Alexey Kovalev. Department of Physics. Collaborators: L. Zabro, Y. Tserkovnyak, G.E.W. Bauer, J. Sinova

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1 Nanomechanical Spinpolarizer p Alexey Kovalev Department of Physics Texas A&M University Collaborators: L. Zabro, Y. Tserkovnyak, G.E.W. Bauer, J. Sinova

2 Outline We study the effects of time dependent strain on transport properties in a long semiconductor rod Predict a piezo-spin effect Propose several realizations of the mechanical spin polarizer based on the state t of the art nano-electro-mechanical (NEMS) systems Theoretical etical conclusions are supported by the tight binding model numerical simulations Conclusions

The setup z x y A rod consisting i of a semiconductor

3 The setup z x y A rod consisting i of a semiconductor (GaAs) and insulator parts is excited by some external source into the torsional oscillations. Semiconductor layer of thickness a/2 is on top of insulator layer of the same thickness. Voltage is applied to the semiconductor rod in order to see the effect of the mechanical oscillations.

4 NEMS that can excite our rod QuickTime and a TIFF (Uncompressed) decompressor are needed to see this picture. B QuickTime and a TIFF (Uncompressed) decompressor are needed to see this picture. MWNT shaft Au rotor P. Mohanty et al, Phys. Rev. B 70, (2004). 300 nm Magnetomotive forces can excite torsional oscillations when coil interacts with external A. M. Fennimore et al, Nature 424, 408 ( magnetic field. 2003). Electrostatic forces can excite torsional oscillations of a golden particle attached to MWNT.

5 Hamiltonian with strain-induced spin-orbit interaction ˆ H = h2 2m * +β ˆ ( k2 + γ[ ˆ σ y (u xy k x u yz k z ) + ˆ σ z (u yz k y u zx k x ) + ˆ σ (u k u k ) ]) x zx z xy y [ σ x k x (u yy u zz ) + ˆ σ y k y (u zz u xx ) + ˆ σ z k z (u xx u yy )] u lattice displacement u = ( u / x + u / x ) / 2 i ij ( i j j i G. E. Pikus and A. N. Titkov, in Optical Orientation, edited by F. Meier and B. P. Zakharchenya (North- Holland, Amsterdam, 1984).

6 Hamiltonian with strain-induced spin-orbit interaction H ˆ = h2 * k2 + γ ˆ σ (u k u k ) + ˆ σ (u k u k ) + ˆ σ (u k u k ) y xy x yz z z yz y zx x 2m x zx z xy y ( [ ] ) u lattice displacement u = u / x + u / x ) / 2 i ij ( i j j i u zy = τ(y) χ For a long rod: ; u = τ(y) χ xy ; u = 0 zx x z τ(y) = ϕ(y) ;ϕ(y) = ϕ sin( π 0 L y For a thin plate cross-section ˆ H = h2 2m * y)cos(ωt) lowest torsional mode χ(x,z) x(x a/2)/2 ( k2 + γτx[ ˆ σ y k z ˆ σ k ]+ h.c. ) z y

7 Physics behind the polarization effect When the rod is thin and only one channel is open, we average over the transverse electron wave-function ˆ h 2 γτa 2 H 1D = k ˆ ) 2m * 2 y + 4 σ k z y + V(y) Vector potential A = ± hγaτ(y,t) y; A=0 4 No magnetic field but electric field E = - A/ t = hγa τ ˆ 4 t σ y z Spin-splitting (Piezo-spin effect) U = U 0 sin( π L y)cos(ωt) ˆ U 0 = hω γaϕ 0 4 σ z

8 Polarization of current in the presence of scatterer Delta scatterer in the middle V = υδ(y L /2) Spin-dependent transmission probabilities T ( ) = 2 ( h 2 k 2 * F /2m ± U 0 cos(ωt) ) ( ) 2( h 2 k 2 F /2m * ± U 0 cos(ωt) )+ υ 2 m * /h 2 Landauer formula gives time dependent polarization P = T T U = 0 (υ 2 m * /h 2 )cos(ωt) T + T E (2E + 2 * 2 2U 2 2 F F υ m /h ) U 0 cos (ωt)

9 Polarized current can be further rectified by a shutter gate or alternatively we can apply AC voltage to get DC spin current. Polarizat tion P (u units U0/E EF) Time (units 2π/ω) ω =10GHz;υ 2 m * /h 2 = 0.4E U 4 F ;U 0 / E F = ; γ = m 1 ; L =1μm;ϕ 0 = 0.2;U ev

