Question 19.4: Carrier Concentrations in Intrinsic Ge For germanium at 25°C......
Carrier Concentrations in Intrinsic Ge
For germanium at 25°C, estimate (a) the number of charge carriers, (b) the fraction of the total electrons in the valence band that are excited into the conduction band, and (c) the constant n_{0} in Equation 19-12a.
n = n_{i} = p_{i} = n_{0} \exp (\frac{-E_{g}}{2k_{B}T }) (19-12a)
From Table 19-5, ρ = 43 Ω · cm, ∴ σ = 0.0233 Ω^{-1} ⋅ cm^{-1}
Also from Table 19-5,
E_{g} = 0.67 \ eV, μ_{n} = 3900 \ \frac{cm^2}{V ⋅ s} , μ_{p} = 1900 \ \frac{cm^2}{V ⋅ s}
2k_{B}T= 2(8.617 × 10^{-5} \ eV/K)(273 \ K + 25 K) = 0.05136 \ eV at T = 25°C
(a) From Equation 19-10,
σ = nqμ_{n} + pqμ_{p} (19-10)
n = \frac{σ}{q(μ_{n} \ + \ μ_{p})} = \frac{0.0233}{(1.6 × 10^{-19})(3900 \ + \ 1900)} = 2.51 × 10^{13} \ \frac{electrons}{cm^3}
There are 2.51 × 10^{13} electrons/cm³ and 2.51 × 10^{13} holes/cm³ conducting charge in germanium at room temperature.
(b) The lattice parameter of diamond cubic germanium is 5.6575 × 10^{-8} cm. The total number of electrons in the valence band of germanium at 0 K is
Total electrons = \frac{(8 \ atoms/cell)(4 \ electrons/atom)}{(5.6575 × 10^{-8} cm)^3/cell}
= 1.77 × 10^{23}
Fraction excited = \frac{2.51 × 10^{13}}{1.77 × 10^{23}} = 1.42 × 10^{-10}
(c) From Equation 19-12a,
n_{0} = \frac{n}{\exp [-E_{g}/(2k_{B}T)]} = \frac{2.51 × 10^{13} \ carriers/cm^3}{\exp (-0.67/0.05136)}
= 1.16 × 10^{19} \ carriers/cm^3
| Table 19-5 Properties of commonly encountered semiconductors at room temperature | |||||||
| Semiconductor | Bandgap (eV) | Mobility of Electrons (mn) (\frac{ cm^2} {V-s }) | Mobility of Holes (μ_{p}) (\frac{ cm^2} {V-s }) | Dielectric Constant (k) | Resistivity (Ω· cm) | Density (\frac{g}{cm^3}) | Melting Temperature (°C) |
| Silicon (Si) | 1.11 | 1350 | 480 | 11.8 | 2.5 × 10^{5} | 2.33 | 1415 |
| Amorphous Silicon (a:Si:H) | 1.70 | 1 | 10^{-2} | ∼ 11.8 | 10^{10} | ∼ 2.30 | — |
| Germanium (Ge) | 0.67 | 3900 | 1900 | 16.0 | 43 | 5.32 | 936 |
| SiC (α) | 2.86 | 500 | 10.2 | 10^{10} | 3.21 | 2830 | |
| Gallium Arsenide (GaAs) | 1.43 | 8500 | 400 | 13.2 | 4 × 10^{18} | 5.31 | 1238 |
| Diamond | ∼ 5.50 | 1800 | 1500 | 5.7 | > 10^{18} | 3.52 | ∼ 3550 |