Post on 13-Dec-2015
Adiabatic quantum pumping inAdiabatic quantum pumping innanoscale electronic devicesnanoscale electronic devices
Adiabatic quantum pumping inAdiabatic quantum pumping innanoscale electronic devicesnanoscale electronic devices
Huan-Qiang Zhou, Sam Young Cho, Urban Lundin, and Ross H. McKenzie
The University of Queensland
[2] H. -Q. Zhou, U. Lundin, S. Y. Cho, and R. H. McKenzie, cond-mat/0309096 (2003)
[1] H. -Q. Zhou, S. Y. Cho, and R. H. McKenzie, Phys. Rev. Lett. 91, 186803 (2003)
Frontiers of Science & Technology Workshop on Condensed Matter & Nanoscale Physicsand
13th Gordon Godfrey Workshop on Recent Advances in Condensed Matter Physics
OutlineOutlineOutlineOutline
. Landauer theory
. Foucault’s pendulum & Archimedes screw
. “Adiabatic” in quantum transport
. Scattering state & scattering matrix
. Parallel transport law
. Scattering/Pumping geometric phases
. Charge/Spin pumping currents
. Conclusions
. How to observe scattering geometric phases
Archimedes ScrewArchimedes ScrewArchimedes ScrewArchimedes ScrewFoucault’s PendulumFoucault’s PendulumFoucault’s PendulumFoucault’s Pendulum
Berry’s (Geometric) PhaseBerry’s (Geometric) Phase Scattering (Pumping) Geometric PhaseScattering (Pumping) Geometric Phase
Quantum WorldQuantum World
Classical WorldClassical World
EF
Rolf Landauer
Landauer TheoryLandauer TheoryLandauer TheoryLandauer Theory
ConductanceConductance
[R. Landauer, IBM J. Res. Develop. 1, 233 (1957)]
Wire width increasing
Co
nd
uct
ance
(2e
/h) width
2
[B. J. van Wees and coworkers, Phys. Rev. Lett. 60, 848 (1988)]
““Adiabatic” : time scalesAdiabatic” : time scales““Adiabatic” : time scalesAdiabatic” : time scales
d dwell time during scattering event
w Wigner delay time is the differencebetween traveling time with scatteringand without scattering
time period during which the system completes the adiabatic cycle
Instantaneous scattering matrix S(t) at any given (“frozen”) time
d w( )
E
V(x(t))
x
scattering statesscattering states
A
= A exp[ i k x] + B exp[-i k x]L = F exp[ i k x] + G exp[-i k x]R
Scattering MatrixScattering MatrixScattering MatrixScattering Matrix
BGF
outgoing scattering states = scattering matrix . incoming scattering states
At any given “frozen” time t
r
r
t
t=B
FAG
= AG
S
Scattering Geometric PhaseScattering Geometric PhaseScattering Geometric PhaseScattering Geometric Phase[H. -Q. Zhou, S. Y. Cho, and R. H. McKenzie, Phys. Rev. Lett. 91, 186803 (2003)]
r tQUANTUMQUANTUM DEVICEDEVICE
1ei
eiei
External parameters X(t)External parameters X(t)
ei originates from the unitary freedom in choosing the scattering states
Geometric phase !
E.g., gate voltages, magnetic field etc
Parallel Transport LawParallel Transport Law
For the period of an adiabatic cycle
A plays the role of a gauge potential in parameter space
“Matrix geometric phase”
SCREEN
ElectronSource
B: Magnetic fieldS: Area of closed path
INTERFERENCE
P()z
P()z
0
z
B
S
Aharonov-Bohm EffectAharonov-Bohm EffectAharonov-Bohm EffectAharonov-Bohm Effect
A
B
z A B= +
+= A2 B
2+ A B2 COS()
= z2Pz()
Phase shift : = (e/c)= (e/c) BS
B = x A
R. Schuster and coworkers, Nature 385, 420 (1997)
How to observe scattering geometric phasesHow to observe scattering geometric phases[ H. -Q. Zhou, U. Lundin, S. Y. Cho, and R. H. McKenzie, cond-mat/0309096 (2003)]
[ Y. Ji, and coworkers, Science 290, 779 (2000)]
Geometric phase
Gauge potential
Time-reversedTime-reversed Scattering StatesScattering StatesTime-reversedTime-reversed Scattering StatesScattering States
x
r
r
t
tS=
EV(x(t))
r t
scattering statescattering state
x
E
t
time-reversedtime-reversed scattering state scattering state
rV(x(t))
ST=r
rt
t
At any given “frozen” time t
PumpingPumping Geometric Phase Geometric PhasePumpingPumping Geometric Phase Geometric Phase[P. W. Brouwer, Phys. Review B 58, R10135 (1998)]
For the time-reversed scattering states
Gauge potential
Pumped charge[c.f.] Brouwer formula for charge pumping
[H. -Q. Zhou, S. Y. Cho, and R. H. McKenzie, Phys. Rev. Lett. 91, 186803 (2003)]
[M. Switkes and coworkers, Science 283, 1905 (1999)]
1X
2X
Observable QuantitiesObservable QuantitiesObservable QuantitiesObservable Quantities
Q1
Q2
Q = Q1 + Q2
Pumped charge is additive
C1
C2
Initial state
I = I1 1 + I2 2
1
2
= 1 + 2
Charge current
Spin current
IC = I+ + I-
IS = I+ - I-
Current
scattering statesscattering states
Scattering states for Scattering states for spin pumpingspin pumpingScattering states for Scattering states for spin pumpingspin pumping
A+A-
G+G-
B+B-
F+F-
For spin dependent scatteringFor spin dependent scattering
At any given “frozen” time t
1
0
0
1+A+ A- eikx +
1
0
0
1+B+ B- e-ikx=L
A+
A-
G+
G-
B+
B-
F+
F-
S++ S+ -
S- + S- -
=
Magnetic atom
4 x 4 matrix
Magnetic atom
Adiabatic Spin Pumping CurrentAdiabatic Spin Pumping CurrentAdiabatic Spin Pumping CurrentAdiabatic Spin Pumping Current[H. -Q. Zhou, S. Y. Cho, and R. H. McKenzie, Phys. Rev. Lett. 91, 186803 (2003)]
ConclusionsConclusionsConclusionsConclusions
Adiabatic quantum pumping has a natural representation in terms of gauge fields defined on the space of system parameters.
We found a geometric phase accompanying scattering state in a cyclic and adiabatic variation of external parameters which
characterize an open system with a continuous energy spectrum.
Scattering geometric phase & pumping geometric phase are both sides of a coin !!
UA F
Stokes’ theoremStokes’ theoremLine integrationLine integration
2X
1X
1dX 2dX; 1dX 2dX
A : Gauge potentialF : Field strength
Initial state
Matrix Geometric PhaseMatrix Geometric PhaseMatrix Geometric PhaseMatrix Geometric Phase UU
F = dA – A A^
Closed systems Open systems
Wave function Row(column) vectors n of
the S matrix
n-th energy level with Mn degeneracies
n-th lead with Mn channels
Discrete spectrum
(bound states)
Continuous spectrum
(scattering states)
Parallel transport due to adiabatic theorem
Parallel transport due to adiabatic scattering (pumping)
Gauge potential Gauge potential
and
Gauge group arising from different choices of
bases
Gauge group arising from redistribution of scattering particles
among different channel
Berry’s Phase vs.Berry’s Phase vs.Berry’s Phase vs.Berry’s Phase vs. Scattering (Pumping)Scattering (Pumping)Geometric PhaseGeometric Phase
Scattering (Pumping)Scattering (Pumping)Geometric PhaseGeometric Phase