TORQUE and MAGNETIZATION MEASUREMENTS on the HEAVY FERMION SUPERCONDUCTOR Cecoin5

TORQUE AND MAGNETIZATION MEASUREMENTS ON THE HEAVY FERMION SUPERCONDUCTOR CeCoIn5

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

Hong Xiao

August, 2009 Dissertation written by

Hong Xiao

B.S., Hebei University, China, 1999

M.S., Institute of Solid State Physics, Chinese Academy of Sciences, China, 2002

Ph.D., Kent State University, USA, 2009

Approved by

Carmen Almasan , Chair, Doctoral Dissertation Committee

David Allender , Members, Doctoral Dissertation Committee

Mark Manley ,

Dmitry Ryabogin ,

Mietek Jaroniec ,

Accepted by

Bryon Anderson , Chair, Department of Physics

John R.D. Stalvey , Dean, College of Arts and Sciences

ii Table of Contents

List of Figures ...... vi

Acknowledgements ...... xii

1 Heavy Fermion Superconductivity ...... 1

1.1 Introduction ...... 1

1.1.1 Kondo eﬀect ...... 2

1.1.2 RKKY Interaction ...... 4

1.2 CenMmIn3n+2m (M =Co, Ir, or Rh; n=1 or 2; m=0, or 1) Family . 5

1.2.1 CeIn3 ...... 6

1.2.2 CeMIn5 ...... 7

1.3 Properties of CeCoIn5 ...... 10

1.3.1 Multi-band Picture ...... 11

1.3.2 Non-Fermi Liquid Behavior ...... 13

1.3.3 Quantum Criticality ...... 14

1.3.4 Anisotropic Superconducting Gap ...... 16

1.3.5 Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) Superconducting

State ...... 17

1.3.6 Pseudogap and Nernst Eﬀect ...... 20

References ...... 23

2 Experimental Details ...... 26

2.1 Sample preparation ...... 26

iii 2.2 Experimental setup ...... 26

2.2.1 Torque measurements ...... 26

2.2.2 Resistivity Measurements ...... 28

2.2.3 Magnetization measurements ...... 29

References ...... 30

1 3 Angular-Dependent Torque Measurements on CeCoIn5 Single Crystals 31

3.1 Introduction ...... 31

3.2 Experimental details ...... 33

3.3 Results and Discussion ...... 34

3.4 Summary ...... 48

3.5 Appendix: torque measurements ...... 48

References ...... 51

4 Anomalous Paramagnetic magnetization in the Mixed State of CeCoIn5

Single Crystals 2 ...... 53

4.1 Introduction ...... 53

4.2 Experimental Details ...... 55

4.3 Results and Discussion ...... 55

4.4 Summary ...... 66

References ...... 67 1This chapter is based on following paper: H. Xiao, T. Hu, T. A. Sayles, M. B. Maple and C. C. Almasan, Phys. Rev. B 73, 184511 (2006) 2This chapter is based on following papers: H. Xiao, T. Hu, T. A. Sayles, M. B. Maple and C. C. Almasan, Phys. Rev. B 76, 224510 (2007) H. Xiao, T. Hu, T. A. Sayles, M. B. Maple and C. C. Almasan, Physica B 403, 952 (2008)

iv 3 5 Pairing Symmetry of CeCoIn5 Detected by In-plane Torque Measurements 69

5.1 Introduction ...... 69

5.2 Experimental Details ...... 71

5.3 Results and Discussion ...... 72

5.4 Summary ...... 83

References ...... 85

4 6 Angular Resistivity Study in CeCoIn5 Single Crystals ...... 87

6.1 Introduction ...... 87

6.2 Experimental Details ...... 87

6.3 Results and Discussions ...... 87

6.4 Summary ...... 92

References ...... 94

7 Summary ...... 95

References ...... 99

3This chapter is based on following paper: H. Xiao, T. Hu, T. A. Sayles, M. B. Maple and C. C. Almasan, Phys. Rev. B 78, 014510 (2008) 4This chapter is based on following paper: H. Xiao, T. Hu, T. A. Sayles, M. B. Maple and C. C. Almasan, AIP Conference Proceedings, Volume 850, 719 (2006)

v List of Figures

1-1 Phase diagram resulting from the competition between the Kondo and RKKY

interactions. From Ref. [8]...... 5

1-2 T − P phase diagram of CeIn3. For clarity, the values of Tc have been scaled

by a factor of ten. Inset: The simple cubic unit cell of CeIn3. The In atoms

(not shown) are located at the center of the faces of the cubic unit cell. From

Ref. [9]...... 6

1-3 Structure of CeMIn5(M = Rh, Ir, Co). From Ref. [15]...... 7

1-4 P − T phase diagram of CeRhIn5. From Ref. [16]...... 8

1-5 Phase diagram of CeIrIn5. From Ref. [18]...... 9

1-6 The ambient pressure values of Tc vs. the room temperature value of c=a

(open circles) for various CeMIn5 compounds. Also shown (solid circles)

are the values of c=a determined at room temperature at the pressure Pmax

where Tc(P ) displays a maximum. The line is a least squares ﬁt to the

ambient pressure values. From Ref. [19]...... 10

1-7 Fermi surfaces of CeCoIn5 based on the itinerant 4f band model. From Ref.

[20]...... 12

1-8 Schematic T −P phase diagram. AFM: Neel state; PG: pseudogap state; SC:

unconventional superconducting state; FL: Fermi-liquid; NFL: non-Fermi-

liquid. From Ref. [28]...... 15

1-9 H − T phase diagram of CeCoIn5 determined from resistivity measurements.

The inset shows the H dependence of the quadratic coeﬃcient A of ‰(T ).

From Ref. [26]...... 16

vi 1-10 HQCP from the ﬁts are plotted along with Hc2 and Tc in zero ﬁeld vs. P .

From Ref. [29]...... 17

1-11 Illustration of the vortex structure (solid lines) and the FFLO modulation

(dashed lines) with the ﬁeld parallel (top) and perpendicular (bottom) to the

heat current. From Ref. [39]...... 19

1-12 H − T phase diagrams at low temperatures and high ﬁelds for H ⊥ ab (left)

and for H k ab (right). The colored portions display the FFLO (pink) and

uniform superconductivity (blue) regions. The black and green lines represent

the upper critical ﬁelds that are in the ﬁrst order and in the second order,

respectively. The red dashed and solid lines represent the phase boundary

separating the FFLO and uniform superconducting states. From Ref. [42]. . 21

1-13 Geometry of the Nernst experiments in the vortex liquid state. From Ref. [44]. 22

2-1 Torque lever chip. From Ref. [2]...... 27

2-2 Chip (left) and puck (right). From Ref. [2]...... 28

2-3 PPMS rotator. From Ref. [2]...... 29

3-1 Angular dependence of the paramagnetic torque ¿p measured in the normal

state of CeCoIn5 at diﬀerent temperatures T and applied magnetic ﬁeld H

values. The solid lines are ﬁts of the data with Eq. (3.1). Inset: Schetch of

the single crystal with the orientation of the magnetic ﬁeld H and torque ¿

with respect to the crystallographic axes...... 35

3-2 Field H dependence of A=H, where A is the ﬁtting parameter in Eq. (3.1).

The solid lines are linear ﬁts of the data...... 37

vii 3-3 Plot of the magnetic moment M vs applied magnetic ﬁeld H, with H k c-axis

(solid symbols) and H k a− axis (open symbols), measured at 4, 6, 10, 15,

and 20 K. Inset: Susceptibility ´ vs temperature T , measured with H k c-axis

and H k a-axis...... 39

3-4 Angular dependence of the reversible torque ¿rev, measured in the mixed

state of CeCoIn5 at a temperature T of 1.9 K and an applied magnetic ﬁeld

H of 0.3 T. The solid curve is a ﬁt of the data with Eq. (3.9). Inset:

dependence of the hysteretic torque ¿, measured in increasing and decreasing

angle at the same T and H...... 42

3-5 Magnetic field H dependence of the fitting parameter fl. The solid line is a

guide to the eye. Inset: Enlarged plot of the low ﬁeld region of the data in

the main panel...... 44

3-6 Field H dependence of the anisotropy measured at 1.9 K...... 45

3-7 Composite plot of the temperature T dependence of the anisotropy . The

circles show the results obtained from the upper critical ﬁeld data (Ref. [23]

), the triangles are obtained from the resistivity data shown in the inset of

this ﬁgure, and the squares are from the present torque data measured in an

applied magnetic ﬁeld of 0.3 T. Inset: T dependence of the in-plane ‰a and

out-of-plane ‰c resistivities measured in zero ﬁeld...... 47

3-8 Predicted angular dependence of the torque from London theory (the mag-

nitude of the torque is in arbitary units). Inset shows the orientations of the

c−axis of the crystal, the applied ﬁeld H and the magnetization M. From

Ref. [17]...... 49

viii 4-1 Magnetic ﬁeld H dependence of the dc magnetization Mmes measured at 1.76

K with H k c-axis on a CeCoIn5 single crystal. The solid line is a linear ﬁt

of Mmes(H) in the normal state. Inset: Upper critical ﬁeld parallel to the

||c c-axis Hc2 − temperature T phase diagram. The open squares are data taken

from Ref. [19] while open circles are data extracted from present Mmes(H)

measurements...... 56

4-2 Magnetic ﬁeld H dependence of the magnetization M1 measured at 1.76 K

which is obtained by subtracting the paramagnetic contribution as an ex-

trapolation of the normal state paramagnetism. Lower inset: Plot of M1(H)

measured at 1.76, 1.80, 1.85, 1.90, 1.95, 2.00, 2.05, and 2.10 K. Upper inset:

Magnetic ﬁeld H dependence of the dc magnetization Mmes measured at 2 K

for H k a...... 59

4-3 (a) Plot of ﬁeld H dependence of the function f determined at 1.8 K. The

solid curve is a guide to the eye. Inset: H dependence of the ﬁtting param-

eter A. The solid line is a ﬁt of the data with a simple power law. (b) H

dependence of vortex magnetization Mv (solid diamonds), deviation magne-

tization Mdev (solid reversed triangles), and magnetization M1 data of Fig.

4-2 (open circles) of CeCoIn5 measured at 1.8 K. The dashed line in Mv(H)

is a linear extrapolation of the high ﬁeld data. The solid curves in Mv(H)

and Mdev(H) are guides to the eye...... 63

4-4 Magnetic ﬁeld H dependence of the paramagnetic magnetization Mp. Inset:

H dependence of diﬀerential susceptibility ´ ≡ dM=dH. The solid curve is

a guide to the eye...... 64

ix 5-1 Field H and temperature T dependence of the amplitude An of the normal-

state torque of CeCoIn5 single crystals. (a) H dependence of An at measured

temperatures of 1.9, 3, 4.5, 5, 6, 8, and 10 K. (b) T dependence of An at

measured magnetic ﬁeld of 14 T. Inset: Angular ’ dependent torque ¿n

measured in the normal state at 1:9 K and 7 T in CeCoIn5 single crystals.

The solid curve is a ﬁt of the data with ¿n = An sin 4’...... 73

5-2 (a) Angular ’ dependence of the reversible torque ¿rev measured at 1.9 K

and 3 T on CeCoIn5 single crystals. The solid curve is a ﬁt of the data with

¿ = A sin 4’. (b) Angular dependence of the irreversible torque ¿irr measured

at 1.9 K and 1 T on CeCoIn5 single crystals. Sharp peaks are present at …=4,

3…=4, 5…=4 and 7…=4...... 75

5-3 Plot of the angular ’ dependence of the upper critical ﬁeld Hc2 and lower

critical ﬁeld Hc1 for magnetic ﬁeld parallel to the ab plane, magnetization

M, free energy F , reversible torque ¿rev, and irreversible torque ¿irr for dxy

wave symmetry...... 77

5-4 Field H dependence of the amplitude Am of the reversible torque in the mixed

state of CeCoIn5 single crystals measured at T = 1:8 K. The solid curve is

a guide to the eye. Inset: Temperature T dependence of Am measured at

H = 3 T. The solid curve is a guide to the eye...... 79

5-5 Angular ’ dependent torque ¿ measured at 1.9 K and 14 T on LaCoIn5 single

crystals...... 81

5-6 Plot of out-of-plane resistance Rc vs. applied magnetic ﬁeld H measured at a

diﬀerent in-plane angle ’ between H and the a axis. Inset: Polar plot of the

four-fold symmetric component R4ϕ of the out-of-plane resistance measured

for two H values...... 82

x 6-1 Resistivity ‰ab vs. angle measured at 2.3 K. The solid lines are ﬁts of the

data with Eq. (6.3)...... 89

6-2 Resistivity ‰ab vs. the perpendicular ﬁeld component H cos , measured at

2.3 K and ﬁxed angles ...... 90

6-3 The ﬁeld H dependence of the ﬁtting parameter m1. The solid line is a guide

to the eye...... 91

xi Acknowledgements

This work is dedicated to my lovely daughter Angela Hu.

First, I would like to thank my research advisor, Dr. Carmen Almasan, who guided me through all the diﬃculties to complete this thesis. Without her patience and support, I could never ﬁnish my PhD studies.

I am thankful to all the members of my research group, Dr. Viorel Sandu, Dr. Yankun

Tang, Dr. Tika Katuwal and Dr. Parshu Gywali. We had a wonderful time working together. Their discussion and help are very important to me.

Also, I would like to thank Dr. T. A. Sayles and Prof. M. B. Maple from the University of California, San Diego, who provided high qualify single crystal samples, which made my work possible.

I would like to thank Prof. Almut Schroder, Prof. Vladimir Kogan, and Prof. K.

Machida for fruitful discussions. Also I thank Dr. Alan R. Baldwin who helped a lot to design and modify the devices.

Special thanks go to Dr. Tao Hu, who is my colleague, husband and friend. His encour- agement and support gave me the strength to start and persist working in this ﬁeld. My work beneﬁts a lot from our discussions.

In addition, I thank my parents. I could not ﬁnish my work without their support.

Finally, I would like to acknowledge ﬁnancial support from the National Science Foun- dation.

xii Chapter 1

Heavy Fermion Superconductivity

In this disseration, I studied the electronic and magnetic transport properties of CeCoIn5.

CeCoIn5 is a newly discovered heavy fermion superconductor. It has a superconducting transition temperature of 2.3 K, which is the highest among heavy fermion superconductors.

This system shows many similarities to the high temperature superconductors (HTSC).

However, compared to HTSC, it has much lower upper critical ﬁeld and is free of disorder, which makes it a good candidate to study. The study of the superconductivity in this heavy fermion system will shed light on the understanding of the mechanism responsible for the superconductivity of HTSC, hence help the realization of its application. In the following,

I will give some background introduction of heavy fermion superconductivity.

1.1 Introduction

The heavy fermion system refers to metallic compounds and alloys that contain 4f or 5f ions (usually Ce, Pr, Yb, or U) and which exhibit enormously enhanced eﬀective conduction electron masses at low temperatures [1]. The heavy fermion systems known to date can be classiﬁed into three groups: superconductors (CeCu2Si2, UBe13, UPt3), magnets (NpBe13,

U2Zn17, UCd11), and materials in which no ordering is observed (CeAl3, CeCu6) [2].

In 1975, the first heavy fermion material CeAl3 was discovered [3]. In metals, the specific heat coefficient at low temperatures reflects the effective mass m∗ of the conduction electrons. For free electrons, the specific heat coefficient is proportional to the mass of the

2 electrons. For CeAl3, the specific heat coefficient is about 1600 mJ/mol K , which reflects that m∗ is 2 or 3 orders of magnitude larger than that of metals (the specific heat coefficient

1 2

is about 1 mJ/mol K2 for metals, like Al).

In 1979, Steglich et al. found the ﬁrst heavy fermion superconductor, CeCu2Si2 (tetragonal structure), with a superconducting transition temperature Tc of 0.5 K [4]. Since then, the heavy fermion superconductors have developed to include 21 compounds, including 11

Ce compounds, 8 U compounds, one Pr compound (PrOs4Sb12) and one Pu compound

(PuCoGa5) [5].

Heavy fermion superconductors have some features in common [1]:

1) Tc (superconducting transition temperature) is of the order of 1 K (except PuCoGa5, which has a Tc of 18 K);

2) large linear coeﬃcient of the normal state speciﬁc heat, typically 70-1000 mJ/mol

K2;

3) large Hc2 (upper critical ﬁeld), compared with type I superconductors;

4) unusual temperature T dependence of many physical properties in the superconducting state;

5) unconventional superconductor in terms of the pairing mechanism responsible for the superconducting state;

6) type II superconductor with • as large as 50 or more (• is a Ginzburg Landau parameter which is deﬁned as • = ‚=», where ‚ is penetration depth and » is coherence length);

7) sensitive to impurities and defects;

8) application of high pressure has a strong inﬂuence on the superconducting properties of the heavy fermion superconductors.

1.1.1 Kondo eﬀect

Kondo systems refers dilute magnetic alloys (metals with a small amount of magnetic impurities added). A minimum is observed at low temperatures in the resistivity-temperature 3

curve of dilute magnetic alloys, including alloys of Cu, Ag, Au, Mg, Zn with Cr, Mn, Fe,

Mo, Re, Os as magnetic impurities. The resistivity increases sharply with further decreasing temperature. The temperature dependence of the resistivity of such a system can be written

2 5 as: ‰(T ) = ‰0 + aT + b ln(„=T ) + cT , where ‰0 is the residual resistivity which depends on the concentration of magnetic impurities; aT 2 shows the contribution from the Fermi liquid properties; the term cT 5 is due to electron-phonon scattering at low T ; b ln(„=T ) reflects the interaction between conduction electrons and magnetic impurities. This minimum resistivity was not understood until 1964, when Kondo explained it theoretically [6]. The increase of the resistance at temperatures lower than the location of the minimum resistivity is due to the interaction between the conduction electrons and the localized d or f electrons of the magnetic impurities. The moment of the magnetic impurity works as a scattering center and the scattering rate increases sharply with further decreasing temperature. This not only solved the longstanding puzzle, but also posed a rich and challenging many-body physics problem, referred to as the “Kondo effect”. The interaction responsible for the Kondo effect has played a prominent role in the physics of strongly correlated electron phenomenon in f-electron materials and it is widely believed that the “Kondo effect”, generalized to the case of a lattice of rare earth or actinide ions, is responsible for the nonmagnetic ground state, enormous quasi-particle effective mass, and striking temperature dependence of the electrical resistivity of heavy fermion compounds [5].

The physical properties of the heavy fermion f-electron materials are similar to the intermediate valence rare earth compounds. It is believed that the narrow feature in the density of states at the Fermi level, which is responsible for the low degeneracy temperature of the heavy fermion compounds, is produced by the Kondo eﬀect. For a typical heavy fermion compound, ‰(T ) is only weakly temperature dependent. It often increases with decreasing temperature (reminiscent of the Kondo eﬀect) above a characteristic coherence 4

2 temperature Tcoh. Below Tcoh, ‰(T ) decreases rapidly and then saturates as T at low temperatures as the heavy Fermi liquid ground state develops. The heavy fermion ground state is unstable with respect to magnetic order, superconductivity, and non-Fermi liquid behavior [5].

This Kondo interaction can lead to a complete screening of the magnetic moments and to the formation of a new quasi-particle, called a heavy fermion. Heavy fermion compounds exhibit local moment behavior at high temperatures and nonmagnetic behavior at low temperatures. The Kondo temperature TK is the crossover temperature between the local-

1 moment and heavy fermion regimes, TK ∝ exp(− ρJ ), where ‰ is the host density of states at the Fermi energy and J is the local-moment/conduction-electron exchange interaction energy.

