeP [29].
Helimagnetism takes its name from the helical shape traced out by the mag-
netic moments at each as one moves along the propagation direction. The spins
at neighboring sites differ in direction by a turn angle. One way of viewing FM or
AFM, in fact, is as helimagnets with turn angles of 0? or 180?, respectively [42].
Such a state results from competition between FM and AFM interactions, leading
to a final state somewhere between the two. The helimagnetism in the binary pnic-
tides is actually a bit more complicated than this description. It consists of two
moments, rather than one, in the ab plane, that propagate along the c-axis [29].
These moments have different magnitudes, but there is a constant phase difference
between them, meaning they have the same turn angle when moving to the next
14
Formula TN (K) q/2? ?? (
?) Average moment (?B)
CrAs 264 0.353 -126 1.73
MnP 50 0.112 16 1.58
FeP 120 0.2 169 0.42
Table 2.1: Quantities relevant to the double helical magnetic state of several B31
compounds. q is directed solely along the c-axis in each case. Data primarily from
Ref. [29].
site along the c-axis. Of the four transition metal sites in the B31 unit cell, two are
associated with each of the two moments. A full description of the magnetic state
only requires knowledge of the size of the moments, the propagation wavevector q,3
and the phase difference between the two moments ??. q is the quotient of the crys-
tallographic structural unit cell length and the magnetic structural unit cell length;
put differently, 1/q i is the number of unit cells covered in each crystal direction be-
fore the magnetic structure begins to repeat. If each component of 1/q i is a whole
number, the magnetic structure is termed ?commensurate? with the lattice, but in-
commensurate structures are also common in antiferromagnets. Relevant values for
the helimagnetic state of CrAs, MnP, and FeP from Ref. [29] are given in Table 2.1.
The phase differences close to ?180? in CrAs and FeP mean AFM interactions are
dominant, while the value near 0? in MnP indicates strong FM fluctuations, logical
given the transition to a FM state at higher fields or temperatures. The moments
are also much larger in CrAs and MnP; this can be linked to their higher transition
temperatures, perhaps a sign that FeP could have its magnetic state more easily
suppressed with pressure.
3Where their orientation is important, vector quantities will be in bold lettering throughout
this work. Otherwise, they will be written with normal font.
15
(a) (b)
(c) (d)
Simple spiral
Noncollinear SDW
Figure 2.2: (a) A spiral (or helimagnetic) magnetic ordering depicted to fit results for
FeAs. Note that there are two separate lines of propagation along the c-axis with a
phase difference between them. (b) A collinear spin density wave, in which spins do
not rotate but rather change magnitude along [001]. The actual antiferromagnetic
state was found to be noncollinear SDW, which corresponds to the state depicted in
(b) with the spins rotated to point in between the a- and b-axes. (c) The projection
of the simple spiral and noncollinear SDW spins onto the ab plane; the former is
circle, the latter an ellipse. (d) Neutron scattering data showing the change in
the c-axis component of the ordering vector q with temperature. The value of
about 2.39, not easily written as a fraction, means that the magnetic structure is
incommensurate with the lattice structure. All elements of this figure are taken
from Ref. [46].
16
It was initially thought that FeAs took on the same magnetic structure as the
other three magnetic B31 materials [24], as it was observed to be antiferromagnetic
and had obvious compositional and structural similarities to them. Figure 2.2(a)
illustrates a spiral structure on the FeAs lattice [46]. More in-depth neutron scatter-
ing work, however, has shown that FeAs is better described by a noncollinear spin
density wave (SDW) order [46], similar to that of Fig. 2.2(b) except that moments
are tilted in the ab plane. This means that instead of rotation in the ab plane of the
two moments, they always point in the same direction but have a differing amplitude
at different points along the c-axis. The reason there is a noticeable moment in both
the a- and b-axis directions is because the moments point at an angle between the
two, resulting in a projection along each principle axis [Fig. 2.2(c)].
2.1.2 Near Ferromagnetism and Magnetic Fluctuations
Many materials experience significant magnetic fluctuations, even if they never
reach an ordered magnetic state. They are termed ?nearly magnetic?. In such a case
spins may interact over a finite distance, but there is no coherent structure across
the entire sample. Frequently, nearly magnetic materials are closely connected to
others that are ordered. A simple example is Pd, which sits under ferromagnetic Ni
on the periodic table. As a film or 1D wire, it can display ordered magnetism [47]. In
bulk, however, it is paramagnetic but with indications of strong fluctuations. A more
exoctic case is the spin-triplet superconductor UTe2, which has many commonalities
to other U-containing ferromagnets, but seems to sit just on the wrong side of the
17
FM-PM dividing line [48]. In some ways nearly magnetic materials may be more
interesting than ordered magnets. After all, it has been established that interesting
physics often arises near the suppression of a magnetic transition, as in the case
of CrAs and MnP. Studying materials dominated by short range interactions but
without an overall structure may skip the need to suppress a strong ordered state.
To appreciate how fine the division between FM and nearly FM materials is, it
is useful to look at the Stoner theory of ferromagnetism. It applies to itinerant fer-
romagnets, where FM arises from partial filling of energy bands (as in the d -electron
transition metals), rather than the localized moments coming from the atoms that
lead to magnetism in the rare earths. The condition for a Stoner ferromagnet is
?0?BIg(EF ) ? 1 [42], where ?0 is the permeability of free space, ?B a Bohr magne-
ton, I the ?Stoner parameter?, which represents the Coulomb interaction between
itinerant electrons, and g(EF ) is the density of states at the Fermi energy. This
inequality comes from consideration of the energy savings obtained in the process
of spin-splitting bands in the absence of field.
In a material with an initially equal number of up and down spins (n? = n?)
the energy cost of flipping all spins (from, say, down to up) within a small energy
range ?E of the Fermi energy E is 1F g(EF )(?E)
2. The energy savings come from
2
th?e change in magnetization M = ?B(n? ? n?). The energy of the magnetic field is
1 H?B dV. In a material this is -?0IM2 = -?0?2 I(n ?n )2B ? ? . After flipping the spins,2
n 1? = (n+g(EF )?E) and n
1
? = (n?g(EF )?E). Therefore the overall energy change2 2
?E = 1g(E )[(?E)2F (1? ? ?20 BIg(EF )]. Spontaneous magnetization is only possible2
when ?E is not positive; i.e., when ?0?
2
BIg(EF ) ? 1. The threshold value can be
18
reached (or nearly reached) with a large density of states at the Fermi energy and
a large Stoner parameter. The definition of the latter has been rather vague so far,
and it is an overall complicated quantity. It can be theoretically calculated, however,
and essentially represents the strength of the exchange interaction between electrons.
Multiplied by EF , the Stoner criterion essentially says that ferromagnetism occurs
when there are a large number of electrons at the Fermi energy strongly interacting
with each other, a conclusion that is not too surprising.
Using the Stoner criterion, it is possible to judge how close a material lacking
order lies to a FM instability. One way to do this with experimentally measurable
quantities is the dimensionless Wilson ratio
4?2k2
R = B
?0
W (2.1)
3? (g ? )20 e B ?
Here kB is the Boltzmann constant, ?0 the 0 K spin susceptibility and ge the
electron g-factor. Note that calculating RW involves quantities obtained through
specific heat and magnetization measurements, which makes sense since those quan-
tities are most sensitive to the behavior of electrons in the system. RW can in fact
be reformulated as the Stoner factor Z = 1- 1 . RW is unity (and Z = 0) for a freeRW
electron gas. Larger (smaller) RW (Z) values mean stronger FM fluctuations: in
Pd RW = 6?8 [49], for BaCo2As2, thought to be near a magnetic quantum critical
point, it is 7?10, depending on field orientation [50].
An Arrott plot can be used to determine whether a material is FM, and how
close it may be to the crossover [49]. If a plot of M2 vs. H/M has a positive y-
19
intercept, then there exists a spontaneous magnetization in zero field [12]. Such a
plot is a simple way of visualizing and assessing the presence of long range order.
An example of an Arrott plot, taken from Ref. [51], is given in Fig. 2.3.
2.2 Magnetoresistance
As has just been seen, the interactions between electrons can generate inter-
esting magnetic phenomena. Reversing this in a way, magnetic fields can change the
behavior of the electrons in a lattice. Beginning from the simplest considerations,
the Lorentz force on a charged particle in an electric and magnetic field is
( )
F = q E + v ? B (2.2)
In the presence of only a magnetic field, a free electron will be accelerated in a
direction perpendicular to its velocity and the field. Thus it will execute a circular
orbit in the plane normal to B4. Unsurprisingly, this changes substantially for an
electron in a solid, under the influence of an applied current, the band structure,
and interactions with other particles. Any change to the electron?s behavior in that
case will also change its path through a material. On a large scale, the behavior of
electrons will be altered, and any resultant change in transport properties is known
4This may the best place to mention that throughout this thesis, the symbol B will be used
for ?magnetic field? and the units will be in T. Some authors prefer to use H (officially ?magnetic
field strength?.) and Oe, with B designated the ?magnetic flux density?. The distinction between
the two is that H is the applied external field, while B is the measured field at a point in space,
which may differ from H due to the magnetization M of the medium at that location. In reality,
the difference is often ignored and generally it is assumed B = ?0H. For example, the magnetic
field when taking data at the NHMFL is listed in T, while the instruments at UMD have it in Oe
(? ? 1050 Oe = 1 T). A similar argument for this convention is found in Ref. [52].
20
Figure 2.3: An example of the use of an Arrott plot to find a paramagnetic-
ferromagnetic transition temperature of Gd5Si2.7Ge1.3. Curves below 305 K have
a positive y-intercept when extrapolating from high H/M, indicating remnant zero-
field magnetization and thus ferromagnetism. Axes were rescaled by powers of the
critical exponents ? and ? to obtain linear behavior and make the extrapolation
easier. Taken from Ref. [51].
21
as magnetoresistance (MR). It was found in the 19th Century that the resistance
of metals such as copper or iron would change in field. However, the effect is weak
in most typical metals, especially in small fields and at room temperature. On the
other hand, in other materials and at lower temperatures resistance can be many
orders of magnitude larger or smaller in the absence of field.
There is a basic, relatively straightforward expectation for what should happen
to a metal?s resistance in a magnetic field. Any deviations are that are of interest
in what they can elucidate about a material?s properties. In some cases, this is
because the material itself has an inherent feature that magnetic fields help to reveal.
In others, such as those exhibiting quantum oscillations, applied fields change the
arrangement of electrons in a system in a way that can only be explained with
quantum mechanics. Naturally, stronger magnetic fields amplify these effects, hence
the reason that many experiments to be covered were performed at the National
High Magnetic Field Laboratory. The phenomena to be explored could increase
linearly, quadratically, or even exponentially with increasing field.
2.2.1 The Basics
To get to the simplest approximation of magnetoresistance, one can start with
the foundational approximation of transport in a solid: the Drude model. In this
picture, electrons in a material move with some drift velocity v and scatter randomly,
on average, once in a time interval ? called the ?relaxation time?. If an electric field
is applied, then consideration of 2.2 shows that they will build up a momentum
22
2
mv = qE? . The current density J = qnv can then be reexpressed as J = nq ? E.
m
It is also known from Ohm?s law that J = ?E, where ? is the conductivity. The
Drude conductivity is then
nq2?
?0 = (2.3)
m
with the ?0? subscript denoting the zero field value5. The resistivity ? = ??1.
From here, the next step is to see the effect of an applied magnetic on the
resistance in the current direction (the longitudinal resistance). It will turn out
that in such a model the prediction is that the resistance in the direction of applied
current should be independent of magnetic field, but the derivation is still informa-
tive. Consider an electron with charge -e moving through a material with E and B
applied. Due to interactions, it has an effective mass m? that is not necessarily the
electron rest mass me. Taking Drude physics into account, now the motion can be
described by
(
? dv v
) ( )
m + = ?e E + v ? B (2.4)
dt t
Due to the presence of a cross product, electron motion along the direction of
applied field will be unaffected by B6. The issue is now to determine the current
5Similarly, ?0 in this section refers to the resistivity when B = 0 T.
6When field and current are parallel, there should be no magnetoresistance since v and B are
parallel, and in practice MR with parallel field and current is small for most samples (including
those to be covered here), to the point when it could be attributed to slight misalignment of
the sample, electrical contacts, and/or field. Current jetting or the chiral anomaly in topological
semimetals can result in nonzero longitudinal magnetoresistance. For data shown throughout this
work, field was perpendicular to current.
23
density J in the plane normal to the field. Taking B = Bz? and E = Ex?, the task is
to solve for J ? x? and J ? y? in steady state (dv = 0). Two separate equations emerge
dt
from this:
e?
vx = ? ? (Ex + vyB) (2.5)m
e?
vy = vxB (2.6)
m?
J must satisfy two separate relations: J = -nev and J = ?ijE. At this point it
is important to emphasize that the conductivity ?ij is a tensor, describing direction
of the current, j, produced by an electric field in the i direction. This was skipped
over in the Drude approximation since J and E were always parallel, but in this
example there will be two components, ?xx and ?xy, are relevant. Making the
appropriate substitutions and rearrangements (solving first for vx, then vy) leads to
two somewhat unwieldy equations
ne2?
?
Jx = (m )2Ex (2.7)
1 + e?
m?
B
?ne
2? (ne?m?Jy = ? )2Ex (2.8)m
1 + e??Bm
It has already been seen that ne2?/m? = ?0. The quantity eB/m
? has units of
inverse time and will be called the cyclotron frequency, ?c, and is the angular speed
of the circular orbit the electron makes perpendicular to the field. The conductivities
24
are the prefactors on the right hand side of the above equations and become
?0
?xx = (2.9)
1 + (? 2c?)
?0?c?
?xy = (2.10)
1 + (?c?)2
The magnetic field in z? has led to a change in the conductivity in x? and y?.
This derivation can be repeated with E = Ey? to get the full in-plane conductivity
tensor
?? ? ? ????xx ??xy??? ?0 ?? 1 ?c??? = = ? (2.11)1 + (? ?)2 ? ?c
?xy ?xx ??c? 1
But the interest here is in the change in ?. In the Drude model ?0 = ?
?1
0 , but
because here ? has become a 2 ? 2 tensor, ?xx =6 1/?xx (and the same is true of
other components). After matrix inversion, the resistivity tensor becomes
? ?
1
? = ??? 1 ??c???? (2.12)?0
?c? 1
and so ? Bxx = ?0, and ?xy = ??0?c? = ? . ?xy is called the Hall resistivityne
and will be covered later. More important for now is that ?xx is unchanged. In other
words, there is no magnetoresistivity, despite the change in conductivity in field.
The reason for the lack of change is that this initial derivation was too sim-
plistic. A more realistic picture explains the origin of magnetoresistance by doing
25
away with the assumption that all carriers have the same n, ? , and m?. In a real
metal, multiple bands can contribute to transport, and each can be described inde-
pendently with their own ?xx,i and ?xy,i. The total J will be the sum of each Ji.
Unless each individual current density coincidentally points in the same direction,
| Jtotal | ? | ? Ji |. The current density has to go down, so by Ohm?s law the
resistivity must go up.
It is useful to illustrate this with a slightly more complex example than the
first, with a single hole and electron band involved in transport. It is also simpler
to write the conductivity as comprising an in phase (magnetoresistive) and out of
phase (Hall) component [52]:
?0,e ?0,h
?total = + (2.13)
1? i?c? 1 + i?c?
Where ?0,i is the zero field Drude conductivity for each band, the electron
band denoted by the ?e? subscript and the hole band by ?h?. The difference in sign
of the imaginary component of the denominators is due to the difference in carrier
sign. Substituting 2.3 for each band gives
[ ne? n ]e h?h
?total = e + (2.14)
1? i?eB 1 + i?hB
?
where ?i =
0,i is the mobility of the electron or hole band (which to first
nie
approximation is field independent). The longitudinal resistivity ?xx is the inverse
of the real part of this equation. Skipping to the desired result the formula for the
longitudinal magnetoresistance with one hole and one electron band is [53]
26
1 (nh?h + ne?e) + (nh?e + ne?h)? ?
2
h eB
?xx = (2.15)
e (nh?h + ne?e)2 + (nh ? ne)2(?h?eB)2
There is now a clear field dependence of the longitudinal resistance. But
note that in the case of only one carrier type (? = 0 for the electrons or holes)
resistivity will be independent of field again. It is insightful to look at other limiting
cases. In low field, as long as the carrier concentrations of the two bands are at all
comparable, the second term in the numerator will be larger than the second term
in the denominator, and a B2 dependence results. At high field the second terms in
both the numerator and denominator are large and the field dependence cancels out.
Generically, then, the magnetoresistance of a metal should have an initial quadratic
field dependence, and then saturate. While it might seem simple, this is actually a
good description of the MR in a number of real materials [52].
2.2.2 Large Magnetoresistance
The MR of a metal is often small, especially above cryogenic temperatures. An
empirical way of judging if a significant MR effect should be seen is by calculating
?c? . This quantity describes how much of its cyclotron orbit a carrier completes
before being scattered. As a rule of thumb, a value greater than one is generally
required for significant MR. The term can be rewritten as ?c? =
B?0 , a definition
ne
from which it is clear that not just high field, but good conductivity (low resistivity)
and low carrier concentration can also make MR more noticeable. The last of these
is why many semimetals or semiconductors have a large MR, in addition to the
27
electron-hole compensation effect to be discussed shortly. Many metals, such as
copper, also show negligible MR at room temperature but a much larger increase of
resistivity in field at low temperatures when their conductivity is much higher.
There are more exotic sources of large MR, and such a property can be useful
for applications. The 2007 Nobel Prize in Physics was awarded for the discovery
of giant magnetoresistance in ferromagnetic-nonmagnetic layered materials [54]. In
those materials, the orientation of the spins in ferromagnetic layers can be con-
trolled by an external field. Parallel alignment of adjacent layers will have a much
lower resistance than an antiparallel configuration, and so it a significant change in
resistance will be seen when spins are flipped. Giant MR materials have found ap-
plication as magnetic field sensors in hard disk drives and magnetic random access
memory, among other areas. A variety of other mechanisms can cause an increase in
MR and go by different names, though there is some inconsistency in which terms
refer to specific physical phenomena, and which are more a general comment on MR
size. Beyond ?giant? [54], there are also ?colossal? [55], ?extreme? [56], ?extraordi-
nary? [57], ?titanic? [58], and ?huge? [59] magnetoresistive effects. This thesis will
use the word ?large? to refer to the MR of FeP and CoP, as it is generic enough to
not imply a specific mechanism at the outset.
Equation (2.15) was examined in limiting cases of magnetic field in the previ-
ous section, but what was not explored was its dependence on the relative carrier
concentrations. Looking again, it is clear that a small nh ? ne will result in a large
MR in high field when the B2-dependent terms dominate. This state of nearly equal
numbers of electron and hole carriers is referred to as ?compensation?, and occurs
28
frequently in semimetals, which are prone to combining multiband conduction from
pockets with low carrier concentrations. In the earliest days this was seen in the
pnictogens (As, Sb, and Bi) [52]. More recently it has emerged in the study of
topological semimetals such as WTe2 [53] and metal-pnictogen compounds (forming
in other structures than B31) like NbP, LaSb, and TaAs2 [56, 60,61].
But while it has recently been seen in many topological semimetals,
compensation-driven large MR is not on its own exotic, as it has been seen to
come neatly out of a limiting case of the the two-band model and the increase of
MR for materials with a lower carrier concentration. In a way, the reliability of
the model in an extreme limit is remarkable, and shows that it is actually a useful
description, rather than too basic of an approximation. The reason it appears in
topological semimetals is because, as semimetals, they are likely to have very simi-
lar, and small, electron and hole densities. Other probes of transport in field, such
as Hall effect measurements and examination of the field dependence of large MR,
can help verify potential topological properties [62], to see nothing of direct imaging
of band structure as is possible with angle-resolved photoemission spectroscopy.
At the end of the previous section it was stated that in the two band model,
MR should saturate at high fields. This is common, but there it is also common for
MR not to saturate, or to only do so at certain angles. This is true in the case even
of seemingly simple conductors like Sn or Pb, as shown in Fig. 2.4, which features
data from Ref. [63]. At two angles 30? apart, the magnetoresistance differs by an
order of magnitude or two, and the higher magnetoresistance orientation does not
saturate. This difference comes from whether carriers make open or closed cyclotron
29
(a)
Sn Pb
(b)
B
Figure 2.4: (a) The magnetoresistance of (left) Sn and (right) Pb at two different
angles corresponding to the maximum and minimum of high field magnetoresistance.
In each plot the left vertical axis corresponds to orientation ?A? and the right one to
?B?, and the MR differs by orders of magnitude with orientation. This is because the
MR saturates for closed Fermi surface orbits, but does not for open ones. (b) (left)
Closed and (right) open electron orbits around the Fermi surface for a magnetic field
out of the plane. The closed orbit results in the electron retracing the same path,
while in the open orbit the velocity never makes a full loop. (a) is from Ref. [63],
(b) from Ref. [64].
30
orbits perpendicular to the magnetic field [52].
A (moderately) more precise definition of ?high field? is the condition that
?c? >> 1. It is at this point that the scattering time is much longer than timescale
set by field-induced motion. As a result, magnetoresistive effectives are determined
more by Fermi surface structure than by scattering. It was noted at the outset that
an electron in a magnetic field traverses a closed circuit perpendicular to that field
in both real and momentum space. As B increases, the real space radius of the
cyclotron orbit will correspondingly get smaller. If the field is strong enough, then
electron motion will be almost entirely circular, and will not move along the direction
of the applied electric field E unless it scatters off of an impurity. A strong enough
field results in an orbit radius shorter than the mean distance between impurities
quantified by ? . At that point, higher fields (and smaller orbital radii) have a
negligible effect on scattering, since the electron can complete its orbit without
running into a scattering center that could redirect it along the low-high voltage
path. The resistance will be independent of field above this threshold, and the
resistance will be higher than at zero field but saturate; this was already shown
mathematically with the two band model 2.15.
That is the cased for a closed orbit, where the electron?s velocity, like its
position, can fully rotate and come back to the same direction [Fig. 2.4(a)]. However,
the magnetoresistance will not saturate for an open orbit [Fig. 2.4(b)]. As an electron
moves in real space, it must also obey the trajectories allowed by the band structure
in reciprocal space at the Fermi energy. If the 2D Fermi surface projection does
31
not form a closed loop, the electron will move without interruption in the same
direction. Its velocity will then remain pointed in mostly the same direction and
increase with field. Thus MR will not saturate but rather will remain quadratic in
the high field limit. Because of this, the field dependence at different angles gives
information about the shape of Fermi surface, and whether it is open or closed in the
plane perpendicular to the applied field. Even in cases when the field dependence
does saturate, the threshold field is related to ?c? and thus the effective mass or
scattering time in the corresponding plane. Angular dependence of MR can give
useful information about the Fermi surface, as exemplified by Fig. 2.4.
2.2.3 Linear Magnetoresistance
Equation 2.15 gives a good idea of field dependence of the resistance in many
metals, but significant deviations from these expectations indicate unusual scattering
processes. Of particular interest has been the observation of linear magnetoresis-
tance (LMR). Nonsaturating linear MR is especially sought after for its possible use
in magnetic sensors [65], for which a simple field dependence is optimal. The ap-
pearance of this phenomenon in heavily-investigated materials such as graphene [66],
topological insulators, the topological semimetals, and, originally, simple elemental
metals like Na, Al, and K [67?69] has led to it being the subject of much theoretical
consideration.
One difficulty with LMR is that, like large MR, there are a variety of potential
mechanisms of both classical and quantum origin [70]. Even simple disorder is one
32
known cause of linear MR in single crystals and polycrstals. Inhomogeneities or dis-
order cause fluctuations in carrier mobility and density that can redirect the current
through a section of the sample [71, 72]. This can be explained by considering the
interplay between disorder and ?guiding-center motion? [73]. In this scenario, carri-
ers are described as having an overall average a drift velocity, while at the same time
making orbits in the plane perpendicular to the field, analogous to cycloid motion.
A disorder potential with a correlation length much larger than the cyclotron radius
will confine carrier motion to the plane normal to the field. This leads to LMR, as
the effect on the conductivity is similar to the Hall effect, which is linear in field.
This is the reason large, linear MR can emerge in graphene when the sample size
exceeds the domain size [66], and in 3D materials like Cd3As2 [74].
Another potential driver of LMR is proximity to a quantum critical point.
Typically, a phase transition is dominated by thermal fluctuations around the tran-
sition temperature. However, when the transition is suppressed to 0 K by some other
parameter (magnetic field, chemical doping, applied pressure), there are no thermal
fluctuations and instead quantum fluctuations dominate [75]. As a result there is
no internal energy scale for the material to reference, and the correlation length and
time of the order parameter both diverge. Thus observable quantities should be
scale-invariant, and this can result in a linear dependence of the resistivity on both
temperature and field, an effect that has been seen in the iron pnictides [76] and
cuprates [77]. This is in fact one motivation behind high magnetic field research, as
in both cases MR could be astonishingly linear up to 100 T, a strong indicator of
QC physics. The observation of quasilinear MR in CrAs near an AFM-SC crossover
33
has been seen as hinting at a potential QCP for this very reason [19].
