TORQUE AND MAGNETIZATION MEASUREMENTS ON THE HEAVY FERMION SUPERCONDUCTOR CeCoIn5 A dissertation submitted to Kent State University in partial ful¯llment of the requirements for the degree of Doctor of Philosophy by 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 e®ect .............................. 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 E®ect .................... 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 ¯t 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 coe±cient A of ½(T ). From Ref. [26]. 16 vi 1-10 HQCP from the ¯ts are plotted along with Hc2 and Tc in zero ¯eld vs. P . From Ref. [29]. 17 1-11 Illustration of the vortex structure (solid lines) and the FFLO modulation (dashed lines) with the ¯eld parallel (top) and perpendicular (bottom) to the heat current. From Ref. [39]. 19 1-12 H ¡ T phase diagrams at low temperatures and high ¯elds 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 ¯elds that are in the ¯rst 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 di®erent temperatures T and applied magnetic ¯eld H values. The solid lines are ¯ts of the data with Eq. (3.1). Inset: Schetch of the single crystal with the orientation of the magnetic ¯eld H and torque ¿ with respect to the crystallographic axes. 35 3-2 Field H dependence of A=H, where A is the ¯tting parameter in Eq. (3.1). The solid lines are linear ¯ts of the data. 37 vii 3-3 Plot of the magnetic moment M vs applied magnetic ¯eld 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 ¯eld H of 0.3 T. The solid curve is a ¯t 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 ¯eld H dependence of the ¯tting parameter ¯. The solid line is a guide to the eye. Inset: Enlarged plot of the low ¯eld 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 ¯eld data (Ref. [23] ), the triangles are obtained from the resistivity data shown in the inset of this ¯gure, and the squares are from the present torque data measured in an applied magnetic ¯eld of 0.3 T. Inset: T dependence of the in-plane ½a and out-of-plane ½c resistivities measured in zero ¯eld. 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 ¯eld H and the magnetization M. From Ref. [17]. 49 viii 4-1 Magnetic ¯eld 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 ¯t of Mmes(H) in the normal state. Inset: Upper critical ¯eld parallel to the jjc 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 ¯eld 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 ¯eld H dependence of the dc magnetization Mmes measured at 2 K for H k a...................................... 59 4-3 (a) Plot of ¯eld 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 ¯tting param- eter A. The solid line is a ¯t 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 ¯eld data. The solid curves in Mv(H) and Mdev(H) are guides to the eye. 63 4-4 Magnetic ¯eld H dependence of the paramagnetic magnetization Mp.