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II. SOLID STATE. SURFACE PHYSICS.THEORY OF
CORE EXCITONS IN SOLIDS
M. Altarelli
To cite this version:
I SOLID STATE. SURFACE PHYSICS.
THEORY OF CORE EXCITONS IN SOLIDS (*)
M. ALTARELLI (**)
Department of Physics and Materials Research Laboratory,
University of Illinois at Urbana-Champaign, Urbana, Illinois, 61801 U.S.A.
Resume.
-
La comprehension acfuelle des effets d'interaction electron-trou dans les spectres de photoabsorption en couches internes des solides non mCtalliques est passee en revue. Dans les cris- taux ayant une polarisabilite electronique faible tels que les halogknures alcalins, les gaz rares, etc., cette interaction est plus forte que les effets de l'ktat solide, si bien qu'il existe une ressemblance exacte avec les spectres d'absorption atomique ou ionique correspondants. Les rtsultats rtcents sur le spectre au seuil K de Li' dans LiF qui semblent ecarter la presence de transitions interdites a des energies de photon aussi basses que 54 eV sont discutes. Dans les semiconducteurs l'effet d'ecran des electrons de valence vis-a-vis du trou interne rkduit considerablement l'interaction electron-trou. Du point de vue theorique, la description de I'etat final est, par analogie, celle des niveaux donneurs d'impuretks. Cette image semble &tre fondamentalement correcte bien que de nouvelles etudes experimentales et theoriques soient necessaires pour interpreter les resultats recents obtenus en spectroscopie modulke. Les effets des porteurs libres sur le spectre de cceur sont aussi discutes en liaison avec les travaux theoriques et experimentaux sur les discontinuites X dans les mktaux.Abstract. - The present understanding of electron-hole interaction effects on (bulk) inner-shell photoabsorption spectra of non-metallic solids is reviewed. In crystals with low electronic polariza- bility, such as the alkali halides, solid rare gases, etc ..., this interaction is stronger than any solid- state effects, so that a close similarity exists with the corresponding atomic or ionic absorption spectra. Recent results on the controversial Li' K-spectrum of LiF will be discussed, which seem to rule out the presence of forbidden transitions at photon energies as low as 54 eV. In semiconductors, valence electron screening of the inner hole considerably reduces the effective electron-hole inter- action. Here the theoretical analogy for the description of the final state is to donor impurity states. This picture appears to be basically correct, although more experimental and theoretical work is called for to give a consistent interpretation of recent photoemission and modulation spectroscopy experiments. The effect of free carriers on core spectra is also discussed, in connection with recent experiments and with theories of X-ray edges in metals.
l . Introduction. - The failure of one-electron models to provide more than a qualitative under- standing of the optical properties of solids in the far U.V. and X-ray regions, of interest here, has long been recognized. This conclusion was already empha- sized in Parratt's 1959 review article [l], and it is reinforced by more recent surveys of data [2]. This failure is particularly evident near thresholds marking the onset of transitions from a given core level, where usually sharp structures, either in the form of line resonances or steep, atomic-like thresholds are observed. This, of course, does not necessarily mean that the one-electron description is adequate far from thresholds, but only that its shortcomings are not as obvious there.
In this article we shall focus our attention on the
(*) Work supported by NSF under grants DMR-76-22352 and
DMR-76-01058.
(**) Address after September 1, 1977 : Istituto di Fisica N G. Mar- coni s, UniversitA di Roma, 00185 Roma, Italy.
problem of including the electron-hole final-state interaction, or excitonic effect, in the theory of optical response near core-to-conduction thresholds of non-metallic crystals. It may not be superfluous to remind that exciton states, familiar from the well- developed spectroscopy of solids in the fundamental gap region, are not one-electron states, but true many-body states of the whole system. An exact calculation of the exciton spectrum would amount to obtaining the two-particle Green's function by solving a Bethe-Salpeter equation for the fully interacting system [3, 41. It is clear that this task is hopeless in general, and one is forced to consider various approximations, valid in some limiting cases. These limiting cases (Frenkel excitons, Wannier excitons, etc.) have been used for a long time [5]
in describing the optical response near the funda- mental gap, and they can be, often in a rather straight- forward manner, adapted to the case of core level spectroscopy.
