GOAL:

• The density of electrons (no) can be found precisely if we know1. the number of allowed energy states in a small energy

range, dE: S(E)dE “the density of states”

2. the probability that a given energy state will be occupied by an electron: f(E)

“the distribution function”

no = bandS(E)f(E)dE

Energy Band Diagram

conduction band

valence band

EC

EV

x

E(x)

S(E)

Eelectron

Ehole

note: increasing electron energy is ‘up’, but increasing hole energy is ‘down’.

Reminder of our GOAL:

•The density of electrons (no) can be found precisely if we know

1. the number of allowed energy states in a small energy range, dE: S(E)dE

“the density of states”

2. the probability that a given energy state will be occupied by an electron: f(E)

“the distribution function”

no = bandS(E)f(E)dE

Fermi-Dirac Distribution

The probability that an electron occupies an energy level, E, is

f(E) = 1/{1+exp[(E-EF)/kT]}

– where T is the temperature (Kelvin)– k is the Boltzmann constant (k=8.62x10-5 eV/K)– EF is the Fermi Energy (in eV)

– (Can derive this – statistical mechanics.)

f(E) = 1/{1+exp[(E-EF)/kT]}

All energy levels are filled with e-’s below the Fermi Energy at 0

oK

f(E)

1

0

EF

E

T=0 o

K

T1>0

T2>T1

0.5

Fermi-Dirac Distribution for holes

Remember, a hole is an energy state that is NOT occupied by an electron.

Therefore, the probability that a state is occupied by a hole

is the probability that a state is NOT occupied by an electron:

fp(E) = 1 – f(E) = 1 - 1/{1+exp[(E-EF)/kT]}

={1+exp[(E-EF)/kT]}/{1+exp[(E-EF)/kT]} -

1/{1+exp[(E-EF)/kT]}

= {exp[(E-EF)/kT]}/{1+exp[(E-EF)/kT]}

=1/{exp[(EF - E)/kT] + 1}

The Boltzmann ApproximationIf (E-EF)>kT such that exp[(E-EF)/kT] >> 1 then,

f(E) = {1+exp[(E-EF)/kT]}-1

{exp[(E-EF)/kT]}-1

exp[-(E-EF)/kT] …the Boltzmann approx.

similarly, fp(E) is small when exp[(EF - E)/kT]>>1:

fp(E) = {1+exp[(EF - E)/kT]}-1

{exp[(EF - E)/kT]}-1

exp[-(EF - E)/kT]

If the Boltz. approx. is valid, we say the semiconductor is non-degenerate.

Putting the pieces together:for electrons, n(E)

f(E)

1

0

EF

E

T=0 o

K

T1>0

T2>T1

0.5

EV EC

S(E)

E

n(E)=S(E)f(E)

Putting the pieces together: for holes, p(E)

fp(E)

1

0

EF

E

T=0 o

K

T1>0

T2>T1

0.5

EV EC

S(E)

p(E)=S(E)f(E)

hole energy

Energy Band Diagramintrinisic semiconductor: no=po=ni

conduction band

valence band

EC

EV

x

E(x)

n(E)

p(E)EF=Ei

where Ei is the intrinsic Fermi level

Energy Band Diagramn-type semiconductor: no>po

conduction band

valence band

EC

EV

x

E(x)

n(E)

p(E)

EF

]/)(exp[0 kTEENn FCC

Energy Band Diagramp-type semiconductor: po>no

conduction band

valence band

EC

EV

x

E(x)

n(E)

p(E) EF

]/)(exp[0 kTEENp VFV

The intrinsic carrier density

kTEVCi

ikTE

VC

g

g

eNNn

neNNpn2/

2/00

is sensitive to the energy bandgap, temperature, and m*

2/3

2

*

22

kTm

N dseC