CHAPTER 3
Elements of solid state physics
• Semiconductors, insulators and metals
• Semiconductors
• Insulators
• Metals

Prof. Dr. Beşire GÖNÜL

Semiconductors, Insulators and Metals
The electrical properties of metals and insulators are well known to all of us.
Everyday experience has already taught us a lot about the electrical properties of metals and insulators.
But the same cannot be said about
“semiconductors”.
What happens when we connect a battery to a piece of a silicon; would it conduct well ? or would it act like an insulator ?

The name “semiconductor” implies that it conducts somewhere between the two cases (conductors or insulators) Conductivity :



σmetals ~1010 /Ω-cm s/c σinsulators ~ 10

-22

/Ω-cm

The conductivity (σ) of a semiconductor (S/C) lies between these two extreme cases.

.:: The Band Theory of Solids ::.


The electrons surrounding a nucleus have certain welldefined energy-levels.



Electrons don’t like to have the same energy in the same potential system.



The most we could get together in the same energy-level was two, provided thet they had opposite spins. This is called Pauli
Exclusion
Principle.

Allowed band Forbidden band Allowed band Forbidden band Allowed band 1

2

4………………N

Number

of

atoms



The difference in energy between each of these smaller levels is so tiny that it is more reasonable to consider each of these sets of smaller energy-levels as being continuous bands of energy, rather than considering the enormous number of discrete individual levels.



Each allowed band is seperated from another one by a forbidden band.



Electrons can be found in allowed bands but they can not be found in forbidden bands.

.:: CALCULATION


Consider 1 cm3 of Silicon. How many atoms does this contain ?



Solution:
The atomic mass of silicon is 28.1 g which contains Avagadro’s number of atoms.
Avagadro’s number N is 6.02 x 1023 atoms/mol .
The density of silicon: 2.3 x 103 kg/m3 so 1 cm3 of silicon weighs 2.3 gram and so contains

6.02  1023
 2.3  4.93  1022 atoms
28.1

This means that in a piece of silicon just one cubic centimeter in volume , each electron energy-level has split up into 4.93 x 1022 smaller levels !

.:: Semiconductor, Insulators, Conductors ::.

Full band All energy levels are occupied by electrons

Empty band

All energy levels are empty
( no electrons)

Both full and empty bands do not partake in electrical conduction.

.:: Semiconductor energy bands at low temperature ::.

Electron energy



Forbidden energy gap [Eg]

Full valance band



Recall that a full band can not conduct, and neither can an empty band. 

Empty conduction band

At low temperatures the valance band is full, and the conduction band is empty.

At low temperatures, s/c’s do not conduct, they behave like insulators. 

The thermal energy of the electrons sitting at the top of the full band is much lower than that of the Eg at low temperatures.

Conduction Electron :








Assume some kind of energy is provided to the electron (valence electron) sitting at the top of the valance band.
This electron gains energy from the applied field and it would like to move into higher energy states.
This electron contributes to the conductivity and this electron is called as a conduction electron.
At 00K, electron sits at the lowest energy levels. The valance band is the highest filled band at zero kelvin. Empty conduction band

Forbidden energy gap [Eg]

Full valance band

Semiconductor energy bands at room temperature








When enough energy is supplied to the e- sitting at the top of the valance band, e- can make a transition to the bottom of the conduction band.
When electron makes such a transition it leaves behind a missing electron state.
This missing electron state is called as a hole.
Hole behaves as a positive charge carrier. Magnitude of its charge is the same with that of the electron but with an opposite sign.

Empty conduction band

Forbidden energy gap [Eg]

energy

e- e - e- e+ + + +
Full
valance band Conclusions ::.


Holes contribute to current in valance band (VB) as e-’s are able to create current in conduction band (CB).



Hole is not a free particle. It can only exist within the crystal. A hole is simply a vacant electron state.



A transition results an equal number of e- in CB and holes in VB. This is an important property of intrinsic, or undoped s/c’s. For extrinsic, or doped, semiconductors this is no longer true.

Bipolar (two carrier) conduction

Electron energy



empty

occupied

After transition

After transition, the valance band is now no longer full, it is partly filled and may conduct electric current. Valance Band
(partly filled
 The conductivity is due to band) both electrons and holes, and this device is called a bipolar conductor or bipolar device.

The Concept of Effective Mass :
Comparing
Free e- in vacuum



If the same magnitude of electric field is applied to both electrons in vacuum 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 effective-mass.

In an electric field mo =9.1 x 10-31
Free electron mass

An e- in a crystal
In an electric field
In a crystal m = ?

m*

effective mass

To find effective mass , m*
We will take the derivative of energy with respect to k ;
2
dE k  dk m

- m* is determined by the curvature of the E-k curve
2
d2E

2 m dk

Change

m*

m*

instead of



- m* is inversely proportional to the curvature

m

2 d 2 E dk 2

This formula is the effective mass of an electron inside the crystal.

Direct an indirect-band gap materials :
Direct-band gap s/c’s (e.g. GaAs, InP, AlGaAs)



E

e-

+

VB

For a direct-band gap material, the minimum of the conduction band and maximum of the valance band lies at the same momentum, k, values.
When an electron sitting at the bottom of the CB recombines with a hole sitting at the top of the VB, there will be no change in momentum values.



CB



Energy is conserved by means of emitting a photon, such transitions are called as radiative transitions.

k



Indirect-band gap s/c’s (e.g. Si and Ge)


E

For an indirect-band gap material; the minimum of the CB and maximum of the VB lie at different k-values.
When an e- and hole recombine in an indirect-band gap s/c, phonons must be involved to conserve momentum.

CB

Phonon e- 

Eg

+

k




VB

Atoms vibrate about their mean position at a finite temperature.These vibrations produce vibrational waves inside the crystal.
Phonons are the quanta of these vibrational waves. Phonons travel with a velocity of sound .
Their wavelength is determined by the crystal lattice constant. Phonons can only exist inside the crystal.



The transition that involves phonons without producing photons are called nonradiative (radiationless) transitions.



These transitions are observed in an indirect band gap s/c and result in inefficient photon producing.



So in order to have efficient LED’s and LASER’s, one should choose materials having direct band gaps such as compound s/c’s of GaAs, AlGaAs, etc…