Professor, Department of Electrical Engineering,
Laser Technology Program,
Indian Institute of Technology, Kanpur
Prof. Utpal Das
http://www.iitk.ac.in/ee/faculty/det_resume/utpal.html
Lecture 5: Semiconductor Basics
(Band Gap and density of states)
Semiconductor Optical Communication
Components and Devices
Brilloun Zone representation (after L. Brilloun).
Thus energy bands form in E or x for which f(E) or S(x) = cosk(a+b) has a
solution and between these bands where it is not possible to find a value of E or x are
called forbidden gaps. Discrete number of such bands are separated by band gaps.
Now for
If one plots the allowed values of energy as a function of k, we obtain the E-k
diagram for the one dimensional lattice, as shown in the previous slide.
Cos k a+b = ±1, π 2π
k=0 or ± , ± =Cos(mπ),a+b a+b
m = 0, ±1, ±2, ......for these band edges
When the expanded E-k diagram of the Periodic potential perturbation in question is
plotted, one notices the deviation from the free space E-k diagram (given by the dotted
white line),
Free Particle Solution: How in particular can the periodic potential solution with an
adjustable k approach the free particle solution with a fixed k in the limit where E >> Vo?
Brillouin zone
In this regard it must be remembered that the wave function for an electron in a crystal
is the product of two ejkx and Q(x) where Q(x) is the wave function in the unit cell.
Q(x) is also a function of k. Increasing or decreasing k by 2p/(all length) modifies both
ejkx and Q(x) in such a way that the product of ejkxQ(x) approach the free particle.
The E-k (energy-momentum) relationship is a Fourier Transform of the
original lattice and hence is called the Reciprocal Lattice of the Crystal
Brillouin zonesThe k-value associated with given energy band is called a Brillouin Zone.
1st Brillouin Zone 2nd Brillouin Zone
One representation is to look at energies for k between ±p/(a+b) in basically 2p/(all length)
range. In the eigenvalue equation notice that increasing or decreasing Cos{k(a+b)} by
2p/(a+b) has no effect on the allowed electron energy. Or E(k) is periodic with a period of
2p/(a+b). i.e. electrons having k values in multiples of 2p/(a+b) are indistinguishable in the
crystal in terms of their energies.
π π- to +
a+b a+b
2π π π 2π- to - & to
a+b a+b a+b a+b
Therefore all the electron energies can be represented within –p/(a+b) ≤ k ≤ + p/(a+b)
changing the k values by 2np/(a+b). Where n is an integer. This representation of the
electron is called the reduced Brillouin zone representation.
As the number of electrons in the system increases the bands starts to be filled up from
the lowest available energies. Normally most of the bands are completely full of
electrons as allowed by Pauli's Exclusion Principle. At low temperature, there could be a
band completely empty. The one below it is usually completely full, called the valence
band. None of these electrons can now conduct electricity.
If now a condition arises that some of the electrons from the completely filled band
can be excited into the completely empty band, then current can be conducted by the
electrons in the empty band called the conduction band.
From Bloch theorem there are only two k values associated with each allowed energy,
one for motion in the positive k-direction and the other for the negative k-direction.
Also note that dE/dt = 0 at k = ± p/(a+b) & k = 0. This is a property of all E-k plots.
Shapes of constant energy surfaces in Ge, Si and GaAs. For Ge there are eight half ellipsoids of
revolution along the (111) axes and the Brillouin zone boundaries are at the middle of the ellipsoids. For
Si there are six ellipsoids along the (100) axes with the centre of the ellipsoids located at about three-
fourths of the distance from the Brillouin zone center. For GaAs the constant energy surface is a
sphere at zone center.
(a) Brillouin zone for Diamond and Zinc Blende lattice.
Brillouin zone for Wurtzite lattice. The most important
symmetry points and symmetry lines are also
indicated: Γ: 2π/a (0,0,1). Zone center: L: 2π/a (1/2, 1/2,
1/2). Zone edge along (111) axes (Λ): X: 2π/a (0,0,1).
Zone edge along (100) axes ∆: K: 2π/a (3/4,3/4,0). Zone
edge along (110) axes (∑).
0
0
0 1/2
1/2
1/2
1/2
1/4
3/4
3/4 3/4
3/4
1/4
1/4
(a) Ge (b) Si (c) GaAs
U Q
Z W
X
K
L
ky
kz
kx
/
-4
-2
0
2
4
6
InAs
E (
eV
)
L X X
K
Fundamentals
of Solid State
Electronics by
C. T. Sah,
World
Scientific.
Direct Band gap semiconductors
U Q
Z W
X
K
L
ky
kz
kx
/
-4
-2
0
2
4
6E
(eV
)GaAs
L X X
K
-4
-2
0
2
4
6
E (
eV
)
L X X L
InP
K
-4
-2
0
2
4
6
Si
E (
eV
)
L X X
K
-4
-2
0
2
4
6
Ge
E (
eV
)
K
L X X
-4
-2
0
2
4
6
GaP
E (
eV
)
K
L X X
Indirect Band gap semiconductors
U Q
Z W
X
K
L
ky
kz
kx
/
6-Fold Degeneracy
of the X-valley CB
8-Fold Degeneracy
of the L-valley CB
0
1
2
3
4
5
6
3 4 5 6 7
Ban
dG
ap E
ner
gy
(eV
)
Lattice Constant (Angstroms)
‘Italics’ - Indirect bandgap
‘Roman’ - Direct bandgap
4H-SiC
6H-SiC
2H-SiC
GaN
ZnO
AlN
Diamond
InN
ZnS
ZnSe
CdS
CdSe
3C-SiC
BP
AlN
GaN
InN
GaP
AlP
Si
Ge
GaSb
InSb
InAs
GaAs
InP
ZnS
MgS
BN
MgSe
MgTe
AlAs
ZnSe
CdS
CdSe
AlSbCdTe
ZnTe
Hexagonal Structure
Cubic Structure
Zn0.5Mn0.5Se
HgS
HgSeHgTe
Review Questions:
1. Take a=5nm and b=4nm with me*=0.07mo. Check for the k=0 and
k=p/(a+b) energy difference again. This would be the first mini-band of
a superlattice. What is your comment with respect to that found in P2
above.