Quantum%20transport%20at%20nano-scale

1
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 Chung, Zarand, Woelfle,
PRB 77, 035120 (2008), selected by Virtual
Journal of Nanoscience and Technology Jan. 8
2008 Chung, Glossop, Fritz, Kircan,
Ingersent,Vojta, PRB 76, 235103 (2007) Chung, Le
Hur, Vojta, Woelfle nonequilibrium transport near
the quantum phase transition (arXiv0811.1230)

Chung-Hou Chung ???
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.)
2
Outline

• 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

3
Quantum dot---A single-Electron-Transistor (SET)
Single quantum dot
Goldhaber-Gorden et al. nature 391 156 (1998)
Coulomb Blockade
4
Quantum dot---charge quantization
5
Kondo effect in quantum dot
Kondo effect
conductance anomalies
Glazman et al. Physics world 2001
L.Kouwenhoven et al. science 289, 2105 (2000)
6
Kondo effect in metals with magnetic impurities
logT
(Kondo, 1964)
electron-impurity spin-flip scattering
(Glazman et al. Physics world 2001)
For TltTk (Kondo Temperature), spin-flip
scattering off impurities enhances Ground state
is Resistance increases as T is lowered
7
Kondo effect in quantum dot
(J. von Delft)
8
Kondo effect in quantum dot
9
Kondo effect in quantum dot
Anderson Model

• New energy scale Tk Dexp(-pU/ G)
• For T lt Tk
• Impurity spin is screened (Kondo screening)
• Spin-singlet ground state
• Local density of states developes Kondo resonance

?d ? Vg
local energy level charging energy
level width All tunable!
U
G 2pV 2?d
10
Kondo Resonance of a single quantum dot
Spectral density at T0
Universal scaling of T/Tk
L. Kouwenhoven et al. science 2000
M. Sindel
particle-hole symmetry
phase shift
Fredel sum rule
11
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

12
Perturbative Renormalization Group (RG) approach
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 lt Tk, J diverges, Kondo screening
J
0
13
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

14

I. Quantum phase transition in coupled
double-quantum-dot system
15
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.

16
Von der Zant et al. (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

17
Quantum phase transition in coupled
double-quantum-dot system
G. Zarand, C.H. C, P. Simon, M. Vojta, PRL, 97,
166802 (2006)
C.H. C and W. Hofstetter, PRB 76 045329 (2007)
R1
L1
K
L2
R2

• 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

18
2-impurity Kondo problem
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)
1
2
K
X
Heavy fermions
R/2
-R/2
H H0 Himp
H0
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Quantum criticality of 2-impurity Kondo problem

• Particle-hole symmetry V0

H ? H H under
Affleck et al. PRB 52, 9528 (1995)
Jones and Varma, PRL 58, 843 (1989)
Jones and Varma, PRB 40, 324 (1989)
Kc is smeared out, crossover
Misleading common belief ! We have corrected it!
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Quantum Phase Transition in Double Quantum dots
P-H Symmetry
triplet states
L1
R1
Izumida and Sakai PRL 87, 216803 (2001)
Vavilov and Glazman PRL 94, 086805 (2005)
K
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 K,
  Magnetic field B
• 2-channel Kondo physics, complete Kondo
  screening for B K 0

K
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Transport properties

• Current through the quantum dots

• Transmission coefficient

• Linear conductance

22
NRG Flow of the lowest energy
Phase shift d
d
Kondo
KltKC
JC
Kondo
p/2
KgtKC
Spin-singlet
Spin-singlet
0
K
Kc
Two stable fixed points (Kondo and spin-singlet
phases )
Jump of phase shift in both channels at Kc
One unstable fixed point (critical fixed point)
Kc, controlling the quantum phase transition
23
Quantum phase transition of a double-quantum-dot
system
JRKKYK
Chung, Hofstetter, PRB 76 045329 (2007)
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Restoring of Kondo resonance in coupled quantum
dots Singlet-triplet crossover at finite
temperatures T
NRG Result
Experiment by von der Zant et al.
T0.003
T0.004

• 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.

