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A Theoretical Analysis of the Spin Susceptibility Tensor and Quasiparticle Density of States for Quasi-One-Dimensional Superconductors

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A Theoretical Analysis of the Spin Susceptibility Tensor and Quasiparticle Density of States for Quasi-One-Dimensional Superconductors A Theoretical Analysis of the Spin Susceptibility Tensor and Quasiparticle Density of States for Quasi-one-dimensional Superconductors A Thesis Presented to The Academic Faculty by Cawley D. Vaccarella In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Physics Georgia Institute of Technology November 2001 Copyright © 2001 by Cawley D. Vaccarella A Theoretical Analysis of the Spin Susceptibility Tensor and Quasiparticle Density of States for Quasi-one-dimensional Superconductors Approved: Carlo^ A^R. Sa de Melo, Chairman Helmut J. Biritz Walter A. de Heer Thomas Nitiprey (G.I.T. Department of Mathematics) Date Approved 111 IS iii my Mother Brigitte for her Faith and Support in All my endeavors iv I am indebted to my advisor Carlos Sa de Melo for the time spent in discussion of the many topics and ideas that needed to come together in order to make this a successful research. I would like to express thanks to Richard Duncan for his help on the numerical portion of this thesis. I would also like to thank Robert Cherng for his collaboration on the analysis of the critical frequencies of the density of states and spin susceptibility and his contribution in obtaining the energy gap. vi vii Contents List of Figures xii I Introduction 1 1.1 Background 1 1.2 Recent Experiments on Quasi-one-dimensional Superconductors ... 4 1.2.1 Anisotropy of the Upper Critical Field in (TMTSF)2PF6 5 1.2.2 Thermal Conductivity of Superconducting (TMTSF)2CI04: Ev­ idence for a Nodeless Gap 10 1.2.3 Evidence for Triplet Superconductivity in (TMSTF)2PF6 from 77SE Knight Shifts 13 1.3 Summary 14 II Background and Motivation for the Description of Cooper Pairs 16 2.1 Cooper Pairing 16 2.2 Spin Structure of Paired States 19 2.2.1 Triplet Pairing 20 2.2.2 Singlet Pairing 23 2.3 The Energy of Quasiparticle Excitations 24 2.3.1 The Diagonalization of the Hamiltonian 24 III THE QUASIPARTICLE DENSITY OF STATES DOS 28 viii 3.1 Comparison Calculation 29 3.1.1 Mineev's Approach 29 3.1.2 A More General Approach 32 3.2 The General DOS for a Lattice 35 3.3 The DOS for the Singlet S Symmetry 37 3.4 The DOS for the Singlet Dxy Symmetry 39 3.5 The DOS for the Triplet P Symmetries 42 3.6 The Low Frequency Limit 47 3.6.1 Low Frequency Dxy Case 47 3.6.2 Low Frequency Gapped P Cases 48 3.6.3 Low Frequency Gapless P Cases 50 3.7 Summary 51 IV THE ELECTRON SPIN SUSCEPTIBILITY 53 4.1 Some Conventions 53 4.2 The Kubo Formula: Linear Response Theory 55 4.3 Electron Spin Susceptibility Tensor 56 4.4 The Unitary Electron Spin Susceptibility 60 4.5 Uniform Spin Susceptibility Tensor for the Unitary Triplet P-wave State 70 4.5.1 Kinks in the Gapped P-Symmetry 77 4.5.2 Kinks in the Gapless P-Symmetry 82 4.5.3 The Low Frequency Limit 84 4.6 Calculation of the General Nonunitary Electron Spin Susceptibility 88 V Conclusion 102 A The Energy of Quasiparticle Excitations 106 A.l The Hamiltonian and Symmetry Properties 106 ix A.2 Mean Field Theory and Hamiltonian Diagonalization 108 B FREQUENCY SUMMATIONS 116 C Summation on Spin Indices 119 Cl the Singlet Case 121 C.2 the Triplet Case 123 D Symmetry Properties of the Susceptibility Tensor 127 D.l the Singlet Case 127 D.2 the Triplet Case 130 D.3 the Nonunitary Triplet Case 132 E Special Limits 136 E.l The Singlet Uniform Dynamical 137 Spin Susceptibility: Xmn(°t ~^ E.2 The Singlet Nonuniform Static Spin Susceptibility: Xmn(<liw ~* °) 137 E.3 The Singlet Uniform Static Spin Susceptibility: Xmnfa -> 0,w -> 0) 138 E.4 The Triplet Uniform Dynamical Spin Susceptibility: Xmnfa -> 0,w) 140 E.5 The Triplet Nonuniform Static Spin Susceptibility: X™n(q,w -> 0) 141 E.6 The Triplet Uniform Static Spin Susceptibility: Xmn(q -> 0,w 0) 143 F The Critical Frequencies 145 F.l Low Frequency 147 F.l.l The Gapped Symmetries 148 x F. 1.1.1 Px+y+z Symmetry 148 F. 1.1.2 S Symmetry 150 F.1.2 The Gapless Symmetries 151 F.1.2.1 Py+Z Symmetry 151 F. 1.2.2 Dxy Symmetry 151 F.2 High Frequency 153 159 Vita 160 xi List of Figures 1 View of the crystal structure of (TMTSF)2CI04 along the direction