Spin Currents in Mesoscopic Systems

Transcription

1 Spin Currents in Mesoscopic Systems Philippe Jacquod - U of Arizona I Adagideli (Sabanci) J Bardarson (Berkeley) M Duckheim (Berlin) D Loss (Basel) J Meair (Arizona) K Richter (Regensburg) M Scheid (Regensburg) P Stano (Bratislava)

2 The electron our main character Spin-1/2 fermion -> exclusion principle λ F << l,l -> Perturbation theory in λ F /l, λ F /L -> Screening of e-e interactions k B T/E f << 1 -> Sommerfeld expansion -> Fermi liquid theory (effective, no interaction) Quantum mechanics + special relativity -> Dirac equation Low energy approximation -> Pauli equation

3 Origin of stronger SOI : Breaking spatial symmetries (i) [010] J Sinova [001] [100] Structural inversion symmetry no Rashba-SOI Broken structural inversion symmetry Rashba-SOI (ii) Bulk inversion symmetry no Dresselhaus-SOI Broken bulk inversion symmetry Dresselhaus-SOI These SOI s are significantly larger Dresselhaus 55, Rashba 60

4 SOI is bad for spintronics: Spin relaxation No current-induced polarization, nor SHE without spin relaxation (Spins in a magnetic field Larmor precess around the field) J Sinova Dyakonov-Perel 71 With both Rashba and Dresselhaus SOI, Two length scales: l so : spin rotates by 2π l so =h/mα l dp : spin is randomized Something special seems to happen at α=β...

5 SOI is good for spintronics: Magneto-electric effects Generating spin currents/accumulations with DC currents/voltages 2D Disordered sample with linear SOI Apply finite charge current <J> -> <p> is finite SOI~finite effective Zeeman field disorder leads to spin relaxation (Dyakonov/Perel) generation of spin accumulation! Levitov, Nazarov, Eliashberg 85 Aronov & Lyanda-Geller 89 Edelstein 90 generation of spin current! Extensions: spin Hall effect: Sinova et al. 04, many others... ballistic systems: Bardarson, Adagideli and PJ 07; Nazarov 07; Krich and Halperin 08

6 Outline Generating spin currents by passing electric currents through SOI-coupled constrictions Scattering approach to spin transport Mesoscopic spin currents : universality and beyond -chaos and random matrix theory -geometric correlations -gauge theory and Onsager reciprocity How can one measure all that? Nondestructive spin to charge conversion

7 Scattering approach to transport (after Landauer, Büttiker, Imry) Electric current C Schoenenberger Generalization to spin (α,β=x,y,z) and electric (α=0) currents spin currents defined in leads (no spin relaxation) Adagideli, Bardarson and PJ

8 Scattering approach to transport Electric current C Schoenenberger Generalization to spin (α,β=x,y,z) and electric (α=0) currents with spin accumulations

9 Scattering approach to transport Electric current C Schoenenberger Generalization to spin (α,β=x,y,z) and electric (α=0) currents and spin-dpdt transmission coefficients Pauli matrix measure spin at exit and entrance

10 Scattering approach to transport Electric current C Schoenenberger Generalization to spin (α,β=x,y,z) and electric (α=0) currents and spin-dpdt transmission coefficients Pauli matrix measure spin at exit and entrance Charges are injected and measured Charges injected, polarization measured Polarization injected and measured

11 Mesoscopic currents : RMT universality I in I out Pass an electric current through SOI-coupled qdot Spin current is Iµ out SOI-coupled quantum dot Look at statistics of spin-resolved transmission coefficients E.g.: Bardarson, Adagideli and PJ 07; Nazarov 07; Krich and Halperin 08

12 Mesoscopic currents : RMT universality I in I out Pass an electric current through SOI-coupled qdot Spin current is Iµ out SOI-coupled quantum dot Random matrix theory of quantum transport Chaotic cavity -> S as a unitary random matrix 3 Dyson s (circular) ensembles of random unitary matrices 1 with Spin Rotational Symmetry and Time Reversal Symmetry 2 without TRS 4 without SRS, with TRS <- OUR ENSEMBLE HERE (τ D >>τ SO )

13 Mesoscopic currents : RMT universality I in I out Pass an electric current through SOI-coupled qdot Spin current is Iµ out SOI-coupled quantum dot Look at statistics of spin-resolved transmission coefficients No spin current on RMT average But universal spin conductance fluctuations (USCF)

14 Mesoscopic currents : semiclassical universality Trajectory-based semiclassics (Fisher, Lee; Baranger, Jalabert, Stone) Amplitude/ stability No SOI->no spin rotation Sum over classical trajectories Classical action Rem: (important!) we want to calculate i.e.: double sum over classical trajectories

15 Mesoscopic currents : RMT/semiclassical universality Trajectory-based semiclassics (Mathur-Stone 92) Spin rotation along cl. path γ τ d >> τ so

