The distribution (with respect to energy) of electrons in the conduction band is given by the density of allowed quantum states times the probability that a state is occupied by an electron. This statement is written in equation form as
n(E) gc(E)fF (E) (4.1)
where fF (E) is the Fermi–Dirac probability function and gc(E) is the density of quan- tum states in the conduction band. The total electron concentration per unit volume in the conduction band is then found by integrating Equation (4.1) over the entire conduction-band energy.
Similarly, the distribution (with respect to energy) of holes in the valence band is the density of allowed quantum states in the valence band multiplied by the prob- ability that a state is not occupied by an electron. We may express this as
p(E) gv(E)[1 fF (E)] (4.2) The total hole concentration per unit volume is found by integrating this function over the entire valence-band energy.
To fi nd the thermal-equilibrium electron and hole concentrations, we need to determine the position of the Fermi energy EF with respect to the bottom of the conduction-band energy Ec and the top of the valence-band energy Ev. To address this question, we will initially consider an intrinsic semiconductor. An ideal in- trinsic semiconductor is a pure semiconductor with no impurity atoms and no lattice defects in the crystal (e.g., pure silicon). We have argued in the previous chapter that, for an intrinsic semiconductor at T 0 K, all energy states in the valence band are fi lled with electrons and all energy states in the conduction band are empty of electrons. The Fermi energy must, therefore, be somewhere between Ec and Ev. (The Fermi energy does not need to correspond to an allowed energy.)
As the temperature begins to increase above 0 K, the valence electrons will gain thermal energy. A few electrons in the valence band may gain suffi cient energy to jump to the conduction band. As an electron jumps from the valence band to the con- duction band, an empty state, or hole, is created in the valence band. In an intrinsic
semiconductor, then, electrons and holes are created in pairs by the thermal energy so that the number of electrons in the conduction band is equal to the number of holes in the valence band.
Figure 4.1a shows a plot of the density of states function in the conduction-band gc(E), the density of states function in the valence-band gv(E), and the Fermi–Dirac probability function for T 0 K when EF is approximately halfway between Ec and Ev. If we assume, for the moment, that the electron and hole effective masses are equal, then gc(E) and gv(E) are symmetrical functions about the midgap energy (the energy midway between Ec and Ev). We noted previously that the function fF (E ) for E EF is symmetrical to the function 1 fF (E) for E EF about the energy E EF. This also means that the function fF(E) for E EF dE is equal to the function 1 fF (E ) for E EF dE.
Figure 4.1 | (a) Density of states functions, Fermi–Dirac probability function, and areas representing electron and hole concentrations for the case when EF is near the midgap energy; (b) expanded view near the conduction-band energy;
and (c) expanded view near the valence-band energy.
gc(E)fF(E) n(E)
Area n0 electron concentration
gv(E)(1 fF(E)) p(E)
Area p0 hole concentration gc(E)
gv(E) Ev
Ec
EF E
fF(E) 0
fF(E)
fF(E) 1 (a)
(b)
(c) gc(E)
Ec
fF(E)
0
gv(E) Ev
[1 fF(E)]
E E
Figure 4.1b is an expanded view of the plot in Figure 4.1a showing fF (E) and gc(E) above the conduction-band energy Ec. The product of gc(E) and fF (E) is the distribution of electrons n(E ) in the conduction band given by Equation (4.1). This product is plotted in Figure 4.1a. Figure 4.1c, which is an expanded view of the plot in Figure 4.1a shows [1 fF (E)] and gv(E ) below the valence-band energy Ev. The product of gv(E) and [1 fF (E)] is the distribution of holes p(E) in the valence band given by Equation (4.2). This product is also plotted in Figure 4.1a. The areas under these curves are then the total density of electrons in the conduction band and the total density of holes in the valence band. From this we see that if gc(E ) and gv(E) are symmetrical, the Fermi energy must be at the midgap energy in order to obtain equal electron and hole concentrations. If the effective masses of the electron and hole are not exactly equal, then the effective density of states functions gc(E) and gv(E) will not be exactly symmetrical about the midgap energy. The Fermi level for the intrinsic semiconductor will then shift slightly from the midgap energy in order to obtain equal electron and hole concentrations.
4.1.2 The n0 and p0 Equations
We have argued that the Fermi energy for an intrinsic semiconductor is near midgap.
