Electron-hole liquids in semiconductors

Prog Quant Electr, Vol 6 pp 141 217

0079-6"/27/80/1101-0141 $0500/0

© Pergamon Press Ltd 1980 Pnnted m Great Bntmn

ELECTRON-HOLE

LIQUIDS

IN SEMICONDUCTORS

A A ROGACHEV A F Ioffe Phystco-Techmcal Institute of the Academy of Sciences of the U S S R, 194021, Lemngrad, U S S R

CONTENTS 1 Introduction 2 Collective Properties of Exotons m Sermconductors 3 Fundamental Experiments Estabhshmg the Existence of ExcRon Condensation 4 Phase Dmgram and Electron-Hole Drop Ra(hl 5 The Net Charge of the Electron-Hole Drops 6 MagneUc Properties of the Electron-Hole L]qtud 7 Magnetohydrodynamlc Effects m the Electron-Hole Llqmd m Germanmm 8 Condensatmn IOnetlcs m UHF Field 9 Electron-Hole Liqmd m Polar Sermconductors 10 Dragging of Electron-Hole Drops by Phonons 11 Multlexclton Bound Complexes m Semiconductors References

141 141 145 151 167 169 184 187 188 199 205 214

1 INTRODUCTION

Twelve years of very intensive investigation of the electron-hole condensed phase m semiconductors puts it Into the rank of the most studled"hqmds" The umque properties of this hquid are determined by the small effective mass of the electrons and holes For this reason, the electron-hole hquld is the most "quantum" of all known ones Many other properties of the electron-hole hqmd are close to those of the monovalent metals, with the essential d|fference that the small effective masses of the paracles make it tmposslble to form an ordered crystalhne lattice due to zero point osctllatmns The electron-hole hqtad exhibits ordenng of a purely quantum nature, which makes the electron and hole mean free path much greater than the average mterpart|cle d~stance Consequently, the e-h hqmd conductivity appears to be comparable to that of many normal metals. At the same tame, the e-h hqmd dens|ty is about 10 orders of magmtude smaller than the density of metals The ex|stenee of many extrema m the conductmn and valence bands of semiconductors perm|ts control of the number of statistically mdependent types of particles, and thus to mvestlgate the relative contribution of the exchange and correlatmn to bmdmg energy As a mmlature model of a real substance, the e-h hquld is especially instrumental m studying strong magnetic field effects Due to the small characteristic energy of the e-h hqmd, the magnetac fields needed for such experiments are about 10,000 tanes less than m the case of ordinary metals The present paper is restr|cted to the review ofe-h drops m unstrained crystals, s|nee an excellent account of the strata-contained hqmds is given elsewhere.(~3s. 150) The work of the author was immensely facilitated by the exxstence of many other reviews on this subject t 11o, 117.~36, ~37,229) The present review Is too short, however, to cover all the problems of the e-h hqmds m semiconductors, and therefore some topics are not considered here At the same ttrne, some other topics were considered too closely, which reflects the part|cular field of interest of the author In such a fast expanding branch of science it is virtually impossible to be up to date In all of the important |terns, and all the onussmns that will be noted should be attributed to the unmtent|onal author's faults

2 COLLECTIVE PROPERTIES OF EXCITONS IN SEMICONDUCTORS

Investigation of excltons m senuconductors estabhshed them as a quasmtom type of elementary excitations freely moving in the crystal lattice. As a natural result of these lnvest~gat|ons was the growing interest about the character of the exclton-excRon interactions JpOE 6/3 - A

141

142

A A ROGACHEV

It is reasonable to suggest that excatons can form molecules, (138 156) or even condense into excIton " d r o p s " - - i e macroscopic e - h "dots" possessing some properties of liquids or solids (169170) TO detect the existence of the e - h drops (EHD) In semiconductors, Agranovltch and Toshich (3} as early as 1966 proposed to observe the additional scattering of light caused by the refractive index modulation in the crystal regions occupied by the drops* Another experiment proposed by them was the observation of the decrease in the rate of energy transfer in the crystal due to the condensation of excltons In this early paper the many-excaton system was treated as an excltonic insulator Other authors emphasized that the excatons having an integral spin ought to obey the Bose statistics Consequently, at sufficiently low temperatures and high e x o t o n densities the Bose condensation may occur(63 157 158)__1e the state where a considerable part of excltons occupies the lowest available energy level in the exclton band The critical temperature of Bose condensation is determined by the well-known relation T~=

3313 h2 (g) 2/3 ----k me~

(1)

where rnex is the excIton effective mass, n is the exciton density and g is the degeneracy factor For germanium at T = 2 K and n = 10 ~v cm-3, the distance between excltons IS about equal to the exciton Bohr radius The interaction of excltons at such distances IS obviously essential, and therefore the exoton gas cannot be regarded as an ideal one It follows from the weakly nomdeal Bose-gas theory (5~ 64) that at low temperatures the Bose gas is stable only when the interatomic interaction IS predominantly repulsive At large distances it is certainly not the case, because the Van der Vaals attraction should prevail However, at smaller distances interaction must eventually become repulsive, since an exclton is a compound particle consisting of two fermlons The problem is whether the exchange repulsion is large enough before the Mott transition has occurred Up to now, no satisfactory theoretical answer has been given In many theoretical works a tentative assumption was made that the metal-dielectric phase transition may be in some way avoided Depending on the relation between the attractive and repulsive energies, the following types of ground state of manyexelton system are feasible (1) Attraction of excltons leads to usual condensation in the coordinate space The repulsive forces stabilize the density of the system after the Mott transition Further increase of the density IS hmlted by the growth of the Fermi energy Running a few steps ahead, it should be mentioned that this IS the ease for germanium, silicon and all other semiconductors for which the e - h liqmd has been observed (2) For larger h - e effective mass rauo, the system behaves as the molecular dielectric similar to the hquld or solid hydrogen (3) The Fermi repulsion of exotons prevails at comparatively large interexcltonic &stances at which metalizataon still does not take place In this case, the system behaves in some respect as weakly nomdeal Bose gas in the presence of repulsion It was discussed in detail by Keldysh et al ,(12s) who showed that the effects of the deviation of excltons from Bose statistics are as essential as the effects caused by the exciton-exclton scattering For this reason, the weakly nomdeal Bose-gas theory strictly speaking cannot be applied to the exelton system The main difference is that the final result depends not only on the amplitude of the exciton-exclton scattering, but on the amplitude of the e x o t o n - h o l e and excaton-electron scattenng as well

* To the author's best knowledge,th~sis the first paper m which the idea of exclton condensauon was expressed m print This type of speculation was most probably publiclydiscussed even earlier For example, the idea of the e-h gas condensationinto a metallic drop was put forwardby the author m April 1963on a seminar at the IoffePhyslcoTechmcalInstitute, m the courseofthe &seusslonon the forbiddengap shrinkage under the influenceofhigh density e-h plasma (14)A year later, an 8-rammowewas taken winchdepictsthe author discussingthis problemin the same way as shownm Fig. 2 of this paper The eqmhbrmmdensity of the e-h hqmd was described asa trade-offbetween the potential - ~(nl/ae2/e)and lunetlc (h2/2rn)(3rc2n)2/3 energies It was a sheer guessworkat that time, because the existing theones of the dense electron gas did not predict a bound plasma state The only property of the many excaton systems theoretically discussed at that time was the posslbdlty of Bose condensaUon

Electron-hole llqmds m semiconductors

143

IO0

80

6O

- A

4o _J

2O

-

~,~4

hv,

eV

FIG 1 G e r m a m u m forbidden-gap s h n n k a g e under lugh mjectmn level (1) Low level of rejection (n ~. 1016cm3), (2) n = 1 4 x 101: cm 3, (3)n = 2 x 1017 cm 3, (4) Stlbmm-doped sample nd = 1 5 x 10 Is cm 3

All attempts to observe Bose condensatmn experimentally have not up to now led to a concluswe result. For the Wannler-Mott excatons, a metalltzatlon and condensataon m the coordinate space, rather than Bose condensation, were observed In thin case, the excaton gas condenses mto"drops" consmtmg of degenerate e - h plasma, the equlhbnum density of which is the result of the "interplay" of the kinetic and Coulomb energ]es The first experimental study of htgh density e - h plasma m semiconductors has been carried out by Asmn et a1.4~ using rejection in p - n junctions The e - h lurmnescence spectra were measured at different currents through the p - n junction The authors showed that with increasing e - h pan: concentration m the &ode base, the long-wavelength rode of the lurmnescence spectrum shifted towards lower energies (Fig 1) The shift reached the value of 1 5 x 10 -2 eV for the rejected carrier denmty of 101 s cm- a The spectrum edge shift was attributed to the shrinkage of the forbidden gap m that part of the semiconductor where large concentration of nonequthbnum electrons and holes exasted (Fig. 2) It was pointed out that this shrinkage takes place m both classical and quantum hmlts. If the career density m small, thin shnnkage can result from the Debye screanmg, and Its value is determined by AEg = - (e2/ers,), where rsc = (ekT/8rte2n) w2, n is the e - h pan: density At strong Fermi degeneratmn of the e - h gas, the forbidden gap shrinkage is determined by the sum of the exchange and correlatmn energms The dependence of the forbidden gap shrinkage on concentration may be approximated as AEg = -~(e2nWa/e) where ~ m about umty. Thus, at small densities the depth of the "well" created by the earner self energy increases with concentration faster than the Fermi energy of electrons and holes, which is proportmnal to n 2/3 T h e value of AE~ determined expenmentally within a factor of two coincides with the pre&ctlon of the Geel-Mann-Bruechnerm2) theory of lugh denmty electron gas In the high density limit (at rs << 1, where r~ = ( ~ n ) - W 3 r ~ 1, rh = (eh2/e2m*), nt71 = m f 1 + mh 1, m e and m h are the effectwe masses of electrons and holes) when the Fermi energy is sufficiently higher than the Coulomb interaction energy, the existence of metalhc type conductwlty seems to be trwal Keldysh et al ,¢~27~however, came to a conclusion that the screened Coulomb lnteractmn results m an e - h coupling near the Fermi surface The binding energy of these pmrs decreases exponentially with nsmg electron and hole denmtles, but remains fimte at any concentrataon. Thus, at suflicaently low temperatures, the conductwlty should vanmh The physical nature of the energy-gap appearance m the wctmty of the Ferm! surface m thin case m qmte smaflar to a Cooper pmr formatmn in the superconductwlty theory, with the Important difference that the electron-phonon interactmn Is replaced by the screened Coulomb attraction. Further mvesugatmns by Kozlov et al ~37) showed that at the large denmty hmlt the gap value m

144

A A ROGACHEV

extremely sensmve to the career effectwe mass amsotropy If the lsoenergeUc surfaces of electrons and holes are not slmdar to each other, the gap disappears at fimte concentrations The possibility of the existence of an excltomc dielectric m the e-h system in semiconductors is not at present experimentally confirmed Conduction band

~7

/ /'~1/ 7771/777~ - -

hi/max

_

l

Ferm~ level

hi/ram

_

Valence band

FIG 2 Forbidden gap m the sample partly occupied by degenerate e-h plasma Electron and hole quasi-Fermi levels may he m the forbidden gap of the host crystal, making the h~gh-denslty state energettcally stable

In semiconductors with redirect optical transmons, such as germanium and silicon, the hfetmae ofexcltons can exceed by four or five orders ofmagmtude the ttme of their formation and thermalmatlon These semiconductors are convement objects for the study of collective properties of excltons also, due to the lack of mduced hght emission Investigation of germamum at hqmd hehum temperature at high excitation levels shows the existence of the metal-insulator translt~on m the many-excaton system (12,30,179) In order to achieve better homogeneity m the exclton &stnbutlon m the sample, and to improve heat &sslpatlon, the experiments were carried out on germamum samples 10/~m thick unmersed rata llqmd hehum The samples were illuminated with 1/zsec-long hght pulses At exclton concentrations less than 10 ~6 cm-3, conductwlty was determmed only by the carriers which were not bound into excltons The magmtude of this conductlwty ~s almost independent of the excltatlon level At a concentration of 2 x 1016 cm-3 the conductwlty sharply increases (Fig 3) and its magmtude m the region of steep growth strongly fluctuates For h~gher concentrations the conductwlty rises smoothly, whde fluctuations typical of the first region disappear and the experimental dependence ofconductwlty on concentration follows the law a ,~, n3/T 2 At concentration n = 2 1017 c m - 3 and T = 2 K the carrier mobility reaches 10 6 c m 2 V - 1 s e c - 1

The dependence of conductwlty on the e-h pmr density measured expermaentally is close to the theoretical dependence obtained by Baber (42) for the degenerate e-h gas EFeEFhn

tr = elm ,~, ~

n 7/3

,.~ T ~

(2)

where EFe and EFh are the electron and hole Fermi energies Thus, at comparatwely small average concentration (L "~ 2) metalhzatlon of the exclton system already takes place The data presented m Fig 3 does not answer the question about the nature of the metal-insulator transmon In prmcaple, th~s data can be interpreted from two following viewpoints more or less homogeneous transmon to metalhc type conduct|vlty, ortheformat|onofhlghdenslty"metalhc"drops* The fact that eq (1)couldbe * The posslbthty of existence of condensed excaton phase was noted by L V Keldysh(129)m the closing address at the International Conference on the Physics of Semiconductors held m Moscow 1968 In more detail, the condensation of exc~tons was discussed m the paper by L V Kddysh and A A Rogachev, read at the meeting of the General Phymcs and Astronomy Branch of the Academy of Sciences of the U S S R, September 1968 Later, this paper was quoted by Kddysh (130)

Electron-hole hqmds in semiconductors

I

2K

145

i ~

I0-

I

[ I I0 i6

2 I0 t6 n,

5 I016 cm 3

FIG 3 Low-temperature conductw]ty of pure germamum as a function of e-h pair density (22)

apphed for the explanaUon of the dependence ofconductwlty on concentration may be used as an argument in support of the umform increase of the current earner density m a constant volume On the other hand, one can suppose that with increasing excltataon intensity a gradual fflhng of the sample with exclton condensate drops occurs The average e - h pmr concentrataon of 2 x 1016cm -3 corresponds to the onset of the current flow between contacts through the drop cloud (percolation threshold) as it was suggested by Benolt a la Gmllaume et al (ss) This assumption was supported by the results of Asum et al ,(33) who showed that the earner mobdlty obtaaned from conductwlty measurements m strong magnetac fields remamed constant for the whole excitation range where metalhc conductivity exists The latter is possible only for the constant local carrier concentration m the metalhc phase The conductwlty character will be considered later in the &scusslon of the current mstabdlty m the condensed e - h phase

3 FUNDAMENTAL

E X P E R I M E N T S E S T A B L I S H I N G THE E X I S T E N C E O F E X C I T O N CONDENSATION

The optical properties of the semiconductor are drastically affected by the presence of the e-h drops Screening of the Coulomb potentml by free carriers leads to the extmct~on of the

exclton absorption hnes and to a sharp decrease of the absorption m the continuous spectrum r e g i o n (~6'17'1sl) In germamum, the excaton maximum m the direct opUcal absorption reg]on disappears at an electron concentration of 1016 cm-3 at 77 K (17) At high carrier concentrations, the Moss-Burstem shift of the absorption edge due to the fiUmg of the valence band edge becomes essential Thus, the area occupied by the e-h drops must be more transparent for photons near the edge of the &rect transitions compared to the regions with small free career concentration This observation was used m the first successful expenmental detect]on of e-h drops (i s) To prove the existence of two phases--the dielectric exclton gas and the "drops" of metahzed condensate with constant density--it is not necessary to observe separate drops The problem of detection of the drops may be overcome by using the advantage of the fact that the spectra of the &rect absorption edge modulataon caused by the presence of the e - h drops should have a constant shape up to the moment when the sample ~stotally filled with condensate Only the amphtude of modulation depends on the

146

A A ROGACHEV

g

2

~ o~o

° J/ 0 883

I

I

0 889

I

0 895

0 901

I 0 907

hv

FIG 4 Absorption modulanon spectra of &rect t r a n s m o n s m germanium measured for &fferent average e-h pair density (/cm 3) (1) - 3 5 x 10 Is, (2) - 5 5 x 1015, (3) - 1 x 1016, (4) - 2 x 1016 All densmes are gwen as they were presented in the original paper by Asnm et al To reconcde these densities with modern values of e-h drop density equal to 2 2 x 1017 cm 3, these densmes as well as those shown m Fig 5 should be upgraded by a factor of 10 (Is)

drop number, or, strictly speaking, on their total volume The expertments were carrted out with germanmm samples 5-10#m thick immersed m hqmd hehum (is 19) Exotons were created by tllummatmg the samples wlth hght pulses 1 psec long Simultaneously, a monochromatic hght with photon energy close to the &rect absorpnon edge was passed through the sample The spectra of the modulation of the probing hght due to the large excaton density in the sample are shown m Ftg 4 Ftgure 5 shows the dependence of modulation magnitude on the excatatton intensity The modulation spectra shape remains practically unchanged upon varying the amphtude ofmodulanon by more than one order of magmtude Stmllar experiments carried out at 77 K showed that both the amphtude and the

50

20 IO

°5I-

1 i i o 15

II

I 5 io 15

I~,

r 2

I io 16

I

cm 3

FIG 5 The &rect absorption edge modulation as a function of the average e -h pair densRy (1) T = 4 2 K , (2) T = 2 K Sohd hnes relate to the case of pulsed hght intensity are changed Dotted hnes are results of Ume-resolved measurements when delay Ume between the end of the hght pulse and absorption measurement is changed Experiment demonstrates the hysteresis nature of the condensation process (~9)

Electron-hole hqmds m semiconductors

147

9-8 7 6 5 0m

/

4 3

/

2 I , 0 88

I 0 90

I

I

I

I

I

L

0 92

0 94

0 96

0 98

I O0

I 02

hv ~

eV

FIG 6 Direct absorption edge spectra recorded at different excitation levels I (1) I = 0, (2) I=0031 .... (3) I = 0 1 1 .... (4) 1=021 .... (5) I = 0 5 1 .... (6) l=lm.x Exclton maximum dmappears when the whole sample is filled with e-h llqmd (curve 4)t4s~ Points show unpubhshed calculation by R Zlmmermannand M Rosier,who take into account Moss-Burstem shift and static screening of Coulomb potential by electrons and holes for e-h pmr density equal to 2 × 1 0 1 7 / c m 3

spectral shape of the modulation simultaneously change with the exciting hght intensity ~19) Such a behavior can be expected when there is no phase transition and the carrier concentration varies gradually The evidence of the existence of the 'two phases with very different free carrier concentrations can be obtained also through the study of the absorption edge in a direct manner without the apphcatlon of the modulation technique Figure 6 shows the absorption spectra of g e r m a m u m measured several #sec after the excitation light pulse end c25~ Curves 2 and 3 are interesting in the following respect a pronounced exclton m a x u n u m is accompanied by a substantial change of the continuous spectrum which spreads over a region of m a n y exclton binding energies Such behavior is unusual, because with gradually increasing free carrier concentraUon the exc~ton m a x i m u m disappears at first, and only at many times greater concentrations the absorption coefficient m the continuous spectrum substantially decreases over a w~de spectra range A httle later, the existence of the e - h drops was also observed in the investigation of the lntnnslc luminescence of pure germanium ~16a~In g e r m a m u m and silicon a wide luminescence line was found several meV apart from the free exclton hne on the low energy side of the spectra Originally, this hne was identified as the excaton molecule recombination line ClO5~ A new factor which led to the reconsideration of the nature of this line m germanium was ~ts anomalous temperature dependence Figure 8 shows the sharp increase of the amplitude of this hne with decreasing temperature The threshold-hke appearance of this broad line with decrease of temperature was interpreted as a mamfestatlon of the first-order phase transltmn Attempts to create a model allowmg coexistence of molecular and hquld phases were made~20 24 181}even after the fact of the condensed excaton phase existence was established on the basis of the data not connected with luminescence Now it is beyond any doubt that the discussed lines m both germanium and silicon do orgmate from the e-h condensed phase But the first luminescence data were rather ambiguous We have taken the liberty of superimposing the theoretical temperature dependence of the excaton and blexciton line Intensities (dashed curves) with the experimental data of Pokrovsky and Svlstunova (Fig 8) to show that using the binding energy of the blexc~ton (unknown at that time) as a fitting parameter it is possible, in pnnclple, to explaan luminescence data m the framework of the exclton molecule model In calculating the blexciton density, the simplest assumption is to

148

A A ROGACHEV

suppose that the excitation volume is Independent of temperature Then biexciton density m a y be obtaaned from the following set of equations

nex

2nR

Tex

TB

--+

=G

n B = ~ - nex exp

(3)

Nex

where %x and ~R are the exciton and blexclton lifetimes, G IS the general rate, N n and Ne~ are the densities of state in the exclton and blexclton bands correspondingly E M is the binding energy of the blexclton Since the width of the luminescence line ascribed to the condensed phase is equal to the sum of the Fermi energies of electrons and holes one can determine the e-h pair density in the conde~lsate* This method of determining the condensate density is more reliable than, e g., the method based on the observation of the exclton m a x i m u m disappearance in the absorption spectrum (Fig 6), since in the latter case one needs to know the e - h parr concentration m the sample The experiments presented above prove the existence of the e-h condensed phase and give an estimate of its dens]ty, but, unfortunately, do not permit any conclusions to be made about the d r o p h k e character of the condensed phase I

I

I

I

I

I

I

I

Sillc0n ~3K

g

ehL (TO/LO) 0

-qb-

C

FE (TO/LO)

C

E

ehL (TA)

.J

I00

I 04

08

h~,

112

eV

FIG 7 Radiative recombination of pure silicon (lO5 11o)

The first expertment wh]ch demonstrated the &screte nature of the e - h drops and gave the estimation of their radn was the observation of the gigantic photovoltage pulses (21 23) These pulses were observed in the point p - n junction in germanium at liquid helium temperature and large excatatlon levels The total charge (i e the product of amplitude and pulse duration) for each pulse was about 10 s e, which means that under the given experimental conditions each drop contains an average of l0 s e-h pairs The observation of these pulses appeared to be a convenient method for the detection of the e-h drops in semiconductors, and was later applied to the Investigation of drops' radn distribution and their motion under the influence of the "phonon wind" The idea of the

* Renormahzatlon of the effectwe masses due to many-body effects is about equal to 10% and leads to relatwely small correction of the density

Electron-holehqtads m setmconductors

149

104 5 A

A

2

:~ IO3 ~

5

8 c ~

IO2

~

5

.J

2

2

1

I

I

I

I

I

0 ~,6

0 38

0 40

0 42

0 44

0 46

I/T

K

FIG 8 Temperature dependence of exczton (circles) and condensed phase (trmngles) luminescence hnes m germamum u6s) Dotted hne (excltons) and sohd hne (blexcltons) calculated by eq (3) assuming Eb = 3 5 x 103 eV

experiment zs rather simple At small excltatzon levels when only excatons and free electrons and holes are created m the sample, the p - n junction current is proportional to the total carrier concentration The electrons and holes bound into excatons also make a contnbut~on to the photocurrent smce the electric field ofthe p-njunchon destroys the excatons When the e-h drop reaches the strong field region of the p-njunctlon xt evaporates rapzdly, which gwes rise to a glganUc current pulse The charge of the pulse is equal to the number of the e-h pairs collected by the p-n junct]on The mam results of these experiments were confirmed by Gudlaume et al, (56) by Chrlstensen et a1,(72) Hvam e t a / , (Is) and by Groosman et al, (97) where continuous rather than pulsed lllumlnatzon was used. Such experiments were made at very wide range of excitation levels m the temperature range 1 7-4 2 K The drop radn estimated from these experiments vaned from 2 to 20 #m Figure 9 Illustrates the results of one of these expertments A sample of pure germanmm 1 mm thick was tmmersed m a hqmd hehum bath and illuminated from one side by a pulsed hght source with 1/~sec pulses On the opposite side of the sample a point p - n junctzon was formed At small excitation mtensmes the observed szgnal was due to the carriers and excltons &ffusmg through the sample The larger intensity of the pulsed hght led to the increase m the current pulse number and shifted the Ume of their arrival closer to the hght pulse

5/u. L sec)

-6

FIG 9 0 s c f l l o g r a m s of p - n junctzon current taken with increasing (from 1 to 7) excitation level (2a)

150

A A ROGACHEV

As was noted in the first part of this review, light scattering by the exciton drops was the first effect suggested for the demonstration of their existence ca) The e-h drops scatter the light due to the difference between their refractive index and that of the host crystal The main reason for this scattering is the large carrier concentration in the drops, which, in accordance with the Drude theory, reduces the dielectric constant In the high-energy region of the spectra The optical constants of the e-h drops were determined by Worlock et al c236}The angular distribution of scattered light intensity depends on the drop radius The Reylelgh-Gans theory shows that (see, e g, van de Hulst C22°}) if the light wavelength in the substance is less than the drop radius the angular diagram of the forward scattering is a universal function of the argument KRO, where K is the light wave vector in the substance, R is the drop radius and 0 is the scattering angle The first successful experiment on light scattering by e-h drops in germanium was reported by Pokrovskli et al {17ot To create the quaslvolume carrier excitation in this experiment the germanium sample was illuminated by an incandescent lamp from the mechanically pohshed side A light with a large absorption coefficient does not practically create free carriers due to the strong surface recombination In this case, only that part of the photons is active for which the absorption coefficient is about the reciprocal sample thickness Figure 10 presents the scattering diagrams obtained when a well-collimated laser beam was placed parallel to the sample surface at two depths x = 0 5 and 1 0 m m The scattering angular diagram found experimentally was satisfactorily described under the assumption that all drops located at a given distance from the surface have the same radius (7 6 and 3 4#m, respectively) The presence of the small plasma drops In the sample leads to a resonant absorption of the radiation at the characteristic frequency This effect IS very similar to the light absorption by sodium drops in ether or in sodium chloride crystals reviewed by Savostianova c192} I o

08 m c

:~06 O .ID O

O4

O2

0

2

5

4

5

KR0 FIG 10 Unwersal angular function o f h g h t scattering by e - h drops m g e r m a n m m Experimental data correspond to R = 7 6 p r o (circles) and R = 3 9 ~ m (squares)¢17o}

If the drop radius is much smaller than the wavelength of the infrared radiation in the substance (/o'), the absorption maximum of the spherical drops is at the frequency {o, which is equal to toy~x/3 where 09p is the plasma frequency For larger drop radii the absorption maximum shifts towards longer wavelengths The e-h absorption spectra in germanium obtained by Vavllov et al {224 225) is shown in Fig 11 Independence of the spectrum shape of the excitation level implies that the e-h drop dimensions do not exceed 2 pm even for highest excitation levels achieved in this experiment According to the Drude theory, the absorption coefficient is determined by ~((O)

3x/~

2

~'O)2

{4}

Electron-hole hqmds m semiconductors I

I

151

I

--I I-06-

04

02

7 5 mW 5

I0

h~,

FIG 11 Far infrared absorpnon by

15

meV

e-h drops m gennamum at three d~fferent mtensmes of exciting light (22s)

where e is the d)electnc constant, c m the velocity of llght, f Va m the part of the volume filled with drops,

c°v co0 - x/3,

[47re2N° I 112 c% = t---~-g-- ]

,

m* is the reduced mass of electrons and holes, 7 m the colhsmn frequency Attempts to fit the experimental data to eq (3) using frequency-independent 7 as the fitting parameter were not very successful The reason was that the absorption curve which m strongly asymmetric imphes a frequency-dependent V Besides, an anomalously large value of the e - h colhsmn rate 7 was required to explain the large half-width of the absorptmn curve Later, Murzm et a1.61) proposed that the short wavelength part of the spectrum is due to the transitions between the light and heavy hole bands This proposal was supported by the more elaborated calculations of Rose et al ,(182) who achieved an excellent fit between the theoretical calculatmns and expermaental data with two fitting parameters zl and z2--the lntraband and mterband hole relaxatmn times respectively As will be discussed m the next sectmn, the drop radii observed experLmentally usually are in excess of 2 #In For such drops, however, ~t is necessary m the scattering cross-section calculations to take into account more than one partial mode This problem was solved by Mle many years ago, and the modern treatment of the problem may be found in many textbooks (see, e g., Born et al (66) For larger drops the absorption maxtmum shifts towards the long-wavelength side of the spectra, as shown in Fig 12 The carrier concentratmn of the e - h hquld in germamum was originally determined from the far-infrared absorptmn data as n = 2 x 1017 c m - 3 WRh a more refined treatment of the same experimental data Rose et al "a2) came to the value of 2 2 x 1017 c m - 3 at 1 5 K which Is in excellent agreement with the data dmcussed in the following sectmn 4

PHASE DIAGRAM AND E L E C T R O N - H O L E

D R O P RADII

The process of the e - h drop formation, at least m such semiconductors as germamum and silicon, can be presented in the following way When a sample is illuminated by light with photon energy larger than the semiconductor forbidden gap, electrons and holes are created which lose their energy due to the emission of optical and acoustical phonons, and come to equihbnum with the lattice At liquid helium temperature, this process takes about 10-10 sec After that (or during the "cooling" process) electron and holes brad into excatons The binding cross-section is of the order of 10-1°-10-11 crn 2, and correspondingly the binding time is less than 10- a sec Since the time ofexcaton formation from free electrons and holes is sufficiently shorter than exclton lifetime, it may be assumed that hght &rectly creates excatons Later, we consider the vahdlty of this assumpnon in more detail

