Spin-Resolved Transport Properties in Inhomogeneous Graphene Nanostructures

Transcription

1 Spin-Resolved Transport Properties in Inhomogeneous Graphene Nanostructures Dario Bercioux DB & A. De Martino, Phys. Rev. B 8, 654 () L. Lenz & DB, EPL 96, 76 () DB, D.F. Urban, F. Romeo, & R. Citro (), arxiv:4:58

2 SOI: Band Mixing Mechanisms First Order Processes: Effect of curvature + Atomic Stark effect p z + + p x p z + + p x Second Order Process: Intrinsic Spin-Orbit Coupling related to the band structure A SO SO B A mk mk Huertas-Hernando et al., Phys. Rev. B 74, 5546 (6)

3 Engineering of SOI in graphene SOI can be largely enhanced by using intercalated Au atoms or heavy ad-atoms (indium, thallium) and substrate engineering!! 5 K Varykhalov et al., Phys. Rev. Lett., 576 (8)! 5 3 K C. Weeks et al., PRX, () x 3

4 SOI: Spectral Properties (a).5 (b).5 (c) Energy Energy k x (k y =) k x (k y =) Energy k x (k y =) Spin state perpendicular to motion direction Rashba, Phys. Rev. B 79, 649(R) (9); DB & De Martino, Phys. Rev. B 8, 654 () 4

5 The spin-precession length l SO `SO = lim K K + E `SO =!=!=."!=.5"!=.9" l # SO l SO L. Lenz & DB, EPL 96, 76 () E 5

6 The Spin-Double Refraction N region k y SO region k y!/ (a) " + (b)! + φ E k + k - k x ξ + ξ - k x Refraction angles " #!/4 " -! -!/4!/ Incident angle!!/4!/ Momentum conservation along the interface Refraction angles Momentum along the x direction Limit angles of refraction Ramaglia, DB, et al., Eur. Phys. J. B 36, 365 (3) Ramaglia, DB, et al., J. Phys.:Condens. Matter 6, 943 (4) DB and De Martino, Phys. Rev. B 8, 654 () 6

7 The SO Barrier 5-5 N N SO Spin-up injection Transmission in the spin-up spindown channels of the N region (a) (b) `SO = 5-5 `SO = DB and De Martino, Phys. Rev. B 8, 654 () 7

8 SO Barrier: Unpolarized Case Full Transmission Polarization along z-axis (a) (b) P z Polarization ϕ Injection angle DB and De Martino, Phys. Rev. B 8, 654 () 8

9 SHE with Boundary N SO Time-reversal symmetry imposes the other edge state on the opposite valley: E(k x ) [t ] (a) k x [! - ] E(k y ) (b) k x [! - ] DB & De Martino, Phys. Rev. B 8, 654 (); Kane & Mele, Phys. Rev. Lett. 95, 468 (5) k y 9

10 Periodic system of SOI barriers (Poster 6) L. Lenz & DB, EPL 96, 76 () Δ(x), λ(x) x a b l From generalizing the Transfer Matrix Method to the case of spin cos(apple`) + cos(apple `) =Tr[T ] B. H. J. McKellar & G. J. Stephenson, Jr., Phys. Rev. C 35, 6 (987); M. Barbier, et al., Phys. Rev. B 8, () For the case of normal incidence we have show that ( cos(apple`) = F + (k x ) cos(apple`) = F (k x ) K (") =s " k y F (k x )= K " + cos(k a) cos(k x b) + " K sin(k a)sin(k x b)

11 Periodic system of SOI barriers at ky= L. Lenz & DB, EPL 96, 76 () Δ(x), λ(x) x a b l = =/4 =/4 =/4 =/ =/4 E k k k ε κ κ κ

12 Adiabatic quantum pumping y -! 3!!! +! 4 3 L T L SO x N T SO N The system Hamiltonian H =v F (s ~ ) ~p + (x) (s y x s x y ) + (x)(s z z )+V T (x)+v SO (x) The pumping parameters V T (x, t) =V T(x)+ V T sin(!t) V SO (x, t) =V SO(x)+ V SO sin(!t + ') (x, t) = (x)+ gv SO (x, t) DB, D.F. Urban, F. Romeo, & R. Citro (), arxiv:4:58 Huertas-Hernando et al., Phys. Rev. B 74, 5546 (6)

13 Spin and charge pumping currents LT=5 nm; LSO=5/75/ nm; EF=7.5 mev; λ=.7 & 3 mev; Δ= &.56 mev (a) # C (b)! S ! c! c - " c " c (c) # C (d) " S ! c! c -! c! c π/8 π/4 3π/8 π/ -.5 π/8 π/4 3π/8 π/! in! in DB, D.F. Urban, F. Romeo, & R. Citro (), arxiv:4:58 3

14 Pumping currents angle average 4 x -4 (a) x -3 x - (b) (c) -4-8! C! S x (d) -5 x -3 (e) -3 x - (f) V SO (mev) V SO (mev) V SO (mev) DB, D.F. Urban, F. Romeo, & R. Citro (), arxiv:4:58 4

15 Conclusions & Outlook Possibility to have a finite SOI in graphene due to interaction with other atoms, curvature effects, Stark effect etc. Spin-Double refraction leading to spin-dependent transmission Edge states due to the spin-hall effect Superlattice of SOI barriers Effect of adiabatic pumping in superstructures High order terms in Rashba SOI for ribbons 5

16 Acknowledgments Excellence Initiative of the German Federal & State Governments DFG program SFB/TR & BE 4564/- Dr. Daniel Urban (Uni-Freiburg) Lucia Lenz (FRIAS) Prof. Hermann Grabert (FRIAS + Uni-Freiburg) Alessandro De Martino (Uni-Cologne) Roberta Citro (Uni-Salerno) Francesco Romeo (Uni-Salerno) Thank you for your attention! 6