Fermi surfaces under specific conditions can give rise to LMR as well. If the
conduction is dominated by a pocket with a linear dispersion, the resulting field
dependence can also be linear [78]. This is because the effective mass for a linear
dispersion is zero, resulting in a very large ?c? . This theory has often be used
to explain the observation of large, nonsaturating LMR in topological semimetals,
which have small pockets with a linear (termed ?Dirac? or ?Weyl?) dispersion. But
this effect is not confined only to materials with a single pocket, and applies equally
well to Fermi surfaces that have other larger pockets that follow the semiclassical
description above. As long as the smaller pocket dominates conduction, and field is
high enough, linear MR is achievable. The requirement on magnetic field involves a
discussion of Landau level quantization that moves the description from semiclassical
to quantum. For that reason it will be saved for Ch. 5 on FeP, which shows angle-
dependent linear MR, since a full discussion of Landau level quantization will not
be discussed until covering quantum oscillations.
2.2.4 The Hall Effect
The Hall effect was briefly touched on in the first derivation of magnetore-
sistance, where it was noted that in the most basic consideration of MR a voltage
perpendicular to both field and current will arise7. The corresponding Hall resis-
tivity ? = ? Bxy for electron carriers, with an opposite sign for hole carriers. Thene
7This is known as the ?Hall voltage? after its discoverer, and all quantities associated with the
transverse voltage arising from a magnetic field are generally called ?Hall (term)?.
34
process to arrive at that result can be restated in a way that focuses on the Hall
effect and perhaps makes it more intuitive.
In a magnetic field, carriers that normally move linearly between the two
current leads will be deflected by the Lorentz force 2.2. Again it is simplest to
define x? as the current direction, y? as the direction between the two voltage leads,
and z? as the direction of applied field. In steady state, F = 0. Breaking the force
equation into perpendicular components and noting that in general Eij = Vij/xij,
in the y? direction 0 = Vy ? vxBz, or Vy = vxBz , where w is the sample width.w w
From the Drude model Ix = q ? ntw ? vx, with t the sample thickness. Substituting
for vx and dividing the voltage by the current gives a resistance equal to
1 B.
qnt
So, in a single band model the transverse (Hall) voltage has a linear dependence on
magnetic field, with a positive or negative slope depending on if the carriers are holes
(q = e) or electrons (q = ?e). Defining a ?Hall coefficient? R = ? 1H results in thene
equation ?xy = RHB, where ?xy = Rxyt is the Hall resistivity. Note that, somewhat
confusingly, despite the employed symbol the Hall coefficient is not a resistance.
Because multiple bands often participate in conduction, RH can be a more useful
quantity than n (which implicity assumes a single band), and here will be given
in units of cm3/C. In the case of a simple, single band material, measurements of
the Hall coefficient provide immediate information about the sign and density of
the carrier, because electrons and holes are deflected in the same direction by a
magnetic field since both their velocities and charges have opposite signs. The sign
of the transverse voltage difference will then also depend on the carrier sign.
Of course, as with magnetoresistance, materials rarely behave at the limit of
35
simplicity, and frequently have multiple carriers of various types and concentrations.
The same two band model used for MR (2.14) can also be applied to the Hall effect.
The only change is the reciprocal of the imaginary part of (2.14), rather than the
real part, is used [52,79]:
B (n 2 2 2h?h ? ne?e) + (nh ? ne)(?h?eB)?xy = (2.16)
e (n 2 2 2h?h + ne?e) + (nh ? ne) (?h?eB)
Again it is instructive to look at extreme cases. When one type of carrier
dominates, the mobility of the less dominant carrier is effectively zero. The second
terms in the numerator and denominator can be neglected and ?xy retains its simple
linear field dependence. But when carrier concentrations are comparable, nh ? ne is
very small, and the second term in the denominator cannot be neglected. Therefore
the Hall resistance will have a significant field dependence, and can even change
sign with field [79]. In a middling case, a higher field will lead to a gradually
less linear dependence. Fitting a nonlinear Hall resistance and an abnormal MR
can give information about the concentrations and mobilities of electron and hole
carriers. There are four variables (the two ? and n values) to solve for with just
two equations, but usually reasonable assumptions can be made to get qualitatively
informative answers.
2.3 Quantum Oscillations
Magnetoresistance can shed light on the interactions experienced by carriers as
they move through a solid. But it is also possible for a magnetic field to lead to a new
36
ordered phenomenon of these carriers. A notable example of this is the oscillatory
behavior that arises in a variety of quantities and found to be periodic in inverse
field. These ?quantum oscillations? have been observed since the 1930s, initially in
semiconductors thanks to their large, compensated MR [80]. Subsequent theoretical
explanation showed how oscillations could be analyzed to give information about the
Fermi surface geometry, carrier concentration, effective mass, and other quantities,
in a subfield referred to as ?Fermiology?.
2.3.1 Origin
The Hamiltonian of a plane wave traveling through a solid is
[ p?2 p?2 p?2 ]2x yE? = x?+ z
? 1 ? 1
y? + 1 z? ? (2.17)
(2m ) 21 (2m 22) (2m
?
3) 2
where E is the energy, ? the electron wave function, p?i the momentum operator
in each direction, and mi the effective masses in each direction. To deal with a
magnetic field B = Bz? requires introducing a magnetic vector potential A for which
B = ? ? A. This gives some freedom in defining A, and a convenient choice is
A = (0, Bx, 0), called ?Landau gauge?. The momentum operator p then becomes
p + eA. As long as the conjugate position operator does not appear in 2.17, the
momentum operator pi can be rewritten as h?ki. Additionally, there are no cross
terms when squaring each term in the brackets. Making all of these changes, (2.17)
becomes
37
[ p2 (h?k + eBx)2 h?2k2]y
E? = x + + z
2m? 2m? 2m?
? (2.18)
1 2 3
Introducing a constant energy shift E ? = E ? (h?2k2z/2m3) and redefining
x0 = ? (h?ky/eB) simplifies the equation to
[ p2? x (eB)2 ]E ? = ? + ? (x? x0)2 ? (2.19)2m1 2m2
which has the same form as the one-dimensional harmonic oscillator with fre-
1
quency ? = eB/(m1m2) 2 and energies E
? = (l+ 1)h??. This results in a total energy
2
2
E = (h?k) + (l+ 1)h??. It can be shown with more effort that ? for this geometry is
2m3 2
the same is the same as the cyclotron frequency ? eBc = ? .m
The implication of this result is that the energy of electron motion in the plane
perpendicular to the field is quantized, just like the energy levels of the quantum
harmonic oscillator. In this case they are known as ?Landau Levels? (LLs). The
in-plane momentum-space areas of these orbits are similarly quantized. One way
of picturing them is as tubes that extend infinitely in the kz direction but with
cross sections whose radii are determined by the Fermi surface shape and strength
of the magnetic field. The cross sections will expand with field but be cut off when
the intersect with the edge of Fermi surface, since carriers cannot have energies
greater than the Fermi energy. Figure 2.5, from Ref. [81], illustrates this in the
case of a circular Fermi surface, but it also applies to more complex geometries.
The separation in energy between two LLs is ?E = h?? = h?eB? . The effectivem
mass can be replaced with its original definition in the context of band structure
38
m? = h?
2 ?A . If the separation of the energies is much smaller than the energies
2? ?E
themselves (which, since the latter are on the order of EF , is almost always true)
then ?A ? ?A . Substituting in for ?E and rearranging terms gives the separation
?E ?E
in k-space between two Landau tubes:
2?eB
?A = (2.20)
h?
As noted, the area of these tubes in the plane perpendicular to the field is
determined by the field strength; increasing field increases the area. The interesting
part comes when the area surpasses the largest cross sectional area of a Fermi pocket,
Aext. In Fig. 2.5, that is when the radius of the tube becomes equal to the radius of
the circle. At that point the density of states of this band must go to zero, as the
carrier energy cannot exceed EF (ignoring for now the slight softening of this limit
from the Fermi-Dirac distribution). Thus a LL will be depleted every time the field
changes by B h?F = Aext. BF is the frequency of quantum oscillations, which will2?e
evidently be periodic in 1/B.
The reason this behavior is oscillatory is the depletion of carriers from a LL in
increasing field as it passes through the Fermi level. LLs are theoretically infinitely
narrow, but in practice broadened by temperature, impurities, etc. With increasing
field, the DOS increases as the band edge, peaks as it passes through EF , then
decreases and goes to zero as the second half of the band passes through, before
increasing again after a period 1/BF when it hits the edge of the next LL. The
density of states at EF , g(EF ) will then show oscillatory behavior periodic in inverse
39
Figure 2.5: The quantization of energy levels into ?Landau tubes? with different n
values by a magnetic field along the z -axis. The size (and thus both in-plane area
and quantum oscillation frequency) are bounded by where they touch the spherical
Fermi surface. Taken from Ref. [81]
40
field. A final thing to note is that QOs imply a closed Fermi surface at a particular
orientation; if it were open, then there could not be a cyclotron frequency as the
electron would never complete a circuit.
2.3.2 Detection and Analysis
QOs come from a change in the density of states at the Fermi level, so they
can be detected in any physical property that depends on g(EF ). The two most
commonly presented, and not coincidentally probably the two most straightforward
to measure, are oscillations in electrical resistance and magnetic susceptibility. In
this case they are referred to as Shubnikov-de Haas (SdH) or de Haas-van Alphen
(dHvA) oscillations, respectively. However, oscillations can also be seen in thermal
conductivity, sound velocity, and even in sample length. This thesis will only ex-
amine SdH and dHvA oscillations. In fact, it will focus primarily on oscillations of
the magnetic torque. The reason for this is again simply one of practicality: the
cantilevers used for torque measurements are in general more sensitive than the
equivalent resistance measurement, and have a simpler background to subtract.
2.3.3 Utility
The general procedure to extract quantum oscillation frequencies from mag-
netic torque was this: a polynomial (typically a third order one was sufficient) was
subtracted from the raw signal. The difference between the two is ideally a purely
oscillatory component centered around zero. Those data were interpolated so as to
41
be evenly spaced in inverse field. A fast Fourier transform (FFT) was then per-
formed on them to find the oscillation amplitude of different frequencies. When
comparing amplitudes for data at different angles or temperatures, the same field
range was used for fitting and interpolating (with the same number of points), since
changing either would affect the resulting amplitude. In some cases in other work,
the original signal itself has been fit globally through a general description of the
oscillation amplitude called the Lifshitz-Kosevich (LK) formula. Analysis of the
angular, temperature, and field dependence of oscillations can give an abundance of
information about a material.
As noted, the QO frequency (in units of magnetic field) is set by the extremal
area of an orbit around a Fermi surface pocket. As a result, it can be directly
equated to a cross sectional k -space pocket size. Many materials will show multiple
oscillation frequencies. This can result from either different extremal paths around
the same pocket, or from multiple pockets located at different points of the Brillouin
zone. A simple way of picturing the former is to imagine a cylinder with its axis
along z? and a radius that changes with height. If field is applied along the cylinder
axis, carriers will have different x?y plane path lengths depending on their location.
These can be referred to as ?neck? and ?belly? orbits and this line of thinking can
readily be extended to more complex shapes.
By changing the angle at which field is applied to the sample, one can then map
out the extremal area of a pocket and partially reconstruct the full shape. These
can be compared to calculations from density functional theory (DFT) using the
Supercell k -Space Extremal Area Finder (SKEAF) program [82]. But even without
42
theoretical calculations, a general idea of the Fermiology can often be obtained. If
a Fermi surface pocket is circular, the frequency will not change with angle. In
contrast, a cylindrical pocket will have a frequency that goes as 1/cos ?, where ?
is the angle between the applied field and the z-axis of the cylinder. Often, seeing
that a pocket?s angular dependence approximates this, and thus is relatively two
dimensional, can be useful information. Note that in this case the frequency will
diverge when field is applied perpendicular to the cylinder axis and cyclotron orbit
is open, the same condition leading to nonsaturating magnetoresistance. Having a
circular pocket (or something close to it) also makes it possible to estimate the carrier
concentration of that pocket. I(f kF)is constant in all directions then Aext = ?k
2
F and
3
the carrier concentration n = kF 22 . Substituting these into into the equation for3?
the oscillation frequency gives
1 (2eF ) 3
n = 2 (2.21)
3?2 h?
The oscillation amplitude will decrease with increasing temperature, as the
Fermi-Dirac distribution blurs the EF cutoff and the LLs broaden. Very roughly,
oscillations will be visible when h??c > kBT . Tracking the decrease in amplitude
with temperature gives an estimate of effective mass through one of the terms in
the LK formula, here to be called the LK factor
?m*T/(Bme)
RT = (2.22)
sinh(?m*T/(Bme))
where me is the electron rest mass and ? = 2?
2ckB/eh? ? 14.69 T/K with
43
c the speed of light and kB the Boltzmann constant [80]. This effective mass is an
average of the effective mass for every direction in the orbital plane. Larger masses
will decay more quickly with temperature, while smaller ones survive to high tem-
peratures, in the most extreme cases to perhaps 100 K or more, but for the materials
here about 10-20 K. With the effective masses it is possible to calculate the Som-
merfeld coefficient ?, which can also be obtained from heat capacity measurements.
In the case of multiple frequencies, their contributions can be added linearly. Com-
parison of ? values obtained through QOs and measurements of specific heat can
give insight into whether all carriers have been observed in QOs analysis.
The LK formula also includes what is called the Dingle factor,
RD = exp[??m*TD/(Bme)] (2.23)
where TD is the Dingle temperature [80]. This accounts for the increase in
oscillation amplitude with field, which is exponential in 1/B. This is why high mag-
netic fields can be so useful in observing QOs. TD is proportional to the scattering
rate ? as
h?
TD = ? (2.24)
2?kB
? is also the inverse of the scattering time ? . From this quantities it is then
possible to extract the mean free path
h?kF 1
` = ? (2.25)m ?
44
by assuming the cross section is circular. If that assumption does not hold,
then similar to m? this only approximates the average ` throughout the plane. In
this work, the location of peaks of the ?decaying envelope? of oscillatory behavior
were used to determine TD. This can only be done if m
? is known. It also requires
that a single frequency dominates the spectrum, as otherwise the combination of
TD complicates the exponential decay.
There is more knowledge to be gained from QOs. For example, the number
of harmonics of a fundamental frequency give an idea of sample quality, as they
are in essence the number of times a carrier can orbit a pocket before being scat-
tered. Frequencies that are the sums or differences of fundamentals also paint a
more intricate picture of the Fermi surface, where carriers may be able to move
between parts of the Fermi surface and trace out partial orbits of multiple pockets.
Frequencies substantially larger than the first Brillouin zone can indicate ?magnetic
breakdown?, where the energy introduced by the field is so large that carriers are
able to jump into the next BZ. Splitting of oscillation frequencies can also be used
to obtain the g-factor of each pocket. However, the concepts covered in the previous
few paragraphs are enough to understand the QO analysis that will be presented in
Chs. 4, 6, and 7. Landau quantization is a powerful phenomenon, and the resulting
Fermiology can give deep insight into the electronic structure of metals.
45
Chapter 3: Experimental Methods
There are many steps to get to the point of obtaining useful data when studying
a material. First of all, the compound of interest needs to be produced, hopefully
reproducibly and with high quality. Next, the proper experimental technique must
be found, and the sample prepared appropriately. Finally, the collected data need
to be analyzed, so that a logical (and, ideally, correct) conclusion can be reached.
This section will cover sample growth and measurement techniques.
3.1 Single Crystal Growth
The first step to measuring a material is to make it. This does not simply mean
getting the correct composition and lattice structure, but also getting the sample
in an appropriate shape to facilitate measurements. Typically, when doing a basic
characterization of a material?s properties, the ideal is a single crystal. Single crystals
maintain a single crystallographic alignment over their entire volume, which for the
work in this thesis could be up to a few millimeters in each direction [Fig. 3.1].
Single crystals are preferable over powders or polycrystals (which are essentially
single crystals too small for individual measurement) for many reasons. The ideal
single crystal will have a uniform composition and a consistent orientation meaning
46
(a) (b)
(c) (d)
Figure 3.1: Single crystals grown by various techniques. (a) CuxTiSe2 grown by
chemical vapor transport with I2 as the transport agent. (b) Elemental Te grown
by chemical vapor transport without a transport agent. (c) TiSe2 grown in a high
pressure Ar environment. (d) Fe0.6Co0.4As grown out of Bi flux. The grid in (a) and
(b) is made up of 1 ? 1 mm2 squares, the scale bars in (c) and (d) are 1 mm long.
47
that there are no grain boundaries that might lead to an extrinsic response in a
measurement. In essence, when measuring a single crystal one can be more confident
that any behavior is coming from the sample itself. Observations can then be linked
to other properties such as sample alignment, which is much more random in a
powder or polycrystal. Another consideration is sample purity; there are many
phases that can be quenched with small amounts of disorder. In fact, it will be seen
that in the case of FeP and FeAs that differences in disorder density between different
single crystals have a substantial impact on measured properties, with important
features sometimes disappearing for inferior samples. For that reason, much of the
effort involved in the work to be presented was spent on sample synthesis, and every
measurement to be discussed, outside of powder x-ray diffraction, was conducted on
single crystals.
Obtaining a powder of the right composition for the materials to be discussed
was often as simple as mixing the raw elements in proportions corresponding to the
final ratios and heating them at high temperatures (say, 700-800 ?C) for a few days.
Growing single crystals can be much more challenging and there are a variety of
techniques that might need to be employed for a crystal, with different advantages
and disadvantages. This thesis will cover crystal growth extensively, and beyond
refinements to previously known techniques will also include synthesis methods not
tried before.
48
3.1.1 Liquid Solvent Flux
Solvent flux relies on the use of additional quantities of a material with a lower
melting point than the desired compound [83]. The idea is that the components of
the desired material will melt more easily in the liquid flux. This principle can be
simply illustrated with table salt (NaCl) and water (H2O). The melting point of
NaCl is 801 ?C. However, at room temperature salt can be dissolved in water. This
is the principle behind flux growth, where relatively low melting point materials
(often elements like Sn, Ga, Pb, or Sb) can be used to dissolve materials with a
much higher melting point. The flux may or may not be a component of the desired
compound.
For many combinations of two or three elements, phase diagrams are available
showing what phases can be formed from different elemental ratios, and the tem-
perature at which the combinations melt [84]. Two examples are given of As-Fe and
As-Bi phase diagrams [Fig. 3.2], which were consulted before attempting to grow
FeAs from Bi flux. In the phase diagrams, the white region represents solid phases
and the purple region liquid ones (though in some cases purple denotes a width
of formation, and thus an imprecise or inconsistent stoichiometry). Vertical lines
represent concentrations corresponding to stable compounds below their melting
temperature, which are connected via horizontal lines. It is possible to determine
at what temperature a specific phase will form or melt based on elemental ratios.
For example, looking at Fig. 3.2(a) it can be seen that a 45:55 mixture of As:Fe
will begin to form both FeAs2 and FeAs at 824
?C, and melt at about 1010 ?C.
49
(a)
(b)
Figure 3.2: (a) The complex As-Fe binary phase diagram. If the two elements
are mixed in an equal ratio, the combination will melt congruently at 1030 ?C.
However, upon cooling not just FeAs, but also Fe2As and FeAs2, will be formed.
Heating the two but staying below 824 ?C, it is possible to form FeAs via a
solid state reaction without any alternate phases In contrast, As and Bi do not
form any compounds together (b), and the melting point decreases monotoni-
cally with increasing Bi concentration. Phase diagrams come from [84] by way
of https://matdata.asminternational.org/apd/index.aspx.
50
Even a 1:1 ratio of the two elements will form some material with a 1:2 and 2:1
concentration if heated hot enough. In contrast, As and Bi do not form any com-
pounds together [Fig 3.2(b)]. Their phase diagram is much simpler and depicts only
the melting point of the mutual combination, which increases as more As is added.
Phase diagrams can be consulted to get a sense of the proper reactant:flux ratio and
temperature profile, based on how much the combined elements need to heated to
ensure that they all melt, and at what temperature they can be cooled to have solid
crystals and liquid flux. However, when combining a large number of elements for
which complete phase diagrams are not available, there is often still some educated
guessing involved.
Flux growth entails combining the reactants and flux in a crucible (alumina,
Al2O3, is sufficiently nonreactive for the elements dealt with here, and many oth-
ers), then sealing the combination in a quartz tube in an inert gas environment
with pressure low enough so that as it increases with temperature the quartz will
not burst. The combination is put into a furnace which is programmed to reach a
temperature high enough so that all components melt based on consultation with
phase diagrams. This upper limit is the melting temperature of quartz, a little above
1200 ?C. From the peak temperature, the growth is slowly cooled; as temperature
decreases, less material can be dissolved in the flux, so gradually more of it will solid-
ify out. Ideally, the cooling step is slow enough and over a large enough temperature
range that large crystals, using all of the reactant material, form. There are multiple
ways to separate crystals from flux. One option is, while the ampule is still above
the melting point of the flux, to very quickly take it out of the furnace and spin it
51
(a)
(b) (c)
Figure 3.3: (a) Examples of box furnaces (one large, two small) used for flux growth.
(b) A sealed quartz ampule containing two alumina crucibles. The bottom crucible
contains all the reactants, the top crucible has quartz wool to separate liquid from
solid during centrifugation. Quartz wool above and below the crucibles prevents
them from hitting the ends of the ampule too hard and potentially breaking it.
(c) A centrifuge used to separate liquid flux from solid crystals. Counterweight is
provided by coins in the bucket opposite the sealed ampule.
52
in a centrifuge, thereby separating the crystals and liquid. This is what was done
for flux growths in this work. During this process the centrifuge was kept on for
only about five seconds until reaching a maximum angular speed of 2000 rev/min.
Another option is to cool the combination down to room temperature, and separate
the solid crystals and now solidified flux either mechanically or through the use of
chemical etchants that preferentially attack the flux.
3.1.2 Chemical Vapor Transport
In contrast to the flux method, which is based in solid-liquid equilibrium, in
chemical vapor transport (CVT) material is converted to the gas phase [85]. The
reactants are again sealed in a quartz ampule, but this time without a crucible and
at a pressure as close to vacuum as possible. The aim is to avoid any unwanted gases
in the ampule, and to make it easier for the reactants to boil. Beyond the elements
that will make up the final crystal, a ?transport agent? may also be included. This
is a material with a high vapor pressure and/or low boiling point that makes it
easier for the reactants to form gaseous compounds. A typical choice is one of the
halogens, either on its own or as part of a compound. Of these the most common
is iodine, since it less harmful than bromine or chlorine (which is often included as
HCl or TeCl4) and is solid at room temperature. However, there are many other
possibilities. Chalcogen-containing materials, for example, can often self-transport
[86,87], and even water vapor or O2 are options in some cases. For elements with high
melting points, it is even possible to use their oxides and enable solid-gas reaction
53
Fe-P-I(g) + I2 (g)
FeP(s)
Fe(s) + P(s) + I2 (s)
T1 T2
Figure 3.4: A diagram depicting the formation of FeP single crystals through CVT.
The raw elements, including the transport agent I2, are mixed together and sealed
at vacuum at one end of a tube. The formation of gas from the solid reactants is
endothermic, and so more favorable at higher temperatures. Applying a temperature
gradient (T1 > T2) results in the consumption of solids at the hot end, and
formation of single crystals of FeP from the gas phase at the cold end.
by an oxidation-reduction reaction [88]. In fact, CVT was first recognized in nature
rather than the lab, as the mechanism by which hematite (Fe2O3) crystallized upon
interaction with HCl gas during the eruption of volcanos [85].
CVT takes advantage of the fact that a change in temperature will change a
chemical reaction rate. The CVT ampule is placed in a temperature gradient, with
the reactants at one end of the tube. They will then form a gas (reacting with the
transport agent, if there is one) which will fill up the entire volume of the ampule.
At the other end of the tube, the gas condenses and forms single crystals of the
desired compound. The position of the reactants at the hot or cold end depends
on whether the initial solid to gas reaction is endo- or exothermic. The majority of
such reactions are endothermic and consume heat, and thus have a higher rate at
higher temperature. In that case, the reactants are placed at the hottest point in
the tube and crystals form at the cold end, where condensation into a solid phase
is more thermodynamically favored. For exothermic reactions, it is the opposite.
54
(a)
(b)
Figure 3.5: Photographs of two different CVT growths of WP using WO3, P, and I2
as starting material. (a) An ampule before heating, with the reactants at the end
closer to the center of the single zone tube furnace, which is the hottest point. (b)
An ampule after the growth process, with the furnace still slightly above room tem-
perature. The pink color indicates the presence of iodine vapor, and most (though
in this case not all) of the material has moved to the colder end, which is noticeably
darker than it was initially.
55
There are many potential variables in CVT. Crystals will often be formed
only through the use of certain transport agents. The temperature and gradient
are also crucial. The intermediate gas phase frequently will not have the same
stoichiometry as the final desired product, so an incorrect choice can easily change
the stoichiometry of the crystals or form undesired phases. Work with FeSe in a
two-zone furnace has demonstrated the sensitivity of crystal quality to small changes
in temperature gradients based on ampule position, length, and even the angle the
furnace is tilted at [89].
Vapor transport growths described in this work were done in single zone hor-
izontal tube furnaces. There was a heater and thermocouple in the middle of the
furnace. As a result there was a natural hot-cold gradient from the middle to either
end of the furnace over a distance of 15 cm. The middle temperature could be
set with the furnace temperature controller, but the gradient could vary somewhat
with temperature; at the high (> 800 ?C) temperatures used here, the tempera-
ture difference between the middle and end of the furnace was about 200 ?C. The
temperature difference could be adjusted by varying the length of the ampules, and
the temperature at specific locations in the furnace could be fond with a handheld
thermocouple rod placed into the furnace that gave immediate readings.
