ABSTRACT
We have seen in Chapter 1 that in metals, the outermost band is usually nearly half-filled Let us define the energy at the bottom of the partially filled band as E = 0 when E = EB (Figure 41)
In metals at around 0 K, the highest energy level at which electrons are present is known as the Fermi energy level (EF) Above this level, the states are available for electrons, but no electrons occupy these states Thus, in metals at around 0 K, the probability of finding an electron at E > EF is zero The Fermi energy values for different metals are shown in Table 41 This table also shows the value of the work function (ϕ), which for metals refers to the energy needed to remove the outermost electron from E = EF For example, the EF value for copper (Cu) is 70 eV (Table 41) Thus, as shown in Figure 41, the work function (ϕ) of copper is 465 eV
The Fermi energy level for metals at 0 K is given by
E E
f= =
2 3( ) (/π or metals)
(41)
where me is the rest mass of the electron = h/2π, h is the Planck’s constant, and n is the free electron concentration in the metal
Example 41 illustrates the calculation of the Fermi energy level for a metal
SOLUTION a. We first convert the concentration of electrons per cubic centimeter to per cubic millimeter
The Fermi energy of metals does vary slightly with temperature and is given by
E T E k T
( ) =
1 12
− π
(42)
Note that since the EF,0 for metals is much higher than the kBT (Table 41) As a result, the EF of metals does not change significantly with temperature The Fermi energy level of semiconductors, especially those that are extrinsic, does change appreciably with temperature (see Section 46)
An electron with energy E = EF represents the highest-energy electron in a metal This is the significance of the Fermi energy level in metals If vF is the speed of an electron with energy EF,0, then
2 2
0mv EF,0 = F,
(43)
From the typical values of EF,0 for metals, we can see that at T = 0 K, the highest speed of electrons with energy EF,0 is ∼106 m/s (see Equation 43) As mentioned in Chapter 3, if electrons are considered classical particles, then the speed of electrons at 0 K should be zero The effective speed of electrons (ve) or the root mean speed of electrons in a metal is given by
5 2mv Ee ≅ F,0
(44)
The effective speed of electrons (ve) in a metal is relatively insensitive to temperature and depends on EF,0
At low temperatures, the conduction band of semiconductors is empty, and the valence band is completely filled Thus, the probability of finding an electron at T = 0 K at the valence bandedge (E = Ev) is 1 As the temperature increases, some electrons make a transition across the bandgap into the valence band The probability of finding an electron at or above the conduction bandedge (E ≥ Ec) is zero At higher temperatures (eg, 300 K), the probability of finding an electron in the valence band is still 1, but there is a small (but nonzero) probability of finding an electron in the conduction band as well, that is, E ≥ Ec Thus, as we go from E = Ev to E = Ec and higher in the conduction band, the probability of finding an electron begins at 1 and eventually approaches zero well into the conduction band In general, the probability function for a system of classical particles is given by the Boltzmann function:
f E A E
k T ( ) = −
exp B
(45)
where A is a constant, kB is the Boltzmann constant, and T is the temperature Electrons in materials are also constrained by the Pauli exclusion principle, which states that
no two electrons can have the same energy level (ie, same set of quantum numbers) With this restriction on the occupancy of states, the probability of finding an electron occupying energy level E is given by the so-called Fermi-Dirac distribution function, which is defined as
f E E E
k T
( ) =
1 exp F
+ −
(46)
where E is the energy of the electron, f (E) is the probability of an electron occupying the state with energy E, kB is the Boltzmann constant, T is the temperature, and EF is the Fermi energy level
The Fermi energy level is defined as the energy level for which the probability of finding an electron is 05 (Figure 42) Note that this level is in the middle of the bandgap, and based on the band diagram of a semiconductor, no electrons are allowed between E = Ev and E = Ec Thus, the Fermi-Dirac function gives a numerical value for any value of energy (E) However, this value of the function corresponds to the probability of finding an electron only if the energy level is allowed
