CONDUCTANCES IN THE TWO-IMPURITY ANDERSON MODEL By WILLIAM BRIAN LANE A ...

143 Pages·2008·1.22 MB·English

CONDUCTANCES IN THE TWO-IMPURITY ANDERSON MODEL
By
WILLIAM BRIAN LANE
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2008
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(cid:176)c 2008 William Brian Lane
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For Amy: my wife, friend, editor, and treasure.
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ACKNOWLEDGMENTS
There are many people to thank, perhaps more than my memory and words can do
justice:
• The members of my supervisory committee for their insight and understanding:
Kevin Ingersent, Christopher Stanton, Arthur Hebard, Robert Coldwell, and James
Keesling.
• John Klauder, Kevin Ingersent, James Fry, James Dufty, Arthur Hebard, Dimitri
Maslov, Konstantin Matchev, and Pradeep Kumar for their classroom instruction at
the University of Florida.
• Paul Simony, Steve Browder, William Mendoza, Bashir Sayar, Robert Hollister, Pam
Crawford, Sanjay Rai, Marcelle Bessman, and Marilyn Repsher for their classroom
instruction and professional collaboration at Jacksonville University.
• Mark Meisel and Steve Hill for their dedication, wisdom, and care.
• Darlene Latimer, Nathan Williams, Kristin Nichola, Donna Balkcom, and Yvonne
Dixon for their faithful service to the Department of Physics.
• Many of the calculations that went into this dissertation were performed on the UF
HPC Cluster; many thanks go to Charles Taylor and the HPC Staff.
• Matt Glossop, Mengxing Cheng, Luis Dias da Silva, Nancy Sandler, and Sergio Ulloa
for their collaboration and insight.
On a more personal note, I would like to issue these thanks, as well:
• My mother, for her love, support, belief in me, and words of kindness and discipline.
• My brother, for his love and friendship.
• Richard Parker, Steve Gregg, James Walden, Tobey Sorrels, Richard Horner, Rick
Borque, Dan MacDonald, Keith Jackson, Dan Brinkmann, Scott Moffatt, Gardner
Gordon, Ed Barnard, Ken Kurdziel, Dana Focks, and Ralph Coleman for their
spiritual care and wisdom.
• My church families at Eastside and Creekside.
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Lastly, I would like to thank my lovely wife, Amy, who has seen me through this
adventure.
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TABLE OF CONTENTS
page
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
CHAPTER
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.1 The Kondo Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.1.1 Resistivity Minimum and the Success of the Kondo Model . . . . . 12
1.1.2 The Anderson Model . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.1.3 Further Attempts at Perturbative Techniques . . . . . . . . . . . . . 19
1.2 The Numerical Renormalization Group . . . . . . . . . . . . . . . . . . . . 22
1.2.1 The Renormalization Group Concept . . . . . . . . . . . . . . . . . 22
1.2.2 Application to the Kondo Problem . . . . . . . . . . . . . . . . . . . 23
1.2.3 Iterative Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.2.4 Fixed Points and Results . . . . . . . . . . . . . . . . . . . . . . . . 29
1.3 Surface and Quantum Dot Realizations of the One-Impurity Kondo Effect 31
1.3.1 Scanning Tunneling Microscopy Studies . . . . . . . . . . . . . . . . 31
1.3.2 Quantum Dot Studies . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.4 Systems of Multiple Impurities . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.4.1 Theoretical Studies of Two-Impurity Models . . . . . . . . . . . . . 40
1.4.2 Multiple-Impurity STM Studies . . . . . . . . . . . . . . . . . . . . 44
1.4.3 Double Quantum Dot Studies . . . . . . . . . . . . . . . . . . . . . 46
1.5 Study Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2 BACKGROUND MATERIAL . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.1 Application of the NRG to the Anderson Model . . . . . . . . . . . . . . . 57
2.1.1 Discretization and Eigensolution . . . . . . . . . . . . . . . . . . . . 57
2.1.2 Calculation of Thermodynamic Properties . . . . . . . . . . . . . . 59
2.1.3 Calculation of Spectral Functions . . . . . . . . . . . . . . . . . . . 61
2.1.4 Fixed Points and Results . . . . . . . . . . . . . . . . . . . . . . . . 62