3.1.3 Growth Details
Having introduced the techniques, the specifics for the growth of each system
to be explored will now be given. FeAs and CoAs samples were grown by both flux
56
and CVT in very similar conditions. For the case of flux growth, Bi in a 20:1 ratio
with prereacted FeAs or CoAs powder was put into an alumina crucible and sealed
in a quartz tube in partial Ar atmosphere. The growth was heated at 50 ?C/h to
900 ?C and remained there for two hours. The furnace was then cooled at a rate of
5 ?C/h to 500 ?C, at which point the ampule was spun in a centrifuge to separate
crystals from flux. For CVT, powders were loaded with polycrystalline I2 into an
evacuated quartz tube, which was placed in a temperature gradient such that the
end initially containing the reactants was at 830-950 ?C and the cold end at 600-
750 ?C. Generally, it seemed that CoAs grew better at temperatures on the lower
end of this range, while the Fe-based compounds formed larger and more quickly
at higher temperatures. In both cases CVT led to large, polyhedral crystals, while
those grown with flux had more regular shapes: needlelike in the case of the arsenides
and more three dimensional for CoAs. The needlelike samples grew mostly along
the b-axis, which is to be expected since it is the shortest principal axis.
It was found that high quality FeP samples could only grow through CVT,
possibly because of the lighter mass and lower boiling point of P (281 ?C) compared
to the sublimation point of As (615 ?C). Several different fluxes were tried, the
only partially successful one was Sb. However, samples were small and showed
magnetoresistance similar to that of Sb, indicating residual flux in the crystals even
after polishing. Later attempts to grow CoP were done only with the CVT for this
reason. The best quality FeP samples were found to grow from CVT using elemental
Fe and P with I2. Attempts made with prereacted FeP powder led to samples with
a higher low temperature resistivity, a sign of increased impurity scattering, though
57
in the case of CoP the two sets of starting conditions gave similar results.
Interestingly, energy-dispersive x-ray spectroscopy (EDS) measurements on
both kinds of samples, from which elemental ratios can be extracted, very consis-
tently showed a 55:45 Fe:P concentration in the high quality crystals grown from
the elements, but a ratio very close to 1:1 in powder-grown crystals. It may be the
case that P vacancies (or Fe self-doping) contribute to enhanced crystal quality, and
may also possibly influence the band structure of Fermi energy position to increase
the density of states at EF . CoP samples grown also showed about 10% excess Co
(or deficient P) when grown from either the elements or powders, and the choice of
reactants did not seem to have a significant impact on crystal quality. No attempts
were made to synthesize FeAs or CoAs directly from their elemental components,
due to the hazards associated with vapor transport of elemental arsenic.
In the case of mixed Fe1?xCoxP and Fe1?xCoxAs, flux growth was initially
attempted for the arsenides. However, samples produced in that way were too small
and so CVT was used instead. The temperature was varied between that of the
iron and cobalt compounds based on the x value. Often, single crystals were found
at both ends of the ampule upon opening. Those at the cold end (which had fully
transported) typically had compositions similar to nominal x value from EDS, while
those still at the hot end were often Co deficient.
58
3.2 Measurements at the University of Maryland
Using facilities at the University of Maryland, it was possible to measure chem-
ical composition and structure, electrical resistance, heat capacity, or magnetic sus-
ceptibility at ambient pressure.
3.2.1 Structural and Compositional Characterization
With different tools, it was possible to assess that samples had the correct
structural as well as elemental composition. Powder x-ray diffraction (XRD) mea-
surements were made with a benchtop Rigaku Miniflex 600 diffractometer. The
theory and implementation of x-ray diffraction are well-described elsewhere, such
as Ch. 6 of Ref. [43]. Succinctly, as the angle of both the sample and detector
are changed, peaks in diffracted intensity will appear when the Bragg condition
2dsin? = m? is satisfied. Here d is the distance between two diffraction planes,
? is the angle from the horizontal at which x-rays meet the sample, m is a natural
number, and ? the x-ray radiation wavelength. Cu K? (? = 1.5406 A?) radiation
was used in the Rigaku Miniflex system. Every compound will have a unique XRD
pattern, with the peak positions determined by the crystal structure and lattice
constants, and the intensities determined by the space group and elemental com-
position. With knowledge of a material?s structure and composition it is possible
to predict the x-ray pattern, and vice versa. Single crystal samples were oriented
through a combination of single crystal x-ray diffraction at UMD with the Miniflex
system and Laue photography at the NIST Center for Neutron Research.
59
Energy-dispersive energy spectroscopy makes it possible to measure the ele-
mental composition of sample using x-rays emitted after interaction with an electron
beam. These measurements were performed over a 15 keV energy range with a Hi-
tachi S-3400 variable pressure scanning electron microscope located at the FabLab
on the UMD campus. High energy electrons from the beam will excite core elec-
trons in the sample. Higher level electrons will drop down to fill the created holes,
emitting an x-ray in the process. A detector reads the energy of these x-rays, which
can be compared to known values corresponding to specific transitions for specific
elements. The intensity of emitted x-rays at different energies can be used to deter-
mine the ratio of elements in the sample. EDS results are completely independent
of crystal structure. This is potentially a problem when multiple phases could be
present, as it can be hard to determine by elemental ratios alone if the sample
is composed exclusively of one compound, or multiple with their own individual
weights. However, it is beneficial if there are potential vacancies or excesses of an
element within the material, or when carrying out substitution studies, as in the
case of Fe1?xCox(P,As).
3.2.2 Electrical Transport
Electrical resistance was measured at UMD in a Quantum Design 9 T, 14 T, or
14 T DynaCool Physical Properties Measurement System (PPMS). The wiring con-
figuration for transport measurements is shown in Fig. 3.6. Gold or silver wires were
attached to the surface of samples using EPO-TEK H20E silver epoxy or DuPont
60
(a) (b) Vb
B
holes electrons
I+ V+ V- I- I+ Va I-
Longitudinal Resistance Hall Resistance
Figure 3.6: Gold wires attached with silver paste to the surface of Bi2Se3 single
crystals, configured to measure (a) longitudinal and (b) Hall resistance. Arrows in
(b) show how positively charged (hole) and negatively charged (electron) carriers
will be deflected in the presence of a magnetic field directed out of the page from
their straight paths along the applied current direction, leading to a transverse
voltage difference. Va - Vb will be positive for hole carriers and negative for electron
carriers. If both are present, then the individual carrier concentrations and mobilities
determine the sign.
61
4929N silver paste thinned with 2-butoxyethyl acetate, and contact resistances were
generally around 1-3 ?. Samples were then placed onto a puck designed for use in
the PPMS, and the wires were soldered to pads that provided electrical connections
to send in current and measure voltage. Voltage was converted to resistance using
Ohm?s law, R = V/I. Specific pucks were also used that allowed one dimensional
rotation in magnetic field in the PPMS. Both the longitudinal and Hall resistance
were measured. The difference in setup between the two is in the arrangement of
the voltage wires [Fig. 3.6]. For a standard resistance measurement, the four wires
are arranged parallel across a sample. The outer two serve as a source and sink
of current, and the voltage difference across the middle two is measured. The lon-
gitudinal resistivity ?xx can be obtained from a combination of the resistance and
geometric factors: ? = R w ? txx xx , where w, t, and l are the sample width, thick-l
ness, and length (separation between voltage leads). The resistivity is an inherent
property of a material and thus is of greater relevance than the resistance, which
depends a sample?s shape. Ideally, samples used for longitudinal resistance mea-
surements are long, thin, and narrow, which will maximize the measured resistance
and ensure that current is directed primarily along a single direction between the
two current leads, and does not have any perpendicular component that distorts the
resistance-resistivity conversion.
The Hall effect was covered theoretically in the previous chapter. The Hall
voltage is the potential difference arising along the y? direction for current in the
x? direction and magnetic field in the z? direction1. When a current is applied to a
1Planar Hall measurements, where field is not perpendicular to the current-voltage plane, are
62
sample, electrons and holes move in opposite directions with opposite charge. In
the presence of a magnetic field, carrier paths will curve as they acquire a y?-axis
component dictated by the Lorentz force. For the two carrier types, their opposite
charges and velocities cancel and both carrier types are accelerated in the same
perpendicular direction. As a result there will be a buildup of either positive or
negative charge at one of the voltage leads, which are now on opposite ends of the
sample perpendicular to the current leads [Fig. 3.6(b)].
The complexities of interpreting this signal were covered in the previous chap-
ter. Experimentally, one issue is that for a metal Rxy can often be much smaller
than Rxx. For that reason it is important to align the two voltage leads with as little
lateral offset as possible, to avoid picking up any longitudinal resistive component.
This can be challenging, so often the Hall response must be antisymmetrized for
positive and negative fields. Since the longitudinal magnetoresistance is symmetric
in field, doing this will ideally cancel out a longitudinal component, leaving just the
Hall resistance. In cases with very small Rxy/Rxx, even this may not result in a
clean signal.
To convert resistance to longitudinal or Hall resistivity, dimensions were mea-
sured using pictures of the samples taken through a microscope and software to
convert microscope magnification to length. The photographs in Fig. 3.6 show some
of the uncertainty inherent to single crystal transport measurements. While sam-
ples pictured are relatively nicely shaped, their thickness and width are not constant
across their length. The need to measure the distance between voltage leads also
of interest in some topological systems, but will not be covered here.
63
leads to some error. The leads used in these studies were either 50 ?m or 25 ?m,
while the separation could often be on the order of hundreds of ?m. Beyond that,
the silver paste itself spreads out upon making contact to the sample surface. It is
difficult to know exactly where the best contact is being made, so for consistency
distance was measured from the middle of each voltage contact in calculating the ge-
ometric factor. For this reason, resistivity values shown here should be understood
to have an inherent uncertainty of perhaps 5%. For that reason, the increase or
decrease of resistance, scaled to its room temperature or low temperature value, is
often a more reliable measure of quantities such as impurity scattering. The residual
resistivity ratio (RRR) is ?(300 K)/?base (typically 1.8 K here), and is commonly
used as a marker of sample quality, with higher RRRs corresponding to lower im-
purity scattering at low temperature for (presumably) the same room temperature
resistivity.
3.2.3 Heat Capacity
( )
The heat capacity C = dQp was measured by the relaxation method eitherdT p
in the 14 T or 14 T DynaCool PPMS systems. Samples were affixed with Apiezon
low temperature N grease to a thermally isolated stage, and the chamber was set
to high vacuum (< 10 mTorr, and as low as 1 ? 10?4 Torr at low temperature).
During a measurement, a known quantity of constant heat is applied for a specified
length of time, and the cooling of the entire setup measured for the same amount
of time afterward. The temperature of the platform obeys a differential equation:
64
C dTtotal = ?Kw(T ? Tb) + P (t). Ctotal is the total heat capacity of the sampledt
and platform, Kw the thermal conductance of the wires connected to the platform,
and P (t) the power applied to the stage?is constant during heating, zero at all
other times [90]. The solution of this equation is an exponential function with a
characteristic time ? = Ctotal/Kw. Kw is known, so the total heat capacity can
be solved for with a model incorporating the data from the heating and cooling
periods. This fitting is done by the PPMS software itself. Since it is only possible to
obtain the total heat capacity, before measuring the sample addenda measurements
must be made. That means that a heat capacity measurement must be made in
the same temperature range with just the stage and the grease before measuring
the sample. For measurements in field, addenda measurements at each field are
necessary. The addenda heat capacity can then be subtracted from the total to get
Cp for the sample alone. When thermal contact is not as good, another term can be
added to the differential equation to account for thermal conductance between the
platform and sample. The PPMS software automatically fits both equations, and
the value for the heat capacity comes from the fit with a smaller deviation.
Heat capacity was converted to specific heat by measuring sample mass and
using the compound?s molar mass. At low temperatures (T << ?D), the Debye
model predicts that the specific heat will obey the relation
Cp = ?T + ?T
3 (3.1)
in the absence of contributions from magnetism, superconductivity, or other
65
ordered phases. It is convenient then to divide by temperature and plot Cp/T vs.
2 ?
2k2 N
T , which should be linear, with ? the y-intercept and ? the slope. ? = B A is
3EF
the Sommerfeld coefficient. Here NA is the Avogadro number. The Fermi energy
h?k2
can also be expressed as F? , meaning that there is a linear relation between ?m
and the effective mass. The Sommerfeld coefficient is also a general indicator of a
larger density of states at the Fermi energy-metals will have larger ? values than
insulators. Very large values, on the order of 100 mJ/mol K or more, are seen in
the heavy fermions and signify significant electron-electron correlations that lead to
large (effective carri)er masses. ? can be used to calculate the Debye temperature1
12?4k
? = B
NAnf.u. 3
D , with nf.u. the number of atoms in the formula unit.5?
3.2.4 Magnetic Susceptibility
Measurements of the magnetic susceptibility ? can reveal magnetic properties
of materials. In such measurements, a magnetic field is applied and the sample?s
response measured. In such a way the magnetization M = B/?0 - H can be mea-
sured. Magnetic susceptibility was measured in two ways: using a superconducting
quantum interference device (SQUID) or a vibrating sample magnetometer. The
SQUID MPMS was either an MPMS-XL or MPMS3, both of which had a maxi-
mum field of 7 T. The MPMS3 also had a VSM option, as did the PPMS DynaCool
with a maximum field of 14 T. Susceptibility was converted to magnetization by
measuring sample mass and using the known chemical formula.
The DC SQUID option of the MPMS works by moving the sample through
66
superconducting coils, which are wound so as to be sensitive only to disturbances of
the magnetic field within them [91]. These coils are inductive coupled to a SQUID,
a superconducting loop featuring two Josephson junctions on opposite sides. Mag-
netic flux ? through the SQUID changes the phase difference of the two Josephson
junctions, causing the voltage to oscillate. In that way, the SQUID is very sensi-
tive to small magnetic fields, and the resolution of the newer MPMS system was
< 10?8 emu [91]. A VSM places the sample in a magnetic field and oscillates it
vertically, and uses a pickup coil to convert the change in ? in the middle of the os-
cillation minimum and maximum to a voltage. For the PPMS and MPMS3 systems,
the sample is oscillated by 1-3 mm at a rate of 40 Hz [90]. The induced voltage
V = d? = d? dz . This in turn is equal to 2?f ?mC ?Asin(2?ft), where C is a known
dt dz dt
coupling constant, m the sample?s magnetic moment, A the oscillation amplitude (a
length), and f the oscillation frequency. Therefore, in knowing the voltage one can
extract the magnetic moment. The system is able to resolve signals below 10?6 emu.
While the sensitivity is not as great as that of a SQUID, one advantage is that there
is no field limit on the VSM systems, and so they could be used up to 14 T. In
contrast, the maximum field of the SQUID is 7 T.
3.3 High Magnetic Field
While temperature is almost certainly the most commonly tuned parameter in
basic materials research, magnetic fields offer the chance to cover a much larger scale.
The typical temperature range of the experiments in this thesis is about 1-300 K.
67
In contrast, magnetic fields can be reliably controlled from 10?3 to 35 T, spanning
four orders of magnitude. In Ch. 2 it was seen that by interacting directly with
the magnetic moments of particles, magnetic fields can alter interactions and induce
new phenomena, a relevant example being the destruction of the superconducting
state at a ?critical? field. Magnetic fields can also reveal fundamental properties,
as in the case of quantum oscillations and the related concept of Fermiology. For
this reason studying the behavior of materials in a magnetic field is another crucial
element in understanding them.
The United States? National High Magnetic Field Laboratory (NHMFL) is
capable of reaching the highest nondestructive magnetic fields on the planet2. There
are several different user facilities: the High B/T Facility in Gainesville, Florida, the
Pulsed Field Facility at Los Alamos National Laboratory in New Mexico, and the DC
Field Facility in Tallahassee, Florida. This thesis will focus on work done at the last
of these, over several weeklong allocations of magnet time that were granted through
different experimental proposals. The DC Field Facility has magnets which, as the
name suggests, provide a constant, steady magnetic field; Fig. 3.7(a) shows what one
of the magnet cells looks like. This is in contrast to the Pulsed Field Facility, where
the magnets have higher maximum fields but reach them for far less than a second.
The High B/T Facility has weaker magnets, but specializes in very low temperature
measurements. High temperature superconductors such as the cuprates have critical
fields exceeding the 100 T that the Pulsed Field Facility can reach nondestructively.
2It vies for records related to high magnetic fields, under various conditions, with facilities in
China and Europe.
68
(a)
(b)
Figure 3.7: (a) Cell 6 of the NHMFL. The top of the cryostat is on the platform, the
41.5 T magnet itself is underneath, behind the darkened panel. (b) (left) YPtBi and
(right) CoAs electrical transport samples on a sample platform for measurement on
a rotating probe at the NHMFL. Photograph in (b) courtesy of D. Graf, NHMFL.
69
There are more exotic materials can see superconductivity strengthened in magnetic
field, such as the recently discovered spin-polarized superconductor UTe2 [4], where
increasing field eliminates a first superconducting state before another emerges at
still higher fields. High magnetic fields can lift the shroud of superconductivity or
other field-antagonistic phases that obscure inherent sample properties. In FeP, the
magnetic field response increases the resistance by a factor of over 100. It even
causes the resistivity to increase with cooling at low temperature, i.e. to exhibit
nonmetallic behavior. All of these effects are enhanced, or perhaps only noticeable,
at high magnetic fields, hence the reason facilities dedicated to such work exist.
The resistive magnets have maximal fields on the order of 30-35 T, though very
recently a new magnet has been opened to users with a 41.5 T maximal field. Also
available is a ?hybrid? magnet in which a 33.5 T resistive magnet is surrounded by
an 11.5 T superconducting magnet, resulting in a 45 T total field. For comparison,
the highest field available to most nonspecialized facilities is about 20 T, due to
impracticalities associated with both the electricity and cooling water a resistive
magnet consumes. The highest field for instruments at the University of Maryland
used for this thesis was 14 T.
The longitudinal electrical resistance and magnetic torque of samples were
measured in several different DC magnets at the Tallahassee facility, with maxi-
mum fields of 30.5-35.1 T. Base temperature using a 3He refrigeration system varied
from about 350-450 mK during different measurement periods. There were minimal
differences in operation of each magnet. A 16 or 32 pin connector platform attached
to the end of a probe made allowed for multiple samples to be measured at once.
70
The platform could also be rotated through a single axis. Field was generally swept
at 2 T/min, and measurements were typically done in constant temperature and
angle with variable field. This is the typical approach as changing temperature in
steady field would be costly in terms of both electricity and time, and at low tem-
peratures the thermometers are not well calibrated in field. Occasionally, field was
held constant while the sample platform was rotated.
3.3.1 Electrical Transport
Transport measurements at the NHMFL were for the most part very similar
to those done at in the PPMS systems. Contacts were made on the samples by
attaching Au wires with Ag paste or epoxy, and the samples were mounted with
GE varnish to platforms made of sapphire, G-10 (fiberglass), or some other insulating
material, as seen in Fig. 3.7(b). These could then be positioned on the probe
platform itself with the desired orientation.
3.3.2 Magnetic Torque
It is also possible to measure magnetic torque in DC fields using piezoresis-
tive cantilever magnetometers [92, 93]. This is a very sensitive probe of the torque
? = m ? B. The sample is attached to two small legs, the cantilevers, extending
out from a silicon chip [Fig. 3.8]. A layer of the chip is heavily doped with boron
to make it conductive. Interaction between the sample moment and magnetic field
will result in a torque, which will cause the sample to move and the cantilevers to
71
(a)
(b)
Figure 3.8: (a) Diagram of a piezoresistive torque cantilever, taken from Ref. [93].
A sample is mounted on a stage, which will bend due to the torque experienced in
field. The change in resistance of the Si:B due to the piezoresistive effect can be
obtained with a Wheatstone bridge. (b) An FeAs crystal mounted on a cantilever
for measurement at the NHMFL DC Field Facility. Sample size is approximately
75 ? 100 ?m2. Photograph courtesy of D. Graf, NHMFL.
72
bend. This bending leads to a change in the length and strain of the Si:B, and thus
resistance through the piezoresistive effect. That resistance change can be measured
with a Wheatstone bridge through the two leads coming off of each leg (four total).
Measuring this change in resistance is a proxy measurement of magnetic torque
with arbitrary units (Ohms or Volts). In theory, this can be converted to N?m with
knowledge of sample mass, and calibration of the chip. However, this was not done
since only the relative change in the torque was relevant for quantum oscillations
measurements3.
The doping level of the Si chip means that small strains can generate a large
voltage, making this an extremely sensitive technique, with sensitivities on the order
of 10?12-10?14 N?m readily achievable [92]. Indeed, torque is in many cases preferable
to resistance for the detection of magnetic transitions or quantum oscillations; the
signal is often cleaner and the background simpler to subtract. In FeAs and CoAs,
for example, quantum oscillations were seen in torque at fields as low as 5 T, but not
at all in resistance even above 30 T. Torque samples are much smaller than those
required for resistance measurements, with dimensions on the order of 100 ?m or
less. Crystals that are too large can bend the cantilever, either with a large mass or
with too strong of a field response. This can lead to a nonlinear resistive response
and corrupt the measurement. There are some drawbacks to the torque method,
such as the difficulty of converting to non-arbitrary units. Additionally, the torque
signal is typically very small when field is aligned along crystal axes and the cross
3While the torque is perpendicular to the magnetic field, for the purposes of quantum oscilla-
tion measurements only the field direction is relevant, because the formation of Landau levels is
determined by the direction of the B, not ? .
73
product disappears. Still, the torque method was very useful due to its extreme
sensitivity and ability to measure samples too small to fit four wires onto.
74
Chapter 4: Crystal Quality and Quantum Oscillations in FeAs
4.1 Introduction
The first material to cover, and in fact the first material studied chronologi-
cally, is FeAs. It is, after all, the most direct link to the iron-based high-temperature
superconductors, which as mentioned include tetrahedrally-coordinated Fe-As lay-
ers. The interest lay in what happens in a simple binary, with octahedral coor-
dination. The spin density wave transition at TN = 70 K has been well estab-
lished [24, 30, 94?96]. However, unlike the iron pnictides, CrAs, or MnP, FeAs does
not superconduct with either chemical substitution [25,26,97] or pressure [94]. Ev-
idently, there is something in the the electronic structure and/or magnetic interac-
tions of FeAs to set it apart.
FeAs orders in a noncollinear spin density wave (SDW) state consisting of
unequal moments along the a- and b-axes with propagation along the c-axis [96,98].
There are some differences from the helimagnetic state that is mostly unchanged in
MnP, CrAs, or FeP. Both spin amplitude and direction are modulated, and there
is possible canting into the propagation direction. Despite a good deal of work
on the properties of FeAs there still exists uncertainty about many aspects of its
electronic structure and what drives its magnetic order. Theoretical work predicted
75
the paramagnetic and AFM Fermi surfaces to differ substantially, with the AFM
Fermi surface consisting of a single electron pocket at the ? point surrounded by
four identical hole pockets [99]. However, the AFM state associated with this Fermi
surface is a more conventional arrangement of alternating up or down spins, rather
than the experimentally observed SDW [98?100]. If the magnetic ordering does not
match experiment, than it is hard to take predictions for its properties as fact. Hall
effect measurements have shown the coexistence of both hole and electron carriers
over a wide temperature range, but disagree over the dominant low temperature
carrier [40,95,101], which may be directionally dependent.
The following chapter will present a method to grow binary FeAs crystals using
Bi flux, which produces samples with a larger residual resistivity ratio than has been
previously reported [24,94?96,98,101,102], which generally indicates better crystal
quality. These crystals show quantum oscillations in magnetic torque measurements
at high fields, which can be analyzed to give a more complete picture of the electronic
structure of FeAs below TN , allowing for comparison to previous theoretical and
experimental results.
4.2 Experimental Details
In previous studies, FeAs single crystals were made via CVT with I2 [24,94?96,
98,101] or from Sn flux [102]. In an effort to improve sample quality, attempts with
other fluxes such as Sb, In, and Sn were made. Ultimately Bi was found to work, in
part because it conveniently does not form compounds with either Fe or As, and any
76
residual flux stayed on the surface of crystals and could be polished away, rather than
embedding itself within the samples as was the case with Sn. At low temperatures,
the resistivity is much smaller in Bi flux crystals, and accordingly the RRR much
higher. RRRs consistently exceeded 70 with a maximum of 120, compared to 20?40
with other growth methods [95, 101]. Given that RRR (and correspondingly, the
residual resistivity value) is often accepted as a measure of crystal quality, it seems
that Bi flux growth results in the highest quality FeAs single crystals yet produced.
The crystal morphology of Bi flux samples is distinct from the polyhedral or
platelike samples resulting from with CVT or Sn flux. Crystals grown in Bi flux
are needlelike (Fig. 4.1, inset), with typical dimensions of 0.03 ? 0.03 ? 0.8 mm3.
Powder x-ray diffraction measurements give the lattice parameters a = 5.44 A?,
b = 3.37 A?, and c = 6.02 A?, in line with previous results1 [24, 95, 96]. The long
direction of the crystal was always the b-axis, as verified by Laue photography and
single crystal XRD and inferred from the initial increase in resistivity with decreasing
temperature that is unique to measurement along [010] [95, 101]. The b-axis was
also almost always the longest growth direction for crystals of the other MnP-type
compounds explored in this work. For the sample used in oscillations measurements,
the orientation of the a-axis was similarly confirmed with XRD and Laue, making
the c-axis the remaining perpendicular direction. Composition was confirmed by
EDS as almost exactly 1:1 for samples from different growths. There was no sign of
Bi contamination in EDS, XRD, or transport measurements. One drawback of the
1Note, however, that axis conventions for this space group have changed over time. In this
thesis the convention is always b < a < c.