C4-96 M. ALTARELLI
There are however a few differences which is important to keep in mind. First of all, a state with a hole in one of the core levels is never stationary, as it is always coupled, via the electron-electron inter- actions, to nearly degenerate electronic configurations. Therefore, it is not strictly possible to describe exact eigenstates of the system in terms of a hole in a single given core band and an electron in the conduction band. However, it is often possible to include these effects as a hole lijetime, in a phenomenological way. Also, there are no truly bound exciton states, but at best sharp, well defined resonant states. Another difference between fundamental gap and core excitons has to do with screening. If we consider the basic electron-hole interaction, it is readily seen that the Coulomb part is screened at a frequency of the order of the exciton binding energy, i.e., essentially by the static dielectric function, whereas the exchange is screened at a frequency of the order of the relevant energy gap [6]. For the case of band gap excitons, this is still a small frequency, but for deep core level excitons it is larger than the plasma frequency, so that screening is totally ineffective in electron-hole exchange matrix elements.
We shall now proceed to review recent work on specific materials and to assess our level of under- standing. The rest of the paper is organized as follows :
section 2 is devoted to insulators, section 3 to semi- conductors and finally section 4 contains concluding remarks.
2. Insulators : the atomic limit. - In this section we shall consider the case of large-gap insulators. Prototype materials of this category are the closed shell solids such as the alkali halides and the solid rare gases. The polarizability of these crystals is very low, and when a hole is created in one of the cores, the major relaxation effects are intra-atomic, so that the atomic or ionic spectrum of the excited species should provide a good first-order approximation to the situation in the solid (this applies only to the low-lying final states, with wave-functions sampling a region of atomic dimensions). These ideas were applied to argon, at the Hamburg 1974 V.U.V. conference, with reasonable success [7], and we now concentrate our attention on the alkali halides, about which there has been considerable controversy. The simplest representative of these compounds is LiF. Consider the K-absorption threshold of the Li+ ion [8] for the family of the four Li halides. The similarity of the four spectra is evident, the most obvious common feature being the sharp peak, whose position, within 2 eV, is independent of the halogen ion. The free Li' ion absorption spectrum [9], furthermore, has a prominent allowed transition (1s' 'S + 1s 2p 'P) at 62.3 eV ; it also has a forbidden one (ls2 'S -+ 1s 2s 'S) at 60.8 eV, in fair agreement with the low-energy shoulder particularly evident in LiF and LiI. It is therefore self-suggesting to
ascribe the main peak to a one-site allowed exciton centered on the Li' ion, and the low-energy shoulder to the forbidden s exciton, which, in the solid, becomes weakly allowed by odd-parity phonon assistance. Structures on the high energy side of the main peak, however, are more sensitive to the halogen ion, as the effective radius of the final state increases, and the analogy to the ionic spectrum is not as helpful. This simple interpretation of the threshold absorp- tion of LiF is not universally accepted. Using a first- principle band-structure calculation as a starting point, and superimposing a large correction for the localized nature of the hole, Kunz et al. [l01 suggested that the latter should crudely amount to a down- ward, nearly rigid, shift of the correlated conduction band by .about 15 eV. This largely ad hoc procedure led the authors to the prediction of a weak forbidden
absorption tail marking the onset of transitions from the K level around 54 eV. Clearly, the limitation of this approach is in considering band structure features as relevant at all, even after acknowledging very large electron-hole effects. Nevertheless, the prediction of the 54 eV onset received claims of experimental support [l 1, 121.
THEORY OF CORE EXCITONS IN SOLIDS C4-97
Comparison of theory and experiment jor transition
energies in the Li+ K-spectrum of LiF, from refe-
rence [14]. Values in parentheses are for corresponding
free ion transitions, from reference [9]. All energies
are in eV.
Theory Exp.
