In semiconductors, as discussed earlier, there are no mobile carriers at zero temperature.
As temperature is raised, electrons from the valence band are thermally excited into the conduction band, and in equilibrium there is an electron density n and an equal hole density p, as shown in Fig. 3.14a. To calculate the electron and hole densities in a pure semiconductor (i.e., no defects are present) we first recall some important expressions for the density of states. The density of states has the form
N(E) = ^cf\EE^
7r2n
where m*dos is the density of states mass and Ec is the conduction bandedge. A similar expression exists for the valence band except the energy term is replaced by (Ev — E)1'2 and the density of states exist below the valence bandedge Ev. In Fig. 3.14 we show a schematic view of the density of states.
3.5. Mobile carriers 121
•a •&#
(a)
(b)
f=J[.O
(c)
Figure 3.14: (a) A schematic showing that electron and hole densities are equal in a pure semiconductor, (b) Density of states and Fermi occupation function at low temperatures, (c) Density of states and Fermi function at high temperatures when nt and pi become large.
In direct gap semiconductors m*dos is just the effective mass for the conduction band. In indirect gap materials it is given by
where m\m%rri^ are the effective masses along the three principle axes. For Si counting the six degenerate X-valleys we have
For the valence band we can write a simple expression for a density of states masses, which includes the HH and LH bands
mdos - \ m
* 3 / 2 , * 3 / 2 \2/3
In pure semiconductors, electrons in the conduction come from the valence band and n = p — m = pi, where ni and pi are the intrinsic carrier concentrations. In general the electron density in the conduction band is
n = / Ne(E)f(E)dE
JEC
E-L, • , (3-20)
' {• CO
JEC
In Fig. 3.14b we show how a change of temperature alters the shape of the Fermi function and alters the electron and hole densities. For small values of n (non-degenerate statistics where we can ignore the unity in the Fermi function) we get
n = Nc exp [(EF - Ec) /kBT] (3.21) where the effective density of states Nc is given by
J
A similar derivation for hole density gives
p=Nv exp [{Ev - EF) /kBT] (3.22) where the effective density of states Nv is given by
3/2
Nv= 2 l '"hM"2JL J
We also obtain
np = 4
( ^TU~ I (m*m^ )3 / 2 e xP (-Eg/kBT) (3.23)
3.5. Mobile carriers 123
MATERIAL
Si (300 K) Ge (300 K) GaAs (300 K)
CONDUCTION BAND EFFECTIVE DENSITY (Nc)
2.78 x 1019 cm-3 1.04xl019cm-3 4.45 x 1017 cm-3
VALENCE BAND EFFECTIVE DENSITY (Nv)
9.84x10*8 cm-3 6.0 xlO!8 cm-3 7.72 X 10*8 cm-3
INTRINSIC CARRIER CONCENTRATION (H; = Pj)
1.5 xlO!O cm-3 2.33 x 1013 cm-3
1.84x106 cm-3 Table 3.3: Effective densities and intrinsic carrier concentrations of Si, Ge, and GaAs. The numbers for intrinsic carrier densities are the accepted values even though they are smaller than the values obtained by using the equations derived in the text.
We note that the product np is independent of the position of the Fermi level and is dependent only on the temperature and intrinsic properties of the semiconductor. This observation is called the law of mass action. If n increases, p must decrease, and vice versa. For the intrinsic case n = n2- = p = pi, we have from the square root of the equation above
rii = pi =
3/2
e x p
(-
77* /rTj'M ^ 94"ằ
Thus the Fermi level of an intrinsic material lies close to the midgap. Note that in calculating the density of states masses m*h and m*, the number of valleys and the sum of heavy and light hole states have to be included.
In Table 3.3 we show the effective densities and intrinsic carrier concentrations in Si, Ge, and GaAs. The values given are those accepted from experiments. These values are lower than the ones we get by using the equations derived in this section. The reason for this difference is due to inaccuracies in carrier masses and the approximate nature of the analytical expressions.
