1. Adiabatic quantum pumping in nanoscale electronic devices Adiabatic quantum pumping in nanoscale 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 Physics and 13 th Gordon Godfrey Workshop on Recent Advances in Condensed Matter Physics

2. OutlineOutline. 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

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

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

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

6.  E V(x(t)) x scattering 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 Matrix B G F outgoing scattering states = scattering matrix. incoming scattering states At any given “frozen” time t r r t t = B F A G = A G S

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

8. Quantum Device

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

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

11. How 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

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

13. Pumping 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)]

14. 1 X 2 X Observable Quantities Q1Q1 Q2Q2 Q = Q 1 + Q 2 Pumped charge is additive C1C1 C2C2 Initial state I  = I 1  1 + I 2  2 1 1 22  =  1 +  2 Charge current Spin current I C = I + + I - I S = I + - I - Current

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

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

17. ConclusionsConclusions 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 !!

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

19. Closed systemsOpen systems Wave function Row(column) vectors n   of the S matrix n-th energy level with M n degeneracies n-th lead with M n channels Discrete spectrum (bound states) Continuous spectrum (scattering states) Parallel transport due to adiabatic theorem Parallel transport due to adiabatic scattering (pumping) 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. Scattering (Pumping) Geometric Phase Scattering (Pumping) Geometric Phase