The Coulomb Blockade in Quantum Boxes Avraham Schiller Racah Institute of Physics Eran Lebanon (Hebrew University) Frithjof B. Anders (Bremen University) * Funded by the ISF Centers of Excellence Program

Charging energy in Quantum boxes Quantum box: Small metallic grain or large semiconductor quantum dot with sizeable Charging energy E C but dense single-particle levels  Charging energy: Energy for charging box with one electron

T = 0, t = 0 T > 0 and/or t > 0 g

Weak single-mode tunneling to the lead High temperature: Perturbation theory about the classical limit Low temperature: PT about well-defined charge configuration PT breaks down near degeneracy points! Effective low-energy model: two-channel Kondo model (Matveev 1991) T << E C

Near perfect transmission (single-mode tunneling) Treated within an effective model where the QB and the lead are coupled adiabatically by a 1D geometry with weak backscattering There are Coulomb-blockade oscillations also near perfect transmission (Matveev 1995)

No unified treatment of all regimes! Different models and treatments were tailored for the different regimes Certain treatments are based on effective models involving high-energy cutoffs which are not fully determined The crossover behavior between the different regimes is not accessible Some regimes, e.g. strong tunneling amplitudes, remain unexplored A unified nonperturbative treatment of all physical regimes is clearly needed!

Point-contact tunneling model g Excess charge inside box

Our approach: Use wilson’s numerical renormalization group (NRG) Problem: The NRG is designed to treat noninteracting conducton electrons. In this case the box electrons are interacting! Solution: Introduce collective charge operators: Map Hamiltonian onto: The constraintcan be relaxed for!!!

Temperature evolution of the Coulomb-blockade staircase Coulomb staircase fully develops only well below E C Capacitance C(T) =-d /dV B

T = 0: Comparison with 2 nd order perturbation theory Excellent agreement with PT at weak coupling at charge plateaus NRG and mapping work!

Increasing the tunneling amplitude: breakdown of perturbation theory Reentrance of Coulomb-blockade staircase for t  tt T = 0, d = 100

Origin of rapid breakdown of perturbation theory and reentrance of CB The relevant physical parameter is the single-particle transmission coefficient In the noninteracting case, the latter is given by

Near perfect transmission Prediction of 1D model :

Near perfect transmission Euler’s constant Reflectance Prediction of 1D model :

Near perfect transmission Single fitting parameter R Extracted R versus noninteracting 1 - T Prediction of 1D model :

Two-channel Kondo effect at degeneracy points Two-channel Kondo effect expected to develop at degeneracy points Characterized by log(T) divergence of the junction capacitance: Kondo temperature Log(T) divergence for all values of t 

Conclusions An NRG approach was devised for solving the charging of a quantum box connected to a lead by single-mode tunneling, applicable to all temperatures, gate voltages and tunneling amplitudes. Rapid breakdown of perturbation theory is found, followed by reentrance of the Coulomb-blockade staircase for tunneling amplitudes exceeding perfect transmission. Two-channel Kondo effect is found at the degeneracy points for all tunneling amplitudes, directly from the Coulomb-blockade Hamiltonian. The tunneling Hamiltonian is capable of describing all regimes of the Coulomb blockade, including the vicinity of perfect transmission.

Two-channel Kondo effect in charge sector (Matveev ‘91) Focus on E C >>k B T and on vicinity of a degeneracy point Introduce the charge isospin Lowering and raising isospin operatorsChannel index

Two-channel Kondo effect Impurity spin is overscreened by two identical channels A non-Fermi-liquid fixed point is approached for T<<T K