Density of States and Fermi Energy Concepts. How do Electrons and Holes Populate the Bands? The...

Density of Statesand

Fermi Energy Concepts

How do Electrons and Holes Populate the Bands?

dEEgc )(The number of conduction band

states/cm3 lying in the energy

range between E and E + dE

(if E Ec).

The number of valence band

states/cm3 lying in the energy

range between E and E + dE

(if E Ev).

dEEgv )(

Density of States Concept

General energy dependence of

gc (E) and gv (E) near the band edges.

How do Electrons and Holes Populate the Bands?

Density of States Concept

Quantum Mechanics tells us that the number of available states

in a cm3 per unit of energy, the density of states, is given by:

Density of States in Conduction Band

Density of States in Valence Band

How do electrons and holes populate the bands?

Probability of Occupation (Fermi Function) Concept

Now that we know the number of available states at each energy, then how do the electrons occupy these states?

We need to know how the electrons are “distributed in energy”.

Again, Quantum Mechanics tells us that the electrons follow the

“Fermi-distribution function”.

Ef ≡ Fermi energy (average energy in the crystal)k ≡ Boltzmann constant (k=8.61710-5eV/K)T ≡Temperature in Kelvin (K)

f(E) is the probability that a state at energy E is occupied. 1-f(E) is the probability that a state at energy E is unoccupied.

kTEE feEf /)(

1

1)(

Fermi function applies only under equilibrium conditions, however, is universal in the sense that it applies with all materials-insulators, semiconductors, and metals.

Fermi-Dirac Distribution


Ef


Probability of Occupation (Fermi function) Concept

At T=0K, occupancy is “digital”: No occupation of states above Ef and

complete occupation of states below Ef .

At T>0K, occupation probability is reduced with increasing energy.f(E=Ef ) = 1/2 regardless of temperature.

kT = 0.0259eV @300KkTEE feEf /)(

1

1)(

21

21



At T=0K, occupancy is “digital”: No occupation of states above Ef and

complete occupation of states below Ef .

At T>0K, occupation probability is reduced with increasing energy.f(E=Ef ) = 1/2 regardless of temperature.

At higher temperatures, higher energy states can be occupied, leaving more lower energy states unoccupied [1 - f(Ef )].

kTEE feEf /)(

1

1)(

kT = 0.0259eV @300K

21

21



If E Ef +3kT

Consequently, above Ef +3kT the Fermi function or filled-state probability decays exponentially to zero with increasing energy.

kTEEkTEE ff eEfe/)(/)(

)(and1


Example 2.2

The probability that a state is filled at the conduction band edge (Ec) is precisely equal to the probability that a state is empty at the valence band edge (Ev).Where is the Fermi energy locate?

Solution

The Fermi function, f(E), specifies the probability of electron occupying states at a given energy E.The probability that a state is empty (not filled) at a given energy E is equal to 1- f(E).

VC EfEf 1

kTEEC FCeEf /1

1

kTEEkTEEV VFFV eeEf // 1

1

1

111

kT

EE

kT

EE FVFC

2VC

F

EEE

The density of electrons (or holes) occupying the states in energy between E and E + dE is:


Probability of Occupation Concept

0 Otherwise

dEEfEgc )()(Electrons/cm3 in the conduction

band between E and E + dE

(if E Ec).

Holes/cm3 in the conduction

band between E and E + dE

(if E Ev).

dEEfEgv )()(

Fermi function and Carrier Concentration



Probability of Occupation Concept

Fermi-Dirac distribution function describing the probability that an allowed state at energy E is occupied by an electron.

The density of allowed states for a semiconductor as a function of energy; note that g(E) is zero in the forbidden gap between Ev and Ec.

The product of the distribution function and the density-of-states function


Typical band structures of Semiconductor

Ev

Ec

0

Ec+

EF

VB

CB

E

g(E)

g(E) (E–Ec)1/2

fE)

EF

E

Forelectrons

For holes

[1–f(E)]

Energy band diagram

Density of states Fermi-Diracprobability

function

probability of occupancy of a state

nE(E) or p

E(E)

E

nE(E)

pE(E)

Area = p

Area

Ec

Ev

ndEEn E )(

g(E) X f(E)Energy density of electrons in

the CB

number of electrons per unit energy per unit volumeThe area under nE(E) vs. E is the electron concentration.

number of states per unit energy per unit volume


Allowed electronic-energy-state systems for metal and semiconductors.

States marked with an X are filled; those unmarked are empty.

Metals vs. Semiconductors

Metal Semiconductor

Ef

Ef

Metals vs. Semiconductors

Ef

Ef

Allowed electronic-energy states g(E)

Metal Semiconductor

The Fermi level Ef is at an intermediate

energy between that of the conduction band edge and that of the valence band edge.

Fermi level Ef immersed in the

continuum of allowed states.

Note that although the Fermi function has a finite value in the gap, there is no electron population at those energies. (that's what you mean by a gap)

The population depends upon the product of the Fermi function and the electron density of states. So in the gap there are no electrons because the density of states is zero.

In the conduction band at 0K, there are no electrons even though there are plenty of available states, but the Fermi function is zero.

At high temperatures, both the density of states and the Fermi function have finite values in the conduction band, so there is a finite conducting population.

Fermi function and Carrier Concentration



Energy Band Occupation


Intrinsic Energy (or Intrinsic Level)

Ef is said to equal Ei

(intrinsic energy) when…

equal number of electrons and holes.

IntrinsicEqual numberof electronsand holes

n-typeMore electrons than Holes

p-typeMore holes than electrons


Additional Dopant States


Pure-crystal energy-band diagram


n-type material


p-type material

Energy band diagrams

np = ni2

Note that donor and acceptor energy levels are not shown.

Ec

Ev

Ef i

CB

Ef p

Ef n

Ec

Ev

Ec

EvVB

(a) intrinsic (c) p-type (b) n-type

Intrinsic, n-Type, p-Type Semiconductors

CB

g(E)

E

Impurities forming ba

nds

EFp

Ev

Ec

EFn

Ev

Ec

CB

VB

Degenerated n-type semiconductorLarge number of donors form a band that overlaps the CB

Degenerated p-type semiconductor


Heavily Doped Dopant States

Density of States and Fermi Energy Concepts. How do Electrons and Holes Populate the Bands? The...

Documents

Transcript of Density of States and Fermi Energy Concepts. How do Electrons and Holes Populate the Bands? The...