CONDUCTANCES IN THE TWO-IMPURITY ANDERSON MODEL

By

WILLIAM BRIAN LANE

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2008

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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.1.1 Resistivity Minimum and the Success of the Kondo Model . . . . . 121.1.2 The Anderson Model . . . . . . . . . . . . . . . . . . . . . . . . . . 161.1.3 Further Attempts at Perturbative Techniques . . . . . . . . . . . . . 19

1.2 The Numerical Renormalization Group . . . . . . . . . . . . . . . . . . . . 221.2.1 The Renormalization Group Concept . . . . . . . . . . . . . . . . . 221.2.2 Application to the Kondo Problem . . . . . . . . . . . . . . . . . . . 231.2.3 Iterative Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.2.4 Fixed Points and Results . . . . . . . . . . . . . . . . . . . . . . . . 29

1.3 Surface and Quantum Dot Realizations of the One-Impurity Kondo Effect 311.3.1 Scanning Tunneling Microscopy Studies . . . . . . . . . . . . . . . . 311.3.2 Quantum Dot Studies . . . . . . . . . . . . . . . . . . . . . . . . . . 35

1.4 Systems of Multiple Impurities . . . . . . . . . . . . . . . . . . . . . . . . . 401.4.1 Theoretical Studies of Two-Impurity Models . . . . . . . . . . . . . 401.4.2 Multiple-Impurity STM Studies . . . . . . . . . . . . . . . . . . . . 441.4.3 Double Quantum Dot Studies . . . . . . . . . . . . . . . . . . . . . 46

1.5 Study Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2 BACKGROUND MATERIAL . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.1 Application of the NRG to the Anderson Model . . . . . . . . . . . . . . . 572.1.1 Discretization and Eigensolution . . . . . . . . . . . . . . . . . . . . 572.1.2 Calculation of Thermodynamic Properties . . . . . . . . . . . . . . 592.1.3 Calculation of Spectral Functions . . . . . . . . . . . . . . . . . . . 612.1.4 Fixed Points and Results . . . . . . . . . . . . . . . . . . . . . . . . 62

2.2 Extension to Two-Impurity Systems . . . . . . . . . . . . . . . . . . . . . . 662.2.1 Transformation to One-Dimensional Form . . . . . . . . . . . . . . . 672.2.2 Discretization and Eigensolution . . . . . . . . . . . . . . . . . . . . 702.2.3 Special Cases: Identical Impurities and R = 0 . . . . . . . . . . . . 732.2.4 Calculation of Thermodynamic and Spectral Properties . . . . . . . 74

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3 PARALLELIZATION OF THE NRG PROCEDURE . . . . . . . . . . . . . . . 76

3.1 Parallelization of the NRG Eigensolution . . . . . . . . . . . . . . . . . . . 763.2 Parallelization of the Matrix Element Calculation . . . . . . . . . . . . . . 78

4 STM STUDIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.1 Review of Single-Impurity Behavior . . . . . . . . . . . . . . . . . . . . . . 824.1.1 Single-Impurity STM Setup . . . . . . . . . . . . . . . . . . . . . . 824.1.2 Results for Single-Impurity STM . . . . . . . . . . . . . . . . . . . . 84

4.2 Two-Impurity STM Studies . . . . . . . . . . . . . . . . . . . . . . . . . . 874.2.1 Two-Impurity Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . 874.2.2 Thermodynamic and Spectral Results . . . . . . . . . . . . . . . . . 914.2.3 Two-Impurity STM Conductance . . . . . . . . . . . . . . . . . . . 944.2.4 Varying Impurity Parameters . . . . . . . . . . . . . . . . . . . . . . 97

5 ASYMMETRIC DOUBLE QUANTUM-DOT DEVICES . . . . . . . . . . . . . 103

5.1 Double Quantum Dot Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.1.1 Model and Simplifications . . . . . . . . . . . . . . . . . . . . . . . 1045.1.2 Special Case: U2 = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.2 Side-Coupled DQD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.2.1 Special Case: U2 = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.2.2 Extended Case: U2 > 0 . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.3 Parallel DQD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.3.1 Special Case: U2 = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.3.2 Extended Case: U2 > 0 - Phase Diagram and Susceptibility . . . . . 1165.3.3 Extended Case: U2 > 0 - Spectral Function and Conductance . . . . 124

6 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6.1 Scanning Tunneling Microscopy Studies . . . . . . . . . . . . . . . . . . . . 1336.2 Double Quantum Dot Studies . . . . . . . . . . . . . . . . . . . . . . . . . 1346.3 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 Tχimp(T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

2-2 Single-impurity Ad(ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3-1 Wall-clock time vs. NP for iterative eigensolution with Nkeep = 3000. . . . . . . 79

3-2 Wall-clock time vs. NP for calculation of operator matrix elements . . . . . . . 81

4-1 Single-Impurity STM Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4-2 Single-Impurity STM Conductance . . . . . . . . . . . . . . . . . . . . . . . . . 84

4-3 Single-Impurity STM Conductance, tc = 0 . . . . . . . . . . . . . . . . . . . . . 85

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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 td/tc = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4-7 Fit of G(V ) with td/tc = 0.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

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, td/tc = 0.1, positive voltages . . . . . . . . . . . . . 96

4-13 Two-impurity conductance, td/tc = 0.1, negative voltages . . . . . . . . . . . . . 97

4-14 Two-impurity conductance, td/tc = 0.4, positive voltages . . . . . . . . . . . . . 98

4-15 Two-impurity conductance, td/tc = 0.4, negative voltages . . . . . . . . . . . . . 99

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 U2 = 0 . . . . . . . . . . . . 108

5-3 Zero-T conductance for side-coupled DQD with U2 = 0 . . . . . . . . . . . . . . 109

5-4 Zero-T A11(ω) for side-coupled DQD with U2 > 0 . . . . . . . . . . . . . . . . . 110

5-5 Zero-T conductance for side-coupled DQD with U2 > 0 . . . . . . . . . . . . . . 111

5-6 Zero-T conductance for side-coupled DQD with ε2 > 0 . . . . . . . . . . . . . . 112

5-7 Parallel-dot phase diagram for U2 = 0 . . . . . . . . . . . . . . . . . . . . . . . . 113

5-8 Observation of upper QPT in Tχimp vs. T for U2 = 0 . . . . . . . . . . . . . . . 114

5-9 Linear relationship between TK and δε+1 . . . . . . . . . . . . . . . . . . . . . . 115

5-10 Zero-T A11(ω) vs. ω > 0 for Kondo-phase parallel DQD, U2 = 0 . . . . . . . . . 116

5-11 Zero-T A11(ω) vs. ω < 0 for Kondo-phase parallel DQD, U2 = 0 . . . . . . . . . 117

5-12 Zero-T A11(ω) vs. ω > 0 for local-moment-phase parallel DQD, U2 = 0 . . . . . 118

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5-13 Zero-T A11(ω) vs. ω < 0 for local-moment-phase parallel DQD, U2 = 0 . . . . . 119

5-14 Approximate phase diagram for parallel DQD, U2 > 0 . . . . . . . . . . . . . . . 120

5-15 Critical point ε+1c vs. U2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5-16 Scaled ε+1c vs. U2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5-17 Traces of Tχimp vs. T , U2 = 10−3 . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5-18 Zero-T limit of χimp vs. δε+1 , U2 > 0 . . . . . . . . . . . . . . . . . . . . . . . . . 124

5-19 Kondo temperature vs. δε+1 for parallel DQD with U2 > 0 . . . . . . . . . . . . 125

5-20 Kondo temperature TK vs. ε1 for parallel DQD without local-moment phase . . 126

5-21 Zero-T A11(ω) vs. ω > 0 for Kondo-phase parallel DQD, U2 = 10−3 . . . . . . . 127

5-22 Zero-T conductance G/G0 vs. ε1 for parallel DQD . . . . . . . . . . . . . . . . . 128

5-23 Zero-T conductance G/G0 vs. δε+1 for parallel DQD . . . . . . . . . . . . . . . . 129

5-24 Zero-T conductance G/G0 vs. ε1 for parallel DQD with no local-moment phase . 130

5-25 Zero-T conductance G/G0 vs. ε2 for parallel DQD with ε1 = −U1/2 . . . . . . . 131

5-26 Zero-T conductance G/G0 vs. ε2 for parallel DQD . . . . . . . . . . . . . . . . . 132

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Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy

CONDUCTANCES IN THE TWO-IMPURITY ANDERSON MODEL

By

William Brian Lane

May 2008

Chair: James Kevin IngersentMajor: Physics

In a number of systems of interest that involve magnetic atoms and their analogous

quantum dot manifestations, there arises a competition between Kondo screening and

various types of magnetic ordering (direct and induced). This competition can be studied

in detail using scanning tunneling microscopy to probe clusters of magnetic adatoms

on metallic surfaces and has direct implications for systems of double quantum dots. In

both of these cases, an observable quantity of interest is the electrical conductance, which

can be calculated by applying the numerical renormalization group to the two-impurity

Anderson model. Depending on their separation and the strength of their exchange

interaction, pairs of magnetic adatoms may exhibit ferromagnetic or antiferromagnetic

alignment of the impurity local moments, in some cases leading to a two-stage Kondo

screening process, effectively isolated impurity screening, or a complete suppression of the

Kondo effect. These behaviors have different signatures in the differential conductance.

A class of double quantum dot devices composed of a Kondo-like dot and a weakly

interacting dot is predicted to display a splitting of the Kondo resonance and a pair of

quantum phase transitions. These behaviors introduce unique signatures in the device

conductance when the level energy on either dot is varied by tuning the appropriate gate

voltage. This work demonstrates that double quantum dots can provide a controlled

experimental setting in which to study quantum phase transitions in a strongly correlated

system.

11

CHAPTER 1INTRODUCTION

The Kondo effect is an emergent many-body phenomenon that, even after years of

study, continues to unfold newer levels of insight and application in the realm of condensed

matter physics [54]. In its most basic form, the Kondo effect is based upon the interaction

between a localized magnetic impurity and the spins of the conduction electrons of the

host metal. As the impurity engages in a spin-exchange “dance” with the conduction

electrons, the electrons become strongly correlated through the impurity. Below a low

temperature scale TK (called the Kondo temperature) the higher-energy conduction

electrons can be thought to “cluster” around the impurity, forming a net spin singlet with

the impurity, masking it from the electrons that lie closest to the Fermi energy, which then

effectively return to their non-interacting state.

Today, the Kondo effect is being investigated experimentally by scanning tunneling

microscope studies of surface impurities and in the context of tunable quantum dot

devices. In this work, I present theoretical results that demonstrate how a number of

interesting behaviors may be identified in conductance measurements in such experiments.

Scanning tunneling microscope studies of a pair of surface impurities are shown to exhibit

a variety of signatures that permit the study of the competition between the Kondo effect

and magnetic ordering that arises between the two impurities. Certain highly asymmetric

quantum-dot devices are shown to exhibit a splitting of the Kondo resonance and a pair of

quantum phase transitions, with intriguing manifestations in the device conductance.

1.1 The Kondo Problem

1.1.1 Resistivity Minimum and the Success of the Kondo Model

Evidence of the Kondo effect was first observed in the 1930’s in measurements of

the resistance of nominally pure samples of Au. It was expected that the resistance of

such samples would decrease monotonically with temperature as T 5 (due to the scattering

of electrons from phonons; see [3]). However, in some samples, results showed that the

12

T (arb. units)

R (

arb.

uni

ts)

Experimental data

Previous theory: R = aT5

Kondo’s result

Figure 1-1. Schematic plot of resistance R vs. temperature T in experiments (blackcircles) that first observed what is now called the Kondo effect. The minimumin R is unexplainable by the original prediction (green curve). Kondo’s result(red curve) successfully described the minimum, but produced a diverging R asT → 0.

resistance reached a minimum value at a certain temperature Tmin (on the order of 4 to

40 K), and then increased as temperature was further decreased [1, 9]. This behavior is

summarized schematically in Figure 1-1. The unexpected minimum was attributed to the

presence of impurities, as when the impurity concentration cimp was varied, it was found

that Tmin ∝ c1/5imp and the depth of the minimum R(T = 0)−R(T = Tmin) ∝ cimp. Study was

then undertaken to understand what aspect of these impurities was causing this minimum.

One possible explanation was that the resistance minimum might be attributed to the

potential scattering of the conduction electrons off the impurities. Such a process can be

modeled by the Hamiltonian

H = Hc +∑

~k,~q,j

exp (i~q · ~Rj)K~q c†~k+~q,σc~k,σ, (1–1)

13

where

Hc =∑

~k

ε~kc†~kσ

c~kσ (1–2)

is the impurity-free conduction band Hamiltonian. The c†~kσ(c~kσ) operators are standard

fermion operators that create (annihilate) conduction band electrons of momentum ~~k and

spin σ. The conduction band has a bandwidth of 2D (such that |ε~k| < D), with a density

of states ρ(ε) =∑~k

δ(ε − ε~k). (In sections to follow, it will often be assumed that ρ(ε) = ρ0

for |ε| < D and zero elsewhere.) Repeated spin indices (σ in this case) are implicitly

summed over, ~Rj is the position vector of impurity j, and K~q is the impurity scattering

potential at wavevector ~q.

To calculate the impurities’ contribution to the resistivity Rimp, an average is

performed over the impurity positions, such that only K ≡ K~q=0 is important. A

perturbation expansion in K is then employed, using the Sommerfeld expansion [24],

producing the result [54]

Rimp =3πcimpmeK

2

2e2~εF

[1 + O

((kBT/εF )2

)], (1–3)

where kB is Boltzmann’s constant, me is the mass of an electron, e is the magnitude of the

electronic charge, and εF is the Fermi energy of the host metal. A combined resistivity of

R(T ) = aT 5 + Rimp does successfully explain a non-zero R(T = 0); however, because of

the minuscule magnitude of (kBT/εF )2 (typically of order 10−8 in the temperature range

of interest) in Eq. (1–3), R(T ) does not display the minimum seen in the experiments

[54]. It was thus concluded that the resistance minimum must be due to some other

impurity-related mechanism.

This mysterious mechanism was not unveiled until 1964, when Kondo made a

number of observations [10]: First, samples that exhibited a resistivity minimum at

low temperatures also displayed a zero-field magnetic susceptibility χ (defined as χ =

dM/dH|H=0, where M is the magnetization of a material and H is an applied magnetic

field) that obeyed a Curie-Weiss law of the form C/(T + θ) at higher temperatures,

14

indicating the presence of localized magnetic moments (hence the use of the term

“magnetic impurity”). Second, as the size of the magnetic moments increased, so did the

depth R(T = 0) − R(T = Tmin) and the location Tmin of the minimum [9], suggesting that

the minimum was caused by an interaction between the spins of the impurities and the

spins of the conduction electrons. Lastly, Kondo noted that while R(T = 0)− R(T = Tmin)

was proportional to cimp, the ratio (R(T = 0)− R(T = Tmin))/R(T = 0) did not vary with

cimp, indicating that this spin interaction was a single-impurity effect.

Based on the conclusion that the resistivity minimum was caused by the spin

interaction between the conduction electrons and individual localized magnetic moments,

Kondo applied the s-d exchange model developed by Zener [2], now more commonly

referred to as the Kondo Hamiltonian:

HK = Hc +J

2~S ·

∑

~k,~k′

c†~kσ~τσ,σ′c~k′σ′ . (1–4)

Here, τ jσ,σ′ , j = x, y, z, are the Pauli spin matrices and ~S is the impurity spin. The

quantity J is the strength of the exchange interaction between the spins of the impurity

and the conduction electrons; if J is negative (positive), the interaction is ferromagnetic

(antiferromagnetic). (Note that this Hamiltonian may also be expanded to include

potential scattering, but this feature is not necessary to obtain Kondo’s results.) With this

Hamiltonian, Kondo carried out a perturbative expansion to third order in J to obtain the

resistivity contributed by a single impurity: [10]

Rimp =3πmeJ

2S(S + 1)

8e2~εF

[1− 2Jρ0(εF ) ln(kBT/D)] . (1–5)

Kondo thus arrived at the total resistivity

R(T ) = aT 5 + cimpR0 − cimpR1 ln kBT/D, (1–6)

15

which, if J > 0 (the antiferromagnetic case), displays a minimum that satisfies Rmin −R(T = 0) ∝ cimp at a temperature Tmin = (R1cimp/5a)1/5, matching the experimental

observations (see red curve in Figure 1-1).

Kondo’s solution was a phenomenal success. He used his perturbative expression for

R(T ) to fit the results from a wide variety of experiments, including some that implied

that J < 0 (ferromagnetic case) [10]. His model was also used to successfully produce

perturbative expressions for a magnetic impurity’s contributions χimp(T ) and Cimp(T ) to

the zero-field magnetic susceptibility and the specific heat [54]:

χimp(T ) =(gµB)2S(S + 1)

3kBT

[1− ρ0J + (ρ0J)2 ln(kBT/D) + c2(ρ0J)2

], (1–7)

Cimp(T ) = kBπ2S(S + 1)(ρ0J)4 [1− 4ρ0J ln(kBT/D)] , (1–8)

where c2 is a constant that depends on the conduction band density of states.

However, there were a number of remaining questions to be addressed: How do

these local moments arise within the host metal? How does the spin exchange interaction

take place? What happens at very low temperatures T ¿ Tmin? The last question was

prompted by the irony that though the ln T term in Kondo’s expansion explained the

resistance minimum, this term diverges as T → 0, and Kondo’s solution was not valid for

very low temperatures. Thus arose a quest to find a non-perturbative solution, dubbed

“the Kondo problem.”

1.1.2 The Anderson Model

The answer to the first two questions—how the local moments arise and the

mechanism of spin interaction—is answered rather straightforwardly by examining the

Anderson model [7], which was developed before Kondo employed the s-d exchange model.

The Anderson model considers the scattering of conduction electrons off transition metal

or rare earth impurities whose d levels lie within the host metal’s conduction band.

The hybridization Vd~k between an impurity’s d-state φd (defined in a coordinate system

that places the impurity at the origin) and the conduction electron states ψd (Wannier

16

functions centered at the impurity site) is given by:

V~kd = 〈φd|Hc|ψd〉. (1–9)

A pair of electrons in the impurity’s d-state experience a Coulomb interaction

U =

∫φ∗d(~r1)φ

∗d(~r2)

e2

|~r1 − ~r2|φd(~r1)φd(~r2)d~r1d~r2, (1–10)

which is typically on the order of 1 to 7 eV.

The Anderson Hamiltonian is

HA = Hc + εdd†σdσ + Ud†↑d↑d

†↓d↓ +

∑

~k

(V~kdc†~kσ

dσ + V ∗~kd

d†σc~kσ), (1–11)

where d†σ (dσ) is a standard fermion operator that creates (annihilates) an electron in the

impurity’s d-state of energy εd. All energies are defined such that εF = 0.

The shape of the density of states ρ(ε) away from the Fermi level and the ~k-dependence

of V~kd are features not crucial to the model [7, 54]. Thus, for the remainder of this section,

the density of states will be taken as ρ(ε) = ρ0 for |ε| < D and 0 elsewhere, and V~kd will be

taken as a constant Vd. To gain an understanding of the general features of the Anderson

model, it is helpful to first consider three simplified cases: (1) U, Vd = 0, (2) U = 0, Vd > 0

and (3) U > 0, Vd = 0. For each case, we examine the change ∆ρ in the electronic density

states caused by an Anderson impurity.

(1) In the case of U = Vd = 0 (an isolated, non-interacting impurity), ∆ρ(ε) is a Dirac

delta-function at ε = εd. This indicates the localized nature of the impurity electron, which

for this case is isolated from the conduction electrons.

(2) In the case of U = 0, Vd > 0 (a non-interacting impurity), the Anderson

Hamiltonian is quadratic in fermionic operators and can be diagonalized exactly. The

effect of Vd > 0 is to give the impurity state a finite lifetime, broadening the delta-function

17

for the Vd = 0 case to a Lorentzian given by

∆ρ(ε) =∆/π

(ε− εd)2 + ∆2, (1–12)

where

∆ = πρ0|Vd|2, (1–13)

and εd satisfies

εd − εd − ρ0|Vd|2 ln

∣∣∣∣D + εd

D − εd

∣∣∣∣ = 0. (1–14)

With the result (1–12), the impurity’s contribution to the magnetic susceptibility is

χimp = 2µ2B∆ρ(εF ), (1–15)

which does not lead to a Curie-Weiss behavior in χ(T ). Using Eq. (1–12) to calculate Rimp

does not result in a resistance minimum at lower temperatures, either. Thus, an impurity

model with U = 0 cannot explain the Kondo effect.

(3) Next, consider the case of U > 0, Vd = 0 (an isolated interacting impurity). Such

an impurity is described by an empty state |0〉 of energy 0, two singly occupied states | ↑〉and | ↓〉 of energy εd, and a doubly occupied state | ↑↓〉 of energy 2εd + U . Thus, if εd < 0

and εd + U > 0, the singly occupied states will be energetically favored, producing a spin-12

localized on the impurity.

For the general case of U, Vd > 0, the conditions for local-moment formation are

modified because the impurity levels acquire a width of order ∆. For example, the singly

occupied level is broadened to the range from εd − c∆ to εd + c∆, where c ∼ 4. Thus, in

general, a local moment arises if −εd À ∆ and εd + U À ∆. In such a case, the Anderson

model maps onto the Kondo model using the Schrieffer-Wolff transformation [12] to obtain

the spin coupling J and potential scattering K:

J = 2

(1

|εd| +1

|U + εd|)

V 2d (1–16)

18

K =1

2

(1

|εd| −1

|U + εd|)

V 2d . (1–17)

The change in the density of states ∆ρ(ε) then takes a form similar to Eq. (1–12), but

with ∆ replaced by a resonance width of order

kBTK ≈ De−1/(ρ0J). (1–18)

(This expression for the Kondo temperature TK will be explained in more detail below.)

This resonance is called the Abrikosov-Suhl resonance or the Kondo resonance. Thus, the

Kondo model is actually a limiting case of the more general Anderson model.

1.1.3 Further Attempts at Perturbative Techniques

The low-temperature divergence of the perturbative results in Eqs. (1–5) through

(1–8) remained a longstanding problem. In 1965, Abrikosov [11] tried to eliminate the

divergence by carrying out Kondo’s perturbative expansion further and summing all

higher-order terms proportional to [ρ0J ln(kBT/D)]n. For the impurity contribution to the

magnetic susceptibility, he arrived at [cf. Eq. (1–7)]

χimp(T ) =(gµB)2S(S + 1)

3kBT

[1− ρ0J

1 + ρ0J ln (kBT/D)+ c2(ρ0J)2

]. (1–19)

This result represented progress, as the ferromagnetic case (J < 0) remains well-behaved

as T → 0. Unfortunately, in the antiferromagnetic case (J > 0), this expression diverges at

the Kondo temperature TK [Eq. (1–18)] instead of at T = 0.

In 1966, Yosida approached the Kondo problem with a variational method, and found

the ground state energies for a singlet (labeled s) and triplet (labeled t) configuration [13]:

E(0)s = −De−4/(3ρ0J), (1–20)

E(0)t = −De4/(ρ0J). (1–21)

Thus, for the antiferromagnetic case, a ground state in which the impurity and conduction

electrons form a singlet is energetically preferred. For the case of a spin-12

impurity, similar

19

variational calculations predicted a finite χimp(T = 0) = (gµB)2/4kBTK [14], holding out

the promise of a low-temperature solution.

From 1967 to 1970, Anderson and collaborators [16, 17, 19–21] made a series of

developments that eventually resulted in an approach known as “poor man’s scaling.”

This method predicted that as T → 0, the “effective interaction” J (described below) goes

to ∞, such that the local moment is in a fully compensated singlet state (as predicted by

Yosida above), producing non-magnetic behavior and a finite χimp(T = 0).

The poor man’s scaling approach begins with the observation that the divergence

problem arises from terms involving ln (kBT/D), in which each decade of energy

approaching εF contributes equally to the properties at T = 0. Thus, no energy scales

may be ignored in perturbation theory (as might be permitted if the divergences grew as

D/kBT or (D/kBT )2). However, the contributions of the energy scales much greater than

kBT can be handled by considering how they renormalize the exchange coupling J .

