Spin-orbit coupling: Atom versus semiconductor crystal

We reconsider a key point in semiconductor physics, the splitting of the valence band states induced by the spin-orbit interaction, through a novel approach which uses neither the group theory formalism, nor the usual $\mathbf{L}·\mathbf{S}$ formulation valid for atoms but conceptually incorrect for periodic lattices; the angular momenta $\mathbf{L}$ and $\mathbf{J}$ have no meaning due to the absence of spherical symmetry. We show that for zinc-blende structures, the valence band eigenstates resulting from spin-orbit coupling are uniquely determined by: (i) the equivalence of the ($x,y,z$) crystal axes, and (ii) the threefold degeneracy of the valence band. The fact that these two conditions are also fulfilled by atomic $p$ states allows us to understand why the spin-orbit eigenstates for threefold atomic and valence electrons have exactly the same structure, albeit with drastic differences in the potential and electronic symmetries. We also come back to the commonly accepted understanding of the exciton-photon interaction in terms of bright and dark excitons having total angular momenta $J=(1,2)$, respectively, and present a simple derivation of this interaction which only relies on spin conservation.

Figure 1

Relevant semiconductor states according to the group theory irreducible representations and their denomination within atomic notations.

Figure 2

(a) Absorption of a photon excites an electron from the valence band $v$ to the conduction band $c$ while keeping its electron spin $s_z$. (b) In terms of electron and hole, the absorption of a photon creates an electron-hole pair with zero total spin.

Figure 3

(a) Interband Coulomb interaction conserves the spin $s$. (b) Electron-hole pairs having a zero total spin can couple to photon.

Figure 4

Components of the $O^{\prime \prime }$ velocity in the $F$ frame (a) and the $O$ velocity in the frame $F^{\prime \prime }$ (b).

Figure 5

Motion of the ($F, F^{\prime \prime }$) frames when $uv=0$.

Figure 6

(a) The frame $F^{\prime \prime }$ and (b) the frame $F$ are related by a rotation around the $z$ axis when $uv\ne 0$.