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Excitation Spectrum of Excitons in a Crystal Lattice*)
E. F. Gross
§ 1. Introduction
It is well known that in crystals broad regions of continuous light absorption are observed, located in various parts of the spectrum. This absorption is usually called the intrinsic, or fundamental, absorption of the crystal lattice. On the long-wavelength side, the region of continuous absorption has an edge (boundary), which in some crystals breaks off very sharply. The absorption of light at the edge increases rapidly as one moves along the spectrum toward higher frequencies, and the absorption coefficient \(x\) in the region of intrinsic absorption usually reaches a very large value, of the order of \(10^5\)—\(10^6\ \mathrm{cm}^{-1}\). The position of the long-wavelength absorption boundary in the spectrum depends on the substance of the crystal and is determined, as is known, by the width of the forbidden band in the crystal.
The long-wavelength edge of fundamental absorption is associated in many substances with the internal photoelectric effect. In substances possessing photosensitivity, the photoeffect is excited chiefly by wavelengths falling at the long-wavelength edge of the fundamental absorption. This is clearly seen, for example, from Fig. 1, which shows
Fig. 1. Absorption curve (a) and curve of the spectral distribution of photoconductivity (b) in a Cu\(_2\)O crystal, according to A. V. Ioffe and A. F. Ioffe.\(^1\)
the curve of the spectral distribution of the internal photoeffect in a cuprous oxide crystal according to the data of A. V. Ioffe and A. F. Ioffe.\(^1\) The maximum of the photoeffect is located at the long-wavelength edge of the fundamental
*) The present review includes, mainly, works published before 1956. The author intends to devote a separate review article to the results of works published later.
absorption. This empirical rule was established already by Gudden and Pohl2 in 1923. However, the cause of the sharp decrease in photosensitivity in the region of strong absorption in the main band remains unclear to this day. Gudden3 considered this phenomenon puzzling. As is known, in 1931 Ya. I. Frenkel introduced4, in order to explain the absorption of light in the main band without the occurrence of photoconductivity, the concept of the exciton.
In this connection, spectroscopic studies of light absorption in crystals in the regions near the long-wavelength boundary of the main absorption are of great interest. Local electron levels situated in the forbidden band of the crystal near the lower edge of the free band, as well as near the upper edge of the filled band, and associated with defects of the crystal structure, can produce additional absorption of light in the region near the long-wavelength boundary of the intrinsic absorption of the lattice. Spectroscopic studies of light absorption in this region could, as it seemed to us, bring great clarity to the phenomena of the photoeffect and reveal the existence of local electron levels in the forbidden band of the crystal.
Therefore, at the Physico-Technical Institute of the Academy of Sciences we undertook systematic studies of the absorption spectra of crystals in order to obtain detailed information on the energy levels of electrons in crystal lattices.
In choosing the object of study we focused first of all on cuprous oxide crystals (\(\mathrm{Cu_2O}\)). The choice of this object was determined by the following considerations. The electrical properties of cuprous oxide, as a “classical” semiconductor, had already long since been studied in detail. Recently, in 1947–1950, V. P. Zhuse and S. M. Ryvkin carried out at the Physico-Technical Institute thorough investigations of the photoelectric properties of this crystal. In addition, cuprous oxide was a convenient object from the purely optical point of view. From cuprous oxide one can readily prepare large transparent plates of various thicknesses, since this material is well processed on optical machines.
Fig. 2. Scheme of electron levels in \(\mathrm{Cu_2O}\) crystals according to data obtained by Zhuse and Ryvkin for photoconductivity.
Labels in the figure: free band; filled band; \(a\), \(b\), \(c\), \(d\).
On the basis of detailed quantitative studies of the internal photoeffect in cuprous oxide near the long-wavelength edge of absorption of the \(\mathrm{Cu_2O}\) lattice, V. P. Zhuse and S. M. Ryvkin5 came to the conclusion that the photoconductivity of \(\mathrm{Cu_2O}\) is of an impurity character. Local electron levels near the free band (adhesion levels) and oxygen acceptor levels near the filled band of the crystal play an essential role here. At the same time, Zhuse and Ryvkin were the first to point out6 the role of excitons in the phenomenon of the internal photoelectric effect. For the external photoeffect, the exciton mechanism had been proposed by Apker and Taft7.
Depending on the position of the electron levels in the crystal, absorption of light may be observed in various spectral regions. In the \(\mathrm{Cu_2O}\) crystal, on the basis of the photoconductivity studies by Zhuse and Ryvkin5, 6, one should expect that absorption of light may be associated with electron transitions schematically shown in Fig. 2. The energy difference between the levels in electron transitions
of type $a$ or $b$ are relatively larger than for transitions of type $v$ or $g$. The energies of the $v$ and $g$ transitions are small, and therefore the light absorption associated with them may be expected in the long-wavelength (infrared) part of the spectrum. The absorption of light associated with transitions of type $a$ and $b$ should be observed near the edge of the fundamental absorption band (for $\mathrm{Cu_2O}$, about $6300\ \text{\AA}$), on its long-wavelength side.
The absorption of light in $\mathrm{Cu_2O}$ has been studied little. The visible region of the spectrum was investigated by A. V. and A. F. Ioffe[^1], M. Pigarev and S. Golub[^8], and Mench[^9]. From these investigations it was known that a $\mathrm{Cu_2O}$ crystal has very strong absorption in the short-wavelength part of the spectrum (the fundamental absorption of the $\mathrm{Cu_2O}$ lattice) and a sharp decrease of absorption at the long-wavelength boundary, near $\lambda = 6300\ \text{\AA}$. Mench showed[^9] that, as the temperature is lowered, the absorption spectrum of cuprous oxide shifts toward shorter wavelengths. All these investigations were carried out on spectral instruments with small dispersion.
§ 2. INVESTIGATION OF THE STRUCTURE OF THE LONG-WAVELENGTH EDGE OF THE FUNDAMENTAL ABSORPTION OF A CUPROUS OXIDE CRYSTAL AT ROOM TEMPERATURE
In order to be able to investigate the long-wavelength edge of the fundamental absorption more thoroughly, in our experiments we used spectral instruments whose dispersion was approximately 10 times greater than the dispersion of the instruments with which the earlier investigations had been performed. The first experiments on the study of light absorption near the edge of the fundamental absorption of the $\mathrm{Cu_2O}$ lattice were carried out with a three-prism spectrograph manufactured by C. Zeiss, with a camera of $F = 840\ \mathrm{mm}$. The dispersion of this instrument in the region $\lambda = 6300\ \text{\AA}$ was about $25\ \text{\AA}/\mathrm{mm}$. The large dispersion of the spectral instrument made it possible to observe the edge of the intrinsic absorption of the $\mathrm{Cu_2O}$ crystal on an enlarged scale. Owing to this, it was possible to notice phenomena that had escaped observation with instruments of small dispersion.
The cuprous oxide for investigating absorption spectra was prepared by prolonged oxidation at a temperature of $1025^\circ\mathrm{C}$ of high-purity electrolytic copper ($99.99\%$ Cu). From the large-crystalline plates of cuprous oxide obtained in this way, polished $\mathrm{Cu_2O}$ plates of various thicknesses, from several tens to several hundreds of microns, were prepared.
Our first experiments on investigating the edge of the fundamental absorption in cuprous oxide were carried out at room temperature ($t = 20^\circ\mathrm{C}$). In these experiments (carried out by the author together with N. A. Karryev), it was found[^10] that the edge of the intrinsic absorption of a $\mathrm{Cu_2O}$ crystal has a complex structure. The absorption of light near the edge does not increase continuously with decreasing wavelength, but has a step-like character. Two sharp boundaries are observed, where the absorption curve undergoes a break and forms, as it were, a step. These boundaries at $t = 20^\circ\mathrm{C}$ are located: 1) at $\lambda = 6371\ \text{\AA}$ ($1.945\ \mathrm{eV}$) and 2) at $\lambda = 6284\ \text{\AA}$ ($1.972\ \mathrm{eV}$), so that the width of the step is about $87\ \text{\AA}$ ($0.027\ \mathrm{eV}$). The absorption of light in the region of the step between these boundaries is relatively small. Beyond the boundary of the step ($\lambda = 6284\ \text{\AA}$), toward shorter wavelengths, the absorption begins to increase rapidly, and the absorption curve rises steeply upward. All these phenomena are clearly visible in Fig. 3, where a microphotogram of the absorption spectrum at the edge of the fundamental band is presented.
At both edges of the step one more interesting phenomenon is observed: before each boundary of the step (on the long-wavelength side), on
against the absorption background a bright narrow line is observed. These lines are directly adjacent to the boundaries where sharp kinks are observed on the absorption curve. Thus, the bright lines are located at \(\lambda 6371\) Å and \(\lambda 6284\) Å and represent narrow emission lines\(^*\).
As the temperature of the crystal is lowered, the edge of the principal absorption of the \(\mathrm{Cu_2O}\) crystal, together with the step and its sharp boundaries, shifts toward the short-wavelength side of the spectrum. The absorption of light on the step depends very strongly on temperature and decreases very rapidly (apparently according to an exponential law) as the crystal is cooled. The boundaries (kinks) of absorption at the edges of the step shift and remain sharp when the crystal is cooled down to the temperature of liquid nitrogen (\(T = 77.3^\circ\) K). At \(T = 77.3^\circ\) K the boundaries of the step are situated: 1) at \(\lambda 6164\) Å (\(2.010\) eV) and 2) at \(\lambda 6086\) Å (\(2.034\) eV). The distance between the first and second boundaries, i.e. the width of the step, at \(T = 77.3^\circ\) K is equal to \(78\) Å (\(0.024\) eV). Thus, the width of the step changes little with temperature.
Fig. 3. Microphotogram of the “step” at the edge of the principal absorption of a \(\mathrm{Cu_2O}\) crystal at \(20^\circ\) C.
The absorption in it, however, weakens at \(T = 77.3^\circ\) K so strongly that it becomes scarcely noticeable at this temperature. Narrow bright lines at the edges of the step are well observed at low temperatures, especially in thick \(\mathrm{Cu_2O}\) plates (0.5 mm and thicker).
The results described above were obtained independently of us by the Japanese authors Hayashi and Katsuki\({}^{11}\). These authors reported in their work that they observed only the boundaries (edges) of absorption (absorption edges) in the absorption spectrum of cuprous oxide.
We succeeded in advancing further in the study of the structure of the edge of the principal absorption of the \(\mathrm{Cu_2O}\) crystal and in discovering a new interesting phenomenon.
§ 3. STRUCTURE OF THE EDGE OF THE PRINCIPAL ABSORPTION OF A \(\mathrm{Cu_2O}\) CRYSTAL AT \(T = 77.3^\circ\) K (LIQUID NITROGEN). HYDROGEN-LIKE SERIES OF NARROW ABSORPTION LINES
As the temperature is lowered, the edge of the principal absorption of the \(\mathrm{Cu_2O}\) crystal, together with the region of step-like absorption, as indicated above, shifts to the short-wavelength side of the spectrum (to its orange part), and on the continuous background of the step, which weakens strongly at low temperatures, there appears a very narrow absorption line\({}^{12}\). This line at a temperature \(T = 77.3^\circ\) K becomes extremely sharp and thin, so that its width becomes comparable with the width of narrow lines in atomic spectra. Its position depends strongly on temperature, and upon cooling the \(\mathrm{Cu_2O}\) crystal it, together with the step, moves into the short-wavelength part of the spectrum. At a temperature \(T = 77.3^\circ\) K this line
\(^*\) Bright lines at the edges of the step were also observed by Nikitin and collaborators\({}^{39}\).
