Computational methods applied to Superconductivity and Magnetism By Alan Kyker BS

145 Pages·2012·1.39 MB·English

Computational methods applied to Superconductivity and Magnetism
By
Alan Kyker
B.S. (University of Tennessee at Knoxville) 1988
DISSERTATION
Submitted in partial satisfaction of the requirements for the degree of
Doctor of Philosophy
in
Physics
in the
OFFICE OF GRADUATE STUDIES
of the
UNIVERSITY OF CALIFORNIA
DAVIS
Approved:
Committee in Charge
2006
i
To Dale
ii
Abstract
Computational methods applied to Superconductivity and Magnetism
by
Alan Kyker
Superconductivity and magnetism are two phenomena where the microscope quantum
world manifestsmacroscopic visible behavior. Themethodsof studyingthese condensates
generally follow a quantum mechanical (bottom up) or phenomenological (top down). In
principal,bottomupmethodsaresu(cid:14)cienttodescribeallbehavior,butinpracticethecal-
culations become intractable. The e(cid:11)ect of electronic structure on the formation of FFLO
phases was studied usinga modi(cid:12)edBCSformalism. Features of the Fermi surfaces which
promoted the formation of FFLO phases were identi(cid:12)ed. Magnetically induced orbital
currents, vortex dynamics and multi-order parameter superconductors were studied using
the phenomenological formalism of Ginzburg and Landau. A new topological structure
was identi(cid:12)ed in multi-order parameter superconductors with Josephson coupling. The
electronic structure tools the were developed for studying FFLO phases were then applied
TiBe . The source of the anomalous temperature dependent susceptibility was identi(cid:12)ed.
2
Classical magnetism was studied using transfer matrix methods. A method was devel-
oped for extracting the density of states for long, narrow, nearest neighbor, 2D Ising and
Edwards-Anderson spin systems.
iii
iv
Contents
List of Figures vii
1 Theory of ~q = 0 Pairing Superconductivity 1
6
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Cooper Pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 BCS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Bogoliubov-Valatin transformation . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 The Gap equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.6 BCS Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.7 FFLO Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.8 Applications of Nesting Density to FFLO calculations . . . . . . . . . . . . 23
1.9 1D Fermi Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.10 2D Fermi Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.11 3D Fermi Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.12 ZrZn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2
1.13 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.14 Free energy calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.15 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2 Fermi Velocity and Incipient Magnetism in TiBe 39
2
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.2 Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.3 Method of Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.4 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.5 Analysis of Velocity Distribution and Susceptibility . . . . . . . . . . . . . . 53
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3 Macroscopic Theory of Multi Order Parameter Pairing in Superconduc-
tiity 67
3.1 Multi Order Parameter Landau Theory . . . . . . . . . . . . . . . . . . . . 68
v
3.2 Ginzburg-Landau Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.3 Multi-Order Parameter Ginzburg-Landau Theory . . . . . . . . . . . . . . . 76
3.4 Solving the Ginzburg-Landau Model . . . . . . . . . . . . . . . . . . . . . . 78
3.5 Boundary Conditions and Simulation Controls. . . . . . . . . . . . . . . . . 80
3.6 The Vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.7 Flux quantization of one order parameter . . . . . . . . . . . . . . . . . . . 85
3.8 Fractional (cid:13)ux quantization with two order parameters . . . . . . . . . . . . 88
3.9 The bi-quadratic term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.10 The Josephson term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.11 J-wall excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.12 Not any knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.13 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4 Classical Spin Systems 101
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.2 Transfer matrix applied to spin systems. . . . . . . . . . . . . . . . . . . . . 103
4.3 One spin at a time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.4 Finite T computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.5 Partition function polynomial and T = 0 calculations . . . . . . . . . . . . . 111
4.6 2D Ising phase transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.7 Spin Glass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.8 Trapping local energy minimum . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.9 Frustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.10 2D 1 glass simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
(cid:6)