At a given temperature (eg, T = T1), as E increases, f (E) decreases, regardless of whether E − EF is positive or negative The probability of finding an electron decreases, as shown in Figure 42 This probability becomes zero only as T approaches infinity
In Figure 42, we can see that at T = 0, f (E) = 1 if E < EF This means that every available level below EF is filled at T = 0 K This situation is the same as that for metals at T = 0 K Also, note that at T = 0 K, f (E) = 0 if E > EF This means that every energy level below EF is empty At the Fermi energy level, E = EF, f (E) = 05 However, the Fermi-Dirac function gives us only a mathematical value Since no electrons are allowed in the bandgap of an intrinsic semiconductor, the actual number of electrons in the bandgap is zero
When E > EF, as the temperature (T) increases, the exp[(E − EF)/kBT] decreases and the value of f (E) increases This means that the probability of finding an electron at higher energy levels becomes higher as the temperature increases
When E < EF, as the temperature increases, the term exp[(E − EF)/kBT] increases and the value of f (E) decreases That is, the probability of finding an electron at lower energy levels becomes lesser as the temperature increases, because chances are better that the electrons have moved up to higher energy levels
An electron creates a hole in the valence band when it is promoted to the conduction band Because a hole can be considered to be a missing electron, the probability of finding a hole is given by
f E f E E E
k T
( ) ( )
exp hole = − =
+ −
1 1 1
− F
(47)
Another interesting situation to consider is one in which (E − EF) is far greater than kBT These energy levels are closer to either the valence or conduction bandedges If (E − EF) >> kBT, then exp[(E − EF)/kBT] >> 1 and Equation 46 can be simplified as:
f E E E
k T
E E
k T ( ) =
=
1
exp
exp F
B− − −
(48)
Note the negative sign in Equation 48 This equation is a form of the Boltzmann equation for classical particles Thus, the “tail” of the Fermi-Dirac distribution (ie, when E is much greater than EF) can be described by the Boltzmann distribution equation The importance of this is as follows: In the conduction band of a semiconductor, when the number of states available is significantly greater than the number of electrons, the chance of two electrons occupying the same energy level (ie, having the same quantum numbers) is not very high Therefore, we need not worry about the limitation on electrons having the same quantum number imposed by the Pauli exclusion principle for the electrons in the conduction bandgap We can use the Boltzmann equation (Equation 48) to calculate the concentration of electrons in the conduction band and the concentration of holes in the valence band (Section 43)
The conductivity of a semiconductor depends upon the carrier concentrations and their mobilities In this section, we will learn to calculate the carrier concentrations If we know the probability of finding an electron at a given energy level in the conduction band, and if we know the number of states available, we can then calculate the total number of electrons in the conduction band The following analogy may help
Consider a high-rise hotel building We want to know how many guests are in the building at a given time Assume that we know how many rooms there are on each floor If we know the probability of finding guests in their rooms, then we can calculate the total number of people in the hotel We can use the same logic to calculate the number of electrons in a conduction band
Consider the band diagram for a typical intrinsic semiconductor (Figure 43) We also plot the Fermi-Dirac function, which describes the probability of finding an electron, and the density of states function Note that the vertical axis is the electron energy (E)
The electron energies in the conduction band range from E = Ec to E = ∞ If at any energy level E, the density of states (ie, the number of energy levels available) is given by N (E), the product
FIGURE 4.3 Band diagram and Fermi-Dirac function for an intrinsic semiconductor (From Pierret, R F 1988 Semiconductor Fundamentals Volume Reading, MA: Addison-Wesley With permission)
f (E) N (E) gives us the number of electrons at any energy level in the conduction band We integrate the product f (E) N (E) over the range E = Ec to E = ∞ to get the total concentration of electrons in the conduction band When the energy of an electron becomes E E qc= + · χ, the electron becomes free (Figure 41) Therefore, we should use E qc + · χ as the upper limit of integration However, as E tends to ∞, the function f (E) approaches zero rapidly Therefore, we will use E = ∞ as the top limit for the integral
