OPTICAL SPECTRUM OF AN EXCITON

In 1931 Ya. I. Frenkel first introduced the concept of the exciton.^1 He showed that an excited state arising at some atom of an ideally periodic crystal cannot be localized and must necessarily move through the crystal in the form of a peculiar excitation wave, called an exciton.

The state of an ideally periodic crystal in which all atoms except one are at normal levels, while one atom is excited, possesses permutation degeneracy, since the energy \(E\) of the crystal does not change if, in the same excited state, instead of

of the original atom there will be some other atom of the crystal. Therefore the state of the crystal with energy \(E\) will be a superposition of states in which only one of the sites of the crystal lattice is excited. The excitation energy, absorbed by some definite atom of the crystal, will then pass from atom to atom until one of the atoms that has received this energy returns to the normal state, emitting the quantum it has received. The transfer of energy through the crystal occurs by migration, i.e., without intermediate exchange by electrons (ionization and recombination) or photons (emission and reabsorption of light). Since the absorbed quantum does not ionize the atom, but only excites it, luminescence associated with excitons is not accompanied by photoconductivity. Absorption and emission of light occur here in different atoms separated from one another by a distance considerably greater than the lattice period. Therefore such luminescence has a distinctly pronounced crystalline character.

Within the framework of the band theory of crystals, an exciton may be described as a pair of particles—an electron and a positive hole—bound by forces of mutual attraction[^2]. Such a model is close to the hydrogen atom. Therefore the exciton should have a spectrum similar to the spectrum of the hydrogen atom.

A theoretical treatment of the absorption of light by excitons was recently carried out by S. I. Pekar and I. M. Dykman[^3]. For the frequencies of the exciton absorption spectrum they obtained the following formula:

\[ \nu_k = \nu_0 - \frac{R'}{k^2}. \tag{1} \]

Here \(\nu_k\) is the frequency of the \(k\)-th absorption line; \(\nu_0\) is the series limit in the exciton spectrum, corresponding to the energy of its photodissociation (for \(k=\infty\), \(\nu_{\infty}=\nu_0\)); \(k=1,2,3,\ldots\), and

\[ R' = R \frac{\mu}{m} \frac{1}{n^4}, \]

where \(R\) is the Rydberg constant, \(m\) is the mass of the electron in vacuum, \(n\) is the refractive index, and \(\mu\) is determined by the relation

\[ \frac{1}{\mu} = \frac{1}{\mu_1} + \frac{1}{\mu_2}, \]

in which \(\mu_1\) and \(\mu_2\) are the “effective” masses of the electron and the hole. Formula (1) is easily transformed into the form

\[ \Delta \nu_k = \nu_{\infty} - \nu_k = \frac{R'}{k^2}, \qquad k=1,\,2,\,3,\ldots \tag{2} \]

In studying the absorption of light in cuprous oxide crystals (\(\mathrm{Cu_2O}\)), E. F. Gross and N. A. Karryev for the first time experimentally observed a new optical effect—the absorption of light by excitons[^4].

The electrical properties of \(\mathrm{Cu_2O}\) have been thoroughly studied by a number of authors. In particular, V. P. Zhuse and S. M. Ryvkin showed that the transfer of the energy of absorbed light to photoelectrons in cuprous oxide crystals has an exciton character[^5]. The scheme of the electronic levels in \(\mathrm{Cu_2O}\) obtained in these studies indicated the possibility of absorption of light in transitions of electrons from the fundamental band to adhesion levels located near the conduction band, and also from acceptor levels of oxygen, lying near the fundamental band, into the conduction band. Since the energies of such transitions are close to the energy of fundamental absorption of the \(\mathrm{Cu_2O}\) crystal lattice (accompanied by the transition of electrons

from the main zone into the conduction zone), then the presence of such electronic transitions should appear near the edge of the fundamental absorption, located for Cu\(_2\)O at about 6300 Å.

The study of the absorption of light by Cu\(_2\)O crystals in the visible region of the spectrum near 6300 Å was carried out with a three-prism Zeiss spectrograph with a camera \(F = 840\) mm. The large dispersion of the spectrograph in this region of the spectrum (about 25 Å/mm) enabled Gross and Karryev to observe new optical phenomena unnoticed in a number of works with instruments having a smaller dispersion\(^6\). It turned out, first of all, that the edge of the fundamental absorption of the lattice has not a monotonically increasing, but a step-like character (Fig. 1). Two distinct steps were found, located in the intervals 6371–6284 and 6284–6040 Å (at 20°), the large width of which undoubtedly testifies

Fig. 1.

