Coulomb Blockade and

Non-Fermi-Liquid Behavior

in a Double-Dot Device

Avraham Schiller

Racah Institute of Physics

Eran Lebanon (Rutgers University)

Special Thanks to Yuval Oreg

Frithjof B. Anders (Bremen University)

Outline:

The single-channel Kondo effect

The Kondo effect in ultra-small quantum dots

Two-channel Kondo physics in a lead-dot-box device

Conclusions

The two-channel Kondo effect

The Coulomb blockade in quantum boxes

From spin to charge two-channel Kondo effects

Entanglement of spin and charge

Discontinuity in the conductance

T(K) T(K)

Resistivity minimum: The Kondo effect

De Haas & ven den Berg, 1936Franck et al., 1961

1/5impmin cT

Fe in Cu

Enhanced scattering at low T

The Kondo Effect: Impurity moment in a metal

A nonperturbative energy scale emerges JTK /1exp

Below TK impurity spin is progressively screened

All initial AFM couplings flow to a single strong-coupling fixed point

A sharp resonance is formed at the Fermi energy for T<TK

Local-moment formation: The Anderson model

d|

d + U

nUnnH dimp

hybridization withconduction electrons

V

The Anderson model: spectral properties

EFd d+U

Kondo resonance

A sharp resonance of width TK develops at EF for T<TK

Unitary scattering for T=0 and <n>=1

d

UVL

Q.D. LeadLead

VR

Ultra-small quantum dots as artificial atoms

Electrostatically-defined semiconductor quantum dots

Goldhaber-Gordon et al., Nature 1998

Quantum dot

Plungergate

Temperature depedence Field dependence

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

Extra channel index

One- versus two-channel Kondo effect

Property One channel Two channel Non-Fermi-liquid

)0( TS

TC /

)0( T

0

KT/1

KT/1 )/log( TTK

)/log( TTK

)2log(2

1Residual entropy

Diverging coefficient

Diverging susceptibility

Requirements for the realizationof the two-channel Kondo effect

No scattering of electrons between the bands

Two independent conduction bands

Equal coupling strength to the two bands

No applied magnetic field acting on the impurity spin

Is realization of the two-channel Kondo effect at all possible?

The Coulomb blockade in quantum box

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

Charging of a quantum box

Two-channel Kondo effect in the 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 dictionary for theCharging of a quantum box

Two-channel Kondo Charging of a quantum box

Spin index

Channel index

Exchange interaction

Magnetic Field

Bandwidth

J

H

D

Isospin index

Physical spin

Tunneling matrix element

Deviation from deg. point

Charging energy

2t

eV

EC

Diverging susceptibility Diverging capacitance C

Spin two-channel Kondo effect in a lead-dot-box device

In an ordinary two-lead device:

Inter-lead spin-exchange spoilsthe two-channel Kondo effect

(Oreg and Goldhaber-Gordon ‘03)

In an ordinary two-lead device:

Inter-lead spin-exchange spoilsthe two-channel Kondo effect

Spin two-channel Kondo effect in a lead-dot-box device(Oreg and Goldhaber-Gordon ‘03)

In an ordinary two-lead device:

Inter-lead spin-exchange spoilsthe two-channel Kondo effect

Quantumbox

Inter-lead spin-exchange is

blocked in a lead-dot-box

device, for kBT < EC !

Spin two-channel Kondo effect in a lead-dot-box device(Oreg and Goldhaber-Gordon ‘03)

Quantumbox

Quantumbox

Quantumbox

Quantumbox

Quantumbox

Quantumbox

Tunneling is blocked by the Coulomb blockade

A second screening channel is dynamically generated

for temperatures below the charging energy

A spin two-channel Kondo effect should develop

if JBox and JLead are tuned to be equal

Our goal: A detailed quantitative theory of this scenario

Note: The above scenario assumes the formation of a stable local moment on the dot, and quantized charge on the box !

Extension to regimes with charge fluctuations

Lead—Quantum dot—Quantum box setting

(Courtesy of D. Goldhaber-Gordon)

Leads

Quantumbox

Quantum dot

The model

2, ,

ˆBBoxCddd

BL kkkk

NNEnUnddccH

BL

kkcddct

,

Hybridization widths:2 t

Method of solution: Wilson’s numerical renormalization group

(E. Lebanon, AS, F.B. Anders, PRB 2003)

Strategy: Fix L and tune B in search of a two-channel Kondo effect

)/ln(20

)()(

2

TTTk

gT K

KB

B Hallmark of spin two-channel Kondo effect:

Definition of TK

Symmetric dot: 2d + U = 0

Two-channel point is found for any U, including U = 0

DECL 1.0

0BN

Spin two-channel effect persists in the mixed-valent regime

TK versus U for a symmetric dot

Exponentially small

TK is significantly enhanced in the mixed-valent regime

Analytic estimatefor stable moment

Can become of the order of the charging energy EC

PerfectTransmissionfor 2CK

Dependence of TK on the gate voltage NB for U = 0

Prediction of bosonization treatment near perfect transmission

Spin 2-channel Kondo effect related to perfect transmission through dot

Shift in Coulomb staircase

B

DC

E

A

Two-channel line and charging curve

for ``realistically large’’ U/EC = 20

DECL 1.0

0BN

2-channelline

Lead BoxDot

tL

Origin of shift in Coulomb staircase

tB

dddccd

local 2

1

2

1

2

111

Note: shift in staircase occurs for CLB E,

Diagonalize first the link between the dot and box

Site immediately coupled to the box is only half occupied

2d/U

Entanglement of spin and charge within the two-channel Kondo effect

BC dN

Nd

E

eTC

ˆ

2)(

2

0BN

Both magnetic susceptibility and charge capacitance diverge logarithmically,but with different Kondo scales (i.e., slopes)

Continuous transition from spin to charge 2-channel Kondo effect

Zero-temperature conductance

V

DECL 1.0 0BN DU

Discontinuous jump in the conductance across the two-channel point

Scaling of the conductance with distance from critical point

DU d 22

DECL 1.0

CKBB2

CKBB2

Conclusions

Quantum-dot devices offer a unique opportunity to study the two-channel Kondo effect.

Exploiting the Coulomb blockade, one can measure the two-channel Kondo effect in a double-dot device.

Among the exotic features found:

A continuous transition from a spin to a chargetwo-channel Kondo effect.

Entanglement of spin and charge.

A discontinuity in the T = 0 conductance.

Enhancement of the Kondo temperature away from the local-moment regime.