Spin Orbit Coupling - Solid State Physics - Lecture Slides, Slides of Solid State Physics

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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 for

terminology & general physics effects

only.

•^

The

Spin-Orbit Coupling

term in the Hamiltonian:

Comes from relativistic corrections to the Schrödinger Eqtn.

-^

It’s explicit form is

H

so^

= [(

)/(4m

2 co

2 )][

V(r)

p]

V(r)

^

The crystal potential

p = - i

ħ 



The electron (quasi-) momentum σ^

^ the

Pauli Spin Vector

-^

The Spin-Orbit coupling Hamiltonian:

Hso

=^

λ L

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 /(2m

) + Vo^

(r) +ps

λ L

S

-^ Solve the Schrödinger Equation with this

H.

-^ Use pseudopotential or other methods.–^ Get bandstructures as before

H

so^

=^

λ L

S

Spin-Orbit Coupling’s most important &

prominent effect is:

Near band minima or maxima at high symmetry points in BZ:

H

so^

SPLITS ORBITAL DEGENERACY!

•^

The most important

of these effects occur near the

valence band maximum at the center of the BZ at

Γ^

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

GaAs

near

the

point, showing

Heavy Hole,Light Hole, &Split-Off^ valence bands.