MAGNETORESISTIVITY and QUANTUM CRITICALITY in HEAVY FERMION SUPERCONDUCTOR Ce1−Xybxcoin5

MAGNETORESISTIVITY AND QUANTUM CRITICALITY IN HEAVY FERMION SUPERCONDUCTOR Ce1−xYbxCoIn5

A dissertation submitted to Kent State University in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy

Derek J. Haney

August, 2016 Dissertation written by

Derek J. Haney

B.S., University of Akron, Akron, OH, 2010

M.A., Kent State University, Kent, OH, 2013

Approved by

Carmen Almasan, Ph.D., Chair, Doctoral Dissertation Committee

James Gleeson, Ph.D., Members, Doctoral Dissertation Committee

Almut Schroeder, Ph.D.,

Robin Selinger, Ph.D.

Accepted by

James Gleeson, Ph.D., Chair, Department of Physics

James L. Blank, Ph.D., Dean, College of Arts and Sciences

ii TABLE OF CONTENTS

LIST OF FIGURES ...... v

LIST OF TABLES ...... xiv

Acknowledgments ...... xv

1 Introduction ...... 1

1.1 Heavy Fermions ...... 1

1.1.1 Kondo Eﬀect ...... 3

1.1.2 Ruderman-Kittel-Kasuya-Yosida Interaction ...... 7

1.2 Quantum Criticality ...... 9

1.2.1 Fermi Liquid Theory ...... 11

1.2.2 Non-Fermi Liquid Behavior Due to Quantum Criticality . . . 13

1.3 Magnetoresistivity in Heavy Fermions ...... 15

1.4 CeCoIn5 ...... 16

1.4.1 CeIn3 and CeM In5 ...... 16

1.4.2 Ce1−xYbxCoIn5 ...... 26

2 Experimental Details ...... 32

2.1 Sample Growth and Preparation ...... 32

2.2 Physical Property Measurement System ...... 33

2.3 Helium-3 Option ...... 36

2.4 Electrical Transport Measurement ...... 38

iii 2.5 Pressure ...... 40

2.6 Magnetoresistivity ...... 45

2.7 Heat Capacity Measurement ...... 46

3 Pressure Studies of the Quantum Critical Alloy Ce0.93Yb0.07CoIn5 . . . . . 49

3.1 Introduction ...... 49

3.2 Experimental results and discussion ...... 51

3.3 Conclusions ...... 64

4 Quantum Criticality and Gap Structure in Ce1−xYbxCoIn5 ...... 67

4.1 Introduction ...... 67

4.2 Experimental results and discussion ...... 69

4.3 Conclusions ...... 79

5 Magnetoresistivity Study of the Kondo Impurity to Kondo Lattice

Crossover in Ce1−xYbxCoIn5 ...... 80

5.1 Introduction ...... 80

5.2 Experimental results and discussion ...... 81

5.3 Conclusions ...... 93

6 Summary and Outlook ...... 94

BIBLIOGRAPHY ...... 99

iv LIST OF FIGURES

1.1 Magnetic resistivity per mole cerium of CexLa1−xCu6 for diﬀerent dop-

ings. Data per Onuki and Komatsubara (1987) [5]...... 5

1.2 Indirect exchange interaction J at interatomic spacing a between local

moments due to Friedel oscillations of the conduction electron spin

density...... 8

1.3 Doniach phase diagram, showing TK and TRKKY with changing ex-

change interaction J, and illustrating the resulting phases of magnetic

order, Fermi liquid, and (possible) superconductivity around a critical

Jc...... 10

1.4 (a) Low temperature T − g phase diagram of typical quantum crit-

ical system where g is a tuning parameter, such as ﬁeld H, pres-

sure P , or doping x. (b) Exponent value ε of temperature depen-

ε dence of ρ(T ) ∝ T for YbRh2Si2 for diﬀerent temperatures and ﬁelds.

Applied ﬁeld B is parallel to the c-axis of the crystal. Graph from

Custers, et al. 2003) [33]...... 14

1.5 Unit cell crystal structure of CeM In5 (M = Co, Rh, or Ir). Figure

from Ref. [43]...... 18

v 1.6 Figure compiled from Ref. [14]. (a) Temperature dependent resistiv-

ity ρ and magnetic susceptibility χ = M/H of CeCoIn5. Open circles

represent χ with H k c and open squares represent χ with H ⊥ c. The

inset to (a) shows zero-ﬁeld-cooled χ (circles, left axis) measured in

10 Oe and resistivity (triangles, right axis) in the vicinity of the super-

conducting transition. (b) Speciﬁc heat C/T for CeCoIn5 at applied

ﬁelds of H = 0 and H = 50 kOe. The nuclear Schottky contribution

(due to a large nuclear quadrupole of In) has been subtracted. The

inset to (b) shows the resulting entropy as a function of temperature. 20

1.7 An H − T phase diagram of CeCoIn5 from Ref. [49] showing the ex-

istence of a QCP at approximately HQCP = 5.1 T. The solid squares

show the temperature/ﬁeld at which ρ crossed over from NFL ρ ∝ T

to FL ρ ∝ T 2. The ﬁlled circles show the location of a peak in the ﬁeld

dependence of the magnetoresistance of CeCoIn5. The inset shows the

divergence of A, which is from the low-temperature, high-ﬁeld Fermi

2 liquid resistivity ρ(T ) = ρ0 + AT ...... 23

vi 1.8 From Ref. [53]. Phase diagram for CeCoIn5 at low temperatures, dis-

playing the superconducting phase and the antiferromagnetic phase

within the vortex cores. (a) T − P phase diagram at zero ﬁeld. Nega-

tive pressure measurements from Ref. [54], where a negative chemical

pressure was obtained by doping CeCoIn5 with cadmium on the in-

dium sites. (b) H − P phase diagram at zero temperature, illustrating

the quantum critical line. The red line is from Hu and collaborator’s

calculations [53], and the dotted yellow line is the result of applying

theory from Ref. [55] and using parameters derived by Hu et al. (c)

H − T phase diagram at zero pressure showing the superconducting

dome with the AFM region within it. The inset shows the same at

chemically negative pressure (due to cadmium doping), showing that

at negative pressures, the superconducting dome is contained within

the AFM region rather than the other way around...... 25

1.9 From Ref. [59]. Superconducting transition temperature Tc and coher-

ence temperature Tcoh plotted as a function of residual resistivity ρ0

for diﬀerent doping levels of Ce1−xRxCoIn5. Closed symbols represent

Tc and open symbols represent Tcoh. The grey upward-pointing tri-

angles on the left represent x = 0, i.e., pure CeCoIn5. Regardless of

the nature of the dopant, Tc and Tcoh are each suppressed to zero with

increasing x (with x ∝ ρ0)...... 27

vii 1.10 From Ref. [62]. (a) Field dependence of transverse magnetoresistance

of CeCoIn5 for diﬀerent temperatures. Inset shows that of (nominal)

Ce0.6Yb0.4CoIn5, which lacks the strong coherent behavior at low ﬁeld

that CeCoIn5 displays. (b) Temperature dependence of Hmax, which

is the ﬁeld value of the peaks in (a), which is where coherent behavior

gives way to single-ion behavior. (c) H − x phase diagram illustrating

the behavior of HQCP vs. doping. Inset shows Tc and Tcoh at diﬀerent

doping levels of Ce1−xYbxCoIn5 and Ce1−xLaxCoIn5...... 28

2.1 (a) A cross-sectional view of the PPMS sample chamber showing its

components. (b) A cutaway view of the sample chamber showing the

puck with sample at the bottom of the sample chamber...... 35

2.2 A schematic showing the use of the T-shaped platform in order to align

the c axis of the sample parallel to the applied magnetic ﬁeld. . . . . 37

2.3 (a) A schematic of the four-terminal sensing method. (b) A photograph

of a sample in our lab in which we are applying the four-terminal

sensing method...... 39

2.4 A photograph of a half-assembled pressure cell...... 41

2.5 Photographs of a wired feedthrough. (a) Image of the feedthrough,

showing the wires coming out of the back. (b) Close-up of the sample

on top of the platform. (c) Side view of the platform with the tin

manometer on the left and the sample on the right...... 44

2.6 A side view of the heat capacity setup...... 47

viii 3.1 (a) Resistivity ρa of Ce0.93Yb0.07CoIn5 as a function of temperature T

for diﬀerent pressures P (0, 2.7, 5.1, 7.4, and 8.7 kbar). The arrow at

the maximum of the resistivity data marks the coherence temperature

Tcoh. (b) Evolution of Tcoh as a function of pressure P . Inset: Super-

conducting critical temperature Tc as a function of pressure P . The

solid lines are guides to the eye...... 53 √ 3.2 (a) Fits of the resistivity ρa(P,T ) = ρa0(P ) + A(P )T + B(P ) T for

diﬀerent pressures on Ce0.93Yb0.07CoIn5 over the temperature range

3 K ≤ T ≤ 15 K. (b) Pressure P dependence of the linear T contribu- √ tion A and T contribution B, obtained from ﬁts of the resistivity

data shown in panel (a). (c) Pressure P dependence of the residual

resistivity ρa0, obtained from the ﬁts...... 56 √ 3.3 (a) Resistivity ρa of Ce0.93Yb0.07CoIn5 as a function of T , in the

temperature range 1.8 K ≤ T ≤ 5 K. The solid lines are linear ﬁts of √ ∗ the data with ρa(P,T ) = ρa0(P ) + B (P ) T for 1.8 K ≤ T ≤ 5 K. In- √ ∗ set: Pressure P dependence of the coeﬃcient B . (b) ρa vs T of

Ce0.92Yb0.08CoIn5 measured in zero magnetic ﬁeld and at 4 T. The 4 T

data has been oﬀset upwards by 5 µΩ cm for visual clarity...... 58

ix 3.4 Magnetic ﬁeld H dependence (plotted as function of H2) of magne-

toresistivity (MR) ∆ρa/ρa(H = 0) ≡ [ρa(H) − ρa(H = 0)]/ρa(H = 0)]

of Ce0.93Yb0.07CoIn5 measured at two diﬀerent temperatures and am-

bient pressure. The dashed line in the main ﬁgures marks Hmax, cor-

responding to the coherence giving way to single-ion Kondo behavior.

Inset: MR data vs H2 measured under 5.1 kbar. The red line shows

the quadratic regime of MR...... 61

3.5 Temperature T dependence of the maximum in magnetoresistivity Hmax

for diﬀerent pressures P . The solid lines below 10 K are linear ﬁts to

the data...... 63

3.6 Pressure P dependence of residual resistivity ρa0 (obtained through

the ﬁtting of the resistivity data as discussed in the text), normalized

to its value at zero pressure (right vertical axis) and P dependence of

inverse slope of Hmax(T ) normalized to its value at zero pressure (left

vertical axis)...... 65

4.1 (a) Normalized resistivity ρ(T )/ρ(300 K) of Ce1−xYbxCoIn5 (x = 0.09

and 0.16) as a function of temperature T . The arrow at the max-

imum of the resistivity data marks the coherence temperature Tcoh.

Inset: magnetic ﬁeld suppression of superconducting transition in the

x = 0.09 crystal, shown in the resistance...... 70

x 4.2 (a) Temperature T dependence of the resistance of x = 0.09 single crys-

tal measured in the temperature range 0.50 K ≤ T ≤ 30 K, and under

the application of magnetic ﬁelds 10 T ≤ H ≤ 14 T. (b) The low

temperature data of (a) plotted as a function of T 2. The solid lines

are linear ﬁts to the data at the lowest temperatures, and the arrows

indicate the point where the data deviate from the straight line ﬁts. . 71

4.3 Magnetoresistivity ∆ρ/ρ0 ≡ [ρ(H) − ρ(H = 0)]/ρ(H = 0)] as a func-

tion of transverse applied ﬁeld H for (a) Ce0.91Yb0.09CoIn5 and (b)

Ce0.84Yb0.16CoIn5, measured at diﬀerent temperatures T ...... 73

4.4 Heat capacity C/T vs. temperature T of Ce0.91Yb0.09CoIn5 measured

with magnetic ﬁeld H k c over the temperature range 0.50 K ≤ 10 K

and applied magnetic ﬁelds 0 ≤ H ≤ 4.5 T...... 75

4.5 Electronic contribution to heat capacity Ce/T vs. temperature T of

Ce0.91Yb0.09CoIn5 measured with applied magnetic ﬁeld H k c over the

temperature range 0.50 K ≤ 10 K and 10 T ≤ H ≤ 14 T. A 1/T 3 nu-

clear Schottky contribution has been subtracted from the data. The

Fermi liquid temperature TFL is extracted as the temperature at which

the data cross from logarithmic temperature dependence to no temper-

ature dependence with decreasing temperature...... 77

xi 4.6 H − T phase diagram of Ce0.91Yb0.09CoIn5 with magnetic ﬁeld H k c-

axis. The area under dotted line represents the superconducting re-

gion. The straight lines are the linear ﬁt of the data extrapolated to

zero temperature. Hmax is the peak in MR. TF L,ρ and TF L,HC are the

temperature at which the data cross from a Fermi liquid to a non-Fermi

liquid, measured by resistivity and heat capacity, respectively. Like-

wise, Tc,ρ and Tc,HC are the superconducting transition temperature

measured by resistivity and heat capacity, respectively...... 78

2 5.1 Magnetoresistivity of CeCoIn5 at T = 10 K plotted against H . The

black line is the measured magnetoresistivity. The blue line is the nega-

tive component, calculated as the slope of the high-ﬁeld linear portion.

The red line is the positive component, calculated by subtracting the

negative curve from the physically measured data. The dashed line

shows the value of (∆ρ/ρ0)sat...... 83

5.2 (a) The negative quadratic component of the magnetoresistivity ∆ρ/ρ0

∗ ∗ ((∆ρ/ρ0)neg) of CeCoIn5, scaled as normalized ﬁeld H/H , where H =

kB ∗ (T + TK ) and TK = 3.45 K. (b) A plot of H (with coeﬃcients) gµB versus temperature T . The lines are all linear ﬁts, with a unit slope

and an intercept at Kondo temperature TK . (c) TK as a function of x. 86

5.3 (∆ρ/ρ0)sat for diﬀerent dopings of Ce1−xYbxCoIn5 plotted versus tem-

perature T . (∆ρ/ρ0)sat is the value at which the magnetoresistivity

(measured as percentage change in resistivity with ﬁeld) saturates. . . 88

xii ∗ ∗ 5.4 (a) A plot of H (with coeﬃcients) versus T , where H scales (∆ρ/ρ0)neg

∗ ∗ kB according to H/H and H = (T + TK ) . The lines are all linear gµB

ﬁts, with a unit slope and an intercept at Kondo temperature TK . Per-

formed over a pressure range of 0 kbar ≤ P ≤ 7.4 kbar. (b) TK , plotted

as a function of P . The line is a guide to the eye...... 91

5.5 (∆ρ/ρ0)sat for Ce0.93Yb0.07CoIn5 for pressures 0 kbar ≤ P < 7.4 kbar

plotted versus temperature T . (∆ρ/ρ0)sat is the value at which the

positive component of the magnetoresistivity (∆ρ/ρ0)pos saturates. . . 92

xiii LIST OF TABLES

2.1 Conversion between the nominal (xnom) and the actual (x) Yb doping

levels in Ce1−xYbxCoIn5 ...... 33

xiv Acknowledgments

The writing of this dissertation involved the concerted eﬀort of far more people than I alone. They have stood by me and supported me for the entire time that I have known them, and without any of them, this work would not be possible.

I would ﬁrst like to thank my wife, Kathy. She has been such a blessing in my life, supporting me and giving me strength through this most stressful period. She has always brought joy into my life, even when I had reason to be anything but joyful. She has been patient with the time it has taken me, watching the children while I spent hours upon hours of our potential family time in front a computer screen typing away.

She has dedicated herself to our family, and without her, I could have no success. I know that with her beside me, we can push through any trial, and this is what I look forward to doing more than anything else.

I would like to thank my advisor, Dr. Carmen Almasan. She has been a wise mentor, teaching me most everything I know about physics research and presenting said research to the community. She has vastly improved my abilities in writing and in speaking. Her experience has given her tremendous insight, and with this she has guided me through my research and given me direction. Most of all, she has had immense patience with my shortcomings, for which I am most grateful.

I would like to acknowledge my labmates during my research, Xinyi Huang and

Yogesh Singh. We have been a team from the start, working together in the lab and on our papers. We have had countless illuminating discussions, opening up doors that were not previously considered. Special thanks go to Yogesh who is a far stronger

xv physicist than I am, and without him, I would have foundered long ago and surely fallen away.

I would like to thank our colleagues at Kent State who supported this research.

Most signiﬁcantly is Dr. Maxim Dzero, who has lent his experience and knowledge to this body of research, patiently correcting my lack of understanding on too many things. Thanks also go to Dr. Alan Baldwin, who has designed and built much of the equipment and software in our lab, and to Wade Aldhizer who has machined several components that were necessary for our research.

I would also like to thank my mother, Kelly Haney. When I was a child, she was the one to sit with me during my homework, supporting me and working with me.

Most importantly, she pushed me, refusing to allow me to walk away from something just because it was hard. If it were not for her being more stubborn than I am, I would never have ﬁnished those frustrating open-ended math problems, which are now enjoyable puzzles.

