quantum transport at nano-scale
DESCRIPTION
Quantum transport at nano-scale. Zarand, Chung, Simon, Vojta, PRL 97 166802 (2006) Chung, Hofstetter, PRB 76 045329 (2007), selected by Virtual Journal of Nanoscience and Technology Aug. 6 2007 - PowerPoint PPT PresentationTRANSCRIPT
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
Coulomb blockade d+U
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)
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
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
Coulomb blockade d+U
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