Transcript Slide 1

Correlations in quantum dots:
How far can analytics go?
Vyacheslavs (Slava) Kashcheyevs
Collaboration:
Amnon Aharony (BGU+TAU)
Ora Entin-Wohlman (BGU+TAU)
Avraham Schiller (Hebrew Univ.)
December 18th, 2006
Outline
• Physics of the Anderson model
– Relevant energies and regimes
– The plethora of methods
• Equations of motion
– Self-consistent truncation
– Gauging quality in known limits
• Anderson model for double quantum dots
– Charge location as a pseudo-spin
– Diverse results in a unified way
Quantum dots
Lead
QD
Lead
Electron gas
plane
GaAs
Gates
AlGaAs
• Tune: gate potentials, temperature, field…
• Measure: I-V curves, conductance G…
• Aharonov-Bohm interferometry,
dephasing, coherent state manipulation…
Building models
• Quantum dots
– Mesoscopic:
many levels involved, statistical description
– Microscopic: few levels, individual properties
• Tunneling Hamiltonian approach
Anderson model: the leads
μ
Lead
• A set of energy levels
Lead
Anderson model: the dot
μ
Lead
QD
• An energy level ε0
Lead
Anderson model: the dot
f (ε0)
n
2
T
1
μ
μ
Lead
QD
• An energy level ε0
Lead
ε0
Anderson model: the dot
f (ε0)
n
2
T
1
μ
μ
Lead
QD
• An energy level ε0
Lead
ε0
Anderson model: the dot
n
Spin-charge
separation or
“Mott transition”
U f (ε0)
2
U
T
1
μ
μ
Lead
QD
• An energy level ε0
Lead
μ
Interactions!
ε0
Anderson model: tunneling
VR
VL
n
U
2
1
Γ
Γ
μ
Lead
• Tunneling
QD
Lead
μ
ε0
Quantum fluctuations:
– of charge Γ > T
– of spin U > Γ > T
Tunneling rate
Γ ~ ρ|V|2
Anderson model: spin exchange
n
U
2
1
μ
Lead
QD
μ
ε0
Lead
• Fix the spin on the dot
• Opposite spin in the leads can lower energy!
Anderson model: spin exchange
n
U
2
1
μ
μ
ε0
Virtual transition:
Lead
QD
Lead
• Fix the spin on the dot
• Opposite spin in the leads can lower energy!
Anderson model: spin exchange
n
U
2
1
μ
μ
Virtual transition:
Lead
QD
Lead
• Fix the spin on the dot
• Same spin can’t go!
ε0
Effective Hamiltonian: Kondo
Lead
QD
Lead
• Ferromagnetic exchange interaction!
Effective Hamiltonian: Kondo
<S>
½ – O(J)
1/2
?
0
Lead
QD
J
h
Lead
• Ferromagnetic exchange interaction!
← can fix S with h >> J
The Kondo effect
<S>
½ – O(J)
1/2
?
~ h/TK
0
Lead
QD
Lead
TK
J
h
singlet-triplet splitting
• |↑↓> and |↓↑> are degenerate (Sz=0 of S=0,1)
• except for virtual excitations ~ J !
Outline
• Physics of the Anderson model
– Relevant energies and regimes
– The plethora of methods
• Equations of motion
– Self-consistent truncation
– Gauging quality in known limits
• Anderson model for double quantum dots
– Charge location as a pseudo-spin
– Diverse results in a unified way
Methods
• Perturbation theory (PT) in Γ, in U
in U – regular & systematic; not good for U>> Γ.
in Γ – breaks down at resonances & in Kondo regime
• Fermi liquid: good at T<TK, exact sum rules for T=0
• Equations of motion (EOM) for Green functions
exact for U; a low order (Hartree mean field) gives local moment
as good as PT when PT is valid
? can we get Kondo at higher orders?
• Renormalization Group
perturbative (in Γ) RG => Kondo Hamiltonian + PM scaling
perturbative (in U) RG => functional RG (semi-analytic)
Wilson’s Numerical RG – high accuracy, but numerics only
• Bethe ansatz:
exploits integrability
exact (!) solution, many analytic results
integrability condition too restrictive, finite very T laborious
Outline
• Physics of the Anderson model
– Relevant energies and regimes
– The plethora of methods
• Equations of motion
– Self-consistent truncation
– Gauging quality in known limits
• Anderson model for double quantum dots
– Charge location as a pseudo-spin
– Diverse results in a unified way
The Green functions
• Retarded
Zubarev (1960)
step function
• Advanced
• Spectral function
grand canonical
Dot’s GF
• Spectral function → Density of states
conductance at T=0
is proportional ~ ρ(μ)!
