1. New hints from theory for pumping spin currents in quantum circuits Michele Cini Dipartimento di Fisica, Universita’ di Roma Tor Vergata and Laboratori Nazionali di Frascati, INFN Advanced many-body and statistical methods in mesoscopic systems II Brasov, Romania, September 1, 2014

2. 2 Model : Michele Cini and Enrico Perfetto, PRB 84 245201 (2011); Michele Cini and Stefano Bellucci, J. Phys.: Condens. Matter 26, 145301 (2014) Michele Cini and Stefano Bellucci, Eur. Phys. B 14, 87, 106 (2014) Advanced many-body and statistical methods in mesoscopic systems II

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5. Method for numerical simulations : 5 Transport theory: Partition-free approach-M. Cini, PRB 22,5887 (1980); Phys. Rev. B89,239902 (2014). Advanced many-body and statistical methods in mesoscopic systems II

6. Definition of Laterally connected ring Symmetrically connected rings: no magnetic moment The ring is tangent to the circuit 6

7. Physical model with spin and spin-orbit interaction 7 We also need the adjacency graph for spin-orbitals Advanced many-body and statistical methods in mesoscopic systems II

8. Spinless case Advanced many-body and statistical methods in mesoscopic systems II 8

9. 9 Tight-binding model with time-dependent magnetic flux Simplest case: spinless model with B perpendicular to ring M.Cini and E. Perfetto, Phys. Rev. B 84, 245201 (2011)

10. 10 Simplest case: spinless model with B perpendicular to ring We may avoid leaving the ring excited by letting it swallow integer fluxons. Then H is the same at beginning and at the end. Finding: the only effect is pumping! B  chirality  direction of pumping

11. 11 pumping by an hexagonal ring – insertion of 6 fluxons Pumped charge staircase Time dependent flux can be used to pump charge. What kind of pumping is that? Advanced many-body and statistical methods in mesoscopic systems II

12. 12 Introduced by Thouless (1983): for a 1d spatially periodic system and time-periodic adiabatic system with H(x+a)=H(x) and H(t+T)=H(t). Quantum Pumping: several kinds are known. q t Flux of curvature =Berry phase

13. 13 q t Flux of curvature =Berry phase 1) Berry phase needs at least 2 parameters Strong implications: Single-valued   Q n =integer (Chern number) 2) Charge at each cycle is quantized

14. 14 Other case:Quantum Pumping in linear systems P.W Brower (1998) has shown that in linear systems one gets two-parameter pumping Cohen (PRB 2003) established properties of pumping in linear response theory If the response is linear, one needs at least two parameters R 1 and R 2 H=H(R 1,R 2 )

15. 15 Charge is not quantized- not an adiabatic result (if pulse time grows charge decreases). We got 1-parameter pumping (only flux varies) Classically, magnetic moments of rings would be linear with applied bias. Because of quantum effects, the magnetic response of quantum rings is cubic (Cini, Perfetto and Stefanucci, Phys. Rev. 2010). Present case: Quantum Pumping from rings Avron,Raveh and Zur (Rev. Mod. Phys.) analyzed the adiabatic response of circuits with rings in adiabatic approximation and found no pumping. The one-parameter pumping is in line with that.

16. 16 Nonadiabatic pumping: inserting a flux quantum in the ring in a time of the order of 10 h/ hopping matrix element we shift about an electron. characteristic hopping time of the system= h/hopping matrix element

17. 17 Equivalent bond concept

18. Introducing spin and magnetic interactions Advanced many-body and statistical methods in mesoscopic systems II 18

19. 19 250 atom ring Half filling (E F =0) Rotating B experiment Advanced many-body and statistical methods in mesoscopic systems II

20. 20 Besides charge, some spin is pumped too Advanced many-body and statistical methods in mesoscopic systems II

21. 21 Rotating ring experiment Advanced many-body and statistical methods in mesoscopic systems II charge spin

22. 22 Rotating ring experiment

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24. 24 Effects of the spin-orbit interaction Advanced many-body and statistical methods in mesoscopic systems II