Anderson considered states that were near the band edge—i.e., states of energy

D − |δD| < |ε| < D, where δD < 0. He then constructed an effective Hamiltonian by

considering states of the form ψ = ψ0 + ψ1 + ψ2, where ψ1 represents states in which

no electrons are in the upper |δD| of the band and no holes are in the lower |δD| of the

band, ψ0 represents states in which at least one hole is in the lower |δD| of the band, and

ψ2 represents states in which at least one electron is in the upper |δD| of the band. This

effective Hamiltonian consists of blocks such as H21, which scatters an electron into the

upper |δD| of the band, and H01, which scatters a hole into the lower |δD| of the band.

Working to leading order in |δD|/D, Anderson arrived at an effective Hamiltonian

H11 with the same form as HK , but with the spin coupling J replaced by J~k,~k′ ≡ J + δJ~k,~k′ .

The renormalization δJ~k,~k′ , accounting for scattering of an electron between two states in

the low-energy sector 1 via one or more transitions involving a virtual intermediate state

20

in a high-energy sector 0 or 2, is given to second order in J by

δJ~k,~k′ = −ρ0J2|δD|2

[1

E −D + ε~k+

1

E −D + ε~k′

], (1–22)

where E is the energy eigenvalue of the effective Hamiltonian. This effective Hamiltonian

is valid over a reduced bandwidth D ≡ D − |δD|.For low-energy excitations, one can set E, ε~k, and ε~k′ to zero in Eq. (1–22) to arrive

at the second-order scaling equation

dJ

d ln D= −ρ0J

2. (1–23)

Reducing D to D by absorbing the energy levels near the band edge into the renormalization

of J to J will cause J to increase. Thus, the higher-energy excitations still contribute to

the solution of the Kondo model, but they do so by renormalizing J .

In this rescaling of D and J , if 0 < ρ0J ¿ 1, the Kondo temperature TK(J,D)

functions as a scaling invariant:

De−1/(ρ0J) = De−1/(ρ0J) ≈ kBTK . (1–24)

This property is extremely useful; for example, the impurity contribution to the magnetic

susceptibility can be written

χimp(T ) =1

TF

(T

TK

), (1–25)

where F (X) is a universal function.

Thermodynamic results based on Eq. (1–23) still exhibit a divergence at T = TK .

Carrying the rescaling out to third order in J produces

d(ρ0J)

d ln D= −(ρ0J)2 +

1

2(ρ0J)3 + O((ρ0J)4), (1–26)

resulting in a more accurate expression for the Kondo temperature:

kBTK ≈ D√

ρ0Je−1/(ρ0J)+O(ρ0J). (1–27)

21

It is justified to integrate out high-energy band degrees of freedom only until D has

been reduced to D = akBT , where a is of order 10. (At temperature T , real thermal

excitations span all energies |ε| . akBT .) In effect, J becomes a function of T , given by

1

ρ0J+

1

2ln

(1

ρ0J

)− ln a = ln

(T

TK

). (1–28)

Applying Eq. (1–28) to the magnetic susceptibility gives

χimp(T ) =(gµB)2

4kBT

[1− ρ0J(T ) + O[ρ0J(T )]2

](1–29)

=(gµB)2

4kBT

[1− 1

ln(T/TK)− ln[(ln (T/TK))]

2(ln[(T/TK)])2+ O

(1

[ln (T/TK)]2

)]. (1–30)

Equation (1–30) still contains a logarithmic divergence at T = TK . Thus, even after

Anderson’s scaling methods, the low-temperature behavior remained a mystery.

The poor man’s scaling method provided vast progress toward solving the Kondo

and Anderson problems, but still broke down due to its perturbative approach. Anderson

had shown that it was reasonable for J → ∞ as T → 0 and for χimp(T → 0) to

be finite, though he was unable to arrive at a well-defined low-temperature expression

for χimp. However, the concept of scaling would prove to be essential to the successful

non-perturbative solution found in the numerical renormalization group.

1.2 The Numerical Renormalization Group

1.2.1 The Renormalization Group Concept

The goal of the renormalization group (RG) is the study of a Hamiltonian H( ~K)

characterized by a set of couplings ~K = (K1, K2, . . .). An RG transformation R maps

H( ~K) into another Hamiltonian H(K ′) of the same form, but with a different set of

couplings:

R[H( ~K)] = H( ~K ′), (1–31)

or, more compactly,

R( ~K) = ~K ′. (1–32)

22

This new Hamiltonian is then valid over a reduced energy scale. The transformation R is

usually characterized by α, the ratio of the new energy scale to the old, such that

Rα(Rα′( ~K)) = Rαα′( ~K). (1–33)

Making a sequence of transformations can be thought to generate a trajectory or flow line

in ~K-space (somewhat analogous to slope fields in coupled differential equations). These

trajectories begin and end at fixed points ~K∗ that satisfy

Rα( ~K∗) = ~K∗. (1–34)

A fixed point is classified as stable or unstable according to whether the local direction

of RG flow is toward or away from ~K∗ (again, analogous to fixed points in coupled

differential equations).

As a simple example of the RG concept, in the poor man’s scaling approach to the

Kondo problem of the previous section, the elimination of higher-energy levels transforms

J into J . The new Hamiltonian has the same form, but is defined for a reduced energy

scale D and has a renormalized J . Anderson demonstrated that the ferromagnetic Kondo

model had a stable fixed point at J = 0, and made the reasonable extrapolation that

J → ∞ was the stable fixed point of the antiferromagnetic Kondo model, but was unable

to prove it.

1.2.2 Application to the Kondo Problem

In 1975, Wilson [23] combined the renormalization group with the poor man’s scaling

method to examine the S = 12

Kondo model. His two most important results were proof

that J → ∞ is the only stable fixed point for the antiferromagnetic Kondo model and

an expression for the effective Hamiltonian HJ=∞ near this fixed point which he used to

calculate thermodynamic behavior at T ¿ TK .

To arrive at these results, Wilson considered the problem of divergence that had

plagued perturbative studies of the Kondo model. He reasoned that the ln(kBT/D)

23

Figure 1-2. Logarithmic discretization of the energy space −1 < ε/D < 1, where the Fermienergy εF = 0.

divergences occurred in expressions for thermodynamic quantities because all energy scales

were contributing to the calculation—e.g., scales of order D, D/10, D/100 . . .. To account

for this energy cascade, Wilson divided the conduction band −D < ε~k < D (where εF is

taken to be 0) into intervals (see Figure 1-2) given by

DΛ−(m+1) < |ε~k| < DΛ−m (1–35)

for m = 0, 1, 2, . . .. Here, Λ > 1 is a chosen quantity termed the discretization parameter.

With this energy discretization in place, the conduction electron basis is then changed

from the set of operators c†~k,σ, |ε~k| < D to a set a†m,q,σ, b

†m,q,σ, where a†m,q,σ(b†m,q,σ)

creates an electron of positive (negative) energy in a state ψm,q,σ(ε/D) given by

ψm,q(ε) =Λm/2

(1− Λ−1)1/2exp(iωmqε), Λ−(m−1) < |ε| < Λ−m, (1–36)

where ωm = 2πΛm/(1 − Λ−1) and the integer q is a Fourier harmonic index. One can

picture the ψm,q=0(ε) states as concentric shells centered around the impurity, each of

radius rm ∼ k−1F Λm/2, where kF is the Fermi wavevector.

Upon rewriting HK with the new a†m,q,σ and b†m,q,σ operators, it can be seen that the

impurity couples directly only to states with q = 0. In Wilson’s numerical renormalization

group (NRG) approach, states with q 6= 0 are ignored, incurring an error proportional

to (1 − Λ−1)/2π, which is negligible as long as Λ is not much greater than 1. (Typically,

1 < Λ . 3 produces acceptable results.)

24

Figure 1-3. Impurity coupled to a chain of electron states. In Wilson’s discretization, thecoupling λn ∼ Λ−n/2.

After dropping the states with q 6= 0, a Lanczos transformation is applied to the set

of operators a†m,q=0,σ, b†m,q=0,σ, resulting in a set f †n,σ, defined as linear combinations of

a†m,q=0,σ and b†m,q=0,σ.

With the new f †n,σ operators, the Kondo Hamiltonian becomes

HK =∞∑

n=0

[εnf †n,σfn,σ + λnf

†n,σfn+1,σ + λ∗nf

†n+1,σfn,σ

]+ 2J ~S · ~s0,σ,σ′ . (1–37)

This Hamiltonian can be pictured as forming a linear chain (see Figure 1-3), in which the

impurity is connected only to site 0 (whose spin operator is written as ~s0,σ,σ′ = 12f †0,σ~τf0,σ′)

and each site is coupled only to its nearest neighbors.

Working with a density of states ρ(ε) = 1/(2D),−D < ε < D, Wilson arrived at

expressions for the couplings

εn = 0 (1–38)

λn =D(1 + Λ−1)

2Λ−n/2ξn, (1–39)

where ξn is a set of constants of order unity, given by

ξn =1− Λ−n−1

[(1− Λ−2n−3) (1− Λ−2n−1)]1/2, (1–40)

such that ξn → 1 as n →∞.

Here is where Wilson’s procedure differs from Anderson’s perturbative scaling

technique: The NRG scheme does not attempt to follow the coupling J as D is reduced,

nor does it rely on ρ0J to be small. Instead (as we shall see in the next section), it tracks

25

the evolution of the low-lying many-body eigenstates of HK on a characteristic energy

scale D ≈ DΛ−n/2 as n →∞.

1.2.3 Iterative Procedure

To apply the techniques of the RG, Wilson considered the Hamiltonian HN for a

partial chain consisting of the impurity plus the N + 1 innermost electron sites,

HN =D(1 + Λ−1)

2

N∑n=0

Λ−n/2ξn

(f †n,σfn+1,σ + f †n+1,σfn,σ

)+ 2J ~S · ~s0,σ,σ′ , (1–41)

such that the sequence of Hamiltonians H0, H1, H2, . . . obeys the recursion relation

HN+1 = HN +D

2(1 + Λ−1)Λ−N/2ξN(f †N,σfN+1,σ + h.c.). (1–42)

This relationship enables an iterative solution of the chain. First, the simple Hamiltonian

H0 = 2J ~S · ~s0,σ,σ′ is diagonalized, obtaining eigenstates |N, Q, S, Sz, m〉 labeled by

the quantum numbers Q (charge measured from half-filling), S (total spin), Sz (spin

z-component), and m (which distinguishes states with all other quantum numbers the

same). These eigenstates are then multiplied by the states of site 1 (|0〉,| ↑〉,| ↓〉, and | ↑↓〉)to construct a basis of states in which H1 is written. The new Hamiltonian H1 is then

diagonalized, obtaining eigenstates that are multiplied by the states of site 2 to write H2,

etc. This iterative procedure provides the basis of the RG transformation.

One of the primary goals of this procedure is to compare the set of energy eigenvalues

(also referred to as the energy spectrum) of HN to that of HN+1 to find a fixed point

of the RG transformation. As the characteristic energy scale of each HN decreases as

DΛ−N/2, in order to make the comparison between successive spectra, a dimensionless

Hamiltonian HN is used:

HN ≡ 2Λ(N−1)/2

D(1 + Λ−1)HN (1–43)

26

Transforming the recursion relation Eq. (1–42) in this manner leads to the RG transformation

HN+1 = R(HN) (1–44)

= Λ1/2HN + ξN(f †N,σfN+1,σ + h.c.). (1–45)

Diagonalizing HN yields the set of energy eigenvalues EN(~m) and their eigenvectors

|N, ~m〉 (where |N, ~m〉 is a shorthand for |N,Q, S, Sz,m〉). Computationally, the RG

transformation can be carried out by calculating

HN+1 =∑

~m

EN(~m)|N, ~m〉〈N, ~m| (1–46)

+ ξN

∑

~m,~m′

[〈N, ~m|f †N,σ|N, ~m′〉|N, ~m〉〈N, ~m′|fN+1,σ + h.c.

].

Thus, the RG transformation of Eq. (1–44) can be thought of as acting on the set of

eigenvalues of HN and matrix elements of fN,σ:

R[(EN(~m), 〈N, ~m|fN,σ|N, ~m′〉)] = (EN+1(~m), 〈N + 1, ~m|fN+1,σ|N + 1, ~m′〉). (1–47)

The iterative procedure is carried out on a computer—hence the name “Numerical

Renormalization Group.” Each Hamiltonian HN is block diagonal in the conserved

quantum numbers of the model. For example, it can be shown that [HK , ~S] = 0, and so

there will be no non-zero matrix elements between states of different total spin or different

spin z-component. This block-diagonal nature permits a faster eigensolution. As described

in Chapter 2, it is straightforward to obtain thermodynamic and spectral quantities as

functions of temperature by keeping track of the energy eigenvalues and operator matrix

elements at each iteration.

The NRG method’s most significant detriment is that the dimension of the Hamiltonian

HN is 22N+3. Since the time to diagonalize a matrix grows as the cube of its dimension,

the computational effort to obtain a complete solution of HN becomes prohibitive beyond

N ≈ 5. Thus, in practice, the summations over ~m and ~m′ are restricted to the Nkeep states

27

10 100 1000N

keep

1

10

100

1000

10000 runtime / runtime(Nkeep

= 10)

Ekeep

/ Ekeep

(Nkeep

= 10)

Figure 1-4. Wall-clock time for the NRG procedure and the highest energy level retainedEkeep vs. Nkeep. The wall-clock time and energy levels are scaled by theirvalues for Nkeep = 10. Note how the runtime (the cost of the calculation) growsmuch more quickly than the energy level retention (the benefit).

of lowest energy, or to all states with energy lower than some level Ekeep, thus reducing the

subsequent Hamiltonian to a manageable size.

To illustrate the state truncation approximation, Figure 1-4 shows the runtime

required for the NRG procedure to be carried out on a 2.2-GHz AMD Opteron processor

for various values of Nkeep and the resulting truncated value of Ekeep. The runtime and

Ekeep results are scaled by their values for Nkeep = 10, which are 0.3 seconds and 1.68,

respectively. Proportionately, the runtime grows faster than Ekeep. (These results are

actually for the NRG study of the Anderson model, as described in Section 2.1.1, but they

illustrate the point well.)

This state truncation is an acceptable approximation, since the low-temperature

behavior is largely determined by the lower-lying energy eigenstates (EN(~m) . 10), and a

28

compromise can usually be reached between computational costs and maintaining an Nkeep

sufficiently high to ensure accuracy. Generally, more complex problems (e.g., multiple

impurities models or models with fewer conserved quantum numbers) have a richer energy

spectrum and thus require a larger Nkeep to maintain a sufficiently high Ekeep to obtain

acceptable results.

1.2.4 Fixed Points and Results

A fixed point of the RG transformation has been found when the renormalized energy

eigenvalues EN(~m) remain unchanged upon increasing N (as seen in Figure 1-5). It

turns out that the transformation R as given in Eq. (1–47) has no fixed points, since the

energy spectra for odd and even N are manifestly different in form [30]. Thus, it is the

transformation R2 that yields fixed points, and comparisons are made between EN(~m) and

EN+2(~m).

Wilson demonstrated that the ferromagnetic Kondo model has a stable fixed point

H∗J=0 (confirming the results from poor man’s scaling), and that the antiferromagnetic

Kondo model has an unstable fixed point H∗J=0 and a stable fixed point H∗

J=∞. (Technically,

each of these “fixed points” is a pair of fixed points—one for even N and one for odd

N—but the thermodynamic properties are the same.)

Wilson constructed H∗J=∞ to successfully arrive at a finite low-temperature expression

for χimp in the antiferromagnetic case [23]:

χimp(T ) =(gµB)2(0.4128± 0.002)

4kBTK

, T ¿ TK . (1–48)

This constant result showed definitively that the impurity local moment disappears as

T → 0 due to the impurity spin being fully compensated. The number 0.4128, dubbed

the Wilson number w, has since been obtained exactly from methods based on the Bethe

Ansatz [28, 29], in which the Kondo model is reduced to a subsidiary spin problem as

solved by Yang [15], producing the exact result w = eC+1/4/√

2π (where C ' 0.577216 is

Euler’s constant), which agrees with the NRG result to within the printed accuracy.

29

0 50 100 150

N (even)

0

0.5

1

1.5

2

2.5

3

E(N

)|S| = 0, |Q| = 01/2, 00, 11/2, 11, 00, 21/2, 10, 0

Figure 1-5. Energy levels E(N) vs. iteration number N (even) for the NRG procedureapplied to the Anderson model. The energy eigenstates are labeled by spinand charge. (Due to spin and particle hole symmetries, states of opposite spinor opposite charge are degenerate.) The energy levels are measured from theground state energy, such that the lowest energy level at each iteration isalways 0. Plateaus in the spectrum indicate the presence of a fixed point ofthe renormalization group transformation. At iteration 6, an unstable fixedpoint is approached, but the levels flow away from it by iteration 96. A stablefixed point is reached at iteration 110, signifying the onset of the Kondo effect.

Wilson also reproduced the high-temperature (T À TK) results from poor man’s

scaling [Eq. (1–30)], and obtained an expression for the crossover regime [23]

χimp(T ) ' 0.68(gµB)2

4kB(T +√

2TK), 0.5TK < T < 16TK , (1–49)

which is identified as a Curie-Weiss law for a reduced (or partially screened) local moment.

The Numerical Renormailzation Group is similarly applied to the Anderson model; I

will defer that description until Chapter 2.

30

1.3 Surface and Quantum Dot Realizations of the One-Impurity Kondo Effect

Wilson’s RG approach to the Kondo and Anderson problems had answered many

unresolved questions, confirmed the screening of the localized magnetic moment at low

temperatures, and explained the observed effects of magnetic impurities in bulk materials.

The question then arose of whether it was possible to observe a single impurity (rather

than a dilute set of impurities) exhibiting the Kondo effect. It was not until more recently

that such study became possible with development of scanning tunneling microscopy and

quantum dots.

1.3.1 Scanning Tunneling Microscopy Studies

Scanning tunneling microscopy (STM) [46] permits the study of metallic surfaces at

the atomic level. As a small metal tip is swept over the surface to be studied, the current

in the tip caused by electrons tunneling to the surface is monitored, allowing analysis of

the surface’s structure and the local electron density of states. One of the most interesting

applications of STM has been the direct observation of single magnetic impurities.

By using STM to study individual isolated magnetic atoms adsorbed onto a

non-magnetic surface, several experiments [57, 59, 62, 89, 90] have observed signatures

of the Kondo effect in the differential conductance G(V ) = dI/dV , where V is the bias

voltage of the sample relative to the tip. The differential conductance is proportional

to the rate at which electrons tunnel from the tip into a state of energy εF + eV on the

surface [57]. This tunneling rate is related to the impurity spectral function Ad(eV/~, T ),

which, is defined such that Ad(ω)dω is the probability that an electron in the impurity

level has an energy between ~ω and ~(ω + dω). (For convenience, in the remainder of

this work, ~ will be taken to be 1.) For T < TK and ω < kBTK , Ad(T, ω) exhibits the

aforementioned Kondo resonance: a Lorentzian [similar to the shape of Eq. (1–12)] of

width TK . Thus, when an STM tip probes a magnetic impurity, G(V ) is expected to show

a physical manifestation of the Kondo resonance at temperatures T < TK .

31

Figure 1-6 shows the results of an STM study of an isolated Co atom adsorbed onto

a gold (111) surface [57]. When the STM tip was positioned far away (12 A) from the

Co atom, a featureless G(V ) was observed; when the STM tip was positioned directly

above the Co atom, G(V ) exhibited a resonance around zero bias (corresponding to

energy excitations in the sample around εF ). The structure of the resonance is unlike the

Lorentzian shape of the Kondo resonance in Ad(ω). Figure 1-7 shows similar results for

an STM study of an isolated Ce atom on Ag(111) [59]. The feature at zero bias is more

symmetric than for Co, but still contains a dip (sometimes called an “antiresonance”)

rather than a peak.

The form of G(V ) for both of these cases is the lineshape predicted by Fano [8] who

worked with a model similar to the Anderson model in the field of atomic ionization.

Fano explained the lineshape by considering interference between tunneling into a discrete

state and a continuum of states. Similarly, the lineshape of G(V ) in the STM studies has

been attributed to quantum mechanical interference between tunneling of electrons from

the tip directly to the surface (a continuum of states for all positions ~r on the surface)

and tunneling from the tip to the impurity (the localized state associated with d†σ) and

thereafter to the surface through the Anderson model’s hybridization Vd.

Applying Fano’s study to the U = 0 Anderson model, the rate of transitions from the

tip into a final state of energy ε is given by [57]

R(ε) = R0(ε)[q + f(ε)]2

1 + f(ε)2, (1–50)

where R0 is the transition rate for an impurity-free system, and

f(ε) =ε− εd

∆/2. (1–51)

As G(V ) is proportional to R(ε = eV ), it displays this Fano lineshape, with a resonance

of width TK . The lineshape is characterized by the Fano parameter q ≤ 0, which depends

on the surface’s electronic structure and the relative probabilities of the two tunneling

32

Figure 1-6. STM differential conductance dI/dV for a tip located over bare Au(111) andover a single Co atom. The Co trace is fitted with a Fano lineshape of width2kBTK [57]. From Madhavan et al., Science 24 April 1998 280: 567-569.Reprinted with permission from AAAS.

paths [73], such that q = 0 indicates no tunneling to the impurity, q = −∞ indicates

no tunneling directly to the surface, and q = −1 indicates equal tunneling through both

channels.

To compare with the experimental results, Eq. (1–50) can be applied to the U 6= 0

Anderson model with the modification

f(ε) =ε− εd − ReΣ(ε)

ImΣ(ε), (1–52)

33

Figure 1-7. STM differential conductance dI/dV for a tip located over a single Ce atom onan Ag(111) surface. The data are fitted with a Fano lineshape of width 2kBTK

[59]. Reprinted with permission from J. Li, W.-D. Schneider, R. Berndt, andB. Delley, Phys. Rev. Lett. 80, 2893 (1998). Copyright 1998 by the AmericanPhysical Society.

where Σ(ε) is the impurity electron’s self-energy. For energies near the Kondo resonance

and for T < TK , f(ε) is

f(ε) =ε− α

kBTK

, (1–53)

where α is a constant.

Figures 1-6 and 1-7 contain fits for the Co and Ce data using Eqs. (1–50) and (1–53).

For the Co data, the fit gives q = −0.7 [57], indicating comparable tunneling through

both channels; for the Ce data, the fit gives q ≈ 0 [59], indicating almost no tunneling

to the impurity. What is fascinating about the q = 0 result is that, even though no

electrons tunnel directly to the impurity, the impurity’s presence is still observed in the

differential conductance. This behavior is evidenced by the fact that, even when q = 0,

34

the tunneling transmission rate [Eq. (1–50)] still acquires a non-uniform shape due to the

presence of the impurity electron’s self-energy. Physically, this property can be understood

by considering second-order processes, in which electrons that have tunneled from the tip

to the surface then tunnel from the surface to the impurity and back again to the surface

before returning to the STM apparatus. The extra tunneling causes the electrons to pick

up a phase shift relative to the electrons that do not tunnel, and destructive interference

takes place.

The Fano resonance has been observed in the differential conductance in many

other STM experiments [62, 89, 90], including magnetic impurities adsorbed onto carbon

nanotubes, which exhibit a one-dimensional density of states, further demonstrating the

robustness of the Fano-resonance behavior. We will observe that the NRG confirms the

Fano-resonance behavior in Chapter 4.

STM studies also permit an examination of the spatial extent lK of the screening

cloud that causes the Kondo effect. Such a quantity is not directly calculated in the

standard NRG procedure, but can be estimated experimentally by moving the STM

tip laterally away from the impurity. Figures 1-8 and 1-9 show G(V ) for various lateral

distances between the tip and impurity. The magnitude and asymmetry of the Fano-shape

feature decrease as the lateral distance increases, indicating a reduction of q caused

by a drop in the rate of electrons tunneling to the impurity. Sufficiently far from the

impurity, the featureless conductance is recovered. Section 6.1 describes a novel approach

to calculate lK with an extension of the NRG [96].