- -
Li' 1s --+ 'S exciton 61.3 60.8 (60.8)
Lif Is --+ 'P exciton 62.2 61.9 (62.3)
Li' 1s -+
r,,
band edge 63.3 64.4 tive features of the halogen ion spectra [15]. It is generally, observed that S-core levels of the halogen do not give rise to sharp structures. This is easily understood by observing that, e.g., a Cl- ion, from which a deep s hole is removed, becomes an object extremely similar to an argon atom, which cannot bind the ejected p-like electron. When the final electron is in an S-like state, as in the L,,, spectrum of Cl-, on the other hand, a contribution to the moderate exciton binding energy (1-2 eV according to Ref. [15]) comes from the Madelung c c l g p of themetal ions, in analogy to the binding of the F-center. It is, however, clear that the final electron strongly overlaps the cations, so that an approach similar to that of reference [l41 seems appropriate to describe the halogen spectra. We are aware of no quantitative attempt to describe these spectra in detail.
3. Semiconductors : the donor limit. - &e now proceed to discuss the more interesting case of semi- conductors. They have received considerable experi- mental attention in the last few years, and the high- resolution techniques of modulation spectroscopy were recently applied to core transitions as well. The important feature of these materials is the large polarizability of valence electrons, which, since the electron-hole Coulomb interaction is statically screen- ed, leads to a reduction of Coulomb effects of the order of 10, the typical value of the static dielectric constant in the semiconductors of interest. After the publication of the first data [l61 on the
~,,,,,
threshold of Si, near 100 eV, Dexter and the author [l 71 observed that the photoexcited electron, subject to the screened Coulomb potential of the very loca- lized positive hole, should reach final states closely resembling those of electrons in the field of phosphorus donor impurities (P has the same core as Si, therefore it introduces an impurity potential closely approxi- mated by a point charge). In particular, the ground state is expected to be a bound exciton, with a bindingenergy of approximately 0.04 eV, the binding energy
in Si : P being 45 meV. There are two effects that tend to reduce the binding energy in the core exciton case with respect to the donor case. One is the finite extent of the hole wave function [18], and the other is the repulsive electron-hole exchange interaction.
Both these effects may, for the ground state, be of order 10-20
%,
in some cases, so that they are not completely negligible in principle ; however, on the scale of the experimental accuracy, they can be ignored, and the similarity between core exciton and the better investigated donor impurity levels can be exploited.To test the validity of this approach, we first consider the experimental data of Aspnes et al. [l 9,201 on the electroreflectance of Gap, GaAs, GaSb in the region corresponding to transitions from the G a 3d core levels.
The prominent feature in these spectra is associated with transitions to the X minima of the conduction band. If the relative position of one electron levels is established with the help of XPS data, the binding
energy of the ground state excitons at the X point
can be inferred [20], although with very large uncer- tainties, and are shown in table 11. When UPS data are used 'for the same procedure [21], slightly larger values are obtained, for an additional binding of 100 meV or so.
On the same table, the experimental binding energies of donor states are listed. In the case of Gap the X points are the absolute minima of the conduction band [22], so that the donor states attached to them are true bound states and are experimentally well known. Germanium has the same core as the gallium atom it replaces, so that this is the case we are inte- rested in. In the case of GaAs and GaSb the X minima are not the absolute ones, and the observation of donor states associated with them is more indirect. Also, it is important to notice that we have been unable to locate data on Ge donors, and we list the information available on other donors. Those reported for GaSb are on the Sb site, so their value is just indicative of the order of magnitude.
It is apparent that, for the Ga 3d spectra, there is a reasonable correspondence between donor and exciton binding energies. I t seems unlikely that the accuracy of exciton binding measurements will improve significantly, so that the comparison can hardly be made more quantitative. However the point we wish to stress is that binding energies of hundreds of meV are what one should expect.
