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102 CHAPTER 7. SEMICONDUCTORS is formally identical to what you would obtain for a set of n c (T ) degenerate energy levels at energy E c , i.e. bunched up at the band edge. For the number of holes in the valence band, we obtain, equivalently: p = n v (T )e − µ−Ev k B T (7.11) (7.10) and (7.11) give the concentration of electrons and holes at a temperature T , in terms of the chemical potential µ, as yet unknown. It is useful to notice that the product np = n c (T )n v (T )e − Eg k B T (7.12) is independent of µ. Here, E g = E c − E v ≃ 1eV is the size of the energy gap. This result is also called the law of mass action. We will be able to use it when carriers are introduced to the system by doping. For an intrinsic semiconductor, the electron and hole densities are equal, and can be obtained my taking the square root of (7.12) n i = p i = (n c (T )p v (T )) 1/2 e − Eg 2k B T (7.13) and substituting back in either the equation for n ((7.10)) or p (7.11)) yields the chemical potential µ = 1 2 E g + 3 4 k BT log(m ∗ h/m ∗ e) . (7.14) The chemical potential thus sits mid gap at zero temperature, and shifts slightly away from that position if the carrier masses are different. Note that the activation energy to create intrinsic carriers (either electrons or holes) is always exactly half the optical energy gap. 7.3 Doped semiconductors What differentiates semiconductors from insulators is the fact that the energy gap E g is sufficiently small in semiconductors to allow significant carrier concentrations at room temperatures by thermal activation alone. However, carriers can also be created in semiconductors by adding impurity atoms in a process called doping. Donor levels. Consider the effect in a Si crystal of replacing a single atom by an As atom. As is a group V element and therefore provides 5 electrons in the place of the 4 of the Si it replaced. Formally, it appears like a Si atom with one extra electron, and one extra positive charge in the nucleus. We now ask whether the added electron stays tightly bound to the extra positive charge. Suppose the electron wanders away from the impurity site. It will of course see an attractive Coulomb force from the charged As impurity. Because the As atom carries a single positive charge, the energy levels are calculated in the same way as those of the Hydrogen atom. We take into account the influence of the surrounding material, in which the extra electron moves, by making two corrections: (i) the Coulomb potential is screened by the dielectric constant of Si (ɛ ≈ 12) so is much weaker than in free space; and (ii) the band mass of the electron is
7.3. DOPED SEMICONDUCTORS 103 smaller than the free electron mass, so the kinetic energy of an electron in a given momentum state is larger. The net effect is that the binding energy of the 1s impurity state is now ∆ d = e 4 m ∗ c 2(4πɛɛ o h¯) 2 = m∗ c/m e ɛ 2 × 13.6 eV (7.15) which can be very small in comparison to the band gap, and often comparable or smaller than thermal energies. Such donor impurities easily donate electrons to the conduction band. The binding energy of the electronic states of the hydrogen atom express the energy difference between the lowest vacuum state and the respective bound states. The hydrogen-like bound impurity states we calculated now are referenced to the bottom of the conduction band, because the electron unbinds from the impurity by occupying a conduction band state, just as an electron unbinds from the Hydrogen atom by assuming a plane-wave vacuum state. As donors in Si have an ionisation energy of 50 meV; donors in GaAs have an ionisation energy of about 6 meV, approximately 50 Kelvin. When a donor atom is ionised, it releases its formerly bound electron into the conduction band. Acceptor levels. A trivalent impurity (e.g. B in Si) appears like a Si atom with an added negative charge, and with a missing electron. It is the mirror image of the case of the donor impurity, and corresponds to a positive charge (a ’hole’) circling a negatively charged nucleus. As in the case of donor atoms, the binding energy of the hole is reduced by the combined effects of high permittivity and low band mass in semiconductors. When the hole unbinds, the impurity accepts an electron from the valence band. The accepted electron is used to complete the covalent bond with the neighbouring atoms, and renders the site negatively charged. So, while ionising a donor atom releases an electron into the conduction band, ionising an acceptor atom absorbs an electron from the valence band, which leaves a hole in the valence band. n- and p-type materials. Even for very low densities of impurities, since the donor or acceptor energies are so much smaller than the gap, impurities in semiconductors are often the principal source of electrically active carriers. If donor atoms predominate, the carriers are predominantly electrons, and the material is said to be n-type. If holes are the dominant carrier type, the material is called p-type. Experimentally, these regimes may be distinguished by a measurement of the Hall effect, whose sign depends on the type of carrier. Impurity ionisation. We just quote here the results for thermal ionisation of the carriers in simple limits. If there are no acceptors, then the carrier concentration at low temperatures is n = (n c N d ) 1/2 e − ∆ d 2k B T (7.16) where N d is the donor density and the factor n c = 2(m ∗ ek B T/2πh¯ 2) 3/2 is the effective density of electron states within an energy k B T of the band edge. Notice again that the activation energy is half the binding energy. Since ∆ d is small and n c (T ) is usually large compared to N d , donor atoms are fully ionised down to very low temperatures, and n ≈ N d . This is called the extrinsic regime. At a still higher temperature, the intrinsic carrier generation by thermal activation from the valence band into the conduction band takes over. If there are only acceptors and no donors, then a similar formula can be obtained for holes by inspection. When both donors and acceptors exist, the problem is in general more complicated.
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