Table I
Position of the lines of the yellow exciton series in the absorption spectrum of Cu\(_2\)O at \(T = 77.3^\circ\)K
| Quantum numbers \(n\) | Wavelengths \(\lambda_n\), Å | Energy, eV | Frequency \(\nu_n\), cm\(^{-1}\) | \(\Delta \nu_n\), cm\(^{-1}\), obs. | \(\Delta \nu_n\), cm\(^{-1}\), calc. |
|---|---|---|---|---|---|
| 1 | 6125.3 | 2.0234 | 16325.7 | 1134.3 | 785 |
| 2 | 5792.7 | 2.1396 | 17263.2 | 196.8 | 196.3 |
| 3 | 5756.6 | 2.1530 | 17371.2 | 88.8 | 87.2 |
| 4 | 5743.8 | 2.1578 | 17410.1 | 49.9 | 49.1 |
| 5 | 5738.1 | 2.1599 | 17427.5 | 32.5 | 31.4 |
| 6 | 5734.1 | 2.1615 | 17439.4 | 20.6 | 21.8 |
| . | . | . | . | . | . |
| . | . | . | . | . | . |
| . | . | . | . | . | . |
| \(\infty\) | 5727.4 | 2.1640 | 17460 | 0 | 0 |
is located at \(\lambda = 6125.3\) Å (see Table I, \(n = 1\)). Figure 4 gives a photograph showing the simultaneous existence of a broad step and a narrow line on it (see § 5).
When the temperature is lowered, not only does the coefficient of light absorption at the step decrease strongly, but the continuous spectrum beyond the step is also weakened extremely strongly, so that the boundary of the onset of strong absorption moves into the short-wavelength part of the spectrum faster than the step moves. As a result of the weakening of the continuous spectrum with decreasing temperature, a series of separate absorption lines, located before the edge of strong absorption, is gradually revealed in the yellow part of the spectrum. These lines at a temperature \(T = 77.3^\circ\)K become very narrow. We were able to detect 5 narrow lines, successively converging with increasing frequency, i.e., altogether six lines including the first line on the background of the step. The positions of the lines depend strongly on temperature, so that the entire series of lines shifts into the short-wavelength part of the spectrum upon cooling of the crystal. The positions of all six lines observed by us in the spectrum at \(T = 77.3^\circ\)K are indicated in Table I (\(n\) from 1 to 6). Figure 5 shows the spectrogram of the yellow series of absorption lines in Cu\(_2\)O.
When the temperature is raised to \(0^\circ\)C, the lines broaden and become barely noticeable against the background of continuous absorption, which increases strongly in intensity with increasing temperature and advances from the short-wavelength side*). The broadening of the lines occurs asymmetrically, more into the long-wavelength part of the spectrum. The short-wavelength side of the lines remains sharper.
The system of converging narrow lines converges to the boundary beyond which the continuous absorption spectrum begins. The boundary of the continuum is located near \(\lambda = 5727.4\) Å (Table I, \(n = \infty\)). The discrete lines together with the continuous spectrum beyond them form a sequence having the appearance of a series observed in line absorption spectra of the atom (ion) near the series limit.
We attempted to find a regularity in the alternation of narrow lines in the absorption spectrum of a cuprous oxide crystal analogous to a series law.
*) Even at room temperature one can notice traces of the first terms of the series, \(n = 1\) and \(n = 2\).
Fig. 4. Spectrograms of the “step,” of the narrow line, and of emission lines at the edge of the fundamental absorption of a Cu\(_2\)O crystal at \(T = 77.3^\circ\) K.
a) Cu\(_2\)O plate of thickness \(d = 0.5\) mm; b) \(d = 1.7\) mm; c) comparison of the Cu\(_2\)O lines with the lines of the iron spectrum.
Fig. 5. Spectrogram of the yellow exciton series in a Cu\(_2\)O crystal at \(T = 77.3^\circ\) K.
It turned out that the frequencies \(\nu_n\) of the new lines satisfy a simple series relation:
\[ \nu_n=A-\frac{B}{n^2}=\nu_\infty-\frac{B}{n^2}=\left(17460-\frac{785}{n^2}\right)\ \mathrm{cm}^{-1};\quad n=1,2,3,4,5,6,\ldots \tag{1} \]
here \(n\) is the quantum number; \(A\) and \(B\) are constants. The quantity \(A=\nu_\infty\) has the value of the series limit \((n=\infty)\), i.e., it corresponds to the red limit of the photodissociation of the system of charges in the crystal lattice, which causes the appearance of the series of lines in the spectrum of \(\mathrm{Cu_2O}\).
Table I gives the experimentally observed values and the values calculated from formula (1) for the frequencies \(\nu_n\) of the lines of the series. There, too, are compared (in wave numbers) the observed and calculated differences \(\Delta\nu_n\) between the frequency \(\nu_\infty\) of the series limit and the frequencies \(\nu_n\) of the lines of the series:
\[ \Delta\nu_n=\nu_\infty-\nu_n=\frac{B}{n^2},\qquad n=1,2,3,4,5,6. \]
As is seen from Table I, the agreement between the observed and calculated values of the frequencies \(\nu_n\) is very good. The exception is the discrepancy between the numbers for the first line of the series \((n=1)\), which is especially clearly manifested in the differences \(\Delta\nu_1\).
Relation (1) and Table I show that the series of lines observed in a \(\mathrm{Cu_2O}\) crystal has a hydrogen-like character. The continuous spectrum beyond the limit of the series of lines should be associated with the detachment of an electron occurring as a result of photodissociation of the system of charges that gives rise to the series of lines in the spectrum of the crystal.
The existence of narrow absorption lines in crystals at low temperatures has been observed many times, for example, in molecular crystals (Obreimov and Prikhot’ko \(^{13,14}\), Shpol’skii \(^{15}\)), in alkali-halide salts colored also with metal impurities (for example, in KI—Tl; Yuster and Delbecq, Pringsheim \(^{16}\)), in salts of rare earths, etc.
However, until now groups of lines satisfying series regularities have never been observed in crystals.
§ 4. OPTICAL SPECTRUM OF EXCITON EXCITATION IN A CRYSTAL LATTICE. THE YELLOW EXCITON SERIES IN A \(\mathrm{Cu_2O}\) CRYSTAL
The investigations described above establish a new and interesting fact: in a solid body one can observe series of narrow lines with a large number of members, regularly converging to a series limit and analogous to those observed in free atoms and ions.
Hence the question naturally arises as to how the hydrogen-like series arises in the crystal lattice of cuprous oxide and what causes it. The position of the series of lines in the spectrum immediately near the long-wavelength edge of the intrinsic absorption of the lattice and the large absorption coefficient in the lines of the series allow one to consider that the phenomenon is connected with the basic lattice of the cuprous-oxide crystal, and not with its local disturbances and deviations from stoichiometric ratios*). The series regularity (1) of the hydrogen-like atom
*) It is known that in a cuprous-oxide crystal there is usually an excess of oxygen, which can be removed by prolonged annealing of \(\mathrm{Cu_2O}\) at high temperatures in vacuum. Despite such treatment of the \(\mathrm{Cu_2O}\) crystal, the series of absorption lines was preserved in the spectrum. This shows that the origin of the series is not connected with an excess of oxygen in the crystal.
in the alternation of the lines of the series shows that the system of electric charges in the crystal, which gives rise to the hydrogen-like series, is bound by Coulomb forces. The position of the series limit near the edge of the fundamental absorption of the lattice gives grounds to believe that the lines of the series are probably associated with transitions of electrons to levels situated in the immediate vicinity of the lower edge of the free band.
All these considerations led us to the idea that the hydrogen-like series of narrow lines in cuprous oxide is caused by absorption of light associated with the excitation of excitons in the $\mathrm{Cu}_2\mathrm{O}$ lattice.
The concept of the exciton as an excited state of the lattice that can move (wander through the crystal) was created by Ya. I. Frenkel in 1931. Frenkel was the first to call attention to the possibility of the existence in a crystal lattice of a special excited state of the electrons which, having arisen in some one cell of the crystal, is transferred without radiation (by a resonance mechanism) from cell to cell and thus moves (migrates) through the crystal.
After Frenkel’s work, excitons and their properties were studied by Wannier^17, Mott^18, Slater and Shockley^19, Frank and Teller^20, and Zeitz^21. Recently various properties of excitons have been considered by Zeitz^22, Heller and Marcus^23, Anselm and Firsov^24, Samoilovich and Korenblit^25, Samoilovich and Kononova^26, and also by Ipatova.
Wannier^17 and Mott^18 regard the exciton as a quasi-hydrogen atom*), in which the electron and the hole, bound by Coulomb forces, rotate about their common center of gravity. The electron and the hole in the exciton state move together (migrate) through the crystal and, while remaining bound, are not current carriers and do not participate in electrical conduction.
The bound exciton states of the electron and hole are characterized by discrete energy levels of a hydrogen-like atom. The absorption spectrum of such a system should therefore consist of discrete lines**), corresponding to transitions of the system, under the action of light, into various excited states of the exciton. In these states the exciton is not a current carrier. Photoconductivity should arise when light of higher frequency can dissociate the exciton and bring the crystal into a state with a free electron and a hole, when the latter become current carriers.
It follows from this that the discrete energy levels of the excited states of the exciton must be situated below the edge of the free conduction band, while the red limit of photodissociation of the exciton must coincide with its lower edge (as determined from optical data***). Hence it is clear that the discrete lines of light absorption corresponding to transitions of electrons into exciton states must be located in the spectrum on the red side of the edge of the fundamental absorption and converge to the series limit located at the edge of the fundamental absorption.
*) The mass of the positive hole is close to the mass of the electron, and therefore it is more natural to regard the exciton as analogous not to the hydrogen atom, but to positronium.
**) The possibility of the existence of narrow lines in the absorption spectra of crystals was also shown by Peierls^27.
***) The determination of the band edge from spectroscopic data may not coincide with the thermal determination.
Thus, theoretical considerations imply, for the frequencies of the exciton absorption spectrum, a series dependence characteristic of a hydrogen-like atom:
\[ \nu_n=\nu_\infty-\frac{R'}{n^2}, \tag{2} \]
where \(\nu_n\) is the frequency of the \(n\)-th absorption line in the exciton spectrum, \(\nu_\infty\) is a constant having the value of the frequency of the series limit in the exciton absorption spectrum, i.e., corresponding to the photodissociation energy of the exciton, \(n\) is a quantum number taking the integer values \(1, 2, 3,\ldots\), and \(R'\) is a quantity determined by the relation
\[ R'=\frac{R\frac{\mu}{m}}{n_0^4}, \]
where \(R\) is the Rydberg constant, \(n_0\) is the refractive index, \(m\) is the mass of the electron in vacuum, and \(\mu\) is the reduced effective mass of the exciton,
\[ \frac{1}{\mu}=\frac{1}{\mu_1}+\frac{1}{\mu_2}, \]
where \(\mu_1\) and \(\mu_2\) are the effective masses of the electron and the hole.