10 Multi-channel rod Gauge transformation ˆ h 2 H = 2m * + h2 2m * k x + f x ˆ σ z ψ = e ψ if (x,y,t ) σ ˆ z ih H if (x,y,t ) ˆ e σ z ψ t ( ) ( ) ( (x,y,t ) ˆ eif σ z ψ )= ˆ H ˆ = e if (x,y,t ) σ ˆ z H ˆ if (x,y,t ) ˆ e σ z 2 + h2 2m * k y γτx ˆ σ z + f y ˆ σ z + h ( k z + γτx[ sin(2 f ) ˆ σ y + cos(2 f ) ˆ σ ]) 2 + h f x t ˆ For simplicity only one channel open along the x-axis ˆ = h2 2 h2 H γτa 2D sin(2 ˆ ˆ 2m k * y + k 2m * z + f ) σ y + cos(2 f ) σ x 4 d < l so [ ] 2 σ z f (x,y,t) t ˆ σ z 2 + U(y) ˆ σ z + V(y,z) U(y) = U 0 sin( π L y)cos(ωt) ˆ σ z U 0 = hω γaϕ 0 4

11 Quasi 1D case When Δ so d << U 0 l so so inter-band transitions can be disregarded Delta potential along the y-axis, constant along the z-xis V(y,z) = υδ(y L /2) Polarization from Landauer formula P = M m M m (T m T m ) (T m + T m ) T m ( ) = 2( h 2 k 2 m /2m * ± U 0 cos(ωt) ) ( ) m 0 2( h 2 k 2 m /2m * ± U 0 cos(ωt) )+ υ 2 m * /h 2

12 Polarization as function of width arization P (units U0/EF) Pola Width (d/λf) ω =10GHz;υ 2 m * /h 2 = 0.4E U 4 F ;U 0 / E F = ; γ = m 1 ; L =1μm;ϕ 0 = 0.2;U ev

13 Simulations based on tightbinding model H ˆ = h2 k 2 + h2 k + γτa sin(2 ˆ σ + cos(2 ˆ 2D σ 2m * y 2m * z ( f ) y f ) x 4 [ ] H = ε ijσ c + c + ijσ + t c ijσ i+1 jσ c ijσ + c + ij +1σ +it so 2 + U(y) ˆ σ + V(y,z) z ) ( c ijσ ) ijσ ijσ ( c + ) i+1 jσ c ij σ [ sin(2 f )σ y ] σ σ + + c i+1 jσ c ij σ [ cos(2 f )σ x ] σ σ σ ijσ ( ) ε ijσ on-site energy, includes U and V b lattice spacing + h.c. d < l so t = 2 h 2 h2 2m * b 2 ; t so = 2m * bl so

14 Simulations based on tightbinding model Conductor Lead p Lead q R(A Σ ) [ H R(A p (i, j) ] ij Σ ) p (i, j) lead p self energy conductor Hamiltonian lead q self energy G R(A ) R(A ) = [ EI H Σ p ] 1 A Σ T = Tr Γ G R Γ G A p(q ) [ p(q ) p(q )] pq p q R Γ p(q ) = i Σ p(q ) [ ]

15 Numerical results for delta wall U0/EF) zation P (units U Polariz Time (units 2π/ω) ω =10GHz;υ 2 m * /h 2 = 0.4E U 4 F ;U 0 / E F = ; γ = m 1 ; L =1μm;ϕ 0 = 0.2;U ev

16 Numerical results for Anderson disorder Random on-site potential in the range {-W/2;W/2} Can be related to effective 2D mean free path l 2D = (6λ F E F 2 )/(π 3 a 2 W 2 )

17 l 2D = 30L l 2D =14L l 2D = 8L 2D P (units U0/EF) Time (units 2π/ω) ω =10GHz;υ 2 m * /h 2 = 0.4E U 4 F ;U 0 / E F = ; γ = m 1 ; L =1μm;ϕ 0 = 0.2;U ev

18 l 2D = 3.6L l 2D = 2L ω =10GHz;υ 2 m * /h 2 = 0.4E 4 F ;U 0 / E F = 2 10 ; γ = m 1 ; L =1μm;ϕ 0 = 0.2;U ev

19 Conclusions We report on piezo-spin effect in semiconductor structures (mechanical motion generating spin imbalance) We derive quasi 1D expression for polarization that captures the qualitative ti behavior For general case we apply numerical simulations based on tightbinding model and find that it should be possible to observe effect in realistic structures (e.g. g GaAs)

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