Phase transitions in electron-hole liquids

P Nozt~res / Phase transtttons tn electron-hole hqutds

cultles are avoided if one confines the liquid in a finmte region by an externally applied stress (which acts h k e gravity bringing water to the bottom of a pitcher). In this technique, developed by Jeffries et al [4], a pusher creates a maximum strain inside the sample, around which the carriers coalesce in a single big drop sitting in a parabolic potential well (thereby avoiding the surface). The experimental geometry is well controlled, and its allows expllcit calculatlons. Moreover, one may scan luminescence in space, thus galnlng access to the radial distribution of the drop. Such a technlque is the most reliable at the moment [5]. Another important feature of luminescence spectroscopy is its intensity. The"enhancement factor" compared to a free particle process gives access to electron-hole correlations inslde the liquid. At the moment, experimental determination of ~ is st111 very unsatlsfactory. An absolute measurement of the radiative quantum yield appears very difficult. Comparlng intensities of excitons and the liquid drop is somewhat easier - but on the one hand the excmton density is not really controlled - and on the other, excltons are complicated beasts whose luminescence is not quantltatively understood. Although a powerful tool, lumlnescence is also dangerous. Sophlstlcated llne-shape analysis relles on questionable theoretical models (neglect of mass renormallzatlon, of Auger broadening, etc.) and often on somewhat uncontrolled experimental conditions. One should therefore look for independent confirmation. Far infrared spectroscopy has proved to be a very useful tool in studylng plasma resonances of the droplets, internal transitions of the excltons (used as a density measurement). In conjunction with the blg drop geometry, it should provide rellable measurements of the reflectlvlty, surface properties, etc... The response to hlgh magnetic flelds is also an interesting problem. Because the Ferml energy EF of the electron-hole plasma is so small, it is comparable to spln and cyclotron splittlngs in attalnable fields. One thus expects glant magnetoeffects, In contrasts to ordinary metals. Oscillatory effects analogous to the de Haas-van Alphen effect have been predicted and observed in Ge. A simple minded analysis in terms of effective masses suggests a 10 % many body enhancement of m~ in the drops, for both the electrons and holes. Various magnetoplasma resonances have been studied vla microwave or far infrared absorption. Such experlments are clearly very rlch - however, they are hard to exploit. The Landau level pattern is compllcated even for a slngle carrier, a fortlorl for a dense system with strong correlation effects. Until now, information collected is mostly qualitative, relying on questionable theoretical models. 3.

EXPERIMENTAL SITUATION IN DIRECT GAP MATERIALS

The problem is much harder for optically active materials, in vlew of the very short radiative lifetime, £n the nS range. As a result, the carriers are no_._~tthermalized : their energy depends

17

on the excitation mechanism. To the extent that one can define a temperature~ the latter may be reduced by optical excftation just above threshold, but it will always remain well above the lattice temperature. Values of Tef f around 50°K are not uncommon. Besides, dlffuslon and denslty profiles raise major problems. In the usual case of surface excltation, carriers are blown by huge temperature gradients : the resultlng plasma expanslon must be described as a non-linear hydrodynamlc problem, taking into account recombination, energy relaxatlon and the rate equations that govern the formation of clusters. Recently, such an expanslon has been observed in Ga As through space resolved luminescence [6]: the theoretical analysls remalns rather primitive. It is not clear whether liquid gas condensatlon can occur at such high values of Tel f. Even if It does, drops would have hardly any tlme to nucleate : any phase diagram should correspond to a non equllibrlum steady state. A number of attempts have been carried out to describe such steady states vla rate equations [7]: apparently any condensation w111 at best mnvolve a "fog" of very tiny clusters. Experimentally, luminescence is not a very convenient tool in optlcally active materials, due to stimulated emission and saturatlon. Gain spectroscopy is preferable, using a two beam setup : an exciting beam creates the carrlers, whlle a probing beam measures the absorptlon/galn spectrum G. The thickness d of the excited region must be small, using either a thin film or a strongly absorbed excltlng beam. If Gd << I, one avoids saturatlon effects. In order to achieve large carrier densities without damaging the crystal, very short pulses must be used, whlch call for fast time resolved spectroscopy. Recently, such a method has been used in Cu CI in the plcosecond range : it yields evidence for the disappearance of excltons due to screening by the hot exclted plasma [8]. Gain appears at the chemlcal potentlal ~ where the populatlon is inverted : if other losses are small, that threshold can be measured fairly accurately (anyhow, it can he checked by excztatlon spectroscopy, in which extra carrlers are pumped above ~ by the probing beam). On the other hand, the band bottom is blurred, and its determination requires line-shape fitting with the accompanying uncertalnties. The denslty and temperature are likely inhomogeneous and not really controlled. Moreover, the theoretical llne-shape is unclear, due to momentum conservation. Wh~le the latter was irrelevant in the phonon asslsted transitions of indirect gap materials, it is crucial in purely radiative processes. Emplrlcally, it is found that ignoring k conservation altogether gives surprisingly good results. That has the merit of simplicity - but is it really justmfled ~ There exists a number of involved many body llneshape calculations [9], taking into account electron-hole correlations, collision broadening, etc... Their accuracy is hard to assess, since they incorporate only a selected class of corrections. While quahtatively well understood, the

