Spin-Orbit Coupling

1. Spin-Orbit Coupling

2. Spin-Orbit CouplingFirst Some General Comments • AnImportant(in some cases) effect we’ve left out! • We’ll discuss it mainly for terminology & general physics effects only. • TheSpin-Orbit Coupling term in the Hamiltonian: Comes from relativistic corrections to the Schrödinger Equation. • It’s explicit form is Hso = [(ħ2)/(4mo2c2)][V(r)  p]σ V(r)  The crystal potential p = - iħ  The electron (quasi-) momentum σ the Pauli Spin Vector

3. 0 1 1 0 0 -i i 0 1 0 0 -1 • The cartesian components of the Pauli Spin Vector σ are 2  2 matrices in spin space: σx = ()σy = ()σz = () Hso 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 Hso = λLS This can be derived from the previous form with some manipulation!

4. The Spin-Orbit Coupling Hamiltonian: Hso = 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-. Hso adds to the Hamiltonian from before, & is used to solve the Schrödinger Equation. The new H is: H = (p)2/(2mo) + Vps(r) + λLS Now, solve the Schrödinger Equation with this H. Use pseudopotential or other methods & get bandstructures as before.

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

6. Hso = λ LS • The most important effect occurs at the top of the valence band at Γ= (0,0,0). In the absence of Hso,the bands there are p-like & triply degenerate. • Hso 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.

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

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

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

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