My most extreme gratitude goes to God. It is of course His creation and His design that I am writing about in this dissertation. He is the one who designed me and created me to be curious about what He has made, and He has brought me to this point in my life. He has provided me such opportunity and so many assets, none of which I deserved, for which I am forever grateful and can never fully repay. I can only hope to use the blessings He has given me to give whatever I can back to Him and to His people.

xvi TO KATHY

xvii CHAPTER 1

Introduction

1.1 Heavy Fermions

“Heavy fermion” is the name given to a class of inter-metallic materials in which, at low temperatures, the conducting quasiparticles exhibit an eﬀective mass signiﬁ- cantly larger than the mass of a free electron, up to 3 orders of magnitude larger [1].

These materials tend to be comprised of a lattice of 4f or 5f ions, such as Ce or

U. The increase in effective mass is made evident by the greatly increased linear coefficient of the electronic specific heat γ, because γ is directly proportional to the effective mass m∗ [1]. The linear-in-temperature specific heat coefficient of a heavy fermion materials can reach values on the order of 1 J/mol K2, whereas in a conventional metal, like Al, γ is about 1 mJ/mol K2 [2]. Likewise, the magnetic and charge susceptibility of a heavy fermion material is directly proportional to the effective mass of the quasiparticles [1].

Heavy fermionic behavior was originally discovered in 1975 by Andres, Graebner, and Ott [3] in CeAl3, in which they observed exceptionally large Pauli susceptibility and linear-in-temperature speciﬁc heat coeﬃcient, which they assumed were due to a coherent 4f bound state. To the surprise and disbelief of many, a few years later, Steglich et al. discovered superconductivity in another heavy fermion material, CeCu2Si2 [4]. This was astonishing because it had previously been believed that magnetic ions were incompatible with superconductivity, yet all measurements

1 3+ of CeCu2Si2 showed the Ce ions to be in a magnetic 4f conﬁguration (Ce ) [1].

1 Since then, a great variety of heavy fermion materials have been discovered, such

as simple metal CeCu6 [5, 6], insulators Ce3Pt4Bi3 [7, 8] and CeNiSn [9, 10, 11],

antiferromagnets CeIn3 [12] and CeRhIn5 [13], and superconductor CeCoIn5 [14, 15].

Below are some shared characteristics of heavy fermion materials and heavy fermion

superconductors:

(1) a large normal state linear-in-temperature speciﬁc heat coeﬃcient γ at low T ,

(2) a transformation from a Curie-Weiss susceptibility at high temperatures (indicative of free local moments) to a large paramagnetic Pauli susceptibility at low temperatures (indicative of no local moments),

(3) a Wilson ratio (ratio between the zero temperature susceptibility and γ) of

W ≈ 1, even though the density of states between materials can vary more than 2 orders of magnitude [16, 17],

(4) a Landau-Fermi liquid at low temperatures with a quadratic temperature

2 dependence (ρ(T ) = ρ0 + AT ), where A is much larger than in conventional metals, and A/γ2 ≈ 1 × 10−5 µm cm[mol K2/mJ] [18],

(5) London penetration depths 20 to 30 times larger than in conventional superconductors [19], and superconducting coherence lengths shorter than in conventional superconductors, making heavy-fermion superconductors extreme type-II superconductors, and

(6) large upper critical ﬁelds (UBe13 has a superconducting transition temperature of 0.9 K and an upper critical ﬁeld of about 11 T, which is about 20 times larger than that of conventional superconductors with the same transition temperature [1]).

2 1.1.1 Kondo Eﬀect

The mechanism that leads to the formation of these heavy quasiparticles has its

foundation in the Kondo eﬀect [20, 21]. This is the process in which a magnetic

ion is screened by the conducting electrons at low temperatures, forming a spinless

scattering center. Using a perturbative calculation with the magnetic impurity as

a local spin coupled to the conduction electron spin density via the spin-exchange

interaction J, Kondo was able to explain why a resistance minimum is observed when a nonmagnetic metal (such as Cu or Au) is doped with a small amount of a magnetic metal, such as Fe or Ce3+. At high temperatures, electron-phonon interactions are the dominant source of electron scattering, and decreasing the temperature suppresses this scattering, hence lowering the resistivity. With further decreasing temperature, the magnetic moments interact more and more rapidly with the conducting electrons, scattering them, increasing resistivity logarithmically. This is the reason for the observed minimum in resistivity. In such a system, the resistivity is modeled by

2 5 2 ρ(T ) = ρ0 + aT − b ln(µ/T ) + cT , where ρ0 is the residual resistivity, aT is the

Fermi liquid contribution, cT 5 is due to electron-phonon scattering at low T , and

−b ln(µ/T ) is the contribution arising from the Kondo eﬀect.

An early problem with Kondo’s theory was that it predicted inﬁnite resistivity at zero temperature, while it was empirically shown that resistivity actually approached a constant value. Kondo’s theory stopped being valid below a temperature TK , known as the Kondo temperature. Theorists struggled with this problem, known as the

Kondo problem, through the 60’s and 70’s until the Anderson impurity model (of which the Kondo eﬀect is a low energy speciﬁc case) was solved exactly using a

3 Bethe-ansatz approach by Wiegmann (as P.B. Vigman) [22] and, independently, An-

drei [23]. They were able to show with exact results that the resistivity due to the

impurity approaches a constant value as T approaches zero. Through the eﬀort of many theorists, it is understood that what physically happens is the following: below the Kondo temperature TK , the magnetic moments become fully screened, forming a spinless, elastic scattering center, known as a Kondo singlet, and the logarithmic increase slows to a constant.

At high temperatures, T TK , the exchange interaction is quite weak, and the local moment is asymptotically free. At these temperatures, the susceptibility obeys a Curie law (χ ∝ 1/T ). As the temperature decreases below TK , and as the electrons screen the local moment to form a Kondo singlet, the susceptibility transitions into a Pauli susceptibility at a constant value of approximately 1/TK . The speciﬁc heat

CV /T increases with decreasing temperature, starting around TK , and saturates at a

R ln 2 value on the order of , where R ln 2 is the full spin entropy. Below TK , the only TK energy scale in the system is TK . For T TK , a narrow peak forms in the density of states at the Fermi level with a width on the order of TK , which is called Kondo or

Abrikosov-Suhl resonance.

An example of the Kondo eﬀect can be seen in the magnetic resistivity data for

CexLa1−xCu6 at low dopings in Fig. 1.1 [5]. The magnetic resistivity is the contribution to the compounds resistivity due only to the magnetic Ce3+ ions introduced. It

is calculated as ρm = (ρCexLa1−xCu6 − ρLaCu6 )/x, subtracting out the electron-phonon

interactions shared by CexLa1−xCu6 and LaCu6. At x = 0.094, the magnetic Ce ions

are dilute and can be considered isolated. As such, Kondo’s theory applies, and we

can see the logarithmically rising resistivity with decreasing temperature.

300

x=0.094 Ce La Cu

x 1-x 6

250

0.29

200

0.5

0.73

150

cm / mol Ce) mol / cm 0.9

100 ( m

0.99

1.0

0.01 0.1 1 10 100

T (K)

Figure 1.1: Magnetic resistivity per mole cerium of CexLa1−xCu6 for diﬀerent dopings. Data per Onuki and Komatsubara (1987) [5].

5 But Kondo’s initial theory only extended to small amounts of isolated magnetic

ions. As more and more magnetic ions are added to the lattice, the behavior changes,

as can be seen in Fig. 1.1, which shows the progression of the magnetic resistivity

per magnetic ion with increasing amounts of the magnetic Ce ions. Lanthanum is

isovalent to cerium, but is nonmagnetic. As the nonmagnetic La3+ ions are removed

and replaced with magnetic Ce3+ ions, the system continuously transforms from a dilute Kondo system to a dense Kondo lattice. At high temperatures, the resistivity at the high doping levels behaves very similarly to that at low doping levels, increasing logarithmically with decreasing temperature. But at about T = 10 K, rather than continuing to increase, a maximum forms in the resistivity, and then the resistivity drops quite sharply, dropping down to near zero with the T 2 dependence typical of a

Fermi liquid. As the magnetic moments become screened at about TK , the screening

clouds of electrons around the impurities are superposed, allowing the sites to scatter

elastically and coherently with each other as opposed to the inelastic, incoherent

scattering above TK [1].

It is this coherence and the interaction of the conduction electrons with the screened magnetic moments that lead to the formation of the heavy fermion quasiparticles. In most heavy fermion materials, such as CeCu2Si2, CeAl3, and CeCoIn5, the coherence leads to the sharp drop in resistivity that we see in Fig. 1.1 in CeCu6.

Interestingly, in some heavy fermion materials, aptly named “Kondo insulators”, the coherence leads to the formation of a ﬁlled band with an insulating gap, dramati- cally increasing the resistivity at low temperatures. Examples of this can be found in Ce3Pt4Bi3 [7, 8] and CeNiSn [9, 10, 11]. We also see the same transformation from a Curie-Weiss susceptibility at high temperature to a constant paramagnetic

6 susceptibility at low temperature that we see in the dilute system.

1.1.2 Ruderman-Kittel-Kasuya-Yosida Interaction

Within a metal, localized magnetic moments interact with each other indirectly

through the conduction electrons. This interaction is known as the the Ruderman-

Kittel-Kasuya-Yosida, or RKKY, exchange interaction. In short, one local moment

interacts with a conduction electron, which then goes on to interact with another local

moment, thus indirectly coupling the two moments. It was initially proposed in 1954

by Ruderman and Kittel [24] and later expanded on by Kasuya [25] and Yosida [26]

to give us the form we know today.

As the conduction electrons pass a local moment, whether it be a nuclear magnetic

moment or a localized inner d- or f- electron spin, the moment induces Friedel oscillations in the electron spin density. A second moment, located near this ﬁrst moment, couples to this wave in the spin density. In this way, long range magnetic order can be established in the system, even if the direct exchange interaction between the local moments is weak. In materials with dilute amounts of magnetic impurities, the spins tend to form in a glassy magnetic state, in a ﬁxed, random orientation. In materials with dense magnetic lattices, the local moments tend to form either ferromagnetic or antiferromagnetic (AFM) order. Whether the coupling is ferro- or antiferromagnetic depends on the spacing between the local moments and the momentum of the conduction electrons, due to the oscillations of the indirect exchange coupling constant over distance, as can be seen in Fig. 1.2. Most heavy fermion materials order anti- ferromagnetically. If the spacing is too far apart, the interaction is negligible due to the attenuation of the oscillations in the conduction electron spin density, and the moments act independently of each other.

7 J

0 a

Figure 1.2: Indirect exchange interaction J at interatomic spacing a between local moments due to Friedel oscillations of the conduction electron spin density.

In the 1970’s, Mott [27] and Doniach [28] proposed separately that in a heavy fermion material, there is a dense lattice of local moments (which Doniach named a

“Kondo lattice”), all experiencing the Kondo effect while also interacting with each other via the RKKY interaction. Due to the Kondo effect’s tendency to quench magnetic moments via conduction electron screening and the RKKY interactions tendency to order them, also via conduction electron interactions, these are competing effects. The outcome of this competition determines the nature of the ground state.

They determined that there are two energy scales within the Kondo lattice: the Kondo

−1/JD(EF ) 2 temperature TK ∝ e and the RKKY scale TRKKY ∝ J D(EF ), where J is the exchange interaction between the local moments and the conduction band, and

D(EF ) is the density of states of the conduction band at the Fermi energy. As shown in Fig. 1.3, for low values of JD(EF ), TRKKY > TK , creating a magnetic order below a non-zero N´eeltransition temperature TN (though with a reduced magnetic moment due to the Kondo eﬀect). For high values of JD(EF ), TK > TRKKY , and the system

8 becomes nonmagnetic, with each local moment resonantly scattering electrons. In

between these two extremes, there is a critical value of JD(EF ) at which TN is

driven to zero, forming a zero-temperature phase transition, i.e., a quantum critical

transition, which will be discussed in Section 1.2. Heavy fermion materials tend to

be near this critical value, making them predisposed towards quantum criticality and

superconductivity.

1.2 Quantum Criticality

A conventional phase transition occurs at a ﬁnite temperature, with thermal ﬂuc-

tuations as the driving force between diﬀerent phases, such as magnetic phases, states

of matter, or crystal structures. By adjusting a tuning parameter such as pressure,

magnetic ﬁeld, or doping, one can drive this phase transition down to a transition at

T = 0 K. At T = 0 K, the ﬂuctuations that drive the transition are no longer thermal in nature but quantum mechanical. As such, a zero-T phase transition is referred to

as a quantum critical transition or quantum phase transition.

Unlike a classical, ﬁnite-temperature transition, which can be passed through by changing temperature, a quantum critical transition is navigated by increasing/decreasing one of the tuning parameters. But, like a classical transition, it can be of either ﬁrst order (occurring abruptly) or second order (occurring continuously).

A second-order quantum critical transition is accompanied by a signiﬁcant amount of quantum ﬂuctuations as one approaches it and transitions through it. A point in a phase diagram where a second-order quantum critical transition occurs is referred to as a quantum critical point (QCP).

QCPs are of particular interest because the quantum critical ﬂuctuations have a peculiar eﬀect on the properties of the system while the system is in the vicinity of

9 Figure 1.3: Doniach phase diagram, showing TK and TRKKY with changing exchange interaction J, and illustrating the resulting phases of magnetic order, Fermi liquid, and (possible) superconductivity around a critical Jc.

10 a QCP. These are not simply theoretical eﬀects at absolute zero, but at physically

measurable, nonzero temperatures, as high as T = 20 K in CeCoIn5. In many heavy-

fermion systems, superconductivity has been noted to form around a QCP where the

antiferromagnetic (AFM) transition has been driven to zero temperature, as shown

in Fig. 1.3. Speciﬁc examples include antiferromagnets CeIn3 and CePd2Si2 which

have pressure-induced superconductivity appearing where the N´eeltemperature is

suppressed to zero [29] and within the ferromagnetic region of UGe2 where the Curie temperature disappears with pressure [30, 31]. This implies that the superconducting pairing could be mediated by excitations associated with the QCP, in this case the quantum critical spin ﬂuctuations due to the AFM transition.

1.2.1 Fermi Liquid Theory

At suﬃciently low temperatures, most metals are well modeled by the Fermi liquid theory, also known as Landau-Fermi liquid theory, named after physicist Lev Landau, who introduced it in 1957 [32]. The Fermi liquid theory describes the properties of a system of interacting fermions (as opposed to the non-interacting fermions in a Fermi gas). Relevant to the topic of this dissertation, the sea of electrons in a metal form such a system. Other notable systems in which the theory enjoys success are the system of He-3 atoms in liquid He-3 and the system of nucleons in an atomic nucleus.

As stated, a Fermi gas is a system of non-interacting fermionic particles. However, in reality, the electrons in a metal do interact with each other and with their environment. Landau considered the situation in which the interactions between the fermions is initially zero, and then the interactions are turned on adiabatically. The ground state of the Fermi gas adiabatically transforms into the ground state of the Fermi liquid. The electrons are replaced by excitations called quasiparticles (so called because

11 they are more of a mathematical construction as opposed to a literal transformation

of the electrons). These quasiparticles have the same spin, charge, and momentum of

the electrons they represent, but their mass and magnetic moment are renormalized

due to the interactions with the other fermions in the system. The quasiparticles can

be thought of as “dressed” versions of the electrons they represent—dressed by the

interactions—causing them to behave as though they were heavier and slower than a

“bare” electron. Importantly, the interactions between particles manifest themselves

as the dressing of the electrons, and we can treat the quasiparticles as non-interacting.

Then, standard Fermi gas theory is applied to these non-interacting quasiparticles.

The Fermi liquid has many characteristic properties, according to theory and ver-

iﬁable by experiment. Speciﬁcally, we are applying these to the electronic properties

of a metal, but they are applicable to other fermion systems as well, such as liquid

He-3.

(1) Density of states: ∗ m pF g(F ) = , (1.1) π2~3

∗ where F is the Fermi energy, m is the eﬀective mass of the electrons (i.e., the mass of the quasiparticles), and pF is the Fermi momentum.

(2) Linear-in-temperature electronic speciﬁc heat:

∗ ∗ m pF 2 m Ce = 3 kBT = γT = γ0T, (1.2) 3~ me

∗ where m is the eﬀective mass of each of the quasiparticles, me is the mass of the bare

electron, and pF is the Fermi momentum. Ce is linear in temperature (i.e., Ce/T is

constant in temperature) for a Fermi liquid. Because there is a phonon contribution

12 3 of Cp ∝ T to the speciﬁc heat, in a low-temperature metal, one would expect to see

2 C(T )/T = Ce/T + Cp/T = γ + βT . (1.3)

(3) Magnetic susceptibility:

∗ 2 g(F ) m χ0 χ = µ0µm a = a , (1.4) 1 + F0 me 1 + F0

a where µm is the magnetic moment of each particle, and F0 is a Landau parameter resulting from the perturbation theory used in deriving the properties of the Fermi liquid.

(4) Quadratic-in-temperature electrical resistivity:

2 ρ(T ) = ρ0 + AT (1.5)

1.2.2 Non-Fermi Liquid Behavior Due to Quantum Criticality

Some of the more interesting cases are not where Fermi liquid successfully de-

scribes the behavior of the material, but rather those where it does not. Quantum

criticality can make itself known in a material by exhibiting what is called non-Fermi

liquid (NFL) behavior, so called because it does not behave according to Fermi liquid

theory discussed in Section 1.2.1. This occurs in a (counter-intuitively) fan-shaped

region above the quantum critical point, as shown in Fig. 1.4. Figure 1.4(a) shows a

generic phase diagram, with g being the tuning parameter associated with the QCP,

such as ﬁeld, pressure, or doping. On one side of the QCP, the zero-temperature

phase transition, there is the ordered phase with broken symmetry. In heavy fermion

systems, this would typically be an antiferromagnetic phase. On the other side is the

unordered phase, such as a paramagnetic state. In between is the NFL region where

quantum criticality dominates the behavior of the system.

