UNIVERSIDADE DE SÃO AULO Instituto de Física de São Carlos Spin orbit interaction in semiconductors J. Carlos Egues Instituto de Física de São Carlos Universidade de São aulo egues@ifsc.usp.br International School of hysics Enrico Fermi: Quantum Spintronics and Related henomena Varenna, June 9
Lectures,3 From Dirac to Bloch electrons (folding down) Where it all comes from: the.p approach Kane model: bul and heterostructures Effective models: from 3D to D (& D) - single-band Rashba model - two-band model: interband coupling - Band inversion and edge states ( TI ) - Zitterbewegung, SHE, Carlos Egues, Varenna
Carlos Egues, Varenna Important point from last lecture: Single vs two-subband Rashba model One Dirac cone Two Dirac cones
Carlos Egues, Varenna Usual Rashba model Exact solution: H H p p p p m s x y x y y x R ( sr ) s( ) s R, s m m -dependent spinor! ir e ( ) y, e i i ( ) A se i x R m R m (, ) s x y
[] Carlos Egues, Varenna Spin expectation values: i ( ) x e cos ( ) i ( ) e y sin ( ) z x Spins on the xy plane! y y x (, x y ) F F araboloids F x y [] x (, ) s x y
Carlos Egues, Varenna Two-subband Rashba model H i i i i i i i i Not a quantum wire! H px py orb spin z spin m c p p p p p p orb x y y x z x y y x x x y y x
Carlos Egues, Varenna Two-subband Rashba model H i i i i i i i i,, x y orb spin z spin m c H p p p p p p p p orb x y y x z x y y x x x y y x Not a quantum wire!
Carlos Egues, Varenna Ref. [] Eigenvectors, Eigenvalues
Carlos Egues, Varenna Ref. [] Crossings Dirac cones Eigenvectors, Eigenvalues
Carlos Egues, Varenna Ref. [] Anticrossings Dirac cones Eigenvectors, Eigenvalues
Folding down Carlos Egues, Varenna
Carlos Egues, Varenna Folding down ) Eigenvalue problem: H ) Split the Hamiltonian into blocs: ( & Q subspaces) H H Q H HQQ H H H H Q Q Q Q Q Q Q Q (I) (II) (II): H ( ) ( ) Q I HQ Q Q I HQ HQ (III) (I): H H ( I H ) H Effective Hamiltonian Q Q Q H, H H H I H H Q Q Q (not a Schroedinger Equation) Motion in : particle in an effective potential due to coupling to Q Veff
Carlos Egues, Varenna Renormalization of H, H H H I H H Q Q Q Note that subspaces and Q are not completely decoupled because but hence d r 3 3 Q Q ( I H ) H Q Q Q d r 3 H Q( I H Q) ( I H Q) H d r Q 3 I HQ( I HQ) ( I HQ) H Q d r
Carlos Egues, Varenna Renormalization of 3 I HQ( HQ) ( HQ) H Q d r ( ) ( ) 3 I ( ) ( ) d r I ) ( I ( ) ( ) ( ) 3 dr 3 dr ( ) renormalization ( ) I ( ) ( ) ( )
Carlos Egues, Varenna Full decoupling ( ) Substituting into : H H ( ) ( ) ( ) H ( ) ( ) ( ) H ( ) [ I ( )] Summarizing: H H Q HQ H Q Q Q H ( ) ( ) [ ( )] H I Q H H H H H Q Q independent of energy to nd order H ( ) I ( ) ( ) ( ( H ) H ) Q Q
An example: auli from Dirac Carlos Egues, Varenna
Carlos Egues, Varenna Dirac equation Dirac equation (spin /) + arbitrary potential (coulomb, crystal pot.) c p m c V() r The 4x4 Dirac matrices obey A particular ( standard ) representation: i j j i I ij, i i I i, i I i i Identity matrix i x, y, z, I i Hence mc I V ( r) I c p c p mc I V ( r) I Q Q
Carlos Egues, Varenna From Dirac to auli mc I V ( r) I c p c p mc I V ( r) I Q Q ( 4 Free particle (i.e., V = ) eigenenergies: c p m c ) mc Let us redefine the origin of energy: (now ) V ( r) cpz c( px ipy ) V ( r) c( px ipy ) cpz cpz c( px ipy ) mc I V ( ) r Q Q c( p ip ) cp m c I V ( r) x y z Q Q mc
