Spin orbit interaction in semiconductors
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1 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 International School of hysics Enrico Fermi: Quantum Spintronics and Related henomena Varenna, June 9
2 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
3 Carlos Egues, Varenna Important point from last lecture: Single vs two-subband Rashba model One Dirac cone Two Dirac cones
4 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](/docs-images/88/115783523/images/5-1.jpg "( ) z x Spins on the xy plane!")
5 [] 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
6 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
7 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!
8 Carlos Egues, Varenna Ref. [] Eigenvectors, Eigenvalues
9 Carlos Egues, Varenna Ref. [] Crossings Dirac cones Eigenvectors, Eigenvalues
10 Carlos Egues, Varenna Ref. [] Anticrossings Dirac cones Eigenvectors, Eigenvalues
11 Folding down Carlos Egues, Varenna
12 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
13 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
14 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 ( ) ( ) ( )
15 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
16 An example: auli from Dirac Carlos Egues, Varenna
17 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
18 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
19 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
20 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
21 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 m 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
22 Carlos Egues, Varenna Effective Hamiltonians Spin orbit interactions via the Kane model (Kane 57)
23 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
24 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
25 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
26 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
27 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!
28 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 m m m m 4, a ( ) n, n ( r ) e u ( r ) ( ) ( ) ( ) ir u r a n l u r l n, n, l
29 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, p x,p y,p z lh u u 7, split-off Valence bands, a ( ) n, n ( r ) e u ( r ) ( ) ( ) ( ) ir u r a n l u r l n, n, l
30 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
31 Carlos Egues, Varenna Diamond structure Unit cell : C, Si, & Ge
32 Carlos Egues, Varenna Diamond structure Unit cell : C, Si, & Ge Tetrahedral bonding
33 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
34 rimitive Cell (Wigner Seitz) Carlos Egues, Varenna
35 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
36 8x8 Kane model (bul) S S 3/, 3 / 3/, / 3/,- / 3/, 3/ /, / /,- / A z z i m z Eg Eg z 3 3 z 3 3 z (Kane parameter) ˆ x X S p z z Eg Eg Eg Eg Use of crystal symmetry! x i y / m Bands: det(a)= Carlos Egues, Varenna
37 Carlos Egues, Varenna Exact diagonalization (bul) S S /, / /,- / 3/, 3 / 3/, / 3/,- / 3/, 3/ z z z z z Eg 3 3 z Eg 3 3 A Eg z Eg 3 z Eg 6 3 Eg Note that C and D commute!
38 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, ()
39 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
40 Actual numerical solution Carlos Egues, Varenna
41 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 Kane model Bul energy bands: 8x8 Kane model E g =.35 ev =.8 ev (/nm) Carlos Egues, Varenna
42 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 Kane model E g =.59 ev =.34 ev (/nm) Carlos Egues, Varenna
43 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 E g =.66 ev =.9 ev (/nm) Carlos Egues, Varenna
44 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 =.8 ev E g = -.33 ev * m lh m 3 E g * me m 3 Eg E g (/nm) Carlos Egues, Varenna
45 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)
46 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)
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