Chapter 6 Exciton-phonon interaction

SEMICONDUCTORS AND SEMIMETALS, VOL. i

CHAPTER

6

Exciton-Phonon Interaction

I. INTERACTION BETWEEN ExciTONS AND NoNPOLAR OPTICAL PHONONS II. POLARIZATION INTERACTION OF FREE ExciTONS WITH PHONONS III.

EFFECTS OF TEMPERATURE AND PRESSURE ON ExciTON STATES

135 136 139

1. Theoretical Background 2. Experimental Results

139 146

IV. IsoTOPic EFFECT ON ELECTRON EXCITATIONS

156

1. Renormalization of Energy of Band-to-Band Transitions in the Case of Isotopic Substitution in LiH Crystals 2. The Dependence of the Energy Gaps of A2B^ and A^B^ Semiconducting Crystals on Isotope Masses 3. Renormalization of Binding Energy ofWannier-Mott Excitons by the Isotopic Effect 4. Luminescence of Free Excitons in LiH Crystals

156 158 168 174

I. Interaction between Excitons and Nonpolar Optical Phonons Foundations of the theory of exciton-phonon interaction were laid in the 1950s. The interaction between an electron and a nonpolar optical phonon in a crystal can be described simply in terms of a deformation potential (see, e.g., Bir and Picus, 1972). The net effect of the lattice displacement on the electron is assumed to be a small shift in the electronic energy band of the crystal. The constant of proportionality between this energy shift and the lattice displacement is defined as the deformation potential. The excitonphonon (deformational potential) Hamiltonian can therefore be expressed as (Ansel'm and Firsov, 1956)

^EP = , hnTj IfjiNojQ

a

«/c+ +q^klbq

+ b-q].

where W^ and W^^ denote the deformation potentials of the electron and hole, respectively; a^ and a^ are the creation and annihilation operators of an exciton with wave vector fc; btq and b^ are the creation and annihilation 135

136

VLADIMIR G. PLEKHANOV

operators of an optical phonon with momentum hq; fi is the reduced mass of the atoms in the unit cell; N is the number of unit cells in the crystal; a is the lattice constant of the crystal; and hcoQ is the energy of the optical phonon.

II. Polarization Interaction of Free Excitons with Phonons Apart from pioneering the study of Ansel'm and Firsov (1956), the interaction of excitons with longitudinal optical phonons was considered by many authors (Knox, 1963; Haken, 1976; Thomas, 1967; Firsov, 1975; Permogorov, 1982). In ionic crystals, there are two main mechanisms of interaction of excitons with lattice vibrations. One — the mechanism of the short-range deformation interaction — is caused by modulation of the wave function of excitons by longitudinal vibrations. The magnitude of this interaction is characterized by the deformation potential (see earlier). The deformation interaction strongly affects the energy spectrum and dynamics of excitons of relatively small radius (e.g., the ground state of excitons in AHC and crystals of inert gases; Knox, 1963). As the radius of the exciton increases, this interaction becomes less important, since the wave vector of actual phonons is ^ oc r~^^ (Toyozawa, 1958), where r^^ is the exciton radius, and the number of such phonons is proportional to q^. The second mechanism — the polarization or Frohlich interaction (Frohlich, 1954) — is caused by the Coulomb interaction of the charge carriers forming the exciton with macroscopic field created by longitudinal optical oscillations (see, e.g., Pekar, 1951; Firsov, 1975). If the exciton radius is much greater than the lattice constant, then the exciton-phonon interaction can be regarded as the sum of independent interactions of electrons and holes with phonons (see Klochikhin, 1980). The interaction operator of charge and mass m (m^ or m^,), neglecting the dispersion of the latter, is (Frohlich, 1954) ^ei = Z ^ ^ e x p ( / q r ) ( b ! , +

fc,),

(1)

where

"'

\q\[

'

V )

^'^

V is the volume of the system, and 7 are the coordinates of the particles. In this expression, we introduce the main parameters that determine the interaction of the electron (hole) with optical vibrations: the polaron "radius":

^*/i = ( 2^^'^^LO I

.

(3)

6

EXCITON-PHONON INTERACTION

137

and the dimensionless Frohlich constant of interaction, 1/1

M

eo 2

le,h — a VCoo

^0/

(4)

^^l.0^e,h

The first of these quantities characterizes the size of the polarization region of the lattice by the extra charge, and the second describes the strength of the electron-phonon interaction (see Knox, 1963; Haken, 1976). As follows from Eq. (1), the interaction operator W^ does not depend on the mass of the quasiparticle, and is the same for electrons and holes. Accordingly, the interaction Hamiltonian of Wannier-Mott excitons with optical phonons has the same form as Eq. (1), the only difference is that exp(fqr) is replaced with exp(iqgr) — exp(iq^r), whereF^,;, are the coordinates of the electron (hole). In the center-of-mass system, the interaction operator becomes (Ansel'm and Firsov, 1956) ^EXL = Z ^^[exp(/^,r) - exp(/(5f;,r)](/7 Iq + b^).

(5)

Replacing m^;, by the reduced mass fx by analogy with Eq. (1), we can define the characteristic size of the polarization region r^^ = [(r*)^ + (rj)^]^/^ and the interaction constant g^^. Making use of the characteristics of the Wannier-Mott exciton (r^^E^), one can express the latter as

g2^ = (^Y^.l='Af^^l)^

(6)

where r^^ = aJ^EniQi^, and a^ = h^/niQe^ = 0.53 A is the Bohr radius of the hydrogen atom. The scattering of excitons by LO phonons is determined by the magnitude and the wave vector dependence of the matrix element «EXL = < ^ , J H E X L I ^ A . > ,

(7)

where ^^^ and ^;t2 ^^^ Ih^ wavefunctions of the initial and final states of the exciton with the wave vectors k^ and k2 = k^ + q. The properties of the matrix element of exciton-phonon scattering, as first noted in Bulyanitza (1970), depend crucially on the parties of the initial ?.^ and final Xj states. If the parity is the same (scattering occurs within the same band, as l 5 - l s or 2s-2s, etc., or in the case of interband scattering ls-2s, ls-3s, etc.), this mechanism of exciton-phonon scattering is forbidden, because H^xl ~^ ^ when q-^0. When excitons are scattered in a ground state band (Is-Is), as

138

VLADIMIR G. PLEKHANOV

shown in the paper of Ansel'm and Firsov, the matrix element has the form

11

EXL OC

\q\

1+

^e^c

1+

Qh^e

(8)

When q is small (it is the small values that are of special importance; see later), the matrix element is proportional to

^EXL

^

m.

m.

rUf^ + m^

qr.

(9)

As follows from Eq. (9), the Frohlich mechanism of intraband scattering is absolutely forbidden when the effective masses are equal. This is because the centers of the distribution of masses and charges coincide at m^ = my^, so the polarization interactions of electrons and holes cancel out completely. In the general case, the matrix element, Eq. (8), arrives at maximum near qr^^ - 1, and then falls off rapidly (Fig. 1). Similar behavior is displayed by the matrix elements of the scattering processes between the same symmetry {is -js, ip -jp, etc.). In such cases, the matrix element attains its maximum at the value of the inverse radius (r„) of the corresponding exciton state, that is, (10) According to the results of the paper by Bulyanitza, when scattering occurs between the bands of different symmetry {is - kp) when q^O, the matrix element tends to be a constant (nonzero) value, and such processes are therefore allowed. As in our first case, however, the matrix element H^Sii

FIG. 1. Dependence of matrix element of intraband scattering on qr^^ for the mass ratio mjm, = 3.5. (After Plekhanov, 1997b.)

6

EXCITON-PHONON INTERACTION

139

falls off rapidly as ^ > r~^^ increases. The dependence of matrix elements on q for some cases of allowed and forbidden scattering for CdS is discussed in Permogorov (1982). The behavior of matrix elements of Eq. (8) is definitive for the structure and properties of luminescence spectra of free excitons and Raman scattering in the resonance region. The existing theory of exciton-phonon interaction describes the case of isotropic band with parabolic dispersion of exciton states. Its successful apphcation to LiH (with the high anisotropy of the valence band, see Fig. 7 in Chapter 5), once again testifies to the fact that the dispersion of the exciton band is indeed parabolic (Plekhanov and Altukhov, 1981) in the range of low kinetic energies of exciton. Note also that, according to Permogorov (1982), the exciton band exhibits parabolic dispersion even for such anisotropic crystal as CdS, which is supported by the successful application of the theory of exciton-phonon interaction to the analysis of luminescence spectra and RRLS of free excitons in CdS crystals.

III. EflFects of Temperature and Pressure on Exciton States 1.

THEORETICAL BACKGROUND

Even after the first works on the spectroscopy of large-radius excitons (see Gross, 1976) it became clear that the location of the edge of fundamental absorption (and hence the exciton structure) in a solid depends on the temperature. Further studies also revealed (see also Knox, 1963; Cardona, 1969) that the temperature dependence of the absorption edge may be caused by two factors: the expansion of the lattice and the lattice vibrations. Vibrations of the lattice will cause not only displacement but also broadening of the energy levels of electron excitations. As first shown by Fan (1951), the change in the energy of band-to-band transitions AEg in most substances is caused primarily by the displacement of the energy levels rather than by their broadening, as had been assumed before. It was demonstrated that in the mechanism of deformation potential of electron-phonon interaction the quantity AE^ is directly proportional not only to the square of deformation potentials of the valence band and the conduction band, but also to the sum of q^^^ and q^^^, where q^^^ is the maximum value of the wave vector of phonon. If the displacement of electron bands (or exciton levels) associated with the Frohhch mechanism of electron-phonon interaction is taken into account, the matrix element is inversely proportional to the wave vector of phonons [see Eqs. (3) and (8)]. There is an important distinction between displacement and broadening of energy levels. The point is that in the case of displacement the theory considers virtual transitions, whereas in the case of broadening the

140

VLADIMIR G. PLEKHANOV

transitions are real. Real transitions require conservation of quasimomentum and energy, whereas only quasi-momentum must be conserved in the case of virtual transitions. In the mechanism of deformation potential this circumstance is especially important for acoustic vibrations, being associated with the directly proportional dependence of the matrix element of electron transition in the magnitude of the wave vector of phonons. And since virtual transitions may involve phonons with not necessarily small values of q, it becomes clear why in the mechanism of deformation potential of electron-phonon interaction the displacement of energy levels greatly exceeds their broadening (for more details see Fan, 1967). The contemporary microscopic treatment of this problem (see, for example, Heine and van Vechten, 1976) singles out four contributions to the temperature dependence of E^ (see also Zollner et ai, 1992; Plekhanov, 1993): 1. The Debye-Waller factor in the Fourier expansion of the periodical part of crystal potential; 2. The term related to intraband transitions, now commonly referred to as the Fan term (Cohen and Chadi, 1980); 3. The contribution from band-to-band transitions, which currently is hard to evaluate; 4. The contribution from thermal expansion of the crystal lattice.