152

A A ROGACHEV

£

1,1

0

5

I0

15 to~ meV

20

25

F1G 12 Extinction (absorptmn and scattering) coeffioent of electron drops m ge rma nmm calculated by Rose et al ~182~ ignoring e h scattering reside the drops

The drop will be in equilibrium with exoton gas If the difference between the exclton flow into a drop and the back-flow due to evaporation from the drop is equal to the rate of the e - h pair annihilation m the drop (169~ 4 3No ~rcR ~

=

lr.R2~(nex

--

ns)

(4)

where N O is the e - h pair density m a drop, v = (8kT/rcm*) 1/2 is the thermal velocity of excatons, m* = me + mh, ns = Nex exp(Eb(R)/kT), Eb(R) Is the exclton binding energy with the drop, and Nex IS the effective density of states in the exoton band Due to the surface energy contribution, E b depends on drop radius R 2~ Eb(R ) = E b - - RNo

(5)

where ct is the surface tension coefficient The drop of radius R is in equilibrium with the exciton gas if 3 RN o - + ns nex - 4 ~d

(6)

This formula should be used if drop radius is smaller than the exclton mean free path lex The latter is not known with sufficient accuracy either In germanium or in silicon Apparently, the exoton free path in germanium is close to 5 x 10 -4 cm at 2 K In the case when lex <'- R, the excatons captured by drops in the dlffUSlOn-lmmed process and the exclton concentralaon In the VlCamty of the drop may be essentially smaller than the average exc~ton concentration nex Equation (6) m this case changes into "81~ 1 No R2 + n~ ne~ - 3 Dr d

(7)

Equations (6) and (7) show the most trnportant differences between condensation ofexcltons and ordinary hqmds Tbas difference is caused by the finite electron and hole lifetime m the drop The e - h drop can exist only In the condition where some supersaturation takes place The value of this supersaturation is greater for larger drops The excaton concentration has

Electron-hole hqulds m semiconductors

153

minimum at a defimte drop radms Actually, an Increase in R leads to an mcrease of the first term m eq (6) or (7), and conversely a decrease of R increases the second term The condensation kinetics was studied by Sliver(199 200) and by Westerwelt (230 2al) The minimum concentration of free exclton is reached when R satisfies the condition

3No 4 za This condition for

26~N~, [ ( Eb kTNoR2 exp - ~-~

26

kT-No R

)1 = 0

(8)

(26/kTNoR) <,. 1 can be written as [8 ~d6Nex exp (-- Eb/kT) l 1/2 g = 13 kTN2 i

(9)

However, drops with radii which correspond to the minimum of the exciton concentration are not stable, because any decrease of the drop size due to fluctuations leads to evaporation In such cases, the excaton concentration slightly increases and, correspondingly, the stationary drop radius also increases At low temperatures, the drop radius determmed by eqns (8) and (9) become comparable with the exclton radius, which defines the h i l t s of applicability ofeq 8 It follows from eq 8 that for germanium at T = 4 2 K, R = 2 x 10 -4 cm and at T = 2 K, R = 1 5 x 10- 5 cm Thus, the account of the finite e-h pair hfetlme leads to a conclusion that only rather small drops are stable in the steady state On the contrary, for the usual hquids a steady state would be that state in which one big drop exists, since in this case the surface energy has a minimum value Up to now It has been assumed that the steady state can be reached during the time of the expenment The rate of approaching the steady state depends on the rate of the embryos formation The embryos are the small drops formed either on the condensation centers (inhomogeneous nucleation) or due to the large fluctuations of the e x o t o n gas density (homogeneous nucleation) After the start of the condensation process the exclton concentration decreases, and consequently the rate of the new embryos' formation falls sharply Therefore, in the experimental conditions the formulae (8) and (9) are strictly speaking never valid. Later in this section this problem will be discussed in some detail Let us assume for now that the corrections in eq (6) due to the finite hfettme are small, and therefore ne:, may be considered to be dependent only on temperature Then, at sufficiently low temperature the excaton concentration is determined by

nex = C T 3 / 2 e x p ( ~ )

(10)

where C is a constant Well below the critical point (Tcr ~ 6 K for germanium) the liquid phase density only slightly depends on temperature The value of the e-h hquId density No(T = 0) = (2 6 _ 0 1) x 10 i 7 c m - 3 was determmed from the lummescence line shape analysis by Thomas et al (21a) This analysis includes both effective-mass renormahzation and the small contribution of LO-phonon replica. With nsing temperature the carrier density in the liquid phase decreases (214.,218) Stmultaneously there is an increase in the free exclton concentration At the critical teml~erature of the gas-liquid transition, the densities in the gas and liquid phases become equal to each other As a rule--e g in the case ofmonovalent metals and m e r c u r y - this critical point and the critical point of the metal-insulator transition in the gas phase approximately coincide In the multivalley semiconductors the behavior of the phase diagram m the vlcmlty of the critical point has a distortion, mentioned by Thomas (z 17) and Thomas et al (218) The many-valley character of the conduction band and the comphcated structure of the valence band substantially increase the density of the gas phase due to the larger density of states In the exclton band Meanwhile, it is assumed that the critical density of the metal-dielectric transition in the excaton gas is relatively independent of the band structure, and depends mainly on the exclton bmdmg energy The bound exlcton states disappear when the screening radius becomes approximately equal to the exclton Bohr radius Thomas et al (218) used the improved Mott criterion (159)

rhqo = 1 19

(12)

154

A A ROGACHEV

where r h = h2~/e2m * lS the exmton Bohr radms, and m* IS equal to 0 071 m 0 for germanium At this value of the reduced mass, r h = l 1 4 A The reciprocal screening radius qo in this estimate can be calculated in either T h o m a s - F e r m i or D e b y e - H u c k e l approximations In the first case the Mott transition critical concentration is determined by the expression nM =

(r~

= 1 7 x 1014cm -3

(13)

In the second case (1 19)%kT n M --

= 2 6 x 1015cm -3

8neZr2

(14)

at T = 6 5 K " However, even in the latter case the experimental crmcal concentration n~ = (6 _+ 1) × 10~6cm -3 appears to be substantmUy higher than the expected M o t t t r a n s m o n point The phase dxagram of e - h h q m d in germanmm, which essentially is a s u m m a r y of the experimental data, ~z~8~ ]s presented in F~g 13 An attempt to fit this diagram by means of proper density and energy scahng to the phase diagram of a simple hqtud appears to be unsuccessful, due to the anomalously small density of the gas phase near the critical point T h o m a s et al (217 218) attributed this difference to the so-called Mott &stortlon The disputed energy regmn ls shown by the dashed curve at the b o t t o m ofFlg 13 Strlctlyspeakmg, nelther the classical D e b y e - H u c h e l approximation nor the T h o m a s - F e r m i approximation can be apphed m the Mort transition point with proper justification By th~s reason, T h o m a s et al t218~hmlted the possible M o t t transmon region at T = 0 K from the low concentration side with the value gwen by the T h o m a s - F e r m ! approxlmaUon (eq (13)) and from the large concentration side with the value 3 1 -= 2 × 1016cm -3 nn - 4n (2rh) a

(15)

Equation (15) gives the concentration at which the exclton wave functions overlap sufficiently to cause exclton lomzation, not accounting the free carrier effects The expermaental investigation of the e-h-hqmd-excaton gas-phase &agrams is usually made by measurement of temperature and c a r e e r concentraUon at the onset of the I

I

I

Ge

T I

/

.5

I

~ Matt ?.,o.,o. e-h liquid

j 0 I--

Nos'l'afes

3

/

/

~'%.

/

I I 1013

I I I

I014 e-h

F1G 13

x

Matt trons~*lon~

tt

I

I

I0 I$

I016

pair

density,

t ! I

i I017

10111

cm 3

Phase dmgram of e - h hquld in germanium Density region m which Mott transmon m exoton gas xs expected to take place denoted as a "Mott d~stortlon" ~2ts)

Electron-hole liquids m semiconductors

155

condensed phase luminescence In the vicinity of the crmcal point, in addition to excitons a large amount of free electrons and holes and, probably, exclton molecules, charged excltons (trlons) and so on are presented Since the hquld-phase density near the critical point is essentially lower than at T = 0 K, and, besides, temperature broadening of the hnes becomes comparable with condensed-phase line width, decomposition of the radiation spectrum into separate components is a somewhat arbitrary procedure However, the flat shape of the phase diagram top provides a comparatively accurate determination of the crystal temperature This occurs due to the fact that the most rapid change m the luminescence spectrum at constant excitation level and changing temperature takes place just near the critical temperature T~ For sdicon the critical temperature of condensation is equal to 25 K and the critical concentration nc ~ 1 5 x 1018 crn -3 i03 The condensation process in germanium and silicon is presently being investigated in great detail, and one may consider that at least a quahtatlve understanding of the kinetics of the process at high enough temperatures has been achieved Equation (6) gives two values of the drop radius which are m equilibrium with the exclton gas ofgwen density The smaller radius Rmm IS determined by the effect of the surface tension, and corresponds to at an unstable state The exciton condensation induced by the newly-formed embryo leads to some decrease of exciton concentration in the sample, which, however, is not an obstacle to the drop growing into a new stable state with a radius a bit smaller than the larger one determined by eq (6) The condensation rate, however, falls drastically, and some increase of exciton concentration is needed for new embryo creation Therefore, if pumping intensity increases slowly one can expect that only the number of drops will be increasing, while their radii will remain constant Since the small embryo is' unstable, the rate of the new drop formation is proportional to e x p ( - 4 r r f R m , J k T ) In the presence of concentration centers the free energy barrier may be absent, if the exciton binding energy is large enough to form an embryo of the radius of R .... or larger by means of successwe capture of excitons If this condition IS not satisfied, the nucleation on the condensation centers also involves a process in which the Gibbs potential barrier has to be overcome The height of this barrier determines whether the lnhomogeneous nucleation would have slgnLficant advantage over the homogeneous one The great number of experimental facts available now have a satisfactory explanation in the framework of homogeneous nucleation theory A rather interesting phenomenon related to the drop formation mechanism in germanium was found by Lo et al (142) They showed that in a very pure germanium there are two pumping intensity thresholds one, at which the e - h drops are created, that corresponds to a rising pumping intensity, and another at which the e-h drops disappear with a decreasing pumping intensity This observation Illustrates the above° mentioned conclusion that in equIhbrIum a smaller density ofexcitons IS needed to maintain the radius of the existing drops at a certain value than to create new drops This phenomenon was investigated in detail by Westervelt tEa° 231) and Etienne et al (89~ A slow increase of the excitation intensity on the ascending branch leads to the increase of the drop number, while their radius remains practically constant The drop radius on the ascending branch is larger than that determined by eq (6), since to keep the rate of embryo formation constant some additional oversaturatlon is needed In the same way, a decrease in the excitation intensity leads to a decrease of the drop radii, and at the point where the exciton condensed phase disappears the drop radius Is close to the minimum stable one given by eq (8) At low temperature the hysteresis disappears, which is probably due to the fact that even the drops slightly exceeding the minimal stable size are still so small that they cannot be destroyed by fluctuations of the recombination and collection rates of period shorter than the time of embryo formation Since the condensate luminescence disappearance threshold P_ for sufficiently high temperatures corresponds practically to the exclton density from eq (8), the dependence log(P_)vs ( l / T ) is more convenient for deterrmning the thermodynamic drop binding energy ~91~ Previously, (see, e g, Pokrovskll tl 711 the actual slope of the log (P + ) vs ( T - 1) dependence was measured, and considerable discrepancy has been observed between the "thermodynamic" and "optical" binding energy in germanium The optical binding oiiergy was measured as a distance between the e-h and the free exciton luminescence lines The

156

A A ROGACHEV

(a) 30

--

> E T:125K

20

--

I0

--

E o-c LcJ

P+'L--~J" °i 02

0

04

P,

06

(

cm-z

mW

///

30 > E T:I45K >, 420

E

P-a

/j~

Q "1" Ld

I

I

I

02

I

I mW

cm

°07

E

I

/ /u0

zo

c

I 08

z

(c)

3O

t

06

P~

~,

I

04

3

121

0

I 0

I

5;J I

04

I

I

08 Pt

I 2 mW

cm

I

I 16

z

FIG 14 Hysteresis curves of e - h drop luminescence m a very pure germanium (he - n , ,.~ 3 × 109/cm 3, dislocation density 104/cm 2) (a) T = 1 64K, Pm.x = 4 7 4 m W / c m 2, (b) T = 1 45 K, Pm=x = 4 2 m W / c m 2, and (c) T = 1 25 K, Pm=x = 1 58 W/cm 2 (232)

Electron-hole h q m d s m sermconductors

157

"thermodynamic" energy determined m th~s way appeared to be significantly less than the "optical" energy (16 and 22 K, correspondingly) Figure 14 shows the e-h drop luminescence at the ascending and descending branches of excataUon The theoreueal dependence of the drop radms on exotatmn intensity for both the ascending and descending branches of hysteresis is presented m F~g 15 (2sl) The foUowmg e - h drop parameters were used m calculation 6 = (2 7 - 0 0 6 4 T 2) x 10-~ergem -2, E b = (22 8 - 0 19TZ)K, N O = (2388 - 0 0 2 5 T ~) x 1017ern -~

E

f

f

iO 2

I01

up

0

i0 °

o I w i0-1

iO-a

I 0

t

I 20

t

I

T,

I

I 30 K

I

I

I

I 40

~ , , 0

I 20

i

T,

=

= I 30

=

=

I

I 40

K

FIG 15 D r o p r a d m s m g e r m a n m m o n a s c e n d m g a n d d e s c e n d m g b r a n c h e s o f l u m m e s c e n c e h y s t e r e s l s curves for homogeneous (a) and m h o m o g e n e o u s (b) nucleatmn calculated by Westervelt (23~) DlffUs]on-hmlted radms is shown by the dashed curve

The drop size on the descending hysteres]s branch was investigated expertmentally by Etienne et al (ss) by means of luminescence spectra shift measurements At T = 1 43 K, R lS equal to 0 025 #m, and at 2 0 K, R = 0 1 #m, which lS m a reasonable agreement with the data presented in F~g 15 The first mvesUgators of the hysteresis phenomena observed that the luminescence intensity retains its value on the ascending or descending branches for more than 1 hr without swxtchmg to the other branch The hysteresis existence mdzcates not only the metastable character of the eqmhbnum between the e x o t o n gas and the drops, but also that the drops are essentially motionless Indeed, a drop of 2/~m radms diffusing as a free particle is displaced over 1 hr by several era, which is more than enough to leave the excatat~on regmn and to dlssooate or recornbme on the crystal surface The free exc~ton coneentraUon m the excitation zone therewith anereases, and, correspondingly, the rate of new drop formation should increase Recently, Westervelt et a/(2~3) directly demonstrated that the e-h drops remain stationary for a time as long as 104 see The experiment was performed as is shown m Fig 16 A superpure germanium sample (8 x 8 x 4 mm 3) was exoted by a 1 5 kun hght source (nearly uniform exotatlon) Its power (P = 2 x 10-3 m W c m - 2 at 2 1 K) corresponds to the absence of a condensed phase on the ascending branch, and at the same time it lS larger than the power at which the condensate breaks up on the descending branch A narrow strip m the middle of the sample was illuminated wRh an argon mn laser, which intensity was suffioent to cause condensatmn. The excaton system remained on the descending hysteresis branch after the laser switch-off (Fig. 14) If the drops were mobde, ]t should be expected that the region occupied by them ought to expand with tune Over several hours no changes In the spatial dlstnbutmn o f e - h drops were observed (Fig 17) The drop d~ffuslon coefficient esttmated from these expertments was found to be Dp ~< 10 9 cm 2 see- 1 Such a small value of the dsffus~on coeffioent md~cates that m the expenment drops were captured by Lmpunty centers or by dlsloeatmns The lack of the drop moUon in the dislocation free samples mdscates, as It was suggested by Westervelt et al, that the drops are JPQE 6/3 - B

158

A A ROOACHEV tI I ~

Volume XCltOtlOn

f

~ Momentory I l o s e r str=pe [ e x c l ' l ' o t ion

I

I

II

/

Top view

x r

-,

ATolens ]mm

W

H

-~

w

FIG 16

FWHM

ExpenmentalconfiguratlonusedbyWesterveltetal(233)toprovethatdropsaremotlonlessm hysteresis experiments

0 2

2 IOK

~,

0 )

6 6 0 sec

0

8 2 0 0 sec

u~

8m

==

E

J

a

O'1"

'6'OOsec l

I -08

I

I

-04

I

0

X~

04

I 08

mm

FIG 17 Luminescence profiles of radiation m the experiment shown m Fig 16 The profiles were taken after some delay (delay time is shown near the corresponding curves) following switching off narrow strip of laser exc]tatzon

Electron-hole hqtads m semiconductors

159

bound on donor or acceptor centers. The binding energy of a drop as a whole on the hydrogen-hke tmpunty center is (5 - 7) × 10 -3 eV (lsl 203) The intensity ratio of the LA phonon-asslsted to the no-phonon transltions due to arsemc impurity in germanmm was mvesUgated by Volkov et al (22 ~) The intensity of the phononassisted radiation Is proportional to the drop volume, whde the no-phonon radlaUon is proportmnal to the number of Impurity centers ms,de the drops Therefore, the rauo between intensities of the no-phonon and phonon-ass~sted rad~auon may be considered as a measure of the drop number m the sample provided each drop contains only one impurity center Actually, a more comphcated situation was studied where due to a large concentratmn of m~purltles (1016-1017 cm- 3) the e - h drops were bound by clusters oflmpunty centers. From prewous conslderatmns ~t follows that ~fthe e - h drops were able to move, hysteresis would not be observed. Th~s conclusion was supported expenmentaUy by Etienne et al ,(ol) who observed an unusual shape of the hysteresis curve (Fig 18) In contrast with the results of Westervelt(232) hysteresis has not been found at low as well as at h~gh temperatures but was well defined at the mechum temperature range The reason was that at temperatures h~gher than the hehum ~-pomt the heat chss~pated wa the copper holder, which resulted m the generation of phonon stream from the exotat~on spot towards the holder This stream was "blowing away" the e - h drops captured by defects In the other case, when the sample was freely mounted m hqmd hehum, no chrected stream arose, and the drops remained moUonless Thus, the nucleation model describing the e - h drop formaUon as a process lumted by the embryo formation ratd ~99 2oo eso 23~) ~s m qual~tatwe agreement with many experimental facts However, a good deal of experiments related to the drop s~ze measurement at low temperatures are inconsistent with th~s theory Before going into a detaded dlscuss~on of these facts, let us consider the fundamentals of the Westervelt theory, which ~s essentially a further development of the Sdver theory on the basis of the embryos formauon rate equation by Becker and Doting (so),

iO z'

¢=

\÷Jl

\\~+ o

P_

I0 z

P+

Exc=taf=on,

A U

\\

,,\ \

\

• \

+

+ \o \* \ eo \

\

I 02

I

I O4

4. 4II

\

\

\

I

I 06

I/T,

+ O,II

I

I 08

K i

FIo 18 Temperature dependence of the formation P+ and breakup P_ excltaUon mtensmes of hysteresis luminescence curve Sohd hne corresponds to the slope of 16 K, dashed hne to 23 K The experiment was made m the condmons when a flux of unequd~brmm phonons was created m the sample above 2- point ~91) * A contemporary presentaUon of the theory of the homogeneous condensation is g~ven by Abraham (2)

160

A A ROGACHEV

The following physical model Is proposed In a spatlally-umform free excltons (FE) gas the drops containing v e-h pmrs are formed from free excRons by means o f a successwe capture of excRons m a c c o r d a n ~ with the reaction EHDt~_I~ + FE = E H D ,

(16)

There is also an opposite reaction describing the evaporation of one exclton EHD~ = EHD{~_ 1~ + FE

(17)

As opposed to the case of classical hqmds, the e-h drop radms can be decreased also by recomblnatmn of the e-h pairs reside the drop with a rate of v/z d, where Zd IS the e-h pmr hfetlme Next, it is assumed that all drops of size v > 1 are energeUcally stable and can be a part of the excaton capture and evaporaUon chain due to which large drops are formed. Since the embryo creatmn is thought of as a rare event, the e m b r y o - e m b r y o colhslons are neglected. Besides, no prowslon is taken for the posslbdlty that the embryo may disappear by coalescing into a larger drop Omission of this process m a y bejusttfied only at the very beginning of the condensation process At sufficiently high e-h drop concentrauon this process may be a major obstacle for new d r o p format|on We shall consider this m some detail later m the section The rates of these three processes (capture, recomblnatmn and evaporation) determine the evolution of both the size distribution, gv c m - a, and the net rate J~ c m - 3 sec- ~ at which drops grow from size (v - 1) to v g, and J~ are determined by the following set ofequatzons

J~ = g~,_l~Tr ~ _ l ~ v n ~ - g, nR~l~n~exp

~-v

Ng/3kr

+

dg~ - J~ - J ~ - ~ l dt

(18)

where

/8kTI1, 2

,Im*kTIJ,2

, E,

E b Is the excaton binding energy m a large drop, and 7 is the degeneracy factor of the excaton In germamum, ~ = 1611 + e x p ( - A J k T ) ] and m slhcon 7 = 2411 + e x p ( - A c , / k T ) ] The excaton ground state sphtting Ac, is 10 -3 eV in germanium and about one third of this m silicon So at low temperatures, the degeneracy factor in g e r m a n m m is 16, with a good accuracy In the steady state, dg,/dt = 0, and also Jv = d = J+ - d does not depend on time and on v as well The dynarmc equlhbrmm state is defined as J + - d_, where J + and J _ are the rates of formation and destruction of drops, respectively J + and J _ calculated m such a way can be apphed also to the description of the ttme dependent processes if the excRaUon rising tmae is large compared to the excaton lifetime determined by the capture of the e-h drops ~'= 1/OrR2-~Nd) (where R and N d are the radius and concentration of drops), and consequently the time of removal ofexcltaUon is much larger than the e-h pair hfeume m the drop Besides, the total number of particles m the drops and the exclton density are related to each other by the conservation equation G = -nex NdV - + -rex

(19)

Td

where G is the e-h pa~r generation rate As has been already mentioned, the drop radal on the ascending and descending branches of the luminescence intensity curves are presented m F~g 15 The drop formation and destruction rates m g e r m a n m m as a function of supersaturation degree x = In (ne~,/n~)are shown m Fig 19 As the excitation level rises, at some power P+ the excaton density reaches the value n + at which the embryo formation rate becomes sufficient

Electron-hole hqtuds m semiconductors

161

iOt;. 200K

/

104

JJO "4

o

, I

'

I

I

I

1

I

2

I

3

X 10 )2

I 50K

t/> ,p

/

10 4

10 4

I

0

I

I

2

4

I

X i012

/ 125K

10 4 J÷

IO - 4

, I

0

I

I

2

4

J_

I

X FIG 19 F o r m a t m J+ and break up J _ rates ofe-h drops m germamum calculated by Westervelt(2al) as a function of supersaturatmn degree x = ln(ne:,/ns) of exclton gas

for creation of a detectable number of drops The drop formaUon rate increases rather sharply with an increase of the supersaturation degree x (Fig 19) Providing that the excRatlon level vanes slowly enough, only the number of the drops grows with increasing excatatlon, while their radius remains practically constant and equals to 3 nL-n ~

R+ = ~ v - - - - ~ 0 zd

(20)

If the excitation intensity is decreasing, the drop number remains constant, at first, while their radms decreases down to the value defined by eq (8)--1 e the mlmmum stable radms With further decrease of the excitation intensity, destruction of a certain amount of drops maintains the excaton density at a level close to n~-x At excatatlon power less than P_ the condensed phase disappeared This conclusion ofthe nucleation theory was checked by means of a rather interesting expertment, in which the drop radms was not measured directly The experimental idea is as follows If the excitataon power is decreased but stall maintained above the value needed for initiation of the drop destruction, and then is increased again, no hysteresis will be observed, because the number of drops remains constant and only their sine changes But If the excRatlon intensity is reduced beyond some threshold value when a decrease m the number of drops occurs, the power increase wdl be accompamed by an increase of both the drop number and their size, and the total emlsslon intensity o f e - h drops will be diminished and correspond to some level between the ascending and descending branches. The experimental results are shown m Fig. 20 The fact that the drop number is constant m the begmumg of the descending branch was confirmed m a somewhat different way by Etienne et al (a9)

162

A A ROOACHEV

> E

c

8

¢:

c

E

4

L-

P-/

/ I I

,

I.O bJ

I

I

I

I

04

Time, I

I

08

,

s I

I ~'

m W cm -2

FIG 20 Luminescence intensity as a function of exciting power P2 in the excitation circle shown in the insertion Luminescence lntenmty is measured dunng the period of ttme labelled as " a d d " Pm,~ iS chosen to be equal to 4 72 m W / c m 2, well above the formation threshold P +, a n d P = 1 74 m W / c m 2 is just below it If P2 is taken below some critical value Pc a number of drops, b, will be destroyed and luminescent intensity at exotaUon level P will be smaller For P2 > Pc this luminescence intensRy in dependent of P2 (232)

However, it should be noted that actually the agreement between the theory and the expenment is not so good as was menUoned by Westervelt et al (232) The results of the drop radms measurements from the e - h drop luminescence hne shift (9°) show that at T = 1 85 K, when the hysteresis m well pronounced, the drop radius stgmficantly grows on the ascending hysteresis branch As follows from Fig 21, the drop rachus does not remmn constant e~ther in the vicinity of their destructmn threshold. And what is more, the drop concentratmn on the ascending hysteresis curve even has a tendency to decrease (F]g 21c) In this case, the calculated and measured supersaturatmn degree values at both the formatmn and breakup thresholds differ significantly (90) AS

080

(C) s

060

s

.." A

...J" s

A[ P3

o 4C

..,"

"

O2O

E ::L

...-cA

(=)

040

u~ O3O

P:l

-" "

0 20

pa --~

•

010

(el

+

015

IP?

_ ~.-

+

~p

÷

.

0O9

o

_ .~A

~o

:o FJ,

----

,'~o

,go

orbll"rory un)t"s

Fzc 21 e-h drop radms m germanmm as a function ofexcttauon power Arrows show the formation and breakup thresholds (a) T = 1 37 K , (b) T = 1 62 K , (c) T = 1 85 K (9o)

Electron-hole hqulds m semiconductors

163

The question whether the drop radms is constant on the descen&ng hysteresis branch is of tmportance m the experimental determination ofthe surface tension coefficient a Westorvelt et al determined this coefficient at T = 0 K as (2.6 _+ 0 3) × 10- 4 erg c m - ~, and the t l n d m g excitation energy in a drop E B = (1.9 + 0 2) × 10- 3 eV, while Etienne et al (9o) argued that a = 3 × 1 0 - * e r g c m -2 and E B = 2 × 10-3 eV, would be more appropriate. Investigations of embryo formaUon In silicon show that In general there is an agreement between the expernnents and the nucleation theory.a3 19s) The essential &fference is that m silicon the Ume of the hysteresis existence does not exceed 10- 3 sec, which apparently results from a large e - h drop mobility However, experimental data about the drop size in slhcon at present are practically absent At low excitation levels, the nucleation theory predicts drops of submlcron size at low temperature, which was experimentally confirmed from the luminescence line shift r~asurements

The light scattenng expermaents give, as a rule, slgn~cantly larger drop sizes For example, in the first expertrnents on the light scattenng by e - h drops, Pokrovskii et aL (170) found (at T = 2 1 K) that near the sample surface the drop radms is 7 6/an, but at some distance from the surface it is 3 4 #m. Using at least two orders of magnitude greater surface exatatlon by an argon ion laser (P = 0 1 W, 2 = 5145A) focussed Into a narrow strip, Worlock et a/(236) found radius values of 4/am and 2 5/am ( T = 2 K) The larger radius value is for the probing light beam passing near the surface, and the smaller one corresponds to 1 mm depth W]th the laser beam focussed into a spot of a smaller size m s~mllar experimental s e t up, (22a) no dependence of drop size either on the laser beam power or on the distance from the sample surface has been detected. The drop radms at T = 2 K was determined as 2 + 0 5/am. Bagayev et a/(46) employed a H e - N e laser (2 = 1 52/am, T = 2 K ) of a rather modest power ((3 8-8) × 1 0 - 3 W ) for a quaswolume excatatlon and obtained a drop radms of 3-5/am at T = 2 K Besides, it has been found that the drop radius depends also on the rise ttme of the pulse f r o n t - - t h e steeper the front, the smaller the drop radius This conclusion is m a qualltatwe agreement with predictions of the nucleation theory, but the quantitative descrepancaes are indeed very large Westervelt (23 t) calculated the drop radms on the ascending hysteresis branch as 1/am at T = 2 I~ It is questionable whether the nucleation theory m its present form may be applied to e - h drops in sdicon at low temperatures, since the theoretical drop radius is about several tenths of an angstrom Ttus value is far beyond the hmlts of apphcablhty of the theory, which demands it to be much larger than the excaton radius It is possible in this case that the "bottleneck" of the embryo formation is the transition from a multaexcatatlon complex containing at most six excatons to a larger drop, smce a drop contmnlng an Intermediate number of drops may be virtually unstable

8u E

4

y+/+++++

0

I-

GT,2

I0 ~¢m "3

t I0

Power dens)ty,

I0 z

in mW mrn"z

FIG 22 Drop radms m germanium measured by means ofp-n junction current pulses technique by G r o s s m a n et al (97) Sohd curve Is the calculaUon by eq (25) assuming that exc~tatlon power of 10raW/ram 2 corresponds to G~ = 2 x 101S/cm 3

164

A A ROGACHEV

Altuhov et al (I2) pointed out that in the very moment of its appearance, the e - h drop luminescence line m silicon shifts towards the short wavelength side of the spectrum by 1 x 10 -a eV This shift enables to estLmate the mmLmum drop size as 300A at 2 K Detailed analysis of the discrepancies between dtfferent drop size measurements is beyond the scope of the present review However, it should be mentioned that at some conditions (immobile drops, low excitation level) it is actually possible to observe on the descending hysteresis branch drops with small radius, which IS in reasonable agreement with the Westervelt theory (2al) On the other hand, the experiments with significantly higher excitation power show much larger drop sizes Westervelt suggested that drop radii obtmned in such conditions are most likely determined by mechamsms other than nucleation At the very beginning of the condensation process, when the number of drops is sufficiently small (1 e at low excitation levels), one may neglect the embryo capture by the already formed drops, as it has been previously pointed out in this section When the drop density increases, the rate of embryo formation slows down due to the capture In order to demonstrate the unportance of embryo capture by the e - h drops in the process of stat|onary drop radius formatmn, let us make a stmple calculation assuming from the very begannmg that the supersaturation degree x = In (nex/n~) is very large, so that 41r g ~ N o x ~ 1 3

~__~ -_ 41rR2a kT kT

(21)

If we put R e = r h, the condition (21) will be satisfied in silicon at T = 2 K and n > 10 ~3 cm - 3 Let us assume also that each free exclton may act as a condensation center Hence, after time t such a drop will have radius R R = "3"~nex , v ~ -° (t + At)

(22)

where A t = 4 rh N O 3 v nex

is introduced m order to account the initial embryo radius R e -- r h As the embryo size increases, its thermal vdocity decreases and corrcsponchngly the probability to be captured by the drop also decreases It is d e a r that only that part of embryos will survive which exist long enough to grow up to such a size where their capture probablhty becomes negligible Due to hfetune fluctuations some embryos will exist much longer than their average hfetLme and develop into large size drops The average number of embryo capture occurring during time t IS equal to

Flr l3 nexl 3

011/2 dt--/~ LT IgvN00] S ~ (t + At) 3/2

.r-~=

(23)

where N d = (3Gza)/4nRaNo), G is the exciton generation rate, and z d is the e - h pair lifetime in a drop This Integral can be calculated with the upper hmlt extended to ~ Then ]2 =

j.1/2 ]721~.3/2 "h "" "'0

The noncapture probability for the average number of captures/~ is approxtmately equal to e - " Assuming an equality between the embryo formation rate and the drop coalescence rate, and onntting the numencal constants of the order of unity, one obtains R2

Crr2D

(

N S / 2 R 6 5~ -1

="/2~a/----------~ ] "'0' ' In h ~ where l0 = ~ ,

/

and zp is the time for which a drop loses its momentum

(25)

Electron-hole hqmds m semiconductors

165

!

