Parametric quantum pumping

2. Parametric quantum pumping M. Moskalets Dpt. of Metal and Semiconductor Physics, NTU "Kharkiv Polytechnical Institute", Ukraine in collaboration with M. Büttiker Dpt. de Physique Théorique, Université de Genève, Switzerland 2006

3. Introduction: what is a quantum pump experiment Quantum-coherent mechanism of a current generation Scattering matrix formalism adiabatic approximation Magnetic field symmetry of a pumped current Simple examples Conclusion

4. Introduction Quantum pumpis a mesoscopic device generating dc current in response to a local and periodic in timeperturbation in the absence of any dc/ac bias

5. Introduction Mesoscopic conductor Vdc R Idc = Vdc / R L << 

6. Introduction Quantum pump V1 ~ cos(ωt + φ1) V2 ~ cos(ωt + φ2) R(t) Idc ~ sin(φ1 - φ2) L << 

7. Introduction Adiabatic transport Experiment 2DEG M. Switkes, C.M. Marcus, K. Campman, A.S. Gosard, Science 283, 1905 (1999) driving gates confining gates Idc ~ sin  Idc Idc ~  The current changes from sample to sample 

8. Idc — ? V=V0 IL(out) I(in) 1 I(in) IR(out) T 1-T IR(out) = 1 IL(out) = 1 1-T T 1 Physical mechanismStationary scatterer Idc = 0

9. Physical mechanismDynamic scatterer V=V0 + V1cos(t) R+2  T+2  T+2 R+2  T+1 R+1 R+1 T+1 ћ  T0 R0 T0 1 R0  1  R-1  T-1 T-1 R-1  T-2 R-2  T-2 R-2

10. Physical mechanism How can one get T   T  ?

11. E ± ћω Physical mechanism Weak oscillating potential (scatterer with internal structure) V1(t) V2(t) E L (E) Vj(t) = Vcos(ωt+φj), j=1,2

12. E ± ћω Physical mechanism Weak oscillating potential (scatterer with internal structure) V1(t) V2(t) E L Vj(t) = Vcos(ωt+φj), j=1,2

13. E ± ћω Physical mechanism Weak oscillating potential (scatterer with internal structure) V1(t) V2(t) E L Vj(t) = Vcos(ωt+φj), j=1,2

14. Physical mechanism Weak oscillating potential E E + ћω absorption Ve-iφ1 Ve-iφ2 + eik(+)L ei(k+/v) L eik L |…|2 T+ ~ 1 + cos(1 - 2 + ωL/v)

15. Physical mechanism Weak oscillating potential E + ћω E absorption Ve-iφ1 Ve-iφ2 + eik(+)Lei(k+/v) L eik L |…|2 T+ ~ 1 + cos(1 - 2 - ωL/v)

16. Physical mechanism Weak oscillating potential absorption: T+ = T+ - T+ emission: T- = T+ whole current

17. Physical mechanism General condition for an adiabatic generation of a dc current Dynamical breaking of a time reversal symmetry Spatial asymmetry V1(t) ~ cos(ωt) , V2(t) ~ cos(ωt + φ) L t  - t V1 V2 V1(t) ~ cos(ωt) , V2(t) ~ cos(ωt - φ)

18. Physical mechanism Quantized energy exchange and interference of photon-assisted scattering amplitudes are physical phenomena responsible for asymmetric scattering leading to generating of a dc current E E  ћω quantized energy exchange E + ћω A1 interfering amplitudesA1 &A2: E A2 V1 V2

19. Scattering matrix approach (Landauer-Büttiker approach) α = 3 . . Nr - 1 Ψ2(in) Ψ2(out) S(t) ΨNr(out) Ψ1(in) Ψ1(out) ΨNr(in)

20. Scattering matrix approach second quantization operators: bg ad b = Sa Sgd g d

21. Scattering matrix approach The scattering matrix is a set of single-particle quantum mechanical transition amplitudes 1. Stationary case: The energy is conserved, therefore the scattering matrix depends on only a single energy (a scattered particle energy) S = S(E) 2. Dynamical case: The (Floquet) scattering matrix depends on two energies SF = SF(E(out);E(in)), E(out) = E(in) +nћω