1.1.2 RKKY Interaction

The RKKY(Ruderman-Kittel-Kasuya-Yosida) interaction refers to the coupling mechanism of nuclear magnetic moments or localized inner d or f shell electron spins in a metal by means of an interaction through the conduction electrons. It is the indirect exchange interaction between two local spins. The RKKY interaction is a basic ingredient for many phenomena in strongly correlated systems, for example, magnetic impurities in quantum wires, normal-state magnetism in high-temperature superconductors, or magnetic ordering in heavy fermion materials [7].

2 The RKKY coupling energy in temperature units is of the order of TRKKY ≈ ‰J =kB.

Kondo compensation of local moments and RKKY coupling between these moments are competing effects. For small J, TRKKY À TK , the RKKY interaction dominates and thus long-range magnetic ordering occurs. For large J, TK À TRKKY , the Kondo effect compensates the local moments and no magnetic order is expected. The phase diagram due to the competition between the Kondo effect and the RKKY interaction is shown in Fig. 5

1-1 [8].

Figure 1-1: Phase diagram resulting from the competition between the Kondo and RKKY interactions. From Ref. [8].

1.2 CenMmIn3n+2m (M =Co, Ir, or Rh; n=1 or 2; m=0, or 1) Family

Since 1998, a new family of Ce-based heavy fermion compounds was discovered. Crystals in this family form as CenMmIn3n+2m (M =Co, Ir, or Rh; n=1 or 2; m=0, or 1) [9, 10,

11]. This family of heavy fermion superconductors exhibits interesting and rich features that make them very attractive to be studied, such as paramagnetism, antiferromagnetism, exotic ambient-pressure and pressure induced superconductivity, non-Fermi liquid behavior, quantum criticality behavior, competition between Kondo and RKKY interactions, etc.

Collectively, these new systems enhance our knowledge of superconductivity, magnetism and heavy fermion (HF) behavior, and provide a suitable experimental environment to search for possible structure-property relationships in heavy fermion materials [12].

The CenMmIn3n+2m family (M = Rh, Ir, Co) has three subgroups: (1) m = 0, CeIn3, cubic antiferromagnet at ambient pressure; (2) n = m = 1, Ce-115 compounds CeMIn5. (3) n = 2, m = 1, Ce2TIn8. These latter Ce-218 compounds exhibit the same phenomena seen in Ce-115 materials, including paramagnetic and antiferromagnetic (AFM) ground states, non-Fermi liquid (NFL) behavior, pressure-induced and ambient pressure superconductivity 6

[13].

1.2.1 CeIn3

In conventional superconductors, the vibrations of the crystal lattice, i.e. the phonons, are responsible for the binding of electrons into Cooper pairs. However, in the case of the heavy fermion superconductors, for example, CeIn3, it is believed that the charge carriers are bound together in pairs by magnetic spin-spin interactions. It is not clear yet whether magnetic interactions are relevant for describing the superconducting and normal-state properties of other strongly correlated electron systems, including the high temperature copper oxide superconductors. The existence of magnetically mediated superconductivity in these compounds could help us to clarify this question [9].

Figure 1-2: T − P phase diagram of CeIn3. For clarity, the values of Tc have been scaled by a factor of ten. Inset: The simple cubic unit cell of CeIn3. The In atoms (not shown) are located at the center of the faces of the cubic unit cell. From Ref. [9].

Figure 1-2 shows the temperature-pressure (T − P ) phase diagram of CeIn3. (Figure is taken from Ref. [9].) CeIn3 is an antiferromagnet at ambient pressure with Neel temperature 7

(transition temperature from paramagnetism to antiferromagetism ) TN = 10 K. TN is found to decrease slowly and monotonically with increasing pressure P . The eﬀective critical pressure Pc , where the Neel temperature TN tends to absolute zero is about 26 kbar (2.6

GPa). Close to Pc, the normal state resistivity assumes a non-Fermi liquid form, i.e. varies

1.6±0.2 as T . In a very narrow region near Pc, superconductivity emerges. The maximum value of Tc is around 200 mK. The inset shows that CeIn3 has a simple cubic structure. Ce moments are aligned antiferromangetically (AF) in adjacent (111) ferromagnetic planes.

1.2.2 CeMIn5

All members of the CeMIn5 family crystallize in the HoCoGa5 crystal structure (space group P4/mmm), composed of alternating layers of CeIn3 and MIn2. Figure 1-3 shows the crystal structure of CeMIn5. (The ﬁgure is taken from Ref. [15].) The key structural unit of the series is the distorted cuboctahedron CeIn3. Such structural arrangement implies that CeMIn5 family members are quasi-2D variants of CeIn3.

Figure 1-3: Structure of CeMIn5(M = Rh, Ir, Co). From Ref. [15].

CeRhIn5 is a HF antiferromagnet with Neel temperature TN = 3.8 K. The electronic coeﬃcient of speciﬁc heat is 420 mJ/(molCeK2). The pressure induced transition from 8

antiferromagnetic metal (AFM) to superconductor (SC) takes place at a critical pressure

Pc = 1.63 GPa [16]. Plotted in Fig. 1-4 is the T −P phase diagram of the HF antiferromagnet

115 CeRhIn5 constructed via In nuclear quadrupole resonance (NQR) measurements. (The

ﬁgure is taken from Ref. [16].) In the high-T region, the nuclear spin-lattice relaxation rate

∗ 1=T1 becomes almost T independent above the temperature marked as T , which indicates that Ce-4f moment ﬂuctuations are in a localized regime above T ∗ [17]. The uniform mixed phase of AFM and SC has been established in the range P = 1.53 - 1.9 GPa. The AFM and SC merge at TN = Tc = 2.0 K at P = 1.9 GPa. The uniform mixed phase of AFM and

SC crosses over to a single phase of SC in a very narrow pressure range, 1.9 - 2.1 GPa.

Figure 1-4: P − T phase diagram of CeRhIn5. From Ref. [16].

CeIrIn5 is a HF superconductor at ambient pressure with Tc = 0.4 K. The electronic coeﬃcient of speciﬁc heat is 750 mJ/(molCeK2). Figure 1-5 shows the phase diagram of

CeIrIn5. (The ﬁgure is taken from Ref. [18].) In this system, the Rh substitution for

Ir acts as a negative chemical pressure that increases antiferromagnetic correlations. In

CeRh1−xIrxIn5, the ground state continuously evolves from an antiferromagnetic metal 9

(x < 0.5) to a superconductor (x > 0.5). Tc shows a maximum at x ≈ 0.7 and a cusplike minimum at x ≈ 0.9, forming the ﬁrst dome (SC1). In CeIrIn5 (x = 1), Tc increases with pressure and exhibits a maximum (Tc = 1 K) at P ≈ 3 GPa, forming a second dome (SC2)

[18].

Figure 1-5: Phase diagram of CeIrIn5. From Ref. [18].

CeCoIn5 is also a HF superconductor at ambient pressure with the coeﬃcient of speciﬁc

2 heat of 290 mJ/(molCe K ) at 2.4 K. The critical temperature Tc = 2.3 K, which is the highest known Tc for a HF system. Whereas CeCoIn5 is isostructural with CeRhIn5 and

CeIrIn5, its cell constants diﬀer signiﬁcantly from the corresponding ones of CeRhIn5 and

CeIrIn5. As a result, the distortions of the cuboctahedron CeIn3, which is the key structural unit in all three materials, are diﬀerent in CeCoIn5 from the ones in CeRhIn5 and CeIrIn5.

This fact makes their properties diﬀerent.

A strong correlation between the ambient pressure c=a ratio and Tc in the CeMIn5 compounds has been observed (increasing c=a increases Tc), as shown in Fig. 1-6. (The ﬁgure is from Ref. [19].) In addition, CeMIn5 shows greatly enhanced Tc when compared with the parent compound CeIn3. It is still an open question whether the reduced dimensionality is responsible for the signiﬁcant increase of Tc in the materials CeMIn5. This is important 10

to the understanding of the physics of superconductivity not only in the heavy fermion materials but also in the broader class of correlated electron systems.

Figure 1-6: The ambient pressure values of Tc vs. the room temperature value of c=a (open circles) for various CeMIn5 compounds. Also shown (solid circles) are the values of c=a determined at room temperature at the pressure Pmax where Tc(P ) displays a maximum. The line is a least squares ﬁt to the ambient pressure values. From Ref. [19].

1.3 Properties of CeCoIn5

The recently discovered heavy fermion superconductor CeCoIn5 has generated a lot of interest, partly due to the many analogies present between this compound and the high transition temperature superconductors. Like the cuprates, CeCoIn5 has a layered crystal structure, a quasi-two-dimensional electronic spectrum, and a superconducting phase appearing at the border of the anti-ferromagnetic phase. Furthermore, the superconductivity is unconventional; i.e., it displays proximity to quantum critical points, giant Nernst eﬀect in the normal state, non-Fermi liquid behavior with the possibility of a pseudogap, Pauli limiting eﬀect, and possibly multi-bands in the superconducting state. Study of both the normal state and mixed state of CeCoIn5 could provide further understanding of not only 11

the underlying physics in heavy fermion materials but also of the mechanism of superconductivity of high temperature superconductors.

CeCoIn5 forms in the HoCoGa5 crystal structure with alternating layers of CeIn3 and

CoIn2. The crystal exhibits a tetragonal symmetry, with characteristic lattice parameters a = 4:62 A˚ and c = 7:56 A.˚ The Kondo effect quenches the magnetic moments of the localized 4f electrons by spin polarization of the conduction electrons. The RKKY interaction enhances long-range magnetic order. In CeCoIn5, these two effects compete with each other and form quasi-particles with enhanced effective mass, i.e. heavy fermions at low temperatures. The electronic specific heat coefficient C/T is about 300-1000 mJ K−2 mol−1 (see

Ref [20]).

In addition, CeCoIn5 is in the superclean regime. High Tc cuprates are type II superconductors and the presence of vortices dominates H − T phase diagram. The vortex physics in the mixed state of high Tc superconductors is very complicated, and it is always a fascinating subject of research. The heavy fermion material CeCoIn5 is also a type II superconductor, but much less anisotropic compared to the high Tc cuprates. The c-axis p ∗ ∗ anisotropy parameter = mc =ma reported is about 1 - 3 in this material [11, 21, 22]. It is of great interest to know how the vortices behave in a magnetic ﬁeld and how this behavior compares to the high Tc superconductors. Magnetotransport measurement is a technique that detects vortex dissipation in a direct way. We performed angular dependent magnetoresistivity measurements at T = 2.3 K in the low ﬁeld region. See the detailed discussion in Chapter 6.

1.3.1 Multi-band Picture

To determine the dimensionality and nature of the electronic structure in CeCoIn5, scientists performed measurements of de Haas-van Alphen oscillations in both the normal and mixed states and revealed the quasi two dimensional nature of the Fermi surface and 12

the presence of a small number of electrons exhibiting 3D behavior [20, 23]. Figure 1-7 shows the calculated Fermi surfaces of CeCoIn5 based on the itinerant 4f band model.

(The figure is from Ref. [20].) It has also been found that the cyclotron masses have magnetic field dependences [20]. Rourke et al. [24] performed point-contact spectroscopy measurements on CeCoIn5 and the spectra show Andreev-reflection characteristics with multiple structures which depend on junction impedance. They concluded that there are two coexisting order parameter components with different amplitudes, which indicate a highly unconventional pairing mechanism, possibly involving multiple bands. Tanatar et al. [25] performed thermal conductivity and specific heat measurements and revealed the presence of uncondensed electrons, which can be explained by an extreme multiband scenario, with a d-wave superconducting gap on the heavy-electron sheets of the Fermi surface and a negligible gap on the light, three-dimensional pockets.

Figure 1-7: Fermi surfaces of CeCoIn5 based on the itinerant 4f band model. From Ref. [20]. 13

We performed torque measurements on single crystals of CeCoIn5 both in the normal state and the superconducting state. We found that the anisotropy parameter calculated from the torque data is ﬁeld and temperature dependent, which provides further evidence that the picture in this unconventional superconductor is a multi-band scenario. See the detailed discussion in Chapter 3.

1.3.2 Non-Fermi Liquid Behavior

In the absence of a phase transition to a cooperative state, it is expected that the low-lying excitations of the system should be well described by the Fermi liquid theory.

In the picture of Fermi liquid theory, down to very low temperatures, (1) ‰ ∝ T 2 (the electrical resistivity varies as T 2 from its zero-temperature value; (2) The coeﬃcient of speciﬁc heat is a constant.; (3) ´(T ) = const. (magnetic susceptibility tends to a constant

Pauli-like value). However, the Fermi liquid theory breaks down in a number of heavy fermion systems, i.e. the low-temperature thermodynamic and transport properties deviate substantially from the predictions of the Fermi liquid theory. (For example, ‰ ∝ T and

C(T )=T diverges as T → 0), although there is no evidence for a phase transition (at least at nonzero temperature).

While there is no long-range magnetic order present in CeCoIn5, it has been shown that this system is close to the anti-ferromagnetic order, which leads to a deviation from the

Fermi liquid behavior in CeCoIn5 due to an abundance of spin ﬂuctuations [26]. The non-

Fermi liquid (NFL) behavior can be observed in many ways. The temperature dependences of C=T , ´, and ‰ as T → 0 are expected to be qualitatively different for a NFL relative to those of a Landau Fermi liquid. For example, its magnetic susceptibility ´ diverges at low temperatures as T −0.42 (for magnetic field perpendicular to the plane). The specific heat data show ∆C=T ∼ ln T temperature dependence between 2.3 and 8 K [27]. Also a

T -linear resistivity was discovered for T < 20 K [11]. 14

1.3.3 Quantum Criticality

Conventional phase transitions occur at nonzero temperatures, when the growth of ran- dom thermal fluctuations leads to a change in the physical state of a system. A quantum critical point is a special class of continuous phase transition that takes place at the absolute zero of temperature, typically in a material where the phase transition temperature has been driven to zero by the application of a pressure or field or through doping. There is great benefit in examining stoichiometric quantum critical systems because disorder may profoundly influence the behavior at a quantum critical point. CeCoIn5 is a good candidate to study quantum critical behavior because it is one of a relatively small number of such systems that are relatively disorder free.

Sidorov et al. measured the temperature-dependent resistivity under pressure on CeCoIn5

n and ﬁtted the data with ‰(T ) = ‰0 + AT [28]. It has been found that there is a crossover

∗ in the pressure dependence of ‰0, n, A and ∆Tc=Tc at a critical pressure P ≈ 1:6 GPa.

This indicates that the system undergoes a crossover near P ∗ from a quantum-critical state

(P < P ∗) to a Fermi-liquid state (P > P ∗). A temperature-pressure phase diagram for

CeCoIn5 based on this was constructed, as shown in Fig. 1-8. (The ﬁgure is taken from

Ref. [28].) There is no long-range ordered state observed in CeCoIn5 at atmospheric or higher pressures. It is speculated that an anti-ferromagnetic QCP in CeCoIn5 may be at an inaccessible slightly negative pressure. At P < P ∗, the system shows non-Fermi liquid

∗ ∗ behavior. Tc reaches a maximum at P . At P > P , Tc drops quickly and the system is a

Fermi-liquid at temperatures just above Tc.

Paglione et al. [26] performed a systematic study of the low-temperature electric resistivity of CeCoIn5 in magnetic ﬁelds up to 16 T and temperature down to 25 mK. It was found that as the magnetic ﬁeld increases, the non-Fermi liquid behavior, ‰ ∝ T , is

2 suppressed and a Fermi liquid state, with ‰ = ‰0 + AT , develops . The slope of the ﬁtted 15

Figure 1-8: Schematic T − P phase diagram. AFM: Neel state; PG: pseudogap state; SC: unconventional superconducting state; FL: Fermi-liquid; NFL: non-Fermi-liquid. From Ref. [28].

‰ vs. T 2 curves, i.e. the coefficient A, is a measure of the strength of the electron-electron interactions. The field dependence of A is shown in the inset to Fig. 1-9 (The figure is taken from Ref. [26].), which displays critical behavior best fitted by the function A ∝ (H −H∗)α,

∗ with H = 5.1 T, coinciding with the upper critical ﬁeld Hc2. This provides evidence that the system can be driven through a quantum critical point by magnetic ﬁeld tuning.

Ronning et al. reported resistivity measurements in the normal state of CeCoIn5 under pressure up to 1.3 GPa [29]. They have found that, although the field-tuned QCP coincides with Hc2 at ambient pressure, it moves inside the superconducting dome to lower fields with increasing pressure. This study showed that the superconductivity is not directly responsible for the NFL behavior in CeCoIn5; instead, the data suggest an anti-ferromagnetic QCP scenario [29]. Figure 1-10 shows the change of the QCP with pressure, together with upper critical field Hc2 and Tc. By applying hydrostatic pressure, the quantum critical field is successfully separated from the superconducting upper critical field. It is concluded that the quantum critical behavior is most likely associated with an as yet undetected antiferromagnetic QCP. 16

Figure 1-9: H − T phase diagram of CeCoIn5 determined from resistivity measurements. The inset shows the H dependence of the quadratic coeﬃcient A of ‰(T ). From Ref. [26].

1.3.4 Anisotropic Superconducting Gap

The type of pairing symmetry of unconventional superconductors is essential to the understanding of their pairing mechanism and of the origin of their superconductivity.

Presently, it is a subject of great interest and debate for the CeCoIn5 heavy fermion superconductor.

It has been established that CeCoIn5 has an anisotropic superconducting gap. Movshovich et al. [30] performed low temperature speciﬁc heat and thermal conductivity measurements on CeCoIn5 and revealed power-law temperature dependencies of these quantities below

2 Tc. The low temperature speciﬁc heat in CeCoIn5 includes T terms, consistent with the presence of nodes in the superconducting energy gap. The thermal conductivity data follow

3 a T dependence at low T in CeCoIn5, consistent with the prediction for an unconventional superconductor with lines of nodes. Combined with the fact that CeCoIn5 is a singlet superconductor (115In and 59Co nuclear-magnetic-resonance measurements [31] and torque 17

Figure 1-10: HQCP from the ﬁts are plotted along with Hc2 and Tc in zero ﬁeld vs. P . From Ref. [29]. measurements [32] have revealed a suppressed spin susceptibility, which implies singlet spin pairing), it is established that the superconducting order parameter of CeCoIn5 displays d-wave symmetry. However, the direction of the gap nodes relative to the Brillouin zone axes, which determines the type of d-wave state, is still an open question and an extremely controversial issue. There is experimental evidence for both dx2−y2 (see Ref. [33]) and dxy

(see Ref. [34]) wave symmetry. This extremely controversial issue needs to be resolved through an experimental technique that allows the direct measurement of the nodal positions. Since torque measurements directly probe the nodal positions on the Fermi surface with high angular resolution, we performed in-plane torque measurements in the normal state and the mixed state of CeCoIn5: Our data point unambiguously towards dxy wave symmetry in CeCoIn5. See the detailed discussion in Chapter 5.

