ECE 340 Lecture 8 : Fermi Level and Equilibrium Carrier...

ECE 340Lecture 8 : Fermi Level and

Equilibrium Carrier Distributions

Class Outline:

•Density of States•Fermi Level•Equilibrium Carrier Distributions

M. J. Gilbert ECE 340 – Lecture 8 9/09/1 1

Things you should know when you leave…

Key Questions• What is the density of states?• What does the Fermi function tell

us?• What is the Fermi level?• How do I find the equilibrium

electron and hole concentrations?• When can I use the simplified

equations for the concentrations?


Extrinsic MaterialsHow tightly bound is the extra electron or hole?

• We can use the Bohr’s hydrogen model to get an idea.

• Electrons move in Si and not in a vacuum.– Different relative

permittivity.• The electron mass must be

represented by the effective mass

r

Donor Acceptor

e-

h+

( ) 220

2

4*

32 r

nB

qmEεεπ

−=

Donor in Si P As SbBinding energy (eV) 0.045 0.054 0.039

Acceptor in Si B Al Ga InBinding energy (eV) 0.045 0.067 0.072 0.16


Extrinsic MaterialsVisualizing donors on the band diagram…

Ed

EaEv

Ec

Ev

Ec

Δx

Let’s take a look at Silicon with Phosphorus impurity atoms:

Ed

Ev

Ec

Eg = 1.12 eV

0.045 eV


Density of StatesWe are counting states…

kykx

kz

K

K + dK

Δk

( ) kKN D ∆= 33 82π

The total number of states in a small region of volume Δk can be written as (states / volume):

Consider free electrons, the volume of the sphere between the two spheres:

dkkk 24π=∆

The separation dK corresponds to an energy difference (for free electrons):

dEE

mdk

Emk

12

1

2

*

2

*

=

=

Above:•Constant energy surfaces in 3D k-space.•Δk is the volume of the sphere contained between the two spheres of radius k and k +dk.


Density of StatesBut it is in k-space…The number of states in the shell is given in terms of the density of states in energy by :

( )dEEN

We now equate the two expressions for the total number of states:

( ) ( )dkkdEEN d2

33 48

2 ππ

=

Plug in what we know about how the free electron model changes in k as we change E then integrate to get the final result:

( ) EmmEN d*

32

*

3 2π

=En

ergy

Ec

Ev

Available Valence Band States

Available Conduction Band States

Eg


Density of StatesThree things to know about the density of states (DOS)…

1. DOS is like seating at a stadium with the number of seats a given distance from the field being equivalent to the number of states available in the range of energy E to E + dE.

2. In general, the number of states available in the conduction and valence band will not be equal.

• Differences in effective masses.

3. Considering closely spaced energies E and E + dE in the respective bands, we can state:

• gc(E)dE is the number of conduction band states lying in the energy range E to E + dE (if E > Ec ).

• gv(E)dE is the number of valence band states lying in the energy range E to E + dE (if E < Ev ).

Ener

gy

Ec

Ev

Eg

( ) ( )

( )( )

32

**

32

**

2

2

π

π

EEmmEg

EEmmEg

vppv

cnnc

−=

−= E ≥ Ec

E ≤ Ev

Units are typically states/cm3-eV


Fermi Level Fine, but how many states are ACTUALLY filled?• The Fermi function tells us the probability of how many of the existing states

at any given energy will be filled with an electron.• It was derived based on three assumptions

– Each allowed state has a maximum of one electron (Pauli Exclusion Principle).– The probability of each allowed quantum state is the same.– All electrons are indistinguishable.

• When is it valid?– Under equilibrium conditions.– In ANY material (semiconductor, metal, insulator, etc…)

f(E)

1

0Ef E

Mathematically speaking the end result is a probability distribution function:

( ) ( )TkEE

b

f

e

Ef −

+

=

1

1½

T = 0 K

Fermi Energy


Fermi LevelWhat about the temperature dependence?

• What observations can we make about temperature?

– At E = Ef, f(E) = ½.

– At E ≥ Ef + 3kbT, f(E) decays rapidly to zero and most states will be empty.

– At E ≤ Ef - 3kbT, 1-f(E) rapidly decays to zero. Therefore, most states will be filled.