If we now turn to the results of the experiments described above, it turns out that the experimental data are in very good agreement with the notions of the exciton. Furthermore, the experimental data make it possible to determine, for \(\mathrm{Cu_2O}\), from the values of the constant \(B\) of the exciton series, the reduced exciton mass \(\mu\), and from it to calculate either the effective mass \(\mu_1\) of the electron or the effective mass \(\mu_2\) of the hole, if one of these quantities is known. Taking \(\mu_2 \cong 1^*)\) and \(n_0=2.5\), from the data for the exciton spectrum we obtain the value \(\mu_1=0.4\).
Thus, from the exciton spectrum reasonable values are obtained for the effective masses, which indicates good agreement of the theory with experiment.
On the basis of the foregoing it must be concluded that the hydrogen-like series of narrow lines observed by us \(^{10,12}\) experimentally in the yellow part of the absorption spectrum of a cuprous oxide crystal is the optical spectrum of the exciton and constitutes direct proof of the existence of excitons in a crystalline lattice.
These results were obtained by us as early as the spring of 1951.**) Several months later, the fact of the existence of a hydrogen-like series of narrow lines in a solid, established by our experiments \(^{10,12}\) with a cuprous oxide crystal, was confirmed by the work of the Japanese physicists Hayashi and Katsuki \(^{28}\), who observed in cuprous oxide, in the yellow part of the spectrum, a series of lines, but erroneously attributed it to a polaron (self-trapped electron). In 1954 our experiments with cuprous oxide were repeated in France and received new confirmation in the work of Nikitine and a group of collaborators \(^{29}\).
*) The value \(\mu_2 \cong 1\) may be taken as the mean of the value \(\mu_2=1.5—1.8\), obtained from data for the \(F\)-center in a \(\mathrm{Cu_2O}\) crystal, and the value \(\mu_2=0.5\), obtained from thermoelectric electromotive force and Hall-effect data. These data were kindly communicated to us by S. I. Pekar and V. P. Zhuze.
**) The results of these investigations were reported on 19. IX. 1951 at the Physico-Technical Institute of the Academy of Sciences of the USSR in Leningrad and on 12. XII. 1951 at the Physics Institute of the Academy of Sciences of the Ukrainian SSR in Kiev.
§ 5. COMPARISON OF THE EXCITON SPECTRUM WITH THE SPECTRUM OF THE HYDROGEN ATOM.
FREE AND POLARIZING EXCITONS
The excitation of an exciton in a crystal lattice under the action of light quanta may be represented by the scheme shown in Fig. 6, a. Alongside it (Fig. 6, b) is shown the energy scheme of the hydrogen atom, the scale of which is reduced by a factor \(R/R'\) for comparison with the energy scheme of the exciton. A comparison of the two schemes shows that there is a substantial difference between the scheme of excitation of an exciton from the ground level in the crystal
\[ \nu_n=\nu_\infty-\frac{R'}{n^2} \]
\[ (n=1,2,3,4,\ldots) \]
and the scheme of excitation of the hydrogen atom from the ground (normal) state of the hydrogen atom
\[ \nu_n=R\left(\frac{1}{1^3}-\frac{1}{n^3}\right) \]
\[ (n=2,3,4,\ldots). \]
Fig. 6. Scheme of the levels of the exciton and the hydrogen atom.
Unlike the hydrogen atom, in order to create an exciton, i.e., to form in the crystal a hole bound to an electron into a system similar to an unexcited hydrogen atom, it is necessary to expend the energy \(\nu_1\)—the energy of formation of the exciton in the one-quantum state \(n=1^*)\). The theory does not give this quantity, but \(\nu_1\) can be determined experimentally from the exciton spectrum, just as the quantity \(\nu_\infty\) of exciton photodissociation can. For cuprous oxide,
\[ \nu_1=2.023\ \text{ev};\quad \nu_\infty=2.164\ \text{ev}. \]
The frequency of the first member of the exciton series determines the smallest energy of formation of an exciton. As is seen from the scheme in Fig. 6, a, an exciton can be created in excited states, for which greater energies are required.
From the foregoing it follows that comparison of the exciton spectrum with the spectrum of the hydrogen atom must be made not by the frequencies \(\nu_n\), but by the differences \(\Delta\nu_n\) between the frequency of the series limit \(\nu_\infty\) and the frequencies \(\nu_n\) of the exciton-series lines:
\[ \Delta\nu_n=\nu_\infty-\nu_n. \]
From comparison of the numbers in the last column of Table I it is seen that there are significant deviations from hydrogen-likeness for the first member
*) If the number of excitons in the state with quantum number \(n=1\) were sufficiently large, then a series of transitions from the exciton state \(n=1\) to excited states with quantum numbers \(n=2,3,4,\ldots\) could be observed. This series of lines, analogous to the Lyman series of the hydrogen atom, should lie in the infrared part of the spectrum. It is possible that in some crystals with a very narrow forbidden band the concentration of thermally excited excitons may be sufficiently large for this “infrared series” of the exciton to be observed.
series of the exciton \(n=1\)*). It follows from this that the Coulomb law of interaction between a hole and an electron in a crystal, as in a dielectric medium, introduced by Mott\(^{18}\),
\[ V=-\frac{e^2}{r n_0^2}, \]
where \(n_0\) is the refractive index, is not satisfied for small exciton orbits. The corresponding energy levels lie deeper than the hydrogenic ones, as is seen in the diagrams of Fig. 6.
The first member of the exciton series in \(\mathrm{Cu}_2\mathrm{O}\) is distinguished not only by its position, but also by its exceptional narrowness, sharpness, and extremely low intensity in comparison with the other members of the series. In order to observe it well, it is necessary to take thick \(\mathrm{Cu}_2\mathrm{O}\) plates. In Figs. 4, \(a\) and \(b\) are shown spectrograms obtained with \(\mathrm{Cu}_2\mathrm{O}\) plates of thickness \(0.5\) and \(1.7\) mm. The first member of the series \(n=1\) is seen here quite distinctly. Here also the “step” and the emission lines at its edges, mentioned above, are visible. In Fig. 4, \(c\) is shown the spectrogram of the first member of the series alongside a portion of the iron comparison spectrum, from which it is seen that the width of the first member of the series is comparable with the width of the atomic lines of Fe. The very weak absorption of the first member of the series and its small width show that the transition from the ground level of the \(\mathrm{Cu}_2\mathrm{O}\) crystal to the first exciton level has a very small probability and is forbidden by the selection rules. The strong absorption in the other lines of the exciton series shows that in cuprous oxide excitons are formed with much greater probability in excited states.
According to the ideas of Ya. I. Frenkel\(^{4}\), in a crystal there may be excitons of two principal types, which he called “free” and “attached.” These two types of excitons have recently been considered by A. S. Davydov\(^{30}\) for the case of molecular crystals (“free” and “localized” excitons) and by I. M. Dykman and S. I. Pekar\(^{31}\) and I. M. Dykman\(^{32}\) for ionic crystals (“nonpolarizing” and “polarizing” excitons).
In excitons of the first type the excitation is transferred so rapidly in the crystal from cell to cell that the interaction of the electron and hole with the vibrations of the ions cannot take place. The exciton does not produce inertial polarization of the crystal. Since in this case there is no interaction of the exciton with the vibrations of the ions, the lines of the excitation spectrum of the free, nonpolarizing exciton remain narrow.
In excitons of the second type the transfer of energy from cell to cell of the lattice occurs slowly. The exciton creates a local inertial polarization of the crystal and, like a polaron, can move through the crystal together with it. As a result of interaction with the vibrations of the lattice ions, the lines of the excitation spectrum of the “localized,” “polarizing” exciton broaden into wide bands.
According to the calculations of Dykman and Pekar\(^{31}\), nonpolarizing excitons are formed in ionic crystals in which the ratio of the effective masses of the electron and the hole \(\mu_1/\mu_2\) is small
\[ \left(\frac{1}{10}<\frac{\mu_1}{\mu_2}<10\right). \]
The excitation spectrum of excitons of this type forms a hydrogen-like series of narrow lines.
*) It should be noted that the assignment of the line \(n=1\) to the exciton spectrum is conditional. This line arises from transitions of electrons within one elementary cell of the \(\mathrm{Cu}_2\mathrm{O}\) crystal. Therefore formula (2) is not applicable to such a transition. In this connection the question arises whether this line may be assigned to the exciton spectrum.
Polarizing excitons exist only in those ionic crystals in which \(\mu_1/\mu_2\) is large or, conversely, small \(\left(\mu_1/\mu_2 > 10\ \text{or}\ \mu_1/\mu_2 < 1/10\right)\).
The excitation spectrum of polarizing excitons does not form a hydrogen-like series and consists of broad bands with widths on the order of tenths of an electron-volt. This case is realized in alkali-halide crystals. The broad absorption bands found by Hilsch and Pohl in these crystals, located in the ultraviolet part of the spectrum, can be interpreted as exciton bands. Dykman’s calculations \(^{32}\) of the exciton absorption of light in a KCl crystal gave good agreement with experiment.
The very small width not only of the first but also of the other lines of the yellow exciton series in a cuprous-oxide crystal leads to the conclusion that the exciton in a \(\mathrm{Cu_2O}\) crystal moves rapidly, migrates through the crystal, i.e., is a free, nonpolarizing exciton.
§ 6. THE GREEN EXCITON SERIES IN A \(\mathrm{Cu_2O}\) CRYSTAL AT THE TEMPERATURE \(T = 77.3^\circ\mathrm{K}\) (LIQUID NITROGEN)
Further investigations were aimed at a deeper study of the phenomena discovered in the absorption of light in cuprous oxide and described above.
First of all, it was of interest to advance further into the region of continuous absorption of cuprous oxide beyond the boundary of the series of narrow exciton lines. Investigation of this region presents great difficulties because of the very large absorption coefficient. Even through comparatively thin \(\mathrm{Cu_2O}\) plates, light in this wavelength region practically does not pass. Nevertheless, with very long exposure, already in our first experiments we discovered \(^{12}\) an inhomogeneity in the continuous absorption spectrum of \(\mathrm{Cu_2O}\) beyond the boundary of the yellow exciton series. The strong continuous absorption, beginning immediately beyond the series of lines, gradually weakens somewhat on moving into the short-wavelength region and reaches a certain broad minimum, located at \(T = 77.3^\circ\mathrm{K}\) approximately near \(5500\ \text{Å}\) (\(2.25\ \text{eV}\)). This indicated some inhomogeneous structure of the free band in the cuprous-oxide crystal. Therefore, together with B. P. Zakharchenya, we undertook \(^{33}\) new studies of this phenomenon. In order to advance further into the region of continuous absorption and to study the phenomenon more deeply, it was necessary to reduce the absorption of light in the \(\mathrm{Cu_2O}\) samples under investigation, i.e., to obtain \(\mathrm{Cu_2O}\) plates as thin as possible. By careful polishing it was possible to bring the cuprous-oxide plates to a thickness of about 20 microns.
In the absorption spectrum of thin \(\mathrm{Cu_2O}\) plates at the temperature of liquid nitrogen (\(T = 77.3^\circ\mathrm{K}\)), beyond the absorption minimum on the short-wavelength side of the yellow series, a new, “green” series of narrow lines opens up, converging in the short-wavelength direction toward a boundary beyond which continuous absorption is observed. We were able to observe four lines of the second series on the spectrogram. The first members of the series are also well observed visually. The absorption of light in the lines of the green series is considerably greater than in the lines of the yellow series, and they are broader. Measurements showed that the frequencies \(\nu_n\) of these four lines also well satisfy the serial regularity of a hydrogen-like atom:
\[ \nu_n = \nu_\infty - \frac{B}{n^2} = \left(18507 - \frac{1246}{n^2}\right)\ \mathrm{cm}^{-1} \tag{2} \]
\[ (n = 2, 3, 4, 5, \ldots). \]
We were unable to detect, at the temperature of liquid nitrogen, either visually or on the spectrograms, a line that could be interpreted as the first member \(n=1\) of the green series.