For germanium, Zp was calculated by Keldysh et al. (132) Substituting for N O and z d the values cated by Westervelt,(231) one obtains for germamum R = 2 2 # m and for sdlcone R = 0 1 #m. In case ofgermamum, Gza is taken as 2 x 1015cm- 3 and for sthcon, 1016 cm- 3 The drop radius as a functmn of excatataon power was mvestagated by Grossman et al {97)by means of the grant p - n junctmn photocurrent fluctuatmn techmque The results of their expermaent are shown m Fig 22 The dashed curve represents the value of drop radms calculated by eq (25), where the generatmn rate G was used as a fitting parameter As one can judge from this figure, eq (25) satisfactorily describes the drop size dependence on the excatatmn level In silicon there is apparently a back-flow of e - h pmrs due to energy released by the Auger recombmatmn reside the drop, and consequently the drop radius should be smaller than that defined by eq (25) In the derwahon ofeq (25) it was assumed that all drops with v > 1 are stable and a large stze drop may be obtained by successwe capture of excitons Therefore the h]ghest possible rate of embryo formauon was assumed If, for certmn v, the drop is unstable, then some fluctuatmn mechamsm is needed to bridge the gap How the system passes through this regmn of instability is one of the main problems of condensatmn theory In the present calculaUons th]s problem is bypassed by assuming the absence of such a gap Any reduction m the embryo development rate increases the drop s]ze In any case, the embryo generatmn rate drops sharply with exclton densRy decrease, and the drop radms is determined by that value of exclton density for which the new drop creatmn rate becomes small enough for the maintenance of the new steady state

0

02

04

06 p,

08

I0

12

mlTI

FIG 23 Drop-cloud profiles m germamum determined by scanning of the sample by narrow He-Ne

laser beam (2 = 3 39#m)(153)

One should note that the measurements of the drop radms at high excitation(22a) were performed for different exc~taUon levels rather than at different e - h pmr density The dispersion of the e-h cloud w a s f o u n d (77" 153,172) at high excitation level This fact can be explained by the growth of the volume occupied by the e - h cloud vath exclton mtenmy wh]le the average density of the e - h pairs m the cloud remains of the order of 10is cm -a The spataal ¢hstnbutmn of the e - h drop density m the wclmty of the focussed laser spot obtained from the absorption measurements of the H e - N e laser (2 = 3 36 #m) probing radmtaon(153) is shown m Fig 23 Figure 24 shows the process of cloud size growth with nsmg excRaUon level The data in Flg~ 24 were obtazned from the investigation of the luminescence

166

A A ROGACHEV

•

a

-

"J

'°f2

"

"% •

o

°

/I

I

.., ! •

I

I

06

I

04

I

I

02

0 I,

FIG 24

Expansion

/

o,oo

;i!

102|

08

270J

o f e-h d r o p c l o u d in g e r m a n i u m

I

02 mm

/ 1

"'° 14 mW

I

04

I

06

I

08

with the increase of excltatmn

i n t e n s i t y (172)

distribution profiles (172} The e - h drop motion away from the excitation region was attributed to the "phonon wind" by Bagayev et al (45) The nature of this phenomenon will be discussed in detail in Section 9 of the review The fact that the volume occupied by drops changes with increasing excitation level explains, at least quahtatlvely, one puzzling feature of the e - h drop luminescence behavior which was pointed out in many of the earlier works on exclton condensation It was found by Pokrovskn et al (169} that the e - h drop luminescence line intensity I a depends on the free exciton hne intensity Iex as I a ~ Ie~ This fact was used as an evidence that the I a line belongs to the condensed phase As pointed out in the first section of this review, this lme was onginally identified by Haynes (105} as the exeaton molecule line, and therefore a quadratic rather than cubic dependence should be expected between I d and Iex The explanation of the cubic dependence proposed by Pokrovsku et al (169) was based on the assumption that the number of drops is constant, which is to be sure a rather tenuous supposition from the nucleation theory point of view The latter suggests an almost constant drop radius, rather than constant drop number Another fault of the model based on the assumption of constant drop number was the impossibility of assessing both the temperature and excitation dependencies of the I a and I ~ lines Meanwhile, this model was used in some earlier papers (see, e g, Vavllov et al (225) In this situation, the author decided that the lesser of the two evils is to invent a third one--namely, the coexistence of metallic and molecular phases (181} The large amount of other expermaental facts, however, forced the abandonment of the assumption of the biexcltonic nature of the 1 08 eV line in stlicon and the 0 709 eV line in germamum Recently, Edelstein et al (ss} and Thewalt et al (211) obtmned an indication for the e x o t o n molecule exmtence in silicon The weak broad line near the free excaton line was attributed to a biexciton From the line's position in the spectrum Thewalt et a/(21°} determined the binding energy as 1 2 x 10-3eV The experimental data showed that the radiation linewidth increased with rising temperature, but remained finite even at very low temperatures The biexcaton line broademng mechanisms in indirect semiconductors were summarized by Rogachev (1s0 is 1) In the process of radiatwe annihilation of one of the excltons forming the

Electron-hole hqmds m semiconductors

167

molecule, a part of the energy can be transferred to the other excaton due to which a wide radmtlon band rather than a narrow line is observed Thts process is possible, since the radlatwe transition involves a momentum conserving phonon. The radiation band w~dth ~s determined by the biexcaton binding energy and is of the order ofh2/(m~xr2y), where rey IS the effectwe blexcaton radius For the long wavelength part of this band the most probable is the mechanism Initially proposed by Haynes," o5} in accordance wRh which annftaftatmn of one of the excatons leads to the iomzatlon of the other The characteristic intensity decrease of the long-wavelength"tall" with decreasing photon energy may be explamed by this process The presence of such a "taft" Is typical for the recombmat~on radiatmn of any of the manyelectron systems The first example of such behavior was observed by Jones et al (x1s) In the soft X-ray emission spectra of metals. The "taft" of the same nature was found by Asnm et al (24) near the 0 709 eV line in germanium However, the original assumption was that this radiation line belonged to the blexclton 5 THE NET CHARGE OF THE ELECTRON-HOLE DROPS For germanium the work function of the electron in the e-h drops is larger than that of the holes Consequently, the hole thermoemlSslon exceeds that of the electrons, and to make their fluxes equal to each other the drop should acqmre a negative charge, as was calculated by R i c e ( 1 7 6 } and Renecke et a/(175} The difference between the work functions of electrons and holes, A/~, is due mainly to the fact that the Fermi energy of electrons is essentially smaller than that of holes More sophisticated theoretical calculations by Kaha et al (119 120) show that the forbidden gap shrinkage inside the drop should be dwlded evenly between the conduction and valance bands The simplest esttmatlon of the drop charge, Q, when the image force lowenng of the potentlal barrier ~s neglected, may be obtained by equating A# to the drop surface potential Q=

A#eR e2

(26)

For a drop radms of 5 # m the best available theory gives Q = - 2 7 e (~2o) If the drop is m thermodynamic equihbnum wRh the e-h gas the problem becomes more comphcated, since one has to take into account the space-charge region screening of the drop charge If the screening radius r a = x/(ekT/8~te2ne) (where ne IS the e-h pair density in the gas, equal to 2 x 1012 cm - 3 at T = 4 2 K for germanium) is much smaller than the drop ra&us, the drop charge increases, since the potential difference between the drop and the e-h gas remains wrtually constant For this case, Rice{176) obtained

Q

2kTeR~2smh{A#) ro

(27)

The value of A# estimated by him was - 0 17R*, and correspondingly, Q = - 5 2 5 e at R = 5 x 10 -4 cm and T = 4 2 K More revolved calculations made by Kaha et al (~2o) gwe Q = - 700e One of the results of the Kalm et al (119 12o)theory is that for an uniaxlal stress in the Ell 1 ] direction exceeding 300 kg c m - 2 the drop charge changes sign At this pressure all electrons transfer into one of the four germanium conductmn band valleys, while the valence band sphttmg remains insignificant At larger pressures (103 kg c m - 2) the valence band sphtting occurs and the electron Fermi energy again becomes smaller than that of the holes A new change of sign should take place The problem of the e-h drop charge was investagated experimentally by Pokrovskn et al,(~72 173) Hvam e t a / , (116) Nakamura et a / , {162) and U g u m o n et al (219) Pokrovskn et al "72 17a) observed a spaUal shift of the luminescence region due to an applied electric field. The sample was excated by a focussed 100 m W H e - N e laser beam The experimental results are presented in Fig. 25 The shift of the luminescence region towards the anode enabled the estimataon of the drop charge value as - 10e, which was in a good agreement with the theory predictions The theoretically predicted charge sign reversal was observed at a pressure of 230 kg c m - 3

168

A A ROOACHEV 50

I

I

I

I

0--I e--2 A-- 3

20

0

~o

/,

C

8

/j

.J

/

!

J 6

I o

4

I 0

-o t.

I

I

02

04

06

mm

FiG 25 Shift of e-h cloud in g e r m a n i u m u n d e r the influence of the apphed electric field (a) E = 0, (b) E = 2 5 V/cm, (c) E = - 2 5 V/cm (172)

Nakamura e t al made the same kind of measurements, with the only difference that a p - n .lunctlon was used as a drop detector, but they did not observe a sign change Moreover, it was shown by Ugumorl e t al (219) that the &rectlon of the e - h drop &splacement depends on the sample doping The exc]tatlon source was a Q-switched YAG laser with 10- a sec pulses The sample temperature was maintained at 1 5 K In relatwely pure samples (n d < 1013 cm -3) and in p-type samples (n A = 8 x 1014 cm- 3) the drop charge was measured as negatwe The most bewddermg result was that m n-type samples (1013 cm -s < n d < 2 x 1014cm -3) the drop charge sign was posltwe The charge value was not determined, because of nonlinear dependence of the &splacement on the apphed voltage The authors gwe no explanation why the sign of the drop charge m acceptor-doped samples ]s opposite to the sign of the donordoped ones It is worth menuomng that no charge was observed by Hvam e t al (116) The good agreement of the expenmental data by Pokrovskn e t al (172) w]th theory is rather surprising The theory actually deals with the case when the density of free carners surrounding the drop is neghglble, or when there is thermodynamic equlhbrmm between the drops and the e - h gas The condmons of the experiment were qmte different from this Under continuous excitation, there is a relatwely large density of free electrons and holes The capture rate of free carriers is charge-dependent m such a way that the positively charged drop captures more electrons than holes, and vice versa So there is a tendency for the drop to be neutral The question Is whether the existing concentration of free carriers is large enough to overwhelm the effect of thermal evaporation 9 At 2 K the equlhbrlum free carrier density is about 109 cm- 3, therefore if the density of nonequthbnum free-careers markedly exceeds this value the capture process will dominate m the charge balance To estLmate the free carrier density under stationary excltataon, let us consider a set of equataons which describes the exclton formation by free carriers and the capture of excatons by the e - h drops dne dt dnex

6Dn 2 - nNdR21)n~ + G

- 6Vn 2 - ltNdR2Vnex

dt ~nNaRaNoz~ 1 = G

where 6 ]s the exclton formation cross-sect]on, and

N d

(28) iS the concentration of the drops

Electron-hole hqutds m semmonductors

169

Here we also assume the local neutrahty condmon--1 e. n = p = ne This set of equations, however, does not account for the thermal dissooatlon of exotons and drops, and for the sake of simplicity we assume that the thermal velocmes of excatons, electrons and holes are equal If the thermal dissoclauon processes are unportant the free carrier density is close to the eqtullbnum one The solution of eq (28) for small G is = [ G I '/2

ne

~vv!

(29)

At large G the free carrier density n e tends to the value ne

4RN°

3 Vra

(30)

The eqs (29) and (30) have a simple physical meaning at low excitation levels one may neglect the electron and hole capture by the drops. On the contrary, at high excitation levels electrons and holes arc trapped by the drops before they have managed to couple into excRons The exclton formation cross-section 6 m germamum has not yet been measured. To estimate the mmLmum value of free carrier density it is reasonable to suppose that 6 does not differ markedly from the highest known experimental value of the electron capture crosssection of the charged shallow donors, 6 = 10-l0 cm-2 (a re~v'lewis given by Abakumov et al (1)) Substituting this value into eq (29) we obtain (for G = 102°cm-3sec-~), ne = 7 x 10 ~~ c m - 3, which greatly exceeds the eqmhbrmm electron and hole concentration at 2 K Thus, one may conclude that at low temperatures under stationary excitation the carrier thermal emission cannot be responsible for the drop charge. In the case of pulse excatatlon, when all free carriers are bound into excatons and/or e - h drops, In pnnctple, at the moment lUSt after a pulse, the conditions can be achieved for the carrier thermal emission to be significant However, It should be taken into account that m the e - h drops the Auger process is the dominant recombination mechanism The mean path of the Auger electron m germanium at which it loses Its kinetic energy is of the order of 1 #m The fast particles created by the Auger-electron having energy close to the electron or hole work function have a free path of about 100A Hence, if only 0 01 of the Auger-electron energy is transferred to the e - h plasma, at least 10- 3 of all of the recombination acts inside a drop o f about 10/~m m diameter would lead to the emission of electrons and holes from the drops But even m this case, the nonequfllbnum electron and hole density wdl be of the order of 101° c m - 3 or larger This estimate is m correspondence with the cyclotron resonance data of Hensel et al (lo6) Beanng m mind the dhfficultms which are present m the interpretation of the results of experiments m which the drop charge was observed, one should not exclude the posslblhty that the charge of the e-h drops Is due to the d~fference between the thermal velocatms or stroking coeflicmnts of electrons and holes, rather than to the different evaporation rates The t r m a l reason for the drop movement when some voltage is applied to the sample could be due to the exclusion and accumulation type contact effects 6 MAGNETIC PROPERTIES OF THE ELECTRON-HOLE LIQUID An e - h hqmd in semmonductors is essentially a model of a metal zn which, because of hlgh values of the dielectric constant e and the small effechve masses, the charactenstm atomic umt of length for the s y s t e m - - l e the effectwe Bohr radms r h = eh2/e2m * (where m*- 1 = m~- t + m~- 1 and me and mh are the effectwe masses of the electron and hole)--lS several tens of a lattice constant Therefore the characteristic energies of such hqmd are three or four orders of magmtude smaller than in normal metals This model provides a unique possibility for the study of the strong magnetic field mfltmnce on the electron structure of metals A umversal measure of the magnetm field strength for any atomic system is the ratio of the effective Bohr radius to the characteristic magnetic length, ln = (ch/eH) 1/2 In the case of metals, the magnetac field influence may be described by the two dimensionless parameters (1) the ratio of the paramagnetm gl~H/E r, and (2) the diamagnetic hcoJE r splitting to the FerlTli energy; where/~ m the Bohr magneton and coc = eH/mc is the cyclotron frequency For

170

A A ROGACHEV

InCr mslc Ge

Vofgt" -- conhgurohon H II [IOO] unpolorlzed T: 2K / ~xlO

//,

\J\

'f

i 6'i=Ai~~ ,I-/%~H165 kG~ L~ 14/ / \ 30/E Hi'r°~ J~iEHDt'Jt~ F~E~.:H~k:~HD, r.i 695

700

705

710

7i5

Photon energy,

FiG

26 e-h

drop

luminescence

in

720

725

730

meV

germamum m the configuration) (2o7)

strong

magnetic

fields

(Volght

semiconductors the typical values of the g-factor and coc are about equal to or even greater than that of the normal metal. Therefore, it should be expected that in qmte moderate fields of a few tesla, the e - h liquid has to behave as normal metal would m a field of 104-105 T Recent developments m the study of the magneUc propert]es of e - h drops m germanium and sd]con were d]scussed by Stormer et al (207) and Altukhov et al (7) The first experimental study of the e - h drops luminescence m strong magneUc fields by Alekseev et al (4) showed substant]al &amagnet]c shift accompanied by the sphttlng of the line on two components The evolut]on of complete e - h drop luminescence spectrum m magnetic field is shown in F]g 26 (Stormer et al (207)) The spectra consist of TALA, and TO phonon replicas The more comphcated line structure was found m experiments performed vath unproved spectral resolution by Martin et al (lsl) Although this structure undoubtedly results from Landau sphttlng of the conduction and valence bands, the meamngful ldenttfication of the observed maxima with definite types of transitions is hardly possible, due to the extremely complicated character of the valance band splitting An additional important Information about the splitting of the bands as well as the magnetic properties of the e-h liquid was obtained in the study of the circular and linear polarized luminescence.(26 2s 206)Figure 27 shows the spectra of polarized luminescence taken in both Faraday and Voigt configurations, with one magnetic field being parallel to the [100] axis of germanmm crystal For this direction all four conduction band valleys are in the equivalent positions, so that the magnetic field does not hft the valley degeneracy For any other magnetic field orientation, cyclotron effective masses of different valleys are not equal and diamagnetic shift puts the different valleys into unequal energy positions Re&stnbutlon of electrons among the valleys caused changes in the ground state and binding energies of the drop This effect was experimentally observed by Vltlns et al (226) At a magnetic field o f H = 18 8 T they found the binding energy E~ = 1.4 _ 0 4meV and 0 8 + 04meV for H parallel to the [ l l l ] - a n d [110] axes, respectavely The absorption of recombination radiation in mdtrect semiconductors such as germanium and silicon is very weak Together with the large refraetwe Index it leads to a multiple scattering by crystal boundarms, which result m considerable depolarization of the radiation The depolarization factor may be defined as the ratio of the expertmentally observed to the initial polarization value Thhs factor is very sensRwe to the sample surface treatment For samples with carefully polished and etched surfaces the depolanzataon factor may be as high as 0 3 or 0 5 The data presented in Fig. 27 was not corrected for depolarization The

Electron-hole hqmds m semiconductors

EHDLA f pureGe , Forodoy /' conhgurot,on H II I00..//

,

171

/" ---Tf ---o- -I~ / --o" Volgf ,' / ~, ~t \ conf,gurohon// \,\

'~

i\

o f

14

14

-

z

c

~ )

,

~4 kH i

705

h

,

,

,

m..:,

710

O,

~ , i

,

,

'~H40 ,

,

,

,

,

I T~,i i I I I I

715 ;'05 710 Photonenergy, rneV

715

FIG 27 P o l a n z e d luminescence of e-h d r o p s in magneUc field Line's m a x i m a are n o r m a h z e d to the c o n s t a n t height (207)

luminescence spectra for different polanzaUons may be readdy calculated provided that the sphttmg of conduction and valence band levels is known The conduction band splitting is rather smaple The energy levels of every particular valley form a ladder of paramagnetlcaUyspilt Landau levels For the [100] direction, the sphttmg Is the same for all valleys (see Fig 28) The calculation of the valence band sphttlng is an Immense mathemattcal problem, even without taking into account the many-body effects The valence band sphttmg was calculated by J Hensel and S Susukl m an unpubhshed work, and some of the results are shown m Fig 28 as they were presented by Stormer et al (207) The effective masses and the carrier density may be used to fit the width and the structure of the observed lines In such a way, it was found that for the e - h hquld in germanium the earner masses are about 10~o heavier than the oneparhcle band masses It should be mentioned, however, that the mass renormallZatlon is just one of the manybody effects The relatwe posmon of the magnetlcally-spht levels also should be calculated m the framework of a many-body problem Later m this section, the g-factor enhancement wxll be discussed, which is essentmlly an example showmg that the one-electron approxmaatmn may be more than 100~o In error when it is used to predict the magnitude of magnetic sphttlngs Figure 29 presents the magnetic field dependence of the forbidden gap, the excaton binding energy, the chemical potentials and the forbidden gap value mslde the drop The kmettc energy reduction gwes probably the leading contribution for the increase of the drop's density m the magnetic field. Expenmental results by Stormer et al (207) are shown m Fig 30 The gigantic magnetostnctlon of the e-h drops may be readily understood from the scahng conslderatmns mentioned at the begannlng of this section. Magneto-oscillations in the total luminescence intensity were the first experimental confirmation of the metalhc nature of the e-h drops in germamum obtained from magnetooptical expermaents (,3,44) Karuzskn et al (125) observed that the hne width of the e - h drop luminescence is also an oscdlatmg functmn of H, but has an opposite phase Experimental investigation of these osctllatlons presents a serious problem, since in the presence of a much stronger gradual decrease of luminescence mtenslty with nsmg magnetic

172

A A ROGACHEV

IO

E

E

~

5

W

~A

•

5

E

E ca

w I0

w

2 Wavenumloer,

I I0 6 cm

O

)

t

Density

3

2 of s t a t e s

i017 m e V i c m

F I G 28

Conduction

3

( u p p e r p a r t ) a n d v a l e n c e ( l o w e r p a r t ) b a n d s p h t t l n g in t h e m a g n e t i c field (20-7)

730

E

720

0_ 710

O

5 Mogneflc

I0 flux

15 dens)ty,

20

T

FiG 29 Magnetic field dependence of free exclton posmon, chemical potential/~ (LA) and forbidden gap width inside and outside the drop (straight solid line) (2o.7)

173

Electron-hole hqmds in semiconductors

,~ ,0

,~

T -

T,25

L

P~I

t~/Y J

I

0

i

i

i ,5

i

l

i

i

I I0

I

I

I

I

I 1.5

i

i

i

i 20

FIG 30 e-t= drop density as a function of the magnetic field (2o~)

field their amphtude does not exceed 10% Taken together these results are m agreement with the observation of Keldysh et al ,(13 ~) that the density of the drop reaches its maxtmum when the electron Fermi level ]s crossed by the Landau level Assuming that drop hfetune is determined by Auger recombmatmn, more strmghfforward mformatmn about density varmtlons may be obtained from hfet]me measurements (lz6) T~me-resolved measurements (58) show very interesting magneto-oscillation behawor (see Fig 31). Oscillations vamsh at certam delaytlme t = 12/zsec, and then reverse m phase Since drop recombmatmn is determined by Auger process Its rate should be proportional to N3o,while radmtive decay is proportmnal to No2 For that reason, recombination intensity under stationary excitatmn will pass through a mmunttm every ttme the Fermi level is crossing the Landau level In the tune-resolved expermaents, anmedmtely after the end of the excitatmn pulse the intensity of radmt]on (since It is propomonal to N~) should have a maxunum when the levels are crossing On the contrary, after a delay of the order o f ~ the muatmn will be stmflar to that of the stationary excltatmn and, as a result, the oscfllaUons change phase Variations of the e - h drop density observed experimentally appeared to be about an order of magmtude less than it was predicted by the theoreucal calculatmns by Kelchsh et al (tat)

t-. 7p. sec

22 ~ " " - -

~

--I

3a

I v ,

.... I

I

H.