13 Figure 1.4: (a) Low temperature T − g phase diagram of typical quantum critical system where g is a tuning parameter, such as field H, pressure P , or doping x. (b) ε Exponent value ε of temperature dependence of ρ(T ) ∝ T for YbRh2Si2 for different temperatures and fields. Applied field B is parallel to the c-axis of the crystal. Graph from Custers, et al. 2003) [33].

Figure 1.4(b) shows an example of this phenomenon in heavy fermion compound

YbRh2Si2, per Custers, et al. (2003) [33]. In this case, the tuning parameter (x-axis)

is the applied magnetic ﬁeld. This is a color-coded diagram showing the temperature

dependence of the resistivity, obtained from d ln ρ/d ln T . That is, with ρ ∝ T ε, the

color gives the value of ε. The two blue regions show that ρ(T ) behaves as T 2, as

predicted by Fermi liquid theory. The orange-wedge in the middle, centered at a QCP

at a ﬁeld HQCP = 0.66 T, represents a region wherein ρ(T ) ∝ T , i.e., NFL behavior

due to the inﬂuence of the quantum critical ﬂuctuations.

Following are some common NFL temperature dependences in quantum critical,

heavy fermion systems [2, 1]:

14 (1) Near-linear temperature dependence of resisitivity:

ρ(T ) ∝ T ε , (1.6)

with ε in the range of 1.0 to 1.2 (as we saw in Fig. 1.4(b)),

(2) Logarithmically diverging speciﬁc heat coeﬃcient γ at the QCP:

C 1 T γ = V ∝ ln 0 , (1.7) T T0 T

(3) Logarithmically diverging susceptibility χ at the QCP:

T χ ∝ ln 0 , (1.8) T

(4) Diverging quasiparticle mass m∗ at the QCP.

1.3 Magnetoresistivity in Heavy Fermions

In diluted Kondo systems, with isolated magnetic impurities in an otherwise nonmagnetic lattice, application of magnetic ﬁeld tends to create a negative, quadratic- in-ﬁeld change in resistivity. Schlottmann used a Bethe-ansatz approach in the early

1980’s [34, 35, 36] to give an exact solution for the magnetoresistivity of the single-

∗ ion Kondo system. He also showed that, if plotted as ∆ρ/ρ0 vs. H/H (where

∗ ∆ρ/ρ0 ≡ [ρ(H) − ρ(H = 0)]/ρ(H = 0)] and H = kB(T + TK )/gµB), the magnetore-

sistivity curves should all scale onto each other, showing that the underlying physics

of the Kondo single-ion regime is dominated by a single energy scale, i.e., TK . Scaling

∗ the magnetoresistivity allows one to calculate H and therefore TK .

In the case of a lattice of magnetic impurities (as opposed to dilute magnetic impurities), the magnetoresistance becomes more complicated, as can be seen in several heavy fermion systems [37, 38, 39, 40]. At high temperatures, one still sees the negative quadratic magnetoresistivity exhibited in the single-ion Kondo regime. This is

15 because the thermal ﬂuctuations dominate over the weakened magnetic interactions

between the magnetic ions, allowing the ions to behave as though they were isolated.

But at low temperatures, as coherence eﬀects begin to take over, a positive magne-

toresistivity begins to manifest itself. As will be discussed in detail in Chapter 5, in

the case of Ce1−xYbxCoIn5, this positive magnetoresistivity can give insight into the coherent and quantum critical characteristics of the material.

1.4 CeCoIn5

The primary material of focus of this dissertation is the heavy fermion compound

CeCoIn5, speciﬁcally when doped with Yb on the Ce sites, i.e., Ce1−xYbxCoIn5. As such, we will spend some time here discussing the history and the characteristics of

CeCoIn5 and closely related materials.

1.4.1 CeIn3 and CeM In5

CeIn3 is a cubic heavy-fermion system that exhibits long-range antiferromagnetic order at low temperatures with a N´eeltemperature TN ≈ 10 K. With pressure, TN is suppressed to zero at about 25 to 30 kbar, forming a quantum critical point. It was found that CeIn3 develops heavy electron superconductivity in a narrow region around this point, reaching a maximum Tc of approximately 200 mK [41].

In 1999, Monthoux and Lonzarich suggested that a quasi-2D, antiferromagnetic heavy fermion material ought to be a superior superconductor to a 3D one [42]. This led to the introduction of layers to the CeIn3 system. Notably, M In2 layers were added (where M is either Co, Rh, or Ir, i.e., transition metals in Group 9) forming

CeM In5, often referred to as Ce-115 materials. CeM In5 forms in a HoCoGa5 crystal structure (tetragonal, space group P4/mmm) with alternating layers of CeIn3 and

16 M In2 as shown in Fig. 1.5 [43].

The ﬁrst report on CeRhIn5 was published in the year 2000 [13]. Like CeIn3, it is a heavy fermion material and is antiferromagnetic and non-superconducting at ambient pressure. Also like CeIn3, application of pressure suppresses TN to zero, with a superconducting dome forming around the QCP. But in this case, the highest reachable critical temperature is Tc = 2.1 K, an entire order of magnitude larger than that of CeIn3, conﬁrming the hypothesis that reducing the dimensionality of the material could support the pair-bonding mechanism.

Shortly after this, ambient-pressure superconductivity was found in CeCoIn5 [14] and CeIrIn5 [44], making them the ﬁrst Ce-based heavy fermion superconductors at atmospheric pressure since the initial discovery of heavy fermion superconductivity in CeCu2Si2 over two decades earlier. In fact, at 2.3 K, the critical temperature of

CeCoIn5 is currently the highest known among Ce-based heavy fermion superconductors at atmospheric pressure.

CeRhIn5 CeRhIn5 is a heavy fermion antiferromagnet with N´eeltemperature TN = 3.8 K

2 and electronic coeﬃcient of speciﬁc heat γ = 420 mJ/molCe K [45, 46, 47]. At ambient pressure, CeRhIn5 is not a superconductor, but at a critical pressure of Pc = 1.63 GPa superconductivity forms. In the pressure range 1.53 GPa < P < 1.9 GPa, antiferromagnetism and superconductivity have been shown to exist together inside the superconducting dome. For P < 1.9 GPa, TN decreases with pressure and Tc increases with pressure until P = 1.9 GPa, where TN = Tc = 2.0 K, and TN disappears inside the superconducting dome with further pressure. After this, the mixed AFM/SC phase rapidly gives way to a single phase of superconductivity by P = 2.1 GPa, which

17 Figure 1.5: Unit cell crystal structure of CeM In5 (M = Co, Rh, or Ir). Figure from Ref. [43].

18 reaches a maximum Tc = 2.2 K at P = 2.5 GPa. Interestingly, application of ﬁeld just below this peak pressure reintroduces antiferromagnetism to the system, possibly originating within the vortex cores. Superconductivity disappears again at approximately

P = 5.0 GPa.

CeIrIn5 Unlike CeRhIn5, CeIrIn5 is a heavy fermion superconductor at ambient pressures,

2 with Tc = 0.4 K and electronic coeﬃcient of speciﬁc heat of γ = 750 mJ/molCe K [48].

However, CeIrIn5 does not exhibit antiferromagnetism. Application of pressure causes

Tc to increase until a maximum of Tc = 1 K at P = 3 GPa, and then further pressure causes superconductivity to be fully suppressed at P ≈ 5.5 GPa.

CeCoIn5 CeCoIn5 was discovered in 2001 by Petrovic et al. [14]. It is a heavy fermion superconductor at ambient pressure with a critical temperature of Tc = 2.3 K and an upper critical field of Hc2(T = 0) = 5 T. CeCoIn5 has a linear coefficient of specific

2 heat of γ = 290 mJ/molCe K at T = 2.4 K. The unit cell constants of CeCoIn5

(a = 4.62 A˚ and c = 7.56 A)˚ differ significantly from those of CeIrIn5 and CeRhIn5, causing distortions in the CeIn3 layers within it, affecting its properties.

Figure 1.6 shows some of the initial measurements that were performed on CeCoIn5 by Petrovic. Figure 1.6(a) shows the resistivity and susceptibility at ambient pressure.

The resistivity has characteristics very similar to that of other heavy fermions: weakly temperature dependent at high temperatures, initially decreasing with decreasing temperature, and then increasing logarithmically according to the Kondo eﬀect. Then at a temperature of about T = 40 K, coherence develops in the system, and resistivity decreases sharply with further decreasing temperature. Below T = 20 K, ρ is linear

19 Figure 1.6: Figure compiled from Ref. [14]. (a) Temperature dependent resistivity ρ and magnetic susceptibility χ = M/H of CeCoIn5. Open circles represent χ with H k c and open squares represent χ with H ⊥ c. The inset to (a) shows zero-field- cooled χ (circles, left axis) measured in 10 Oe and resistivity (triangles, right axis) in the vicinity of the superconducting transition. (b) Specific heat C/T for CeCoIn5 at applied fields of H = 0 and H = 50 kOe. The nuclear Schottky contribution (due to a large nuclear quadrupole of In) has been subtracted. The inset to (b) shows the resulting entropy as a function of temperature.

20 with temperature, indicating NFL behavior due to quantum critical ﬂuctuations, and

ﬁnally at T = 2.3 K, a superconducting state is formed. The inset shows a detailed

drop in ρ (triangles) to zero at T = Tc.

The susceptibility in Fig. 1.6 for H k c (circles) and H ⊥ c (squares) was measured

at 1 kOe and shows anisotropy, and for H k c, the transition into the low-temperature

coherent state is visible as χ exhibits a transformation from the Curie-Weiss χ ∝ 1/T to the paramagnetic constant χ. The susceptibility in the inset (circles) shows the superconducting transition as χ shows complete magnetic screening.

Figure 1.6(b) shows C/T for CeCoIn5 (with a Schottky contribution from the nuclear quadrupole moment of In subtracted) at zero ﬁeld and at H = 50 kOe. At zero ﬁeld, the superconducting transition is clearly seen as an abrupt, sharp peak.

At H = 50 kOe, the superconducting transition is suppressed, and C/T is seen to increase to approximately 1 J/mol K2 at near-zero temperatures, showing that indeed

CeCoIn5 is a heavy-fermion material. The inset to Fig. 1.6(b) shows the entropy as a function of temperature as a result of the changing C/T .

While there is no long-range magnetic order in CeCoIn5 at ambient conditions, it is close to antiferromagnetic order, near a quantum critical point, causing NFL behavior [49]. The NFL behavior reveals itself in the temperature dependences of

C/T , χ, and ρ. ρ(T ) exhibits linear behavior for T < 20 K [14]. At low temperatures, its magnetic susceptibility diverges as χ ∝ T −0.42 for magnetic ﬁeld H k c. And for

2.3 K < T < 8 K, ∆C/T ∝ ln 1/T [50].

Pressure measurements of up to P = 4.2 GPa were done on the temperature- dependence of the resistivity of CeCoIn5 by Sidorov et al. in 2002 [15]. They were

n able to ﬁt the resistivity as ρ(T ) = ρ0 + AT and found a crossover in ρ0, n, A, and

21 ∆Tc/Tc (where ∆Tc is the width of the superconducting transition) at a critical pressure P ∗ ≈ 1.6 GPa, indicating a change from non-Fermi liquid behavior below P ∗ to

∗ ∗ a Fermi liquid state above P . Interestingly, Tc reaches a maximum at P .

Paglione et al. performed field measurements of the resistivity at low temperatures in 2007 [49], in fields up to 16 T and temperatures as low as 25 mK. They showed that the NFL behavior (revealed in the linear resistivity) is suppressed with increasing field,

2 and a Fermi liquid state develops, with ρ(T ) = ρ0 + AT . Extrapolating to T = 0 the

2 temperatures TFL below which linear resistivity gives way to ρ ∝ T , and using the

QCP divergence of the coeﬃcient A of the T 2 dependence of ρ (with A diverging as

α A ∝ (H − HQCP ) ), Paglione showed the existence of magnetically mediated QCP at

HQCP ≈ 5.1 T, seeming to coincide with the upper critical ﬁeld Hc2(0), as shown in

Fig. 1.7. The inset of Fig. 1.7 shows the divergence of A as the QCP is approached.

Ronning et al. also performed pressurized resistivity measurements on CeCoIn5,

up to pressures of 1.6 GPa and ﬁelds of 9 T, and down to temperatures as low as

40 mK [51]. Performing the same type of measurements at various pressures that

2 Paglione et al. did at ambient pressure (TFL where ρ(T ) crossed from T to T and

the divergence of the FL region’s T 2 coeﬃcient A), Ronning shows that the quan-

tum critical ﬁeld HQCP is only coincidentally approximately equal to Hc2(T = 0)

at ambient pressure. With increasing pressure, both HQCP and Hc2 decrease, but

HQCP does so at a faster pace, decreasing to zero at approximately 1.1 GPa, whereas

Hc2(P = 1.1 GPa) ≈ 4 T. This allowed them to conclude that the NFL behavior ob-

served in CeCoIn5 was not due to the superconducting ﬂuctuations, but quantum

critical ones, likely due to an antiferromagnetic transition.

One controversial point in CeCoIn5 is the existence of antiferromagnetism at low

22 Figure 1.7: An H−T phase diagram of CeCoIn5 from Ref. [49] showing the existence of a QCP at approximately HQCP = 5.1 T. The solid squares show the temperature/field at which ρ crossed over from NFL ρ ∝ T to FL ρ ∝ T 2. The filled circles show the location of a peak in the field dependence of the magnetoresistance of CeCoIn5. The inset shows the divergence of A, which is from the low-temperature, high-field Fermi 2 liquid resistivity ρ(T ) = ρ0 + AT .

23 temperatures. Due to the presence of the superconductivity, the antiferromagnetism

is diﬃcult to detect, and this caused some doubt as to whether or not the NFL be-

havior was due to antiferromagnetic ﬂuctuations. Young et al. provided evidence of

magnetism contained within the vortex cores within the superconducting dome [52].

This was done by performing nuclear magnetic resonance (NMR) measurements on

it in the normal and the superconducting states for H ⊥ c and H ≈ Hc2(0) = 11.8 T.

Also, Hu et al. performed an investigation on the vortex cores of CeCoIn5 [53], showing an extreme upturn in the ﬂux-ﬂow resistivity of the cores with decreasing T and

H. They attributed this to critical spin fluctuations within the vortex cores due to the presence of (or proximity to) an AFM transition. Performing the measurements at different temperatures, Hu and collaborators observed a phenomenological relationship between the superconducting transition and Néeltransition with different pressures and fields, and they were able to determine a low temperature phase diagram for CeCoIn5 that included the quantum critical AFM transition, as shown in

Fig. 1.8 (ﬁgure from Ref. [53]).

It has been established that CeCoIn5 has an anisotropic superconducting gap. Low temperature speciﬁc heat and thermal conductivity measurements in CeCoIn5 [56] re-

2 veal power law dependences below Tc. The speciﬁc heat has T contributions, imply- ing the presence of nodes in the superconducting energy gap, and the low-temperature thermal conductivity exhibits a T 3 dependence, which is consistent with line nodes in an unconventional superconductor. Also, CeCoIn5 is a singlet superconductor, as demonstrated by the suppressed spin susceptibility shown in 115In and 59Co NMR measurements [57] and torque measurements [58]. Together, these facts establish that the superconducting order parameter of CeCoIn5 displays d-wave symmetry.

24 Figure 1.8: From Ref. [53]. Phase diagram for CeCoIn5 at low temperatures, displaying the superconducting phase and the antiferromagnetic phase within the vortex cores. (a) T − P phase diagram at zero ﬁeld. Negative pressure measurements from Ref. [54], where a negative chemical pressure was obtained by doping CeCoIn5 with cadmium on the indium sites. (b) H − P phase diagram at zero temperature, illustrating the quantum critical line. The red line is from Hu and collaborator’s calculations [53], and the dotted yellow line is the result of applying theory from Ref. [55] and using parameters derived by Hu et al. (c) H − T phase diagram at zero pressure showing the superconducting dome with the AFM region within it. The inset shows the same at chemically negative pressure (due to cadmium doping), showing that at negative pressures, the superconducting dome is contained within the AFM region rather than the other way around.

25 1.4.2 Ce1−xYbxCoIn5

Many experiments have been performed on CeCoIn5 with various rare-earth metals

doped on the Ce site (Ce1−xRxCoIn5, with R = rare-earth metal) [59]. Interestingly, they universally suppress Tc and the coherence temperature Tcoh regardless of whether

R is magnetic or nonmagnetic, as can be seen in Fig. 1.9 (from Ref. [59]). This illustrates that disorder is the dominant force in suppressing both Tc and Tcoh, and it is spin-independent. Also, for most of the dopants, the linear resistivity of CeCoIn5

n is driven to sublinear ρ = ρ0 + T , with n decreasing with increasing x. This may

indicate proximity to a magnetic transition, as is seen in CeRhIn5 just above its own

antiferromagnetic transition at TN = 3.8 K.