Fig. 7. Spectrograms of the yellow and green series in a \(\mathrm{Cu_2O}\) crystal (a) and of the enlarged green series (b) at \(T = 77.3^\circ\ \mathrm{K}\).
Figure 7 shows a portion of the spectrum on which both series are visible, and presents the green series in enlarged form.
Table II
Positions of the lines of the green exciton series in the absorption spectrum of \(\mathrm{Cu_2O}\) at \(T = 77.3^\circ\mathrm{K}\)
| Quantum numbers \(n\) | Wavelengths \(\lambda_n\), Å | Energy, eV | Frequencies \(\nu_n\), cm\(^{-1}\) | \(\Delta\nu_n\), cm\(^{-1}\), obs. | \(\Delta\nu_n\), cm\(^{-1}\), calc. |
|---|---|---|---|---|---|
| 1 | — | — | — | — | 1246 |
| 2 | 5496 | 2,255 | 18 195 | 312 | 311 |
| 3 | 5444 | 2,277 | 18 369 | 138 | 138 |
| 4 | 5426 | 2,284 | 18 429 | 78 | 77 |
| 5 | 5419 | 2,287 | 18 454 | 53 | 50 |
| . | . | . | . | . | . |
| . | . | . | . | . | . |
| . | . | . | . | . | . |
| \(\infty\) | 5404 | 2,294 | 18 507 | 0 | 0 |
Table II gives the frequencies \(\nu_n\) of the lines of the green series, as well as the differences \(\Delta\nu_n\) between the frequencies \(\nu_\infty\) of the series limit and the frequencies \(\nu_n\) of the observed lines,
\[ \Delta\nu_n = \nu_\infty - \nu_n = (18507 - \nu_n)\ \mathrm{cm}^{-1}. \]
The differences \(\Delta\nu_n\) obtained from the experiment are compared in Table II with the values of \(\Delta\nu_n\) calculated from the relation
\[ \Delta\nu_n = \frac{B}{n^2} = \frac{1246}{n^2}\ \mathrm{cm}^{-1} \]
\[ (n = 2,\ 3,\ 4,\ 5). \]
As is seen from Table II, the frequencies of the second series also satisfy very well the series regularity of a hydrogen-like atom. The limit of the green series is shifted relative to the limit of the yellow series
toward higher frequencies by an amount \(\delta = 0.130\) eV. We assume that the second hydrogen-like series in cuprous oxide is caused, just like the first series, by excitons in the \(Cu_2O\) crystal lattice.
One of the possible explanations for the appearance of two series in the absorption spectrum may be the assumption that there exist two overlapping free bands in the \(Cu_2O\) crystal, as shown in Fig. 8. Each series converges to its own conduction band.
Fig. 8. Possible excitation scheme of the yellow and green series in \(Cu_2O\).
It is noteworthy that the frequency difference between the limits of the yellow and green exciton series, \(\delta = 0.130\) eV, coincides within the errors of measurement with the energy of one of the infrared absorption bands in \(Cu_2O^{10}\), equal to \(0.138\) eV. It is possible that these infrared absorption bands belong to vibrations of the \(Cu_2O\) lattice.
The green series in cuprous oxide was also observed by Hayashi and Katsuki \(^{28}\), who likewise attribute it to an exciton. The yellow (first) series in \(Cu_2O\), however, Hayashi and Katsuki associate with a polaron (self-trapped electron), which is erroneous for the following reasons:
1) The dissociation energy of a polaron is of the order of tenths of an electron-volt \(^{34}\), and therefore the absorption spectrum of light by a polaron should be located in the infrared region, and not in the visible part, as is the yellow series in \(Cu_2O\), with an excitation energy of the order of 2 eV.
2) By its polarization nature a polaron is closely connected with lattice vibrations, and therefore the optical spectrum of its excitation should consist mainly of one broad band, as in \(F\)-centers \(^{34,35}\), and not of very narrow lines, as in the yellow series.
3) When the temperature is lowered, the absorption of light caused by polarons should decrease very strongly, since the concentration of polarons in the crystal decreases according to an exponential law \(^{34}\). Nothing of the kind is observed for the yellow series of \(Cu_2O\).
§ 7. THE EXCITON SPECTRUM IN \(Cu_2O\) AT TEMPERATURE
\(T = 4.2^\circ K\) (LIQUID HELIUM)
New lines
In order to investigate the exciton series more thoroughly, new experiments were undertaken on the absorption of light in a cuprous oxide crystal with deeper cooling of the crystal to the temperature of liquid helium, \(T = 4.2^\circ K\). In these new experiments, in which, besides B. P. Zakharchenya, N. M. Reinov took part, a spectral instrument of greater dispersion was used. The dispersion of the prism spectrograph used in these investigations was approximately 7 Å/mm in the spectral region near \(\lambda = 5800\) Å, i.e. three times greater than the dispersion of the spectrograph used previously.
Upon cooling the cuprous oxide crystal to the temperature of liquid helium, the following phenomena were observed \(^{36}\). The width of the absorption lines of both the yellow and the green series decreased when the temperature was lowered to \(T = 4.2^\circ K\), and the lines became considerably narrower than at the temperature of liquid nitrogen. Both series of lines shifted further into the short-wavelength part of the spectrum.
At \(T = 4.2^\circ\) K, still another new boundary (step) was found\({}^{36}\), located at \(\lambda = 5841\) Å \((2.1219\ \text{eV})\), where the continuous absorption undergoes a jump, and one more very weak line on this step at \(\lambda = 5817\) Å \((2.1306\ \text{eV})\).
Table III
Positions of the lines of the yellow and green exciton series in \(\mathrm{Cu_2O}\) at \(T = 4.2^\circ\) K
| Quantum numbers \(n\) | Yellow series: wavelengths \(\lambda_n\), Å | Yellow series: energy, eV | Yellow series: frequencies \(\nu_n\), cm\(^{-1}\) | Green series: wavelengths \(\lambda_n\), Å | Green series: energy, eV | Green series: frequencies \(\nu_n\), cm\(^{-1}\) |
|---|---|---|---|---|---|---|
| 1 | 6095.8 | 2.0332 | 16 404.7 | 5817 | 2.131 | 17 191 |
| 2 | 5770.8 | 2.1477 | 17 328.6 | 5469 | 2.266 | 18 285 |
| 3 | 5735.0 | 2.1611 | 17 436.8 | 5419 | 2.287 | 18 454 |
| 4 | 5722.8 | 2.1657 | 17 473.9 | 5402 | 2.294 | 18 512 |
| 5 | 5716.8 | 2.1618 | 17 492.3 | 5394 | 2.298 | 18 539 |
| 6 | 5713.1 | 2.1694 | 17 503.6 | · | · | · |
| · | · | · | · | · | · | · |
| · | · | · | · | · | · | · |
| · | · | · | · | · | · | · |
| \(\infty\) | 5706.6 | 2.1719 | 17 523.3 | 5380 | 2.303 | 18 587 |
Table III gives the wavelengths of the absorption lines of the yellow and green exciton series in \(\mathrm{Cu_2O}\) at a temperature \(T = 4.2^\circ\) K. The frequencies of the lines of both the yellow and the green series at \(T = 4.2^\circ\) K (apart from the first members) are well described by the hydrogen-like series dependence:
\[ \text{for the yellow series}\quad \nu_n = 17\,523.3 - \frac{780.7}{n^2}\ \mathrm{cm}^{-1}; \]
\[ n = 1, 2, 3, \]
\[ \text{for the green series}\quad \nu_n = 18\,587 - \frac{1200}{n^2}\ \mathrm{cm}^{-1}; \]
\[ n = 1, 2, 3\ldots \]
At liquid-helium temperature, in the absorption spectrum of a \(\mathrm{Cu_2O}\) crystal we found\({}^{36}\) another 7 weak, thin absorption lines situated between the lines of the yellow exciton series, as shown in Fig. 9. The frequencies of the new lines are indicated in Table IV.
Fig. 9. Diagram of the arrangement of weak absorption lines between members of the yellow exciton series at \(T = 4.2^\circ\) K.
Figure 10 presents photographs of the members of the yellow series, between which these weaker lines, observed at \(T = 4.2^\circ\) K, are visible. The new lines correspond to different absorption coefficients and therefore cannot be detected with sufficient distinctness simultaneously on a single spectrogram. Therefore Fig. 10 presents three spectrograms, \(a\), \(b\), and \(c\), which (for detecting the new lines) were obtained with different exposures: 20 minutes, 5 minutes, and 5 seconds, respectively. The new lines in Fig. 10 are marked by arrows.
Attention is drawn to the characteristic distribution of the intensities of the members of the first series, which appears especially clearly for the member
Fig. 10. Spectrograms of members of the yellow series at \(T = 4.2^\circ\text{K}\) with weak lines between them, obtained with exposures:
a) 20 minutes, b) 5 minutes, c) 5 seconds.
Visible labels in the spectrograms:
a) \(n=2\), 3, 4, 5; \(5734.5\,\text{\AA}\), \(5724.4\,\text{\AA}\).
b) \(n=2\), 3, 4; \(5737.0\,\text{\AA}\), \(5731.5\,\text{\AA}\), \(5734.5\,\text{\AA}\).
c) \(n=2\), 3; \(5762.5\,\text{\AA}\), \(5758.0\,\text{\AA}\), \(5751.5\,\text{\AA}\).
Table IV
Position of new lines in the exciton spectrum in \(\mathrm{Cu_2O}\) at \(T = 4.2^\circ\text{K}\) and \(T = 1.3^\circ\text{K}\)
| Designations | Wavelengths at \(T=4.2^\circ\text{K}\), \(\text{\AA}\) | Frequencies at \(T=4.2^\circ\text{K}\), \(\text{cm}^{-1}\) | Wavelengths at \(T=1.3^\circ\text{K}\), \(\text{\AA}\) | Frequencies at \(T=1.3^\circ\text{K}\), \(\text{cm}^{-1}\) |
|---|---|---|---|---|
| \(n''' - 2\) | 5758.7 | 17 365.0 | 5758.8 | 17 367.1 |
| \(n'' - 2\) | 5753.2 | 17 381.6 | 5752.2 | 17 383.7 |
| \(n' - 2\) | 5746.3 | 17 402.5 | 5745.1 | 17 406.1 |
| \(n''' - 3\) | 5731.8 | 17 446.5 | 5731.8 | 17 446.5 |
| \(n'' - 3\) | 5729.0 | 17 455.0 | 5729.5 | 17 453.5 |
| \(n'' - 3\) | 5729.0 | 17 455.0 | 5728.0 | 17 458.1 |
| \(n' - 3\) | 5726.5 | 17 462.7 | 5727.3 | 17 460.2 |
| \(n' - 3\) | 5726.5 | 17 462.7 | 5726.5 | 17 462.7 |
| \(n' - 3\) | 5726.5 | 17 462.7 | 5726.0 | 17 464.2 |
| \(n' - 3\) | 5726.5 | 17 462.7 | 5725.4 | 17 466.0 |
| \(n' - 4\) | 5719.8 | 17 483.1 | 5719.3 | 17 484.7 |
| \(n' - 4\) | 5719.8 | 17 483.1 | 5718.1 | 17 488.3 |
| \(n' - 4\) | 5719.8 | 17 483.1 | 5714.7 | 17 498.7 |
| \(n' - 4\) | 5719.8 | 17 483.1 | 5713.9 | 17 501.2 |
\(n=2\) in the photograph of Fig. 10c, obtained with the smallest exposure. Here it is clearly seen that the absorption of light is distributed about the line very asymmetrically: gradually decreasing toward longer wavelengths, it drops off sharply toward shorter wavelengths.
