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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. Adiabatic quantum pumping in nanoscale electronic devices. Huan-Qiang Zhou, Sam Young Cho ,

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Adiabatic quantum pumping in nanoscale electronic devices

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

  2. 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

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

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

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

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

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

  8. Quantum Device

  9. 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

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

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

  13. [c.f.] Brouwer formula for charge pumping Pumping Geometric Phase [H. -Q. Zhou, S. Y. Cho, and R. H. McKenzie, Phys. Rev. Lett. 91, 186803 (2003)] [P. W. Brouwer, Phys. Review B 58, R10135 (1998)] For the time-reversed scattering states Gauge potential Pumped charge [M. Switkes and coworkers, Science 283, 1905 (1999)]

  14. X X 1 2 Q1 Q2 Observable Quantities t1 C1 Initial state t = t1 +t2 C2 t2 Pumped charge is additive Q = Q1 +Q2 Current I t = I1 t1 + I2 t2 IC = I+ + I- Charge current IS = I+ - I- Spin current

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

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

  17. 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 phase are both sides of a coin !!

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

  19. Scattering (Pumping) Geometric Phase Berry’s Phase vs.

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