2.2 Extension to Two-Impurity Systems . . . . . . . . . . . . . . . . . . . . . . 66
2.2.1 Transformation to One-Dimensional Form . . . . . . . . . . . . . . . 67
2.2.2 Discretization and Eigensolution . . . . . . . . . . . . . . . . . . . . 70
2.2.3 Special Cases: Identical Impurities and R = 0 . . . . . . . . . . . . 73
2.2.4 Calculation of Thermodynamic and Spectral Properties . . . . . . . 74
6
3 PARALLELIZATION OF THE NRG PROCEDURE . . . . . . . . . . . . . . . 76
3.1 Parallelization of the NRG Eigensolution . . . . . . . . . . . . . . . . . . . 76
3.2 Parallelization of the Matrix Element Calculation . . . . . . . . . . . . . . 78
4 STM STUDIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.1 Review of Single-Impurity Behavior . . . . . . . . . . . . . . . . . . . . . . 82
4.1.1 Single-Impurity STM Setup . . . . . . . . . . . . . . . . . . . . . . 82
4.1.2 Results for Single-Impurity STM . . . . . . . . . . . . . . . . . . . . 84
4.2 Two-Impurity STM Studies . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.2.1 Two-Impurity Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.2.2 Thermodynamic and Spectral Results . . . . . . . . . . . . . . . . . 91
4.2.3 Two-Impurity STM Conductance . . . . . . . . . . . . . . . . . . . 94
4.2.4 Varying Impurity Parameters . . . . . . . . . . . . . . . . . . . . . . 97
5 ASYMMETRIC DOUBLE QUANTUM-DOT DEVICES . . . . . . . . . . . . . 103
5.1 Double Quantum Dot Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.1.1 Model and Simplifications . . . . . . . . . . . . . . . . . . . . . . . 104
5.1.2 Special Case: U = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 106
2
5.2 Side-Coupled DQD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.2.1 Special Case: U = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 107
2
5.2.2 Extended Case: U > 0 . . . . . . . . . . . . . . . . . . . . . . . . . 108
2
5.3 Parallel DQD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.3.1 Special Case: U = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 112
2
5.3.2 Extended Case: U > 0 - Phase Diagram and Susceptibility . . . . . 116
2
5.3.3 Extended Case: U > 0 - Spectral Function and Conductance . . . . 124
2
6 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.1 Scanning Tunneling Microscopy Studies . . . . . . . . . . . . . . . . . . . . 133
6.2 Double Quantum Dot Studies . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.3 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
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LIST OF FIGURES
Figure page
1-1 Resistivity minimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1-2 Logarithmic discretization of the energy space . . . . . . . . . . . . . . . . . . . 24
1-3 Impurity coupled to a chain of electron states . . . . . . . . . . . . . . . . . . . 25
1-4 Wall-clock time for the NRG procedure . . . . . . . . . . . . . . . . . . . . . . . 28
1-5 Evolution of energy spectra during NRG process . . . . . . . . . . . . . . . . . . 30
1-6 Differential conductance for a single Co atom . . . . . . . . . . . . . . . . . . . 33
1-7 Differential conductance for a single Ce atom . . . . . . . . . . . . . . . . . . . 34
1-8 Differential conductance for a single Co atom with varying tip position . . . . . 36
1-9 Differential conductance for a single Ce atom with varying tip position . . . . . 37
1-10 Conductance of a quantum dot for various dot occupancies . . . . . . . . . . . . 39
1-11 Differential conductance for Co atoms . . . . . . . . . . . . . . . . . . . . . . . 45
1-12 Differential conductance for a pair of Ni atoms . . . . . . . . . . . . . . . . . . . 46
1-13 Differential conductance for various Ce configurations . . . . . . . . . . . . . . . 47
1-14 Setup for coupled double quantum dot experiment . . . . . . . . . . . . . . . . . 48
1-15 Coulomb blockade valleys for a DQD . . . . . . . . . . . . . . . . . . . . . . . . 49
1-16 Differential conductance vs. interdot coupling . . . . . . . . . . . . . . . . . . . 50
1-17 Side-coupled DQD displaying two-stage Kondo screening behavior . . . . . . . . 51
1-18 Conductance vs. gate voltage for a side-coupled DQD . . . . . . . . . . . . . . . 52
2-1 Single-impurity Tχ (T) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
imp
2-2 Single-impurity A (ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
