quantum transport at nano-scale

Quantum transport at nano-scale

Chung-Hou Chung 仲崇厚 Electrophysics Dept.

National Chiao-Tung University

Hsin-Chu, Taiwan

Collaborators: Matthias Vojta (Koeln), Gergely Zarand (Budapest), Walter Hofstetter (Frankfurt U.)

Pascal Simon (CNRS, Grenoble), Lars Fritz (Harvard),

Marijana Kircan (Max Planck, Stuttgart),

Matthew Glossop (Rice U.) , Kevin Ingersent (U. Florida)

Peter Woelfle (Karlsruhe), Karyn Le Hur (Yale U.)

Zarand, Chung, Simon, Vojta, PRL 97 166802 (2006)Chung, Hofstetter, PRB 76 045329 (2007), selected by Virtual Journal of Nanoscience and Technology Aug. 6 2007Chung, Zarand, Woelfle, PRB 77, 035120 (2008), selected by Virtual Journal of Nanoscience and Technology Jan. 8 2008Chung, Glossop, Fritz, Kircan, Ingersent,Vojta, PRB 76, 235103 (2007)Chung, Le Hur, Vojta, Woelfle nonequilibrium transport near the quantum phase transition (arXiv:0811.1230)

• Introduction

• Quantum criticality in a double-quantum-dot system

• Quantum phase transition in a dissipative quantum dot

• Nonequilibrium transport in a noisy quantum dot

• Conclusions

Outline

Quantum dot---A single-Electron-Transistor (SET)

Single quantum dot

Goldhaber-Gorden et al. nature 391 156 (1998)

Coulomb blockade d+U

d

Vg

VSD

Coulomb Blockade

Goldhaber-Gorden et al. nature 391 156 (1998)

Quantum dot---charge quantization

Kondo effect

Kondo effect in quantum dot

even

odd

conductance anomalies

L.Kouwenhoven et al. science 289, 2105 (2000)

Glazman et al. Physics world 2001


d

Vg

VSD

Kondo effect in metals with magnetic impurities

For T<Tk (Kondo Temperature), spin-flip scattering off impurities enhances

Ground state is

Resistance increases as T is lowered

electron-impurity spin-flip scattering

logT

(Kondo, 1964)

(Glazman et al. Physics world 2001)


(J. von Delft)


Anderson Model

local energy level :

charging energy :

level width :

All tunable!

Γ= 2πV 2ρd

U

d ∝ Vg

New energy scale: Tk ≈ Dexp-U )

For T < Tk :

Impurity spin is screened (Kondo screening)

Spin-singlet ground state

Local density of states developes Kondo resonance

Spectral density at T=0

Kondo Resonance of a single quantum dot

phase shift

Fredel sum rule

particle-hole symmetry

Universal scaling of T/Tk

L. Kouwenhoven et al. science 2000M. Sindel

P-H symmetry

/2

Numerical Renormalization Group (NRG)

Non-perturbative numerical method by Wilson to treat quantum impurity problem

Anderson impurity model is mapped onto a linear chain of fermions

Logarithmic discretization of the conduction band

Iteratively diagonalize the chain and keep low energy levels

K.G. Wilson, Rev. Mod. Phys. 47, 773 (1975)

W. Hofstetter, Advances in solid state physics 41, 27 (2001)

Perturbative Renormalization Group (RG) approach: Anderson's poor man scaling and Tk

HAnderson

•Reducing bandwidth by integrating out high energy modes

•Obtaining equivalent model with effective couplings

•Scaling equation

< Tk, J diverges, Kondo screening

J J

J J

J

Anderson 1964

Quantum phase transitions

c

T

gg

Non-analyticity in ground state properties as a function of some control parameter g

True level crossing: Usually a first-order transition Avoided level crossing which becomes sharp in the infinite volume limit: Second-order transition

• Critical point is a novel state of matter

• Critical excitations control dynamics in the wide quantum-critical region at non-zero temperatures

• Quantum critical region exhibits universal power-law behaviors

Sachdev, quantum phase transitions,

Cambridge Univ. press, 1999

I.

Quantum phase transition in coupled double-quantum-dot system

Recent experiments on coupled quantum dots

• Two quantum dots coupled through an open conducting region which mediates an antiferromagnetic spin-spin coupling

• For odd number of electrons on both dots, splitting of zero bias Kondo resonance is observed for strong spin exchange coupling.

