Phonon self-energies in semiconductors: anharmonic and isotopic contributions

PERGAMON

Solid State Communications 117 (2001) 201±212

www.elsevier.com/locate/ssc

Phonon self-energies in semiconductors: anharmonic and isotopic contributions M. Cardona*, T. Ruf Max-Planck-Institut fuÈr FestkoÈrperforschung, Heisenbergstrasse 1, 70569 Stuttgart, Germany

Abstract During the past ten years, semiconducting crystals with varying isotopic compositions have been grown. They can be used for obtaining detailed information on their lattice dynamics, including dispersion relations, eigenvectors and the self-energies related to anharmonicity and isotopic disorder. Examples involving diamond, zincblende, and wurtzite-type semiconductors are discussed. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: A. Semiconductors; D. Phonons; D. Anharmonicity; E. Inelastic scattering PACS: 63.20.-e; 63.20.Ry; 63.20.Mt

1. Introduction Quantized harmonic vibrations of crystal lattices were introduced by Einstein [1] and by Debye [2] in order to explain the temperature dependence of the speci®c heat of insulators. They were labeled as phonons around 1932, probably by Frenkel [3]. The harmonic nature of phonons, necessary to represent them as particles with in®nite lifetime, is only an approximation based on the expansion of the lattice energy in powers of the components of the atomic displacements uj. Keeping only terms linear and quadratic in uj leads to the harmonic approximation. The phonons can then be represented as quantized vibrations of harmonic oscillators with angular frequencies v `,q, where q is a wavevector and ` labels the corresponding branch of the dispersion relations v ` vs q. The total energy of the lattice, and its dependence on uj, can nowadays be obtained ab initio [4], within the so-called adiabatic approximation, by computing the total energy of the crystal vs. uj up to terms of fourth-order in uj. The terms of third and fourth-order in uj are called anharmonic. In their presence, the harmonic phonons are not exact eigenstates of the vibrational Hamiltonian, i.e. they have ®nite lifetime t which is related to the spectral width of the phonon vs. * Corresponding author. Tel.: 149-711-689-1710; fax: 149-711689-1712. E-mail address: [email protected] (M. Cardona).

frequency G (full width at half maximum FWHM) by the frequency (or energy)±time uncertainty relation

G´t . 1

…1† 21

According to Eq. (1) a width G of 1 cm corresponds to t ˆ 5 ps: Anharmonicity of the interatomic potentials was ®rst introduced by GruÈneisen in order to interpret the thermal expansion of insulators [5]. It was used by a number of distinguished theorists to describe the line shapes of phonon spectra observed by infrared spectroscopy (ir). For a discussion of the early work see Ref. [6]. In modern language phonons are referred to as quasiparticles, with a complex self-energy S ˆ S r 1 iS i induced in insulators by anharmonic phonon±phonon interactions and, in crystals with several isotopes of a given element, also by isotopic mass disorder. In metals and heavily doped semiconductors one must also take into account the self-energy which corresponds to the interaction of the phonons with the conduction electrons. The latter has received considerable attention since the discovery of high-Tc superconductors: Sweeping their temperature through Tc produces measurable changes in the self-energies of some of the phonons. We shall not discuss here metals any further. The interested reader may consult Ref. [7]. The purpose of this paper is to discuss the anharmonic and isotopic disorder contributions to the self-energy of phonons, in semicondutors of the tetrahedral variety (i.e. with the diamond, zincblende, or wurtzite structures) with

0038-1098/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0038-109 8(00)00443-9

202

M. Cardona, T. Ruf / Solid State Communications 117 (2001) 201±212

phonon. We discuss next global or integrated effects, to which all phonons contribute in a weighted average way. These effects, are the thermal expansion and the thermal conductivity. We shall not discuss here the thermal conductivity, a ®eld in which there has been considerable progress recently in connection with isotopically substituted Ge and Si [11]. The thermal expansion is also related to the third-order anharmonicity of the interatomic potentials. It can be written in terms of the mode GruÈneisen [5] parameters g `,q which are de®ned as: Fig. 1. Self-energy diagrams representing anharmonic contributions to the phonon linewidths and shifts (a,b,c). Also, corresponding contributions of the isotopic mass ¯uctuations (d and e). Contribution (d) vanishes if the average mass is used for the harmonic crystal (VCA). Contribution (b) is always real and often is not called self-energy, but simply energy-shift, see Ref. [49].