As a rule, the majority of experimental studies concerned with this problem (see, for example, Cohen and Chadi, 1980, and the references therein) consider two contributions: the contribution from electron-phonon interaction, expressed as either the Debye-Waller or the Fan term, and the contribution from thermal expansion of the lattice. A consistent study of the effects of temperature on the energy of band-to-band transitions for a large number of semiconductor compounds (Zollner et ai, 1992; Logothetidis et al, 1985, 1991; Cardona and Gopalan, 1989) indicates that theory can be brought into good agreement with experiment when at least the first three factors are taken into account. Note also that, apart from the microscopic approach to the temperature dependence Eg{T), the attempts to explain the empirical formula of this dependence have continued for over three decades. First, we should note the well-known Varshni formula, widely used in experimental works (Varshni, 1967), and the currently no less popular Manoogian-Leclerk relation (see Manoogian, 1982; Quintero et al, 1991, and references therein). The dependence of the electronic energy gap EQ on the isotopic mass Mj^ (where k is the atomic species) at a constant temperature can be separated into two contributions, similar to those responsible for the temperature

6

EXCITON-PHONON INTERACTION

141

dependence of the gaps (see, e.g., Pankov, 1971):

The first term is the contribution of the electron-phonon interaction (EP) at constant volume. The second one is due to the change of the lattice constant or, equivalently, of the crystal volume with the isotopic mass. Anharmonic corrections to the crystal volume at low temperature depend on the atomic masses through the "zero-point" vibrational ampHtudes. The origin of this term is equivalent to that of the thermal expansion (TE) at low temperature; by analogy we call it the zero-point thermal expansion term. As mentioned earlier, the renormalization of the unperturbed band energy 8kn (of the state |/cn> with wave vector k and band index n) by the electron-phonon interaction (see Fig. 2) can be written as EkniT) = Eln + A?^ + A|^ + iTl,.

(12)

where /^^ is the shift of the band energy induced by the Debye-Waller (DW) term. The complex self-energy (SE) term has a real part A|^, which gives rise to an energy shift of the band states and an imaginary part r^„, which causes a lifetime broadening of these states.

_a 0.)

k.n

(a) Q.j

. r\ Q.j

FIG. 2. (a) Feynman diagrams for the self-energy of electrons due to interaction with phonons, (b) Feynman diagrams for two-phonon Raman scattering, and (c) renormalization vertex corresponding to the sum of Debye-Waller (DW) and self-energy (SE) or Fan term. (After Cardona and Gopalan, 1989.)

142

VLADIMIR G. PLEKHANOV TABLE I

PARAMETERS NEEDED FOR THE EVALUATION OF THE THERMAL EXPANSION CONTRIBUTION TO THE TEMPERATURE SHIFTS OF BANDGAPS

m/dp)r Material

Gap

Diamond Diamond Diamond LiH Ge

E'o Ei £i

X^-X^

r-x

(meV/GPa)

J3(GPa)

a(300K) (10-^K-i)

7.0" 6.4' 10^ -0.11'^ -14^

442

LO

312' 75

3 5.90

''M. Cardona and N. E. Christensen, Solid State Commun. 58, 421 (1986). ' P . E. Van Camp, V. E. Van Doren, and J. T. Devreese, Phys. Rev. B34, 1314 (1986). '^ O. Madelung, in Numerical Data and Functional Relationships in Science and Technology, O. Madelung, ed. (Springer, Berlin, 1987), Vols. 17a and 22a. '^Y. Kondo and K. Asaumi, J. Phys. Soc. Japan 57, 367 (1988). "M. W. Guinan and C. F. Cline, J. Nonmetals 1, 11 (1972). ^K. J. Chang, S. Froyen, and M. L. Cohen, Solid State Commun. 50, 105 (1984).

In the following description of the different terms, Eqs. (11) and (12), we are very close to Zollner et al. (1992). By the thermal expansion (TE) the lattice constant increases and thus the band gaps shrink, if they have a positive pressure coefficient dE/dp. The shift for the gaps are found to be (Cohen and Chadi, 1980)

i) =-Mmm. dTjj^

,13)

\dpjj

where a(T) is the temperature-dependent thermal expansion coefficient and B is the bulk modulus. This term can be evaluated very easily using the values listed in Table I and is small compared to those due to the electron-phonon interaction as described later (see also Plekhanov, 1993). The Debye-Waller term arises from the simultaneous interaction of an electron (with wave vector k in band n) with two phonons of the wave vector Q and modej (electron- two phonon interaction). In the rigid-ion approximations Zollner et al. assumed that the potential V^ of the atom of type a moves rigidly with the atom in a phonon vibration. Then the DW contribution to the shifts of an electronic state nk for frozen-in lattice displacement L/y^ of one atom of type a located at the atomic site R^^ is given by Cohen and Chadi (1980) ..J,

.

V V B{nXT,(x,n\k')(S)B{n\k'J,cc\n,k) ,r^ll' ' n,k Iccl a

~y^nk-

-^

-^

^n'k')

(14)

6

EXCITON-PHONON INTERACTION

143

where the sum runs over all intermediate electron states n\l\ all lattice vectors T, and the basis (a) of the lattice. The angular brackets with superscript t denote the thermal or temporal average. The energies necessary to distort the lattice (phonon energies) have been assumed to be much smaller than the usual electronic bandgaps and thus are neglected in the denominator of Eq. (14). Thus B{n,kJ,oc,n\k')

= {nU'lVV^r - Ri,)\n'k'}

(15)

is the matrix element of the gradient of the potential V^ of one atom of type a located at the atomic site R^^. The notation used here is somewhat symbolic but very intuitive, to emphasize the influence of the isotopic mass entering the phonon displacements Uj^. For explicit expressions apphcable for a lattice with basis, see Gopalan et al. (1987). ZoUner et al. used an empirical pseudopotential band structure (Cohen and Chadi, 1980) and assumed that the matrix element of the atomic potential with the true wave functions is the same as that of the pseudopotential evaluated with the pseudowave functions (Sham, 1961) (rigid pseudoion method). To the second order in phonon displacement, Eq. (14) for the Debye-Waller term is equivalent to simply multiplying the structure factors S{G) used in the band-structure calculations by Debye-Waller factors exp( —2W) with W= |Gp/12. This term is the dominant contribution to the temperature shifts of bandgaps. It has been evaluated for several materials (Gamassel and Auvergne, 1975), but usually overestimates the shifts when considered without the other electron-phonon term discussed next. The third contribution to the temperature shifts of electronic states is the real part of the self-energy term, which arises from the interaction of an electron with one phonon taken to the second order in perturbation theory (see the second term in Fig 2a). This term is usually somewhat smaller than the Debye-Waller term, but opposite in sign. Therefore it should be taken into account in a reahstic calculation of temperature shifts, although it requires a Brillouin-zone integration to be evaluated. According to Zollner et al, it can be expressed as

(16) The imaginary part of the self-energy causes lifetime broadenings of critical points and is responsible for intervalley scattering processes (see also Zollner et aU 1991). For numerical reasons, Zollner et al. (1992) Fourier-transformed Eqs. (14) and (16) to the phonon representation (thus replacing the sum over lattice

144

VLADIMIR G . PLEKHANOV

sites by an integration over all phonons in the first Brillouin zone) and labeled the contribution of a single phonon with wave vector Q and branch j (with the occupation number NQJ = 1) as (dEnk/8NQj)fr, where K stands for DW or SE. From these electron-phonon coupHngs Zollner et al. defined the dimensionless electron-phonon spectral functions:

9^''.("J.") = £(i|)/(n-%).

(17)

in such a way that the temperature shifts due to the DW or SE terms are given by dQg'F,{nrk,n)(N^

{AE^k)f. =

+

(18)

For the integration over all phonon wave vectors Q, the cited authors used the tetrahedron method with 89 points in the irreducible wedge of the Brillouin zone. For the numerical evaluation of the terms in Eqs. (14) and (16) Zollner et al assumed that only the phonon amphtudes and not their energies depend on the temperature and isotope concentration. The calculated energy of the EQ gap in natural diamond ^^C as a function of temperature (assuming EQ = 7.3 eV at 0 K), including all three contributions, is shown by the solid line in Fig. 3. The contribution of thermal expansion (dashed-dotted line) and self-energy (dashed-double-dotted fine) are displayed separately. It can be seen that the shifts due to thermal

7.35 I

'

I

•

I

• I

7.30 ^

7.25

u ° 7.20 Diamond

7.15 7.10

J

0

.

I

I

I

>

I

I

I

I

i_

100 200 300 400 500 600 700 T(K)

FIG. 3. Shifts of the £j gap in natural diamond ^^C (soHd Hne) including thermal expansion (dashed-dotted Hne), self-energy (dashed-double-dotted line), and Debye-Waller term (not shown separately). (After Zollner et al, 1992.)

6

EXCITON-PHONON INTERACTION

145

expansion are small, especially because of the very small pressure dependence of the E'o gap commonly found in semiconductors (Madelung, 1987). Furthermore, the TE contributions are rather small for all gaps in diamond because of the large bulk modulus (442 GPa; see Van Camp et ai, 1986). The DW contribution is dominant up to about 400 K, but at 700 K the SE accounts for 40% of the shifts. The only experimental data for the shifts of the £o gap in diamond known to us are the reflection measurements of Clark et al. (1964) reporting a shift of the direct gap of 100 meV between 133 and 295 K. This resuh is much larger than calculated shifts (only 15 meV), but it is not clear whether the assigment of the experimental peak to £0 transition is correct. By the way, we should note that in silicon, the £0 and £1 critical points are almost degenarate (Lautenschlager et a/., 1987). The temperature dependence of the indirect gap E^ (0 K) = 5.41 eV in diamond ^^C is shown in Fig. 4. It can be seen that the DW term dominates the shifts, in contrast to the results for the £'0 critical point (see Fig. 3). Figure 4 also shows the experimental data of Clark et al. (1964) (dashed Une), which are in good agreement with the calculated results. We can see that at given temperature, isotope effects enter in Eqs. (14) and (16) through the mean-squared phonon amplitude


[1 +

2NQJ{T)1

(19)

5.45

5.25

0

100 200 300 XOO 500 600 700

T(K) FIG. 4. Shifts of the £,(0 K) = 5.41 eV gap in natural diamond ^^C (solid line) including thermal expansion (dashed-dotted line), self-energy (dashed-double-dotted Hne), and DebyeWaller term (not shown separately) The dashed hne shows the experimental data of Clark et al, (1964). (After Zollner et al, 1992.)

146

VLADIMIR G. PLEKHANOV

where M„ is the mass of one atom of t y p e ^ and NQJ{T) the occupation number of the phonon with wave vector Q, branch 7, and energy Q.QJ. Neglecting the small isotope dependence of the matrix elements of Eq. (15), the Debye-Waller and self-energy terms cause not only temperature shifts but also an isotope-dependent renormalization of the electron energies at zero temperature. At 0 K (when N = 0), we have 'ocM,-i/^

(20)

since the phonon frequency is proportional to M~^^^.

2.