77
Bi flux growth method is that the small, thin samples are ill-suited for Hall effect
or single crystal susceptibility measurements, as they are too light and narrow.
4.3 Transport Results
The temperature-dependent resistivity for single crystal FeAs is shown in
Fig. 4.1. The 300 K resistivity value for Bi flux crystals is about 300 ?? cm, similar
to what has been seen previously [95,101]. The initial increase in ? as temperature
decreases, with a maximum near 150 K, signifies that the measurement is conducted
with I ? b-axis. A kink at 70 K marks the SDW onset at the same temperature as in
other transport, susceptibility, and heat capacity measurements [24,30,94,95]. The
inset to Fig. 4.1(a) shows the resistivity plateauing below 20 K at about 2.5 ?? cm.
With increasing field, magnetoresistance in FeAs evolves from the B2 depen-
dence expected at low fields to a more linear relation [Fig. 4.1(b)]. This linearity
continues without saturation up to 31.5 T. Such a crossover has previously been re-
ported to occur at about 6 T for measurements at 10 K [101]. In Bi flux samples it
occurs at roughly 10 T below 1 K with B ? I. The other measurements only went to
15 T, but MR was slightly larger over that field range. The data in Fig. 4.1(b) may
show the onset of oscillatory behavior just below 30 T, but there were not enough
data for possible analysis and no sign of oscillations appeared at other angles, where
MR was also smaller.
78
(a)
B = 0 T
I || b
RRR = 120
FeAs
T (K)
(b)
500 ?m
B || a ? I
?B2 T = 460 mK
?B
B (T)
Figure 4.1: (a) Resistivity vs. temperature for an FeAs crystal grown from Bi flux.
The high RRR and low residual resistivity (inset) indicate very good crystal quality.
(b) Magnetoresistance as a percentage of 0 T resistivity for FeAs up to 31.5 T. Fits
of low and high field data to quadratic and linear functions, respectively, show a
transition in field dependence of MR around 10 T. Inset: an FeAs crystal wired for
longitudinal resistance measurements, showing the needlelike geometry particular
to Bi flux growth.
79
MR (%) ? (?? cm)
B (a)
?
? b
c
a
B || a
B || b
B || c
B || [110]
B (T)
(b)
B || a B || b
T = 400 mK T = 460 mK
B || c B || [110]
T = 410 mK T = 390 mK
1/B (T-1)
Figure 4.2: (a) Raw torque cantilever data for FeAs for several field orientations.
Inset: a schematic showing how field angle was changed in the two measurements
on the same sample. Measurements to 31.5 T went from B ? a-axis to B ? b-axis
(? = 0?, sweeping ?). Those to 35 T went from B ? a-axis to B ? c-axis (? = 0?,
sweeping ?). (b) The residual oscillatory signal of the raw data. Amplitudes are
arbitrary but consistent relative to those in (a), and have been enhanced by a factor
of 100.
80
2
Residual Amplitude (a.u. x 10 ) Torque Amplitude (a.u.)
4.4 Quantum Oscillations
As explained in Ch. 3, quantum oscillations arise from the formation of quan-
tized Landau levels in a material in the presence of a magnetic field. Changing the
field strength causes these bands to pass through the chemical potential, and the
resulting change in occupancy produces an oscillatory signal that can be detected
in a wide variety of density of states-dependent quantities [80, 103, 104]. QOs were
seen in the magnetic torque measurements of an FeAs single crystal at the NHMFL.
Torque data show multiple frequencies across different angles of applied field, as ev-
ident in Fig. 4.2(a) which shows the raw torque signal at several field orientations.
Oscillatory behavior was clear in the torque signal as low as 10 T at some angles.
Two sets of measurements were made on the same crystal with 31.5 T and
35 T magnet systems as it was rotated in two different planes relative to magnetic
field, as illustrated in the schematic in Fig. 4.2(a). Data to 31 T were taken at 24
angles with ? = 0? and changing ?. In this configuration ? = 0? signifies B ? a-axis
and ? = 90? is B ? b-axis. Up to 35 T, 16 measurements were made with ? kept at
0? while ? was changed, corresponding to B ? a-axis at ? = 0? and B ? c-axis at
? = 90?.
To extract the oscillatory component a 3rd order polynomial was subtracted
from the raw data; Fig. 4.2(b) shows examples of the presence of different frequencies
at different angles, and changes in the amplitude of the residual torque signal. Fast
Fourier transforms (FFT) were then performed on the residual data to obtain a
frequency spectrum [Figs. 4.3(a) and 4.3(b)].
81
(a) ? = 170?
?1 ?2
?
100?
98?
95?
B || b 90?
86?
80?
70?
60?
B || a 45?
? = 0?
(b)
? = 105?
B || c
?
? ?
B || a ? = 0?
F (kT)
(c) (e) 
? ? EF = 0 
EF = 55 meV 
(d) (f) 
? ?
? (?) ? (?)
Figure 4.3: FFTs at base temperature (350?550 mK) of oscillatory signals for all
angles, offset for clarity. In (a) field goes from parallel to a-axis to parallel to b-axis,
in 5? increments at higher angles. In (b) field moves between the a- and c-axes in
7? increments over the entire range. (c)-(f) show observed peaks for each oscillation
band as a function of angle as well as theoretically generated frequencies based on
AFM Fermi surface calculation of Parker and Mazin [99] (orange line) and the same
calculation with EF raised by 55 meV in work by Limin Wang (blue). Note that
that AFM state is not the same as the experimentally observed one. For ?, ?, and
? fits to an ellipsoidal Fermi surface are also given (black lines). For (c), where peak
splitting occurred the average (black) of the ?1 and ?2 frequency peaks (red) was
used for the fit.
82
F (kT)
Amplitude (a.u.)
4.4.1 Angular Dependence
By plotting FFT data for all angles of the two runs as in Figs. 4.3(a) and
(b), it is clear that although frequency values change substantially with angle, they
can be grouped to one of five extremal Fermi surface orbits. Harmonics of these
five frequencies also appear at integer multiples, indicating low scattering as carriers
move in their orbits. In the ? scan [Fig. 4.3(a), B in the ac plane] two low frequencies
(denoted ?1 and ?2) around 500 T, and one higher frequency peak near 1500 T (?),
were observed. The proximity of the two ?i peaks indicates that they arise from the
same Fermi surface pocket, with two slightly displaced extremal orbits. The ?1??2
frequency difference was roughly 150 T, independent of temperature or angle.
For the measurement varying ? [Fig. 4.3(b), B in the ab plane] there are two
peaks: one with a frequency of about 300 T for angles closer to 0? (?) and a higher
frequency peak with F ? 2000 T (?). However, the ? peak diverged to much higher
frequencies exceeding ? near B ? c-axis with a substantially reduced amplitude. As
Fig. 4.3(b) shows, the amplitude decreased substantially as this change occurred.
The observation of two main orbits (one of which shows some frequency splitting)
in both the a?b and the a?c planes should stem from two distinct Fermi surface
pockets. This fits with the theoretical prediction of one unique electron and hole
pocket each in the magnetic state [99] as well as experimental evidence suggesting
multiple carriers in this regime [40,95,101].
83
4.4.2 Fermi Surface Shape
For an ellipsoidal Fermi surface, the frequency should vary with angle as
F = ? F0 , where F0 is the maximum frequency, ? the angle and  the
cos2? + 1 sin2?

eccentricity of the cross sectional ellipse in the plane of rotation [105]. Figs. 3(c),
(d), and (f) show fits of peak frequency to this equation for ?, ?, and ?. At angles
with split ?1 and ?2 frequencies, their average value was used for the fit. Divergence
from fits makes it clear that the pockets are not perfectly ellipsoidal, however the
qualitative agreement shows that ?, ?, and ? correspond to orbits around three di-
mensional parts of the Fermi surface with a generally ellipsoidal shape. In contrast,
? shows a slight increase in frequency at lower angles, until roughly ? = 70? when
frequency increases by an order of magnitude before plateauing. This behavior is
closer to that of cylindrical or two dimensional pockets, although ? does not fit well
to the inverse cosine dependence expected from a perfect cylinder.
Fig. 4.4 shows two theoretical Fermi surfaces for antiferromagnetic FeAs ob-
tained with density functional theory (DFT). Calculations were done for the ?AF2?
state, calculated by Parker and Mazin to be most favorable at low temperatures [99],
in which Fe atoms align antiferromagnetically with both nearest and next-nearest
neighbors. This same arrangement was favored in the calculations of Frawley et
al. [98], whereas Griffin and Spaldin [100] differed in having a ferromagnetic ar-
rangement of next-nearest neighbors. However, neither of these orderings matches
the experimental SDW structure [46]. The top surface in Fig. 4 uses the original
AF2 Fermi level, while the bottom one is from the same calculation but with the
84
Fermi level raised by 55 meV. In both cases, calculations were done by Limin Wang,
a postdoc in the Paglione group at UMD. The shift changes the size of the pockets
but not their shapes or locations, establishing the robustness of this Fermi surface
geometry and therefore also of the expected angular dependence of oscillation fre-
quencies. In either case there is an electron pocket at the central ? point and four
identical hole pockets at (ka, kb, kc) = (?0.25, ?0.3, 0). Note that all pockets shown
are contained within the first Brillouin zone and would have closed orbits. The non-
saturating magnetoresistance may come either from an open orbit not theoretically
predicted, or a closed path that saturates at a field larger than 31 T, perhaps due
to a low residual resistivity.
Theoretical quantum oscillation frequencies were generated, again by Limin
Wang, from the DFT calculations using the Supercell K-space Extremal Area
Finder (SKEAF) program [82] are plotted together with the experimental data in
Figs. 4.3(c-f). Two bands, one electron-like and one hole-like, were expected for each
plane of field rotation, a 1:1 correspondence to what was obtained in measurements.
Based on expected frequencies and angular dependence ? and ? correspond to hole
pocket oscillations, with ? and ? belonging to the electron band. As noted, raising
the Fermi level does not change the angular dependence, but the accompanying
change in pocket size gives closer agreement to the observed oscillation frequencies
in most cases. For ? and ? expected angular dependence matches well to data, and
in fact the splitting seen in the hole band is also present in the unshifted Fermi
surface calculation in the range ? = 30??90?. This reinforces the roughly ellipsoidal
pockets inferred from experimental angular dependence. For ? the divergence at
85
higher angles does not happen in the theory, and the expected frequency increase
is much smaller. For ? a variation of frequency with angle is seen, however the
locations of the maximum and minimum oscillation frequencies are reversed. This
indicates that the electron pocket area is larger in the kb?kc plane than in the ka?kb
plane, the opposite of the band structure prediction. Overall the DFT Fermi surface
appears to give an accurate description of electron and hole pocket shape for field
rotated between the a- and b-axes (? = 0?, changing ?), but not the a- and c-axes
(? = 0?, changing ?). kc corresponds to the propagation direction of the SDW, while
the moments lie in the ka?kb plane. The fact that this is also the field direction with
the strongest divergence from calculation points to a connection between disagree-
ment of DFT and experiment over both magnetic ordering (as was already known)
and band structure (as shown here). Oscillations data do, however, support the two
carrier picture put forth by other groups [40,95,101].
The roughly ellipsoidal Fermi surface makes it acceptable to calculate carrier
concentration from frequency values, a process which assumes a circular cross sec-
tion. Applying (2.21) gives ranges of 2.2 ? 1019 ? 1.5 ? 1021 cm?3 for the hole pocket
and 3.5 ? 9.6 ? 1020 cm?3 for the electron pocket, based on maximum and minimum
observed frequencies. These are comparable to the values n = 8 ? 1018h cm?3 and
ne = 1 ? 1021 cm?3 found by Khim et al. [101] through a fit of MR data. The
hole pocket has a much more dramatic angular dependence, and for a small range
of angles near B ? c-axis even exceeds the electron value. Assuming comparable
scattering rates, this anisotropy could account for the sign change in RH at low
temperature seen by Segawa and Ando [95] but not Khim et al. [101]
86
4.4.3 Temperature Dependence
To calculate effective mass via the LK factor (2.22), temperature dependence
was taken at three field orientations: ? = ? = 0? (B ? a-axis), ? = 0?, ? = 98?
(near B ? c-axis) and ? = 0?, ? = 135? (B ? [110]). Oscillatory signals for these
orientations are shown in Fig. 5.3(b). The second angle gives an idea of the effective
mass along the c-axis, but ? was not set to exactly 90? since the torque signal was
much reduced directly along the principal axis. Fig. 4.5 gives an example of the clear
suppression of FFT amplitude of ?1, ?2, and ? with temperature for B ? [110].
Temperature dependent amplitudes for three different field orientations are
shown in Fig. 4.5(b-d). Table 4.1 gives the extracted effective masses. With B ? a-
axis [Fig. 4.5(b)], only the ? hole pocket (F = 315 T) is seen. As with oscillation
frequencies, experimental effective masses can be compared to those generated with
DFT for the original or 55 meV shifted Fermi level [82]. A fit to (2.22) at this angle
gives an effective mass of 3.1me, larger than the theoretical predictions of 1.78me
(EF = 0 eV) and 1.138me (EF = 55 meV) for the hole pocket in the same orientation.
For ? = 98? [Fig 4.5(c)] only ?, at 1.61 kT, appears. Given the absence of any other
frequencies, amplitude can be directly extracted from the oscillatory data, and the
effective mass is m* = 1.2me. For ? = 90
? the predicted electron band masses are
0.668me and 0.812me (0 eV) or 0.6322 and 0.940me (55 meV). At 135
? [Fig. 5(d)],
? (now at 2.77 kT) survives only up to 1.8 K. Due to the presence of the lower ?
frequencies in the residual signal, amplitude is taken from the FFT. The LK fit gives
m* = 3.2me, nearly a factor of three larger than its value at ? = 98
?. This again
87
EF = 0
ka
k kbc
FeAs, AF2 State
EF = 55 meV 
ka
k kbc
Figure 4.4: DFT calculated Fermi surfaces of FeAs in the predicted ?AF2? magnetic
state [99], consisting of an electron pocket (pink) at the ? point and four identical,
symmetrically oriented hole pockets (yellow). The bottom has had the Fermi energy
raised by 55 meV, which changes the size of the pockets but not their location or
general shape.
88
exceeds predictions of 1.124me (0 eV) or 1.252me (55 meV).
At ? = 135?, ?1 and ?2 are found at 412 and 536 T. Using the average value
of the amplitude of the two peaks the effective mass comes out to m?, ave = 3.8me.
The individual peaks have similar values, further supporting the idea that they arise
from the same band. This number is similar to the value of 3.1me obtained for the
same pocket for B ? a-axis. The 0 eV Fermi level prediction is for two peaks with
masses 1.561me and 2.023me, while that for 55 meV is one peak of 1.301me. As with
all other measured angles, the experimental effective masses are larger than those
predicted.
From heat capacity measurements, the Sommerfeld coefficient of the specific
heat is 6.65 mJ 2 [30]. This can also be calculated from oscillations data using themol K
relation2
?k2BNA?SH = (4.1)
3EF
The inverse lattice parameters give the in-plane Brillouin zone area (?k2F ) for
each field orientation to calculate EF . The contribution from each band is then
proportional to m?. The sum of the contributions of each pocket (keeping in mind
that there are four hole bands) is ?SH = 7.7, 16, and 0.29
mJ
2 for field along [100],mol K
[110], and [001], respectively. The smaller ?SH along [001] indicates that not all the
orbits for that orientation may have been observed. However, the average of these
three values is 8.0 mJ 2 , close to the previously reported value. Therefore it seemsmol K
2The Sommerfeld coefficient is has the subscript ?SH? in this chapter, standing for ?specific
heat?, to distinguish it from the ? oscillation band. In all other parts of this thesis it is simply ?.
89
unlikely that there is a significant missing contribution to the density of states at EF
that would solve the theory-experiment discrepancy. Instead, the enhanced effective
masses make it probable that there are electron correlations in FeAs unaccounted for
by DFT. It has recently been proposed that spin-orbit coupling may have significant
influence on the FeAs band structure in the magnetic state [98]. Although this
correction is not normally included in calculations for Fe-based compounds, it (or
other effects) may account for some of the observed disagreement.
4.4.4 Dingle Temperature and Scattering
The Dingle factor in the oscillation amplitude is given by (2.23), and the Dingle
temperature is proportional to the scattering rate ? (2.24). By approximating a
circular Fermi surface again, it is then possible to extract the mean free path (2.25).
Since the Fermi surfaces seem to be ellipsoidal, the mean free path values extracted
from TD allow for a reasonable comparison but are not meant to be exact. TD was
calculated based on fits of peaks in the decaying ?envelope? of the residual signal
as a function of inverse field [Fig. 4.5(e-g)] and are listed in Table 4.1 along with
estimates of `. For ?ave the average frequency of ?1 and ?2 (474 T) was used. It is
only possible to solve for TD if the effective mass is known, limiting the analysis to
only the three angles for which temperature dependent measurements were made.
Additionally, since calculation of TD relies on a clear exponential decay of amplitude
it is typically necessary to have one dominant peak at a specific angle to extract
a Dingle temperature. For that reason it is not possible to calculate TD for each
90
?2
? (a) B || [110]1
0.39 K
0.59 K
0.82 K
1.52 K
1.72 K ?
5.11 K
F (kT)
B || a (b) ? (e)
B || a, 400 mK
?
B || b (f)
(c) ?
B || b, 460 mK
?
B || [110] (g)(d)
?
? ave1 B || [110], 390 mK
?2
?ave
?
T (K) 1/B (T
-1)
Figure 4.5: (a) FFTs at multiple temperatures for B ? [110], showing the decrease
in amplitude with temperature of the ? (500 T) and ? (2800 T) peaks. (b-d) Fits
to the LK factor at different field angles. Plots of peak amplitudes vs. inverse
field at base temperature were fit to a decaying exponential to extract the Dingle
temperature (e-g). Base temperature varied slightly between measurements. Due
to its small amplitude relative to ?1 and ?2, it was not possible to extract TD for ?
at B ? [110].
91
Amplitude (a.u.) Amplitude (a.u.)
peak at each angle. This is not an issue for B ? c-axis, where only ? appears and
TD = 5.5 K. However, due to their very similar frequencies it is hard to separate ?1
and ?2, and so the best that can be done is to give TD = 2.2 K as the average value
for the ? oscillation at ? = 135?. The ? oscillation is only a small modulation of the
signal for this orientation [Fig. 4.2(c)]. Again there is anisotropy in the hole pocket,
as TD goes from 2.2 K to 5.1 K as field moves from [110] to the a-axis. A change in
scattering with orientation is not surprising given the previously noted differences in
both the longitudinal and Hall resistance for measurements along different crystal
axes [95].
The Dingle temperatures of FeAs are similar to those seen in BaFe2As2. In
that material TD can be calculated for two out of three observed bands, and for
both is in the range 3?4 K [106]. In another 122 material, KFe2As2, TD is between
0.1?0.2 K for five different pockets [107]. For the K compound, RRR values up
to 2000 are possible [108], while for BaFe2As2 RRR less than 10 is typical [109],
though it can be raised to nearly 40 with annealing [110]. FeAs seems to resemble
BaFe2As2 much more than an exceptionally ?clean? material like KFe2As2.
4.5 FeAs: Conclusions
Bi flux growth results in higher quality FeAs crystals than previous attempts
with Sn flux or I2 CVT. This improvement makes it possible to observe quantum
oscillations in magnetic torque at high fields. Measurements in two different planes
reveal five unique peaks, corresponding to one electron and one hole band in each
92
Orbit Type B ? [hkl] F (T) m* /me TD (K)
?1 h [110] 412 3.6 ?
?2 h [110] 536 3.9 ?
?ave h [110] ? 3.8 2.2
? h [100] 316 3.1 5.1
? e [110] 2765 3.2 ?
? e [001] 1615 1.2 5.5
Table 4.1: Parameters extracted from fits of FeAs quantum oscillation amplitude to
the LK and Dingle factors at several field orientations.
direction (with the hole band split for field in the ac plane). These peaks can be in-
dexed using a DFT-calculated Fermi surface for antiferromagnetic FeAs [99]. Three
peaks near 500 T (split peaks ?1 and ?2 and ?) stem from extremal orbits around
the predicted four identical hole pockets, and two others (? and ?, one in each plane)
near 2 kT come from the electron pocket at the ? point. The ? oscillation band has
a two dimensional shape and cannot be easily assigned a simple geometry. The other
three observed oscillations show a three dimensional, qualitatively ellipsoidal angu-
lar dependence, as expected from calculations, with slight disagreement in pocket
size. Comparison of the specific heat coefficient ?SH seems to indicate that there is
not a significant missing contribution.
The observation of two distinct frequencies overall validates the multiband
notion of transport in the low temperature SDW state indicated by previous exper-
iment [40,95,101]. There is good agreement with the calculated Fermi surface when
field is swept in the ac plane, but disagreement for ab plane rotation. Most notable
is a significant increase in the cross sectional area of the hole pocket near the ka?kc
plane, where it becomes larger than the electron pocket. Extracted effective masses
93
for both hole and electron pockets are larger than predicted by calculations, indicat-
ing that the degree of electron correlations is greater than theoretical expectations.
It was already known DFT results did not match the magnetic state of FeAs. This
study makes it clear that the band structure is on the right track, but still awaits a
full theoretical description.
94
Chapter 5: A Magnetoresistance Study of FeP
5.1 Introduction
A bit surprisingly, given its composition, FeP may actually have more in com-
mon with MnP and CrAs than FeAs. Its magnetic structure is better described
as helimagnetic [29], rather than a spin density wave. The two moments rotate in
plane while propagating along the c-axis, rather than changing their amplitude but
pointing in the same direction, as in FeAs [46]. The 120 K Ne?el temperature is also
a bit higher than 70 K in FeAs, where for CrAs it is about 260 K in CrAs and the
initial PM-FM transition in MnP is at about 290 K [21]. Crystal growth and trans-
port properties also differ between the two materials. FeP cannot be grown from
flux, and has a much higher RRR and MR than FeAs. The change in the pnictogen
atom clearly makes a big difference. It may be that the volatility of phosphorus
generates vacancies that change the Fermi level or contibute to magnetism, as can
be the case in transition metal dichalcogenides with deficiencies on the chalcogen
site [6]. Or the change in spin orbit coupling, since the mass of P is less than half
of that of As, can be the driver.
This chapter will present transport measurements of FeP single crystals at 3He
temperatures in magnetic fields up to 35 T. The focus of the high field measurements
95
in this case is not on quantum oscillations, as a thorough study of that has already
been done [111]. Instead the interesting result is the very large, nonsaturating
magnetoresistance in all directions, which can be enhanced several hundred times
over its zero field value. This is similar to the behavior seen in sibling compound CrP
as well as the 1:1 rare earth-pnictide binaries [56,112,113], which form in a different
(cubic) structure. It can be attributed to a nearly compensated Fermi surface and
the change in carrier mobility with temperature. Most interestingly, MR is linear
when field is applied along the c-axis, which also when it has its minimum high field
value and Shubnikov-de Haas oscillations vanish. While such behavior is similar
to the quasilinear MR seen in CrAs near the 0 K helimagnetic-superconducting
phase diagram boundary [19], in FeP this comes at ambient pressure. The likely
explanation for LMR is quantum Abrikosov MR, in which conduction arises from a
single Fermi surface pocket [78]. The observation in LMR in a magnetic, good metal
stands as a contrast to the XMR seen in CrP and the rare earth-pnictide binaries.
Temperature dependence shows that there is a ?turn on? temperature to the MR
that can be attributed to high sample mobility. Angle sweeps at fixed field indicate
the complex nature of the Fermi surface as noted in previous quantum oscillations
work [111].
5.2 Methods and Characterization
CVT growth of FeP single crystals with I2 is well-established [31, 85, 111,114,
115]. The procedure was unchanged from other materials, with a hot zone temper-
96
ature of 850 ?C. Single crystals of FeP were found at the cold end of the furnace
after 10-14 days, but on occasion some would also be found at the hot end, indicat-
ing that the reaction did not strictly follow the thermodynamics determined by the
temperature gradient.
Powder x-ray diffraction (XRD) measurements of the resultant crystals showed
single phase FeP with lattice parameters of a = 5.10 A?, b = 3.10 A?, and c = 5.79 A?,
in line with previous reports [27, 111]. Of note is that FeP has the shortest b-axis
of all the 1:1 transition metal pnictides, with only CoP having shorter a- and c-
axes. Energy dispersive x-ray spectroscopy of single crystals consistently showed a
composition of 55% Fe, 45% P, slightly off stoichiometry. Attempts at CVT growth
were also made using prereacted FeP powder. However, samples grown from powder
had lower residual resistivity values, suggestive of lower crystal quality. Interestingly,
EDS measurements on these crystals showed 50:50 Fe:P ratio. Phosphorus vacancies,
or iron self-doping enhance the conducting properties of this material, perhaps by
changing impurity concentration, carrier density, or the position of the Fermi level.
Such behavior has been seen in both TiSe2 and Bi2Se3, where changes in Se vacancy
level significantly impact carrier concentration (obtained from Hall measurements)
and temperature-dependent resistivity [6, 116]. DFT predictions have shown that
the Fermi surface in MnP-type compounds can be restructured with small changes
in electron count [32]. A slight increase in this number may result in a significant
increase in carrier density or Fermi pocket size.