d
3-1 Wall-clock time vs. N for iterative eigensolution with N = 3000. . . . . . . 79
P keep
3-2 Wall-clock time vs. N for calculation of operator matrix elements . . . . . . . 81
P
4-1 Single-Impurity STM Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4-2 Single-Impurity STM Conductance . . . . . . . . . . . . . . . . . . . . . . . . . 84
4-3 Single-Impurity STM Conductance, t = 0 . . . . . . . . . . . . . . . . . . . . . 85
c
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4-4 One-Impurity STM Conductance - positive voltages . . . . . . . . . . . . . . . . 86
4-5 One-Impurity STM Conductance - negative voltages . . . . . . . . . . . . . . . 87
4-6 Fit of G(V) with t /t = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
d c
4-7 Fit of G(V) with t /t = 0.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
d c
4-8 Two-impurity STM setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4-9 Strength of RKKY interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4-10 Two-impurity magnetic susceptibility . . . . . . . . . . . . . . . . . . . . . . . . 92
4-11 Two-impurity spectral function . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4-12 Two-impurity conductance, t /t = 0.1, positive voltages . . . . . . . . . . . . . 96
d c
4-13 Two-impurity conductance, t /t = 0.1, negative voltages . . . . . . . . . . . . . 97
d c
4-14 Two-impurity conductance, t /t = 0.4, positive voltages . . . . . . . . . . . . . 98
d c
4-15 Two-impurity conductance, t /t = 0.4, negative voltages . . . . . . . . . . . . . 99
d c
4-16 Fitted two-impurity conductance, positive voltages . . . . . . . . . . . . . . . . 100
4-17 Fitted STM two-impurity conductance, negative voltages . . . . . . . . . . . . . 101
4-18 Differential conductance for various two-impurity systems . . . . . . . . . . . . . 102
5-1 Double quantum dot schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5-2 Dot 1 spectral function for side-coupled DQD with U = 0 . . . . . . . . . . . . 108
2
5-3 Zero-T conductance for side-coupled DQD with U = 0 . . . . . . . . . . . . . . 109
2
5-4 Zero-T A (ω) for side-coupled DQD with U > 0 . . . . . . . . . . . . . . . . . 110
11 2
5-5 Zero-T conductance for side-coupled DQD with U > 0 . . . . . . . . . . . . . . 111
2
5-6 Zero-T conductance for side-coupled DQD with (cid:178) > 0 . . . . . . . . . . . . . . 112
2
5-7 Parallel-dot phase diagram for U = 0 . . . . . . . . . . . . . . . . . . . . . . . . 113
2
5-8 Observation of upper QPT in Tχ vs. T for U = 0 . . . . . . . . . . . . . . . 114
imp 2
5-9 Linear relationship between T and δ(cid:178)+ . . . . . . . . . . . . . . . . . . . . . . 115
K 1
5-10 Zero-T A (ω) vs. ω > 0 for Kondo-phase parallel DQD, U = 0 . . . . . . . . . 116
11 2
5-11 Zero-T A (ω) vs. ω < 0 for Kondo-phase parallel DQD, U = 0 . . . . . . . . . 117
11 2
5-12 Zero-T A (ω) vs. ω > 0 for local-moment-phase parallel DQD, U = 0 . . . . . 118
11 2
9
5-13 Zero-T A (ω) vs. ω < 0 for local-moment-phase parallel DQD, U = 0 . . . . . 119
11 2
5-14 Approximate phase diagram for parallel DQD, U > 0 . . . . . . . . . . . . . . . 120
2
5-15 Critical point (cid:178)+ vs. U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
1c 2
5-16 Scaled (cid:178)+ vs. U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
1c 2
5-17 Traces of Tχ vs. T, U = 10−3 . . . . . . . . . . . . . . . . . . . . . . . . . . 123
imp 2
5-18 Zero-T limit of χ vs. δ(cid:178)+, U > 0 . . . . . . . . . . . . . . . . . . . . . . . . . 124
imp 1 2
5-19 Kondo temperature vs. δ(cid:178)+ for parallel DQD with U > 0 . . . . . . . . . . . . 125
1 2
5-20 Kondo temperature T vs. (cid:178) for parallel DQD without local-moment phase . . 126
K 1
5-21 Zero-T A (ω) vs. ω > 0 for Kondo-phase parallel DQD, U = 10−3 . . . . . . . 127
11 2
5-22 Zero-T conductance G/G vs. (cid:178) for parallel DQD . . . . . . . . . . . . . . . . . 128
0 1
5-23 Zero-T conductance G/G vs. δ(cid:178)+ for parallel DQD . . . . . . . . . . . . . . . . 129
0 1
5-24 Zero-T conductance G/G vs. (cid:178) for parallel DQD with no local-moment phase . 130
0 1
5-25 Zero-T conductance G/G vs. (cid:178) for parallel DQD with (cid:178) = −U /2 . . . . . . . 131
0 2 1 1
5-26 Zero-T conductance G/G vs. (cid:178) for parallel DQD . . . . . . . . . . . . . . . . . 132
0 2
10