Charge Carrier Statistics
Density of States
The density of states is given in general by the equation:
g E  
m 2mE
 23
The term g(E) is the number of states with E between E and E + dE per unit volume
(crystal volume) per dE:
g E  
1
d States
Volume
dE
Applying this to the conduction and valence band in general gives:
g c E  
g v E  
mn* 2mn* E  E c 
E  Ec
 23
m*p 2m*p Ev  E 
E  Ev
 23
where m n* and m *p depend on the semiconductor:
GaAs:
mn*  me*
Si:
m 6
2
Ge:
m 4
2
*
n
*
n
3
3
m m 
*
l
*2
t
1
3
m m 
*
l
*2
t
1
3
 


2
2
 


2
2
 mlh*
 
2
 m
 
2
 m
*
m   mhh

*
p


2
2
2
*
m   mhh

*
p
 
 
*
m*p   mhh

3
3
3
*
lh
*
lh
3
3
3
3
3
3
where ml* and mt* are the longitudinal and tranverse effective mass respectively (see
*
Pierret page 79) and mlh* and m hh
are the light and heavy hole effective mass
respectively.
Fermi Function
“The Fermi function f(E) is a probability distribution function that tells one the ratio of
filled to total allowed states at a given energy E” (Pierret pg. 96) and is given by:
1
f E  
 E  E F  KT
1 e
where K is Boltmann’s constant, T is the temperature of the system, and EF is termed the
Fermi Energy that will be investigated in more detail later.
The graph of f(E) at several temperatures is shown below:
Donors and Acceptors
Consider a silicon crystal and a phosphorus atom as a substitutional impurity. The
valency five phosphorus atom will form four bonds with the surrounding Si atoms but
will have one valence electron left over. This valence electron is weakly bound to the
phosphorus atom and can easily escape to the conduction band of Si to aid in conduction.
This phophorus atom is therefore called a donor since it donates one electron to the
conduction band.
Likewise, consider a boron atom as a substitutional impurity in a Si crystal. The valency
three boron atom will form four bonds with the surrounding Si atoms but needs to acquire
an additional electron to reach a valency of eight. The boron atom can easily acquire this
electron from a neighboring Si atom producing a hole in the valence band that can then
travel throughout the crystal and contribute to conduction. The boron “accepts” an
electron and hence produces a hole in the Si and is therefore called an acceptor.
This donor and acceptor concept is schematically represented by the following figure
(Pierret Fig. 4.13):
The term intrinsic is used to denote values for pure or undoped semiconductors. A
intrinsic Fermi energy level Ei will be defined in later sections that represents the Fermi
energy level of undoped semiconductors. Also, intrinsic electron and hole concentrations
will be defined as ni and pi well.
Equilibrium Concentration Relationships
The following definitions will be used:
n – Electron concentration. Total number of electrons per cm3 in the conduction band.
p – Hole concentration. Total number of holes per cm3 in the valence band.
ni – Intrinsic carrier concentration. Both electron and hole concentrations in an intrinsic
semiconductor.
N D - Number of ionized donors per cm3
N D - Total donor concentration
N A - Number of ionized acceptors per cm3
N A - Total acceptor concentration
The conduction band electron concentration and the valence band hole concentration are
given by:
n
Etop
Ec
p
Ev
Ebottom
g c E  f E dE
g v E 1  f E dE
For nondegenerate semiconductors (where the fermi energy in more than 3KT away from
both the valence and conduction band), the equations for n and p simplify to:
n  N C e  E F  E C  KT
p  N V e  EV  EF  KT
where NC and NV are the effective density of states in the conduction and valence bands
respectively and are:
 2mn* kT 

