Localized Particles

2.3.1. Donors, Acceptors, and Deep Traps

When a type V atom such as P, As, or Sb, which has five electrons in its outer or valence electron shell, is a substitutional impurity in Si it uses four of these electrons to satisfy the valence requirements of the four nearest-neighbor silicons, and the one remaining electron remains weakly bound. The atom easily donates or passes on this electron to the conduction band, so it is called a donor, and the electron is called a donor electron. This occurs because the donor energy levels lie in the forbidden region close to the conduction band edge by the amount AE0 relative to the thermal energy value kBT, as indicated in Fig. 2.12. A Si atom substituting for Ga plays the role of a donor in GaAs, A1 substituting for Zn in ZnSe serves as a donor, and so on.

A type III atom such as A1 or Ga, called an acceptor atom, which has three electrons in its valence shell, can serve as a substitutional defect in Si, and in this role it requires four valence electrons to bond with the tetrahedron of nearest-neighbor Si atoms. To accomplish this, it draws or accepts an electron from the valence band, leaving behind a hole at the top of this band. This occurs easily because the energy levels of the acceptor atoms are in the forbidden gap slightly above the valence band edge by the amount AEa relative to ksT, as indicated in Fig. 2.12. In other words, the excitation energies needed to ionize the donors and to

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add electrons to the acceptors are much less than the thermal energy at room temperature T = 300K, that is, AED, AEA kBT, so virtually all donors are positively ionized and virtually all acceptors are negatively ionized at room temperature.

The donor and acceptor atoms that we have been discussing are known as shallow centers, that is, shallow traps of electrons or holes, because their excitation energies are much less than that of the bandgap (A£D, AEk Eg). There are other centers with energy levels that lie deep within the forbidden gap, often closer to its center than to the top or bottom, in contrast to the case with shallow donors and acceptors. Since generally Eg » kBT, these traps are not extensively ionized, and the energies involved in exciting or ionizing them are not small. Examples of deep centers are defects associated with broken bonds, or strain involving displacements of atoms. In Chapter 8 we discuss how deep centers can produce characteristic optical spectroscopic effects.

2.3.2. Mobility

Another important parameter of a semiconductor is the mobility ft or charge carrier drift velocity v per unit electric field E, given by the expression ft = \v\/E. This parameter is defined as positive for both electrons and holes. Table B.9 lists the mobilities fit and fih for electrons and holes, respectively, in the semiconductors under consideration. The electrical conductivity a is the sum of contributions from the concentrations of electrons n and of holes p in accordance with the expression where e is the electronic charge. The mobilities have a weak power-law temperature dependence T", and the pronounced T dependence of the conductivity is due principally to the dependence of the electron and hole concentrations on the temperature. In doped semiconductors this generally arises mainly from the Boltzmann factor exp(—EJkBT) associated with the ionization energies E, of the donors or acceptors. Typical ionization energies for donors and acceptors in Si and Ge listed in Table B.10 are in the range from 0.0096 to 0.16 eV, which is much less than the bandgap energies 1.11 eV and 0.66eV of Si and Ge, respectively. Figure 2.12 shows the locations of donor and acceptor levels on an energy band plot, and makes clear that their respective ionization energies are much less that Eg. The thermal energy kBT = 0.026 eV at room temperature (300K) is often comparable to the ionization energies. In intrinsic or undoped materials the main contribution is from the exponential factor exp(-Eg/2kRT) in the following expression from the law of mass action a = (nefte +pefih)

where the intrinsic concentrations of electrons nx and holes p, are equal to each other because the thermal excitation of nl electrons to the conduction band leaves behind the same number pK of holes in the valence band, that is nt = pr We see that the expression (2.15) contains the product memb of the effective masses me and mh of the electrons and holes, respectively, and the ratios mjm0 and mh/ma of these effective masses to the free-electron mass m0 are presented in Table B.8. These effective masses strongly influence the properties of excitons to be discussed next.

2.3.3. Excitons

An ordinary negative electron and a positive electron, called a positron, situated a distance r apart in free space experience an attractive force called the Coulomb force, which has the value -e2/47ts0r2, where e is their charge and c0 is the dielectric constant of free space. A quantum-mechanical calculation shows that the electron and positron interact to form an atom called positronium which has bound-state energies given by the Rydberg formula introduced by Niels Bohr in 1913 to explain the hydrogen atom

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