The Two Channel The Two Channel Kondo EffectKondo Effect

(The breakdown of the Fermi liquid (The breakdown of the Fermi liquid paradigm in quantum dots: theory and paradigm in quantum dots: theory and

experiment)experiment)

Department of Condensed Matter Physics

Weizmann Institute of ScienceHRI, 8 Feb 2008

Outline• Ancient and modern history of the Kondo effect • Shot noise of 5/3 in the Kondo limit (theory and

experiments)• The Fermi liquid and the Non Fermi liquid

theories• The two-channel Kondo (2CK) theoretical

predictions 90 136602 (2003).

• Experimental observation of the two channel Kondo state 446, 167 – 171 (2007).

• Summary

Resistance of metals at low temp

Temperature (K)

104R(T)R(273)

1 2 3 4 5

26.4

26.5

impurityphonon RTcR 5

De Hass, de Bour, de Berg (1934)

M. Sarachik

et al. 1964

Iron’s Concen. in Mo.8Nb.2

Fe is an example for a 3d transition metals

k

kkk ccconductionH

The s-d modelThe s-d model FERMI SEA

(Conduction band)

Local spin of conduction electrons Impurity

spin

J σ•S, J>0

TD

LogJν 242

TD

LogνJJcRTcR 32sdimpurity

5phonons

Kondo Solution

1/5sdmin cT

Band Width

Density of states

The Kondo Problem

Perturbation theory fails at the Kondo temp.

J2=J3Log(D/T) TK=D e-1/J

What happens below TK?

Developments

• Scaling and Renormalization Group• Exactly solvable points• Coulomb Gases• Numerical Renormalization Group• Bosonization (Schotte and Schotte)• Bethe ansatz solution (Andrei, Wiegman-

Tsvelik )

Kondo ResonanceConduction spin Impurity spin

J σ•S, J>0

A universal bound state is formed

between the local spin and the conduction electrons

Densi

ty o

f st

ate

s

Energy

kT

EF

Fermi-Edge Singularity )FES(

FERMI SEA(Conduction band)

EF

Eg•Attraction between valance hole and conduction electrons.

• Orthogonality “catastrophe”

Sudden potential +

Many body effects

EF+Eg

Ab

sorp

tion

Single Channel Kondo and Many FES

Time

J σ•S=JzσzSz+ J-σ+S-+ J+σ-S+

Densi

ty o

f st

ate

s

Energy

kT

EF

Origin of Spin-Spin interactionThe Anderson Model s-d

model

•Electron processes

JzσzSz

J-σ+S-

Origin of Spin-Spin interactionThe Anderson Model s-d

model

•Electron processes•Hole processes

initholeinitelec

*

EE1

EE1

ttJ

t- couplingd- impurity level

U-charging energy

d

U+2d

t

• Theory: a small quantum dot forms the local impurity state [Ng & Lee, Glazman & Raikh (1988)]

Experiments 1CK in a quantum dot

Strongly coupled and small Strongly coupled and small dotsdots

1 μm

Exp:Yang Ji, M. Heiblum et al. Sci. 290,

779. TK=D e-1/J

J~t2/U

1998

2000

In quantum dots we can tune and control many parameters

Phase measurements

Gerland, Costi, von Delft & Oreg PRL 84, 3710 2000

Exp:Yang Ji, M. Heiblum et al. Sci. 290, 779 2000

0 µV; -440 µV; -80 µV; -1100 µV

Exp: Zafalon, et al. 2007

2004

Fractional Noise in Fractional Noise in the the

Kondo LimitKondo Limit

Eran Sela, YO, von Oppen, KochPRL 2006

Nozières FL Hamiltonian

αεπνTK

ε ε ε εβnσ

πν2TK

=(1-e 2iδσ)/(2πiν)=-δσ/πν δσ=

αεTK

βnσ

νTK

Elastic part

Floating of the Kondo resonance

δn δε

δσ(ε=0, n=0) = δσ(δε, n=νδε)

δσ=αεTK

βnσ

νTKα=β

Keldysh vs FGR

Relation of ψ particles to L-R movers

• Scattering from R L causes backscattering current• Both α and β contribute to the backscattering of R L• In the unitary limit: no scattering between L and R

ψ is free

S D

Right movers

Left movers

For symmetric coupling ψ=(L+R)/√2

Long rigorous derivation …

α β2β1β2 α

t

αβ1 β2

Symmetry breaking effects

1. Left-Right symmetry of the dot2. SU(2) symmetry (magnetic field)3. Particle-hole symmetry

δθ

δh

δr

2222

2

12

52

Kb T

VV

h

eI

12222 hr

Nature Physics 2007

Fermi Liquid vs. Non Fermi Liquid

Fermi Liquid (FL)• Concept of quasi particles.• 1/(Life time) << Typical energy•

•

•

Fermi Liquid vs. Non Fermi Liquid

TT 2/

pF

p

N(p)1

S

D

dI/dVsd

Ө= Max[T,Vsd]

dI/dVsd

Ө=Max[T, Vsd]