1.3.2 Quantum Dot Studies

A quantum dot (QD) is a nanostructure that spatially confines a specified number

of electrons. The electronic properties of a QD are similar to those of atoms, featuring

discrete energy levels, internal Coulomb repulsion, and a coupling to the sea of electrons

in the external host. When attention is focused on the highest unfilled (or partially filled)

level in the QD, these parameters are akin to εd, U , and Vd from the Anderson model,

35

Figure 1-8. STM differential conductance dI/dV for tip located at various lateral distancesfrom a single Co atom on Au(111) [57]. The resonance’s asymmetry andamplitude decrease as the distance increases. From Madhavan et al., Science24 April 1998 280: 567-569. Reprinted with permission from AAAS.

36

Figure 1-9. STM differential conductance dI/dV for tip located at various lateral distancesfrom a single Ce atom on Ag(111) [59]. The feature at V = −70mV isattributed to the Ag surface state, and is modified by the proximity of theimpurity. Reprinted with permission from J. Li, W.-D. Schneider, R. Berndt,and B. Delley, Phys. Rev. Lett. 80, 2893 (1998). Copyright 1998 by theAmerican Physical Society.

37

respectively. The values of εd, U , and Vd can be controlled by tunable gate voltages,

permitting manipulation of this artificial atom. Thus, QD devices have become a versatile

tool for studying magnetic impurity systems, permitting the experimental observation of

previously inaccessible regimes and behaviors.

Tuning εd permits control over the number of electrons in the dot. When a QD is

tuned to have an odd number of electrons (i.e., it meets the conditions for local-moment

formation in the Anderson model as described in Section 1.1.2), it must have a non-zero

total spin, and therefore can display the Kondo effect. Because of the two-dimensional

geometry of most QDs (such as semiconductor QDs formed at the interface between GaAs

and AlGaAs), the Kondo effect causes an increase in the conductance G(V ) through the

dot [55, 64] as T decreases.

These predictions are confirmed in the results of a QD experiment [64] shown in

Figure 1-10. Part (a) of the figure shows G(V ) vs. the gate voltage V corresponding to

εd, for temperatures ranging from 25 mK (light blue curve) to 1 K (orange curve). As

the gate voltage is increased over the range shown in the figure, the occupancy of the dot

increases from N to N + 4 (where N is even). These regions form a Coulomb blockade

pattern, indicated by peaks and valleys in the conductance. For even occupancy, the

conductance decreases with temperature, indicating that no Kondo effect takes place; for

odd occupancy, the conductance increases with temperature, indicating the occurrence of

the Kondo effect and thus the presence of a local moment on the dot. Similar results were

found in [55], where the enhanced conductance was shown to disappear for temperatures

above the Kondo temperature.

Figure 1-10(b) shows G(V ) vs. T for three values of V that produce odd occupancy

N + 1. As V (i.e., εd) changes, so does TK . If the data are instead plotted as G(V ) vs.

T/TK , as in part (c), the results lie on top of each other, indicating that G(V ) is given by

a universal function F (T/TK) as seen in Eq. (1–25).

38

Figure 1-10. (a) Conductance G(V ) vs. gate voltage V for temperatures ranging from 25mK (light blue curve) to 1 K (orange curve). Coulomb blockade valleys arelabeled by electron occupancy, where N is even. (b) For odd occupancyN + 1, G(V ) vs. T for three values of V (corresponding to arrows in a). (c)G(V ) vs. T/TK for the same three values of V , indicating the universalscaling of the conductance in the Kondo regime [64]. Reprinted withpermission. Copyright 2001 by Physics World (2001January pp33-38).

39

It is fascinating to note that, when the Kondo effect takes place, this quantum dot

becomes nearly “transparent,” as G(V ) nearly reaches the quantum mechanical maximum

value of 2e2/h.

Quantum dots are thus an exciting arena of study, offering a vast range of possibilities

for the creation of tunable atom-like structures. For example, another possible application

of QDs to the Kondo effect is the study of a dot with spin S > 1/2 [63].

1.4 Systems of Multiple Impurities

Since Kondo’s breakthrough in 1964 and Wilson’s application of the RG in 1975,

the physics of a single isolated magnetic impurity in a simple metal host has become

well understood. However, matters become more subtle when one considers a system of

multiple magnetic impurities that are permitted to interact with each other. The direct

or indirect exchange of electrons among multiple impurities can lead to the ferromagnetic

(parallel) or antiferromagnetic (antiparallel) alignment of the impurities’ spins, modifying

the Kondo screening behavior. The competition between magnetic ordering and Kondo

screening has important implications for heavy fermion systems, for small magnetic

devices, and for future quantum computers that will utilize magnetic moments as quantum

bits.

As a first approach to understanding systems of multiple impurities, one can gain a

great deal of insight by considering a pair of magnetic impurities. There has been much

theoretical and numerical work performed to studying the two-impurity Kondo model and

the two-impurity Anderson model. Experimentally, the study of multiple-impurity systems

has been pursued in the venues of STM and double quantum dots (DQDs).

1.4.1 Theoretical Studies of Two-Impurity Models

As modeling a system of an arbitrary number of impurities is a very demanding task,

much attention has been devoted to two-impurity models, which, as we shall see, still

capture much of the essential physics of a many-impurity system. The Kondo Hamiltonian

40

[Eq. (1–4)] is extended into its two-impurity version:

HK2 = Hc +∑

i

Ji

2~Si ·

∑

~k,~k′

ei(~k′−~k)·~ric†~kσ~τσ,σ′c~k′σ′ . (1–54)

Here, the new index i = 1, 2 labels the impurities and ~ri is the position vector of impurity

i. Potential scattering terms have been omitted from Eq. (1–54) for simplicity.

As with the single-impurity Kondo model, the two-impurity Kondo model can be

thought of as a special case of the two-impurity Anderson model, when each impurity

favors the formation of a local moment. The two-impurity Anderson model will be

described in greater detail in Chapter 2.

Various theoretical and numerical methods have been employed to study the

two-impurity Kondo and Anderson models, including perturbative scaling [32], the NRG

[34–36, 41, 42, 49, 74, 80], quantum Monte Carlo methods [33, 37], variational methods

[40], and conformal field theory [43, 45]. These studies have highlighted the importance of

a number features that are not present in the single-impurity Kondo model: the presence

of two effective conduction-band channels, a greater complexity of the electron density of

states, and a competition between the Kondo effect and magnetic ordering effects.

The first new feature, the presence of two effective conduction-band channels arising

from the coupling to the impurities at locations ~r1 and ~r2, will be explored in more detail

in Section 2.2.1. Essentially, it is computationally more convenient to define conduction

electron states that are symmetric and antisymmetric about the midpoint between the two

impurities, called the “even” channel and “odd” channel, respectively. While this division

of the conduction band is a theoretical construct, it does have physical implications, such

as the two-stage Kondo screening process described below and found in [27].

The second new feature, a greater complexity of the electron density of states, will

also be illustrated in more detail in Section 2.2.1. The variable impurity separation

R = |~r1 − ~r2| means that one cannot simply ignore the ~k-dependence arising from the

41

form of εk. As described below, confusion and disagreement have resulted in the past from

ignoring this complexity. I therefore pay careful attention to this issue in the present work.

The final new feature in the two-impurity Kondo model is the competition between

the Kondo effect (which seeks to lock each impurity into a net spin singlet with the

conduction electrons) and magnetic interactions between the local moments of the

impurities (which seek to lock the impurities into a singlet or triplet with each other). The

two impurities may directly interact with each other, reflected in a term such as

HI = −I ~S1 · ~S2 (1–55)

being added to the Hamiltonian in Eq. (1–54). Here, I is the strength of the magnetic

interaction, with I > 0 causing an ferromagnetic (parallel) alignment of the impurity

moments and I < 0 causing an antiferromagnetic (antiparallel) alignment of the impurity

moments. In order to examine how this interaction competes with the Kondo effect, |I|must be compared with kBT 1−imp

K , where T 1−impK is the single-impurity Kondo temperature.

Even if such a direct interaction is not present, there is still an indirect Ruderman-Kittel-

Kasuya-Yosida (RKKY) interaction to consider [4, 5]. The RKKY interaction is mediated

by the conduction electrons, and arises from the Friedel oscillations in the conduction

electron density around each impurity [24]. As we shall see in greater detail in Chapter 4,

the RKKY interaction IRKKY varies in strength and magnetic nature (ferromagnetic for

IRKKY > 0 or antiferromagnetic for IRKKY < 0) depending on the separation between the

two impurities. The interaction strength is, again, to be compared with the single-impurity

Kondo temperature.

The majority of the previous theoretical work has focused on identical impurities

(J1 = J2) without a direct magnetic interaction. The competition between the RKKY

interaction and the Kondo effect has been predicted to cause a number of behaviors that

were not present in the single-impurity Kondo model.

42

For example, when IRKKY À T 1−impK , at higher temperatures the ferromagnetic

RKKY interaction causes the two impurity moments to form a combined spin-1 system

[32, 35, 37, 49]. This spin-1 is then screened in a two-stage Kondo effect. In this two-stage

Kondo effect, the spin-1 reduces to a spin-12, which then reduces to a spin-0. (For examples

of this behavior, see Figures 1-17 and 4-10.) These two stages can be explained by the

presence of the even and odd conduction-band channels (see above discussion and [27]),

each of which exhibits its own Kondo temperature. Chapter 4 will explore how this

two-stage screening process may be observed in an STM experiment.

Another class of behavior arises when |IRKKY| ¿ T 1−impK . For this regime, the RKKY

interaction is not strong enough to combine the impurity moments into a spin-1 before the

Kondo effect screens each moment individually. Thus, the two impurities are effectively

isolated from each other [32, 49]. This behavior will also be observed in the context of

STM experiments in Chapter 4.

A very different behavior arises when −IRKKY À T 1−impK . For this regime, the RKKY

interaction causes the impurity spins to antiferromagnetically align into a spin singlet

state, such that no Kondo effect can take place [32, 35, 37, 49]. This behavior will also be

observed in the context of STM experiments in Chapter 4.

One last feature of the model that has been explored in great detail is the transition

between the independent-impurity regime (|IRKKY| ¿ T 1−impK ) and the impurity-singlet

regime (−IRKKY À T 1−impK ). The original poor man’s scaling analysis [32] predicted a

smooth transition between the two regimes. This smooth transition was later supported

by quantum Monte Carlo methods [33, 37]. However, an NRG study [35] predicted the

presence of an unstable critical point separating the two regimes. At first, the discussion

of the discrepancy centered around the differences in the methods used (for example,

the NRG work had not used ~k-dependent couplings between the impurities and the

conduction-band channels), but finally a connection between the two results was verified

by variational methods [40], conformal field theory [43, 45], and a modified version of

43

the NRG (which preserved the ~k-dependence of the couplings) [49]. The later bodies of

work explain that the unstable critical point only occurs when the two-impurity model

employed is symmetric under a very specific particle-hole transformation (regardless of

whether the couplings are taken to be ~k-dependent). While this symmetry is physically

unlikely, a system close to particle-hole symmetry still exhibits a signature of the critical

point.

As noted above, the majority of the previous work has focused on the special case of

identical impurities, which is both computationally simpler than non-identical impurities

and physically relevant, since one often deals with impurities of the same element (as in

the STM experiments described below). In the newer context of double quantum dots,

however, one can tune the dots to be non-identical, revealing a rich tapestry of new

behaviors, as will be explored in Chapter 5.

1.4.2 Multiple-Impurity STM Studies

Using STM techniques, it is possible to arrange multiple impurities in various

configurations. The simplest case to study is that of two impurities at various separations.

Figure 1-11 plots the results of an extension of the single Co atom study [60] in

which a pair of Co atoms are examined. The top two traces are the same as the results

in Figure 1-6. The bottom trace shows G(V ) when a second Co atom is added, revealing

the disappearance of the Kondo resonance. The authors of the study attribute this

disappearance to the value of εd shifting away from the Fermi level due to the addition

of the second impurity. This shift of εd causes a drastic reduction of J [see Eq. (1–16)],

thereby lowering TK below 6 K, which is the temperature at which the experiment took

place, making the Kondo resonance vanish from the data.

Figure 1-12 shows the results of a similar experiment [70] involving two magnetic

Ni atoms adsorbed onto Au(111). The four traces show G(V ) for different impurity

separations. The Kondo resonance (labeled as peak “A” in each trace) remains unchanged

until the atoms are moved to within 3.4 A of each other, at which point the resonance

44

Figure 1-11. STM differential conductance for Co atoms on a non-magnetic Au(111)surface [60]. Reprinted with permission from W. Chen, T. Jamneala, V.Madhavan, and M. F. Crommie, Phys. Rev. B 60, R8529 (1999). Copyright1999 by the American Physical Society.

shrinks, indicating a drop in TK , again thought to be caused by a modification of εd. (This

time, the resonance does not disappear, since the new Kondo temperature is still above

the temperature at which the experiment took place.)

Returning to the Ce study described in Section 1.3.1, Figure 1-13 [59] shows the

differential conductance for a single Ce atom, a cluster of Ce atoms, and a film of Ce

atoms deposited onto the same Ag(111) surface. While the general lineshape is the

same, the width 2kBTK of the dip and the asymmetry parameter q change, indicating a

modification of the tunneling path interference pattern.

45

Figure 1-12. STM differential conductance for a pair of Ni atoms on a non-magneticAu(111) surface [70]. Reprinted with permission from V. Madhavan, T.Jamneala, K. Nagaoka, W. Chen, J. L. Li, S. G. Louie, and M. F. Crommie,Phys. Rev. B 66, 212411 (2002). Copyright 2002 by the American PhysicalSociety.

1.4.3 Double Quantum Dot Studies

Similar modification to the Kondo effect has been observed in DQD devices (two

quantum dots connected to each other and to the same external structure), which have

been proposed as a possible two-qubit system to be used in quantum computation. Figure

1-14 shows the setup of a DQD device [76] composed of dots L and R coupled by a

conducting region C. The dot energy parameters εα, Uα, and ∆α (where α = L,R labels

the dots) and the inter-dot tunneling rate are tuned by the gate voltages VgL, VgC , and

VgR. The differential conductance of each dot Gα(V ) = dI/dVα is measured with the left

and right leads.

46

Figure 1-13. STM differential conductance dI/dV for various Ce configurations on Ag(111)[59]. Reprinted with permission from J. Li, W.-D. Schneider, R. Berndt, andB. Delley, Phys. Rev. Lett. 80, 2893 (1998). Copyright 1998 by theAmerican Physical Society.

As in the study described in Section 1.3.2, the occupancy of each dot can be tuned by

the gate voltages VgL and VgR that control the resonant energies εL and εR, respectively.

Coulomb blockade valleys were found in the differential conductance plots of Figure 1-15,

parts A and C. The valleys are labeled by the number of electrons in dot R, where M is

odd. Parts A and B correspond to dot L having an even number (N − 1) of electrons, and

thus no local moment. As in the single QD study, the Kondo resonance is present around

zero bias only when dot R has an odd number of electrons, forming a local moment on dot

R. However, when dot L contains an odd number of electrons (parts C and D), the Kondo

resonance on dot R is suppressed. The authors of the study attribute this suppression

47

Figure 1-14. Setup for coupled double quantum dot experiment [76]. Dots L and R arecoupled by the conducting region C. Gate voltages determine energy levels,occupancies, and coupling of the dots; leads measure the conductance of eachdot. From Craig et al., Science 23 April 2004 304: 565-567. Reprinted withpermission from AAAS.

to the exchange of electrons across the central region, leading to either a spin-0 state

(antiferromagnetic alignment) between the two dots with no Kondo effect, or to a spin-1

state (ferromagnetic alignment) with a weaker Kondo effect having a TK smaller than the

temperature of the experiment. Whichever behavior is the cause, the magnetic interaction

between the dots significantly modifies the Kondo effect.

This modification can be observed to develop in Figure 1-16, which plots the

conductance of the left dot GL(VL) for various strengths of the coupling between dot

R and the conducting region. When dot L and dot R both have an odd number of

electrons (part A), increasing the coupling (and therefore increasing the magnetic ordering

effect) causes the Kondo resonance to become suppressed and to split into two peaks.

This suppression and splitting behavior agrees with theoretical predictions [71, 82, 88] and

has also been observed in parallel-coupled double quantum dot experiments [77]. When

dot L has an odd number of electrons and dot R an even number (part B), increasing

the coupling causes slight quantitative modification to the Kondo resonance, but the

qualitative features remain the same.

48

Figure 1-15. Differential conductance of dot R for the DQD setup shown in Figure 1-14.(A) Coulomb blockade valleys for dot R when dot L has an even number(N − 1) of electrons. Valleys are labeled by the electron occupancy number ofdot R, where M is odd. (B) Differential conductance dI/dVR when dot L hasN − 1 electrons and dot R has M − 1, M , and M + 1 electrons. The Kondoresonance is observed around zero bias when dot R has an odd number ofelectrons. (C) Coulomb blockade valleys for dot R when dot L has an oddnumber (N) of electrons. Valleys are labeled by the electron occupancynumber of dot R, where M is odd. (D) Differential conductance dI/dVR whendot L has N electrons and dot R has M − 1, M , and M + 1 electrons. TheKondo resonance previously observed on dot R has been suppressed [76].From Craig et al., Science 23 April 2004 304: 565-567. Reprinted withpermission from AAAS.

49

Figure 1-16. Differential conductance of dot L for the DQD setup shown in Figure 1-14,for various strengths of the coupling between dot R and the conductingregion, (A) when dots L and R both have an odd number of electrons and(B) when dot L has an odd number of electrons and dot R an even number[76]. From Craig et al., Science 23 April 2004 304: 565-567. Reprinted withpermission from AAAS.

By constructing DQD devices of different geometries and fine-tuning the device

parameters with gate voltages, one may study a great variety of physical behaviors

previously inaccessible in experiments. This new breadth of possibilities has stimulated a

large number of theoretical and numerical studies.

For example, a study utilizing the NRG and slave-boson mean-field theory [85]

working with a side-coupled DQD (for which a central dot is connected to the leads and

to a side dot) predicts a two-stage Kondo screening effect (cf. Section 1.4.1), as seen in

Figures 1-17 and 1-18. The upper panel of Figure 1-17 shows Tχimp vs. T/TK , indicating

the screening of the DQD spin in two stages. The lower panel shows how this two-stage

process is evidenced in the conductance (given in units of G0 = 2e2/h). The screening

of the central dot at T ∼ TK enhances the conductance, which rises nearly to G0. The

screening of the side dot at a much lower T ∼ 10−4TK leads to a suppression of the

50

conductance. This suppression is attributed to a splitting of the Kondo resonance in the

central dot spectral function.

Figure 1-17. Side-coupled DQD displaying two-stage Kondo screening behavior [85].Model parameters are ε1 = ε2 = −0.25, U1 = U2 = 0.5, ∆1 = 0.035, andinter-dot tunneling amplitude λ = 0.003. The upper panel shows the squareof the impurity magnetic moment µ2 [kBTχimp/(gµB)2 in the language of thepresent work] vs. T/TK . The lower panel shows how the two-stage process isevidenced in the conductance (given in units of G0 = 2e2/h). Reprinted withpermission from P. S. Cornaglia and D. R. Grempel, Phys. Rev. B 71,075305 (2005). Copyright 2005 by the American Physical Society.

Figure 1-18 [85] shows the conductance for this side-coupled device as ε = ε1 = ε2

is varied at various temperatures. For T = 0 (the bottom panel), at the particle-hole

symmetric point ε = −U/2 where the total device occupancy is even, the conductance

vanishes, whereas there is perfect conductance at ε ≈ 0 and ε ≈ −U , where the total

occupancy of the device is odd. Thus, here we see the behavior of a Coulomb blockade

valley, as observed in experimental studies.

51

Figure 1-18. Conductance vs. ε = ε1 = ε2 (controlled by gate voltage) for a side-coupledDQD [85] at various temperatures. Model parameters are U1 = U2 = 0.25,∆1 = 0.125, and inter-dot tunneling amplitude λ = 0.025. Note that theconductance features vanish when T & TK . Reprinted with permission fromP. S. Cornaglia and D. R. Grempel, Phys. Rev. B 71, 075305 (2005).Copyright 2005 by the American Physical Society.

In the middle panels of Figure 1-18, the second stage of the Kondo screening effect

is not taking place, and so the conductance at ε = −U/2 is enhanced. In the top

panel, the temperature is on the order of TK , and so the features of the conductance

spectrum disappear. (This agrees with the experiment in [55], in which the conductance

enhancement disappears when T exceeds TK .)

Other experimental and theoretical studies have used DQD devices to examine

Coulomb-blockade behavior [72], observe interference between electron paths [75],

52

construct Aharonov-Bohm interferometers [58, 68, 69, 87], observe the two-channel

Kondo effect [95], construct two-level QD devices [75], probe the length of the Kondo

screening cloud [91], and observe the Kondo effect in a QD side-coupled to a quantum wire

[66, 91].

While there has been much theoretical work done, the majority of these studies has

focused on devices composed of identical quantum dots or devices for which both dots

are in the Kondo regime. Recently, various efforts [88, 91, 94, 98] have been made to

study a class of DQD devices in which one dot is in the Kondo regime and the other is

weakly interacting (i.e., has a small Coulomb energy U2, and therefore no local moment).

Attention has been paid to the special limit of U2 = 0 which, though difficult to achieve

experimentally, can be modeled as a single Anderson impurity (Dot 1) connected to a

conduction band via a nonconstant hybridization [88]

∆(ε) ≡ π∑

~k

|V~kd|2δ(ε− ε~k) (1–56)

=

[λ√

∆2 + (ε− ε2)√

∆1

]2

(ε− ε2)2 + (∆2)2, (1–57)

where λ is the inter-dot tunneling amplitude and ∆i = πρV 2i is the effective hybridization

width of dot i with the leads.

The effects of this nonconstant hybridization have been studied [88, 94, 98] in the

specific geometries of the side-coupled DQD (in which case the Kondo-like Dot 1 couples

to the leads only through the non-interacting Dot 2) and the parallel DQD (in which

case both dots connect directly to the leads, but not to each other). In the side-coupled

configuration, for sufficiently strong inter-impurity tunneling λ, the Kondo resonance

in Dot 1 splits, producing noticeable changes in the device conductance. In the parallel

configuration, there arises a pair of quantum phase transitions separating Kondo-screened

phases and a local-moment phase in which the impurity remains unscreened down to zero

temperature. Since the condition U2 = 0 is experimentally unreasonable to obtain, it is

53

necessary to explore the effects of small positive U2 on these behaviors. In Chapter 5, I

will employ the two-impurity Anderson model (since the effective one-impurity model is

limited to U2 = 0) to study how these behaviors are modified for U2 & 0 and eventually

destroyed for sufficiently large U2.

1.5 Study Overview

The data presented in Section 1.4 show that systems of coupled magnetic impurities

and their DQD analogs display a rich variety of behaviors warranting deeper investigation.

A system of two coupled Kondo-like impurities has been shown to exhibit interesting

properties, but it is difficult to observe those properties experimentally in bulk hosts.

Studying a two-impurity system on a surface with STM techniques should allow one

to probe effects such as two-stage Kondo screening and impurity singlet formation.

Also, while systems of two identical (or at least two Kondo-like) impurities have been

explored in depth, systems of highly inequivalent impurities have received little attention.

This broader range of impurity configurations is achievable in systems of DQDs. The

remainder of this study will follow along these two paths of interest: The manifestation of

known two-impurity behavior in STM studies and the properties of systems of two highly

inequivalent impurities in the form of DQDs.

Before embarking upon these paths, in Chapter 2, I will review the transformation

of the single-impurity and two-impurity Anderson Hamiltonians into forms that can be

studied using NRG techniques. I will describe the NRG procedures used to diagonalize

these Hamiltonians and to calculate the thermodynamic and spectral properties of

interest. In the case of the two-impurity Anderson Hamiltonian, I will focus on the special

limits of identical impurities and zero impurity separation, as these computationally

simplified cases will be of use in Chapters 4 and 5.