I

I

I

kG

FIG 31 e-h drop luminescencemtonmty as a ftmct]on of the mag,etlc field tak¢~ from ddt'eR~t do]ay after the end of excitation pulse The curves were arl~tranly shifted m vemcal dlrcctmn (ss) JPQE 6/3-

C

174

A A ROGACHEV

As was mentioned above, the lummescence intensity I 0 measured ~mmedlately after the end of the excitation is increasing with rising density of the drop, whde the drop hfetlme becomes shorter Thus, I o depends only on the quantum efficiency of luminescence, while the integrated intensity is a function of both the quantum efficiency and the total lwet~me % This fact enabled Betzler et a/(Ss) to esUmate the quantum efficaency as equal to 0 25 Their calculations are based on the assumptmn that the intensity of radlatwe recombination is proportional to N 2 It ~s not exactly so, because stnctly speaking it should be proportmnal to the square of the absolute value of the e-h relatwe motion wavefunctlon taken at zero I~(0)[2N0 The dependence of this value on N is considerably weaker than N 2 (71) But It probably &d not affect the final result significantly, because the same correction should be apphed to the rate of Auger recombinatmn which may be regarded as proportional to [~(0)lZN g The value of the radiatwe recombination coefficient was found to be B = 3 × 10- ,4 cm 3 sec- 1, which IS quite reasonable for indirect semiconductors The linear and mrcular polarization spectra contain information on the magnetization and electron (hole) spin orlentatmn degree For a weak magnetm field, the average moment of electrons (Sz> and holes Q~) is determined by the well-known relations (s3) ( Sz) =

(Jz)=

3 gl~H 8 EFe ' 15 gI#H 8 EFh

(31)

In EHD, the g-factor of electrons, g, and holes, gl, differ considerably from that of the free particles, due to the many-body effects The free electron mean g-factor in the amsotropm 1 case is equal to g = 3(gl) + 2gl) (for germanmm gll = 0 9, g± = 1 92 and ~ = 1 58, for silicon gll = g J- = g = 2 00) The value of g, depends on the valence band parameters k, 7,, 72 and ~/3 The analytical expression for g, was obtained by Dyakonov et a/(s3) in spherical approxlmatmn (that is 72 = ~3)- This approximation may not be accurate enough, because the g-factor value is extremely sensitive to the choice of the valence band parameters The carcular polarization of the germanium LA line (0 709 eV) for a weak magnetic field in Faraday configuration is equal to Pc,re = -( - Oz>)

(32)

because for thls hne the mare contribution is due to the transition vla F z point, and the LA phonon has a zero angular momentum (28) In silicon the e-h drop luminescence is determined by the transltmns wa the F and A bands The magnitude of Pc,~c depends on the relative contribution of the matrix elements involved (,66) p .... remains proportional to ((S~> - Q,>)

P~,~c = 4~( - Q~>)

(33)

The sign and the magnitude of~b~are different for various phonon-assisted and no-phonon transitions In a stronger magnetic field, when condltmn g,#H <
2Q,> - 2 (2Q~> - 3 P.... = ¼+ Oa,> + (Sz> (5Qz> _ 4Q~>)

(34)

For g,#H < EFh, eqs (31) and (32) are val|d for an arbitrary g#H and EF~ The mare paramagnet|c properties of the e-h drops m sdlcon and germanmm appeared to be rad|cally different (s 2s) It was found that m gerrnamum the slgn of the e-h drop luminescence is determined by the orientation of hole spins, m silicon, however, it corresponds to the electron orientation

Electron-hole hqmds m semiconductors

175

Thin dflference was attnbuted to the stronger spm-orbR mteracUon m germamum than m sdlcon and consequently much faster electron spin relaxaUon. As a result of the short spin relaxaUon time m germamum, the electron spins are m thermal equfltbnum w~th the crystal lamce. Accordingly, the quantum oscfllauons of polanzaUon were observed on the high energy rode of the e-h drop lununescence spectrum. The magmtude of the oscdlatlons appeared to be many Umes htgher than that calculated for an 1deal Fermi gas, ueglectlng the electron-electron interaction. The electron g-factor enhancement happens to be most mgmficant near the Fermi surface The effective hole g-factor also exceeds Rs free hole value In mhcon the eleetro~ g-factor an~sotropy m uegllg~ble, and the g-factor value ~svery close to that of the free electron, which m&cates a very weak spin-orbit coupling As a result, a much longer spin relaxation time m expected Therefore, the electrons do not change in a mgmficant way their spin onentatlon dunng the condensation, and in the drop they retain the onentaUon obtmned in the free exc~ton state All experimental results available now show that the e-h exchange interaction is the main mecMmsm of electron spin relaxaUon m SfllCOrLIn the process ofexpermaental measurement of the electron or hole spin moments, the pru'aary concern IS the calculaUon or measurement of the depolanzaUon factor and the ~b~ factor entenng eq (33) In principle, both of these factors can be determined by companson of the experimentally observed and the calculated polarization of the exc~ton radiation m a strong magneUc field, when the sphttlng of the exmton ground state ~sgreater than several kT and polarization of the exc~ton hne saturates In some cases, such as the phononless or long~tudlnal phonon-asSlsted tranmtlons, the theoretical value of ~N m equal to --1 Therefore taking rote account only the paramagnetlc sphttmg (see F~g 32), the polanzatlon of the LA-ass~sted exc~ton radlat~on m germanium for small H ~s determined from eq (32) m which

1 g#H

~')/2~! •

_1/2~

=

4 kT

Q'> =

5g~#H 4 kT

( ~ /..t-gH

<)"> | n

~;/u.g,,H

80Ii)~

)

-'/2

+9 2

h

+'(~ ~--

"3/z

('g H "3'2 +3,'2

(35)

L~c2 t

[

pg.H

~ ~ co, e,. ~

3/z

)'3 ~ Co,O,H

-'/2 +%

t

+3/2

(a) (b) (c) (d) FIG 32 A dmgram of exclton level sphttmg for germamum m magneUc field (g > 0, gl < 0) (a) paramagneUc sphttmg only, (b) paramagnetlc and crystal sphttlng (for an exclton whose electron m occupying the [IIl] valley and H [J [III ], (c) the same as (a) and (b) with addRmn of diamagnetic shift of electron levels 6, (d) the same as (b) wtth the addition ofdlamagneUc sphttmg of hol_elevels which m effectwely acting to oppose the crystal sphttmg (01 is the angle between [III] and [III] or equtvalent to those axes) (7)

For a stronger field, the polarization saturates and m equal to Pc,,c = - (gl/[gl[) It does not hold, however, m the presence of conduction or valence band diamagnetic sphttlng The dlagmagneUc sphttmg manifests ~tselfin the reduction of the saturaUon value or even m the decreasing of the polanzaUon at higher fields It should be menUoued, however, that the vah&ty of the low-field eqs (35) does not depend on dmmagneUc effects. The physical reason why the &amagnetac sphttmg of the conducUon band reduces ctrcular polarization m the lugh-field region m rather sunple When not all of the conduction band valleys are m eqmvalent posmon vath respect to the magneUc field direction, the cyclotron masses of different valleys are unequal Consequently their energy

176

A A ROGACHEV

and the populatmn should also be dffferenL The unequal populaUon of the conduction band valleys m a magnet]c field causes the hnear polanzaUon of the exclton humnescence m the same way as, say, umax]al stress does. Consequently, for the &rect]ons of observauon for whmh the recombmatmn radmtmn ]s almost completely huearly polarized, the circular polanzatmn should be smaller The mare reason for the hnear polanzat]on of exclton luminescence m the sphttmg of the e x a t o n ground state due to the electron effective mass amsotropy (the so-called crystal sphttmg, Ac,, which separates the states with J = ~ from the states with J = 1, for germanmm Ac, m equal to 1 x 10- 3 eV) The crystal sphttmg also leads to a dependence of spin sphttmg of both the J = ~2 and J = ½ states on the magnetic field onentatmn about the axes of the conductmn band extrema (see Fig 32b) Moreover, m high magnetic fields the dmmagneuc effects proportional to H 2 should be taken into account The hole states with J = ½have lower energy than the J = ~ states when H > 30kG (see F]g 32d) All the above-mentmned reasons lead to a comphcated dependence of polarization on H m the h]g_h-field reg]on, as shown m Fig 33 (27)

60--

~

CI

4O

I

I

20

I

I

40

I\

B, T FIG 33

ThecEcularpolanzatmnofLA ¢xcRonin germaniumfor &fferentmagneticfield orlentatlon (a) H[I [IOO], (b) HII [IIO], (c) HII [III], T=42K (7)

To determine the depolanzatlon factor for germanium, the low magnetic field region should be used where Pc,,cis lsotroplc, and may be calculated from eqs (35) and (33) assuming that gl = - 1 6 (100) For slhcon the &amagnetlC effects are neghgible, and the crystal sphttmg is much smaller than m germanmm (Act = 0 3 x 10- 3 eV (4s) Thus, in magnetic fields stronger than 3-4 T the crystal sphttmg may not be taken into account Accordingly, for sfl]con the P¢,,csaturates in a strong magneUc field In the low-field region, eqs (34) and (35) are st]ll apphcable, m spite of the fact that m the saturatton region the polarization is not equal to ~bN,but is determined by another combination of matrix elements (166) P]kus calculated that for an exclton bound to a neutral donor m Slllcon q5N = --1, and m the saturation region P~,, = - 1 The measurement of the polanzatmn of thin hne unmedmtely gives the depolanzauon factor, and consequently one may determine SN for all phonon rephcas of the exclton lme In such a way ~t was found that ~bTO= 0 4, CPLO= 0 45, ~bLA= 0 5 Using these values of th~ and the data presented m Fig 34 the g-factor of the holes g i m free and bound excltons was found to be equal to 1 2 for both cases The paramagneUc susceptibility Z of the many-electron system ]s proportional to the effective g-factor g* of the electrons and holes involved. In the H a r t r e e - F o c k approx]matmn, the paramagueUc suscepUblhty is equal to (see, e g., lhnes (16~)) Z= Z0

1-

(36) 3 EF !

where Xo m the Paul] paramagnet~c susceptlbflRy, Eex m the exchange energy and Ev is the Fermi energy EquaUon (36) strongly overestnnates the contribution of the exchange energy since it does not allow for the Coulomb repulmon of the hke particles with parallel spins The

Electron-hole hqmds m semiconductors

?5

25

JN

-15

177

Io

2o

30

40

5o

I\

-°°I- \ . .

-30 FIG 34 C]rcular polarlzatmn of some of the exclton hnes m silicon for H II [III ] and T = 1 9 K (1) free exclton TO hne, (2a, b) donors (phosphorous) bound exciton (2a) TO hne, (2b) phononless hne, (3) acceptor (boron) bound TO hne Sohd hnes are the experimental data, dotted, are calculated m which hole g-factor ~s used as a fitting parameter (s)

essentml tmprovement of the theory may be achieved by calculating the exchange energy with screened Coulomb potential (235) In th~s case the exchange energy is equal to (7) E~? = Eex ~,(l) qS(l) = 1 - ~l 2 + (2-194/4+ ½12)In 1 + g

- ~l arctan ~

(37)

where l = (r~. kF)- 1, IS the screenmg length, and k F is the Fermi wave vector Then, Instead of eq (36), we have Z___=;~o 1

3

~

t

(38)

In the "metalhc" density range (rs .~ 1), r~. is thought to be about equal to the average d~stance between the charged particles, i e r~, may be approximated as r~c. = n-1/3 This choice of r , . Is somewhat arbarary, but a prowdes rather good agreement with the experimental data on monovalent metals and the results of more soplusticated calculations In the e - h hqmd, both the electrons and holes take part m the screening and thus r~. = 2No) l/s, where No is the densay of e-h pairs m the drop Electron and hole g-factor enhancement was estimated m the framework of th]s model by Altukhov et al (~) They calculated that m sthcon g*/g = 1 23 for electrons and 1 42 for holes, and corresponchngly 1 3 and 1 5 for germanmm Since the spm-lamce relaxation time of the electrons and holes in germamum is much shorter than their hfettme, the electron spins are in thermal equdibnum wah the lattice This sauaUon ~s sketched m Fig. 35a. In th~s case, the maxtmum value of the polarization has to be observed on the long-wave s]de of the drop luminescence band The

T

l"sd<< "rd

{a)

~'sd >T~

(b)

FIG 35 The Fermi chstnbutlon for eqmhbnum (a) and nonequfllbnum (b) onentatmn of electrons m the e-h drops

178

A A ROGACHEV

3o~-

8o[--

I

zo I

60

2

I° f ~40

3

| I,o,

__

5

20

4

60--

I

/

::'5

o

.... [J

I

]

I

I

,

4i

t

s

-2

~ . 1

FIG 36 Circular polarization of exoton and e-h drop luminescence m the magnetic field (a) n I1[IOO], (b) n II [III ] (1) free exclton hne polanzat]on, T = 4 2 K, (2) long-wavelength part ofthe drop hne, (3) m]ddle of the drop line, (4) short-wavelength part of drop hn¢, (5) drop hne average polarization, T = 2 K Electron level dtagram ~s shown on the bottom (2s)

carcular polarization degree of the luminescence for &fferent parts of the drop hne ]s shown m Fig 36a, b The average polarization of the drop luminescence is shown by dashed hnes The spectral distribution of polarized and unpolanzed magnetolummescence for H parallel to the [100] axis is shown in Fig 37 EquaUons (31) and (32) suggest that when the absolute values of the electron and hole gfactors do not &ffer slgmficantly, the spin orientation of the hole provides the mam contribution to the polarization Thus, the electron term in eq (32).should be regarded as a small correction Equation (32) and the experimental value of polanzaUon may be used to determine the effective hole g-factor, g* Taking the free electron g-factor to be equal to 1 58 and its enhancement factor in the drop to be equal to the theoretically esttrnated value of 1 3, eq (32) gwes g~ = - 2 2 The free hole g-factor was calculated by spherical approxunat]on (ss) The Luttmger Hamfltoman parameters (14a) ])1, ])2, ])3 and k were measured by Hensel e t al (la7 According to their cyclotron resonance measurements,

Electron-hole h q m d s m semlconductors

)0

179

--

2 05

--

c_ .J

0 706

0 708

hll,

0 710 ~ - ' 0

712

eV

FIG 37 LA-lummescence hne ofe-h drops m g e r m a n m m Sohd hne, I + + I - , dotted hne, I + - I - , HII [IOO], T = 1 9 K (1) H = 18kG, (2) H = 5 5 k G (zs)

= 1/5(272 + 373) = 5 64, Vl = 13 08, k = 3 41 With this set of parameters, the calculation gave gl = - 1 2, which leads to an enhancement factor of gl*/g~ = 1 83 The coincidence should be regarded as reasonable, keeping in mind that the spherical approximation is not quite accurate for g-factor calculation It is well known that the magnetic susceptibility of the Fermi gas has a component oscillating with the magnetic field These oscillations are due to the crossing of the Fermi level by the paramagnetlcally-spht Landau levels Oscillations of this type were expern'nentally observed in the ctrcular polanzataon of the high-energy side of the e-h drop luminescence line I n g e r m a n i u m t26 28) ( s e e F i g 3 8 ) T h e oscillations probably originate from the conduction band sphttlng, because the experimentally observed oscdlatlon position is in a good agreement with the theoretically expected one In magnetic fields less than 3T, the quantlzatlon in the valence band may not be resolved because of the small separation of the corresponding Landau levels The cyclotron frequency is determined by the cross-sectional area of the electron Fermi surface with the plane perpendicular to H In general, for an arbitrary orientation of the magnetic field, there are four different cyclotron effective masses in the conduction band of germanium But if the magnetic field is oriented along one of the main crystal axes, some of the valleys appear to be In the equivalent positions For H [I El00] orientation, all four conduction band valleys are eqmvalent If the magnetic field is directed along the [111 ] and [110] axes there are two kinds of valleys, with light and heavy cyclotron masses The density of states in each of the valleys depends on the effective mass in the direction of the magnetic field in such a way that the "light" Landau bands have greater density of states Hence, the osollatlons associated with the crossing of the Fermi level by the "light" Landau band has greater Intensity The amphtude of the oscfllatmns IS determined by the paramagnetic sphttmg of the Landau levels, and thus by the electron g-factor. The amplitude of the oscillating component of polarization was calculated for a 2 x 10- 3 eV wide spectral region near the Fermi surface The broadening of the Landau levels due to fimte temperature was not taken into account The oscillation amplitude calculated that way for g* equal to the free electron g-factor in the conduction band of germamum (g = g = 1 58) appeared to be much smaller than the experimental value An enhanced value due to the electron-electron interaction g-factor (g* = 2 1) is also too small to account for the observed amphtude "of oscillations Moreover, the theoretical dependence of the polarization oscillataons cannot be fitted with the experimental one with any magnetic field independent g-factor value, since otherwise the average polarization would be too h)gh for greater g-factors

180

A A ROGACHEV

14

R

II0oo) T.

/ \

= 9" K

12

N

s

4

ssr ""~,x

-

2 0

04

08

12

16 B,

20

24

I

28

T

FIo 38 Polarization oscdlatmn on the short-wavelength side of the e-h drop luminescence hne (H[I [IOO]) Dotted hne, calculation lgnonng many-body effects (2s)

Altukhov et al (7) supposed that m order to obtmn a better fit, the same type of quantum oscdlations of the g-factor as m the case of the two dunensmnal conductlv~ty(12) should be accounted for This effect may be explmned as follows m the quantum 1Lm~t,the contnbutmn of the electrons from the same Landau band to the exchange energy is greater than the contnbutmn of other Landau bands The crossing of the Ferm! level by the Landau level reduces the populaUon of the upper Landau spin sublevel, and thus reduces the exchange energy of electrons occupying tills level The difference m the exchange energy between the Landau "up" and "down" spin sublevels results m an increase of the paramagnet~c sphttmg Tills mechanism of g-factor modulatmn is dlustrated m Fig 39. A sharp increase of the gfactor value takes place at the moment when the Ferm~ level passes between two spin Landau sublevels g~H

Ferm= level

--'<>--O--

~*~H FIG 39 g-factor enchancement near electron Ferm~ level

So we conclude that there is a satisfactory explanation of all available experunental data if a thermal eqmhbnum m the sp~n system of the e-h drops m germanmm is assumecL Somewhat dtffercnt behawor was observed m sthcon. Due to the weaker spm-orbm mteractmn, the doctron spm relaxation time m silicon ~s many orders of magmtude longer than m germanium At the same tune, the lif©tunes of exc~tons and e-h drops in slhcon are shorter than m germamum Therefore, the electrons do not change the spin onentaUon they acqmrod m the exc~ton state, and retam ~t m the e-h drop Experunents show (5'a) that m e-h drops the spin rclaxaUon tune is much longer than the e-h parr lffettme, which gives nse to a now phenomenon--namely a nonoqmhbnum onentatmn of the electron spins m e-h drops Under certam favorable conchtaons, the noneqmhbnum onentaUon may be much greater in magmtude than the oquihbnum one The more intense e-h drop lununescence hne m silicon is formed by the supsrposmon of TO and much weaker LO rephcas The O~-factor

Electron-hole hqmds m sermconductors

181

connecting average electron and hole spin moments with the arcular polanzaUon degree of the recombination radiauon for th~s hne ~s equal to

I,.o

Oro + i r °

i~o

1 + ~I l o fro

(39)

where ILo/Iro = 0 11 <1o3)The degree of circular polanzatton of LO-LO e-h lurmnescence line calculated by eq (33) under equdlbnum conditions should be about 10 - 2 of the free exclton value This value may be increased by a factor of 1 5 taking into account the g-factor enhancement due to many-body interaction. The stgn of the e-h drop luminescence polanzahon should be the same as for the exclton hne However, much stronger polanzatlon of the e-h drop luminescence with a sign opposite to the polanzaUon of free exclton luminescence was observed (5,s) Furthermore, the polarization appeared to be dependent on the excltaUon power (see Fig 41) The important feature as seen from Fig 41 is that the maxtraum of polarization ~s on the lugh-energy side of the drop hne as opposed to the eqmhbrxum spin onentaUon when the maxtmum has to be on the low-energy side of the spectrum These facts indicate the existence of noneqmhbnum onentaUon of electrons m the e-h drops which retain their free exoton onentaUon Figure 35b shows the Fermi distribution for the case ofnoneqmhbnum onentaUon The bottom shift of the spm"up" and spin "down" bands are not the same because of the fact that the exchange energy of every particular type of electrons depends on their density, and is almost independent of the denmy of electrons with opposite spin <239) -Il-

?5

l

Q2

ioo

I

J ~ v ,

eV

I oeo

-25

-5

I

FIG 40 Luminescence (1' 2' 3') and polar~atton (1,2,3) spectra o f e - h drops m sthcon Curve 1 corresponds to 1', 2 to 2', and 3 to 3' Excitation level is the lowest for 1 - 1' Electron onentatlon degree Pa = (n+ - n_)/(n+ + n_) is equal to 04 for curve 1,024 for 2, 009 for 3 (s)

15 ~

5

4

I0 o~ 5 0 0 -5

"

B,

T

FIG 41 Circularpolanzatlondegreeofexoton(4)ande-htummescenceasafunctaonofthemagnettc field 1, 2, 3 correspond to the same experimental condlUons as shown m Fig 40 (s)

182

A A ROGACHEV

The existence of huge n o n e q u t h b r m m polarization suggests that the spin relaxatmn time of electrons m drops exceeds the e - h pmr hfettme z a = 1 5 × 10 -7 sec On the other hand, dependence of the polarization degree of e - h lummescence on free excaton hne intensity shows that under experunental condmons the spin relaxat|on tune of the electron m the exclton L~ is comparable to the exclton hfetlme V~x,which is determined by the capture rate of free excatons mto drops Hence, the o n e n t a t m n of electrons m the excltons does not reach the equilibrium value pO = _ th(gl~H/2kT) and their spin m o m e n t may be written as 1 n+ - n_

=

IPe

=

1

po

(40)

2n+ q- n_ -- 2 1 + (l + lPex[)__o Le 7Je x

where Pe is the expertmentally observed orientation degree of electrons in the free excltons, and n÷ and n_ are the density of excatons with spin "up" and spin "down" electrons The hfetmae of the excaton %~ can be estunated from mtensmes ofexcaton and d r o p hnes Iex and I d as l , x ~°

%~ = Zd I~ r-~d

(41)

where I ~ and I d are the experunentally measured intensities of free exclton and drop hnes, z~ is the e-h pair hfetlme in the drop, z 0~ is the radmtwe hfetlme of the exclton, and z ° is the radmtwe hfetune m the drop The raUo ofz°x/Z ° is known with rather poor accuracy Vashlsta et al (233) gwe it between 2 and 4 Altukhov et al (5 s) found this ratio to be dependent on the magneUc field, and equal to 7 5 in a field of 5 T and T = 2 K The degree of electron o n e n t a u o n m drops Is P~ Pd -- - l + - - Zd Tsd

(42)

where Ld lS the electron spin relaxation tune in the drop Equation (41) shows that as the excitation level increases, Zexwdl decrease and the values of Pe and P d wtll decrease accordingly Experimental data shown in Fig 40 may be used to calculate the electron polarization degree in the drops by means ofeqs (38), (41) and (42) neglecting the e q m h b n u m onentatlon of holes The latter m a y be accounted as a small correction if better accuracy is needed The highest expenmentally observed electron onentatlon degree (curve 1 in Fig 40) ~s as high as 0 5, which is an order of magmtude hagher than the theoretically expected equlhbnum orientation m the drops, and assuming it to be equal to the free e x o t o n orientation, the spin relaxation tune may be calculated by means of eq (40) Altukhov et al (s) determined it as Le = 3 x 10-Ssec The very existence of n o n e q u l h b n u m polarization unposes certain hmltatlon on the possible choice of the s p l n - l a m c e relaxation mechanism m exc~tons and e-h drops Since for H = 5 T and T = 1 9 K the degree of electron orientation in excltons is IPel < IP°I = 0 9 and Pd m a y be as high as 0 5, and therefore it follows from eq (42) that Ld >> Zd = 1 5 X 10- 7 sec Smnce %x ~ 10- s sec, it is necessary that the ratio L J L a should be less than 0 1 m order to make the n o n e q m h b n u m orientation of electrons m drops observable But for any mechamsm of electron spin relaxation assocmted w~th the electron m o m e n t u m scattering %JZ~a ~s greater than umty Accorchngly, for the scattenng by ~mpunues we have L/ZF .~ (E/E~)- 2, where E is the kmetac energy and E~ is the forbidden gap width, then for [Pdl ~ 1 the ratm (LJLd),.,p IS about equal to (EF/kT) 1/2 .~ 10 For the phonon scattering in sdlcon %/zp ~ (E/Eg)-' and (z~JZ~)pho..~ (EF/kT) a/2 > 102 Thus, whichever abovementioned mechamsm dominates, the electron spin relaxatmn Ume m the free excltons

Electron-hole hqmds m semiconductors

183

should be much longer than in the drops. Numerical esttmates for this mechanism gwe (Le),,v > 10-4 sec, (%,)pho, > 10-s see The intervalley scattering by short-range potentml leads to the %~ of the same order of magnitude, and again %JLa >> 1, since the probability of intervalley transmons increases wtth electron energy. The only known electron spin relaxation mechanism in sdlcon which can be responsible for the faster electron spin relaxation in exeltons and leachng to L~ < 10-7 sec is the e-h exchange mteracUon (82'84'165) The rate of the e-h spin relaxaUon depends on the exchange splitting between J = 2 and J = 1 sublevels of the exciton ground state, and also depends on the hole spin relaxataon Ume, wluch is almost equal to the hole momentum relaxation ttrne ~ph-In the presence of a magnetic field, the e-h exchange becomes weaker m as much as the paramagnetic splitting hw = (g - g i ) # H exceeds the e-h exchange splitting and It ~s gwen by the following equations for Aex < hT~ 1, Tse =

Zsh(~) ~1+ (O)'l:sh)2]

(43)

and for h09 > A~ and 09 > Zs~1,

(44) for Aex > ho) and A~ > h~A ~ = ~h

(45)

The experimental value of~s~ = 3 x 10 = s sec tmphes that % >> Zsh because Lh << 10 = lO sec, and besides, for the exclton, Ae~ << hz~ 1 Excaton momentum scattenng time calculated by Anselm and Flrsov theory (13) is 10 - l i sec at T = 2K, whtch gwes Aex = 10-5 eV (eq 43) When Xsh = 10-11 sec, however, eqs (44) and (45) show that owing to the strong increase of L~ in a magnetac field, the saturation ofpolanzation should be observed m fields higher than 3 T, which is not m agreement with the data presented in Fig, 41 In order to reach an agreement with the linear dependence of polanz~Uon on magnetic fields up to 5 T, one should make an assumption that T~h ~ 10-~2 SeC, wtuch does not at present seem qmte realistic B~r et al (62) calculated that the spin relaxaUon time of electrons m the drops with almost complete electron orientation (IPdl ~ 1) Is given by

2 × I0 -4A2 E2h

~"~

h

~x E3

(~)

where E x is the excaton bin&ng energy. As Ie, I is reduced, E~ increases When Ex = 14 7 meV, Erh = 16meV and Aex ~ 10 -5 eV, this equation gwes ~u = 10-*sec So, Altukhov et al (8) concluded that the e-h exchange sphttmg Aex = 10- 5 eV provides the necessary relationship between the exclton hfetxme and the electron spin relaxation time m free exeltons, and in e-h drops, which is requited for an observation of large nonequdibnum orientation of the electrons m e-h drops and exeitataon-mtenslty-dependent p o l a n ~ t a o n of excltons At high excitation levels in the spectral region 1.094-1 096 eV between the TO lines of e-h drops and free exciton, there is a weak recombination radiation which cannot be assoemted with either free or hound excaton.(5) The spectra of this luminescence and its circular polarization m a magnetic field are shown in Fig 42 The degree of polarization of the radmt~on is 0 7 of the TO exciton. Intensaty of'this luminescence falls abruptly when the temperature rises from 1 5 to 5 K. Thts radmtion was attributed to free complexes, which (as the positive sign of the polarization shows) may be either an exeiton, captured electron or blexczton wath the total moment of two holes J = 2 Recently, Edelstam et al (s5) and Thewalt et al (221} managed to observe in this spectral regton a well defined spectral line, which was attnbuted to the biexcaton (see SecUon 3).

184

A A ROGACHEV

150 75

-4FLO I

I~ -3 75

1095 hv) eV

-750 FIG 42 Luminescence (1) and polarization degree spectra (2) of the spectral region near the free exclton hne m sflleon (s)

7 MAGNETOHYDRODYNAMIC

EFFECTS IN THE ELECTRON-HOLE GERMANIUM

LIQUID IN

The e-h hquld m semiconductors is essentially a metal, with the highest known conductivity to density ratio Together with extremely small surface tension, it makes the e - h llqmd an exceechngly attractwe object for magnetohydrodynamlc studies However, at present there are only a few papers dealmg with this subject The reason probably is that the e - h hqmd In unstrmned crystals of germanium was mmnly investigated under such experimental condmons when it existed as a cloud of droplets In this cloud, mterdroplet distance is many tunes greater than their radius, and hence the only possible conductmty m such a system is the conductivity due to the free electron and hole components of the gas phase surrounding the droplets The density increase should lead to creation of current conducting channels inside the drop cloud Unfortunately, there Is a number of pnncapal obstacles which do not allow the realization of such a mtuatlon under usual experunental condmons where a crystal thickness is cons)derably greater than a drop ra&us It was mentioned m Section 1 that m thin crystals a genmne e - h hqmd conductmty may be observed Asnm et al (36) have pointed out that m this case the behavior of the drop system may be determined by the interaction of drops with the crystal surface, rather than with the phonon wind (see Section 9) In prmclple, there are several ways for the drop to interact with the surface One of them is the attraction of the drop to the surface due to the deformation of crystal lattice by the drop Mahler et al (14s) and Schukm et al (193)theoretically estimated the energy of such an interaction for a typical drop ofR = 1/~m m germanium, when it Is near the surface (within a distance smaller then the drop radius), to be equal to 1 eV or more The other possible reason is the attraction of the drops by the surface state charge This energy may be the same order of magmtude as the deformation energy The movement of the drops along the sample surface is hardly possible, due to fluctuataonsin surface state density as well as surface charge irregularities In thin germanium crystals, a rapid rise of d c conductmty in the e-h drop system was observed at such excitation levels when the drops occupmd a significant part of the sample volume Benoltala Gmllaume et al (55) proposed that the conductmty threshold ~s determined by the percolataon mechanism, i e the formation of e - h hqmd conductwe channels between the contacts The expermaental proof of such behavior was obtaln~d by Asmn et al (32 33 34.)The experunents were carried out on samples of pure germanium (na + n, < 1012cm-2), 10 #m thick The current contacts were produced by melting of In-As alloy Due to the formation of As-doped recrystalhzatlon layer on the boundary of pure gertaamum, such contacts show ohnuc propemes for current earners of both signs in the eh hqmd up to electric fields of ~ 1 Veto- 1 The samples were fllamanated by a GaAs laser with 2/~sec pulses and peak power up to 15 W The concentration of e-h pairs was determined by photoconducuvlty

Electron-hole hqmds m sermconductors

185

measurements at room temperature. The accuracy of this procedure was estunated as being within a factor of two (22) The drift of the e-h llqtud conductwe channels in crossed electric and magnetic fields was demonstrated by Asnin et al (33) The Ampere force acting on the current channel is proportional to 6±(H) H E, where 6±(H) is the e-h llqmd transverse conductwlty. For an lSOtOplCe-h system, f± (H) was calculated by Gantmakher et al (102) The magnetoreslstanee of the e-h hquld has a rather interesting pecuharlty. Without an apphed magnetic field the e-h eonductlwty is detgrrmned to a large extent by the e-h scattering Reslstavlty of the e-h plasma, however, tends io infimty m very weak magnetic fields, if the e-h scattering is the only scattering mechanism in the system. In practice, the magnetae field needed to cause a sufficient change in conductivity is determmed by the condltmn that the cyclotron radius rc = (ch/eH) 1/2 is smaller than the sample stze The strong magnetic field influence on eonductwlty is due the fact that e-h scattering does not change the total momentum of the e-h system Therefore, since the magnetic field deflects current earners of both signs m the same direction, the e-h scattenng cannot oppose this deflection For a simplified model, assuming equal phonon scattenng tunes zp for electrons and holes, the conductivity dependence on the magnetic field may be written as fix(H) = f0(1 + 6o1 o92z2H) -1

(47)

where co1, o92 are the electron and hole cyclotron frequencels, zefy = ~/(zvz), and z is the e-h scattering tune One more sunphfymg assumption here is z v >> z Under the influence of the Ampere force, the current channel will move with velocaty e2

V=

2

E H Cmemh

%fJ"

2

(48)

1 + oglU2"Ceff

At E = 0 1 V c m - 1 and H = 1 kG, V = 5 x 104 cm sec- 1 Current channels drifting with this velocity will cover a distance I ~ 0.1 cm for tlme t = 2 x 10- ~ sec To visualme this effect, Asnln et al (33,a6) employed the same procedure used for the first expertmental detection of e-h drops (see Section 2). The essence of this method is that sample regions occupied by drops are more transparent than the rest of the sample for photon energies close to the direct band gap In the particular experiment, a 10/zm-thlck sample was placed into a crossed electrle and magnetic feld m Faraday geometry. The sample was scanned by probing hght (2 = 1 4/zm) focussed into a vertical slit 0 05 cm wide As shown on the insert of Fig 43, when the electric field is directed along the Y-axis with polarity depleted as E +, the current channel should drift in the X-axis direction, and the number of drops under the slit increases As a result, the modulation amplitude will rise In the opposite case (E = E - ) , modulation decreases Figure 43 shows that in the discussed experunent the drop cloud did demonstrate such a behavior The magnitude of the shift Increased with electric field and decreased at higher magnetic fields in accordance with eq (48) The sign of the effect changed with reversed direction of the electric or magnetic fields The scanning of the sample showed that increase of the modulation amphtude on the left side of the contacts was accompanied by a decrease of the signal on the rxght side It was found also, that the shape of the modulation spectra remained constant while the modulation amplitude changed, due to the e-h llqmd drift Therefore, the effect is induced by the drift of conductive channels having constant density of current carriers At E = 0.36 Vcm -1 and H = 1 6 k G the velocity of the eonductwe channel drift was found to be 3 x 104 cm see- l, which is in reasonable agreement with calculations employing eq (49) The percolation mechanism of d c conductivity in the e-h drop system provides a posslblhty of obserwng a speofie magn~tohydrodynamle effect in the e - h h q u l d - - t h e pinching of current channels reduced by their own magnetic field (3s.36) The pinch effect in the case of the e-h liquid is due to the magnetic attraction of current lines inside the channel, which leads to instability of the ehanne)because at some current level the surface tensmn falls to equalme the magnetic pressure This type'of effect differs from the normal pinch-effect, where the magnetic forces are acting against rising gas pressure. In pnnciple, lnstablhty of this kind may be

186

A A ROGACHEV 2 I0 "s sec

FIG 43 AbsorpUon modulation signal m the presence of the electric field Experimental c o n d m o n is shown in the insertion Dotted line shows tllummated region of the sample VerUcal strip is the proNng light beam p o s m o n Electric field E (m ¥ / c m ) is (1) g = O , (2a) E = 0 5, (2b) E = 0 16, (2c) E = 0 0 2 , (3a) E ffi - 0 5, (3b) E = - 0 16, (3¢) E = - 0 0 2 , T = 1 8K(34)

observed m any hqmd metal, but to the author's best knowledge it was never observed, due probably to the rather small c o n d u c t m t y to surface tension raho For typical magmtudes of the surface tension coefficient and the curvature radius of the hqmd's surface in the channel (3 = 2 x 1 0 - * e r g c m -2, R = 1 0 - 3 - 1 0 - 4 c m ) , the surface curvature-reduced pressure (capdlary pressure) turned out to be two orders of magnitude smaller than the e-h gas pressure When the magnetic field pinches the current channel the surface tension force leads to addmonal pressure, determined by the Laplace formula pt, = ~[(1/rl) - (l/r2)], where rl and r 2 are the outer and inner curvature radn of the channel surface An order of magmtude estimate ofthe magnetic pressure may be written as Pn = (1/c2)62E2~ Equahmng PL with Pn, the crmcal pinching electric field Ec, may be determined E,2, =

~c2(1/rl) -- (l/r2) tr2rZ

(49)

Using rt = 10-*cm, r2 = 10-3cm, a = 1 0 4 f l - l c m - ~ , one obtains Ec, ,~ 1Vcm -~ At E > Ec,, mmal warping leads to current channel mstabdlty, resulting m the &sruptlon of the channel, which manifests itself by the lnstabihty of the current The evolution of such mstablhty is shown m Fig 44 With rmmg electric field, instability gradually develops into nearly regular oscillations The oscdlatlon frequency depends to some degree on external circuit parameters (C, L, R) and is of the order of 1 MHz. The oscdlatton period may be thought of as the time reqtured to move the e-h pear wathln the e-h hquld to a distance approximately equal to the channel radius under the influence of magnetic pressure Asnm et a/(36) estimated this time as t = 1 #see for typical e-h hqmd parameters, which is m accordance with the experimental data It was shown that the mstabdity of this kind ~s accompamed by N-shape negative conducttwty. The current-voltage curves exhibiting the negative conductivity may be obtained if the current is measured after a certain delay following the end of the excitation pulse. Such curves are shown m Ftg 45

187

Electron-hole hqmds in semiconductors

_

~

~

2 p. see I

!