From the theoretical point of view, the calculation of binding energies of donors or core excitons asso- ciated with conduction band minima off the center of the zone presents considerable difficulties which have until recently gone unnoticed [23-251. These have to do with the problem of correctly accounting for the large coupling between the various degenerate valleys which is introduced by the potential of the core hole (or the impurity atom). In calculating the matrix element of the hole or impurity potential between Bloch waves in valleys i and j it is straight-
TABLE I1
Comparison of core exciton binding energies (inferred from comparison of photoemission and optical experiments), donor experimental binding energies (the impurity and, in parentheses, the atom it replaces, are indicated) and donor theoretical binding energy Wom Ref. E241 ; the ZnSe value is from standard one- valley eflective-mass theory)
Exciton binding Donor binding Donor binding Threshold from PES (meV) exp (meV) theory (meV) Gap Ga 3d-X, 170
+
150 (") Ge(Ga) 201 (h) (Ga) 90 GaAs Ga 3d-X, 90 f 200 (") Si(Ga) 70-100 (') (Ga) 5 1 GaSb Ga 3d-X, 90 f 250 (") Te(Sb) 80 (J), 185 (k) (Ga) 41Se(Sb) 200 ( l ) ZnSe Zn 3p+T6 100
f
300 ( b ) Ga(Zn) 28 (h) (Zn) 28 PbSe-
800 (C) Pb 5d+C6 PbTe W 0 ( d ) Si 2 ~ + A l 600 $- 200 (") P 45 ( h ) 47.5 900 & 400 (f) 300 ( g )c)
Ref. [20]. (") Ref. [29]. (') Ref. [45'].(b) Ref. [27]. (f) Ref. [30]. (3 Ref. [46].
(c) Ref. 1281, PES vs. reflectlvlty. ( g ) Ref. 1311.
(5
Ref. [47].(*) Ref. [28], PES us. electron energy loss. ( h ) Ref. [44]. ( l ) Ref. [48].
where G are the reciprocal lattice vectors,
0
is the Fourier transform of U(r) and the coefficients c(ki ; k j ; G) come from the Fourier expansion of the periodic part of $:, t,hkJ :The standard approximation, consisting in retaining only G = 0 and setting c(ki, k j ; 0) = 1, so that
was shown 123-251 to be completely unjustified, as G # 0 (umklapp) terms dominate the summation of equation (1). Notice that assumption (3) is equi- valent to replacing the Bloch function by plane waves. This total loss of information on the Bloch waves in the impurity or core exciton problem is particularly severe for 111-V compounds. Here the conduction band Bloch functions near the minima have different symmetry character around cation and anion sites, as emphasized by Morgan [26], with important differences in the structure of the impurity states, depending on the site. In our treat- ment [24] the Bloch function structure is fully retained and the resulting levels depend on the site on which the impurity or the hole is located. In the final column of table 11, we show results obtained for a point charge sitting on the Ga site, taking into account the momen- tum and frequency dependence of the dielectric function. We have however neglected local field effects, i.e. we have assumed that the relation between
the screened potential
0
and the bare potential,U,,
israther than the more correct expression involving the dielectric matrix :
=
C
C1(Ak - G, Ak - G') O b ( ~ k - G'). (5)G'
It is clear that theory underestimates the binding energy of the ground state, and it seems l~kely that this is due to the approximation of equation (4), which ignores the reduction in screening associated with the lower valence charge density at the Ga site. Another effect not contained in the calculation is the finite size and non-spherical symmetry of the d-core hole charge. Pantelides [l81 included similar effects in his calculations for Si, but unfortunately his results are not meaningful, as they are obtained under the assumption of equation (3).
Also shown in table I are results for the Zn 3p threshold in ZnSe, for which an exciton binding at the
r
conduction band minimum on the order of 100 f 300 meV, quite compatible with the observed 28 meV binding of the Ga donor, was estimated [27].A puzzling situation is found for the Pb 5d threshold
THEORY OF CORE EXCITONS IN SOLIDS C4-99
a nearly vanishing binding [28]. It should be men- tioned that these assignments rely on the identification of critical features in the optical or energy loss spectra above threshold, and their comparison with empirical pseudopotential band structure calculations. We shall briefly mention excitonic effects above threshold later. It is important to observe that the electronic dielectric constants of PbSe and PbTe at room temperature are 22.9 and 32.8 respectively, so that smaller effects are expected in these materials than in other semiconductors considered so far. It appears that more experimental and theoretical work is needed to clarify the nature of the threshold in these materials.
