Correlations in quantum dots: How far can analytics go? ♥ Slava Kashcheyevs Amnon Aharony Ora Entin-Wohlman Phys.Rev.B 73, (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

VGVG 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 GaAsAlGaAs Gates Lead Electron gas plane

Correlations: Coulomb blockade QD Lead V bias VGVG Peaks in linear conductance G = I / V bias as function of V G

Coulomb blockade

Correlations: continued G, e 2 /h VGVG T = 800 mK T = 15 mK van der Wiel et al., Science 289, 2105 (2000) high T low T oddevenoddeven S=1/2S=0S=1/2S=0 Characteristic temperature T K (V G ) The Kondo effect

Kondo “ice sheet” formation Singly occupied, spin-degenerate orbital QDLead Charging energy U

Kondo “ice sheet” formation QDLead 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 ε 0 +Uε0ε0 ε 0 is linear in V G Fix Fermi level at 0 E ε0ε0 ε 0 +U ε↓ε↓ 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 E F –Strong renormalization: U,Γ~T K Perturbation theory in Γ –Singular (spin-half state at Γ=0) –Misses both CB and Kondo FL PT in Γ Temperature Mag. field ~ ~ U Γ = πρ|V k | 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=2e 2 /h

Equations of motion Example: 1 st equation for

Full solution for U=0 bandwidth D Γ Lead self-energy function Lorenzian DOS Large U should bring ε 0 +Uε0ε0 ω=0 Fermi 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 position Zeeman 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 E d /Γ Looking at DOS: E d / Γ Energy 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 and near Fermi energy, parameters in the solution combine as Smaller than the true Kondo T: 2e 2 /h conduct. ~ 1/log 2 (T/T K ) DOS at the Fermi energy scales with T/T K * 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) – ~ T K (in the Kondo regime)

Results: magnetic susceptibility ! Bethe susceptibility in the Kondo regime ~ 1/T K Our χ is smaller, but on the other hand T K * <<T K ?!

Results: magnetic susceptibility Γ TK*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

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