1. el el 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

2. Basics of k.p-theory for bulk
Problem: Band structure at k = 0 is known. How to
determine for k-vectors near k = 0?
𝐸 𝑐 𝒌 , 𝜓 𝑐 𝒌,𝒓 = ?
𝐻 𝑘 = 𝒑+𝒌 𝑚 0 +𝑉(𝒓)
Perturbation theory:
V(r) periodic
𝐻 𝒌=0 + ℏ 𝑚 0 𝒌∙𝒑+ ℏ 2 𝑘 2 2 𝑚 0 𝑢 𝑐𝑘 𝒓 = 𝐸 𝑐 𝒌 𝑢 𝑐𝑘 (𝒓)
𝐸 𝑐 𝑘 ≅ 𝐸 𝑐 ℏ 2 𝑘 2 2 𝑚 ℏ 2 𝑚 𝑛≠𝑐 𝑢 𝑐 0,𝒓 𝒌∙𝒑 𝑢 𝑛 0,𝒓 𝐸 𝑐 0 − 𝐸 𝑛
= 𝐸 𝑐 ℏ 2 𝑘 2 2 𝑚 𝑐 ∗

3. k.p theory for bulk (cont'd)
ℏ 2 𝑚 𝑛≠𝑐 𝑢 𝑐 0,𝒓 𝒌∙𝒑 𝑢 𝑛 0,𝒓 𝐸 𝑐 0 − 𝐸 𝑛
Advantage:
main contribution from top val. bands
Only 2 parameters determine mass:
Very few parameters that can be calculated ab-initio
or taken from experminent describe relevant electronic
structure of bulk semiconductors
Can be generalized for all bands near the energy gap:

4. k.p theory for bulk (cont'd)
ℏ 2 𝑚 𝑛≠𝑐 𝑢 𝑐 0,𝒓 𝒌∙𝒑 𝑢 𝑛 0,𝒓 𝐸 𝑐 0 − 𝐸 𝑛
Advantage:
main contribution from top val. bands

5. 𝒑 2 2𝑚 +𝑉 𝒓 +𝑈(𝒓) 𝜓 𝒓 =𝜀𝜓 𝒓 𝜓 𝒓 = 𝐹 𝑛 𝒓 𝑢 𝑛0 𝒓
Envelope Function Theory:
method of choice for electronic structure of mesoscopic devices
Problem: How to solve efficiently...
𝒑 2 2𝑚 +𝑉 𝒓 +𝑈(𝒓) 𝜓 𝒓 =𝜀𝜓 𝒓
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
𝜓 𝒓 = 𝐹 𝑛 𝒓 𝑢 𝑛0 𝒓 x
Periodic
Bloch Function u
Result: Envelope equation (1-band) builds on k.p-theory...
𝐸 𝑐 −𝑖𝛻 +𝑈 𝒓 −𝜀 𝐹 𝑛 𝒓 =0

6. 𝐸 𝑐 −𝑖𝛻 −𝑒𝑈 𝒓 𝐹 𝜈 𝒓 = 𝐸 𝜈 𝐹 𝜈 (𝒓)
Example for U(r): Doped Heterostructures +
Ec (z) + + + + EF EF + +
neutral
donors + + + Ec
Unstable
Charge transfer
Thermal equilibrium
Resulting electrostatic potential follows from ...
𝐸 𝑐 −𝑖𝛻 −𝑒𝑈 𝒓 𝐹 𝜈 𝒓 = 𝐸 𝜈 𝐹 𝜈 (𝒓)
Fermi distribution function
𝛻 2 𝑈 𝒓 = 4𝜋𝑒 𝜀 0 𝑁 𝐴 𝒓 − 𝑁 𝐷 𝒓 − 𝜈 𝑓 𝜈 𝐹 𝜈 𝒓 2
Self-consistent “Schrödinger-Poisson” problem

7. Quantization in heterostructurescb
Band edge discontinuities
in heterostructures lead to
quantized states
Material A B A vb cb
electron
Schrödinger eq. (1-band):
hole vb