16 Mesoscopic currents : RMT/semiclassical universality Trajectory-based semiclassics (Mathur-Stone 92) Spin rotation along cl. path γ Reason : (i) Spin average factorizes from orbital average (ii) SU(2) average gives (fully broken SRS) Side-remark: leading order RMT spin conductance flucs. Also reproduced -> at this level, universality prevails Adagideli and PJ (unpublished)

17 Mesoscopic spin currents: beyond universality Next-order corrections : Expansion of ballistic Green s function to leading order in 1/k F L and k α /k F =mα/hk F U is no longer pure spin rotation - it now includes orbital effects -split trajectories into N γ SOI-bended segments -specialize to Rashba SOI Adagideli, PJ, Scheid, Duckheim, Loss, Richter, PRL 10

18 Mesoscopic spin currents: beyond universality Next-order corrections : Expansion of ballistic Green s function to leading order in 1/k F L and k α /k F =mα/hk F U is no longer pure spin rotation - it now includes orbital effects -split trajectories into N γ SOI-bended segments -specialize to Rashba SOI Adagideli, PJ, Scheid, Duckheim, Loss, Richter, PRL 10

19 Mesoscopic spin currents: beyond universality Spin conductance in the diagonal approximation, leading order in SOI (i) no spin (iv) spin measurement (ii) spin generation (iii) spin rotation Adagideli, PJ, Scheid, Duckheim, Loss, Richter, PRL 10

20 Mesoscopic spin currents: beyond universality Spin ballistic regime : k α l, k α L << 1 Geometric Correlations : polarization of current depends on vector connecting entrance and exit leads (~Edelstein)

21 Mesoscopic spin currents: beyond universality Spin ballistic regime : k α l, k α L << 1 Geometric Correlations : spin currents are fundamentally different from charge currents - the latter are not affected by GC s.

22 Gauging away weak SOI in confined systems Linear, inhomogeneous SOI with covariant derivative / SU(2) gauge field Helmoltz decomposition is necessary nonzero for spatially varying SOI, α(x), β(x) Coming up: There is a gauge trsf. that reduces SOI to U(1)xU(1) to leading order!! Aleiner and Fal ko 01; Brouwer, Cremers, Halperin 02; Adagideli, Lutsker, Scheid, PJ, Richter, 11

23 Gauging away weak SOI in confined systems Linear, inhomogeneous SOI with covariant derivative SU(2) gauge transformation with Aleiner and Fal ko 01; Brouwer, Cremers, Halperin 02; Adagideli, Lutsker, Scheid, PJ, Richter, 11

24 An exactly solvable example Rashba SOI with unidirectional gradient Perform spin rotation Adagideli, Lutsker, Scheid, PJ, Richter, 11

25 An exactly solvable example Rashba SOI with unidirectional gradient block Hamiltonian with pseudo-magnetic field

26 An exactly solvable example Charge conductance ~has B-linear dependence - agrees with Brouwer et al. Spin conductance ~difference in charge conductance at opposite B-fields

27 Spin currents vs. Onsager reciprocity Spin conductance ~difference in charge conductance at opposite B-fields Onsager relations: G ij (B)=G ji (-B) (Büttiker 86) vanishes for 2-terminal does not vanish for multi-terminal! does not vanish (with true B-field)! Large, linear in SOI currents for broken TRS/multiterminal

28 Spin currents vs. Onsager reciprocity Vs. opening of a third terminal Regular ~ huge effect Chaotic ~ smaller Vs. external flux (breaking of TRS) Regular ~ periodic Chaotic ~ average

29 Measuring mesoscopic spin currents : it s the symmetry Single-channel QPC polarization Focus on symmetry I QPC (B) vs. I QPC (-B) Consider 3-terminal quantum dot -> no reciprocity restriction -> no TRS restriction (1) If I µ QPC (0) = 0 spin species behave independently B µ ->-B µ <-> Expect even current at QPC: I QPC (B) = I QPC (-B)

30 Measuring mesoscopic spin currents : it s the symmetry Single-channel QPC polarization Focus on symmetry I QPC (B) vs. I QPC (-B) Consider 3-terminal quantum dot -> no reciprocity restriction -> no TRS restriction (2) If I µ QPC (0) = 0, and I QPC (0) = 0 B blocks, -B blocks Expect odd current at QPC: I QPC (B) = -I QPC (-B)

31 Measuring mesoscopic spin currents : it s the symmetry Set V g such that T=1/2 Set V QPC such that IQPC=0 (initially) Apply in-plane B-field V g

32 Measuring mesoscopic spin currents : it s the symmetry Set V g such that T=1/2 Set V QPC such that IQPC=0 (initially) Apply in-plane B-field

33 Conclusions Magnetoelectrically generated spin currents exist in mesoscopic ballistic systems They have universal fluctuations They have finite average - breakdown of RMT universality! What matters: Homogeneous / inhomogeneous SOI Onsager reciprocity relations They can be measured with polarized QPC