SpinOrbit Coupling SpinOrbit Coupling First Some General Comments

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 Hso = [(ħ 2)/(4 mo 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 space: σx = ( 0 1 1 0 ) σy = ( 0 -i i 0 ) σz = ( ) 1 0 0 -1 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!

• The Spin-Orbit Coupling Hamiltonian: Hso = λL S λ A constant “The Spin-Orbit Coupling Parameter”. • Sometimes 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/(2 mo) + Vps(r) + λL S • Now, solve the Schrödinger Equation with this H. Use pseudopotential or other methods & get bands as before.

Hso = λL S Spin-Orbit Coupling’s most important & prominent effect is: Near band minima or maxima at high symmetry points in BZ: Hso 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)

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.

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 Ga. As near the Γ point, showing Heavy Hole, Light Hole, & Split-Off valence bands.