22. Scattering matrix approach Stationary case (energy is conserved) Dynamical case (energy can be changed by an integer number of quanta ћω) 1. Operators for scattered particles 2. Unitarity:SS† = S†S = I (a current conservation) 3. (anti)Commutation relations: (not more than one particle in each state) [aa†(E), ab(E)] = da,b d(E- E) [ba†(E), bb(E)] = da,b d(E- E)

23. ? S(E) SF (En ,E) Scattering matrix approach Adiabatic approximation 1. StationaryS(E)isNr X Nrunitary matrix; 2. FloquetSF(E,En) isNr X Nr X nmaxunitary matrix having much more number of elements [nmax >>1, SF(E,E± nmaxћω )  0] Can one connect these matrices at least in the limit af a small driving frequencies ω  0? (that could greatly reduce the amount of calculations needed)

24. Scattering matrix approach Adiabatic approximation The answer is yes but with some restrictions

25. Scattering matrix approach Adiabatic approximation To this end we introduce a matrixSin(E,t) and expand it in powers of ω

26. Scattering matrix approach Adiabatic approximation 1. Zeroth-order approximation (it satisfies the unitarity condition) HereS(E,t)is a “frozen” scattering matrix which describes scattering on a stopped at a time moment t and thus stationaryscatterer. It depends on time as follows where{P} is a set of varying parameters. The form of a zero order term allows us to introduce the following criterion of smallness of a driving frequency (that correlates with the definition based on the transmission time)

27. However the unitarity condition requires an additional term A satisfying the following equation Scattering matrix approach Adiabatic approximation 2. First-order approximation: Naively one think that the first-order term (linear in S) is

28. Scattering matrix approach Adiabatic approximation Thus to find the Floquet scattering matrix SF(E,En)havingNrNrnmax elements in the limit of small frequencies ω  0 it is necessary to find a “frozen” scattering matrixS and the matrix A each of them has only NrNr elements

29. Scattering matrix approach Adiabatic approximation ieiT1/2 Example: (no magnetic field) ei`R1/2ei ei`R1/2e-i -a a

30. Scattering matrix approach Adiabatic approximation Why we have introduced the matrixА ? • This matrix describe an asymmetry in scattering • The reflection coefficient does not depend on А(i.e., sometimes the first term is enough) • The matrices S and A have completely different symmetry properties. That allows us to describe correctly the (magnetic field) symmetry of a current generated

31. Scattering matrix approach Adiabatic approximation Symmetry properties Stationary scattering matrix: Residual matrix:

32. Pumped current Dc current a b

33. Pumped current Adiabatic dc current generated current due to external bias

34. Pumped current The current conservation (nothing comes from outside) b How can it be ?

35. Pumped current A quasi-particle picture α Iαβ V(t) β

36. Pumped current A point-like scatterer Idc = 0  L R S(t) we calculated: there is no a quantum pump effect

37. Pumped current A point-like scatterer I - ?  L R S(t) Are there any currents here? Yes, the ac currents are present

38. Pumped current A time-dependent current an adiabatic limit generated current due to external bias due to oscillating charge on a scatterer (being a full time derivative the last term does not contribute to a dc current, but it is necessary to satisfy a charge continuity equation) without external bias (Brouwer, 1998)

39. Pumped current A point-like scatterer I(t)  0  L R S(t) due to external bias due to oscillating charge on a scatterer

40. Pumped current Two point-like scatterers SL(t) SR(t)   L R Idc 0

41. Magnetic field symmetry Two terminal case L R S(t) H γ, θ, R, T all are even in H;  is odd in H. All the quantities depend on time V

42. Conclusion • We developed a scattering matrix approach valid up to linear in driving frequency terms • We have shown that the periodically and adiabatically driven mesoscopic scatterer generates ac currents • We analyzed a magnetic field symmetry of a pumped current and found the odd in magnetic field and linear in an applied voltage dc current which can not be attributed to a classical rectification effect