EXPERIMENTAL RESULTS

The temperature shift of the E^ in diamond was investigated by Collins et al (1990). The results of ColHns et al are depicted in Fig. 5. According to Collins et al, the temperature dependence of the indirect gap in diamond has the following form: E(T)

=E +

dcofico) {n{cD, T)+H-

a(c,, + 2c,,)AV{T)/3V

(21)

Here n{(D, T) is the Bose-Einstein occupation number, and f{oS)d(o is the difference in the electron-phonon coupling for the conduction-band minima and the valence-band maximum for those modes in the frequency range co to CO + do), the energy gap at 0 K being E' + \\d(Df{oS). The final term in Eq. (21) allows for the temperature-dependent lattice expansion, where AF(r)/K is the fractional volume expansion to temperature T, c^^ and c^2 are elastic constants, and a is the change in energy per unit compressional hydrostatic stress, measured as a = 5 ± 1 meV G P a ~ \ The volume expansion accounts for only about 6% of the temperature dependence of E^^ (Fig. 5). Most of the measured temperature dependence arises from the term j dojf{a>)n{(JO, T). The functional form of f(co) is not known. However, CoUins et al. found that a precise fit can be made to the temperature dependence of E^^ using f(co) = ccog(co\ where g{(jo) is the density of phonon states (DoUing and Cowley, 1966) for ^^C diamond, and the constant c of proportionahty is given by the fit to Eg^{T). At 0 K, changing the isotopes gives a contribution to ^^E -^^E of

^•-\

r-

f(co)do)=n.5±2me\.

(22)

As was shown by Collins et al, although the best fit to the data in Fig. 5

6

147

EXCITON-PHONON INTERACTION

V

.S

-60

0)

C

-100 h

300

500

Temperature (K) FIG. 5. Squares: experimental data of Clark et al, (1964) for the temperature dependence of the indirect energy gap of diamond ^^C. The calculated shift (thick line) is the sum of contributions from the lattice expansion (curve a) and the electron coupling (curve b), using f{€o) = ccog{(o). (After Colhns et al, 1990.)

is obtained when /{(o) = ccogioj), adequate fits are obtained for /(co) = c'(o^-^g{co) to /(co) = c''cD^'^g{(j)) producing most of the uncertainty shown in Eq. (22). The 20% uncertainty contributes a further 0.1 meV to the uncertainty in Eq. (22). A second contribution A2 to the isotopic dependence of E^^ comes (as was mentioned) from the volume change produced by changing the isotope. The difference in molar volume of ^^C and ^^C is 1/2

sy-^^V = -

Y,hcOi

2(cii + 2c^2) i

- 1

(23)

where the frequencies are for ^^C, 7, is the Gruneisen parameter of the ith

148

VLADIMIR G . P L E K H A N O V

750

Temperature (K) FIG. 6. Temperature dependence of the lowest indirect energy gap of Ge. Points show the trend of experimental data from McClean (1960), the fine curve "a" shows the effect of volume expansion and the full curve shows the total fit of Eq. (21) with/(w) = cco^g(co) {p = 0.4). (After Davies et al, 1992.)

mode, and the density of phonon states is known (Dolling and Cowley, 1966). Taking into account y = 1.15-1.6, ^^ftcoLQ-^^^^Lo — 50cm"\ A2 is equal to 3 ± 1.3 meV (Collins et al, 1990). Thus it was shown that the indirect energy gap of diamond changes by 13.6 ± 0.2 meV when the isotope composition changes from ^^C to ^^C. Most of this shift is caused by the isotopic dependence of the electronphonon coupHng, giving a shift estimated from the temperature dependence of energy gap as A^ = 13.5 + 2meV. An additional smaller contribution, estimated to be A2 = 3 ± 1.3 meV, comes from the difference in molar volume of ^^C and ^^C diamond. The temperature dependence of the lowest indirect energy gap of Ge is shown in Fig. 6 (Davies et al, 1992). Here also the two main contributions to the isotope dependence of the indirect energy gap are from electronphonon coupling and from the dependence of the atomic spacing on the isotope. The volume change on changing the isotope from the natural isotopic content (effective mass number 72.6) to nominal ^"^Ge (with an average mass oi A = 73.9) has been measured at low temperature (Buschert

6

EXCITON-PHONON INTERACTION

149

et aU 1988). 72.6Y_73.9Y

72.6V

= (14.9 ± 0.3) X 1 0 - ^

(24)

The change in energy with volume of the indirect gap is also known (Schmid and Christensen, 1990): a = VdE/dV = - 3800 meV.

(25)

Consequently the change in exciton energy with mass number A from the volume change is only about 12% of the total shift (dE/dA) ,oi = 0.044 me V,

(26)

and most of the shift arises from the isotope dependence of the electronphonon coupling. A simple estimate of this contribution (as in the case of diamond) to the energy shift can be made from measurements of the temperature dependence of the indirect energy gap. Evaluation of Eqs. (22) and (23) yields, for the range p = 0-0.8 (Davies et a/., 1992) an electronphonon contribution (dE/dA),. ph = 0.22-0.30 meV.

(27)

The total predicted shift is the sum of the contributions in Eqs. (26) and (27) and is (dE/dA),^, = 0.26-0.34 meV.

(28)

The observed shift in the indirect energy gap at the isotope effect in Ge of Aoji = 0.36meV/amu (see also Agekyan et al, 1989; Cardona, 1994). The electron-phonon interaction renormalization of a bandgap £„/^ in CdS can be expressed as (Zhang et al, 1998) £(M, T) = £o ^Cd^^Cd

'^(^cd. T) +

a

2 I

cos^s

n{(D^, T) + - , (29)

where SQ represents the unperturbed gap energy and the additional two terms on the right-hand side correspond to the contributions from the acoustic phonons (Cd vibrations) and optic phonons (S vibrations), respectively. Using an average acoustic-phonon frequency of co^d ^ 60cm~^ and an average optic-phonon frequency of a>^ c^ 270cm~\ a least-squares fit to experimental data of the temperature dependence of the A and B excitons

150

VLADIMIR G . PLEJCHANOV -

T -

-r

•

I

(

r

T

'

'*^-^. •

\

-

^

2.56

••"*•-.. ^*X. 1

^'-^v i

\

2.54

\

N

\ 2.52

1 \

-1

N !

H

2.50

2.48

\ O

1

Ad

\

J.,

100 200 Temperature (K)

300

FIG. 7. Temperature dependence of the A and B excitons in CdS. The filled circles and open squares display wavelength-modulated reflectivity data (Anedda and Fortin, 1976), while the diamonds and triangles represent the measured gap energy of the A bandgap reduced by the exciton binding energy of 27 meV (Benoit a la Guillaume et al, 1969; Seller et al, 1982). The solid lines are least-squares fits to the data performed with Eq. (29), the average phonon frequencies COQ^ :^ 60cm~^ and co^ ~ 270 cm ~^ The dashed and the dotted fines represent the individual contributions of acoustic phonons (Cd vibrations with average frequency co^d) and optic phonons (S vibrations with average frequency (0^) to the shift of the B exciton, respectively. (After Zhang et al, 1998.)

in CdS (Benoit a la Guillaume et al, 1969; Anedda and Fortin, 1976; Seller et al, 1982) as shown in Fig. 7, yields a = 0.0134 and b = 0.1310 eV^ amu for the A exciton and a = 0.0159 and b = 0.0999 eV^ amu for the B exciton, respectively. The zero-temperature isotopic-mass-dependent renormalization of the electron energies can also be predicted using Eq. (29). Using the approximate expression cocd(S) ^ l/^/Mcdcs) one can find: dE Cd

4cOrAM ^Cd^ Cd'

dE dMs

4a)s M |

(30)

6

EXCITON-PHONON INTERACTION

151

TABLE II ISOTOPE SHIFTS OF THE EXCITONIC ENERGIES IN CdS FOR Cd SUBSTITUTION MEASURED BY ZHANG et al (1998) COMPARED WITH THE RESULTS FOR S SUBSTITUTION EXTRACTED FROM REFLECTIVITY SPECTRA IN KREINGOL'D et al (1984) AT T ~ 6 K. THE SLOPES ARE GIVEN IN UNITS OF /zeV/ amu

PR

PL Exciton

r^(^) 68 ±13

dE/dMs

r,{A) 68 ± 20

r^(^)

n{B)

61 ±20 740 ± 100

40 ±22 740 ± 100

In this manner, Zhang et al. obtained dE/dM^^ = 36 /leV/amu (42 /leV/ amu) and dE/dM^ = 950 /leV/amu (724 /leV/amu) for the A{B) exciton, respectively. Given the simpHfied treatment of the CdS lattice dynamics by means of only two averages, the results are in surprisingly good agreement with the results for isotope substitution of the Cd and S atoms Usted in Table II (Kreingol'd et al, 1984; Zhang et a/, 1998). In all the preceding cases, the deformation potential interaction via acoustic phonons was considered. Since the calculations include only short-range (deformation-potential) electron-phonon interaction, it may be concluded that the long-range Frohlich interaction (especially actualized for polar materials CdS and GaAs) is not important for the phenomenon treated in this section. The similar long-wavelength structure of spectra of mirror reflection of pure (LiH, LiD) and isotopically mixed crystals (see later and also Fig. 9 on p. 157) enables us to attribute it to the excitation of the first and the second exciton states (Klochikhin and Plekhanov, 1980). As the temperature increases, the exciton reflection spectra of pure (see Fig. 2) and mixed crystals shift toward the longer wavelengths as a whole. The exciton structure of reflection spectra broadens. In the temperature range from 2 to 200 K, the line of the ground state of the exciton broadens approximately fivefold. At T ^ 130-140 K the peak caused by excitation of the exciton in the state n = 2s becomes indiscernible in the spectrum (Fig. 8). The reflection spectra of LiH^Di_^ crystals, including LiD, behave in a similar way. The figure shows the temperature dependence of the energy of long-wavelength maximum in the reflection spectrum for three crystals. For all three crystals, the dependence E^^ ^ f{T) in the temperature range T ^ 140 K is weU approximated by a hnear function. The temperature coefficients of Hnear shift found in this way are given in Table III. We see that the temperature coefficient dE/dT is larger for the heavier isotope. Assuming that the temperature shift is caused by the interaction of excitons

152

VLADIMIR G . PLEKHANOV

FIG. 8. Temperature shift of location of maximum of Is exciton state in the reflection spectrum of crystals LiH, curve 1; LiH0.25D0.75' curve 2; and LiD, curve 3. Experimental values shown by points, calculated values by solid Hues. (After Plekhanov, 1997b.)

with optical vibrations, we have dE^_AdE^

If'slf^

£i(0)-£:2(Q)

(T)

(31)

where dEJdT=dE„^u(UD)/dT,dE2/dT=dE„=u(UH)/dT; T= ftcoLo(LiD)//c^; and £i(0) and £2(0) are the values of the maximum of exciton band of LiD and LiH, respectively, at T = 0K. Substituting the values of the quantities involved Ihco^o = 104 meV (Plekhanov, 1997b)] we find that the magnitude oidE„ = is/dT for LiD is 0.34 meV/K. Comparing this with the experimental value (0.28 meV/K), we see that there is a rather large discrepancy, which is discussed a httle later. In accordance with the arguments developed earlier, the temperature dependence of £„ = is (as well as that of E^) is shaped by two contributions: the lattice expansion, and the exciton-(electron)-phonon interaction, which is a sum of three terms. The first of these can be found by evaluating the baric shift of £„ = i,. It is well known that the effects of pressure on the spectra of fundamental absorption is generally associated with changes in the width of the forbidden band, in the probabilities of transitions, and in

6

EXCITON-PHONON INTERACTION

153

the eflfective masses of carriers (see, e.g., Bir and Pikus, 1972). For nondegenerate states, usually only the change in Eg is of practical importance. Then one can expect that the exciton series shifts as a whole, without any significant changes in the binding energy and the intensities of individual lines (see also Gross, 1976). When the pressure P is relatively small (when the pressure-induced change in the energy of a given state is much less than the distance to the adjacent levels), the band shift may be assumed to be a linear function of the pressure. In this approximation, the pressure dependence of Eg (and hence £)„ = iJ can be expressed as E(P) = E(0) - iE\ - E\)krP,

