How is the electrical conductivity of an intrinsic and extrinsic semiconductor expressed?

In one line

The conductivity of a semiconductor is \(\sigma = e(n\mu_e + p\mu_h)\), the sum of contributions from electrons and holes; for intrinsic material \(n = p = n_i\), and for doped material the majority carriers dominate.

Explained simply

In a metal only electrons carry current. In a semiconductor there are two kinds of carriers: electrons in the conduction band and holes in the valence band. Both drift in an electric field, in opposite directions, but because they carry opposite charges both add to the current in the same direction, like two teams pushing a cart from opposite ends: one pushing and the other pulling.

How it works

1. Electron current density: \(J_n = ne\mu_eE\); hole current density: \(J_p = pe\mu_hE\).
2. Total: \(J = (ne\mu_e + pe\mu_h)E = \sigma E\), so \(\sigma = e(n\mu_e + p\mu_h)\).
3. Intrinsic: \(n = p = n_i\), so \(\sigma_i = n_ie(\mu_e + \mu_h)\).
4. n-type: donor concentration \(N_D \gg n_i\), so \(n \approx N_D\) and \(p\) is tiny: \(\sigma_n \approx N_De\mu_e\).
5. p-type: \(p \approx N_A\): \(\sigma_p \approx N_Ae\mu_h\).
6. Resistivity \(\rho = 1/\sigma\).
7. With rising temperature, \(n_i\) increases very rapidly (more electrons cross the gap), so intrinsic conductivity increases, opposite to metals.

Key formula

\(\sigma = e(n\mu_e + p\mu_h)\). \(\sigma\): conductivity (S/m or (Ω m)⁻¹), \(e = 1.6\times10^{-19}\) C, \(n, p\): electron and hole concentrations (m⁻³), \(\mu_e, \mu_h\): mobilities (m²/V s).

Quick example

Intrinsic Si: \(n_i = 1.5\times10^{16}\) m⁻³, \(\mu_e = 0.135\), \(\mu_h = 0.048\) m²/V s. \(\sigma_i = 1.5\times10^{16}\times1.6\times10^{-19}\times0.183 \approx 4.4\times10^{-4}\) S/m, so \(\rho \approx 2.3\times10^3\) Ω m. Doping with \(N_D = 10^{21}\) m⁻³ gives \(\sigma \approx 10^{21}\times1.6\times10^{-19}\times0.135 = 21.6\) S/m, about 50,000 times larger.

Common mistakes

• Subtracting the hole contribution because holes move the other way; the currents add.
• Using \(\mu_h\) for n-type majority carriers.
• Using cm⁻³ concentration with m²/V s mobility without converting.

How MU asks it

Asked as "Derive the expression for conductivity of a semiconductor" (5 marks) and numericals: "find the conductivity/resistivity of intrinsic or doped Si/Ge" (3 to 5 marks). Write the current densities, combine to get \(\sigma\), give the intrinsic and extrinsic forms, and keep SI units throughout.