Attempts to grow single crystals were also made using Bi, Sn, Pb, and Sb
as fluxes. Only Sb produced crystals, but they were small and showed traces of
97
residual flux in transport measurements. Growth temperature and the purity of
starting material also had a significant effect on both residual resistivity and the
magnitude of MR. The crystals grown with CVT were large and often polyhedral,
but nevertheless showed a clear preference for growing along the b-axis, the shortest
principle axis.
The zero-field transport behavior of FeP looks qualitatively like to that of FeAs
[Fig. 5.1], with similar values of ?(300 K) and a plateau in resistivity below 20-30 K.
One difference is that FeP never has an increase in ? with the initial cooling from
room temperature. The most obvious transport feature is a 120 K kink marking
the onset of helimagnetism [Fig. 5.1, inset]. Residual resistivity is very small, as
low as 0.2 ?? cm in some samples. The residual resistivity ratio (RRR), defined as
?(300 K)/?(1.8 K), can be over 1500, indicative of very good crystal quality. This
is similar to the highest values attained for MnP [117] and higher than the 400-500
seen in the Cr-based B31 materials [18, 112]. This is again a difference from FeAs,
where even the best quality Bi flux samples had RRRs that were about an order of
magnitude lower.
5.3 Magnetoresistance
5.3.1 Field Dependence
Field sweeps were made at various angles and temperatures for multiple sam-
ples at the NHMFL. The first, called S1 [Fig.5.2(a)], was aligned with two different
[101] reflections perpendicular to current. However, angles as listed in figures have
98
300
2.0
1.5
1.0
200 0.5
0.0 0 10 20 30
TN = 120 K
100
FeP
0 0 100 T (K) 200 300
Figure 5.1: Resistivity (at 0 T) as a function of temperature for an FeP crystal with
a RRR of approximately 1000. It is generally similar to that of FeAs, with a feature
at 120 K denoting the Ne?el temperature. The inset is of low temperature data,
showing the plateau below about 30 K. The variation in the lowest temperature
data is due to the resistance being close to the noise floor of the PPMS.
99
? (?? cm)
300
(a) FeP (b) 90?
250 50
40 65?
30
200 45?20 B || c 90?10 60? 30?
150 00 10 20 30 45? 15?
      S1        S2
100 90?: B || c
30? 90?: B || b 90?
90?: B || a 20? 90?: B || a
50 90?
0 0 10 20 30 0 10 20 30
B (T) B (T)
Figure 5.2: High DC field transverse magnetoresistance at several angles for FeP
crystals S1 (a) and S2 (b), which had different orientations. The temperature was
about 400 ? 50 mK for all measurements, but the variation should not have an
impact on the data in this region. The inset to (a) is a closeup of the B ? c-axis
data, with a linear fit (pink) over the entire data range. Note that MR for the
common B ? a-axis orientation is larger for S2, which can be linked to its higher
RRR value. S1 was actually oriented along two different [101] crystal directions,
but data were shifted by a constant angle here for simplicity.
100
MR
been shifted so that 0? corresponds to B ? c-axis and 90? was B ? a-axis for simplic-
ity. For the second, S2 [Fig. 5.2(b)], the same angles were B ? b-axis and B ? a-axis,
respectively. In both cases field was always applied perpendicular to current. The
RRR was about 1300 for S1 and 1400 for S2. Measurements on a third sample
(not shown) found the expected suppression of magnetoresistance accompanying a
decreasing Lorentz force as field was rotated into the same direction as the current.
Both S1 and S2 show nonsaturating magnetoresistance at all angles, and of-
ten quantum oscillations. The Shubnikov-de Haas oscillations do not show all the
known oscillation bands [111], particularly those at high frequency, but this can be
explained by the decreased resolution of electrical transport compared to the mag-
netic measurements in the reference. For both crystals, the maximum MR is when
B ? [100]. The larger MR of S2 can be connected to its higher RRR, as the two
are often highly correlated in large MR metals [58, 118]. Most angles show similar
behavior, with an initial quadratic dependence that becomes more linear at high
applied field, like what was seen in FeAs [Fig. 4.1(b)].
The curve for B ? c-axis stands out for several reasons. Foremost of all,
it is strikingly linear from very low fields, with a sublinear deviation only barely
noticeable below 0.5 T. This orientation also results in the lowest MR of any of the
angles measured in either rotation plane, though the increase is still about 40 times
by 31 T. Furthermore, in the vicinity of this angle there are no quantum oscillations.
This could be attributed to the presence only a single small Fermi surface pocket,
and condensation of all carriers into the lowest Landau level at very low fields [78],
a notion that will be expanded upon in the Discussion.
101
60 S1
40
20
T = 2 K 00?: B || a
B = 14 T 90?: B || c
0 -45 0 45 90 135 180 225
S2
250
200
150 T = 0.6 KB = 35 T 00?: B || b
90?: B || a
0 30 60 90
? (?)
Figure 5.3: Angle-dependent magnetoresistance for FeP samples (a) S1 and (b) S2
from Fig. 5.2 at 14 T and 35 T, respectively. Angular magnetoresistance oscillations
are clearer in the higher field, lower temperature data. Data in figure have two fits,
the blue line is a fit to a simple 2D Fermi Surface, MR ? cos ?. The red line is two
an anistropic 3D Fermi surface, MR ? 1(cos2?+ ??2sin2?) 2 , where ? is the ratio of
effective masses. The peak just off of the a-axis was excluded from fitting.
102
MR
5.3.2 Angular Dependence
Resistance was also measured at constant field and temperature while rotating
the crystals in 14 T and 35 T fields [Fig. 5.3]. There are many detailed features in
the angle dependence even between the principal axes, which are especially obvious
with higher field and lower temperature. Such features are known as angular mag-
netoresistance oscillations, and are clear indicators of an anisotropic Fermi surface
structure. MnP, which also has a large MR, has shown a similarly complex struc-
ture when rotated in field [119]. It is difficult, however, to extract anything more
quantitative from these oscillations. Most notable, however, is that the maximum in
MR comes about 5? off of the a-axis at 14 T. This could perhaps be attributed to a
slight error in the rotator, but there is an obvious underlying sinusoidal dependence
centered on crystal axes that is different from the asymmetric MR peak.
This periodicity can be fit to two different models to determine the dimen-
sionality of the Fermi surface. The two dimensional model [blue line in Fig. 5.3(a)]
is simply a cosine. The three dimensional model [red line in Fig. 5.3(a)] accounts
for the anisotropy of the effective mass in the planes perpendicular to the field at
1
the peak and valley of MR, denoted ?, and has the overall form (cos2?+??2sin2?) 2 .
The peak located about 5? away from 0? and 180? was excluded from the fitting.
It can be seen that neither model is an especially good fit to the real curve, even
ignoring the peak. This indicates that neither the simplest, two dimensional model,
nor a three dimensional, slightly more elaborate ellipsoidal picture, account for the
angular dependence. Like with FeAs, there is perhaps a generally ellipsoidal form
103
to the pockets, but nevertheless their shapes are more complicated than that.
5.3.3 Temperature Dependence
Field sweeps at various temperatures and temperature sweeps in constant field
were made for specific angles up to 14 T at UMD. The temperature sweeps show a
similar behavior to that seen in CrP and several extreme MR rare-earth pnictide bi-
naries, with a ?turn on? temperature, T*, below which magnetoresistance increases
significantly [58, 112]. Figure 5.4 shows the data for the angle with maximum MR,
about 75? in Fig. 5.3(a), near B ? a. At and above 5 T, this actually leads to an
increasing resistance with decreasing temperature. Judging by the minimum of the
14 T temperature sweep, T* ? 35 K. This temperature is in the same 30-50 K
range seen in the previously mentioned materials. However, one feature unique to
FeP is that resistance does not saturate below T* at high enough fields. In fact, for
B = 14 T the increase in resistivity below 25 T, after subtraction of the zero field
temperature dependence, fits well to a straight line with a slope of about 0.5 ?? cm .
K
The zero field resistivity was also practically temperature independent below 30 K.
A power law fit of the form ? = A + BTn gives a good fit when n = 3.6 for data
between 2 K and 60 K. This is similar to the the value of the value n = 3 found for
CrAs [120], indicating that that neither electron-phonon (n = 5) or electron-electron
(n = 2) scattering dominate in the low temperature regime.
The continued increase in resistance in field down to lowest temperatures is
quite distinct from the compensated rare earth pnictides, where the increase below
104
25 0S1
15 m = -0.48 ?? cm K-1 
20 10
14 T 5
15 00 25 50
12 T
10 10 T
08 T T*
5 05 T
02 T
0  
  
  
0 25 50 75
T (K)
Figure 5.4: Temperature dependence of the resistivity of FeP sample S1 at the
maximal magnetoresistive angle in the a-c plane from Fig. 5.3(a), ? = 5?. The
black points are at 0 T, the colored and labeled points in applied field. The inset
shows the difference between the 14 T and 0 T data, or the ?magnetoresistive ??,
with a linear fit to the data below 25 K and the obtained slope.
105
? (?? cm)
?MR (?? cm)
T? promptly gives way to a plateau [113]. Instead, in FeP even for fields 5 T or
below the resistance continues to increase down to 1.8 K. In this region the zero field
resistance has plateaued, indicating that the MR is no longer simply a function of
?0 and thus violates Kohler?s rule [121,122]. This means that a different scattering
process has emerged below T*, that must generate the sinusoidal anisotropy of the
MR and the anomalies seen at unexpected angles.
5.4 Hall Effect
Hall effect measurements can also help to illuminate the changes in FeP with
temperature, field, and orientation. The Hall resistance was measured for three high
RRR samples with different orientations [Fig. 5.5(a)], two of which were shown in
Figs. 5.2 and 5.3: B ? [101] (S1), [010] (S2), and [001] (S3). The Hall coefficient RH
was extracted by antisymmetrizing ?14 T field sweeps. It is known from QO work
that there are various Fermi surface pockets in this material [111] and thus likely
both electrons and holes. Still, the value of RH was negative for all orientations
both above and below TN. There is, however, a noticeable temperature dependence,
with |RH| maxima in the B ? [001] and [101] samples in the vicinity of TN, and
|RH| minima in all three samples at lower temperatures between 25-75 K. Sample
S1 (B ? [101]) showed slight nonlinearity below 50 K [Fig. 5.5(b)], but the other two
were linear in field over the entire temperature range. The two-band model (2.16)
can be used to extract the Hall coefficient of the nonlinear data. The only way for
?xy to be nonlinear in that model is if the B
2 term is nonzero. Thus, (p?n) must be
106
nonzero, and the material cannot be perfectly compensated. This single equation has
three unknowns, so some assumptions must be made. From the higher temperature
RH values it is possible to estimate n = 5 ? 1021 cm?3. A value of 2.5 ? 1021 cm?3,
slightly less than n, is chosen for p. This is because the two carrier densities must be
somewhat comparable to result in a nonlinear slope, and the negative RH is a sign
that electrons are always the more abundant carrier. It was found that values of
1-4 ? 1021 cm?3 give equally good fits, indicating that the fit is robust to moderate
variation in p. Thus it is the ratio of the mobilities is more relevant, and can be
solved for. ?n is always larger, unsurprising given the consistently negative RH.
But ?n/?p drops dramatically from 50 K to 2 K with values that are also robust
to the choice of p [Fig. 5.5(b), inset]; the hole mobility is becoming increasingly
comparable to the electron mobility. A significant temperature dependence of the
carrier mobilities is unsurprising when a material has a low resistivity [112,121,122].
The beginning of nonlinearity in the Hall effect is about the same tempera-
ture at which MR becomes appreciable in longitudinal resistance. The increasing
presence of the hole band, due primarily to a change in mobility rather than carrier
density, is then likely a source for the large magnetoresistance as well. This is also
the case for WTe2 and the cubic rare earth pnictides [53,113]. However, the change
in those materials is much more extreme, and ?xy actually changes sign at high field.
Even at 2 K in FeP, electrons are still eight times more mobile than holes, and the
nonlinearity of the Hall resistivity is much more subtle and only observed for one of
three measured orientations. The negative RH at all orientations and temperatures
is proof that electron transport is still dominant in FeP. It is not as well compen-
107
0
(a)
-5
-10
 S1, B || [101]
 S2, B || b
-15  S3, B || c0 50 100 150 200
T (K)
0.0
(b)
-0.5 S1, B || [101]
100
-1.0
10
-1.5 0 25 50 1
T (K)
 5 K  20 K  100 K  150 K  200 K
0 2 4 6 8 10 12 14
B (T)
Figure 5.5: (a) The Hall coefficient as a function of temperature for field along three
different crystal orientations. (b) Hall resistivity from antisymmetrization of ?14 T
field sweeps for B ? [101] at representative temperatures. Solid lines are fits to the
data, and which become nonlinear below 50 K. The inset shows ?n/?p, the ratio of
the electron to hole carrier mobility obtained from (2.16), at low temperatures where
?xy is no longer linear. The slight oscillations visible in the data are an artifact from
ansymmetrization.
108
?xy (?? cm) RH (10-4 cm3/C)
?n/?p
sated as the previously mentioned materials, or classic examples like Bi [52], but
the electron and hole concentrations are close enough that it can lead to large, non-
saturating MR when the mobilities are also comparable. This, along with carrier
concentrations an order of magnitude larger, may be why the MR, while sizeable,
does not reach the values of 105 or more seen with other compounds at similar field
and temperature [56,58,113].
5.5 Discussion
The magnetotransport of FeP is interesting for two separate reasons: because
it is large with clear orientation dependence, and because it is linear for B ? c-axis.
Focusing initially on the former, there are as mentioned various generators of large
MR. Somewhat unique to its observation here is that FeP is antiferromagnetically
ordered and a very good metal. The latter, along with the strong anisotropy of
MR magnitude, rules out disorder. Samples with small amounts (< 10%) of Co
doping on the FeP site have a magnetoresistance of less than 5% at 10 T with a
more quadratic field dependence. This is substantially lower even than FeP samples
with similarly low RRR values, another sign that disorder is actually antagonistic to
large MR. Rather the low residual resistivity is benificial, as it increases ?c? . Large
MR has been found in isostructural MnP [119] and CrP [112], though the highest
field reached in those studies was just 8 T and 14 T, respectively. This result seems
to indicate that magnetic ordering is not a significant factor in producing large
MR, since CrP is paramagnetic and MnP passes through several different stages
109
of magnetic ordering with field at low temperature [21]. FeP and MnP can also be
contrasted with FeAs and CrAs, which have similar magnetic structures [29] but MR
merely on the scale of sample resistivity at 14 T [13,19]. There is also no significant
magnetoresistance in the vicinity of TN , indicating little change in scattering when
going from strong magnetic fluctuations to an ordered state.
Instead, electron-hole compensation is the most likely explanation, with
semimetallic bismuth being the classic example [52]; more recently nearly per-
fect compensation has been pertinent to the family of rare earth-pnictide 1:1 bi-
naries [56, 123, 124] as well as CrP. Compensated or nearly compensated materials
also should see MR saturation either only at very high fields or not at all, and up to
35 T MR does not saturate in any direction for FeP. Additionally, from Hall effect
measurements conduction is electron-dominated at all temperature and alignments.
For comparison, FeAs has multiple changes in carrier sign [95] but has a much
smaller MR [13, 101]. However, quantum oscillations clearly show the contribution
from multiple Fermi surface pockets [111], some of which surely come from hole
bands. It cannot be a coincidence that nonlinearity in the Hall effect at one orien-
tation comes near the ?turn on? temperature [58, 112]. Compensation explains the
independence of MR from magnetic order, as its size depends primarily on the dif-
ferent in hole and electron carrier density, which is a product of the band structure.
Slight differences in dispersion or shifts of the Fermi energy for different elements
would tweak the degree of MR.
The angle-dependent linear magnetoresistance is an even more intriguing re-
sult. Such behavior has been observed in the Fe-pnictide superconductors, where
110
theory attributes it to the changes that occur in the Fermi surface as those materi-
als enter the spin density wave regime [125]. The field at which magnetoresistance
crosses from a quadratic to linear dependence is proportional to the SDW gap and
the scattering rate. A typical value for the parent pnictides was estimated to be
2 T. The relatively high 120 K transition temperature, large RRR, and low Dingle
temperatures seen in FeP would all contribute to a similarly low crossover field. The
slightly lower TN and RRR in FeAs then fit with its quadratic-linear crossover at
5 T [Fig. 4.1].
There are also parallels to observations in CrAs under pressure. With the
application of about 1 GPa, superconductivity with Tc = 2 K emerges [18, 34] as
TN decreases. Near the critical pressure for competition between magnetism and
superconductivity, the magnetoresistance becomes increasingly linear [19]. This is
attributed to non-Fermi liquid physics arising at a quantum critical point. However,
quantum criticality is associated with linear scaling of resistance with both field and
temperature [75]. Under pressure CrAs has a T2 dependence just above Tc [34], and
for FeP resistivity is similarly never linear. This suggests that the binary pnictides
are not quantum critical materials. This was already suspected for CrAs due to
the first order nature of its magnetic transition, which is accompanied by a large
magnetostriction [44]. But despite not following typical quantum critical physics,
there is still evidence of unconventional superconductivity in CrAs that is often
linked to the presence of magnetic fluctuations [44, 120]. So it is still of interest
to look at the effect of pressure on FeP, which may also become a superconductor,
perhaps at lower pressure due to its lower TN .
111
The mostly likely potential source of LMR is the Abrikosov quantum picture
[78]. In this scenario, conduction comes from a single Fermi surface pocket, for which
above a certain field strength all the carriers will be in the lowest Landau level. If this
pocket has a (Dirac-like) linear dispersion, then there will be a linear scaling of MR
with field. This explanation is also applicable in the case of CrAs. With B ? c-axis
only a single, small Fermi surface pocket contributes to conductivity. It has a ?semi-
Dirac? dispersion, meaning it is linear in only a single k -space direction, just below
the Fermi energy. There are, unfortunately, no angle-dependent measurements on
CrAs at ambient or applied pressure to see if its behavior is as angle-sensitive as
LMR in FeP. But a comparison can be made to CrP, which has the same semi-
Dirac point at an even lower energy. The MR in that compound is very large
(much larger than that of CrAs) with significant anisotropy, but is not linear for
any field orientation, which can be explained by the semi-Dirac point being further
from EF [112]. Evidently large and linear MR have two separate origins in the
Cr pnictides. The difference in magnetotransport between CrP and CrAs despite
similar RRR values is another sign that large MR is not solely a product of high
sample purity. FeP, then, just happens to have the right combination of properties
to exhibit both linear and large MR. DFT calculations from Limin Wang show
that there is a semi-Dirac point at about -100 meV. This is further away than that
predicted for CrP (-47 meV), but the evidence from EDS for nonstoichiometry in
high RRR FeP samples could lead to a change in EF that potentially moves it closer
to the partially linear dispersion.
Previous quantum oscillations results detected multiple frequencies with field
112
along the FeP c-axis [111]. But based on the lattice parameters and the size of
the effective Brillouin zone below TN , only frequencies below about 1000 T can
potentially correspond to orbits of a single pocket. All higher frequencies represent
orbits whose areas are larger than the ka? kb plane of the first Brillouin zone. That
means they must stem from ?magnetic breakdown?, a situation in which the energy
of the field enables carriers to skip between multiple pockets and zones [52], that has
been suggested to occur in both MnP and FeP [111,119]. Only one of the observed
frequencies for B ? c-axis is below 1000 T. Therefore it seems that FeP, like CrAs
under pressure, has conduction through only a single pocket when field is applied
along [001]. In fact, only for B ? c-axis is there a single frequency below 1000 T,
and all others are above 3000 T, far exceeding orbits that stay only in the first BZ.
For S1, SdH oscillations appear at many angles for fields up to 35 T, but
not B ? [001]. Oscillatory behavior is not possible if all carriers are in the lowest
Landau level; this is the same condition for linear MR, and thus explains both
observed phenomena for this orientation. As was noted in the work on CrAs, the
3
critical field for all carriers to be in the lowest LL goes as (Ec - EF ) 2 , where Ec is the
energy of the crossing point [19]. Thus a very small pocket, at low temperatures,
will require a very small field to enter the quantum limit. The vanishing effective
mass in the case of a linear dispersion further lowers this requirement.
113
5.6 FeP: Conclusions
The magnetoresistance of FeP is large and nonsaturating at all angles. Its
dependence on sample quality and the results of Hall measurements indicate that
it is related to electron-hole compensation, though to a lesser degree than in other,
semimetallic systems. Large MR has a ?turn on? temperature of around 35 K similar
to that seen in binary pnictides in the MnP family and others. At most angles MR
has a quadratic dependence, evolving to linear by 35 T. However, with field along
the c-axis MR is linear from very low field, and quantum oscillations vanish. This
is ascribed this uniqueness to the highly anisotropic Fermi surface, as evidenced by
angular MR measurements, in which only a single, small pocket is responsible for
conduction in this orientation, and carriers quickly enter the lowest Landau level.
Quantum criticality seems less likely due to the nonlinear temperature dependence
in this system and the related pressurized CrAs, but that the presence of a semilinear
dispersion might be required. Hints of unconventional superconductivity in CrAs
under pressure are also an intriguing line of further study in this material.
114
Chapter 6: Near Ferromagnetism in CoAs
6.1 Introduction
While at first glance CoAs has obvious structural and compositional similar-
ities to FeAs and FeP, there are some distinguishing features. Near 1000 ?C, CoAs
transitions to the hexagonal NiAs structure, a property also seen in some other ma-
terials from this family but neither of the Fe-based binaries. Moreover, the electron
count differs, because Co is one spot to the right on the periodic table, and has
seven, not six, 3d electrons. The Fe-Co line is the border of magnetic ordering in
this family. From left to right on the periodic table, CrAs, MnP, FeP, and FeAs are
all ferro- or antiferromagnetic at low temperatures. In contrast, CoAs, CoP, and
NiAs are all paramagnetic [40].
There were previously no low temperature reports on CoAs single crystals be-
yond neutron work to characterize the structure and show that there was no long
range magnetic order [28]. Despite that, the magnetic susceptibility had been ob-
served to have an unexplained nonmonotonic temperature dependence [28,40]. CoAs
merited investigation as a PM but potentially magnetically unstable comparison to
the complicated ordering of the Cr, Mn, and Fe-based materials. As already noted,
interesting behavior such as superconductivity can emerge near the boundary of
115
magnetic transitions [126].
This chapter will parallel that on FeAs. CoAs crystals can be grown by both
CVT and bismuth flux, and quantum oscillations were seen in flux samples sam-
ples in torque measurements at the NHMFL. As in the case of FeAs, predicted
QO frequencies and masses can be compared to theoretical values coming from
DFT calculations. Here, experiment does an even better job of matching theory,
resulting in an experimentally verified Fermi surface picture. This can be com-
bined with resistivity, Hall effect, heat capacity, and magnetization measurements
to provide a comprehensive overview of the properties of binary CoAs. Interestingly,
DFT calculations actually favor, though only very slightly, a FM ground state, de-
spite the confirmation that CoAs is indeed paramagnetic. However, features in the
temperature-dependent magnetic susceptibility and heat capacity are signs of the
presence of magnetic fluctuations. The conclusion is that CoAs is nearly ferromag-
netic, and future work to manipulate magnetic fluctuations with pressure, chemical
substitution, or other techniques may stabilize interesting ground states.
6.2 Crystal Growth
As noted in Chapter 2, synthesizing CoAs single crystals is very similar to
growing FeAs, and can be done with Bi flux or I2 CVT at comparable tempera-
tures. One difference is that flux-grown crystals CoAs are platelike, with typical
dimensions of 0.4 ? 0.2 ? 0.1 mm3, in contrast to the needlelike FeAs Bi flux single
crystals and the much larger and irregularly shaped crystals that result from CVT
116
[Fig. 6.1(a)]. The axis perpendicular to the basal plane is always c, the longest crys-
tal axis. Growths whose maximum temperature was 1000 ?C and which were spun
at 925 ?C had a more hexagonal shape [Fig. 6.1(b)]. Several MnP-type compounds
transition between the orthorhombic structure and the hexagonal NiAs structure at
high temperatures, including CoAs (T ?S ? 975 C) [23]. Tremel et al. theorized
that the orthorhombic structure was more stable than the hexagonal one in the bina-
ries only for d2 to d6 transition metals, and that starting at d7 another orthorhombic
structure, the NiP type, should be favored [22]. Yet CoAs bucks this trend (as does
CoP), and retains the unit cell seen in the Cr-Fe pnictides. Regardless of appearance,
room temperature powder XRD of the hexagonally-shaped crystals shows them to
be orthorhombic, despite of their appearance, and the conclusion is that they form
in the hexagonal structure with the corresponding morphology, before trasitioning
during cooldown. CVT was the method used for single crystal growth in past studies
of CoAs [28,127]. CVT crystals are much larger than those grown out of flux, with
dimensions exceeding 1 mm. Like the Fe-based materials, they have an irregular,
polyhedral shape, making principal axes more difficult to identify. But if orientation
can be established, they are much easier to use for Hall effect, heat capacity, and
magnetic measurements than flux samples. Powder XRD measurements of CVT
and flux crystals give the same lattice parameters: a = 5.28 A?, b = 3.49 A?, and
c = 5.87 A?, which also match previous results [28, 40].
117
(a)
(b)
Figure 6.1: (a) Comparison of CoAs crystals grown by I2 CVT (upper two samples)
and Bi flux (lower sample). The CVT samples are much larger, but with a much
more irregular shape, sometimes composed of multiple single crystals fused together.
