Nonequilibrium Singlet/Triplet Kondo
Effect in Carbon Nanotubes
Jens Paaske
The Niels Bohr Institute
&
Nano-Science Center Copenhagen
Collaborators:
• P. Wölfle (Karlsruhe)
• A. Rosch (Cologne)
•N. Mason (Urbana)
•C. Marcus (Harvard)
•J. Nygård (Copenhagen)
NDIS-Workshop, Dresden, April 2006
Outline
• Experimental motivation
• Basics of the zero-bias Kondo-effect
• Singlet-Triplet nonequilibrium Kondo-effect in a carbon nanotube
- Application of poor man’s scaling with frequency dependent couplings
• More detail on decoherence-effects
• Sketches of previous applications of scaling with coupling-functions
• Application to electromechanical exchange-tunneling in molecular transistor
Zero-bias S=1/2 Kondo-effect in quantum dots
Enhanced transmission at low
temperatures (T<TK):
2
G(V = 0, T = 0) = 2 eh
Scaling with TK =
1
2
√
4ΓL ΓR
(ΓL +ΓR )2
ΓU eπε0 (ε0 +U )/(ΓU ) :
G(V = 0, T ) = G(V = 0, T = 0)g(T/TK )
Basic mechanism of the Kondo-effect:
Kondo-regime:
max(−ε0 , U + ε0 ) Γ ≡ πνF |t|2
Lateral GaAs/AlGaAs quantum-dot
”...
”
Carbon nanotube quantum-dot
...unpublished data by J. Nygård et al. (SWCNT with N=2):
Bias-spectroscopy of Single-Wall Carbon Nanotube
(Data by J. Nygård et al.)
Shell-filling (J < δ)
Energy scales:
• Level spacing
• Charging energy
• Orbital splitting
∆ ∼3 meV
EC ∼ ∆
δ ∼ 0.5 ∆
• Exchange energy J < 0.05 ∆
S=1/2
S=0
S=1/2
Lineshape
Singlet-Triplet Kondo-effect at finite bias?
Zeeman splitting
of the triplet:
Exchange-tunneling at finite bias
(Kondo-regime)
EC πνF
α=L,R
i=1,2
|tα,i |2
Relevant two-electron states and spin-operators
(J < δ )
Singlet: Es=0
Triplet: ET=δ-J
Singlet: Es’=δ
Singlet: Es’’=2δ
Effective low-energy Kondo model
(via Schrieffer-Wolff transformation)
H=
k,σ
i=1,2
α=L,R
H int
= νF−1
0
(εk − µα )c†αikσ cαikσ + Hdot
+ H int .
{
k,k ,σ,σ i,j=1,2
α,α =L,R
gij
α α
+ τij3 T + τij1 P
ij · τσ σ
δij S
1 +
τij |s
s | + τij− |s s| δσ σ
+pij
δ
|s
s|
+
ij
αα
2
ij
+qα α δij
|m
m| δσ σ }c†α ik σ cαjkσ
m=−1,0,1,s
Initial couplings:
∗
g ii
α α = νF tiα tiα /EC
√
12
21 ∗
g α α = g α α = 2νF tiα t∗iα /EC
3 ii
pii
α α = τii gα α
21 ∗
12
p12
=
p
=
g
αα
αα
α α
q ij
α α = 0
Poor man’s scaling with frequency dependent couplings*
Perturbative corrections to exchange interaction at conduction
electron bandwidth D and incoming energy ω:
( ”Spin-propagator” )
δ
ω
=
+
ω
( Conduction electron )
ω
Demand invariance of physical observables under infinitesimal change in D:
ii (ω)
∂gα
α
∂ ln D
=
−
α
ii
gαii α (α V /2)gα
α (α V /2)θ(D − |ω − α V /2|)
1 ii
ii
+ gα
α (α V /2)gα α (α V /2 − J )θ(D − |ω + J − α V /2|)
2
1 ii
ii
+ gα
α (α V /2)gα α (α V /2 + J )θ(D − |ω − J − α V /2|)
2
1 iī
īi
3
3
+ gα α (α V /2)gα α(α V /2 + τii (J − δ))θ(D − |ω + τii (δ − J) − α V /2|)
2
*)
Cf. A. Rosch, J. Paaske, J. Kroha and P. Wölfle, Phys. Rev. Lett. 90, 76804 (2003).
Coupled RG-equations for J=B=0:
∂gαii α (ω)
∂ ln D
= −
4gαii α (α V /2)gαii α (α V /2)θ(D − |ω − α V/2|)
α +gαiī α (α V /2)gαīi α (α V /2
∂gαiī α (ω)
∂ ln D
−
τii3 δ)θ(D
− |ω +
τii3 δ
(1)
− α V /2|) ,
iī
3
ii
ii
= −
3gα α (α V /2 + τii δ) + pα α (α V /2) − qα α (α V /2)
(2)
α ×gαiī α (α V/2 + τii3 δ)θ(D − |ω − τii3 δ − α V /2|)
iī
īī
īī
īī
+gα α (α V /2) 3gα α (α V /2) + qα α (α V /2) − pα α (α V /2)
× θ(D − |ω − α V /2|)} ,
∂pii
α α (ω)
∂ ln D
= −4τii3
α gαiī α (α V /2)gαīi α (α V/2 − τii3 δ)θ(D − |ω + τii3 δ − α V /2|),
(3)
∂pαiī α (ω)
∂ ln D
=
∂gαiī α (ω)
,
∂ ln D
(4)
∂qαii α (ω)
∂ ln D
1 ∂pii
α α (ω)
= −
.