One of these rare-earth dopants that stands out is ytterbium (Yb). The super-

conducting transition and coherence temperature are both robust to Yb substitu-

tion [60], much more so than for the other rare-earth dopants, as is illustrated in

the inset of Fig. 1.10(c), where the Tcoh and Tc are shown for diﬀerent doping lev-

els of Ce1−xYbxCoIn5 and of Ce1−xLaxCoIn5. This has been theorized by Dzero and

Huang [61] to be due to a healing eﬀect due to correlations between the Yb impurities.

When discussing Ce1−xYbxCoIn5, it is important to keep in mind that the actual

Yb concentration x is about a factor of 3 lower than the initially reported nominal Yb doping xnom (x ≈ xnom/3) for xnom . 0.5 [63]. This needs to be mentioned because several studies were reported using the nominal Yb concentration based on stochiometric calculations, which were not shown to be inaccurate estimations of x until 2014. Using nominal values of concentration in some places and actual in others would create confusion. When referencing studies that used xnom rather than x, both values will be given in order to increase clarity and eliminate ambiguity. While this

26 Figure 1.9: From Ref. [59]. Superconducting transition temperature Tc and coherence temperature Tcoh plotted as a function of residual resistivity ρ0 for diﬀerent doping levels of Ce1−xRxCoIn5. Closed symbols represent Tc and open symbols represent Tcoh. The grey upward-pointing triangles on the left represent x = 0, i.e., pure CeCoIn5. Regardless of the nature of the dopant, Tc and Tcoh are each suppressed to zero with increasing x (with x ∝ ρ0).

27 Figure 1.10: From Ref. [62]. (a) Field dependence of transverse magnetoresistance of CeCoIn5 for different temperatures. Inset shows that of (nominal) Ce0.6Yb0.4CoIn5, which lacks the strong coherent behavior at low field that CeCoIn5 displays. (b) Temperature dependence of Hmax, which is the field value of the peaks in (a), which is where coherent behavior gives way to single-ion behavior. (c) H − x phase diagram illustrating the behavior of HQCP vs. doping. Inset shows Tc and Tcoh at different doping levels of Ce1−xYbxCoIn5 and Ce1−xLaxCoIn5.

28 correction to the Yb concentration does reduce the perceived robustness of Tc and Tcoh

of Ce1−xYbxCoIn5, the CeCoIn5 system is still more robust to doping with Yb than with other rare earth metals, with Tc persisting up to approximately x = 0.4, whereas other rare earth dopants fully suppress Tc at approximately x = 0.15 to x = 0.2.

Like CeCoIn5, Ce1−xYbxCoIn5 is subject to quantum critical behavior due to its proximity to an AFM QCP. This was observed by Hu et al. in 2013 [62]. They performed magnetoresistance measurements on different doping levels of Ce1−xYbxCoIn5, and noted two distinct ranges of behavior with field, as shown in Fig. 1.10 (figure from

Ref. [62]). Figure 1.10(a) shows the ﬁeld dependence of the magnetoresistance (MR)

⊥ ⊥ ⊥ ∆ρa /ρa ≡ [ρa (H) − ρa(0)]/ρa(0), where ρa (H) is the resistivity along the a axis under transverse ﬁeld H. At low H and T , where the coherent Kondo lattice state is formed, MR increases with ﬁeld, while at high H and T , in the noncoherent, single-

2 ion state, MR decreases as H . The peak at ﬁeld Hmax is the ﬁeld at which the coherent Kondo lattice gives way to the single-ion state. Figure 1.10(b) shows the T dependence of Hmax, and Hu and collaborators noted that it is linear for temperatures below 20 K, which is the temperature below which NFL behavior is observed in

CeCoIn5 and Ce1−xYbxCoIn5, and they were able to demonstrate that extrapolation of Hmax to T = 0 gives the location of the AFM QCP. Using this, these authors created the H − T phase diagram in Fig. 1.10(c), showing that quantum criticality in Ce1−xYbxCoIn5 is suppressed with increasing Yb doping, until about xnom = 0.20

(x = 0.07), where HQCP is driven to zero.

The low-temperature resistivity of Ce1−xYbxCoIn5 is shown to closely follow √ ρa(x, T ) = ρa0(x) + A(x)T + B(x) T , where ρa0(x) is the residual resistivity, A(x)T

29 √ is the linear contribution due to the quantum critical ﬂuctuations, and B(x) T is attributed to the mixed-valence state of Yb [62, 64]. In the parent compound CeCoIn5,

B = 0, and only the linear contribution is present. But with increasing Yb concentra- √ tion, the linear contribution is suppressed and is fully replaced by the T contribution

at xnom = 0.20 (x = 0.07), supporting the conclusion that the quantum critical ﬂuc-

tuations in Ce1−xYbxCoIn5 are fully suppressed above xnom = 0.2 (x = 0.07).

Using x-ray absorption and photoemission spectroscopy techniques, the Yb ions have been shown to change valence across the doping range of Ce1−xYbxCoIn5 [65, 66].

At the lowest dopings, near x = 0, the Yb ions have a valence of +3, in a magnetic state with 13 f electrons. As the Yb concentration increases, the average Yb valence quickly decreases to an intermediate-valence state of +2.3 at xnom = 0.2 (x = 0.07), after which it remains constant throughout the doping range. The nonmagnetic state of Yb, with 14 f electrons and valence of 2, has a larger volume than the magnetic state. Hence, this intermediate-valence state induces a strain in the lattice. Theory by Dzero and Huang [61] uses strain-dependent hybridization between f electrons in the Yb ions and the conduction electrons to show that this lattice strain leads to an increase in the pair correlation radius between the Yb ions. The correlations are attractive and have a healing eﬀect on the formation of coherent behavior, causing

Tcoh and Tc to decrease at a lower rate than if the correlations were not attractive.

Measurements by Singh et al. [64] experimentally support Dzero and Huang’s theory.

Although Tcoh initially decreases slowly with increasing Yb, resistivity and heat capacity measurements show that after about xnom = 0.65, coherence is actually reestab- lished in the system, though apparently this is via coherence between the conduction electrons and the Yb impurities, as opposed to the Ce ions at lower concentrations

30 of Yb [64]. This is shown in Tcoh which actually increases after xnom = 0.65. It is also seen in the magnetoresistivity. For 0.2 < xnom ≤ 0.7, the positive component in the magnetoresistivity that is seen in lower dopings is missing, until it reappears in xnom = 0.75, albeit much weaker than in the low dopings. Finally, the heat capacity shows a decrease in γ with increasing Yb concentration until xnom = 0.65, after

which it again increases due to the strengthened coherence eﬀects. All of these are

consistent with the Fermi surface reconstruction above xnom noted in Ref. [67].

31 CHAPTER 2

Experimental Details

2.1 Sample Growth and Preparation

The single crystals of Ce1−xYbxCoIn5 were grown by the group of our collaborator,

M. B. Maple, of the University of California, San Diego [66], in a manner similar to the method used by C. Petrovic in 2001 to make CeCoIn5 [14]. Ce, Yb, and Co were

combined in stoichiometric amounts along with excess In in an alumina crucible,

which was then sealed in a quartz tube with 150 psi of argon gas. The samples

were heated to 1050 ◦C at a rate of 50 ◦C/hr and were kept there for 72 hours in

order to homogeneize the components. Then, the samples were cooled in a two-

stage process: ﬁrst, they were cooled rapidly to 800 ◦C, and from there, they were slowly cooled to 450 ◦C, at which point a centrifuge was used to spin oﬀ the excess In

ﬂux. The resulting crystals are shiny, smooth rectangles with approximate dimensions

2.1×1.0×0.16 mm3. The unit cell lattice dimensions were obtained via x-ray powder diﬀraction measurements, and the doping level of the samples was determined by applying Vegard’s law to the measured unit cell volume [63]. There is found to be some discrepancy between the nominal doping level xnom and the actual doping level x. For xnom . 0.4, x ≈ xnom/3. As xnom increases past approximately 0.4, x becomes closer to xnom until xnom = x = 1. Unless stated otherwise, all dopings discussed will be actual doping level x rather than the nominal doping xnom. Table 2.1 shows the conversion between the nominal and the actual doping levels in Ce1−xYbxCoIn5.

The samples were lightly sanded to remove any impurities on the surface and then

32 xnom x

0 0 0.10 0.03 0.20 0.07 0.25 0.09 0.40 0.16 0.50 0.23 0.65 0.38 0.70 0.46 0.75 0.54 0.775 0.58

Table 2.1: Conversion between the nominal (xnom) and the actual (x) Yb doping levels in Ce1−xYbxCoIn5 were soaked in HCl acid for an hour in order to remove any remaining indium ﬂux.

Finally, they were rinsed using ethanol and dried.

2.2 Physical Property Measurement System

All measurements were performed using the Physical Property Measurement Sys- tem (PPMS) developed by Quantum Design. The PPMS is a cryostat and workstation for automated use in performing a variety of measurements such as electron transport, heat capacity, and torque magnetometry. During measurement, the sample is held in a sample chamber inside the PPMS dewar. The sample chamber is ﬁlled with helium gas and pumped down to a few torr.

We were able to perform automated changes to the sample environment and automated measurement sequences using the MultiVu software designed to control the

PPMS. Flowing helium gas along the sample chamber, we could control the temperature of the sample to anything within the range 1.8 K to 400 K with temperature stability ≤ 0.2 %. With Quantum Design’s Helium-3 option, one could control down

33 to 0.35 K. We were also capable of applying a ﬁeld as high as 14 T in the vertical direction, parallel to the c axis of the crystals. With the PPMS and the MultiVu software used to control it, one can write a script or sequence for the machine to follow in taking the measurement.

Figure 2.1 shows the inside of the sample chamber contained within the dewar of the PPMS. The sample is placed on a thermally conducting puck that is then placed in the bottom of the sample chamber. At the bottom of the sample chamber are twelve pins that plug into the puck and allow for the electrical connections that are necessary to measure the resistivity of the sample or to use the thermometer and the heater on the bottom of heat capacity platform. To ensure good thermal conduction between the sample and the puck, the sample is ﬁxed to the puck with either Crycon grease (in the case of resistivity measurements) or Apiezon N grease (in the case of heat capacity measurements).

The lower portion of the sample chamber is constructed out of copper to ensure uniform temperature in the vicinity of the sample. Surrounding the same lower portion is the superconducting electromagnet, which creates a uniform field in the vicinity of the sample. Surrounding the sample chamber is a low pressure space called the cooling annulus. An impedance system is used to control the flow of helium gas through the annulus. At the base of the sample chamber are two thermometers and a heater used in temperature control. The two thermometers are for different temperature ranges: a Cernox thermometer for T < 100 K, and a platinum thermometer for T > 800 K. In the crossover region between 80 K and 100 K, a weighted average is used. There is another Cernox thermometer up higher on the neck of the sample chamber to monitor the temperature gradients.

34 Figure 2.1: (a) A cross-sectional view of the PPMS sample chamber showing its components. (b) A cutaway view of the sample chamber showing the puck with sample at the bottom of the sample chamber.

35 2.3 Helium-3 Option

The standard PPMS conﬁguration allows one to measure down to about 1.8 K. In order to measure to lower temperatures, we made use of the Helium-3 option, which is a special insert that utilizes Helium-3 (He-3 or 3He) to go to lower temperatures than the more common 4He used in the standard PPMS conﬁguration. With the He-3 option, we could measure down to about 0.5 K. The low temperature is obtained via evaporative cooling using a pot of liquid He-3 in a closed system.

The He-3 used in our lab has a vertical configuration, meaning the platform that the samples are fixed to is perpendicular to the floor rather then parallel to the floor, as the standard PPMS configuration is. This caused the c axis of the samples to be perpendicular to the field, but we needed the applied field to be parallel to the c axis in order to be consistent with measurements done using the standard PPMS configuration. In order to correct this, we constructed a T-shaped platform out of a sapphire substrate (Al2O3) for the sample to rest on. The flat top of the ‘T’ was fixed to the platform with a thin layer of Apiezon N grease, and the sample was fixed to the side of the ‘T’, again with Apiezon N grease. The resulting configuration placed the c axis parallel to the applied magnetic field, as desired, shown in Fig. 2.2.

The ‘T’ was included in the background measurement that is necessary with each heat capacity measurement, and it was subtracted accordingly. Zero-ﬁeld measurements with and without the ‘T’ showed that the data retained its quality and accuracy despite the additional mass and the decrease in direct thermal conductance between the sample and the platform.

36 Figure 2.2: A schematic showing the use of the T-shaped platform in order to align the c axis of the sample parallel to the applied magnetic ﬁeld.

37 2.4 Electrical Transport Measurement

The resistivity of the samples was measured using the four-terminal sensing method,

also known as the Kelvin resistance method. Four leads are attached to the sample,

as illustrated in Fig. 2.3(a). Measurements of the sample’s thickness t, width w, and

the distance between the two inner leads LV is taken. A known current I is made to

ﬂow between the outer leads. The inner leads are attached to a high impedance voltmeter, which measures the voltage drop V across the inner leads. The advantage to the Kelvin method is that the only resistance measured is of the sample between the inner leads, as opposed to the simpler two-terminal sensing method which includes the contact resistance and the resistance of the wires. This is because the voltage is measured across the inner leads (thereby ignoring the resistance of the outer leads), and the current ﬂowing between the point of contact of the inner leads and the voltmeter is negligibly small (allowing us to neglect the resistance of the inner leads).

With V measured, the sample resistance R is simply calculated as,

R = V/I .

If we assume that the sample is thin enough that the current is approximately uni-

formly distributed through the sample between the voltage leads, the resistivity ρ is

calculated using the standard equation,

A wt ρ = R = R , L LV

where A = wt is the cross sectional area of the sample, and L = LV is the distance

between the inner leads.

Current ﬂow, voltage measurement, and resistance calculation were all performed

by the PPMS. Resistivity was later calculated using the resistance and the dimensions

38 Figure 2.3: (a) A schematic of the four-terminal sensing method. (b) A photograph of a sample in our lab in which we are applying the four-terminal sensing method.

39 of the sample, as described in the previous paragraph.

As stated, four gold wire leads were aﬃxed to the top of the sample, as illustrated in Fig. 2.3(b). The gold wires have a diameter of 50 µm and a purity of 99.995 %, and they were ﬁxed to the sample using a two-part, electrically conductive, silver- based epoxy, EPO-TEKr H20E. The two parts were mixed together in a 1:1 ratio by weight, and the ends of the gold wires were dipped in the epoxy and laid on top of the sample, parallel to each other, as shown in Fig. 2.3(b). To cure the epoxy, the samples were heated at 200 ◦C for 5 minutes.

2.5 Pressure

For pressure measurements, we made use of the Pcell 30 (see Fig. 2.4), designed by easyLab Technologies Ltd (now Almax easyLab Group Ltd) for use with the PPMS.

The Pcell 30 is a pressure cell, and as the name suggests, it is capable of applying a hydrostatic pressure of 30 kbar to the sample. The liquid pressure-transmitting medium inside the pressure cell is a mixture of one part pentane and one-part isopen- tane (by weight). Pressure was applied at room temperature using a hydraulic press, and the pressure at low temperatures was measured using a tin manometer placed inside the cell with the sample: the tin’s superconducting transition temperature would be measured at zero ﬁeld and compared to a temperature-pressure phase diagram to determine the pressure. Wires passed from the sample inside the pressure cell to the outside via a feedthrough sealed with Stycast 2850FT resin, cured with catalyst 11.

Figure 2.4 shows a half-assembled Pcell. The white cap in the middle contains the sample and the pressure-transmitting ﬂuid. The wires coming out of it have been soldered to the sample and tin manometer on the inside and will be soldered to the puck on the bottom of the ﬁgure after assembly. The puck mates with the bottom of

40 Figure 2.4: A photograph of a half-assembled pressure cell.

41 the sample chamber, allowing the PPMS to measure the resistance of the sample and of the tin. The thin, dark-grey rod in the top half is the piston that pushes against the white cap creating the extreme pressure.

The wired feedthroughs were assembled in-house as follows. Four, 36-gauge twisted- wire pairs were each spun around on an assembly designed to provide a consistent twist along their entire length: 70 full rotations over a length of approximately 20 cm, except for the region which will be passing through the feedthrough, which is left un- twisted, so as to allow pressure-sealing epoxy to completely ﬁll the space between the wires. The wires have an insulating coating, and this is scraped oﬀ on each end with a scalpel to a length of 1 to 2 mm and tinned with solder. A small rectangular plastic platform is cut (roughly 3 mm × 1.5 mm) in order to create a staging area for the sample (as seen in Fig. 2.5(b)), and also to galvanically separate the sample and the tin manometer while they are inside the pressure cell (as can be seen in Fig. 2.5(c)).

The two wire pairs corresponding to the sample’s current and its voltage reference are tied onto the sample platform via holes in the four corners, with the bare ends all ending on the same face of the platform. It is important that the pairs are tied in such a way that one pair is tied to one end of the platform, and the other pair is tied at the other end. The ends of the other two pairs, the tin manometer’s current and its voltage reference, are twisted together in such a way as to allow for a two-point resistance measurement on the manometer. That is, one wire of the voltage-reference pair and one wire of the current pair are twisted together so as to be able to meet as one at one end of the manometer, while the other two wires are twisted together to meet as one at the other end of the manometer. The wires are painted diﬀerent colors in order to to be able to distinguish which is which, even after they are all sealed in

42 the feedthrough.