25. How to make a magnetic current Spin-up electrons move to left and spin down electrons to right. A pure spin current does not move charge, but magnetization. It is even (same in both wires) 25 Spin up Spin down

26. Same spin current pumped in both wires (it is even). 26 No charge current

27. Same current pumped in both wires 27 No charge current

28. The stationary pure spin current produces no magnetic field; the electric field has a special pattern (seen in a plase orthogonal to the wire): wire electric field

29. Advanced many-body and statistical methods in mesoscopic systems II 29 The analytical theory of magnetic current generation

30. Model is bipartite if ring has even number of sites the adjacency graph for spin-orbitals is also bipartite Advanced many-body and statistical methods in mesoscopic systems II 30 Vertical bonds due to in plane B

31. Ground state properties of our model at half filling Theorem: each site is exactly half filled at any time This arises from the fact that the system is bipartite. Bipartite system  sign change of red orbitals changes the sign of H. But sign change of red orbitals is a gauge.

32. Ground state properties of our model at half filling Theorem: each site is exactly half filled. No time-dependent flux in ring, It follows that the charge is totally pinned on each site. 32 But how does the spin current arise? Recall the equivalent bond concept of the spinless model with dynamical flux

33. B(t) pumps spin current because spin-up electrons can do a trip to the down-spin sector where they gain opposite phases. This works like a time-dependent and spin dependent phase. The mechanism for spin current generation 33 spin-up ring states =effective bond spin-down ring states =effective bond

34. pure spin current, the same on both wires 6pA 4pA The effect is robust! What happens at finite temperatures? pure spin current, the same on both wires 34 Advanced many-body and statistical methods in mesoscopic systems II

35. What happens if we depart from half filling? Charge current is small 8pA 0.2pA 0.8pA 8pA Spin current Charge current 35

36. Thought experiment: (M. Cini, submitted for publication) 1) Store magnetization in reservoirs 2) Isolate magnetized reservoirs 3) Connect magnetized reservoirs with wire: a spin current is generated Advanced many-body and statistical methods in mesoscopic systems II 36

37. 37 spin current Advanced many-body and statistical methods in mesoscopic systems II

38. 38 Advanced many-body and statistical methods in mesoscopic systems II

39. 39 Delay proportional to length of connection Oscillatorty polarization Advanced many-body and statistical methods in mesoscopic systems II

40. 40 30 atom ring, Cubes replaced by 4-atom rings connected by 200 atom leads t1=30 t2=35 Currents observed at centre of storage-ring connection External current is purely spin The frequency of the external oscillations is reduced by dividing by 3 the external wire band width (here internal wires are 50 atoms long, external wires 100 atoms long) Advanced many-body and statistical methods in mesoscopic systems II

41. 41 30 atom ring, Cubes replaced by 4-atom rings connected by 200 atom leads t1=30 t2=35 Currents observed at centre of storage-ring connection Note delay- External current is purely spin The frequency of the external oscillations is reduced by dividing by 3 the external wire band width (here internal wires are 50 atoms long, external wires 100 atoms long) Advanced many-body and statistical methods in mesoscopic systems II Temperature dependence is mild!

42. 42 Conclusions A time-dependent magnetic field in the plane of the bipartite ring at half filling pumps a pure spin current (= magnetic current) in the external circuit, driven by the relativistic spin-orbit interaction. Ballistic rings asymmetrically connected to wires and pierced by a time- dependent magnetic field can be used to pump charge. Semiclassical approximations are qualitatively wrong. The phenomenon is purely quantum and nonlinear (violates Brower theorem). Rotating magnetic fields pump spin-polarized currents- No spin- orbit interaction is involved. Rotating rings in fixed magnetic fields pump spin- polarized currents with or without the effects of spin-orbit coupling This current keeps its polarization totally at room temperature and partially if the carrier concentration deviates from half filling. It can be stored as magnetization and later released in a controllable way. Advanced many-body and statistical methods in mesoscopic systems II Thank you for your attention!