Transient properties of a quantum dot in the Kondo regime

Peter NordlanderLaboratory for Nanophotonics

Department of Physics and Astronomy, Rice University, Houston TX, USA

In collaboration with D.C. Langreth, M. Plihal, A Goker, and A. Izmaylov

RICELANPLANP

Work supported by NSF, and the Robert A. Welch Foundation

NEQDIS06, Dresden April 2006

Single electron transistor (SET)Tunneling

barrier

IVB

VdVT

VT

Tunnelingbarrier

Quantumdot

Source Drain

VBεF

εF

Vd

VT

The SET provides several means of tunability of a quantum dot in the Kondo regime

Time dependent modulation of Vd, VB, and VT allow the investigation of the nonadiabatic properties of theKondo state

D. Goldhaber-Gordon et Al,Nature 391(1998)156

S.M. Cronenwett et Al,Science 281(1998)540

Outline• Introduction• Nonequilibrium NCA

– Dot level modulation• Response to AC modulation of the dot level for

low bias• Response to a sudden shift of the dot level with

finite constant bias– Response to a sudden shift of bias across

the leads– Response to a sudden shift of the coupling

of the dot to its leads• Conclusions and References

Mechanism (changing Vd)

εF εFεd ∼ εF

High conductanceVd<0

εF εF

Low conductance, unlessa Kondo state is formedVd>0

εd << εF

For a sufficiently small dot, the SET can be switched between conductingand nonconducting by the addition or removal of a single electron

Single electron transistor! Small size, low power consumption ……….

How fast can it be switched?

Linear response (VB=0)Wingreen, Meir 1992

Formula valid in time-dependent case and when many-electron interactions are present. Conductance depend onspectral function around the εF

∫

∂∂

−Γ

=ε

εερ FDdot

dot fdeTG )(2

)(2

h

Spectral function of the dot level.

If many-electrons interactions (No Kondo) are neglected, the device function is trivial

G

N-2

N-1

N

N+1

U

Low TemperatureHigh Temperature

2Γ

Vd

Effect of many-electron interactionsWhen dot contains an odd number of electrons, the interaction with theleads can result in the Kondo effect (spin fluctuations)

εF]exp[DTK

dot

dotF

Γ−

−∝εε

εdot ΓdotD A Kondo resonance is induced

just above the Fermi level forTƒ 10TK

Mixed –valence regimeCharge and spin fluctuations

Empty impurity Kondo regimeSpin fluctuations

Nature of the Kondo state

The slow spin flips are screened byelectron-hole pair excitations.

The Kondo resonance a coherent superposition of electron-hole pairs of excitation energies around kBTK

The Kondo state results when the occupation of a localized degenerate (spin) level is constrained (Coulomb repulsion).

The spin of the dot level can be flipped by the exchange of the dot electron with an electron from the Fermi level.

Gunnarsson & Schonhammer, 1983

The Kondo state is suppressed by thermal excitation of electron-hole pairsThe Kondo state need long time (many spin fluctuations) to form

Transport through a Kondo state

T=0.005 T=0.02 T=0.08(TK=0.0022)

The size and shape of the Kondo resonance depend on T

The Kondo resonance increases the conductance of the SET. For T<TK, the dot becomes transparent!

The enhancement of the conductance depend on T/TKThe Kondo temperature depends on exp[-(εF-εd)/Γ]

Transport through SET in the Kondo regime

N-2

N-1

N

N+1

Vd

T=0.002

T=0.2

εF -εd

Vd

G Low TemperatureHigh Temperature

Equilibrium conductanceas a function of dot level

The Kondo effectenhances the conductance whenodd number of electrons in the dot

odd even

Kondo enhanced conductance

S.M. Cronenwett et Al,Science 281(1998)540 D. Goldhaber-Gordon et Al,

PRL 81(1998)5225

Anderson model

εF,Left εF,Right

]..)([)(21)()(

,,,,,,

,,,,

''',

∑∑

∑∑+++

+=

+

≠

leadk lead

kleadk

leadk

kleadk cHcctVnt

nnUnttH

σσσσ

σσσ

σσσσσσ

σσσ

ε

ε

All parameters are expressed in units of Γ (which is tunable)T is temperature, t denotes time ∑ −=Γ

k leadkF

leadklead tVt )(|)(|2)( ,,

2,,, σσσ εεδπ

The Coulomb parameter is assumed large, εσ+U>>εF+kBT

We assume half-filled elliptic band, D=18

We assume a symmetric dot i.e. Γleft= Γright

Dot level is doubly degenerate (spin)

Total current is the sum of the left and right currents

Most units will be assumed to be one

Solution of time dependent NCA

>=<

>=<+

+

)'()()',(

,)'()()',(

tbtbTttiB

tctcTttiG

C

C

-∞ +∞

t t’

C∑ =+=→→ +++++++

σσσσσσσσσσσ 1,, bbccQbccccbcccc kkkkAuxiliary boson

)',(),()'()',()( ttGtttdttttGtt

iC∫ Σ+−=

−∂∂

σσσσ δεG and B satisfies Dyson Eqs.