(I). C.M. Macrus et al. Science, 304, 565 (2004)

•A quantum dot coupled to magnetic impurities in the leads

• Antiferromagnetic spin coupling between impurity and dot suppresses Kondo effect

•Kondo peak restored at finite temperatures and magnetic fields

Von der Zant et al. (PRL, 2005)

Quantum phase transition in coupled double-quantum-dot system

L1

L2 R2

R1

C.H. C and W. Hofstetter, PRB 76 045329 (2007)

G. Zarand, C.H. C, P. Simon, M. Vojta, PRL, 97, 166802 (2006)

Non-fermi liquid

KcK

T

Spin-singletKondo

• Critical point is a novel state of matter

• Critical excitations control dynamics in the wide quantum-critical region at non-zero temperatures

• Quantum critical region exhibits universal power-law behaviors

Affleck et al. PRB 52, 9528 (1995) Jones and Varma, PRL 58, 843 (1989)

Sakai et al. J. Phys. Soc. Japan 61, 7, 2333 (1992); ibdb. 61, 7, 2348 (1992)

R/2-R/2

X

H0 =

Himp

Heavy fermions

2-impurity Kondo problem

Quantum criticality of 2-impurity Kondo problem

Affleck et al. PRB 52, 9528 (1995)

Jones and Varma, PRL 58, 843 (1989)

Jones and Varma, PRB 40, 324 (1989)

Kc = 2.2 Tk

Jump of phase shift at Kc K < Kc, = /2 ; K >KC ,

Quantum phase transition as K is tuned

• Particle-hole symmetry V=0H H’ = H under

Non-fermi liquid

KcK

T

Spin-singletKondo1 2

even

odd

• Particle-hole asymmetry Kc is smeared out, crossover

Misleading common belief ! We have corrected it!

Quantum Phase Transition in Double Quantum dots: P-H Symmetry

• Two quantum dots (1 and 2) couple to two-channel leads

• Antiferrimagnetic exchange interaction K, Magnetic field B

• 2-channel Kondo physics, complete Kondo screening for B = K = 0

L1

L2

R1

R2

Izumida and Sakai PRL 87, 216803 (2001)

Vavilov and Glazman PRL 94, 086805 (2005)

Simon et al. cond-mat/0404540

triplet states

Hofstetter and Schoeller, PRL 88, 061803 (2002) singlet state

K

K

Transport properties

• Transmission coefficient:

• Current through the quantum dots:

• Linear conductance:

JC

NRG Flow of the lowest energy Phase shift

0

KKc

K<KC

K>KC

Two stable fixed points (Kondo and spin-singlet phases )

One unstable fixed point (critical fixed point) Kc, controlling the quantum phase transition

Jump of phase shift in both channels at Kc

Kondo

Spin-singlet

Kondo

Spin-singlet

Crossover energy scale T* k-kc

• J < Jc, transport properties reach unitary limit:

T( = 0) 2, G(T = 0) 2G0 where G0 = 2e2/h.

• J > Jc spins of two dots form singlet ground state,

T( = 0) 0, G(T = 0) 0; and Kondo peak splits up.

• Quantum phase transition between Kondo (small J) and spin singlet (large J) phase.

Quantum phase transition of a double-quantum-dot system

J=RKKY=KChung, Hofstetter, PRB 76 045329 (2007)

NRG Result Experiment by von der Zant et al.

Restoring of Kondo resonance in coupled quantum dotsSinglet-triplet crossover at finite temperatures T

• At T= 0, Kondo peak splits up due to large J.

• Low energy spectral density increases as temperature increases

• Kondo resonance reappears when T is of order of J

• Kondo peak decreases again when T is increased further.

T=0.003

T=0.004

Singlet-triplet crossover at finite magnetic fields

J=0.007, Jc=0.005, Tk=0.0025, T=0.00001, in step of 400 B

J close to Jc, smooth crossover

Antiferromagnetic J>0 Ferromagnetic J<0

J >> Jc, sharper crossover

B in Step of 0.001

J=-0.005, Tk=0.0025

EXP: P-h asymmetry

NRG: P-h symmetry

Quantum criticality in a double-quantum –dot system: P-H Asymmetry

V1 ,V2 break P-H sym and parity sym. QCP still survives as long as no direct hoping t=0

Non-fermi liquid

KcK

T

Spin-singletKondo

G. Zarand, C.H. C, P. Simon, M. Vojta, PRL, 97, 166802 (2006)

even 1 (L1+R1) even 2 (L2+R2)K

_

Quantum criticality in a double-quantum –dot system

K

_

No direct hoping, t = 0 Asymmetric limit: T1=Tk1, T2= Tk2

2 channel Kondo System

QC state in DQDs identical to 2CKondo state

Particle-hole and parity symmetry are not required

Critical point is destroyed by

charge transfer btw channel 1 and 2

Goldhaber-Gordon et. al. PRL 90 136602 (2003)

