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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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## Spin-Orbit Coupling

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**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**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 = λLS This can be derived from the previous form with some manipulation!**The Spin-Orbit Coupling Hamiltonian:**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.**Hso = λLS**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)**Hso = λ LS**• 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 kp 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.**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.

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