Carlos Egues, Varenna From Dirac to auli mc I V ( r) I c p c p mc I V ( r) I Q Q ( 4 Free particle (i.e., V = ) eigenenergies: c p m c ) mc Let us redefine the origin of energy: (now ) V ( r) I c p c p mc I V ( r) I Q Q H H Q H V ( r) HQ c p HQ HQ H Q Q Q H m c I V ( r) I Q mc
Carlos Egues, Varenna From Dirac to auli (cont.) Folding down recipe: H H Q HQ H Q Q Q H independent ( ) ( of energy HQ) to H nd order Q H H I ( ) ( ) [ ( )] Q H H H H H Q Q ( ) I ( ) ( ) ( ( H ) H ) Q Q H I V( r ) I c p c m c V( r) p I ( ) I c p c [ m c V( )] p r ( ) I c ( ) I c p 8mc 4 p 4mc 4
Carlos Egues, Varenna From Dirac to auli (cont.) p I p p H I c V ( r) I c p c p I c c 4 4 4 8m c mc V ( r) 8m c 8m c After some manipulations (exercise/notes), we find the auli equation H 4 p V( ) V m r p 4mc 4m c p 8mc usual K + V mass correction Spin orbit int. Darwin term ( energy independent!) H V Interestingly, Taylor expanding the classical relativistic electron energy 4 mc m c m c cp m c m 4 c p c p p p c 4 h.o.t 4 mc 3 mc 8 mc m 8 mc
Carlos Egues, Varenna Effective Hamiltonians Spin orbit interactions via the Kane model (Kane 57)
Carlos Egues, Varenna Bloch electrons Solids ( crystal ): crystalline lattice ( ions ) + electrons R naxˆ Equivalent form: n=3 n= n= For each a periodic potential: a H pu ( r) u ( r), m m ( pˆ i) Find solutions in the unit cell! ( u ( r ) is periodic in R ) n ( r), n, u (non-interacting) V( r) V( r R) ˆ p H V ( r) m Bloch s theorem H i R ( r R) e ( r) i r ( r) e u ( r), u( r) u( R r) there are several solutions n=3 n= n= n: band index (Bloch function) (periodic part of the Bloch function) eriodicity Bloch s theorem Energy bands (continuum) ( r) e n, n, ir u n, ( r) Non parabolic in general
Carlos Egues, Varenna.p approach: bul Single-particle Schrödinger s equation: no spin-orbit case Inserting ( r ) e u ( r) ir n, n, (Bloch s theorem) in the above: H pu ( r ) u ( r ), n n u n m m H H H ( ) Hu V r m u ( r ) a ( ) ( ) n l u r l n ( r) u ( R r) n u n n n ( pˆ i) (Assumed nown!) ) Expand: ( mixing of bands, complete set, finite in practice) l u ( r ) u ( ) m m r ( bra-et notation) ) Multiply Schrödinger s equation from the left by and integrate: l H, pˆ H V( r ), V( r ) V( R r ) m bare electron mass : crystalline periodic potential ( ) u ( ), m H ul um pul al u n m ul al m m l l l lm um p ul al ( ) l m m Matrix equation: (still exact!) Energy bands wavefunctions
Carlos Egues, Varenna Matrix equation ( H V), H l l l (Assumed nown) c, c l l l l l l l l ( H V ) cl l cl l l l m c ( ) cv l l l m l l l l ( l ) ml Vml cl, Vml m V l l V V N c V c V3 V N V c N VN N N Ac Solutions: det A 3 3 N c, c, c,, c,,,, N
Carlos Egues, Varenna The correct starting point Crystalline periodic potential H p V( r) p 4 m 4mc 4m c 8 V p m c V H p V( ) V m r 4mc p p V( r) V p Still periodic! m 4m c
Carlos Egues, Varenna Spin orbit Add the term H so 4m c V p 4m c V p V dv r 4m c r p (Dirac) to the Hamiltonian: dv L s L s r dr Matrix equation (still exact!) (e.g. central potential:, usual coupling) dr r mc Straightforwardly l l lm um p u l al ( ) l m V m 4mc Bare inetic energy: wrong curvature (i.e., mass ) for valence bands. Note that now Hu, n nu n contains s-o effects -dependent spin-orbit terms: neglected in the 8x8 Kane model (zero for the conduction band ) [no cubic Dresselhaus or cubic Rashba; (Krich & Halperin RL/7)]. H H V ( r ) Vp m 4mc Spin-orbit interaction only here!