1.3.5 Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) Superconducting State

In 1964, Fulde and Ferrell and Larkin and Ovchinnikov developed theories of inhomo- geneous superconducting states [35, 36]. Superconductivity can be destroyed by magnetic

ﬁeld in two ways: (1) when the Zeeman energy (which is related with the interaction of 18

the spins of the electrons with magnetic fields) exceed the energy of the superconducting coupling of electrons into Cooper pairs. The former interaction results in Pauli paramagnetism, where electrons align their spins along the magnetic field. The latter interaction favors the spin up and down coupling in singlet superconductivity. The superconductor is in Pauli limiting if the superconductivity is suppressed by this reason. The upper bound √ 2∆0 of Hc2 is determined by the Pauli field Hp, which is defined as Hp = , where ∆0 is gµB the zero temperature value of the superconducting gap, g is electron’s effective g factor, and „B is Bohr magneton. (2) If the superconductivity is destroyed when the kinetic energy of the supercurrent around the normal cores of the superconducting vortices in type

II superconductors becomes greater than the superconducting condensation energy, then it

0 is called orbital limiting. The orbital limiting ﬁeld Hc2 due to the kinetic energy of the superconducting currents around the vortex cores, deﬁnes Hc2 in the absence of Pauli limiting. It is commonly derived from the slope of the experimentally determined H − T phase

0 boundary at Tc, as Hc2 = 0:7(dHc2=dT )|Tc [37]. The relative strength of Pauli and orbital

0 limiting is determined by the Maki parameter fi = Hc2=Hp. The FFLO theory predicts that in a purely Pauli limited superconductor, (i.e. for fi > 1:8, calculated by Ref. [38]), the magnetic field acting on the Cooper pair’s spin can induce pairs with nonzero total momentum and, consequently, a spatially modulated order parameter [35, 36].

Larkin-Ovchinnikov (LO) structure is a collection of periodically spaced planes of nodes of the superconducting order parameter that are perpendicular to the direction of the applied ﬁeld. The LO order parameter is described as ˆ(r) = ˆ0 cos(qr) and it is oscillating in space along the direction of the vector q k H (See Fig. 1-11 for an illustration.) [39].

The heavy fermion superconductor CeCoIn5 satisﬁes the requirements of the theory for the formation of the FFLO state. For example, it is very clean and the electronic mean- free path signiﬁcantly exceeds the superconducting correlation length [30, 37]. Its Maki 19

Figure 1-11: Illustration of the vortex structure (solid lines) and the FFLO modulation (dashed lines) with the ﬁeld parallel (top) and perpendicular (bottom) to the heat current. From Ref. [39]. parameter ﬁ ≈ 3:5 is greater than 1.8.

There is also experimental evidence that the FFLO state exists in CeCoIn5 [37, 40, 41,

42]. Radovan et al. gave the the first thermodynamic evidence for the existence of the FFLO state [40]. They reported measurements of the heat capacity and magnetization on CeCoIn5 single crystals and showed that superconductivity in CeCoIn5 is enhanced for particular orientations of an external magnetic field. The enhanced superconductivity is due to a new configuration of the superconducting state (FFLO state), where regions of superconductivity alternate with walls of spin-polarized unpaired electrons. This configuration lowers the free energy, which allows superconductivity to remain stable. Within the Bardeen-Cooper-

Schrieﬀer (BCS) picture, the FFLO state corresponds to Cooper pairs having a ﬁnite center- of-mass momentum due to coupling across the Zeeman-energy-split Fermi surface for up- and down-spin electrons.

Bianchi et al. also performed specific heat measurements on CeCoIn5 and found that with increasing magnetic field, there is a clear evolution of the character of the specific 20

heat anomaly from a mean-field-like step at lower magnetic field to a very sharp peak at a higher magnetic field (acquiring symmetric character) [37]. The specific heat data indicate that the change from second to first order occurs at a critical magnetic field H ≈ 10 T and a critical temperature T0 ≈ 1 K. They also observed the development of the second low-temperature TF F LO anomaly in the low-temperature (≤ 300 mK)/high-field (≥ 10 T) corner of the H − T plane. The TF F LO anomaly can be described as a step followed by a gradual decrease of the specific heat with decreasing temperature. So, it is a behavior characteristic of a second-order phase transition.

The low temperature and high ﬁeld H − T phase diagrams for both magnetic ﬁeld con-

ﬁgurations are shown in Fig. 1-12. (The ﬁgure is from Ref. [42].) The realization of the

FFLO state in this quasi-two-dimensional system can help us understand the magnetically mediated electron pairing and the superconducting mechanism of other layered systems, including high-temperature superconductors. The properties of CeCoIn5 in the relatively high temperature and low field region of the H − T phase diagram might also be affected by the Pauli limiting effect present in the system. We performed magnetization and torque measurements to study the paramagnetic and diamagnetic response in the mixed state and address how does the Pauli paramagnetism affect the mixed-state thermodynamics of

CeCoIn5. This is essential to the understanding of the interaction between superconductivity and magnetism in heavy fermion materials. See the detailed discussion in Chapter

1.3.6 Pseudogap and Nernst Eﬀect

Two general features of a superconductor appear at the superconducting critical temperature Tc. One is the formation of an energy gap and the other is the Meissner effect, i.e. the expulsion of magnetic flux. There is strong evidence that in underdoped copper oxides, an energy gap, i.e. the pseudogap, opens up at a temperature significantly higher than Tc . 21

Figure 1-12: H −T phase diagrams at low temperatures and high fields for H ⊥ ab (left) and for H k ab (right). The colored portions display the FFLO (pink) and uniform superconductivity (blue) regions. The black and green lines represent the upper critical fields that are in the first order and in the second order, respectively. The red dashed and solid lines represent the phase boundary separating the FFLO and uniform superconducting states. From Ref. [42].

The pseudogap is closely related to the gap that appears at Tc because the variation of the gap magnitudes around the Fermi surface and their maximum amplitudes are very similar.

However, the Meissner eﬀect is absent in the pseudogap state. The nature of the pseudogap state and its relation to the superconducting state are of great importance in understanding copper oxide superconductivity [43].

The appearance of a transverse electric ﬁeld E in the presence of a thermal gradient

∆T and magnetic field H is called the Nernst effect. If one applies a temperature gradient in the x direction and a magnetic field in the z direction, then an electric signal will be generated in the y direction. See Fig. 1-13 ([44]) for an illustration. The Nernst coefficient

Ey ” is deﬁned as ” = ∆TH . In type II superconductors, there is a vortex liquid state close to the upper critical 22

Figure 1-13: Geometry of the Nernst experiments in the vortex liquid state. From Ref. [44].

ﬁeld line Hc2(T ). The vortices readily ﬂow in response to an applied temperature gradient.

The vortex motion generates an electric field E = B × v which is perpendicular to both the vortex velocity v and B. There are reports of the Nernst effect (detection of vortex- like excitations) in high Tc superconductors at temperatures significantly above Tc, for example, in La2−xSrxCuO4 and Bi2Sr2−yLayCuO6, which shed light on the understanding of the pseudogap state [43, 44, 45].

In CeCoIn5, Bel et al. [46] measured the thermoelectric coeﬃcients and reported that a large Nernst signal emerges with a magnitude drastically exceeding what is expected for a multiband Fermi-liquid metal. Furthermore, in the mixed state, this signal overwhelms the one associated with the motion of superconducting vortices. The Nernst eﬀect observed in

CeCoIn5 may provide information on the origin of the anomalous Nernst signal observed in the normal state of high Tc cuprates due to the intriguing analogy between the cuprates and CeCoIn5. References

[1] Harry B. Radousky, Magnetism in Heavy Fermion Systems, world Scientiﬁc, (2000). [2] G. R. Stewart, Rev. Mod. Phys. 56, 755 (1984). [3] K. Andres, J. E. Graebner, and H. R. Ott, Phys. Rev. Lett. 35, 2291 (1975). [4] F. Steglich, J. Aarts, C. D. Bredl, W. Lieke, D. Meschede, W. Franz, and H. Schafer, Phys. Rev. Lett. 43, 1892 (1979). [5] M. Brian Maple, J. Phys. Soc. Jpn. 74, 222 (2004). [6] J. Kondo, Prog. Theor. Phys. 32, 37 (1964). [7] Reinhold Egger and Herbert Schoeller, Proceedings of the 21st International Conference on Low Temperature Physics, Prague, August 8-14, (1996). [8] Image is taken from powerpoint presentation of Jiunn-Yuan Lin. [9] N. D. Mathur, F. M. Grosche, S. R. Julian, I. R. Walker, D. M. Freye, R. K. W. Haselwimmer, and G. G. Lonzarich, Nature, 394, 39, (1998). [10] J.D. Thompson, R. Movshovich, Z. Fisk, F. Bouquet, N.J. Curro, R.A. Fisher, P.C. Hammel, H. Hegger, M.F. Hundley, M. Jaime, P.G. Pagliuso, C. Petrovic, N.E. Phillips, and J.L. Sarrao, Journal of Magnetism and Magnetic materials 226-230(2001) 5-10. [11] C Petrovic, P G Pagliuso, M F Hundley, R Movshovich, J L Sarrao, J D Thompson, Z Fisk, and P Monthoux, J. Phys.: Condens. Matter 13, L337-L342, (2001). [12] E. G. Moshopoulou, J. L. Sarrao, P. G. Pagliuso, N. O. Moreno, J. D. Thompson, Z. Fisk, and R. M. Ibberson, Proceedings ICNS, supplement of Applied Physics A. Material Science & Processing (2001). [13] M. F. Hundley, A. Malinowski, P. G. Pagliuso, J. L. Sarrao, and J. D. Thompson, Phys. Rev. B, 70, 035113 (2004). [14] R.S. Kumar, H. Kohlmanna, B.E. Lighta, A.L. Corneliusa, V. Raghavanb, T.W. Dar- lingb, and J.L. Sarraob, Physica B 359-361, 407 (2005). [15] Image is taken from powerpoint presentation of M. Tanatar. [16] S Kawasaki, M Yashima, T Mito, Y Kawasaki, G-q Zheng, Y Kitaoka, D Aoki, Y Haga, and Y. Onuki, J. Phys.: Condens. Matter 17 S889, (2005). [17] S. Kawasaki, T. Mito, G.-q. Zheng, C. Thessieu, Y. Kawasaki, K. Ishida, Y. Kitaoka, T. Muramatsu, T. C. Kobayashi, D. Aoki, S. Araki, Y. Haga, R. Settai, and Y. Onuki, Phys. Rev. B 65, 020504(R) (2001).

23 24

[18] Y. Kasahara, T. Iwasawa, Y. Shimizu, H. Shishido, T. Shibauchi, I. Vekhter, and Y. Matsuda, Phys. Rev. Lett. 100, 207003 (2008).

[19] Ravhi S. Kumar and A. L. Cornelius Phys. Rev. B 70, 214526 (2004).

[20] R. Settai, H. Shishido, S. Ikeda, Y. Murakawa, M. Nakashima, D. Aoki, Y. Haga, H. Harima, and Y. Onuki, J. Phys. Condens. Matter. 13, L627 (2001).

[21] S. Majumdar, M. R. Lees, G. Balakrishnan, and D. McK. Paul, Phys. Rev. B 68,012504 (2003).

[22] T. P. Murphy, Donavan Hall, E. C. Palm, S. W. Tozer, C. Petrovic, Z. Fisk, R. G. Goodrich, P. G. Pagliuso, J. L. Sarrao, and J. D. Thompson, Phys. Rev. B 65, 100514(R) (2002).

[23] D. Hall, E. C. Palm, T. P. Murphy, S. W. Tozer, Z. Fisk, U. Alver, R. G. Goodrich, J. L. Sarrao, P. G. Pagliuso, and T. Ebihara, Phys. Rev. B 64, 212508 (2001).

[24] P. M. C. Rourke, M. A. Tanatar, C. S. Turel, J. Berdeklis, C. Petrovic, and J. Y. T. Wei, Phys. Rev. Lett. 94, 107005 (2005).

[25] M. A. Tanatar, Johnpierre Paglione, S. Nakatsuji, D. G. Hawthorn, E. Boaknin, R. W. Hill, F. Ronning, M. Sutherland, Louis Taillefer, C. Petrovic, P. C. Canﬁeld and Z. Fisk, Phys. Rev. Lett. 95, 067002 (2005).

[26] Johnpierre Paglione, M. A. Tanatar, D.G. Hawthorn, Etienne Boaknin, R.W. Hill, F. Ronning, M. Sutherland, Louis Taillefer, C. Petrovic, and P. C. Canﬁeld, Phys. rev. Lett. 91, 246405 (2003).

[27] J. S. Kim, J. Alwood, G. R. Stewart, J. L. Sarrao, and J. D. Thompson Phys. Rev. B 64, 134524 (2001).

[28] V. A. Sidorov, M. Nicklas, P.G. Pagliuso, J. L. Sarrao, Y. Bang, A.V. Balatsky, and J. D. Thompson, Phys. Rev. Lett. 89, 157004 (2002).

[29] F. Ronning, C. Capan, E. D. Bauer, J. D. Thompson, J. L. Sarrao, and R. Movshovich, Phys. Rev. B 73, 064519 (2006).

[30] R. Movshovich, Phys. Rev. Lett. 86, 5152 (2001).

[31] N. J. Curro, B. Simovic, P. C. Hammel, P. G. Pagliuso, J. L. Sarrao, J. D. Thompson, and G. B. Martins, Phys. Rev. B 64, 180514(R) (2001).

[32] H. Xiao, T. Hu, C. C. Almasan, T. A. Sayles, and M. B. Maple, Phys. Rev. B 76, 224510 (2007).

[33] K. Izawa, H. Yamaguchi, Y. Matsuda, H. Shishido, R. Settai, and Y. Onuki, Phys. Rev. Lett. 87, 057002 (2001).

[34] H. Aoki, T. Sakakibara, H. Shishido, R. Settai, Y. Onuki, P. Miranovic, and K. Machida, J. Phys.: Condens. Matter 16, L13 (2004). 25

[35] P. fulde and R. A. Ferrell, Phys. Rev. 135, A550 (1964).

[36] A. I. Larkin and Y. No. Ovchinnikov, Zh. Eksp. Teor. Fiz. 47, 1136 (1964).

[37] A. Bianchi, R. Movshovich, C. Capan, P.G. Pagliuso, and J. L. Sarrao, Phys. Rev. Lett. 91, 187004 (2003).

[38] L. W. Gruenberg and L. gunther, Phys. Rev. Lett. 16, 996 (1966).

[39] C. Capan, A. Bianchi, R. Movshovich, A. D. Christianson, A. Malinowski, M. F. Hundley, A. Lacerda, P. G. Pagliuso, and J. L. Sarrao, Phys. Rev. B 70, 134513 (2004).

[40] H. A. Radovan, N. A. Fortune, T. P. Murphy, S. T. Hannahs, E. C. Palm, S. W. Tozer, and D. Hall, Nature (London), 425, 51 (2003).

[41] C. Martin, C. C. Agosta, S. W. Tozer, H. A. Radovan, E. C. Palm, T. P. Murphy, and J. L. sarrao, Phys. Rev. B, 71, 020503(R) (2005).

[42] K. Kumagaia, T. Oyaizua, Y. Furukawaa, H. Shishidob, and Y. Matsuda, Physica B, 403, 1144 (2008)

[43] Z. A. Xu, N. P. Ong, Y. Wang, T. Kakeshita, and S. Uchida, Nature (London) 406, 486 (2000).

[44] Yayu Wang, Z. A. Xu, T. Kakeshita, S. Uchida, S. Ono, Yoichi Ando, and N. P. Ong, Phys. Rev.B 64, 224519 (2001).

[45] C. Capan, K. Behnia, J. Hinderer, A. G. M. Jansen, W. Lang, C. Marcenat, C. Marin, and J. Flouquet, Phys. Rev. Lett. 88, 056601 (2002).

[46] R. Bel, K. Behnia, Y. Nakajima, K. Izawa, Y. Matsuda, H. Shishido, R. Settai, and Y. Onuki, Phys. Rev. Lett. 92, 21702 (2004) Chapter 2

Experimental Details

2.1 Sample preparation

Single crystals of CeCoIn5 were grown by our collabrator, the group of Prof. M. B.

Maple, by using a ﬂux method [1]. Stoichiometric amounts of Ce and Co were mixed with excess In in an alumina crucible which were encapsulated in an evacuated quartz ampoule.

The compound solidiﬁes with a two-stage cooling process: (1) initial rapid cooling from

1150 ◦C, where the molten material is homogenized, to 750 ◦C (2) a slower cool to 450 ◦C.

The obtained crystals are well-separated, faceted platelets. Single crystals of CeCoIn5 with shiny surfaces were selected to perform the experiments. The surfaces of the crystals were etched in concentrated HCl for several hours and then thoroughly rinsed in ethanol. This removes the indium present on the surface. Note that indium is a superconductor with a superconducting transition temperature of 3.4 K.

2.2 Experimental setup

Experimental investigations included electrical transport and magnetic measurements.

The main experiments were torque measurements and magnetization measurements. CeCoIn5 single crystals were measured in changing temperature T , magnetic ﬁeld H, and sample position .

2.2.1 Torque measurements

Torque measurements were performed using a torque magnetometer in conjunction with a Quantum Design PPMS (Physical Property Measurement System) Horizontal Rotator.

The rotator enabled angular-dependent torque data to be gathered over the complete range

26 27

Figure 2-1: Torque lever chip. From Ref. [2]. of environmental conditions that are available for the PPMS probe. The PPMS platform allows high precision temperature control (accurate to 0.01 K) between 1.8 K ≤ T ≤ 400 K and application of magnetic ﬁelds up to 14 T.

The torque magnetometer measures the magnetic torque, ¿ = m × B, experienced by a sample of magnetic moment m in an applied magnetic ﬁeld B. The torque magnetometer uses a piezoresistive technique to measure the torsion, or twisting, of the torque lever about the lever’s symmetry axis. (See Fig. 2-1 for an illustration.) The torque lever chip together with the puck (Fig. 2-2), is mounted on a PPMS Horizontal Rotator (Fig. 2-3). The sample is mounted on the chip by applying a small amount of Apiezon N (or M) grease to the center of the torque-lever sample stage. The mass of the samples should be less than 28

Figure 2-2: Chip (left) and puck (right). From Ref. [2].

10 mg and the dimensions should be no greater than 1.5 mm × 1.5 mm × 0.5 mm. The maximum torque signal that can be measured is 10−5 N·m. Signals larger than this value will produce a nonlinear response and break the torque lever chip. The torque lever twists when a magnetic ﬁeld is applied to the sample. Two constant piezoresistor grids patterned onto the legs of the torque lever in the region of high stress sense the torque. A Wheatstone bridge circuit is integrated on the chip and measures the change in the resistance of the piezoresistors, produced by the diﬀerence of the magnetic torque in each leg due to the mechanical stress.

2.2.2 Resistivity Measurements

We performed resistivity measurements on CeCoIn5 single crystals with the PPMS. A multiterminal technique was used, which has the advantage of allowing the simultaneous determination of both the in-plane and out-of-plane components of the resistivity tensor.

In detail, we measured the electrical dissipation across the top face and bottom face of the single crystal, i.e. Vtop and Vbot at the same time. A LR-700 bridge was used to measure each individual voltage. We determined the in-plane and out-of-plane resistivities, i.e., ‰c 29

Figure 2-3: PPMS rotator. From Ref. [2].

and ‰ab, by using a ﬂux transformer method [3].

2.2.3 Magnetization measurements

We used a Quantum Design MPMS (Magnetic Properties Measurement System) to perform the magnetic measurements. The MPMS platform allows high precision temperature control of the sample chamber between 1.76 K ≤ T ≤ 400 K and application of magnetic

fields up to 5 T. Typical measurements are magnetization M vs. applied magnetic field H, temperature T , or angle . The superconducting Quantum Interference Device (SQUID) detector system can measure a change in the magnetic flux trapped in second derivative

−7 2 detection coils to an extremely high precision, well below `0 = 2.07 × 10 Gcm . References

[1] C. Petrovic, P. G. Pagliuso, M. F. hundley, R. Movshovich, J. L. Sarrao, J. D. Thompson, Z. Fisk, and P. Monthoux, J. Phys. Condens. Matter. 13, L337 (2001).

