Intrinsic semiconductor
Generally speaking the conductivity of metals is due to free electrons in them and these are largely absent from semiconductors. There does, however, appear to be a characteristic energy associated with the conduction process. If sufficiently energetic photons (quanta of e.m. radiation) are incident on a semiconductor its conductivity rises enormously. The photon's energy can be equated with the energy needed to transfer an electron from a bound state (the electron is said to be in the valence band) to a free or conducting state (then it is said to be in the conduction band). This characteristic energy is known as the band-gap energy or just the band gap.
Measurements in magnetic fields (Hall-effect measurements) show that there are two types of charge carrier: a negatively charged one identified with the electron and a positively charged one called a hole. When a photon causes an electron to transfer from valence band to conduction band it leaves behind a hole in the valence band. This hole can move under the influence of an electric field, that is it is a charge carrier that acts like a positively charged electron.
Silicon has a band gap of about \(1.1 ~eV\) (electron volts, \(1 ~eV = 1.6 \times 10^{-19} J\)), meaning that an electron (charge, \(-q = -1.6 \times 10^{-19} C\)) changes its potential by \(1.1 V\) on crossing the band gap. Germanium has a band gap of \(0.7 ~eV\) and diamond has one of \(5.4 ~eV\). Thermal energy can also cause the creation of electron-hole pairs that causes conduction. In a pure semiconductor kept in the dark, thermally generated charge carriers are the sole means of conduction. The numbers of them per unit volume (the intrinsic carrier concentration, \(n_{i}\) are given by
\[n_i = N~exp\left(\frac{-E_g}{2k_BT}\right)~~~~~~~~~~~~~~~~~~~~~~~~(1)\]
where \(N\) is a constant for a given semiconductor, \(E_{g}\) is the band gap energy in Joules, \(k_{B}\) is Boltzmann's constant (\(1.38 \times 10^{-23}~J/K\)) and \(T\) is the temperature in \(K\). The factor of \(2\) in the denominator of the exponent means that in classical terms, the electron starts off in the middle of the band gap. This is because electrons are not classical particles. Be that as it may, no matter how carefully prepared the material is, there will always be thermal carrier generation and associated conductivity.
The conductivity in any material is given by \(\sigma = nq \mu\), where \(\mu\) is the mobility of the charge carrier. The mobility is the speed (\(v\)) that a charge carrier acquires per unit electric field (\(E\)), that is \(\mu = v/E\). \(q\) is the magnitude of the electronic charge and n the number of charge carriers per unit volume. For an intrinsic semiconductor
\[\sigma =nq(\mu_{e} +\mu_{h})~~~~~~~~~~~~~~~~~~~~~~~~(2)\]
where \(n\) is the number of electron-hole pairs and \(\mu_{e}\) and \(\mu_{h}\) are the respective electron and hole mobilities. It is worthwhile putting some numbers into equations 1 and 2. The material constants for silicon are \(N = 3 \times 10^{25}/m^{3}\), \(E_{g} = 1.1~~eV\) = \(1.76 \times 10^{-19}~J\), \(\mu_{e} = 0.14 m^{2}V^{-1}~s^{-1}\) and \(\mu_{h} = 0.05~m^{2}V^{-1}s^{-1}\), so at \(300~K\), equation \(2\) yields \(n_i= 2 \times 10^{16}~m^{-3}\) and equation \(3\) gives \(\sigma= 6 \times 10^{-4}~S/m\) or \(p = 1/\sigma = 1650~\Omega m\). A bar of intrinsic silicon \(100~mm\) long and of square cross-section, \(10\) mm by \(10\) mm, would then have, by equation 1, a resistance of \(1.7 M\Omega\). Compare this with a copper bar of the same size, which would have a resistance of \(17 \mu\Omega\). Equation 2 also shows that the carrier density and hence a will rapidly increase with temperature (by \(7 \%/K\)).
