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

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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

NTHU, May 8, 2007

- 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

ed+U

Coulomb blockade

ed

Kondo effect

Single quantum dot

Vg

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

VSD

odd

even

conductance anomalies

Glazman et al. Physics world 2001

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

Kondo effect in metals with magnetic impurities

(Kondo, 1964)

logT

electron-impurity scattering

via spin exchange coupling

(Glazman et al. Physics world 2001)

At low T, spin-flip scattering off impurities enhances

Ground state is spin-singlet

Resistance increases as T is lowered

(J. von Delft)

AndersonModel

New energy scale: Tk ≈ Dexp(-pU/G)

For T < Tk :

Impurity spin is screened (Kondo screening)

Spin-singlet ground state

Local density of states developesKondoresonance

d ∝ Vg

local energy level :

charging energy :

level width :

All tunable!

U

Γ=2πV 2ρd

= p/2

Kondo Resonance of a single quantum dot

Spectral density at T=0

Universal scaling of T/Tk

M. Sindel

L. Kouwenhoven et al. science 2000

particle-hole

symmetry

phase shift

Fredel sum rule

1

2

V

t

V

Interesting topics/questions

- Non-equilibrium Kondo effect

- Kondo effect in carbon nanotubes

- Double quantum dots / Multi-level quantum dot:
- Singlet-triplet Kondo effect and Quantum phase transitions

g

g

c

Quantum phase transitions

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

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

True level crossing: Usually a first-order transition

Sachdev, quantum phase transitions,

Cambridge Univ. press, 1999

- 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

Recent experiments on coupled quantum dots

(I). C.M. Macrus et al.

Science, 304, 565 (2004)

- 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.

cond-mat/0508395, (PRL, 2005)

- 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

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

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

L1

R1

L2

R2

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

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

Numerical Renormalization Group (NRG)

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

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

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

- Logarithmic discretization of the conduction band

- Anderson impurity model is mapped onto a linear chain of fermions

- Iteratively diagonalize the chain and keep low energy levels

triplet states

L1

R1

Izumida and Sakai PRL 87, 216803 (2001)

Vavilov and Glazman PRL 94, 086805 (2005)

Simon et al. cond-mat/0404540

Hofstetter and Schoeller, PRL 88, 061803 (2002)

L2

singlet state

R2

- 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

R1

R2

L2

even 2 (L2+R2)

even 1 (L1+R1)

T

Non-fermi liquid

1

2

Kondo

Spin-singlet

J

Jc

J-Jc

Specific heat coefficient g -2

2-impurity Kondo problem

Quantum phase transition as J is tuned

For V1 = V2 and with p-h symmetry

Jc = 2.2 Tk

Affleck et al. PRB 52, 9528 (1995)

Jump of phase shift at Jc J < Jc, d = p/2 ; J >JC , d = 0

Jones and Varma, PRL 58, 843 (1989)

Jones and Varma, PRB 40, 324 (1989)

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

ibdb. 61, 7, 2348 (1992)

J-Jc

Crossover energy scale T*

NRG Flow of the lowest energy

Phase shift d

d

Kondo

J<JC

JC

Kondo

p/2

J>JC

Spin-singlet

Spin-singlet

0

Jc

J

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

Jump of phase shift in both channels at Jc

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

Quantum phase transition of Model System (I)

- 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.

Singlet-triplet crossover at finite temperatures T

NRG Result

Experiment by von der Zant et al.

T=0.003

T=0.004

- 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.

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).