The wires are all fed through the feedthrough, with the platform oriented verti- cally, as seen in Fig. 2.5(c). The feedthrough hole is then ﬁlled with Stycast 2850FT epoxy that has been mixed with catalyst 11 at 4.5 parts catalyst to 100 parts Stycast.

After thoroughly mixing, the epoxy is placed in a vaccuum for 30 minutes in order to remove air that had been added during the mixing process. After the epoxy is in- jected into the feedthrough and thoroughly worked in between the wires, it is baked at 100 ◦C for 3 hours.

After the epoxy is cured, the sample and manometer can be added. The manometer is created by winding some tin wire into a narrow helix, about 2 mm long. The manometer’s superconducting transition temperature will be used to determine the pressure, and having a higher resistance will allow for a more accurate reading of the transition. Hence, good practice is to increase the resistance as much as possible by using a longer wire and coiling it tightly, while not letting the coils touch each other as this will create a shorter electrical path, reducing the resistance. The winding of the helix is most easily accomplished by wrapping it around a very narrow cylinder, such as a pin, and then sliding it oﬀ of the end. The manometer is then soldered onto the wires underneath the platform.

The sample is also ﬁxed to the platform. This is most easily accomplished under a microscope with two pairs of ﬁne tweezers. The ends of each of the four copper wires that had been tied to the platform are bent into a loop, and the four gold leads of the sample are tied onto those loops and soldered there. It is recorded which copper pair the current leads and the voltage leads were soldered on to lest they become mixed up during measurements. After soldering, the gold leads are wrapped up and

43 Figure 2.5: Photographs of a wired feedthrough. (a) Image of the feedthrough, showing the wires coming out of the back. (b) Close-up of the sample on top of the platform. (c) Side view of the platform with the tin manometer on the left and the sample on the right.

44 separated from each other so as not to touch during measurement. After this, the

sample/feedthrough is carefully inserted into a Teﬂon pressure cap that is ﬁlled with

the pressure transmitting ﬂuid, and the pressure cell is assembled around it. Finally

pressure can be applied at room temperature using a hydraulic press, applying a force

that is then locked in with a locknut.

Before each measurement, the pressure inside the sample is calculated by measur-

ing the superconducting transition temperature of the tin manometer at zero ﬁeld.

The pressure dependence of the critical temperature has been determined to be well

2 −2 modeled by Tc(P ) = Tc(0) − aP + bP , with Tc(0) = 3.732 K, a = 4.89 × 10 K/kbar, and b = 3.80 × 10−4 K/kbar2 [68]. It is important that the transition be measured at zero field because even a few Oersted can significantly lower the tin’s critical temperature and thus affect the pressure calculation. This means compensating for the remanent field that can be left in the PPMS superconducting magnet, which can be as high as 15 Oe. This is easily accomplished manually by tuning the temperature down to the transition temperature, and then tuning the magnetic field until the resistance is as low as possible (due to raising the critical temperature). The magnetic field where the superconducting transition is as high as possible is the field at which the remanent magnetization is compensated for. The field is then latched here and a typical resistivity measurement is performed to determine the superconducting transition temperature and thus the pressure.

2.6 Magnetoresistivity

Magnetoresistivity or magnetoresistance (MR) is the phenomenon in which the electric resistivity of a material changes with application of magnetic ﬁeld. In this

45 work, magnetoresistivity is deﬁned as the percent change in resistivity:

∆ρ (H) ρ (H) − ρ (H = 0) a = a a , (2.1) ρa(H = 0) ρa(H = 0) where ρa(H) is the electrical resistivity along the a axis under ﬁeld H. All MR measurements were performed with H parallel to the c axis and perpendicular to the current, which ﬂowed along the ab plane. Thus, we call this a transverse MR measurement.

If the four leads on the sample are not perfectly aligned or parallel, it is possible that the MR measurement could pick up a Hall effect voltage due to the field pushing the charge carriers and creating a voltage across the sample, perpendicular to both the field direction and current direction. This effect was subtracted as follows. Ap- plying the field in the opposite direction (H k−c) would create the same Hall voltage except negative, but the MR would be unaffected. Therefore, if we perform the same measurement twice, once with the field pointing in each direction and perform a symmetry calculation as below on the two field orientations, the Hall component will be subtracted out leaving only the MR. ρ (+H) + ρ (−H) ρ (H) = ρ (H) = a(measured) a(measured) , (2.2) a a(symmetry) 2 where +H and −H denote the field pointing in opposite directions.

2.7 Heat Capacity Measurement

The heat capacity of the samples was measured using the PPMS’s Heat Capacity

dQ Option. It was calculated according to (CP ) = ( dT )P using a known applied heat dQ and measuring the temperature change dT . The sample is placed on a platform

suspended by four delicate wires within a puck frame (eight wires if using the stan-

dard puck as opposed to the He-3 puck). Under the platform is a small heater and

46 Figure 2.6: A side view of the heat capacity setup. thermometer, each attached to two of the aforementioned wires (supplying current and measuring resistance), as shown in Fig. 2.6. The heater is used to apply a pulse of known power to the sample until the temperature increases by 2%, as monitored by the thermometer. After being heated, the sample is allowed to cool for the same amount of time as the heating, with the four wires serving as thermal conductors, connecting the platform to the puck frame, which acts as a thermal bath/sink. The measurement is performed with the system pumped down to a high vacuum of less than 0.01 mTorr in order to ensure that nearly all thermal conduction occurs through the wires, which have known thermal conducting properties.

In order to isolate the heat capacity of the sample, one needs to subtract that of the platform, the grease used to hold the sample, and the ‘T’ described in the

He-3 section above. To do so, the measurement is performed twice: once without the sample and once with. The two measurements are subtracted leaving only the sample’s heat capacity.

To isolate the eﬀect the magnetic ions had on the heat capacity, it was necessary to remove the phonon contribution due to lattice vibrations. To do so, we also measured

47 the heat capacity of the compound LaCoIn5, which is the iso-structural, nonmagnetic analog of Ce1−xYbxCoIn5, and subtracted it from our measurements.

48 CHAPTER 3

1 Pressure Studies of the Quantum Critical Alloy Ce0.93Yb0.07CoIn5

3.1 Introduction

The reader will recall that the heavy-fermion alloys Ce1−xYbxCoIn5 possess a number of intriguing and often counterintuitive physical properties: (i) upon an increase in the concentration of ytterbium atoms, the critical temperature (Tc) of the superconducting transition decreases only slightly compared to other rare-earth sub- stitutions [60, 59] and superconductivity persists up to concentration x ≈ 0.55 (actual doping x ≈ 0.28); (ii) the value of the out-of-plane magnetic ﬁeld (H) corresponding to the quantum critical point (QCP) (separating the antiferromagnetic (AFM) and the paramagnetic (PM) states at zero temperature) approaches zero as xnom → 0.2

(x → 0.07) [62]; (iii) there is a crossover in the temperature (T ) dependence of resistivity (ρa) measured along the a axis: for xnom < 0.2 (x < 0.07), resistivity remains predominately linear in temperature, while for xnom > 0.2 (x > 0.07), resistivity has a square-root temperature dependence [64]:

√ ρa(x, T ) = ρa0(x) + A(x)T + B(x) T, (3.1)

with residual resistivity ρa0(x) ∝ xnom(1 − xnom) (in accord with Nordheim law [69,

70]), B(x) → 0 as xnom → 0 and A(x) → 0 as xnom is gradually increased from zero

to xnom ≈ 0.2; (iv) there is a drastic Fermi-surface reconstruction for xnom ≈ 0.55

1This chapter is based on the following article: Y. P. Singh, D. J. Haney, X. Y. Huang, B. D. White, M. B. Maple, M. Dzero and C. C. Almasan, Phys. Rev. B 91, 174506 (2015)

49 (x ≈ 0.28), yet (Tc) remains weakly aﬀected [67]. More recently, penetration depth

measurements [71] have shown the disappearance of the nodes in the superconducting

order parameter for xnom ≥ 0.2.

The emergent physical picture which describes the physics of these alloys is based on the notion of co-existing electronic networks coupled to conduction electrons: one is the network of cerium ions in a local moment regime, while the other consists of ytterbium ions in a strongly intermediate-valence regime [72, 73]. This picture is supported by recent extended x-ray absorption ﬁne structure spectroscopic measurements [65], as well as photoemission, x-ray absorption, and thermodynamic measurements [66, 74]. Moreover, our most recent transport studies [64] are generally in agreement with this emerging physical picture. In particular, for xnom ≈ 0.6 we observe the crossover from coherent Kondo lattice of Ce to coherent behavior of Yb sub-lattice, which is in agreement with recent measurements of the De Haas-van

Alphen eﬀect [67], while superconductivity still persists up to xnom ≈ 0.75 of ytter-

bium concentration. Nevertheless, it remains unclear which of the conduction states

- strongly or weakly hybridized - of the stoichiometric compound contribute to each

network.

In order to get further insight into the physics of the Ce1−xYbxCoIn5 alloys, we study the transport properties under applied magnetic ﬁeld and pressure for the alloy with actual concentration xact ≈ 0.07. One of our goals is to clarify the origin of the

square-root temperature dependence of resistivity and to probe the contribution of

the heavy-quasiparticles to the value of A(x) and B(x) [see Eq. (3.1)]. To address this issue, we study the changes in the residual resistivity and the coeﬃcients A and B [see Eq. (3.1)] with pressure. Our results show that while both the residual

50 resistivity and the coeﬃcient A decrease with pressure, B shows very weak pressure dependence. This indicates that the AFM quantum ﬂuctuations are suppressed with pressure and that the light quasiparticles involved in the scattering mechanism that √ gives the T dependence originate from the electrons from the small Fermi surface

that hybridize with Yb ions. We ﬁnd that the Kondo lattice coherence and the

superconducting critical temperature increase with pressure in accord with general

expectations [64, 75].

Another important aspect of the present work concerns the evolution of the phys-

ical quantities aﬀected by the presence of the ﬁeld-induced quantum critical point.

In our recent work [62, 64], we have shown that the temperature dependence of the

magnetic ﬁeld Hmax at which magnetoresistivity has a maximum is a signature of

system’s proximity to ﬁeld-induced QCP. Consequently, here we study the depen-

dence of Hmax on pressure. We ﬁnd a remarkable similarity between the dependence

−1 of the residual resistivity and (dHmax/dT ) on pressure. Yet, this result is not sur-

prising because it is well understood that the tendency towards antiferromagnetic

ordering originates from the partial screening of the f-moments by conduction elec-

trons. Hence, a strong pressure dependence of the relevant physical quantities such

as A and Hmax is expected.

3.2 Experimental results and discussion

Figure 3.1(a) shows ρa data as a function of temperature of a Ce0.93Yb0.07CoIn5

single crystal measured under pressure. The qualitative behavior of resistivity is the

same for all pressures used in this study: the resistivity initially decreases as the

sample is cooled from room temperature, then it passes through a minimum in the

51 temperature range 150 K to 200 K, followed by an increase as the temperature is further lowered. This increase is consistent with a logarithmic temperature dependence, in accordance to the single-ion Kondo effect. With the onset of coherence effects at the Kondo lattice coherence temperature (Tcoh) (defined as the peak in the resistivity data), the resistivity decreases with further decreasing the temperature below Tcoh, while at even lower T , superconductivity sets in at Tc.

The onset of coherence is governed by the process in which the f-electrons of

1 0 Ce can resonantly tunnel into the conduction band, i.e., f f + e. Because the cell volume Ω changes due to these resonant processes, i.e., Ω(f 1) − Ω(f 0) > 0, the electronic properties are strongly susceptible to the application of external pressure.

Thus, we expect that pressure increases the local hybridization of Ce0.93Yb0.07CoIn5 and, hence, increases the coherence temperature. Figure 3.1(b) shows that, indeed, the disordered Kondo lattice Tcoh increases with increasing pressure, just as it does for pure CeCoIn5 and the other members of the Ce1−xRxCoIn5 (R = rare earth) series [74].

The inset to Fig. 3.1(b) shows the pressure dependence of Tc. For small values of pressures, clearly Tc ∝ Tcoh as they linearly grow with pressure [see Fig. 3.1(b) and its inset]. This is expected since at low temperatures the coherence temperature of superconducting heavy-fermion metals plays the role of a renormalized bandwidth and, therefore, provides the ultraviolet cutoﬀ for the superconducting instability.

It is well known [76, 77, 78, 79] that large and small Fermi surfaces co-exist in the stoichiometric CeCoIn5. The quasiparticles from the large Fermi surface are composed of the f-states as well as conducting d-states due to the hybridization between Ce f- and d-orbitals, and hence have heavy eﬀective mass. Consequently, the transport and

30 coh Ce Yb CoIn

0.93 0.07 5

P (kbar)

0 -cm)

2.7 ( a 5.1

7.4

8.7

(a)

0 50 100 150 200 250 300

T (K)

Ce Yb CoIn

0.93 0.07 5 60

2.1 40

(K)

coh 2.0 c T T

20 1.9

0 2 4 6 810

(b) P (kbar)

0 2 4 6 8 10

P (kbar)

Figure 3.1: (a) Resistivity ρa of Ce0.93Yb0.07CoIn5 as a function of temperature T for diﬀerent pressures P (0, 2.7, 5.1, 7.4, and 8.7 kbar). The arrow at the maximum of the resistivity data marks the coherence temperature Tcoh. (b) Evolution of Tcoh as a function of pressure P . Inset: Superconducting critical temperature Tc as a function of pressure P . The solid lines are guides to the eye. 53 thermodynamic properties of these quasiparticle states strongly depend on pressure

since hybridization involves quantum mechanical tunneling between f 0 and f 1 valence

states, changing the volume of the Ce ions. In contrast, the quasiparticle states on

the small Fermi surface have zero spectral weight contribution from the Ce f-states

and, therefore, have light eﬀective mass and must show weak pressure dependence.

An open question is, do the electrons from the small Fermi surface hybridize with

ytterbium ions, or do only the electrons from the large Fermi surface hybridize with

both cerium and ytterbium ions? As just discussed, the former (latter) scenario would

give a pressure independent (dependent) coeﬃcient for the temperature dependence

of the scattering processes. Therefore, to address this question, we study the changes

in the temperature-dependent part of resistivity under pressure.

As we have already discussed in the Introduction, we have previously shown

that there are two distinct contributions to the scattering of the quasi-particles in √ Ce1−xYbxCoIn5 alloys: a T contribution and a linear-in-T contribution. This latter

one is due to quantum critical ﬂuctuations and it is observed only at small Yb doping

(xnom ≤ 0.2, xact ≤ 0.07) [see Eq. 3.1]. In what follows we trace out the changes in

the coeﬃcients A and B with pressure for the Ce0.93Yb0.07CoIn5 alloy, for which both

of these contributions are present at least over a certain temperature range and under

ambient pressure. The goal is to determine the eﬀect of pressure on quantum critical √ ﬂuctuations and on the scattering mechanism that gives the T dependence in resis-

tivity. Figure 3.2(a) shows that the data are ﬁtted very well with Eq. 3.1 (the solid

lines are the fits to the data) for 3 K ≤ T ≤ 15 K and for all pressures studied. From these fits we obtain the pressure dependence of the fitting parameters ρa0, A, and

B, which allow us to probe the relative contribution of heavy- and light-quasiparticle

54 states to scattering.

Figure 3.2(b) shows the pressure dependence of the parameters A and B extracted from the ﬁtting of ρa(T ) of Fig. 3.2(a), which, as discussed above, are the weights of the linear-in-T and square-root-in-T scattering dependences, respectively. Notice that A decreases while B remains relatively constant with increasing pressure. The suppression of A with pressure indicates that the AFM quantum ﬂuctuations are suppressed with increasing pressure. Also, the insensitivity of B to pressure suggests √ that the inelastic scattering events leading to the T dependence in this temperature

range involve light eﬀective mass quasiparticles from the small Fermi surface. Hence,

these ρa(T ) data for 3 K ≤ T ≤ 15 K show that there are two distinct contributions

to scattering originating from the two Fermi surfaces: AFM quantum ﬂuctuations of

the heavy quasiparticles (with a linear-in-T scattering behavior) and quasiparticles √ from the small Fermi surface (with a T scattering behavior).

Moreover, the value of the coeﬃcient B(P = 0, x) increases with ytterbium di- lution [62] and it remains essentially unchanged under the application of pressure at temperatures well above Tc. These observations strongly suggest that the value of B(P = 0, x) is governed by the quasiparticle excitations from the Fermi pockets near the M-points of the quasi two-dimensional Brillouin zone. Recall that according to the recent thermopower measurements and subsequent theoretical studies [80, 79] of the parent compound CeCoIn5, the Fermi pockets near the M-points remain un-

gapped giving rise to the nonzero thermal conductivity in the superconducting state.

If we now consider the results of the recent penetration depth measurements that

show the disappearance of the nodes in the superconducting order parameter for

xnom ≈ 0.2 [71], we conclude that with Yb doping: (a) both Fermi surfaces must

55 Ce Yb CoIn

0.93 0.07 5 20

P (kbar)

15 -cm)

0 ( 2.7 a

5.1

7.4

(a)

8.7

0 5 10 15

T (K)

2.4

1/2 B ( B

= + AT + BT 0.45

a a0

2.2

0.40

2.0 -cm/K -cm/K) 0.35

1.8

0.30 1/2

1.6 A( )

(b) 0.25

1.4

6 -cm)

5 ( a0

(c)

0 2 4 6 8 10

P (kbar) √ Figure 3.2: (a) Fits of the resistivity ρa(P,T ) = ρa0(P ) + A(P )T + B(P ) T for different pressures on Ce0.93Yb0.07CoIn5 over the temperature range√ 3 K ≤ T ≤ 15 K. (b) Pressure P dependence of the linear T contribution A and T contribution B, obtained from ﬁts of the resistivity data shown in panel (a). (c) Pressure P dependence of the residual resistivity ρa0, obtained from the ﬁts.

