Contents

• Origin of Energy gap

• Bloch Theorem

• Kronig Penny Model

UNIT IV

Band Theory of Solids

• E-K Diagram and Brillion Zones

• Effective Mass

• Discussion on Intrinsic and Extrinsic Semiconductor

• Hall Effect

• Assignment and Tutorial sheetPPT slides presented by: Dr. Amit sharma

Applied Physics-I

Origin of Energy Gap

E vs K for free electron E vs K for an electron in linear lattice

Band Theory of Solids

Bloch Theorem

• Let us consider an electron moving in X direction in one dimensional crystal having periodic potential

V(x)=V(x+a)

The Schrödinger wave equation for the moving electron is:The Schrödinger wave equation for the moving electron is:

The solution of the eqn is ψ(x) = eiKx uk(x) (1)

where uk(x) = uk(x + a)

Here equation 1 is called Bloch theorem.

Kronig - Penney Model

• The Kronig-Penney model demonstrates that a simple one-dimensional periodic potential yields energy bands as well as energyband gaps.

• The potential assumed is shown as below• The potential assumed is shown as below

• E-K Diagram and Brillouin Zones

From the E-K diagram is clear that the electron has allowed energy

values in the region or zone extending from k= -π/a to k= +π/a. This

zone is called first Brillouin Zone. Higher zones can be defined

accordingly from the graph.

Effective MassEffective Mass ::

If the same magnitude of electric field is applied to both electrons in

vacuum and inside the crystal, the electrons will accelerate at avacuum and inside the crystal, the electrons will accelerate at a

different rate from each other due to the existence of different

potentials inside the crystal.

The electron inside the crystal has to try to make its own way.

So the electrons inside the crystal will have a different mass than that

of the electron in vacuum.

This altered mass is called as an effectiveeffective--massmass..

Taking derivative of energyenergy with respect to k k and taking and taking m*m*(Effective mass)(Effective mass)

instead of minstead of m;;

2

2 2

2

d E k

d k m

d E

md k

=

=

h

h

2 2

2*

md k

md E d k

= h

Intrinsic semiconductor • Carrier concentration and Fermi Level

T is the absolute temperature, p is hole concentration in valance band and n is electron concentration in conduction band.Nc , Nv are the effective density of states in the conduction band and in the valence band respectively.

Carrier concentration and Fermi level of intrinsic semiconductor (Derivation)

Fermi Level of N and P type Semiconductors

NV is the effective density of states in the valence band.NA is the concentration of acceptor atoms.

Hall Effect• The Hall effect is the production of a voltage difference (the Hall

voltage) across an electrical conductor, transverse to an electric current in the conductor and a magnetic field perpendicular to the current. It was discovered by Edwin Hall in 1879.

• The Hall coefficient is defined as the ratio of the induced electric field to the product of the current density and the applied magnetic field. to the product of the current density and the applied magnetic field.

RH=Ey /jx B

Applications of Hall Effect

• The sign of current carrying charges can be determined.

• Carrier concentration can be measured.

• The mobility can be measured.

• Power flow in electromagnetic waves can be determined with the help of Hall effect.help of Hall effect.

Assignment

1. Distinguish between intrinsic and extrinsic semiconductors. Indicate the conduction and valance bands donor and acceptor states in energy level diagram.

2.Derive an expression for carrier concentration in extrinsic semiconductors. What would be the position of Fermi level ? Explain.semiconductors. What would be the position of Fermi level ? Explain.

3.Explain the origin of formation of energy bands. How does the band theory of solids lead to the classification of solids into conductors, semiconductors and insulators.

4. What is Hall effect ? Derive an expression for Hall coefficient. Discuss some important applications.

Assignment continued

5.Discuss KRONIG-Penny model . How it explains the formation of energy bands separated by forbidden energy gap in solids.

6 Define and find the effective mass of electron. Show that it is different from the rest mass of the electron. Give physical meaning of different from the rest mass of the electron. Give physical meaning of rest mass.

Tutorial Sheet

1.Calculate the Hall coefficient of sodium assuming bcc structure of sodium of cell side 0.428nm.

2.Find the Hall coefficient of copper assuming 5x10 28 atoms per meter cube.

3. What is the concentration of holes in Si crystals having donor 3. What is the concentration of holes in Si crystals having donor concentration of 1.4x1024 /cm 3 when the intrinsic carrier concentration is 1.4x1018 /cm 3 ? Find the ratio of electron to hole concentration.

4. Calculate the position of Fermi level at 300K for germanium crystal containing 5x1022 arsenic atoms /m 3 .

5. For Kronig-Penney potential with P<<1, Calculate the lowest energy at k=0.

6.Find the ratio between the kinetic energies of an electron in two dimensional square lattice (a) when kx = ky =π/a and (b) kx = π/a, ky =0.