25
Singlet-triplet crossover at finite magnetic
fields
J0.007, Jc0.005, Tk0.0025, T0.00001,
in step of 400 B
NRG P-h symmetry
EXP P-h asymmetry
Ferromagnetic Jlt0
Antiferromagnetic Jgt0
J close to Jc, smooth crossover
J gtgt Jc, sharper crossover
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Quantum criticality in a double-quantum dot
system P-H Asymmetry
G. Zarand, C.H. C, P. Simon, M. Vojta, PRL, 97,
166802 (2006)
V1 ,V2 break P-H sym and parity sym. ? QCP still
survives as long as no direct hoping t0
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Quantum criticality in a double-quantum dot
system
No direct hoping, t 0
Asymmetric limit
T1Tk1, T2 Tk2
2 channel Kondo System
QCP occurs when
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
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Transport of double-quantum-dot near QCP (only K,
no t term)
K
K
NRG on DQDs without t, P-H and parity symmetry

At KKc
Affleck and Ludwig PRB 48 7279 (1993)
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The only relevant operator at QCP direct hoping
term t
dim 1/2
(wr.t.QCP)
RG

most dangerous operators off-diagonal J12
typical quantum dot
At scale Tk,
may spoil the observation of QCP
30
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
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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
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Side-coupled double quantum dots
Chung, Zarand, Woelfle, PRB 77, 035120 (2008),

• 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 gt 0.
• Kosterlitz-Thouless transition, Jc 0

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2 stage Kondo effect
1st stage Kondo screening
Jk Kondo coupling
2nd stage Kondo screening
dip in DOS of dot 1
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NRGSpectral density of Model (II)
U1 ed-0.5 G0.1 Tk0.006 L2
Kosterlitz-Thouless quantum transition
No 3rd unstable fixed point corresponding to the
critical point
Crossover energy scale T exponentially depends
on J-Jc
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Dip in DOS of dot 1 Perturbation theory
J 0
d1
wnlt Tk
J gt 0 but weak
self-energy
vertex
sum over leading logarithmic corrections
36
Dip in DOS perturbation theory
U1, ed-0.5, G 0.1, L2, J0.0005, Tk0.006,
T8.2x10-10

• Excellence agreement between Perturbation theory
  (PT) and NRG for T ltlt w ltlt Tk

• PT breaks down for w T

• Deviation at larger w gt O(Tk) due to interaction
  U

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Summary I

• Coupled quantum dots in Kondo regime exhibit
  quantum phase transition

quantum critical point

• The QCP of DQDs is identical to that of a
  2-channel Kondo system

• 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

K-T transition
38

II. Quantum phase transition in a dissipative
quantum dot
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Quantum dot as charge qubit--quantum two-level
system
charge qubit-
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Quantum dot as artificial spin S1/2 system
Quantum 2-level system
41
Dissipation driven quantum phase transition in a
noisy quantum dot
Noise charge fluctuation of gate voltage Vg
K. Le Hur et al, PRL 2004, 2005, PRB (2005),
Noise SHO of LC transmission line
Impedence
H Hc Ht HHO
N1/2 Q0 and Q1 degenerate
Caldeira-Leggett Model
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Spin Boson model
43
Delocalized-Localized transition
delocalized
localized
h N -1/2
K. Le Hur et al, PRL 2004,
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Charge Kondo effect in a quantum dot with Ohmic
dissipation
Hdissipative dot
non-interacting lead
N1/2 Q0 and Q1degenerate
Anisotropic Kondo model
de-localized
Jz -1/2 R
localized
gJ
Kosterlitz-Thouless transition
45
Generalized dissipative boson bath (sub-ohmic
noise)
Ohmic

Sub-Ohmic
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Generalized fermionic leads Power-law DOS
Anderson model
Quantum phase transition in the pseudogap
Anderson/Kondo model
d-wave superconductors and graphene r 1
Fradkin et al. PRL 1990
Local moment (LM)
J
X
Kondo
Jc
0
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Delocalized-Localized transition in Pseudogap
Fermi-Bose Anderson model
C.H.Chung et al., PRB 76, 235103 (2007)
Pseudogap Fermionic bath
Sub-ohmic bosonic bath
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Phase diagram
Field-theoretical RG
49
Critical properties via perturbative RG
exact
50
Critical properties via NRG
51
Spectral function
Critical properties via NRG
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Spectral function
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Summary II

• Delocalized-localized quantum phase transition
  exists in the new paradimic
• pseudogap Bose-Fermi Anderson (BFA) model
  relevant for describing
• a noisy quantum dot

