High-temperature non-adiabatic quantum pump effect

2. High-temperature non-adiabatic quantum pump effect M.V. Moskalets Dpt. of Metal and Semic. Physics, NTU “Kharkiv Polytechnic Institute” Kharkiv, Ukraine Geneva, 2006

3. outline • Introduction • quantum pump • scattering approach • Double-barrier pump • exact Floquet scattering matrix • pumped current • external bias • some figures • Summary

4. quantum pump it is a device generating dc current in response to ac local drive without any external dc/ac bias The system is able to produce a directed current if both spatial and time reversal symmetries are broken: S(t) ≠ S(-t) i) adiabatic (slow) drive: traversing the system an electron sees a frozen scatterer with local in time scattering matrix S(t) ≠ S(-t)  S(X1(t); X2(t+t);…) ii) non-adiabatic (fast) drive: while traveling inside a system an electron sees the same potential at different time moments S(X(t1); X(t2);…)  S(t) ≠ S(-t)

5. scattering approach to quantum transport bg = Sgdad bg ad Sgd g d

6. SF,11 1 double-barrier scatterer with periodic drive SF,21 SF(En,E) SF,11 a eikx + b e-ikx SF,21 L R 1 d

7. . . . left barrier . . . wave function: En an bn a0 E E b0 L a-m E-m b-m x x=0 En = E + nћ

8. (we collect all the terms having the same dependence on time) . . . boundary conditions . . . at x=0: at x=d:

9. . . . approximations . . . ћ << E L x << E x (the same is for the right barrier) we introduce

10. . . . transformed boundary conditions . . . performing inverse Fourier transformation:

11. step 1: step 2: . . . solution by recursion . . .

12. back in time: back in time: q=1: q=0: t-2 t-3 L L R R t t t- t- x=0 x=0 x=d x=d . . . Floquet matrix the sum over paths: (the retardation channels)

13. unitarity (retarded) physical meaning (advanced) t +  t -  t t t + 3 t - 3 E Sin(t,E) Sout(E, t) E t + 5 t - 5 … … in- / out- scattering matrices definition

14. microreversibility and symmetry properties let the parameters depend on time as follows: then the scattering matrices are subject to the following symmetries: Floquet scattering matrix: in/out scattering matrices: ( H is a magnetic field )

15. dc pumped current intra-channel interference inter-channel interference 1) periodicity in frequency:  = 2/F(a level spacing) 2) I(0) is temperature independent 3) I(0) dominates over I(i) at high frequencies: I(0)/I(i) ~ F at high temperatures/frequencies only I(0) matters

16. why is d/dt ? it is a cumulative effect of the Fermi sea for each scattering channel (, q,n) each photon-assisted channel is biased by its own eVn eVn S(t) indices number the channels:  - orbital, q - retardation, n - photon-assisted

17. dc bias: ac bias: quantum pump under an external bias 1 = , 2 =  + eV, eV, kBT << , kBT >> ћ-1 conservation laws: 1 t out in

18. decomposition of an excessive current two-terminal pump I1 I2 V1(t) V2(t) ac bias:

19. figures rectification of external ac currents: the dependence on a frequency I1 I2 V2(t)

20. I1 I2 figures the asymmetry of a dynamical transmission:

21. figures a pumped charge: the effect of the temperature I1 I2

22. I1 I2 figures a high-temperature pumped charge: the effect of a uniform oscillating potential U U=0 U=5

23. figures a high-temperature pumped current: I1 I2

24. summary (a high temperature/frequency limit) • Photon-assisted interference occurring within the same spatial path is relevant • The pumped current is independent of the temperature • Pumped current oscillates with frequency: the spectrum of these oscillations has a peak corresponding to internal dynamics • The rectification of external ac currents and the generation of a pumped current are determined by different scattering matrices