18

P Nozt~res / Phase transtttons

problem thus remalns opeu as far as a quantitative analysis is concerned. In practice, a dense electron-hole plasma has been observed in many direct gap materials. The best studied cases are GaAs and CdS : typical measured densities are respectlvely n = 4.1016 and 2.1018cm -3. Similar results hold in ZnO, CdSe, InP, GaSb, ZnSe. In (I.VII) materials, the exclton and biexclton binding energles are much larger, and the available densltles are usually short of reaching ionlzatlon. The plasma state has been observed in CuCI [8] wlth n ~ 1020cm -3. These values are only indicative ; they vary conslderably from author to author, because of dlfferent experlmental conditions. (It is clear that such plasmas are not a saturated system with flxed density). Each experiment should be consldered on its own, and quoting a density n does not mean much. Empirically, it ~s found that the density n is nearly independent of pumping power, growing linearly with the effective temperature Teff. Such a behavlour can only originate from klnetlc arguments : a p r e h m ~ n a r y explanation is glven in ref.[6] in terms of a hydrodynamic model. As of now, there is no clear evidence that a liquid gas phase separation occurs in dlrect gap materials. It seems unlikely, in vlew of the high Tef f. In order to observe it, one should better control the density and temperature proflies, working for instance with thin films as in ref. [8]. 4.

THEORETICAL PICTURE

(ground state)

Standard theorles treat the IIquld drop as a degenerate normal two components Fermi llquid. Starting from a straightforward Hartree-Fock (HF) approximation, correlations are included vla the random phase approximation (RPA) or improvements thereof. The best calculation along those llnes is that of Vashlshta and Singwl [IO]. The results are illustrated on Fig. 2 for the simple hypothetlcal case of a semiconductor wlth isotroplc, non degenerate bands, wlth equal electron and hole masses. These results are deeply affected by

I

.

i|

\

HF I

~ %

NC

Figure 2 : The energy per particle as a function of density, measured by the partlcle distance Bohr radii : ~-~= .. q-~(rsao ~ )3. E O in the exelton energy.

