Capri spring school, April 2009

With collaborators:

P. Mehta - Princeton C. Bolech - Rice A. Jerez - NJIT, Rutgers G. Palacios - Rutgers N. Andrei - RutgersSpecial Thanks: C. Yee -Rutgers

Sung Po Chao

Nonequilibrium transport in the Anderson model of a biased Quantum Dot

Goldhaber-Gordon et al, Cronenwet et al.van der Wiel et al.

Conductance vs gate voltage - Kondo enhancement in odd valleys

Differential conductance vs bias - Nonequilibrium dynamics

T varies in the range 15-800mK

Quantum Dot in and out of Equilibrium

Equilibrium

Nonequilibrium

(Lee & Ng,Glazman & Raikh)

Hamiltonian to describe the system: Single Impurity Anderson model

Scattering Bethe Ansatz approach: (P.Mehta, N.Andrei 06)1. Diagonalize the Hamiltonian out of equilibrium:

2 . Compute expectation value:

Current and Dot Occupation

Current and dot-occupation operators:

Expectation value expressed in terms of Bethe Ansatz density :

with and given by

Conductance in equilibrium: Numerical results

• Conductance vs. gate voltage

- Direct calculation from current. - Verifies Friedel SR - TBA vs SBA

Von Delft, notes

Conductance out of equilibrium: Numerical results..2 cases

Case 1:

Case 2: From Kondo to mixed valence

(0 refers to equilibrium Fermi surface of the leads)

Conductance out of equilibrium: Numerical results case 1

• Obtain Tk from quadratic fit for Kondo peak nearby V=0. Observed universality for small

voltage region.

• Conductance vs. bias voltage V:

• Conductance nearby zero voltage:

• Conductance for voltage larger than Kondo temp:

Conductance out of equilibrium: Numerical results case 2

• Conductance vs. bias voltage :

• Contour plot of conductance vs source-drain voltage and gate voltage:

• The Kondo effect forms as is decreased, destroyed as the bias voltage is increased

(M Grobis et al,PRL 100,246601)

Summary and Outlook

• Computed Nonequilibrium transport properties of Anderson impurity model.

• Outlook: Finite temperature/finite magnetic field. Muti-terminal quantum dot/quantum wire