N C  2
2

 h

3
 2m *p kT 

N V  2
 h2 


3
2
2
When the semiconductor is undoped, n=p=ni and EF = Ei which leads to the following
equations for n and p:
n  ni e  EF  Ei  KT
p  ni e  Ei  EF  KT
Intrinsic carrier concentration ni, Intrinsic Fermi energy level and the np product
If the semiconductor is not doped, then the concentration of electrons in the conduction
band n is equal to the concentration of holes in the valence band p, hence n=p=ni where
ni is defined as the intrinsic carrier concentration. Using the equations above, it is easy to
show that the product np is:
np  ni2  N C NV e  EV EC  KT  N C NV e
 EGap KT
where Egap=EC-EV. Thus we have an equation for the intrinsic carrier concentration:
ni  NC NV e
 EGap KT
Besides the product np, we can equate n and p to obtain the intrinsic Fermi energy level:
n p
N C e  EF  EC  KT  N V e  EV  EF  KT
Ei 
EC  EV kT  N C

ln 
2
2  NV
Ei 
*
EC  EV 3kT  m p

ln *
m
2
4
 n







Carrier concentration as a function of temperature and determination of EF
The Fermi energy level is a function of the carrier concentration, thus the carrier
concentration of the semiconductor will be analyzed as a function of temperature.
The ratio of ionized donor atoms to total donors and the ratio of ionized acceptor atoms to
total acceptor atoms is given by:
N D
1

 EF  ED  KT
N D 1  gDe
gD  2
N A
1

 E A  E F  KT
N A 1  g Ae
gA  4
where ED is the donor energy level, EA is the acceptor energy level, gD and gA are the
degeneracy factors.
Now armed with all of these equations describing carrier concentrations as a function of
multiple variables, it is now valuable to analyze the carrier concentration and Fermi
energy level as a function of temperature.
The plot of the carrier concentration versus temperature is shown below:
There are three regions of the above figure: Freeze out, Extrinsic and Intrinsic regions.
The carrier concentration and Fermi energy level will be derived for each region.
Freeze out and low tempertature region
Consider a donor doped semiconductor doped with a concentration of ND donors. For
very low temperatures close to zero Kelvin, all the donor atoms retain their electrons
which do not have enough energy to be excited to the conduction band. All the states in
the valence band are occupied by electrons which do not have enough energy to be
excited across the relatively large bandgap (relative to the gap required by the donor
atom’s electrons). At slightly higher but still low temperatures, the donor atom’s
electrons will start to be excited to the conduction band according to the equation for
N D
. Since all the valence band states are happily occupied by electrons, the conduction
ND
band carrier concentration n is given by:
n  N D 
ND
1  g D e  EF  ED  KT
Using the relation n  N C e  EF  EC  KT , we can express the above equation as:
n  N D 
where N  
ND


1  n 
N
 

N C  EC  ED  KT
. The carrier concentration n is then easily solve for:
e
gD
 N 
n
 
2  2

N
2


  N  N D 


1
2
The Fermi energy level is now easily solved for using n  N C e  EF  EC  KT to get:

 N
E F  EC  kT ln 
 2NC

1


2

4
N

D 
1 
 1 


N  


Extrinsic Region
For moderate temperatures, practically all the donor and acceptor atoms will be ionized.
Writing the charge neutrality equation:
p  n  ND  N A  0
and using the equation np  ni2 , we have:
2

N D  N A  N D  N A 
2
n
 
  ni 
2
2



1
2
2

ni2 N A  N D  N A  N D 
2
p

 
  ni 
n
2
2



1
2
when the semiconductor is donor-doped such that N D >>ni , then we have:
n  ND
p
ni2
ND
and the Fermi energy level is easily shown to be:
N 
E F  Ei  kT ln  D 
 ni 
when the semiconductor is acceptor-doped such that N A >>ni , then we have:
p  NA
n
ni2
NA
and the Fermi energy level is easily shown to be:
N 
E F  Ei  kT ln  A 
 ni 
Intrinsic Region
For high temperatures, all the donor and acceptor atoms will be ionized yet the intrinsic
carrier concentration grows to the point where it is the primary source of conduction and
valence band charge carriers. Hence, we have:
n  ni
p  ni
E F  Ei