(32)

where VydPjr

dV'\dVjT

krydP

Here kj is the isothermal compressibiUty and the hydrostatic deformation potential (E^); and superscripts c and v stand for the conduction band and the valence band. The deformation potential E^ in LiH we estimate from the results of the paper by Kondo and Asaumi (1988). This study was concerned with measuring the energy of the maximum of the long-wavelength peak of mirror reflection of LiH crystals as a function of external hydrostatic pressure (P:^330kbar). The absence of exciton structure (except for the n = Is state) in the reflection spectrum (see also Fig. 3 in Chapter 5) at room temperature leads to the assumption that the binding energy of the exciton does not depend on the pressure, and therefore the function E„ = is = f{P) uniquely defines Eg ^ /(P), and vice versa. In the paper of Kondo and Asaumi it was shown that in the pressure range 40 < P ^ 330 kbar this shift is linear with the baric coefficients dE/dP = —1.1 meV/kbar. According to Guinan and Cline (1972), /c^ = 3 x 10"^ bar" K Then E^ = 0.36 eV for LiH. Apart from this low value of £^, one should also note the nonmonotonic dependence E„=is = f{P)' at P ^ 40 kbar, according to the calculations of the band structure (Perrot, 1976; Kulikov, 1978), the quantity £„=is (and hence Eg) increases and, after P ^ 40 kbar, decreases hnearly.* This behavior of the exciton (n = Is) maximum in the reflection spectrum may be attributed to two factors: 1. Removal of the ban with respect to K on the electron transition W^-X^, which is estimated theoretically to lie 0.03 eV below the direct X^-X^ transition in LiH (for more details see Plekhanov et a/., 1976). •Observe also that, according to the results of Ghandehari et al. (1995) the baric coefficient takes on two values: 0 ^ P ^ 300 Kbar dE/dP = - 4 . 6 + 2 meV/kbar, and dE/dP = -0.237 ± 0.001 meV/kbar in the range 300 ^ P ^ 2510 kbar. Note that, in contrast to LiH, the value of Eg in CsH starts to decrease immediately after application of external pressure.

154

VLADIMIR G. PLEKHANOV

2. Excitation, as the pressure increases, of the higher p-states in the conduction band, as noted earher in Plekhanov et al. (1984) and Hama and Kawakami (1988). These two factors may also work simultaneously, which, on the other hand, may be responsible for the low value of E^ characteristic of the electron transitions occurring at points other than the T point of the Brillouin zone (cf. Wolford, 1987). Note also that for most semiconductor compounds with the structure of diamond (or zinc blende) the rate of change {dEJdP) of the quantity E^ at points F, X, and L is, on the average, 12, —1.5, and 5meV/kbar, respectively (Pankov, 1971; Moss et al, 1973), and thus differs not only in magnitude but also in sign (for more details, see Bir and Pikus, 1972). The value of E^ found enables one to estimate the contribution of the lattice expansion into the change in E^^w It constitutes 12% at low temperatures and 20% at room temperature of the entire shift for LiH crystals. We see that the main change in E„ = is(Eg) comes from the Debye-Waller term and the term that accounts for the internal energy (the Fan term). As demonstrated in papers on semiconductors (see earlier), the inclusion of the latter two terms (which, incidentally, have opposite signs) along with the lattice expansion brings the theory into good agreement with the experimentally observed dependence E^ = f{T). The microscopic calculation of the temperature shift of E^ indicates (Harrison, 1970) that this shift can only be related to those terms in the expansion of the potential energy that are quadratic with respect to displacement (see later and Zollner et al, 1992; Cohen and Chadi, 1980). In the one-oscillator Einstein model, the dependence of E^ on T can be represented as

E(T) =E(0)-A coth(^\

(34)

where E(0) is the value of Eg(E„ = is) at T = 0, and hco^f^ is the effective frequency of the phonon in the model under consideration. The temperature dependence of the location of the maximum of the exciton peak in the reflection spectra of pure and mixed crystals, calculated according to Eq. (34), is shown in Fig. 8 by a solid line. As follows from Fig. 8, there is good enough agreement between theoretical curves and experimental data. The calculated energy of actual phonons ho^^f, as can be seen from Table III, falls into the range of acoustic vibrations. The latter seems to point to the domination of the mechanism of deformation potential of electron-(exciton)-phonon interaction. This is a reasonable conclusion if we recall that the energy of the longitudinal optical phonon is 104 meV

6

EXCITON-PHONON INTERACTION

155

TABLE III VALUES OF PARAMETERS CALCULATED BY FORMULA (34) AND VALUES; OF TEMPERATURE COEFFICIENT OF LINE SHIFT

Crystals LiH LlHo.251^0.75 LiD

£,(0)(meV)

A (meV)

hoj^ff (meV)

4961 5018 5061

12+ 1 15 + 1 17 + 1

11 + 1 12+ 1 13 + 1

dE„^JdT

(meW/K)

0.19 + 0.01 0.22 + 0.01 0.25 + 0.01

for LiD, and 140 meV for LiH. This apparently also explains the discrepancy between theoretical and experimental values of the linear temperature coeflScient dE/dT (see earlier). Linear extrapolation (harmonic approximation) of the location of maximum of the long-wavelength exciton peak £„ = is to T = 0K yields £(0) = 4962 meV for LiH and 5066 meV for LiD. These results agree well with the calculated values (see second column in Table III). The difference between these two extrapolations is 104 meV, which practically coincides with the experimental value at T = 2 K; AE^ = ^^(LiD) - £^(LiH) - 103 meV (see, e.g., Plekhanov, 1996a). Assuming that this value is wholly determined by the interaction of electrons with zero oscillations of the lattice, we can evaluate it using the Fan approximation (Fan, 1951):

A£,(exp) = ( 5 E , ( ^ ^ - l ) ,

(35)

where M^ and M2 are the masses of Ught and heavy isotopes, SE^ is the narrowing of the forbidden gap because of the preceding interaction, AF^(exp) is the experimentally observed narrowing of E^ equal to 103 meV. Obviously, in the case of LiH and LiD, the masses M^ and M2 must be replaced with the reduced masses of the elementary cell: /XLIH = 7/8 and i^LiD = 14/9. Substituting these values into Eq. (35), we find that 5Eg = 412 meV. Then the "actual" (the crystal lattice not perturbed by zero oscillations) width of the forbidden gap in LiH is E^ = 4992 + 412 = 5404 meV, which is greater than £^ at 2 K for LiD (5095 meV, see Table III) by more than 300 meV (more than twice the energy ftc^Lo for LiH). This mismatch between experiment and theory (harmonic approximation) is too large. Observe that for isotopic substitution in ZnO (Kreingol'd, 1978), Ge (Agekyan et a/., 1989), and diamond (Collins et a/., 1990) a similar theoretical evaluation of SE^ by Eq. (35) is in good agreement with the experiment. Today, the reason behind such striking disagreement in the case of LiH is not quite clear. There are, however, at least two features that fundamentally distinguish LiH from ZnO, Ge, and C — to wit, isotopic

156

VLADIMIR G. PLEKHANOV

substitution greatly changes the scattering potential (Plekhanov, 1995d), and zero oscillations give a substantial contribution to anharmonizm. Indeed, the energy of zero oscillations in the Debye approximation for a two-atom crystal is EQ ^ (9/8^^0) where 0 is the Debye temperature. Given that 0 = 1190 + 80K for LiH (Yates et al, 1974), we find that £o ;^ 115 meV, which is close to the energy of the LO phonon and much greater than £^ — in other words, it is not at all small. To conclude this section, note that the different temperature dependence of exciton peaks of n = Is and 2s states leads to the temperature dependence of the binding energies of Wannier-Mott excitons — this problem has not received adequate treatment. More specifically, the energy £^ in LiH crystals (Klochikhin and Plekhanov, 1980) decreases with increasing temperature, whereas E^, increases for excitons of the green and yellow series in CU2O crystals (Itoh and Narita, 1975).

IV. 1.

Isotopic Effect on Electron Excitations

RENORMALIZATION OF ENERGY OF BAND-TO-BAND TRANSITIONS IN THE

CASE OF ISOTOPIC SUBSTITUTION IN LiH CRYSTALS

Isotopic substitution affects only the wavefunction of phonons; therefore, the energy values of electron levels in the Schrodinger equation ought to have remained the same. This, however, is not so, since isotopic substitution modifies not only the phonon spectrum, but also the constant of electronphonon interaction (see earher). Therefore, the energy values of purely electron transitions in hydride and deuteride molecules are found to be different (Herzberg, 1945). This effect is even more prominent when we are dealing with a solid (Kapustinsky et a/., 1937). Intercomparison of absorption spectra for thin films of LiH and LiD at room temperature revealed that the long-wavelength maximum (as we now know, the exciton peak; Plekhanov et al, 1976) moves 64.5 meV toward the shorter wavelengths when H is replaced with D. For obvious reasons, this fundamental result could not then receive consistent and comprehensive interpretation, which does not behttle its importance today. As we see later, this effect becomes even more pronounced at low temperatures. The mirror reflection spectra of mixed and pure LiD crystals cleaved in liquid helium are presented in Fig. 9. For comparison, in the same diagram we also plotted the reflection spectrum of LiH crystals with clean surfaces. AH spectra have been measured with the same apparatus under the same conditions. As the deuterium concentration increases, the long-wavelength maximum broadens and shifts toward the shorter wavelengths. As can clearly be seen in Fig. 9, all spectra exhibit a similar long-wavelength

6

157

EXCITON-PHONON INTERACTION

L

1

5.15

1

L_

\

5.05

E.eV

FIG. 9. Mirror reflection spectra of crystals: LiH, curve 1; LiH^Di_^, curve 2; and LiD, curve 3 at 4.2 K. Light source without crystal, curve 4. Spectral resolution of instrument indicated in the diagram. (After Plekhanov, 1997b.)

Structure. This circumstance enables us to attribute this structure to the excitation of the ground (Is) and the first excited (2s) exciton states. The energy values of exciton maxima for pure and mixed crystals at 2 K are presented in Table IV. The binding energies of excitons £^, calculated by the hydrogenlike formula, and the energies of interband transitions Eg are also given in Table IV. The ionization energy, found from the temperature quenching of the peak of reflection spectrum of the 2s state in LiD is 12 meV. This value agrees fairly well with the value of A£2s calculated by the hydrogenlike formula. Moreover, £^ = 52 meV for LiD agrees well with the energy of activation for thermal quenching of free-exciton luminescence in these crystals (Plekhanov, 1990b). Going back to Fig. 9, it is hard to miss the growth of A12 (Plekhanov, 1996a), which in the hydrogenlike model causes an increase of the exciton Rydberg with the replacement of isotopes (Fig. 10). When hydrogen is completely replaced with deuterium, the exciton Rydberg (in the WannierMott model) increases by 20% from 40 to 50 meV, whereas E^ exhibits a TABLE IV VALUES OF THE ENERGY OF MAXIMA IN EXCITON REFLECTION SPECTRA OF PURE AND MIXED CRYSTALS AT 2 K AND ENERGIES OF EXCITON-BINDING E^, BAND-TO-BAND TRANSITIONS E„

Energy (meV)

LiH

1^1^0.82^0.18

L I H Q 40^0.60

LiD

Li^H(78K)

Eu E2S

4950 4982

4967 5001

5003 5039

5043 5082

4939 4970

E, E,

42

45

48

52

41

4992

5012

5051

5095

4980

158

VLADIMIR G . PLEKHANOV

FIG. 10. Binding energy of Wannier-Mott excitons as function of reduced mass of ions based on values of reduced mass of ions for ^LiH, ^LiH, ^LiD, ^LiD, and LiT. (After Plekhanov, 1996b.)

