Correlations in quantum dots: How far can analytics go? ♥ Slava Kashcheyevs Amnon Aharony Ora...
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Transcript of Correlations in quantum dots: How far can analytics go? ♥ Slava Kashcheyevs Amnon Aharony Ora...
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