tn

electron-hole hqutds

the actual band degeneracles and anlsotroples (qualitatively, a single heavy hole band with many llght electron valleys reduces the kinetic energy without affecting the valence exchange energy). When these peculiarlties are included In the theory, agreement wlth experiment is reasonably good for Ge and 81 (~ 10-20 %, comparable to discrepancies between experiments). Empirically, it is found that the sum of the exchange and c o r relation energies is insensitive to band structure, depending only on the total density [IO]. Thls somewhat surprising result is useful, since the band complexities are left only in the calculable kinetic energy. (Note that the result does not hold for Eex and Ecorr separately, hence the need for coherent approximations I)• In polar materials, binding is affected by the lattlce polarizability, as stressed originally by Kjeldysh and S I h n . The relevant parameter is the ratio of the phonon energy w o to the exclton blnding go. When ~o << go, the plasma involves dressed "polarons", with an effective mass and screened interactions ; in the opposite limit, the phonons are baslcally decoupled. There exist several theoretical treatments of the effect. The problem has been reviewed by Benl and Rice [II], who carrled extensive calculations for a wlde varlety of materials. Agreement wlth known experimental data is reasonable, although not as satisfactory as for Ge and Si (especially for GaAs). All these theories break down in the limit of low densities, as they do not take properly into account the correlatlons that ultlmately lead to a dilute gas of excltons (hiexcitons). That is not so bad for strongly bound plasmas, like Ge or Si, in which electron hole pairing is only a small correction. It becomes cruclal if the liquid is weakly bound - or even not bound at all as in Cu20 or CuCI. Even in Ge and Si, it has been polnted out recently that one can allgn the electron and hole spins, either by a strong magnetic field or by circularly polarized optical pumping. Such spin allgnment suppresses blexclton formatlon (as In polarlzed H+) - it also reduces the correlation energy, and thus the binding. Indeed, • f a large stress is applled in such a way as to ellmlnate the multi valley structure, Timofeev has shown that the liquid gas condensation dlsappears [12]. It then becomes important to have a theory whlch interpolates properly between the dense plasma and the d11ute exclton gas. Such a theory is possible In the ground state when the excltons are Bose condensed then electron-hole pairing enters vla an order parameter, which can be treated wlthln a mean fleld approxlmatlon, similar to the BCS theory of superconductivity. This point was recognized lon~ ago by Kjeldysh, Kozlov and Kopaev [13]. An interpolatlon scheme was devised by Sllln [14]. Recently, the problem has been taken agaln by Comte and the author [15], in the simple model of an isotroplc, non degenerate semlconductor. If one ignores screening, a simple variational ansatz ylelds the curve marked "NC" on Fig. 2. As compared to Hartree Fock ("HF"), that result accounts for electron hole palrlng, but not for screening, in contrast to "RPA" whlch does the reverse. A self consistent treatment of

P Nozl~res /Phase transznons ~n electron-hole liqutds

screening and pairing appears possible in this language []6]. A very rough estimate suggests a two hump curve for E(rs) leading to a flrst order transition with a finite vapour pressure at zero temperature. These estimates are only preliminary. More reliable numerical calculations are highly desirable, as well as a generalization to spin polarized electron hole gases. The results should be compared to the Vashishta Singwl treatment of short range correlations, for varylng mass ratios a = me/m h. An znterestlng new development is the possibility of additzonal phase separation in the presence of a large stress, suggested by Kirczenow and Slngwl [16]. In (I]I) stressed Ge, the conduction valleys split into a single lower "cold" valley and three upper "hot" valleys an energy A above. Let ~] and ~2 be the corresponding electron-hole palr chemical potentials reckoned from their respective band edges. The mixture of cold and hot pairs is unstable if ( ~ / - ~ p < O where C is the "hot" concentration and ~ = B2 - ~]" For intermediate stresses, ~(C) has the shape depicted on Fig. 3. On a time scale short compared to the Intervalley relaxation tlme rv, C is constant and a phase separatlon should occur, with a plateau AB obFigure 3 tained through the usual Maxwell construction. As intervalley relaxation proceeds, ~ relaxes towards its equilibrium value (-A) : the concentration relaxes toward A if ~ > -A and vlce-versa. Such a phase separation has been observed by Ba3a j e t al [17] through a careful line-shape analysis of time resolved luminescence. More direct evidence was obtained by Timusk and Zarate [18], who observed two plasmon absorption lines in far I.R. absorpt'i'on. Apparently, the effect is somewhat marginal. It is not observed in llO stressed Ge, nor in Si. In Oe, phase separation only occurs for fairly large stresses, such that only one valley is populated in thermal equilibrium. One may also imagine materials in which it would occur at moderate stresses, such that n] and n 2 are both ~ O at equilibrium. If now T v is made very short (through the addition of neutral impurities, for instance), one should observe a discontinuous behaviour of C (and of the density N) as the stress is varied : the system will jump discontinuously from A to B when (-A) goes through the Maxwell plateau. Such "first order chemlcal transitions" have been discussed In another context [20]: they occur at constant ~ (cf. vaporization at constant p or V in an ordinary gas). 5.