2

3

FIG 44 Oscfllograms o f e x o t o n hght pulse (1) and photoconductwlty current for E < Ec, (2) and E > Ec, (3, 4)(3~)

20

E IO

i

i

i

I

O 07' 04 06 08 E)

V c m "I

FIG 45 Negatwe resistance of e-h hqmd m germamum at different average density of e-h pmrs (/cm 3) (1) 3 3 - 101°, (2) 2 4 x 1016, (3) 1 5 x 1016 Time-resolved measurements with 4#see delay after the excltauon pulse end (36)

The pinch-effect character of instability was demonstrated by stabdlzang of the current in longitudinal magnetic field A five-fold increase of the critical field E~, was observed for magnetic fields as small as 5 k G 8 C O N D E N S A T I O N KINETICS IN U H F FIELD

As a rule, a decrease of the e-h radiation intensity m the heating U H F field is observed (a9.149) An mterestlng effect was found by Ashlonadze et al (3a) m studying the U H F field effect on the kinetics of the e-h drop luminescence. Depondmg on the experimental conditions, and especially on the prehlstory of the same tllumlnatlon, not only a decrease of the e-h drop luminescence intensity with an apphed U H F field may be observed, but also in some cases the lummescence m a y rise as well The sample was placed in the electnc field max~rnum of the 8 m m wavegmde and illuminated through a hole in one of ~ts walls by Fte-Nv laser beam The m a x i m u m UHF-field intensity was 20 V c m - 1 As shown In Fig 46, curves 1 and 1' are the ascendlngand descending hysteresis branches without the U H F field If the excataUon power Is fixed m the A point when drops have been already created, the drop

188

A A ROGACHEV

mtenmty increases wRh U H F field switch-on (see Fig 46, curve 3) This effect is revermble when the radmtmn power decreases the system returns agmn to point A It was &fferent m the case when the UHF field was apphed first and then the excltatmn mtens~ty was increased (curves 2 and 2') Since the e-h drop heating by the U H F field is lnslgmficant, due to small UHF rachataon absorptmn by drops as shown by J C. Hensel et al ,(~os) it may be assumed that the U H F field affects only the free carriers In this case, an increase of the electron and hole velocmes and a decrease of the exclton generatmn rate (or tmpact mmzatlon of the exclton(2°) takes place Thus, m applying the U H F field the exclton denmy falls and consequently the free carrier denmty rises Ashkmadze et al (as) supposed that the drop luminescence mtenmy increase is due to the increase of the carrier flow into drops The UHFfield reduced increase of the threshold excltaUon power at the onset of the e-h drop luminescence was explained as a slow-down of embryo formation process due to an increase ofthmr temperature as a result of free-carrier trapping It was also suggested that the increase of free carrier denmy may be responsible for the drop radms increase observed on the ascending hysteresis branch (Fig 21)

{o)

I'

I

g

~

AI

J

( b)

A

.........

q 4

2'

c

c

S

._1

I0

20

30

40

r-T--;-T--,--e POHF

G

FIG 46 H ysterests of e-h drop luminescence m the presence of U HF field (1) 1' UHF field off, (2) 2' UHF field on, (3) Luminescenceintensity as a function of UHF power Pu.r It ~sreversible ff A is the starting point (4) Temporarily svotchmg off the excRatton (5) When Punt goes to zero, the luminescence intensity is returmng to A (3s)

9 ELECTRON-HOLE LIQUID IN POLAR SEMICONDUCTORS

The e-h hqmd has been recently observed m a number of polar semmonductors, such as AgBr, CdS, CdSe, ZnO, ZnS, GaAs, GaP, SIC Interest m these semiconductors was sUmulated by a remark made by L V Keldysh and A P Slhn, that interaction of carriers with LO phonons can stabihze the e-h hqmd (13a) Prior to the &scusslon of the specific problems oftbe e-h hqmd in polar semmonductors, a brief outlook at the fundamentals of the e-h hqmd theory should be gwen The binding energy EB of an exclton m the e-h drop is defined as the difference between the energms of the e-h hqmd ground state and of the exclton binding energy Ex E B = E k + E e x +Eco r - E

x

(50)

whore E~ and Eex are the kmetm and exchange energies, respectwely The first two terms in eq (50) ropresent the ground-state energy m the Hartree-Fock approramaUon EuF = Ek + Eex The &fference between theexact value of energy and Eur m called the correlaUon energy Ecor At suflicaentty large e-h hqmd densRy, Euv gives the mare contnbutmn to the ground-state energy, wtule E,o, may be considered as a small correction However, m the case of a bound state formed by particles interacting vm Coulomb potential, the kmeuc energy has to be half of the potentml energy For this reason, the e-h h q u ~ theory based on the energy expansmn vahd for high densmes proved to be ineffective (t o~)

Electron-hole hqmds m sennconductors

189

Since the H a r t r e e - F o c k approximation m a starting pomt for any more sopinsticated calculations, consider some properties of the e-h system which follow from tins approximation. For the simplest case, when both the valence and conduction bands consist of one valley wRh msotroplc effecUve masses, Euv is 3 h2

EHF = 7 x ~

tu/~

2 (3~ 2n)/3

3 _

e2(3n2n) 1/3 (51)

2~

where 8 is the dielectric constant, n is the e-h pair concentration,/J- ~ = m~ ~ + m ; 1, me and mh are the effecUve masses of electrons and holes Expressing the energy m atomic units Ex = (e2/28rh), and taking r~ = (3/4~n)l/3r~ t (where r h = (~h2/e2#) is the exclton Bohr radius), eq (51) can be written as EuF =

2.21 2 rs

1832

(52)

rs

Consider now a selmconductor wRh v-valleys In both the conduction and valence bands which are separated widely enough in k-space so that electrons and holes near the bottom of the different c-valleys can be considered as staUstmcally independent particles. In this case, EHF will stdl be defined by eq (52) if n is the electron density in each valley Thls result has a simple physical meaning m the Hartree-Fock approxlmaUon the electrons occapymg the same valley "do not know" about the presence of electrons in other valleys The condensed phase density in a many-valley sermconductor will be correspondingly v times larger than in a single-valley one If there is a different number of valleys m the bands the binding energy m the condensed state, m principle, even in the Hartree-Fock approximation may be larger than the exclton binding energy The main idea here is simple enough In this case the exciton binding energy can be slgmficantly decreased wRhout a consMerable increase m the kinetic energy of the condensed phase For this purpose one has to take a semiconductor having small electron to hole effective mass ratio and large number ofvalleys m the conducUon band. Due to the small electron density in each valley the electron contrthutlon to EHv in such a semiconductor will be small It ~s easy to show that in the H a r t r e e - F o c k approximation the e-h liquid Is stable with respect to the process of decomposition into free exc~tons, provided the number of valleys in one of the bands is at least 12 Umes larger than in the other Since tins sltuatlon seems to be unreal it can be said that in the H a r t r e e - F o c k approximation the e-h liquid is not stable The example discussed above is just an dlustrative one, since the densities involved are too small to justify the apphcabdlty of the H a r t r e e - F o c k approxlmaUon. The energy Env for germanium and silicon has been calculated by Brmkman et al., (~7) wRh the allowance for an~sotropy of electron and hole effective masses and the interaction between the bands of light and heavy holes as well

EHF =

0.468 2

1 136

rs

rs

EHV=

0 717 2 1-;

1 157 r,

(Go) (SO

(53)

where rh = (ehZ/e2g), rn~ 1 = ~(m~l I + 2m~'~),

and

g-1 = m£1 + m~l,

mob = 0075me(Go), 0234mo(S0

The maximum values of Erie corresponding to eq. (53) are equal to - 0.689 and - 0.467 for germanmm and sdlcon, respectively, winch is essentaally less than the value - 0 . 3 8 for a simple-band semwonductor However, it xs the correlation energy which is responsible for the exctton condensate stability. The exchange energy incorporates only correlations in spatial dlstrlbuuon of like particles with parallel spins The other types of correlation--namely: of electrons and holes, of electrons (holes) wRh anttparallel spins, and of electrons (holes) from chfferent valleys--are accounted for m the correlation energy JPQE 6/3 - O

190

A A ROGACHEV

In the e - h hquld m many-valley semmonductors, which is essentially a multlcomponent plasma, the corr[latlon energy was found to be of greater Importance Within the exchange "hole" surrounding an electron in sdmon there are about 20 pamcles of another nature (holes and electrons from other valleys) In many-valley semmonductors the screening abdlty of these particles is enchanced, since Increasing numbers of the valleys reduces the Fermi energy The first successful attempts to calculate the e - h hqmd binding energy m germanium and slhcon were based on two modifications of the random phase approxlmatlon the Nozleres-Pmes method 163 (74) and the Hubbard method 113 (67)Apphcatlon of the random phase approximation (RPA) leads for large k to serious errors, since particles with opposite and parallel spins are treated m the same way However, particles w~th parallel spins cannot approach each other by virtue of Pauh's exclusion principle and, therefore, should not contribute to the correlation energy at large k Hence, some Improvement of RPA ~s needed to ehmmate thls defect of the theory In the multlcomponent plasma the role of the spin correlation IS not so ~mportant as m the one-component electron gas For this reason, approximations based on RPA appeared to be successful for calculaUons of binding energy m germanmm and slhcon In the case of small k the RPA correlation energy was used as m the Nozleres-Pmes approximation For large momentum transfer, k >> kv, (kv is the Ferm~ wave number) the second-order perturbation theory was used Then interpolation between these two extreme cases was made In the Hubbard approximation, instead of RPA polarizability rrRva (k, ~0), a new function nn was introduced (for the single component system) Irn(k, 0~) =

nRpA (k, 09) 0 5k 2 1 + k2 + k------~F rtRvA(k' 09)

(54)

This function decreases the polarlzablhty at large k Minor discrepancies in calculations of the e--h drop binding energy by Brmkman et al (67) and Combescot et al (73) stem mainly from the fact that Brmkman et al calculated the correlation energy for lsotroplc effective masses rather than from differences of approximations employed Recently, G Benl et al (sa) took into account the amsotropy and obtained vlrtuaUy the same results as Combescot et ab Table 1, taken from the above-mentioned paper by Bern et al, summarized the results of calculations of the e - h hquld ground-state energy in germanium obtmned by various authors It should be noted that, if the results of calculations of the binding energy and the condensed phase density are in many cases in good agreement w~th the experiment, dtfferent theories give remarkably different predmuons of the e - h correlation function geh(r) It is important to know this function, e g, m calculating the radlatwe recombination rate, since the latter is proportional to geh(0), which Is the e-h paar correlation function geh(r) taken for zero mterpartlcle separation Approximations based on different variations of the RPA do not allow for the multiple e-h scattering whmh determines g~h(r) Vashlsta et al (221-223) modified the STLS method of K S Slngwl et al (2°~) to make it appropriate for the mulUcomponent plasma and, thus, avoided to a considerable degree the above-mentioned hm~taUon They derived a self-consistent set of equations for correlation functions of various carriers Great difficulties m numermal calculations forced them in the first stage to take into account only the e - h correlation These calculations appeared to be extremely successful m the predmtlon of both binding energy and correlanon function g~h(r) In the following paper, Vashlsta et al (222) presented the calculations which included all types of correlations The g~h(0) value obtained m these calculations appeared to be essentially larger than that calculated by RPA Moreover, g~h(0) calculated within the limits of the self-consistent approach rises rather sharply with decreasing e - h density, while the RPA gwes a comparatavely weaker density dependence. The effects of multiple scattering of electrons and holes accounted for in the self-consistent method are partmularly Important for the calculation of the binding energy and geh(0) m semiconductors with simple-band structure, for which the condensed phase density Is considerably smaller than in many-valley semiconductors The ra&atlve recombination intensity is proportional to geh(0), which presents an opportunity for the experimental determmanon of th~s value For umaxml

Electron-hole h q m d s m semiconductors

191

deformaUon of germamum along the main crystal axes, the band structure may be changed gradually m such a way that instead of four valleys !n the conduction band and two valleys in the valence band. only one valley is left in each Comparison of the experimental and theoretical data in the case of the unmxml deformaUon of germanium was published by Chou et al (Tt) They showed that calculations based on the random-phase approximation grossly underestimate the value and give a wrong density dependence of g~h(O)when the number of valleys in the conduction and valence bands decreases TABLE 1 Experimental and theorettcal ground-state energtes (m meV) of the e - h hquld m Ge and Sl (s*) HA means Hubbard approximation including amsotropy, FSC means fully self-consistent method RS(I 83) RPA FSC

V D S ~222)

Ge Sl

BR (67)

C N t74)

HA

FSC

53 203

61 210

58 208

59 220

657

665

HA

FSC

Experlment (~°)

61 217

62 229

60+02 229+05

B R (54)

In polar sermconductors the charge carriers are large-radius polarons The Ionic component of the lattice polarizablhty increases the dielectric constant at low frequencies 0) 2 -- 0) 2

(55)

where ~-0and e.~ are the low-frequency and optical &electric constants, respectively, and cot is the LO-phonon frequency The lattice polarization region surrounding slow electrons or holes has a radius

[h21I/2 r,., = ~

1

(56)

where me.hIS the effective mass of electrons and holes This value defines the polaron effective radius Interaction of a charge carrier with the induced polarization charge decreases its energy by a value E;'*

=

1 [ ~ ° - ' ~ / e2

- 2 / -S-AT-o/

= - ~,.hh0)l

(57)

where ~ is the dlmensaonless polaron couphng constant The value of the semiconductor forbidden-gap shnnkage IS equal to the sum of polaron shifts of electrons and holes AE, = E~ + E~

(58)

Since the movmg charge carrier is followed by the lattice polarization region surrounding lit is larger than thebare carrier mass me.h The effective mass of it. the polaron effective mass me.h a polaron with small momentum in the case of rumple lsotropic bands is equal to 1 m*.h = m,.h(1 + ~,.h)

(59)

It should be noted that the effectwe mass usually determined from the cyclotron resonance experiments is essentially the polaron effective mass m*h The polaron sluft for the anisotropic case was calculated by Matsuura et al (152) If the exciton radius in the polar semiconductor ~s larger than the sum of the electron and hole polaron radii, the exciton binding energy can be determined by a hydrogen-like expression In wluch the electron and hole effective masses are taken to be equal to their polaron mass. and the dielectric constant lS equal to ~o e4#* Ex = 2e~h 2

(60)

192

A A ROGACHEV

where (~*)-1 = (m*)-1 + (m*)-1 is the reduced exclton mass The binding energy of an eXCltOn in a polar semiconductor Is the energy required to separate ~t into two free polarons--~ e eq (60) determines the excRon binding energy with the zero level of energy counted from the bare forbidden gap reduced by the sum of polaron shifts As the d~stance between the electron and the hole decreases, the lattice polarization regions with opposite sxgns overlap The effectwe dlelectnc constant tends m th~s case to coo, and the polaron shifts of electrons and holes vamsh Besides, the effective masses of electrons and holes decrease from the free polaron effectwe mass to the bare effectwe masses The decrease of the effectwe dielectric constant increases the exc~ton binding energy, while the reduction of polaron shift and the decrease of reduced exc~ton mass makes the binding energy smaller In some papers devoted to calculation of the e - h drop binding energy m polar semiconductors, the statement was made that these tendencies almost completely cancel each other, and, therefore, the exc~ton level appears to be approximately at the same position as it would be determined by eq (60) However, this statement has to be taken with some reservations Theoretical calculation of the exclton binding energy in polar semiconductors was carried out by Mahantl et al ,t146~ Sak,(lS6) Pollman et al (174~ and Kane (~23) Since the polarlzatmn does not exactly follow a fast electron, the interaction, strictly speaking ~s not local, and to find the binding energy ~t ~s necessary to solve the Bethe-Solplter equation In some hmltmg cases, the solution of the Bethe-Solplter equation can be reduced to the solution of the Coulomb problem w~th a local potentml by introducing a number of ad&tlonal terms m the Hamlltonlan The significant breakthrough m the exclton binding energy calculation was the creation of proper varmtlonal procedure The variational wave funcUon proposed by Pollmann et al (i 74)m the hmlt of small excRon binding energy (re.h << rh) leads to eq (60), but w~th the bare reduced mass instead of the polaron mass This hmltauon was avoided by Kane t l 23) His varmtlonal calculation shows that m many polar semiconductors in which the e - h hquld was observed (such as AgBr, CdS, CdSe, ZnO) the main contribution to exclton binding energy was made by the decrease of the effectwe dielectric constant caused by the hole motion In all of the above-haentloned compounds, holes are considerably heavier than electrons This decrease ~s so large that the excaton binding energy in some semiconductors ~s greater than the donor binding energy (mh ~ oo) As follows from Table 2 (see p 198), there are many cases when the theoretical values of the exc~ton binding energy are inconsistent wRh the experimental data In theoretical calculations of the e - h llqmd binding energy it seems preferable to compare the hquld phase ground-state w~th the theoretical value of the exclton binding energy, rather than with experimental one In the case of a b~g &screpancy between the theoretical and experimental values ofexclton binding energy, one may suppose either omlsslon of some ~mportant type of interaction m the theory or an error in the experiment Such interaction should also be important in calculations of the e - h hquld binding energy Therefore, determination of e - h hqmd binding energy by substractmg the experimental value of the exlton binding energy from the theoretical value of the hqtud ground-state energy may be meaningless in the cases where the theory cannot properly explmn the experimental value of exclton binding energy Following this hne of reasoning m the Benl-Rlce (s4) theory of e - h hqmd ground-state m the polar semiconductor, the theoretical values of exclton binding energy calculated by Mahantl et al (146) should be used The various theories wRh &fferent degrees of soptustxcataon lead their authors to the conduslon that the electron-phonon interaction in polar semiconductors makes the e - h hqmd more stable with respect to the decay into free excRons Keldysh and Sdln (133) have calculated the exchange energy of degenerate e-h gas m a polar semiconductor replacang the frequency-independent d~electnc constant by 8(o) defined from eq. (55). They have found that m the hm~ts of both the small and large densRles the exchange energy ~s larger than m the case ofe = 80, but smaller than at e = e~ Then, on the grounds of calculations by Sak, (~s6) they made an assumption that the exc~ton binding energy ~s very close to the value related to e = e0 Thus, they came to conclusion that the e - h llqmd binding energy wdl be larger If the interaction with the longitudinal optical phonons is taken into account

Electron-hole hqmds m semiconductors

193

It should be mentioned, however, that using the same type of arguments one can arrive at just the opposite conclusion If the e-to-h effective mass ratm ~ssmall enough, ~t ~sconceivable that e-h as well as h-h interaction would be wa the dielectric constant, which ~s close to So, while the relatively fast electrons wdl interact via the d~electnc constant, close to e~, and consequently the e-e mteracuon will be enhanced compared with the case of dlsperslonless d~electnc constant e = eo So, m the Hartree-Fock approx~maUon, the electron-phonon couphng m the case where mh >> me decreases the binding energy rather than increasing ~t The question whether the electron-phonon couphng does increase the binding energy of the e-h hqmd may be answered only after careful examination of correlation energy The e-h hquld correlation energy m polar semiconductors was calculated by Bern et al ,(52,84) and by Rosier et al (185) In early mvestagat~ons, the single plasma pole approximation has been used (185) A substantial d~sadvantage of this approxamatmn is that the coefficient D4 an the Lundqmst formula for the multacomponent plasma ~s to a large extent determined arbitrarily Rosier et al (185) and Bena et al (54) g~ve the RPA calculatxon of the ground-state energy and the gas-hqmd transmon phase dmgram (1as) The most elaborate calculations have been earned out by Benl and Race, who used the RPA with Hubbard correctaon and took rote account the valence subbands mteractaon They emphastzed that af the Hamiltonlan which accounts for the optacal phonon xnteractlon exphc~ty as used, the input parameters should be bare masses rather than experamentally observed cyclotron masses, since the latter are essentially polaron masses The bare and polaron effective masses are related to each other by eq (59) Unexpectedly, th~s obwous improvement of the theory resulted m greater d~sagreement between the theory and experiment Thus, an the case of cadmxum sulphide the calculated binding energy was found to be 1 meV, while the prewous calculation(185) gave 12 meV, which was m excellent agreement w~th the experimental data available at that t~me The e-h hquxd appeared to be unbound m such semiconductors as CdSe, ZnS, ZnO However, the blndxng energy as high as 30 meV was found for AgBr It should be mentmned that Bern and Race determined the binding energy by companng the calculated values ofthe e-h llqmd ground-state energy with the expertmental values of the exc~ton binding energy The exper~mentally-determlned excaton binding energy value an AgBr of 16 meV seems to be surpnsmgly small compared with the theoretical value of 49 5 meV (see Table 2, p 198) The reason for such a b~g dascrepancy as not clear at present Surprising also ls the fact that the experimental binding energy xs smaller than that obtained with d~electnc constant equal to e.o, and w~th effective masses equal to the bare electron and hole effective masses This energy (20 meV) may be considered as the low hmlt of the excaton binding energy m AgBr Comparison of the theoretical values of the e-h hqmd ground-state energy and the exc~ton binding energy shows that the e-h hqmd m AgBr should be unstable (see Table 2) The results of calculatmn of the e-h hquld ground state energy m AgBr are shown in Fig 47 The upper arrow mdxcates the polaron shift, and the lower one the deepest excaton level Curve (a) corresponds to the case when the lnterachon w~th phonons ~snot taken rote account--~ e. the effectave electron and hole masses are the bare effective masses, e = ~o, and the bare forbidden gap ls taken as energy zero Curve 0a) corresponds to the approx~matmn ~*--~ e the effectwe masses are taken equal to the polaron masses, e = Co, and the polaron sluft reduced forbidden gap ~staken as the energy zero. Curve (c) ~scalculated by Bern and Race on the basas of their dynarmc screening theory The results of calculaUon of the e-h drop binding energy for some polar semiconductors together wath corresponding experamental data are presented m Table 2 In polar semiconductors with s~mple-band structure, such as CdS, CdSe, ZnS and ZnO, the e-h hqmd theoretically ~s unbound. At least the theory g~ves a very weak bmdm& Tlus fact ~s not m hne w~t~ the experimental data, wtuch will be discussed later m th~s sectmn The probable ~eason for this, as pointed out by Bern and Rice, (54) as that the theory does not allow for the multiple electron and hole scattering, whach s~gmficantly increases the bmchng energy m semiconductors w~th s~mple-band structure. Besides, the binding energy is the result of subtractmn of two large numbers the hqmd ground-state energy, and the sum of the polaron shift and the exc~ton binding energy To achieve an

194

A A ROGACHEV

agreement with the experiment it is necessary to change one of these values by approximately 103/o, which is apparently within the experimental error in determination of input parameters In the case of such semiconductors as InP and GaAs, m which the e-h hqmd ground-state energy is well below the optical phonon energy, the Bem-R~ce theory gwes much smaller value of the binding energy compared with that obtamed from the e* approximation

-70

\ AgBr

-80

\

-

-90

/\

>

°j

- I00 w -~10

-120

-130

I

I

2

5

4

rs F m 47 Ground-state energy ofe-h hqmd m AgBr The upper arrow shows polaron-shffted band gap energy The lower arrow shows the exclton experimental binding energy (a) e = e~ and effectwe masses of electrons and holes are bare masses (b) e = co, effective masses are polaron masses and polaron shifts are added to the band gap (e~, approximation) (c) Bern and Rice improved RPA calculations (54)

Investigating the reason behind this descrepancy, Benl and Rice(s4) came to conclusion that m the case when the ground-state energy of the condensed phase is much smaller than the LO phonon energy, the e* approximation is preferable Most of the experimental data on ground-state energy and density of the e-h hqmd in polar semiconductors has been obtained from luminescence measurements at large excitation levels In gallium phosphide, the radiation line attributed to radiative recombination of the e-h liquid is a sum of three phonon replicas (TA, LA TO) The results obtained by different authors in the study of binding energy and liquid-phase density differ remarkably because different procedures were used to decompose the overlapping ra&atlon bands The e-h hqmd luminescence spectrum analysis by Blmberg et al (61) as well as time-resolved measurements by Bontemps et al (65) showed that the e-h hqmd line is superimposed from its high energy side vath the e-h plasma radtation line, which is centered at about 2 3eV This line is attributed to radiative recombination of electrons and holes that cannot be bound into exlctons due to the screening by high density e-h plasma The time-resolved experiments by Bontemps et al (6s) showed that the llqmd phase was forming during the excitation pulse (6 nsec long) directly from the e-h plasma rather than from the excaton gas The free exciton hnes were detected in the spectra recorded 20 nsec after the exciting hght pulse, which m&cates that the plasma density dropped below the level defined by the Mott criterion, eq (12) At low temperatures (2 K) and moderate excitation levels, the 2 29 line previously attributed to the plasma remained in the spectra

Electron-hole hqmds m semiconductors

195

For an explanaUon of this apparent contra&ctlon, a proposal was made by Bontemps et al t65) that m this s~tuaUon ~t ~s virtually another hne, which should be ascribed to blexc~ton luminescence rather than to the e-h plasma The blexe~ton binding energy in th~s case should be 5-10 meV, which is considerably larger than the theoretical value Importance of the "camel-back" type structure of the conduction band longitudinal energy &spers~on described by Carter et al (68)was emphasized by Blmberg et al (61)This structure slgmficantly increases the binding energy due to a rather substanUal increase of the effectwe density of states m the conduction band and, correspondingly, the electron Fermi energy lowering As a result, the condensed phase density appears to be almost twice as large The radiation hne shape should also be changed, which affects the procedure of radiation hne decomposition into individual phonon rephcas Theoretical spectra were calculated as a superposltlon of three phonon rephcas--I(TA) I(LA) I(TO) = 0 34 1 0 0 4 The e-h llqmd binding energy was determined as 17 5 + 3 meV, which Is slgmficantly larger than any value previously reported The critical temperature obtained from the time-resolved experiments was estimated as 55 K (65) The weakest point of any analysis revolving decompos~tlon of the e-h hquld hne is the assumption that the same phonon replica intensity ratio is vahd for both the free exc~ton and the e-h hquld Kardontchlk et al (124) have obtained experimental evidence that TO to LO phonon rephca intensity ratio can differ by a factor of two for the e-h hqmd compared w~th that of excltons These &fficultles m decomposition of the radmtlon hne into mdwldual phonon components may be overcome by investigation of phononless luminescence m mtrogendoped galhum phosphide R Schwabe et al (194)have shown that at low nitrogen concentration (less than 1018 cm- a), the lsoelectromc nitrogen ~mpurlty had a comparatwely small effect on the energy position of the e-h hqmd ra&at~on line However, at the mtrogen concentration of 4 3 x 10 Is cm-a the condensate binding energy increased by 4 meV The effect of mtrogen impurity, however, ~s not hm~ted only by the appearance of phononless radiative recombination and by the small change of condensed phase energy At large enough nitrogen concentration a possibility appears that the exc~tons captured by mtrogen atoms wall move within the corresponding ~mpurlty band (124) At mtrogen concentrations less than 2 x 101Tcm-3, only luminescence of the exc~tons bound to mtrogen atoms or the so-called A-hne is observed At mtrogen concentrations h~gher than 2 x 1017 cm -s and at the same excatat~on power the radmt~on hne related to the phononless recombination m the e-h hqmd was observed This line exists both at the sample excitation by light with the photon energy larger than the galhum phosphide forbidden gap, and at resonance exclton creaUon using a dye laser tuned to the Ahne frequency (hv = 2 317 eV) The condensed phase hne intensity is hnear with excitation power for about four orders of magnitude The experiment shows that the condensed phase ~s formed by excltons captured in the ~mpurlty band rather than by free exc~tons The mtrogen concentration of 2 = 101Tcm-3, at which the condensed phase hne appears, corresponds to mtrogen atom separation of approximately twice the exclton radms It ~s not known at present whether the d~ffus~on of exc~tons m the impurity band ~s thermally activated A radmtlon line w~th many features typical for the e-h hqmd has been observed by D B~mberg et al (s9) in cubic SdlCOn carbide This crystal has rather large polar couphng (Co = 9 72, e,o~ = 5 62) Three conduction band valleys of sdlcon carbide are located in the Xpoints The electron effectwe mass is amsotroplc (ml,/m ± = 2 8) The e-h hquld radxat~on spectrum comprises four phonon rephcas (LO, TO, LA, TA) The well-resolved TA hne m the spectra was used to determine the binding energy (see Fig 48) The binding energy was found to be equal to 19 5 _ 4 meV, which ~sm good agreement with the calculations of Benlet al (53) The comphcated structure of the conduction band of this compound, as m the case of galhum phosphide, has a considerably stronger effect on the binding energy than the optical phonon couphng The largest e-h hqmd binding energy has been found in AgBr by D Huhn et al ,(1 t4) and appeared to be as large as 55 meV This crystal is probably the only known semiconductor

196

A A ROGACHEV

,B- SiC Tt~th, I 8K I=

V

Experiment To Theory (T,IK)

~-

(TO) EHL //EHL (LA)

EHL (LO) r ~ ~

~

c=

P'TA

I

XTA

I

/

.