Finally, let us consider the 2p threshold of silicon. At least 3 different values of the exciton binding energy have been obtained by comparing the optical absorption data of reference [l61 with photoemis- sion results. Bauer et al. [29] recently obtained a value of 600
+
200 meV; an even larger value, 900+
400 meV, is quoted by Margaritondo and Rowe [30], whose procedure, unlike most of the data assembled in table 11, requires the knowledge of the position of the Fermi level with respect to the band edges in the bulk. However, earlier data [31] indicate a smaller binding energy, on the order of 300 meV. These large values have been interpreted as indication of a very localized final state, perhaps with atomic character, like those described in the preceding section for insulators. The same conclusion was inferred by a null result of a recent attempt to detect electroreflectance spectra at this soft X-ray threshold, and a lower limit of 300 meV was obtained for the binding energy [29].However, other experimental evidence is on the contrary indicating exciton effects on a much shallower energy scale. In figure 1, the spectra of the SiH, molecule and of crystalline Si are shown on a wide energy range above the L,,, threshold [32]. In the molecule, the localization of the lowest excited states results in the large pile-up of oscillator strength in the threshold region, dramatically evident in the prominent peak. No such prominent line is seen in the solid spectrum, although the broadening, as displayed by the slope of the absorption step, is about 0.1 eV. An exciton binding of 0.3 eV or more should necessarily result in a well resolved line. More- over, the oscillator strength being proportional to the electron probability density at the excited atom,
(
F(0)12,
the amount of oscillator strength in the threshold region gives a measure of the final state localization. The large peak in the molecular spectrum, which is about 2.5 eV wide, contains more oscillator strength than the first 6 eV of the semiconductor spectrum, supporting the claim of a less localized final state.Furthermore, the hypothesis of donor-like final states is successful in accounting for the threshold lineshape 1171 and the threshold height is in order
J I OoO [do ' l l L l ' 120 140 160 id0 ' 208 PHOTON ENERGY ( E V )
-
-5
1 ' 1 ' 1 ' 1 ' 1 0 =3 SILICON PHOTON ENERGY (€V)FIG. 1. - Absorption cross section (right scale) of gaseous SiH, and crystalline Si near and above the L,l~lll threshold (from Ref. [32]).
of magnitude agreement with the observed one, if the optical matrix element is obtained by OPW calculations, and if proper account is taken of the enhanced overlap of the electron and hole resulting from deviations from hydrogenic behaviour at short distances. In figure 2 the hydrogenic envelope function corresponding to a binding energy of 40 meV, used in reference [17], is compared to the ,more accurate ground state wavefunction obtained from the treat- ment of reference [24] ; the probability of electron- hole overlap is increased by a factor of seven.
If the broadening parameter deriving from core hole lifetime effects is extracted from the experi- mentally observed absorption spectrum, it appears very hard to accomodate a ground state binding energy larger than the one we propose.
It is clear therefore that more experimental and theoretical work is needed to clarify the relation among the various photoemission results, and between photoemission and optical spectroscopy.
Other interesting results exist for this threshold
M. ALTARELLI
FIG. 2. - (a) Th? ground state hydrogenic envelope function used in the lineshape calculation of reference [17]; (6) a more accurate
(Ref. [24]) ground state envelope function.
a series of samples with varying phosphorus doping content [32], spanning the insulator to metal transition (up to 1OZ0 cm-3, or rs
5
1). The absorption threshold is totally unaffected by the presence of free carriers, within the lifetime and experimental broadening, up to these high concentrations.There are two theories describing electron-hole final states effects for insulating and metallic crystals, respectively. For the insulating case, the Elliott exciton theory applies [33], which, e.g., was used in our discussion of the Si L,,, threshold lineshape [17]. For metals, as will probably be extensively discussed elsewhere in this Conference, there is an asymptoti- cally exact theory of threshold behaviour [34], whose range of validity is the subject of much contro- versy [35].
In metallic systems, as pointed out with great clarity by Friedel [36], optical absorption at a certain frequency results from the superposition of various different processes, because all electrons in the Fermi sea are affected by the hole potential, whose sudden appearance following the photon absorption can project, i.e. induce transitions of Fermi sea electrons into final excited states. In figure 3 three examples of processes leading to absorption are shown [36]. Figure 3a has nearly vanishing amplitude, as a result of Anderson's [37] orthogonality catastrophe, and when the energy is very close to threshold, this sup- pression is in part compensated by the shake-ofS processes of figure 3c. Also for energies within a cutoff, on the order of E,, from threshold, the addi-
tional replacement processes of figure 36 produce
an absorption enhancement which is the analog of the exciton enhancement of the insulating case. The complicated nature of the absorption process, which in essence is a result of the indistinguishability of electrons and of the Pauli principle, amounts to redistributing the oscillator strength over an energy scale of order E,. For absorption at energies farther from threshold than the cutoff, it is easy to see, from compketeness arguments, that shake-ofS processes (Fig. 3c) almost exactly compensate the orthogonality
FIG. 3. - Schematic representation of optical (solid line) and projection (dashed) transitions between one-electron levels in metallic systems (from Ref. [36]). (a) Direct; (b) replacement and (c) shake-off lowest order processes; (d) a higher order shake-off
and replacement process.