Ef + 3kbT

Ef - 3kbT

T = 300 KEf = 0.5 eV


Fermi LevelLet’s look at a quick example…

• Let’s say that the probability that a state is filled at the conduction band edge is equal to the probability that a state is empty in the valence band. Where is the Fermi level?f(E)

1 - f(E)Probability that a state is filled

Probability that a state is emptyf(Ec) = 1 – f(Ev)

( ) ( )TkEEc

b

fc

e

Ef −

+

=

1

1 ( ) ( )

( )TkEE

TkEEv

b

vf

b

fv

e

e

Ef

−

−

+

+

−=−

1

11

111

TkEE

TkEE

b

vf

b

fc −=

−2

vcf

EEE +=

Ec

Ev

Ef


Fermi LevelVisualizing the Fermi level in intrinsic material…

• In intrinsic material:– Concentration of holes in valence band is equal

to the number of electrons in the conduction band.

– Electron probability tail, f(E), is symmetric with hole probability tail, 1 - f(E).

– No states in Eg despite non-zero occupation probability.

– Fermi level (Ef) lies in the middle of the energy gap.

Ec

Ev

Ef

Ener

gy

Ec

Ev

EgEf

Ener

gy

Ec

Ev

Eg

Carrier Distributions:


Fermi LevelVisualizing the Fermi level in n-type material…

• In n-type material:– The Fermi level now lies closer to

the conduction band.– There are many more electrons that

there are holes.– The difference between Ef and Ev

provides a measure of the strength of the doping.

Ev

Ef

Ener

gy

Ec

Ev

Eg

Ef

Ec

Ener

gy

Ec

Ev

Eg



Fermi LevelVisualizing the Fermi level in p-type material…

• In p-type material:– The Fermi level now lies

closer to the valence band.– There are many more holes

that there are electrons.

Ev

Ef

Ener

gy

Ec

Ev

Eg Ef

Ec

Ener

gy

Ec

Ev

Eg



Equilibrium Carrier DistributionsLet’s put together our knowledge of the density of states and the Fermi function…

We want to know how many carriers we have in our semiconductor for ANY energy or temperature.

For electrons

For holes

( ) ( )∫=top

c

E

Ec dEEfEgn

( ) ( )[ ]∫ −=v

bottom

E

Ev dEEfEgp 1

Focus on the electron relation.

∫ −

+

−=

top

c b

f

E

E TkEE

cnn dE

e

EEmmn

1

232

**

πPlug in the relationships for the density of states and the Fermi function:

Make some substitutions and bring the constants out:

( )TkEE

b

c−=η

( )Tk

EE

b

cfc

−=η Etop ~ ∞ ( )

ηηπ ηη d

eTkmm

nc

bnn ∫∞

−+=

0

21

32

23

**

1


Equilibrium Carrier DistributionsThat integral is awful! Isn’t there an easier way?

( )ηη

π ηη de

Tkmmn

c

bnn ∫∞

−+=

0

21

32

23

**

1

( )ηη

π ηη de

Tkmmp

v

bpp∫∞

−+=

0

21

32

23

**

1

Fermi-Dirac integral of order ½.

Fermi-Dirac integral of order ½.Nc Nv

23

2

*

23

2

*

22

22

=

=

π

π

TkmN

TkmN

bpv

bnc

Effective conduction band density of states.

Effective valence band density of states.

23

0

*,

319

,11051.2

×=

mm

cmN pn

vc

3kbT

3kbT

Non-degenerate Semiconductor

Degenerate Semiconductor

Degenerate Semiconductor

( )

( )TkEE

v

TkEE

c

b

fv

b

cf

eNp

eNn−

−

=

=Non-degenerate Semiconductor


Equilibrium Carrier ConcentrationsThese are simple closed form expressions, but are they the simplest?

• There are other forms which you may find more useful in homeworks or tests.

• We can find the carrier concentrations relative to Ei , the Fermi level for an intrinsic semiconductor.

For intrinsic semiconductors, we know the following: n = p = ni and Ei = Ef

Then the relations for n and p become:

TkEiE

vi

TkEE

ci

b

v

b

ci

eNn

eNn−

−

=

= Solve for Nc and Nv

TkEE

iv

TkEE

ic

b

vi

b

ic

enN

enN−

−

=

=

By combining equations, we arrive at two very important and useful results:

TkEE

i

TkEE

i

b

fi

b

if

enp

enn−

−

=

= Solve for NiTk

E

vcib

g

eNNn 2−

=


Equilibrium Carrier ConcentrationsA note on the effective mass…

kx

kz

ky

kx

ky

kz

ml

mt

mt

----- -- ----

•In silicon, most of the electrons reside in the X valley as it is lowest in energy.

•The X direction has six equivalent valleys with different masses in each of the possible directions.

•We use the effective mass which is the geometric mean of the masses (density of states effective mass).

•GaAs conduction band does not have this complication.

ml = 0.98 m0mt = 0.19 m0

( ) 031

232

* 6 mmmm tln =

ECE 340 Lecture 8 : Fermi Level and Equilibrium Carrier...

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