FINITE

TEMPERATURES

: CRITICAL POINT AND MOTT

TRANSITION At flnlte T, the drops evaporate, first into exeltons and possibly biexcltons. As the temperature and density grow, these complexes dissociate into

19

free electrons and holes, elther through thermal ionizatlon and/or through disappearance of the bound states (Mort transition). Determination of the corresponding phase diagram is difficult, both on experlmental and theoretical grounds. In view of denslty inhomogeneities, the vapour is not necessarily saturated everywhere. The most reliable results are obtained in the strain well geometry, in which the density profile is controlled : it can be included explicitly in the flttln~ Granted that this problem is solved, one must disentangle the l£quld and gas contributions to the broad observed lines. That implies a many parameter fit based on simple independent partlcle models. Such models are apparently good for the dense liquid and d11ute excitons ; they are certainly very bad for the gas plasma near the crltical polnt, in which correlation effects are very large. Current interpretation of luminescence spectra should thus be considered with caution. The llquld denslty nL(T) is inferred from the llne-shape ; in principle, the same procedure works for the gas plasma if it is at least partly degenerate. For classical excltons, the density can only be calculated, uslng an ideal gas model and measured values of the hlndlng energy. Such "experimental" values are all right if T << T c ; they are very doubtful near T c (alternate "onset" measurements of the vapour pressure are not reliable due to uncertainties in diffusion, recombination, nucleation, etc...). In vlew of experimental uncertainties and questionable interpretation, discrepancies between various authors remain sizeable (~ 25 % in SI). In Ge and Sl, it is well established that the critical point corresponds to an ionized degenerate plasma. In the simple theory of Combescot, the free energy is written as F = E o - I/2 7T 2, in which y is calculated for free particles. Desplte its oversimphficatlons~--~is model is remarkably successful. The more involved "droplet fluctuatlon" model of Relnecke and Ying takes some account of crlttcal fluctuations : it will be descrlbed at thls conference [3]. Agreement with experiment is reasonable in view of the very simplistic theoretical models. Real difficulties arise below Nc, in the Mort transition region where excltons are progressively dissociated. At intermediate densities the gaseous plasma line-shape is distorted by Auger tails, which makes data analysis somewhat doubtful. Standard approximations rely on static screening, characterized by a wave vector qD (Thomas Fermi in the degenerate low temperature limit, Debye H~ckel in the classical case). Accordlng to the Mort criterion, the exclton bound state disappears when qD ao > 1.19. When applied to Ge or Si, such a criterion predicts a Mott transition at very low density, two orders of magnitude below N c. This result is clearly m e a n ingless, since the screening length qD -! ~ a o would then be much smaller than the interpartlcle spacing d. It simply means that the Thomas Fermi (Debye H~ckel) formulae are applied in a strong coupling regime where they do not hold. The