In Chapter 3, I will give an overview of my efforts to improve the computational

efficiency of the NRG iterative procedure by applying parallel processing techniques to

the diagonalization of the iterative Hamiltonians HN and the calculation of operator

54

matrix elements, such as 〈N, ~m|d†σ|N, ~m〉. To parallelize the diagonalization process, I

have employed two methods that take advantage of the block-diagonal nature of HN : In

the first method, larger matrix blocks are diagonalized by all processors working together

using Scalable LAPACK [101] routines; in the second method, smaller matrix blocks

are diagonalized by individual processors using LAPACK [100] routines. For the sample

calculation shown in Chapter 3, the two methods result in a minimum relative wall-clock

time of 65% for 4 processors, reflecting a mediocre scalability of wall-clock time with

increasing the number of processors. To parallelize the calculation of operator matrix

elements, I have employed a method in which each processor calculates a “chunk” of

matrix elements at a time, again taking advantage of the block-diagonal nature of HN .

For the sample calculations shown in Chapter 3, this method results in a much better

scalability of wall-clock time with increasing number of processors.

In Chapter 4, I will apply the NRG methods of Chapter 2 to model STM studies of

surface impurities. I will begin by recapitulating the setup and the results for a single

surface impurity [73], and then present new results for a pair of identical surface impurities

without a direct interaction. By varying the impurity separation, I will demonstrate how

many of the behaviors described in Section 1.4.1 are manifested in the STM conductance

spectrum as the magnitude and sign of the RKKY interaction change. In particular, we

will see that impurity configurations that display a two-stage Kondo effect (IRKKY À TK)

produce a conductance spectrum given by a sum of two Fano lineshapes (each having its

own q and TK); effectively isolated impurities (|IRKKY| ¿ TK) produce a conductance

spectrum given by a single Fano lineshape; and impurities locked into a spin singlet state

(−IRKKY À TK) produce a generally featureless conductance spectrum.

In Chapter 5, I will apply the NRG methods of Chapter 2 to model the aforementioned

class of highly-asymmetric DQD devices, beginning with the U2 = 0 special case described

above and expanding to include U2 & 0. This new range of device parameters will

require the use the two-impurity model of Chapter 2, instead of the effective one-impurity

55

model [88] discussed above. I will focus on the side-dot and parallel DQD configurations

described above. In the side-coupled case, I will explore the effects of positive U2 on the

splitting of the Kondo resonance and the resulting changes in the conductance minimum

[98]. In the parallel-dot case, I will explain how the pair of QPTs evolve and eventually

disappear as U2 is increased from zero.

In Chapter 6, I will summarize the behaviors observed in Chapters 4 and 5, and

present ideas for future work.

56

CHAPTER 2BACKGROUND MATERIAL

2.1 Application of the NRG to the Anderson Model

In this chapter I will review the single-impurity Anderson Hamiltonian and show

how it is developed into a form that can be studied using NRG techniques [30, 31]. I will

then outline the procedure for using these NRG techniques to calculate thermodynamic

and spectral properties such as the impurity magnetic susceptibility and operator spectral

functions. I will then describe the fixed points that arise in the NRG treatment and how

they are manifested in the thermodynamic and spectral properties. Lastly, I will describe

the extension of the NRG techniques to the two-impurity Anderson model.

2.1.1 Discretization and Eigensolution

The single-impurity Anderson Hamiltonian [7] consists of (as ordered below) a

conduction-band term, an impurity term, and a hybridization term:

HA =∑

~k

ε~kc†~kσ

c~kσ + εdd†σdσ + U(d†↑d↑)(d

†↓d↓) +

∑

~k

(V~kdc†~kσ

dσ + V ∗~kd

c~kσd†σ). (2–1)

All energies are measured with respect to the Fermi level (ε = ε − εF , |ε| ≤ D) and

repeated spin indices (σ, in this case) are implicitly summed over. The c†~kσ(c~kσ) operators

are standard fermion operators that create (annihilate) conduction band electrons of

momentum ~k and spin σ, and the d†σ (dσ) operators create (annihilate) impurity electrons

of energy εd, spin σ, and on-site Coulomb repulsion U .

It is standard to assume for simplicity that the hybridization parameter V~kd is

a constant, Vd, and that the conduction band density of states is a constant over a

bandwidth of 2D: ρ(ε) = ρ0 for |ε| < D and 0 elsewhere. With these assumptions,

εd = −U/2 is a point of particle-hole symmetry, i.e., invariance under the transformation

dσ → −d†σ and c~k,σ → c†~k,σ. In the case −εd À ∆ ≡ πρ0V

2d and εd + U À ∆, this model

describes an impurity with a well-defined spin-12

for kBT ¿ ∆, and exhibits the Kondo

effect at sufficiently low temperatures.

57

The study of the Anderson model using NRG methods [30] employs the same

discretization procedure as was described for the Kondo model in Section 1.2.2 and

[23]. The Anderson Hamiltonian is thus transformed into a linear chain form similar to

Eq. (1–37):

HA =∞∑

n=0

[εnf

†n,σfn,σ + λnf †n,σfn+1,σ + λ∗nf †n+1,σfn,σ

](2–2)

+ εdd†σdσ + U(d†↑d↑)(d

†↓d↓) +

√2Vd

(d†σf0,σ + h.c.

).

Again choosing a density of states ρ(ε) = ρ0 = 1/2D, |ε| < D, the couplings εn and λn are

the same as in Eqs. (1–38) and (1–39).

As with the NRG treatment of the Kondo problem, the chain Hamiltonian (2–2) is

solved via an iterative diagonalization procedure. The iterative dimensionless Hamiltonian

is

HN = Λ(N−1)/2

N−1∑n=0

Λ−n/2ξn

(f †n,σfn+1,σ + h.c.

)(2–3)

+ δNd†σdσ + UN

(d†σdσ − 1

)2+ ∆

1/2N

(d†σf0,σ + h.c.

),

where ξn is given by Eq. (1–40), and

δN =

(2Λ(N−1)/2

1 + Λ−1

)(2εd + U

2D

)(2–4)

UN =

(2Λ(N−1)/2

1 + Λ−1

)U

2D(2–5)

∆N =

(2Λ(N−1)/2

1 + Λ−1

)22∆

πD. (2–6)

The eigenstates of HN , again abbreviated |N, ~m〉, are labeled by charge Q, spin S, and

spin z-compoment Sz, as HN commutes with the charge and spin operators, defined as [30]

QN =N∑

n=0

(f †n,σfn,σ − 1

)+

(d†σdσ − 1

), (2–7)

~SN =1

2

N∑n=0

f †n,σ~τσ,σ′fn,σ′ +1

2d†σ~τσ,σ′dσ′ . (2–8)

58

As described in Section 1.2.3, the NRG procedure begins by diagonalizing the initial

Hamiltonian H0:

H0 = δ0d†σdσ + U0

(d†σdσ − 1

)2+ ∆

1/20

(d†σf0,σ + h.c.

). (2–9)

Using the eigenvalues E0(~m) and eigenstates |0, ~m〉, the RG transformation

HN+1 = R(HN) (2–10)

= Λ1/2HN + ξN(f †N,σfN+1,σ + h.c.) (2–11)

is employed to construct H1, which is then diagonalized, etc.

2.1.2 Calculation of Thermodynamic Properties

At each iteration N in the NRG procedure, one can calculate certain properties of

the impurity system along an exponentially decreasing energy or temperature scale. The

two properties of primary interest in this dissertation are the impurity contribution to the

magnetic susceptibility χimp(T ) and the impurity spectral function Ad(ω).

For the calculation of χimp(T ), there is an exponential temperature scale given by [30]

TN(β) =D

2kBβ(1 + Λ−1)Λ−(N−1)/2, (2–12)

where β is a chosen parameter. (We will see the criteria for choosing β below.) If during a

run of the NRG algorithm one utilizes two values of β such that TN(β1) = TN+1(β2) = T ,

one can compare the two resulting values of χimp(T ) as a measure of the NRG algorithm’s

accuracy (and thereby evaluate the choices of Λ and the number of states kept).

The calculation of χimp(TN) is given by [30]

kBTNχimp(TN)/(gµB)2 ≈[TrS2

Nz exp(−βHN)

Tr exp(−βHN)− Tr(S0

Nz)2 exp(−βH0

N)

Tr exp(−βH0N)

], (2–13)

59

in which SNz is the z-component of the spin operator defined in (2–8), and the superscript

“0” refers to an operator for an impurity-free system, i.e.,

H0N = Λ(N−1)/2

[N−1∑n=0

Λ−n/2ξn

(f †n,σfn+1,σ + f †n+1,σfn,σ

)], (2–14)

~S0N =

1

2

N∑n=0

f †n,σ~τσ,σ′fn,σ′ . (2–15)

To evaluate the traces in Eq. (2–13), one utilizes the fact that the energy eigenvalues

of HN (H0N) do not depend on Sz. Thus, the traces reduce to sums over k and S, where

k represents all quantum numbers other than S and Sz. Carrying out the sums over Sz

produces the results

Tr exp(−βHN) =∑

k,S

(2S + 1) exp(−βEN(k, S)

), (2–16)

TrS2Nz exp(−βHN) =

∑

k,S

1

12(2S + 1)

[(2S + 1)2 − 1

]exp

(−βEN(k, S)), (2–17)

Tr exp(−βH0N) =

∑

k,S

(2S + 1) exp(−βE0

N(k, S)), (2–18)

Tr(S0Nz)

2 exp(−βH0N) =

∑

k,S

1

12(2S + 1)

[(2S + 1)2 − 1

]exp

(−βE0N(k, S)

). (2–19)

Thus, at the end of each iteration N , TNχimp(TN) may be evaluated using only the energy

eigenvalues of HN and H0N .

Here is where a trade-off arises in the NRG procedure. Using Eq. (2–13) incurs an

error on the order of β/Λ (due to the contribution from chain sites with index n > N

that are not incorporated until subsequent iterations). Thus, choosing a small value of

β would seem to be preferred. However, in order for Eqs. (2–16) through (2–19) to be

evaluated accurately, all energy eigenvalues up to several times 1/β must be included (so

that exp(−βEN

) ¿ 1 for the omitted states). Because the higher energy eigenstates at

iteration N − 1 are truncated (see Section 1.2.3), these larger eigenvalues at iteration N

may not be calculated accurately. Thus, β is chosen to keep the O(β/Λ) error reasonable

60

without requiring too many eigenstates to be kept. For Λ = 2.5 or 3 (the values used in

the calculations for this study) 0.3 < β < 0.6 works well.

2.1.3 Calculation of Spectral Functions

As mentioned in Section 1.3.1, the impurity spectral function Ad(ω, T ) is needed

in order to calculate the impurity’s contribution to transport properties such as the

conductance. The impurity spectral function can be evaluated using the NRG procedure

by employing the formula [44]

Ad(ω, T ) =1

ZN(β)

∑

~m,~m′|〈N, ~m|d†σ|N, ~m′〉|2

(e−βEN (~m) + e−βEN (~m′)

)×

δ (ω − (EN(~m′)− EN(~m))) , (2–20)

where ~ has been taken to be 1, β satisfies Eq. (2–12) with TN → T , δ(x) is the Dirac

delta-function, and ZN(β) is the partition function for an N -site chain [73]:

ZN(β) =∑

~m

e−βEN (~m). (2–21)

The matrix elements 〈N, ~m|d†σ|N, ~m′〉 are evaluated recursively by a transformation

similar to Eq. (2–10):

〈N, ~m|d†σ|N, ~m′〉 =∑

~n,~n′,iN ,i′N

U∗N(~m,~niN)UN(~m′, ~n′i′N)δiN ,i′N 〈N − 1, ~n|d†σ|N − 1, ~n′〉. (2–22)

Here, UN(~m,~niN) is a unitary matrix composed of the eigenvectors |N, ~m〉 of HN

expressed in the basis of |N − 1, ~n〉 ⊗ |iN〉, where iN = 0, ↑, ↓, ↑↓ labels the basis state

of site N . Thus, Ad(ω) can be evaluated at the end of each iteration N using the energy

eigenvalues EN(~m), the eigenvectors UN(~m,~niN), and the recursively-calculated d†σ matrix

elements. It turns out that the matrix-element calculation is vastly more time consuming

than the diagonalization process.

Since the energy spectrum EN(~m) ranges from around 1 (due to the finite value of N)

to K (due to the truncation of eigenstates), Ad(ω, T ) can be evaluated accurately only at

61

ω = ωN(q) (similar to TN in Section 2.1.2), given by

ωN(q) =(1 + Λ−1)Λ1/2

2DΛq−N/2, (2–23)

where q ∼ 1. For the results in this study, I have used q = 1.25.

In practice, the Dirac delta-functions in Eq. (2–20) are replaced with logarithmic

Gaussian functions [93]:

δ (ω − (EN(~m′)− EN(~m))) → e−b2/4

bω√

πe−[ln(ω/(EN (~m′)−EN (~m)))/b]2 , (2–24)

where b is width parameter, typically chosen between 0.3 and 0.7. (The calculations in

this work have used b = 0.5.) The set of exponentially-spaced points Ad(ωN) is then

connected by a spline curve [51]: a piecewise polynomial function with a continuous second

derivative. All of the spectral function results in this dissertation are for T = 0, for which

case the summation in Eq. (2–20) is restricted to terms in which ~m and/or ~m′ are/is a

ground state of HN , and ZN(β) reduces to the number of ground states.

2.1.4 Fixed Points and Results

As the parameter space of the Anderson model is larger than that of the Kondo

model, the transformation R2 [see Eq. (2–10)] for the Anderson model has more fixed

points than for the Kondo model. Each of these fixed points is described by an effective

Hamiltonian H∗N obtained by inserting special values (0 or ∞) for εd, U , and ∆ into the

general HA [Eq. (2–2)]. At each fixed point, the excitation spectrum can be related by

suitable phase shifts to that at the free-electron fixed point H0, whose iterative form

H0N is given in Eq. (2–14). (As mentioned in Section 1.2.4, each of these fixed points is

technically a pair of fixed points, one for even N and one for odd N .)

The free-orbital fixed point H∗FO can be obtained by setting εd = U = ∆ = 0 in Eq.

(2–2). The impurity is thus decoupled from the metal, and has a set of four degenerate

states: |0〉, | ↑〉, | ↓〉, | ↑↓〉. The iterative Hamiltonian H∗FO,N = H0

N , and the fixed point

energy spectrum is that of H0N with an additional four-fold degeneracy arising from

62

the four possible states of the decoupled impurity. This fixed point is characterized by

Tχimp = 1/8 [30, 31]. [Henceforth, I set kB/(gµB)2 = 1.]

The mixed-valence fixed point H∗MV can be obtained by setting εd = ∆ = 0 and

U = ∞ (notice that this fixed point only occurs for particle-hole asymmetric cases). The

doubly-occupied state | ↑↓〉 is thus inaccessible, and the states |0〉, | ↑〉, | ↓〉 are degenerate

and decoupled from the conduction band. The iterative Hamiltonian H∗MV,N = H0

N , and

the fixed point energy spectrum is that of H0N with an additional three-fold degeneracy

arising from the three possible states of the decoupled impurity. This fixed point is

characterized by Tχimp = 1/6 [30, 31].

The local-moment fixed point H∗LM can be obtained by setting ∆ = 0 and −εd =

U = ∞ (cf. conditions for local-moment formation in Section 1.1.2). Thus only the

singly-occupied states | ↑〉, | ↓〉 are accessible, and the impurity behaves as a free local

moment. The iterative Hamiltonian H∗LM,N = H0

N and the fixed point energy spectrum is

that of H0N with an additional two-fold degeneracy arising from the two possible states of

the decoupled impurity. This fixed point is thus analogous to the J = 0 fixed point from

the RG solution of the Kondo problem and is characterized by Tχimp = 1/4 [30, 31].

The strong-coupling fixed point H∗SC can be obtained by keeping εd and U fixed and

finite and setting ∆ = ∞. The iterative Hamiltonian H∗SC,N is given by a modified version

of H0N in which the sum on n is taken from 1 to N − 1 (instead of from 0 to N − 1).

Thus, the energy spectrum of H∗SC,N is given by H0

N−1. In this case, the impurity and the

f †n=0,σ states are locked into a spin singlet and decouple from the rest of the conduction

band. This fixed point is thus analogous to the J = ∞ fixed point from the RG solution

of the Kondo problem and is characterized by Tχimp = 0 [30, 31]. Physically, switching

from the excitation spectrum of H0N to that of H0

N−1 amounts to a π/2 phase shift of the

low-energy conduction electrons. This phase shift reflects the fact that such conduction

electrons cannot hop on or off site 0 of the NRG chain because that site is locked into a

singlet with the impurity spin. It is this maximal (unitary) scattering that is responsible

63

for the impurity contribution to the resistance reaching its maximum value at T = 0. It is

found that, as long as ∆, U > 0, H∗SC is the only stable fixed point.

Lastly, there is a pair of similar fixed points: the empty-impurity (or full-impurity)

fixed point can be obtained by setting ∆ = U = 0 and εd = ∞ (or εd = −∞), in which

case only the state |0〉 (or | ↑↓〉) is allowed. The iterative Hamiltonian H∗EI,N (or H∗

FI,N) is

equal to H0N , and the fixed point spectrum is that of H0

N with a shift of −1 (or +1) in the

charge quantum number Q of each state. Both of these fixed points are characterized by

Tχimp = 0 and are related to the strong-coupling fixed point [30, 31].

As the NRG procedure is carried out, the Hamiltonians HN (N = 0, 1, 2, . . .) will

follow a trajectory governed by their proximity to these fixed points. This trajectory

picture is useful in understanding the thermodynamic results of the NRG method by

comparing the temperature TN to |εd|, U , and TK . Whenever, as TN decreases (with

increasing N), Tχimp passes near one of its special fixed-point values, the iterative

Hamiltonian HN is near the corresponding fixed point.

For example, Figure 2-1 plots the magnetic susceptibility for a set of NRG calculations

for three values of εd, U , and ∆. These calculations were performed using Λ = 3.0 and

keeping Nkeep = 500 states at the end of each iteration. Following convention [30], I will

identify the Kondo temperature by the condition

TKχimp(TK) = 0.0701, (2–25)

which agrees with Eq. (1–27).

Curve A demonstrates a case in which U À −εd À ∆, displaying a number of the

RG fixed points over various temperature regimes. At higher temperatures T > U , there

is a free-orbital regime in which Tχimp ≈ 1/8. As temperature decreases to |εd| . T . U ,

there is a mixed-valence regime in which Tχimp ≈ 1/6. When T decreases further to

TK < T . |εd|, there is a local-moment regime in which Tχimp ≈ 1/4. (If ∆ were

64

1e-10 1e-08 1e-06 0.0001 0.01 1k

BT/D

0

0.05

0.1

0.15

0.2

0.25

k BT

χ imp(T

)/(g

µ B)2

A

C

B

Figure 2-1. Tχimp(T ) vs. T for an Anderson impurity with (A) εd = −0.01D, U = D,∆ = 0.001D, (B) εd = −U/2, U = D, ∆ = 0.05D, and (C) εd = −U/2, U = D,∆ = D.

zero, this regime would persist down to zero temperature.) Lastly, when T nears TK

(kBT/D ≈ 2.6× 10−10), the system enters the strong-coupling regime and Tχimp → 0.

Curve B demonstrates a case in which particle-hole symmetry (εd = −U/2) is obeyed,

and U À ∆. Again, there is a free-orbital regime at T > U , a local-moment regime at

TK . T . |εd|, and a strong-coupling regime at T . TK (kBT/D ≈ 1.0 × 10−5), but the

mixed-valence regime has disappeared due to the particle-hole symmetry.

Curve C demonstrates another particle-hole-symmetric case, but with ∆ = U .

Here, the free-orbital regime is followed directly by the strong-coupling regime, with no

local-moment behavior in between.

Figure 2-2 plots the T = 0 impurity spectral function corresponding to the parameters

for curve B in Figure 2-1. The main features are the Hubbard bands at ~ω = εd and

65

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1ω

0

2

4

6

Ad(ω

)

Figure 2-2. Zero-temperature spectral function Ad(ω) vs. ~ω/D for the parameters ofcurve (B) of Figure 2-1.

εd + U and the Lorentzian-shaped resonance of width kBTK centered around ~ω = 0,

signaling the occurrence of the Kondo effect. Note that, because of particle-hole symmetry,

Ad(−ω) = Ad(ω).

2.2 Extension to Two-Impurity Systems

As seen in Section 1.4, a variety of new physics arises when one considers systems of

multiple magnetic impurities. While it is challenging to consider an arbitrary number

of impurities, much of this new physics is manifested in the less computationally

demanding case of a pair of impurities. In this section, I will outline the extension of

the NRG procedure to the two-impurity Anderson Model [39, 41, 42], and then explain the

computation of thermodynamic and spectral properties.

66

2.2.1 Transformation to One-Dimensional Form

To study a system of two magnetic impurities, we begin with the two-impurity

Anderson Hamiltonian:

HA2 = Hc + Himps + Hhyb, (2–26)

Hc =∑

~k

ε~kc†~kσ

c~k,σ, (2–27)

Himps =∑

i

[εid

†i,σdi,σ + Ui(d

†i,↑di,↑)(d

†i,↓di,↓)

]+ λ(d†1,σd2,σ + h.c.), (2–28)

Hhyb =∑

~k,i

(V~k,ie

i~k·~rid†i,σc~k,σ + h.c.)

, (2–29)

where i = 1, 2 labels the impurities, ~ri = ∓12Rz is the position of impurity i, and λ

represents a direct electron tunneling between the two impurities. Let us assume that

V~k,i = Vi, where V1 and V2 are real constants. Now the conditions for particle-hole

symmetry (invariance under di,σ → −d†i,σ and c~k,σ → c†−~k,σ) are εi = −Ui/2, λ = 0, and that

the density of states satisfies ρ(ε) = ρ(−ε).

The first step in developing HA2 into a form suitable for NRG study is to rewrite

it in a one-dimensional form. This rewriting will replace the vectors ~k and ~ri with a

dimensionless energy ε = ε~k/D and the impurity separation R = |~r1 − ~r2|, respectively.

Let us begin by focusing on the hybridization term,

Hhyb =∑

~k,i

Vi

(d†i,σe

i~k·~ric~k,σ + h.c.)

. (2–30)

First, the summation over ~k is replaced by an integral using

∑

~k

→ Ω0

(2π)3

∫d3~k, (2–31)

67

where Ω0 is the volume of the unit cell. This replacement will also require a transformation

of the conduction electron operators

c~k,σ →√

(2π)3

Ω0

c~k,σ, (2–32)

giving

Hhyb =∑

i

Vi

[d†i,σ

(∫d3~kei~k·~ri

√Ω0

(2π)3c~k,σ

)+ h.c.

]. (2–33)

The quantity in parentheses is identified as the field operator which annihilates a

conduction electron at site ~r, and whose Hermitian conjugate creates a conduction

electron at site ~r:

ψσ(~r) =

∫d3~kei~k·~r

√Ω0

(2π)3c~k,σ. (2–34)

Thus, the hybridization term is given simply by

Hhyb =∑

i

Vi

[d†i,σψσ(~ri) + h.c.

]. (2–35)

In order to rewrite Hhyb in a one-dimensional form, we define even (e) and odd (o)

parity operators

ψe,σ =1√2

[ψσ(~r1) + ψσ(~r2)] , (2–36)

ψo,σ =1√2

[ψσ(~r1)− ψσ(~r2)] . (2–37)

Replacing the ψσ(~ri) operators with the ψp,σ operators (p = e, o), Hhyb becomes

Hhyb =V1√

2

[d†1,σ (ψe,σ + ψo,σ) + h.c.

]+

V2√2

[d†2,σ (ψe,σ − ψo,σ) + h.c.

