Spin-Orbit Coupling

Spin-Orbit Coupling First Some General Comments An Important (in some cases) effect we’ve left out! We’ll discuss it mainly for terminology & general physics effects only. The Spin-Orbit Coupling term in the Hamiltonian: Comes from relativistic corrections to the Schrödinger Equation. It’s explicit form is H so = [(ħ 2 )/(4m o 2 c 2 )][  V(r)  p]  σ V(r)  The crystal potential p = - iħ   The electron (quasi-) momentum σ  the Pauli Spin Vector

The cartesian components of the Pauli Spin Vector σ are 2  2 matrices in spin space: σ x = ( ) σ y = ( ) σ z = ( ) H so has a small effect on electronic bands. It is most important for materials made of heavier atoms (from down in periodic table). It is usually written H so = λL  S This can be derived from the previous form with some manipulation! i i

The Spin-Orbit Coupling Hamiltonian: H so = λL  S λ  A constant  “The Spin-Orbit Coupling Parameter”. Sometimes, in bandstructure theory, this parameter is called . L  orbital angular momentum operator for the e -. S  spin angular momentum operator for the e -. H so adds to the Hamiltonian from before, & is used to solve the Schrödinger Equation. The new H is: H = (p) 2 /(2m o ) + V ps (r) + λL  S Now, solve the Schrödinger Equation with this H. Use pseudopotential or other methods & get bandstructures as before.

H so = λL  S Spin-Orbit Coupling’s most important & prominent effect is: Near band minima or maxima at high symmetry points in BZ:  H so Splits the Orbital Degeneracy. The most important of these effects occurs near the valence band maximum at the BZ center at Γ = (0,0,0)

H so = λ L  S The most important effect occurs at the top of the valence band at Γ = (0,0,0). In the absence of H so, the bands there are p-like & triply degenerate. H so partially splits that degeneracy. It gives rise to the “Spin-Orbit Split-Off” band, or simply the “Split-Off” band. Also, there are “heavy hole” & “light hole” bands at the top of valence band at Γ. YC use the k  p method & group theory to discuss this in detail.

Schematic Diagram of the bands of a Direct Gap material near the Γ point, showing Heavy Hole, Light hole, & Split-Off valence bands.

Calculated bands of Si near the Γ point, showing Heavy Hole, Light Hole, & Split-Off valence bands.

Calculated bands of Ge near the Γ point, showing Heavy Hole, Light Hole, & Split-Off valence bands.

Calculated bands of GaAs near the Γ point, showing Heavy Hole, Light Hole, & Split-Off valence bands.