Question

Derive an expression for the conductivity of a semiconductor.

Or

Show that resistivity of semiconductor depends upon their densities and their mobilities.

Or

Explain why the conductivity of intrinsic semiconductor increases with rise in temperature.

Answer 1

Consider a block of a semiconductor of length l and area of cross-section A. Let an electric field E be applied across its ends and a potential difference V is acting across its ends as shown in figure 14.91. The electrons and holes in the semiconductor are moving due to this applied electric field. Both electrons and holes contribute to the current. The total current I can be written as the sum of the contribution of electron and hole currents.

∴ I = I_e + I_h

Where I_e = current due to electrons, and I_h = current due to holes. Both electrons in the conduction band and holes in valence band are moving randomly as electrons in a metal.

But using the relation

I_e = n_e Av_e e, and for the hole

I_h = n_h Av_h e, we get

I = n_eA_eV_e + n_hA V_he

Where n_e is the magnitude of the electron charge, n is the electron density and nh is hole density and ve and vh are electron and hole drift velocities. A low applied electric fields, semiconductors obey Ohm’s law so that

where µ_e and µ_h is the mobility of electrons and holes respectively (the mobility µ is defined as the drift velocity per unit electric field).

The conductivity σ \(( = \frac{1}{\rho})\) given by

σ = \(\frac{1}{\rho}\) = e(n_e v_e + n_h v_h) ...........(i)

The eq. (i) shows that the resistivity of a semiconductor depends on the electron and hole densities and their mobilities. With the increase of temperature of the semiconductor, the value of n_e and n_h also increases.Hence the conductivity a of semiconductor increases with the increase in temperature.