p theory for bu seconductors The attce perodc ndependent partce wave equaton s gven by p + V r + V p + δ H rψ ( r ) = εψ ( r ) (A) 4c In Eq. (A) V ( r ) s the effectve attce perodc potenta caused by the ons and core eectrons: V ( r + n a ) = V ( r ); = x, y, z (A) where na s an arbtrary vector n Bravas attce. It can be shown that 4 c ( V ) p (A) n Eq. (A) corresponds to spn-orbt nteracton. Spn-orbt nteracton We can transfor A by ( V ) p = ( V p) = S ( V p). 4 c 4 c c If we woud be consderng free atos (centra fed potenta) we coud wrte dv r V =. dr r Ths gves the faar spn-orbt ter dv dv c r S p = S r dr L. dr r c The reatvstc correctons are denoted by δ H r n Eq. (A). In ost cases these correctons are fary sa.
Boch's theore and boundary condton Accordng to Boch's theore the soutons of Eq. (A) satsfy d Tdψ ( r) = ψ ( r + d ) = e ψ ( r ), (A4) where d s an arbtrary vector n the Bravas attce. Condton (A4) pes ψ n r ( r) = Nu ( r ) e (A5) n where N s for norazaton and ( r ) the attce perodc atoc Boch state: u n u n ( r + d ) = u ( r ) (A6) n Perodc boundary condton and Eq. (A4) w restrct the vaues of : ψ nκ ( r + N a ) = ψ nκ ( r ) ( N ; = x, y, z), (A7) where a ; = x, y, z are the prtve vectors of the Bravas attce. Consequenty wave vector ust fuf N a = π n ( n =,,..; = x, y, z) (A8) and the vaues of the wave vector are = ( n / N ) a, where n s an nteger. Sovng for atoc Boch states Insertng Boch state (A5) n (A) we obtan p + V ( r) + ( V ) p + + ( V ) u p + n 4c 4c r. = ε u ( r) n n (A9) We assue that the oca snge partce Hatonan H ( = ) has a copete ortonora set of egen- functons u n.e. H ( = ) u n = ε n u n. (A) An arbtrary ( we behavng ) attce perodc functon can then be wrtten as a seres expanson usng egenfunctons u n.
Matrx foruaton Next use seres expanson u = c ( ) u (A) n =,, N n Eq. (A9) and fnd atrx equaton for deternng the unnown coeffcents c ( ). Ths gves atrx e egenvaue equaton (for a =,... N ) ε n ε n + δ n + u n + ( V ) u c = p. (A) 4c Sovng atrx equaton (A) gves the egenstates of wave equaton (A9). When s sa non-dagona ters are sa and the owest order souton for egenstate.e. u n u and c ( ) = δ n. (A4) The correspondng egenvaue s ε n = ε n +. (A5) Perturbaton theory If the nondagona ters are "sa" we can prove egenvaue (A5) by second order perturbaton theory ε u n H I u u H I u n n = ε n + u n u n + n ε n ε where, (A6) H I = p + ( V ). (A7) 4c Egenvaue (A6) foows of course drecty by usng H ( = ) as the zeroth order Hatonan and usng W = + H as a perturbaton. I
Effectve ass The second order egenenerges can be wrtten where ε n n n n ε n ε = ε + + (A8) = p + ( V ). (A9) 4 c It s seen that the egenvaue depends n the vcnty of the Γ pont quadratcay on the wave vector coponents. Eq. (A8) s often wrtten as ε n = ε n + α β ; α, β x, y, z αβ = (A) αβ µ n where α β αβ π nπ n µ n = δ αβ + (A) ε ε s the effectve ass tensor. n n Kane s theory p foruaton n Eq. A s based on perturbaton theory. A ore exact approach capabe of ncudng strong band to band nteractons s provded by Eq. A. Incudng a copete set of bass states n Eq. A s not feasbe nuercay. However, we can prove the p theory drastcay f we ncude n A those bands u that are strongy couped and correct ths approxaton by treatng the nfuence of dstant (energetcay) bands perturbatvey. 4