In deriving the equations for the thermal-equilibrium concentration of electrons n0
and the thermal-equilibrium concentration of holes p0, we will not be quite so restric- tive. We will see later that, in particular situations, the Fermi energy can deviate from this midgap energy. We will assume initially, however, that the Fermi level remains within the bandgap energy.
Thermal-Equilibrium Electron Concentration The equation for the thermal- equilibrium concentration of electrons may be found by integrating Equation (4.1) over the conduction band energy, or
n0
gc(E)fF (E) dE (4.3)
The lower limit of integration is Ec and the upper limit of integration should be the top of the allowed conduction band energy. However, since the Fermi probability function rapidly approaches zero with increasing energy as indicated in Figure 4.1a, we can take the upper limit of integration to be infi nity.
We are assuming that the Fermi energy is within the forbidden-energy band- gap. For electrons in the conduction band, we have E Ec. If (Ec EF) kT, then (E EF) kT, so that the Fermi probability function reduces to the Boltzmann ap- proximation,1 which is
fF(E) ____ 1 1 exp __ (E EF)
kT
exp (E EF)
__ kT (4.4)
4 . 1 Charge Carriers in Semiconductors 109
1The Maxwell–Boltzmann and Fermi–Dirac distribution functions are within 5 percent of each other when E EF 3kT (see Figure 3.35). The notation is then somewhat misleading to indicate when the Boltzmann approximation is valid, although it is commonly used.
Applying the Boltzmann approximation to Equation (4.3), the thermal-equilibrium density of electrons in the conduction band is found from
n0
Ec
4__ (2 m n* )32
h3 ______E Ec exp __ (E kT EF) dE (4.5) The integral of Equation (4.5) may be solved more easily by making a change of variable. If we let
E__ Ec
kT (4.6)
then Equation (4.5) becomes n0 4___ (2 m n* kT)32
h3 exp __ (Ec EF) kT
0
12 exp () d (4.7) The integral is the gamma function, with a value of
0
12 exp () d 1 _ 2 __
(4.8)
Then Equation (4.7) becomes
n0 2 __ 2 m n* kT
h2 32 exp __ (EckT EF) (4.9)
We may defi ne a parameter Nc as
Nc 2 __ 2 m n* kT
h2 32 (4.10)
The parameter m n* is the density of states effective mass of the electron. The thermal- equilibrium electron concentration in the conduction band can be written as
n0 Nc exp (Ec EF)
__ kT (4.11)
The parameter Nc is called the effective density of states function in the conduction band. If we were to assume that m n* m0, then the value of the effective density of states function at T 300 K is Nc 2.5 1019 cm3, which is the order of magnitude of Nc for most semiconductors. If the effective mass of the electron is larger or smaller than m0, then the value of the effective density of states function changes accordingly, but is still of the same order of magnitude.
Objective: Calculate the probability that a quantum state in the conduction band at E Ec kT2 is occupied by an electron, and calculate the thermal-equilibrium electron concentration in silicon at T 300 K.
Assume the Fermi energy is 0.25 eV below the conduction band. The value of Nc for silicon at T 300 K is Nc 2.8 1019 cm3 (see Appendix B).
■ Solution
The probability that a quantum state at E Ec kT2 is occupied by an electron is given by fF(E) ____ 1
1 exp __ E EF
kT exp __ (E kT EF) exp ____ (Ec (kTkT2) EF)
EXAMPLE 4.1
4 . 1 Charge Carriers in Semiconductors 111
or
fF(E) exp ____ (0.25 (0.02592))
0.0259 3.90 105
The electron concentration is given by n0 Nc exp __ (Ec EF)
kT (2.8 1019) exp __ 0.02590.25
or
n0 1.80 1015 cm3
■ Comment
The probability of a state being occupied can be quite small, but the fact that there are a large number of states means that the electron concentration is a reasonable value.
■ EXERCISE PROBLEM
Ex 4.1 Determine the probability that a quantum state at energy E Ec kT is occupied by an electron, and calculate the electron concentration in GaAs at T 300 K if the Fermi energy level is 0.25 eV below Ec.