Fig. 2.

to the presence of electronic transitions connected with local levels in the Cu\(_2\)O crystal lattice.

As the temperature is lowered, the step-like absorption shifts into the short-wavelength region of the spectrum, while its intensity decreases. Against the weakened background of the first step a very narrow absorption line appears (at \(k = 1\) in Fig. 1), whose width does not exceed the usual width of lines in atomic spectra. Upon further cooling of the crystal this line also shifts into the short-wavelength region. At \(T = -200^\circ\) it is located at \(\lambda = 6125.3\) Å. With decreasing temperature the second step (corresponding to stronger absorption) also shifts into the short-wavelength region of the spectrum, and near its edge, on the side of longer wavelengths, a whole series of narrow absorption lines is found, successively approaching one another with increasing frequency. A microphotogram of these lines is shown in Fig. 2. The lines indicated in this figure are located at \(\lambda = 6125.3;\ 5792.7;\ 5756.6;\ 5743.8;\ 5738.1;\ 5734.1\) Å (\(T = -200^\circ\)). The lines converge to the series limit at \(\lambda = 5727.4\) Å. As the temperature is raised to \(0^\circ\), the lines broaden and cease to be noticeable against the background of the continuous spectrum advancing from the short-wavelength side. Subsequent calculations showed that the lines are arranged in a regular manner, so that for the difference \(\Delta \nu\) between the frequency of the series limit and the frequencies of the individual lines there is the simple relation

\[ \Delta \nu_k = \frac{B}{k^2}, \qquad k = 1,\ 2,\ 3,\ldots, \tag{3} \]

where \(B\) is a constant quantity equal to \(785\ \text{cm}^{-1}\).

It is obvious that the indicated lines form a hydrogen-like spectrum. The form of this spectrum is shown in Fig. 3. Relation (3) for the series of absorption lines coincides with the theoretical formula (2), obtained by Pekar and Dykman for the exciton spectrum. All this confirms that the hydrogen-like series of lines discovered by Gross and Karryev in the absorption spectrum of Cu₂O crystals is the optical spectrum of an exciton in this crystal. The boundary of the exciton spectrum is associated with the separation of the electron from the hole under the action of light, i.e., with the photodissociation of the exciton.

The experimental data obtained make it possible to determine the energy of formation of the exciton (from the wavelength of the first line of the spectrum) and the energy of its photodissociation. In addition, substituting into formula (2) the found value \(R' = B = 785\ \text{cm}^{-1}\), one can calculate the effective mass of the electron \(\mu_1/m\), since all the remaining quantities are known. According to the data of Gross and Karryev, the energy of formation of an exciton in a Cu₂O crystal is \(2.014\ \text{eV}\), the energy of photodissociation is \(2.154\ \text{eV}\), and

\[ \frac{\mu_1}{m} = 0.6, \]

which agrees well with the theoretical values of these quantities.

The absorption spectrum in the region of the long-wavelength edge of the fundamental absorption band of the lattice was also investigated by the indicated authors on CdS single crystals. In this case step-like absorption was found, and at a temperature of \(-200^\circ\) two narrow lines were found near the edge of the principal absorption band, apparently belonging to the optical spectrum of an exciton in CdS.

Fig. 3.

The experiments of Gross and Karryev are direct experimental proof of the existence of excitons in the crystal lattice. They also make it possible to determine a number of important characteristics of excitons, which is of great significance for the further development of our concepts of the various physical processes occurring in real crystals.

V. Leshkovtsev

Cited Literature

1. Ya. I. Frenkel, Phys. Rev. 37, 17, 1276 (1931).
2. R. Peirls, Ann. d. Phys. 13, 905 (1932); Ya. I. Frenkel, ZhETF 6, 7 (1936); C. H. Wannier, Phys. Rev. 52, 191 (1937).
3. I. M. Dykman, S. I. Pekar, DAN SSSR 83, No. 6 (1952).
4. E. F. Gross, N. A. Karryev, DAN SSSR 84, No. 2, No. 3 (1952).
5. V. P. Zhuze, S. M. Ryvkin, DAN SSSR 77, No. 2 (1951).
6. G. Mönch, Zs. f. Phys. 78, 728 (1932); M. Pigarev, S. Golub, Sow. Phys. 6, 603 (1934); A. V. Ioffe, A. F. Ioffe, ZhETF 6, 737 (1936); G. Blankenburg, K. Kassel, Forschungen und Fortschritte 26, 33 (1950); M. Hayashi, K. Katsuki, Phys. Abstr. A 54, 199 (1951).