Carlos Egues, Varenna Truncating the = set 8x8 Kane model : u, u, u3, u4, u5, u6, u7, u8 u p u u p u u p u 3 8 m m m m a( ) u p u u p u u p u a ( ) 3 8 m m m m a3( ) u p u u p u u p u a ( ) a5( ) a6( ) a7( ) a8( ) u8 p u u8 p u u8 p u3 8 8 m m m m 3 3 3 3 3 8 m m m m 4, a ( ) n, n ( r ) e u ( r ) ( ) ( ) ( ) ir u r a n l u r l n, n, l
Carlos Egues, Varenna ictorically u u, n, s states Conduction band Hu u n n n E g hh u u u u 3, 4, 5, 6 3 4 5 6 p x,p y,p z lh u u 7, 8 7 8 split-off Valence bands, a ( ) n, n ( r ) e u ( r ) ( ) ( ) ( ) ir u r a n l u r l n, n, l
Carlos Egues, Varenna Tight binding view: choosing u n Atoms Solids n=3 n= n= n=3 n= n= Narrow bands higher DOS F ( ) density of states (DOS) a gaps periodic potential: V( r) V( r R) energy bands
Carlos Egues, Varenna Diamond structure Unit cell : C, Si, & Ge
Carlos Egues, Varenna Diamond structure Unit cell : C, Si, & Ge Tetrahedral bonding
Carlos Egues, Varenna Zincblend structure Unit cell cation sub lattice (FCC) Anion sub lattice (FCC) (also tetrahedral bonding) Mn: Zn: Se: e.g.: ZnSe, GaAs (II-VI, III-V) e.g.: Zn x Mn -x Se
rimitive Cell (Wigner Seitz) Carlos Egues, Varenna
Carlos Egues, Varenna Truncated basis set: 8 = functions E g Band structure ( point) s states p x,p y,p z J=L+s : Known solutions : spin orbit coupling 3i 4m c X V x p hh lh split-off y V y p x Y u ( r ) is( r) S u ( r ) is( r) S u 3( r ) X i Y u 4( r ) X i Y Z 6 3 u 5( r ) X i Y Z 6 3 u 6( r ) X i Y u 7( r ) X i Y Z 3 3 u 8( r ) X i Y Z 3 3 X Y Z xf ( r) yf ( r) zf ( r) p x p y p z
8x8 Kane model (bul) S S 3/, 3 / 3/, / 3/,- / 3/, 3/ /, / /,- / A z z i m z 3 6 3 Eg Eg z 3 3 z 3 3 z (Kane parameter) ˆ x X S p 3 6 3 3 z z 6 3 3 3 Eg Eg Eg Eg Use of crystal symmetry! x i y / m Bands: det(a)= Carlos Egues, Varenna
Carlos Egues, Varenna Exact diagonalization (bul) S S /, / /,- / 3/, 3 / 3/, / 3/,- / 3/, 3/ z z 3 3 3 6 z z 3 3 6 3 z Eg 3 3 z Eg 3 3 A Eg z Eg 3 z Eg 6 3 Eg Note that C and D commute!
Carlos Egues, Varenna A useful theorem A B M C D If the matrices C and D commute (CD = DC), then det( M ) det( AD BC ) Silvester, Math Gazette, 46-467 ()
Carlos Egues, Varenna Bul effective masses det( M ) det( AD BC ) Doubly degenerate ( hh ) characteristic polynomial: six bands remaining bands: e, lh, split-off 3 E g Eg Eg For small we find (Kane): c( c 3 E g g ( ) E m hh g hh m m 3 3( Eg) ) ( ), m E m m m 3 E E! lh m 3 Eg mlh mlh m 3 Eg 4 c c g g Wrong sign for the hh mass! (8x8 Kane model) ( ) ( ) 4 lh, 4 so( ) ( ), m 3 so E g mso mso m 3 Eg Evan O. Kane, hys. Chem. Solids,, 49 (957) E g
Actual numerical solution Carlos Egues, Varenna
Energy (ev) I. Vurgaftman et al., AR 89, () Evan O. Kane, hys. Chem. Solids,, 49 (957) InSb m e /m m hh /m m lh /m m so /m Exp..35 -.63 -.5 -. Kane model.34. -.3 -.555 Bul energy bands: 8x8 Kane model E g =.35 ev =.8 ev (/nm) Carlos Egues, Varenna
Energy (ev) I. Vurgaftman et al., AR 89, () Evan O. Kane, hys. Chem. Solids,, 49 (957) GaAs Bul energy bands: 8x8 Kane model m e /m m hh /m m lh /m m so /m Exp..67 -.3496 -.9 -.79 Kane model.53. -.88 -.43 E g =.59 ev =.34 ev (/nm) Carlos Egues, Varenna
Energy (ev) Evan O. Kane, hys. Chem. Solids,, 49 (957) E. G. Novi et al., RB 7, 53 (5) CdTe Bul energy bands: 8x8 Kane model Exp. ~.96 m e /m m hh /m m lh /m m so /m Kane model.88. -.3446 -.678 E g =.66 ev =.9 ev (/nm) Carlos Egues, Varenna
Energy (ev) E. G. Novi et al., RB 7, 53 (5) Evan O. Kane, hys. Chem. Solids,, 49 (957) Note band inversion! Bul energy bands: 8x8 Kane model HgTe m e /m m hh /m m lh /m m so /m Exp. -- -- Kane model.46. -.3 -.45 =.8 ev E g = -.33 ev * m lh m 3 E g * me m 3 Eg E g (/nm) Carlos Egues, Varenna
Energy (ev) Carlos Egues, Varenna Band inversion & edge states: TI CdTe HgTe CdTe Metallic surface (or edge if D) Gap closes at interface HgTe Gapless edge/surf. states: localized at interface robust (TR protected, if single Kramers pair) (/nm) Crucial ingredient: Spin orbit interaction (.p) anratov et al. 85 (/nm)
Carlos Egues, Varenna Summary Lec Folding down: From Dirac to auli (s-o from the Dirac eq.) Bloch electrons & spin orbit (periodic pot.) Where it all comes from: the.p approach Kane model for bul & examples Inverted band structures & edge states (anratov et al. 85)