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.