QCP occurs when

Transport of double-quantum-dot near QCP (only K, no t term)

At K=Kc

Affleck and Ludwig PRB 48 7279 (1993)

NRG on DQDs without t, P-H and parity symmetry

KK

The only relevant operator at QCP: direct hoping term t

charge transfer between two channels of the leads

dim[

(wr.t.QCP)

Relevant operator

Generate smooth crossover at energy scale

RG

most dangerous operators: off-diagonal J12

At scale Tk, typical quantum dot

may spoil the observation of QCP

How to suppress hoping effect and observe QCP in double-QDs

assume

effective spin coupling between 1 and 2

off-diagonal Kondo coupling

more likely to observe QCP of DQDs in experiments

The 2CK fixed point observed in recent Exp. by Goldhaber-Gorden et al. Goldhaber-Gorden et al, Nature 446, 167 ( 2007)

At the 2CK fixed point,

Conductance g(Vds) scales as

The single quantum dot can get Kondo screened via 2 different channels:

At low temperatures, blue channel finite conductance; red channel zero conductance

• Two coupled quantum dots, only dot 1 couples to single-channel leads

• Antiferrimagnetic exchange interaction J

• 1-channel Kondo physics, dot 2 is Kondo screened for any J > 0.

• Kosterlitz-Thouless transition, Jc = 0

Side-coupled double quantum dots

1 2

V JevenChung, Zarand, Woelfle, PRB 77, 035120 (2008),

2 stage Kondo effect

1st stage Kondo screening

Jk: Kondo coupling

D Tk dip in DOS of dot 1

2nd stage Kondo screening

Jk 4V2/U

J: AF coupling btw dot 1 and 2

c 1/

Kosterlitz-Thouless quantum transition

NRG:Spectral density of Model (II)

80J

Kondo spin-singlet

No 3rd unstable fixed point corresponding to the critical point

Crossover energy scale T* exponentially depends on |J-Jc|

U=1

d=-0.5

=0.1

Tk=0.006

Log (T*)

1/J

Dip in DOS of dot 1: Perturbation theory

self-energy

vertex

sum over leading logarithmic corrections

n< Tk

12

when Dip in DOS of dot 1

d1

J = 0

J > 0 but weak

Dip in DOS: perturbation theory

• Excellence agreement between Perturbation theory (PT) and NRG for T* << << Tk

U=1, d=-0.5, J=0.0005, Tk=0.006, T*=8.2x10-10

• PT breaks down for T*

• Deviation at larger > O(Tk) due to interaction U

Summary I

• Coupled quantum dots in Kondo regime exhibit quantum phase transition

• correct common misleading belief: The QCP is robust against particle-hole and parity asymmetries

•The QCP is destroyed by charge transfer between two channels

• The effect of charge transfer can be reduced by inserting additional even number of dots, making it possible to be observe QCP in experiments

quantum critical point

x

JcJ

Kondo spin-singlet8 T* J-Jc

L2

L1 R1

R2

K-T transition

8 J

Kondo spin-singlet

• The QCP of DQDs is identical to that of a 2-channel Kondo system

II.

Quantum phase transition in a dissipative quantum dot


d

Vg

VSD

Quantum dot as charge qubit--quantum two-level system

charge qubit-

Quantum dot as artificial spin S=1/2 system

Quantum 2-level system

Dissipation driven quantum phase transition in a noisy quantum dot

Noise ~ SHO of LC transmission line

Noise = charge fluctuation of gate voltage Vg

Caldeira-Leggett Model

K. Le Hur et al, PRL 2004, 2005, PRB (2005),

Impedence

H = Hc + Ht + HHO

N=1/2Q=0 and Q=1 degenerate

Spin Boson model

K. Le Hur et al, PRL 2004,

Delocalized-Localized transition

h ~ N -1/2

/

delocalized localized

Charge Kondo effect in a quantum dot with Ohmic dissipation

Jz = -1/2 R

Kosterlitz-Thouless transition

localized

de-localized

g=J

Hdissipative dot

non-interacting lead

N=1/2Q=0 and Q=1degenerate

Anisotropic Kondo model

Generalized dissipative boson bath (sub-ohmic noise)

{Ohmic

Sub-Ohmic

Generalized fermionic leads: Power-law DOS

Anderson model

Fradkin et al. PRL 1990

d-wave superconductors and graphene: r =1

0JX

Jc

Local moment (LM)Kondo

Quantum phase transition in the pseudogap Anderson/Kondo model

Delocalized-Localized transition in Pseudogap Fermi-Bose Anderson model

Pseudogap Fermionic bath Sub-ohmic bosonic bath

C.H.Chung et al., PRB 76, 235103 (2007)