20

P Nozl~res / Phase transitions m electron-hole hquids

screening length cannot be smaller than d, and any Mort transition can only occur when Nao3 ~ l, i.e. near the critical density N c. Since in that case the excitons strongly overlap, the very idea of a bound state is doubtful, and the idea of a vanishing palr binding does not mean much. The theoretical problem of exczton dlsappearance thus remains open. There exist many attempts to go beyond the static screening picture, includIng dynamic corrections, mass renormalzzatlon, etc...[9]. These calculations are not totally convincing, as they miss a number of important features (exchange, selfconslsteney of screening and pairlng, Auger broadening of excitons and single particles, etc...). A standing questlon is the existence of a first order ionlzatzon transition, occurring at a lower density than the h q u l d gas critical polnt N c. Such a transition was suggested by Rice []9], followlng an old proposal of Landau and Zeldovlch. It might also exist xn llquid Hg or in metala~nnonium solutions. Assuming that the excztons are well defined entities, it may be viewed as a chemical separation of excztons and electronhole pairs. Let ~lx and lJeh be the corresponding chemlcal potentials, n x and neh the densities. The condition for local chemical znstablhty reads 2 3~x ~ e h ~x < 0 ~nx ~ e h ~neh (Strangely enough, thls condition is often quoted incorrectly in the literature). The theory in this respect is very confused. Some claim that the instability occurs for a purely classical system [20]. Others believe that quantum corrections to Debye Huckel screening are necessary [21]. Since anyhow the treatment of screening at high densities Is doubtful, it is impossible to conclude. Experimentally, the observed broadening of the exciton line in Si as the density increases has been attributed to Mort ionization [25]. No first order transltion was observed. Since the corresponding densities are << No, a Mott transition is impossible : the observed broadening might be due to Auger processes. Similar experiments in Ge [23] were interpreted in terms of blexcztons, trions, etc... A discontinuous phase transition was claimed in GaP by Maaref et al [24]: it has slnce been questioned. Very recently, Schowalter et al [25] performed very careful lineshape analysis of a strain well confined plasma in Go. They interpret their data zn terms of the phase diagram of Fig. 4, which displays two first order transitions. Such an Interpretatlon is somewhat speculative. More direct evidence might be obtained from infrared reflection experiments. If two phase transztlon exist, the strain well plasma should dlsplay tw__~ospherleal interfaces, izquld-plasma and plasma-exciton. Such sharp interfaces will produce reflection if their thickness zs smaller than the optical wave length. Such measurement would complement spatial scan-

A

!

I

I

0

"-'60 P

5o

~3o 20 1014

I

I015

I

I

1016

I017

Density(crn-3)

Figure 4 nlng of the fluorescence light. Altogether, the hehaviour of electron-hole plasmas at finite temperature if far from settled. There is a need for well controlled experimental set-uus. Luminescence should be complemented with infrared measurements. Most of all, one should be careful with theoretical interpretation.

6.

BOSE CONDENSATION OF EXCITONS AND BIEXCITONS

In the realistic band structures, it can only exlst at low densities (Fermi surface mismatch destroys the excitonlc insulator insfability at high density). Consequently, it can be observed only if there is no liquid drop condensation at zero nressure (otherwise, the superfluid T c lies inside the phase separation curve). This condition is met in the spin polarized experiments of Tzmofeev et al []2], and in some direct gap materials, such as Cu20. Next, exclton Bose condensation Is zn competltzon with molecular formation. If biexcitons are stable, they will be the condensing entities at very low density ; when N grows, biexcitons dissociate and pair condensation turns into particle condensation [26], with a different broken symmetry. The corresponding phase diagram is sketched on Fig. 5.