• Occupation number → Local charge & spin
• Friedel-Langreth sum rule (T=0)
Equations of motion
• Example: 1st equation for
Full solution for U=0
Lead self-energy function
spin-flip
excitations
Lorenzian DOS
Large U should bring
hole
excitations
electron
excitations
Γ
bandwidth D
ε0 ω=0
Fermi
ε0 +U
Full hierarchy
…
A general term
m = 0,1, 2… lead operators
Dworin (1967)
n=
0 – 3 dot operators
Contributes to a least m-th order in Vk
and (n+m–1)/2-th order in U !
Certain order of EOM truncation
↔ certain order of perturbation theory!
Decoupling
Decoupling
“D.C.Mattis scheme”:
Theumann (1969)
• Use mean-field for at most 1 dot operator:
• Use
values
• Demand full self-consistency
Meir, Wigreen, Lee (1991)
 Linear = easy to solve
 Fails at low T – no Kondo
 Significant improvement
 Hard-to-solve non-linear integral eqs.
The self-consistent equations
Zeeman splitting
Level position
Self-consistent functions:
The only input
parameters
EOMs: 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
:
Outline
• Physics of the Anderson model
– Relevant energies and regimes
– The plethora of methods
• Equations of motion
– Self-consistent truncation
– Gauging quality in known limits
• Anderson model for double quantum dots
– Charge location as a pseudo-spin
– Diverse results in a unified way
EOMs: Results
Ed / Γ
Energy ω/Γ
• Zero temperature
• Zero magnetic field
•
& wide band
even
Level renormalization
odd
Changing Ed/Γ
Looking at DOS:
Fermi
Results: occupation numbers
• Compare to
perturbation theory
Gefen & Kőnig (2005)
• Compare to
Bethe ansatz
Wiegmann & Tsvelik (1983)
Better than 3% accuracy!
Check: Friedel-Langreth sums
• No quasi-particle
damping at the
Fermi surface:
• Fermi sphere volume
conservation
Good – for nearly empty dot
Broken – in the Kondo valley
Results: melting of the peak
2e2/h conduct.
At small T and
near Fermi energy,
parameters in the
solution combine as
~ 1/log2(T/TK)
Smaller than
the true Kondo T:
DOS at the Fermi energy scales with T/TK*
Experiment: van der Wiel et al., Science 289, 2105 (2000)
EOMs: 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
Outline
• Physics of the Anderson model
– Relevant energies and regimes
– The plethora of methods
• Equations of motion
– Self-consistent truncation
– Gauging quality in known limits
• Anderson model for double quantum dots
– Charge location as a pseudo-spin
– Diverse results in a unified way
Double dots: a minimal model
• Two orbital levels
• Two leads
• Inter-dot
– repulsion U
– tunneling b
• Aharonov-Bohm flux
(wide band)
Interesting properties
• Charge oscillations / population switching
• Transmission zeros / phase lapses
• “Correlation-induced” resonances
Konig & Gefen PRB 71 (2005) [PT]
Sindel, Silva, Oreg & von Delft
PRB 72 (2005) [NRG & HF]
Meden & Marquardt PRL (2006)
[fRG & NRG]s
Singular value decomposition
• Diagonalize the tunneling matrix:
• Define new degrees of freedom
• The pseudo-spin is conserved in tunneling!
Map to Anderson model
z
θ
x
Rotated magnetic field!
Special case: an exact result
• Degenerate levels:
• Spin is conserved → Friedel rule applies:
Need a way to get
magnetization M !
Local moment,
Mapping to Kondo Hamiltonian
• Schrieffer-Wolff for U >> Γ,h (local moment)
• Anisotropic exchange
Silvestrov & Imry PRL (2000)
Martinek et al PRL (2003)
• Effective field
• Anisotropy is RG irrelevant
Main results
• An isotropic
Kondo model in
external field
• Use exact
Bethe ansatz
• Key quantities
• Return back
Local moment
here:
Main results: anisotropic Γ’s
• Both competing scales depend on ε0
h ≈ TK => M=1/4
fRG
h=0
Double dots: conclusions
• Looking at the right angle makes
old physics useful again
• Singular value decomposition (SVD)
reduces dramatically the parameter space
• Accurate analytic expressions for linear
conductance and occupations
• Future prospects:
– more levels
– add real spin
– non-equilibrium