catastrophe (to all orders in the hole potential). The lowest order replacement transitions are not possible at such energies, so that, in first approxi- mation, the free carriers have no effect on the spectrum. A weak enhancement, corresponding to the Coulomb enhancement far above threshold in Elliott's theory, comes from higher order replacement and shake-off processes, such as that depicted in figure 3d.
The experimental conditions in reference [32] are such that no details on the scale of E,, always smaller than 100 meV, are discernible, due to the lifetime broadening, so that no effect of free carriers is detected, contrary to naive one-electron type of formulations, which predict a suppression of the absorption spectrum in presence of free-carrier screening. The failure of this approach has been also noticed at fundamental gaps : in optically pumped GaAs the Coulomb enhancement of absorption above the gap persists (Fig. 4) to high pumping intensity, corresponding to metallic conditions (r, < 1) [38]. The only effect of free carriers appears to be an increase of broadening ;
in the soft X-ray threshold case, this effect is obscured by the already present lifetime broadening.
THEORY O F CORE EXCITONS IN SOLIDS C4-I01 1.0 L.I. xxxxx 0.05 PO ( a ) 0 1.53 1.52 1.51 PHOTON ENERGY (eV)
TON)
PHOTON ENERGY (eV)
FIG. 4. - Optical density of GaAs in the fundamental gap region
under optical pumping (from Ref. [38]). (a) L.I. low intensity jumping, PO z 5 X 10' W/cm2. The dashed curve illustrates the
absorption due to parabolic bands in the absence of exciton effects.
(b) Data for high excitation intensity. Values of carrier densities are estimated to be :for Po,n = 7 X 10'5cm-3,for6 P,,, 5 X 10'6cm-3,
for 20 P,, n = 1.5 X IOl7 cm-3.
perhaps in systems where higher Fermi energies are obtainable, would be very interesting in elucidating the details of the evolution from exciton lines to X-ray edge singularities, which has been theoretically described only in unrealistic, oversimplified models [39].
[l] PARRATT, L. G., Rev. Mod. Phys. 31 (1959) 616.
[2] BROWN, F. C., in Solid State Physics, F . Seitz, D. Turnbull and
H. Ehrenreich, editors (Academic, New York), 1974, Vol. 29, p. 1.
[3] N O Z ~ R E S , P., Theory of Interacting Fermi Systems (Benjamin, New York), 1966, chapter 6.
[4] SHAM, L. J. and RICE, T. M., Phys. Rev. 144 (1966) 708. [5] See e.g. KNOX, R. S., Theory of Excitons (Academic, New
York), 1963.
[6] HOPFIELD, J. J., in Quantum Optics, R. J . Glauber, editor (Academic, New York), 1969, p. 340.
[7] ALTARELLI, M., ANDREONI, W. and BASSANI, F., in Proceedings of the IV International Conference on VUV Radiation Physics, Hamburg, 1974 (Pergamon Vieweg, Braunsch- weig), 1974, p. 357 ; Solid State Commun. 16 (1975) 143. [8] HAENSEL, R., KUNZ, C. and SONNTAG, B., Phys. Rev. Lett. U)
(1968) 262.
[9] MOORE, C. E., Atomic Energy Levels, NBS Circ. No. 467 (U.S.G.P.O. Washington, D.C.), 1949, vol. 1. [l01 KUNZ, A. B., MICKISH, D. J. and COLLINS, T. C., Phys. Rev.
Lett. 31 (1973) 756.
[l l] SONNTAG, B., Phys. Rev. B 8 (1974) 3601.
1121 GUDAT, W., KUNZ, C. and PETERSEN, H., Phys. Rev. Lett. 32 (1974) 1370.
4. Concluding remarks. - This final section is devoted to an attempt to ~ndicate open problems in the theory of core excitons, and topics where further work is needed.