4.11 2D continuous bond distribution glass simulation . . . . . . . . . . . . . . . 124
4.12 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Bibliography 129
vi
List of Figures
1.1 Quasiparticle density of states at Fermi level . . . . . . . . . . . . . . . . . 14
1.2 BCS phase diagram with q~= 0 . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3 Quasiparticle nesting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4 Gap-less density of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5 Integral kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.6 1D FFLO phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.7 2D nesting density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.8 2D FFLO phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.9 Tight binding Fermi surface . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.10 3D tight binding FFLP phase diagram . . . . . . . . . . . . . . . . . . . . . 29
1.11 ZrZn nesting density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2
1.12 ZrZn dominate Fermi surface . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2
2.1 Full TiBe non-magnetic band structure . . . . . . . . . . . . . . . . . . . . 46
2
2.2 Expanded TiBe non-magnetic band structure. . . . . . . . . . . . . . . . . 47
2
2.3 TiBe DOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2
2.4 Fermi Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.5 Fermi velocity spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.6 Fermi velocity moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.7 (cid:31)(q~) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.8 Temperature dependant DOS near Fermi Energy and susceptability . . . . 61
2.9 B dependant DOS at the Fermi level . . . . . . . . . . . . . . . . . . . . . . 62
3.1 Vortex Smoke Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.2 Single vortex in 2D simulation . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.3 Two non-equilibrium 3D vortices . . . . . . . . . . . . . . . . . . . . . . . . 87
3.4 Current and (cid:12)eld of single vortex in single order parameter simulation . . . 88
3.5 Phase graph of condensate with and without vortex . . . . . . . . . . . . . 89
3.6 Current and (cid:12)eld of single vortex in two order parameter simulation . . . . 90
3.7 2D simulation of condensate competition . . . . . . . . . . . . . . . . . . . . 92
vii
3.8 Phase frustration caused by vortex separation in two condensate supercon-
ductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.9 Wall of frustration caused by vortex separation in two condensate super-
conductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.10 Vortex knot components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.1 Long narrow spin system with transfer matrix cuts. . . . . . . . . . . . . . . 104
4.2 Long narrow spin system with overlapping cuts. . . . . . . . . . . . . . . . . 107
4.3 Log of density of states for 12 by 70 FM Ising model. . . . . . . . . . . . . . 115
4.4 Speci(cid:12)c heat for a range of FM Ising systems. . . . . . . . . . . . . . . . . . 116
4.5 Frustrated plackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.6 Log of density of states for 12 by 70 FM Ising model. . . . . . . . . . . . . . 122
4.7 Speci(cid:12)c heat for a range of FM Ising systems. . . . . . . . . . . . . . . . . . 123
4.8 Odd eigenvalue projection of 18 spin wide Gaussian spin strip . . . . . . . . 127
viii
Chapter 1
= 0
Theory of ~q Pairing
6
Superconductivity
1
CHAPTER 1. THEORY OF Q~ = 0 PAIRING SUPERCONDUCTIVITY 2
6
1.1 Introduction
The work in this chapter is derived from the publication
\Fermiology and Fulde-Ferrell-Larkin-Ovchinnikov Phase Formation"; A. B. Kyker, W.
E. Pickett, and F. Gygi, Phys. Rev. B 71, 224517 (2005).
Almost half a century ago Ginzburg addressed the question of possible super-
conductivity in ferromagnetic material[1], and studied the problems posed by orbital su-
percurrents within a material with intrinsic magnetic (cid:13)ux. About a decade later, and
armed with BCS theory[2], Fulde and Ferrell (FF)[3] and separately Larkin and Ovchin-
nikov (LO)[4]addressedtheseparate questionwithhow aBCSsuperconductorcopeswith
an intrinsic spin splitting, which breaks the degeneracy of spin up and spin down Fermi
surfaces. Both FF and LO concluded that (neglecting orbital current e(cid:11)ects) that there
is a superconducting phase (the \FFLO phase") above the usual upper critical (cid:12)eld H
c2
where superconductivity persists based on q~ = 0 (non-zero momentum) pairs and the
6
order parameter becomes inhomogeneous.
Since that time there has been a considerable number of papers exploring the
competition between, and possible coexistence of, the superconducting and magnetic long
range order parameters.[5] Full treatment requires consideration of both orbital and spin
e(cid:11)ects, and for the most part theories have tended to suppose that one is dominant in a
particular systemandconcentrate on thatone. Thusinvestigations have focusedeither on
the orbital e(cid:11)ects such as spontaneous vortex phases, or on the exposition of the FFLO
phase without complications from vortex behavior. Much has been accomplished with