IT nor~ n I

Figure 5

P Nozl~res / Phase transmons zn electron-hole liquids

Note that a molecular liquid bound by Van der Weals attraction is unlikely, in view of the extreme quantum nature of the system. Finally, one faces major kinetic problems. The excitons must be cooled below T c within a lifetime, an arduous task in direct gap materials (at high densities, heating due to Auger recomblnatxon is an extra difficulty). Moreover, we know essentially nothing on the nucleation dynamlcs of Bose Einstein condensation. One can help nucleation by pumping biexcitons directly via coherent two photon absorptlon. In that case, Bose condensation is put 'by hand", and one studies its persistence rather than its nucleation. Such a line of experiment seems very promising. Usually, excltons have internal degrees of freedom : spin, valley index in indirect gap materials, electric dipole moment in optically active materials. An important feature of Bose Einstein condensation is the coherence of internal states for all condensed partlcles (as met for instance in superfluzd 3He). On the other hand, one does not expect energy superfluidity, contrary to ~ w l d e spread belief. The number of excitons is not strictly conserved, because of various perturbatlons, (such as Auger virtual production of two excitons) ; as a result, the phase of the order parameter is locked, which precludes the existence of supercurrents [27]. Experimentally, Bose condensation should show up as a discrete line xn the exclton luminescence spectrum. Various claims have been made in the past , in AgBr for instance : they remain very controversial. The most convincing evidence was obtained recently zn Cu20 and CuCI [28]. In Cu20 , biexciton formation is opposed by a mass ratio me/m h ~ 1 and by znterband exchange ; in CuCI, instead, biexcztons do exist. In both cases, actual Bose condensation was not observed due to insuff£czent coohng, but sizeable departures from the classical izneshape were found, well explained in terms of an ideal gas Bose Einstein statlstzcs. These results suffer from the uncertalnties common to all hneshape fittlng - yet they provlde an encouraglng evidence for the existence of Bose condensation. In principle, observatlon of the condensate coherence would provide much more direct evidence for superfluzdzty. One may think of several "theorist" experiments In thls respect. In indxrect gap materlals, for instance, exciton condensation will produce a static stress on the lattice with a wavevector equal to that of excltons. Such a stress should give rise to a structural phase transitlon - a point emphasized long ago by Kohn zn the context of excztonic insulators. Bose condensatlon would then show up as additional Bragg peaks, monitored by exciton cooling. The effect is certainly very weak, but zt may be worth some thought. Even more striking, assume that one may condense optlcally active excltons (the transition should be nearly forbidden zn order to maintain a reasonably long llfe-

21

time). Then the condensed pairs should have parallel dipole moments, and the luminescence radiation should be coherent. Contrary to an exciton laser in which coherence originates zn the photon part (controlled by the optical structure), it here arises from the exciton itself. In practlce, coherence should extend only on short distances, due to defects zn the superfluxd state. Nevertheless, It should enhance radiative recombination drastically, so that the density wi11 surfrzde the Bose condensation threshold nc(T). Recombination is then monitored by cooling. Admittedly, this Is only wishful thinking - the main point Is to emphaslze the interest of experiments that would give direct access to superfluld coherence, rather than relying always on hne-shape analysls. CONCLUSION From this very brief survey, zt zs clear that the problem of electron-hole condensation in semiconductors is still quite active. If a theorist's plea zs allowed, zt would call for more controlled experimental sltuations (using especially the strain well geometry), and for more diversified experiments. As for the theory, zt should be used with great care : very elaborate fits are doubtful if they rely on oversimplified models : agreement wxth experlment is not necessarily an argument ! REFERENCES [I] Among recent reviews, the special volume devoted to that problem remains the standard reference : Rice T.M. (Theory), Hensel J.C., Phillips T.G., Thomas G.A. (Experiment), Solid State Physics, Vol. 32, F. Seitz, D. Turnbull, H. Ehrenrelch, ed. (Academic Press, N.Y. 1977). More recent reviews are due to Gbbel E.O., Mahler G., Festkorperprobleme XIX, 105, Springer 1979, and to Rogachev A.A., Prog. Quantum Electronics Vol. ~, 141 (1980) (Pergamon). Two articles in the volume "Optical properties of Solids", vol. 2, Balkanskl M., ed. (North Holland 1980) are specifically devoted to high density electron hole systems. The paper by Thomas G.A. and Timofeev V.B. surveys the existing data on electron-hole complexes in various materials. Voos M., Leheny R.F. and Jag Deep Shah focus on the problem of luminescence. The specific problem of direct gap materials zs reviewed by Kllngshlrn C., Haug H., Phys. Reports 70, 315 (1981). Finally, Bose condensation of excztons is extenslvely described In Hanamura E., Haug H., Phys. Reports 33, 209 (1977) and more recently by Mysyrowicz A., Hulzn D., Chase L.L., to be published in "Collective Excitations in Solids", ERICE School 1981 (Plenum). [2] Bimberg D., Bludau W., Llnnebach R., Bauser E., Solid State Comm. 37, 987 (1981). [3] Forchel A., Laurich B., Wagner J., Schmid W., Reineeke T.L., Phys. Rev. B 25 (1982). Gourley B.L., Wolfe J.P., Phys. Rev. B 24

22

P Nozl~res / Phase transtt~ons tn electron-hole hqutds

(19811 5970. See the review by Reinecke T.L. at this conference. [4] Markiewicz R.S., Wolfe J.P., Jeffries C.D., Phys. Rev. BI5 (1977) 1988. [5] The use of controlled stresses also allows direct measurement of the exciton mobilltv : Tamor M.A., Wolfe J.P., Phys. Rev. Lett. 44 (1980) 1703.

