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

EC but dense single-particle levels

Charging energy:

QVC

QQE B

0

2

2)(

0

2

2C

eEC

Energy for charging box with one electron

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

QVC

QQE B

0

2

2)(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 << EC

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

BBB

kkBkLk

BL kkkk QV

C

QcctccH ˆ

2

ˆH.c.

0

2

,',, ,

,

)(ˆk

BkBkBkB cceQ

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:

n

nnneQ̂

n

nnQ 1ˆ

Map Hamiltonian onto:

QVC

QccQtccH B

kkBkLk

BL kkkk

ˆ2

ˆH.c.ˆ

0

2

,',, ,

The constraint BQQ ˆˆ can be relaxed for 0 !!!

Temperature evolution of the Coulomb-blockade staircase

Coulomb staircase fully develops only well below EC

Capacitance

C(T) =-d<Q>/dVB

T = 0: Comparison with 2nd order perturbation theory

CEDd /

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

tT = 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 22

2

)(1

)(4

t

tT

Near perfect transmission

)2sin()(cosln2 2

BBB NNeRR

Nn

Prediction of 1D model :

Near perfect transmission

)2sin()(cosln2 2

BBB NNeRR

Nn

Euler’s constant

Reflectance

Prediction of 1D model :

Near perfect transmission

)2sin()(cosln2 2

BBB NNeRR

Nn

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:

)/ln(20

)(2ˆ

TTTk

eTC K

KBBdV

Qd

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 EC>>kBT and on

vicinity of a degeneracy point

Introduce the charge isospin

NNNNz 112

eVcccctccH zqk

kLqBqBkLBL k

kkk

,,, ,

Lowering and raising isospin operatorsChannel index

NN 1

Two-channel Kondo effect

2,12,1 ,

)0(

sSJccΗ impk

kkk

Impurity spin is overscreened by two identical channels

rT 0

A non-Fermi-liquid fixed point is approached for T<<TK