§ 8. THE EXCITON SPECTRUM IN Cu\(_2\)O AT THE TEMPERATURE \(T=1.3^\circ\) K
(LIQUID HELIUM)
Further investigations of the absorption spectrum of a Cu\(_2\)O crystal were undertaken with a spectrograph with a diffraction grating giving a dispersion of about \(4\ \text{Å}/\text{mm}\). In addition, the Cu\(_2\)O crystal was cooled to still lower temperatures, which can be reached by evaporating liquid helium at low pressures.
At the temperature \(T=1.3^\circ\) K, when the lines of the exciton spectrum become still narrower, it was possible to notice\({}^{37}\) indications, apparently, of a fine structure of the first four members of the yellow exciton series, beginning with the second.
Fig. 11. Spectrograms of the yellow exciton series in Cu\(_2\)O at: a) \(T=1.3^\circ\) K; b) \(T=77.3^\circ\) K.
At \(T=1.3^\circ\) K, owing to the narrowing of the lines and a very strong weakening of the continuous spectrum, we were able to observe well\({}^{37}\) the higher members of the yellow exciton series near the series limit up to and including its 9th member\({}^{*}\).
In Fig. 11 a spectrogram of the yellow exciton series at \(T=1.3^\circ\) K is presented. It is compared with a spectrogram of the same series at the temperature \(T=77.3^\circ\) K, obtained with the same Cu\(_2\)O plate of thickness 160 microns. From these spectrograms it is seen how strongly it weakens at
\({}^{*}\) Attention is drawn to the magnitudes of the exciton orbits, which should correspond to the higher members of the series (\(n \simeq 10\)) observed in the experiment at \(T=1.3^\circ\) K. For such large quantum numbers the radii of the exciton orbits in the crystal are enormous. Taking \(\mu_1=\mu_2=m\), we obtain for the radii of the electron and hole orbits in the exciton
\[ r_{n\,\text{exciton}}=2r_{1\mathrm{H}}\cdot n^{2} n_0^{2}\simeq n^{2}n_0^{2}\text{Å}, \]
where \(r_{1\mathrm{H}}\) is the radius of the first orbit in the hydrogen atom, and \(n_0\) is the refractive index of the crystal. Putting \(n_0^{2}\simeq 6\) and \(n=10\), we obtain
\[ r_{10\,\text{exciton}}=600\ \text{Å}. \]
upon going from \(T=77.3^\circ\mathrm{K}\) to \(T=1.3^\circ\mathrm{K}\), a continuous spectrum superposed on the series, and how the lines of the series narrow at the temperature \(T=1.3^\circ\mathrm{K}\).
Table V
Position of the members of the yellow exciton series in \(\mathrm{Cu_2O}\)
at \(T=1.3^\circ\mathrm{K}\)
| Quantum numbers \(n\) | Observed values of frequencies \(\nu_n\), in \(\mathrm{cm}^{-1}\) | Calculated values of frequencies \(\nu_n\), in \(\mathrm{cm}^{-1}\) | \(\Delta\nu\), in \(\mathrm{cm}^{-1}\) |
|---|---|---|---|
| 1 | 16 406,8 | 16 814,0 | 407,2 |
| 2 | 17 331,3 | 17 346,9 | 15,6 |
| 3 | 17 438,3 | 17 445,6 | 7,3 |
| 4 | 17 477,3 | 17 480,1 | 2,8 |
| 5 | 17 494,4 | 17 496,1 | 1,7 |
| 6 | 17 504,5 | 17 504,8 | 0,3 |
| 7 | 17 510,0 | 17 510,0 | 0 |
| 8 | 17 513,4 | 17 513,4 | 0 |
| 9 | 17 515,8 | 17 515,8 | 0 |
| . | . | . | . |
| . | . | . | . |
| . | . | . | . |
| \(\infty\) | 17 524,5 |
In Table V (second column) are given the measurements, observed by us, of the members of the yellow exciton series at \(T=1.3^\circ\mathrm{K}\). The measurements refer to the centers of the lines of the series, without taking account of their fine structure.
The narrowing of the absorption lines at \(T=1.3^\circ\mathrm{K}\) makes it possible to measure the lines of the series, especially its higher members (\(n=7\) and \(n=8\)), with relatively high accuracy. Measurements of these members of the series, for which hydrogen-like behavior should be well obeyed, can be used to calculate the constants \(\nu_\infty\) and \(R\) of the serial frequency dependence*). The expression obtained is
\[ \nu_n = 17\,524.5 - \frac{710.5}{n^2}\qquad (n=1,2,3,\ldots). \]
The frequency values of the yellow series calculated from this formula are presented in the third column of Table V. The differences \(\Delta\nu\) between the observed and calculated frequency values show deviations from hydrogen-like behavior. From Table V it is seen that deviations are observed not only for the first member of the series \(n=1\), but are also noticeable on the other lines of the series \(n=2,3,4,5\), the magnitude of these deviations decreasing with increasing quantum numbers. Thus, the smaller the exciton orbits, the larger the deviations from hydrogen-like behavior. This may be connected, on the one hand, with the fact that neglecting the microstructure of the crystal and considering it macroscopically as a medium with a certain dielectric constant is possible only for large exciton orbits. On the other hand, the deviations for small orbits may be explained by the fact that the hole in the crystal is not a point charge.
At \(T=1.3^\circ\mathrm{K}\) we discovered\(^{37}\) still new lines, located between the members of the series \(n=4\) and \(n=5\) and between \(n=5\) and \(n=6\), in addition to those lines which we observed\(^{36}\) at \(T=4.2^\circ\mathrm{K}\). Moreover, because of the narrowing of the lines at \(T=1.3^\circ\mathrm{K}\), some of the weak lines previously observed by us between the members of the yellow series split into several
*) These calculations were performed by V. P. Zakharchenya.
very narrow fine lines. All the lines observed by us at \(T = 4.2^\circ\) K and \(T = 1.3^\circ\) K, located between the members of the yellow exciton series, are compared in Table IV*).
As our investigations of the Stark effect in the yellow exciton series show \(^{38,43}\) (see § 9), the lines observed between the members of the exciton series at the temperature of liquid helium (Table IV) appear under the influence of an electric field also at the temperature of liquid nitrogen and increase in intensity as the field increases. Therefore they may be assigned to forbidden electronic transitions that appear under the influence of an external electric field.
At present it is still difficult to give a definite interpretation of the energy levels between which such forbidden transitions occur. Various assumptions are possible here.
Probably the existence of these levels is connected with the incomplete hydrogen-like character of the exciton energy scheme. In this connection the question arises: would it not be more correct to represent the exciton as similar not to the hydrogen atom or positronium, but to a more complex atom? In free atoms (for example, alkali metals) only levels with large quantum numbers are hydrogen-like. In the exciton, however, owing to the large radii of the electron orbits in the crystal, a sufficiently good hydrogen-like character of the energy levels can also be observed at smaller quantum numbers, which is in good agreement with experiment. The optical excitation spectrum of the exciton may then be complicated and will resemble the spectrum of a complex atom (or ion) more than the simple spectrum of the hydrogen atom.
Deviations from hydrogen-like behavior in the structure of the exciton should lead to the removal of degeneracy with respect to the quantum number \(l\) (which determines the orbital angular momentum of the amount of motion) and to the appearance of new levels in the exciton energy scheme. The appearance of new forbidden lines in the exciton spectrum between the members of the yellow series should then be associated with violations of the selection rules \(\Delta l = \pm 1\).
§ 9. THE STARK EFFECT ON THE LINES OF THE EXCITON SPECTRUM
a) “Homogeneous” electric field
It is well known that the splitting of lines of atomic spectra in magnetic and electric fields is of great help in the analysis of complex spectra. We therefore undertook a study of the action of external electric and magnetic fields on the lines of the exciton series in cuprous oxide. These investigations, interesting in themselves, could, as we assumed, make it possible to understand the complex spectrum of cuprous oxide and help interpret the new lines discovered by us at the temperature of liquid helium.
It is known that the splitting of the lines of the hydrogen spectrum in the Stark effect is very large. Therefore it could be expected that the influence of the electric field on the exciton spectrum would also be considerable.
It is further known that the distance \(\Delta\) between the two extreme sublevels into which a term with quantum number \(n\) is split in the Stark effect is equal to:
\[ \Delta = Bn(n - 1)F, \]
*) In the region of the first member of the series \(n = 1\) at the temperature of liquid helium we noticed a weak, indistinctly expressed line at \(\lambda = 6039\) Å (2.0523 eV).
where \(F\) is the electric-field strength and \(B\) is a certain coefficient. It follows from this that the splitting \(\Delta\) in the Stark effect is approximately proportional to \(n^2\), i.e., it is larger the larger the diameter of the electron orbit. This effect should play a particularly large role for an exciton, since the electron orbits in an exciton are \(n_0\) times larger than in the hydrogen atom (\(n_0\) is the refractive index; for \(\mathrm{Cu_2O}\), \(n_0 \simeq 2.5\)). Therefore, one should expect a large Stark effect for an exciton.
When an electric field was applied to a cuprous-oxide crystal, we indeed discovered \({}^{38}\) new phenomena leading to very large changes in the exciton spectrum.
In the first stage of our experiments we carried out studies on the yellow exciton series at the temperature of liquid nitrogen (\(T = 77.3^\circ\mathrm{K}\)). To study the Stark effect we used two voltage sources (20 and 70 kilovolts), supplied to electrodes fastened to \(\mathrm{Cu_2O}\) plates (Fig. 12, a). The behavior of the exciton absorption lines in an electric field was studied with the aid of a prism spectrograph with a dispersion in the yellow part of the spectrum of \(7\ \text{\AA}/\mathrm{mm}\).*)
Fig. 12. Appearance of the electrodes for studying the Stark effect in a \(\mathrm{Cu_2O}\) crystal: a) linear electrodes for obtaining a “homogeneous” electric field; b) pointed electrodes, by means of which a nonuniform field was obtained.
In the \(\mathrm{Cu_2O}\) plates with which we worked, we succeeded \({}^{38}\) in tracing the influence of an electric current on five members of the yellow exciton series (\(n = 1, 2, 3, 4, 5\)). The largest changes, as is to be expected from the theory of the Stark effect, are undergone by the higher members of the series with quantum numbers \(n = 5, 4\), and 3. The influence of the electric field is at first detected only on the line \(n = 5\) and becomes noticeable already in comparatively weak fields (approximately \(5\ \mathrm{kV/cm}\)**). As the electric field increases, the line \(n = 5\) first broadens, and then, at approximately \(6\ \mathrm{kV/cm}\), a triplet appearing in its place becomes visible. In a stronger field the broadening of the line \(n = 4\) becomes noticeable, and between the members of the series \(n = 4\) and \(n = 5\) a new line appears, which corresponds to the line \(n' - 4\), observed between these members of the exciton series at the temperature of liquid helium \({}^{38}\) (Table IV). Further, when the field is increased to approximately \(8\ \mathrm{kV/cm}\), instead of the line \(n = 4\) two components appear, which together with the line \(n' - 4\), increasing in intensity, form a triplet. The separation between the components of the triplet increases with increasing applied field. At the same time the components of the triplet broaden with increasing field.