Singlet-triplet crossover at finite magnetic fields

Jc=0.00042

Tk=0.0002

Effective S=1/2 Kondo effect

Glazman et al. PRB 64, 045328 (2001)

Hofstetter and ZarandPRB 69, 235301 (2002)

B in Step of 0.001

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

NRG: P-h symmetry

EXP: P-h asymmetry

Ferromagnetic J<0

Antiferromagnetic J>0

J close to Jc, smooth crossover

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

J >> Jc, sharper crossover

splitting is linearly proportional to B

1

2

V

J

even

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

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

- 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

Anderson's poor man scaling and Tk

HAnderson

- Reducing bandwidth by integrating out high energy modes

Anderson 1964

J

J

J

J

- Obtaining equivalent model with effective couplings

- Scaling equation

w < Tk, J diverges, Kondo screening

J

0

4V2/U

J: AF coupling btw dot 1 and 2

Tk

D

rc

1/G

2 stage Kondo effect

1st stage Kondo screening

Jk: Kondo coupling

2nd stage Kondo screening

dip in DOS of dot 1

1/J

J

0

8

Kondo

spin-singlet

NRG:Spectral density of Model (II)

U=1

ed=-0.5

G=0.1

Tk=0.006

L=2

Kosterlitz-Thouless quantum transition

No 3rd unstable fixed point corresponding to the critical point

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

Dip in DOS of dot 1: Perturbation theory

1

2

when

w

Dip in DOS of dot 1

J = 0

d1

wn< Tk

J > 0 but weak

self-energy

vertex

sum over leading logarithmic corrections

Dip in DOS: perturbation theory

U=1, ed=-0.5, G= 0.1, L=2,J=0.0006, Tk=0.006, T*=1.6x10-6 Tk

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

- PT breaks down for w T*

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

1

2

2

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

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

Jk1

Two-impurity, S=1, underscreened Kondo

I

Jk2

Jk1

J

( Jk2 = 0 )

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

I > Ic: Tcimp = 0 spin-singlet

Ic ~ Jk1 Jk2

D

J

1

U=1

ed=-0.5

G=0.1

Tk=0.006

L=2

Dot 2

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

‘

‘

Linear AC conductivity

R1

1

1

Jk1

R2

even 2 (L2+R2)

L2

even 1 (L1+R1)

J

Jk2

2

2

Jk

J

n

J

x

J-Jc

T*

8

J

8

0

Jc

0

Kondo

spin-singlet

Kondo

spin-singlet

Comparison between two models

Model (I)

Model (II)

2 impurity, S=1, Two-channel Kondo

2 impurity, S=1, One-channel Kondo

complete Kondo screening

quantum critical point

underscreened Kondo

K-T transition

R1

R2

L2

Conclusions

- Coupled quantum dots in Kondo regime exhibit quantum phase transition

Model system (I):

2-channel Kondo physics

Quantum phase transition between Kondo and spin-singlet phases

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

Model system (II):

1-channel Kondo physics, two-stage Kondo effect

Kosterlitz-Thouless quantum transition,

Provide analytical and numerical understanding of the transition

- Our results have applications in spintronics and quantum information

T

g

g

c

Outlook

Quantum critical and crossover in transport properties near QCP

Non-equilibrium transport in various coupled quantum dots

Quantum phase transition in quantum dots with dissipation

Localized-Delocalized transition

Quantum phase transition out of equilibrium

_

G

G

1

2

K

even 2 (L2+R2)

even 1 (L1+R1)

T

Non-fermi liquid

Kondo

Spin-singlet

K

Kc

Quantum criticality in a double-quantum –dot system

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

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

2-impurity Kondo problem

Quantum phase transition as K is tuned

Quantum critical point (QCP) at Kc = 2.2 Tk

Broken P-H sym and parity sym. QCP still survives as long as no direct hoping t=0

Hoping term t is the only relevant operator to suppress QCP

_

G

G

1

2

K

Quantum criticality in a double-quantum –dot system

No direct hoping, t = 0

Asymmetric limit:

T1=Tk1, T2= Tk2

QCP occurs when

2 channel Kondo System

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

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

Transport of double-quantum-dot near QCP

No direct hoping, t=0

Finite hoping t

QCP survives without P-H and parity symmetry

QCP is destroyed, smooth crossover

suppress hoping effect observe QCP in QD Exp.

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

Thank You!

Highly motivated MSc and/or Ph D students

are welcome

to participate the collaboration in my research

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