[15] Comte C., Noz1~res P., J. de Physique 43 (1982) 1069. [16] K1rczenow G., Slngwl K.S., Phys. Rev. Lett. 41 (1978) 326 and 42 (1979) I004. [17] Ba3aj J., Tong P.M., Wong G.K., Phys. Rev. Lett. 46 (1981) 61. [18] Timusk T., Zarate H.G., Bull. Am. Phys. Soc. 26 (1981)

[6] Romanek K.M., Nather H., Fischer J., Gobel E. O., J. Lumin. 24 (1981) 585.

[19] Rice T.M., Physics of Highly Excited States in Solids, Ueta M., Mshina Y. (eds.1 Springer 1976, p. 144.

[7] See for instance Koch S.W., Haug H., Phys. Stat. Sol.b 95 (1979) 155 and Haug H., Abrah---~ P.F., Phys. Rev. B23 (1981) 2960.

[20] Gltterman M., Steinberg V., Phys. Rev. A 20 (1979) 1236 ; J. Chem. Phys. 69 (1978) 276-3.

[8] H u h n D., Antonetti A., Chase L.L., Martin J. L., Migus A., Mysyrowicz A., to be published. [9] Haug H., Tran Thoai D.B., Phys. Stat. Sol. b 85 (1978) 561 ; Z=~mmermann R., Kilimann K., Kraeft W.D., Kremp D., R~pke G., Phys. Stat. Sol. b 90 (1978) 175, and subsequent papers. [I0] See a recent review by Singwi K., Proceedings of the NATO Advanced Study Institute, Antwerp, July 1981. [ll] Eeni G., Rice T.M., Phys. Rev. B 18 (1978) 768. [12] Kukushin I.V., Kulakovskii V.D., Timofeev V.B., JETP Lett. -34 (1981) 34. =-w (See the revlew at this conference) ; Zimmermann R. R~ssler M., Solid State Comm. 25 (1981) 651. [13] Kjeldysh L.V., 27 (1968) 521 K-~eldysh L.V., State 6 (1965)

Kozlov A.N., Sov. Phys. JETP ; Kopaev Yu.V., Soy. Phys. Solld 2219.

[14] Silin A.P., Soy. Phys. Solld State 19 (19771 77. Similar results were obtained-i~ a different language by Zirmmsrmann R., Phys. Stat. Sol. b 76 (19761 191.

[21] Kremp D., E b e h n g W., Kraeft W.D., Phys. Star. Sol. b 69, K59 (1975). [22] Jagdeep Shah, Combescot M., Dayem A.H., Phys. Rev. Lett. 38 (1977) 1497. [23] Thomas G.A., Rice T.M., Solid State Comm. 23 (1977) 359. [24] Maaref H., Barrau J., Brousseau M., Collet J., Mazzachi J., Pugnet M., Phys. Stat. Sol. b 88 (1978) 261. [25] Schowalter L.J., Steranka P.M., Salamon M.B., Wolfe J.P., to be published ; Schowalter L.J., Ph.D Thesls (Univ. of lllinols). [26] Nozt~res P., Salnt-James D., J. de Physique 43 (19821 I133. [27] Nakajlma S., Physics of Highly Excited States In Sollds, Ueta M., Nishlna Y. (eds.) Springer 1976, p. 130. See also NagaAka, same volume, p. 137. [28] H u h n D., Mysyrowlcz A., Benolt ~ la Gulllaume C, Phys. Rev. Lett. 45 (1980) 1970.