n N E f E dE E E
= =
∫ ( ) ( ) c
(49)
The concentration of electrons in the conduction band is shown in Figure 43 as the shaded area under the curve, representing the integral shown in Equation 49
We will now use the effective density of states approximation We assume that most, if not all, electrons reside at the bottom of the conduction band at E = Ec This is reasonable, because electrons in the conduction band tend to minimize their energies The effective density of states at the conduction bandedge (Nc) is given by
N
m k T c
B= 2 2 2
π (410)
Thus, Equation 49 now becomes
n
m k T f E=
2 2 2
( )B cπ (411)
Note that because we assumed that most electrons in the conduction band reside at the bottom, we multiplied the effective density of states by the value of the probability function at E = Ec
From the definition of the Fermi-Dirac function (Equation 46), we get
f E E E
k T
( )
exp c
= +
1 −
(412)
Since (Ec − EF) >> kBT, the exponential term is much larger than 1, and hence f (Ec) is written as
f E E E
k T
E E
k T ( )
exp
=
=1
− − −
(413)
Substituting the value of f (Ec) from Equation 413 into Equation 411, we get
n
m k T E E
k T =
2
2 2
Bπ − −
(414)
By keeping the first term as Nc, we get
n N
E E
k T =
c
exp− −
(415)
The negative sign outside is shown so that the term inside is positive, because Ec is above EF We can rewrite Equation 414 as follows to eliminate the negative sign in the exponential term:
n
m k T E E
k T =
2
2 2
Bπ −
(416)
In Sections 45 and 46, we will see that we can calculate the concentration of electrons for intrinsic as well as extrinsic semiconductors using Equation 416 if we know the position of the EF relative to Ec
We can now derive a similar expression for calculating the concentration of holes in the valence band Remember that the conduction band is nearly empty and has few electrons, the concentration of which we have calculated (Equation 416) Now the valence band is nearly completely filled with valence electrons in the covalent bonds of the semiconductor atoms When some of these bonds break, the electrons move to the conduction band This creates holes in the valence band The concentration of holes (p) in the valance band is given by
p N E f E dE E
= =
∫ ( )[ ( )]1 0
(417)
Notice that the density of states N (E) is now multiplied by 1 − f (E), which is the probability of finding a hole We again use the effective density of states at the valence bandedge (N (Ev)); that is, we assume that most holes remain at the top of the valence band Recall that electron energy increases as we move up the band diagram, whereas the hole energy increases as we go down the band diagram The effective density of states at E = Ev is given by
N m k T
2 2 2
π
(418)
Rewriting Equation 417, we get
p
m k T f E=
−2 2
1 2
( ( ))B vπ (419)
From the Fermi-Dirac function
1 1 1
− − −
f E E E
k T
E
( )
exp
exp
=
+
= + v F
− −
−
E
k T
E E
k T
+
1exp
(420)
∴ −
−
− 1
f E
E E
k T
E E
k T
( )
exp
exp ( )
=
+
(421)
Note the denominator of the exponential term now contains (EF − Ev) and not (Ev − EF) Again, if (EF − Ev) >> kBT, then the Fermi-Dirac function simplifies to a Boltzmann function
since exp[(EF − Ev)/kBT] >> 1 The denominator in Equation 421 becomes 1
Thus, Equation 421 now becomes
1− −
∴ − − −
f E E E
k T
f E E E
( ) exp
( ) exp
=
=1 k TB
(422)
The concentration of holes in the valence band from Equation 419 now becomes
p m k T E E
k T =
− −
2
2 2
π
(423)
For the sake of convenience, we can represent the first term as Nv and can write Equation 423 as
p N
E E
k T =
v F v
exp− −
(424)
The negative sign outside is shown so that the term inside is positive, because EF is above Ev This equation gives us the concentration of holes in either an intrinsic semiconductor or an extrinsic semiconductor If we know the difference between the valence bandedge and the Fermi energy level, temperature, and the effective mass of holes, we can calculate the concentration of holes (p)
Let us multiply the equations for electron and hole concentrations From Equations 415 and 424, we get
n p N N
E E
k T
E E
k T × − − − −=
c v
exp exp =
N N E E
k T c v
exp− −
(425)
Since (Ec − Ev) = Eg, we get
n p N N
E
k T ×
− =
c v
exp
(426)