]. (2–38)

Jones [34] showed that, if one considers an isotropic conduction band ε~k = ε|~k| and

transforms the basis over ~k into a basis over the dimensionless energy ε, only four linear

combinations of the c~k,σ operators couple to the impurities at each energy ε. These linear

combinations are denoted cε,p,σ, and obey

c†ε,p,σ, cε′,p′,σ′

= δ (ε− ε′) δp,p′δσ,σ′ . (2–39)

68

If the decoupled linear combinations of c~k,σ are neglected, the parity field operators

become

ψp,σ =√

ρ0D

∫ +1

−1

dε wp(R, ε) cε,p,σ, (2–40)

where

w2e,σ(R, ε) =

ρ(Dε)

ρ0

(1 +

sin kεR

kεR

), (2–41)

w2o,σ(R, ε) =

ρ(Dε)

ρ0

(1− sin kεR

kεR

), (2–42)

and

ρ(ε) =Ω0

2π2Dk2dk

dε(2–43)

is the density of states per unit cell per spin direction.

Thus, the conduction band is pictured as consisting of two effective channels of

electrons, labeled by parity p. To complete the picture, even and odd impurity operators

are defined:

de,σ =1√2(d1,σ + d2,σ), (2–44)

do,σ =1√2(d1,σ − d2,σ). (2–45)

Using the cε,p,σ and dp,σ operators, Hhyb arrives at its final one-dimensional form

Hhyb =

√ρ0D

2

∑p

∫ +1

−1

dεwp (R, ε)× (2–46)

[(V1 + V2) d†p,σcε,p,σ + (V1 − V2) d†−p,σcε,p,σ + h.c.

],

where −p indicates the parity opposite to p.

Using the new set of parity operators, the conduction-band term is similarly into

Hc =∑

p

∫ +1

−1

dεc†ε,p,σcε,p,σ + . . . , (2–47)

69

where “. . .” represents the contribution of states that are completely decoupled from the

impurities and are neglected. The impurity term transforms into

Himps =1

2

(ε1 +

1

2U1 + ε2 +

1

2U2

) (d†e,σde,σ + d†o,σdo,σ

)+

1

2

(ε1 +

1

2U1 − ε2 − 1

2U2

) (d†e,σdo,σ + d†o,σde,σ

)

+1

2U1

[d†e,σde,σ + d†o,σdo,σ + d†e,σdo,σ + d†o,σde,σ − 1

]2

+1

2U2

[d†e,σde,σ + d†o,σdo,σ − d†e,σdo,σ − d†o,σde,σ − 1

]2

+λ(d†e,σde,σ − d†o,σdo,σ

)− 1

2(U1 + U2). (2–48)

2.2.2 Discretization and Eigensolution

To transform HA2 [defined in Eqs. (2–26) through (2–29)] into a discretized form, it is

necessary to define a new basis of conduction electron wavefunctions ψa,p,m,q(ε) (associated

with energies ε > 0) and ψb,p,m,q(ε) (associated with energies ε < 0). These states, which

are centered around the impurity sites, are generalizations of the states ψm,q(ε) defined

in Eq. (1–36). Again, m is the index of an energy bin, as depicted in Figure 1-2 and q

is the Fourier harmonic index. As in the single-impurity case, states with q 6= 0 do not

couple to the impurity, and can be ignored as a reasonable approximation as long as the

discretization parameter Λ is not too much greater than 1.

The states with q = 0 are given by

ψa,p,n,0(ε) = A−1n,pwp (R, ε) , Λ−(n−1) < ε < Λ−n, (2–49)

ψb,p,n,0(ε) = B−1n,pwp (R, ε) , −Λ−n < ε < −Λ−(n−1), (2–50)

where

A2n,p =

∫ Λ−n

Λ−(n−1)

dε w2p(R, ε) (2–51)

B2n,p =

∫ −Λ−(n−1)

−Λ−n

dε w2p(R, ε) . (2–52)

70

As in the discreitzation of the single-impurity problem, the states ψa,p,n,q=0(ε) and

ψb,p,n,q=0(ε) are associated with the set of operators fn,p,σ. The innermost state f0,p,σ

(i.e., the one that couples directly to the impurity) is given by

f0,p,σ =1√2Fp

∫ +1

−1

dε wp(R, ε) cε,p,σ, (2–53)

where

F 2p =

1

2

∫ +1

−1

dε w2p(R, ε) . (2–54)

Using the fn,p,σ operators, HA2 is thus transformed into a linear chain form, with the

conduction band and hybridization terms given by

Hc =∑n,p

[εp,nf

†n,p,σfn,p,σ + λp,n

(f †n,p,σfn+1,p,σ + h.c.

)](2–55)

Hhyb =

√ρ0D

2

∑p

Fp

[(V1 + V2) d†p,σf0,p,σ + (V1 − V2) d†−p,σf0,p,σ + h.c.

], (2–56)

and Himps given by Eq. (2–48). Here, the couplings εp,n and λp,n are more complicated

than their single-impurity forms in Eqs. (1–38) and (1–39), and depend on R through

wp (R, ε).

Finally, an iterative dimensionless Hamiltonian analogous to Eq. (2–9) is defined:

HN = Hc,N + Himps,N + Hhyb,N , (2–57)

71

where

Hc,N =Λ(N−1)/2

(1 + Λ−1)D

N∑n=0

∑p

[εp,nf

†n,p,σfn,p,σ + λp,n

(f †n,p,σfn+1,p,σ + h.c.

)], (2–58)

Himps,N =1

2δsumN

(d†e,σde,σ + d†o,σdo,σ

)+

1

2δdiffN

(d†e,σdo,σ + d†o,σde,σ

)

+1

2U sum

N [d†e,↑de,↑d†e,↓de,↓ + d†o,↑do,↑d

†o,↓do,↓

+d†e,↑de,↑d†o,↓do,↓ + d†e,↑do,↑d

†e,↓do,↓ + d†e,↑do,↑d

†o,↓de,↓

+d†o,↑de,↑d†e,↓do,↓ + d†o,↑de,↑d

†o,↓de,↓ + d†o,↑do,↑d

†e,↓de,↓

−d†e,↑de,↑ − d†e,↓de,↓ − d†o,↑do,↑ − d†o,↓do,↓]

+1

2Udiff

N [d†e,↑do,↑d†o,↓do,↓ + d†o,↑de,↑d

†o,↓do,↓

+d†o,↑do,↑d†e,↓do,↓ + d†o,↑do,↑d

†o,↓de,↓ + d†o,↑de,↑d

†e,↓de,↓

+d†e,↑do,↑d†e,↓de,↓ + d†e,↑de,↑d

†o,↓de,↓ + d†e,↑de,↑d

†e,↓do,↓

−d†e,↑do,↑ − d†e,↓do,↓ − d†o,↑de,↑ − d†o,↓de,↓]

+λN

(d†e,σde,σ − d†o,σdo,σ

), (2–59)

Hhyb,N =∑

p

Fp

[(∆sum

N

2

)1/2

d†p,σf0,p,σ +

(∆sum

N

2

)1/2

d†−p,σf0,p,σ + h.c.

], (2–60)

where

δsumN =

2Λ(N−1)/2

1 + Λ−1

(ε1 + 1

2U1 + ε2 + 1

2U2

D

), (2–61)

δdiffN =

2Λ(N−1)/2

1 + Λ−1

(ε1 + 1

2U1 − ε2 − 1

2U2

D

), (2–62)

U sumN =

Λ(N−1)/2

1 + Λ−1

(U1 + U2

D

), (2–63)

UdiffN =

Λ(N−1)/2

1 + Λ−1

(U1 − U2

D

), (2–64)

∆sumN =

(2Λ(N−1)/2

1 + Λ−1

)22ρ0 (V1 + V2)

2

D, (2–65)

∆diffN =

(2Λ(N−1)/2

1 + Λ−1

)22ρ0 (V1 − V2)

2

D. (2–66)

72

Thus, by applying a renormalization group transformation similar to Eq. (2–10), the

two-impurity Anderson model may be solved iteratively.

2.2.3 Special Cases: Identical Impurities and R = 0

The presence of the two electron channels illustrates the greater computational

complexity of the two-impurity Anderson model over the single-impurity Anderson

model. The bases of the iterative Hamiltonians HN square in size compared to the

single-impurity model, requiring more states to be retained after each NRG iteration in

order to achieve the same degree of accuracy. Another complication is the modification of

the conduction-electron density of states by the addition of the factor ± sin (kεR)/(kεR) in

Eqs. (2–41) and (2–42). Due to this factor, the densities of states of even- and odd-parity

states are inequivalent, and the two-impurity model becomes particle-hole asymmetric

even if ρ(ε) = ρ(−ε). In addition to the more complex nature of the calculations, there are

now eight parameters to investigate (εi, Ui, Vi, λ, and R). However, these complications can

be simplified by considering one of two special cases: identical impurities or R = 0.

In the special case of identical impurities (ε1 = ε2, U1 = U2, V1 = V2), Eq. (2–26)

exhibits parity symmetry—i.e., it remains unchanged under the transformations d1 ↔ d2

(or de → de, do → −do in parity-operator language) and ~r1 ↔ ~r2. As such, parity is a

conserved quantum number, as there are no matrix elements of HN that connect states

of opposite parity (cf. the “sum” and “diff” terms in HN). This property causes HN

to be broken up into smaller matrix blocks. In addition to this simplification, there are

three fewer parameters to investigate. Because one often considers impurities of the same

element, this special case is very relevant. For example, I will utilize parity symmetry in

the context of surface impurities in Chapter 4.

73

Another simplified case is the limit of zero impurity separation. For ~r1 = ~r2 = ~0,

w2e (0, ε) = 2ρ(Dε)/ρ0 and w2

o (0, ε) = 0. Thus, the hybridization term simplifies to

Hhyb =

√ρ0D

2

∫ +1

−1

dε 2ρ(Dε)/ρ0 × (2–67)

[(V1 + V2) d†e,σcε,e,σ + (V1 − V2) d†o,σcε,e,σ + h.c.

].

Therefore, the odd conduction electron states decouple from the impurity. Because of

this decoupling, the odd conduction electron states (i.e., all of the fn,p=o,σ states) may be

ignored completely, thereby reducing the basis of the iterative Hamiltonians HN .

Even though there is now effectively only one conduction band, the model is still

much richer than the one-impurity Anderson model: The impurity part of the Hamiltonian

Himp,N in Eq. (2–59) contains four (rather than two) distinct impurity operators and seven

(rather than three) coupling parameters. While the limit of R = 0 may seem physically

unreasonable for magnetic impurities in a metal, it is a good approximation whenever the

two impurities are separated by a distance R satisfying kF R ¿ 1, where kF is the Fermi

wavevector. This condition is assumed to apply in the studies of double quantum dots in

Chapter 5.

2.2.4 Calculation of Thermodynamic and Spectral Properties

The methods for obtaining the magnetic susceptibility and the impurity spectral

function outlined in Sections 2.1.2 and 2.1.3 can also be applied to the two-impurity

model. Applying Eq. (2–13) to the two-impurity model will calculate the contribution of

both impurities to the magnetic susceptibility.

To calculate spectral functions using a formula analogous to Eq. (2–20), the matrix

elements of d†σ are replaced by some combination of the matrix elements of d†1,σ and d†2,σ,

permitting the calculation of a variety of spectral functions. For example, if one wishes to

calculate the linear conductance through a double quantum dot device, it is necessary to

74

employ the Landauer formula [6, 18]

G(T ) = G0

∫ +∞

−∞dω (−df/dω) [−ImT(ω)] , (2–68)

where G0 = 2e2/h is the conductance quantum, f(ω, T ) = [exp (ω/T ) + 1]−1 is the

Fermi-Dirac function, and T(ω) is the transmission matrix, whose imaginary part is given

by

− ImT(ω) = π

∆1A11(ω) + ∆2A22(ω) +√

∆1∆2 [A12(ω) + A21(ω)]

. (2–69)

The spectral functions of interest may be calculated using the NRG formula

Aij(ω, T ) =1

ZN(β)

∑

~m,~m′〈N, ~m|di,σ|N, ~m′〉〈N, ~m|d†j,σ|N, ~m′〉 ×

(e−βEN (~m) + e−βEN (~m′)

)δ (ω − (EN(~m′)− EN(~m))) . (2–70)

75

CHAPTER 3PARALLELIZATION OF THE NRG PROCEDURE

As described in Section 2.2.3, one of the drawbacks of using the NRG to study

the rich physics of the two-impurity Anderson model is the scale of the computational

requirements. The bases of the iterative Hamiltonians HN are squared in size compared

to the single-impurity model. Because of the greater complexity of the system (such as

the presence of two inequivalent channels), a larger number of states must be retained

at the end of each iteration in order to achieve an acceptable level of accuracy. The

computational rigor becomes even greater when studying three-impurity systems [61, 84].

In order to improve on the computational efficiency of two-impurity studies (and

more complicated systems), I have sought to adapt the iterative NRG algorithm to

utilize parallel processors, implemented using the Message Passing Interface (MPI, [99]).

The two most time-consuming aspects of the algorithm are the diagonalization of the

matrices HN and the recursive calculation of operator matrix elements [see Eq. (2–22)].

As mentioned above, the matrix element calculations require much more computer time

than the Hamiltonian diagonalization. It is shown below that the calculation of operator

matrix elements also reaps a greater benefit from the use of parallelization.

3.1 Parallelization of the NRG Eigensolution

I have achieved parallelization in the eigensolution of HN in two ways: The first

method takes advantage of the block-diagonal nature of HN by assigning each of the

individual blocks of the Hamiltonian to a different processor to diagonalize individually

using standard LAPACK routines [100]. This method employs a master-slave arrangement

of the processors, in which one node (the master node) assigns the matrix blocks to

the other nodes (the slave nodes) and manages the results; since the master node does

not perform any calculations itself, it sits idle for most of the run. Because of the

straightforward division of the independent diagonalization tasks, this procedure falls

under the category of “embarrassingly parallel” algorithms, meaning that as long as there

76

is no significant overhead, the wall-clock time should scale roughly as 1/NP , where NP is

the number of processors in use. This method does require a good deal of inter-processor

communication: At the beginning of each iteration, the nodes must be updated with

the energy eigenvalues and eigenstates of the previous iteration [in order to evaluate the

recursive relation in Eq. (2–10)]; during each iteration, the master node must combine the

results from the diagonalization of the matrix blocks and communicate up-to-date status

on which states to truncate.

The second method involves using Scalable LAPACK (ScaLAPACK, [101]) routines to

diagonalize a single large matrix on multiple computer nodes. ScaLAPACK requires that

the matrix be divided up among the processors, resulting in a great deal of communication

between the processors during the calls to ScaLAPACK routines. For the case considered

below, this communication quickly creates a noticeable overhead as NP is increased.

In practice, I utilize these two methods together to minimize the computer wall-clock

time, employing the first for smaller matrix blocks and the second for larger matrix blocks.

The most efficient balance of the two methods depends on the computer hardware being

used. On clusters with slower inter-processor communication (such as the UF Physics

Department’s dragon cluster), it is more advantageous to primarily use the first method,

while on clusters with faster inter-processor communication (such as the UF HPC cluster),

it is more advantageous to apply the second method.

Figure 3-1 illustrates the improvement in performance for iteration N = 5 for an

NRG calculation that retained 3000 states at the end of each iteration. These runs were

performed using different numbers of 2.2-GHz AMD Opteron processors connected by a

gigabyte-ethernet network. The vertical axis is the computer wall-clock time, scaled by

the wall-clock time for a single-processor run, and the horizontal axis is the number of

processors NP involved in the calculation. Initially (NP = 2, 3), there is a steep decrease

in the amount of wall-clock time. There is markedly better improvement between 2

and 3 processors than there is between 1 and 2. This difference is due to the fact that,

77

during the NP = 2 run, the master node sits idle while the one slave node diagonalizes

the matrix blocks of dimension less than 100, whereas during the NP = 3 run, two

slave nodes diagonalize the matrix blocks of dimension less than 100. After the initial

improvement in performance, the wall-clock time saturates, remaining nearly constant

for 4 ≤ NP ≤ 7. Upon using NP > 7, the wall-clock time begins to increase, due to

the increasing overhead of communication between the processors. This interpretation

is evidenced by the linear-like increase in the wall-clock time with increasing NP . The

inefficient performance of this calculation is attributed to inefficiencies in ScaLAPACK,

which rarely sees remarkable improvement for matrices with dimensions less than several

thousand [101].

In general, it turns out that the iterative eigensolution process does not benefit

significantly from the application of parallel processing techniques, as evidenced by

the minimum relative wall-clock time of 65% in Figure 3-1. While there is improved

performance for larger values of Nkeep, it is rarely necessary to keep more than 3000

states (in current NRG endeavors). Also, the benefit accrued by using multiple processors

is usually outweighed by the amount of time spent waiting for many processors to all

become available at once in a typical high-performance cluster environment in which many

users’ jobs compete for computational resources. However, future studies utilizing the

NRG procedure that require a higher number of states may benefit from the preceding

parallelization techniques.

3.2 Parallelization of the Matrix Element Calculation

The calculation of the matrix elements of operator such as d†σ [see Eq. (2–22)] is

extremely computationally demanding. The necessary computer time grows as the cube of

the Hamiltonian dimension, and is even larger than the time required for the eigensolution

process. However, I have succeeded in greatly reducing the wall-clock time for the matrix

element calculation by, again, adapting the algorithm to run on parallel processors. In

this implementation, each processor works on a “chunk” of matrix elements independently

78

0 2 4 6 8N

P

0.6

0.7

0.8

0.9

1

wal

l-cl

ock

time/

wal

l-cl

ock

time(

NP=

1)

Figure 3-1. Wall-clock time vs. number of processors NP for eigensolution of HN=5.Matrix blocks with dimension less than 100 were diagonalized by individualprocessors using LAPACK routines; matrix blocks with dimension greater than100 were diagonalized by all processors using ScaLAPACK routines. Thevertical axis is scaled by the wall-clock time for a calculation utilizing a singleprocessor.

(made possible by the block-diagonal nature of HN and of the matrix element arrays),

which causes the wall-clock time to decrease rather quickly as the number of processors

increases. This method employs a master-slave arrangement of the processors, in which

one node (the master node) assigns the chunks to the other nodes (the slave nodes) and

manages the results. Since the master node does not perform any calculations itself, it sits

idle for most of the run.

Figure 3-2 shows the benefits of applying the above procedure to the calculation

of the matrix elements of d†σ at the end of a single iteration. The vertical axis is the

wall-clock time scaled by the wall-clock time for a single processor run. The horizontal

79

axis is the number of processors NP and the traces are labeled by Nkeep. Here there is a

rather significant improvement in performance, with 6 processors reducing the wall-clock

time by 90%. The data are plotted on a log-log scale and are fitted with power laws to

illustrate the nature of the improvement in performance.

For Nkeep = 2000, the wall-clock time scales as 1/NαP , with α ≈ 1.5. For Nkeep = 4000,

the wall-clock time does not scale as smoothly as in the Nkeep = 2000 case, but the

wall-clock time still follows the general trend of 1/Nαkeep, with α ≈ 1.2. These values

of α would seem to exceed the optimum value of α = 1 that one can expect from an

“embarrassingly parallel” algorithm. However, the variation in the wall-clock data for the

Nkeep = 4000 runs indicate the amount of error associated with analyzing this performance

behavior. For example, a run that utilizes processors located on the same node will

have much faster inter-processor communication than a run that utilizes processors

located on different nodes. Another possible factor is the difficulty in guaranteeing that

a parallel calculation has access to 100% of the computational power of all NP nodes; if

other jobs utilize the same processor, the computational performance will decrease and

wall-clock time will increase. This situation is typically guarded against by job scheduling

software, but is not always avoided. This second error is more likely to occur during

a run that takes a longer amount of time, which may explain why the fluctuations in

the wall-clock time are so much greater for the larger value of Nkeep. It would therefore

be a worthwhile endeavor to see how this algorithm scales in a more controlled parallel

processing environment.

In spite of such errors and difficulties, these results represent a much larger

payoff than was seen for the eigensolution phase of the NRG procedure. This superior

improvement is attributed to the fact that the parallelized matrix-element algorithm

requires much less inter-processor communication than the parallelized eigensolution

does. Notwithstanding these improvements, the benefits achieved by utilizing many

processors are often outweighed by the amount of time spent waiting for those processors

80

1 10N

P

0.01

0.1

1

wal

l-cl

ock

time/

wal

l-cl

ock

time(

NP=

1)N

keep = 2000

Nkeep

= 4000

Figure 3-2. Wall-clock time vs. NP for calculation of d†σ operator matrix elements. Tracesare labeled by number of states kept and fitted with power laws. The verticalaxis is scaled by the wall-clock time for a calculation utilizing a singleprocessor.

to become available. When many independent NRG calculations are required (such as

for the conductance plots in Chapter 5), it is typically more efficient to perform these

calculations simultaneously on independent processors. However, future NRG endeavors

or improvements in cluster management may bring about a greater benefit from the

parallelization techniques described above.

81

CHAPTER 4STM STUDIES

4.1 Review of Single-Impurity Behavior

4.1.1 Single-Impurity STM Setup

As already seen in Chapter 1, magnetic impurities produce interesting effects when

studied using scanning tunneling microscopy. In Figure 4-1, we see the setup for the

study of a single magnetic impurity adsorbed onto a non-magnetic surface with the STM

tip situated directly over the impurity. As depicted schematically by the dashed arrows,

electrons in the STM tip can either (1) tunnel (with matrix element td) into the impurity,

and then tunnel (with matrix element Vd) from the impurity into the surface, or (2) tunnel

(with matrix element tc) directly into the surface at a location adjacent to the impurity.

Tunneling to other locations on the surface is ignored in this model as the tunneling

current density decreases exponentially with tip-surface separation.

The STM differential conductance can be calculated using the iterative NRG method.

Assuming that the tip is centered directly over the impurity, the essential physics of the

conductance is displayed in the formula [73]

G(V ) =dI

dV=

4πe2

~ρtµ

−2Aa(eV/~), (4–1)

where µ is defined below, e is the magnitude of the electron charge, V is the voltage of the

surface relative to the tip (such that V = 0 corresponds to alignment of the two Fermi

energies), ρt is the tip density of states (taken as a constant about εF ), and Aa(ω) is the

spectral function of the tunneling operator

a†σ = µ[tdd

†σ + tcψ

†σ(~rimp)

]. (4–2)

The field operator ψ†σ(~rimp) creates an electron in a surface state adjacent to the impurity

and µ =√|td|2 + |tc|2 is a normalization factor introduced to maintain

∫ +∞−∞ Aa(ω)dω = 1.

82

Figure 4-1. Schematic STM setup indicating possible tunneling paths from the STM tip tothe metal surface.

Following [73], I assume that the surface-state operator ψ†σ(~rimp) is equivalent

to the innermost Wilson-shell operator f †0,σ. While this equivalency is a significant

assumption, relaxing it should only change the quantitative details of the results, and not

the qualitative features. Using the NRG procedure, we can calculate the matrix elements

of the operators d†σ and f †0σ in the basis of the eigenstates of each HN and evaluate Aa at

ω = ωN (see Section 2.1.3 for details) using a formula analogous to Eq. (2–20):

Aa(ω, T ) =µ2

ZN(β)

∑

~m,~m′|〈N, ~m|

(tdd

†σ + tcf

†0,σ

)|N, ~m′〉|2 × (4–3)

(e−βEN (~m) + e−βEN (~m′)

)δ (ω − (EN(~m′)− EN(~m))) .

Here, we see that interference terms proportional to tdtc will come into play, producing

the asymmetric Fano lineshape observed experimentally (see Sections 1.3.1 and 1.4.2). As

mentioned in Section 2.1.3, all of the results in this chapter are for T = 0, in which case

the summation in Eq. (4–3) is restricted to terms in which ~m and/or ~m′ is a ground state

of HN , and ZN(β) reduces to the number of ground states.

83

4.1.2 Results for Single-Impurity STM

Here I will review NRG results for an STM study of a single isolated magnetic surface

impurity. To ensure that the impurity develops a well-defined spin-1/2 local moment and

a noticeable Kondo temperature, I have worked with model parameters (measured in units

of D) εd = −U/2 = −0.5 and ∆ = 0.051, with a top-hat density of states ρ(ε) = ρ0 for

|ε| < D and 0 otherwise. In the NRG procedure, I have used a discretization parameter

Λ = 3.0 and kept the lowest-lying 500 energy eigenstates at the end of each iteration.