When the field is increased in the interval \(8\text{--}10\ \mathrm{kV/cm}\), three more new lines appear between the members of the series \(n = 3\) and \(n = 4\), which correspond to the lines \(n' - 3\), \(n'' - 3\), and \(n''' - 3\), discovered by us at the temperature of liquid helium (Table IV).
A further increase in the electric field leads to very large changes in the exciton spectrum. The lines \(n'' - 3\) and \(n''' - 3\) in-
*) The influence of an electric field on the exciton spectrum was also observed by A. A. Kalinyak and L. G. Fedorovich \({}^{52}\).
**) We could judge the average value of the electric field only from the applied potential difference and the distance between the electrodes.
increase in intensity and, with increasing field, broaden and move together from the line \(n=3\) of the exciton series into the violet part of the spectrum. The member of the series \(n=3\), on the contrary, as the field increases, shifts into the red part of the spectrum and, broadening, merges with a broad band appearing on the long-wavelength side of the line \(n=3\) and increasing in intensity with increasing field. This band, by its position in the spectrum, corresponds to the lines \(n' - 2\) and \(n'' - 2\), observed at liquid-helium temperature (Table IV). Simultaneously with this band, as the field increases, between the members of the exciton series \(n=3\) and \(n=2\) there appears another well-observed broad new line, which corresponds to the line \(n''' - 2\), observed in liquid helium (Table IV).
We were able to notice only a small broadening and a certain weakening of the intensity of the member of the series \(n=2\) under the influence of an electric field of the order of \(20\)—\(25\ \text{kV/cm}\). Even at the maximum fields of the order of \(50\ \text{kV/cm}\) used in our experiments, we could not detect any changes in the first member of the exciton series \(n=1\), despite its exceptional narrowness.
Along with the above-mentioned changes in the exciton spectrum, we discovered a very interesting phenomenon. As the electric field increases, the higher members of the exciton series, together with their components, broaden and, together with their satellites (new lines), weaken and gradually disappear: first the member of the series \(n=5\); then, at a larger field, \(n=4\); then, at a still larger field, \(n=3\). Simultaneously, as each member of the series gradually disappears, the continuous spectrum advances in its place from the side of the series limit. We were able to observe that at sufficiently strong fields practically only the first two members, \(n=1\) and \(n=2\), remained from the entire exciton series*). We explain this phenomenon of the disappearance of the higher members of the exciton series under the influence of an electric field by the tearing of the electron by the field from the exciton level, i.e., by ionization (dissociation) of the exciton by the electric field.
This effect in the exciton is similar to the phenomenon of ionization of the hydrogen atom in a strong electric field, first predicted by Oppenheimer\({}^{39}\) and discovered for the hydrogen atom in the work of Rausch von Traubenberg\({}^{40}\). The theory of the phenomenon for the hydrogen atom was developed by Lanczos\({}^{41}\). The possibility of destroying the exciton by an electric field was considered by Seitz\({}^{42}\). To observe ionization of the hydrogen atom by an electric field, fields of the order of \(10^6\ \text{V/cm}\) are necessary, as shown by the experiments of Rausch von Traubenberg. For the exciton, the value of the critical field at which a certain line of the series disappears from the spectrum because of ionization by the field is, as our experiments show, reduced to a value of the order of \(10^4\ \text{V/cm}\), owing to the large orbits of the electron and hole in the exciton and the influence of the dielectric constant of the crystal. According to Lanczos’ calculations, the critical electric field for ionization of the hydrogen atom from a level with quantum number \(n\) varies as \(1/n^4\) (as the square of the ionization energy of the level). Hence it is clear that ionization by an electric field, as should be expected, will occur more readily from levels with larger quantum numbers (there disappear
*) By its external manifestations, the disappearance of the higher members of the exciton series with increasing field might, it would seem, be interpreted as the result of heating of the \(\mathrm{Cu_2O}\) crystal by an electric current increasing with increasing field. This, however, is not so. Special experiments established that the disappearance of the higher members of the series up to \(n=3\) inclusive upon heating the crystal is associated with a very large displacement of the entire series of lines, including the first members \(n=1\) and \(n=2\), into the red part of the spectrum, which is not observed in an electric field.
lighter lines in the spectrum near the series limit), which is in fact observed experimentally both for the hydrogen atom and for the exciton.
The changes in the exciton spectrum observed by us, occurring under the influence of an increasing electric field, are schematically shown in Fig. 13. In Fig. 14
Fig. 13. Schematic representation of changes in the spectrum of the yellow exciton series in a Cu\(_2\)O crystal, occurring under the influence of an increasing external electric field at \(T = 77.3^\circ\) K.
are given spectrograms of the Stark effect in the exciton spectrum of a Cu\(_2\)O crystal, in which one can see the broadening of the members of the exciton series, their splitting into separate components, the appearance of new lines, and the gradual disappearance of the higher members of the series as the field is increased. By “new” lines we mean here those 7 lines which also appear in the absence of an external electric field when the crystal is cooled to the temperature of liquid helium (Table IV). Although the wavelengths of these lines in liquid nitrogen and liquid helium differ because of the inequality of the temperatures, their positions relative to the members of the series are the same in both cases, which leaves no doubt as to their identity.
Fig. 14. Spectrograms of the Stark effect in a “homogeneous” electric field \(E\) on the lines of the yellow exciton series in a Cu\(_2\)O crystal at \(T = 77.3^\circ\) K (for various values of \(E\)).
As already noted above, we regard the new lines as “forbidden” lines arising as a result of the appearance, under the influence of-
of an electric field, which are forbidden by the selection rules for transitions between exciton levels in a Cu\(_2\)O crystal.
We also investigated the state of polarization of the exciton lines in an electric field. The experiments showed that in the exciton spectrum in an electric field there are observed (as is usual in the Stark effect) both components polarized parallel to the field (\(\pi\)-components) and components polarized perpendicular to the field (\(\sigma\)-components). Thus, the line \(n''' - 3\) is polarized parallel to the field, while the line \(n'' - 3\) is polarized perpendicular to the field. \(\pi\)- and \(\sigma\)-components are also observed for the member of the series \(n = 4\). The results of the polarization studies should be regarded as preliminary, since these phenomena require more detailed study with a spectral instrument of greater dispersion.
Quantitative determinations of the magnitudes of the displacement of the exciton spectral lines in electric fields are very difficult, since determining the field strength acting in the crystal is associated with very large errors because of the potential drop at the electrodes that exists in Cu\(_2\)O plates. In addition, the field distribution in Cu\(_2\)O plates is often nonuniform (see below). Having selected cuprous-oxide plates in which the electric field was comparatively uniform and the Stark splitting was the same over the entire length of the spectrograph slit, we carried out quantitative measurements of the displacements of the exciton lines as a function of the mean electric-field strength \(E\), determined from the potential difference and the distance between the electrodes (Table VI). On
Table VI
Displacement of exciton lines in Cu\(_2\)O in an electric field \(E\)
| \(n = 3\) | \(n = 3\) | \(n''' - 3\) | \(n''' - 3\) | \(n = 4\) | \(n = 4\) | \(n' - 4\) | \(n' - 4\) |
|---|---|---|---|---|---|---|---|
| \(E\), kV/cm | \(\Delta\lambda\), Å | \(E\), kV/cm | \(\Delta\lambda\), Å | \(E\), kV/cm | \(\Delta\lambda\), Å | \(E\), kV/cm | \(\Delta\lambda\), Å |
| 10 | \(+0.4\) | 9 | 0 | 7 | \(+1.0\) | 7 | 0 |
| 14 | \(+1.2\) | 12 | \(-0.7\) | 12 | \(+1.8\) | 10 | \(-0.5\) |
| 18 | \(+2.1\) | 14 | \(-1.1\) | 13 | \(+2.0\) | 12 | \(-0.9\) |
the basis of this table one can obtain an idea of the magnitudes of the displacement of exciton lines in an electric field. From these data, however, it is impossible to draw conclusions about the laws obeyed by the displacements of the lines in the field: whether the dependence of the displacement on the field is linear, as in the hydrogen atom, or quadratic. To resolve this question, further investigations with a spectrograph of greater dispersion and with deeper cooling of the crystal are necessary.
The results of the experiments described above show that the Stark effect on exciton lines in a Cu\(_2\)O crystal differs strongly from the Stark splitting of the lines of the hydrogen atom. This indicates the inadequacy of the hydrogen-like exciton model proposed by Mott.
b) Nonuniform electric field
The observations described above of the effect of an external field on the lines of the Cu\(_2\)O absorption spectrum were carried out by us in a “uniform”*) electric field. In order to investigate more thoroughly the discovered
*) Insofar as this is attainable in Cu\(_2\)O crystals with the electrodes shown in Fig. 12, a.
Panel b). Visible labels: $n=2$, $n=3$, $n=4$; axis marked $E$.
Panel a). Visible labels: $n=2$, $n=3$, $n=4$; axis marked $E$; mark $0\ \mathrm{cm}^{-1}$.
Fig. 15. Spectrograms of the Stark effect in an inhomogeneous electric field \(E\) on the lines of the yellow exciton series in a \(\mathrm{Cu_2O}\) crystal at \(T = 77.3^\circ\ \mathrm{K}\): \(a\)—with the aid of tips; potential difference \(8\ \mathrm{kV/cm}\); \(b, c, d, e\)—with the aid of linear electrodes; potential differences respectively \(10, 12, 15, 20,\) and \(25\ \mathrm{kV/cm}\).
phenomena known to us, it is better and more convenient to trace the behavior of exciton lines in a gradually increasing electric field, and, in order to reach large fields, we undertook investigations of the Stark phenomenon in inhomogeneous electric fields[^43].
As is known, observation of the Stark effect in inhomogeneous fields in atomic spectra is carried out by the Lo-Surdo method[^44] at the cathode of a discharge tube, where there is a strong potential drop and large gradients of the electric field. To create strong inhomogeneous electric fields with large gradients in a cuprous-oxide crystal, we made use of a point. One of the electrodes by means of which a potential difference was applied to the $\mathrm{Cu}_2\mathrm{O}$ plate was linear and sufficiently long; the other was a point. The electrode scheme is shown in Fig. 12, b. With such an arrangement of the electrodes the potential drop was to be concentrated near the point, where the field was to be strong and inhomogeneous.
By forcing light to pass through the $\mathrm{Cu}_2\mathrm{O}$ crystal near the point and projecting the point onto the slit of the spectrograph, we could observe the exciton absorption spectrum in the region of the strong electric field near the point. With the aid of this method we were able to observe[^43] in the exciton spectrum the characteristic pattern of splitting of lines in an electric field, similar to that observed in atomic spectra by the Lo-Surdo method. In Fig. 15, a is given a photograph of the exciton spectrum of $\mathrm{Cu}_2\mathrm{O}$ in an inhomogeneous electric field, obtained with the aid of a point (the potential difference between the electrodes is equal to $8\ \mathrm{kV/cm}$).