4 ∂ ln D
(5)
Perturbative renormalization-group flow
1. Determine the flow of coupling-constants with bandwidth D.
(Solve coupled Diff. Eqs.)
2. Solve for renormalized coupling-functions.
(Integrate RG-equations)
3. Calculate broadening of excited states (spin-relaxation*).
(2nd order PT with renormalized coupling-functions)
4. Recalculate coupling-functions using broadened step-functions:
(Implemented as θ(D −
ω 2 + Γ(V, δ, T )2 ) )
Approximations valid
for ln(δ/TK) >> 1
*)
Cf. J. Paaske, A. Rosch, J. Kroha and P. Wölfle,
Phys. Rev. B 70, 155301 (2004).
Stages of the RG-flow as D → 0
Early stage of RG (D>V):
Late stage of RG (D<V):
Reducing D
L-R tunneling
only possible
with V ~ δ
D=0
D=0.00105 D0
D=0.0011 D0
D=0.0012 D0
D=δ=0.0015 D0
D=0.002 D0
D=0.004 D0
T=0
V=0.001 D0
ω/δ
Renormalized conductance
1. Solve stationary quantum Boltzmann equation
for nonequilibrium occupations {nT, nS’, nS}:
Wγγ nγ =
γ
W γ γ nγ
with constraint
γ
2π
Wγ γ (V, δ, T ) =
∞
dω
−∞
nγ = 1
γ=s,−1,0,1,s
2
|gαγ ;γ
(ω)|
f (ω−µα )(1−f (ω+εγ −εγ −µα ))
,σ ;α,σ
σ,σ =↑,↓
α,α =L,R
Spin-relaxation rate:
2. Calculate current from 2nd order PT:
2πe
I=
∞
dω
and
Γss = Γs,s
+
v
1 (
Wγ ,s +
Wγ ,s )
2 γ =s
γ ;γ
|gR,σ ;L,σ (ω)|2 f (ω−µR )(1−f (ω+εγ −εγ −µL ))nγ
−∞ σ,σ =↑,↓
γ,γ =s,−1,0,1,s
γ =s
− (L → R)
Comparing to experiment
δ
687 mK
δ
tL2 = 0.030
tR2 = 0.077
tL1 = 0.023
tR1 = 0.105
Peak-height
a(1 + (21/0.22 − 1)(T /TK )2 )−0.22 , a = 0.11, TK = 1.0 K
(S=1/2)
π2
b( 2 )/ ln2 [
dI/dV(V = δ)
(S=1)
T 2 + (cΓ(V = δ))2 /TK ], b = 0.68, c = 1.18
0.16
−1
−1 −1
−1/(2(g11 +g22 )−0.79(g11
+g22
) )
TK ≈ D0 e
0.15
0.14
Γ ≈ 340 mK
0.13
≈ 4 mK
0.12
100
150
200
300
T/mK
500
700
Saturation at spin-relaxation-rate Γ >> TK
*Steady-state nonequilibrium decoherence for ln(V /TK ) 1
Nonequilibrium PT for single S=1/2 dot:
*)
Cf. J. Paaske, A. Rosch, J. Kroha and P. Wölfle,
Phys. Rev. B 70, 155301 (2004).