∫∞

∞−

<<< Σ+Σ=

−∂∂ )',(),()',(),()',()( ttGttttGtttdttGtt

i ARσσσσσσεProjected upon the real axis

The coupled Dyson’s equation are solved numerically on a discrete grid

Instantaneous spectral function and current are obtained from G and B

Method can be used for arbitrary εd (t), Γ (t) and VB(t)

AC modulation of the dot level (VB=0)

T=0.005TK=0.001

ε

Equilibrium SPF and Conductance as a function of εd

εdot-εF

tosΩ+= Cεε(t)ε acdotd

εd(t)=-5+4cosΩtTK=10-6

Dot A

G(Ω→∞) smallfor finite T

Since the dot level oscillates in time, the Kondo state is not fully formed.The conductance depends on the average spectral function over a period.In the limit of Ω→∞, the conductance is given by G(T,εdot)

εd(t)=-2.5+2cosΩt

TK=10-3

Dot B

G(Ω→∞) large for finite T

Conductance as a function of Ω

Averaged spectral function Dot A

Satellites develop at ≤NΩ around the Kondopeak causing a logarithmic fall-off of G

Satellites develop at ≤NΩ around εdot

When εdot+NΩ get close to εF, the Kondo effectincrease the conductance causing the nonmonotonic G(Ω)

N=2 N=1

Photon assisted tunneling, PAT

The same calculation neglecting many-electron effects would have G(Ω)= almost constant

Application of a sudden voltage to the dot

t<0 t=infinity

TK <<T

TK ~T

Vd

The sudden bias moves the dot level from the non-Kondo regime (TK<<T) to a state in the Kondo regime (TK~T)

The instantaneous conductance of the dot reflects the timescalesfor formation of the Kondo state

How does the spectral function of the dot level evolve in time?

Sudden dot level change response (VB=0)Instantaneous spectral functionTK=0.0022, T=0.0025

Fast time scale τ0~1/Γ

Slow time scaleτ~1/TK(T)

Thermal fluctuations reduce the Kondo effect

The Kondo resonance takes a long time to form, τ∼1/TK

For times t smaller than τ, the state can be viewed as adecohered Kondo state

Equivalency between finite time and equilibriumthermal decoherence

h2TtπTcoth(t)Teff =

Long time limit, UniversalitySystem 1, S1: εdot=-2.0, TK=0.0022System 2, S2: εdot=-2.225, TK=0.0011

)/(ln1632

2

0 KTTGG π

= Abrikosov, 1965

Both the transient and final conductancesatisfy universality, i.e. scaling with TK

Deviations from universality becauseof admixture of mixed valent state

Timescale for VB=0

2TtπTcoth(t)Teff =The approach to equilibrium follows

( )πTt]exp[

dlnTdlnG2

(T)GdTdGT(t)T

(T)G(t))(TG(T)G

)G(G(t))G( Eq.

tEq.

Eq.

eff

tEq.eff

Eq.Eq.

−→−

→−

=∞−∞

∞→∞→

No TK !

The rate constant for the approach to equilibrium is 1/τ = πT

Korringa rate: A spin flip requires the creation of a zero energy e-h pair. Phase space restriction=> ∫ =− Tffd )](1)[( εεε

Solid line: πTDot •: S1Ring o: S2

Deviation from simple formula for T<TK

1/τ→πTK and possibly at large T (lnT/TK)

Reshaping of spectral functions continuebeyond the time where the conductance has saturated for all T studied suggestingthe presence of a longer timescale

Experimentally accessible quantity is the integrated currentInstantaneous current during a pulse

T=0.0025TK=0.0022

τon

G(t,τ o

n)

∫+

=offon

onon tGdtGττ

ττ0

int );()(

τon

on

on

ddG

ττ )(int

The derivative of Gint(τon) with respect to τon is a measure of the Time scale for formation of theKondo state

Sudden change of VB (εdot(t) =const)

VB(t)TK =0.0025(VB=0)

VB (t)=VB[Θ(t)-Θ(t-τ)]

VB=0.2

A finite bias introduce a split Kondo resonanceand extra broadening of the two Kondo resonances

Instantaneous current I(t;τ) Solid line: I(t,τ). Fast rise time of order τ1 ~2π/VB (for VB>TK)

Dotted line: I(τ,τ)-I(t+τ,τ). Slower decay of order τ3. The timescaleof the order 1/TK and 1/TK(VB)

Dashed line conductance G(t) for VB=0.Rise time of order 1/TK(VB=0)t

Time scales for sudden switching of VB

τ0

τ1

τ3

τ2

∫+

=offon

onon tIdtQττ

ττ0

int );()(

on

on

ddQ

ττ )(int

Split Kondo Peak (SKP) oscillations!Quantum beating between the two Kondo resonances.Oscillation frequency ω=2π/VB