Thus, for a given semiconductor, the product of the concentration of electrons and the concentration of holes at a given temperature is constant This is known as the law of mass action for semiconductors It is similar to the ideal gas law: pressure × volume is a constant For example, for silicon (Si) (Eg ≈ 11 eV at 300 K), the value of n × p is constant, whether we have intrinsic, n-type, or p-type silicon For an n-type silicon crystal, created by doping with antimony (Sb), the electron concentration in the conduction band is higher compared to ni The increase in the concentration of electrons in an n-type semiconductor is compensated by a decrease in the concentration of holes in the valence band Similarly, for a p-type semiconductor, holes are the majority carriers, that is, p >> n To maintain a constant value of n × p, the concentration of electrons (n) decreases
For an intrinsic semiconductor, ni = pi Applying Equations 415 and 424 to an intrinsic semiconductor, we get
N E E
k T N
E E
exp exp− − − −
=
(427)
In this equation, EF,i is the Fermi energy level of an intrinsic semiconductor Taking a natural logarithm of both sides, we get
E E E k T
N
ln= + +
2 ( )
(428)
Substituting for Nc and Nv from Equations 410 and 418, respectively, we get
E E E k T
m
ln= + +
4 ( )
(429)
The first term in Equation 429 represents the middle of the bandgap Thus, the Fermi energy level of an intrinsic material (EF,i) is very close to the middle of the bandgap (Figure 44)
E E k T
m
ln− =
(430)
For intrinsic semiconductors with a larger effective mass of holes, the Fermi energy level is slightly above the middle of the bandgap When the effective mass of holes is higher, the value of Nv is higher Thus, the EF,i moves away from the valence band and is therefore slightly above the middle of the bandgap (Figure 45) For intrinsic materials with a larger effective electron mass, the EF,i is slightly below the middle of the bandgap The effective masses also change with the temperature,
FIGURE 4.4 Fermi energy level position for a typical intrinsic semiconductor (From Pierret, R F 1987 Advanced Semiconductor Fundamentals Reading, MA: Addison-Wesley With permission)
as does the bandgap (Eg) The variation of EF,i as given by Equation 430 is shown in Figure 45 Note that the dependence of Eg on the temperature, as given by the Varshni parameters, is not shown in the figure The slight deviation that causes the EF,i to shift from the middle of the bandgap as a result of the difference between the effective mass of holes and electrons is illustrated in Example 42
SOLUTION From Equation 4.30, we get
For intrinsic semiconductors, we can derive the electron and the hole concentrations using Equation 416 and Equation 423, respectively, as follows:
n N
E E
=
exp− −
(431)
p N E E
=
exp−
−
(432)
As mentioned previously, the negative sign is included so that the terms inside the brackets remain positive, because Ec is above EF,i and EF,i is above Ev In Equations 431 and 432, ni and pi are the electron and hole concentrations, the subscript i indicates the intrinsic semiconductor, EF,i is the Fermi energy level for an intrinsic semiconductor, and Nc and Nv are the effective densities of states for the conduction and valence bandedges, respectively
We now multiply Equation 431 by Equation 432 to get
n p N N E E
k T
E E
× − − − −=
exp exp
T N N
E E
k T
=
exp− −
or since Ec − Ev = Eg, the bandgap energy
n p N N
E
× −=
exp
(433)
This is the same as Equation 426, which confirms that the n × p value is a constant for a given semiconductor at a given temperature It does not matter whether the semiconductor is extrinsic or intrinsic
We have now shown that n × p value remains constant for a given semiconductor at a given temperature:
n p n p× ×= i i (434)
This is a very important equation, since it allows us to make a connection between the properties of intrinsic and extrinsic semiconductors
Another way to rewrite Equation 433 is as follows:
n N N
E
=
exp−
2 (435)
We can see quantitatively from Equation 435 that the concentration of thermally generated carriers changes with the temperature
We discussed the values of ni and pi for different semiconductors in Chapter 3 The intrinsic carrier concentrations are strongly temperature-dependent and are inversely related to the bandgap (Eg) The values of electron or hole intrinsic concentrations for silicon, germanium (Ge), and gallium arsenide (GaAs) are shown in Figure 46
As we can see in Figure 46, the value of ni for Si at 300 K is ∼15 × 1010 electrons/cm3 The bandgap of the semiconductors (Eg) also changes with the temperature This depen-
dence is given by the Varshni parameters For silicon, the change in densities of states Nc and Nv
(Equations 410 and 418) and the variation in the bandgap (Eg) with a temperature in the range 200-500 K are given by the following simplified equations (Green 1990; Bullis and Huff 1994):
N
T c =
2 86 10 300
. .