,

,

2 I

,

E~ ITA ,- "--q'.

22

~

23

Energy, eV FIG 48 e-h hqu]d luminescence m cubic SIC X, free exc]ton TA phonon rephca, /~,A, chemical potent:al position of TA phonon rephca of e-h hqmd luminescence (49)

X

XX

(b)

X XX ~ T / 18 5* K

eo

==

T jl

¢:

~I~ 26" K

•

.,V^

'Ir

~1

l

•

=_o E

26" K

bJ

~ I

27

°K I

I

26 25 E, eV

I

24

I

27

I

I

26 25 E, eV

I

24

FIG 49 Lurmnescence of AgBr at pulsed exc:tatlon Pr*.x= 50kW/cm 2 on the left s]de, and P=u = 5 mW/em 2 on the nght The arrows mdtcate the posltaons of the free exclton X and exc]tomc molecule X X L:qtud-phase temperature determined as the best fit of expenmental and calculated (etrcles) hne shape Is as follows 50K ( T - 18 5K), 70K (T= 26K) and 97K ( T = 45K) H19)

Electron-hole llqmds m semiconductors

197

with redirect optical transmons where condensation occurs from the gaseous phase, which Is mixture of free excltons and exclton molecules The AgBr luminescence spectra are shown m Fig 49 A wide radmt]on line (hv = 2 6 eV) ascribed to the e-h drop radmtive recombmatmn ~s observed in the luminescence spectrum, along with the hue wRh a maximum at hv = 2 68 eV, which corresponds to the LA rephca of free excRon radmtmn To fit the calculated radmtmn spectra to the observed ones it Is necessary to assume that the drop temperature ~s much higher than that of the crystal A new radmtlon hue associated w~th anmhllation of excxton molecules occurs in the spectra at high excitation levels and h~gh temperatures.(l x4) The presence of the hue at high temperatures seems to be strange, since at elevated temperatures the molecule dmsoclation rate increases However, the molecule formation rate (which is propomonal to n~) increases more rapidly with nmng temperature than the thermal dlssoclatmn rate of molecules, provided that the molecule binding energy EM is less than the condensed phase binding energy Eb Indeed, from rumple statmtical conmderatmns, assuming that excltons are m eqmhbrlum with the condensed phase, the b]exc~ton to exclton concentration ratm ~s r/M

=

~ e ~ / k /

ne x

N~

=

~.r,=Ne~

e-(nb-E=lhJ

(61)

Nex

where Nex and N M a r e the exc]ton and bmxcRon densmes of states, respectwely. Since the exclton molecule binding energy (EM = 7 meV) Is conmderably less than the hqmd binding energy (Eb = 55 meV), the ratm aM n& 1, as follows from eq (61), uses with temperature This conmderatmn is not vahd for low temperatures, when the evaporation from drops can be neglected and nex is almost temperature-independent, and bmxcRon concentratmn should decrease with increasing temperature This conclusion is inconsistent with the experimental data presented in F~g 49 It is conceivable that at low temperatures the prevadmg mechamsm of excuton molecule destructmn ~s their capture by the e-h drops

LAx

IOC -

GAP

--

Experurnent

/ ~ /

//

(Tv=~= I 8 K ) /

I

/// /

---

, ~

Theory (3

phonons

----Theory(L-Ap~n ,K,

50 -

N.. lEOr I!TOrl/]

o,+{~r/

vY/,I

It'+r_2__...I /

J94

',~I~ ^'A

11//

22m4

,

h~

E~ "

_L,

-:#,A

22S4 Photon energy,

12~4


J",~ £'

I

1

X,~J~

XNp

'i 2 h+ ~

eV

FIG 50 e-h hqmd luminescence in GaP Theoretical hue shapes calculated wRh "camel's back" structure of the conduction aband <6t)

The condensed-phase binding energy Eb experimentally determined as 55meV is significantly larger than the theoreUcal value of 30 meV obtained by Bern et ai (s+)It should be mentmned that th~s value IS obtmned by subtracuon of the experimental exctton binding energy from the theoretical value of the ground-state energy As has been already pomted out, the exc~ton energy was found to be anomalously small (16 meV) But ff the theoretical excRon binding energy Ex were used for calculatmn of Eb, the e-h hquid binding energy would be negatwe (see Table 2)

198

A A ROGACHEV TABLE 2

Semtconductors AgBr CdS

eo

E~~'p E~ ~°~ E~~ n~xp Ref E~ ~°" ntoh~°" ( E'~'h~°" e~ (× 10-3eV) (× 10-3eV)(x 10-3eV)( × 10~.e,-3)(exp)(×10-3eV)(× 10~8_3)(× 10-3eV)

106 468 858 526

16 27

495 352

CdSe

94

62

15

18 1

ZnO ZnS GaAs GaP

859 86 131 1102

40 52 111 907

59 36 47 18

467 448 495

SIC

972 562

55 13 12 13 0 30 22

8 26 1 3 04 0 73 1

114 139 80 86 132 197 202

1 175_+3 9 14 12 6 195+4

001 86 10 6 5 16 8

205 61 65 196 194 146 59

17+3 ,

9+2

60

30 1 -4 -21 -5 028 119

10 39

-35 -72

054

-7 1

20 49 0024 71

-87 -138 0

192 A=0 138 A=oo

EZ~e°' is the theoretical binding energy of the e-h hquid calculated by Bern and Rice (53 54 EtxheorIS the theoretical binding energy ofexcatons calculated by Kane (123 Ebtheor is the difference between the theoretical ground-state energy ofhqmd and the experimental exclton binding energy E~ *he°"is the difference between the theoretical ground-state energy and the theoretical exc~ton binding energy

T h e s e m i c o n d u c t o r s with & r e c t o p t i c a l t r a n s l t m n s have, as a rule, a simple c o n d u c t i o n b a n d structure, a n d the t h e o r y gives either a very small or negative b i n d i n g energy T h e e - h d r o p s or the e - h h q m d m such s e m i c o n d u c t o r s s h o u l d be t r e a t e d as m e d i a with n e g a t w e a b s o r p t i o n coefticlent T h e negative a b s o r p t i o n coefficient m a y be as large as 104 c m - 1, so i n d u c e d hght emission d e t e r m i n e s n o t only the r e c o m b i n a t i o n r a & a t l o n s p e c t r u m shape, b u t also the kinetics of the e - h liquid f o r m a t i o n Since the luminescence s p e c t r a are d i s t o r t e d by s t i m u l a t e d emission at high e x c i t a t i o n levels, to o b t a i n q u a n t i t a t i v e results it is necessary to investigate t o g e t h e r with the luminescence the light a b s o r p t i o n ( e n c h a n c e m e n t ) s p e c t r a as well The cooling time of h g h t - g e n e r a t e d carriers m a y be close to their lifetime Therefore, the e - h system t e m p e r a t u r e m a y be m u c h higher t h a n t h a t of the lattice H i l d e r b r a n d et al (112) have s h o w n t h a t the e - h p l a s m a t e m p e r a t u r e m such s e m i c o n d u c t o r s as G a A s a n d I n P d e p e n d s n o t only on the exciting r a & a t l o n p o w e r b u t also o n the difference of the p h o t o n energy a n d the s e m i c o n d u c t o r f o r b i d d e n g a p value S i m u l t a n e o u s investigation of a m p l i f i c a t i o n s p e c t r a a n d luminescence e x c i t a t i o n spectra of the e - h p l a s m a b y c h a n g i n g laser wavelength by H i l d e r b r a n d e t al (112) s h o w e d t h a t the e - h d r o p l l q m d can be f o r m e d If the c u r r e n t c a r r i e r t e m p e r a t u r e is greater t h a n Tely > 7 K The l n e x p h c a b l e result of this s t u d y was t h a t the chemical p o t e n t i a l level was l o c a t e d a p p r o x i m a t e l y 8 m e V below the f o r b i d d e n gap, which is twice the exclton b i n d i n g energy Thts d i s c r e p a n c y is r a t h e r serious, since the B r m k m a n - R l c e t h e o r y (67) does n o t give at any c o n c e n t r a t i o n such a d e e p l o c a t i o n of the F e r m i level S t o p a r c h m s k l l (2°5) investigated the r a & a t l v e r e c o m b i n a t i o n o f p u r e g a l l i u m a r s e m d e at very low excitation levels H e o b s e r v e d a wide r a d i a t i o n hne n e a r the l o n g - w a v e l e n g t h edge of the exciton hne T h e width of this line d i d n o t d e p e n d o n the excltaUon level for a r a t h e r n a r r o w range of the e - h p m r d e n s i t y (2 x 1015-1016 c m - 3) This e n a b l e d ldenUficatlon of the line with the e - h h q m d T h e b i n d i n g energy of the h q m d p h a s e d i d n o t exceed m this case 0 3 meV, which is m g o o d a g r e e m e n t with B e r n - R i c e t h e o r y S i m u l t a n e o u s investigation of lurmnescence a n d light ampllficaUon s p e c t r a in C d S a n d C d S e b y B a u m e r t e t al (49) s h o w e d t h a t the a m p h f i c a t m n a n d luminescence s p e c t r a e x h i b i t e d r e m a r k a b l e similarity in the spectral region o c c u p i e d b y the line usually a s c r i b e d to the

Electron-holehqmds m semiconductors

199

bmxclton recombination (line M) and the line rdated to recombination of interacting excitons (lme P) The line related to the e-h plasma appears at considerably higher excitation levels in the same spectral region The light amphficatlon due to induced transitions is possible in the e-h plasma or moderate denmy exclton gas m the region of optical transitions which are accompanied by Interaction with free carriers or excitons, as was shown by Orcenberg et al (95)However, these transmons are described by a second order perturbation theory, and their amphficaUon factor should be several orders of magnitude smaller than in the direct transition region The difference between the amplificataon factor m the case of interacting excltons and the degenerate e-h plasma enabled Lysenko et al (14s) to observe the e-I, plasma using a sharply focused laser beam Because of small active volume, the induced light amplification does not occur in the exciton interaction spectral region The luminescence and amphficat~on spectra observed m CdS and CdSe by Shlonola(19v) bear resemblance to the spectra in the case of galhum arsenide at large excitation levels The short-wavelength edge of the luminescence spectra and the absorption coefficient sign change point were determined with a good accuracy The position of these characteristic points weakly depends on the excitation level, but the width of radiation and amphficatlon lines rise with increasing excitation level The range of excitation mtensmes within which the spectrum line width does not change is either very narrow or is not present at all As in the case ofgalhum arsemde, the appearance of the degenerate plasma hne is accompanied by the vanishing of the free eXCltOn lines, as was shown by Shlonoya(19~) and Muller et al(16o) The term "electron-hole hquid" is accepted in the papers related to the subject matter of this section, rather than "electron-hole drop" This is caused by the lack of data on the existence of drops and their radii in all semiconductors, with the exception ofgermanmm and slhcon It is concewable that in semiconductors with small carrier lifetime, two types of e-h drops may exist (1) Large drops or large regions of degenerate e-h plasma formed on the e-h pairs generation site As was reported by Benolt a la Gmllaume et al ,(sT)such large drops probably exist in gallium phosphide An alternative explanation is that in this case a cloud of smaller drops actually was observed (2) Small droplets of several exclton radii being in a steady state with the exciton gas or the e-h plasma surrounding them The absence of exeiton lines may be considered as an indication that the e-h plasma density is large enough As was mentioned in Section 3, the possibility of drop formation well above the Mott transition is questionable, at least for simple-band semiconductors 10 DRAGGING O~F ELECTRON-HOLE DROPS BY PHONONS Early investigations of e - h drop spatial dmtnbutaon have shown that the drop could migrate up to 1-2 mm from the point of its generation The reason for drop displacement remained unclear for a long tune Investigating gigantic photocurrent fluctuations in p - n junctions, Asnm et al ~23) found that the e-h drops excited on one side of a 1 mm-thmk germanmm sample could be detected on the other side Delay between the end of the excatation pulse and the drop detection was reduced as the excitataon level was increased (Fig 7) Hvam et al (116) determined the size of the e - h drop cloud measuring the stgnal from the p-n junction as a function of distance between )he p-n junction and the laser light spot It was round that in the initial stage the drop cloud expanded rapidly, followed by slower motion with a velocity of about 10a cm/sec Scanmng the regton occupied bythe e-h drops with an Infrared radiation detector, Pokrovsloi et al (172) showed that the luminescence area expanded with increasing excitation power (Fig 24) Mattos et al (15a) investigated the drop spatial chstribution by means of scattering of a narrow H e - N e laser beam (2 = 3.39 #m) It was found that when a semiconductor is excited by a sharply-focused axgon-lon laser beam, the e - h drop cloud forms a hernmphere, the rachus of which grew as the eacltataon level was increased, and reached 1 mm at laser power equal to 0 1 W. Since the e-h drops of 1 lan radius have an effective mass of about 106 m o and the drop momentum relaxation tLme may be

200

A A ROGACHEV

e s t a n a t e d 10 - 9 sec, the chffuslon coeffiaent should not exceed 10-3 cme/sec at T = 2 K Therefore slgndicant drop nugratlon from the excRatlon site winch was repeatedly observed expenmentally may be explained only by the existence of some drift moUon mechamsm Balslev et al {4~)suggested that the drop moves due to the excRatlon concentration gradient (exoton drag) However, this model gives drop velocmes two orders ofmagmtude less than those expertraentally observed. Moreover, the experLment shows that as the temperature decreases, which results m strong decrease of the free exoton density, the drag effect, contrary to the "exc~ton drag" model pre&ct~ons, is mtensflied. The dynamic cloud model proposed by Combescot(Ts) assumes that the large Fermi gas pressure existing m the e - h plasma during the action of the excmng hght pulse is the source of drop mot10n After the pulse-end, the drop moves balhstlcally, only undergoing scattering by free excitons However, the mteracUon of the e - h drop wlth phonons ignored m this model should stop It during a tLmewhich Is several orders of magnitude less than the experimentally observed drop displacement tLme (t = 10#sec) As long as the electron-phonon interaction in semiconductors IS strong enough, the drops may be considered as moving m a v|scous me&urn and consequently the drift velocity is defined by the foUowmg snnple relation

F vd = ~ .

(62)

where F Is the force apphed to a drop, M is the drop mass, Zp is the drop momentum relaxation time Now it IS firmly estabhshed that the mare reason for drop motion is the "phonon wind" mechamsm proposed by Keldysh (134) Nonequfllbnum phonons are generated as a result of energy &sslpatlon during thermohzatlon and recombination of the e - h pairs produced by light The acoustical phonons play a dormnant role, because the opUcal phonon hfettme wRh respect to the decay into acoustical phonons is small (about 10-11 sec) Since the phonons with wave-number greater than 2k v cannot be absorbed by the e - h drops, and the densRy of final states for the phonons with k << k r Is small, the dominant role m the drag process is to be played by the phonons wxth k approxtmately equal to kr A smaple esttmate shows that the number of such phonons generated during deceleration of fast electrons or holes m pure crystals is very small This, probably, explmns why m some early papers the low-energy phonon absorption by drops was not considered as the mam drag mechamsm Dunng absorption of the low-energy phonons, the force actmg upon the e - h drop (per one e - h paar) wdl be equal to 6 k v Wv F - - (63) where ~ ~s the average effe~lve cross-sect|on of mteracUon of an electron (hole) with a phonon of energy he%, and Wn is the phonon energy flux densRy The value of 0 depends on the particular type of electron-phonon lnteractmn (abso~Uon or scattenng) The contribution of both the h|gh-energy T A phonon scattering and the low-energy acoustical phonon absorption to # has been considered m the pioneering papers (45,~34) The source of low-energy phonons was not specified, and it was assumed that the maan contribution to the dragging o f e - h drops by the phonon wind is due to hole-scattenng by high-energy TA phonons According to the estlmste made by Bagaev et al ,(4~) it is these phonons with k equal approxtmately to 3 × 107 cm winch are accumulated in the crystal as a result of the energy &sslpatlon by fast electrons and holes, since further decay of these phonons at hqmd hehum temperature is a very slow process Bagaev et al (45) put forward the idea that high-energy phonons may be effectwe m the exc~ton-phonon scattenng process The excaton ground state ~s spht due to electron effecUve-mass amsotropy The electro~wave-funcUon amsotropy has approxmaately the same effect on the exoton ground-state sphtUng as the stretching of the germamum crystal along the [III] darectmn. ModutaUon of th|s sphtUng by a sound wave increases substantmlly the nonelasUc phonon scattenng by excatons It was also assumed that this drag mechamsm may be apphed to the e - h drops Such an assumpt|on ~s d|fficult to justify, smce each hole m the hqmd interacts stmultaneously with electrons from different

Electron-hole h q m d s m semiconductors

201

valleys, which makes the combmed effect to be almost tsotropie, Besides, interaction cut off at k .~ kF (see Ref 110) was not taken into account, resulting m strong overestunation of the mteractaon cross-section. The form-factor omitted m the calculation by Bagaev et al should be estmaated as 10- 5 (see Ref 104) The expertmental mvestigatton o f e - h drop dragging by the phonon wind camed out by Barbaev et al (4s) consisted in observataon of the drop-cloud shift This cloud was produced by means of bulk He-Ne-laser excitation (2 = 1 52/am) and was propelled by the noneqmfibnum phonon flux This flux has ltsongm m the crystal region excated by a Nd 3+ YAG-laser (2 = 1 06 #m) placed 7 mm apart from the bulk He-Ne laser excatatton region. The drop movement was deduced from observations of the probing 40 mW He-Ne (2 = 3 39/zm) laser-beam absorption About 1 mm drop-displacement was observed and the drop velocaty was estunated as l0 s cm/sec Analysis of these results was based on the assumption that the phonon wand was caused by high-energy TA-phonons Notwithstanding the obvtous inconsistency of the theory, this point of vtew was later supported by Astemirov et al (4o 4~) Asmn et al (29) proposed that the e - h drops themselves are mtenswe source of nonequdlbnum low-energy acoustical phonons, and have shown experunentally that the absorption of these phonons by drops is the mare reason for drop draft This conclusion* Is consistent wath one of the two posstble phonon wmd sources (highenergy or low-energy phonons) proposed by Ketdysh (~34) Dunng the Auger-recombmaUon in germanium and sthcon, fast electrons with energy equal to Eg are generated They may lose the energy by emitting optical phonons and by collision wath electrons and holes in the drop The electron free-path due to scattering by optical phonons, as has been shown by Crowell et a / , (76) IS about 10-6era Therefore, the fast particle moves m a mg-zag manner, which increases the ttme of its presence reside the drop The rate of the energy loss from the colhslons with electrons and holes increases as the fast parUdes decelerate For this reason, 0 3 eV electrons m silicon and about 0 1 eV in germamum lose their energy inside the drop predominantly due to colhslons wath electrons and holes In any event, the nummum energy transferred by fast electrons to the degenerate e - h plasma cannot be less than hOgopt s i n c e the e--h scattenng at concentrattons typical for the e - h drops in germanium and sfltcon ~s consaderably more effectwe than scattering by acoustical phonons Est~matlon of drop overheating may be obtained by equaltzmg the energy lost by the drop due to acoustical phonon emass~on and the energy gamed at the Auger-electron deceleration If each Augerelectron transfers to the drop an energy equal to 0 leg, then _1 = A T mv z 0 1E~z d T ~p

(64)

where zp = (mv2,/k,T)L, and L is the energy relaxation tune Equation (64) for the e - h drops m germamum gwes T = 0 1 K at T = 2 K It is valid, provided that the drop stze is small compared wath the mean free path of the acoustical phonon with k ~ k F reside the drop At high enough temperatures, excess energy is carried away from the drops by low-energy acoustical phonons with k ~< 2kr At lower temperatures, the hmmng wave-number of emitted phonons is k = (kBT/hv~) T h e absorption cross-sect~on of such phonons can be obtained m a stmple way from the calculatton by Bagaev et al (45) D e2m e2kFe +

apt, =

D2m,,kF,,

8rrhSpvsNo

(65)

where De his the deformation potenttal of electrons and holes, N Ois the e - h pmr density m the drops, v~is the sound velocaty, p is the crystal density, a-a..~Zph-e h IS the phonon relaxation tune on electrons and holes In the case of germanium, De = 12 eV, D h = 5 eV, v~ = 5 × 105 cm/sec and p = 4 5 g/cm- 3, which gwes aph = 10-15 crn2 * It ~s necessary to mention that the p h o n o n wind or p h o n o n drag regardless o f e - h drops ~s a very old ~dea indeed The problem was menttoned for the first u m e by Sommerfeld and Bethe ~2o4) Later, Gurevtch (~s 99) mtroduced a phonon drag term m the thermoelectric effect theory, and emphasized tts tmportance The theory of photon-dragenchanced thermoelectric coefficient m semiconductors was given by Frederlkse (92) and H e m n g , " ~1) who showed that the effect Is due to low-energy p h o n o n s SubstanUal increase of the thermoelectric coefficient related to the phonon drag was observed by Gebale et al (93) m g e r m a n m m

202

A A ROGACHEV

Equation (65) Is valid only for phonons with k ~ kr At smaller k, the screening effects become essential. In the hmmng ease of small k, the absorpUon cross-section is determined by the squared sum of the deformation potentmls De and Dh Using eqs (64) and (65) one may estimate the drag velocity for different excitataon conditions A strong absorption of the low-enorgy L A phonons by the e - h drops has been observed by Hensel et al (109) The phonons were generated by thermal pulses and propagated ballistically through the sample in the [III] direction The time of the phonon arrival to the opposite sample face was registered by means ofa superconductwlty boleometer The e - h drops were produced inside the crystal by He-Ne laser bulk excitation (2 = 1 52 #m) As follows from the data presented m Fig 51, the TA phonons were not in practice absorbed. Simultaneously, the crop migration due to the phonon flux action was observed. The drop velocity was found to be as high as 10*cm/sec The drop momentum relaxation time was estimated as 2 x 10-9 sec~ which was in good agreement with the other experimental data available An evidence that the phonon drag is due mainly to the phonons generated by drops heated via the process of Auger-electron energy relaxation inside the drops has been obtained m lnvestagataons drop velocity oscillations m a longitudinal quantuang magnetic field (29 30) Interaction of the drop with the low-energy phonons is proportional to the Fermi surface density of states Therefore, it reaches maximum value every tame the Landau level crosses the Fermi level Since the propelling force and the breaking force change m the same way, the drop velocity at constant phonon flux wdl be independent of magnetac field This suggests that the pbonon flux oscillations is the main cause of drop velocity oscillations The latter is feasible only if the drops themselves are the phonon source, as was shown by Asmn et al (29) The phonon flux emitted by a drop oscdlates proportionally to the density of states on the Fermi level, and may be regarded as the mare factor which leads to the oscillations of the drift velocity of that part of the drops which is moving away from the excitation region Besides, there is a certain redistribution of phonon flux between electrons and holes, since, due to a strong Coulomb interaction of electrons and holes, their temperatures are equal Such phonon flux redistribution should also contribute to the drop velocity oscillations, due to the fact that the deformation potential of the valence band is considerably smaller than that of the conduction band, and as a consequence the phonons emitted by holes are weakly absorbed by holes of the moving drops I

I

I

I

I

I

i

,

I

I

,ore

att[.~]

It

I~,-.-~).] - - ~

LI

m

I~! bOl°

I 52/~m

'°ll\ [~ I 0

I

I

~ I

I

2

I

4 I ~

Pump

I 6

I

I 8

/J. SeC

FUG 51 Heat pulse phonon absorption by e-h drops m germanium (t09)

Electron-hole hqmds in semiconductors

203

BII

,*I--

3~

F

F ~I-

)

I)

,.j

I

.....3".,..

'.:.

I

I

B---'

%-.

v .,.,I

,v--.~

,,. I

I

I

l

l

I

I

25

BII ()i,>

/

2O

%

2

%\ "',

I

~

::k 3'2' I' I I

e5

l

/" I O'

I

l

I

I

I

,,,d"

, : , . . i , , . ) ., I

I

I

I

I

I

I

BII )5

•"

%

io

"t %o e

5

II I I

) I 0

I 2 0

I

I 30

I

40

I ',% $0

B, T FIG 52 e-h drop transit time as a functaon of longitudinal magnetic field Solid line, experimental, dotted, theoretical calculation for conduction band only, T = 1 8 K Landau level numbers placed near the corresponding minima Numbers marked by a stroke correspond to the "hght" Landau bands whmh have the biggest m= and thus the highest denmy of state (ao)

The results of experimental observations of the e-h drop velocity oscillations in the longitudinal magneuc field are shown in Ftg 52 The drop transit time was measured as a function of the longitudinal magnetic field when the drop moved a distance of 1 mm from the excltauon regmn to the p-n junctmn The velocity oscillation maxima corresponds to the crossing of the Fermi level by the Landau levels Quantmatlon of the valence band states m magnetic fields used in the expenments was probably too small to be resolved The extrema posmon and amphtudes of the drift velocity oscillations are in good agreement with the calculations by Asnln et al (ao) Direct expenmental demonstraUon proving the phonon flux oscillations to be caused by the drop velocity osclllatmns has been obtmned in the study of the combined effect of the phonon flux from drops, and that produced by the thermal phonon generator (34) The latter was made m a form of a metal film evaporated on the crystal surface Both the intensity and duration of the phonon flux were regulated by changing the strength of current pulses passing through the film Since the phonon flux from the thermal generator does not del~nd on the magnetic field, its actmn must supress the drop velocity oscillations providing that the drag model given above is valid Figure 53 shows the results of the experiment in which the prechcted oscillation amphtude decrease was observed with the increase of the phonon flux from the thermal generator The intensity ofhght pulses creating the e-h drops was reduced m such a manner that with increasing heat pulse intensity, and with the drop velocity at n = 1, the oscillation maximum is maintained at constant level Another independent demonstratmn of the fact that the phonon wind action ~sdue mmnly to low-energy phonons has been given by Doehler et al (81) They determined the velocity distribution in the drop cloud from the Doppler shift measurements of the light scattered by the drops (2 = 3 39/~m) at chfferent photon energtes of the excitation It has b ~ n found that reduction of the exciting light photon energy by one half did not lead to a noticeable change

204

A A ROGACHEV

)0@1

04 ~

7"

t 0.