The damping of SKP oscillations occur on a “new” longer time scale τ2

Measurement of dQint/dτprovides estimates ofτ1 and τ2

The integrated current

Sudden dot level, Finite constant VBEquilibrium spectral function as a function of VB and T

VB T=0.0015

TK <<T

TK ~T VB=0-0.08 T=0.001-0.2

VB=0.04

VB=0.2

Instantaneous spectral function

t<0

t =∞

t=∞t=179t=83t=55t=41t=27

The magnitude and shape of thesplit Kondo resonance depend onVB and T.

The timescale for formation willalso depend on VB and T

T),(VT1τB

EffK

=

Instantaneous conductance (I(t)/VB)

T=0.0015 T=0.005

SKP oscillations are small but remain for longer time τ2

The rise time and saturationConductance depend strongly on VB for VB>4T

Nonuniversal contributions (ringing) of the conductance on timescale Γ

Instantaneous conductance as a function of T

T=0.00075T=0.0015T=0.003

T=0.005

T=0.015

T=0.03

VB=0.01 VB=0.04

The rise time depend strongly on T for T>0.25VB

The magnitude and lifetime of the SKP oscillations decrease with increasing T

Extraction of the rise time

4Vπ

V)(T,1

τ =

Rosch et Al, 2001

Finite T consistent with T=0

)TVπTF(

V)(T,1

)TVTF()](f-)[1(fdε

41

τ

l'll'l,

=⇒

=∫∑ εε

Solid lines: 1/τ(T,V)

For large V

4VT⇔

Long time limit, GEq.(T,VB)

Rise time definedas time τ when G(τ)=0.99GEq.(T,VB)

Korringa rate for finite bias. Thephase space is increased to:

Excellent agreement with numerical simulations for T>TK

SKP oscillations (S1)T=0.00075 G(t,VB)-G(∞)G(t,VB)

VB=0.04

VB=0.06

VB=0.08

VB=0.11

VB=0.15

Damping of SKP oscillationsoccur on a much slower timescale than the rise time.

Damping depend on VB

VB=0.04G(t,T) G(t,T)-G(∞)

T=0.00075

T=0.0015

T=0.003

T=0.005T=0.015

Damping depend on T

Instantaneous spectral functiont∈ [42,672]t∈ [0,42]

Conductance rises and saturates. Split Kondopeak not fully formed.

Conductance is crudemeasure of equilibration

SKP oscillations. Formation of split Kondo Resonances

Conductance only weaklyTime dependent

SKP damping rateCalculated SKP damping rates τ2 (divided by 2!?)

Our lowest temperature result agrees wellwith Rosch, Kroha and Wölfe, 2001

Lower temperature simulations may be needed to get rid of the factor of 2

Our TDNCA approach is presently beingextended to low T regime

The integrated current Measurement of dQint/dτprovides estimates ofrise time τ1 and SKP damping rate τ2

∫+

=offon

onon tIdtQττ

ττ0

int );()(

on

on

ddQ

ττ )(int

Sudden change of tunnel barrier (Γ change)

t<0 t=infinity

εdot=const

Γ=Γ1 +Θ(t)(Γ2 -Γ1)

TT1K <<

TT2K ≈

The sudden increase in Γ moves the dot from a non-Kondo regime (TK<<T) to the Kondo regime (TK~T)

The instantaneous conductance of the dot reflects the timescalesfor formation of the Kondo state

Instantaneous conductance (Γ change)

S1: T=0.02TK=0.0025

Short timescale

Red: Dot level changeBlack: Gamma switching

Approach to equilibrium on the same timescale as for sudden shift of the dot level

ConclusionsThe application of time dependent voltages on the gates of an SET in the Kondo regime provides novel insight into the Kondo problem

•AC modulation of the dot level•Nonmonotonic dependence of G on Ω•G determined by interplay between pat and Kondo effect

• Pulsed modulation of the dot level•Kondo resonance is formed on a timescale τ=1/TK

•For finite bias τ=1/T*(T,V)•SKP oscillations

•Pulsed bias across the leads•Very large amplitude SKP oscillations•Transient currents influenced by many different timescales

•Sudden change of the tunneling barrier•Timescale for formation of Kondo state same as for dot level shift

ReferencesPN, M. Pustulnik, Y. Meir, N.S. Wingreen, D.C. Langreth, PRL83(1999)808

PN, N.S. Wingreen, Y. Meir, D.C. Langreth, PRB 61(2000)2146M. Plihal, D.C. Langreth, PN, PRB 61(2000)R13341 M. Plihal, D.C. Langreth, PN, PRB 71(2005)165321

A. Izmaylov, A. Goker, B. Friedman, PN, to be published