Phase diagram

Field-theoretical RG

Critical properties via perturbative RG

exact

Critical properties via NRG

Spectral function

Critical properties via NRG

Spectral function

Summary II

• Kosterlitz-Thouless quantum transition between localized and delocalized phases in a noisy quantum dot with Ohmic dissipation

• Delocalized-localized quantum phase transition exists in the new paradimic pseudogap Bose-Fermi Anderson (BFA) model: relevant for describing a noisy quantum dot

• Excellent agreement between perturbative RG and numerical RG on the critical properties of the BFA model

• For metallic leads, our model maps onto Spin-boson model

III. Nonequilibrium transport near quantum phase transition

Nonequilibrium transport in Kondo dot

Decoherence (spin-relaxation rate) cuts-off logrithmic divergence of Kondo couplings suppresses coherence Kondo conductance

Steady state nonequilibrium current at finite bias V generates decoherence spin-flip scattering at finite V, similar to the effect of temperatures

Energy dependent Kondo couplings g in RG

Keldysh formulism for nonequilibrium transport

Nonequilibrium transport near quantum phase transition in a dissipative quantum dot

Effective Kondo modeldissipative quantum dot

: Dissipation strength

Dissipative spinless 2-lead model

New mapping:

valid for small t, finite V, at KT transition and localized phase

2-lead anisotropic Kondo model

New!

1

2

t tNew idea!

2-lead setupBias voltage VNonequilibrium transport

Fresh Thoughts: nonequilibrium transport at transition

What is the role of V at the transition compared to that of temperature T ?

What is the scaling behavior of G(V, T) at the transition ?

Important fundamental issues on nonequilibrium quantum criticality

Will V smear out the transition the same way as T? Not exactly! Log corrections

Is there a V/T scaling in G(V,T) at transition? Yes!

t t

Steady-state currentSpin Decoherence rate

K. Le Hur et al.

Zarand et al

New mapping: 2-lead anisotropic Kondo

Nonequilibrium perturbative RG approach to anisotropic Kondo model

•Decoherence (spin-relaxation rate) from V

•Energy dependent Kondo couplings g in RG P. Woelfle et. al.

G=dI / dV

Single Kondo dot in nonequilibrium, large bias V and magnetic field B

Paaske Woelfle et al, J. Phys. Soc. ,Japan (2005) Paaske, Rosch, Woelfle et al, PRL (2003)

Exp: Metallic point contact

Paaske, Rosch,Woelfle, et al, Nature physics, 2, 460 (2006)

Delocalized (Kondo) phase P. Woelfle et. al. 2003

At KT transition? In localized phase?

Scaling of nonequilibrium conductance G(V,T=0)

Localized phase:

At KT transition

(Equilibrium V=0)New!

G noneq =dI noneq / dV

(Non-Equilibrium V>0)

Black--Equilibrium Color--Nonequilibrium

V<<TEquilibrium scaling

V>>T Nonequilibrium scaling

New!

V/T scaling in conductance G(V,T) at KT transition

e eV V

Nonequilibrium Conductance at critical point

Large V, G(V) gets a logrithmic correction

V and T play the “similar” role but with a correction

Small V, nonequilibrium scaling G(V, T=0) ~ G(V=0,T) equilibrium scaling

At KT transition:

New!

eq

log

Charge Decoherence rate

Spinful Kondo model: Spin relaxzation rate due to spin flips

Spin Decoherence rate

Dissipative quantum dot: charge flip rate between Q=0 and Q=1

Nonequilibrium :Decoherence rate cuts off the RG flow

Nonlinear function in V !

Equilibrium :Temperature cuts off the RG flow

Nonequilibrium transport at localized-delocalized transition

Chung, Le Hur, Woelfle, Vojta (unpublished, work-in progress)

Important fundamental issues of nonequilibrium quantum criticality

What is the role of V at the transition compared to that of temperature T ?

What is the scaling behavior of G(V, T) at the transition ?

Will V smear out the transition the same way as T?

Is there a V/T scaling in G(V,T) at transition?

Nonequilibrium RG scaling equations of effective Kondo model

V<<TEquilibrium scaling

V>>T Nonequilibrium scaling

V and T play the similar role but with a logrithmic correction

New!

Outlook

Non-equilibrium transport in various coupled quantum dots

Quantum critical and crossover in transport properties near QCP

Quantum phase transition out of equilibrium

V

c

T

g g

• Kondo effect in carbon nanotubes

Optical conductivity

Linear AC conductivity

Sindel, Hofstetter, von Delft, Kindermann, PRL 94, 196602 (2005)

1

Spin-boson model: NRG results

N.-H. Tong et al, PRL 2003

quantum transport at nano-scale

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