3

5

03

8 I03 ~a

_o

02'

I

I

08

09

0

I

I

I0

II

8,

I

I

)2

I 3

T

"\

0 )

I

I

05

~0

P'r I F I o 5 3 n = 1 oscdlatlon amphtude as a function of heat pulse mtenmty P In the insertion, drop velocaty zs shown as a function of longitudinal magnetic field for different heat to hght pulse mtenmty ratios (1) 0 , (2) 0 1 , (3) 0 8 (a4)

m the drop velocity Doehler et al came to the conclusion that the phonons emitted during thermohzatlon o f e - h pairs chd not play a mgntticant role m the drop dragging Greenstem et al (gs) came to the same conclusion. They succeeded m obtaining photographs of the e-h drop cloud m the germamum crystal and found that both raze and shape were independent of the exclton photon energy The photographs were obtained by scanning of the crystal with optical systems of a small aperture m which germanium photodlode was used as the radiation detector The cloud shape differed nolaceably from sphencal and had an amsotropy corresponding to that of the deformation potential It was extended along the [III ] direction for which the deformauon potentml and kv have maxtma Excatatlon of the samples by short and powerful hght pulses forms the socalled "hot spot", emitting a wide spectrum of the phonons including those with k ,,~ kF which effectwely propel the e - h drops It can be assumed that It is just this mechamsm which is responmble for the drop cloud dlspermon observed m early works of Hvam et al (1t 6) and Damen et al (78) The specific characteristic of the e-h drop motion m germamum, as has been pomted out by Ashkmadze et al ,(37) ~s the

I

I

I

I

I

I

l

1 1 I

B

I

I

I

f

i

I

I

I

I

I O0

~ u

0 95

E _.1

0 9O I I I I I I I I I [ I I I I I I I I I hv

FIo 54

20

I0

0

=0 709

eV

e-h drop lumanescence mtenmty as a function of heat pulse power "Phonon wind" created by this pulse Is blowing drops away to the surface with high recombination velocity (31)

Electron-hole hqmds m semtconductors

205

existence of the threshold value of the force that sets the drop m motion, below winch the drop remams immobile. In other words, the drop moUon becomes noticeable only after the exoting radiation power exceeds a defimte value It has also been shown that if the excmng rachation power is mamtamed below tins threshold and an addiuonal phonon flux is created by a thermal pulse, the drop moUon becomes possible, and exinblts a threshold character (34) The drop velocRy rises hnearly wRh thermal pulse power increase. An additional feature is the existence of the minnnum velocity Vm,,= 5 X 103 cm/sec, corresponding to the drop motion onset. The exmtence of the threshold radmt~on power has been confirmed also m the expermaents in winch the phonon wmd"blew away" the e - h drops towards the surface w~th a large surface recombination velocity The drops vamshed when they reached the surface, which was detected as the lununescence intensity decrease (31) As seen from the data presented in Fig. 54 the luminescence quenching exhibits the threshold Existence of the drag force threshold Is, apparently, a direct consequence of the drop capture by Lrnpur~tycenters observed by Westervelt et al (233 234) 11 M U L T I E X C I T O N B O U N D C O M P L E X E S IN S E M I C O N D U C T O R S

The topic of multicxc~ton complexes is closely related to the subject matter of the present review They are characterlstac of complex-band-structure semiconductors, and have no analogs in real atonuc systems It is well known at present that, e g, m silicon one may observe complexes composed of as many as sxx exc~tons. Investlgat~on of the multlexCltOn complexes may be useful m resolving the problem of how one exoton or exc~ton molecule transforms into macroscopic exoton drops Since the mult~exc~ton complexes in stllcon with number of exc~tons m > 6 were never observed experimentally, m = 6 may be considered as the upper lunlt of the complex stability m this crystal Larger formations are, apparently, unstable A new stability region is achieved when the small e-h droplets are formed. An answer to the question how wide is the uistabIhty gap is of extreme tmportance in the embryo formation theory, since the size of this region defines the magnitude of exc~ton densRy fluctuations needed to form an embryo Small e-h drops cease to be stable when they reach some crmcal size, defined by the contribution of the surface tension energy and by the increase of the klnetac energy of electrons and holes due to the drops' size limRations An attempt was made to calculate the binding energy of the bound multleXClton complexes(2s~'z3s) and small e-h drops, (Shore et al (~gs)and Rose et al (1s4)) using the density functional theory The latter shows that the small-drop binding energy oscillates closely following the consecutive filling up of s, p and other electron and hole shells A surprismg fact is that drops of more than m = 10 e - h pmrs appear to be stable, having probably some small instability reg~onnear m = 40 (for gcrmamum) The drops consisting of more than 100 e - h pmrs possess practically all the properties of bulk e - h liquids Another tmportant result of these calculatmns ~sthe conclusmn that the interpretataon of the line shift in terms of surface tensmn coefficmnt ~ is questionable even for the b~gger drops The mulUexclton complexes were chscovered by Kammshl et a/(121122) and Pokrovskli(1:1) m silicon doped by boron and phosphorous as a new series of narrow lines appearing at the lower-energy end of the exclton spectrum These lines were attributed to the exc~ton-~mpunty complexes contmmng more than one bound exc~torL The intensity dependence of the new line on the excitation level and the luminescence kinetics was In good correspondence with that expected for multlexclton complexes (~ss) The main experLrnental fact was that new radmtlon lines appeared on the long-wavelength side of the spectra as excitation lntensRy increased The energy chstance between the free excRon hne and the new lines m the spectrum m early papers was supposed to be equal to the exc~ton binding energy m the complexes As a result, the first contrachetmn arose: the bindmg energy increased with the number of captured excatons and was s)~,nificantlylarger than the exeRon blnchng energy in the e - h drop Doubts m the vahdity of the imtially suggested mtetpretatmn of the new hne nature increased when it was found that the Zeeman sphtung ofthese lmes consisted of equal numbers of components and d~qered instgnflicantly for hnes with d~erent m (Fig. 55) The position and the number of Zeeman components m the spectra were the same as for the sphttmg of luminescence hne of one exclton bound on a neutral phosphorous atom The only JPQE 6/3 - E

206

A A ROGACHEV

H"=O rnf4

Si

I I

0860~

,

P

[.5 x I0 ~3 cm"3]

..

4 2K H= 53 kG

I 0854

I

o

(D°X)

c

g E .J

I 10816

~

I T "

/ ~ - T

--

.~f2 I

J=3/2

10810

3/2 1/2

654

3 2 I

0"0"11"/1"0"0"

m=l

o-llllll 6

I 0784

15141

3

2

I

J,I/2~

'-1/2

I 0776

X,

/~m

FxG 55 Zeeman sphttmg of phononless luminescence hnes of multmxclton bound complexes The dmgram shows transmons m the complex formed by one exc~ton bound to the neutral donor m

slhcon(189)

difference m the spectra correspon&ng to various numbers of captured excltons was the different rat]o of the Zeeman component lntensmes, whmh m&cated a different thermala_atlon degree of corresponding Zeeman states This factor later appeared to be declswe for the explanat]on of thin hne nature m the framework of the multlexoton complexshell model However, httle cons]deraUon has been mmally given to tillS fact Th]s oversunple picture of the mulUexc~ton complex sphttang, as well as the data on the umaxml stress spectroscope, led Sauer et al (189) to abandon the multlexclton complex model To explain the Zeeman sphttmg data, Morgan (Is 2) proposed the multmxclton complex model, m whmh it was assumed that the recomb|nataon hnes observed were related to the rachatwe decay of the first captured exclton The rest of the excltons only changed the binding energy of the first one It has also been assumed that the mteractmn of the first captured exoton with all others should be weak, and the state of these exotons does not change as a result of the first bound exc~ton recombmaUon. Morgan's model hears many internal contrathctmns It does not g~ve an answer to the question why the first bound exc~ton holds an exceptmnal posmon m the recombmat|on process, and the recombmatmn ra&atmn hnes of other exc~tons are absent The problem of the dLfferent thennal|zat|on degree appears also to be unresolved Investigation of excItons bound on neutral impurities of gallium and alum|mum m slhcon has shown that the state of two holes m these complexes had a tnplet structure. Th|s structure is due to the h - h mteractmn, and the effect of the central cell potential The state w~th J = 0 has the lowest energy, wh|ch corresponds to the Morgan" s,) predtct~on that the impurity center field has the strongest effect on the J = 0 state Th|s triplet sphttmg exms only for the state of two holes (so called J - J couphng). In the case of one acceptor-bound exclton, two holes are m the mmal state and the tnplet states must he thermahzed, which ~sjust the fact observed m the expertrnents of Thewalt,(2°s 209) L|ghtowlers et al (1,0) and Lyon et al ,(144) whtle m the case of two bound exc~tons the tnplet state ~s final and thermohzatmn must be

Electron-hole hqmds m semiconductors

207

absent This conclusion agrees with experiment and with the prodictlons of the multmxctton complex-shell model, rather than with the Morgan model, whmh predicts the same behavior of one and two bound excitorl complex hnes. An important step in the understanding of multteXClton complex nature has been made by Dean et aL(7s) who showed that in nitrogendoped cubm sihcon carbide (//-SIC) there is a set of line~ similar to the hnes found in sfllcon by Pokrovskii et al (t71) Dean et al (TS) were able to explain some of expenmontal data by introducing the concept of successive filling of shells resultmg from the symmetry of Impurity centers with electrons and holes In particular, they have shown that it was possible m such a way to explain the similarity of Zeeman spectra of the complexes with different captured excRon number nt Since their model dealt only with transitions to the impurity center ground state, it gave strong overestimation of the binding energy, which was increasing for larger numbers of captured excltons The most successful modification of the shell model was proposed by Karczenow (136)As in Dean's model, in this model a multiexcmton complex is formed by successive filling of the shells, the number of states in which are defined by the structure of the semiconductor conductmn and valence bands The first hole shell m silicon has four states because there are four J = 3/2 spin projections The electron shell contains, correspondingly, 12 states (two states for each conduction band valley) Interaction of electrons with the central-cell field may split the elect~on shell The number of states in each subshell m defined by the impurity center symmetry (Td) According to thin the central cell field in silicon can split the electron shell into three groups of states, with F 1, F 3 and F 5 symmetnes containing 2, 4 and 6 states, respectively The sequence o f f 5, F 3 and r 1 states is determined by the central cell potential sign It is also supposed that in the case of donor multmxcRon complexes this sequence will be the same as in the case of neutral donor. However, the distance between subshells may be different in these two cases, and depend on the captured excaton number An important assumption in this model is that the interaction of electrons and holes from the same or from different shells does not result in additional shell splitting In any event, this spirting is small compared with the distances between the shells Very small splitting of the lines of phosphorous multmxclton complexes in silicon, which can be regarded as the splitting discussed above, was observed by Parsons, (16.) by employing high-resolution spectroscopy

[rt 4r3 5 , 4 r e ] BMEC4

,3F3,5,4Fs ]

, :3FB] BMEC :5

:r, .2r~., ,3r.] 2r~,~ :2r. ]

BMEC 2

[2 r. ,r~., .2r.]

~'

az

7. 2

.r~,, .r~]

BE

a'

8

y'

r. ]

7,J' r, r,

D°

FIG 56 Transmon of mult]excRon complexes based on substRuttonal donor m slhcon (]36)

208

A A ROGACHEV

Different transiUons considered for the subsUuonal donor mtflUexclton complex m the framework of the shell model are shown m Fig. 56 Two electrons m the ground state of the first bound excRon completely fill the F 1 shell, and one hole is present m the Fa shell. Only the F s - F ~ transalon (the so called ~-hne) is possible from the ground state The state where one electron occupies the F 3 or F s subshells will be the first excited state of the complex. Since the F 3 and F 5 states, generally speal~ng, have different-energy two rathaUon lines F s - F 3 (~/-hne) and F s - F 5 (~"-hne) appear The|r observation ~s possible only at sutiicaenfly high temperature One of the electrons m the muluexc~ton complex containing two excatons falls into the F 3 5 subshell Further, for the sake of sunphc~ty it wdl be assumed that the F a - F s splitting is unresolved The transmon between the Fs hole shell and an electron from the F 3 5 subshell leaves the complex m the ground state. Th~s line m Klrczenow's model (~36) is denoted as fl~ Another llne attributed to the complex with two excltons corresponds to the transmon after which the complex remains in the excated state, smce one of two states m the F~ shell IS empty Filling of the hole shell takes place m a sanflar way up to m = 4 (or m = 3 in the case of the acceptor mulUexc~ton complex), when the first hole shell becomes completely filled. Then the next hole shell has to be built up K~rczenov model is virtually only the way to classify the transalons It does not answer the quesUon whether the complex remams bound when the next hole shell is being filled. Later, the exper|inental facts will be discussed, which demonstrate the existence of complexes hawng at least two holes m the shell following the F s shell After KJrczenow's paper, {136} a large number of papers were pubhshed m support of shell-model basic features Thewalt(208 209)has noticed a general agreement between the shell model predmhons and the set of lines attnbuted to the mulUexc~ton complex bound on phosphorous m silicon The F s -+ F~ transmons (a-lines) can be seen in the phononless transmons, since the Ft state has a maximum electron density in the central cell As has already been pointed out, after these transmons the complex remains m the excited state provided that it contmned in the lmUal state more than two electrons (m >/2) The F s - - , r' 3 s transmons (//-lines) are possible (at T = 0) only when m t> 2 Since the electron density in the F 3 s subshell vanishes m the central cell, the fl-lmes may be observed only m the phonon replicas (Fig 57) // (24 n3 ~3Q2~2

I

g

i

--

SI P (IxlOISCm-3)

Np = a' Rephca Repl¢o

42K ~2

":1

<,

~LO

,

- 200-

I 82

I 94

II

Photon enery,

I

meV

FIG 57 T~me-resolved luminescence spectra of phosphorus complexes m sd]con (2~°}

The ~tm and fl'~-1 hnes considered above may be observed at low temperatures As temperature increases, the occurrence of new lines connected w~th the excited states of the complex ~s possible The first excited state of the mulUexc~ton complex m the shell model ~s that m which one or two empty places m the FI subshell are available Because ~t ~s not known at present whether the multlexclton complex with the empty F 1 subshell ~s stable or not, m the following discussion only such excited states shall be consflered where one FI electron ~s

Electron-hole lgtmds m semiconductors

209

present m the complex. The excated hole states are not considered, since the central-cell field can spilt only the level of two holes (the so-called J - J couphng) The next hole-excRed states may be m the 2s or 2p shells. Kirczenov denoted the transmons between two excited states as y-transRlons. In all cases except m = 1 they correspond to the F s - F 3 5 transrhon, and may, respectively, be observed only in the phonon rel~lcas The y 1-transition Is the only exception. The bound exctton-exctted state may relax in two ways. After recombmaUon with the F 1 electron, the donor remains m the F 3 5 excated state. Since these transmous take place between two excited states, R Is denoted as the ~-transmon* Due to small sphttmg of the F 3 and F5 states, there extst strictly spealong two y-transmons, ~' and ~" Unlike other ~transmons, these transluons may be observed in the phononless spectra. Another way for relaxatmn of the excRed state ~s the recombmatmn of a hole w~th the F 3 s subshell electron After this transition, which may be observed only in the phonon rephcas (the 6-transition), the donor remmns in the ground state To achieve a noticeable filling of the mulUexcRon complex excRed states in sthcon it is necessary to have the temperature of about 20 K or more At this temperature, the lines corresponding to m > 1 dtsappear The u, ~ and y hnes in the radmtive recombInatmn spectra of an excRon bound to a donor are well distingmshed in Fig 58, taken from Thewalt's paper (209) //

// I

ETO

SI ( P I 2 x l O I 6 c m 3) 19 K

8

BTOa l TO ~| 8= :,1. I TO • TO II,,\

'Jl I a I

TA

_1

( F3 ~

2S I

1083

I

I

I

I

I

I

1095

i.• ., I

|

1116

I

I,,* .r

rl~ P

l I

I

)

~x18

~ I

I

I

I

I

I

t

1132

Photon energy,

I

)

I 1156

meV

FIG 58 Phosphorus-doped slhcon luminescence spectra showing various transluons predicted by

shell model for the complex contmmng one bound excaton (2o9}

The observation of new radiative lines m the TO phonon-asslsted radiation spectrum in phosphorous-doped silicon, which were absent in the phononless spectra, was a good verLtiCatlon of the shell model conclusions (Fig, 57) These new lines were interpreted as fl hues m the Ktrczenow model The fact that these lines are actually the fl lmes--i e ~= and /~,-1 lines belong to the same multlexciton complex--was confirmed experimentally by Thewalt, t21°} who mvestagated the me-resolved luminescence spectra. At least it can be confirmed that the ~3 and f12 and, probably, the ~4 and f13 lines decay with about the same time constants (Fig 57) MultIexcxton complexes connected w~th other subsUtut~onal donors in sihcon, such as anUmony and arsemc, were investigated by Elhot et al (87) The effect of the unlaxial stress on the multIexcRon complexes was used for classification of the electron and hole states (2~2) The uniaxlal stress lifts the degeneracy of the Fs hole shell by sphttmg it into two subshells j= = ±½ and j= = +_~ with two states in each If the stress is applied along the [III] direction, the electron shell remmns unspht As a result of the hole shell sphttmg all • and fl hues acquire the doublet structure The doublet hue thermohzation depends on the filling of the hole levels m the LmtIal state m a way predicted by the shell model In particular, the thermohzations of the f12 and ~3 lines appear to be the same, which corresponds to the earher observation that these two lines belong to the same complex {210) Since the hole shell exhibits twofold sphttlng at any direction of the stress, the ~-line doublet structure should be observed under applicauon of stress along the [IIO] and [IOO ] directions. A new feature arising from the stress applied along the [IOO ] and [IIO ] directions has to be the//-line sphtting, since the F 5 and F 3 shells of the n-npunty center at large m are split in

210

A A ROGACHEV

the same way as the F5 and F 3 s t a t e s of the substitutional donor The greatest sphtting occurs along the [IOO ] chrecUon. This sphttmg has not been found by Sauer et al (19o) However, Thewalt's measurements have shown the presence of such sphttmg, which ~s in agreement with the shell-model predmtmns A review of umaxlal stress ~pectroscopy studies of mulUexc~ton complexes was gwen by Thewalt. (212) The most chrect and mfonnatwe way of studymg the electron and hole shell filhng is the mvestlgaUon of the magneUc field reduced circular polanzatmn spectra The advantage of this method is that the average spin of the parades involved may be determined even m the case when the Zeeman line sphttlng m the magneUc field cannot be resolved due to phonon-mduced wldemng of the hnes Besides, it appeared that theoretical caleulatmn of integrated polanzatmn ofZeeman-spht hnes ~s much sanpler than calculat|on of the radmtmn mtensmes and the polanzauon of each separate component The first experiment along this hne was published by Altukhov e t al,(6) at the tune when the multlexclton complex model was challenged by experimental results on Zeeman sphttmg, discussed above The expermaents were carried out with boron-doped slhcon m the Faraday cordiguraUon The magneUc field was darected along the [III ] ax~s of the crystal The experwnent has shown that the polanzatmn sign was the same as for complexes with m = 1 and m = 2, but in the case of m = 3 it changed to the opposite The radmt~on polarlzatmn degree of the complex with m = 2 was found to be approxunately twice as small as that of the bound exclton As was shown m Section 5 ofthls rewew, the carcular polarization mduced by the orientation of electron and hole magneUc moments m sfllcon have opposite signs The polanzatmn slgn of m = 1 and m = 2 complexes corresponds to predominant hole orientation, while m the case of m = 3 it is determined by the electron orientation The decrease of the polanzataon in the case of m = 2 compared with that of the bound exclton (m = 1), and the opposite polanzatmn sign of the m = 3 radmtmn hne, were explained by the increase of the hole shell filling. According to the shell model, the hole shell of the m = 3 acceptor complex containing four holes is closed and the total hole spin moment is equal to zero The radmtion polar|zat|on m this case is determined excluswely by the F 1 and F 3 s electron orientation Figure 59 shows the spectra of TO-phonon-assisted and phononless

hi I~ I_ (~

NP

I ~

I

/\,,Z

I L_

I

A

S, B

--'4

I",-)9°K

,o,o

I/.v I

I

~

I"I.L..

If I

H

i

I

h~, ev

I

A

TO A2

'' ~ ' ~ ' ~ "

~./

\ I

N"t''-;7/~

V

I~-I_ hv, ev

FIG 59 Luminescence spectra (I+ + I_) and circular polanzed luminescence (I÷ - I ~ ) (TOphonon-ass~sted and phononlcss (NP) transmons) of boron complexes m slhcon, H = 50kG, n. = 3 x 101S/crn 3, HI[ Jill] "°)

Electron-hole hqmds m semiconductors

211

(NP) kLmmescence (I+ + I_ ) and polarized lununescence (I + - I_ ) of boron multlexoton complexes m a magneUc field of 5 T chrected along the [III] crystal ax~ .1°) Zeeman splitting of the radmtion TO components ~s resolved m the polanzatmn spectra as ~r+ and ~r- hne components. The phononless radiatmn polarization has a s:gn opposite to that of the TOphonon-asslsted radmtmn polargatmn, because the TO phonon takes away a part of the angular momentum, and, as a result, the TO-phonon-asslsted and phononless optical transRmns obey d~fferent selecUon rules, as was shown by l~kus (166) Comparison of the theoreUcal and expertmental values of the average ra&aUon polanzatmn degree ~s demonstrated m F~g. 60. The average polanzat:on degree of the mulUexcaton complex luminescence m the magneUc field ~s defined by the onenta~aon of electrons and holes m the corresponchng shells, and for the TO-phonon-ass~sted radmtton, also by some constant equal to the radmtmn polarization degree of the A ~o hne (m = 1) at high magnetic field strength, when polanzatmn ~s saturated. (s)

/

4

A~,o

a .=

a0 od 0

"0 "0

=, O-"

0

~..~

H,

kG

'~A=

-0

"04

FIG 60 MagneUc field dependence of csrcular polar:zatlon degree of TO and N P luminescence hnes Sohd curves, experimental TheoreUcal calculatmns are shown by dotted hnes (for m = 1, 2, 3) and by points (m = 4) Circles shows experimental results for weak hnes Pex is experimental polanzatmn degree not corrected for depolarization {lo~

In the case of the phononless radiation the polar:zatlon sign for the F s - F 3 5 transitions coincides w~th that of the TO-phonon-assisted radiation, whde for the F s - F 1 transmons it ~s opposite to that Therefore, the phononless radiation polanzaUon bears information on the relative contribution of the F 3 5 and F 1 states into the radmtion and, thus, permits the determination of the sphttmg of the F 3 s and FI states. Now, the existence of this sphttmg :s a point of controversy In a very recent paper, Thewalt et al ,~213~on the bas~s of the umaxml stress spectroscopy studies, claimed that the valley-orbit sphttmg m the born complexes IS essentmlly equal to zero Polarized spectroscopy measurements by Altukhov et al., (1°) on the contrary, g~ve some re&cations that the F 3 5 state is the lowest, and F1-F 3 s sphttmg is increasing with complex number m. This conclusion was derived from the analym of the experimental data presented m F~g 60 The sphttmg of these states are equal respectively, to A 1 = 0 05 meV, A2 = 0 13 meV, and A 3 - 0 16 meV Sphttmg of the F 1 and F 3 s states may result from the interact:on of electrons with the accepter central cell Since the mteractmn between an electron and an accepter :s repulsive, the probabtilty of the electron getting rote

212

A A ROGACHEV

the central cell will increase wRh the complex number, due to the Coulomb interaction screening The theoretical curves m FI~ 60 were calculated wRh the lsotrop~c hole g-factor equal to 12 and the sphttmg between the two hole states wRh J = 0 and J = 2 equal to Ao2 = 0 1 meV, assuming the J = 0 state to be the lowest. Tbe polarlzaUon degree for the complexes vath m = 3 and m = 4 was found to be the same within the In-mrs o f t b e expernnental accuracy. The theoretical curves for A 3 and A 4 (Fig. 60) were calculated under the assumption that the A r ° hne resulted from recombination of four electrons from the F 1 and F s s shells with four holes from the d o s e d tuner F s shell Thus, the A 4 hne is v|rtually a hne of a new type, since it corresponds to the recombination process which l ~ v e s both the hole and electron shell m the excited state. In th~s case, throe holes (and, correspondingly, one empty state) remmn m the tuner shell, and one hole m the outer shell If the A 4 hne resulted from the recombination of a hole from the outer shell and an electron from the F 1 or F 3 5 subsbells, the radmtlon polarization would have an opposRe mgn and its value would be close to the polarization of the A 1 (m = 1) radmtlon The outer hole shell in the multlexoton complex is, probably, formed by two-fold degenerate r 7 state stmflar to the corresponding accepter state (141) Investlgat|on of phosphorus d o n o r muluexclton complexes shows the sLmdar behavior of polarization (11) Since the total moment of the two electrons m the F1 subshell is equal to zero, the ~,,, radiation polarization degree is determined by the average hole moment, which decreases with increasing m For th~s reason, the average a-radmt~on polarization degree has to decrease with increasing m and to become zero at m = 4--2 e when the hole shell is closed Thus, investigation of the circular p o l a n z a t m n spectra gives a sLmplc and rehable ldentlficaUon of the % and % hnes If the % and ~6 hnes do result from the recombination of electrons and holes from the closed F 1 and F s shells m the complexes w~th m = 5 and m = 6, the average polarization degree of these hncs has to be equal to zero Experimental data presented by Altukhov e t al (11) are m an excellent agreement wRh that prediction Figure 61 shows the phononless r a & a t m n spectra o f t b e phosphorus-doped mhcon crystal samples (% = 3 x 101 ~ cm ~) for the Faraday configuration (H [[ [III] ax~s) at the magnetic field of 3 T

I.* I_ a~

a4

SI P 9°K "~ J"" 30kG 1

xl~t

x2

xl

ao

ItI_

I+ I.

