1
J. Humlíček FKL II
8. Electrons and holes in equilibrium
In a semiconductor crystal, there is usually a small concentration of freely moving charges, electrons and holes, formed by the wavefunctions
from small ranges of k-vectors around the bottom of the conduction band and the top of the valence band. The gas of free quasiparticles is in a
thermal contact with “the rest” of the crystal, in particular with the vibrational movement of the lattice; we treat the latter as the thermal reservoir
having a given temperature T. The free carrier concentration can vary due to changes of temperature or in an external electromagnetic field.
Consequently, the change of internal energy U of the gas is composed of the acquired heat, thermodynamic work A performed by the gas, and the
work needed to change the number N of the carriers:
,dU TdS A dN    (8.1)
where S is entropy and a  chemical potential. Statistical equilibrium of the gas in contact with the thermal reservoir is described by the
grandcanonical distribution of the probability w of finding the gas in a state with the energy Ea and the number of particles Na (this probability is
proportional to the number of possibilities of achieving this state),
1
( , ) exp , exp .
a
a a a a
a a a
a N
E N E N
w E N Z g
Z kT kT
     
      
   
 (8.2)
The normalizing factor Z is called statistical sum, or partition function; ga is the degree of degeneracy of the state with the energy Ea. The
grandcanonical potential is related to the partition function by
ln( ) .kT Z   (8.3)
2
Fermion gas
The gas consisting of N (identical, non-interacting) fermions is subject of Pauli exclusion principle. Its Hamiltonian je is the sum of singleparticle
terms,
1
ˆ ˆ ˆ, .
N
i i i i i
i
H H H E 

  (8.4)
The state vector of the fermi gas is antisymmetric with respect to exchange of particles. The energy levels Ei can be degenerate; we have
therefore add an index to denote the corresponding states, i. The probability
,
, ,
,
, ,
1
exp ,
exp
i i
i
i i
i i
i
i i
i
i
i i
n
E
w n
Z kT
E
Z n
kT
 

 


 
  
 
 
  
 

(8.5)
depends on the number ni,i of occupied states i with the energy Ei. In the model of gas, the occupation of various states is independent of each
other and the corresponding probabilities are multiplied, as well as the corresponding factors in the partition function. The allowed number of
occupied single-particle function for fermions is restricted to either 0 or 1; the mean (expectation) value of the number is therefore
,
,
, , ,
1
= ,
exp 1
i
i i i
i i
i i
i i i FD
n i
E
n n w f
E kT
kT


  


  
          
 
 (8.6)
since
3
, ln 1 exp .i
i
i
E
kT
kT

   
      
  
(8.7)
The mean value of Eq. (8.6) has also the meaning of the probability of the occupation of the state i,i; it is usually called Fermi-Dirac statistics,
 
 
1
= .
exp 1
FDf x
x 
(8.8)
The mean number of particles with the energy Ei (i = 1,..., gi ) is
.i
i i FD
E
n g f
kT
 
  
 
(8.9)
Approximating the actual energies by a continuous spectrum, the number of particles having the energies between E and E + dE is
( ) ,FD
E
f D E dE
kT
 
 
 
(8.10)
where D is the density of states. The mean value of the number of particles becomes
( ) ,FD
E
N f D E dE
kT


 
  
 
 (8.11)
and the mean value of energy is
4
( ) .FD
E
E Ef D E dE
kT


 
  
 
 (8.12)
Parabolic minimum of band energies
The quadratic dispersion
 
22 22
2
yx z
x y z
kk k
E k
m m m
 
    
 
(8.13)
results in the density of states within the volume V (in practical terms, in the close neighborhood the minimum)
2 3
2
( ) .
x y zm m m
D E V E

 (8.14)
This density includes the double (spin) degeneracy. Denoting
* 1/3
( ) ,x y zm m m m (8.15)
equation (8.11) results in the concentration of electrons in the gas equal to
5
* 3/2
2 3
0
* 3/2 1/2
2 3
0
2 ( )
2 ( )
, .
exp( ) 1
FD
N m E
n f EdE
V kT
m kT t dt
z
t z kT






 
   
 
 
 


(8.16)
The integral in (8.16) cannot be expressed using elementary functions. It belogs to a class of Fermi-Dirac integrals (for j = 1/2)
0
( ) , 1 ,
exp( ) 1
j
j
t dt
F z j
t z

