Transcript Correlations in quantum dots

Correlations in quantum dots:
How far can analytics go? ♥
Slava Kashcheyevs
Amnon Aharony
Ora Entin-Wohlman
PhD seminar on
May 18, 2006
Phys.Rev.B 73,125338 (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
Lead
QD
Gates
Lead
Electron gas
plane
GaAs
AlGaAs
• 2D electron gas
– extended
– Ordered
– Coulomb interaction
is not too important
• 0D quantum dot
– localized
– no particular symmetry
– Coulomb interaction
is dominant
Correlations: Coulomb blockade
Vbias
Lead
Lead
QD
VG
Peaks in linear conductance G = I / Vbias as function of VG
Coulomb blockade
Coulomb blockade
Coulomb blockade
Correlations: continued
G, e2/h
low T
2
1
S=1/2
odd
S=0
even
S=1/2
odd
S=0
even
high T
VG
0
T = 15 mK
The Kondo effect
T = 800 mK
Characteristic
temperature
TK (VG)
van der Wiel et al., Science 289, 2105 (2000)
The Kondo effect
Kondo “ice sheet” formation
• Singly occupied,
spin-degenerate
orbital
Charging
energy U
Lead
QD
Lead
Kondo “ice sheet” formation
• Singly occupied,
spin-degenerate
orbital
• Transport via
spin flips
• Opposite spins tend
to form a bond
Lead
QD
Lead
• 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
The model: quantum dot
Fix Fermi level at 0
2
1
0
ε0
ε0 +U
E
ε0 +U
QD
ε0
ε↓
ε↑
Allow for Zeeman splitting
ε0 is linear in VG
The model: leads and tunneling
• Set of non-interacting
levels for the leads
leads
• Tunneling between the dot and the leads
tunn
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)
Glazman&Raikh, Ng&Lee (1988) – quantum dots
Lines of attack I: standard tools
• Perturbation theory in U
U
– Regular (from U=0 to finite U)
– Ground State is a singlet
Γ = πρ|Vk|2
• Fermi liquid around GS
• Perturbation theory in Γ
– Singular (spin-half state at Γ=0)
– Misses both CB and Kondo
*
Mag. field
– Narrow resonant peak at EF
~~
– Strong renormalization: U,Γ~TK
S=1/2
PT in Γ
FL
S=0
* Temperature
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
Zubarev (1960)
step function
• Advanced
• Spectral function
grand canonical
Dot’s GF
• Density of states
at Fermi level for T=0
and
• Conductance
for G=2e2/h
• Local charge (occupation number)
Equations of motion
• Example: 1st equation for
Full solution for U=0
Lead self-energy function
Kondo
quasi-particles
Lorenzian DOS
Large U should bring
hole
excitations
electron
excitations
Γ
bandwidth D
ε0 ω=0
Fermi
ε0 +U
Full hierarchy
…
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
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!)
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: Fermi liquid sum rules
No “drowned” electrons rule!
• 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
Results: melting of Kondo “ice”
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*
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
TK*
Γ
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