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

8

7

3so

2

3so lh

so

hh

1

15hh

lh

el el6

GAPE

Non-relativistic solidWith spin-orbit coupling included

p3-As

s1-Ga

E

kr

6-fold

4-fold

2-fold

Atom

Basics of k.p-theory for bulk

(0), (0, )c cE r

Problem: Band structure at k = 0 is known. How to determine for k-vectors near k = 0?

Perturbation theory: V(r) periodic

𝐸𝑐 (𝒌 ) ,𝜓𝑐 (𝒌 ,𝒓 )=?

𝐻𝑘=(𝒑+𝒌 )2

2𝑚0

+𝑉 (𝒓 )

[𝐻𝒌=0+ℏ𝑚0

𝒌 ∙𝒑+ℏ2𝑘2

2𝑚0]𝑢𝑐𝑘 (𝒓 )=𝐸𝑐 (𝒌 )𝑢𝑐𝑘 (𝒓 )

𝐸𝑐 (𝑘 )≅ 𝐸𝑐 (0 )+ℏ

2𝑘2

2𝑚0

+ ℏ2

𝑚02∑𝑛≠ 𝑐

|⟨𝑢𝑐 (0 ,𝒓 )|𝒌 ∙𝒑|𝑢𝑛 (0 ,𝒓 ) ⟩|𝐸𝑐 (0 )−𝐸𝑛 (0 )

2

¿𝐸𝑐 (0 )+ℏ2𝑘2

2𝑚𝑐∗

k.p theory for bulk (cont'd)

Advantage: main contribution from top val. bands

* 02

0

21

c

gap

mm

Pm E

Only 2 parameters determine mass: 22

0 0

2 2 2, 20gap

PE eV

m m a

h

ℏ2

𝑚02 ∑𝑛≠ 𝑐

|⟨𝑢𝑐 (0 ,𝒓 )|𝒌 ∙𝒑|𝑢𝑛 (0 ,𝒓 ) ⟩|𝐸𝑐 (0 )−𝐸𝑛 (0 )

2

Very few parameters that can be calculated ab-initioor taken from experminent describe relevant electronicstructure of bulk semiconductors

Can be generalized for all bands near the energy gap:

k.p theory for bulk (cont'd)

Advantage: main contribution from top val. bands

* 02

0

21

c

gap

mm

Pm E

22

0 0

2 2 2, 20gap

PE eV

m m a

h

ℏ2

𝑚02 ∑𝑛≠ 𝑐

|⟨𝑢𝑐 (0 ,𝒓 )|𝒌 ∙𝒑|𝑢𝑛 (0 ,𝒓 ) ⟩|𝐸𝑐 (0 )−𝐸𝑛 (0 )

2

Envelope Function Theory:method of choice for electronic structure of mesoscopic devices

Envelope Function F

Periodic Bloch Function u

Non-periodic external potential:slowly varying on atomic scale

2 2 *( ) /(2 )cE k k mr

h ( )U r

Problem: How to solve efficiently...

Periodic potential of crystal:rapidly varying on atomic scale

Ansatz: Product wave function ...

x

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

𝜓 (𝒓 )=𝐹 𝑛 (𝒓 )𝑢𝑛0 (𝒓 )

Example for U(r): Doped Heterostructures

++

Ec

+ + EF ++ + + + + Ec (z)EF

1E

Thermal equilibriumCharge transfer

Resulting electrostatic potential follows from ...

Fermi distribution function

Self-consistent “Schrödinger-Poisson” problem

Unstable

neutraldonors

𝛻2𝑈 (𝒓 )=4𝜋 𝑒

𝜀0 [𝑁 𝐴 (𝒓 )−𝑁𝐷 (𝒓 )−∑𝜈

𝑓 𝜈|𝐹𝜈 (𝒓 )|2]

Quantization in heterostructures

A B A

cb

vb

Band edge discontinuitiesin heterostructures lead toquantized states

electron

hole

Schrödinger eq. (1-band):

Material

cb

vb

*

1r r r

r l l ll

U F EFm