-0.004 -0.002 0 0.002 0.004eV/D

0

0.2

0.4

0.6

0.8

G(V

)

td/t

c = 0.0

0.10.20.3

Figure 4-2. Conductance (arbitrary units) vs. bias voltage V for an STM tip locateddirectly over a single magnetic impurity. Model parameters are (in units of D)εd = −U/2 = −0.5, ∆ = 0.051, with td/tc as labeled in the legend. The NRGcalculations were performed for Λ = 3, retaining Nkeep = 500 states after eachiteration. The lineshapes are similar to those in Figures 1-6 and 1-7.

Figure 4-2 plots the STM differential conductance (in arbitrary units) for this system

for various values of td/tc (the absolute values of td, tc and ρt only change the vertical

scale), essentially reproducing the results of [73]. (The NRG data points have been

84

-0.004 -0.002 0 0.002 0.004eV/D

0

2

4

6

G(V

)

tc = 0.0

Figure 4-3. Conductance (arbitrary units) vs. vias voltage V for an STM tip locateddirectly over a single magnetic impurity. Model parameters are as for Figure4-2, except tc = 0. In this case, the conductance is directly proportional to theimpurity spectral function Ad(ω) (cf. Figure 2-2).

fitted with spline curves.) As td/tc increases, and the fraction of electrons that tunnel

via the impurity becomes larger, the conductance lineshape becomes asymmetric due to

interference between the two tunneling paths. Figure 4-3 shows the conductance for tc = 0,

in which case there is no direct tunneling to the surface and G(V ) has the same form as

Ad(ω) (cf. Figure 2-2).

It is important to note that all of the interesting features of the lineshape occur on

a small energy scale determined by kBTK , which in this case is 1.2 × 10−5D. This small

energy scale is even more visible in Figures 4-4 and 4-5, which show the same conductance

spectra on a logarithmic voltage scale.

85

1e-08 1e-06 0.0001 0.01eV/D

0

0.2

0.4

0.6

0.8

G(V

)td/t

c = 0.0

0.10.20.3

Figure 4-4. Positive-bias data from Figure 4-2, replotted on a logarithmic voltage scale.

As examples, the NRG results for td/tc = 0.0 and 0.3 are successfully fitted with a

Fano lineshape in Figures 4-6 and 4-7, respectively. The fits were obtained using [73]

G(V ) = g0 + aq2 − 1 + 2xq

1 + x2, (4–4)

where g0 is a background conductance, a is a constant, x = eV/kBTK , and q is Fano’s

asymmetry parameter [cf. Eq. (1–50)]. (Note that the NRG results show a broader

resonance than the fits; this is typical of NRG spectral results, which tend to be less

accurate at higher energy scales.) The fitted values of q for td/tc = 0.0 and 0.3 are 0.0 and

−0.84, respectively. These values are similar to those found for the STM studies of single

Ce and Co atoms in Section 1.3.1, which showed similar lineshapes.

86

1e-081e-060.00010.01-eV/D

0

0.2

0.4

0.6

0.8G

(V)

td/t

c = 0.0

0.10.20.3

Figure 4-5. Negative-bias data from Figure 4-2, replotted on a logarithmic voltage scale.

4.2 Two-Impurity STM Studies

4.2.1 Two-Impurity Set-up

Figure 4-8 shows the setup for an STM study of two impurities separated by a

distance R. I take these impurities to be identical (i.e., they have the same values of

εi = εd, Ui = U , and Vi = V ). I also assume that these values do not change with

R (as εd did in the experiments described in Section 1.4.2), and that there is no direct

tunneling from one impurity to the other [i.e., λ = 0 in Eq. (2–28)]. Thus, the impurities

only exchange electrons via the conduction band. I also assume that the STM tip is

situated directly above one of the impurities (impurity 1 in the diagram), such that the

tip’s electrons tunnel only into impurity 1 and the surface surrounding it, and not into

impurity 2. This assumption means that I can use the same tunneling parameters td and

87

-0.004 -0.002 0 0.002 0.004eV/D

0

0.2

0.4

0.6

0.8

G(V

)td/t

c = 0.0

Figure 4-6. NRG results for the single-impurity STM conductance G(V ) for theparameters shown in Fig. 4-2 with td/tc = 0 (squares) fitted to a Fanolineshape (line). The best-fit value of the Fano parameter q in Eq. (4–4) isq = 0.

tc for comparison with the one-impurity results. (Chapter 6 discusses a possible method

for considering a tip at any location.)

The Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction [4, 5] is an indirect

coupling that arises between local magnetic moments in a metal, and will play a role

in the two-impurity system described above. The RKKY interaction, which leads to an

additional term

HRKKY = −IRKKY~S1 · ~S2 (4–5)

in the effective low-energy Hamiltonian, is mediated by the conduction electrons. Even

when there is no direct exchange interaction between the impurities, their spins can

become aligned or anti-aligned, depending on the magnitude and sign of IRKKY. For

the two-impurity Anderson model with εd = −U/2, a constant density of states, and a

88

-0.004 -0.002 0 0.002 0.004eV/D

0

0.2

0.4

0.6

0.8G

(V)

td/t

c = 0.3

Figure 4-7. NRG results for the single-impurity STM conductance G(V ) for theparameters shown in Fig. 4-2 with td/tc = 0.3 (squares) fitted to a Fanolineshape (line). The best-fit value of the Fano parameter q in Eq. (4–4) isq = −0.84.

Figure 4-8. Schematic STM setup for a study of two impurities.

89

0 2 4 6k

FR

0

0.005

0.01

0.015

f(k F

R)

FM

AFM FM

Figure 4-9. Plot of f(kF R) entering Eq. (4–6), calculated for the top-hat density of statesand band dispersion ε~k ∼ k3 employed in this chapter. Regions offerromagnetic and antiferromagnetic interaction are labeled FM and AFM,respectively.

dispersion ε~k ∼ k3, the strength of the RKKY interaction is given by [52]

IRKKY = 2 ln 2 D(ρ0J)2f(kF R), (4–6)

where ρ0J = 8∆/πU ¿ 1 is the spin-exchange coupling entering the Kondo model and

f(kF R) is an oscillating function of the separation R multiplied by the Fermi wavevector

kF , plotted in Figure 4-9.

When IRKKY > 0 the impurities exhibit a ferromagnetic interaction (favoring the

formation of an impurity spin triplet), and when IRKKY < 0 the impurities exhibit an

antiferromagnetic interaction (favoring the formation of an impurity spin singlet). In

an STM experiment, changes in the magnitude and sign of the RKKY interaction may

be observed by moving the impurities using the STM tip. We shall see that the change

between ferromagnetic (FM) and antiferromagnetic (AFM) RKKY interactions exerts a

key influence on the shape of the STM conductance spectrum.

90

As a guide to the STM study, I follow previous work [32, 35] and compare IRKKY with

the characteristic energy scale of the Kondo effect, given by kB times the (R-independent)

one-impurity Kondo temperature [54]:

kBT 1−impK = D

√ρ0J exp(1.58ρ0J − 1/ρ0J). (4–7)

To plot the STM conductance, I use an extension of Eq. (4–1)

G(V ) =dI

dV=

4πe2

~ρtµ

−2Aa1(eV/~), (4–8)

where Aa1(ω) is the spectral function of the impurity-1 tunneling operator

a†1,σ = µ[tdd

†1σ + tcψ

†σ(~r1)

]. (4–9)

As in Section 4.1.1, where I took ψ†σ(~rimp) ∝ f †0,σ, here, I take the operator ψ†σ(~r1) to be

(up to a prefactor) f0,1,σ ≡√

2Fef†0,e,σ +

√2Fof

†0,o,σ, where Fp is defined in Eq. (2–54).

Again, this assumption will only change the quantitative details of the conductance

results, but not the qualitative features.

4.2.2 Thermodynamic and Spectral Results

To give an idea of the behavior of a system of two impurities with varying separation,

Figures 4-10 and 4-11 plot the susceptibility and the d†1,σ-spectral function Ad1 for several

values of kF R, along with the one-impurity results for comparison. (Note that the

one-impurity susceptibility results have been multiplied by 2 to represent two impurities

at infinite separation.) As in Section 4.1.2, these results are for (measured in units of

D) εd = −U/2 = −0.5 and ∆ = 0.051, with a top-hat density of states ρ(ε) = ρ0 for

|ε| < D and 0 otherwise, and a dispersion ε~k ∼ k3. With these model parameters held

constant, I have evaluated Eqs. (4–6) and (4–7) for five different impurity separations:

kF R = 1.5, 2.0, 2.5, 2.84 and 3.3. These values give IRKKY/T 1−impK = 400, 190, 50, −0.23,

and −19, respectively. As before, I have used discretization parameter Λ = 3.0, but for

91

1e-14 1e-12 1e-10 1e-08 1e-06 0.0001 0.01 1k

BT/D

0

0.1

0.2

0.3

0.4

0.5

k BT

χ imp(T

)/(g

µ B)2

kFR = 1.5

2.02.52.843.31-imp. results

Figure 4-10. Impurity susceptibility Tχimp vs. temperature for two impurities, each havingU = −2εd = D and ∆ = 0.051D, and five different impurity separations Rspecified in the legend.

these calculations, I kept 2000 states at the end of each iteration (since the two-impurity

problem’s Hamiltonians have a much larger basis).

For a separation kF R = 1.5, the RKKY interaction is strongly ferromagnetic, as

shown by the ratio IRKKY/T 1−impK = 400. At relatively high temperatures (kBT/D ∼ 10−4),

the ferromagnetic RKKY interaction causes the two impurity spins to point in the same

direction, creating a spin-1 singlet state for which kBTχimp/(gµB)2 = 12, as seen in Figure

4-10. This combined spin-1 is then screened out by the Kondo effect in two stages, seen

as the two separate falls in Tχimp (occurring at kBT/D ∼ 10−5 and kBT/D ∼ 10−12)

separated by a relatively flat region. This two-stage screening is also manifested in Figure

4-11, where there are two rises in the spectral function, occurring over the same energy

ranges at which the drops in Tχimp occurred, again indicating a two-stage Kondo effect.

92

1e-14 1e-12 1e-10 1e-08 1e-06 0.0001 0.01 1ω

0

4

8

12

Ad1

(ω)

kFR = 1.5

2.02.52.843.31-imp. results

Figure 4-11. Impurity-1 spectral function Ad1 vs. frequency ω for the same cases as shownin Figure 4-10.

The temperature and energy scales over which this two-stage screening process occurs

correspond to two Kondo temperatures T pK , where p = e, o labels the even and odd

conduction band channels [32].

For a separation kF R = 2.0, the susceptibility indicates that the ferromagnetic RKKY

interaction again creates a spin-1 singlet state at relatively high temperatures, but now

the two energy scales over which the Kondo screening occurs are much closer to each

other, and there is no flat region between the two falls in Tχimp. (As kF R increases from

1.5 to 2.0, this flat region progressively narrows in width.) This proximity of the two

energy scales is also seen in the spectral function for kF R = 2.0, in which it is difficult to

distinguish two separate rises toward the zero-frequency peak.

For a separation kF R = 2.5, the susceptibility resembles the one-impurity results

(which have been scaled up by a factor of 2). This resemblance indicates that the RKKY

93

interaction is too weak to align the impurity spins, such that the two spin-12

moments

are Kondo-screened independently. This independent-screening effect is also evidenced in

the spectral function, which matches up very closely with the one-impurity results. (This

independent-impurity behavior is surprising given the estimated ratio of IRKKY/T 1−impK =

50; see below for possible explanations.)

Finally, the two antiferromagnetic cases (kF R = 2.84 and 3.3) show very different

behavior than the ferromagnetic cases did. The susceptibility plot shows that the

antiferromagnetic RKKY interaction causes the two impurity spins to point in opposite

directions, creating a spin-0 state for which Tχimp falls with decreasing temperature

even faster than it does in the one-impurity Kondo effect. Because of this dominant

development of a spin-0 system, the Kondo resonance in the spectral function is very

strongly suppressed for the strongly antiferromagnetic cases. As we shall see in the next

section, this suppression enters into the STM conductance spectrum, as well.

As mentioned earlier, the independent-impurity behavior found for kF R = 2.5 seems

inconsistent with the estimated ratio of IRKKY/T 1−impK = 50. A similar inconsistency occurs

in the case for kF R = 2.84, which has a predicted ratio of IRKKY/T 1−impK = −0.23, yet

shows behavior in which the the antiferromagnetic RKKY interaction dominates over the

Kondo effect. It is most likely that these inconsistencies (which occur as one approaches

IRKKY = 0) indicate (1) an inaccuracy in the perturbative expressions for IRKKY and

T 1−impK in Eqs. (4–6) and (4–7), and/or (2) errors arising from the NRG discretization.

Also, such errors may be exacerbated by the proximity of the unstable quantum critical

point described in Section 1.4.1. Whatever the reason, the five cases studied in this section

do effectively display the range of qualitative behaviors expected in the two-impurity

Anderson model.

4.2.3 Two-Impurity STM Conductance

Figures 4-12 through 4-15 plot the STM differential conductance for each of the

five values of kF R considered in the previous section, along with the results for a single

94

impurity. Figures 4-12 and 4-13 show results for td/tc = 0.1, which corresponds to

approximately 99% of the electrons that leave the tip tunneling directly into the surface

and 1% tunneling into the impurity; Figures 4-14 and 4-15 show the results for td/tc = 0.4,

which corresponds to approximately 86% of the electrons that leave the tip tunneling

directly into the surface and 14% tunneling into the impurity. For the ferromagnetic cases,

increasing td/tc causes an increase in the zero-bias value of G, as was seen in the case of a

one-impurity study (cf. Figures 4-4 and 4-5). For the antiferromagnetic cases, increasing

td/tc causes almost no change in the conductance spectrum, indicating the isolated nature

of the spin singlet state of the two impurities.

For the cases with kF R = 1.5, G(V ) exhibits similar features to the one-impurity

case, but, as was seen in the susceptibility and spectral function, the changes in G(V )

occur over two easily distinguished energy scales. For the cases of kF R = 2.0, G(V ) shows

lineshapes similar to those for kF R = 1.5, but the two energy scales are more difficult to

distinguish. As seen in the susceptibility and spectral function, the results for kF R = 2.5

closely resemble those for the one-impurity case. For the two strongly antiferromagnetic

cases, the conductance spectrum is essentially featureless for |eV/D| < 10−5 due to the

suppression of the Kondo resonance seen in the spectral function.

These results indicate that the competition between Kondo screening and the

RKKY interaction is clearly revealed in the conductance spectrum. Thus, it is possible

to study this competition in STM experiments simply by moving the two impurities. We

also see that it is possible to study one- and two-stage Kondo screening effects in such

experiments. For a relatively strong ferromagnetic RKKY coupling, the effective Kondo

temperature [as defined in Eq. (2–25)] drops with separation, and (as a result) the lowest

energy scale of the conductance lineshape also decreases with separation. For a relatively

strong antiferromagnetic RKKY coupling, the suppression of the Kondo resonance causes

the lineshape to remain featureless for smaller energies.

95

1e-12 1e-10 1e-08 1e-06 0.0001 0.01eV/D

0

0.1

0.2

0.3

0.4

0.5

G(V

)k

FR = 1.5

2.02.52.843.31-imp. results

Figure 4-12. STM differential conductance vs. positive bias voltage for the sametwo-impurity cases as shown in Figure 4-10, with tunneling into impurity 1described by td/tc = 0.1.

The two-impurity conductance results may also be fitted with Fano lineshapes,

though the two-stage Kondo screening effect requires a sum of two Fano lineshapes to be

used. For example, Figures 4-16 and 4-17 show the NRG results for td/tc = 0.4, kF R = 1.5

fitted with the formula [cf. Eq. (4–4)]

G(V ) = g0 + a1q21 − 1 + 2x1q1

1 + x21

+ a2q22 − 1 + 2x2q2

1 + x22

, (4–10)

where xi = eV/kBTK,i, with TK,1 and TK,2 being the temperatures at which the two stages

of the screening process occur. The best fits to the data in Figures 4-16 and 4-17 are

achieved with kBTK,1/D = 3.75× 10−5, kBTK,2/D = 3.40× 10−12, q1 = 1.0, and q2 = −0.2.

(As with the single-impurity results, the NRG produces resonances slightly broader than

the fit does.)

96

1e-121e-101e-081e-060.00010.01-eV/D

0

0.1

0.2

0.3

0.4

0.5

G(V

)k

FR = 1.5

2.02.52.843.31-imp. results

Figure 4-13. STM differential conductance vs. negative bias voltage for the sametwo-impurity cases as shown in Figure 4-10, with tunneling into impurity 1described by td/tc = 0.1.

4.2.4 Varying Impurity Parameters

In addition to the impurity separation R, it is important to consider the variation of

the above behaviors with other impurity-system parameters which influence the energy

scales T 1−impK and IRKKY. For example, as ∆ increases, TK increases more strongly than

IRKKY [cf. Eqs. (4–7) and (4–6)]. This behavior is illustrated in Figure 4-18, which plots

the differential conductance for a two-impurity system with ∆ given in the legend. For

∆ = 0.05 and 0.1, IRKKY À T 1−impK and the conductance has the form of a sum of two

Fano lineshapes, indicating a two-stage Kondo screening effect. For ∆ = 0.15 and 0.2,

IRKKY ¿ T 1−impK and the conductance has the form of a single Fano lineshape, indicating

independent-impurity screening. (Note that varying ∆ cannot change the sign of IRKKY.)

97

1e-14 1e-12 1e-10 1e-08 1e-06 0.0001 0.01eV/D

0

0.5

1

1.5G

(V)

kFR = 1.5

2.02.52.843.31-imp. results

Figure 4-14. STM differential conductance vs. positive bias voltage for the sametwo-impurity cases as shown in Figure 4-10, with tunneling into impurity 1described by td/tc = 0.4.

Similar changes may be found by decreasing U , since T 1−impK and IRKKY depend on

ρ0J = 8∆/πU .

It is also important to consider the effects of allowing electrons to tunnel directly

between the two impurities. For λ > 0, impurity states created by d†e,σ and d†o,σ have

energies εd + λ and εd − λ, respectively. The resulting changes to the energy spectrum of

HN lead to a modification of the RKKY interaction IRKKY → IRKKY − 4λ2/U [92]. Thus,

in the cases in which the Kondo effect is observed (IRKKY > 0 or −IRKKY ¿ T 1−impK ), a λ

of sufficiently large magnitude (for example, |λ| ≈ ∆ in the case of U ¿ D and R = 0 [92])

causes impurity-singlet formation and destroys the Kondo effect. Thus, the conductance

signatures in Figures 4-12 through 4-15 should still be observed in the general case λ 6= 0,

98

1e-141e-121e-101e-081e-060.00010.01-eV/D

0

0.5

1

1.5G

(V)

kFR = 1.5

2.02.52.843.31-imp. results

Figure 4-15. STM differential conductance vs. negative bias voltage for the sametwo-impurity cases as shown in Figure 4-10, with tunneling into impurity 1described by td/tc = 0.4.

with the modification that the comparison of IRKKY−4λ2/U with T 1−impK determines which

of the three types of behavior is observed.

The results in this chapter demonstrate that STM devices may be used to study

the competition between magnetic ordering (induced directly or indirectly) with the

Kondo effect in a simple two-impurity system. The different behaviors of two-stage Kondo

screening, independent-impurity screening, and impurity-singlet formation yield different

signatures in the STM conductance spectrum. Such signatures permit the experimental

observation of these behaviors that have been studied in depth with various theoretical

methods [32–37, 40–43, 45, 49, 74, 80].

99

1e-14 1e-12 1e-10 1e-08 1e-06 0.0001 0.01

eV/D

0

0.5

1

1.5

G(V

)

kFR = 1.5

Figure 4-16. NRG results for the two-impurity STM conductance G vs. positive biasvoltage for the parameters shown in Figure 4-2 with kF R = 1.5 andtd/tc = 0.1 (squares), fitted to the sum of two Fano lineshapes [Eq. (4-13)].

100

1e-141e-121e-101e-081e-060.00010.01

-eV/D

0

0.5

1

1.5

G(V

)

kFR = 1.5

Figure 4-17. NRG results for the two-impurity STM conductance G vs. negative biasvoltage for the parameters shown in Figure 4-2 with kF R = 1.5 andtd/tc = 0.1 (squares), fitted to the sum of two Fano lineshapes [Eq. (4-13)].

101

1e-14 1e-12 1e-10 1e-08 1e-06 0.0001 0.01 1eV/D

0

0.1

0.2

0.3

0.4

G(V

)

∆ = 0.050.10.150.2

Figure 4-18. STM differential conductance vs. positive bias voltage for a two-impuritysystem with U = −2εd = D, kF R = 1.5, tunneling into impurity 1 describedby td/tc = 0.1, and values of ∆ (in units of D) given in the legend.

102

CHAPTER 5ASYMMETRIC DOUBLE QUANTUM-DOT DEVICES

As seen in Sections and 1.3.2 and 1.4.3, quantum dot (QD) devices permit the study

of a rich variety of physical behaviors thanks to the experimental control of their geometry

and energy level configuration. Due to their discrete energy levels, QD devices have rightly

been described as artificial atoms (or artificial molecules in the case of coupled QDs).

In keeping with this comparison, when a QD’s energy levels are tuned such that it is

energetically favorable for the dot to have an odd number of electrons (see Section 1.3.2),

the dot interacts with the lead electrons in much the same way that a magnetic impurity

interacts with conduction electrons, thus displaying the Kondo effect [55, 64, 76]. When

multiple Kondo-regime dots are connected to each other, an artificial multiple-impurity

system is created, permitting the careful study of many behaviors that were previously

difficult to access experimentally.

As described in Section 1.4.3, most of the previous theoretical work on DQDs has

focused on devices for which both dots are in the Kondo regime. A different class of

highly asymmetric DQD devices has begun to be explored [88, 91, 94, 98], revealing novel

properties, including a splitting of the Kondo resonance and a pair of quantum phase

transitions. In this new class of devices, one of the dots (hereafter referred to as Dot 1)

is tuned to be in the Kondo regime, while the other (hereafter referred to as Dot 2) is

effectively non-interacting and is tuned near a Coulomb blockade peak [i.e., at a maximum

in G(V ) such as in Figure 1-10], such that it does not display a Kondo resonance. (Thus,

this device exhibits properties very different from the two-impurity behaviors discussed in

Section 1.4.1.)

In terms of the two-impurity Anderson model, this description means that Dot 1 will

meet the conditions for local-moment formation (−ε1 À ∆1, ε1 + U1 À ∆1), while Dot

2 must be tuned such that ε2 ≈ 0 (to be near resonance with the leads) and constructed

such that U2 = 0 (to be non-interacting and not develop a local moment). This last

103

condition, while difficult to achieve exactly, can be approximately met by considering a

large dot, since the Coulomb interaction U2 will vary inversely with the size of the dot. A

similar construction was developed to study the two-channel Kondo effect in a quantum

dot [95].

As we shall see below, the condition U2 = 0 also reduces the computational

rigor of studying this asymmetric DQD numerically, permitting the use of an effective

single-impurity model [88, 94, 98]. However, as it is impossible to realize a perfectly

non-interacting dot in experiments, it is important to consider the effects of a small but

non-zero interaction on Dot 2. This chapter will focus on the limit in which U2 remains

small compared to other bare parameters of the two-impurity Anderson model, in which

case no significant local moment is expected to form on Dot 2.

In this chapter, I will first describe the application of the Anderson model to the

DQD device of interest. I will then focus on two device configurations—the “side-dot”

and “parallel” geometries—that exhibit physical properties of particular interest. For each

configuration, I will summarize the main features of the U2 = 0 special case, and then

demonstrate how these features are modified by a non-zero U2.

