electronic transport and quantum phase transitions of quantum dots in kondo regime chung-hou chung...
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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
• 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