56 be gapped below Tc due to the proximity pairing eﬀect, and (b) the absence of the nodes in the superconducting order parameter for xnom ≥ 0.2 suggest that the order parameter may have exotic symmetry, either d + is or d + id, which in principle can give rise to topologically protected surface states [81, 82]. The d-component must be present since the order parameter of the parent compound CeCoIn5 has dx2−y2 symmetry [83, 84], while the conventional s-wave superconductivity can be ruled out due to monotonous concentration dependence of Tc. Therefore, intriguingly, Ce1−xYbxCoIn5 may provide an important playground for the realization of the long thought topological superconductivity [81]. However, to verify the realization of speciﬁc scenarios for the symmetry of the superconducting order parameter in Ce1−xYbxCoIn5, one would need a detailed understanding of the electronic properties in both normal and superconducting states [82].

Figure 3.2(c) shows the pressure dependence of the residual resistivity ρ0 extracted from the fitting of the data of Fig. 3.2(a). As discussed in the Introduction, the residual resistivity in this system depends on the impurity concentration in accordance with Nordheim’s law. In systems with proximity to a quantum critical point, there will also be a contribution to residual resistivity from the quantum critical fluctuations. Since tuning with pressure does not introduce any impurity scattering in the system, the decrease in residual resistivity with increasing pressure indicates that the scattering due to AFM quantum spin fluctuations is suppressed by pressure, hence the system is driven away from the QCP. Indeed, quantum fluctuations in this family of heavy fermion superconductors are known to be suppressed by pressure because the AFM order in the Ce-lattice is suppressed [85, 86, 87].

Ce Yb CoIn

0.93 0.07 5 P (kbar)

2.7

5.1 -cm)

7.4 8

)

4.8

8.7 1/2

4.4 ( a 4.0

4 -cm/K

3.6

3.2

23456789

(a) B* ( B*

0 P (kbar)

Ce Yb CoIn

0.92 0.08 5

H = 0 T

H = 4 T

-cm) 16 ( a

(b)

1 2 3

1/2 1/2

T (K ) √ Figure 3.3: (a) Resistivity ρa of Ce0.93Yb0.07CoIn5 as a function of T , in the temperature range 1.8 K ≤ T ≤√ 5 K. The solid lines are linear fits of the data with ∗ ρa(P,T ) = ρa0(P ) + B (P ) T for√ 1.8 K ≤ T ≤ 5 K. Inset: Pressure P dependence ∗ of the coefficient B . (b) ρa vs T of Ce0.92Yb0.08CoIn5 measured in zero magnetic field and at 4 T. The 4 T data has been offset upwards by 5 µΩ cm for visual clarity.

58 √ Figure 3.3(a) shows ρa data vs T around the superconducting transition tem-

perature (1.8 K ≤ T ≤ 5 K). This figure shows that from just above Tc to about 4 K, √ the ρa(T ) data follow very well a T dependence (solid lines are linear fits to the data √ ∗ ∗ with ρa(P,T ) = ρa0(P ) + B (P ) T ). The pressure dependence of the coefficient B

is shown in the inset to Fig. 3.3. Notice that B∗ is signiﬁcantly suppressed with

increasing pressure. This pressure dependence of B∗ suggests that the scattering just

above Tc is largely governed by ﬂuctuating Cooper pairs originating from the heavy

Fermi surface. This observation is in agreement with the ﬂuctuation correction to

resistivity due to pre-formed Cooper pairs composed of heavy quasiparticles. Indeed,

for a 3D Fermi surface, and in the case of a strong coupling superconductor with √ relatively small coherence length [88], one expects a T ﬂuctuation contribution to

resistivity [89]. Therefore, these ρa(T ) data show that the strong SC ﬂuctuations of √ the heavy quasiparticles give the T dependence just above Tc and that the linear-

in-T contribution of Eq. 3.1 that is due to the system’s proximity to the ﬁeld-induced

QCP, is masked by these strong SC ﬂuctuations. The superconducting ﬂuctuations, nevertheless, decrease as the system moves away from Tc to higher temperatures. In-

deed, as discussed above, the resistivity data reveal that other scattering mechanisms

dominate at temperatures above about 4 K [see Fig. 3.2 and its discussion]. √ Alternatively, the T dependence of the resistivity just above Tc is also consistent

with the composite pairing theory in a 3D system [90], which predicts an incoherent

transport of composite Cooper pairs above the superconducting critical temperature √ with the resistivity growing as T . It is important to emphasize that the size of the

composite pairs is only a few lattice spacing, i.e., the electrons in a composite pair

are tightly bound. From this point of view, the transport of composite pairs is not

59 governed by ﬂuctuation corrections to conductivity, which are usually discussed in

the context of conventional superconductors. Nevertheless, the decrease in B∗ with increasing pressure is also consistent with this theory because the composite pairs incorporate the heavy quasiparticles. √ Figure 3.3(b) shows ρa data vs T around the superconducting transition temperature (1.8 K ≤ T ≤ 5 K), measured at ambient pressure in zero magnetic ﬁeld and 4 T. √ The temperature at which the data deviate from the T dependence decreases with

applied ﬁeld, showing that, as expected, the Cooper pair ﬂuctuations are suppressed

by magnetic ﬁeld.

Next, we present the results of the transverse (H ⊥ ab) magnetoresistivity (MR)

measurements, deﬁned as ∆ρa/ρa(0) ≡ [ρa(H) − ρa(H =0)]/ρa(H =0)], on

Ce0.93Yb0.07CoIn5 in applied ﬁelds up to 14 T, for temperatures 2 K ≤ T ≤ 60 K, and

applied pressures up to 8.7 kbar. The main panel of Fig. 3.4 and its inset show such

MR curves measured at ambient pressure and 5.1 kbar, respectively. The 9 K MR data

in both panels show non-monotonic H dependence: the MR increases with increasing

ﬁeld, displays a maximum at a ﬁeld Hmax, and decreases with further increasing H,

with an H2 dependence at high ﬁelds (see inset to Fig. 3.4) that is typical of a single-

ion Kondo system. This positive MR behavior at low H values is due to the formation

of the coherent Kondo lattice state. Hmax represents the value where the coherent

state gives way to the single-ion state due to the fact that magnetic ﬁeld breaks the

coherence of the Kondo lattice [91, 92, 93, 94, 95, 96].

In a conventional Kondo lattice system, as T increases, Hmax moves toward lower

field values, signifying that a lower field value is sufficient to break coherence at

these higher temperatures due to thermal ﬂuctuations, with a complete suppression

60 2 Ce Yb CoIn

0.93 0.07 5

1.0

P = 5.1 kbar

2 T = 9 K

-2 m ax (H=0) (%) a

-4

0.5 (H)/

a -6 (H=0)(%) a

0 50 100 150 200 250

2 2

H (T )

0.0 (H)/

a Ce Yb CoIn

0.93 0.07 5

P = 0 kbar

T = 9 K

-0.5

T = 50 K

0 20 40 60 80 100

2 2

H (T )

Figure 3.4: Magnetic field H dependence (plotted as function of H2) of magnetoresistivity (MR) ∆ρa/ρa(H = 0) ≡ [ρa(H) − ρa(H = 0)]/ρa(H = 0)] of Ce0.93Yb0.07CoIn5 measured at two different temperatures and ambient pressure. The dashed line in the main figures marks Hmax, corresponding to the coherence giving way to single-ion Kondo behavior. Inset: MR data vs H2 measured under 5.1 kbar. The red line shows the quadratic regime of MR.

61 of the positive contribution to MR, hence Hmax = 0, at T ≈ Tcoh (red solid squares

in Fig. 3.4). On the other hand, as we have recently revealed [62], Hmax(T ) in the

Ce1−xYbxCoIn5 alloys with concentrations xact ≤ 0.07 shows deviation from the conventional Kondo behavior and exhibits a peak, below which Hmax decreases with decreasing temperature. This is shown in in Fig. 3.5, which is a plot of the temperature dependence of Hmax for four diﬀerent hydrostatic pressures. We have attributed the decrease in Hmax(T ) with decreasing T to quantum spin ﬂuctuations that dominate the MR behavior below about 20 K [62]. Notice that Hmax(T ) shows linear

behavior below 10 K (see Fig. 3.5). A linear extrapolation of this low T behavior to

zero temperature gives HQCP [62]. Notice that HQCP ≈ 0.2 T in Ce0.93Yb0.07CoIn5 at ambient pressure, as previously reported [62], showing that this Yb doping is close to the quantum critical value xc for the Ce1−xYbxCoIn5 alloys.

Three notable features are revealed by Fig. 3.5: (i) the application of pressure does not change qualitatively the Hmax(T ) dependence, (ii) there is no noticeable change in the value of HQCP with pressure for P ≤ 8.7 kbar, most likely because of the already

small value of HQCP (HQCP = 0.2 T) at ambient pressure, and (iii) both the value

of Hmax and the position in T of the Hmax(T ) peak shifts to higher temperatures

with increasing pressure; as a result, the slope dHmax/dT for T < 10 K increases with

pressure.

According to Doniach’s phase diagram [28], the Kondo temperature TK and the

magnetic exchange interaction temperature TRKKY of Ce Kondo lattice increase with

increasing pressure. Hence, the increase in Hmax with pressure is a result of increased

Tcoh, and the shift in the peak of Hmax(T ) to higher T with pressure is a result of the

increase of both TRKKY and Tcoh with pressure. The increase in the slope dHmax/dT

Ce Yb CoIn 10

0.93 0.07 5

8 (T)

max H

P (kbar)

2.7

5.1

7.4

0 5 10 15 20 25 30 35 40 45 50 55 60

T (K)

Figure 3.5: Temperature T dependence of the maximum in magnetoresistivity Hmax for diﬀerent pressures P . The solid lines below 10 K are linear ﬁts to the data.

63 with increasing pressure means that a larger applied ﬁeld is required to break the

Kondo singlet. We note that both quantum spin ﬂuctuations and applied magnetic

ﬁeld contribute to the breaking of Kondo coherence at temperatures T < 10 K. There-

fore, a larger dHmax/dT at higher pressures can be understood in terms of weaker

quantum spin ﬂuctuations since a larger ﬁeld is required to break the Kondo singlet

compared with the ﬁeld required for smaller dHmax/dT where spin ﬂuctuations are

stronger.

We show in Fig. 3.6 the inverse of this slope as a function of pressure, normalized

to its zero pressure value. We also show in the same ﬁgure (right vertical axis) the

residual resistivity as a function of pressure, also normalized to its zero pressure

value. Notice that these two quantities scale very well, indicating that the same

physics dominates their behavior with pressure, i.e., the suppression of quantum

critical ﬂuctuations with increasing pressure.

3.3 Conclusions

In this chapter, we studied the Ce0.93Yb0.07CoIn5 alloy (xnom = 0.2) using trans-

port and magnetotransport measurements under hydrostatic pressure. Our resistivity √ data reveal that the scattering close to Tc follows a T dependence, consistent with

the composite pairing theory in a 3D system [90] or with a ﬂuctuation correction, with

a coeﬃcient that decreases with increasing pressure. This latter result implies that

the scattering in this T range is largely governed by the heavy quasiparticles from the heavy Fermi surface, hence it may reﬂect the scattering of composite pairs [90] as a result of superconducting ﬂuctuations. At higher T , our data reveal the presence

of two scattering mechanisms: one linear in T with a coeﬃcient A that decreases √ with increasing pressure and the other one with a T dependence with a coeﬃcient

64 (

1.0 1.0 dH (P)

Ce Yb CoIn dT

0.93 0.07 5 ax m

0.9 0.9 ) -1 (P=0) ( a0 dH (P=0) dT ax m

0.8 0.8

(P)/ -1 dH

max

a0 ( (P))

dT )

0.7 0.7

a0 -1

0.6 0.6

0 2 4 6 8

P (kbar)

Figure 3.6: Pressure P dependence of residual resistivity ρa0 (obtained through the ﬁtting of the resistivity data as discussed in the text), normalized to its value at zero pressure (right vertical axis) and P dependence of inverse slope of Hmax(T ) normalized to its value at zero pressure (left vertical axis).

65 B that is pressure independent. Given that the strong pressure dependence of the

A parameter directly relates to the strongly hybridized conduction and cerium f-

electron states, we believe that the linear temperature dependence of the resistivity

is governed by the scattering of heavy-quasiparticles, while the scattering processes √ leading to the T -term in resistivity are governed by the scattering of light electrons

from the small Fermi surface. Since the linear T dependence is a result of quantum

spin ﬂuctuations, the decrease of A with increasing pressure implies that quantum

ﬂuctuations are suppressed with pressure. This conclusion is conﬁrmed by the fact

that residual resistivity also decreases with pressure.

We also performed magnetoresistivity measurements under applied hydrostatic

pressure in order to study the evolution of quantum critical spin ﬂuctuations with

pressure. First, our magnetoresistivity data reveal that this Ce0.93Yb0.07CoIn5 alloy is close to the quantum critical value xc for the Ce1−xYbxCoIn5 alloys. Second, these data conﬁrm our ﬁndings from resistivity measurements that quantum critical

fluctuations are suppressed with increasing pressure. Finally, we also analyzed the temperature and pressure dependence of the magnetic field Hmax at which the MR reaches its maximum value. At low temperatures, Hmax grows linearly with temperature. Interestingly, we find that the slope dHmax/dT also grows with applied pressure, similar to the pressure dependence of the coherence temperature. This result suggests that the MR is largely governed by the heavy-electrons from the large Fermi surface.

66 CHAPTER 4

Quantum Criticality and Gap Structure in Ce1−xYbxCoIn5

4.1 Introduction

As discussed in Chapter 1, one of the most interesting problems in the physics of f-electron materials is their quantum critical behavior. Quantum critical behavior in these materials results from a zero-temperature transition, from a magnetically ordered to a disordered phase, resulting in a quantum critical point (QCP) in a

T − δ phase diagram (‘δ’ is a tuning parameter). The fluctuations originating from the resulting QCP severely affect the finite-temperature properties of the material in the vicinity of QCP. Importantly, heavy-fermion materials exhibit deviation from their Fermi-liquid properties in presence of a QCP. For many of the heavy-fermion materials, these fluctuations are also believed to play a supporting role in the super- conductive pairing of electrons.

Heavy-fermion material Ce1−xYbxCoIn5 has recently emerged as a remarkable system. Many of the properties observed in this material do not conform to those of similar heavy-fermion compounds, and have led the way to the proposals of new, exciting possibilities (e.g., composite pairing (CP) mechanism and topological superconductivity [97], two different Fermi surfaces contributing in transport [98], and doping dependent gap character [71]). The fact that Yb ions act as the hole coun- terpart of Ce ions presents a unique scenario where a recent theoretical proposal suggests a significant modification in the Fermi surface of Ce1−xYbxCoIn5 with Yb doping. Due to this change in Fermi surface topology, the superconducting order

67 parameter changes from nodal to nodeless at a critical doping level around x = 0.037.

On the other hand, our previous ﬁndings suggest that there is a critical doping in the vicinity of x ≈ 0.07 that exhibits zero-ﬁeld quantum criticality without any further tuning.

The gap symmetry in Ce-based heavy-fermion superconductors is generally believed to be nodal. However, the recent proposals of nodeless gap symmetry in

CeCu2Si2 [99] and a doping-dependent change from a nodal to nodeless nature of gap symmetry in Ce1−xYbxCoIn5 [71] calls for a fresh look on the superconductivity in these systems. One interesting question in the case of Ce1−xYbxCoIn5 is the relationship between the quantum spin ﬂuctuations and the nodal structure of the superconducting gap. In many cases the spin ﬂuctuations lead to the formation of a nodal gap [100].

The goal of this project is to determine the critical doping corresponding to the zero-ﬁeld quantum critical point. This will allow us to probe the relative evolution of quantum criticality and nodal gap structure in Ce1−xYbxCoIn5 as the Ce Kondo lattice is diluted with Yb substitution.

This study was performed on single crystals of CeCoIn5, with Yb doping x = 0.09 and x = 0.16. We performed resistivity (ρ) measurements between 0.50 and 15 K and transverse (H ⊥ ab) magnetoresistivity (MR) measurements as a function of temperature between 2 and 300 K, and applied magnetic ﬁeld up to 14 T. A zero ﬁeld resistivity measurement was also performed in the temperature range 1.8 to 300 K.

The heat capacity measurements were performed under magnetic ﬁelds up to 14 T, applied parallel to the c axis (H k c) of the crystals and for temperatures as low as

0.50 K.

68 4.2 Experimental results and discussion

Heavy-fermion materials exhibit a characteristic resistivity behavior in which a logarithmic increase is observed due to the Kondo eﬀect below a certain temperature.