(b) A Bi flux crystal from a higher temperature growth with a more hexagonal shape,
a product of the high temperature orthorhombic-hexagonal structural transition. In
both photos the grid size is 1 mm ? 1 mm.
118
6.3 Physical Properties
The resistivity of single crystal CoAs [Fig. 6.2(a)] is 60-80 ?? cm at room
temperature and displays a featureless, slightly sublinear temperature dependence
before saturating for T < 30 K. The residual resistivity ratio (RRR), defined as
?(300 K)/?(1.8 K), is up to 70 for flux crystals compared to about 40 for the best
CVT samples. Typical resistivities at 1.8 K are roughly 1-2 ?? cm, with slightly
higher values for CVT samples. As with FeAs, the lower residual resistivity of the
smaller Bi flux crystals is interpreted as an indication that they are of higher quality
than those grown with vapor transport, though the effect is not as dramatic in this
case. Given the more regular shape of flux-grown crystals in both cases, they may
be less likely to have two intergrown crystals with different orientations that would
lead to domains and grain boundaries. Measurements of the magnetoresistance up
to 31.5 T showed anisotropy, but the MR was only 0.5-2.8, with similar quadratic-
to-linear behavior to FeAs.
Hall effect measurements were performed between ?9 T on the wider vapor
transport crystals, with the antisymmetric component of the MR in Hall geometry
used to calculate RH [Fig. 6.2(a), inset]. The antisymmetrized curves are linear
with a positive slope over the entire temperature range, indicating hole-dominated
conduction. Data sets for multiple samples show a peak near 100 K. Similar sharp ex-
trema have been observed in RH measurements of Sb [128], CrB2 [129], and of course
FeAs [95]. In the previous chapter FeP exhibited a nonmonotonic temperature de-
pendence. The peak is clear evidence for multiband transport with a significant
119
(a)
I2 CVT
Bi flux
T (K)
(b)
9
9
8 0 T 10 T
1 T 14 T
8 5 T
7 7
0 10 20
0 10 20 30 40
T 2 (K
2)
Figure 6.2: (a) The resistivity of CoAs single crystals grown by Bi flux and I2
vapor transport. Inset: temperature dependence of the Hall coefficient. (b) Low
temperature molar heat capacity. The line is a fit of the 0 T data to C/T = ?+?T 2
for 4.5 K < T < 7 K. Inset: a closeup of the same data, as well as that for various
applied fields, at lowest temperature. Lines connect points and are not fits.
120
C/T (mJ K-2 mol-1) ? (?? cm)
R  (cm3H /C)
temperature dependence of carrier density and/or mobility, though the linearity of
the Hall resistivity makes this more difficult to assess. The theoretical calculations
and quantum oscillations measurements to be presented support the notion that
multiple carrier types are present. In FeAs and CrB2 the RH peaks occur at the
onset of antiferromagnetism. However, no other measured properties of CoAs show
features near the location of the RH maximum.
Single crystal heat capacity data were taken at low temperature. Figure 6.2(b)
shows the data in zero field with a straight line fit to the standard low temperature
heat capacity model (3.1) for 4.5 K < T < 7 K. The fit yields ? = 6.91 mJ
K2 mol
and, from ?, a Debye temperature ?D = 397 K. These values are close to those
seen in FeAs (? = 6.652 mJ2 , ?D = 353 K) [30] and CrAs (7.5
mJ and 370 K)
K mol K2 mol
[130]. For MnP, ? is estimated to be 5.4?7.6 mJ2 , with a large uncertainty due toK mol
magnetic contributions [131]. The closeness of these values shows the electronic and
phononic similarities of this group of compounds, even with differences in magnetic
order. Closer to 1.8 K, there is a subtle bump followed by a drop in C/T . Similar
behavior was observed in near ferromagnets CaNi2 and CaNi3 [132], and a low
temperature enhancement in C/T can be an indicator of spin fluctuations [133,134].
This feature is much less noticeable in CoAs than other materials, but it is still
present in measurements in fields up to 14 T, where there is a slight positive deviation
from linearity in C/T below 10 K2, followed by a lower temperature drop [Fig. 6.2(b),
inset].
Measurements of magnetization were also done on an unaligned CVT crystal.
The susceptibility of single crystal CoAs has a small moment that increases slightly
121
(a)
ZFC
FC
B = 50 mT
T (K)
(b)
2 K
10 K
25 K
50 K
100 K
300 K
2 K
B (T)
Figure 6.3: (a) Zero field cooled (ZFC) and field cooled (FC) magnetic susceptibility
data of a CoAs single crystal. (b) Field dependence of magnetization between ?14 T
at various temperatures. Inset: a closeup of the 2 K data between ?1 T. The initial
upsweep is red, the downsweep blue, and the return sweep light green.
122
M (10-3 ?B/Co) ? (10
-4
 emu mol-1 Oe-1)
upon initial cooling, with a broad maximum around 225 K [Fig. 6.3(a)], followed
by a minimum near 35 K. The magnitude and features are generally similar to
previous reports for polycrystal [28, 40], though the minimum at low temperatures
is not as sharp as the kink seen by Saparov et al. This could indicate that the kink
reported in that work is actually from an impurity phase. Also, the low temperature
susceptibility does not exceed the higher temperature value for single crystals. A
similar broad peak just above 200 K is also seen in FeAs [95], which has a similar
overall shape of temperature-dependent susceptibility above TN to CoAs. Motizuki
qualitatively explained the observed maxima in both compounds as stemming from
the temperature dependent spin fluctuations, which saturate in amplitude above
the temperature of the peak [135]. The difference would then be that in FeAs these
fluctuations eventually settle into an ordered state. In CoAs on the other hand,
like the low temperature bump in heat capacity, the susceptibility peak indicates an
unfulfilled tendency to magnetic order.
At low fields, the field-cooled curve shows a larger low temperature upturn than
the zero field-cooled curve, but the difference is small and disappears above 0.2 T.
The size of the upturn is also sample dependent, and so is most likely the result
of paramagnetic impurities, without which ? would simply plateau with further
temperature decrease. This would also explain why the upturn is larger for powder
samples in other work, as those are more likely to have corruption from other phases.
Changing the magnitude of the applied field did not vary the value of ? or location
of the peaks between 0.05 and 7 T, nor was there much change the susceptibility
values. Field-dependent magnetization up to 14 T [Fig. 6.3(b)] is nonsaturating
123
50
300 K 025 K
40
200 K 010 K
100 K 002 K
30
20
10
0
0.0 0.5 1.0 1.5 2.0 2.5
B/M (103 T/(?B/Co))
Figure 6.4: An Arrott plot showing B/M vs. M2 for CoAs at various temperatures.
The lack of a positive y-intercept indicates that there is no spontaneous moment at
zero field, and therefore no ferromagnetism.
124
M2 (10-6  (?B/Co)
2)
at all temperatures from 300 K to 2 K, and nonlinear at low field for T ? 10 K.
There is negligible hysteresis at 2 K [Fig. 6.3(b), inset], indicating a lack of clear
FM behavior. An Arrott plot using M(B) data at 2 K similarly shows no sign
of long range magnetic ordering [Fig. 6.4]. All signs point to CoAs being PM,
the same conclusion reached in previous magnetic and 4.2 K neutron diffraction
measurements [28, 40]. A weak FM moment was observed in a powder sample by
another group [136], but could be the result of the inclusion of 57Fe in those samples
for later Mo?ssbauer study.
6.4 Theoretical Calculations
The electronic structure of CoAs was obtained via first-principles density
functional theory calculation of the paramagnetic state. As with FeAs, all theo-
retical work was done by Limin Wang. The calculation was conducted using the
WIEN2K [137] implementation of the full potential linearized augmented plane wave
method within the Perdew-Burke-Ernzerhof generalized gradient approximation us-
ing the lattice parameters obtained from powder XRD. The k -point mesh was taken
to be 11 ? 17 ? 10. Figure 6.5 shows the PM band structure, density of states
and Fermi surface of CoAs. The Fermi surface consists of two hole pockets and two
electron pockets, and the bands around the Fermi level are dominated by the Co d
orbitals. The electron pockets have a ?Czech hedgehog? shape centered at the Y
point. The concentric hole pockets occupy the center of the first Brillouin zone but
spread into other zones, resembling connected hourglasses.
125
Four possible magnetic ground states were considered: paramagnetic, ferro-
magnetic, and two distinct antiferromagnetic orderings?one in which Co atoms
align ferromagnetically with nearest neighbors and antiferromagnetically with next
nearest neighbors, and another where they are antiferromagnetic with both. These
are the same orderings considered for FeAs [99], which do not completely match
that compound?s SDW. DFT results show a preference for FM over PM in CoAs
by 20 meV/Co atom, with the two AFM scenarios at much higher energies. This
energy difference translates to about 230 K, the location of the local maximum in
susceptibility. The calculated moment for the theoretical FM state is 0.28?B, smaller
than in any of the AFM binary pnictides [Table 2.1], with only FeP having a value
at all close. Another theoretical study claimed that the MnP type structure can
naturally lead to a FM instability [32]. But measurements here and by others show
no indication of long range magnetic ordering in CoAs. Additionally, the magneti-
zation at 140 kOe and 2 K is still two orders of magnitude smaller than the expected
moment. Neutron diffraction measurements saw no purely magnetic reflections and
set an upper limit of 0.1?B at 4.2 K on any potential FM moment [28], ruling out
a moment close to the 0.28?B prediction. The energy difference between ferro- and
paramagnetism is in truth small. It is possible that the onset of magnetic order-
ing occurs at an even lower temperature than that reached in these experiments or
previous ones, but measurements with an adiabatic demagnetization refrigeration
setup showed no resistive transitions down to 100 mK. The small energy difference
is within the inherent uncertainty of DFT due to the need to arbitrarily choose
approximations without full knowledge of the relevant parameters. Experimentally,
126
(a) (b)
Total
Co d
As p
EF
E (eV)
?    Y  S     ?  X U  Z    V  R      ?
(c) (d)
kc electron kc hole
ka ka
kb kb
Figure 6.5: DFT-calculated (a) band structure (where different colors distinguish
different bands), (b) density of states, (c) electron and (d) hole Fermi surfaces of
paramagnetic CoAs.
127
E (eV)
N(E) (states/eV/unit cell)
CoAs is by all indications paramagnetic; DFT simply reinforces the idea of its nearly
ferromagnetic properties.
One way to quantify near ferromagnetism is the dimensionless Wilson ratio RW
(2.1). For CoAs, RW = 6.2, comparable to the values for the known near ferromag-
net Pd (RW = 6?8) [49] and BaCo2As2 (7?10, depending on field orientation) [50],
which is thought to be near a magnetic quantum critical point. RW can be recon-
figured as the Stoner factor Z = 1- 1 , where Z? 1 signifies stronger ferromagnetic
RW
correlations. ZCoAs = 0.84, similar to near ferromagnets CaNi2 (0.79) and CaNi3
(0.85) [132], which showed a low temperature enhancement in C/T. Both theoretical
and experimental results point to strong low temperature FM fluctuations in CoAs.
DFT calculations were also made for paramagnetic FeAs with the same meth-
ods and a comparison to CoAs is in Fig. 6.6. The electronic structures of PM FeAs
and CoAs differ only by a rigid band shift, as demonstrated by the fact that raising
the Fermi level of the calculated FeAs band structure [Fig. 6.6(a)] and density of
states plot [Fig. 6.6(b)] nearly reproduces the CoAs equivalent in both cases. The
shift is about 1 eV, which is logical given that Co has an extra electron compared
to Fe. To explore this relationship further the PM FeAs band structure was recal-
culated using the CoAs lattice parameters. It should be noted that while the a-
and c-axes are smaller for CoAs compared to FeAs, the b-axis is actually longer,
and that in general there is no simple dependence of the lattice parameters in the
MnP family with element selection [Table 1.1]. There is a negligible difference in
PM FeAs band structure calculated using the two unit cell sizes, indicating that the
70 K spin density wave onset in FeAs and corresponding lack of ordering in CoAs
128
(a)
CoAs
FeAs (+ 0.9 eV)
?     Y   S      ?  X   U  Z      V  R        ?
(b)
CoAs
FeAs
E (eV)
Figure 6.6: (a) A comparison of the band structures of paramagnetic FeAs (after
having its Fermi level shifted up by 0.9 eV) and CoAs, with a schematic of the
Fermi surface convention used. (b) A density of states comparison, with FeAs again
shifted by 0.9 eV. In both plots a dashed green line indicates the original Fermi level
of FeAs.
129
N(E) (states/eV/unit cell) E (eV)
has a more complicated origin than just unit cell size and bond distance.
The predicted PM Fermi surface of CoAs is very different from that of FeAs
[99], and low temperature AFM is also highly unfavored in CoAs. The relatively
empty dispersion in the region between -1 and 0 eV and significant dropoff in the
density of states at EF are probably responsible for this, as the density of states near
EF is thought to have a large impact on the behavior of spin fluctuations [135], as is
seen in Stoner theory. A significant difference in magnetic ordering occurs in other Fe
and Co binaries. FeSe and CoSe can both be synthesized in a tetragonal structure.
FeSe is a PM, potentially spin-fluctuation mediated superconductor (Tc = 8 K) [138],
while below 10 K CoSe is a spin glass [12]. Like the arsenides, their band structures
and densities of states have essentially a 1 eV shift between them [139]. CoSb (in the
hexagonal NiAs structure) goes from PM to a spin glass phase with Fe substitution,
while FeSb itself is AFM [140]. For both pnictides and selenides of Fe and Co, rigid
band shifts coming from the extra cobalt electron have a large effect on ground state
magnetism.
6.5 Quantum Oscillations
Measurements of longitudinal resistance and magnetic torque were made at
the NHMFL up to 31.5 T on single crystals grown from Bi flux. Magnetotrans-
port was featureless and reached about three times the sample resistance without
a significant change in field dependence, and no oscillations were visible. However,
oscillations were readily observable in the more sensitive torque signal as low as
130
6 T. Figure 6.7(a) shows the torque at selected orientations of the sample relative
to applied field. Various oscillation frequencies emerge in the data over the entire
angular range, though some correspond to harmonics or the sum of independent
fundamental frequencies. As with FeAs, experimental results could be compared to
DFT band structure predictions with the SKEAF program [82].
6.5.1 Fermi Surface Geometry
The change in oscillation frequency spectrum with applied field angle reflects
the geometry of the Fermi surface pockets. In the measurement ? = 0? corresponded
to B ? c-axis, while ? = 90? was B ? b-axis. The c-axis was confirmed by single
crystal XRD on the platelike sample. Given the ambiguity of the orthorhombic
structure for a square plate crystal, the second axis determination was made on the
basis of comparison to theoretically predicted frequencies. The observed angular
dependence matched very well to predictions for B ? b-axis but not at all for B ? a-
axis. Data were taken in 5? intervals from 0? to 100? and at 120?, 150?, and 180?
[Fig. 6.7(b)].
There were four distinct frequencies predicted by SKEAF, but in the experi-
mental data only three independent sets of oscillations were observed. Each one can
be assigned to one of the predicted electron bands and the two hole bands based on
similarities between calculated and observed frequencies [Fig. 6.7(c)]. There is some
slight disagreement in exact frequency value, but the angular dependence matches
well. The unobserved band corresponds to a second electron band, and does not
131
(a) (b)
180?
150?
  0?: B || c 120?100?
90?: B || b
H2a, H2b
H1
0? 90?
30? 120? +5?
45? 150?
60? 180?
0?
B (T) E H1a, E1b 2 F (kT)
(c)
H1 H2 E1
? (?)
Figure 6.7: (a) Magnetic torque of a CoAs single crystal at various angles at base
temperature (400-500 mK), where B ? c-axis at 0? and the b-axis at 90?. (b) Fast
Fourier transforms of the oscillatory component of the torque for each angle, offset
for clarity. Unlabeled peaks are either higher harmonics or sums of the fundamental
frequencies. Data between 0? and 100? were taken in 5? increments. (c) A com-
parison of observed oscillation frequencies (red squares) and those expected from
the paramagnetic DFT band structure (blue lines) for three of the four predicted
oscillation bands. H1 and H2 are hole bands, E1 is an electron band. The latter two
have some peak splitting.
132
F (kT) Amp. (a.u.)
Amp. (a.u.)
match any of the angular dependent data. It is not atypical for a predicted band
to be absent in measurements [82]. The missing band has the highest predicted
effective mass, which would reduce the oscillation amplitude and make it more diffi-
cult to detect. Additionally, the predicted frequencies correspond to cross sectional
areas much larger than the first Brillouin zone, and an erratic angular dependence
indicative of a potentially unrealistic orbit, another known phenomenon with the
SKEAF program. More important is that all experimentally observed frequencies
can be indexed to a theoretical band. The extremal orbits predicted around the elec-
tron and hole pockets all have a nontrivial shape, and correspondingly the expected
oscillation frequency also varies widely with angle.
The fact that, in spite of this complexity, there is still agreement in the plane
in which rotational measurements were done indicates that the paramagnetic Fermi
surface in Fig. 6.5 reflects the true CoAs Fermi surface as far as can be determined.
The three bands are denoted H1, H2, and E1, with H and E signifying hole and
electron, respectively. The observed hole pockets show consistently higher oscilla-
tion frequencies than the electron pocket, meaning that they are larger, explaining
the positive Hall coefficient. The linearity of the Hall signal, despite the presence of
multiple carriers, can be attributed to a much greater number of hole carriers indi-
cated by the larger oscillation frequencies of the hole bands. Carrier concentration
scales as with oscillation frequency as F 1.5, so frequencies roughly three times as
large for the two hole bands compared to the one electron band would mean nh is
an order of magnitude larger than ne.
To extract oscillation frequencies a third order polynomial was fit to the raw
133
data and subtracted to obtain a residual oscillatory signal. Figure 6.8(a) gives ex-
amples at 100? and 180?. The frequency spectrum after a FFT is in Fig. 6.7(b). The
oscillation frequencies, and therefore Fermi surface cross sectional areas, increase as
the field more closely aligns with the b-axis. Accordingly, both the electron and
hole Fermi surfaces have their largest cross sections in the ac-plane [Fig. 6.5]. Fig-
ure 6.7(c) shows the frequency at which peaks were observed at different angles (red
squares), as well as the angular dependence predicted via SKEAF (blue lines) in
the range 0??90?. E1 shows multiple peaks in this range. This and the increase in
frequency closer to 90? are both in line with theory. The increase in frequency also
comes with a decrease in amplitude, and it is not until 70? that peaks are again clear.
At this point only H1 and H2 are observed, in the 7?8 kT range, with intermittent
lower peaks potentially corresponding to E1.
There are two angular ranges for which fewer than three bands appear: H1
does not appear for 0? < ? < 40?, and E1 does not appear at almost all angles
above 60?. In both cases the disappearance of frequencies can be explained by
experiment-related factors, rather than disagreement with theory. Until 40?, pre-
dicted H1 frequencies are less than 35 T. Such a low frequency is hard to pick out in
the FFTs, which cover a large, higher frequency range, and at fields comparable to
oscillation frequency it is possible that the ?quantum limit?, where all carriers are
in the lowest Landau Level, is reached, and no oscillations will be seen anyway. At
high angles, the predicted effective mass for E1 increases, exceeding 4me. This will
decrease oscillation amplitude, and so it is unsurprising to see this frequency band
become less prominent in the data.
134
6.5.2 Effective Mass
Temperature-dependent measurements were made to determine effective
masses in the ka?kc and ka?kb planes using the LK factor (2.22). At 100
? [Fig. 6.8(a),
left] H1 and two split peaks stemming from the H2 band appear. While the am-
plitudes show a decay with temperature, the fits to the LK formula are not great.
In any case, they give m?H1, exp. = 2.3me, m
? ?
H2a, exp. = 2.6me, and mH2b, exp. =
2.4me. The subscripts a and b denote the lower and higher of the two split fre-
quencies. Theoretical predictions gave m? ?H1, th. = 2.99me and mH2, th. = 1.91me
at 90??the frequency splitting of H2 was not predicted. At 180
? [Fig. 6.8(a),
right] two split E1 frequencies as well as one H2 frequency are seen. Here the
effective mass fits are much better [Fig. 6.8(b), right], and the H2 signal survives
to at least 15 K, indicative of lighter masses at this angle: m?E1a, exp. = 0.70me,
m?E1b, exp. = 0.36me, and m
?
H2, exp. = 0.46me, compared to predicted values of
m? = 1.43m , m? ?E1a, th. e E1b, th. = 1.44me, and mH2, th. = 0.33me. Overall the predicted
and observed masses do not show close agreement, which speaks to the failure of
DFT to capture the shape of the band dispersion. Encouragingly, however, the pre-
diction of smaller masses for B ? c-axis compared to B ? b-axis is borne out by the
data.
As with FeAs, the effective mass gives an alternate method of calculating the
Sommerfeld coefficient (4.1). For B ? [010], the total ? comes out to 3.9 mJ2 ,K mol
close to the value of 6.83 mJ2 directly measured in heat capacity experiments.K mol
For B ? [001], ? = 16.9 mJ
K2
, a much larger value due to the reduced effective
mol
135
masses in the ab plane. In either case no contributions at the Fermi level seem to
be missing, despite not observing the fourth, heavier, DFT-predicted band.
6.5.3 Dingle Temperature
With the masses known, the Dingle temperature (2.23) and scattering rate
(2.24) can be assessed. Again, it is only feasible to do Dingle analysis for peaks
with a large relative amplitude at an angle for which m? is known. Thus TD could
only be obtained for H2 at 100
? (by averaging the split peaks) and 180?, since the
exponential decay of H2 dominates the oscillatory signal [Fig. 6.8(a)]. Fitting the
position of peaks with inverse field results in TD, H2 = 3.66 K and 14.1 K at 100
?
and 180?, corresponding to ?H2 = 3.0 ? 1012 s?1 and 1.2 ? 1013 s?1, respectively.
The angular dependence of H2 clearly shows it is not spherical, but it is possible to
average the values for the two different field directions to obtain a rough estimate
of ` = 500 A? for H2. The hole pocket has a significant directional dependence in
terms of both effective mass and Dingle temperature between the ac and abplanes,
as the values listed in Table 6.1 indicate. In the AFM states of CrAs, MnP, FeP, and
FeAs, the ab-plane features two noncollinear rotating magnetic moments [21,29,46].
This could be a sign that any magnetic fluctuations in CoAs are occurring in the ab
plane.
136
(a)
470 mK 440 mK
100? 180?
near H || b H || c
B (T) B (T)
(b)
H,1  8.4 kT (x10) H,2  570 T (?10)
H2a, 7.2 kT E1a, 210 T
H2b, 7.9 kT E1b, 380 T
T (K) T (K)
(c)
H 2 (averaged) H2
1/B (T-1) 1/B (T-1)
Figure 6.8: (a) Residual oscillatory signal for 100? (near B ? b-axis) and 180?
(B ? c-axis). (b) Temperature dependence of the amplitude of the observed peaks
at both angles, with fits to the Lifshitz-Kosevich factor. To ease comparison between
different bands, on the left H1 amplitude is increased by a factor of 10 and on the
right H2 decreased a factor of 10. (c) A plot of the peak amplitude versus inverse
field for H2 at both angles, with accompanying exponential decay fits to solve for the
Dingle temperature. Only H2 was used since it was the most prominent frequency
in both cases. Data for 100? are an average of the two observed peaks.
137
Amp (a.u.) Amp (a.u.) Amp (a.u.)
Band B ? [hkl] F (T) m* /me TD (K)
H1 [010] 8390 2.3 ?
H2a [010] 7200 2.6 ?
H2b [010] 7900 2.4 ?
H2,ave [010] ? ? 3.66
H2 [001] 570 0.46 14.1
E1a [001] 210 0.70 ?
E1b [001] 380 0.36 ?
Table 6.1: Parameters extracted from CoAs torque oscillations data with field
applied in different directions. Note that what is called [010] actually corresponds
to an angle 10? off of the b-axis. The average m? of H2a and H2b was used in
calculating TD for H2,ave.
6.6 CoAs: Conclusions
The flux-grown crystals of CoAs have a lower residual resistivity and more
consistent orientation, but the vapor transport samples can grow much larger, en-
abling bulk measurements such as magnetization or heat capacity. Data show hole-
dominated conduction with no indication of long range magnetic ordering down to
1.8 K. This is in line with previous polycrystal work, even as predictions slightly
favor weak moment ferromagnetism. Quantum oscillations are present in torque
starting from 6 T up to 31.5 T, and their angular dependence is in line with the
geometry of the calculated paramagnetic Fermi surface.
Transition metal pnictide compounds and the iron-based superconductors have
shown unique magnetic arrangements and often superconductivity upon suppression
of ordered magnetism. CoAs is paramagnetic, but the nonmonotonic temperature
dependence of its magnetic susceptibility and a low temperature enhancement of
138
heat capacity point to possible magnetic fluctuations. The notion of a ferromag-
netic instability is also supported by DFT calculations favoring long range ordering
at zero temperature, and other theoretical work has claimed that the inherent band
structure of B31 materials may make them susceptible to ferromagnetic Stoner in-
stabilities [32]. Furthermore, there are many similarities to antiferromagnetic FeAs.
Both show a maximum in paramagnetic susceptibility, and their calculated elec-
tronic structure differs only by a one electron rigid band shift, with the effect of unit
cell size apparently negligible.