5.1 Double Quantum Dot Setup

5.1.1 Model and Simplifications

The asymmetric DQD device is represented schematically in Figure 5-1 and is

described by the Hamiltonian [88]

H =∑

~k,α

ε~k,αc†~k,α,σc~k,α,σ +

∑i

[εid

†i,σdi,σ + Ui(d

†i,↑di,↑)(d

†i,↓di,↓)

]+

∑

~k,i,α

(V~k,i,αei~k·~rid†i,σc~k,α,σ + h.c.) + λ(d†1,σd2,σ + d†2,σd1,σ). (5–1)

The left and right leads are labeled by α = L,R, respectively, and the dots are labeled by

i = 1, 2. I assume that the impurity separation R = |~r1 − ~r2| satisfies kF R ¿ 1 (since kF

tends to be relatively small in semiconductor heterostructures), such that the dots may

104

Figure 5-1. Schematic of an asymmetric double quantum dot. Attention will be focused onthe side-coupled configuration V1 = 0 and the parallel configuration λ = 0.Each configuration will first be explored in the limit of U2 = 0 (Dot 2 large),and then U2 small but non-zero will be considered.

be treated as Anderson impurities with zero separation (hence, the odd conduction band

channel may be dropped from the NRG analysis as described in Section 2.2.3).

To simplify the model, it is assumed (as in previous chapters) that V~k,i,α is real and

independent of ~k. It is also assumed that the two leads are identical (ε~k,L = ε~k,R) and

couple symmetrically to the quantum dots (Vi,L = Vi,R), in which case each dot hybridizes

with the operators c†~k,L,σand c†~k,R,σ

only in the linear combination c†~k,σ= (c†~k,L,σ

+c†~k,R,σ)/√

2

[88]. In effect, Dot i couples to a single lead with a hybridization Vi =√

2Vi,α. Thus, the

Hamiltonian in Eq. (5–1) reduces to Eq. (2–26) with R = 0. Finally, in the work to follow,

the density of states in each of the leads is taken to be ρ(ε) = ρ0 for |ε| < D and zero

otherwise. Henceforth, I take D = 1 as the unit of energy and for notational simplicity set

~ = 1.

In an experimental realization of this DQD device, ε1 and ε2 would be controlled by

tuning the voltages of plunger gates on Dot 1 and 2 (respectively), while λ, V1, and V2

would be controlled by tuning the voltages on gates that define the tunneling barriers

105

separating the dots from one another and from the leads. In the sections that follow, I

focus on fixing U1, V1, U2, V2, and λ, and examining the changes in the zero-temperature

linear conductance G of the DQD device as ε1 or ε2 is varied. The linear conductance is

calculated using the Landauer formula [6, 18], given in Eq. (2–68). This formula requires

four impurity spectral functions A11(ω), A22(ω), A12(ω), and A21(ω), which are calculated

using the NRG formula in Eq. (2–70). All calculations below were performed for Λ = 3.0,

retaining Nkeep = 500 states after each iteration.

5.1.2 Special Case: U2 = 0

Even with the above simplifications, the preceding Hamiltonian is quite difficult to

study. As a further simplification, previous work [88, 94, 98] has begun by considering the

special case of U2 = 0. It has been shown [88] that for U2 = 0, the above two-impurity

Anderson model can be mapped onto that of a single impurity (in this case, Dot 1)

described by Eq. (1–13) with a nonconstant hybridization function [88]

∆(ε) ≡ π∑

~k

|V~kd|2δ(ε− ε~k) (5–2)

=

[λ√

∆2 + (ε− ε2)√

∆1

]2

[(ε− ε2)2 + (∆2)2], (5–3)

where ∆i ≡ πρ0V2i . Because this special limit maps onto an effective one-impurity

problem, it may be studied with relative computational ease.

This nonconstant hybridization gives rise to a number of novel effects, including

zero-field splitting of the Kondo resonance and a pair of quantum phase transitions. As we

shall see, both of these effects are observable in the linear conductance of the DQD device.

These effects are most straightforwardly observed in two special cases: the side-coupled

DQD (∆1 = 0) and the parallel DQD (λ = 0). I shall now examine these two special

cases, beginning each by considering the limit of U2 = 0 and then expanding to discuss the

effects of small but finite U2.

106

5.2 Side-Coupled DQD

5.2.1 Special Case: U2 = 0

In the special case ∆1 = 0 and λ > 0, Dot 1 is connected to the leads only through

Dot 2. In the U2 = 0 limit, Eq. (5–3) produces a Lorentzian hybridization function of

width ∆2 centered on ω = ε2. Figure 5-2 shows that, when ε2 = 0 (i.e., Dot 2 is precisely

on resonance), for sufficiently small values of λ, the Dot-1 spectral function A11(ω)

displays a shape similar to that seen in the case of a constant hybridization: the Hubbard

bands at ω ≈ ε1 and ε1 + U1, and the Kondo resonance of width 2kBTK (cf. Figure 2-2). If

λ is increased, however, the Kondo resonance splits. This splitting is attributed to the fact

that the relatively low value of ∆(ω) for |ω| & kBTK causes the Kondo resonance to rise,

but the enhanced value of ∆(ω = 0) causes A11 to fall as ω → 0 to satisfy the Fermi-liquid

relation A11(ω = 0) ≤ 1/[π∆(0)] [88]. For still higher values of λ, the splitting increases,

indicating that TK grows as λ is increased.

Such a splitting of the Kondo resonance is similar to that observed for a Kondo

impurity under the influence of a magnetic field, which causes a suppression of the Kondo

effect due to a destruction of the Kondo ground state at lower temperatures. In the case

at hand, however, one finds upon examining the energy spectra calculated by the NRG

procedure [88] that the Kondo ground state is preserved; therefore, the Kondo effect is

still taking place, at a TK that increases with λ. A similar splitting effect has also been

predicted to occur in a side-coupled DQD device with identical impurities [85].

The zero-field splitting of the Kondo resonance is also evidenced in the linear

conductance of the side-coupled DQD device, as shown in Figure 5-3. As also seen in

Figure 1-18, the zero-temperature conductance vanishes at the particle-hole symmetric

point ε1 = −U1/2, regardless of the value of λ; this feature can be explained by considering

the fact that G(T = 0) ∝ sin(η22), where η22 is the phase shift of electrons scattering from

Dot 2 [98]. At the particle-hole symmetric point, η22 makes a discontinuous jump from −π

to 0 (regardless of the value of λ), such that G(T = 0) = 0. Away from the particle-hole

107

-0.4 -0.2 0 0.2 0.4ω

0

2

4

6A

11(ω

)λ = 0.030.0630.078

Figure 5-2. Dot 1 spectral function for a side-coupled DQD device withε1 = −U1/2 = −0.25, ε2 = U2 = 0, ∆2 = 0.02, and λ as labeled in the legend.For sufficiently large λ, we observe a novel splitting of the Kondo resonancethat does not destroy the Kondo ground state.

symmetric point, Dot 1 no longer has a Kondo resonance, and G approaches its maximum

value G0 = 2e2/h. Increasing λ causes the Kondo effect to strengthen (indicated by an

increase in TK as predicted by the zero-field splitting of the Kondo resonance), pushing the

upturns in G away from the particle-hole symmetric point [98].

5.2.2 Extended Case: U2 > 0

If we now consider U2 > 0 while holding ε2 = 0, we see in Figure 5-4 that for

sufficiently small U2, there is still a visible splitting of the Kondo resonance, with a

new particle-hole asymmetry introduced by the fact that the DQD no longer satisfies

ε2 = −U2/2. As U2 increases, this asymmetry becomes more pronounced, shifting spectral

weight from the positive-ω side to the negative-ω side. Once U2 becomes strong enough

108

-1 -0.5 0 0.5ε

1

0

0.5

1

G/G

0U

2 = 0, λ = 0.03

U2 = 0, λ = 0.063

U2 = 0, λ = 0.078

U2 = 0, λ = 0.1

Figure 5-3. Zero-temperature conductance vs. Dot 1 energy level for a side-coupled DQDwith U1 = 0.5, ε2 = U2 = 0, ∆2 = 0.02, and λ as labeled in the legend.

(U2 ∼ 0.1 in the case at hand), the peak on the positive-ω side disappears, leaving a single

peak that moves toward ω = 0.

The asymmetry in the Dot-1 spectral function manifests itself in the conductance

traces in Figure 5-5. For U2 . 10−4 (not shown), there is no noticeable deviation from

the U2 = 0 behavior, indicating the robustness of the effective one-impurity model.

For U2 = 0.01, where the Dot-2 interaction has become comparable to other energy

scales of the device (such as U1, ∆1, and ∆2), a noticeable asymmetry develops in G vs.

ε1. Upon further increase in U2, the conductance develops a peak on the order of G0

on the right-hand side. This is comparable to the peaks seen in the identical-impurity

side-coupled device in Figure 1-18. The absence of a left-hand peak is most likely due to

the inequivalence of the two dots in this device. Note that for all cases with U2 > 0 shown

in Figure 5-5, the conductance drops all the way to zero at a single value of ε1. Just as for

109

-0.2 -0.1 0 0.1 0.2

ω

0

2

4

6

A11

(ω)

U2 = 0

0.010.030.10.5100

Figure 5-4. Zero-temperature Dot-1 spectral function for a side-coupled DQD withε1 = −U1/2 = −0.25, ∆1 = 0, ε2 = 0, ∆2 = 0.02, λ = 0.067, and the values ofU2 specified in the legend. The interaction on Dot 2 introduces a particle-holeasymmetry that becomes more pronounced as U2 increases. For sufficientlylarge U2, the right-hand peak in A11(ω) disappears.

U2 = 0, the zero in G occurs at the point where the Dot-2 phase shift η22 jumps between

−π and 0.

An important question to consider is whether the deviations from the U2 = 0

behavior are unique to devices with U2 > 0 or are merely the result of broken particle-hole

symmetry on Dot 2. This issue can be investigated by comparing Figure 5-5 with Figure

5-6, which plots the conductance for a side-coupled DQD with U2 = 0 and ε2 6= 0 (for

which Dot 2 breaks particle-hole symmetry even though it is still noninteracting). For

relatively small deviations from particle-hole symmetry (i.e., U2 and |ε2| smaller than

the other device parameters), there is no qualitative difference between the device with

U2 > 0 and ε2 = 0 and the device with U2 = 0 and ε2 6= 0. For more significant deviations

110

-1 -0.5 0 0.5ε

1

0

0.5

1

G/G

0U

2 = 0

0.010.020.040.10.20.30.40.5

Figure 5-5. Zero-temperature conductance vs. Dot 1 energy level for a side-coupled DQDwith U1 = 0.5, ε2 = 0, ∆2 = 0.02, λ = 0.063, and the values of U2 specified inthe legend.

from particle-hole symmetry (i.e., when U2 and ε2 are comparable to the other device

parameters), there is still qualitative agreement on some features (the peak of G ≈ G0 and

the minimum of G = 0) but the conductance for the device with U2 = 0 and ε2 6= 0 shows

a plateau that is not present in the device with U2 > 0 and ε2 = 0. Thus, while there is

some similarity in the behavior of the two devices, they do produce noticeably different

conductance spectra, signifying that the U2 > 0 side-coupled DQD device offers a new

realm of behavior to explore.

5.3 Parallel DQD

In the special case of λ = 0 and ∆i 6= 0, the dots are not directly connected to each

other, but, as was the case in Section 4.2.1, the dots still interact with each other via the

conduction electrons. The primary feature of this special case is a pair of quantum phase

111

-1 -0.5 0 0.5ε

1

0

0.5

1

G/G

0ε

2 = 0

0.0050.020.040.050.060.080.10.2

Figure 5-6. Zero-temperature conductance vs. Dot 1 energy level for a side-coupled DQDwith U1 = 0.5, U2 = 0, ∆2 = 0.02, λ = 0.063, and the values of ε2 specified inthe legend.

transitions (QPTs) that can be observed by tuning ε1 and ε2. As in the side-coupled case,

first the behavior of the dots in the U2 = 0 limit will be reviewed, and then the general

case of U2 > 0 will be explored.

5.3.1 Special Case: U2 = 0

In the special limit of U2 = 0, the effective hybridization for Dot 1 [Eq. (5–3)]

vanishes at ε = ε2 with a power-law ∆(ε) ∝ (ε − ε2)2. Thus, when Dot 2 is in resonance

with the leads (ε2 = 0), the hybridization vanishes at the Fermi energy, ε = 0. The

presence of such a power-law vanishing of the hybridization (or pseudogap) in the

Anderson impurity model is known to introduce a pair of quantum phase transitions

between Kondo-screened phases and a local-moment phase in which the impurity degree of

freedom remains unquenched at absolute zero [47, 48, 53, 56, 67, 81]. In the DQD device

112

being considered, the finite-temperature manifestations of these QPTs may be studied

experimentally by tuning ε1 for fixed values of U1, ∆1, ε2, and ∆2. (Later, it will be shown

that these QPTs can be observed by tuning ε2, as well.) The QPTs are summarized in the

phase diagram in Figure 5-7.

Figure 5-7. Schematic zero-temperature phase diagram for the U2 = 0 parallel-dot systemas a function of ε1.

If one begins with a large, positive ε1 and moves through ε1 = 0 to negative values,

the behavior of the DQD device exhibits an evolution from an empty-impurity regime (EI)

to a mixed-valence regime (MV) and then to a Kondo regime (K). Each of these regimes

(together termed the “strong-coupling regime”) possesses the property that the impurity

magnetic susceptibility χimp obeys Tχimp → 0 as T → 0. [However, in contrast to the

conventional Anderson model, for which χimp(T = 0) generally remains finite—approaching

0.103/TK in the Kondo limit [30]—the power-law form of the effective hybridization results

in a strong-coupling value χimp(T = 0) = 0 [56]. This aspect of the strong-coupling regime

will be important below when U2 > 0 is considered.] As can be seen in Figures 5-8 and

5-9, the Kondo temperature TK defined via the condition TKχimp(TK) = 0.0701 is linearly

113

proportional to δε+1 ≡ ε1 − ε+

1c, where ε1 = ε+1c denotes the location of the quantum critical

point, at which there is no Kondo effect and Tχimp = 1/6 all the way to T = 0.

1e-12 1e-10 1e-08 1e-06 0.0001 0.01T

0

0.05

0.1

0.15

0.2

0.25

Tχ im

p

δε1

+ = 10

-2

10-4

10-6

10-8

10-10

0

-10-10

-10-8

-10-6

-10-4

-10-2

Figure 5-8. Tχimp vs. T for a U2 = 0 parallel-coupled DQD near its upper QPT at ε1 =ε+1c, calculated for U1 = 0.5, ∆1 = 0.05, ε2 = 0, ∆2 = 0.02, λ = 0, and the

values of δε+1 = ε1 − ε+

1c specified in the legend.

If ε1 is decreased below ε+1c, the device enters a local-moment regime (LM) in which

Tχimp approaches 1/4 as T → 0, indicating the presence of an unscreened local moment

on Dot 1. In this LM regime, we can define a crossover temperature T ∗ (analogous to TK

in the strong-coupling regime) via the (somewhat arbitrary) criterion T ∗χimp(T∗) = 1/5.

In the vicinity of the QPT, it is found that T ∗ ∝ δε+1 . (The same linearity is observed for

other choices of the crossover value of Tχimp.)

Once ε1 is decreased past the particle-hole symmetric point ε1 = −U1/2, however,

T ∗ begins to decrease, until it becomes zero at a second critical point, ε1 = ε−1c. Similar

to the behavior in the upper half of the local-moment regime, in the vicinity of the QPT,

T ∗ varies linearly with δε−1 ≡ ε1 − ε−1c. If ε1 is decreased past this second critical point,

114

1e-08 1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1|ε

1 - ε

1c

+|

1e-10

1e-08

1e-06

0.0001

0.01

1

TK

U2 = 0

Figure 5-9. Kondo temperature TK vs. δε+1 ≡ ε1 − ε+

1c for a U2 = 0 parallel-coupled DQD.Other dot parameters are U1 = 0.5, ∆1 = 0.05, ε2 = 0, ∆2 = 0.02, λ = 0.

Kondo screening again takes place, and the device enters a second Kondo regime (K).

Further decrease of ε1 will bring Dot 1 into a mixed valence regime (MV) and finally a

full-impurity regime (FI), all of which possess the property that Tχimp → 0 as T → 0.

This lower strong-coupling regime is similar to the upper one, with TK ∝ δε−1 .

The QPT at ε1 = ε−1c, which was not discussed in [88], is related to the QPT at

ε1 = ε+1c by particle-hole inversion, as evidenced by the fact that |ε+

1c + U1/2| = |ε−1c + U1/2|when ε2 = −U2/2. (This rule applies to the U2 > 0 case, as well.) Because these two

quantum critical points are related via such a simple symmetry, they exhibit many of the

same features. Thus, I shall focus my discussion on the behavior around ε+1c.

Figures 5-10 through 5-13 show the spectral function A11(ω) of Dot 1 for various

values of ε1 in the upper strong-coupling regime and the upper half of the local-moment

regime. In the strong-coupling regime (Figures 5-10 and 5-11), the main feature of the

115

1e-10 1e-08 1e-06 0.0001 0.01ω

0.0001

0.01

1

100

10000

1e+06

1e+08A

11(ω

)δε

1

+ = 10

-1

10-2

10-3

10-4

10-5

10-6

10-7

10-8

Figure 5-10. Zero-temperature Dot 1 spectral function A11(ω) vs. ω > 0 for aparallel-coupled DQD with U1 = 0.5, ∆1 = 0.05, ε2 = U2 = 0, ∆2 = 0.02, λ = 0,and various values of ε1 located in the strong-coupling regime, specified in thelegend through δε+

1 = ε1 − ε+1c.

spectral function is a resonance or quasiparticle peak centered at ω ' kBTK . This peak,

which is the pseudogap generalization of the Kondo resonance, has a width proportional to

TK , a height inversely proportional to TK , and therefore an integrated area that remains

constant as TK vanishes at the QPT. For |ω| ¿ TK , A11(ω) vanishes as ω2. This power-law

behavior is attributed to the pseudogap nature of the hybridization, which also vanishes

as ω2. There are similar features in A11 in the local-moment regime (Figures 5-12 and

5-13), except that now the resonance occurs on the negative frequency side, centered at

ω ' −kBT ∗.

5.3.2 Extended Case: U2 > 0 - Phase Diagram and Susceptibility

I now examine the effects of allowing U2 > 0 in the parallel DQD device, focusing on

the limit of small U2 (meaning 0 < U2 < U1, ∆1, ∆2) and (initially) keeping ε2 = 0. This

116

1e-101e-091e-081e-071e-061e-050.00010.0010.010.1|ω|

0.0001

0.01

1

100

10000

1e+06

1e+08A

11(ω

)δε

1

+ = 10

-1

10-2

10-3

10-4

10-5

10-6

10-7

10-8

Figure 5-11. Negative-frequency spectral functions corresponding to the data shown inFigure 5-10.

more general system contains a richer phase diagram, given the larger parameter space to

explore. We will begin by considering the schematic phase diagram in Figure 5-14, which

focuses on varying ε1 and U2.

The first effect of non-zero U2 that should be noted is a change in the locations ε+1c

and ε−1c of the critical points. Figure 5-14 shows that the critical points (at which TK → 0)

shift toward each other as U2 increases. The special property |ε+1c + U1/2| = |ε−1c + U1/2|

does not apply, due to the fact that the DQD device breaks the particle-hole symmetry

condition ε2 = −U2/2.

For U2 ¿ U1, ∆1, ∆2, the phase boundaries follow the form

ε±1c = ε±1c(U2 = 0) + A±(U1, ∆1, ε2, ∆2)U2 (5–4)

where A+(−)(U1, ∆1, ε2, ∆2) is negative (positive). This relationship is demonstrated in

Figures 5-15 and 5-16, which show the relative change in ε+1c with increasing U2. These

117

1e-10 1e-09 1e-08 1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1ω

0.0001

0.01

1

100

10000

1e+06

1e+08A

11(ω

)δε

1

+ = -10

-1

-10-2

-10-3

-10-4

-10-5

-10-6

Figure 5-12. Zero-temperature Dot 1 spectral function A11(ω) vs. ω > 0 for aparallel-coupled DQD with U1 = 0.5, ∆1 = 0.05, ε2 = U2 = 0, ∆2 = 0.02, λ = 0,and various values of ε1 located in the local-moment regime, specified in thelegend through δε+

1 = ε1 − ε+1c.

plots show that A+ is proportional to (∆2)−1 but only weakly dependent on ∆1. Such

behavior is also found for A−, confirming the similarity of the two QPTs.

As observed in Figure 5-14, when U2 becomes comparable to the other energy scales

(U1, ∆1, and ∆2), the critical points approach each other more quickly than seen in the

linear behavior for smaller U2. At a certain value of U2 (approximately 0.046, for the

case illustrated in Figure 5-14), the critical points merge, and the local moment regime

disappears as U2 is increased further. In the discussion to follow, I will first focus on values

of U2 that are small enough to preserve the local-moment regime, and then discuss the

behavior of the system when the local-moment regime has disappeared.

Considerable insight can be gained into the behavior of the U2 > 0 parallel DQD

device near its quantum critical points by examining the temperature variation of Tχimp.

Figure 5-17 shows Tχimp vs. T for a sequence of ε1 values and a fixed U2 = 10−3. (This

118

1e-101e-091e-081e-071e-061e-050.00010.0010.010.1|ω|

0.0001

0.01

1

100

10000

1e+06

1e+08A

11(ω

)δε

1

+ = -10

-1

-10-2

-10-3

-10-4

-10-5

-10-6

Figure 5-13. Negative-frequency spectral functions corresponding to the data shown inFigure 5-12.

value of U2 has been chosen since it gives rise to very well-separated energy scales that will

provide insight into behavior for larger U2.)

First, as ε1 approaches ε+1c from the upper strong-coupling regime, Tχimp remains

near 1/6 (as in the U2 = 0 case) until Dot 1 becomes Kondo-screened and Tχimp falls to

zero at TK , with TK ∝ δε+1 as before. As in the case of U2 = 0, χimp → 0 as T → 0.

Around a crossover value ε+1 > ε+

1c, a region of new behavior unfolds. This new region is

characterized by a rise in Tχimp from 1/6 toward 1/4. However, as long as ε1 > ε+1c, Tχimp

never reaches 1/4, and is eventually screened down to zero.

Another new development in this region is that χimp → 0.103/TK (instead of 0)

as T → 0, mimicking the conventional single-impurity Kondo effect [30]. This shift in

χimp(T = 0) is shown in Figure 5-18 and is also evidenced by how the red traces in Figure

5-17 fall to zero less steeply than the black traces. It is important to note that the change

in χimp(T = 0), while very rapid, is a continuous evolution as ε1 passes below ε+1 . Thus,

119

0 0.01 0.02 0.03 0.04 0.05U

2

-0.5

-0.4

-0.3

-0.2

-0.1

0ε 1

ε1

+

ε1c

+

ε1c

-

ε1

-

Strong-Coupling Phase

Local-Moment Phase

Strong-Coupling Phase

Figure 5-14. Approximate phase diagram on the U2-ε1 plane for a parallel-coupled DQDdevice with U1 = 0.5, ∆1 = 0.05, ε2 = 0, and ∆2 = 0.02. Solid lines indicatethe quantum phase transitions separating the strong-coupling andlocal-moment phases. Note the shift of the critical energies ε+

1c and ε−1c as U2

increases, resulting in the disappearance of the local-moment regime forU2 & 0.046. Dashed lines (labeled ε+

1 and ε−1 ) represent crossovers fromregions of behavior essentially identical to that for U2 = 0 (ε1 > ε+

1 andε1 < ε−1 ) to regions of novel behavior located between the solid and dashedlines. Arrows indicate values of U2 for which TK and G are plotted against ε1

in later figures.

ε+1 is to be considered a crossover, and not a new critical point. (There is some degree of

arbitrariness in the definition of ε+1 .) Figure 5-19 shows that this new region ε+

1 > ε1 > ε+1c

exhibits a very rapid drop of TK with δε+1 , departing from the universal behavior of the

U2 = 0 limit.

Introduction of an interaction on Dot 2 not only changes the behavior in the

strong-coupling phase near the transition, but also changes the behavior at the transition

120

1e-10 1e-09 1e-08 1e-07 1e-06 1e-05 0.0001 0.001 0.01

U2

1e-08

1e-06

0.0001

0.01

1

100

(εI(U

2) -

ε I(U2 =

0))

/ ε I(U

2 = 0

)∆

1 = 10

-4, ∆

2 = 10

-4

10-4

, 10-3

10-4

, 10-2

10-4

, 10-1

10-3

, 10-4

10-2

, 10-4

Figure 5-15. Critical value ε+1c vs. U2 for a parallel-coupled DQD with U1 = 0.5, ε2 = 0,

λ = 0, and ∆1 and ∆2 as specified in the legend. Note the linear variation ofε+1c with U2, and the weak dependence on ∆1.

