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:
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Comes from relativistic corrections to the Schrödinger Equation.
Hso = [(ħ2)/(4mo2c2)][V(r) p]σ
V(r) The crystal potential
p = - iħ The electron (quasi-) momentum
σ the Pauli Spin Vector
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).
Hso = λLS
This can be derived from the previous form with some manipulation!
Hso = λLS
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) + λLS
Now, solve the Schrödinger Equation with this H. Use
pseudopotential or other methods & get bandstructures as before.
Spin-Orbit Coupling’smost important & prominent effect is:
Near band minima or maxima at high symmetry points in BZ:
HsoSplits the Orbital Degeneracy.
Γ = (0,0,0)
p-like & triply degenerate.
Schematic Diagram of the bands of a Direct Gap material near the Γ point, showing Heavy Hole, Light hole, & Split-Off valence bands.
Calculatedbands of Si near the Γ point, showing Heavy Hole, Light Hole, & Split-Off valence bands.
Calculatedbands of Ge near the Γ point, showing Heavy Hole, Light Hole, & Split-Off valence bands.
Calculatedbands of GaAs near the Γ point, showing Heavy Hole, Light Hole, & Split-Off valence bands.