Transcript The band structure - An-Najah National University

The crystal structure of the III-V
semiconductors
Diamond and Zincblende Lattices
Unit cells for silicon (Si) and gallium arsenide (GaAs) Silicon - diamond lattice GaAs
- zincblende (cubic zinc sulfide) lattice (most other III-V and many II-VI
semiconductors have zincblende lattice)Diamond and zincblende lattice based on
tetragonal pattern of bonds from each atom to nearest neighbors-two interlocking
facecentered- cubic lattices lattice parameter (or constant), a- repeat length of the
unit cells
e. g., GaAs, a = 5.65 Å (Angstroms) = 0.565 nm.
The band structure ?
First Brillouin zone E vs. k band
diagram of zincblende semiconductors
One relevant conduction band is
formed from S- like atomic orbitals
“unit cell” part of wavefunction is
approximately spherically symmetric.
The three upper valence bands are
formed from (three) P- like orbitals
and the spin-orbit interaction splits off
lowest, “split-off” hole (i. e., valence)
band. The remaining two hole bands
have the same energy (“degenerate”)
at zone center, but their curvature is
different, forming a “heavy hole” (hh)
band (broad), and a “light hole” (lh)
band (narrower)
Compound Semiconductors
(alloys)
For optoelectronics, most devices are fabricated of“compound
semiconductors” particularly III-V materials made from
•Group III (Al, Ga, In) and
•Group V (N, P, As, Sb) elements
•Sometimes Si and Ge (Group IV) are used as photodetectors
•Sometimes II-VI (e.g. ZnSe) and IV-VI materials (e.g., PbTe)
Alloys of compound semiconductors used extensively to adjust the basic materials
properties, e.g., lattice constant, bandgap,refractive index, optical emission or
detection wavelength
EXAMPLE –
InxGa1- xAs (where x is the mole fraction of indium)
InxGa1- xAs is not strictly crystalline because not every unit cell
is identical (random III site location), but we treat such alloys as
crystalline to a first approximation
The Human eye response
Lasers and LEDs for displays or lighting must emit in the 430-670 nm
wavelength region (bandgaps of 3.0-1.9 eV).
Technologically Available
Materials
Some of the applacations
Large Area, Full Color Displays
LED Traffic Lights
the first principles calculation
guess first  i
i
compare charge convergence
new  i
Empirical tight binding
| i
Hv= <
i
  a
| 

ˆ | 
 | H
v
|Hv-ESv|= 0
The Hamiltonian in sp3d2
xc
yc
zc
d1c
d2c
0
0
sa
xa
ya
za
d1a
d2a
sc
sa
Esa
0
0
0
0
0
Vss*g0
xa
0
Epa
0
0
0
0
g1*-Vscpa
Vxx*g0
Vxy*g3
Vxy*g2
Vxad1c*g1 yVxad1cg1
ya
0
0
Epa
0
0
0
g2*-Vscpa
Vxy*g3
Vxx*g0
Vxy*g1
g2*-Vxad1 yVxad1cg2
za
0
0
0
Epa
0
0
g3*-Vscpa
Vxy*g2
Vxy*g1
Vxx*g0
0
kVxad1c*g3
d1a
0
0
0
0
Eda
0
0
0
Vd1d1g0
0
d2a
0
0
0
0
0
Eda
0
0
Vd1d1g0
sc
Vss*g0
0
0
Esc
0
0
0
0
0
xc
Vsapc*g1
Vxx*g0
Vxy*g3
Vxy*g2
Vd1axc*g1yVd1axc*g1
0
Epc
0
0
0
0
yc
Vsapc*g2
Vxy*g3
Vxx*g0
Vxy*g1 g2*-Vd1axc yVd1axcg2
0
0
Epc
0
0
0
zc
Vsapc*g3
Vxy*g2
Vxy*g1
Vxx*g0
0
kVd1axcg3
0
0
0
Epc
0
0
d1c
0
Vxad1c*g1 g2*-Vxad1
0
Vd1d1g0
0
0
0
0
0
Edc
0
d2c
0
yVxad1cg1 yVxad1cg2kVxad1c*g3
0
Vd1d1g0
0
0
0
0
0
Edc
g1*-Vscpa g2*-Vscpa g3*-Vscpa
Vsapc*g1 Vsapc*g2 Vsapc*g3
Vd1axc*g1 g2*-Vd1axc
yVd1axc*g1yVd1axcg2 kVd1axcg3
The equation came from ETB
Volume optimization for InN by wien2K
Volume optimization for InAs by wien2K
Volume optimization for InSb by wien2K
Band structure of InN by wien2k
Band structure of InAs by wien2k
Band structure of InSb by wien2k
Band structure of InN by ETB
Density of states for InN
DOS (arbitrary units)
DOS for InN
-16
-12
-8
-4
0
e ne rgy e V
4
8
12
16
Band structure of InAs by ETB
Density of states for InAs
DOS (arbitrary units)
DOS for InAs
-15
-10
-5
0
5
energy eV
10
15
20
Band structure of InSb by ETB
Density of states for InSb
DOS (arbitrary units)
DOS for InSb
-10
-8
-6
-4
-2
0
energy eV
2
4
6
8
10