We note that the carrier concentration increases exponentially as the bandgap decreases. Results for the intrinsic carrier concentrations for Si, Ge, and GaAs are shown in Fig. 3.15. The strong temperature dependence and bandgap dependence of intrinsic carrier concentration can be seen from this figure. In electronic devices where current has to be modulated by some means, the concentration of intrinsic carriers is fixed by the temperature and therefore is detrimental to device performance. Once the intrinsic carrier concentration increases to ~ 1015 cm"3, the material becomes unsuitable for electronic devices, due to the high leakage current arising from the intrinsic carriers.
A growing interest in high-bandgap semiconductors, such as diamond (C), SiC, etc., is partly due to the potential applications of these materials for high-temperature devices where, due to their larger gap, the intrinsic carrier concentration remains low up to very high temperatures.
1019 1000 500
7TC)
200 100 27 0 -20
1018 1017 16
i
10
io1
io1
IIERD
P5 OS
u
INTRINSIC
12
io11 io10
IO9
IO8
io7
IO6
=dt=S
\
\
\
\
\
\
\
\
— \
\
\
\
\
\
\
\
\ i
\
\
\ S i
\
\
\
\
\
\
1
\
s. \
\ \
\ \
—i—
\
\
GaAs
!L
\
\
\
\
\
\
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 1000/7 (X-7)
Figure 3.15: Intrinsic carrier densities of Ge, Si, and GaAs as a function of reciprocal tem- perature.
3.5. Mobile carriers 125 EXAMPLE 3.2 Calculate the effective density of states for the conduction and valence bands of GaAs and Si at 300 K. Let us start with the GaAs conduction-band case. The effective density of states is
Note that at 300 K, kBT = 26 meV = 4 x 10"21 J.
_ /P.067 x 0.91 x 10~30 (kg) x 4.16 x 10~21 (J)
c " \ 2 x 3.1416 x (1.05 x 10"34 (Js))2
= 4.45 x 1023 m"3 = 4.45 x 1017 cm"3
In silicon, the density of states mass is to be used in the effective density of states. This is given by
mdos = 62/3(0.98 x 0.19 x 0.19)1/3 ra0 = 1.08 m0
The effective density of states becomes Nc = 2
2 ;
_ A.06 x 0.91 x 10~30 (kg) x 4.16 x 10~21 (J) V/ 2 _3
~ ^ 2 x 3.1416 x (1.05 x 10~34 (Js))2 J m
= 2.78 x 1025 m"3 = 2.78 x 1019 cm"3
We can see the large difference in the effective density between Si and GaAs.
In the case of the valence band, we have the heavy hole and light hole bands, both of which contribute to the effective density. The effective density is
For GaAs we use rrihh = 0.45rao, mth = 0.08mo and for Si we use rrihh = 0.5mo, rri£h = 0.15rao, to get
NV(GBLAS) = 7.72 x 1018cm"3
TV^Si) = 9.84 x 1018cm"3
EXAMPLE 3.3 Calculate the position of the intrinsic Fermi level in Si at 300 K.
The density of states effective mass of the combined six valleys of silicon is
The density of states mass for the valence band is 0.55 mo- The intrinsic Fermi level is given by (referring to the valence bandedge energy as zero)
= ^ - (0.0132 eV)
The Fermi level is then 13.2 meV below the center of the mid-bandgap.
E X A M P L E 3.4 Calculate the intrinsic carrier concentration in InAs at 300 K and 600 K.
The bandgap of InAs is 0.35 eV and the electron mass is 0.027m0. The hole density of states mass is 0.4rao. The intrinsic concentration at 300 K is
m = pi = 2 ( ^ J (ml m*h)S/* exp
(0.026)(1.6 x 10"- 1 91 9 x) \
3 /2
2 x 3.1416 x (1.05 x 10"3 4)2
(0.027 x 0.4 x (0.91 x 1 0 ~3 0)2)3 / 4 exp (--
= 1.025 x 1021 m "3 = 1.025 x 1015cm~3 The concentration at 600 K becomes
nl(600 K) = 2.89 x 1015cm~3