special emphasis on the quantum effects observed at low temperatures. The complex self-energy S `;q …v; T† is speci®c to each phonon and, for a given phonon, it is a function of v and of T. We shall consider here the selfenergy effects represented by the ®ve Feynman diagrams of Fig. 1. Diagrams a±c have anharmonic vertices while d and e are related to disorder. Diagram a represents a secondorder perturbation effect of the third-order anharmonicity. The intermediate state consists of two phonons, of frequencies v 1 and v 2. In order to have a ®nite value of S i for a given v `,q, one must have two phonon states which allow the decay of v `,q with energy conservation (real transitions, v ˆ v 1 1 v2 ). S r, however, is associated with virtual transitions. Diagram b is a ®rst-order perturbation term associated with the fourth-order anharmonicity. It has only a real part which cannot be separated experimentally from the real part of a: their dependence on temperature is, usually, rather similar. Ab initio calculations, however, permit the separation of these two contributions to the energy shift induced in v `,q by S r [8,9]. Unless otherwise speci®ed, we shall lump the contribution of b together with the S r of a. Diagram c represents second-order perturbation terms with the fourth-order anharmonicity at the vertices. The corresponding self-energy is complex if energyconserving transitions are possible. The remaining diagrams represent isotopic mass ¯uctuations, although sometimes they can also be used to represent impurities or mixed crystal effects (see, for instance, recent Raman data for AlxGa12xN [10]). Within the so-called virtual crystal approximation (VCA), in which the masses of a given atom are replaced by their isotopic average, we can neglect the d terms. The e terms represent a temperature independent complex self-energy (provided there are states with v ˆ v 1 into which v `,q can scatter. Otherwise, the self-energy is real) which must be added to the anharmonic contributions. The self-energy effects just mentioned are speci®c of each

g`;q ˆ 2

d ln v`;q d ln V

…2†

The corresponding expression for the volume expansion is [12]: DV 1 j ˆ S "v g coth V0 2B0 V0 `;q `;q `;q 2

…3†

where B0 is the bulk modus, V0 the volume at T ˆ 0; and j ˆ "v `;q =kB T; with kB the Boltzmann constant. Eqs. (1) and (2), taken together, also yield a contribution to the shift of the frequency of a given phonon with increasing temperature. In order to evaluate it, one must take the change in volume implied by Eq. (3) and multiply it by the corresponding value of g `,q. For this purpose, either calculated or experimental values of (DV/V0)(T ) and g `,q are used. The harmonic lattice dynamics, corrected by thermal expansion effects, is referred to as the quasiharmonic approximation. Note that the contribution of thermal expansion to frequency shifts is very small at low T, a fact that results from one or two sign reversals of (DV/V0)(T ) at low T produced by negative values of g for TA phonons at the edge of the Brillouin Zone (BZ). Its temperature dependence is very ¯at at low T, compared with that of the a±c diagrams of Fig. 1. For T ! 0 Eq. (2) becomes: DV0 1 ˆ S "v g 2B0 V0 `;q `;q `;q V0

…4†

Eq. (4) represents the zero-point quantum renormalization of the volume which, for a large crystal, manifests itself as a truly macroscopic effect. Because of the dependence of v `,q on isotopic mass, Eq. (3) enables us to calculate the dependence of the volume on the isotopic mass which can be measured when crystals with different isotopic compositions are available [13]. The effect of the volume on the isotopic mass vanishes, according to Eq. (3), for T ! 1 (or, more precisely, for T q TD ; where TD is the Debye temperature). The phonon self-energies have been discussed in many theoretical papers since the early days of quantum mechanics. We mention here only a few of them [6,14± 18]. Most of these publications, however, are concerned with the theoretical formalism as expressed phenomenologically in terms of a few adjustable parameters, sometimes obtained from other independent experiments. The level of understanding reached for the processes involved has been

