Band structure of cubic semiconductors (GaAs) near the center of the Brillouin zone

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Band structure of cubic semiconductors (GaAs) near the center of the Brillouin zone

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Band structure of cubic semiconductors (GaAs) near the center of the Brillouin zone

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Band structure of cubic semiconductors (GaAs)

near the center of the Brillouin zone

E

el

el

s1-Ga

4-fold

hh

p3-As

6-fold

hh

lh

lh

2-fold

so

With spin-orbit coupling

included

Atom

Non-relativistic solid

Basics of k.p-theory for bulk

Problem: Band structure at k = 0 is known. How to

determine for k-vectors neark = 0?

Perturbation theory:

V(r) periodic

Can be generalized for all bands near the energy gap:

Very few parameters that can be calculated ab-initio

or taken from experminent describe relevant electronic

structure of bulk semiconductors

k.p theory for bulk (cont'd)

Advantage:

main contribution from top val. bands

Only 2 parameters determine mass:

k.p theory for bulk (cont'd)

Advantage:

main contribution from top val. bands

Envelope Function Theory:

method of choice for electronic structure of mesoscopic devices

Problem: How to solve efficiently...

Periodic potential of crystal:

rapidly varying on atomic scale

Non-periodic external potential:

slowly varying on atomic scale

Ansatz: Product wave function ...

Envelope

Function F

Periodic

Bloch Function u

x

Result: Envelope equation (1-band) builds on k.p-theory...

+

+

+

+

+

+

+

Example for U(r): Doped Heterostructures

Ec (z)

+

EF

EF

+

+

neutral

donors

Ec

Unstable

Charge transfer

Thermal equilibrium

Resulting electrostatic potential follows from ...

Fermi distribution function

Self-consistent “Schrödinger-Poisson” problem

Quantization in heterostructures

cb

Band edge discontinuities

in heterostructures lead to

quantized states

Material

A

B

A

vb

cb

electron

Schrödinger eq. (1-band):

hole

vb