2% increase, and at 2-4.2 K is A£^ = 103 meV. This quantity depends on the temperature, and at room temperature is 73 meV, which agrees well enough with A£^ = 64.5 meV, as found in the paper of Kapustinsky et al. The continuous change of the exciton Rydberg was earlier observed in the crystals of sohd solutions A3B5 (Nelson, 1982; Nelson et al, 1976; Monemar et al, 1976) and A2B6 (Radautsan et al, 1971; Brodin et al, 1984). Isotopic substitution of the light isotope (^^S) by the heavy one (^^S) in CdS crystals (Kreingol'd et al, 1984) reduces the exciton Rydberg, which was attributed to the tentative contribution from the adjacent electron bands (see also Bobrysheva et al, 1982), which, however, are not present in LiH (for more details see Kunz and Mickish, 1975; Baroni et al, 1985). The single-mode nature of the exciton reflection spectra of mixed crystals LiH^Di_^ agrees qualitatively with the results obtained with the virtual crystal model (see, e.g., Elhott et al, 1974; Onodera and Toyozawa, 1968), being at the same time its extreme realization, since the difference between ionization potentials ( 0 for this compound is zero. According to the virtual crystal model, C = 0 implies that AEg = 0, which is in contradiction with the experimental results for LiH^D^.^ crystals. By now the change in Eg caused by isotopic substitution has been observed for many broadgap and narrow-gap semiconductor compounds (see Section III.2).

2.

THE DEPENDENCE OF THE ENERGY GAPS OF

A2B6

AND

A3B5

SEMICONDUCTING CRYSTALS ON ISOTOPE MASSES

In this section we briefly discuss the variation of the electronic gap (Eg) of semiconducting crystals with its isotopic composition. In the last section the whole row of semiconducting crystals was grown. These crystals are

6

159

EXCITON-PHONON INTERACTION

diamond (Collins et a/., 1990; Collins, 1998), copper halides (Garro et al, 1996a; Gobel et al, 1997), germanium (Haller, 1995), and GaAs (Garro et a/., 1996b). All enumerated crystals show the dependence of the electronic gap on the isotope masses. Before we complete the analysis of these results, we should note that before these investigations, studies were carried out on the isotopic effect on exciton states for a whole range of crystals by Kreingol'd and co-workers (Kreingol'd, 1978, 1985; Kreingol'd et a/., 1976, 1977; Kreiongol'd and KuHnkin, 1986). We don't know why these papers are unknown in Western scientific Hterature. First, the following are the classic crystals CU2O (Kreingol'd et al, 1976, 1977, 1984) with the substitution 0 ^ ^ - > 0 ^ ^ and Cu^^ -^ Cu^^. Moreover, there have been some detailed investigations of the isotopic effect on ZnO crystals, where E^ was seen to increase by 55cm"^ (0^6 ^ o^^) and 12cm-^ (at Zn^^ -> Zn^^) (see Fig. 11) (Kreingol'd, 1978; Kreingol'd and Kulinkin, 1986). In Kreingol'd et al. (1984) it was shown that the substitution of a heavy ^"^S isotope for a hght ^^S isotope in CdS crystals resulted in a decrease in the exciton Rydberg constant (£^), which was explained tentatively (Bobrysheva et al, 1982) by the contribution from the nearest electron energy bands, which, however, are absent in LiH crystals (see also Fig. 7 in Chapter 5). More detailed investigations of the exciton reflectance spectrum in CdS crystals were done by Zhang et al (1998). The photoreflectance spectra of ^^^CdS and "^*CdS are depicted in Fig. 12. Besides the exciton ground state of A and B excitons we clearly see the excited states of the A and B excitons.

T^^nM

FIG. 11. The reflection spectrum of ^^-^ZnO^^ crystal (solid line) and ^^"^ZnOl^^Ol^^ crystal dashed Hne). (After Kreingol'd and Kulinkin, 1986.)

160

VLADIMIR G . PLEKHANOV T

' ^ -

'

'

•

•

1

•

•

1 1 1 1 1 1

1 1 1

1

I I , .

(a) " ' C d S A:n=1 1

2

CO

1

1

I

c Z3 JO

t

I

B: n=1

2

00

>> "to

c

2

A: n=1

00

1 Q-

1

B: n=1 i

i

1 .

2.55

-

I

"

2

2.58

2.59

1 .

CO

>.... > [

2.56

2.57

Energy (eV) FIG. 12. Photoreflectance of (a) ^^^CdS and (b) ""'CdS at 6K. The assignment of the spectral features to various components of the series of A and B excitons is indicated. (After Zhang et al, 1998.)

Polarization measurements indicate the features at about 2.574 and 2.590 eV arise from the A and B exciton transitions to the n = 2 excited states (2s), respectively. This enabled Zhang et al, to directly determine the binding energy (E^) and the corresponding bandgaps (Eg) from a hydrogenlike model (Knox, 1963): E„ = E-

EJn'

(36)

In this manner, it was possible to obtain the binding energies of A excitons of 26.4 ± 0.02 meV and 26.8 ± 0.02 meV in ^^^CdS and ""'CdS, respectively. The corresponding bandgaps £^(r7^-r9^) are, as denoted in Fig, 13 by n = oo, 2.5806 (2) eV and 2.5809 (2)eV, respectively. In the case of B excitons, these values are E^ = 27.1 ± 0.2 meV (27.1 ± 0.2 meV) and Eg(r,^-r,,) = 2.5964 (2) [2.5963 (2)eV] for the ^^^CdS ("^^CdS) sample. Unfortunately, the n = 2 excited states of the A and B excitons could not be observed in other isotopic CdS samples. Better samples are required for such measurements.

6

EXCITON-PHONON INTERACTION

161

For GaAs or ZnSe, isotope substituents of either type should lead to shifts of the EQ gap, which have been calculated to be 430 (420) and 310 (300) /ieV/amu for cation (anion) mass replacement, respectively (Garro et a/., 1996). These values are in reasonable agreement with data measured for GaAs IdEo/dM^^ = 390 (60 ^ueV/amu] (Garro et aU 1996a, b) and preliminary results for isotopic ZnSe obtained by Zhang et al. (1998) based on photoluminescence measurements of the bound exciton (neutral acceptor IJ IdE/dMs, = 140 ± 40 /zeV/amu and dE/dM^^ = 240 ± 40 /^eV/amu]. Such behavior, however, is not found in wurtzite CdS. A previous reflectivity and photoluminescence study of "^^Cd^^S and "^^Cd^'^S shows (Kreingol'd et a/., 1984) that for anion isotope substitution the ground state (n = 1) energies of both A and B excitons have a positive energy shift with the large rate of dE/dM^ = 740 ± 1 0 0 /leV/amu. This value is more than one order of magnitude larger than dE/dM^^ obtained by Zhang et al. (see also Fig. 13). The electronic band structures of semiconductors with a diamond or zinc-blende crystal lattice have a degenerate valence-band maximum for hght and heavy holes at the center of the Brillouin zone (fc = 0, F point) as well as a spht-ofiF valence band with its maximum at the same location in k space. Three conduction-band minima are observed at the high-symmetry points F, L, and X. The lowest energy conduction-band minimum occurs in Ge at the L point and in Si near the X point, forming indirect bandgaps in these two elemental semiconductors. The width and the character (direct or indirect) of this lowest energy gap is of paramount importance for a large number of semiconductor properties and, in turn, for all semiconductor

110 112 114 116 Cd mass (amu)

110 112 114 116 Cd mass (amu)

FIG. 13. Dependence of the ground-state energy of the T\{k) (a) and T\(&) excitons (b) on the isotopic mass of Cd obtained from photoreflectance measurements at 6 K. The sohd hnes represent the best fit with a straight hne. (After Zhang et al, 1998.)

162

VLADIMIR G. PLEKHANOV

devices. Because of this great significance there exists a strong interest in all eflfects that influence the band structure. The isotopic composition aflfects the bandgaps through the electronphonon coupHng and through the change of volume with isotopic mass. Several groups have conducted low-temperature studies of the direct and indirect bandgaps of natural and isotopically controlled Ge single crystals. For the first time, Agekyan et al (1989) used photoluminescence, infrared absorption, and Raman spectroscopy with Ge crystals of natural composition and crystals with 85% ^^Ge and 15% ^'''Ge. They found an indirect (see also Fig. 14) bandgap change A£^ = 0.9 meV and a direct bandgap change A£^ = 1.25 meV with an error of ±0.05 meV. Etchegoin et al. (1992) and Davies et al. (1992) reported photoluminescence studies of natural and several highly enriched, high-quality single crystals of Ge. Measurement of the energies of impurity-bound excitons by Davies et al. enabled the direct determination of bandgap shifts with the crystal isotope mass because the radiative recombination does not require phonon participation. Figure 15 shows the no-phonon energies of excitons E^p bound to P and Cu in several isotopically controlled crystals. As expected from the very large Bohr orbit of the excitons (see Davies et al, 1992 and the references therein), their binding energy depends only on the average isotope mass and not on the isotopic disorder. The rate of bandgap energy change with isotope mass as determined by Davies et al. is dE^^ldA = dE^p/dA = 0.35 ± 0.02 meV/amu.

(37)

Etchegoin et al. (1992) obtained a very similar value. The contribution to the bandgap shift originating in the volume change can be estimated using the results of Buschert et al. (1988) for the lattice-constant change with isotope mass and the published dependence of

Q886

am

E,cV

FIG. 14. Transmission spectra of ^^Ge (1) and "^^Ge (2) in the vicinity of the direct excitons transitions at 1.7 K. (After Agekyan et ai, 1989.)

6

EXCITON-PHONON INTERACTION

163

Mass number FIG. 15. Energies of the no-phonon lines of excitons bound to Cu acceptors (squares) and P donors (circles). (After Davies et al, 1992.)

EiQ on volume (Schmid and Christensen, 1990). They found {dE^c/dAX^^ = 0.132meV/amu.

(38)

This is the smaller contribution to the experimentally determined energygap change with isotope mass. It is in reasonable agreement with the earlier estimates of Agekyan et al The main contribution to dE/dA can be directly related to the change of the energy gap with temperature (see also earlier). This change is described by structure factors that contain electron-phonon interaction terms (Debye-Waller factors) and self-energy terms. For practical calculations these terms are expanded in a power series of the atomic displacements. The leading terms are proportional to the mean-square displacements of each atom. Describing the lattice atoms in terms of harmonic oscillators, one finds (u^} =

h(^-^n]lMcD.

(39)

164

VLADIMIR G . PLEKHANOV

With increasing temperature, both n and increase, leading to the observed reduction of the energy gap. At low temperatures, n = 0 and we deal only with the zero-point oscillation. Combining the dependence of o on M with the preceding equation, one finds (40)

X 1/JM.