of highest conductivity (a-axis) 2 2 Diagram of the crystal structure of (TMTSF)2PF6 showing the principle directions a, b' and c* 3 3 The temperature versus pressure phase diagram for (TMTSF^PFg. No­ tice the proximity of the superconducting phase to the spin density wave (SDW) insulating phase 5 4 The magnetic field versus temperature (H — T) phase diagram for (TMTSF)2PF6 using the junction criterion for magnetic fields aligned along the three principle axis a, b', and c* 6 5 (TMSTF)SPF6 interlayer resistance versus temperature for various mag­ netic fields H || b' at P — 6 kbar. Five different criteria for the putative critical temperature TC(H) are indicated: 0 (onset), J (junction), M (midpoint), X (R -> 0), and Z (R = 0) 7 6 This figure shows the anisotropy inversion for H||a and H||b' for four different resistance criteria: onset, junction, midpoint and R —> 0. 8 7 This figure shows the DOS for the S symmetry for positive frequency in units of temperature (IK) [T = hu/kB]. Notice the gap at T = A0//CB- [tx = fcB(5800K),iy = M1226K),*Z = A:B(48K),^ = M4003K), A0 = M2.281K)] 38 xii 8 This figure shows the DOS for the Dxy symmetry for positive fre­ quency in units of temperature (IK) [T = hu/kB\. Notice in con­ trast to the gapped S that there is no gap in Af(ui) in this case. [t = it (4003K), x = M5800K),ty = M1226K),*, = JfcB(48K),/i B A0 = M3.675K)] 41 9 This figure shows the DOS for the Px symmetry for positive frequency in units of temperature (IK), T = fku/kB. The gapped response begins at T = Hu^in/kB = 1.301K. [A = 1,5 = 0,C = 0,tx = A; (5800K),t = fc (1226K), = fi = A; (4003K), 43 B y B tz kB(A8K), B A0 = JfeB(3.136K)] 10 This figure shows the DOS for the Px+y symmetry for positive fre­ quency in units of temperature (IK), T = fku/kB. The gapped re­ sponse begins at T = hu^in/kB = 0.897K. [A = 1,B = 1,C = 0,tx = 44 A; (5800K),ty = A; (1226K),tz = = B B itB(48K),/x itB(4003K),A0 = JfcB(2.162K)] 11 This figure shows the DOS for the Px+y+z symmetry for positive fre­ quency in units of temperature (IK), T = fku/kB. The gapped re­ sponse begins at T = huj^]n/kB = 0.745K. [A = B = C = ^/2/3,tx = A; (5800K),t = k (1226K),t = = A; (4003K), 45 B y B z itB(48K),/x B A0 = JfcB(2.199K)] 12 This figure shows the DOS for the Py symmetry for positive frequency in units of temperature (IK), T = fko/kB. From the plot it is clear that this is a gapless symmetry. [A = 0, B = 1, C = 0, tx — /cB(5800K), ty = Ml226K),i, = & (48K),/z = M2.703K)] 46 B M4003K),A0 = 13 This figure shows the DOS for the Py+Z symmetry for positive frequency in units of temperature (IK), T = fku/kB. From the plot it is clear that 2 this is a gapless symmetry. [A = 0, B = 3y/2/4, C = V2 - B , tx = fc (5800K),ty = ),t = = A;B(4003K), = fcB(2.091K)] 47 B kB(1226K z kB(48K),n A0 14 This figure shows the level curves corresponding to Dq^ = 0 for the gapless symmetry Dxy. [tx = &B(5800K),ty = /cB(1226K),tz — kB(48K),n = itB(4003K)] 48 xiii 15 This figure shows the level curves corresponding to Dp = 0 for the gapped symmetries Px, Px+y, Px+y+z respectively. [tx = A;B(5800K), ty = kB(1226K),tz = jfcB(48K),/i = A;B(4003K)] 49 16 This figure shows the level curves corresponding to Dp = 0 for the gap­ less symmetries Py, Py+Z respectively. [tx = fcB(5800K),ty = feB(1226K),tz = A;B(48K),/i = A;B(4003K)] 50 17 This figure shows the susceptibility ^{^22} = ^{^33} f°r tne Px sym­ metry for positive frequency in units of temperature (IK), T = hcu/ks- [A = 1,B = 0,C = 0,tx = A;B(5800K),iy = A;B(1226K), tz - A;B(48K),/i = A;B(4003K), A0 = fcB(3.136K)] 78 18 This figure shows the structure of the peak in the susceptibility for the Px symmetry. 79 19 This figure shows the susceptibility 3{xn} for the Px+y symmetry. From the plot it is clear that this is a gapless symmetry. [A = 1,B = 1, C = 0, tx = A;B(5800K), ty = A;B(1226K), tz = kB(48K), /i = A;B(4003K), A0 = fcB(2.162K)] 80 20 This figure shows the susceptibility 3f{x22} for the Px+y symmetry. 81 21 This figure shows the susceptibility ${x33} for the Px+y symmetry. 81 22 This figure shows the susceptibility 3{xn}, ${X22}, ${x33 = 0} for the Py symmetry. [A = 0,B = 1,C = 0,ix = A;B(5800K),ty = A;B(1226K),i, = A;B(48K),/i = A;B(4003K),Ao = A;B(2.703K)] 83 23 This figure shows the level curves corresponding to Dp = 0 for the gapped symmetries Px, Px+y, Px+y+z respectively.
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