Electronic Transport and Quantum Phase Transitions of Quantum Dots in Kondo Regime

Chung-Hou Chung

1. Institut für Theorie der Kondensierten Materie Universität Karlsruhe, Karlsruhe, Germany

2. Electrophysics Dept. National Chiao-Tung University, HsinChu, Taiwan, R.O.C.

Collaborators: Walter Hofstetter (Frankfurt), Gergely Zarand (Budapest),

Peter Woelfle (TKM, Karlsruhe)

Acknowledgement:

Michael Sindel, Matthias Vojta

• Introduction

• Electronic transport and quantum phase transitions in coupled quantum dots: Model (I): parallel coupled quantum dots, 2-channel Kondo, non-trivial quantum critical point

Model (II): side-coupled quantum dots, 1-channel Kondo, Kosterlitz-Thouless quantum transition

• Conclusions and Outlook

Outline

Kondo effect in quantum dot

even

odd

Coulomb blockade

Single quantum dot

conductance anomalies

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

Glazman et al. Physics world 2001

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

d+U

d Kondo effect

Vg

VSD

Kondo effect in metals with magnetic impurities

At low T, spin-flip scattering off impurities enhances

Ground state is spin-singlet

Resistance increases as T is lowered

electron-impurity scattering

via spin exchange coupling

logT

(Kondo, 1964)

(Glazman et al. Physics world 2001)

Kondo effect in quantum dot

(J. von Delft)

Kondo effect in quantum dot

Kondo effect in quantum dot

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

• Double quantum dots / Multi-level quantum dot:

Singlet-triplet Kondo effect and Quantum phase transitions

Interesting topics/questions

• Non-equilibrium Kondo effect

• Kondo effect in carbon nanotubes

V

1 2

V Vt

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

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

(II). Von der Zant et al.

cond-mat/0508395, (PRL, 2005)

Model system (I): 2-channel parallel coupled quantum dots

Model system (II): 1-channel side-coupled quantum dots

Coupled quantum dots

L1

L2 R2

R1

C.H. C and W. Hofstetter, cond-mat/0607772

G. Zarand, C.H. C, P. Simon, M. Vojta, cond-mat/0607255

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)

Transport properties

• Transmission coefficient:

• Current through the quantum dots:

• Linear conductance:

Model System (I)

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

• Antiferrimagnetic exchange interaction J, Magnetic field B

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

2-impurity Kondo problem

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

For V1 = V2 and with p-h symmetry Jc = 2.2 Tk

Non-fermi liquid

JcJ

T

Spin-singletKondo1 2

L2

L1 R1

R2

Affleck et al. PRB 52, 9528 (1995)

Jones and Varma, PRL 58, 843 (1989)Jump of phase shift at Jc J < Jc, = /2 ; J >JC ,

Quantum phase transition as J is tuned

Jones and Varma, PRB 40, 324 (1989)

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

Specific heat coefficient

-2J-Jc

JC

NRG Flow of the lowest energy Phase shift

0

JJc

J<JC

J>JC

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

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

Jump of phase shift in both channels at Jc

Kondo

Spin-singlet

Kondo

Spin-singlet

Crossover energy scale T* J-Jc

• 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 Model System (I)

NRG Result Experiment by von der Zant et al.

Restoring of Kondo resonanceSinglet-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

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

• T = 0 singlet-triplet crossover at finite magnetic fields.

• Splitting of Kondo peaks gets smaller as B increases.

• B J, Kondo resonance restored, T( = 0) 1 reaches

unitary limit of a single-channel S = ½ Kondo effect.

• B > J, Kondo peak splits again.

• B J, T() shows 4 peaks in pairs around = (B J).

Effective S=1/2 Kondo effect

Tk=0.0002Jc=0.00042

Glazman et al. PRB 64, 045328 (2001)

Hofstetter and Zarand PRB 69, 235301 (2002)

Singlet-triplet crossover at finite field and temperature

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

splitting of Kondo peak due to Zeemann splitting of up and down spins

splitting is linearly proportional to B

• 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

Model System (II)

Vojta, Bulla, and Hofstetter, PRB 65, 140405, (2002)

Cornaglia and Grempel, PRB 71, 075305, (2005)

1 2

V Jeven

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

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

More general model of 1-channel 2-stage Kondo effect

Two-impurity, S=1, underscreened Kondo1

2

I

Jk1

Jk2

1 2Jk1

J( Jk2 = 0 )

Vojta, Bulla, and Hofstetter, PRB 65, 140405, (2002)

Ic ~ Jk1 Jk2 D

I < Ic: Timp = 1/4 residual spin-1/2

I > Ic: Timp = 0 spin-singlet

Optical conductivity

Linear AC conductivity

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

‘‘

1

Dot 2

JU=1

d=-0.5

=0.1

Tk=0.006

Comparison between two models

1 2Jk

J

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

L2

L1 R1

R2

2 impurity, S=1, Two-channel Kondo 2 impurity, S=1, One-channel Kondo

1

2

J

Jk1

Jk2complete Kondo screening

underscreened Kondoquantum critical point

K-T transition

8 J

Kondo spin-singlet

x

JcJ

Kondo spin-singlet8

T* J-Jc

Model (I) Model (II)

Conclusions

• Coupled quantum dots in Kondo regime exhibit quantum phase transition

Model system (II):

• Our results have applications in spintronics and quantum information

Quantum phase transition between Kondo and spin-singlet phases

Singlet-triplet crossover at finite field and temperatures, qualitatively agree with experiments

Kosterlitz-Thouless quantum transition,

Provide analytical and numerical understanding of the transition

L2

L1 R1

R22-channel Kondo physics

1-channel Kondo physics, two-stage Kondo effect

Model system (I):

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

Quantum phase transition in quantum dots with dissipation