ZoUner et al (1992) have performed a numerical calculation (see also earlier) of the electronic bands using an empirical pseudopotential method including the necessary lattice dynamics. They found for Ge {dEiG/dA)e-p = 0.41 meV. The total calculated shift of the indirect bandgap energy with isotope mass adds up to (d£^iG/^^)totai = 0.48 meV. This result compares favorably with the experimental values stated in the preceding by Davies et al. and by Etchegoin et a/., who reported {dEic/dA^otai = 0.37 ± 0.01 meV/amu (see also Fig 16). Measurements of the direct bandgap at the F point (k = 0) in the Brillouin zone have also been performed. Although the direct bandgap is technologically less important than the minimum indirect bandgap, determining the dependence of this gap on isotope mass is of the same fundamental significance as the indirect bandgap studies. Davies et al. (1993) used 742.0

739.5 72

74

76

Atomic Mass (amu) FIG. 16. Atomic mass dependence of the indirect gap E^ of Ge at r = 6K. (After Parks et al, 1994.)

6

689.0

165

EXCITON-PHONON INTERACTION

T=6 K

885.5 f 72

72

74

74

7G

atomic mass

atomic mass

FIG. 17. Isotopic mass (in amu) dependence of the (a) EQ and (b) EQ + A^ direct energy gaps obtained from photomodulated reflectivity measurements at T = 6K. The curves are the best to relation EQ = EQ -\- C/^M, where M is the atomic mass and EQ is the energy gap at M = 00. The fitting yields E^ = 959 meV and C = -606meV/amu. (After Parks et al, 1994.)

low-temperature optical-absorption measurements of very thin samples of Ge single crystals with natural composition and three different, highly enriched isotopes. They found dE/dA = 0.49 ± 0.03 meV/amu

(41)

for the temperature extrapolated to zero. Parks et al. (1994) used piezo- and photomodulated reflectivity spectra of one natural and four monoisotopic Ge crystals. These techniques do not require the extreme sample thinning, which is necessary for optical-absorption measurements, and the derivative nature of the spectra emphasizes the small changes. The excellent signal-tonoise ratio and the superb spectral resolution enabled a very accurate determination of the dependence of E^G ^^ isotopic mass (Fig. 17). At very low temperatures an inverse square-root dependence accurately describes the bandgap dependence: 17

ra

+

c

(42)

M

A fit through five data points yields £§G = 959meV

and

C

- 606 meV/amu1/2

166

VLADIMIR G. PLEKHANOV

Written as a linear dependence for the small range of isotopic masses, Parks et al. found dE^^/dA = 0.49 meV/amu, in perfect agreement with the results of Davies et al (1993). Parks et al also determined the isotope mass dependence of the sum of the direct gap and the split-off valence band (AQ) and found ^(EDG + ^o)/^^ ^ 0-^4 meV/amu. The experimental results can be compared to the Zollner et al (1992) calculations, which are found to be of the correct order of magnitude. The theoretical estimates for the contributions of the linear isotope shifts of the minimum, indirect gaps, which are caused by electron-phonon interaction, are too large by a factor of ^1.7 and for the smallest direct gap, they are too large by a factor of ^ 3.2. Substitution of '^^Ga on the "^^Ga increases the bandgap in GaAs (Garro et al, 1996a) on 10.5 cm~^ (see also Table V). The interesting results were communicated in the papers of Cardona and co-workers (Garro et al, 1996a; Gobel et al, 1997), where the dependence of Eg on the isotope effect in CuCl crystals was studied. When the ^"^Cu on the ^^Cu is substituted, the value of Eg in CuCl crystals decreased by 1.24 c m " \ for example, the isotope effect on the electronic excitation has an opposite sign. Considering the series of Ge, GaAs, ZnSe, CuBr, for example, the 3d states of the first constituent play an increasing role in determining the band structure. In Ge, these states can be considered as localized core states (atomic energy level ?^ — 30 eV). In GaAs, however, they have already moved up in energy by 10 eV, and their hybridization with the top of the valence band affects the gap (see, e.g., Harrison, 1970). Proceeding further in the series, this effect becomes more important, and the CuBr and Cu M states even overlap in energy with halogen p-states, with which they strongly hybridize. Therefore, we cannot exclude that the main reason for the opposite sign of the isotopic effect in these compounds may be connected to the different character of the d-electron-phonon interactions in these semiconductors (Plekhanov, 1998). The change of the indirect gap of diamond between pure ^^C and ^^C was determined by Collins et al (1990), using for this purpose the luminescence spectra of diamond. The luminescence spectra of the natural (^^C) and synthetic (^^C) diamond were investigated by Collins et al (1990), R\xi et al (1998), and Collins (1998). Figure 18 compares the edge luminescence for a natural diamond with that for a synthetic diamond. The peaks labeled A, B, and C are due, respectively, to the recombination of a free exciton with the emission of transverse-acoustic, transverse-optic, and longitudinal-optic phonons having wave vector ±k^;^ and quanta (in ^^C diamond) of Clark et al (1964): fto^TA = 87 ± 2, hcDjQ = 141 ± 2, /ICOLO = 163 ± 1 meV. Features B2 and B3 are further free-exciton processes involving the preceding TO phonon with one- and two-zone center optic phonons, respectively.

6

5.00

167

EXCITON-PHONON INTERACTION

5.10

5.30

5.20

5.40

P h o t o n energy (eV)

FIG. 18. Spectra measured at 77 K of the phonon-assisted free exciton cathodeluminescence feature (A, B, and C) and the phonon-assisted bound-exciton features (D) from a natural semiconducting ^^C diamond and a ^^C synthetic diamond. (After Colhns et al, 1990.)

Boron forms an efFective-mass-like acceptor in diamond, and both specimens used in Fig. 18 are slightly semiconducting with uncompensated boron concentrations around 5 x lO^^cm"^ in the natural diamond and 3 X lO^^cm"^ in the synthetic diamond. Peaks labeled D are associated with the decay of excitons bound to the boron acceptors (for details see Colhns et al, 1990). Comparison of the data from the two diamonds shows that the zero-phonon lines D^ and D^ are 14 ± 0.7 meV higher for ^^C than for ^^C diamond, and that the LO and TO phonon energies are lower by a factor of 0.96 (cf. Subsection 2 of Section IV), equal within experimental error to the factor (12/13)^^^ expected to be first order when the lattice is changed from ^^C to ^^C. The low-energy thresholds of the free-exciton peaks A, B, and C are given by Collins et al. (1990): ^th(^) = £,x - ^^TA. ^th(^) = E^^ - hcojo.

and

£,JC)

Egx - hcDi^o-

As was shown by Colhns et al, the predicted thresholds are entirely consistent with the experimental data. From the results of Colhns et al, it was concluded that the dominant contribution arises from electron-phonon

168

VLADIMIR G. PLEKHANOV

5000

5100 5200 5300 Energy (meV)

5400

FIG. 19. Cathode-luminescence spectra of isotopically modified diamond at 36 K. Intrinsic phonon-assisted recombination peaks are labelled in the top spectrum, those from boronbound excitons in that at the bottom. The spectra are normalized to the intensity of the B peak and vertically offset for clarity. (After Ruf et al, 1998.)

coupling, and that there is a smaller contribution due to a change in volume of the unit cell produced by changing the isotope. These two terms were calculated as 13.5 + 2.0 and 3.0 ± 1 . 3 meV, respectively. The more detailed and quantitative investigations of E^ ^ /(x), where x is the isotope concentration, were done by Ruf et al. (1998), who studied five samples of diamond with different concentrations x (Fig. 19). From these data Ruf et al. determined the linear variation of E^ ^ f{x) for diamond (Fig. 20). Linear fits of the experimental data of Ruf et al. (sohd line in Fig. 20) yield a slope of 14.6 ± 0.5 meV/amu, close to the theoretical predictions. All of these results are documented in Table V, where the variation of Eg, Ejj, and /ICOLO are shown at the isotope effect. We should highhght here that the most prominent isotope effect is observed in LiH crystals, where the dependence of £^ = / ( C H ) is also observed and investigated. To end this section, we note that E^ decreases by 97 cm" ^ when ^Li is replaced with ^Li (see Table V).

3.

RENORMALIZATION OF BINDING ENERGY OF WANNIER-MOTT

EXCITONS BY THE ISOTOPIC EFFECT

In the original work of Plekhanov et al. (1976) the exciton binding energy Efj was found to depend on the isotopic composition and this change in E^ was attributed to the exciton-phonon interaction (originally with LO

6

169

EXCITON-PHONON INTERACTION

5380

100 C concentration (%) FIG. 20. Energy of the DQ multiplet in isotopic diamond at 36 K. The filled symbols are for the main component, the open ones for weaker side peaks. The solid lines are linear fits to the data. (After Ruf et al, 1998.)

phonons). The preferential interaction of excitons with LO phonons in LiH (LiD) crystals was later repeatedly demonstrated in the luminescence spectra (Plekhanov, 1990a) and resonant Raman light scattering (Plekhanov and O'Konnel-Bronin, 1978a; Plekhanov and Altukhov, 1985), which consist of a phononless hne (in the former case) and its LO repetitions. The effects of the Frohlich mechanism of exciton-phonon interaction on the energy spectrum of Wannier-Mott exciton has been considered over and over again (Peierls, 1932; Haken, 1976; Segall and Marple, 1967; Segall and Mahan, 1968; Fedoseev, 1973; Ansel'm and Firsov, 1956; Thomas, 1967; Firsov, 1975; Toyozawa, 1958; Frohlich, 1954; Klochikhin, 1980; Bulyanitza, 1970; Fan, 1967; Singh, 1984). Today we know that the main consequences of the electron and hole interaction in excitons with polarization vibrations are the static screening of the lattice charges (introducing EQ) and the change in the effective masses of the particles. Both these effects of electron-(hole)phonon interaction can easily be taken into account, and lead to a change in the exciton Rydberg E^. These corrections do not destroy the hydrogenlike structure of the exciton spectrum. At the same time, the non-Coulombic corrections to the electron-hole Hamiltonian modify the hydrogenlike structure — removing, for example, degeneration of levels with respect to orbital and magnetic quantum numbers (see, e.g., Fedoseev, 1973). The very fact, however, that the problem of renormalization of energy spectra of Wannier-Mott excitons does not result in an exact solution even in the hmiting cases, often gives rise to a situation in which there is no agreement between the results obtained by different authors. Starting with the classical works of Haken (1976), all papers can be divided into two broad classes depending on how they deal with the Coulomb interaction: between "bare"

170

VLADIMIR G. PLEKHANOV

electrons and holes, or between electrons and holes in the polaron state. In other words, first the interaction of band electrons and holes with LO optical phonons is taken into account, and then the Coulomb interaction between electrons and holes clad in the "polarization coats" is considered. As shown in the following, the study of exciton-phonon interaction in crystals with isotopic effect not only provides entirely new information, but also enables us to reconstruct experimentally the values of Fohlich and Coulomb interaction constants. From Fig. 10 we see that when hydrogen is completely replaced with deuterium, the binding energy of the exciton exhibits a 20% increase from 42 to 52 meV (Plekhanov et al, 1976). It is easy to see that in the model of virtual crystals the binding energy of the exciton in LiT crystals (Plekhanov, 1996c) must be equal to 57 meV (see Fig. 10). Hence it follows that in the linear approximation the isotopic dependence of binding energy of Wannier-Mott excitons may be expressed as E, = E,(0)(l + y).

