Quantum criticality in a double-quantum-dot system Chung-Hou Chung Electrophysics Dept. National...
-
Upload
barbra-mcdonald -
Category
Documents
-
view
228 -
download
3
Transcript of Quantum criticality in a double-quantum-dot system Chung-Hou Chung Electrophysics Dept. National...
Quantum criticality in a double-quantum-dot system
Chung-Hou Chung Electrophysics Dept.
National Chiao-Tung University
Hsin-Chu, Taiwan
Collaborators:
Gergely Zarand (Budapest),
Matthias Vojta (TKM, Karlsruhe)
Pascal Simon (CNRS, Grenoble)
G. Zarand, C.H. C, P. Simon, M. Vojta, PRL, 97, 166802 (2006)
• Introduction
• Quantum criticality in a double-quantum-dot system:
particle-hole symmetry
• Quantum criticality in a 2-impurity Kondo system
• Quantum criticality in a double-quantum-dot system:
more general case: no P-H or parity symmetry
• Realization of QCP in a proposed experimental setup
• Conclusions
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
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
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)
Quantum phase transition and non-Fermi liquid state in 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, 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
Coupled Quantum dots
• 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
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:
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=K
C.H. C and W. Hofstetter, cond-mat/0607772
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)
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
Kc = 2.2 Tk
Non-fermi liquid
KcK
T
Spin-singletKondo1 2
Affleck et al. PRB 52, 9528 (1995)
Jones and Varma, PRL 58, 843 (1989)Jump of phase shift at Kc K < Kc, = /2 ; K >KC ,
Quantum phase transition as K 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)
• Particle-hole symmetry V=0
H H’ = H under
even
odd
2-impurity Kondo problem
Smooth crossover
• Particle-hole asymmetry
even
odd
Zhu and Varma, cond-mat/0607426
Sharp phase transition
2-impurity Kondo problem
QCP destroyed crossover P-H asymmetry plus
Zhu and Varma, cond-mat/0607426
V12 : Effective potential scattering terms generated
Relevant operator at K=Kc
Splitting between even and odd resonances
even
odd
Quantum criticality in a double-quantum –dot system
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
Optical conductivity
Linear AC conductivity
Sindel, Hofstetter, von Delft, Kindermann, PRL 94, 196602 (2005)
1
Transport of double-quantum-dot near QCP
At K=Kc
Affleck and Ludwig PRB 48 7279 (1993)
NRG on DQDs without P-H and parity symmetry
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
Conclusions
• Coupled quantum dots in Kondo regime exhibit quantum phase transition
• The QCP is robust against particle-hole and parity asymmetries
•The QCP is destroyed by charge transfer between two channels
• The QCP of DQDs is identical to that of a 2-channel Kondo system
• The effect of charge transfer can be reduced by inserting additional even number of dots, making it possible to be observe QCP in experiments