Section 4.3.4.3.2. Dielectric description

The description of the bulk plasmon in the free-electron gas can be extended to any type of condensed material by introducing the dielectric response function , which describes the frequency and wavevector-dependent polarizability of the medium; cf. Daniels et al. (1970). One associates, respectively, the and functions with the propagation of transverse and longitudinal EM modes through matter. In the small-q limit, these tend towards the same value: As transverse dielectric functions are only used for wavevectors close to zero, the T and L indices can be omitted so that: The transverse solution corresponds to the normal propagation of EM waves in a medium of dielectric coefficient , i.e. to For longitudinal fields, the only solution is , which is basically the dispersion relation for the bulk plasmon.

In the framework of the Maxwell description of wave propagation in matter, it has been shown by several authors [see, for instance, Ritchie (1957)] that the transfer of energy between the beam electron and the electrons in the solid is governed by the magnitude of the energy-loss function , so that One can deduce (4.3.4.14) by introducing a δ function at energy loss for the energy-loss function: As a consequence of the causality principle, a knowledge of the energy-loss function over the complete frequency (or energy-loss) range enables one to calculate by Kramers–Kronig analysis: where PP denotes the principal part of the integral. For details of efficient practical evaluation of the above equation, see Johnson (1975).

The dielectric functions can be easily calculated for simple descriptions of the electron gas. In the Drude model, i.e. for a free-electron plasma with a relaxation time τ, the dielectric function at long wavelengths is with , as above. The behaviour of the different functions, the real and imaginary terms in , and the energy-loss function are shown in Fig. 4.3.4.16 . The energy-loss term exhibits a sharp Lorentzian profile centred at and of width 1/τ. The narrower and more intense this plasmon peak, the more the involved valence electrons behave like free electrons.

Figure 4.3.4.16| top | pdf | Dielectric and optical functions calculated in the Drude model of a free-electron gas with ħωp = 16 eV and τ = 1.64 × 10−16 s. R is the optical reflection coefficient in normal incidence, i.e. R = [(n − 1)2 + k2]/(n + 1)2 + k2] with n and k the real and imaginary parts of . The effective numbers and are defined in Subsection 4.3.4.5 [courtesy of Daniels et al. (1970)].

In the Lorentz model, i.e. for a gas of bound electrons with one or several excitation eigenfrequencies , the dielectric function is where denotes the density of electrons oscillating with the frequency and is the associated relaxation time. The characteristic , , and behaviours are displayed in Fig. 4.3.4.17 : a typical `interband' transition (in solid-state terminology) can be revealed as a maximum in the function, simultaneous with a `plasmon' mode associated with a maximum in the energy-loss function and slightly shifted to higher energies with respect to the annulation conditions of the function.

Figure 4.3.4.17| top | pdf | Same as previous figure, but for a Lorentz model with an oscillator of eigenfrequency ħω0 = 10 eV and relaxation time τ0 = 6.6 × 10−16 s superposed on the free-electron term [courtesy of Daniels et al. (1970)].

In most practical situations, there coexist a family of free electrons (with plasma frequency and one or several families of bound electrons (with eigenfrequencies . The influence of bound electrons is to shift the plasma frequency towards lower values if and to higher values if . As a special case, in an insulator, and all the electrons have a binding energy at least equal to the band gap , giving .

This description constitutes a satisfactory first step into the world of real solids with a complex system of valence and conduction bands between which there is a strong transition rate of individual electrons under the influence of photon or electron beams. In optical spectroscopy, for instance, this transition rate, which governs the absorption coefficient, can be deduced from the calculation of the factor as where is the matrix element for the transition from the occupied level j in the valence band to the unoccupied level in the conduction band, both with the same k value (which means for a vertical transition). is the joint density of states (JDOS) with the energy difference . This formula is also valid for small-angle-scattering electron inelastic processes. When parabolic bands are used to represent, respectively, the upper part of the valence band and the lower part of the conduction band in a semiconductor, the dominant JDOS term close to the onset of the interband transitions takes the form where is the band-gap energy. This concept has been successfully used by Batson (1987) for the detection of gap energy variations between the bulk and the vicinity of a single misfit dislocation in a GaAs specimen. The case of non-vertical transitions involving integration over k-space has also been considered (Fink et al., 1984; Fink & Leising, 1986).