[2] Quantum Design, Physical Property Measurement System Hardware and Options (1999).

[3] C. N. Jiang, A. R. Baldwin, G. A. Levin, T. Stein, and C. C. Almasan, Phys. Rev. B 55, 3390(R) (1997).

30 Chapter 3

1 Angular-Dependent Torque Measurements on CeCoIn5 Single Crystals

3.1 Introduction

The heavy fermion compound CeCoIn5 forms in the HoCoGa5 tetragonal crystal structure with alternating layers of CeIn3 and CoIn2. It is superconducting at 2.3 K, the highest superconducting transition temperature Tc0 yet reported for a heavy fermion superconductor [1]. Considerable progress has been made in determining the physical properties of this material. The superconductivity in this material is unconventional. The presence of a strong magnetic interaction between the 4f moments and itinerant electrons allows the possibility of nonphonon mediated coupling between quasiparticles [2, 3]. Angular-dependent thermal conductivity measurements show dx2−y2 symmetry, which implies that the anisotropic antiferromagnetic ﬂuctuations play an important role in superconductivity [4]. Bel et al. reported the giant Nernst eﬀect in the normal state of CeCoIn5, which is comparable to high Tc superconductors in the superconducting state [5]. Non-Fermi liquid behavior was observed in many aspects [1, 6, 7]. The unconventional superconductivity and the similarity to high Tc superconductors attract great interest to study this system.

Measurements of de Haas-van Alphen oscillations in both the normal and mixed states have revealed the quasi two-dimensional nature of the Fermi surface and the presence of a small number of electrons exhibiting 3D behavior [8, 9]. Through point-contact spectroscopy measurements, Rourke et al. [10] found that in CeCoIn5 there are two coexisting order parameter components with amplitudes ∆1 = 0.95 ± 0.15 meV and ∆2 = 2.4 ± 0.3

1This chapter is based on following paper: H. Xiao, T. Hu, T. A. Sayles, M. B. Maple and C. C. Almasan, Phys. Rev. B 73, 184511 (2006)

31 32

meV, which indicate a highly unconventional pairing mechanism, possibly involving multiple bands. This is very similar to the case of MgB2, where two coexisting s-wave gaps were found by the same technique [11]. Thermal conductivity and speciﬁc heat measurements made by Tanatar et al. [12] have revealed the presence of uncondensed electrons, which can be explained by an extreme multi-band scenario, with a d-wave superconducting gap on the heavy-electron sheets of the Fermi surface and a negligible gap on the light, three-dimensional pockets .

The presence of multibands (gaps) together with the reported field dependence of the cyclotron masses [8] point towards a possible temperature and/or field dependent anisotropy in the superconducting state, which according to the standard anisotropic Ginzburg-Landau p ∗ ∗ ||a ||c (GL) theory is given by ≡ mc =ma = Hc2 =Hc2 = ‚c=‚a = »a=»c (c and a are crystallographic axes, and m; Hc2; ‚; and » are the effective mass, upper critical field, penetration depth, and coherence length, respectively). Specifically, in the multiband scenario proposed by Rourke et al. [10] and Tanatar et al. [12], different gaps may behave differently in a magnetic field, which may lead to a field-dependent . Also an anisotropic gap may result in a temperature-dependent . Reports up to date give values of the anisotropy of CeCoIn5 in the range 1.5 to 2.47. For example, Petrovic et al. [1] have reported an anisotropy of at least 2, as estimated from the ratio of the upper critical fields Hc2 along the c and a directions. Measurements of Hc2() at 20 mK give a value for the anisotropy of about 2.47

[13]. Magnetization measurements of the lower critical ﬁeld for temperatures between 1.5 and 2.1 K give the ratio of the out-of-plane and in-plane penetration depth ‚c=‚a ≈ 2:3 and of the in-plane and out-of-plane coherence length »a=»c ≈ 1:5, which gives an anisotropy of

2.3 and 1.5, respectively [14].

Magnetic torque is a sensitive tool for probing the anisotropy. It has been successfully applied to investigate the highly anisotropic high temperature superconductors and also 33

the less anisotropic materials such as MgB2 [15, 16]. However, all these previous torque measurements were made on materials that have negligible paramagnetism. On the other hand, CeCoIn5 is a magnetic superconductor, so it may have large paramagnetism which cannot be ignored in the study of the mixed state. Here, we report torque measurements on single crystals of CeCoIn5 both in the normal state and the superconducting state.

Our results show large paramagnetism in this material in the normal state. Therefore, we assume that there are two contributions to the torque signal in the mixed state of

CeCoIn5 single crystals: one coming from paramagnetism and the other one coming from vortices. We determined from the reversible part of the vortex signal and found that is not a constant, instead, it is ﬁeld and temperature dependent. This provides evidence that the picture in this unconventional superconductor is not a simple single-band scenario, supporting the conclusions of Rourke et al. [10] and Tanatar et al. [12].

3.2 Experimental details

The mass of the single crystal for which the data are shown is 0.75 mg. Angular dependent measurements of the magnetic torque experienced by the sample of magnetic moment M in an applied magnetic ﬁeld H, were performed over a temperature range 1.9

K ≤ T ≤ 20 K and applied magnetic ﬁeld range 0.1 T ≤ H ≤ 14 T using a piezoresistive torque magnetometer. In this technique, a piezoresistor measures the torsion, or twisting, of the torque lever about its symmetry axis as a result of the magnetic moment of the single crystal. The sample was rotated in the applied magnetic ﬁeld between H k c-axis ( = 0◦)

◦ and H k b-axis ( = 90 ) and the torques ¿inc and ¿dec were measured as a function of increasing and decreasing angle, respectively, under various temperature - ﬁeld conditions.

The contributions of the gravity and puck to the total torque signal were measured and subtracted from it. To measure the background torque due to gravity, we measured the torque signal at diﬀerent temperatures in zero applied magnetic ﬁeld with the sample 34

mounted on the puck. The gravity torque is almost temperature independent and it is negligible at high applied magnetic fields. However, as the applied magnetic field decreases, the total torque signal becomes smaller and the effect of gravity becomes important, hence should be subtracted from the measured torque. To determine the contribution of the puck to the measured torque, we measured the torque without the single crystal on the puck at different magnetic fields and temperatures. The magnitude of the torque of the puck increases with increasing magnetic field. Also, the contribution of the puck to the measured torque is much larger than the effec of gravity. Therefore, the former contribution should always be subtracted from the total measured torque signal.

3.3 Results and Discussion

Previous torque studies of Tl2Ba2CuO6+δ [17] and MgB2 [15, 16] systems have shown that the normal state torque coming from paramagnetism is small compared with the flux- vortex torque, therefore, one could neglect the former contribution to the total torque signal measured in the superconducting state. However, this is not the case for CeCoIn5, for which the paramagnetic torque signal in the normal state is comparable with the total torque signal measured in the superconducting state, as shown later in this section. Hence, one needs to subtract the former signal from the latter one to determine the torque due to vortices. This is similar to the case of the electron-doped high-Tc cuprate Nd1−xCexCuO4, where a large paramagnetic contribution from Nd ions is discussed seperately from a superconducting contribution [18]. Therefore, we first discuss the field and temperature dependence of the magnetic torque in the normal state and show how we subtract this contribution from the measured torque in the mixed state, and then we return to the discussion of the torque signal in the mixed state and to the determination of the field and temperature dependence of the bulk anisotropy.

All the torque curves measured in the normal state, some of which are shown in Fig. 35

CeCoIn

H = 6 T

1.9 K

6.0 K

10 K

20 K

0 Nm)

-6

c (10

p H

-1

-2

-50 0 50 100 150 200 250 300 350 400

(deg)

Figure 3-1: Angular dependence of the paramagnetic torque ¿p measured in the normal state of CeCoIn5 at different temperatures T and applied magnetic field H values. The solid lines are fits of the data with Eq. (3.1). Inset: Schetch of the single crystal with the orientation of the magnetic field H and torque ¿ with respect to the crystallographic axes. 36

3-1, are perfectly sinusoidal and can be well ﬁtted with

¿p(T; H; ) = A(T;H) sin 2; (3.1) where A is a temperature- and ﬁeld-dependent ﬁtting parameter. Indeed, note the excellent

fit of the data of Fig. 3-1 with Eq. (3.1) (solid lines in the figure). The field dependence of

A=H at 1.9, 6, 10 and 20 K is shown in Fig. 3-2. The solid lines are linear ﬁts to the data, which show that A=H is linear in H with a negligible y-intercept and a slope that increases with decreasing T . So A is proportional to H2, i.e.,

A(T;H) = C(T )H2; (3.2) with C a temperature-dependent ﬁtting parameter.

Next we show that the torque measured in the normal state and given by Eq. (3.1) is a result of the paramagnetism. Indeed, the torque of a sample of magnetic moment M placed in a magnetic ﬁeld H is given by

¿p(T; H) = M × H: (3.3)

The resultant magnetic moment M can always be decomposed into a component parallel

Mk and one perpendicular M⊥ to the ab−plane of the single crystal. With the magnetic

ﬁeld H making an angle with the c−axis of the single crystal, Eq. (3.1) becomes:

¿p(T; H; ) = [MkH cos − M⊥H sin ]k: (3.4)

On the other hand, the experimental relationship of the torque, given by Eq. (3.1) with the

ﬁtting parameter A given by Eq. (3.2), becomes

2 ¿p(T; H; ) = A(T;H) sin 2 = 2C(T )H sin cos : (3.5) 37

CeCoIn

1.9 K

6 K

10 K

20 K

4 A/m)

3 -7

2 -A/H (10

0 2 4 6 8 10 12 14

H ( T )

Figure 3-2: Field H dependence of A=H, where A is the ﬁtting parameter in Eq. (3.1). The solid lines are linear ﬁts of the data. 38

Therefore, with C ≡ (C1 − C2)=2, Eqs. (3.4) and (3.5) give

Mk = C1H sin ≡ ´aHk;

where C1 ≡ ´a, the a-axis susceptibility, and

M⊥ = C2H cos ≡ ´cH⊥; (3.6)

where C2 ≡ ´c, the c-axis susceptibility. This shows that the torque measured experimentally is of the form:

´ − ´ ¿ (T; H; ) = a c H2 sin 2 (3.7) p 2

The fact that A=H = (á − ć)H=2 ≡ (Mk − M⊥)=2 [see Eqs. (3.5) and (3.7)] shows that A=H plotted in Fig. 3-2 reflects the anisotropy of the magnetic moments along the two crystallographic directions, a and c, while its linear field dependence shows that the magnetic moments are linear in H, hence the susceptibilities along these two directions are field independent. The temperature dependence of the magnetic moments is given by the temperature dependence of the parameter C. Therefore, Eq. (3.7) shows that the T ,

H, and dependences of the torque measured in the normal state of CeCoIn5 reflect its paramagnetism and the anisotropy of its susceptibility ∆´ ≡ á − ć along the a and c directions.

To check further the consistency of the data and to determine precisely the paramagnetic value of the torque, we also measured the magnetic moment M of the same single crystal of CeCoIn5 using a superconducting quantum interference device (SQUID) magnetometer.

The magnetic moments measured at 4, 6, 10, 15, and 20 K are plotted in the main panel of Fig. 3-3 as a function of the applied magnetic ﬁeld for both H k c-axis ( = 0◦) and

H k a-axis ( = 90◦). The magnetic moments for both ﬁeld orientations are linear in H with Mk < M⊥ for all temperatures measured, consistent with the torque data of Fig. 3-2 39

CeCoIn CeCoIn

5 5

H || c-axis

1.5

H || a-axis

1.2 emu/mol)

8 -2 (10

0.9

6 emu) -4

5 10 15 20

4 T (K) M(10

, 4 K

, 6 K

, 10 K

, 15 K

, 20 K

0 1 2 3 4 5

H (10 Oe)

Figure 3-3: Plot of the magnetic moment M vs applied magnetic ﬁeld H, with H k c-axis (solid symbols) and H k a− axis (open symbols), measured at 4, 6, 10, 15, and 20 K. Inset: Susceptibility ´ vs temperature T , measured with H k c-axis and H k a-axis. 40

and with Eq. (3.6). Note that the units for the magnetic moments of Figs. 2 and 3 are diﬀerent. We change the units and compare the results of the two types of measurements.

For example, the torque measured at 6 K and 5 T gives A=H = −1:42 × 10−7 Am−1.

−7 −1 Since A=H = (Mk − M⊥)=2, ∆M ≡ Mk − M⊥ = −2:84 × 10 Am . The SQUID

−4 measurements give at the same temperature and applied magnetic ﬁeld M⊥ = 7:2 × 10

−7 −1 −4 −7 −1 emu = 7:2×10 Am and Mk = 4:2×10 emu = 4:2×10 Am ; hence, an anisotropy

∆M = −3:03 × 10−7 Am−1. Therefore, the values of the magnetic moments obtained in the two types of measurements are within 5% of each other, a diﬀerence well within our experimental uncertianty.

The inset to Fig. 3-3 is a plot of the susceptibilities along the two directions, calculated from the slopes of M(H) of Fig. 3-3. Clearly ´(T ) shows anisotropy with respect to the

ﬁeld orientation. Note that the susceptibilities for both directions increase with decreasing temperature. The continuous increase of ´(T ) in the investigated temperature range may be related with the non-Fermi liquid behavior due to the proximity to the quantum critical

field [19]. These values of ć and á are consistent with the ones reported by other groups

[7].

The above study of the magnetic torque in the normal state has shown that the contribution of paramagnetism to the torque signal is very large, it has a quadratic H dependence, and also a T dependence [see Eqs. (3.1) and (3.2)]. Therefore, to extract the vortex torque in the mixed state, one needs to account for this paramagnetic contribution and subtract the two background contributions from the measured torque. The gravity and puck contributions to the measured torque were determined and subtracted as explained in the Section

3.2. The resultant torque includes the paramagnetic ¿p and the vortex ¿v contributions and is plotted in the inset to Fig. 3-4. Note that the vortex and paramagnetic torque contributions have opposite signs since the magnetic moment representing the vortex torque is 41

diamagnetic.

The inset to Fig. 3-4 is the angular-dependent torque data measured in the mixed state at 1.9 K and 0.3 T in increasing and decreasing angles. Again, this torque includes

¿p and ¿v. Note that ¿() displays hysteresis. This hysteretic behavior is similar to the behavior in high Tc superconductors and is a result of intrinsic pinning [20]. The reversible component of the torque is determined as the average of the torques measured in increasing and decreasing angle; i.e.,

¿rev = (¿dec + ¿inc)=2 (3.8)

A plot of ¿rev(), obtained from ¿inc() and ¿dec() data, is shown in the main panel of Fig.

3-4. The reversible component of the torque reﬂects equilibrium states, hence it allows the determination of thermodynamic parameters. In the three-dimensional anisotropic London model in the mixed state, the vortex torque ¿v is given by Kogan’s model [21]. We assume that the paramagnetic contribution in the mixed state is given by Eq. (3.1). Therefore, ( ) 2 ||c `0HV − 1 sin 2 ·Hc2 ¿rev() = ¿p + ¿v = a sin 2 + 2 ln ; (3.9) 16…„0‚ab †() H†() where a is a ﬁtting parameter, V is the volume of the sample, „0 is the vacuum permeability, p 2 2 2 1/2 ‚ab is the penetration depth in the ab−plane, = mc=ma, †() = (sin + cos ) ,

||c · is a numerical parameter of the order of unity, and Hc2 is the upper critical ﬁeld parallel

||c 2 to the c-axis [Hc2 (1:9 K) = 2:35 T]. We define fl ≡ `0HV=(16…„0‚ab). To obtain the field dependence of the anisotropy , we fit the torque data with Eq. (3.9), with a, fl, and as three fitting parameters. The solid curve in the main panel of Fig. 3-4 is the fitting result for T = 1.9 K and H = 0.3 T. The value of the fitting parameter a is 20% smaller than what we would expect from the extrapolation of the normal state paramagnetic torque data. Also, a has an H dependence with an exponent of 2:30±0:01, instead of 2. So, either there is an extra contribution from other physics, which has a weak field dependence 42

4 CeCoIn

T = 1.9 K Nm) -8

H = 0.3 T (10

-1

-2

0 90 180

Nm) 0

(deg) -9 (10 rev

-2

-4

0 30 60 90 120 150 180

(deg)

Figure 3-4: Angular dependence of the reversible torque ¿rev, measured in the mixed state of CeCoIn5 at a temperature T of 1.9 K and an applied magnetic ﬁeld H of 0.3 T. The solid curve is a ﬁt of the data with Eq. (3.9). Inset: dependence of the hysteretic torque ¿, measured in increasing and decreasing angle at the same T and H. 43

in addition to the paramagnetism contribution, or maybe the paramagnetic contribution becomes smaller in the mixed state of CeCoIn5. Further experiments are needed to clarify this issue.

Figure 3-5 is a plot of the field dependence of fl. The inset is an enlarged plot of the low field region. Note that fl displays linear behavior up to a certain field with no y-intercept, then it deviates from linearity at H ≈ 0:5 T, and it increases fast in the high

ﬁeld region. Since, on one hand ‚ should be ﬁeld independent, and on the other hand Eq.

(3.9) is valid only for applied magnetic fields much smaller than the upper critical field, i.e. H << Hc2(T ) for a given temperature, we assume that 0:5 T, the field at which fl(H) deviates from linearity, is the cutoff field Hcut for the applicability of the above theory. The slope of fl(H) in the linear H regime gives ‚ (T = 1.9 K) = 787 nm. This value is larger than previous reports, which give ‚ab = 600 nm from measurements using a tunnel diode oscillator [22] and ‚ab = 330 nm from magnetization measurements [14].

Next, we ﬁx ‚ to the three values given above and ﬁt the ¿rev() data with only two

fitting parameters, a and . The resultant field dependence of is shown in Fig. 3-6 for the different ‚ values. The parameter is first decreasing with increasing H, reaches a minimum at H = 0.5 T, and then increases with further increasing field. As mentioned above, the cutoff field is 0.5 T. The data for H > Hcut are not reliable due to the failure of Kogan’s theory in this H region. So we conclude that the anisotropy decreases with increasing field. We note that this field dependence of in CeCoIn5 is opposite with the one for MgB2 [16], in which increases with increasing field. We found that the value of is very sensitive to the value of ‚, i.e., the larger the value of ‚, the larger the value of , with no effect however on its H dependence.

To study the temperature dependence of , we performed torque measurements in the mixed state at 1.9 K, 1.95, 2.00 K in an applied magnetic ﬁeld of 0.3 T, and determined 44

CeCoIn

0.1 Nm ) -7

2 (10 Nm)

-7

0.0 (10

0.0 0.5 1.0

H (T)

0.0 0.5 1.0 1.5 2.0

H (T)

Figure 3-5: Magnetic field H dependence of the fitting parameter fl. The solid line is a guide to the eye. Inset: Enlarged plot of the low field region of the data in the main panel. 45

3.0

CeCoIn

1.50

= 787 nm

2.5

1.45

= 330 nm

1.40

2.0

= 600 nm

0.2 0.4 0.6 0.8 1.0 1.2

H (T)

Figure 3-6: Field H dependence of the anisotropy measured at 1.9 K. 46

from Eq. (3.9). Figure 7 is a composite plot of the temperature dependence of the anisotropy. The squares give (T ) determined from torque measurements as discussed above (with ‚ (1.9 K) = 600 nm, ‚ (1.95 K) = 670 nm, and ‚ (2 K) = 740 nm taken

||c ||a from Ref. [22]), the solid circles give (T ) calculated from the ratio of Hc2 =Hc2 taken p from previous reports [23], while the triangles give (T ) determined from ‰c(T )=‰a(T ) measured in zero ﬁeld [see ‰c(T ) and ‰a(T ) in inset to Fig. 3-7]. Note that the overall trend is a decrease of the anisotropy with increasing T , with a stronger dependence around

Tc(0:3 T) = 2:23 K. The values of obtained from the torque measurements are well within the range previously reported [1, 13, 14]. The fact that the anisotropy depends both on temperature and magnetic ﬁeld could explain its relatively wide range of values reported in the literature.