• Kosterlitz-Thouless quantum transition between
  localized and delocalized phases in a noisy
  quantum dot with Ohmic dissipation

• For metallic leads, our model maps onto
  Spin-boson model

• Excellent agreement between perturbative RG and
  numerical RG on the critical properties of the
  BFA model

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III. Nonequilibrium transport near quantum phase
transition
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Nonequilibrium transport in Kondo dot
Steady state nonequilibrium current at finite
bias V generates decoherence spin-flip
scattering at finite V, similar to the effect of
temperatures
Decoherence (spin-relaxation rate)
cuts-off logrithmic divergence of Kondo
couplings suppresses coherence Kondo
conductance
Energy dependent Kondo couplings g in RG
Keldysh formulism for nonequilibrium transport
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Nonequilibrium transport near quantum phase
transition in a dissipative
quantum dot
dissipative quantum dot
Effective Kondo model
57
2-lead setup Bias voltage V Nonequilibrium
transport
New idea!
Dissipative spinless 2-lead model
New!
New mapping
valid for small t, finite V, at KT transition and
localized phase
2-lead anisotropic Kondo model
58
Fresh Thoughts nonequilibrium transport at
transition
Zarand et al
Steady-state current Spin Decoherence rate G
K. Le Hur et al.
t
t
New mapping 2-lead anisotropic Kondo
Important fundamental issues on nonequilibrium
quantum criticality
What is the role of V at the transition compared
to that of temperature T ?
Will V smear out the transition the same way as
T? Not exactly! Log corrections
What is the scaling behavior of G(V, T) at the
transition ?
Is there a V/T scaling in G(V,T) at transition?
Yes!
59
Nonequilibrium perturbative RG approach to
anisotropic Kondo model

• Decoherence G (spin-relaxation rate) from V
• Energy dependent Kondo couplings g in RG

P. Woelfle et. al.
GdI / dV
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Single Kondo dot in nonequilibrium, large bias V
and magnetic field B
Exp Metallic point contact
Paaske, Rosch, Woelfle et al, PRL (2003)
Paaske Woelfle et al, J. Phys. Soc. ,Japan (2005)
61
Paaske, Rosch,Woelfle, et al, Nature physics, 2,
460 (2006)
62
Delocalized (Kondo) phase
P. Woelfle et. al. 2003
At KT transition?
In localized phase?
63
(No Transcript)
64
Scaling of nonequilibrium conductance G(V,T0)
New!
Black--Equilibrium Color--Nonequilibrium
(Equilibrium V0)
(Non-Equilibrium Vgt0)
G noneq dI noneq / dV
At KT transition
Localized phase
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V/T scaling in conductance G(V,T) at KT transition
New!
VltltT Equilibrium scaling
VgtgtT Nonequilibrium scaling
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Nonequilibrium Conductance at critical point
log
eq
At KT transition
Small V, nonequilibrium scaling G(V, T0)
G(V0,T) equilibrium scaling
Large V, G(V) gets a logrithmic correction V and
T play the similar role but with a correction
New!
67
Nonequilibrium Decoherence rate G cuts off the
RG flow
Spin Decoherence rate G
Spinful Kondo model Spin relaxzation rate due to
spin flips
Charge Decoherence rate G
Dissipative quantum dot charge flip rate
between Q0 and Q1
Nonlinear function in V !
Equilibrium Temperature cuts off the RG flow
68
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 ?
Will V smear out the transition the same way as T?
What is the scaling behavior of G(V, T) at the
transition ?
Is there a V/T scaling in G(V,T) at transition?
69
Nonequilibrium RG scaling equations of effective
Kondo model
70
VltltT Equilibrium scaling
VgtgtT Nonequilibrium scaling
New!
V and T play the similar role but with a
logrithmic correction
71
Outlook
Quantum critical and crossover in transport
properties near QCP
Non-equilibrium transport in various coupled
quantum dots
Quantum phase transition out of equilibrium

• Kondo effect in carbon nanotubes

72
Optical conductivity
Sindel, Hofstetter, von Delft, Kindermann, PRL
94, 196602 (2005)
1
Linear AC conductivity
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Spin-boson model NRG results
N.-H. Tong et al, PRL 2003