Electrical- Band Gap

Question 1

Why is diamond an insulator but Si and Ge semiconductors even though they all have the same crystal structure?

Answer 1

They are all in group IV. There is a decrease in Eg down the group because of an increase in atoms size which results in longer and weaker bonds. Reduced orbital overlap results in reduced splitting between binding and anti-bonding orbitals and therefore a reduction in Eg.

Question 2

How to increase the number of mobile electrons in semiconductors

Answer 2

Raise the temperature to promote more electrons from the VB into the CB

Question 3

Formula for number of carriers based on temperature

Answer 3

n=n0exp(-E/kT)
n0 is constant (total number of electrons)
E is activation energy (eV)
k is Boltzmann constant
T is temperature in K
Applied for semiconductors and insulators

Question 4

How does conductivity vary with temperature for metals and semiconductors?

Answer 4

For metals n is large and constant and q is constant. Mobility μ decreases slightly with temperature so σ also decreases slightly with temperature. For intrinsic semiconductors (and insulators) n is small but rises exponentially with T. μ still decreases slightly with temperature and q is constant but overall σ rises exponentially with T.

Question 5

Graph of lnσ vs 1/T for metal, semiconductor and insulator

Answer 5

Metal straight line positive gradient but almost horizontal at high position. Semiconductor negative gradient straight line steeper and lower down than metal line. Insulator very steep negative gradient straight line under and on left side of semiconductor line.

Question 6

Arrhenius plot formula for semiconductor σ

Answer 6

σ=Aexp(-E/kT)

A is a constant

Question 7

What can be said about the energy levels in the VB of a metal at absolute 0?

Answer 7

The energy levels in the VB are completely full up to the Fermi level and completely empty above it.

Question 8

What does the Fermi function represent?

Answer 8

f(E) represents the probability that an energy level, E, is occupied and can have values between 0 and 1. At T=0K f(E) is 1 up to Ef and 0 above Ef

Question 9

Describe the band picture for a metal at 0K

Answer 9

E is y-axis and f(E) x axis from 0 to 1 going left. Vertical line a 1 up to Ef then horizontal line to 0 at Ef. Stays at 0 above Ef. This is all for the VB

Question 10

Describe the band picture for a metal above 0K

Answer 10

Starts with vertical line at 1 but curves to diagonal before Ef and goes through Ef before 0 and curves to join 0 some point above Ef. Still for VB

Question 11

What does conduction require?

Answer 11

Electrons to increase their energy to some level above Ef where unoccupied levels exist. This requires energy from an external energy source which could be thermal energy

Question 12

Formula for Fermi function

Answer 12

f(E)=1/(exp((E-Ef)/kT)+1)

T is in K

Question 13

What is f(E) at Ef?

Answer 13

1/2

Question 14

What happens to the range over which f(E) drops from 1 to 0 as T increases?

Answer 14

The range increases meaning the band picture line gets further from being horizontal, deviates from 1 sooner and joins 0 later.

Question 15

What order of magnitude is the energy range over which f(E) drops from 1 to 0?

Answer 15

Several kT

Question 16

Describe the band picture for an insulator at 300K

Answer 16

Looks the same as that for metal but there is an energy band gap between the VB and CB. The probabilities can only be realised in the VB and CB because electrons are forbidden to have energy levels within the band gap. In the VB f(E) is essentially equal to 1 and in the CB it is equal to 0 (for 300K). Ef is still there in the centre of the band gap.

Question 17

Describe the band picture for an semiconductor at 300K

Answer 17

Same shape as for insulator but the band gap is no much smaller. Therefor f(E) is not 1 for all of the VB and not 0 for all of the CB. Electron-hole pairs are created in equal concentrations due to electrons being promoted to the CB. The area not 1 in the VB is proportional to the number of electron holes. The area not 0 in the CB is proportional to the number of conduction electrons.

Question 18

Types of semiconductors

Answer 18

Elemental, compound, intrinsic, extrinsic

Question 19

What is an electron hole?

Answer 19

A missing valence electron. Their movement in one direction represents valence electrons moving in the opposite direction. Symbol h+ (superscript)

Question 20

Formula for conductivity of intrinsic semiconductors using nqμ

Answer 20

σ=nq(μe+μh)
n is density of conduction electrons (=density of holes)
q is 1.6x10^-19
μe is mobility of conduction electrons
μh is mobility of holes

Question 21

How do μh and μe compare?

Answer 21

The cooperative motion of the valence electrons (μh) is an inherently slower process than the motion of conduction electrons (μe)

Question 22

What is the number of charge carriers in intrinsic semiconductors proportional to?

Answer 22

exp(-Eg/2kT)
Eg is band gap
Means it depends on overlap of the tails of the Fermi function curve with the VB and CB

Question 23

Temperature dependence of conductivity for intrinsic semiconductors formula

Answer 23

σ=σ0exp(-Eg/2kT)