(To order Im[ΣR ]/V ∼ g 2 for V T )
1. Vertex corrections (Keldysh)
2. Check: dynamical susceptibility for finite B
3. T-matrix with self-energy- and vertex-corrections
1. Calculate renormalized Kelydysh-matrix-vertex
in terms of:
- ph-propagator
- self-energy-broadening
- noneq. spin-distribution-function
>
ab
>
2iΠ
(0)
Π
(0)
2iΠ
(0)
1
1
4Γ
s
cd
1
1
3
3
Λ̃ab
(Ω) = τcd
δab + δcd τab
+ τaa
−2nλ (0)
+
τ
ab
2
2
Ω + iΓs
Ω 2 + Γs 2
Ω + iΓs
2
Γs = 12 (Γ↑ + Γ↓ ) + 2Π> (0) = πgLR
V
2. Calculate transverse dynamical spin-susceptibility with
broadened spin-levels and renormalized emission-vertex:
⊥ R
χ (t) = iθ(t)
[S − (t), S + (0)]
(max(|Ω + B|, Γ) V )
( ”Spin-relaxation rate” )
M B + iΓs
χ (Ω) ≈
B Ω + B + iΓs
⊥ R
2
2
2
(gLL
+ gRR
+ 2gLR
)B
M=
2 V
2gLR
and
3. Calculate leading logarithm (Kondo)
in simplest observable:
T-matrix (local DOS)
π
Im TR
(Ω)
= −
αα
16νF
2
D
z
gαα zgα α + gα α gα α ln
(Ω − µα )2 + Γ1 2
α ,α ∈{L,R}
z
D2
⊥
⊥
⊥
⊥
z
+ gαα 2 gα α + gα α gα α + gα α gα α ln
(Ω − µα )2 + Γ2 2
⊥
⊥
Log-singularities at Ω = µα are cut off by spin-relaxation rates:
2
Γ1 = π ⊥gLR
V
& Γ2 =
π z 2
2 ( gLR
2
+ ⊥gLR
)V
Previous applications of frequencydependent Poor Man’s Scaling
• S=1/2 Kondo-effect: Conductance at V, B >0
(Comparing to experiment by Ralph & Buhrman; Motivation for the RG)
• S=1/2 Kondo-effect: Spectral-function at B>V=0
(Equilibrium: comparing to Numerical RG calculation)
• Vibrational sidebands and S=1/2 Kondo-effect: Spectral-function at V=B=0
(Relevant for flexible single-molecule transistors)
• S=1/2 Kondo-effect: Conductance at V, B >0 *
(Comparing to experiment by Ralph & Buhrman ’96)
Better match for
ln(B/TK ) 1
Notice that FDRG reproduces all details of 3. order nonequilibrium PT
when expanding in initial-couplings. (no trivial requirement!)
*)
Cf. A. Rosch, J. Paaske, J. Kroha and P. Wölfle, Phys. Rev. Lett. 90, 76804 (2003).
3. order PT:
•S=1/2 Kondo-effect: Spectral-function at B>V=0
(Equilibrium: comparing to Numerical RG calculation)
g̃zσ (ω)
=
g̃⊥ (ω)
=
*)
B 1/ ln[B/TK ]
1
1
Θ[|ω + σB| − B]
+ Θ[B − |ω + σB|]
2 ln[|ω + σB|/TK ]
2 ln[B/TK ] ω + σB 

1
2 ln[B/TK ]
B
1
B 
1
1
B 
Θ[|ω + σ | − B]
+
Θ[B
−
|ω
+
σ
|]
−
B
B 2
2
2
ln[B/T
]
4
ln[B/T
]
4
ln[|ω
+
σ
|/T
]
ω
+
σ
K
K
K
2
2
σ
Cf. A. Rosch, T. Costi, J. Paaske and P. Wölfle, Phys. Rev. B 68, 14430 (2003).
*Vibrational sidebands and Kondo-effect in single-molecule transistor
Cotunneling with electromechanical coupling:
•
Cotunneling involves virtual oscillator
displacements, and may leave the oscillator
in an excited state.
•
Amplitude of depicted process:
with Franck-Condon factors like
*)
Cf. J. Paaske and K. Flensberg, Phys. Rev. Lett. 94, 176801 (2005).
Anderson-Holstein
Kondo model
(Lang & Firsov, 1963)
(Schüttler & Fedro, 1988)
With number-state matrix-elements of the exchange-operator :
Differential conductance
•
Tunneling current:
•
Local DOS:
With T-matrix:
(Γ
ΓL>> ΓR)
Parity selection rule:
All sidepeaks at odd
multiples of
must
vanish at ph-sym. point.
Perturbative RG-eqs. for ln(ω0 /TK ) 1
(Spin-operator)
…………..
…………..
ω
ω
=
…………..
+
…………..
(Displacement-operator)
• Tuning
to the ph-symmetric point.
• Assume weak coupling:
Restrict to
ω
RG-solution: renormalized coupling-functions
• Scaling-invariants from initial cut-off
down to
:
,
,
• Intrinsic energy-scale:
[
• Broadening
with vibron-relaxation rate (Golden rule
]
):
Renormalized conductance
Summary
• Evidence for nonequilibrium Kondo-correlations in SWCNT
- Good agreement for dI/dV vs. V with frequency-dependent poor man’s scaling
- Nonequilibrium effects alone do not suffice to explain the data
- Observed low-T peak saturation consistent with calculated spin-relaxation rate
• Established the spin-relaxation rates as cut-off ’s in leading log for T-matrix
- Calculation of Keldysh-matrix vertex-corrections to order g2
- Identification of spin-relaxation rates in the nonequilibrium magnetic susceptibility
- S=1/2 Kondo-effect remains at weak-coupling for V>>TK
• Perturbative demonstration of vibrational sidebands to the S=1/2 Kondo-effect
- Faint sidebands may coexist with zero-bias Kondo-peak
- Zero-bias Kondo-peak is not attenuated, -TK is slightly enhanced
- Parity selection rule prevents odd sidebands at ph-symmetric point
FIN
Number-state matrix-elements
•
Energies of intermediate empty, or
doubly occupied states:
•
Franck-Condon factors:
•
for
•
•
Enhanced Kondo-temperature:
Asymptotic power-series
as
:
Parity selection-rule
• General property:
•
at the particle-hole symmetric point
Invariance under joint
ph-transformation and
inversion of oscillator
implies parity conservation
in low-energy transitions.