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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
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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
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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)
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M. Sarachik
et al. 1964
Iron’s Concen. in Mo.8Nb.2
Fe is an example for a 3d transition metals
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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
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TD
LogJν 242
TD
LogνJJcRTcR 32sdimpurity
5phonons
Kondo Solution
1/5sdmin cT
Band Width
Density of states
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The Kondo Problem
Perturbation theory fails at the Kondo temp.
J2=J3Log(D/T) TK=D e-1/J
What happens below TK?
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Developments
• Scaling and Renormalization Group• Exactly solvable points• Coulomb Gases• Numerical Renormalization Group• Bosonization (Schotte and Schotte)• Bethe ansatz solution (Andrei, Wiegman-
Tsvelik )
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Kondo ResonanceConduction spin Impurity spin
J σ•S, J>0
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A universal bound state is formed
between the local spin and the conduction electrons
Densi
ty o
f st
ate
s
Energy
kT
EF
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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
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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
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Origin of Spin-Spin interactionThe Anderson Model s-d
model
•Electron processes
JzσzSz
J-σ+S-
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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
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• Theory: a small quantum dot forms the local impurity state [Ng & Lee, Glazman & Raikh (1988)]
Experiments 1CK in a quantum dot
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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
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1998
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2000
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In quantum dots we can tune and control many parameters
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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
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2004
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Fractional Noise in Fractional Noise in the the
Kondo LimitKondo Limit
Eran Sela, YO, von Oppen, KochPRL 2006
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Nozières FL Hamiltonian
αεπνTK
ε ε ε εβnσ
πν2TK
=(1-e 2iδσ)/(2πiν)=-δσ/πν δσ=
αεTK
βnσ
νTK
Elastic part
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Floating of the Kondo resonance
δn δε
δσ(ε=0, n=0) = δσ(δε, n=νδε)
δσ=αεTK
βnσ
νTKα=β
Keldysh vs FGR
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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 …
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α β2β1β2 α
t
αβ1 β2
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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
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Nature Physics 2007
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Fermi Liquid vs. Non Fermi Liquid
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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
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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
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The two channel Kondo Hamiltonian
H=Σ εk i†kαikα + Σ εk f†kαfkα
Jiσi•S+Jfσf •S
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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
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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
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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
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2CK
1CK
1CK
Jfinite lead
J infi
nit
e lead
dI/
dV
Θ
J=J
J>J
J<J
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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
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g)e
2/h
(
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dI/
dV
Θ
J>J
J<J
g(0,T)-g(Vsd,T)
Tα
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J≈J
Scaling exponent and
Scaling function
A new metallic state A new metallic state of matter was of matter was
observedobserved
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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.
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Magnetic field induced two-channel Kondo effect in multiple quantum
dotsKikoin & YO
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TK2 vs. TK1
1K1
K2
TT
0
c
2K1
K2 ET
T
K1
c
TE
1
T K2
Ec
δETT cK1K2
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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
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• 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.
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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