The field and temperature dependence of the anisotropy implies the breakdown of the standard anisotropic GL theory, which assumes a single band anisotropic system with a temperature and field independent effective-mass anisotropy. In fact, a temperature- and

ﬁeld-dependent anisotropy was previously observed in NbSe2 [24], LuNi2B2C [25], and MgB2

[16, 26], the latter one having a value of similar with CeCoIn5. Very recently, Fletcher et al. [27] reported measurements of the temperature-dependent anisotropies (λ and ξ) of both the London penetration depth ‚ and the upper critical ﬁeld of MgB2. Their main result is that the anisotropies of the penetration depth and Hc2 in MgB2 have opposite temperature dependences, but near Tc they tend to a common value. (See Fig. 3-4 of Ref

[27]. This result conﬁrms nicely the theoretical calculation of anisotropy based on a two- gap scenario [28]. So, the temperature dependence of in CeCoIn5 and MgB2 could have a similar origin. The ﬁeld dependence of could also be related with a multiband structure.

In this scenario, the response to an applied magnetic ﬁeld may be diﬀerent in the case of the d-wave superconducting gap on the heavy-electron sheets of the Fermi surface and the 47

2.2

60 cm)

(

2.0

CeCoIn 0 20 40 60 80

1.8 T (K)

1.6

0 1 2 3 4

T (K)

Figure 3-7: Composite plot of the temperature T dependence of the anisotropy . The circles show the results obtained from the upper critical field data (Ref. [23] ), the triangles are obtained from the resistivity data shown in the inset of this figure, and the squares are from the present torque data measured in an applied magnetic field of 0.3 T. Inset: T dependence of the in-plane ‰a and out-of-plane ‰c resistivities measured in zero field. 48

negligible gap on the light, three-dimensional pockets.

3.4 Summary

Torque measurements were performed on CeCoIn5 single crystals in both the superconducting and normal states. Two contributions to the torque signal in the mixed state were identiﬁed: one from paramagnetism and the other one from the vortices. The torque curves show sharp hysteresis peaks at = 90◦ ( is the angle between H and the c−axis of the crystal) when the measurements are performed in clockwise and counterclockwise directions.

This hysteresis is a result of the intrinsic pinning of vortices, a behavior very similar to high transition temperature cuprate superconductors. The temperature and magnetic ﬁeld dependence of the anisotropy is obtained from the reversible part of the vortex torque. We

ﬁnd that decreases with increasing magnetic ﬁeld and temperature. This result indicates the breakdown of the Ginzburg-Landau theory, which is based on a single band model and provides evidence for a multiband picture for CeCoIn5, which is highly possible due to its complex Fermi topology [8, 9].

3.5 Appendix: torque measurements

When a material with a moment M is placed in a magnetic ﬁeld H, it feels a torque

¿ = M × H. If H is parallel to a symmetry axis of the single crystal, then M is parallel to

H, hence the torque is zero.

Now, we discuss the vortex torque for large ( is the anisotropy parameter of the material. For large , the vortices are conﬁned to the ab−plane of the crystal and the moment they produce is dependent on the perpendicualr component of the applied magnetic

field (H cos ) and it can only align with the c−axis, or opposite to the c−axis. Figure 3-8 shows the predicted angular dependence of the torque. Angle is defined as that made by the applied magnetic field with the c−axis of the single crystal. When = 0, H is parallel 49

Figure 3-8: Predicted angular dependence of the torque from London theory (the magnitude of the torque is in arbitary units). Inset shows the orientations of the c−axis of the crystal, the applied ﬁeld H and the magnetization M. From Ref. [17]. 50

to the c−axis of the crystal. The magnetic moment produced by vortices is diamagnetic and should be in the opposite direction of the c−axis. As increases, H moves away from c−axis, but the diamagnetic moment produced by the vortices remains in the negative c direction as long as H does not pass through the ab−plane of the crystal. When H reaches the ab−plane, H makes an angle of 90◦ with M, so the magnetic torque ¿ is maximum. As the angle increases further, H passes the ab−plane and the diamagnetic moment produced by the vortices suddenly changes direction to cancel the applied ﬁeld, i.e. M is along the c−direction. Notice that now the magnetic torque ¿ starts to decrease and ¿ reaches zero when gets to 1800. The direction of ¿ for 0 < < 90◦ is opposite to that of ¿ for

90◦ < < 180◦.

In more detail, for Hc1 ¿ H| cos | ¿ Hc2, the equilibrium vortex magnetization can be

2 approximated by the London result M = (`0=8…„0‚ ) ln(·Hc2=H| cos |), where ‚ and Hc2 are the in-plane penetration depth and the upper critical ﬁeld along the c−axis, respectively, and · is a numerical parameter of order unity. Hence, the vortex torque density is given by

`0H sin ·Hc2 ¿v() = MH sin = 2 ln( ): (3.10) 8…„0‚ H| cos | References

[1] C. Petrovic, P. G. Pagliuso, M. F. hundley, R. Movshovich, J. L. Sarrao, J. D. Thomp- son, Z. Fisk, and P. Monthoux, J. Phys. Condens. Matter. 13, L337 (2001).

[2] N. D. Mathur, F. M. Grosche, S. R. Julian, I. R. Walker, D. M. Freye, R. K. W. Haselwimmer, and G. G. Lonzarich, Nature 394, 39 (1998).

[3] R. Movshovich, M. Jaime, J. D. Thompson, C. Petrovic, Z. Fisk, P. G. Pagliuso, and J. L. Sarrao, Phys. Rev. Lett. 86, 5152 (2001).

[4] K. Izawa, H. Yamaguchi, Yuji Matsuda, H. Shishido, R. Settai, and Y. Onuki, Phys. Rev. Lett. 87, 057002 (2001).

[5] R. Bel, K. Behnia, Y. Nakajima, K. Izawa, Y. Matsuda, H. Shishido, R. Settai, and Y. Onuki, Phys. Rev. Lett. 92, 217002 (2004).

[6] V. A. Sidorov, M. Nicklas, P. G. Pagliuso, J. L. Sarrao, Y. Bang, A. V. Balatsky, and J. D. Thompson, Phys. Rev. Lett. 89, 157004 (2002).

[7] J. S. Kim, J. Alwood, G. R. Stewart, J. L. Sarrao, and J. D. Thompson, Phys. Rev. B 64, 134524 (2001).

[8] R. Settai, H. Shishido, S. Ikeda, Y. Murakawa, M. Nakashima, D. Aoki, Y. Haga, H. Harima, and Y. Onuki, J. Phys.: Condens. Matter. 13, L627 (2001).

[9] Donavan Hall, E. C. Palm, T. P. Murphy, S. W. Tozer, Z. Fisk, U. Alver, R. G. Goodrich, J. L. Sarrao, P. G. Pagliuso, and Takao Ebihara, Phys. Rev. B 64, 212508 (2001).

[10] P. M. C. Rourke, M. A. Tanatar, C. S. Turel, J. Berdeklis, C. Petrovic, and J. Y. T. Wei, Phys. Rev. Lett. 94, 107005 (2005).

[11] P. Szab´o, P. Samuely, J. Kaˇcmarˇc´ik, T. Klein, J. Marcus, D. Fruchart, S. Miraglia, C. Marcenat, and A. G. M. Jansen, Phys. Rev. Lett. 87, 137005 (2001).

[12] M. A. Tanatar, Johnpierre Paglione, S. Nakatsuji, D. G. Hawthorn, E. Boaknin, R. W. Hill, F. Ronning, M. Sutherland, Louis Taillefer, C. Petrovic, P. C. Canﬁeld, and Z. Fisk, Phys. Rev. Lett. 95, 067002 (2005).

[13] T. P. Murphy, Donavan Hall, E. C. Palm, S. W. Tozer, C. Petrovic, Z. Fisk, R. G. Goodrich, P. G. Pagliuso, J. L. Sarrao, and J. D. Thompson, Phys. Rev. B 65, 100514(R) (2002).

[14] S. Majumdar, M. R. Lees, G. Balakrishnan, and D. McK. Paul, Phys. Rev. B 68, 012504 (2003).

51 52

[15] Ken’ ichi Takahashi, Toshiyuki Atsumi, Nariaki Yamamoto, Mingxiang Xu, Hideaki Kitazawa, and Takekazu Ishida, Phys. Rev. B 66, 012501 (2002).

[16] M. Angst, R. Puzniak, A. Wisniewski, J. Jun, S. M. Kazakov, J. Karpinski, J. Roos, and H. Keller, Phys. Rev. Lett. 88, 167004 (2002).

[17] C. Bergemann, A. W. Tyler, A. P. Mackenzie, J. R. Cooper, S. R. Julian, and D. E. Farrell, Phys. Rev. B 57, 14387 (1998).

[18] Nariaki Yamamotoa, Takekazu Ishida, Kiichi Okudaa, Kenji Kurahashib, and Kazuyoshi Yamada, Physica C 298, 357 (2001).

[19] G. R. Stewart, Rev. Mod. Phys. 73, 797 (2001).

[20] Takekazu Ishida, Kiichi Okuda, Hidehito Asaoka, Yukio Kazumata, Kenji Noda, and Humihiko Takei, Phys. Rev. B 56, 11897 (1997).

[21] V. G. Kogan, Phys. Rev. B 38, 7049 (1988).

[22] S. O¨zcan, D. M. Broun, B. Morgan, R. K. W. Haselwimmer, J. L. Sarrao, Saeid Kama, C. P. Bidinosti, P. J. Turner, M. Raudsepp, and J. R. Waldram, Europhys. Lett. 62, 412 (2003).

[23] T. Tayama, A. Harita, T. Sakakibara, Y. Haga, H. Shishido, R. Settai, and Y. Onuki, Phys. Rev. B 65, 180504(R) (2002).

[24] Y. Muto, K. Noto, H. Nakatsuji, and N. Toyota, Nuovo Cimento Soc. Ital. Fis. 38B, 503 (1977).

[25] V. Metlushko, U. Welp, A. Koshelev, I. Aranson, G. W. Crabtree, and P. C. Canﬁeld, Phys. Rev. Lett. 79, 1738 (1997).

[26] C. Ferdeghini, V. Ferrando, V. Braccini, M. R. Cimberle, D. Marr´e, P. Manﬁnetti, A. Palenzona, and M. Putti, Eur. Phys. J. B 30, 147 (2002).

[27] J. D. Fletcher, A. Carrington, O. J. Taylor, S. M. Kazakov, and J. Karpinski, Phys. Rev. Lett. 95, 097005 (2005).

[28] V. G. Kogan, Phys. Rev. B 66, 020509(R) (2002). Chapter 4

Anomalous Paramagnetic magnetization in the Mixed State of CeCoIn5 Single

Crystals 1

4.1 Introduction

The interplay between superconductivity and magnetism is a subject of great interest in the study of superconductors. The heavy fermion material CeCoIn5 is a strongly correlated f-electron superconductor, which makes it a good candidate to study this effect. This material displays several novel phenomena. For example, it is in the vicinity of the antiferromagnetic quantum critical point [1, 2, 3]. As a result, its magnetic susceptibility ´ diverges at low temperature T as ´ ∝ T −0.42 (Refs. [4, 5]). Heavy electrons are essential for the development of superconductivity [6, 7, 8]. Angular-dependent thermal transport and specific heat measurements in a magnetic field provide evidence for d-wave pairing symmetry, which indicates singlet pairing [9, 10]. Nuclear magnetic resonance (NMR) measurements report suppressed spin susceptibility in the mixed state as a function of temperature [6]. Large spin fluctuations exist in this system. There is an unusually large specific heat jump at the superconducting transition temperature Tc0, which is due to the superconducting pairing, but also to strong spin fluctuations [11, 12].

A magnetic field suppresses superconductivity by coupling either to the spins or the orbital angular momenta of the electrons. If the spin effect dominates, then the material is in the Pauli limit. On the contrary, if the orbital effect dominates, then the material √ is in the orbital limit. The Maki parameter fi ≡ 2Hc20=Hp (where Hc20 is the orbital

1This chapter is based on following papers: H. Xiao, T. Hu, T. A. Sayles, M. B. Maple and C. C. Almasan, Phys. Rev. B 76, 224510 (2007) H. Xiao, T. Hu, T. A. Sayles, M. B. Maple and C. C. Almasan, Physica B 403, 952 (2008)

53 54

critical ﬁeld in the absence of the Pauli limiting and Hp is the upper critical ﬁeld limited by

Pauli paramagnetism) gives the relative strength of the orbital pair breaking by magnetic

field and Pauli limiting [13]. In the standard Bardeen-Cooper-Schrieffer (BCS) theory, the orbital effect dominates the Pauli limiting effect. This is the case for most superconducting materials. However, heavy fermion materials have large effective mass m*, so the Fermi velocity is very small, hence, the orbital effect is greatly reduced in heavy fermion materials.

In particular, CeCoIn5 has a small Fermi energy, large superconducting gap and a short coherence length. Also it is in the clean limit, with a long mean free path, which is much larger than the superconducting coherence length. The value of the Maki parameter ﬁ ≈

3.6. Hence, CeCoIn5 satisﬁes all the theoretical requirements for the formation of the

Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state [14, 15]. In addition, there is experimental evidence for the existence of the FFLO state [15, 16, 17]. All these findings support the fact that CeCoIn5 is, indeed, in the Pauli limit, which means that the spin effect dominates the orbital effect in this material at low temperatures.

The usual orbital depairing effect forms vortices in CeCoIn5 in the presence of an applied magnetic field, while the Zeeman depairing effect forces the spins to align with the field, hence destroys the spin singlet pairing required for the existence of the Cooper pairs. For these reasons, one would expect an unusual mixed state for CeCoIn5, in which the diamagnetic and paramagnetic contributions could be anomalous in the presence of the Zeeman effect. Therefore, it is important to address the issue of the paramagnetic and diamagnetic response in the mixed state and how the Pauli paramagnetism affects the mixed-state thermodynamics of CeCoIn5. This is essential to the understanding of the interaction between superconductivity and magnetism in heavy fermion materials.

We performed magnetization and torque measurements in the normal and mixed states of CeCoIn5 in order to address the above issues. We successfully separated the paramagnetic 55

and vortex contributions. The paramagnetic magnetization is unusual and it has a nonlinear magnetic field dependence, while the susceptibility ´p in the mixed state increases with increasing field. The increase of the susceptibility with increasing field is due to the fact that heavy electrons contribute to both superconductivity and paramagnetism and the

Zeeman eﬀect is large. The vortex contribution has no anomaly within the investigated temperature range, although Pauli limiting eﬀect is present in this system.

4.2 Experimental Details

The mass of the single crystal for which data are shown here is 5.5 mg and the zero-ﬁeld superconducting transition temperature Tc0 = 2:3 K.

Both dc magnetization and angular dependent torque measurements were performed in normal and mixed states, over a temperature range 1.76 K ≤ T ≤ 20 K in magnetic ﬁelds up to 14 T. The dc magnetization measurements were carried out using a superconducting quantum interference device magnetometer in magnetic ﬁelds applied parallel to the c-axis of the single crystal. The torque measurements used a piezoresistive torque magnetometer.

The single crystal was rotated in the applied magnetic ﬁeld between H k c-axis ( =

0◦) and H k a-axis ( = 90◦) and the torque was measured as a function of increasing and decreasing angle, under various temperature-ﬁeld conditions. Details regarding the background subtraction in the torque measurements can be found elsewhere [18].

4.3 Results and Discussion

Figure 4-1 shows the field dependence of the measured magnetization Mmes at 1.76 K for H k c−axis. This is a representative Mmes(H) curve in the mixed state for temperatures up to 2.10 K. For each Mmes(H) curve, we zero-field-cooled the single crystal to the desired temperature and measured the magnetization in increasing field up to 50 kOe and then decreasing the field to zero. Note that Mmes(H) is irreversible in the low field region and 56

CeCoIn

T = 1.76 K

H || c

0 40

(emu/g) (kOe) mes || c|| c2

M H

0 1 2

T (K)

-1

0 20 40 60

H (kOe)

Figure 4-1: Magnetic field H dependence of the dc magnetization Mmes measured at 1.76 K with H k c-axis on a CeCoIn5 single crystal. The solid line is a linear fit of Mmes(H) ||c in the normal state. Inset: Upper critical field parallel to the c-axis Hc2 − temperature T phase diagram. The open squares are data taken from Ref. [19] while open circles are data extracted from present Mmes(H) measurements. 57

it becomes reversible above a certain H value. Also, the magnetization increases monotonically with increasing H up to a certain value, beyond which it becomes linear in H. We

||c deﬁne this latter H value as the upper critical ﬁeld along the c-axis Hc2 (T ). ||c Plotted in the inset to Fig. 4-1 is the Hc2 (T ) phase boundary. The open squares are data taken from previous reports [19], while the open circles are data extracted from the

||c present measurements of Mmes(H), with Hc2 (T ) defined as just described above. Note that ||c all data fall on the same curve, which confirms that our definition of Hc2 (T ) is correct. Recently we reported large paramagnetism in the normal state of this material [18].

As a result, the magnetization in the mixed state has two contributions: a paramagnetic contribution and a diamagnetic contribution due to the vortices; i.e.,

Mmes = Mp + Mv: (4.1)

Also, as discussed above, in the mixed state of CeCoIn5 one expects that orbital and Zee- man depairing mechanisms coexist. As a result, CeCoIn5 could display a novel mixed state with anomalous paramagnetic and diamagnetic contributions. As a starting procedure to determine these contributions, we ﬁrst assume the simplest case in which the paramagnetic magnetization is the same in the normal and mixed states, i.e., it is linear in H and the spin susceptibility is ﬁeld independent. This has previously been done in the study of

Nd1.85Ce0.15CuO4−y (Ref. [20]). Hence, we ﬁt the linear part of the Mmes(H) curve in the normal state and extrapolate it into the low-ﬁeld region, where the sample is in the mixed state (see the solid line on the main panel of Fig. 4-1). By subtracting the paramagnetic

c c magnetization in the normal state Mn ≡ ńH as determined (ń is the normal state susceptibility in the c direction), we should obtain the field dependence of the diamagnetic magnetization.

The main panel of Fig. 4-2 shows the diamagnetic magnetization M1(H) ≡ Mmes(H) −

c ´nH at 1.76 K and over the whole measured magnetic ﬁeld range, i.e., from 0 to 50 kOe, 58

determined as just discussed above. Note that M1(H) is non-monotonic with two peaks present in −M1(H) curves: the ﬁrst peak is at a very small ﬁeld value (25 Oe for T = 1:76

K) and is very sharp. This peak corresponds to the lower critical field. A second, broader peak, however, appears at higher fields (for T = 1:76 K, this peak is in the field range 10−20 kOe). We show the enlarged non-monotonic part of the diamagnetic response M1(H) in the lower inset to Fig. 4-2 for the measured temperatures of 1.76, 1.80, 1.85, 1.90, 1.95,

2.00, 2.05 and 2.10 K, from bottom to top. As the temperature increases, the second peak becomes ﬂatter, it shifts to lower H values, and at 2.00 K it disappears and the M1(H) curve becomes monotonic. Nevertheless, even at this temperature, the M1(H) curve does not resemble a typical diamagnetic M(H) curve.