[Ans. f (E F
) 2.36
5 10
, n 0
3.02
13 10
3 cm
]
Thermal-Equilibrium Hole Concentration The thermal-equilibrium concentra- tion of holes in the valence band is found by integrating Equation (4.2) over the valence-band energy, or
p0
gv(E)[1 fF (E)] dE (4.12) We may note that
1 fF(E) ___ 1 1 exp E__ F E
kT (4.13a)
For energy states in the valence band, E Ev. If (EF Ev) kT (the Fermi function is still assumed to be within the bandgap), then we have a slightly different form of the Boltzmann approximation. Equation (4.13a) may be written as
1 fF(E) ___ 1 1 exp __ EF E
kT exp __ (EkTF E) (4.13b)
Applying the Boltzmann approximation of Equation (4.13b) to Equation (4.12), we fi nd the thermal-equilibrium concentration of holes in the valence band is
p0
Ev
4 (2 m p* )32
__ h3 ______Ev E exp __ (EkTF E) dE (4.14)
where the lower limit of integration is taken as minus infi nity instead of the bottom of the valence band. The exponential term decays fast enough so that this approxima- tion is valid.
Equation (4.14) may be solved more easily by again making a change of vari- able. If we let
__ Ev E
kT (4.15)
then Equation (4.14) becomes p0 4(2 m p* kT)32
___ h3 exp (EF Ev)
__ kT 0 ( )12 exp ( ) d (4.16)
where the negative sign comes from the differential dE kTd . Note that the lower limit of becomes when E . If we change the order of integration, we introduce another minus sign. From Equation (4.8), Equation (4.16) becomes
p0 2 2 m p* kT
__ h2 32 exp __ (EFkT Ev) (4.17)
We may defi ne a parameter Nv as
Nv 2 2 m p* kT
__ h2 32 (4.18)
which is called the effective density of states function in the valence band. The parameter m p* is the density of states effective mass of the hole. The thermal- equilibrium concentration of holes in the valence band may now be written as
p0 Nv exp __ (EF Ev)
kT (4.19)
The magnitude of Nv is also on the order of 1019 cm3 at T 300 K for most semiconductors.
Objective: Calculate the thermal-equilibrium hole concentration in silicon at T 400 K.
Assume that the Fermi energy is 0.27 eV above the valence-band energy. The value of Nv for silicon at T 300 K is Nv 1.04 1019 cm3. (See Appendix B)
■ Solution
The parameter values at T 400 K are found as:
Nv (1.04 1019) 400 _
300 32 1.60 1019 cm3
and
kT (0.0259) 400 _
300 0.03453 eV
The hole concentration is then
p0 Nv exp __ (EF Ev)
kT (1.60 1019) exp __ 0.034530.27
or
p0 6.43 1015 cm3 EXAMPLE 4.2
The effective density of states functions, Nc and Nv, are constant for a given semiconductor material at a fi xed temperature. Table 4.1 gives the values of the den- sity of states function and of the density of states effective masses for silicon, gallium arsenide, and germanium. Note that the value of Nc for gallium arsenide is smaller than the typical 1019 cm3 value. This difference is due to the small electron effective mass in gallium arsenide.
The thermal-equilibrium concentrations of electrons in the conduction band and of holes in the valence band are directly related to the effective density of states con- stants and to the Fermi energy level.
4 . 1 Charge Carriers in Semiconductors 113
■ Comment
The parameter values at any temperature can easily be found by using the 300 K values and the temperature dependence.
■ EXERCISE PROBLEM
Ex 4.2 (a) Repeat Example 4.2 at T 250 K. (b) What is the ratio of p0 at T 250 K to that at T 400 K? ] 3 10 ; (b) 4.54 3 cm 13 10 2.92 0 [Ans. (a) p
TYU 4.1 Calculate the thermal equilibrium electron and hole concentration in silicon at T 300 K for the case when the Fermi energy level is 0.22 eV below the conduction-band energy Ec. The value of Eg is given in Appendix B.4.
(Ans. n 5.73 0
15 10
3 cm
, p 8.43 0
3 10
3 cm
)
TYU 4.2 Determine the thermal equilibrium electron and hole concentration in GaAs at T 300 K for the case when the Fermi energy level is 0.30 eV above the valence-band energy Ev. The value of Eg is given in Appendix B.4.
(Ans. n 0.0779 cm 0
, p 3
6.53 0
13 10
3 cm
)
TEST YOUR UNDERSTANDING