(43)

where £^(0) is the purely Coulombic binding energy of the exciton (in the frozen lattice), which in our case is equal to 31.5 meV, and the angular coefficient is P = 12.18 meV/M, where M is the reduced mass of ions of hthium and hydrogen (deuterium, tritium) ions; y = ^M/£^(0) (see also Plekhanov, 1996c). From the standard equation for the Coulomb binding energy of the exciton,

.. = 0 ,

(44)

we get the dimensionless constant of Coulomb interaction: ,

-

^

= 0.47.

Comparing the value of rj^ = 0.47 and the constant of FrohUch excitonphonon interaction g^ = 0.33 (Plekhanov and Altukhov, 1981) we see that they are close enough. This implies that both the Frohlich and the Coulomb interactions between electrons (holes) and LO phonons in the exciton must be treated with equal attention, as has already been emphasized in Klochikhin (1980). This paper deals from the start with "bare" electrons and holes, and all renormalizations are calculated in the two-particle configuration. Such an approach enables us to avoid the considerable difficulty that arises when polarons (Sak, 1972) are used as start-up particles. This difficulty is primarily associated with the fact that the momentum of each particle is

6

EXCITON-PHONON INTERACTION

171

conserved when the particles are treated separately, whereas it is the center-of-mass momentum that is conserved when a pair moves as a whole. As demonstrated in Klochikhin (1980), this approach also makes it possible to calculate the higher-order corrections to the exciton-phonon interaction. It was also shown that the use of the pole parts of polaron Green functions in place of complete expressions in Sak (1972) and Mahanti and Varma (1972) leads to a situation where the corrections of the order of rj^g^ and g"^ to the potential energy are lost because the corrections to the vertex parts and Green functions cancel out. The quantity lost is of the same order (g^) as the correction to the residue but has the opposite sign (for more details see Sak, 1972; Mahanti and Varma, 1972). The approach developed in Klochikhin (1980) enabled the calculation of corrections of the order oirj^g^ and g"^, the latter is comprised of the correction to the Frohlich vertex and the correction to the Green functions in the exciton-phonon loop. It is important that the latter have opposite signs and cancel out exactly in the limit £ ^ « hcOi^Q. As a result, because of the potential nature of the start-up Coulomb interaction, the correction to the Coulomb vertex of the order of rj^g^ does not vanish. As a result, the following expression was obtained in Klochikhin's paper for the binding energy E^ of WannierMott excitons when £^ « ^COLQ (the spectrum of exciton remains hydrogenUke):

Eb = hcoLo

n^ -g^ + n^9\c + v)

(45)

where c, v = {mc^^liiY'^ and m^ and m^ are the electron and hole masses. Now E^ depends explicitly on g^ (the Frohlich constant of exciton-phonon interaction), and hence depends on the isotopic composition of the lattice, whereas the standard expression for the binding energy ^h =ft<^Lo(^^- g^) = e'^fi/lsoh^, which describes the exciton spectrum of many semiconductors accurately enough, exhibits no dependence on the isotopic effect. In the case of Eq. (45) the exciton spectrum remains hydrogenlike. When the higher-order corrections are taken into account, Eq. (45) becomes E.= ' ' ' 1 + g \ ^ ^ " ^ i ^ + g^^(c, ^* 2snh^

+ C, ^—^] {m, + mj].

(46)

The order-of-magnitude evaluation of the coefficients ^1,(2 gives Ci « 0.15 and C2 * 002; when g^{m^ + mj « 3.3, the correction of the order of ri^g* is much less than the term of the order of rj^g^ (Klochikhin, 1980).

172

VLADIMIR G. PLEKHANOV TABLE V THE CHANGE OF THE VALUE OF THE EXCITON BINDING ENERGY (AE^), BAND-TO-BAND TRANSITIONS ( A £ ^ ) AT THE 100% ISOTOPIC SUBSTITUTION

Compound 70Qe _, 76Qg GaAs (^^Ga -• ^^Ga) CU2O (^^O -^ i^O) CU2O (i^O -> i«0) CdS (^^S ^ 3^S) CdS C^^Cd -^ ""^Cd) CuCl (^^Cu-^^^Cu) ZnO (^^0-> i«0) ZnO (^^Zn^^^Zn) 12c -^ '^C LiH (H -^ D) LiH (^Li -^ ^Li) All 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

A£^ Indirect (L point) 2.1 meV = 16.94cm" ^ [1,2,3] Direct (E point) 0.85 meV = 10.5 cm" ^ [4] Direct (yellow) 2.2 meV = 18 cm"^ [5] Direct (green) 2.2 meV = 18 cm"^ [5] — Direct 0.003 meV (F point) -0.155 meV = -1.24cm"^ [9] (F point) 6.82 meV = 55cm-i [10,11] (F point) 1.5 meV = 12cm-^ [10,11] Indirect (F point) 13.6 meV = 109.7 c m " ' [12] Direct 103meV = 831 cm"^ [15,16,17] Direct 5 meV = 41 c m " ' [16]

AEf, — — — — - 1 . 6 meV [7] 0.4meV [8] — — — — lOmeV [16] 3 meV [16]

values are in meV or cm~^ V. F. Agekyan et al, Fizika Tverdogo Tela 31, 101 (1989). C. Parks et al., Phys. Rev. B49, 14244 (1994). J. J. Haller, J. Appl. Phys. 11, 2857 (1995). N. Garro et al, Phys. Rev. B54, 4732 (1996). F. L Kreingol'd et al, Pis'ma v ZETPH 23, 679 (1976). F. I. Kreingol'd, Fizika Tverdogo Tela 27, 2839 (1985). F. L Kreingol'd et al, Fizika Tverdogo Tela 26, 3490 (1984). J. M. Zhang et al, Phys. Rev. B57, 9716 (1998). N. Garro et al.. Solid State Commun. 98, 27 (1996). F. I. Kreingol'd Fizika Tverdogo Tela 20, 3138 (1978). F. I. Kreingol'd and B. S. Kulinkin, Fizika Tverdogo Tela 28, 3164 (1986). T. Collins et al, Phys. Rev. Lett. 65, 891 (1990). R. M. Chrenko, J. Appl. Phys. 63, 5873 (1988). K. S. Hass et al. Phys. Rev. B54, 7171 (1992). A. A. Klochikhin and V. G. Plekhanov, Fizika Tverdogo Tela 22, 585 (1980). V. G. Plekhanov, Uspekhi Fiz. Nauk 167, 577 (1997b). A. F. Kapustinsky et al, Physicochimica USSR 1, 799 (1937).

So from Eq. (46) we see that the correction to the purely Coulombic binding energy, Eq. (44), is important. This is primarily because the values of rj^ and g^ are close to each other. Setting m^/m^ = 3.5 and g^/rj^ = I — eje^, and (^ooAo) =(<^To/^Lo = 1/3-5, in Klochikhin and Plekhanov (1980) it was found that £f,(theor) = 48 and 42 meV for LiD and LiH, respectively. Comparing these results with the experimental values (see Table V) we observe good agreement between theory and experiment. Hence it follows a natural conclusion that the isotopic dependence of the exciton binding energy is due primarily to the Frohlich interaction mechanism between excitons and phonons.

6

EXCITON-PHONON INTERACTION

173

FIG. 21. Temperature dependence of polaron contribution to the change in E^ for isotopic substitution in LiH crystal. (After Plekhanov, 1990a.)

In the preceding section we saw that isotopic substitution affects not only Ejj, but also Eg. For the LiH-LiD system at low temperatures the diflference is AEg = 103 meV. Apart from the zero oscillations considered in the previous section (see also Kreingol'd, 1978; Allen, 1994), this change is also contributed to by the polaron shift, which explicitly depends on the temperature. Figure 21 shows the temperature dependence of AE = £„ = i,(LiD) - £„^i,(LiH), which is generally similar to AE^ = f{T). From this diagram, we see that as the temperature rises from 2 to 300 K, the quantity AE decreases from 103 to 73.5 meV. The last result agrees well with the value of 65 meV obtained by Kapustinsky et al. In light of the Frohlich mechanism of exciton-phonon interaction considered in the preceding, the magnitude of the polaron shift may be estimated by the following expression:

A^poi= -Ethco^^o

- 1

^1 -f m^ +

/l + - ^ V mi

This estimate gives about 20 meV (Plekhanov, 1997b), which is about 1/4A£; on the other hand, it agrees well enough with the magnitude of the polaron shift, as derived from the temperature dependence of the shift (see Fig. 21). Although the isotopic change in Eg has been observed in a large number of compounds (dielectrics and semiconductors), the number of studies of isotopic effect on the exciton binding energy is limited to four cases. As demonstrated earlier, the replacement of a light isotope with a heavy one in LiH crystals (Plekhanov et a/., 1976) leads to an increase in £^; the binding energy decreases in CdS crystals (Kreingol'd et a/., 1984) and remains the same in the crystals of germanium (Parks et al, 1994) and diamond (Collins

174

VLADIMIR G. PLEKHANOV

et al, 1990). Quantitative measurements of the isotopic effect on the levels of large-radius exciton allow for experimental reconstruction of the FrohHch and Coulomb constants.

4.

LUMINESCENCE OF FREE EXCITONS IN LiH

CRYSTALS

Because of the low intensity of scattered light, and thanks to the high resolution of modern spectroscopic instruments, the development of highly sensitive techniques for the detection of weak optical signals (photon counting mode, optical multichannel analyzers, optical linear arrays, and other specialized systems (see, e.g., Chang and Long (1982), the luminescence method has become one of the most common techniques for studying excitons in dielectrics and semiconductors. While the structure of spectra of fundamental reflection (absorption) depends on the internal degrees of freedom of the Wannier-Mott exciton, the structure and shape of the luminescence spectrum are determined primarily by its external degrees of freedom. The latter are associated with the translation motion of largeradius excitons as a whole, with the translation mass M = m^ + m^ (Knox, 1963). The results on the luminescence (RRLS) of LiH^D^.^ crystals presented in the following were obtained from the clean surfaces of crystals cleaved directly under hquid superfluid hehum in the cell of optical cryostats (Plekhanov et a/., 1984). The effects of surface habitus on optical spectra (including the luminescence spectra) of excitons in hygroscopic LiH and LiD crystals were briefly described earlier (Plekhanov et a/., 1984; Pilipenko and Gavrilov, 1985; Plekhanov, 1987). The luminescence of LiH crystals was first observed in 1959 by Gavrilov. The broadband luminescence in the red part of the spectrum was attributed to the defects related to the nonstoichiometric composition of crystals (see also Plekhanov et a/., 1988). Later the broadband luminescence of pure and activated (mostly with mercurylike ions with 5^ outer electron configurations LiH crystals were studied in a large number of works (see the review of Plekhanov et a/., 1988, and the references therein). The first results on photoluminescence (Pustovarov, 1976) of LiH crystals near the edge of fundamental absorption, like the first results on cathode-ray luminescence (Zavt et a/., 1976a) were rather quahtative. The results on cathode-ray luminescence spectra of LiH crystals at 6 K are described in greater detail in Zav'yalov et al. (1985); they are analyzed and compared with the photoluminescence spectra in Plekhanov et al. (1988). LO repetitions were discovered in the spectra of x-ray luminescence of LiH crystals cleaved in the inert gas environment (Plekhanov et al, 1977). The discovery of the LO structure of luminescence spectra and luminescence excitation spectra, and later that of RRLS (Plekhanov and O'Konnel-Bronn, 1978b) in LiH (LiD) crystals, has presented the opportunity for spectroscopic

6

5.00

175

EXCITON-PHONON INTERACTION

4.60

E,eV

FIG. 22. Emission spectra of free excitons at 2 K in LiH crystals cleaved in liquid helium. Spectral resolution of instrument indicated on diagram. (After Plekhanov, 1995a.)