At a ﬁrst glance, the second peak in −M1(H) looks like the second magnetization peak that appears in high-temperature superconductors [21, 22], or in the UPt3 heavy fermion material [23]. The reasons for the presence of the second magnetization peak in these materials are an enhanced pinning and/or the presence of a phase transition. However, the second peak observed here in CeCoIn5 is not due to enhanced pinning since M1(H) shows only a very small hysteresis even at the lowest measured temperature of 1.76 K. Also this second peak in −M1(H) is not due to a phase transition since this peak is very broad.

This indicates that the subtraction of the linear in H paramagnetic magnetization in the mixed state, which gives rise to the second peak in −M1(H) and which assumes that the paramagnetic magnetization in the mixed and normal states is the same, is not correct. So, we conclude that there is another contribution to the paramagnetic magnetization. Under these circumstances, in principle, it is very hard to separate the vortex and paramagnetic responses. However, we show here that torque measurements along with magnetization measurements permit the successful determination of both responses.

The above discussion, which points towards the presence of another contribution to 59

CeCoIn

T = 2 K 5 0.0

H || a T = 1.76 K emu/g)

2 -1

||c (10

H (1.76 K) = 3 T 1

-0.5 M

0 20 40

H (Oe) emu/g) -1

0.0

2.10 K (10 1

-2 M emu/g) -1

-0.5 (10 1

-4 M

1.76 K

-1.0

-6

10 20 30

H (kOe)

0 20 40

H (kOe)

Figure 4-2: Magnetic ﬁeld H dependence of the magnetization M1 measured at 1.76 K which is obtained by subtracting the paramagnetic contribution as an extrapolation of the normal state paramagnetism. Lower inset: Plot of M1(H) measured at 1.76, 1.80, 1.85, 1.90, 1.95, 2.00, 2.05, and 2.10 K. Upper inset: Magnetic ﬁeld H dependence of the dc magnetization Mmes measured at 2 K for H k a. 60

the paramagnetic magnetization that is not linear in H, is consistent with the theoretical report of Adachi et al. which shows, based on quasi-classical Eilenberger formalism, that the functional form of the mixed-state paramagnetic magnetization Mp in the presence of both Zeeman and orbital eﬀects is given by [24]:

Mp = Mn[1 + f(H)] ≡ Mn + Mdev; (4.2)

where f(H) is a ﬁeld dependent function and Mdev is the deviation of the mixed-state paramagnetic magnetization from the linear in H behavior, i.e. from Mn. Therefore, in order to determine Mp, hence Mdev, one needs to determine f(H). We determine f(H) from torque measurements in the mixed and normal states, as follows.

The magnetic moment of a sample placed in a magnetic ﬁeld feels a torque ¿ ≡ M × H.

Hence, both the paramagnetic and vortex magnetizations have associated induced torques

¿p and ¿v, respectively. As we have previously shown [18], the reversible torque measured in the mixed state is given by:

¿rev(T; H; ) = ¿p + ¿v; (4.3) where

¿p(T; H; ) = ¿n[1 + f(H)] ≡

á (T ) − ć (T ) n n H2 sin 2[1 + f(H)] ≡ A(T;H) sin 2; (4.4) 2 with A(T;H) a fitting parameter, and ¿v is given by Kogan’s model [25]. Equation (4.4)

a c is valid if the magnetizations Mp and Mp along the a and c crystallographic directions, respectively, have the same H dependence; i.e., if the function f(H) is direction independent. f(H) can then be obtained from Eq. (4.4) as

A(T;H) f(H) = a c − 1; (4.5) χn−χn 2 2 H 61

a c χn−χn 2 in which A(T;H) and B ≡ 2 H are obtained by ﬁtting the torque data in the mixed and normal state, respectively. Note that f(H) = 0 in the normal state due to the fact that Mp = Mn. Therefore, by knowing f(H), one can obtain the mixed-state paramagnetic magnetization Mp from Eq. (4.2) and the vortex magnetization Mv by subtracting Mp from the measured magnetization in the mixed state [see Eq. (4.1)].

We note that the above assumption that the dependence of f(H) on direction is negligible is supported by the present torque data, which can be ﬁtted only with a A(T;H) sin 2 dependence, with no additional angular dependencies. This assumption that the magnetiza-

a c tions Mp and Mp along a and c axis have the same H dependence is, in addition, supported by previous studies. For example, we have previously shown [26] that the field-dependent in-plane normal-state resistivity data measured along the c and a crystallographic directions scale, with the anisotropy as the scaling factor (see Fig. 4-3 of the above reference). This implies that the same field dependence, hence same physics, dominates the charge transport when H is applied along the a and c directions. In another study [27], which points toward the same conclusion, the authors have shown that the difference between the response of a high temperature superconductor in the mixed state when the magnetic field is along the a and c directions is closely related with the field dependence of the upper critical field.

In fact, the authors have shown, through calculations of thermodynamic and electromag- netic properties, the presence of a similar scaling law for several thermodynamic properties, including magnetization. Since the high temperature superconductors are generally even more anisotropic than CeCoIn5, we believe that these results most likely apply also to this latter system; i.e. the spin scattering along the c and a directions of CeCoIn5 has the same

field dependence, but different coefficients, which are related with the anisotropy.

We performed torque measurements on CeCoIn5 single crystals both in the normal and mixed states. From normal-state torque measurements we obtain B = (2:39 × 10−7H2) 62

Nm, where H is in Tesla. We have already shown [18] that ¿v and ¿p can be successfully separated in the mixed state, with ¿v well described by Kogan’s model and ¿p = A sin 2

[see Eqs. (4.3) and (4.4)]. Therefore, we obtain A(H), shown in the inset to Fig. 4-3(a), by

fitting the torque data in the mixed state with Eq. (4.3). A simple fit of these A(H) data with a power law gives A(H) = (1:57×10−7H2.32) Nm, where H is in Tesla (the solid curve in the inset). Note that the magnetic field dependence of the coefficient A, which gives the

ﬁeld dependence of the paramagnetic contribution in the mixed state, is stronger than H2, which is typical for paramagnetism. Also, we note that the plot of A(H) has data points only up to 1.8 T since Kogan’s model, which gives the vortex torque, is valid only for ﬁelds

||c much smaller than Hc2 . (See Ref. [18] for more discussion.) The plot of f(H) for the field range 0 − 18 kOe, obtained from Eq. (4.5), is shown in the main panel of Fig. 4-3(a). As discussed above, by knowing f(H), one can obtain the paramagnetic and vortex magnetizations in the mixed state. Figure 3(b) shows the field dependence of different magnetization curves. The diamonds give the vortex response Mv, obtained by subtracting Mp [given by Eq. (4.2)] from Eq. (4.1). Since the analysis of the torque data is limited to magnetic fields lower than ∼ 18 kOe [18], there are no data

||c points in Mv(H) in the ﬁeld region close to Hc2 . However, a linear extrapolation of the ||c available high ﬁeld data leads exactly to Hc2 [see the dashed line in Fig. 4-3(b)]. This linear extrapolation of Mv(H) is reasonable since the vortex magnetization should be linear in H

||c when H is close to Hc2 . ||c Knowing Mv(H) up to Hc2 , permits the calculation of f(H) for H > 18 kOe [see the data points for H ≥ 20 kOe in Fig. 4-3(a)] from Eq. (4.2) with Mp given by Eq. (4.1).

Finally, knowing f(H) over the whole H range allows the determination, from Eq. (4.2), of Mdev(H), shown by the reversed solid triangles in Fig. 4-3(b), and Mp(H) shown in Fig.

4-4. Note that Mdev is a non-monotonic function of H. The shapes of Mdev(H) and M1(H) 63

(a)

0.0

CeCoIn

T = 1.8 K

-0.2

-0.4

f Nm) -7

-4

-0.6 A (10 A

0 1 2

-0.8

H (T)

0.0 M

dev

M (emu/g)

-0.1

M =M +M

1 v dev

(b)

0 10 20 30

H (kOe)

Figure 4-3: (a) Plot of field H dependence of the function f determined at 1.8 K. The solid curve is a guide to the eye. Inset: H dependence of the fitting parameter A. The solid line is a fit of the data with a simple power law. (b) H dependence of vortex magnetization Mv (solid diamonds), deviation magnetization Mdev (solid reversed triangles), and magnetization M1 data of Fig. 4-2 (open circles) of CeCoIn5 measured at 1.8 K. The dashed line in Mv(H) is a linear extrapolation of the high field data. The solid curves in Mv(H) and Mdev(H) are guides to the eye. 64

1.4

CeCoIn

1.2

T = 1.8 K

1.0 emu/g) -5

||c (10

p H

0.8

0 20 40

0.6

H (kOe) (emu/g) p M

0.4

||c

0.2

0.0

0 10 20 30 40 50

H (kOe)

Figure 4-4: Magnetic ﬁeld H dependence of the paramagnetic magnetization Mp. Inset: H dependence of diﬀerential susceptibility ´ ≡ dM=dH. The solid curve is a guide to the eye. 65

(open circles) are similar, which shows that the anomalous behavior of M1(H) is due to

Mdev(H). This reinforces the suitability of our analysis. Also note that Mp(H) is not linear in H in the mixed state.

The inset to Fig. 4-4 is a plot of the diﬀerential paramagnetic susceptibility ´p ≡ dMp=dH. Note that the paramagnetic susceptibility in the mixed state is magnetic-ﬁeld dependent, while its value is constant, equal with 1:84 × 10−5 emu/g in the normal state.

||c The jump in ´p(H) around Hc2 reflects the superconducting phase transition. ||c The Mp(H) and ´p(H) dependencies below Hc2 can be understood from the fact that the “heavy”electrons of CeCoIn5 contribute to both paramagnetism and superconductivity, and the Zeeman effect is large. Specifically, in the normal state, the large paramagnetic moment comes from the heavy fermion quasi-particles. In the mixed state, the condensation energy favors the formation of Cooper pairs with one spin up and one spin down, (note that

CeCoIn5 has a d-wave symmetry, i.e., singlet spin pairing), while the large Zeeman eﬀect decouples the spins of some of the ”heavy” electron Cooper pairs, which, hence, contribute to paramagnetism. Therefore, the magnetization Mp(H) in the mixed state is suppressed compared with the magnetization Mn(H) in the normal state. The number of decoupled

“heavy” electron spins available to align with H, hence to participate in the mixed state paramagnetism, increases with increasing H. This gives rise to an increase in ´p with increasing H. The ﬁnite value of ´p(H) as H → 0 is consistent with the ﬁnite density of quasiparticles present in the mixed state. As expected, this value of ´p is smaller than the value in the normal state, and it is very close to the value reported by NMR measurements

[6].

In materials in which the electrons responsible for paramagnetism do not participate in superconductivity (e.g., localized d or f electrons), the susceptibility in the mixed state is ﬁeld independent and hence the paramagnetic magnetization is a linear extrapolation of 66

the normal state paramagnetism. Our result of a suppressed paramagnetism in the mixed state is consistent with recent 115In and 59Co NMR measurements [6].

Note that the vortex response in the mixed state has a monotonic ﬁeld dependence [see

Mv(H) in Fig. 4-3(b)] with no anomaly observed for the investigated temperature T = 1:8

K(T=Tc = 0:78). Theorectical calculations of Adachi et al. [24] have shown an anomalous response, i.e. a change in the Mv vs. H curvature below the reduced temperature T=Tc = 0:3 with no anomaly above this reduced temperature. Hence, our experimental result conﬁrms this latter theoretical prediction.

We also measured M(H) for H k a-axis. However, in the temperature and field range investigated (T ≥ 1:76 K and H ≤ 50 kOe), no second peak was obtained after subtracting the linear paramagnetic moment (see upper inset to Fig. 4-2). Nevertheless, note that this M1(H) curve is still anomalous in the sense that there is a change of curvature, which implies that a similar anomalous paramagnetism exists in the H k a direction due to the presence of a large Zeeman effect. However, the larger upper critical field along the a-axis requires even lower temperatures and higher magnetic fields for the full observation of this effect.

4.4 Summary

In summary, we performed magnetization and torque measurements both in the normal and mixed states of CeCoIn5 single crystals to study the paramagnetic and vortex response in the presence of a large Zeeman effect present in this material. The paramagnetic magnetization is suppressed in the mixed state and the spin susceptibility is field dependent, increasing with increasing field. This H dependence is a result of the fact that heavy electrons contribute to both superconductivity and paramagnetism and the Zeeman effect is large in this material. There is no anomaly present in the vortex response in the temperature range investigated. References

[1] F. Ronning, C. Capan, A. Bianchi, R. Movshovich, A. Lacerda, M. F. Hundley, J. D. Thompson, P. G. Pagliuso, and J. L. Sarrao, Phys. Rev. B 71, 104528 (2005).

[2] Johnpierre Paglione, M. A. Tanatar, D. G. Hawthorn, Etienne Boaknin, R. W. Hill, F. Ronning, M. Sutherland, Louis Taillefer, C. Petrovic, and P. C. Canﬁeld, Phys. Rev. Lett. 91, 246405 (2003).

[3] V. A. Sidorov, M. Nicklas, P. G. Pagliuso, J. L. Sarrao, Y. Bang, A. V. Balatsky, and J. D. Thompson, Phys. Rev. Lett. 89, 157004 (2002).

[4] J. S. Kim, J. Alwood, G. R. Stewart, J. L. Sarrao, and J. D. Thompson, Phys. Rev. B 64, 134524 (2001).

[5] G. R. Stewart, Rev. Mod. Phys. 73, 797 (2001).

[6] N. J. Curro, B. Simovic, P. C. Hammel, P. G. Pagliuso, J. L. Sarrao, J. D. Thompson, and G. B. Martins, Phys. Rev. B 64, 180514(R) (2001).

[7] P. Monthoux and G. G. Lonzarich, Phys. Rev. B 63, 054529 (2001).

[8] P. Coleman, C. P´epin, and A. M. Tsvelik, Phys. Rev. B 62, 3852 (2000).

[9] K. Izawa, H. Yamaguchi, Yuji Matsuda, H. Shishido, R. Settai, and Y. Onuki, Phys. Rev. Lett. 87, 057002 (2001).

[10] H. Aoki, T. Sakakibara, H. Shishido, R. Settai, Y. O¯nuki, P. Miranovi´c, and K. Machida, J. Phys.: Condens. Matter. 16, L13 (2004).

[11] Satoru Nakatsuji, David Pines, and Zachary Fisk, Phys. Rev. Lett. 92, 016401 (2004).

[12] Yunkyu Bang and A. V. Balatsky, Phys. Rev. B 69, 212504 (2004).

[13] Kazumi Maki, Phys. Rev. 148, 362 (1966).

[14] A. Bianchi, R. Movshovich, N. Oeschler, P. Gegenwart, F. Steglich, J. D. Thompson, P.G.Pagliuso, and J. L. Sarrao, Phys. Rev. Lett. 89, 137002 (2002).

[15] A. Bianchi, R. Movshovich, C. Capan, P. G. Pagliuso, and J. L. Sarrao, Phys. Rev. Lett. 91, 187004 (2003).

[16] H. A. Radovan, N. A. Fortune, T. P. Murphy, S. T. Hannahs, E. C. Palm, S. W. Tozer, and D. Hall, Nature 425, 51 (2003).

[17] C. Martin, C. C. Agosta, S. W. Tozer, H. A. Radovan, E. C. Palm, T. P. Murphy, and J. L. Sarrao, Phys. Rev. B 71, 020503(R) (2005).

67 68

[18] H. Xiao, T. Hu, T. A. Sayles, M. B. Maple, and C. C. Almasan, Phys. Rev. B 73, 184511 (2006).

[19] T. Tayama, A. Harita, T. Sakakibara, Y. Haga, H. Shishido, R. Settai, and Y. Onuki, Phys. Rev. B 65, 180504(R) (2002).

[20] F. Zuo, S. Khizroev, Xiuguang Jiang, J. L. Peng, and R. L. Greene, Phys. Rev. Lett. 72, 1746 (1994).

[21] L. Miu, T. Noji, Y. Koike, E. Cimpoiasu, T. Stein, and C. C. Almasan, Phys. Rev. B 62, 15172 (2000).

[22] Nurit Avraham, Boris Khaykovich, Yuri Myasoedov, Michael Rappaport, Hadas Shtrik- man, Dima E. Feldman, Tsuyoshi Tamegai, Peter H. Kes, Ming Li, Marcin Kon- czykowski, Kees van der Beek, and Eli Zeldov, Nature 411, 451 (2001).

[23] Kenichi Tenya, Masataka Ikeda, Takashi Tayama, Hiroyuki Mitamura, Hiroshi Amit- suka, Toshiro Sakakibara, Kunihiko Maezawa, Noriaki Kimura, Rikio Settai, and Yoshichika O¯nuki, J. Phys. Soc. Jpn. 64, 1063 (1995).

[24] Hiroto Adachi, Masanori Ichioka, and Kazushige Machida, J. Phys. Soc. Jpn. 74, 2181 (2005).

[25] V. G. Kogan, Phys. Rev. B 38, 7049 (1988).

[26] T. Hu, H. Xiao, T. A. Sayles, M. B. Maple, Kazumi Maki, B. D´ora, and C. C. Almasan, Phys. Rev. B 73, 134509 (2006).

[27] Zhidong Hao and John R. Clem, Phys. Rev. B 46, 5853(R) (1992). Chapter 5

1 Pairing Symmetry of CeCoIn5 Detected by In-plane Torque Measurements

5.1 Introduction

Superconductivity in the heavy fermion superconductor CeCoIn5 is unconventional as exemplified by the non-Fermi liquid behavior [1, 2, 3], the giant Nernst effect present in the normal state [4], its proximity to quantum critical points [5, 6], the Pauli limiting effect [7, 8, 9], and the possible multiband picture in the superconducting state [10, 11,

12]. Unconventional superconductivity is always a subject of great interest. Knowing the pairing symmetry, which is related to the ground state and gap energy, is essential to the understanding of the pairing mechanism and the origin of superconductivity. For a conventional superconductor, which is described by the BCS theory, the pairing is phonon mediated and the pairing symmetry is s wave. For an unconventional superconductor, the quasiparticle gap vanishes at certain points on the Fermi surface. For example, most of the experimental evidence on high temperature superconductors, such as YBa2Cu3O7−δ, indicates that the dx2−y2 wave symmetry dominates in these materials.

It has been established that the superconducting order parameter of CeCoIn5 displays d- wave symmetry. 115In and 59Co nuclear magnetic resonance measurements [13] and torque measurements [14] have revealed a suppressed spin susceptibility, which implies singlet spin pairing. A T 2 term is present in the low-temperature T speciﬁc heat, consistent with the presence of nodes in the superconducting energy gap [15]. Nuclear quadrupole resonance and nuclear magnetic resonance measurements on CeCoIn5 have revealed that

1This chapter is based on following paper: H. Xiao, T. Hu, T. A. Sayles, M. B. Maple and C. C. Almasan, Phys. Rev. B 78, 014510 (2008)

69 70

the nuclear spin lattice relaxation rate 1=T1 has no Hebel-Slichter coherence peak just

3 below the superconducting transition temperature Tc, and it has a T dependence at very low temperatures, which indicates the existence of line nodes in the superconducting energy gap [16].