We succeeded in observing an even more distinct picture of the Stark phenomenon in an inhomogeneous field in special cases. While studying the Stark phenomenon in various $\mathrm{Cu}_2\mathrm{O}$ specimens with electrodes as shown in Fig. 12, a (without a point), we found on the spectrogram that the splitting of the exciton lines in some specimens was not the same along the line $aa$ projected onto the slit of the spectrograph. This indicated an inhomogeneity of the electric field in some $\mathrm{Cu}_2\mathrm{O}$ plates along the line $aa$, connected with certain peculiarities of the structure of the plates.
In one of these $\mathrm{Cu}_2\mathrm{O}$ plates these properties were especially pronounced, and the Stark phenomenon was very large. In Fig. 15, b, c, d, e, f are shown spectrograms obtained by us with this $\mathrm{Cu}_2\mathrm{O}$ plate at potential differences on the electrodes of, respectively, 10; 12; 15; 20; and $25\ \mathrm{kV/cm}$.
The very strong electric fields obtained in the $\mathrm{Cu}_2\mathrm{O}$ crystal by the method of the inhomogeneous electric field allowed us to establish new facts that had escaped observation in “homogeneous” electric fields. We succeeded in detecting the influence of the electric field on the second and first members of the exciton series ($n=2$ and $n=1$). In strong fields the line $n=2$ weakens in intensity, broadens somewhat, and shifts toward the red part of the spectrum, as is seen in Fig. 15, e from the bending of the end of the line $n=2$ at the lower edge of the spectrogram.
We could not detect, even in strong electric fields, either a shift or a splitting of the first member of the exciton series $n=1$, despite its extreme narrowness. Only at the lower edge of the spectrum, corresponding to regions of the $\mathrm{Cu}_2\mathrm{O}$ crystal located near the point, where the field is strongest, did we find a considerable increase in the intensity of the line $n=1$. In this respect the behavior of the line $n=1$ in an electric field is similar to that of lines forbidden by selection rules.
As is evident from the spectrograms in Fig. 15, in the spectrum of the exciton in an inhomogeneous electric field all the characteristic features of the Stark effect appear with particular clarity and vividness.
1) The appearance of components of the lines of the series and an increase in the distances between them with increasing field.
2) The displacement of the lines into the red and violet parts of the spectrum as the field increases.
3) The broadening of the lines with increasing field.
4) The appearance of new forbidden lines, an increase in their intensity, and their displacement into the red or violet parts of the spectrum with increasing field.
5) The gradual disappearance of the higher members of the exciton series as the field increases.
These phenomena are so characteristic and so similar to the splitting of lines in the spectra of free atoms in an electric field that they leave no doubt that the Stark effect is involved here, although, as far as we know, this phenomenon has hitherto never been observed in the spectra of crystals in an external electric field.
§ 10. ZEEMAN EFFECT ON THE LINES OF THE YELLOW EXCITON SERIES
As is known, the splitting of spectral lines of free atoms in a magnetic field depends on the resultant orbital angular momentum quantum number of the electrons \(L\), the resultant spin vector \(S\), and the total angular momentum quantum number \(J\), but does not depend on the values of the principal quantum numbers \(n\) (Preston’s rule). It follows from this that the Zeeman splitting does not depend on the radii of the electronic orbits (the principal quantum numbers \(n\)). Therefore one should expect that the Zeeman splitting of the lines of the exciton spectrum, unlike the Stark effect, will be of the same order of magnitude as the splitting of lines in atomic spectra. The distance between the extreme components of a normal triplet in atomic spectra is
\[ \Delta \nu = \frac{H}{10700}\ \text{cm}^{-1}, \]
where \(H\) is the magnetic-field strength. In fields of about 30,000 oersteds the Zeeman splitting amounts to only about \(3\ \text{cm}^{-1}\), or approximately \(1\ \text{Å}\) for visible light.
Experimental investigations of the Zeeman effect on exciton lines were carried out by us with a spectrograph having a diffraction grating with a dispersion of \(4\ \text{Å}/\text{mm}\). The investigations were performed both in unpolarized light and in polarized light, using polarizing prisms that transmit light with the electric vector directed perpendicular or parallel to the direction of the magnetic field. All these experiments were carried out at the temperature of liquid nitrogen (\(T = 77.3^\circ\text{K}\)).
Under these conditions we observed \(^{45}\) splitting of the first member \(n = 1\) of the yellow exciton series in a magnetic field. The clearest picture of the Zeeman splitting was obtained in investigating the phenomenon in polarized light. With such a position of the polarizing prism-analyzer, when the electric vector is perpendicular to the direction of the magnetic field (\(\sigma\)-components), the Zeeman splitting of the first member of the yellow exciton series has the form of a doublet, the components of which are situated symmetrically with respect to the position of the line \(n = 1\) in the absence of a magnetic field. With the other polari-
zation (π-component) an absorption line is observed, located in the same place as in the absence of the field.
Figure 16 shows photographs of the Zeeman splitting of the first member of the yellow exciton series. The magnitude of the splitting depends on the magnetic-field strength \(H\). Table VII gives numerical values of the distances between the components of the doublet of σ-components, obtained by us in magnetic fields of 26,000 and 32,000 oersteds. From the data of Table VII it is seen that the numerical values of the Zeeman splitting for the exciton are the same as for free atoms. The splitting \(\Delta \nu\), as it should be for the Zeeman effect, is proportional to the magnetic field \(H\).
Fig. 16. Spectrogram of the Zeeman splitting (σ-component) of the first member of the yellow exciton series in a Cu\(_2\)O crystal at \(T=4.2^\circ\) K in a magnetic field of 32,000 oersteds.
In a recently published paper, Samoilovich and Korenblit \(^{25}\) considered the magneto-optical properties of the exciton. Adopting Mott’s representation for the exciton (the quasipositronium model), the authors derived a formula for the Zeeman splitting in the exciton spectrum for the case of strong fields, when the Paschen–Back effect takes place:
\[ \left. \begin{aligned} \Delta \nu &= \frac{\Delta m}{M}\,\frac{eH}{2\mu c}\,\Delta m_l;\\ \Delta m_l &= 0,\ \pm 1,\ldots, \end{aligned} \right\} \tag{3} \]
where \(e\) is the electron charge; \(c\) is the velocity of light; \(H\) is the magnitude of the magnetic field; \(M=\mu_1+\mu_2\);
\[ \Delta m=\mu_1-\mu_2;\qquad \mu=\frac{\mu_1\mu_2}{M}, \]
where \(\mu_1\) and \(\mu_2\) are the effective masses of the electron and the hole.
The Zeeman splitting of the exciton lines, on the one hand, and the optical spectrum of the exciton without the influence of a magnetic field, on the other hand, make it possible to determine separately the effective masses \(\mu_1\) and \(\mu_2\) of the electron and the hole. Indeed, the Zeeman splitting makes it possible to determine the quantity
\[ \left(\frac{1}{\mu_1}-\frac{1}{\mu_2}\right), \]
while the constant of the serial dependence of the exciton lines (without a field) makes it possible to calculate the reduced mass of the exciton, and consequently the quantity
\[ \left(\frac{1}{\mu_1}+\frac{1}{\mu_2}\right), \]
from which \(\mu_1\) and \(\mu_2\) are determined.
Table VII
Magnitude of the Zeeman splitting of the first member of the yellow exciton series in Cu\(_2\)O (σ-components)
| \(H\), in oersteds | \(\Delta \lambda\), in Å | \(\Delta \nu\), in cm\(^{-1}\) | \(\dfrac{\Delta \nu}{H}\,10^6\) |
|---|---|---|---|
| 26,000 | 0.9 | 2.4 | 92 |
| 32,000 | 1.1 | 2.9 | 91 |
Thus the effective masses of the electron and the hole can be determined from spectroscopic data alone.
If it is assumed that the Zeeman splitting \(\Delta \nu\) of the first member of the yellow exciton series in Cu\(_2\)O observed by us is the splitting of a normal triplet and that formula (3) can be applied to it, then, using the data for \(\mu\) obtained from the optical spectrum of the exciton, one can calculate the effective masses of the electron and the hole in a Cu\(_2\)O crystal. They were found to be \(\mu_1=0.54\) and \(\mu_2=0.60\).
Thus the values of the effective masses of the electron and the hole determined spectroscopically do not differ greatly from the data obtained by other methods for determining these quantities.
§ 11. STUDIES OF THE STRUCTURE OF THE EDGE OF THE FUNDAMENTAL ABSORPTION IN OTHER CRYSTALS
The existence of two hydrogen-like series in the absorption spectrum of cuprous oxide naturally raises the question: is a similar structure of the fundamental absorption edge observed in other crystals? Absorption spectra of crystals near the long-wavelength boundary of the fundamental absorption, with high dispersion and at low temperatures, have as yet been studied very little. Data are available for only a few crystals.
Thus, in the spectrum of cadmium sulfide, at the absorption edge at liquid-helium temperature (4.2° K), a complex structure was discovered1, consisting of a group of 11 narrow lines and 4 broader bands (Fig. 17). The frequencies of these lines and bands are compared in Table VIII (the band edges are indicated in braces). The absorption spectrum of CdS is almost as complex as that of cuprous oxide, which indicates a large number of electronic levels in the CdS crystal. The picture is further complicated by the fact that the complex structure of the CdS absorption edge depends on the state of polarization of the absorbed light.
Table VIII
Position of lines and bands near the edge of the fundamental absorption in the spectrum of a CdS crystal
| Wavelength in Å | Frequencies in cm\(^{-1}\) | Energy in eV |
|---|---|---|
| 4889.5 | 20 452 | 2.5348 |
| 4870.8 | 20 530 | 2.5448 |
| 4869.2 | 20 537 | 2.5454 |
| 4868.2 | 20 541 | 2.5459 |
| 4867.1 | 20 546 | 2.5465 |
| 4866.2 | 20 549 | 2.5469 |
| 4865.6 | 20 552 | 2.5472 |
| 4864.9 | 20 555 | 2.5476 |
| 4863.5 | 20 561 | 2.5483 |
| 4861.8 | 20 567 | 2.5492 |
| 4858.0 | 20 585 | 2.5512 |
| 4854.2 | 20 601 | 2.5532 |
| 4853.5 | 20 603 | 2.5536 |
| { 4836 | { 20 679 | { 2.563 |
| { 4820 | { 20 748 | { 2.572 |
| { 4817 | { 20 759 | { 2.573 |
| { 4815 | { 20 770 | { 2.574 |
| { 4809 | { 20 793 | { 2.577 |
| { 4800 | { 20 832 | { 2.582 |
| { 4793 | { 20 862 | { 2.586 |
| { 4791 | { 20 872 | { 2.587 |
Furthermore, through our experiments with B. S. Razbirin and M. A. Yakobson it was established2 that the group of fine, weak, narrow lines on the long-wavelength side of the strong absorption bands is highly variable from crystal to crystal. The number of lines, their frequencies, intensities, and widths are different for different crystals, as can be seen from Fig. 18. By contrast, the strong absorption bands are stable and almost do not change their position from crystal to crystal.
Experiments with CdS crystals of various thicknesses established that these bands become stronger with increasing thickness, whereas the fine variable lines do not change in intensity when the crystal thickness is varied by a factor of 100–1000. Further, upon prolonged annealing of the crystal the fine lines disappeared from the spectrum, while the bands remained. These experiments show that the fine lines are associated with electron levels located at the surface of the crystal and are probably caused by impurities or lattice defects on the surface. The strong absorption bands, on the contrary, are associated with the bulk of the crystal and correspond to the levels of the fundamental lattice of CdS. We associate their origin with excitons in the CdS crystal, which is confirmed by the hydrogen-like serial dependence between the frequencies of these bands discovered by us with B. S. Razbirin3.
