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Intrinsic semiconductors
In semiconductor technology it is common to add small quantities of dopants to a semiconductor to change its electrical properties. Doped semiconductors are called extrinsic semiconductors. Semiconductors without anything added are called intrinsic semiconductors. This week we will consider intrinsic semiconductors.
Reading: Singh 2.1 - 2.5 or Thuselt 2.1 and 2.2
For the exam
- You need to be able to look at a band structure diagram for a semiconductor and be able to identify the conduction band, the valence band, the energy gap \(E_g\), whether the semiconductor is direct or in direct, and to be able to determine the effective masses of electrons and holes. Some examples of band structure calculations are: GaN, 6H SiC, GaAs, GaP, Ge, InAs
- Semiconductors are transparent to photons with energies \(\hbar\omega < E_g\) and they absorb light for \(\hbar\omega > E_g\) as the photon moves an electron from the valence band to the conduction band. Light emission is observed in direct band gap semiconductors when an electron falls from the conduction band to the valence band. The emitted photon energy is \(\hbar\omega = E_g\).
- You need to be able to explain what a hole is.
- For a semiconductor, then density of states at the top of the valence band grows like the square root of energy, $D(E)\approx D_v\sqrt{E_v -E}$, and the density of states at the bottom of the conduction band also grows like the square root of energy, $D(E)\approx D_c\sqrt{E -E_c}$. These approximations are used in the Boltzmann approximation.
- The Fermi function \(f(E)=\frac{1}{1+\exp\left(\frac{E-E_F}{k_BT}\right)}\) describes the probablility that an electron state at energy \(E\) is occupied. Here \(E_F\) is the Fermi energy, \(k_B\) is Boltzmann's constant, and \(T\) is the absolute temperature.
- The electron density is the density of states at an energy times the probability that the states are occupied, integrated over all energies, \(n = \int_{-\infty}^{\infty} D(E) f(E) dE\).
- For semiconductors, the number of electrons in the conduction band is \(n = \int_{E_c}^{\infty} D(E) f(E) dE\) and the number of holes in the valence band is \(p = \int_{-\infty}^{E_v} D(E)(1- f(E)) dE\). Using the Boltzmann approximation these expressions can be written as, \( n=N_c\exp\left(\frac{E_F-E_c}{k_BT}\right)\) and \( p=N_v\exp\left(\frac{E_v-E_F}{k_BT}\right)\). From these formulas you should be able to calculate the density of electrons in the conduction band, the density of holes in the valence band, the intrinsic carrier concentration \(n_i\), and the Fermi energy for an intrinsic semiconductor.
Additional Information
Properties |
Si | Ge | GaAs |
Bandgap Eg | $1.166-\frac{4.73\times 10^{-4}T^2}{T+636}$ eV (indirect) | $0.7437-\frac{4.77\times 10^{-4}T^2}{T+235}$ eV (indirect) | $1.519-\frac{5.41\times 10^{-4}T^2}{T+204}$ eV (direct) |
Effective density of states in conduction band (300 K) Nc | 2.78 × 1025 m-3 | 1.04 × 1025 m-3 | 4.45 × 1023 m-3 |
Effective density of states in valence band (300 K) Nv | 9.84 × 1024 m-3 | 6.0 × 1024 m-3 | 7.72 × 1024 m-3 |
Effective mass electrons
m*/m0 | ml* = 0.98
mt* = 0.19 | ml* = 1.64
mt* = 0.082 | m* = 0.067 |
Effective mass holes
m*/m0 | mlh* = 0.16
mhh* = 0.49 | mlh* = 0.044
mhh* = 0.28 | mlh* = 0.082
mhh* = 0.45 |
Crystal structure | diamond | diamond | zincblende |
Density | 2.328 g/cm³ | 5.3267 g/cm³ | 5.32 g/cm³ |
Atoms/m³ | 5.0 × 1028 | 4.42 × 1028 | 4.42 × 1028 |
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