Nevertheless, the direction of the gap nodes relative to the Brillouin zone axes, which determines the type of d-wave state, namely dx2−y2 or dxy, is still an open question and an extremely controversial issue. For example, angular-dependent thermal conductivity measurements in a magnetic field have revealed four-fold symmetry consistent with dx2−y2 symmetry [17]. Neutron scattering experiments by Eskildsen et al. [18] revealed a square lattice oriented along [110], also consistent with dx2−y2 wave symmetry. However, field- angle-dependent specific heat measurements have found the symmetry of the superconducting gap to be dxy [19]. More recent magnetoresistance [20] and neutron [21] data have shown dx2−y2 symmetry. Furthermore, theoretical calculations by Ikeda et al. [22] strongly sug-

∗ gest dxy wave symmetry when taking into account the available T experimental data of the

∗ FFLO states. [T is the temperature at which the upper critical field Hc2(T ) changes from second order to first order.] In contrast, recent calculations by Tanaka et al. [23] based on the Fermi liquid theory and by Vorontsov et al. [24] based on a unified microscopic approach, support the dx2−y2 gap symmetry.

This extremely controversial issue needs to be resolved through an experimental technique that allows the direct measurement of the nodal positions. The experiments that are phase sensitive usually include surface or boundary eﬀects, while those that detect bulk properties are not phase sensitive. In the study presented here we use torque measurements to clarify the gap symmetry. Torque is a bulk measurement so it provides information on the order parameter of the bulk, not only the surface. (The order parameter of the surface might be diﬀerent from that of the bulk.) It also directly probes the nodal positions on the 71

Fermi surface with high angular resolution since torque is the angular derivative of the free energy. Hence, such an experimental technique is ideal to determine the direction of the gap nodes relative to the Brillouin zone axes and, in fact, it has already been successfully used to identify the nodal positions of untwinned YBa2Cu3O7−δ single crystals [25] and

Tl2Ba2CuO6+δ thin ﬁlms [26]. In addition, recent theoretical calculations by H. Adachi [27] have shown that low-ﬁeld torque measurements can be used to detect the nodal positions of a d-wave superconductor with a small Fermi-surface anisotropy, as is the case of CeCoIn5.

In-plane torque measurements were performed on single crystals of CeCoIn5 both in the normal state and in the mixed state. The reversible part of the angular-dependent mixed- state torque data, obtained after subtracting the corresponding normal-state torque, shows a four-fold symmetry with a positive coefficient. Sharp peaks in the irreversible torque data were observed at angles equal with …=4, 3…/4, 5…/4, and 7…/4, etc. The symmetry of the free energy extracted from the four-fold symmetry of the reversible torque of the mixed state coupled with the position of the sharp peaks in the irreversible torque of the mixed state point unambiguously towards dxy wave symmetry in CeCoIn5. Further support for the dxy wave symmetry is provided by the H and T dependence of the mixed-state reversible torque. Normal-state torque shows a four-fold symmetry. Nevertheless, the T and H dependences of the coefficients of the four-fold torques in the normal and mixed states are different. Hence, these two four-fold symmetries have clearly different origin.

5.2 Experimental Details

A piezoresistive torque magnetometer was used to measure the angular dependence of the in-plane torque of CeCoIn5 both in the normal state and mixed state. The torque was measured over a large temperature range (1.8 K ≤ T ≤ 10 K) and magnetic ﬁeld H range

(1.5 T ≤ H ≤ 14 T) by rotating the single crystal in fixed magnetic field. The angle ’ for the in-plane rotation was defined as the angle made by the field with the a-axis of the single 72

crystal. The contributions of the gravity and puck to the total torque signal were measured and subtracted from it as discussed elsewhere [12].

The experiments were carried out in a Physical Property Measurement System (PPMS).

In such a system with a one axis rotator, it is very diﬃcult to ensure an in-plane alignment of better than about ±3◦. If misalignment exists, i.e. the magnetic ﬁeld is not completely within the ab plane of the single crystal, there should be a sin 2’ term (see Eq. (6) of

Ref. [28]). Indeed, the angular-dependent in-plane torque signal has a sin 2’ term [ﬁ and

’ of Eq. (6) are ’ and −23:70, respectively, in the present case], which we attribute to the misalignment of the single crystal, in addition to the sin 4’ term. In fact, the amplitude of the measured sin 2’ term gives a misalignment ≈ 3:70. The torque data shown in this paper are after subtracting this sin 2’ term.

5.3 Results and Discussion

Previously, we have shown that the b-axis rotation torque signal measured in the mixed state has a paramagnetic component that is comparable with the diamagnetic component

[12]. The former component is a result of the anisotropy of the susceptibilities along the a and c axes. Therefore, such a paramagnetic torque signal is absent in the present measurements in which the torque is measured while rotating the single crystal along the c axis, since ´a ≈ ´b. Nevertheless, T - and H- dependent torque measurements in the normal state reveal that the normal-state torque signal is not negligible, is reversible, and it has a four-fold symmetry [see inset to Fig. 5-1(b)]. The solid curve is a ﬁt of the data with

¿n = An sin 4’. The ﬁeld and temperature dependence of the amplitude An gives the H

4 and T dependence of the normal-state torque. The coeﬃcient An has an H dependence up to 14 T for all measured temperatures from 1.9 K to 10 K [see Fig. 5-1(a)]. As the temperature increases, the slopes in Fig. 5-1(a) decrease. This is consistent with the temperature

4 4 dependence of H =An shown in Fig. 5-1(b); i.e., H =An increases, hence An decreases, with 73

(a) 1.9 K CeCoIn

3 K

m = 7.6 mg

4.5 K

5 K

4 6 K Nm) -7

8 K

(10 10 K n 2 A

0 2 4

4 4 4

H (10 T )

2.0

(b)

H = 14 T

1.6

4 T= 1.9 K /Nm) 4 1.2

H = 7 T

T 2 11 Nm)

-8

0.8 (10 (10 n n

-2 /A 4

0.4 H -4

0 90 180

(deg)

0.0

0 4 8 12 16 20

T (K)

Figure 5-1: Field H and temperature T dependence of the amplitude An of the normal-state torque of CeCoIn5 single crystals. (a) H dependence of An at measured temperatures of 1.9, 3, 4.5, 5, 6, 8, and 10 K. (b) T dependence of An at measured magnetic ﬁeld of 14 T. Inset: Angular ’ dependent torque ¿n measured in the normal state at 1:9 K and 7 T in CeCoIn5 single crystals. The solid curve is a ﬁt of the data with ¿n = An sin 4’. 74

4 increasing T . A straight-line ﬁt of the data with H =An = c(1 + T=T0) gives T0 = 2:2 K, which has a value close to the single ion Kondo temperature TK (reported to be between

10 4 −1 −1 4 −1 1 and 2 K, Ref. [29]) and c = 3 × 10 T N m . Hence, An(H;T ) ∝ H (1 + T=TK ) .

Therefore, in approaching the superconducting transition, An decreases with decreasing H and increases with decreasing T . We subtracted the corresponding normal-state torque from the torque measured in the superconducting state.

The torque data measured in the mixed state of CeCoIn5, obtained by subtracting the corresponding normal-state torque from the measured torque, have both reversible and irreversible components. The reversible torque ¿rev is the average of the torque data measured in clockwise and anti-clockwise directions, while the irreversible torque ¿irr is the average of the antisymmetric components of the torque data measured in clockwise and anti-clockwise directions. Figure 5-2(a) shows the reversible part of the angular-dependent in-plane torque data measured in the mixed state at 1.9 K and in a magnetic ﬁeld of 3 T. Clearly, there is a four-fold symmetry present in the torque data, although the data points are somewhat scat- tered. The solid curve is a ﬁt of these mixed-state data with ¿rev(H;T;’) = Am(T;H) sin 4’.

The coeﬃcient Am is positive since the torque displays a maximum at …=8. Figure 5-2(b) shows the irreversible part of the mixed-state torque data ¿irr(’) measured at T = 1:9 K and H = 1 T for a single crystal with a mass of 2.6 mg. The ¿irr(’) data have sharp peaks at …=4, 3…/4, 5…/4, and 7…/4.

The nodal positions of CeCoIn5 can be obtained from the reversible and irreversible mixed-state torque data, as previously done in the study of YBa2Cu3O7 [25]. Speciﬁcally,

k theoretical calculations predict that the in-plane upper critical ﬁeld Hc2 has a four-fold symmetry for a d-wave superconductor [30]. In the case of dxy wave symmetry, the angular

k variation of the upper critical ﬁeld ∆Hc2 ∝ − cos 4’; hence, it has maxima at …=4, 3…/4, 5…/4 and 7…/4. Figure 5-3 shows the angular dependence of the reversible and irreversible 75

CeCoIn 5 5 T = 1.9 K H = 3 T Nm)

-9 m=7.6 mg

(10 0 rev τ

-5 (a)

0 90 180 ϕ (deg)

CeCoIn T = 1.9 K, H = 1 T, m = 2.6 mg 4 5 Nm) -8 (10

irr 2 τ

(b) 0 0 90 180 270 360 ϕ (deg)

Figure 5-2: (a) Angular ’ dependence of the reversible torque ¿rev measured at 1.9 K and 3 T on CeCoIn5 single crystals. The solid curve is a ﬁt of the data with ¿ = A sin 4’. (b) Angular dependence of the irreversible torque ¿irr measured at 1.9 K and 1 T on CeCoIn5 single crystals. Sharp peaks are present at …=4, 3…=4, 5…=4 and 7…=4. 76

torque obtained by starting from this angular dependence of Hc2, as follows. The lower

k k critical ﬁeld Hc1 is out of phase with Hc2 (see Fig. 5-3) since the thermodynamic critical ﬁeld

2 Hc = Hc1Hc2 is independent of the magnetic ﬁeld orientation. Therefore, the magnetization

M, given by M' −Hc1ln(Hc2=H)=ln•, has the same angular dependence as Hc2 (see Fig.

5-3). The easy axis of magnetization (maximum magnetization) should correspond to free energy F minima. This implies that, for the dxy symmetry, F has minima at …=4, 3…/4,

5…/4 and 7…/4. (see Fig. 5-3). The torque is the angular derivative of the free energy F ; i.e., ¿ = −@F=@’. Hence, the reversible torque data for a material with dxy wave symmetry should display a four-fold symmetry with maxima at …=8, 5…=8, 9…=8, etc (see Fig. 5-3).

Also, the free-energy minima act as intrinsic pinning centers for vortices, so the irreversible torque data for a material with dxy wave symmetry should display peaks at the same angles at which the free energy has minima; i.e., at …=4, 3…/4, 5…/4 and 7…/4.

Notice that the ’ dependence of reversible and irreversible torque data of CeCoIn5 shown in Figs. 5-2(a) and 5-2(b) is the same as the ’ dependence of ¿rev and ¿irr, respectively, shown in Fig. 5-3, obtained from the theoretically predicted angular dependence of the upper critical ﬁelds for a material with dxy symmetry. We hence conclude that the reversible along with the irreversible torque data in the mixed state unambiguously imply that the wave symmetry of CeCoIn5 is dxy. The fact that the angular dependences of the reversible and irreversible torques, with the latter not being aﬀected by the subtraction of the reversible normal-state torque in the mixed state, give the same angular dependence of the free energy indicates that there is no error in the subtraction of the normal-state contribution to obtain the reversible torque in the mixed state; i.e. the assumption we made that the normal-state contribution in the mixed state is just a simple extrapolation of its behavior above Tc is correct. 77

rev

irr

0 45 90 135 180 225 270 315 360

(deg)

Figure 5-3: Plot of the angular ’ dependence of the upper critical field Hc2 and lower critical field Hc1 for magnetic field parallel to the ab plane, magnetization M, free energy F , reversible torque ¿rev, and irreversible torque ¿irr for dxy wave symmetry. 78

To understand further the four-fold symmetry displayed by the present torque measurements in the mixed state, we studied the ﬁeld and temperature dependence of the amplitude

Am. Figure 4 is a plot of the H dependence of Am, which gives the H dependence of the torque, obtained by fitting the angular-dependent torque data measured in different magnetic fields. Note that Am(H) increases with increasing H, reaches a maximum, and then decreases with further increasing H. Also note that the four-fold symmetry vanishes close

k k to Hc2 (Hc2 = 6 T). This ﬁeld dependence of the magnitude of Am is the same as the

ﬁeld dependence of the basal-plane reversible torque in the mixed state of a layered dx2−y2 wave superconductor (see Fig. 8(a) of Ref. [27]) calculated by Adachi et al. [27] based on the quasiclassical version of the BCS-Gor’kov theory with a Fermi surface that is isotropic within the basal plane. The sign diﬀerence between the data of the present Fig. 5-4 and

Fig. 8(a) of Ref. [27], which is for a dx2−y2 wave symmetry, further indicates that the present data reﬂect dxy symmetry since the torque data have opposite signs for the dxy and dx2−y2 wave symmetries.

The inset to Fig. 5-4 is a plot of the temperature dependence of the amplitude Am.

Note that Am, hence the torque, decreases with increasing T and vanishes towards Tc. The fact that both the T and H dependences of the reversible mixed-state torque vanish at the superconducting - normal state phase boundary further indicates that the observed four-fold symmetry is related with superconductivity; hence, it reﬂects the gap symmetry.

We note that the behaviors of Am(H;T ) and An(H;T ) are totally diﬀerent (compare

Figs. 5-1 and 5-4). So the four-fold symmetries present in normal and mixed states have diﬀerent origins. The origin of the four-fold symmetry in the normal state is not yet clear.

Torque measurements on LaCoIn5 single crystals, which also have a tetragonal structure but are not superconducting and the f electrons are absent, give some clues on the normal state four-fold symmetry of CeCoIn5. The angular-dependent torque data for LaCoIn5 are 79

CeCoIn 8 5 T = 1.8 K m = 7.6 mg

8 H = 3 T Nm)

-9 4 6 (10 Nm) m

-9

A 4 (10 m 2 A 2

0 1.8 1.9 2.0 T (K) 0 1 2 3 4 5 6 H (T)

Figure 5-4: Field H dependence of the amplitude Am of the reversible torque in the mixed state of CeCoIn5 single crystals measured at T = 1:8 K. The solid curve is a guide to the eye. Inset: Temperature T dependence of Am measured at H = 3 T. The solid curve is a guide to the eye. 80

shown in Fig. 5-5. Clearly, the sin 4’ symmetry observed in CeCoIn5 is completely absent here. The difference in the normal-state torque between CeCoIn5 and LaCoIn5 could be due to the presence of heavy electrons in the former compound and their absence in the latter one. The crystalline electric field, which is important in heavy fermion systems, might be responsible for the four-fold symmetry in the normal-state torque. Also, the field induced order, possibly quadrupolar order, could be another reason for the four-fold symmetry in the normal-state torque of CeCoIn5. No doubt, the origin of this normal-state four-fold symmetry present in the torque data requires further study. Nevertheless, this is beyond the scope of this dissertation.

Recently, Weickert et al. [20] tried to determine directly the anisotropy of the upper critical ﬁeld in the basal plane from measurements of resistance R as a function of the angle

’ and magnetic ﬁeld. Their results give a dx2−y2 gap symmetry. To shed further light on the discrepancy between these results and the torque results shown here, we also measured the resistance Rc of CeCoIn5 in the c direction with the applied ﬁeld in the ab plane at 1.8 K

(same experimental condition as for the torque measurement reported here). Figure 5-6 is a plot of Rc(H) measured at different ’ values. Notice the Rc(H) curves cross in the transition region such that the angular-dependent magnetoresistance ∆R ≡ R(’) − R(’ = 0) is positive at low dissipation and negative at high dissipation, all the way into the normal state of CeCoIn5. The fact that the sign of the angular magnetoresistance changes in the superconducting transition region and since the thermodynamic upper critical field is ill defined show that one cannot determine Hc2(’) accurately from the R(H;’) curves.

In fact, as shown in the inset to Fig. 5-6, the four-fold part of Rc(’) gives an angular dependence consistent with dxy (black symbols) or dx2−y2 (red symbols) symmetry when the upper critical ﬁeld is chosen in the transition region close to the onset (high dissipation) or close to zero resistance (low dissipation, as Weickert et al. did [20]), respectively. We 81

LaCoIn T = 1.9 K, H = 14 T, m = 2.2 mg

2 Nm)

-7 0 (10

-2

-4

0 90 180 270 360

(deg)

Figure 5-5: Angular ’ dependent torque ¿ measured at 1.9 K and 14 T on LaCoIn5 single crystals. 82

1.5

1.0 0

) 50

90 H = 6 T

-4 11.0 60 H = 4.6 T

135 45 70 5.5 (10 c

R 80

0.0

0 )

90 -6 -5.5

0.5 (10

-11.0 180 0 4 R

-5.5

0.0

5.5

225 315

11.0

0.0

270

4.0 4.5 5.0 5.5 6.0

H (T)

Figure 5-6: Plot of out-of-plane resistance Rc vs. applied magnetic ﬁeld H measured at a diﬀerent in-plane angle ’ between H and the a axis. Inset: Polar plot of the four-fold symmetric component R4ϕ of the out-of-plane resistance measured for two H values. 83

note that if the in-plane angular-dependent resistance Rc(’) in the mixed state gives Hc2(’) of CeCoIn5, then the position of the maximum (minimum) resistance corresponds to the position of the minimum (maximum) Hc2 for the following reason. Tc shifts to higher or lower temperatures as the angle of the applied field is varied in the basal plane. Therefore, by keeping the actual temperature T = Tc fixed, a rotation of the magnetic field results in a variation of the measured resistance if Hc2 changes with direction such that Rc(’) and

Hc2(’) are out of phase [31]. We also note that, in addition to the four-fold symmetry in

Rc(’), shown in the inset to Fig. 5-6, there is a two-fold symmetry in Rc(’), which has previously been observed and associated with background [20].

We want to emphasize that the measurement of resistance is not a thermodynamic measurement, hence the determination of the upper critical field and its angular dependence from such a measurement is not reliable. In fact it is well known that the extraction of Hc2 from the resistive transition is inappropriate for unconventional superconductors such as both hole-doped and the lower Tc electron-doped cuprates. Actually, the magnetoresistance of a superconductor in the mixed state is due to vortex dissipation (as a result of vortex motion), hence it depends on the strength of the vortex pinning centers. Therefore, if one defines Hc2 close to zero resistivity, as Weickert et al. [20] did, the such determined “Hc2(’)” is in fact the irreversibility field Hirr(’); hence, it most likely reflects the pinning anisotropy of vortices instead of the in-plane anisotropy of the thermodynamic upper critical field.

5.4 Summary

In-plane angular-dependent torque measurements were performed on CeCoIn5 single crystals both in the normal and mixed states. Normal-state torque measurements show a four-fold symmetry. The reversible torque in the mixed-state, obtained after subtracting the corresponding normal state contribution, also shows a four-fold symmetry with maxima at …=8, 5…=8, 9…=8, etc. Sharp peaks in the irreversible torque data were observed at …=4, 84

3…=4, 5…=4, etc. These latter peaks correspond to minima in the free energy of a dxy wave symmetry. The mixed state four-fold symmetry and the peak positions in the irreversible torque point unambiguously towards dxy wave symmetry of the superconducting gap. The

field and temperature dependencies of the amplitude of the normal-state torque are different from that of the mixed-state torque, which indicates that the normal-state torque has a different origin. References

[1] C. Petrovic, P. G. Pagliuso, M. F. hundley, R. Movshovich, J. L. Sarrao, J. D. Thomp- son, Z. Fisk, and P. Monthoux, J. Phys. Condens. Matter. 13, L337 (2001).

[2] V. A. Sidorov, M. Nicklas, P. G. Pagliuso, J. L. Sarrao, Y. Bang, A. V. Balatsky, and J. D. Thompson, Phys. Rev. Lett. 89, 157004 (2002).