In addition, we found in some CdS crystals strong distortions of the spectral lines, as can be seen in Fig. 19. We established by special experiments that these phenomena are caused by inhomogeneous deformations of CdS crystals glued to glass plates, arising during their cooling because of the difference between the coefficients of thermal contraction of CdS crystals and glass substrates.
Figure labels: CdS, \(T = 77.3^{\circ}\mathrm{K}\); CdS, \(T = 4.2^{\circ}\mathrm{K}\).
Fig. 17. Spectrograms of the structure of the fundamental absorption edge in a CdS crystal at \(T = 4.2^{\circ}\mathrm{K}\) and \(T = 77.3^{\circ}\mathrm{K}\).
Fig. 18. Diversity in the line absorption spectra of different CdS crystals.
Further, the structure of the absorption edge was discovered in a crystal of mercuric iodide (\(\mathrm{HgI_2}\))\(^*\). In this crystal, at \(T = 77.3^\circ \mathrm{K}\), a narrow intense line \(\lambda = 5330\ \text{Å}\) is observed on the long-wavelength side of the edge of the fundamental absorption, which at \(T = 77.3^\circ \mathrm{K}\) is located near \(\lambda = 5255\ \text{Å}\). At the temperature of liquid hydrogen, Nikitin and co-workers found this same narrow line at \(\lambda = 5298\ \text{Å}\), and also a number of weak lines near the absorption edge.
Interesting results were obtained by Nikitin and Perni \(^{50}\) with a lead iodide crystal (\(\mathrm{PbI_2}\)). At the temperature of liquid helium (\(T = 4.2^\circ \mathrm{K}\)) they found a group of four narrow lines at the absorption edge of this crystal: \(\lambda_1 = 4947\ \text{Å}\); \(\lambda_2 = 4896\ \text{Å}\); \(\lambda_3 = 4838\ \text{Å}\), and \(\lambda_4 = 4845\ \text{Å}\), with the boundary of continuous absorption near \(\lambda = 4829\ \text{Å}\).
Fig. 19. Spectra of strongly inhomogeneous CdS crystals.
The study of the properties of the exciton spectrum described in the preceding chapters, and in particular the hydrogen-like absorption spectrum in \(\mathrm{Cu_2O}\) crystals, naturally leads to the question of phenomena in which other properties of the exciton might manifest themselves. The study of the absorption spectrum provides information mainly about the process of exciton formation in the crystal, about their generation under the influence of light. No less interesting are the phenomena connected with the life of excitons in the crystal lattice, with their motion, migration through the crystal, and also the processes of disappearance, the annihilation of excitons.
Immediately after the discovery of the exciton spectrum in the \(\mathrm{Cu_2O}\) crystal, we made attempts to detect the emission of a hydrogen-like series in the luminescence of \(\mathrm{Cu_2O}\). The appearance of a hydrogen-like series in the emission—
\(^*\) In the work of Nikitin, Kutor, Ziskind, and Perni \(^{49}\), as well as in the experiments of B. P. Zakharchenya and the author.
would point directly to annihilation of the exciton through the emission of light, i.e., to the process inverse to the excitation of excitons under the action of a light quantum. In the case of \(\mathrm{Cu_2O}\) the experiment gave a negative result. We did not detect radiation of the hydrogen-like series, although the known luminescence in the near infrared part of the spectrum was observed with high intensity.
Experiments with other crystals, carried out in our laboratory at the Physico-Technical Institute in Leningrad, showed that at low temperatures the complex structure of the edge of the fundamental absorption is observed not only in absorption, but also in emission. This interesting fact was discovered by us in studies carried out together with M. A. Yakobson and A. A. Kaplyanskii, first for \(\mathrm{CdS}^{51}\) and \(\mathrm{HgI_2}^{52,53}\), and then was confirmed by studies with V. V. Sobolev, B. S. Razbirin, and R. Shakhmamet’ev on crystals of \(\mathrm{CdSe}^{54}\), and \(\mathrm{CuI}\) and \(\mathrm{CuBr}\).
It turned out that when a crystal is excited by the ultraviolet light of a mercury lamp \((\lambda = 3665\ \text{Å})\), radiation is observed whose spectrum consists of narrow lines. Some of these lines almost coincide with the corresponding absorption lines. Thus, the luminescence radiation here has a resonance character. The state of polarization of the luminescence lines of single crystals is the same as the state of polarization of the corresponding absorption lines.
In the luminescence of \(\mathrm{CdS}\) crystals, Grillo discovered\(^{55}\) a group of narrow lines that forms a hydrogen-like series.
In Fig. 20 are presented the absorption and emission spectra of \(\mathrm{HgI_2}\) single crystals. The small displacement of the emission lines relative to the absorption lines may be attributed to the change in the equilibrium state of the crystal lattice that must occur by the moment of emission after its absorption of light.
Fig. 20. Spectrograms of absorption and luminescence of \(\mathrm{HgI_2}\) single crystals in the region of the edge of the fundamental absorption of the crystals \((T = 4.2^\circ\ \mathrm{K})\): \(a\)—luminescence, short exposure; \(b\)—absorption at a small absorption coefficient and in polarized lines; \(c\)—absorption at a large absorption coefficient; \(d\)—luminescence, long exposure.
Recently the results of our investigations on luminescence have been confirmed by observations of other authors.
It seems to us that the emission of lines in luminescence, almost coinciding in the spectrum with the strong absorption lines, apparently,
can be regarded as the emission spectrum of the exciton. The most convincing evidence in favor of this is the observation in the luminescence of CdS of three lines which are observed in absorption and whose frequency satisfies the hydrogen-like dependence^48 (see above). The appearance of these lines in luminescence confirms the assumption that these lines are caused by annihilation of the exciton with emission of light.
Emission of exciton lines is observed when the crystal is irradiated with light from the region of continuous absorption (both when excited by the lines of a mercury lamp, \(\lambda\lambda 3660\), isolated by means of a Wood “black” filter, and when excited by the continuous spectrum of incandescent lamps). This leads to the idea that two mechanisms of exciton excitation in the crystal are possible. One is direct, with excitation of an electron and a hole from the filled band without their separation from one another to the exciton level through direct absorption of light or thermal energy (Fig. 21, process \(a\)). The other is indirect, first with separation of the electron from the hole and its transfer into the free band through absorption of light from the region of the fundamental band, and then with its recombination with the hole into an unfilled band, at the exciton level (Fig. 21, process \(b\)).
In this connection the thought arises that direct recombination of an electron and a hole in a crystal is in general unlikely. Recombination of an electron and a hole occurs in the crystal not directly, but through the intermediate formation of an exciton. A free electron and hole in a crystal, approaching one another, first form an exciton and may exist for some time bound in the exciton state, and then their recombination occurs from the exciton levels into the filled band. In this connection it should be noted that a band-to-band transition with emission of light, as is known, is forbidden by the selection rules.
In addition to the luminescence experiments, we carried out certain investigations of photoconductivity which established a connection between photoelectric phenomena and the structure of the edge of the fundamental absorption.
We obtained^56 data (at \(T = 77.3^\circ\mathrm{K}\)) for the absorption spectra of a number of crystals: \(\mathrm{MoO_3}\), \(\mathrm{Bi_2O_3}\), \(\mathrm{TiO_2}\), \(\mathrm{V_2O_5}\), \(\mathrm{As_2S_3}\), in which the structure of the edge of the fundamental absorption is not observed at all, and the absorption curve at the edge has a monotonic course. It is noteworthy that in the latter crystals the internal photoelectric effect is absent (or very weak). On the contrary, in crystals where an exciton structure of the absorption edge with narrow strong lines is observed (\(\mathrm{Cu_2O}\), \(\mathrm{CdS}\), \(\mathrm{HgI_2}\), \(\mathrm{PbI_2}\)), the internal photoeffect is very strong, and its spectral curve has a maximum located at the absorption edge, where the exciton absorption lines are observed.
From this one may conclude that, for the occurrence of a large internal photoeffect, it is apparently necessary that there exist a structure at the long-wavelength edge of the fundamental absorption with sharp, strong, discrete absorption lines. This points to a connection of the exciton spectrum with photoelectric phenomena: in crystals where excitons can be excited by light, a strong internal photoeffect can arise.
This conclusion is confirmed by the fine structure, discovered by us together with A. A. Kaplyanskii and B. V. Novikov^52,53,57, of the spectral distribution curve of the internal photoeffect in \(\mathrm{HgI_2}\) and \(\mathrm{CdS}\) crystals, with narrow photocurrent peaks corresponding to fine absorption lines.
The experiments here also lead to two possible processes for the generation of photocurrent, associated with the exciton mechanism of photoconductivity in accordance with the two possible paths of exciton formation in the crystal (Fig. 21, \(a\) and \(b\)).
The results described above of studies of the absorption spectra of crystals of various compounds are still too meager for general conclusions to be drawn from them. Nevertheless, in our opinion, they do make it possible to state certain conclusions.
The few data on the absorption spectra of crystals that are available show that there is no obvious similarity among the spectra of different crystals. Well-pronounced hydrogen-like series of lines (as in cuprous oxide) near the absorption edge are, as a rule, not observed in the spectra of crystals. In some cases the spectra of crystals consist of only two or three lines (at the temperature of liquid nitrogen).
An explanation of this may be sought in the fact that excitons in some crystals may be polarizing; then, according to the theory of Dykman and Pekar ^31, the frequencies of the exciton spectrum need not satisfy the series dependence of the hydrogen-like atom. It must be noted, however, that in this case the exciton lines must be broad. Further, it follows from the theory of Dykman and Pekar that in some cases the lower states of the exciton may be nonpolarizing, while the upper, more excited states may be polarizing. This will lead to the first (long-wavelength) exciton lines being narrow, and the following (short-wavelength) ones being broad; moreover, the hydrogen-like character of the spectrum will not hold. Since the broad lines of the upper states, overlapping one another, may merge with the region of continuous absorption of the crystal, in experiment essentially only the first narrow exciton lines will be observed.
Fig. 21. Two mechanisms of exciton excitation in a crystal.
The results of the theory of Dykman and Pekar ^31 are obtained from consideration of Mott excitons with allowance for the interaction of the electron and hole with the vibrations of the ions. From this one obtains a polarizing exciton and its deviations from hydrogen-likeness. A nonpolarizing exciton according to the Dykman–Pekar theory must be hydrogen-like. It can hardly be doubted that excitons in a CuO\(_2\) crystal are nonpolarizing. In analyzing the absorption spectrum of cuprous oxide it was noted that deviations from hydrogen-likeness are also observed in the spectrum of the nonpolarizing exciton of Cu\(_2\)O.
On the basis of all the experimental material available at present, it seems to us—as already noted above—that the exciton should be regarded as a system resembling a complex atom more than the hydrogen atom. The hole in a crystal cannot be represented as a point charge. The complex structure of the exciton, and consequently also the structure of the absorption edge, depend on the substance, on the atoms and ions forming the crystal lattice, and are evidently connected with the structure of the crystal.
The optical spectrum of excitation of an exciton may then be complicated and will resemble the spectrum of a complex atom (or ion) more than the spectrum of the hydrogen atom. This is clearly manifested both in cuprous oxide and still more clearly in the spectra of other crystals investigated.
The representation of the exciton as a hydrogen-like system of a hole and an electron should be regarded only as a first approximation, correctly, but schematically, conveying only the most general features of the phenomenon.
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