New hints from theory for pumping spin currents in quantum circuits

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 statisticalmethods in mesoscopicsystems II Brasov, Romania, September1, 2014

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 statisticalmethods in mesoscopicsystems II

4. Advanced many-body and statisticalmethods in mesoscopicsystems II

5. Method for numerical simulations : Transporttheory: 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 Laterallyconnected ring The ring is tangent to the circuit Symmetrically connected rings: no magnetic moment

7. Physical model with spin and spin-orbit interaction 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 statisticalmethods in mesoscopicsystems II

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

10. Simplest case: spinless model with B perpendicular to ring Wemayavoidleaving the ring excited by lettingitswallowintegerfluxons. Then H is the sameatbeginning and at the end. Finding: the onlyeffectispumping! Bchirality direction of pumping

11. pumping by an hexagonal ring – insertion of 6 fluxons Pumped charge staircase Time dependentflux can be used to pumpcharge. Whatkind of pumpingisthat? Advanced many-body and statistical methods in mesoscopic systems II

12. t q Quantum Pumping: severalkinds are known. Introduced by Thouless (1983): for a 1d spatiallyperiodicsystem and time-periodicadiabaticsystem with H(x+a)=H(x) and H(t+T)=H(t). Flux of curvature =Berry phase

13. t q 2) Charge at each cycle is quantized Single-valuedy Qn=integer (Chernnumber) Strong implications: 1) Berry phaseneedsatleast 2 parameters Flux of curvature =Berry phase

14. Othercase:QuantumPumping 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 responseis linear, oneneedsatleasttwoparameters R1 and R2 H=H(R1,R2)

15. Present case: Quantum Pumping from rings Avron,Raveh and Zur (Rev. Mod. Phys.) analyzed the adiabaticresponse of circuits with rings in adiabaticapproximation and found no pumping. Chargeisnotquantized- not an adiabaticresult (ifpulse time growschargedecreases). Wegot 1-parameter pumping (onlyfluxvaries) Classically, magneticmoments of ringswould be linear with appliedbias. Because of quantum effects, the magneticresponse of quantum ringsiscubic (Cini, Perfetto and Stefanucci, Phys. Rev. 2010). The one-parameterpumpingis in line with that.

16. characteristic hopping time of the system= h/hopping matrix element 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.

17. Equivalent bond concept

18. Introducing spin and magnetic interactions Advanced many-body and statisticalmethods in mesoscopicsystems II

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

20. Besidescharge, some spin ispumpedtoo Advanced many-body and statistical methods in mesoscopic systems II

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

22. Rotating ring experiment

24. Effects of the spin-orbitinteraction Advanced many-body and statisticalmethods in mesoscopicsystems II

25. How to make a magneticcurrent Spin up Spin down 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)

26. No chargecurrent Same spin currentpumpedin bothwires (itiseven).

27. No chargecurrent Same current pumped in both wires

28. The stationary pure spin currentproduces no magneticfield; the electricfieldhas a special pattern (seen in a plaseorthogonal to the wire): electricfield wire

29. The analyticaltheory of magneticcurrent generation Advanced many-body and statistical methods in mesoscopic systems II

30. Model is bipartite if ring has even number of sites the adjacencygraph for spin-orbitalsisalso bipartite Vertical bonds due to in plane B Advanced many-body and statisticalmethods in mesoscopicsystems II

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-dependentflux in ring, Itfollowsthat the chargeistotallypinned on each site. Buthowdoes the spin currentarise? Recall the equivalent bond concept of the spinless model with dynamical flux

33. The mechanism for spin current generation 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. spin-up ring states =effective bond spin-down ring states =effective bond

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

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

36. Thoughtexperiment: (M. Cini, submitted for publication) 1) Storemagnetization in reservoirs 2) Isolate magnetizedreservoirs 3) Connect magnetizedreservoirs with wire: a spin currentisgenerated Advanced many-body and statisticalmethods in mesoscopicsystems II

37. spin current spin current Advanced many-body and statisticalmethods in mesoscopicsystems II

38. Advanced many-body and statisticalmethods in mesoscopicsystems II

39. Oscillatorty polarization Delay proportional to length of connection Advanced many-body and statisticalmethods in mesoscopicsystems II

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 statisticalmethods in mesoscopicsystems II

41. 30 atom ring, Cubesreplaced by 4-atom ringsconnected by 200 atomleads t1=30 t2=35 Currentsobservedat centre of storage-ring connection Note delay- Externalcurrentispurely spin The frequency of the externaloscillationsisreduced by dividing by 3 the externalwire band width (hereinternalwires are 50 atoms long, externalwires 100 atoms long) Temperature dependenceismild! Advanced many-body and statistical methods in mesoscopic systems II

42. Conclusions Ballisticringsasymmetricallyconnected to wires and pierced by a time-dependentmagneticfield can be used to pumpcharge. Semiclassicalapproximations are qualitativelywrong. The phenomenonispurely quantum and nonlinear (violatesBrowertheorem). Rotatingmagneticfieldspump spin-polarizedcurrents- No spin-orbitinteractionisinvolved. Rotatingrings in fixedmagneticfieldspump spin- polarizedcurrents with or without the effects of spin-orbitcoupling A time-dependentmagneticfield in the plane of the bipartite ring athalffillingpumps a pure spin current(= magneticcurrent) in the externalcircuit, driven by the relativistic spin-orbitinteraction. 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. Thank you for your attention! Advanced many-body and statistical methods in mesoscopic systems II