[3] J. S. Kim, J. Alwood, G. R. Stewart, J. L. Sarrao, and J. D. Thompson, Phys. Rev. B 64, 134524 (2001).

[4] R. Bel, K. Behnia, Y. Nakajima, K. Izawa, Y. Matsuda, H. Shishido, R. Settai, and Y. Onuki, Phys. Rev. Lett. 92, 217002 (2004).

[5] Johnpierre Paglione, M. A. Tanatar, D. G. Hawthorn, Etienne Boaknin, R. W. Hill, F. Ronning, M. Sutherland, Louis Taillefer, C. Petrovic, and P. C. Canﬁeld, Phys. Rev. Lett. 91, 246405 (2003).

[6] F. Ronning, C. Capan, A. Bianchi, R. Movshovich, A. Lacerda, M. F. Hundley, J. D. Thompson, P. G. Pagliuso and J. L. Sarrao, Phys. Rev. B 71, 104528 (2005).

[7] H. A. Radovan, N. A. Fortune, T. P. Murphy, S. T. Hannahs, E. C. Palm, S. W. Tozer, and D. Hall, Nature 425, 51 (2003).

[8] A. Bianchi, R. Movshovich, N. Oeschler, P. Gegenwart, F. Steglich, J. D. Thompson, P.G.Pagliuso, and J. L. Sarrao, Phys. Rev. Lett. 89, 137002 (2002).

[9] C. Martin, C. C. Agosta, S. W. Tozer, H. A. Radovan, E. C. Palm, T. P. Murphy, and J. L. Sarrao, Phys. Rev. B 71, 020503(R) (2005).

[10] P. M. C. Rourke, M. A. Tanatar, C. S. Turel, J. Berdeklis, C. Petrovic, and J. Y. T. Wei, Phys. Rev. Lett. 94, 107005 (2005).

[11] M. A. Tanatar, Johnpierre Paglione, S. Nakatsuji, D. G. Hawthorn, E. Boaknin, R. W. Hill, F. Ronning, M. Sutherland, Louis Taillefer, C. Petrovic, P. C. Canﬁeld, and Z. Fisk, Phys. Rev. Lett. 95, 067002 (2005).

[12] H. Xiao, T. Hu, T. A. Sayles, M. B. Maple, and C. C. Almasan, Phys. Rev. B 73, 184511 (2006).

[13] N. J. Curro, B. Simovic, P. C. Hammel, P. G. Pagliuso, J. L. Sarrao, J. D. Thompson, and G. B. Martins, Phys. Rev. B 64, 180514(R) (2001).

[14] H. Xiao, T. Hu, T. A. Sayles, M. B. Maple, and C. C. Almasan, Phys. Rev. B 76, 224510 (2007).

85 86

[15] R. Movshovich, M. Jaime, J. D. Thompson, C. Petrovic, Z. Fisk, P. G. Pagliuso, and J. L. Sarrao, Phys. Rev. Lett. 86, 5152 (2001).

[16] Y. Kohori, Y. Yamato, Y. Iwamoto, T. Kohara, E. D. Bauer, M. B. Maple, and J. L. Sarrao, Phys. Rev. B 64, 134526 (2001).

[17] K. Izawa, H. Yamaguchi, Yuji Matsuda, H. Shishido, R. Settai, and Y. Onuki, Phys. Rev. Lett. 87, 057002 (2001).

[18] Morten Ring Eskildsen, Charles D. Dewhurst, Bart W. Hoogenboom, Cedomir Petro- vic, and Paul C. Canﬁeld, Phys. Rev. Lett. 90, 187001 (2003).

[19] H. Aoki, T. Sakakibara, H. Shishido, R. Settai, Y. O¯nuki, P. Miranovi´c, and K. Machida, J. Phys.: Condens. Matter. 16, L13 (2004).

[20] Franziska Weickert, Philipp Gegenwart, Hyekyung Won, David Parker, and Kazumi Maki, Phys. Rev. B 74, 134511 (2006).

[21] C. Stock, C. Broholm, J. Hudis, H. J. Kang, and C. Petrovic, Phys. Rev. Lett. 100, 087001 (2008).

[22] Ryusuke Ikeda and Hiroto Adachi, Phys. Rev. B 69, 212506 (2004).

[23] K. Tanaka, H. Ikeda, Y. Nisikawa, and K. Yamada, J. Phys. Soc. Jpn. 75 suppl., 250 (2006).

[24] A. Vorontsov and I. Vekhter, Phys. Rev. Lett. 96, 237001 (2006).

[25] Takekazu Ishida, Kiichi Okuda, Hidehito Asaoka, Yukio Kazumata, Kenji Noda, and Humihiko Takei, Phys. Rev. B 56,11897 (1997).

[26] M. Willemin, C. Rossel, J. Hofer, H. Keller, Z. F. Ren, and J. H. Wang, Phys. Rev. B 57, 6137 (1998).

[27] H. Adachi, P. Miranovi´c, M. Ichioka, and K. Machida, J. Phys. Soc. Jpn. 75, 084716 (2006).

[28] I. Aviani, M. Miljak, V. Zlati´c, K. D. Schotte, C. Geibel, and F. Steglich, Phys. Rev. B 64, 184438 (2001).

[29] S. Nakatsuji, S. Yeo, L. Balicas, Z. Fisk, P. Schlottmann, P. G. Pagliuso, N. O. Moreno, J. L. Sarrao, and J. D. Thompson, Phys. Rev. Lett. 89, 106402 (2002).

[30] K. Takanaka and K. Kuboya, Phys. Rev. Lett. 75, 323 (1995).

[31] N. Keller, J. L. Tholence, A. Huxley, and J. Flouquet, Phys. Rev. Lett. 73, 2364 (1994). Chapter 6

1 Angular Resistivity Study in CeCoIn5 Single Crystals

6.1 Introduction

Thermal conductivity measurements indicate that CeCoIn5 is in the superclean regime, in which vortex viscosity may be greatly enhanced and leads to anomalous vortex dynamics

[1]. It is of great interest to know how vortices behave in a magnetic ﬁeld and to compare this behavior with that obtained in high transition temperature Tc superconductors.

6.2 Experimental Details

Angular-dependent resistivity measurements were performed at 2.3 K for H ≤ 1 T. It was found that the resistivity curves scale with the perpendicular field component H cos , which is a result of flux-flow dissipation. We also obtained the explicit functional dependence of the resistivity on field and angle, which is consistent with the scaling observed experimentally.

The single crystal has a Tc0 = 2.3 K in zero ﬁeld. The in-plane resistivity ‰ab was determined using an algorithm described elsewhere [2]. The samples were rotated in an applied magnetic

ﬁeld from H k c−axis ( = 0◦) to H k I k a−axis ( = 90◦).

6.3 Results and Discussions

Typical resistivity curves are shown in Fig. 6-1. The resistivity is largest at = 0◦ and decreases monotonically as increases. At = 90◦, the resistivity has a nonzero minimum, therefore there is still a small amount of dissipation in the system.

In another protocol, we scanned the ﬁeld and measured ‰ab at ﬁxed angles. A plot of ‰ab

1This chapter is based on following paper: H. Xiao, T. Hu, T. A. Sayles, M. B. Maple and C. C. Almasan, AIP Conference Proceedings, Volume 850, 719 (2006)

87 88

vs. the perpendicular ﬁeld component H cos , measured at diﬀerent angles, is shown in Fig.

◦ 6-2. For angles smaller than a critical angle c ≈ 54 , the diﬀerent resistivity curves overlap, i.e. the resistivity depends only on the perpendicular ﬁeld component: ‰(H; ) = ‰(H cos ).

Similar scaling behavior in the mixed state was previously reported in Bi2Sr2CaCu2O8

[3] and MgB2 [4]. In highly anisotropic superconductors like Bi2Sr2CaCu2O8, the magnetic

ﬁeld penetrates in the form of two dimensional pancake vortices, therefore vortex dissipation depends only on, hence scales with, the ﬁeld component perpendicular to the ab−planes

[5]. On the other hand, based on the anisotropic Ginzburg-Landau theory, Blatter et al.

[6] obtained the same scaling expression; i.e., ‰(H; ) = ‰(H; †; ) = ‰(H cos ) for < c,

2 −2 2 1/2 where †θ = (cos + sin ) for superconductors with anisotropy . Therefore, this general relationship explains the scaling of the resistivity observed in the mixed state of both highly anisotropic as well as less anisotropic superconductors. The above scaling implies that the resistivity should depend on ﬁeld and angle as H cos for < c. Also note that the data of Fig. 6-2 show that ‰(H cos ) is nonlinear. Next we obtain an explicit functional dependence for resistivity in the mixed state. Consider one coordinate frame

(x; y; z) associated with the crystallographic axes in which the current I is along the y-axis.

Another coordinate frame (x0; y0; z0) is obtained by rotating the ﬁrst one around the x-axis such that z0 is always along the magnetic ﬁeld H and is the angle between z and z0, hence, the c-axis and H (see inset to Fig. 6-3). Based on the dissipation energy conservation and the continuity equation, we write:

2 2 2 j j 0 0 j yy = y y + z0z0 (6.1) (yy) (y0y0) (z0z0) ab ab ab and

1 cos2 sin2 = + ; (6.2) (yy) (y0y0) (z0z0) ab ab ab respectively. The first term on the right-hand side of Eq. (6.2) is the flux-flow dissipation, 89

5 T = 2.3 K

1.0 T

0.4 T

0.2 T

0.1 T cm)

3 (

0 45 90 135 180

(deg)

Figure 6-1: Resistivity ‰ab vs. angle measured at 2.3 K. The solid lines are ﬁts of the data with Eq. (6.3). 90

0.6

0.5

0.4

T = 2.3K cm)

R10 0.3 ( R20 ab

R30

R40

0.2

R50

R54

R60

0.1 R70

R80

R88

0.0

0.0 0.2 0.4 0.6 0.8 1.0

Hcos (T)

Figure 6-2: Resistivity ‰ab vs. the perpendicular ﬁeld component H cos , measured at 2.3 K and ﬁxed angles . 91

5 j

z'z'

j y' 1 y'y' m

T = 2.3 K

0.0 0.2 0.4 0.6 0.8 1.0

H (T)

Figure 6-3: The ﬁeld H dependence of the ﬁtting parameter m1. The solid line is a guide to the eye. 92

while the second term is a result of quasiparticle dissipation and thermal activation, which explains the nonzero resistivity at = 90◦. Based on the time-dependent Ginzburg-Landau

(GL) theory, Kopnin calculated the ﬂux-ﬂow conductivity for anisotropic superconductors

(y0y0) 2 2 [7]: = uaHc2()n()=2H, with u = » =lE (» is the coherence length and lE is a characteristic length that determines the scale of spatial variations of the gauge-invariant potential `), a is a coeﬃcient given by numerical calculations using the vortex order pa-

2 −2 2 1/2 rameter obtained by solving the GL equation, and Hc2() = Hc2=(cos + sin ) .

With this expression for (y0y0), Eq. (6.2) becomes:

m cos2 ‰(yy) = p 1 + fl sin2 ; (6.3) ab (cos2 + −2 sin2 ) where m1 = 2H‰n=uaHc2 and fl are fitting parameters; fl represents the quasiparticle dissipation. The solid lines in Fig. 6-1 are fits of the data with Eq. (6.3). Note that Eq.

(6.3) describes the experimental data very well except in the vicinity of 90◦. This small discrepancy could be the result of the lock-in transition in which the dissipation is greatly

◦ reduced around 90 . The field dependence of m1 is plotted in Fig. 6-3. Note that m1 is linear in H, which is consistent with its definition. Equation (6.3) is also consistent with the scaling ‰(H cos ). Indeed, it gives the scaling when the second term is substantially smaller than the first term and when −2 sin2 ¿ cos2 , i.e. ¿ tan−1 ≈ 63◦( ≈ 2).

The first condition is satisfied since in the mixed state the flux-flow dissipation is much larger than the quasiparticle dissipation. The second condition is experimentally satisfied, i.e., ‰(H cos ) scales for ≤ 54◦ (Fig. 6-2). This indicates the consistency between the the experimental data and the explicit functional dependence of resistivity given by Eq. (6.3).

6.4 Summary

In summary, angular-dependent resistivity was measured in the mixed state, which shows H cos scaling, a result of ﬂux-ﬂow dissipation. The explicit functional dependence 93

of ‰ab on H and obtained is consistent with the H cos scaling. References

[1] Y. Kasahara, Y. Nakajima, K. Izawa, Y. Matsuda, K. Behnia, H. Shishido, R. Settai, and Y. Onuki, cond-mat/0506071, (2005).

[2] C. N. Jiang, A. R. Baldwin, G. A. Levin, C. C. Almasan, D. A. Gajewsi, S. H. Han, and M. B. Maple, Phys. Rev. B 55, 3390(R) (1997).

[3] H. Raﬀy and S. Labdi, Phys. Rev. Lett. 66, 2515 (1991).

[4] C. Ferdeghini, V. Braccini, M.R. Cimberle, D. Marre, P. Manfrinetti, V. Ferrando, M. Putti, and A. Palenzona, Eur. Phys. J. B 30, 147 (2002).

[5] P. H. Kes, J. Aarts, V. M. Vinokur, and C. J. van der Beek, Phys. Rev. Lett. 64, 1063 (1990).

[6] G. Blatter, V. B. Geshkenbein, and A. I. Larkin, Phys. Rev. Lett. 68, 875 (1992).

[7] N. Kopnin, Theory of Nonequilibrium Superconductivity, Oxford: Clarendon press, pp. 238-245, (2001).

94 Chapter 7

Summary

This dissertation describes research on the magnetic and electronic properties of CeCoIn5 heavy fermion superconductor, both in the normal and mixed states. The recently discovered heavy fermion superconductor CeCoIn5 has generated a lot of interest, partly due to the many analogies present between this compound and the high transition temperature superconductors. Like the cuprates, CeCoIn5 has a layered crystal structure, a quasi-two- dimensional electronic spectrum, and a superconducting phase appearing at the border of the antiferromagnetic phase. Furthermore, the superconductivity is unconventional; i.e., it displays proximity to quantum critical points, giant Nernst eﬀect in the normal state, non-Fermi liquid behavior with the possibility of a pseudogap, Pauli limiting eﬀect, and possibly multi-bands in the superconducting state. Study of both the normal state and mixed state of CeCoIn5 will bring further understanding of not only the underlying physics in heavy fermion materials but also of the mechanism of superconductivity of high temperature superconductors.

Chapter 1 presented a brief introduction to the general topic of heavy fermion superconductivity. Then we introduce the new Ce-based heavy fermion superconductor family, with explicit emphasis on a summation of the properties of CeCoIn5. Also, some challenging issues regarding the properties of both the normal and mixed states of CeCoIn5, which are not resolved yet, are introduced.

Chapter 2 gives a description of the experimental setup and the experimental methods, including both the electrical transport and magnetic measurements.

Chapter 3 describes the ﬁeld and temperature dependence of the anisotropic parameter

95 96

of CeCoIn5. Experiments show that there is probably a d-wave superconducting gap on the heavy-electron sheets of the Fermi surface and a negligible gap on the light, three- dimensional pockets. The presence of multibands (gaps) together with the reported ﬁeld dependence of the cyclotron masses point towards a possible temperature- and/or ﬁeld- dependent anisotropy in the superconducting state. We performed torque measurements on

CeCoIn5 single crystals in both the superconducting and normal states. Two contributions to the torque signal in the mixed state were identiﬁed: one from paramagnetism and the other one from the vortices. The torque curves show sharp hysteresis peaks at =90◦ ( is the angle between H and the c-axis of the crystal) when the measurements are done in clockwise and counterclockwise directions. This hysteresis is a result of the intrinsic pinning of vortices, a behavior very similar to high transition temperature cuprate superconductors.

The temperature and magnetic ﬁeld dependence of the anisotropy is obtained from the reversible part of the vortex torque. We ﬁnd that decreases with increasing magnetic

ﬁeld and temperature. This result indicates the breakdown of the Ginzburg-Landau theory, which is based on a single band model and provides evidence for a multiband picture for

CeCoIn5, which is highly possible due to its complex Fermi topology.

Chapter 4 describes a study of the mixed-state property of CeCoIn5 regarding the im- pact of Pauli-limiting eﬀect. Previous studies have shown that CeCoIn5 is Pauli limited; i.e., the interaction of the magnetic ﬁeld with the electron spins dominates that of magnetic

ﬁeld and the electron orbits. In the presence of the Zeeman eﬀect, an unusual mixed state is expected in which the diamagnetic and paramagnetic contributions could be anomalous.

It is important to address the issue regarding the paramagnetic and diamagnetic response in the mixed state and how does the Pauli paramagnetism aﬀect the mixed-state thermodynamics of CeCoIn5. This is essential to the understanding of the interaction between superconductivity and magnetism in heavy fermion materials. We performed magnetization 97

and torque measurements both in the normal and mixed states of CeCoIn5 single crystals in order to study the paramagnetic and vortex response in the presence of a large Zeeman effect present in this material. The paramagnetic magnetization is suppressed in the mixed state and the spin susceptibility is field dependent, increasing with increasing field. This H dependence is a result of the fact that heavy electrons contribute to both superconductivity and paramagnetism and the Zeeman effect is large in this material. There is no anomaly present in the vortex response in the temperature range investigated.

Chapter 5 resolves the problem of the superconducting order parameter of CeCoIn5.

The type of pairing symmetry of unconventional superconductors is essential to the understanding of their pairing mechanism and of the origin of superconductivity. Presently, it is a subject of great interest and debate for the CeCoIn5 heavy fermion superconductor. It has been established that the superconducting order parameter of CeCoIn5 displays d-wave symmetry. However, the direction of the gap nodes relative to the Brillouin zone axes, which determines the type of d-wave state, is still an open question and an extremely controversial issue. We performed in-plane angular-dependent torque measurements on CeCoIn5 single crystals both in the normal and mixed states. Normal-state torque measurements show a fourfold symmetry. The reversible torque in the mixed state, obtained after subtracting the corresponding normal-state contribution, also shows a fourfold symmetry with maxima at

…/ 8, 5…/ 8, 9…/8, etc. Sharp peaks in the irreversible torque data were observed at …/ 4,

3…/ 4, 5…/4, etc. These latter peaks correspond to minima in the free energy of a dxy wave symmetry. The mixed-state fourfold symmetry and the peak positions in the irreversible torque point unambiguously toward dxy wave symmetry of the superconducting gap. The

field and temperature dependencies of the amplitude of the normal-state torque are different from that of the mixed-state torque, which indicates that the normal-state torque has a different origin. 98

Chapter 6 shows angular-dependent resistivity measurements in the mixed state. We found a H cos scaling, which is due to ﬂux-ﬂow dissipation. The explicit functional dependence of ‰ab on H and obtained is consistent with the H cos scaling.

I think there is still follow-up research that my work leads to. For example, there is no anomaly observed in the vortex response in the mixed state for the investigated reduced temperature T=Tc = 0:78 . However, theoretical work [1] showed that the vortex response in the presence of large Zeeman effect would be different from pure diamagnetism at lower reduced temperature, T=Tc < 0:3. This anomaly could be due to the abnormal vortex core structure in the mixed state, which might be a result of the fact that CeCoIn5 is close to a quantum critical point. We can study the vortex core structure by studying the flux-flow resistivity by I − V measurements and find out the relationship between the vortex core structure and quantum criticality. References

[1] Hiroto Adachi, Masanori Ichioka, and Kazushige Machida, J. Phys. Soc. Jpn. 74, 2181 (2005).