  
  (8.17)
obeying the recurrent relation
1( ) ( ) .j j
d
F z jF z
dz
 (8.18)
Of special importance is the limit
0
( ) exp( ) exp( ) exp( ) ( 1) for ,j
jF z z t t dt z j z

       (8.19)
where  is the Euler function (generalized factorial), (j+1) = j(j) , (1/2) = , (3/2) = .
6
Nondegenerate gas (classical limit)
In the limit of Eq. (8.19), the chemical potential lies deep below the lowest band energy and the gas is denoted as non-degenerate. The
requirement for fulfilling this condition is evidently the possibility to consider the unity in the denominator in the fraction of Fermi-Dirac
statistics of Eq. (8.8) negligible in comparison with the large values of the exponential function. Resulting statistics is the classic (Boltzmann) fB:
=exp exp .FD B
E E E
f f
kT kT kT kT
          
        
       
(8.20)
Equation (8.16) for the concentration is reduced to
* 3/2
3 3
(2 )
2 exp .
8
m kT
n
kT
 

 
  
 
(8.21)
The condition for the validity of Boltzmann statistics can be interpreted in a simple way. Equation (8.21) can be rewritten in the following form,
3 3
3 3
* 3/2 *
1 8 1
exp , ;
2 (2 ) 2 2
T T
N h
n
kT m kT V m kT
 
 
 
 
   
 
(8.22)
the quantity T is de Broglie wavelength of the quasiparticle having the effective mass m* and energy kT. For reaching the classical limit, the
volume per a single particle, V/<N>, is required to be much larger than the volume of the cube with the edge equal to the de Broglie wavelength
corresponding the the thermal energy of kT. This means the negligibility of the quantum effects related to the symmetry particle exchange, and
the gas follows the classical physics. Of course, the pronounced quantum effects remain in the presence of effective masses.
At a prescribed concentration (achieved, e.g., by the level of doping, with the temperature small to prevent the excitation "over the gap"),
equation (8.21) provides the following explicit dependence of chemical potential on temperature:
7
3 3
* 3/2
4
ln .
(2 )
n
kT
m kT



 (8.23)
Reaching the limit of classical statistics is supported by large temperature, large effective mass, and small concentration. At the temperature of
300 K and effective mass of 9.1E-28 g (this is the rest mass of free electron), the value of the exponential function in the classical statistics is
3 3
* 3/2 19 -3
4
exp ;
(2 ) 2.51 10 cm
n
n
kT m kT
 

 
  
 
(8.24)
at the concentration of 1E18 cm-3
, the chemical potential has the value of -0.107 eV, and the values of Boltzmann and Fermi-Dirac statistics for
the energy E = 0 are 0.0159 and 0.0157, respectively. Using the classical statistics is evidently allowed in this situation.
Strongly degenerate gas
Another simplification occurs for large concentrations, low temperatures, and small effective masses. For chemical potential much larger than kT,
the Fermi-Dirac statistics is close to the step function, assuming the value of 1 for E< and 0 for E> ( is the limit value for the chemical
potential at vanishing temperature). In this approximation, the concentration of Eq. (8.16) is
0
* 3/2 * 3/2
1/2 3/2
02 3 2 3
0
2 ( ) (2 )
.
3
kT
m kT m
n t dt


 
  (8.25)
Inversely, the chemical potential at zero temperature is
2/32 2/3 2 2
2/3 2/3
0 * *
(3 ) 3
.
2 8 2
h
n n
m m



 
   
 
(8.26)
8
The mean value of energy in a unit volume is then
0* 3/2 * 3/2
3/2 5/2
0 02 3 2 3
0
2 ( ) 3 (2 ) 3
.
5 3 5
E m m
E dE n
V

 
 
