Real solids

Colliex, C.

doi:10.1107/97809553602060000593

International
Tables for
Crystallography
Volume C
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International Tables for Crystallography (2006). Vol. C. ch. 4.3, pp. 401-403

Section 4.3.4.3.3. Real solids

C. Colliex^a

4.3.4.3.3. Real solids

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The dielectric constants of many solids have been deduced from a number of methods involving either primary photon or electron beams. In optical measurements, one obtains the values of $[\varepsilon_1]$ and $[\varepsilon_2]$ from a Krakers–Kronig analysis of the optical absorption and reflection curves, while in electron energy-loss measurements they are deduced from Kramers–Kronig analysis of energy-loss functions.

Fig. 4.3.4.18 shows typical behaviours of the dielectric and energy-loss functions.

(a) For a free-electron metal (Al), the Drude model is a satisfactory description with a well defined narrow and intense maximum of $[{\rm Im}(-1/\varepsilon)]$ corresponding to the collective plasmon excitation together with typical conditions $[\varepsilon_1\simeq\varepsilon_2\simeq 0]$ for this energy $[\hbar\omega_p]$ . One also notices a weak interband transition below 2 eV.

Figure 4.3.4.18| top | pdf |

Dielectric coefficients $[\varepsilon_1]$ , $[\varepsilon_2]$ and $[{\rm Im}(-1/\varepsilon)]$ from a collection of typical real solids: (a) aluminium [courtesy of Raether (1965)]; (b) gold [courtesy of Wehenkel (1975)]; (c) InSb [courtesy of Zimmermann (1976)]; (d) solid xenon at ca 5 K [courtesy of Keil (1968)].

(b) For transition and noble metals (such as Au), the results strongly deviate from the free-electron gas function as a consequence of intense interband transitions originating mostly from the partially or fully filled d band lying in the vicinity of, or just below, the Fermi level. There is no clear condition for satisfying the criterion of plasma excitation $[(\varepsilon=0)]$ so that the collective modes are strongly damped. However, the higher-lying peak is more generally of a collective nature because it coincides with the exhaustion of all oscillator strengths for interband transitions.
(c) Similar arguments can be developed for a semiconductor (InSb) or an insulator (Xe solid). In the first case, one detects a few interband transitions at small energies that do not prevent the occurrence of a pronounced volume plasmon peak rather similar to the free-electron case. The difference between the gap and the plasma energy is so great that the valence electrons behave collectively as an assembly of free particles. In contrast, for wide gap insulators (alkali halides, oxides, solid rare gases), a number of peaks are seen, owing to different interband transitions and exciton peaks. Excitons are quasi-particles consisting of a conduction-band electron and a valence-band hole bound to each other by Coloumb interaction. One observes the existence of a band gap [no excitation either in $[\varepsilon_2]$ or in $[{\rm Im}(-1/\varepsilon)]$ below a given critical value ] and again the higher-lying peak is generally of a rather collective nature.

Čerenkov radiation is emitted when the velocity v of an electron travelling through a medium exceeds the speed of light for a particular frequency in this medium. The criterion for Čerenkov emission is $[\varepsilon_1(\omega)\gt {c^2\over v^2}=\beta^{-2}. \eqno (4.3.4.26)]$

In an insulator, $[\varepsilon_1]$ is positive at low energies and can considerably exceed unity, so that a `radiation peak' can be detected in the corresponding energy-loss range (between 2 and 4 eV in Si, Ge, III–V compounds, diamond, $[\ldots]$ ); see Von Festenberg (1968), Kröger (1970), and Chen & Silcox (1971). The associated scattering angle, $[\theta\simeq\lambda_{\rm el}/\lambda_{\rm ph}\simeq10^{-5}\,{\rm rad}]$ for high-energy electrons, is very small and this contribution can only be detected using a limited forward-scattering angular acceptance.

In an anisotropic crystal, the dielectric function has the character of a tensor, so that the energy-loss function is expressed as $[{\rm Im}\left(-{1 \over \sum\limits_i\sum\limits_j \varepsilon_{ij} q_i q_j}\right) . \eqno (4.3.4.27)]$

If it is transformed to its orthogonal principal axes $[(\varepsilon_{11}, \varepsilon_{22}, \varepsilon_{33})]$ , and if the q components in this system are [q_1,q_2,q_3] , the above expression simplifies to $[{\rm Im}\left(-{1 \over \sum\limits_i \varepsilon_{ii} q^2_i}\right) . \eqno (4.3.4.28)]$

In a uniaxial crystal, such as a graphite, $[\varepsilon_{11}=\varepsilon_{22}=\varepsilon_\perp]$ and $[\varepsilon_{33}=\varepsilon_\|]$ (i.e. parallel to the c axis): $[\varepsilon({\bf q}, \omega)=\varepsilon_\perp\sin^2\theta+\varepsilon_\|\cos^2\theta, \eqno (4.3.4.29)]$ where $[\theta]$ is the angle between q and the c axis. The spectrum depends on the direction of q, either parallel or perpendicular to the c axis, as shown in Fig. 4.3.4.19 from Venghaus (1975). These experimental conditions may be achieved by tilting the graphite layer at 45° with respect to the incident axis, and recording spectra in two directions at $[\pm\theta_E]$ with respect to it (see Fig. 4.3.4.20 ).

Figure 4.3.4.19| top | pdf |

Dielectric functions in graphite derived from energy losses for E ⊥ c (i.e. the electric field vector being in the layer plane) and for E||c [from Daniels et al. (1970)]. The dashed line represents data extracted from optical reflectivity measurements [from Taft & Philipp (1965)].

Figure 4.3.4.20| top | pdf |

Geometric conditions for investigating the anisotropic energy-loss function.

References

Chen, C. H. & Silcox, J. (1971). Detection of optical surface guided modes in thin graphite films by high energy electron scattering. Phys. Rev. Lett. 35, 390–393.Google Scholar

Daniels, J., Festenberg, C. V., Raether, H. & Zeppenfeld, K. (1970). Optical constants of solids by electron spectroscopy. Springer tracts in modern physics, Vol. 54, pp. 78–135. New York: Springer-Verlag.Google Scholar

Keil, P. (1968). Elektronen-Energieverlustmessungen und Berechnung optischer Konstanten. I. Festes Xenon. Z. Phys. 214, 251–265.Google Scholar

Kröger, E. Z. (1970). Transition radiation, Čerenkov radiation and energy losses of relativistic charged particles traversing thin foils at oblique incidence. Z. Phys. 235, 403–421.Google Scholar

Raether, H. (1965). Electron energy loss spectroscopy. Springer Tracts Mod. Phys. Vol. 38, pp. 85–170. Berlin: Springer.Google Scholar

Taft, E. A. & Philipp, H. R. (1965). Optical properties of graphite. Phys. Rev. A, 138, 197–202.Google Scholar

Venghaus, H. (1975). Redetermination of the dielectric function of graphite. Phys. Status Solidi B, 71, 609–614.Google Scholar

Von Festenberg, C. (1968). Retardierungseffekte im Energieverlustspektrum von GaP. Z. Phys. 214, 464.Google Scholar

Wehenkel, C. (1975). Mise au point d'une nouvelle méthode d'analyse quantitative des spectres de pertes d'énergie d'électrons rapides diffusés dans la direction du faisceau incident: application à l'étude des métaux nobles. J. Phys. (Paris), 36, 199–207.Google Scholar

Zimmermann, S. (1976). The dielectric function of InSb determined by electron energy losses. J. Phys. C, 9, 2643–2649.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 4.3, pp. 401-403