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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 first some general comments
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 = 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, 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.

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)

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.


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