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 for terminology & general physics effects only. • TheSpin-Orbit Coupling term in the Hamiltonian: Comes from relativistic corrections to the Schrödinger Eqtn. • 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 components of thePauli Spin Vector σare 2  2matrices 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). • This is usually written Hso = λLS We can derive this 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  The orbital angular momentum operator for the e-. S  The 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 • Solve the Schrödinger Equation with this H. • Use pseudopotential or other methods. • Get bandstructures as before

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

6. Hso = λ LS • The most important effectoccurs at the top of the valence band atΓ= (0,0,0). In the absence ofHso,the bands there are p-like & triply degenerate. • Hsopartially 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 Diagramof the bands of aDirect Gapmaterial near the Γ point, showingHeavy Hole, Light hole, & Split-Offvalence bands.

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

9. Calculatedbands of Ge near the Γ point, showingHeavy Hole, Light Hole, & Split-Offvalence bands.

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