In Kondo-lattice materials, the increase in resistivity is followed by a sharp drop with decreasing temperature due to the development of coherence eﬀects. Consequently, a resistivity peak is observed in Kondo-lattice materials at a temperature called the coherence temperature (Tcoh). Figure 4.1 shows resistivity ρ(T ) data as a function of temperature for single crystals of Ce1−xYbxCoIn5 with doping levels x = 0.09 and

0.16. The data shown is measured in the temperature range 2 K ≤ T ≤ 300 K and is normalized to the values at 300 K. The peak in the resistivity data is attributed to the development of coherence eﬀect at temperatures below Tcoh. No superconducting transition in resistivity is observed for the x = 0.16 crystal in the measured temperature range, while the x = 0.09 crystal shows a clear superconducting transition at

Tc ≈ 1.9 K. Although the qualitative behavior of the resistivity data is reported to be the same in the present case and elsewhere [62, 64, 98], an in-depth analysis of the x = 0.09 crystal provides a great deal of information regarding the physics of the system that evolves with doping.

The inset to Fig. 4.1 shows the suppression of Tc with magnetic field for the x = 0.09 crystal. As can be seen, a field of H = 5 T suppresses Tc below 0.5 K and exposes the normal state of the material. Based on the resistivity data, the value of the upper critical field Hc2 is about 5 T.

Although no extraordinary feature is observed in the low-temperature resistivity data (inset to Fig. 4.1), application of higher ﬁelds results in the appearance of a slight curvature in resistivity at lowest temperatures as seen in Fig. 4.2(a). Based on

69 Figure 4.1: (a) Normalized resistivity ρ(T )/ρ(300 K) of Ce1−xYbxCoIn5 (x = 0.09 and 0.16) as a function of temperature T . The arrow at the maximum of the resistivity data marks the coherence temperature Tcoh. Inset: magnetic ﬁeld suppression of superconducting transition in the x = 0.09 crystal, shown in the resistance.

70 Figure 4.2: (a) Temperature T dependence of the resistance of x = 0.09 single crystal measured in the temperature range 0.50 K ≤ T ≤ 30 K, and under the application of magnetic fields 10 T ≤ H ≤ 14 T. (b) The low temperature data of (a) plotted as a function of T 2. The solid lines are linear fits to the data at the lowest temperatures, and the arrows indicate the point where the data deviate from the straight line fits. the previous findings in this material, we conclude this low temperature behavior of resistivity to be an indication of the system’s recovery to the Fermi-liquid state as a result of the application of external magnetic field. Indeed, when plotted as a function of T 2, a low temperature Fermi-liquid resistivity (the data lying on the straight lines in Fig. 4.2(b)) is observed in narrow temperature ranges for different high magnetic

ﬁeld values. The deviation from the FL behavior in resistivity occurs at temperatures

(TFL) which vary with ﬁeld (see arrows in Fig. 4.2(b)).

In heavy fermion compounds, one of the possible reasons for a breakdown of Fermi

71 liquid behavior is the presence of a quantum critical point (QCP) located at absolute

zero temperature in the δ − T phase diagram (‘δ’ being a tuning parameter). At the

QCP, the material crosses from the FL to NFL behavior as a function of δ. Exposing

the normal state of heavy fermion superconductors by applying external magnetic

ﬁeld is one way to access the low temperature FL state. Thus the presence of a FL

resistivity at low temperatures in x = 0.09 crystal suggests the vicinity of this doping

level to a QCP.

Further conﬁrmation to this observation is obtained through the magnetoresis-

tance (MR) measurements performed on single crystals of x = 0.09 and x = 0.16. For

Kondo systems, magnetoresistivity behavior is quite well explained, and is primarily

governed by two competing energy scales, i.e., the Kondo temperature TK (favoring

a non-magnetic coherent state) and the Ruderman-Kittel-Kasuya-Yosida (RKKY)

scale (favoring a magnetic ground state). In our group’s recent works [62, 64, 98],

we have discussed that temperature dependence of MR peak can be used to obtain

information regarding the nature of the ground state in HF materials. As shown in

Fig. 4.3, the ﬁeld dependence of the magnetoresistivity for the x = 0.09 single crys-

tal shows deviations from that of the x = 0.16, particularly in the low-ﬁeld region.

Although the high-ﬁeld MR is quadratic and negative in ﬁeld for both dopings, the

x = 0.09 crystal exhibits a low-ﬁeld positive contribution to the magnetoresistivity.

Although the positive contribution in low ﬁeld can be attributed to the development

of coherence eﬀect, as discussed in Chapter 3, the temperature dependence of the MR

peak Hmax in Ce1−xYbxCoIn5 compounds provides convincing evidence of the pres-

ence of a QCP. In the present case (for the x = 0.09 crystals), Hmax also decreases

72 Figure 4.3: Magnetoresistivity ∆ρ/ρ0 ≡ [ρ(H) − ρ(H = 0)]/ρ(H = 0)] as a function of transverse applied ﬁeld H for (a) Ce0.91Yb0.09CoIn5 and (b) Ce0.84Yb0.16CoIn5, measured at diﬀerent temperatures T .

73 with decreasing temperature, thereby suggesting the presence of an additional energy contribution which gets stronger in the limit of T → 0. As has been discussed, this contribution comes from the strong quantum spin ﬂuctuations emanating from a QCP. Absence of any such feature in the magnetoresistance of the x = 0.16 crystal

(see Fig. 4.3(b)) indicates that this doping lies away from the QCP. The lack of any positive component also shows that the coherence is not as strong for this doping, which corroborates the resistivity data of Fig. 4.1, showing a lower Tcoh and higher resistivity below Tcoh.

As stated above, specific heat measurements were also performed on the x = 0.09 crystal, shown in Fig. 4.4. Measurement of specific heat in heavy fermion materials can provide strong evidence for the presence of quantum spin fluctuations. In the presence of these fluctuations, the standard Fermi liquid theory (which suggests a constant value of specific heat at low temperatures) breaks down and a logarithmic- in-T , non-Fermi-liquid (NFL) behavior is observed.

As can be seen in Fig. 4.4, the superconducting transition is gradually suppressed with magnetic field and approaches 0.5 K for a field of 4.5 T. With the further application of field (see Fig. 4.5), a region of saturation (indicating the Fermi liquid state) starts appearing in the electronic contribution to specific heat at low temperature. We note that for H > 10 T, a low-temperature upturn masks the Fermi liquid state. This upturn follows a 1/T 3 nuclear Schottky contribution. In order to fully reveal the Fermi liquid region, we subtracted the Schottky contribution obtained by

3 fitting the low temperature data with CSchottky/T ∝ 1/T . In addition, the NFL state in specific heat exhibits a logarithmic-in-T behavior in the temperature range of 2 K . T . 10 K, as shown in Fig. 4.5. We define the temperature where Ce/T

74 Figure 4.4: Heat capacity C/T vs. temperature T of Ce0.91Yb0.09CoIn5 measured with magnetic ﬁeld H k c over the temperature range 0.50 K ≤ 10 K and applied magnetic ﬁelds 0 ≤ H ≤ 4.5 T.

75 changes from the Fermi liquid region to the logarithmic-in-T NFL state as TF L,HC , and we can extrapolate this to zero temperature in order to determine HQCP and use it to extract the value of the quantum critical field HQCP , as discussed below. In contrast, for the x = 0.16 single crystals, the NFL state is seen in the heat capacity data over the entire temperature and magnetic field range studied, i. e., for temperatures as low as 0.5 K and magnetic fields up to 14 T.

The resistivity, the magnetoresistivity, and the heat capacity data for x = 0.09

single crystals all provide clear signatures of a low temperature Fermi liquid state.

The high temperature (T > TFL) behavior in resistivity [62] and the logarithmic-in-T specific heat suggests that the breakdown of the FL behavior is due to the system being in the vicinity of a quantum critical point. With these clear indications of the x = 0.09 doping level exhibiting quantum critical behavior established, we obtain the value of the quantum critical field corresponding to x = 0.09. In order to extract the value of the quantum critical field HQCP , the temperature corresponding to the recovery of the Fermi liquid state (TFL) is extracted for different field values. The linear extrapolation of the H-dependence of TFL to 0 K into the superconducting phase

2 provides the value of HQCP . TF L,ρ marks the onset of low-temperature T behavior, and is indicated by the arrows in Fig. 4.2(b). This same temperature extracted from the speciﬁc heat data from the point where a saturation is observed, as discussed above, is TF L,HC . Finally, the temperature dependence of MR peak (Hmax) can also

be extrapolated to 0 K to obtain the estimate for the value of HQCP , as discussed in

detail in Chapter 3.

Figure 4.6 shows the H − T phase diagram obtained as just discussed. The superconducting boundary is generated from the SC transitions observed in resistivity

76 Figure 4.5: Electronic contribution to heat capacity Ce/T vs. temperature T of Ce0.91Yb0.09CoIn5 measured with applied magnetic ﬁeld H k c over the temperature range 0.50 K ≤ 10 K and 10 T ≤ H ≤ 14 T. A 1/T 3 nuclear Schottky contribution has been subtracted from the data. The Fermi liquid temperature TFL is extracted as the temperature at which the data cross from logarithmic temperature dependence to no temperature dependence with decreasing temperature.

77 Figure 4.6: H −T phase diagram of Ce0.91Yb0.09CoIn5 with magnetic ﬁeld H k c-axis. The area under dotted line represents the superconducting region. The straight lines are the linear ﬁt of the data extrapolated to zero temperature. Hmax is the peak in MR. TF L,ρ and TF L,HC are the temperature at which the data cross from a Fermi liquid to a non-Fermi liquid, measured by resistivity and heat capacity, respectively. Likewise, Tc,ρ and Tc,HC are the superconducting transition temperature measured by resistivity and heat capacity, respectively.

78 and heat capacity data. As can be clearly seen in Fig. 4.6, the linear extrapolation

of the TFL’s and Hmax provide a value of 0 T for HQCP , suggesting that the x = 0.09

doping level is quantum critical.

Recent penetration depth measurements on Ce1−xYbxCoIn5 establish that the su-

perconducting gap changes its character from nodal to nodeless with increasing Yb

doping [71]. In fact, it was observed that this change starts at the x = 0.015 dop-

ing level, and for doping x ≥ 0.037, the system exhibits a nodeless superconducting gap. In view of these reports, our current findings suggest that, indeed, the nodal gap structure in this material is a result of quantum spin fluctuations. Our group’s previous findings [62] suggest that HQCP decreases with increasing x and is fully

suppressed at x = 0.09.

4.3 Conclusions

In this chapter, we performed transport and thermodynamic studies of the heavy-

fermion superconducting alloys Ce1−xYbxCoIn5 focusing on Yb concentrations x = 0.09 and 0.16. Resistivity, magnetoresistivity, and specific heat measurements all indicate the presence of a zero-field quantum critical point at x = 0.09 Yb doping. Therefore, x = 0.09 is a critical doping for this system. These results are consistent with our previous findings [62] that HQCP decreases with increasing Yb doping and reaches zero just above x = 0.07. Also, they are consistent with penetration depth measurements revealing a reconstruction of the Fermi surface and a change of the superconducting gap changes from a nodal (x ≤ 0.09) to a nodeless (x ≥ 0.09) structure.

79 CHAPTER 5

Magnetoresistivity Study of the Kondo Impurity to Kondo Lattice

Crossover in Ce1−xYbxCoIn5

5.1 Introduction

Heavy fermion systems exhibit a characteristic Kondo coherence state which is established when the high temperature incoherent scattering of electrons gives way to a low temperature Fermi liquid regime. It has been shown in multiple heavy fermion systems that the high temperature incoherent scattering is similar to the case of the typical single-impurity Kondo model despite the presence of local moments on each lattice site. On the other hand, the coherent state is an inherent property of heavy fermion materials and presents itself in the form of markedly diﬀerent physical properties of the material compared to the Kondo impurity regime. For example, there is a substantial drop in electrical resistance and a positive ﬁeld dependence of the magnetoresistivity (MR).

Extensive theoretical and experimental investigations on heavy fermion systems over the years have contributed to the understanding of the coherent state in great detail. Despite the fundamental diﬀerence in lattice make-up between heavy fermion systems and conventional Kondo systems (that is, a lattice of magnetic ions vs. magnetic impurities in a lattice of nonmagnetic ions), the Kondo impurity description holds very well for heavy fermion systems when they are in the non-interacting, single-ion regime at high temperatures and high ﬁelds. However, the crossover from the incoherent to the coherent scattering state is a gradual transition and can spread

80 over a wide range of tuning parameter values. This large window of transition lacks

any particular theoretical or experimental description due to the complexity in nature

of interaction between the Kondo scale (TK ) and the Ruderman-Kittel-Kasuya-Yosida

(RKKY) interactions (TRKKY ), which start dominating at lower temperatures.

In this chapter, we provide a detailed description of the crossover region in going

from the incoherent scattering state to the coherent Kondo regime in heavy fermion

system Ce1−xYbxCoIn5. For this study, we will treat the heavy fermion system as a superposition of Kondo impurity behavior and Kondo lattice behavior, with one dominating over the other based on favorable conditions. The MR behavior in both of the limiting cases has been well studied and understood. We restrict our discussion here to the MR measurements performed for diﬀerent dopings and under applied hydrostatic pressure.

As described previously in this work, published studies on Ce1−xYbxCoIn5 have found that this system exhibits quantum criticality for Yb doping in the range of 0 ≤ x ≤ 0.09 for diﬀerent conditions of pressure and magnetic ﬁeld. In the intermediate doping range, i. e., beyond x = 0.09, the system is found to always be in the incoherent scattering regime, as though it were composed of non-interacting Kondo singlets. In this chapter, we utilize doping levels that exhibit quantum criticality as well as those that do not.

5.2 Experimental results and discussion

We ﬁrst discuss the parent compound CeCoIn5. In the absence of any doping, it is a Kondo lattice system and enters the coherence state below a temperature of

Tcoh ≈ 46 K. Figure 5.1 shows the typical MR behavior of CeCoIn5 for temperatures less than Tcoh. The overall transverse magnetoresistivity (with H ⊥ ab), deﬁned

81 as ∆ρ/ρ(0) ≡ [ρ(H) − ρ(H = 0)]/ρ(H = 0)], shows a positive component due to the formation of the coherent 4f band below Tcoh = 46 K. For higher magnetic ﬁelds the magnetoresistivity decreases and ultimately becomes negative and quadratic in

field, presumably due to the destruction of coherence. This observation finds further support in the fact that for temperatures T > Tcoh, we only see the negative, quadratic-in-field MR. The magnetoresistivity of CeCoIn5 can be separated into two components superimposed on each other as shown in Fig. 5.1. Thus, the magnetoresistivity can be expressed as

∆ρ/ρ0 = (∆ρ/ρ0)pos + (∆ρ/ρ0)neg (5.1)

The two components of the magnetoresistivity were determined as follows. The negative, quadratic component (∆ρ/ρ0)neg was determined by plotting the magnetoresistivity against H2 and determining its slope at higher ﬁelds, where single-ion eﬀects dominate. Then, (∆ρ/ρ0)neg was subtracted from the measured magnetoresistivity in order to obtain the positive component (∆ρ/ρ0)pos. As we discuss below, these two components of the MR represent the two extreme behaviors in heavy fermion compounds, i.e., the single-impurity Kondo behavior in (∆ρ/ρ0)neg and the Kondo coherence behavior in (∆ρ/ρ0)pos.

The quadratic magnetic ﬁeld dependence of (∆ρ/ρ0)neg is typical of a single-ion

Kondo system, in which a magnetic Kondo impurity in a nonmagnetic lattice is screened by the conduction electron spin. An analogous description for the CeCoIn5 system, with a lattice of localized f-moments, holds for high magnetic ﬁelds or high temperatures (T > Tcoh) where the coherence breaks down and the lattice sites are left with non-interacting Kondo singlets.

82 2 Figure 5.1: Magnetoresistivity of CeCoIn5 at T = 10 K plotted against H . The black line is the measured magnetoresistivity. The blue line is the negative component, calculated as the slope of the high-ﬁeld linear portion. The red line is the positive component, calculated by subtracting the negative curve from the physically measured data. The dashed line shows the value of (∆ρ/ρ0)sat.

83 The positive contribution (∆ρ/ρ0)pos shows an initial, rapid increase with increas-

ing magnetic ﬁeld followed by a saturation. This behavior is typical of many metallic

systems. With the formation of a complete coherent state, the heavy fermion mate-

rials enter into the Fermi liquid regime. In such a case, these materials are basically

metallic with very high eﬀective mass. Thus, similar to conventional metals, the ini-

tial increase in positive MR is the result of the fact that increasing ﬁeld increases

the cyclotron frequency of electronic orbits. Once the maximum possible value of

cyclotron frequency is achieved, no further increase in MR is seen [101]. It should

be mentioned that for the Ce1−xYbxCoIn5 in the doping range 0 ≤ x ≤ 0.09, a qualitatively similar MR behavior is observed, while for x ≥ 0.16 only (∆ρ/ρ0)neg is

observed.

From here, we will examine the positive and negative contributions to the overall

MR in more detail. As stated above, its negative component is a manifestation of iso-

lated magnetic impurities, which we are able to reveal by applying ﬁeld. Schlottmann

gave an exact solution for the ﬁeld dependence of the magnetoresistivity in such a

single-ion system [34, 35, 36], which varies as approximately H2. He also showed that,

∗ if plotted as ∆ρ/ρ0 vs. H/H , where

∗ ∗ kB H = H (T ) = (T + TK ) (5.2) gµB

(with TK being the Kondo temperature), the magnetoresistivity curves measured at

diﬀerent temperatures scale.

