Correlations in quantum dots:How far can analytics go? ♥

Slava Kashcheyevs

Amnon Aharony

Ora Entin-Wohlman

Phys.Rev.B 73,125338 (2006)PhD seminar on May 18, 2006

Outline

• The physics of small quantum dots– Zero-D correlations in a nutshell

• The models and methods– Generalized Anderson impurity model

• Equations-of-motion (EOM) technique– What we do & What we get

• Lessons (hopefully) learned

VG

• 2D electron gas– extended– ordered– Coulomb interaction

is not too important

Quantum dots defined by gates

• 2D electron gas– extended– Ordered– Coulomb interaction

is not too important

• 0D quantum dot – localized – no particular symmetry – Coulomb interaction

is dominant

QD

GaAs AlGaAs

Gates

Lead

LeadElectron gas plane

Correlations: Coulomb blockade

QDLead

Lead

Vbias

VG

Peaks in linear conductance G = I / Vbias as function of VG

Coulomb blockade

Coulomb blockade

Coulomb blockade

Correlations: continued2

1

0

G, e2/h

VG

T = 800 mK

T = 15 mK

van der Wiel et al., Science 289, 2105 (2000)

high T

low T

odd even odd evenS=1/2 S=0 S=1/2 S=0

Characteristic temperature

TK (VG)

The Kondo effect

The Kondo effect

Kondo “ice sheet” formation

• Singly occupied, spin-degenerate orbital

QD LeadLead

Charging

energy U

Kondo “ice sheet” formation

QD LeadLead

• Singly occupied, spin-degenerate orbital

• Transport via spin flips

• Opposite spins tend to form a bond

• Each spin flip breaks a “Kondo molecule”, and spins in the leads adjust to make a new one

Outline

• The physics of small quantum dots– Zero-D correlations in a nutshell

• The models and methods– Generalized Anderson impurity model

• Equations-of-motion (EOM) technique– What we do & What we get

• Lessons (hopefully) learned

QD

The model: quantum dot

2

1

0

ε0 +Uε0

ε0 is linear in VG

Fix Fermi level at 0

E

ε0

ε0 +U

ε↓

ε↑

Allow for Zeeman splitting

• Set of non-interacting levels for the leads

The model: leads and tunneling

leads

• Tunneling between the dot and the leads

tunn

Glazman&Raikh, Ng&Lee (1988) – quantum dots

The model

• The Anderson impurity model

• Generalizations– Structured leads: any network of tight binding sites– More levels, more dots– Spin-orbit interactions (no conservation of σ)

P.W. Anderson, Phys.Rev. 124, 41 (1961)

Lines of attack I: standard tools

• Perturbation theory in U – Regular (from U=0 to finite U)– Ground State is a singlet

• Fermi liquid around GS– Narrow resonant peak at EF

– Strong renormalization: U,Γ~TK

• Perturbation theory in Γ– Singular (spin-half state at Γ=0)– Misses both CB and Kondo

FL

PT in Γ

Temperature M

ag

. fie

ld ~ ~

U

Γ = πρ|Vk|2

*

*S=0

S=1/2

Lines of attack II: heavy artillery

• Bethe ansatz solution – large bandwidth + Γ↑=Γ↓ integrability

– gives thermodynamics, but not transport– solvability condition is too restrictive

• Numerical renormalization group

• Functional renormalization group

Outline

• The physics of small quantum dots– Zero-D correlations in a nutshell

• The models and methods– Generalized Anderson impurity model

• Equations-of-motion (EOM) technique– What we do & What we get

• Lessons (hopefully) learned

Equations-of-motion technique

• Define operator averages of interest– real-time equilibrium Green functions

• Write out their Heisenberg time evolution– exact but infinite hierarchy of EOM

• Decouple equations at high order– uncontrolled but systematic approximation

• ... and solve

The Green functions

• Retarded

• Advanced

• Spectral function

grand canonical

Zubarev (1960)

step function

Dot’s GF

• Density of states

• Conductance

• Local charge (occupation number)

at Fermi level for T=0 and

for G=2e2/h

Equations of motion

• Example: 1st equation for

Full solution for U=0

bandwidth D

Γ

Lead self-energy function

Lorenzian DOS Large U should bring

ε0 +Uε0 ω=0Fermi

hole excitations

electron excitations

Kondo quasi-particles

Full hierarchy

…

Decoupling

• Use values

Meir, Wigreen, Lee (1991)

Linear = easy to solve Fails at low T – no Kondo

Decoupling

• Use mean-field for at most 1 dot operator:

“D.C.Mattis scheme”:Theumann (1969)

• Demand full self-consistency

Significant improvement Hard-to-solve non-linear integral eqs.

The self-consistent equations

Self-consistent functions:

Level positionZeeman splitting The only input parameters

How to solve?

• In general, iterative numerical solution

• Two analytically solvable cases:

– and wide band limit: explicit non-trivial solution

– particle-hole symmetry point : break down of the approximation

Results (finally!)

• Zero temperature• Zero magnetic field• & wide band

Level renormalization

Changing Ed/Γ

Looking at DOS:

Ed / ΓEnergy ω/Γ

Fermi

odd

even

Results: occupation numbers

• Compare to perturbation theory

• Compare to Bethe ansatz

Gefen & Kőnig (2005)

Wiegmann & Tsvelik (1983)

Better than 3% accuracy!

Check: Fermi liquid sum rules

• No quasi-particle damping at the Fermi surface:

• Fermi sphere volume conservation (Friedel sum rule)

Good – for nearly empty dot

Broken – in the Kondo valley

No “drowned” electrons rule!

Results: melting of Kondo “ice”At small T andnear Fermi energy, parameters in the solution combine as

Smaller than the true Kondo T:

2e2/h conduct.

~ 1/log2(T/TK)

DOS at the Fermi energy scales with T/TK* As in experiment (except for factor 2)

Results: magnetic susceptibility

• Defined as

• is roughly the energy to break the singlet = polarize the dot

– ~ Γ (for non-interacting U=0)

– ~ TK (in the Kondo regime)

Results: magnetic susceptibility

!

Bethe susceptibility in the Kondo regime ~ 1/TK

Our χ is smaller, but on the other hand TK* <<TK ?!

Results: magnetic susceptibility

Γ

TK*

Results: compare to MWL

Meir-Wingreen-Lee approximation of averages gives non-monotonic and even negative χ for T < Γ

Outline

• The physics of small quantum dots– Zero-D correlations in a nutshell

• The models and methods– Generalized Anderson impurity model

• Equations-of-motion (EOM) technique– What we do & What we get

• Lessons (hopefully) learned

Conclusions!

• “Physics repeats itself with a period of T ≈ 30 years” – © OEW

• Non-trivial results require non-trivial effort

• … and even then they may disappoint someone’s expectations

• But you can build on what you’ve learned

PPTs & PDFs at kashcheyevs