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

Quantum DeviceQuantum DeviceQuantum DeviceQuantum Device

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