An earlier pressure study [127] up to 10 GPa showed a possible structural
transition at 7.8 GPa, and other elements of the structure identified by the authors
indicate a potential sensitivity to lattice shifts at high pressures. Very basic high
pressure measurements were made as a corollary to this study, but at 2 GPa ?(T)
was nearly indistinguishable from ambient pressure results. It is possible CoAs is al-
ready close to the crossover between stabilizing magnetic order and being dominated
by fluctuations. Future work with chemically substituting or applying pressure to
binary CoAs may lead to the discovery of structural, magnetic, or superconducting
transitions.
139
Chapter 7: CoP: A Nonmagnetic Contrast
A look at CoP will round out the four possible Fe/Co-P/As combinations in
the MnP-type family. As with CoAs, single crystal reports on this material were
scant before the work shown here, consisting of just crystal structure analysis and
high temperature magnetic susceptibility [23, 141, 142]. It too is void of magnetic
order, a property that it can now be seen is determined by the occupant of the
transition metal site in this family. It was previously found that, unlike CoAs, CoP
does not undergo a high temperature transition to the hexagonal NiAs structure1.
There are other differences between the two, and in fact some similarities in terms
of transport properties to FeP. Investigation into CoP makes it possible to look at
the effect of changing out the metal and pnictogen atom in the Pnma structure
individually.
7.1 Crystal Growth
Unsurprisingly, CoP can also be grown with vapor transport. One notable
aspect is that polycrystalline powders without impurity phases were much easier to
make than for FeP. There is, evidently, a difference in reactivity of the two elements
1In fact, it has been noted that only arsenides can take on both structures [22].
140
with phosphorus. A previous report on crystal growth compared transport with the
elements in a stoichiometric ratio or prereacted powders (with I2 as the transport
agent in both cases) [142], and saw higher mass transfer with the individual elements.
However single crystals produced from either method are of to similar quality, and
growths were left long enough for all material to react (10-14 days). As was also
noticed for FeP, crystals grown from either starting material showed about a 55:45
Co:P ratio with EDS. Temperatures at the hot end of the tube from 900 ?C to
1025 ?C had no obvious impact on the growth. Since large crystals of sufficient
quality could be grown with CVT, and flux growth was unsuccessful in the case of
FeP, no attempts were made at flux growth of CoP. Lattice parameters from powder
XRD of CVT crystals are a = 5.08 A?, b = 3.28 A?, and c = 5.59 A?, in agreement
with previous reports [23].
7.2 Physical Properties
The electrical transport behavior of CoP is very similar to that of CoAs. The
temperature-dependent resistivity has an initial linear decrease from room temper-
ature (which is seen when electron-phonon scattering dominates) before saturating
at lower temperature [Fig. 7.3(a)]. RRR of the best samples is about 80, higher than
what was found with CoAs crystals grown by any method. The residual resistivity is
about 0.5 ?? cm. This parallels the higher RRR and lower ?0 of FeP in comparison
to FeAs, indicative of a general trend of less impurity scattering in the phosphide
binaries in comparison to the arsenides. That being said, the difference between the
141
40 (a)
30
20
10
CoP RRR = 90     ?0 = 0.5 ?? cm
0 0 100 T (K) 200 300
(b)
8
7
6
5   ? = 4.3 mJ/K2 mol
?D = 336 K
4 0 10 20 30 40
T2 (K2)
Figure 7.1: (a) Zero field resistivity as a function of temperature for a CoAs single
crystal. (b) Low temperature specific heat of a CoP single crystal. The red line is
a fit to the low temperature Debye model [Eq. 3.1], and extracted parameters from
the fit are given.
142
C/T (mJ/K2 mol) ? (?? cm)
two Co compounds in terms of RRR is much less.
Measurements of the specific heat were also made at low temperature. The
data, when plotted as C/T vs. T2, can be fit to a straight line with ? = 4.3 mJ
K2 mol
and ?D = 336 K [Fig. 7.1(c)]. Both values are slightly smaller than in CoAs and FeAs
[11,30], but nevertheless indicate metallic behavior and a similar phonon dispersion
to the other B31 structure materials. The smaller ? can be interpreted as a sign of
slightly weaker correlations. This fits with the fact that CoP is further from long
range magnetism than the three materials discussed thus far.
Magnetoresistance is large in CoP, increasing by nearly 100 times up to 31 T
[Fig. 7.2]. As with FeP, the smallest increase comes for field very close to the c-axis.
The lowest value at 31 T was actually found to be at -5?, but that could be from
a slight misalignment of the rotator or the sample on the rotator. The inset to
Fig. 7.2 shows the MR at 31 T for more angles, and illustrates the steep decline
in MR over a narrow angular range near the c-axis. Data were not taken with
as fine of spacing near B ? a-axis, but the decrease in MR at 86? indicates that
the same phenomenon may happen there as well. This would contrast with FeP,
where the B ? a-axis results in the highest magnetoresistance. None of the curves
showed the linearity of FeP near B ? c-axis, but rather the gradual crossover to
high field linearity seen in FeAs and, away from the c-axis, FeP. Given that CoP
is paramagnetic, it also confirms that the magnetism is not the primary reason for
large MR in FeP, which also makes sense given that AFM FeAs has only a moderate
field-induced increase. Though not as much Hall effect or temperature-dependent
143
100 00?: B || c
80 90?: B || a
75 0-5?  45?
-00? -60?
60 50 -15? -86?-30?
25 B = 31 T
40
0 -30 0 30 60 90
? (?)
20
T = 600 mK
0 0 10 20 30
B (T)
Figure 7.2: Transverse magnetoresistance of CoP at 600 mK and up to 31 T at
different angles, with field rotated between being parallel to the a- and c-axes.
Inset: the 31 T MR at various angles, extracted from different field sweeps rather
than a single angle sweep.
144
MR
MR data was taken on CoP, presumably it too shows some degree of compensation,
resulting in the large increase in scattering in field. Note that MR never saturates, as
was the case with FeAs or FeP. This indicates either some amount of compensation,
or open Fermi surface orbits. While no quantum oscillations were seen in transport
except possibly near B ? c-axis (likely attributable to somewhat noisy data, and
the generally difficulty of observing SdH oscillations in many of these compounds),
many dHvA were seen at all angles in magnetic torque, making the latter scenario
unlikely.
Magnetic measurements of CoP show an almost flat temperature dependence,
with a slight dip around 250 K and divergence at lowest temperatures [Fig. 7.3(a)].
The data shown are with a 5 T applied field due to the small signal, but temperature
sweeps at lower fields down to 0.1 T showed the same general temperature depen-
dence, including the 250 K dip. There was none of the nonmonotonic behavior above
the upturn that could be linked to magnetic fluctuations in CoAs and FeAs [135].
An Arrott plot has a negative y-intercept [Fig. 7.3(b)], indicating no spontaneous
zero field magnetization. Using the specific heat and magnetization data, the Wilson
ratio (2.1) can be calculated, and RW = 1.7 is obtained. This is quite a bit lower
than the value of 6.2 for CoAs, indicating much weaker ferromagnetic fluctuations.
7.3 Quantum Oscillations
As with FeAs and CoAs, magnetic torque measurements were carried out for a
CoP crystal rotated such that field moved in the orthogonal ab and bc planes. This
145
1.3
(a)
1.2
1.1
CoP
1.0
B = 5 T
0 100 200 300
T (K)
5 K (b)
4 10 K
3
25 K
50 K
2
300 K
1
0
0 2 4 6
B/M (103 T/(?B/Co))
Figure 7.3: (a) M(T) of an unaligned CoP single crystal at B = 5 T. Fields down
to 0.1 T had the same temperature dependence. (b) An Arrott plot for the same
sample, showing a lack of ordered magnetism down to 5 K.
146
M2 (10-6 (?B/Co)
2) ?  (emu/mol Oe)
1 (a) 0 (b)
0
00?: B || a 00?: B || c
-1 90?: B || b -1 90?: B || b
-15? 60? 02? 53?
-2 -00? 75? 31? 80?
-30? 90? 47? 90?
-45? -2
-30 10 20 30 0 10 20 30
B (T) B (T)
1.0 (c) 00? 1.0 (d)  0?
0.5 0.5
0.00 5000 10000 0.00 200 400 600 800
0.4 (e) (f)
0.3 45? 1.5 45?
0.2 1.0
0.1 0.5
0.00 500 1000 1500 2000 0.00 500 1000 1500
2.0
(g)
1.5 90? 4 (h) 90?
1.0 2
0.5
0.00 200 400 600 800 00 200 400 600 800 1000 1200
F (T) F (T)
Figure 7.4: Magnetic torque of the same CoP crystal from 0 to 35 T rotated such
that field moves in the (a) ab and (b) bc planes. (c-h) Fast Fourier transforms of
the residual oscillatory signal after subtraction of a polynomial at 0?, 45?, and 90?
for each of the two measurements. Note that (g) and (h) are at the same field
orientation. Some higher frequency peaks have been cut off for ease of seeing lower
frequencies.
147
Amplitude (a.u.) Amplitude (a.u.) Amplitude (a.u.) ? (a.u.)
allowed for a three dimensional analysis of quantum oscillation frequencies, of which
there were many with widely varying frequencies. No band structure calculations
were made for CoP from which theoretical frequencies and effective masses could be
generated. For that reason, it is difficult to extract a detailed picture of the Brillouin
zone from QO measurements alone, nor can it be determined whether fundamental
frequencies come from electron or hole orbits. But still some comments can be made.
7.3.1 Angular Dependence
Raw torque data are shown in Fig. 7.4 for select angles up to 31 T, with more
angles measured beyond those shown. Oscillations are clear at fields as low as 5 T.
Measurements with field rotated in the b-c direction were slightly more noisy. There
are clear differences in the oscillation pattern as orientation is changed, and these
are reflected in the FFTs. The change in oscillation peak frequency as a function
of angle is given in more detail in Fig. 7.5 for the two directions in which B was
rotated. Note that the points connected by lines are only assumed to be the same
orbit due to their similar frequencies; without theoretical input there is no way to
confirm this. In tracking individual orbit frequencies, it does not seem that specific
peak frequencies are changing significantly in value, though some appear over a
limited angular range, which could potentially be put down to signal resolution
(especially for higher frequency peaks, which often had a lower amplitude). This
is quite different from FeAs and CoAs, where a smaller number of fundamental
frequencies showed a clear angle dependence. The CoP Fermi surface generally
148
00?: B || a
10000 90?: B || b
5000
1000
500
(a)
100 -30 0 30 60 90
10000
5000 00?: B || c90?: B || b
1000
500
(b)
100 0 30 60 90
? (?)
Figure 7.5: Quantum oscillation peak frequency as a function of angle for field
rotated in the (a) ab and (b) bc planes. Different colors correspond to presumed
different extremal Fermi surface pocket orbits in each plane, with no correspondence
between (a) and (b). Note that data were taken for a smaller number of angles for
(b), which is the reason for the large gaps in angle.
149
F (T) F (T)
seems to be composed of pockets with a more spherical shape, which therefore
would have minimal angular dependence.
Some of the observed peaks, especially low frequencies near 0? or 90? in Fig. 7.5,
are clearly harmonics of a single fundamental. The FFTs in Fig. 7.6(d) and (g) are at
the same orientation and both show peaks at 200, 400, and 600 T, which are almost
certainly related. Fig 7.6(e) also has its most prominent peak at about 300 T with
others appearing every multiple of 300 T above that with low amplitude. Such clear
harmonics indicate low scattering rates, in line with the low residual resistivity.
The presence of relatively high frequency oscillations (1 kT or more, up to 10 kT)
at some angles is a sign of large area orbits. Both of these point to CoP being very
conductive, with a high carrier density.
7.3.2 Temperature and Field Dependence
Temperature dependence was only done for the a-b rotation measurements,
with B along [100], [110], and [010]. The results for some frequency bands are
shown in Fig. 7.6, and the full slate of parameters is in Table 7.1. For B ? [100] and
[110], only the temperature dependence of the highest amplitude peak was tracked,
since the others peaks disappeared at too low of temperature for adequate fitting.
For B ? [010], the effective masses of six peaks, four at low frequency and two at high
frequency, were obtained. The mass of the four low frequency peaks increases with
each successively higher frequency, as should be the case for harmonics of a single
fundamental. In general, all effective masses were close to the electron rest mass,
150
indicating a lack of strong correlations. That being said, they are n?ot substantially?N k2
lower than those of CoAs. Once again calculating ? = a B2 xix m
?
j , where x3h? i
and xj are the lattice parameters in the plane perpendicular to field, the values are
2.2, 1.4, and 5.3 mJ/K2 mol for [100], [110], and [101]-oriented fields. That the first
two values are smaller than the 4.3 mJ/K2 mol from specific heat is expected, since
there are known oscillation frequencies not included. The last was computed using
only the 180 and 9950 T parameters, as the other four observed peaks are either
harmonics or split peaks. It is approximately equal to the direct measurement, a
good indicator that there are no additional missing orbits and that the other FFT
peaks do indeed correspond to harmonics.
Again, the Dingle analysis is limited to only the most prominent oscillation
frequency at each angle. This is easy enough for field along [100] or [110], but
for [010] the 180 T dominates. TD values for CoP are a bit higher than most of
the others calculated for FeAs and CoAs, a bit surprising given its lower residual
resistivity. The calculated ? values are 5.8 ? 1012, 4.1 ? 1012, and 6.3 ? 1012 s?1
for B ? [100], [110], and [010] respectively. Given the relative angle insensitivity
of most CoP oscillation frequencies, a spherical Fermi approximation may be more
legitimate than in the case of FeAs or CoAs. The calculated ` values from doing so
are 860, 580, and 280 A? for the same orientations. These are actually some of the
longest mean free paths of the three compounds this analysis has been done far, in
part because of the low masses (and in some cases high frequencies) of the orbits.
This could account for why harmonics are much more visible in CoP than the two
arsenides examined in previous chapters.
151
(a)
1.0 B || b
0.5 K  0.7 K  1.7 K  3.1 K
0.5 3.9 K  4.3 K  6.2 K  10  K
0.0
180 T Peak
-0.5 1.0
0.5
-1.0
0.05 0.06 0.07 0.08 0.09
0.04 0.06 0.08 0.10 0.12 0.14 0.16
B-1 (T-1)
(b) 0.2 (c)
3
2
0.1
1
0 0.0
0 200 400 600 800 9600 9800 10000 10200 10400
F (T)
0.3
4
180 T  290 T (d) (e)9950 T  10100 T
3 385 T  605 T 0.2
2
0.1
1
0 0.0
0 2 4 6 8 10 0 2 4 6
T (K)
Figure 7.6: (a) Residual torque amplitude after subtraction of the nonoscillatory
component for CoP at various temperatures for B ? b-axis. The inset is a fit of the
lowest frequency (180 T) peaks in the residual data to the Dingle factor [Eq. 2.23].
The temperature dependence of FFT amplitudes at (b) low and (c) high frequencies,
with colors corresponding to the same temperatures as in (a). Peak amplitudes
as a function of temperature for (d) low and (e) high frequency oscillation bands
(labeled). Solid lines are fits to the Lifshitz-Kosevich factor [Eq. 2.22]
152
Amplitude (a.u.) Residual ? (a.u.)
B ? [hkl] F (T) m* /me TD (K)
[100] 8910 1.21 6.99
[110] 295 0.46 4.99
[010] 180 0.49 7.71
[010] 290 0.61 ?
[010] 385 0.94 ?
[010] 605 1.73 ?
[010] 9950 1.39 ?
[010] 10100 1.61 ?
Table 7.1: Parameters extracted from CoP torque oscillations data with field
applied in different directions. The four low frequencies for B ? [010] are likely all
harmonics of the same fundamental.
7.4 CoP: Conclusions
Looking at CoP crystallizes some of the ideas that came from examining the
different properties of FeAs, FeP and CoAs. It is clear that magnetic fluctuations
have much less relevance to describing this material; it is not ordered, and beyond
that has a much lower Wilson ratio than CoAs. Therefore any properties common
to CoP and any of the other three compounds must not be related to magnetism.
The high quality of CVT crystals is reflected in the low residual resistivity and
significant number of harmonics seen in QO measurements. It seems then that B31
phosphides are ?cleaner? than the arsenides. The lack of magnetism also indicates
that it is the transition metal that plays a significant role in the presence or absence
of magnetic order. On the other hand, the large, anisotropic, and nonsaturating
magnetoresistance in CoP indicates that that of FeP is likely unrelated to magnetic
order, and reinforces compensation, combined with the low residual resistivity, as
153
the most logical explanation. The linear MR of FeP at a specific angle, however, is
not replicated here, another sign that the two large and linear MR have separate
origins in this material class.
Future investigation into the theorized properties of CoP would be useful,
especially if it could describe the Fermi surface that seems to be composed mostly
of low dimensional bands that appear over only a short angular span. CoP serves
an important role as a contrast to the others presented here. Investigation of the
properties of this compound are a useful basis to further understand the interesting
behavior stemming from this family it belongs to.
154
Chapter 8: Fe1?xCoxPn
Given the differences in the low temperature magnetic state of FeAs and FeP
in comparison to CoAs and CoP, it was natural to investigate the effect of Fe-Co sub-
stitution. In the iron pnictides, chemical substitution to suppress the spin density
wave is one way of stabilizing the unconventional high-temperature superconduct-
ing phase [16]. Chemical substitution, along with magnetic field or pressure, is a
common way to generate the same quantum critical phase diagram. For that rea-
son samples of both Fe1?xCoxAs and Fe1?xCoxP were grown and their properties
measured1.
Chemical substitution series in the B31 family have been produced before, in-
cluding with magnetic end members [23,25,26,141]. In fact, the early neutron studies
on CrAs, MnP, and FeP also included work on the CrAs1?xSbx series, showing a
change in TN and antiferromagnetic structure with substitution [29]. The movement
from helimagnetic, orthorhombic CrAs to collinear AFM, hexagonal CrSb made
things especially complicated in that case. In the case of the insulating Ru-based
pnictides of P and As, it was found that with Rh doping emerged a metallic state
and superconductivity, with a maximum Tc = 3.7 K at x = 0.45 in the P series [143].
1Growth of Fe1?xCrxAs was also attempted to look at a series in which both end members were
AFM, but no pure phase crystals were obtained. A Co1?xRhxAs series produced no interesting
behavior.
155
In this case, there was also a ?pseudogap? phase that emerged at lower doping in
which a superstructure (possibly from charge density wave formation) was noticed
in x-ray measurements. The cuprates are the most famous example of a pseudogap
material, whose temperature is heavily dependent on doping level [17]. Ru and Rh
lie just below Fe and Co on the periodic table; the similarities in electron count
with doping between the two are yet another reason to examine the 3d elements.
Combined with the requirement that magnetism be at some point suppressed with
lowered Fe content, Co doping on the Fe site was of high interest.
8.1 Crystal Growth
Synthesis was first attempted for the Fe1?xCoxAs series. It was found that
crystals grown with Bi flux were too small for transport measurements, so CVT was
tried instead. For safety reasons, instead of starting with the elements, prereacted
FeAs and CoAs were combined with I2 in an evacuated quartz ampule. Since the
optimal vapor transport conditions for FeAs and CoAs are slightly different, various
temperatures were tried to find the best way to mix the two. The temperature at
the hot end of the ampule was varied from 850-950 ?C, where the lower temperature
had been used for pure FeP and higher temperatures for CoP. Temperatures of at
least 900 ?C seemed to be better for substituted samples, as lower temperatures
led to little material transporting or a large deviation from expected stoichiometry.
Concentrations were checked with EDS, and all single crystal data presented here
is from samples for which Co doping level was determined individually.
156
Often, crystals appeared at both the hot and cold ends of the CVT ampule
once the growth was cooled. EDS showed that samples at the hotter end typically
had a lower x than those on the cold end, and one that was substantially lower than
the molar ratio of the initial mixed powders. This implies different thermodynamics
for the growth of samples depending on Co concentration, and that it was easier
to transport samples with more Co. This may be connected to the fact that CoP
formed more easily than FeP as a powder, indicating Co reacts more readily with
pnictogen atoms. The value of x for crystals from the cold end of these growths was
generally close to the nominal value, with a variation (in terms of x ) of about 0.05.
Measurements on individual crystals produced more consistent results, indicating
that the variation between samples was real and not the result of uncertainty in the
EDS measurement itself.
Although CVT produced larger samples, they were still not as big as those
obtained for the end members for either the phosphide or arsenide series. The small
mass made heat capacity and magnetization (especially given the relatively small
moments) measurements difficult. Given the variation in the Co doping level, even
powder measurements or those incorporating multiple single crystals were likely to
wipe out a lot of the features that were very sensitive to x value, due to the variation
observed in EDS. Therefore, the only data to be presented on these series is from
resistance measurements.
157
8.2 Transport
As seen in Chs. 4 and 5, a distinct kink in resistivity marks TN in both FeAs
and FeP, which both have a somewhat sigmoidal resistivity curve. In contrast, both
of the Co compounds have a mostly featureless ?(T). With increased Co doping,
there is a continuous evolution from one form to the other, as the curves begin to
look more linear. The RRR for doped samples is much lower than either parent
compound, which is unsurprising, as the mixture of two kinds of transition metal on
a single site should increase disorder. Inherent disorder or differing sample quality
could also explain the variation in room temperature resistivity values, as does the
fact that they differ by a factor of about five for the two parent compounds. The
effect of substitution on scattering is most obvious in the phosphide series, where
RRR can exceed 1000 for FeP but quickly falls below 10 with only a small amount
of doping. [Fig. 8.1].
Notable, however, is that the kink to mark the AFM transition disappears
after very little doping. In fact, it does not appear once there is more than 5% Co in
either case, and instead becomes more washed out and is replaced by an inflection
point. There is no way of determining from resistance alone whether the inflection
point still corresponds to a magnetic transition. However, it was assumed to do so
and was tracked as a function of Co concentration. This was done by looking for the
maximum in d?/dT (equivalent to the inflection point, where d2?/dT2 = 0). The
presumed Ne?el temperature is suppressed by 50% Co concentration in both cases
[Fig. 8.1], and takes on the linear (and at low temperature, constant) shape typical
158
of Co(P,As), but with much higher residual resistivity. For the arsenides, the critical
doping for suppression of the inflection point is slightly lower, likely due to the fact
that TN of the pure compound is 50 K lower.
The suppression of a second order magnetic transition to zero temperature is
a critical point. However, typically quantum critical points also come with resistiv-
ities that are linear in both temperature and magnetic field, due to the dominance
of quantum fluctuations and loss of an internal energy scale [2, 76]. That is not the
case in Fe1?xCoxPn. While at higher temperatures ?(T) is linear, it still shows the
same plateau at the lowest temperatures. The higher temperature behavior sim-
ply gradually looks more like CoAs, where such a dependence likely from a typical
electron-phonon scattering, and the plateau is the impurity scattering baseline. Ad-
ditionally, magnetoresistance is negligible for doped samples, 10% or less at 14 T
and 2 K, despite being massive in the case of the phosphides and appreciable even
for the arsenides. Measurements below 200 mK using adiabatic demagnetization
refrigeration showed no change in resistance (i.e., no superconducting transitions).
The lack of superconductivity and linearity with temperature or field demonstrate
that the suppression of magnetism in the Fe-Co binaries does not lead to the same
kind of quantum critical physics as in other systems.
8.3 Conclusions
Already, there were differences between the MnP-type binaries and the tetrag-
onal, layered iron-pnictide superconductors. The arrangement of As atoms around
159
(a) (b)
x = 0.18
x = 0.08
Fe1-xCoxAs Fe1-xCoxP
300 x = 0 300
x = 0.30
x = 0.53
x = 0
x = 0.52
200 x = 0.04 200
x = 0.02
x = 0.11
100 100
x = 0.23
x = 1 x = 1
0 0
0 100 200 300 0 100 200 300
T (K) T (K)
(c) (d)
70 120
60
100
50
80
40
60
30
40
20
20
10
0 0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
xEDS xEDS
Figure 8.1: Resistivity as a function of temperature for Fe1?xCoxAs (a) and
Fe1?xCoxP (b) with various cobalt concentrations (labeled). Below are phase di-
agrams of presumed TN (based on the ?(T) inflection point) for the (c) As and (d)
P series.
160
TN (K) ? (?? cm)
the Fe atoms was different in the two systems, and pressure studies on FeAs showed
no superconductivity [144]. Now, by having doped Co onto the Fe site, this differ-
ence has been reconfirmed. There is no familiar phase diagram with the suppression
of AFM leading to the emergence of superconductivity [16]. While long range order
is indeed suppressed, that result was unavoidable given that the Co materials are
PM. At some point between 100% Fe or Co, the system has to lose its ordered state.
Neutron scattering measurements could better confirm that the inflection point of
the resistivity does still correspond to a magnetic transition, as it was assumed to,
but may also require growths with a more consistent Co concentration. As is seen
from transport, even a few percent difference in x can change the presumed TN sub-
stantially. Such measurements could also examine if there is any change to magnetic
ordering wavevector, or its strength, with doping. But the most important question
was whether the AFM suppression leads to a QCP and superconducting state as in
the Fe-based superconductors or CrAs, MnP, and Ru1?xRhx(P,As) [19, 21, 34, 143].
The answer to that is definitively negative.
161
Chapter 9: Chapter 9: Conclusion
9.1 Summary of Findings
The five principal chapters of this thesis have dealt with the properties of the
four possible combinations of (Fe/Co)(P/As) in the MnP-type structure, and the
effect of Fe-Co substitution. In doing so, some trends have become clear.