It seems that progress in the following areas would greatly increase our ability to extract useful infor- mation from optical experiments :
(a) Final state effects above threshold. There is evidence that a one-electron picture cannot give a satisfactory interpretation of structure above threshold, even for the fundamental absorption region [40] and that final state effects (together with local field effects) may considerably alter the position and strength of features associated with critical points of the band structure.
(b) Surface exciton effects. There are recent reports of experimentally observed exciton effects at surfaces on insulators [41] as well as semiconductors [42]
and some theoretical work in this new direction [43]. The problem is very difficult, ,but it may provide very useful information about the electronic structure of surfaces.
(c) Finally, in view of the discussion of section 3,
it is perhaps useful to reiterate the need for more experimental and theoretical work in semiconductors to elucidate the present conflicts between different experiments, particularly in silicon, and to explore the transition from insulator to metallic behaviour of optical spectra.
Acknowledgment. - The author wishes to express his gratitude to Drs. D. E. Aspnes, R. S. Bauer, and R. F. Leheny for providing him with their results
prior to publication, and to Professor F. C . Brown for many valuable discussions.
rences
[l31 FIELDS, J. R., GIBBONS, P. C. and SCHNATTERLY, S., Phys. Rev. Lett. 38 (1977) 430.
[l41 ZUNGER, A. and FREEMAN, A. J., Phys. Lett. 60A (1977) 456.
[l 51 For a qualitative discussion of alkali halide soft-X-ray spectra, see
PANTELIDES, S. T., Phys. Rev. B 11 (1975) 2391.
[l61 BROWN, F. C. and RUSTGI, 0. P., Phys. Rev. Lett. 28 (1972) 497.
[l71 ALTARELLI, M. and DEXTER, D. L., Phys. Rev. Lett. 29 (1972) 1100.
[IS] The potential of a 2p hole is computed in
PANTELIDES, S. T., Solid State Commun. 16 (1975) 217. [l91 ASPNES,-D. E., OLSON, C. C. and LYNCH, D. W., Phys. Rev.
B 12 (1975) 2527; J. Appl. Phys. 47 (1976) 602; Phys. Rev. B 14 (1976) 2534.
[20] ASPNES, D. E., OLSON, C. G. and LYNCH, D. W., in Proceedings of the XIII International Conference on the Physics of Semiconductors, Rome 1976, F. G. Fumi, editor (Marves, Rome), 1976, p. 1000.
[21] PETROFF, Y., Reference [20], p. 975 ;
THIRY, P., PETROFF, Y., PINCHAUX, R., CHELIKOWSKY, J. R. and COHEN, M. L., Solid State Commun. 20 (1976)
C4- 102 M. ALTARELLI
[22] Recent evidence of a camel's back structure of the conduction band of Gap, with minima not exactly at X
(DEAN, P. J. and HERBERT, D. C., J. Lumin. 14 (1976) 55) is not relevant for levels wlth blnding energies of hundreds of meV.
[23] SHINDO, K. and NARA, H., J. Phys. Soc. Japan 40 (1976) 1640. [24] ALTARELLI, M., HSU, W. Y. and SABATINI, R. A., J. Phys. C 10
(1977) L-605.
[25] RESTA, R., J. Phys. C . : Solid Srate Phys. i 0 (1977) L179. [26] MORGAN, T. N., Phys. Rev. Lett. 21 (1968) 819.
[27] BAUER, R. S., BACHRACH, R. Z., FLODSTROM, S. A. and MC MENAMIN, J. C., J. Vac. Sci. Technol. 14 (1977) 378. [28] MARTINEZ, G., SCHLUTER, M. and COHEN, M. L., Phys. Rev.
B 11 (1975) 660 ;
MARTINEZ, G., SCHLUTER, M., COHEN, M. L., PINCHAUX, R., THIRY, P., DAGNEAUX, D. and PETROFF, Y., Solid State Commun. 17 (1975) 5.
1291 BAUER, R. S., BACHRACH, R. Z., ASPNES, D. E., MCMENAMIN, J. C., Nuovo Cimento B 39 (1977) 409.
[30] MARGARITONDO, G. and ROWE, J. E., Phys. Lett. 59A (1977) 464.
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