band structure of cubic semiconductors (gaas) near the center of the brillouin zone lh so hh lh el...
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![Page 1: Band structure of cubic semiconductors (GaAs) near the center of the Brillouin zone lh so hh lh el Non-relativistic solid With spin-orbit coupling included](https://reader036.vdocuments.mx/reader036/viewer/2022082411/5697bf921a28abf838c8f04f/html5/thumbnails/1.jpg)
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
![Page 2: Band structure of cubic semiconductors (GaAs) near the center of the Brillouin zone lh so hh lh el Non-relativistic solid With spin-orbit coupling included](https://reader036.vdocuments.mx/reader036/viewer/2022082411/5697bf921a28abf838c8f04f/html5/thumbnails/2.jpg)
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
πΈπ (π ) ,ππ (π ,π )=?
π»π=(π+π )2
2π0
+π (π )
[π»π=0+βπ0
π βπ+β2π2
2π0]π’ππ (π )=πΈπ (π )π’ππ (π )
πΈπ (π )β πΈπ (0 )+β
2π2
2π0
+ β2
π02βπβ π
|β¨π’π (0 ,π )|π βπ|π’π (0 ,π ) β©|πΈπ (0 )βπΈπ (0 )
2
ΒΏπΈπ (0 )+β2π2
2ππβ
![Page 3: Band structure of cubic semiconductors (GaAs) near the center of the Brillouin zone lh so hh lh el Non-relativistic solid With spin-orbit coupling included](https://reader036.vdocuments.mx/reader036/viewer/2022082411/5697bf921a28abf838c8f04f/html5/thumbnails/3.jpg)
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
β2
π02 βπβ π
|β¨π’π (0 ,π )|π βπ|π’π (0 ,π ) β©|πΈπ (0 )βπΈπ (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:
![Page 4: Band structure of cubic semiconductors (GaAs) near the center of the Brillouin zone lh so hh lh el Non-relativistic solid With spin-orbit coupling included](https://reader036.vdocuments.mx/reader036/viewer/2022082411/5697bf921a28abf838c8f04f/html5/thumbnails/4.jpg)
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
β2
π02 βπβ π
|β¨π’π (0 ,π )|π βπ|π’π (0 ,π ) β©|πΈπ (0 )βπΈπ (0 )
2
![Page 5: Band structure of cubic semiconductors (GaAs) near the center of the Brillouin zone lh so hh lh el Non-relativistic solid With spin-orbit coupling included](https://reader036.vdocuments.mx/reader036/viewer/2022082411/5697bf921a28abf838c8f04f/html5/thumbnails/5.jpg)
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...
π (π )=πΉ π (π )π’π0 (π )
![Page 6: Band structure of cubic semiconductors (GaAs) near the center of the Brillouin zone lh so hh lh el Non-relativistic solid With spin-orbit coupling included](https://reader036.vdocuments.mx/reader036/viewer/2022082411/5697bf921a28abf838c8f04f/html5/thumbnails/6.jpg)
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 βSchrΓΆdinger-Poissonβ problem
Unstable
neutraldonors
π»2π (π )=4π π
π0 [π π΄ (π )βππ· (π )ββπ
π π|πΉπ (π )|2]
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Quantization in heterostructures
A B A
cb
vb
Band edge discontinuitiesin heterostructures lead toquantized states
electron
hole
SchrΓΆdinger eq. (1-band):
Material
cb
vb
*
1r r r
r l l ll
U F EFm