We investigated the magnetoresistivity of Ce1−xYbxCoIn5 according to Schlottmann’s

theory. This was done by plotting ∆ρ/ρ0 (or just (∆ρ/ρ0)neg, for 0 ≤ x ≤ 0.09) against H/H∗, where H∗ was chosen so that the magnetoresistivity curves would scale, as seen in Fig. 5.2(a). Furthermore, a plot of H∗ vs. T revealed a linear H∗(T )

84 dependence, as expected based on Schlottmann’s theory. This scaling shows that the

negative component of the magnetoresistivity is consistent with the single-ion theory

as surmised. A linear ﬁt of these data give a slope and intercept. Dividing H∗ by

∗ the slope (kB/gµB) allowed us to plot gµBH /kB = T + TK (see Fig. 5.2(b)), thus experimentally determining the Kondo temperature TK , shown in Fig. 5.2(c).

Because we were able to apply Schlottmann’s theory at these lower dopings using the negative MR component (∆ρ/ρ0)neg, we could use Schlottmann’s theory to determine the Kondo temperature across the entire doping range, as is shown in Fig. 5.2(c).

Notice that TK increases with increasing doping. It appears to be quite low for dopings exhibiting quantum criticality (x ≤ 0.09), increasing with doping from 3.45 K for x = 0 to approximately 6 K at x = 0.07 and x = 0.09. After the quantum critical doping of x = 0.09, TK increases rather sharply above 10 K at x = 0.16, where it remains relatively constant up to x = 0.46. The jump in TK is also visible in Fig. 5.2(b), where the low dopings and the higher dopings are separated into two distinct groups.

Such variations in TK can be understood on the basis of a large Grüneisenratio, which makes TK very sensisitive to the configuration of nearest neighbor ligands around the f-ion [96]. In particular, for heavy fermion systems, the Grüneisenratio diverges in the vicinity of a quantum critical point [102]. Considering this, the lower values of TK for 0 ≤ x ≤ 0.09 can be attributed to a very similar surrounding of the

Ce ions in a dense Kondo lattice (except for slight dilutions by Yb ions). On the other hand, for x > 0.09 the system is in a more dilute Kondo lattice scenario and we see larger TK values that are nearly doping-independent.

Until now, we have focused only on the single-ion Kondo contribution to heavy fermion systems and have shown that despite having a lattice of local moments, heavy

85 Figure 5.2: (a) The negative quadratic component of the magnetoresistivity ∆ρ/ρ0 ∗ ∗ kB ((∆ρ/ρ0)neg) of CeCoIn5, scaled as normalized field H/H , where H = (T + TK ) gµB ∗ and TK = 3.45 K. (b) A plot of H (with coefficients) versus temperature T . The lines are all linear fits, with a unit slope and an intercept at Kondo temperature TK . (c) TK as a function of x.

86 fermion materials still exhibit isolated Kondo singlets for temperatures T > Tcoh, and their transport properties are governed by the incoherent scattering of electrons.

Electronic transport in this regime is further affected by the presence of quantum criticality in Ce1−xYbxCoIn5 system because it affects the Kondo scale differently for doping with and without quantum criticality.

We next turn our attention to the positive component of the MR. Figure 5.3 shows the saturation value (∆ρ/ρ0)sat of the positive MR component plotted as a function of temperature for different dopings. (∆ρ/ρ0)sat represents the amount by which the resistivity increases as the field is applied. (∆ρ/ρ0)sat is only observed in the lowest dopings (x ≤ 0.09) because only these dopings have positive MR. In all three dopings shown, (∆ρ/ρ0)sat appears at approximately Tcoh. At first, it is very weak, indicating that the coherence-related metallic band does not form very strongly at high temperatures. Then, it increases with decreasing temperature. Interestingly, for x = 0.03 and x = 0.07, this trend changes below a certain temperature, decreasing with decreasing temperature, forming a maximum in (∆ρ/ρ0)sat, whereas no such

maximum is observed in CeCoIn5. It should be noted that even though a maximum

is not observed in (∆ρ/ρ0)sat for CeCoIn5, a shoulder is visible at lowest temperature

measured, slowing the rapid increase of (∆ρ/ρ0)sat with decreasing temperature.

The trends in (∆ρ/ρ0)sat could be the result of a combined eﬀect of quantum

criticality and Kondo coherence. First notice that the (∆ρ/ρ0)sat values approach zero

around Tcoh ≈ 45 K for all three dopings. This is evidence that the saturation values

are the result of a metal-like Fermi liquid state formed at low temperatures due to the

coherence eﬀects. Next notice that the parent compound, CeCoIn5, shows the largest

(∆ρ/ρ0)sat, increasing by as much as 36% at the lowest temperatures. This could be

87 Figure 5.3: (∆ρ/ρ0)sat for diﬀerent dopings of Ce1−xYbxCoIn5 plotted versus temperature T . (∆ρ/ρ0)sat is the value at which the magnetoresistivity (measured as percentage change in resistivity with ﬁeld) saturates.

88 understood as the result of the strongest quantum critical ﬂuctuations present in the

parent compound. The quantum critical ﬁeld for the doped samples is progressively

suppressed to 0 T for x = 0.09, and the quantum ﬂuctuations are consequently also

suppressed. In fact, above T = 20 K, there is no signiﬁcant diﬀerence in (∆ρ/ρ0)sat

for all three dopings, indicating that the eﬀect of quantum ﬂuctuations are limited to

T ≤ 20 K irrespective of the doping level. This observation is in line with our group’s

previous report where we have shown in electrical- and magneto-transport that this

indeed is the case [62].

Next, we consider the low temperature range (below the peak) for values of

(∆ρ/ρ0)sat in the doped samples. It is evident that the quantum ﬂuctuations get stronger as the temperature decreases and the system gets closer to the zero-T transition. This results in increased resistivity, and we should see monotonically increasing resistivity with decreasing temperature. However, for a metallic system the saturation value of positive MR depends on the conductivity according to the following relation

[101]:

ωcτ = Bσ0/ne (5.3)

Increasing the applied magnetic ﬁeld increases the cyclotron frequency to the max-

imum value shown by the equation above. Beyond this value no signiﬁcant MR is

observed unless ωcτ > 1. Owing to large scattering at lower temperatures, the σ0

value decreases signiﬁcantly making ωcτ << 1 and (∆ρ/ρ0)sat very small.

Next, we performed the MR measurements under pressure on Ce0.93Yb0.07CoIn5

sample. The maximum applied pressure was 7.4 kbar, and the magnetoresistivity

was again separated into positive and negative components for each pressure. As

above, the Schlottmann analysis was performed on each doping. Pressure does not

89 change the validity of the analysis: with each pressure, (∆ρ/ρ0)neg continues to be quadratic and negative, and scales according to Schlottmann’s theory, exhibiting a

∗ linear relationship in H (T ) that can be used to measure TK . Figure 5.4 shows the results of this analysis. Figure 5.4(a) shows H∗ plotted as a function temperature and Fig. 5.4(b) shows the resulting TK . Pressure does not have a strong eﬀect on TK : it only increases it by 5% when pressure increases from 0 to 7.4 kbar.

Likewise, pressure has a very small eﬀect on value of (∆ρ/ρ0)sat, as shown in

Fig. 5.5, which is a plot of (∆ρ/ρ0)sat for Ce0.93Yb0.07CoIn5 for diﬀerent pressures.

However, a marked increase is observed in the values of the coherence temperature with increasing pressure, as is evident by the high temperature tails in Fig. 5.5 and previous pressure measurements in Ce0.93Yb0.07CoIn5 (see Fig. 3.1(b)).

We conclude that, since there is no change in (∆ρ/ρ0)sat at low temperatures but there are significant changes at high temperatures, the pressure significantly affects the coherence in HF system. We expected that pressure would have an effect on quantum fluctuations too, but since the fluctuations for x = 0.07 doping level are almost suppressed, the magnitude of the effect of the pressure is very small on the quantum fluctuations, as discussed in Chapter 3.

These measurements show that Tcoh increases with pressure while TK is pressure independent. Also, the investigation on each of the doping levels shows that TK increases with doping, while Tcoh decreases with doping. The natural conclusion is that TK and Tcoh are independent of each other. This might seem to be a surprising result, since coherent scattering is closely related to the Kondo eﬀect, but we think that no direct relationship between TK and Tcoh is observed in the present case due to the complexity of interactions introduced by RKKY and quantum criticality at

90 ∗ ∗ Figure 5.4: (a) A plot of H (with coeﬃcients) versus T , where H scales (∆ρ/ρ0)neg ∗ ∗ kB according to H/H and H = (T + TK ) . The lines are all linear ﬁts, with a unit gµB slope and an intercept at Kondo temperature TK . Performed over a pressure range of 0 kbar ≤ P ≤ 7.4 kbar. (b) TK , plotted as a function of P . The line is a guide to the eye.

91 Figure 5.5: (∆ρ/ρ0)sat for Ce0.93Yb0.07CoIn5 for pressures 0 kbar ≤ P < 7.4 kbar plotted versus temperature T . (∆ρ/ρ0)sat is the value at which the positive component of the magnetoresistivity (∆ρ/ρ0)pos saturates.

92 lower temperatures. The insensitivity of TK to pressure might be due to the fact that unlike chemical substitution, small pressure does not signiﬁcantly aﬀect the f- electrons’ surroundings in the present case.

5.3 Conclusions

In this chapter, we have performed magnetotransport studies under ambient and small applied pressure on the heavy-fermion superconducting alloys Ce1−xYbxCoIn5 with 0 ≤ x ≤ 0.46. We demonstrated that the heavy fermion transport properties in the crossover region between the isolated Kondo singlet regime and the coherent

Kondo lattice regime can be well explained as a superposition of these two extremes.

We suggest a MR based method to separate these two regimes which helps in providing a better understanding of the overall behavior of these systems. In lack of a proper theoretical description of the crossover regime, our method is a simple approach to address important questions regarding quantum criticality and Fermi liquid state in these materials. The ﬁndings in this chapter provide excellent agreement with the pressure measurements done on Ce0.93Yb0.07CoIn5 in Chapter 3.

93 CHAPTER 6

Summary and Outlook

The motivation of this body of research was to investigate the quantum critical nature of the superconducting, heavy fermion family of Ce1−xYbxCoIn5. The parent material, CeCoIn5, has a superconducting transition temperature of Tc = 2.3 K, which is uncharacteristically high for a heavy fermion material.

Doping ytterbium on the Ce site provides a unique study case. With the ex- ception of Yb, doping rare-earth metals on the Ce site produces consistent results no matter which rare-earth is used, whether they be magnetic or nonmagnetic.

Ce1−xYbxCoIn5 is particularly robust to dopant concentration, displaying superconductivity and Kondo coherence up to at least x = 0.58 (xnom = 0.775). In fact, recent measurements show that coherence strengthens at high dopings and a second quantum critical doping appears at about x = 0.54. It also displays a non-Fermi liquid

behavior throughout the doping range, even in the intermediate doping range where

the material is not quantum critical.

This uniqueness of Yb could be attributed to the fact that it is in a mixed valence

state throughout the doping range. At extremely dilute concentrations, it exhibits a

valence of +3, but increased doping decreases this valence until about x = 0.07, after

which it remains constant at +2.3 for x > 0.07.

In the course of this research, we investigated the quantum critical behavior of

94 heavy fermion superconductor Ce1−xYbxCoIn5 with variable Yb concentration, pressure, and magnetic ﬁeld. Our primary method of analysis was in the magnetoresistivity, in which we applied a method developed by our research group that connects the magnetoresistivity to the system’s quantum criticality. At any temperature below

Tcoh, the magnetoresistivity shows a nonmonotonic field dependence, with low fields corresponding to a coherent state, and high fields corresponding to a noncoherent,

dρ single-ion Kondo regime. A primary value of interest is Hmax, where dH moves from positive to negative, indicating a transition from the coherent state to the single-ion

state. Extrapolating Hmax(T ) to zero allows us to determine the location of the quan-

tum critical ﬁeld HQCP . Analyzing Hmax also gives us insight into the nature of the

energy scales TK and TRKKY .

For T . Tcoh/2, the resistivity of Ce1−xYbxCoIn5 is closely modeled by √ ρa(x, T ) = ρa0(x) + A(x)T + B(x) T. (6.1)

One of the open questions before these measurements were performed was that of √ √ the origin of the T component. This T dependence is not present in the parent

compound, CeCoIn5, but appears with Yb doping or with application of ﬁeld H, as

shown by Hu, et al. [62]. In applying pressure to Ce0.93Yb0.07CoIn5, we found that √ the coeﬃcient of this T component is pressure independent. As such, we concluded

that it is tied to the scattering of light quasiparticles from the smaller of the two

Fermi surfaces that coexist in the stoichiometric CeCoIn5.

Conversely, the linear contribution, originating from the proximity of the system

to a quantum critical point, is strongly pressure dependent, indicating that it is

tied to the heavy quasiparticles from the larger Fermi surface. These particles are

heavy because they belong to the hybridized f- and d-orbitals, and they are pressure

95 dependent because this hybridization involves quantum mechanical tunneling between

the f 0 and f 1 valence states, changing the volume of the Ce ions.

Our pressure measurements also showed that the quantum critical ﬂuctuations

are suppressed with pressure in Ce0.93Yb0.07CoIn5. This was seen ﬁrst in the residual

resistivity. As pressure increases, the residual resistivity decreases. Since application

of pressure does not remove impurities from the system, we infer that quantum crit-

ical ﬂuctuations at zero temperature are suppressed. Likewise, the linear coeﬃcient

to ρ(T ), which is tied to the quantum critical ﬂuctuations, decreases with temper-

ature, further supporting the conclusion that the quantum critical ﬂuctuations are

suppressed. The magnetoresistivity also supports this conclusion that pressure moves

the system away from quantum criticality in that it shows that a larger ﬁeld is required

to break the Kondo singlet.

Measurements in another doping, Ce0.91Yb0.09CoIn5, show that x = 0.09 is a quantum critical doping, with a zero-field quantum critical point. This is confirmed in three different analyses: (1) linear extrapolation of Hmax to T = 0, (2) linear extrapolation of TF L,ρ to T = 0, where TF L,ρ is the temperature at which the system transitions to a

Fermi liquid state to a non-Fermi liquid state, measured as the temperature where the resistivity moves away from a T 2 temperature dependence, and (3) an independent measurement and linear extrapolation of TF L,HC to T = 0, where TF L,HC is measured using heat capacity measurements, deﬁned as the temperature at which C/T moves away from a Fermi liquid saturated state.

Measurements on a sample of higher Yb concentration (x = 0.16) show no signs of

quantum criticality and weak coherence. There is no peak in magnetoresistivity, and it

is well modeled by Schlottmann’s theory. That is, it exhibits negative, quadratic-in-H

96 magnetoresistivity throughout the measurable temperature range, which is consistent

with the behavior of a single-ion Kondo system. Despite being removed from quantum

critical behavior, Ce0.84Yb0.16CoIn5 still exhibits non-Fermi liquid behavior at low √ temperature with ρ(T ) ∝ T .

Finally, performing an in-depth study on the two main components of the ﬁeld de-

pendence magnetoresistivity reveal one that is dependent on the coherence-related or-

der and another that is dependent on the formation of isolated Kondo singlets. These

measurements allowed us to determine the Kondo temperature TK of the material.

The measurements show that TK increases with doping and is relatively pressure in-

dependent (up to P = 7.4 kbar). These show that in Ce1−xYbxCoIn5, TK and Tcoh are independent temperature scales, which is likely due to the complicating interactions contributed by the RKKY interaction and the quantum critical ﬂuctuations.

It would be beneﬁcial to repeat the measurements at higher pressures in order to determine if a higher pressure would be capable of altering the physics in a more signiﬁcant way. For example, this pressure range was incapable of driving the near- quantum-critical doping of Ce0.93Yb0.07CoIn5 away from quantum criticality, but it is unknown if this is because the pressure is not a driving factor in the quantum criticality of this particular doping or if the pressure was simply not large enough. We would have liked to perform these higher pressure measurements during this particular research cycle, but repeated pressurization failures prevented us from doing so.

Furthermore, it would be instructive to perform pressurized magnetoresistivity measurement across a variety of dopings in the Ce1−xYbxCoIn5 family, both above and below the quantum critical doping of Ce0.91Yb0.09CoIn5. Measurements by Ronning, et al. [51], indicate that pressure suppresses quantum criticality in CeCoIn5, while

97 pressure seems to have little eﬀect on the quantum critical point in Ce0.93Yb0.07CoIn5, and pressure measurements on the newly discovered reentrant quantum critical doping of x=0.54 indicate that they may actually strengthen quantum critical behavior.

Measurements on some intermediate dopings may give some indicators as to the reason for these diﬀerent reactions to pressure. Performing the magnetoresistivity measurements on the parent compound CeCoIn5 (for which we know that pressure suppresses HQCP ) would allow us further prove the ﬁdelity of the analysis and compare it to established literature.

Finally, with this analysis in hand, one can inspect other heavy fermion systems.

It would allow us to gain deeper insight into the quantum criticality and the Kondo coherence in systems such CeRhIn5, CeCu6, and CeAl3 (and their substitution fami- lies), just to name a few.

Concluding this work, CeCoIn5 is still a deep well in which a lot can be learned about heavy-fermion superconductivity, especially in regards to its relationship to quantum criticality. Yb still proves an interesting dopant in the CeM In5 family, providing unusual properties, and more can be done to better understand it. Magne- toresistivity is a simple measurement that contains plenty of information, revealing information on quantum criticality, Kondo coherence, and superconductivity, and it is an excellent experimental supplement to any material that is strongly sensitive to application of magnetic ﬁeld, as the heavy fermion compounds are.

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