j-j couped atoc Boch states We choose bass set u ( r ) to be egenfunctons of operators J and z u J, J ψ J, J ε ( = ) Presentaton of group Td u, S Γ 6 u, Z + ( X + Y 6 ε Γ 8 u5, X + Y ) ε Γ 8 u7, X + Y ) + Z ε Γ 7 u, S Γ 6 u4, X Y ) Z 6 ε Γ 8 u6, X Y ) ε Γ 8 u 8, X Y ) + Z ε Γ 7 J. Legend of Tabe I The conducton bands and 6 vaence bands n Tabe I are the strongy couped and ost portant eectron bands n the copound seconductors. u are egenstates of Hatonan p H = = V ( V + r + ) 4c p and dagonaze the spn-orbt nteracton. Γ 6 corresponds to conducton band, Γ 8 denotes the heavy hoe ( j = ± / ) and Γ 8( j = ± / ) the ght hoe bands. Γ 7 s nown as spt-off band. The orgn of the energy scae s at the band edge of the conducton band e. ε Γ =. 6 The HH- and LH bands are degenerate at =, ε Γ = ε. 8 ε s caed band gap and s the spn orbt spttng. 5
Tabe II 8 band Hatonan Eq. A, A (wthout correcton fro weay couped bands) S,,, S,,, S Pz P Pz P P P, Pz, P, Pz P P S P Pz P Pz, P Pz, P, P Pz Legend of Tabe II Hatonan n Tabe II negects n (A) the spn-orbt ter that depends on the wave vector e the ter ( V ) 4 c (Ths ter s assued to be sa cose to = ). (A) Soe notatons used n Tabe II ± = ± ( x y ) ε = ε ε (band gap) Γ Γ 6 8 = ε ε (spn-orbt spttng) Γ Γ 8 7 (A) P = S p X = S p Y = S p Z (eectrc dpoe aptude) x y z 6
Coupng wth dstant bands The eght bands ncuded n Hatonan of Tabe II are not enough. Addtona eectrons bands are ncuded by perturbaton theory. For exact dervaton see Consder wave equaton H + W ψ = εψ, (A8) where W = p + (the dependent spn-ter s thrown away). The egenvaues ε correspond to the egenstates =,,..8 of the Hatonan H - these are the eght tghty couped bands of Tabe II ( u ( r ) ) Assue another set of egenstates of H ony weay couped to by W. These are caed weay couped or dstant bands. Perturbaton theory Tra souton of ncudng the dstant bands: ψ = c + c (A9) Insert (A9) n Eq. (A8) c ε ε δ + W + c W = c ε ε δ µ W µ + + c µ W =. (A) If s a strongy couped band: c and <<. The second of Eqs. (A) gves c c c W ε ε (A) 7
Correcton to the 8-band Hatonan Insert A n the frst of Eqs. (A) c W W W ε ε ( ε ε ) δ + + =. (A) We concude that the nfuence of the dstant bands can be taen nto account by repaceent p + p + + p ε ε p (A) n wave equaton (A8). It can be shown that the Hatonan of dstant band nteracton W s gven by Tabe III Correcton W S fro dstant bands,,, S,,, S ' c, G H ( G F ) I H *, H F H I I *, ( G F ) H ( F + G) H I S ' c * *, I H G H ( G F ), I I H F H * * * * * *, H I ( G F ) H ( F + G ) 8
Legend of Tabe III B F ( ) = A + ( ) B z G( ) = A ( ) z L + M, L A = B = M, C = D B, D = N H ( ) = D ( ) z x y I ( ) = B( ) D x y x y where X p x X p y L = + M = + ε ε ε ε X p x p y Y + X p y p x Y S p N = x = ε ε c ε ε ε = Average band edge energy of the dstant bands 9