The presence or absence of magnetic ordering is determined by the transition
metal, and by extension electron number. While Fe and Co are both elemental
ferromagnets, the former forms AFM compounds in this structure, and the latter
PM ones. The only two other magnetic materials in this family are composed of Cr
or Mn, both to the left of Fe. The Fe-Co line is the boundary between stabilizing
the strong magnetic fluctuations of CoAs into an ordered state. Electron count
in these materials has also been thought to determine whether hexagonal NiAs
or two orthorhombic distortions (the MnP and NiP-type) is favored [22], and the
topological properties of the Fermi surface [32]. This is due to the change in metal-
metal bonding that occurs as more electrons are added and energy levels change.
The same effect may be related to the development of long range order. As seen in
Ch. 6, the band structures of CoAs and FeAs with the CoAs lattice parameters, both
in the PM state, are nearly identical, indicating that the differences in ground state
162
magnetism are not simply a matter of unit cell size. The change in interactions or
shift in the Fermi level associated with a change in valence electron number must also
be considered. The Fe1?xCox(P,As) substitution series showed that magnetic order
is suppressed with 30-50% Co, though the distinctness of the transition vanishes at
much lower levels (< 5%). The disappearance of ordered magnetism does not lead
to a superconducting or otherwise interesting low temperature state, in contrast to
other members of this family.
On the other hand, though transition metal choice is the key to stabilizing long
range order, it seems that the pnictogen element determines the strength of magnetic
fluctuations. CoAs has a higher Wilson ratio than CoP, among other signs of being
nearly ferromagnetic. And while TN is higher in FeP than FeAs, the moment is
much smaller than the other B31 helimagnets [Table 2.1], although at about 0.5?B
it is comparable to that of FeAs. Additionally, the linear magnetoresistance when
field is applied along the c-axis indicates that it may be near the point at which
the magnetic state is suppressed. A previous study showed that not only does the
SDW state in FeAs survive up to 11 GPa, but that TN changes by less than 5 K
in that range [144], rather than the continuous suppression of TN seen in MnP and
CrAs [21]. Analysis of Cr-based pnictides found a continually weaker moment with
lighter pnictogen atom. This was attributed to the shorter b-axis with the smaller
atoms [18]. Similarly, the phosphides studied here have smaller b-axes than the
arsenides, in large due to the atomic size of P compared to As.
The presence of As or P also impacts the magnetoresistance. CoP and FeP
both have substantially larger MR values at high field than their arsenide coun-
163
terparts. The same has been found by other researchers for the Cr-based equiva-
lents [19,112]. Based on those results and data shown here for FeP, it is possible to
infer that large MR is driven by a combination of moderate electron-hole compensa-
tion and a highly anisotropic Fermi surface. However, it is clear that compensation
is not perfect, since the Hall effect only shows slight nonlinearity in FeP, and the
MR is still orders of magnitude smaller than in semimetals. The larger mass of the
As atom may lead to a spin-orbit coupling effect that changes the band structure
at the Fermi energy, as was suggested in a study of FeAs [98], and may be why
theoretical predictions are unable to describe the material?s behavior. The lower
residual resistivities of the phosphides, 0.5 ?? cm or lower at 1.8 K, indicate re-
duced impurity scattering, even as the temperature dependence of resistivity looks
similar for materials with the same transition metal component. Given that trans-
port behavior in field has been a key driver to claim possible quantum critical or
topological physics in these materials [19,32,112], the knowledge that phosphorus is
the determining element of those phenomena could be significant in pinning down
the source of large MR both theoretically and experimentally, and for future study
of potential topological physics in this family.
9.2 Future Directions
There are several potential paths of followup study in the metal-pnictide bi-
naries. An obvious next step for comparison is a study of the behavior of FeP
under pressure to see if helimagnetism is suppressed and a different phase emerges.
164
The behavior of each lattice parameter with pressure, and any structural transitions
such as the one that seems to occur in CoAs under pressure [127], are a related
line of investigation. The b-axis collapse accompanying the magnetic transition in
CrAs [18] and negative thermal expansion of the CoAs a-axis [23] are hints that the
MnP structure may have some unusual behavior under compression. The negative
thermal expansion may also be a sign of changing strength of magnetic interactions.
The emergence of unconventional superconductivity in CrAs and MnP under pres-
sure makes study of magnetically and structurally similar materials of clear interest.
Such work has already been done for FeAs [144], but not for FeP, CoP, or CoAs,
outside of a cursory measurement at 2 GPa in the last of these that showed mini-
mal change in resistivity. Compared to the pressure-induced superconductors, FeP
has a lower magnetic ordering temperature (recall that while TN = 50 K in MnP,
TC = 291 K) and the Co compounds do not order, meaning all may superconduct
at lower pressures than were needed for the two already studied compounds.
So far, density functional theory calculations have failed to reproduce the
correct low temperature magnetic state for FeAs and CoAs, and quantum oscil-
lation measurements further indicate that the predicted Fermi surfaces, especially
for FeAs, do not match experiment. More effort is clearly needed to understand
the relevant interactions that must be considered so that the proper ordering is
energetically favored. That may also require more precise mapping of the band
structure. While quantum oscillations are a good probe, they cannot give the same
clear picture that as something like angle-resolved photoemission spectroscopy. Im-
proved band structure pictures would also help determine the influence of semi-Dirac
165
points, specifically their proximity to the Fermi energy, on the magnetotransport,
and accordingly what role topology plays in these systems. The differences between
the type of pnictogen in the structure also indicate that spin-orbit coupling, whose
strength depends on atomic mass, may be relevant. Further work with other re-
lated materials that form with even heavy atoms, such as those to be covered in the
Appendix, could shed light on this.
166
Appendix Additional MnP-Type Compounds
Beyond those covered so far in this thesis, there are a whole host of binary
transition metal-pnictogen compounds. Many, including several different doping
series have been characterized in depth in other works [23,25,26,40,141]. However,
due to a combination of a lack of time and interest on the part of researchers, there
are still some for which complete physical property reports are lacking. This section
will briefly go over additional work, which was either not compelling or not complete
enough for to be included as an independent section.
A.1 RhPn
There exist three Rh-based pnictogen compounds: RhAs, RhSb, and RhBi.
In a report on the properties of almost all MnP- or NiAs-type transition metal
arsenides, Saparov et al. specifically mention that RhAs was the only one not stud-
ied, simply saying that it was ?expensive? [40]. However, interesting results have
been noted in the past for Rh-based binaries. In the 1960s, RhAs and RhBi were
reported to be superconductors with transition temperatures of 0.58 and 2.06 K,
respectively [145]. While those are not high temperatures, only one other binary
pnictide has been reported as an ambient pressure superconductors: WP, to be dis-
167
cussed in the next section. A much more recent study on insulating RuPn materials
found that Rh substitution suppressed the metal-insulator transition and led to a
superconducting dome with a maximum Tc of 1.8 K for Ru1?xRhxAs and 3.7 K
for Ru1?xRhxP (note that pure RhP does not form) [143]. Furthermore, lying just
below Co on the periodic table, Rh would add another layer of complexity to the
near-magnetism seen in the Co-based pnictides.
A.1.1 Crystal Growth
It was found that each of RhAs, RhSb, and RhBi differ substantially in the
required growth process. All three involve different fluxes. To grow single crystals
of RhAs, Rh and As were first prereacted to form a binary powder. Bismuth can
be used as a flux for single crystal growth; Sn flux and vapor transport were both
unsuccessful. The optimal growth profile was found to be a 1:20 RhAs:Bi ratio, in
which the material was heated to 1050 ?C, held for 24 h, then cooled at 3 ?/h to
650 ?C and spun in a centrifuge. It was observed that lower spin temperatures often
led to formation of RhAs2. Given the lack of a Rh-As phase diagram, however, the
exact temperature at which the 1:2 phase begins to form is still unknown. Samples
grown in this way have a hexagonal or diamond-like shape, indicating that there is
likely a higher temperature hexagonal-orthorhombic structural transition for RhAs,
as there is for CoAs and other B31 compounds [22,23].
RhSb can be grown from flux, however RhSb2 begins to form at 1000
?C, thus
requiring a very high spin temperature to get the 1:1 phase exclusively. Sn, Bi, and
168
self flux were unsuccessful, but Pb was found to work. The temperature profile used
was to heat to 1160 ?C, hold for 10 h, and spin at 1090 ?C after very slow cooling at
a rate of 1 ?/h. It is likely that this temperature range could be extended to produce
more crystals, with a higher max temperature (but below the melting point of the
quartz ampule, about 1225 ?C) and spin temperature closer to the RhSb2 formation
boundary.
The heaviest member of this family, RhBi, received the least thorough inves-
tigation of optimal growth conditions. However, it was found that self flux could be
used to produce crystals. Rh and Bi in a 2:3 ratio can be heated to 1050 ?C, held
for some time, and cooled at 3 ?C/h to 800 ?C before being put into a centrifuge.
This was intended to avoid the RhBi2 formation temperature of 780
?C, however
growths with this profile still had both the 1:1 and 1:2 phases. However, given the
different morphology resulting from different crystal structures (RhBi2 is triclinic)
it is easy to separate crystals by picking through the growth. However, they were
two small for any measurements beyond XRD and EDS. It is possible that a higher
spin temperature or greater proportion of bismuth flux would favor the formation
of exclusively RhBi in the form of large crystals.
A.1.2 Physical Properties
Resistivity as a function of temperature and field was measured for RhAs and
RhSb single crystals [Fig. A.1(a)]. While ?(300 K) differs between the two, the
overall temperature dependence is similar, being primarily linear until low temper-
169
(a)
80
60
RhAs
RhSb
40
20
0
0 100 200 300
T (K)
6
(b) ? = 3.0 mJ/K2 mol 
?D = 297 K5
4
3
2
? = 1.7 mJ/K2 mol 
?
1 D
 = 306 K
0
0 5 10 15 20
T2 (K2)
Figure A.1: (a) Resistivity in zero field as a function of temperature for RhAs and
RhSb single crystals. (b) Specific heat of two single crystal samples of RhAs and
RhSb. Lines are fits to the Debye low temperature model with obtained parameters
from those fits next to the data.
170
C/T (mJ/K2 mol) ? (m? cm)
ature saturation, much like CoAs and CoP. RRR values were low, below five for all
measured samples and lower in RhSb. This may be related to overall molar mass,
as it was seen that phosphide samples had consistently higher RRRs among the 3d
pnictides. It could also be due to the fact that not as much time was invested in
perfecting growth conditions. In line with the low RRR, MR was very small in com-
parison to other Pnma binaries, reaching values of about 3% in both compounds at
9 T.
No superconducting transition was found to temperatures below 200 mK using
an ADR for RhAs, in contrast to the report of Ref. [145]. The explanation for
this could be that other compounds mentioned in that work have similar elemental
ratios and Tc values to the MnP-type binaries; if samples of those materials were
slightly off stoichiometry, they could have been confused for the 1:1 materials and
also have had a slight change in transition temperature. RhBi was also reported
as a superconductor in that paper. Whether it really is, or suffers from the same
misidentification problem as RhAs, ought to be pursued.
Heat capacity data fit well below about 20 K2 (4.5 K) to the Debye low tem-
perature specific model (3.1) [Fig. A.1]. Values of the Sommerfeld coefficient are 3.0
and 1.7 mJ/K2 mol for RhAs and RhSb, respectively. These are both a bit smaller
than in the 3d -based materials, which is perhaps related to the higher residual re-
sistivity and indicative of smaller Fermi surface pockets and poorer conduction. It
is also a sign that increasing the mass of the transition metal atom does not signifi-
cantly affect electron correlations, which would increase ?. The Debye temperatures
are 297 K and 306 K for the As and Sb compound are also slightly lower than has
171
been seen so far.
A.1.3 Conclusions
While preliminary, there is no indication of ordered magnetism or supercon-
ductivity in the Rh-based pnictides. Of course, more effort to synthesize RhBi would
determine whether that material is a superconductor as reported, or was misiden-
tified as seems to be the case for RhAs. The movement to the 4d transition metal
block should increase the spin-orbit coupling strength, thought to be influential in
the band structure of some MnP-type materials [32, 98]. The decrease in the Som-
merfeld coefficient in RhAs and RhSb is a sign that there is no great enhancement
in correlated electron behavior.
A.2 WP
Having established that RhAs is not a superconductor down to 0.1 K, it seems
that WP the only MnP-type parent compound to superconduct at ambient pressure.
It was in fact only recently discovered, with Tc reported to be 0.8 K [33]. As
mentioned in the Introduction, superconductivity in a nonsymmorphic space group
is interesting on its own, due to potential associations with topological physics.
Previous study on WP is scant, but the lack of any transition in resistivity, and
the elemental composition, indicate that ordered magnetism is unlikely, as opposed
to the results for superconducting CrAs and MnP. Attempts were made to look at
more of the basic properties of WP.
172
A.2.1 Crystal Growth and Characterization
In the only previous extensive work on single crystal WP, samples were grown
by I2 chemical vapor transport using prereacted WP powder [33]. However, the
suspicion was that this was not optimal for two reasons. First, in FeP samples had
a much lower residual resistivity when grown directly from the elements, as opposed
to powder. Second, the vastly different melting temperatures of W (3422 ?C) and P
(590 ?C in the less volatile red form) foretell that it will be very difficult to get them
to react well. Thus inspiration for CVT growth was taken from an earlier work that
used the oxides of heavy transition metals (Hf, Ta, W, and others) to produce single
crystal transition metal phosphides with CVT [88]. WO3 has a lower melting point
(1473 ?C), and it is also thought that the release of oxygen contributes to gas phase
transport, along with I2.
WO3 and red P powders were ground together and placed into a quartz tube
in a 1:1 ratio of W:P, with additional I2. The reactants were placed in the middle
of a furnace at 900-1000 ?C with the cold end at the edge of the furnace and left
for 10-14 days, similar to other CVT growths already covered. After that time, it
was found that powder remained at the hot end, while a mixture of material was
at the cold end [Fig. A.2]. X-ray diffraction of the powder showed that it was WP,
while the cold end was a mixture of WP single crystals, which were black and shiny,
and larger, red-tinged chunks of what was identified by XRD as W8P4O32 (which
can also be written as W2O4[PO4]). Often the WP crystals were found fused to the
W8P4O32, but could easily be polished to leave just the desired material. Lattice
173
parameters were a = 5.73 A?, a = 3.25 A?, and c = 6.22 A?, very close to previous
reports [33, 146].
As is the case with most of the other materials covered so far, CVT-grown crys-
tals generally grew along the b-axis, the shortest crystallographic direction. They
were, however, more platelike than the needlelike crystals reported by Liu et al. [33]
This enabled proper alignment of the samples for investigation of anisotropy in
the orthorhombic system, and facilitated Hall measurements (which require a wide
enough face to place two opposing voltage leads). The naturally flat faces perpen-
dicular to the long growth axis were typically either the a-axis or (101).
EDS on WP showed a similar pattern to FeP grown directly from the elements.
Again, there was an excess transition metal content, with consistently about 53-55%
W (and thus 47-45% P). The prior study reported an EDS-derived composition of
exactly 1:1 [33], which may account for some of the differences in behavior that will
be shown in the next section. As with FeP, a slight change in stoichiometry may
change the density of scattering sites or the Fermi level location, both of which could
subsequently affect other properties.
A.2.2 Normal State Properties
The increased crystal size enabled several kinds of measurements on WP in
the normal state. The resistivity has an almost perfectly linear dependence at high
temperature, before plateauing down to 1.8 K [Fig. A.3(a)]. This is similar to
what was previously seen, however crystals grown by the new WO3 method have
174
C
B
A
Figure A.2: The different materials that form at the cold end of a sealed quart
ampule when starting with WO3, P, and I2 at the hot end. Pieces marked A are
chunks of W8P4O32, those marked B are WP (either single crystals, or multiple
crystals intergrown), and C is a piece of WP fused to a piece of W8P4O32. All three
were commonly found in these growths.
175
substantially lower residual resistivity than those grown from WP polycrystals. This,
as noted, parallels what was seen in FeP, which also had nonstoichiometric EDS
results. The RRR could be several hundred, with ?0 as low as 0.25 ?? cm. The
suspicion, as mentioned, is that WP are not able to form a high quality powder to
their vastly different states at temperatures of roughly 800 ?C under which powders
are typically prepared. Magnetoresistance becomes significant around 50 K, about
the same temperature as for FeP and other ?turn on? materials [58,112]. A field of
9 T is enough to cause an upturn in the resistivity with further cooling, with the
minimum coming at 35 K.
The large magnetoresistance made it difficult to get a clear Hall signal due to
corruption of the symmetric component. Thus, it cannot yet be determined whether
the Hall resistivity is linear up to 14 T. However, temperature sweeps were made at
?14 T with B ? [101], with the difference at the two field extremes used to calculate
a slope, which implicitly assumes a linear Hall resistivity [Fig. A.3(b)]. Conduction
is electron-dominated at all temperatures, but there is a clear temperature depen-
dence to RH , including a peak at 130 K. Given the presence of the MR ?turn on?
temperature, it is likely that multiple bands of both carrier types, with differing
mobilities, are present in WP. However, there is no clear, interesting behavior below
50 K, in the vicinity of the increase in MR. If taken as a carrier density, RH corre-
sponds to n ? 1022 cm?3, indicating a large number of carriers, as there should be
in such a good metal.
The low temperature specific heat also differs from the previous report
[Fig. A.3(c)]. The data, while they can be fit decently by the Debye low tem-
176
(a) (b)
80 -2 B || [101]
60
-4
40
20 0 T        WP -6
9 T RRR = 346
00 100 T (K) 200 300 0 100 T (K) 200 300
4.2 (c) 8 (d) B = 0.5 T
4.0
3.8 6
3.6
3.4
4
3.2
? = 3.0 mJ/K2 mol 
3.0 ?D =  472 K 2
2.8
2.60 5 T2 (K2) 10 0 100 T (K) 200 300
Figure A.3: (a) Resistivity as a function of temperature for a WP single crystal
with current along the b-axis, in zero field and with field along the [101] direction.
(b) Hall coefficient (based on temperature sweeps in ?14 T) for a WP single crystal
with field along [101]. (c) Low temperature specific heat of a different WP single
crystal. The green line is a fit to the Debye model, with extracted parameters noted.
(d) Magnetization of WP powder as a function of temperature. Data shown is field
cooled, but there was no difference with zero field cooling.
177
C/T (mJ/K2 mol) ? (?? cm)
? (10-5 emu/mol Oe) RH (10-4 cm3/C)
perature model, are not smooth down to 1.8 K. This may be experimental error,
or could be from some other contribution to the specific heat, such as the magnetic
fluctuations seen in CoAs-not enough measurements were done to be certain. Re-
gardless, the extracted ? value of 3.0 mJ 2 is lower than most of the other binarymol K
pnictides presented. However, the value is still twice that reported for the needlelike
samples [33]. As speculated, a change in W:P ratio may also change the Fermi level,
and therefore the density of states at EF . If that is the source of the increase in
?, it would also account for superior low temperature conductivity in oxide-grown
samples.
The magnetic susceptibility of a WP single crystal was also measured
[Fig. A.3(d)]. Data shown are only for zero field cooling in low field, but no differ-
ence was seen with field cooling or higher fields. There is some noise and jumpiness
in the data, due to the small moment against the large background of the sample
holder. WP has generally paramagnetic behavior, with increasing ? with cooling.
However, the change is very small across 300 K, only about 6% of the 300 K value.
As suspected, there are no signs of strong magnetic fluctuations in this material.
Two WP samples were also brought to the 41.5 T resistive magnet at the
NHMFL for temperature dependent measurements. One was only measured at
low temperatures, 0.4-3 K, with field along the a-axis [Fig. A.4(a)]. It showed a
quasilinear dependence on field, but with two distinct regions of slope that crossed
over at about 20 T (green and red lines). Data were nearly identical across all
measured temperatures. The second samples had its [011] axis aligned with the
field, and data were taken over a wider temperature range, allowing the drop in MR
178
(a)
8
6
4
T = 0.40 K
2
B || a
0 (b)
40 30 B = 34.5 T
20
30 10
00 20 40
T (K)
20
   T =
0.40 K
10    15 K   26 K
   50 K B || [011]
00 10 20 30 40
B (T)
Figure A.4: (a) Magnetoresistance of a WP sample with B ? a-axis at 0.39 K up to
41.5 T. Data up to 3 K looked identical. Low field blips come from errors associated
with the superconducting transition and low resistance electronics. The green and
red lines are linear fits to low and high field data, respectively. (b) Magnetoresistance
of a different WP crystal with B ? [011] at multiple temperatures. Data did not
vary from base temperature to 8 K, and the next lowest temperature was 15.4 K.
The light blue line is a power law fit of the base temperature data, yielding n = 2.27.
The inset shows MR at 34.5 T (the highest field obtained at higher temperature)
as a function of temperature, including a 7.49 K data point not shown in the main
graphic.
179
MR
with increasing temperature to be seen [Fig. A.4(b)]. Due to issues with magneti
power supplies, the field range was limited to 34.5 T for higher temperatures. The
field dependence is much better fit by a higher order power law in this case, yielding
n = 2.27 for the lowest temperature data. The MR is also much larger in this
orientation, reaching a value of about 45 for 0.39 K, compared to only 9 for the
other sample at the same temperature. This also seems similar to FeP, where the
direction of smaller MR also shows more linear behavior. However, in this case that
is the a-axis. Of course, more extensive angular dependence was not performed to
see how the value with field at other orientations compares, and it may be the case
that MR is even smaller for another direction. The inherent anisotropy, however,
is evident from these data, which make an excellent case for further study with
a rotating probe. No quantum oscillations were observed in either sample, but
this may be due to the overall noisier appearance of the data, and they may be
more obvious in magnetic torque measurements, as has been the case for other B31
materials.
A.2.3 Superconducting Properties
Measurements below 1.8 K were made with a He3 insert to the PPMS, except
in the case of one sample where zero field temperature sweeps were also acquired with
an ADR. WP samples grown from WO3 and P do indeed superconduct at low tem-
perature, but with some differences from the prior report. A sample with a higher
residual resistivity (1.5 ?? cm) had a very sharp zero field transition at 0.85 K,
180
similar to what has been reported [33]. However, three samples with lower residual
resistivities showed broader transitions, but at higher temperatures [Fig. A.5(a);
note that the lower Tc sample has had ? reduced by a factor of 10]. The data for
these samples are much noisier, for two reasons. First of all, the residual resistivities
are an order of magnitude lower. Secondly, the three higher RRR, higher Tc samples
were platelike, whereas the single low Tc samples was needlelike, like those of Liu
et al [Fig. A.5(b)]. While the platelike samples could be polished to try to maxi-
mize resistance, this could not be done to the extreme degrees of needlelike samples
(an as-grown length over 1 mm, and a thickness of less than 50 ?m). As a result
they hit up against the PPMS noise floor. Nevertheless, with enough averaging of
the data there is a clear transition from zero to nonzero resistance, with the zero
coming close to 1.0 K for all three samples. A concern given the low upper critical
field of WP is residual flux in the magnet, which can change Tc and also broaden
superconducting transitions. However, magnetic field was oscillated to zero above
Tc for all samples, and the low Tc sample was measured simultaneously with a high
Tc one with a similar current, indicating that the broadness is an inherent feature
of the sample.
The upper critical field is, like Tc, increased in the platelike samples. In the
case of samples A and B, measured together, 200 mT is still enough to suppress
the transition below 0.5 K, though there may still be a slight dip in the data for B
[Fig. A.5(c)]. A plot of Bc2 vs. Tc is complicated by the broadness of the transitions,
so the initial dip of the resistance from its plateau value is used [Fig. A.5(d)]. For
two samples, A and C, measured in the same orientation, it is clear that the higher
181
0.3 WP, B = 0 T (a)
     A ?  10  
0.2         B
        C
0.1         D
0.0
0.0 0.5 T (K) 1.0 1.5
1.5 175
1.0   B (mBT ()m: (b) 150 (c) 0    50
0.5 10    60 12515   200 A
0.0 20 A 100 B
0.15 C75
0.10 50
0.05
25
0.00 B
0.6 0.8 1.0 1.2 1.4 0
T (K) 0.0 0.5 Tc (K) 1.0
Figure A.5: (a) Zero field temperature sweeps, showing superconducting transitions,
of four different WP single crystals. Sample A has had its resistivity reduced 10
times to fit on the scale. (b) Temperature sweeps in field for WP samples (top) A
and (bottom) B, measured simultaneously. (c) Upper critical field as a function of
temperature for WP crystals A, B, and C, based on the onset of the superconducting
transition, judged by the initial dip in resistivity. A and C have B ? c-axis, for B
the orientation is unknown.
182
? (?? cm) ? (?? cm)
Bc2, onset (mT)
Tc results in a much higher critical field. Furthermore, in the case of sample A, Bc2
is roughly three times larger than for samples with a similar Tc. This may also be
attributable to the improved RRR (while lower than other samples, it is still higher
than those of Liu et al.). It may also be due to a different orientation, as in the
previous study the axis parallel to field was not know.
A.2.4 Conclusions
It is obvious that growth method plays a significant role in determining the
behavior of WP. Vapor transport growth using WO3 and elemental P produces much
better quality samples that WP polycrystalline starting material, based on residual
resistivity. These samples also seem to have higher superconducting transition tem-
peratures, though that seems to be also linked to sample morphology, with a lower
Tc seen in a platelike sample reminiscent of that grow by the alternate technique.
The dependence of Tc on sample purity is an indicator of unconventional supercon-
ductivity, and an obviuos motivator for further study of the only ambient pressure
superconducting parent compound in the MnP series.
183
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