   (8.27)
Note the difference from the classical result, where the volume density of energy is 3nkT/2.  is usually called Fermi energy; for metals, it is the
largest single-electron energy in the limit of zero temperature.
Quantitative orientation can use the free electron mass and the electron density of 1E22 (typical metal); equation (8.26) leads to the Fermi energy
of 7.8 eV. For a heavily doped semiconductor, the typical concentration can be about 1E19 cm-3
; with the effective mass 0.1m0, the Fermi energy
is 0.78 eV. V takové hloubce vodivostního pásu ovšem nemůžeme očekávat platnost předpokládané kvadratické disperze pásových energií.
Intrinsic semiconductor
In an ideal semiconducting crystal at nonzero temperature, thermal excitation of valence states leads to the occupation of conduction band states,
leaving mobile holes in the valence bands. The thermal movement always excites electron-hole pairs. Their concentration is determined by the
ratio Eg/kT and effective masses of electrons and holes, Eg is width of the forbidden gap. We will retain the zero of the energy scale at the bottom
of the conduction band, Eq. (8.13); consequently, the top of the valence band is located at the energy -Eg. We will limit ourselves to isotropic
bands around the Brillouin zone center, with the dispersions (E’ is the energy of valence band)
   
2 2
2 2
, ' ,
2 2
g
n p
k k
E k E k E
m m
    (8.28)
where mn and mp are the effective masses of the negatively and positively charged carriers (elekctrons and holes, respectively). The probability of
the state with the energy E’ being unoccupied by an electron, i.e., it represents a hole, results from Eq. (8.6) as
9
   
1 1
' ' 1 ' 1 .
' '
exp 1 exp 1
FD FDf E f E
E E
kT kT
 
    
    
    
   
(8.29)
According to the assumption of the thermal excitation of the pairs, the concentrations of electrons (n) and holes (p) are equal; they are given by
the following formulas:
3/2
2 3
0
3/2 3/2
2 3 2 3
0
( )2
,
( ) ( )2 ' 2
' ' .



 

 

 
  
 
   
      
   

 
g
n
FD
E
p p g
FD g FD
m E
n f EdE
kT
m m E EE
p f E E dE f EdE
kT kT
(8.30)
In the classical limit (small concentrations, large temperatures and effective masses), the Fermi-Dirac functions may be replaced by the
Bolzmann statistics; using (8.20) and (8.21), the concentrations have the simplified form:
3/2
3 3
3/2
3 3
(2 )
2 exp ,
8
(2 )
2 exp .
8
 

 

 
  
 
 
  
 
n
p g
m kT
n
kT
m kT E
p
kT
(8.31)
The condition of neutrality (n=p) leads to
10
3/4
exp exp ,
2
3
ln .
2 4


    
     
    
 
    
 
p g
n
g p
n
m E
kT m kT
E m
kT
m
(8.32)
The chemical potential at zero temperature (Fermi energy) is located in the middle of the forbidden gap. The hole effective masses are typically
multiples of those of electrons, the chemical potential at finite temperatures is located above the mid-gap, with the values of the order of kT. The
concentration of electrons and holes (called intrinsic) is
3/2
03 3
(2 )
2 exp exp .
8 2 2
n p g g
i T
m m kT E E
n n p n
kT kT


   
        
   
(8.33)
The value of the factor n0T in front of the exponential (the latter is typically rather small) in Eq. (8.33) corresponds to a single electron-hole pair
in the volume of the cube with the edge equal to de Broglie wavelength of a particle, having the mass n pm m and energy kT, see Eq. (8.22). At
the temperature of 300 K and effective masses of both electrons and holes of 9.1E-28 g (i.e., the rest mass of free electrons), this concentration
amounts to 2.5E19 cm-3
, de Broglie wavelength is 4.3 nm; The intrinsic concentration ni with the gap Eg of 1 and 0.1 eV is 1E11 and 3.6E18
cm-3
, respectively.
11
Doped semiconductors
The presence of donor and acceptor levels is schematically shown in the following figure:
Energies in a doped semiconductor.
ED
EA
-EA
-ED
0
-Eg
Eg
ENERGIE
POLOHA
12
At finite temperatures, the conduction-band states are occupied via the thermal excitation of (localized) impurity states, and also via the
excitation of valence states. The charge neutrality requires the equality (the number of electrons in the conduction band) + (the number of
electrons in the acceptor states) = (the number of holes in the valence band) + (the number of holes in the donor states), i.e.,
3/2
2 3
0
3/2
2 3
0
( )2
( )2
.
n A
FD A FD
p g D
FD D FD
m EE
f EdE n f
kT kT
m E E E
f EdE n f
kT kT


 



    
   
   
    
    
  


(8.34)
The consequences of the occupation of different states can be assessed relatively easily in the following situation. Let us assume the chemical
potential within the forbidden band, far from its edges; in addition, we will assume the temperature and effective masses large enough for the
approximation of the Fermi-Dirac statistics in Eq. (8.34) by the Boltzmann exponential. This assumption requires the following inequality to be
satisfied,
exp 1 ,X
kT
 