1r' V xl

,,)

~ [

~I;I_

hz,, eV

Fie 61 Luminescence (I+ + I - ) and circular polarized luminescence (I+ - I _ ) of phosphorus multlexoton complexes m slhcon B = 3T, BJ[ [III], T = 1 9 K (~J)

Electron-hole

hqmds

m

semiconductors

213

The a5 and a6 hnes of the polarized luminescence spectra have a smadar shape and their ~+ and ~ - components are essenttally equal to each other In the lmalts of the experimental error the average polanzatmn of hnes ~s equal to zero, indicating that these hnes belong to the recombination of the electrons and holes from the dosed shells Thus, the as and % hnes differ from other ~ lmes, since the corresponding transitions leave both electron and hole shells m the exated state, as was proposed by Kargenov(13° and Thewalt. (2°s) In the case of normal ~-transmous, when m ~< 4, only the electron shell remains m the exated state Very recently, Thewalt et al (2t3) gave evidence of the existence of these hnes m hthmm-doped silicon In the case of the donor mulUexcaton complex, the next after the F s hole shell has one hole for m = 5 and two holes for m = 6 As menUoned prewously, m the chscussmn of the acceptor complexes, the other shell can be formed by two F7 states It is qmte probable that the filling of the next shells leads to smaller or even negatwe excaton binding energy, and the existence of the donor complexes with m > 6 and acceptor complexes voth m > 5 wdl be tmposslble This hypothesis at present seems hkely, because of the lack ofexpenmental data confirming the existence of the complexes with m > 6 Recently, the complexes connected with the acceptor tmpunty of zinc m galhum phosphide have been found by Sauer et a/(191) Upon decreasing the excatatlon level, the hnes corresponding to large m (see Fig, 62) vamshed rapidly Besides, a characteristic decrease of the luminescence relaxation ttme was observed for larger m. The hfetlme of the first bound exclton was 65 nsec For the following complexes ~t decreases (for m2, ~ = 40 nsec, for m3, = 28 nsec, and for m,, ~ = 25 nsec) E , eV 2 30

2 28 I

2 52

I

I

GoP Zn

N

T= I 6K

z. L"

z.T ,

(m))

(m~) m2

i'

i'i"

'xO5,A

sNP

J~ J £1~1 ~

A

Zn~ P

I.,,2 LJt I

FIG

62

I 545

I

I

.540 5:55 )% nm Multlexclton complex luminescence of z~nc-dope~ galhum phosphlde, pulsed laser exotatmn, z -- 12 nsec, 2 = 488 nm, P = 20 kW (191)

Hence, one may conclude that a rather good understanchng of the nature of bound muluexclton complexes m semiconductors has been achieved. Recently, new recombmaUon hnes have been observed m accepter-doped sthcon, which eventually result from radiatwe recombmahon of excltons captured by the clusters of two or more anpunues (9) When the unptmty concentrauon is approaclung the Mott tranmlon regmn from the insulating side, an mteresUng development ~kes place Accepters form mapurlty bands, where the hole spins are arranged m antfferromagneUc order, and as a result m moderate magneUc fields the hole magneUc moment Is essentmlly equal to zero If some free exc~tons are added rote the system, their holes are captured by the tmpunty band m such a way that the antfferromagnetlc order pers~sts--~ e the photoexoted hole would be mdlstmgmshable from the eqmhbrmm ones The spectral hne of such an tmpunty band exc~ton is situated between the lines of excltons

214

A A ROGACHEV

bound on boron acceptors and the expected posmon of the impurity-conduction-band transmonline PolanzaUon properues of the new line appear to be rather different from those of the acceptor-bound exclton In the latter case, the polarization has a "hole" sign, while in the former the sign of polarization corresponds to the orientation of the electrons This behavior is in excellent agreement w~th the existence of the above-mentioned antiferromagnetic order, and does not change substantially up to the critical concentration of the Mott transition (nd = 5 x l0 is cm 3) Acknowledgements--In the course of preparation of thts review, the author enjoyed the opportumty to dtscuss many subJeCtSwith V M Asmn, P D Altkhov and G E Plkus For the translation of the rewew Into English, the generous help ofG V lt'menkov and A L Gerasimov was indispensable The excellent sloll and remarkable patience of Mrs Ludmda P Rusanoweh were crucial m typmg the manuscript The author is indebted to Miss Nataha I Sablma and V I Stepanov for technical asmstance The mQst profound gratitude the author would hke to express to hts wife for her understanding and patience during the work on this review, as well as for her actwe help m correction of the numerous m~stakes REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

N ABAKUMOV,V I PEREL' and I N YASSIEVICH,Flz Tverd Tela 12, 3 (1978) F ABRAHAM,Homogeneous Nucleation Theory, Acad Press, New York (1974) M AGRANOVlCHand B S TosrlICH, Zh Eksp Teor Ftz 53, 149 (1967). S ALEKSEEV,V S BAGAEV,T I GALKINA,O V GOOOLIN, N A P~NIN, A M SEMENOVand V G STOPACHINSKII,ZhETF P~s Red 12, 203 (1970) P D ALTUKHOV,G E PIKES and A A ROGACHEV,ZhETF Pis Red 25, 154 (1977) P D ALTUKHOV,K N EL'TSOV,G E PIKES and A A ROGACHEV,ZhETF Pls Red 26, 468 (1977) P D ALTUKHOV,V M ASNIN,Y U N LOMASOV,G E PIKESand A A ROGACHEV,The Apphcattons of H~gh Magnetw Fwlds m Semiconductor Physics, p 253, Oxford Umverslty (1978) P D ALTUKHOV,G E PIKES and A A ROGACHEV,F~z Tverd Tela 20, 489 (1978) P D ALTUKHOV,K N EL'TSOVand A A ROGACHEV,ZhETF Pzs Red 31, 30 (1980) P D ALTUKHOV,K H EL'TSOV,G E PIKES and A A ROGACHEV,Fzz Toerd Tela, 22, 599 (1980) P D ALTOKHOV,K N EL'TSOV,G E PIKES and A A ROGACHEV,FIz Tverd Tela, 22, 239 (1980) T ANDOand Y UEMURA,J Phys Sac Japan, 37, t044 (1974) A I ANSEL'Mand Yo A FIRSOV,Zh Eksp Teor FIz 28, 151 (1955) V M ASNINand A A ROGACHEV,Fzz Tverd Tela, 5, 1730 (1963) V M ASNIN,A A ROGACHEVand S M RYVKIN,Flz Tekh Poluprov 1, 1740 (1967) V M ASNINand A A ROGACHEV,Phys Star Sol 20, 755 (1967) V M ASNIN,A A ROGACHEVand G L ERISTAVl,Phys Stat Sol 29, 443 (1968) V M ASNINand A A ROGACHEV,ZhETF P~s Red 9, 415 (1969) V M ASmN and A A ROGACHEV,Proc III lnt Conf Photoconductwzty, p 13, Stanford Umverslty, (1969) V M ASNIN,A A ROGACHEVand N I SABLINA,F~z Tekh Poluprov 4, 808 (1970) V M ASNIN,A A ROGACHEVand N I SABLINA,ZhETF Pzs Red l l , 162 (1970) V M ASNINand A A ROGACHEV,ZhETF P~s, Red 14, 494 (1972) V M ASNIN,A A ROGACHEVand N I SABLINA,FI2 Toerd Tela, 14, 396 (1972) V M ASNIN,B V ZuBov, T M MURINA,A i PROKHOROV,A A ROGACHEVand N I SABLINA,Zh Eksp Teor F)z 62, 737 (1972) V i ASNIN,FtT. TVerd Tela, 15, 3298 (1973) V M ASNIN,A A ROGACHEVand Yu N LOMASOV,ZhETF Pts Red (Lemngrad), 1, 596 (1975) V M ASNIN,G L BIP.,Yu N LOMASOV,G E PIKESandA A ROGACHEV,Zh Eksp Teor F~z 71, 1600(1976) V M ASNIN,G L BIR, Y U N LOMASOV,G E PIKusand A A ROGACHEV,FIz Tverd Tela, 18, 2011 (1976) V M ASNIN,A A ROGACHEV,N I SABLINAand V I STEPANOV,F~z Toerd Tela, 19, 3150 (1977) V M ASNIN,N I SAaUNAand V I STEPANOV,ZhETF Pzs Red 27, 584 (1978) V M ASNIN,B M ASHKINADZE,N I SABLINAand V I ST~PANOV,ZhETF Pzs Red 30, 495 (1979), 30, 752 (1979) V M ASNINand N I MIRTSKHULAVA,Ftz Tverd Tela, 21, 3677 (1979) V M ASNINand N I MIRTSKHULAVA,ZhETF P~s Red 30, 200 (1979) V M ASNIN,N I SABLINAand V I STEPANOV,Flz Tverd Tela, 22, 418 (1980) V M ASNIN,N I MIRTSKHULAVAand A A ROGACHEV,Flz TVerd Tela, 19, 209 (1977) V M ASNIN,N I MIRTSKHULAVAand A A RtmACHEV,Sold State Commun B M ASHKINADZEand I M FISHMAN,Flz Tekh Poluprov 11, 301 (1977) B M ASHKINADZEand I M FISHMAN,F~z ~erd Tela, 20, 1071 (1978) B M ASHKINADZE,N N 7_aNOV'EVand I M FISBMAI%Zh Eksp Teor F~z 70, 678 (1978) T A ASTEMIROV,V S BAGAEV,L I PADUCHIKHand A G POYARKOV,ZhETF Pls Red 24, 255 (1976) T A ASTEMIROV,V S BAGAEV,L I PADUCHIKHand A G POYARKOV,Flz Toerd Tela, 19, 937 (1977) W G BAALS,Proc Roy Sac 158, 383 (1937) V S BAGAEV,T I GALKINA,N A PENIN,V B STOPACHINSKHand M N CHLrRAEvA,Z h E T F P l s Red 16, 120 (1972) V S BAGAEV,Springer Tracts Mod Phys 73, 72 (1975) V S BAGAEV,L V KELDYSn,N N SmEL'OINand V A TSVETKOV,Zh Eksp Teor Ftz 70, 702 (1976) V S BAGAEV,N V ZAMKOVETS,L V KELDYgH,N N SIBEL'D1Nand V A TSVETKOV,Zh Eksp Teor F~z 70, 1501 (1976) J BALSLEVand J M HVAN, Phw Stat Sol (b), 65, 531 (1974) V F V A

Electron-hole hqmds m semiconductors 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113

215

J BALSLEV,Sol~d State Commun 23, 205 (1977) R BAUMERT,E BECKMANN,J BROSER, R BOSERand R RENz J Luminescence, 18/19, 558 (1979) R BECr.~R and W DORn~G, Ann Phys (Germany) 24, 719 (1935) S T BELYAEV,Zh Eksp Teor F)z 34, 417 (1958) G BENI and T M RICE, Phys Rev Lett 37, 874 (1976) G BENI and T M RICE, Physics oJ Semtconductors Con./ Set 43, p 355, The Institute of Physics, Bnstol and London (1978) G BENI and T M RICE, Phys Rev B, 18, 768 (1978) C BENOITA LA GUILLAUME,M VOOSand F SALVAN,Phys Rev B, 5, 3079 (1972) C BENOITA LA GUILLAUME,M VOOS,F SALVAN,J M LAURANTand A BONNOT,C R Acad Sc~ ser B, 272, 236 (1971) C BENOITA LA(}UILLAUME,N BONTEMPS,D HULrNand A MvzYRowtcz, J Luminescence, 18/19, 532 (1979) K BETZLER,B (3 ZHURKIN and A L KARUZSKn, Solut State Commun 17, 577 (1975) D BIMI~RG, M S SKOL~HCKand W J CHOYKE, Phys Rev Left 40, 56 (1978) D BIMBERG,L M SANDER,M S SKOLNICK,U ROSt~R and W J CHOYKE,J Lununescence, 18/19, 542 (1979) D BIMBERG,M S SKOLNICKand L M SANDER,Physzcs of Semiconductors, (1978) Conf Ser 43, p 175, The lnsutute of Physics, Bnstol and London (1979) G L BIR, A G ARosov and G E PIKUS, Zh Eksp Teor F~z 69, 1382 (1975) J M BLATT,K W BOER and W BRANDT,Phys Rev 126, 1691 (1962) N I BOGOLVUBOV,lzvest~ya Akad Nauk SSSR set fiz 11, 77 (1947) N BONTEMPS,A MYZYROWICZand D HULIN, Solut-St Electromcs, 21, 1375 (1978) M BORN and E WOLF, Pnnczples of Optics, Pergamon Press (1964) W F BRINKMANand T M RICE, Phys Rev B, 7, 1508 (1973) A C CARTER, P J DEAN, M S SKOLNICKand R A STRADLING,J Phys C, 10, 5111 (1977) D B CHmNUT,J chem Phys 40, 405 (1964) D B CHESNUT,J chem Phys 41, 472 (1964) H CHOU and G K TONG, Phys Rev Lett 41, 1677 (1978) O CHRISTENSEN and J V HVA~I, Proc 7tvelfth lnt Conj" on Phystcs of Semzeonduetors, Stuttgart, 1974, Teubner, Stuttgart p 56, (1974) J COLLE'L J BARREAU.M BROUSSEAUand H MAAR~, Phys Stat Sol (b) 80, 461 (1977) M CO~4BESCOTand P NOZIERES,J Phys C, 5, 2369 (1972) M Combescot, Sohd-St Commun 30, 81 (1979), Phys Rev B, 12, 1591 (1975) C R CROWELLand S M SZE, Appl Phys 39, 4047 (1968) T C DAMENand J M WORLOCK,Proc III Int Conf L~ght Scattering m Sohds, Campmas, 1975, Flammarlon, Parts p 183, (1976) P J DEAN. D C HERBERT.D BIMBERGand W J CHOYKE, Phys Rev Lett 37, 1635 (1976) A F DITE, V G LYSENKO,V G REV~qKO,T G TRATASand V B TIMO~--L~V,Proc Thzrteenth lnt Conf on Physzcs of Semiconductors, Rome, 1976, North-Holland, Amsterdam p 958, (1976) A D DURANDIN,N N SmEL'DIN,V B STOPACHINSKIIand V A Tsv~rKov, ZhETF Pts Red 26, 395 (1977) J DOEHLERand J ~ WORLOCK, Sohd-St Commun 27, 229 (1978) M I DYAZONOVand V I PEI~L, Zh Eksp Teor Fzz 60, 1954 (1971) M I DYAKONOVand V I PEREL, F,z Tverd Tela, 14, 1452 (1972) M I DYAKONOVand V I PEREL, Zh Eksp Teor Fzz 65, 362 (1973) V M EDELSTEIN,V D KULAKOVSKIIand V B TIMOF~V, Physws ofSemzconductors Conf Ser 43, p 383, The Institute of Physics, Bnstol and London, (1978) V D EC,OROV, G O MILLER, H H WEBERand R. ZIMM]ZR~IA~,Nuovo C~mento, 39, 628 (1977) K R ELLIOTTand T C McGILL, Sohd-St Commun 28, 491 (1978) B ETIENNE, C BENOITA LA GUILLAUMEand M Voos, Phys Rev Lett 35, 5361 (1975) B ETIENNE, C BENOITA LA GUILLAUMEand M VOOS,Phys Rev B, 14, 712 (1976) B ETIENNE, C BENOITA LA GUILLAUMEand M Voos, Nuovo Czmento, 39, 639 (1977) B ETIENNE, J M WORLOCKand M Voos, Solwl.St Electromcs, 21, 1381 (1978) H P R FReDERIKS~,Phys Rev 92, 248 (1953) T H GEBALLEand G W HULL, Phys Rev 98, 940 (1955) M GEEL-MANN and K A BRUECKNER,Phys Rev 106, 364 (1957) M GR~NSTE~ and J P WOLFE, Phys Rev Lett 41, 715 (1978) A A GRINBERG, S M RYVKIN and A A ROOACHEV,FEz TOera Tela, 7, 2206 (1965) B GROSSMAN,K L SHAKL]mand M VOOS,Sold-St Commun 23, 271 (1977) L GUREVICH,J Phys U S S R 9, 4 (1945) L GUREVtCH,J Phys U S S R 10, 67 (1946) J HALPERN and B LAX, J Phys Chem Sol 27, 111 (1966) E HANAMURA,Proc Tenth Int Conf on Physics of Semzconductors, Cambndge, Mass p 504, (1970) V F GANTMAKHERand I V LEVINSON,Zh Eksp Teor F~z 74, 261 (1978) R B HAMMOND,T C McGXLL and J W MAYER,Phys Rev B, 13, 3566 (1976) H HASEGAWA,Phys Rev 118, 1523 (1960) J R HAYNES, Phys Rev Lett 1, 451 (1958) J C HENS~L,T G PmLLIPS and T M RICE, Phys Rev Lett 30, 227 (1973) J C HE~S~ and K ZUZUKI, Phys Rev B, 9, 4219 (1947) J C HENSELand T G PHILLIPS,Proe TwelfthInt Conf on Physics of Semwonductors, Stuttgart, 1974, Teubner, Stuttgart p 51, (1974) J C HENSELand R C DYSV.s, Phys Rev Lett 39, 969 (1977) J C HENSEL,T G PHILLlPSandG A THOMAS,Sold-St Phys~cs,Vol 32,p 88, NewYork, Acad Press(1977) C HERRING, Phys Rev 92, 857 (1953) O HILDERBRAND, E O GOEBEL, l( M ROMANEKand H WEB~R, Phys Rev B, 17, 4775 (1978) J HUBBARD,Proc Rov Soc (Lond) A, 243, 336 (1957)

216

A A. ROGACHEV

114 D HULIN, A MYZYROWICZ,M COMBESCOT,J PELANTand C BENOITA LA GUILLAUM~ Phys Rev Lett 39, 1169 (1977) 115 J V HVAMand O CHRISTENSEN,Solid-St Commurl 15, 929 (1974) 116 J M HVAMand J BALSLEV,Phys Rev B, 11, 5053 (1975) 117 C D JEFFRIES,Scwnce, 189, 955 (1975) 118 H JONES, N F MOTT and H W B SKINNEIL Phys Rev 45, 379 (1934) 119 R K KALIA and P VASHtSTA,Solut-St Commun 24, 171 (1977) 120 R K KALIA and P VASHISTA,Phys Rev B, 17, 2655 (1978) 121 A S KAMINSKnand YA E POKROVSKII,ZhETF Pzs Red 11,301 (1970) 122 A S KAMINSKn,YA E POKROVSKIIand N V ALKI/EV,Zh Eksp Teor F~z 59, 1937 (1970) 123 E O KAr,W., Physics of Semzconductors Conf Ser 43, The Institute of Physics, Bnstol and London, p 1321 (1978) 124 J E KARDONTCHIKand E COHEN, Phys Rev B, 19, 3181 (1979) 125 A L KARUZSKII,K W BETZLEg, B G ZI-PJRKINand B M BALTm~,Zh Eksp Teor Fzz 69, 1088 (1975) 126 A L KARUZSKII,K W BgrZLER, B G ZHURKIN and B M BALTElLZhETF P~s Red 22, 65 (1975) 127 L V KELDYSHand YU V KOPAEV,F~z Toerd Tela, 6, 2791 (1961) 128 L V KELDYSHand A N KOZLOV, Zh Eksp Teor Fzz 54, 978 (1968) 129 L V KELDYSH,Proc Ninth Int Conf on Phystcs of Semzconductors, Moscow, 1968, Nauka, Leningrad, p 1303 (1968) 130 L V KELDYSH,Uspekh~ Ftz Nauk, 100, 514 (1970) 131 L V KELDYSHand A P, SILIN,FIz ~derd Tela, 15, 1532 (1973) 132 L V KELDYSHand S G TIKHOOEEV,ZhETF P~s Red 21, 582 (1975) 133 L V KELDYSHand A P StuN, Zh Eksp Tear F~z 69, 1053 (1975) 134 L V KELDYSH, ZhETF Prs Red 23, 100 (1976) 135 S M K~LSO, Physws of Semiconductors Conf Ser 43, The Institute of Physics, Bristol and London, p 363 (1978) 136 G KIRCZENOW, Canada J Phys 55, 1787 (1977) 137 A N KOZLOVand L A M~KSIMOV,Zh Eksp Teor Phys 48, 1184 (1965) 138 M A LAMPERT,Phys Rev Lett 1, 450 (1958) 139 R F LEHENYand J SHAH, Phys Rev Lett 37, 871 (1976) 140 E C LIGHTOWLERSand M O HENRY, J Pbys C, 10, 247 (1977) 141 N O LIPARY and A BALDERESCHI,Sohd-St Coramun 25, 665 (1978) 142 T K Lo, B J FELDMANand C D JEFFamS, Phys Rev Lett 31, 224 (1973) 143 J M LtYrrINGEIL Phys Rev 102, 1030 (1956) 144 S A LYON, D L SMITH and T C McGILL, Phys Rev B, 17, 2620 (1978) 145 V G LYSENKO, V I REVENKO,T G TRATASand V B TIMOFEEV,Zh Eksp Teor Flz 68, 335 (1975) 146 H MAAREF, J BARRAU, M BROUSS~U, J COLLETand J MAZZASCHI,Solut-St Commun 22, 593 (1977) 147 S D MAHANTIand S M VARMA,Phys Rev B, 6, 2269 (1972) 148 G MAHLERand U SCHgODER, Sol.l-St Commun 26, 787 (1978) 149 A A MANLmKOVand S P SMOL~, ZhETF P~s Red 25, 436 (1977) 150 R S MARKIEWICZ,J P WOLFEand C D Jm~FRim, Phys Rev B, 15, 1988 (1977) 151 R W MARTIN, H L STORMEILW RUHLE and D BB4mERG,J Luminescence, 12/13, 645 (1976) 152 M MATStrORAand S WANG, Solut-St Commun 14, 533 (1974) 153 J V C MATTO~ K L SrlAKL~, M VOOS,T C DAM~q and J M WORLOOq Phys Rev B, 13, 5603 (1976) 154 T N MORGAN,Proc Twelfth Conf on Physics of Semiconductors Stuttgart, 1974, Teubner, Stuttgart, p 391 (1974) 155 T N MORGAN, Proc Thwteenth Int Conj" on Physics of Semiconductors, Rome, 1976, North-Holland, Amsterdam, p 825 (1976) 156 S A MOSKALENKO,Optzka ~ Spektroskoplya, 5, 147 (1958) 157 S A MOSKAL~KO, FIz Tverd Tela, 4, 276 (1962) 158 S A MOSKALENKO,KHADZm and A I BOBRYSHEVX,Flz Toerd Tela, 5, 1444 (1963) 159 N F MOTT, Metal-Insulator Transitions, Taylor and Francis, Ltd (1974) 160 G O MULLERand R Z1MMERMANN,Physics of Semwonductors, Conf Set 43, The Instttute of Physics, Bristol and London, p 165 (1978) 161 V N MURZIN, V A ZAYATSand V L KONONENKO, FIz Toerd Tela, 17, 2684 (1975) 162 A NAKAMURAand K MORIGAKI.Proc OJ~ Serum Phys H~ghly Excited States m Soluts, Tomakomat, 1975, 163 P NOZIERESand D PINES~Phys Rev 111, 442 (1958) 164 R R PARSONS,Solut-St Commun 22, 671 (1977) 165 G E PIKUS and G L BIlL Zh Eksp Teor F~z 67, 788 (1974) 166 G E P[KUS, F~z Tekh Poluprov 19, 1653 (1977) 167 D PINES, Elementary Excitations m Solids, W A Benjamul Inc. New York, Amsterdam (1964) 168 YA E POL~OVSKII and K I SVlSTt~NOVA,ZhETF P~s Red 9, 435 (1969) 169 YA E POKROVSKn,A S KXmNSKUand K I SV]STUSOVA,Proc Tenth lnt Conf on Physics of Semiconductors, Cambridge Mass (CONF-700801), p 504 (1970) 170 YA E POKROVSKnand K I SVISrONOVA,ZhETF Pts Red, 13. 287 (1971) 171 YA E POKROVSKII,Phys star Sol (a)11, 385 (1972) 172 YA E POKROVSKIIand K I SVtSTUNOVA,Proc Twelfth Int Conj" on Physics of Sermconductors, Stuttgart, 1974, Teubn©r, Stuttgart, p 71 (1974) 173 YA E POKnOVSrm and K I SWSTVNOVA,F~z Tekh Poluprov 16, 3399 (1974) 174 J POLLMANNand H BUTTNElLPhys Rev B, 16, 4480 (1977) 175 T L REI~¢KE and S C LING, Sohd-St Commun 14, 38I (1974) 176 T M RICE, Phys Rev B, 9, 1540 (1974) 177 T M RICE, Solut-State Physzcs, Vol 32, p 1, New York, Academic Press (1977) 178 T M RICE, Contemp Phys 3, 241 (1979)

Electron-hole hqmds m semiconductors

217

179 A A ROGACHEV,Proc Ninth lnt Conf on Physics of SemJconductors, Moscow, 1968, Nauka, Leningrad, p 407 (1968) 180 A A RO~ACHEV,lzvestla Akad Nauk SSSR, ser fiz 37, 229 (1973) 181 A A RO~ACHEV,Exc~tons at H~gh Density, Springer Tracts m Modern Physws Vol 73, p 129, Spnnger-Verlag (1975) 182 J H. RosE, H B SHOREand T M RxcK Phys Rev B, 17, 752 (1978) 183 J H ROSEand H B SHORE, Phys Rev B, 17, 1884 (1978) 184 J H ROSE, R Prelrrra~, L M SAND~ and H B SHORE, Sold-St Commun 30, 697 (1979) 185 M RosstJ~t and R ZXMMm~MA~,Phys Star Sol (b) 83, 85 (1977) 186 J SAK, Phys Rev B, 6, 2226 (1972) 187 L M SANDER,H H SHOREand J H ROSE,Sold-St Commun 27, 331 (1978) 188 R SAUm~,Proc Twelfth Int Conf on Physics of Sernzconductors, Stuttgart, 1974, Teubner, Stuttgart, p 42 (1974) 189 R SAUI~ and J WEBE~ Phys Rev Lett 36, 48 (1976) 190 R SAUl, W SCHMIDTand J WEaE~ Sold-St Commun 24, 507 (1977) 191 R SAtwa~,W SCHMIDT,J WEBERand U I~HBEIN,Physics of Semzconductors, 1978, Conf Set 43, The InstRute of Physics, Bristol and London, p 623 (1978) 192 M SAVOSTIANOVA,Z Phys~k, 64, 262 (19301 193 V A SCHUKIN and A V SUBASmEV,F~z Toerd Tela, 21, 1461 (1979) 194 R SCHWABE,F THUS~T, R BINDEMANN,W S ~ T and K JACOBS,Phys Lett 64A, 226 (1977) 195 J SHAH, A H DAYM~ and M COMBmCOT,Sold-St Commun 24, 71 (1977) 196 J SHAH, R F LEH~Y, W R H~DING and D R WIGHT, Phys Rev Lett 38, 1164 (1977) 197 S SHIONOYA,J Lurmnescence, 18/19, 917 (1979) 198 H B SHOREand R S Pl~n~rER, Physics of Sermconductors, 1978, Conf Ser 43, Bristol and London, p 627 (1978) 199 R N SILVEg, Phys Rev B, 11, 1569 (1975) 200 R N SILVa, Phys Rev B, 12, 5689 (1975) 201 K S SINGWI, M P TOSl, R H LAND and A SJOLANDER,Phys Rev 176, 589 (1968) 202 T SKm'TRUP, Sold-St Comrnun 23, 741 (1977) 203 D L SMITH, Sold-St Cornrnun 18, 637 (1976) 204 A ~OMMERFI~Dand H BETHK Handbuch der Physlck, lid 24/2 (1934) 205 V V STOPACmNSKU,Zh Eksp Teor Ftz 72, 592 (1977) 206 H L STORMIng,R W MARTIN and J C HENSEL, Proc Thzrteenth lnt Conf on Physics of Semtconductors, Rome, 1976, North-Holland, Amsterdam, p 000 (1976). 207 H L STORM~ and R W MART~N,The Apphcat~ons of Htgh Magnetic F~eldsm Semiconductor Physics, p 269, Oxford Umvers~ty (1978) 208 M L W THL~VALT,Phys Rev Lett 38, 521 (1977) 209 M L W TH~ALT, Canada J Phys ~.5, 1463 (1977) 210 M L W THL~VALT,Sold-St Commun 25, 671 (1977) 211 M L W THEWALTand J A ROSTOgOWSKX,Sohd-St Commun 25, 991 (1978) 212 M L W THL~VALT,Phystcs of Semtconductors, 1978, Conj" Ser 43, The Institute of Physics, Bristol and London, p 605 (1978) 213 M L W THEWALT,J A ROSTOROWSXIXand C Klacz~Now, Canada J Phys 57, 1898 (1979) 214 G A THOMAS,T G PHILLIPS, T M RICE and J C HENSEL, Phys Rev Lett 31, 386 (1974) 215 G A THOMAS,T M RICE and J C HENS~, Phys Rev Lett 33, 219 (1974) 216 G A THOMAS,Scientific American, 234, 28 (1976) 217 G A Thomas, Nuovo Ctmento, 39, 561 (1977) 218 G A THOMAS, J B MOCK and M CAPIZZI, Phys Rev B, 18, 4250 (1978) 219 T UOUMO~I, K MO~IOAr~ and C NAO~SmMA, Techmcal Report of ISSP, Set A, p 919 (1978) 220 H C VAN DE HULST, Light Scattering by Small Particles, Wiley, New York (1957) 221 P VASHISTA,P BHATTACHARYYAand K S SI~GWI, Phys Rev Lett 30, 1248 (1973) 222 P VASHISTA,S G DAS and K S Slsowt, Phys Rev Lett 33, 911 (1974) 223 P VASH~STA,V MA~IDA and K S Sr~owI, Phys Rev B, 10, 5108 (1974) 224 V S VAVILOV,V A ZAYATSand V N MURZlN, ZhETF P~s Red 10, 304 (1969) 225 V S VAVlLOV, V A ZAYATSand V N MURZXN, Proc Tenth lnt Conf on Physics of Semiconductors, Cambridge, Mass, p 509 (CONF-700801) (1970) 226 J VXTINS,R L AGGARWALand B LAX, Sold-St Commun 30, 103 (1979) 227 N B VOLKOV,B G ZHURK~N,A L KAgUZSKIt,M SHmAand V A FRADKOV,Physics of Semtconductors, 1978, Conf Set 43, The InstRute of Physics, Bristol and London, p 391 (1978) 228 M Voos, K L SHAKL~ and J M WOm.OCK, Phys Reo Lett 33, 1161 (1974) 229 M Voos and C BLmOITA LA GIULLAUME,Optical Properttes of Solds, New Developments, p 143, NorthHolland, Amsterdam (1975) 230 R M W~T~tV~LT, Phys Stat Sol (b), 74, 727 (1976) 231 R M WEST~Vm.T, Phys Stat Sol (b), 76, 31 (1976) 232 R M WEST~aVELT, J L STA~HLIand E E HALLER, Phvs Stat Sol (b), 90, 557 (1978) 233 R M W~STI~VELT,J C CUL~F~gS'rONand B S BLACK, Physu:s of Semiconductors, 1978, Conf Set 43, The InstRute of Phystcs, Bristol and London, p 359 (1978) 234 R M WES~Vm.T, J C CULBO.STONand B S BLACK,Phys Rev Lett 42, 267 (1979) 235 E P WOHLFORTH,Phil Mag 41, 543 (1950) 236 J M WOm~OCK,T C DAMm~,K I SnAr,.t.~ and J P C,-OgDON, Phys Rev Lett 33, 771 (1974) 237 H J WUNSCHK K H ~ B ~ G E g and V E KriA~TSI~V,Phys Star Sol (b), 86, 505 (1978) 238 H J WUNSCHKK I - ~ G m t and V E KI-IA~TSLL=V,Physics of Semiconductors, 1978, Con.]"Set 43, The Insatute of Physics, Bristol and London, p 615 (1978) 239 R ZIMM~MANNand M ROSLER, Sohd-St Commun 25, 561 (1978)