Adiabatic quantum pumping in nanoscale electronic devices Frontiers

Adiabatic quantum pumping in nanoscale electronic devices Frontiers of Science & Technology Workshop on Condensed Matter & Nanoscale Physics and 13 th Gordon Godfrey Workshop on Recent Advances in Condensed Matter Physics Huan-Qiang Zhou, Sam Young Cho, Urban Lundin, and Ross H. Mc. Kenzie The University of Queensland [1] H. -Q. Zhou, S. Y. Cho, and R. H. Mc. Kenzie, Phys. Rev. Lett. 91, 186803 (2003) [2] H. -Q. Zhou, U. Lundin, S. Y. Cho, and R. H. Mc. Kenzie, cond-mat/0309096 (2003)

Outline. Foucault’s pendulum & Archimedes screw. Landauer theory. “Adiabatic” in quantum transport. Scattering state & scattering matrix. Parallel transport law. Scattering/Pumping geometric phases. How to observe scattering geometric phases. Charge/Spin pumping currents. Conclusions

Foucault’s Pendulum Archimedes Screw Classical World Quantum World Scattering (Pumping) Geometric P Berry’s (Geometric) Phase

Landauer Theory [R. Landauer, IBM J. Res. Develop. 1, 233 (1957)] Conductance EF Conductance (2 e 2/h) Rolf Landauer width Wire width increasing [B. J. van Wees and coworkers, Phys. Rev. Lett. 60, 848 (1988)]

“Adiabatic” : time scales t td t d (t w) t time period during which the system completes the adiabatic cycle dwell time during scattering event t w Wigner delay time is the difference between traveling time with scattering and without scattering Instantaneous scattering matrix S(t) at any given (“frozen”) time

Scattering Matrix V(x(t)) A B y scattering states E G F x y L = A exp[ i k x] + B exp[-i k x] y. R = F exp[ i k x] + G exp[-i k x] At any given “frozen” time t B F = r t t r A G A =S G outgoing scattering states = scattering matrix. incoming scattering states

Scattering Geometric Phase [H. -Q. Zhou, S. Y. Cho, and R. H. Mc. Kenzie, Phys. Rev. Lett. 91, 186803 (2003)] e 1 ig reig QUANTUM DEVICE t eig External parameters X(t) E. g. , gate voltages, magnetic field etc eig originates from the unitary freedom in choosing the scattering states Geometric phase g !

Quantum Device

Parallel Transport Law For the period t of an adiabatic cycle “Matrix geometric phase” A plays the role of a gauge potential in parameter space

Aharonov-Bohm Effect R. Schuster and coworkers, Nature 385, 420 (1997) B: Magnetic field S: Area of closed path Phase shift : f = (e/c) BS D B= x. A SCREEN y. A z P(f) z S B Electron Source yz y. B P(f) z y. A + y. B 2 y Pz(f)= z 0 2 2 y y = A + B + 2 y. A y. B COS(f+d) = f INTERFERENCE

How to observe scattering geometric phases [ H. -Q. Zhou, U. Lundin, S. Y. Cho, and R. H. Mc. Kenzie, cond-mat/0309096 (2003)] [ Y. Ji, and coworkers, Science 290, 779 (2000)] Geometric phase Gauge potential

Time-reversed Scattering States At any given “frozen” time t y scattering state V(x(t)) E r S= t t r x y time-reversed scattering state t V(x(t)) E r x ST= r t t r

Pumping Geometric Phase [H. -Q. Zhou, S. Y. Cho, and R. H. Mc. Kenzie, Phys. Rev. Lett. 91, 186803 (2003)] [P. W. Brouwer, Phys. Review B 58, R 10135 (1998)] For the time-reversed scattering states Gauge potential Pumped charge [c. f. ] Brouwer formula for charge pumping [M. Switkes and coworkers, Science 283, 1905 (1999)]

Observable Quantities X 1 t 1 Q 2 C 1 t 2 Initial state t = t 1 + t 2 C 2 X 2 Pumped charge is additive Q = Q 1 + Q 2 Current Charge current Spin current I t = I 1 t 1 + I 2 t 2 IC = I + + I IS = I + - I -

Scattering states for spin pumping y scattering states A+ AB+ B- G+ GF+ F- Magnetic atom For spin dependent scattering y L = A 1 + A 0 eikx + B+ + 0 1 1 0 At any given “frozen” time t A+ G+ A- = G- S++ S+ - S- + S- - 4 x 4 matrix B+ F+ BF- + B- 0 1 e-ikx

Adiabatic Spin Pumping Current [H. -Q. Zhou, S. Y. Cho, and R. H. Mc. Kenzie, Phys. Rev. Lett. 91, 186803 (2003)] Magnetic atom

Conclusions 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. Adiabatic quantum pumping has a natural representation in terms of gauge fields defined on the space of system parameters. Scattering geometric phase & pumping geometric p are both sides of a coin !!

Matrix Geometric Phase U Initial state X 1 U X 2 Line integration Stokes’ theorem A F d. X 1 ; d. X 2 A : Gauge potential d. X 1 d. X 2 F : Field strength F = d. A – A^ A

Berry’s Phase vs. Scattering (Pumping) Geometric Phase Closed systems Wave function Open systems Row(column) vectors na( the S matrix ) of 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 group arising from different choices of bases Gauge potential and Gauge group arising from redistribution of scattering particles among different channel