On the Relationship Between Polarizability and the Width of the Forbidden Energy Gap for Electrons in a Dielectric

A. A. Vorob'ev

Between the magnitude of the electric strength of crystals of alkali-halide salts, on the one hand, and the energy of liberation of electrons in them, on the other, there is an almost linear dependence. A similar dependence has also been observed between the magnitude of the electric strength and the polarization properties of these same molecules. It is of interest to consider the dependence between the energy of liberation of electrons in a crystal and the polarization properties of the particles composing it.

The electric field of a plane electromagnetic wave propagating in a crystal causes polarization of the individual atoms or ions of the crystal. In the first case this mainly involves displacement of the electron shells relative to the atomic nuclei. In ionic lattices, at low frequencies, individual ions are also displaced relative to one another. Owing to the large mass of the ions, they do not have time to follow a high-frequency field.

The electric polarization of the constituent parts of a crystal leads to the appearance of an electromagnetic field which is superposed on the electric field of the incident wave. The directions of the two fields in crystals are generally different, so that anisotropy of the dielectric coefficient and of the refractive index appears.

Absorption of light in crystals is based on quantum transitions of internal electrons.

It consists mainly in excitation, that is, in the transition of electrons to higher energy levels. Since electrons in a crystal cannot possess arbitrary energy, but only energy corresponding to definite energy bands, the absorbed light quantum must have an energy corresponding to the energy differences possible in the crystal. Therefore, in the general case, the absorption spectrum for a crystal will consist of separate bands, whose width depends on the width of the energy bands and sometimes may be further increased owing to the thermal motion of the atoms in the crystal.

Thus, the width of the forbidden energy bands in crystals can be determined from the absorption spectrum. The edge of the intrinsic absorption band of crystals in the ultraviolet region corresponds to the transition of electrons from the fundamental energy band to the conduction band. Such a transition of electrons is accompanied by the appearance of photoconductivity in the crystal. Additional absorption bands indicate the presence in the crystal of additional absorption centers and electron transitions. For example, the existence of the \(F\) band in the absorption spectrum of ionic crystals is explained by the transition of electrons from color centers (\(F\)-centers) into the conduction band.

The greater the polarizability of the particles composing the crystal, the weaker may be their bond with their center. With greater polarizability, electrons can more easily pass into the conduction band. Therefore there may be a relation between the magnitude of the dielectric constant and the intrinsic absorption of light in the crystal.

For impurity semiconductors Mott determined the relation:

\[ \Delta E \sim \varepsilon^{-2}, \tag{1} \]

where \(\Delta E\) is the width of the forbidden energy interval for the electrons of the crystal, also called the activation energy; \(\varepsilon\) is the effective dielectric constant. Its value depends on the polarization surrounding the site of the lattice disturbance. The numerical value of \(\varepsilon\) lies between the static \(\varepsilon_{st}\) and optical \((\varepsilon_e)\) dielectric constants of the crystal.

According to formula (1), a large value of the dielectric constant, i.e., a large polarizability of the lattice near the impurity atom, leads to a smaller value of \(\Delta E\). Experimental data for semiconductors qualitatively confirm formula (1), but these data are few in number.

Table

Values of the dielectric constant \(\varepsilon\), the energy of the edge of intrinsic absorption in the ultraviolet region \(\Delta U_{uv}\), and the energy of electron binding in \(F\)-centers for crystals of alkali-halide salts, in electron volts.

The table gives data for the magnitude of the energy required to transfer electrons from the \(F\)-band and from the fundamental band into the conduction band, in electron volts. These data are compared with the electronic component of the dielectric constant.

In Figs. 1 and 2, for crystals of the alkali-halide series, the dependences are given between the energy \(\Delta U_{uv}\) of the edge of intrinsic absorption in the ultraviolet region and the electronic component of the dielectric constant \(\varepsilon_e^{-2}\). In Fig. 2 this dependence is given correspondingly for the energy of liberation of electrons from \(F\)-centers. These data show that for each alkali metal there is a separate dependence. For potassium and sodium salts the dependence of the value \(\Delta U_F\) on \(\varepsilon_e^{-2}\) is close to the linear one required by formula (1).

In Fig. 3 the dependence is presented between the energy of the edge of intrinsic absorption in the ultraviolet region and the electronic component of the dielectric constant for crystals of alkali-halide salts. The results given in Figs. 1–3 lead to the conclusion that the electron binding energy decreases with increasing polarizability.

ions. A considerable decrease in the activation energy is observed in going from fluoride salts to chloride salts, and a smaller one in going to bromide and iodide salts. These changes should be compared

Fig. 1. Dependence between the energy of the edge of intrinsic absorption in the ultraviolet region and \(\varepsilon_{\mathrm{e}}^{-2}\), where \(\varepsilon_{\mathrm{e}}\) is the electronic part of the dielectric constant.

with the electron affinity energy of the electron for the halide, i.e., the energy acquired in the formation of a negative ion. Calculations of the magnitude of the electron affinity energy for the halides from cyclic processes

Fig. 2. Dependence between the energy for transferring electrons from the \(F\) band to the conduction band and \(\varepsilon_{\mathrm{e}}\).

gave the values in electronvolts indicated in parentheses: \(F\) (4.0); \(Cl\) (3.8); \(Br\) (3.4); \(J\) (3.1). It is important to note that an increase of the activation energy is also observed with increasing dielectric constant, as is the case for fluoride salts and bromide salts. This increase in the activation energy is connected with an increase in the ionization poten-

of an alkali metal. The ionization potential, i.e., the work necessary to obtain a positive ion from a neutral atom, in electron volts, is given below in parentheses: Li (5.371); Na (5.116); K (4.32); Rb (4.159).

Fig. 3. Dependence between the energy of the edge of intrinsic absorption in the ultraviolet region, \(\Delta U_{\mathrm{uv}}\), and the electronic component of the dielectric constant \(\varepsilon_3\) for crystals of alkali-halide salts.

A comparison of the characteristics presented leads to the conclusion that the magnitude of the activation energy—the polarizability of molecules in the lattice—is not determined solely by the work of ionization of the alkali metal or by the magnitude of the electron affinity of the halides. In general, for salts of the alkali-halide series, with an increase in the dielectric constant the electron-excitation energy decreases.

REFERENCES

1. A. A. Vorob'ev, High-Voltage Engineering. GEI, 1945.
2. E. Fermi, Molecules and Crystals, 1947.
3. G. Bush, UFN, X, VII, issue 2, 258–325, 1952.
4. E. Rabinovich and Tilo, Periodic System of the Elements. GTTI, 1933.