Studies of energy relaxation processes in the course of interaction with phonons (Plekhanov, 1981). As demonstrated earlier, most low-energy electron excitations in LiH crystals are the large-radius excitons. Exciton luminescence is observed when LiH crystals are excited in the midst of fundamental absorption. The spectrum of exciton photoluminescence of crystals of lithium hydride cleaved in hquid hehum consists of a narrow (in the best crystals its half-width is A£ ^ lOmeV; Plekhanov et aU 1984; Plekhanov and Altukhov, 1985) phononless emission line and its broader phonon repetitions, which arise due to radiative annihilation of excitons with the production of 1-5 longitudinal (LO) phonons (Fig. 22). The phononless emission line coincides in an almost resonant way with the reflection line of the exciton ground state (Plekhanov, 1990b), which is another indication of direct electron transition. The lines of phonon replicas form an equidistant series biased toward the lower energies from the resonance emission line of excitons. The energy diflference between these lines, as in Plekhanov (1981) is about 140 meV, which is close to the calculated energy of LO phonons in the middle of the Brillouin zone (Verble et al, 1968) and measured by Plekhanov and O'Konnel-Bronin (1978a). The most important distinctions between the exciton luminescence spectrum shown in Fig. 22 and those measured earlier (Plekhanov et a/., 1977) are the following: (a) the presence of a second series of LO repetitions counted from the level of the n = 2s exciton state, (b) comparable intensities of the phononless emission line and its 2LO replica, and (c) noticeable narrowing of the observed lines (see also Plekhanov, 1987). Here note also an overall increase in the intensity over the entire luminescence spectrum. Evidently, the intensity of phononless emission lines of free excitons (proof of the existence of quasimomentum is given in the next subsection) increases because the rate of emissionless recombination on the pure surface decreases. This seems natural for the

176

VLADIMIR G. PLEKHANOV

surface of a specimen cleaved in liquid helium because the surface states (as a rule, of extrinsic origin; Fischer and Stolz, 1982; Schultheis and Balslev, 1983) and their electric fields (recall that the value of the exciton Rydberg is relatively low, E^ = 40 meV) lead not only to broadening of the luminescence Hues, but also to quenching of their intensity, and first of all, quenching of the intensity of the zero-phonon line.

a. Proof of Existence of Quasi-Momentum of Excitons in LiH (LiD) Crystals It is well known that one of the main properties of the exciton is its abiUty to move freely over the crystal lattice (Knox, 1963; Gross, 1976). In the effective-mass approximation (Dresselhaus, 1956), an exciton is regarded as a quasi-particle associated with a certain value of the wave vector (quasimomentum) /c, the change whereof characterizes the motion of the exciton. The proof of motion of the exciton, especially in semiconductor crystals, has been the subject of many papers (see, e.g., Knox, 1963; Gross, 1976; Agekyan, 1977; Segall and Marple, 1967; Segall and Mahan, 1968; Thomas, 1967; Gross et al, 1971; Permogorov, 1975). It was first demonstrated by Gross and co-workers (1971) that the exciton in a crystal may possess a considerable amount of kinetic energy. When the exciton distribution is in equihbrium, the kinetic energy is known to be determined by the temperature of the crystal lattice. The studies of the lineshapes and the temperature dependence of their relative intensity in the spectra of exciton-phonon luminescence have supplied the most comprehensive information concerning the exciton motion (and have proved this motion to be free), which enabled the estabhshment of the exciton distribution law within the band (Plekhanov, 1997b). Indeed, according to the law of conservation of quasimomentum, the phononless emission is possible only for those excitons whose wave vector is of the same order of magnitude as the wave vector of the exciting photons (k ^ 0), and whose kinetic energy is practically zero. Then, the phononless emission line is narrow. The emission processes, however, which are associated with excitation of one or several optical phonons, may involve excitons with arbitrary kinetic energy and wave vector values. In this case, the excess thermal motion wave vector of the exciton is passed on to the phonons. Since the energy of optical phonons, as a rule, does not depend on the wave vector, the emission spectra will contain fines whose long-wavelength edges are displaced (see Fig. 22) by the energy of a whole number of optical phonons with respect to the energy of the phonon ground state — the bottom of the exciton band (see also Gross et al, 1971). The shape of the emission lines, as indicated earlier, reflects the distribution of excitons with respect to their kinetic energy E^^^. Such LO structure has actuafiy been observed in the spectra of intrinsic luminescence

6

177

EXCITON-PHONON INTERACTION

of a large number of semiconductor crystals. The shape of LO phonon repetitions is described by the Maxwell distribution of excitons with respect to kinetic energy: dW

= Py£i/,^exp

(47)

ksT

kin

where W is the probabihty of exciton-phonon interaction, which depends on k in the case of one-phonon scattering, and does not depend on k in the case of two-phonon scattering (Gross et al, 1971; Klochikhin et al, 1976). Figure 23 shows the comparison of experimental first- and second-order lines of exciton luminescence in LiH crystals with the shape described by Eq. (47). In general, the agreement between theory and experiment is reasonably good. At the same time, we notice that the half width of the line is somewhat larger, and its shape on the long-wavelength side differs from that described by Eq. (47). As shown in the following (see also Plekhanov et a/., 1988; Plekhanov, 1994a), this deviation is mainly determined by the magnitude of the longitudinal-transverse splitting — that is, by the strength of exciton-photon interaction (Knox, 1963; Benoit a la Guillaume et al, 1970). Reasonable accuracy in the description of the shape of ILO and 2LO repetitions can be achieved by assuming that the temperature of the exciton gas is ;^200 K (Plekhanov, 1997b). This value is more than three times as high as the temperature of crystal in the cryostat, which suggests that there is no thermodynamic equilibrium between the excitons and the lattice. This is one of the reasons why the emission lines of LO replicas in crystals cleaved in hquid helium exhibit broadening even at the lowest temperature. This circumstance has already been noted in Plekhanov et al. (1984). A detailed study of the temperature evolution of emission spectra of free excitons in LiD crystals (Plekhanov et a/., 1988) revealed a pronounced short-wavelength shift of the hues of the ILO and 2LO replicas with the increasing temperature (see also Fig. 26), which is obviously associated with

LiH

62K \ \ 2L0

>^1L0

1

i

1 \

4.80

1

1

4.70

1 1

^%i-

E,eV

FIG. 23. Lineshapes of ILO and 2LO replicas in exciton luminescence spectrum of LiH crystal at 62 K, and Maxwell approximation of distribution of excitons with respect to kinetic energy (dashed line at 200 K). (After Plekhanov et ai, 1984.)

178

VLADIMIR G . PLEKHANOV

the shift of the Maxwell distribution. This is a clear indication that excitons in these crystals do possess kinetic energy. This conclusion fits in well with the results on the temperature dependence of the relative intensity of ILO and 2LO repetitions. In the temperature range where the exciton distribution is in equilibrium with respect to kinetic energy, this dependence is well described by a linear function (Toyozawa, 1958). Thus we can conclude that the preceding results and their consistent interpretation clearly indicate that the observed emission is the emission of free excitons possessing a considerable amount of kinetic energy. Accordingly, the motion of excitons is coherent and comphes with the law of conservation of quasi-momentum fc.

b.

Band Relaxation of Free Excitons

Detailed information regarding the kinetic energy of free excitons can be extracted from the luminescence excitation spectra of free excitons (Gross, 1976; Gross et al, 1970). This function directly reflects the process of exciton relaxation over the band. The important circumstance is that while the kinetic energy of excitons in the case of exciton-phonon luminescence is usually of the order of k^ T, it can considerably exceed the binding energy of the exciton in the case of luminescence excitation spectra, as first demonstrated in Gross et al. (1970). As an example, let us consider the exciton luminescence excitation spectrum (Fig. 24) of LiH crystals cleaved in hquid hehum. The most striking difference between the results displayed in Fig. 24

5.60

5.40

5.20

E,eV

FIG. 24. Luminescence excitation spectrum of the 2LO replica line in a LiH crystal at 4.2 K and original record in the region of the exciton ground and first excited state. (After Plekhanov, 1987.)

6

179

EXCITON-PHONON INTERACTION

5.30

5.20

5.10

E,eV

FIG. 25. Luminescence excitation spectrum of the ILO replica line in a LiH crystal at 4.2 K (After Plekhanov et al, 1988.)

and the excitation spectra of crystals cleaved in hot air (Plekhanov, 1981) show the presence of fine structure exactly in the exciton region. The observed structure of the excitation spectrum is related primarily to the manifestation of the n = 2s exciton state. In addition, the long-wavelength wing (see arrow I in Fig. 24) features a kind of "singularity" at a distance of 8meV. This singularity may be related to the intervalley splitting (Plekhanov and Altukhov, 1981), given the existence of three equivalent X valleys in the band structure of these crystals (Baroni et ai, 1985) where direct and allowed transition takes place (Plekhanov et al, 1976). As the exciton kinetic energy increases, the observed structure smears out (for more details see Plekhanov, 1987). This may be caused both by the reduced probability of indirect transitions in the course of absorption at large wave vector values (Ansel'm and Firsov, 1956; Yu, 1979), and by the reduced probabiHty of the generation of excitons from recombination of electron-hole pairs (Lipnik, 1964, Zinov'ev et a/., 1983; Abakumov et al, 1980). The results presented in Fig. 24 lead to the conclusion that the excitons in these crystals (as well as in LiD; Plekhanov, 1981) may have very large kinetic energy (up to ^0.5 eV). This is more than 10 times their binding energy. Such a high value of the kinetic energy of excitons once again points to their high temperature, which fits in well with the large half width of the phonon repetitions of exciton luminescence. The high velocity of these hot excitons ensures large travel paths within the crystal, thus facihtating their capture by the defects (impurities) in the LiH lattice. This apparently also explains the low quantum yield of free exciton luminescence in these crystals {Y\ '^ 3-0.1%; Plekhanov, 1997b). A similar structure consisting of LO replicas is also displayed by the excitation spectrum of emission lines of ILO repetition (Fig. 25) in LiH crystals. As in the previous case (see Fig. 24), the highest intensity in the spectrum corresponds to the line removed from the emission line by the energy of the 2LO phonons. The low intensity of the luminescence line of ILO phonons did not enable the measurement of its spectrum over a broad range of energies. It is clear, however, that the intensity of the spectrum decreases and the lines broaden as the kinetic energy of excitons

180

VLADIMIR G. PLEKHANOV

E,eV FIG. 26. Temperature evolution of lineshapes of the ILO and 2LO replica lines in the luminescence spectra of free excitons in LiD crystals. (After Plekhanov et al, 1988.)

increases. In this way, accurate measurements of the spectra of muhiphonon luminescence and its excitation in crystals with clean surfaces enable us not only to demonstrate the existence of the quasimomentum of such excitons, but also to trace the band relaxation of excitons (see also Fig. 26).