  
 
(8.35)
and Eq. (8.34) reduces to
3/2
3 3
3/2
3 3
(2 )
2
8
(1/ )exp 1
(2 ) 1
2 exp .
8
exp 1
n A
A
p g D
D
m kT n
X
E
X
kT
m kT E n
EkT X
X
kT





  
 
 
 
  
    
 
(8.36)
This is an algebraic equation of the order of 4 for the root X, which specifies the chemical potential.
13
Further simplification occurs for a single type of dopants, lat us select donors (meaning no acceptors, nA = 0), and a small ionization energy
compared with the gap (ED « Eg):
3/2
3 3
(2 )
2 .
8
exp 1
n D
D
m kT n
X
E
X
kT



 
 
 
(8.37)
For low temperatures (kT « ED), the value of 1 in the numerator of Eq. (8.37) is negligible, leading to
3/2
3/4
(2 )
exp ,
22(2 )
D
D
n
E
X n
kTm kT


 
  
 
(8.38)
and the chemical potential
3
3/2
(2 )
ln
2 2 2(2 )
D D
n
E nkT
m kT



   (8.39)
and the concentration of conduction electrons
3/4
3/2
2(2 )
exp .
(2 ) 2
n D
D
m kT E
n n
kT


 
  
 
(8.40)
Conversely, for high temperatures (ED « kT), the value of the exponential in the numerator of Eq. (8.37) is negligible, leading to
3
3/2
(2 )
,
2(2 )
D
n
X n
m kT


 (8.41)
14
chemical potential
3
3/2
(2 )
ln
2(2 )
D
n
n
kT
m kT



 (8.42)
and the concentration of conduction electrons
.Dn n (8.43)
This is expected, as the thermal energy ionizes all of the donors.
15
Relations (8.37-43) can be used for the acceptor doping (setting nD = 0), via the substitution of nD , ED, nn by nA , EA, np, setting the zero energy at
the valence band maximum, and the direction of increasing energies inside the valence band.
Let us note a consequence of Eqs. (8.31) and (8.33). The product of the electron and hole concentrations is
3
2
3 3 2
(2 )
2 exp ,
(8 )
n p g
i
m m kT E
np n
kT


 
   
 
(8.44)
which is the square of the intrinsic concentration (on the absence of doping). The free electrons or holes from impurity states reduces the
concentration of the carriers of the opposite sign; this effect becomes more pronounced for smaller intrinsic concentrations.
Semiconductor crystals containing both donor and acceptor levels are called (usually only partly) compensated.
The statistical equilibrium in crystals with anisotropic and degenerate bottom of the conduction band requires the density of states to be modified
(planned for exercises). Similar problems occur for non-quadratic dispersion of valence bands.
The density of states is well known from measurements on many doped semiconductors. Shown below are examples, in particular for Si:P in the
neighborhood of the metal-insulator (MI) transition, occurring at ~3.5E18 cm-3
.
16
Photoluminescence spectra of Si:P at weak (crosses) and strong
(circles) injection, and density of states from transport
measurements (full squares), at 4.2 K.
Different levels of phosphorus doping.
Parametrized density of states (solid lines).
From Altermatt et al., JAP 2006.
17
Energy of the maxima of impurity band (full circles)
and activation energies (empty symbols) in
phosphorus doped Si.
Solid line: parametrized position of the band
maximum.
Dashed/dotted lines: results of several theoretical
treatments.
From Altermatt et al., JAP 2006.
18
Real part of conductivity derived from transmission spectra of Si:P (5.59E16 cm-3
) at different
temperatures. The wavenumber of 363 cm-1
corresponds to the ionization energy of 45 meV.
0 200 400
0.0
0.5
1.0
1.5
1s(A1
)->cb
->2p+/-
1s(A1
)->2p0
->2p+/-
1s(T2
)
1s(E)
->2p0
1s(T2
)
15K
25
45
75
TransN7-Sig1TD
200
65
300K
100
WAVENUMBER (cm
-1
)
Si:P 7.87 
-1
cm
-1
5.59x10
16
cm
-3
1
(
-1
cm
-1
)
1s(E)
19
Intrinsic concentration for several semiconductors.
20
Fermi energy relative to the center of gap for silicon.