ε1 = ε+1c. As seen in Figure 5-17, instead of remaining at 1/6 down to zero temperature (as

it did in the U2 = 0 case), Tχimp rises up to 1/4 and remains there as T → 0.

As ε1 is lowered below ε+1c, Dot 1 exhibits local-moment behavior in which Tχimp =

1/4 down to zero temperature. In contrast to the case U2 = 0, for U2 > 0, it is not

possible to define a crossover scale T ∗ that vanishes continuously as ε1 → ε+1c. Because

of this lack of scale, the phase transition for U2 > 0 is identified as a Kosterlitz-Thouless

type transition [22]. This identification is supported by the fact that there is such a

strong drop in TK on the strong-coupling side of the transition, but no such feature

on the local-moment side. Such a highly asymmetric divergence is characteristic of a

Kosterlitz-Thouless type transition.

121

-6 -5 -4 -3 -2 -1 0log(U

2/∆

2)

-6

-5

-4

-3

-2

-1

0lo

g(ε 1c

+ /ε1c

+ (U2=

0) -

1)

∆2 = 0.1

0.020.01

Figure 5-16. Critical value ε+1c vs. U2/∆2 for a parallel-coupled DQD device with

U1 = 0.5, ε2 = 0, λ = 0, and ∆1 = 0.05.

Approaching ε−1c from the local-moment phase, there is still no characteristic

temperature scale at which Tχimp → 1/4, indicating that the lower quantum phase

transition is also a Kosterlitz-Thouless type. For ε−1c > ε1 > ε−1 , there is again a region

of new behavior in which Tχimp initially rises with decreasing T from 1/6 towards 1/4,

but is eventually Kondo-screened to Tχimp = 0. This screening sets in around a Kondo

temperature TK that drops to zero very rapidly as ε1 → ε−1c from below and exhibits the

conventional Kondo behavior of χimp(T = 0) = 0.103/TK . For ε1 < ε−1 , the universal

U2 = 0 behavior is recovered, with TK ∝ δε−1 = ε−1c − ε1 and χimp(T = 0) = 0. (As with

crossing ε+1 , this change is continuous, and so ε−1 is to be regarded as a crossover and not a

critical point.)

As illustrated in Figure 5-14, at a certain value of U2 (approximately 0.046, in the

case at hand), the two critical points merge and the local-moment phase disappears.

122

1e-20 1e-18 1e-16 1e-14 1e-12 1e-10 1e-08 1e-06 0.0001 0.01T

0

0.05

0.1

0.15

0.2

0.25

Tχ im

p

ε1 > ε

1

+

ε1 = ε

1

+

ε1c

+ < ε

1 < ε

1

+

ε1 = ε

1c

+ε

1 < ε

1c

+

Figure 5-17. Tχimp vs. T for a parallel-coupled DQD with U2 = 10−3. Other dotparameters are U1 = 0.5, ∆1 = 0.05, ∆2 = 0.02, ε2 = 0, and λ = 0. Each tracerepresents a different value of ε1 near ε+

1c. The thick blue line corresponds tothe critical level energy ε+

1c, and the dashed curve corresponds to thecrossover (corresponding to the upper dashed line in Figure 5-14) from theU2 = 0 pseudogap behavior (black traces in the region labeled “ε1 > ε+

1 ”) tonovel behavior (red traces in the region labeled “ε+

1c < ε1 < ε+1 ”).

This disappearance is demonstrated in Figure 5-20, which shows that TK has a non-zero

minimum that increases with U2. (The minimum value of TK for U2 = 0.05 is on the

order of 10−90, and not visible in the plot.) It is noted that the width of the dip in TK

vs. ε1 remains roughly constant with increasing U2 beyond the disappearance of the

local-moment regime. Indeed, Figure 5-14 shows that the crossover energies ε±1 —defined as

the locations of the downturns in TK vs. ε1—seem to become independent of U2 for large

U2. Thus, even though the quantum phase transitions have disappeared for sufficiently

large U2, there remains a signature of their proximity in the dip in TK vs ε1. It should

123

-8 -7.98 -7.96 -7.94 -7.92log(ε

1 - ε

1c

+)

0

0.05

0.1

TKχ(

T=

0)

Figure 5-18. TKχimp(T = 0)/ vs. δε+1 = ε1 − ε+

1c for a parallel-coupled DQD device withU1 = 0.5, ∆1 = 0.05, U2 = 10−6, ∆2 = 0.02, and ε2 = λ = 0. The rapid upturnin χimp(T = 0) at ε1 = ε+

1 (around δε+1 = −7.96 on this graph) signifies the

departure from the U2 = 0 universal behavior. (Note that the value of ε+1 in

this graph does not correspond to its value in Figure 5-19 as a greaternumber of energy eigenstates were kept at the end of each NRG iteration toobtain these results.)

therefore be feasible to probe the quantum phase transitions in an experiment, even when

U2 ≈ 0 is not satisfied.

5.3.3 Extended Case: U2 > 0 - Spectral Function and Conductance

The evolution of the DQD device’s behavior with increasing U2 exhibits signatures in

the dot spectral functions and linear conductance, as well. Figure 5-21 shows the spectral

function A11(ω) for a parallel DQD device with U2 = 10−3. (As in Figure 5-17, this value

of U2 was chosen because it creates well-separated energy scales.) For ε1 > ε+1 (seen in the

black traces), the main feature of the spectral function is a quasiparticle peak centered at

ω ' kBTK , just as it was in the case of the U2 = 0 device. However, in the U2 > 0 device,

124

1e-08 1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1|ε

1 - ε

1c

+|

1e-10

1e-08

1e-06

0.0001

0.01

1

TK

U2 = 0

10-6

10-5

10-4

0.0010.010.020.030.040.045

Figure 5-19. Kondo temperature TK vs. δε+1 = ε1 − ε+

1c for U1 = 0.5, ∆1 = 0.05, ∆2 = 0.02,ε2 = λ = 0. For U2 = 0, TK decreases linearly with δε+

1 . The downwarddeparture from linearity, which sets in around ε1 = ε+

1 , signals the onset ofnovel U2 > 0 behavior.

A11(ω = 0) has a non-zero value, whereas A11(ω = 0) = 0 in the U2 = 0 device. As ε1

crosses ε+1 from above, A11(ω = 0) rises to a height comparable to the quasiparticle peak

(as in the solid blue trace). For ε1 between ε+1c and ε+

1 (seen in the red and green traces),

the quasiparticle peak is absorbed into the resonance centered around ω = 0, and the

Kondo temperature TK may be determined from the width of the zero-frequency resonance

(similar to how it may be identified in the conventional Anderson model). Qualitatively

similar behavior—a quasiparticle peak for ε1 > ε+1 which is absorbed into the ω = 0

resonance for ε+1c < ε1 < ε+

1 —is observed in the spectral function A22(ω) (not shown).

It will now be seen that the asymmetric nature of the quantum critical points for

U2 > 0 will have interesting effects on the conductance. Even when U2 is sufficiently large

to remove the local-moment phase, signatures of the critical points are visible. Figure 5-22

125

-0.5 -0.4 -0.3 -0.2 -0.1 0

ε1

1e-12

1e-10

1e-08

1e-06

0.0001

0.01T

K

U2 = 0.045

0.050.10.50.75

Figure 5-20. Kondo temperature TK vs. ε1 for a parallel DQD with U2 as specified in thelegend. After the critical points ε±1c meet at U2 ≈ 0.046, the local-momentregime disappears, and TK reaches a non-zero minimum. Other deviceparameters are U1 = 0.5, ∆1 = 0.05, ε2 = 0, ∆2 = 0.02, and λ = 0.

shows the zero-temperature conductance G (measured in units of G0 = 2e2/h) as ε1 is

swept for various values of U2 that exhibit the quantum critical points ε±1c. The general

effect of increasing U2 is to suppress the conductance, as seen in a similar theoretical

study of a Kondo-like dot connected to a large grain [91]. When ε1 passes ε±1 , there is

a dip in G. At each critical point, there is a discontinuous jump in G. The thinning of

the local-moment regime with increasing U2 can be seen by the narrowing of the middle

plateau of each trace.

The behavior near ε±1 and ε±1c can be more clearly understood from Figure 5-23, which

plots G/G0 vs. |δε+1 | = |ε1 − ε+

1c| for ε1 in the upper strong-coupling and local-moment

phases. (There is similar behavior for ε1 near ε−1c, which is omitted for simplicity.) By

126

1e-30 1e-25 1e-20 1e-15 1e-10 1e-05ω

1

100

10000

A11

(ω)

δε1

+ = 0.01

0.0020.000920.000820.000720.000670.000620.000570.00052

Figure 5-21. Zero-temperature Dot 1 spectral function A11(ω) vs. ω > 0 for aparallel-coupled DQD with U1 = 0.5, ∆1 = 0.05, ε2 = 0, U2 = 10−3,∆2 = 0.02, λ = 0, and various values of ε1 located in the strong-couplingregime, specified in the legend through δε+

1 = ε1 − ε+1c.

comparing the traces marked by ε1 > ε+1c in Figure 5-23 with their corresponding traces

of TK in Figure 5-19, it is seen that the onset of the drop in TK corresponds to the dip in

G/G0. Note that a decrease in TK leads to a suppression of the conductance.

The discontinuous conductance jumps begin to close as U2 is increased (leading to

their disappearance when the local-moment regime vanishes). The limiting values on

opposite sides of the conductance jumps exhibit an interesting property; defining the left-

and right-hand limit of G at each critical point as

GL± ≡ limη→0+

G(ε1 = ε±1c − η) (5–5)

GR± ≡ limη→0+

G(ε1 = ε±1c + η), (5–6)

127

-0.5 -0.4 -0.3 -0.2 -0.1 0ε

1

0

0.5

1

G/G

0

U2 = 0

0.010.020.030.04

Figure 5-22. Zero-temperature conductance G/G0 vs. ε1 for a parallel-coupled DQD devicewith U2 as specified in the legend. At each critical point, there is adiscontinuous jump in G. The limiting values of each discontinuity obeyGL± + GR± = G0 [definitions in Eqs. (5–5) and (5–6)]. When ε1 passes ε±1 ,there is a dip in G. Other device parameters are U1 = 0.5, ∆1 = 0.05, ε2 = 0,∆2 = 0.02, and λ = 0.

respectively, it is found that GL± + GR± = G0. This property—perhaps the result of some

remaining symmetry across each critical point—will be seen later when ε2 is varied for

fixed ε1 and U2 > 0.

These results show that it is possible to identify the critical points ε±1c and the new

boundaries ε±1 from the conductance in a DQD device that does not satisfy U2 = 0.

Even when U2 is large enough to close the local-moment regime entirely, the quantum

phase transitions leave a signature, as seen previously in the non-zero minimum of TK

(see Figure 5-20). A signature is also found in the zero-temperature conductance, plotted

in Figure 5-24. When U2 is large enough to close the local-moment regime, there is still

128

0.001 0.01 0.1|ε

1 - ε

1c

+|

0

0.5

1

G/G

0

U2 = 0.001, ε

1 > ε

1c

+

0.001, ε1 < ε

1c

+

0.01, ε1 > ε

1c

+

0.01, ε1 < ε

1c

+

0.02, ε1 > ε

1c

+

0.02, ε1 < ε

1c

+

0.03, ε1 > ε

1c

+

0.03, ε1 < ε

1c

+

Figure 5-23. Zero-temperature conductance G/G0 vs. |δε+1 | = |ε1 − ε+

1c| for a parallel DQDdevice with U2 as specified in the legend. Traces marked by ε1 > ε+

1c are inthe strong-coupling regime and may be compared with traces in Figure 5-19.Other device parameters are as in Figure 5-22.

a dip (peak) at ε1 = ε−(+)1 , with a plateau region corresponding to the dip in TK vs. ε1.

(The rough region around ε1 = −0.35 in the U2 = 0.05 trace is the result of a numerical

instability associated with being near a Kosterlitz-Thouless transition.) It is interesting

to note that the shapes of the conductance traces for U2 > ∆1, ∆2 resemble those for the

side-coupled case with U2 > ∆2, λ in Figure 5-5.

Finally, as mentioned earlier, it is also possible to observe these quantum phase

transitions by varying ε2. While a complete phase diagram of ε1, ε2, and U2 is beyond the

scope of the present study, it is insightful to examine a couple of specific cases.

Figure 5-25 shows the zero-temperature conductance as ε2 is swept in a parallel DQD

device with a particle-hole symmetric Dot 1 (ε1 = −U1/2). Because of this particle-hole

symmetry on Dot 1, the traces in Figure 5-25 are symmetric about ε2 = −U2/2. For

129

-0.5 -0.4 -0.3 -0.2 -0.1 0ε

1

0

0.5

1

G/G

0U

2 = 0.045

0.050.10.50.75

Figure 5-24. Zero-temperature conductance G/G0 vs. ε1 for a parallel DQD device with U2

as specified in the legend. The traces may be compared with those in Figure5-20. Other device parameters are U1 = 0.5, ∆1 = 0.05, ε2 = 0, ∆2 = 0.02, andλ = 0. The rough feature in the solid red trace is due to a numericalinstability associated with the Kosterlitz-Thouless transition.

U2 = 0, the device is always in the strong-coupling phase except at ε2 = 0, where there is

an infinitesimally thin local-moment phase and a discontinuous point in the conductance

trace. When U2 is positive, however, the local-moment phase acquires a non-zero width,

and there is a pair of discontinuous jumps in G, signifying the presence of two quantum

critical points ε±2c. It is important to note that the left- and right-hand limiting values of

G again add to G0, confirming that the same quantum phase transitions are observed by

varying ε2 as were observed when ε1 was varied.

Similar behavior is also found when Dot 1 is particle-hole asymmetric. Figure 5-26

shows G vs. ε2 with ε1 ≈ ε+1c for each value of U2. There is a very pronounced asymmetry

in these results, but the same general features apply as in the particle-hole symmetric

130

case: For U2 = 0, there is an infinitesimally thin local-moment regime, marked by a

discontinuous jump in G; for U2 > 0, there is a pair of quantum critical points ε±2 , each

marked by a discontinuous jump in G whose left- and right-hand limiting values add to

G0.

-0.04 -0.02 0 0.02 0.04

ε2

0

0.5

1

G/G

0

U2 = 0

0.01

Figure 5-25. Zero-temperature conductance G/G0 vs. ε2 for a parallel DQD device withε1 = −U1/2 = −0.25, ∆1 = 0.05, ∆2 = 0.02, λ = 0, and U2 as specified in thelegend. For U2 > 0, there is a pair of critical points ε±2c, each causing adiscontinuous jump in G. The limiting values of each discontinuity obeyGL± + GR± = G0, as in the case of G vs. ε1. Note that the traces aresymmetric about ε2 = −U2/2 due to the particle-hole symmetry of Dot 1.

The results seen in this chapter indicate that these highly asymmetric DQD devices

exhibit a rich variety of interesting physics. In the U2 = 0 special limit, the side-coupled

DQD configuration exhibits a splitting of the Kondo resonance and the parallel DQD

configuration exhibits a pair of quantum phase transitions. We have seen that, for

sufficiently small U2, these behaviors persist, although their qualitative features are

131

-0.04 -0.02 0 0.02 0.04

ε2

0

0.5

1

G/G

0

U2 = 0, δε

1

+/ε

1c

+ = -.0071

U2 = 0.01, δε

1

+/ε

1c

+ = -.006

Figure 5-26. Zero-temperature conductance G/G0 vs. ε2 for a parallel DQD device with U2

as specified in the legend and ε1 near ε+1c for each value of U2. For U2 > 0,

there is again a pair of critical points ε±2c, each causing a discontinuous jumpin G. The limiting values of each discontinuity obey GL± + GR± = G0, as inthe case of G vs. ε1. Other device parameters are as in Figure 5-25.

modified. It should be possible to realize these behaviors experimentally by varying the

dot resonant levels by tuning the appropriate gate voltages. For sufficiently large U2, these

behaviors disappear. Thus, such experiments can also observe the evolution of the DQD

device from the U2 = 0 limit to a device with two local-moment dots.

132

CHAPTER 6CONCLUSIONS

It is clear that systems of multiple magnetic impurities and their analogous

manifestations in QD systems are not understood exhaustively. While much theoretical

work has been performed and novel experiments have been designed to probe two-impurity

systems, there are still many more possibilities to consider. In this work, I have demonstrated

that a number of interesting behaviors—such as the competition between magnetic

ordering and the Kondo effect, a splitting of the Kondo resonance, and a pair of quantum

phase transitions—should be experimentally observable in STM conductance studies of a

pair of magnetic adatoms, and in electrical transport through highly asymmetric DQDs.

6.1 Scanning Tunneling Microscopy Studies

In STM studies, the competition between magnetic ordering and the Kondo effect

is manifested in the differential conductance, displaying a variety of behaviors as the

impurity separation is varied. For a sufficiently large ferromagnetic RKKY interaction,

the two impurities form a net spin singlet that experiences a two-stage Kondo effect,

producing a conductance spectrum given by a sum of two Fano lineshapes. For a weaker

RKKY interaction, the two impurities do not become aligned, and are Kondo-screened

independently, producing a conductance spectrum given by a single Fano lineshape,

qualitatively matching the conductance spectrum of a single impurity. For a sufficiently

strong antiferromagnetic RKKY interaction, the Kondo effect is suppressed, leading

to a disappearance of the Kondo resonance and a featureless conductance spectrum.

While these results were obtained for a two-impurity system with several simplifications,

qualitatively similar behaviors should be found in more general cases.

For example, the results of Section 4.2.3 considered the limiting case of an STM

tip positioned directly over one of the impurities. To consider an STM study performed

with the tip at a variable position, it should be possible to utilize an extension of the

method developed by Borda [96] to study spatial correlations in the single-impurity

133

Kondo problem: This method involves setting up an NRG calculation of a two-impurity

system (cf. Section 2.2) with the second impurity site “empty” (i.e., ε2, U2, ∆2 = 0),

permitting the computation of properties at any distance from the impurity of study (by

varying the impurity separation R), thereby enabling a theoretical prediction of the Kondo

screening length lK . The extension of this method to a two-impurity STM study with a

generalized tip position would thus involve setting up a three-impurity calculation with

the third site “empty” and calculating the matrix elements of the impurity-3 operator

d†3σ at successive iterations. Given the great complexity of three-impurity models [61, 84],

such NRG calculations would benefit from the parallel-processing techniques described in

Chapter 3. Studies of single impurities—such as depicted in Figures 1-8 and 1-9—indicate

that the Fano lineshape in the conductance only changes quantitatively as the STM tip is

moved. In contrast, in a generalized two-impurity STM study, it is necessary to consider

three tunneling paths, which may give rise to novel interference effects, producing a more

complicated version of the Fano lineshape.

In Chapter 4, the impurities under consideration also exhibited no dissipative

effects, as would be caused by the presence of phonons or noise which would introduce

decoherence to the electron tunneling paths, thereby modifying the Fano lineshape. These

scenarios may be modeled by coupling the impurities to a bosonic bath. Such coupling

has been studied in single-impurity systems using the NRG [50, 78, 97], and is known

to produce a number of novel effects, including a quantum phase transition accessible

by tuning the coupling between the impurity and the bosonic bath. The study of a

two-impurity system coupled to a bosonic bath has not been approached with the NRG

due to the computational complexity; thus, such NRG studies would benefit from the

parallelization techniques described in Chapter 3.

6.2 Double Quantum Dot Studies

Systems of DQDs offer a tunable two-impurity system, sparking a great deal of

theoretical interest. To take advantage of this tunability, a class of highly-asymmetric

134

DQD devices has been studied in which Dot 1 possesses a well-defined local magnetic

moment and Dot 2 is near-resonance with the leads (ε2 ≈ 0) and is essentially non-interacting

(U2 ≈ 0). It has been shown [88, 94, 98] that, in the special limit of U2 = 0, such a device

may be modeled as a single Anderson impurity with a nonconstant hybridization to the

conduction band. In the side-coupled DQD configuration (∆1 = 0), the nonconstant

hybridization produces a zero-field splitting of the Kondo resonance in which the

Kondo effect is preserved. In the parallel DQD configuration (λ = 0), the nonconstant

hybridization gives rise to a pair of continuous QPTs separating strong-coupling and

local-moment phases. These QPTs may be observed experimentally in the device

conductance G by tuning the gate voltages that control ε1 and ε2 [98].

Even though the special condition of U2 = 0 may be impossible to achieve

experimentally, the results of Chapter 5 indicate that the resonance splitting and quantum

phase transitions may be observed when U2 is non-zero but sufficiently small, with

intriguing modifications to these behaviors. In the side-coupled DQD configuration, a U2

that is non-zero but sufficiently smaller than the other device parameters produces merely

a slight asymmetry in the splitting of the Kondo resonance and in the variation of G with

ε1. As U2 becomes larger than the other device parameters, qualitatively new behavior

develops, including a peak in G vs. ε1.

In the parallel DQD configuration, as U2 increases from zero, the locations ε±1c of the

QPTs shift toward each other and develop an asymmetry indicative of a Kosterlitz-Thouless

type transition. Also, regions of new behavior unfold in the strong-coupling phases,

characterized by a dramatic drop in the Kondo temperature TK as the QPTs are

approached. The critical points ε±1c are manifested in the zero-temperature conductance

G by discontinuous jumps that obey an intriguing sum rule GL± + GR± = G0, where

G0 = 2e2/h is the quantum-mechanical maximum value of G. These same QPTs may be

observed by varying ε2, giving rise to critical points ε±2c which also exhibit discontinuous

135

jumps in G that obey the sum rule. Finally, for sufficiently large U2, the QPTs merge and

the local-moment phase disappears entirely, producing a continuous G vs. ε1.

Further study of the parallel DQD configuration is called for, including the development

of phase diagrams that illustrates the effects of changes in U1, ∆1, and ∆2, which were left

fixed in Chapter 5. It will be useful to see the effects that such changes produce on the

behavior of the critical points ε±1c and ε±2c, and also the effects that they produce on the

value of U2 for which the local-moment phase disappears.

Further work is also needed to understand the physics behind the sum rule exhibited

by the discontinuities in the conductance at the critical points. To develop this understanding,

it will be necessary to examine other device properties, such as the phase shifts ηij

(i, j = 1, 2) of electrons that tunnel through the DQD device. It will also be necessary to

understand the conditions under which this sum rule is obeyed.

To make predictions that are experimentally relevant, it will also be necessary to

explore the effects of non-zero temperature on the conductance signatures reported for

the two DQD configurations in Chapter 5. Calculating the spectral functions in the

Landauer formula at finite temperature is not a straightforward task, but can be handled

to reasonable accuracy [44, 93]. In general, it has been found that the conductance

spectrum remains qualitatively the same for 0 ≤ T . TK and is significantly different

for T > TK [85, 98]. In the parallel configuration, it will be particularly interesting to see

the effects of finite temperature on the discontinuities observed in the zero-temperature

conductance G.

6.3 Epilogue

Even after years of progress, the study of magnetic impurity systems continues

to yield newer levels of insight and application. With the ongoing development of

theoretical methods such as the numerical renormalization group and parallel computation

techniques, and the continuing progression of experimental innovations such as scanning

tunneling microscopy and quantum dot devices, we will continue to push the boundaries of

136

our understanding of these systems and develop novel applications. Truly, the only limit is

our imagination.

137

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BIOGRAPHICAL SKETCH

Brian Lane has been enjoying the study of physics since his junior year in high school.

In 2003, he graduated with his B.S. in physics from Jacksonville University (summa cum

laude). In 2005, he received his M.S. in physics from the University of Florida and was

wed to Amy Knight at Creekside Community Church. Since 2004, he has been developing

this dissertation under the supervision of Professor Kevin Ingersent, while developing

himself as a physics instructor. Upon successful defense of this dissertation, he will

continue to pursue his interests in condensed matter theory and physics education as a

faculty member at Jacksonville University.

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