Non FL• Superconductors.• Luttinger liquid in 1D.• FQHE liquids.• Low D systems + interaction + disorder.• • Multi-channel

Kondo.const-Ө2

const-Ө1/2

J31

YO and David Goldhaber-Gordon PRL 2003

Lead

-1

Lead-2

Lead-3

333231

232221

131211

JJJ

JJJ

JJJ

000

000

00J

f-dot Large

f

2221

1211

J00

0JJ

0JJ

f

i

J00

0J0

000J22

J31J13=J11J33

J31J13~V3V1V1V3=V1V1V3V3~J11J33

V3

V1ECEC >T

The two channel Kondo Hamiltonian

H=Σ εk i†kαikα + Σ εk f†kαfkα

Jiσi•S+Jfσf •S

Multi-Channel Kondo

RG: (Nozieres – Blandin, Zawadowski). NRG. Bethe Ansatz (Andrei, Wiegmann-Tsvelik) Conformal Field Theory (Affleck - Ludwig) “Pseudo Particle” or “Slave Bosons” (Read - Coleman, Ruckenstein - Cox) Bosonization (Kivelson - Emery)

Spin flavor Majorana Fermion

2CK fixed point and a local Majorana

• Total charge ρi↑+ ρi↓+ ρf↑+ρf↓

• Total spin ρi↑- ρi↓ + ρf↑-ρf↓

• Flavor ρi↑+ ρi↓- ρf↑-ρf↓

• Spin flavor ρi↑- ρi↓- ρf↑+ρf↓

Bosonization a series of rotations and translations Refermionization

iΣΨ†sf ∂Ψ sf + J+[Ψ†

sf(0) + Ψsf(0)][S+-S-]

η+ ia-

iη- a+

η=η†

{η,η†}=1

2-Channel vs. 1-Channel 2-Channel vs. 1-Channel KondoKondo

Specific Heat

Conductance

Spin Susceptibility

Entropy

Two ChannelSingle Channel

Log[2]21

KTTa

KTT

Log[1]0a0TT

TK

K

2B

T)(gμb

2TT

21 Kcc

TT

TT K

KLogα

TT

T)(gμ K

K

2B Log β

KTT

21

2CK

1CK

1CK

Jfinite lead

J infi

nit

e lead

dI/

dV

Θ

J=J

J>J

J<J

Observation of the two channel Kondo effectR. M. Potok, I. G. Rau, Hadas Shtrikman, Yuval Oreg and D. Goldhaber-Gordon

Nature 446, 167 – 171 (2007).

f-dot Large

S

D1μm

c

g)e

2/h

(

dI/

dV

Θ

J>J

J<J

g(0,T)-g(Vsd,T)

Tα

J≈J

Scaling exponent and

Scaling function

A new metallic state A new metallic state of matter was of matter was

observedobserved

What is the effect of magnetic field (Kikoin and YO

PRB submitted)? What is the shot noise (1CK Sela, YO, et al. PRL 2006),

2CK? What happens when the temperature is smaller

than the level spacing in the finite reservoir. Can we have more channels? Can we design novel many body states leading

to a new comprehension of strongly correlated systems?

Summary and Summary and F.A.Q.F.A.Q.

Magnetic field induced two-channel Kondo effect in multiple quantum

dotsKikoin & YO

TK2 vs. TK1

1K1

K2

TT

0

c

2K1

K2 ET

T

K1

c

TE

1

T K2

Ec

δETT cK1K2

A simple realization of MCKYO and David Goldhaber-Gordon )PRL 2003(

2221

1211

JJ

JJ

dN-1

N 21

Pistil:small dot

Petal: large dots or leads

l

• Due to correlations induced by the screening cloud sometimes two electrons are scattered in pairs leading to an “effective charge of 5/3”.

Ask Eran Sela.

Suggestions for 2CK Realizations

• Two level systems, “spins are isotropic channels”

• Coulomb blockade peak is a degenerate state

• Quadruple 2CK (Cox) • Non equilibrium (Wen)• Luttinger leads (Kim)

Theo: Zawadowski, von Delft et al, Exp: Dan Ralph and Burman.

The. Matveev et al. (Requires a large dot, and smooth contacts) Exp. Devoret et al. Hard in real systems.Schiller et al.

cE

ck EeET c /

Average level spacing

Kondo temp.

Charging energy

Lead-Dot Coupling const

~100 Publications/attempts to realize MCK