M. Cardona, T. Ruf / Solid State Communications 117 (2001) 201±212

203

larity in S , one can usually assume that D and G are constant, independent of v . In this case, Eq. (4) represents a Lorentzian pro®le. Otherwise rather complicated line shapes may be obtained (see e.g. CuCl [23] and GaP [24]). In a perfect crystal, the diagrams of Fig. 1a±c must conserve wavevector at their vertices. In the usual case in which the phonon under consideration has q ˆ 0; the two v 1 and v 2 phonons in Fig. 1a must have q 1 ˆ 2q2 ; and in Fig. 1c q1 1 q2 1 q3 ˆ 0: No such restriction applies to Fig. 1b. The self-energy is obtained by adding up all contributions which ful®ll these conditions. The temperature dependence appears when one multiplies each contribution by the corresponding Bose±Einstein statistical factors. In order to quantify the measured temperature dependences, it has become customary to assume that in a there is a dominant channel involving only two frequencies v 1 and v 2 (for S i the constraint v ˆ v 1 1 v2 must hold), whereas for c one assumes three dominant frequencies v 1, v 2, v 3 with v ˆ v1 1 v2 1 v3 : The corresponding statistical factors are:

Fig. 2. Temperature dependence of the FWHM of the LO and TO Raman phonons of GaAs. The points and crosses represent experimental data, the solid curve for G LO and the lower solid curve for G TO represent the results of ab initio calculations [8,9]. The upper solid curve for G TO represents those calculations slightly scaled so as to pass through the experimental curves. The dashed curves represent predictions based on the Klemens ansatz: Eq. (7) with A 4 ˆ 0: Replacing this ansatz by v1 ˆ 2v2 yields curves which agree with the ab initio calculations.

rather low until the advent of ab initio calculations based on the electronic band structure and density functional perturbation theory [4,8,9,12,19±21]. Experimental progress has not been very impressive either. Linewidths (FWHM G ˆ 22S i) are often affected by spectrometer resolution and, in the case of LO phonons, by the presence of free carriers. Frequency calibrations are often not suf®ciently precise and stable to measure reliably frequency shifts with temperature. Measurements are usually performed by means of ®rst-order Raman and ir-spectroscopy and therefore can only be used to investigate phonons near the center of the BZ. Recently, however, some measurements of self-energies have been performed by inelastic neutron scattering (INS) using germanium crystals of different isotopic compositions [22]. It follows from the discussion above that for uSu p v; the spectral function of a particular phonon can be written as: I `;q …v† ˆ

1 G…v† 2p ‰v 2 v `;q 2 D…v†Š2 1 ‰G…v†=2Š 2

…5†

where D…v† ˆ S r plus the thermal expansion contribution, and G ˆ 22S i : Except in cases where v `,q lies near a singu-

…1 1 n1 †…1 1 n2 † 2 n1 n2 ˆ 1 1 n1 1 n2

…6a†

…1 1 n1 †…1 1 n2 †…1 1 n3 † 2 n1 n2 n3

…6b†

In order to reduce the number of frequencies needed to ®t D(T ) and G (T ), measured on the basis of Eq. (5), Klemens [17] made the assumptions v 1 ˆ v2 ˆ v=2 for the third order and, correspondingly, v1 ˆ v2 ˆ v3 ˆ v=3 for the fourth-order terms. With these assumptions, G (T ) can be written as:        v v v G…T† ˆ …A 3 1 A4 † 1 2A3 n 1 3A4 n 1 n2B 2 3 3 …7† where n…x† ˆ ‰exp…"x†=…kT† 2 1Š21 : In Eq. (7) we have assumed that no isotopic disorder is present. Otherwise a temperature independent term related to Fig. 1e must be added. On the basis of the diagrams of Fig. 1, it is easy to see that for elemental crystals A3 / M 21 and A4 / M 23/2, where M is the average isotopic mass. Experimentalists often ®t their data for G (T ) (and even for D(T )!) with expressions of the type (6a) and (6b). When using the Klemens ansatz (Eq. (7)), third-order terms (A3) are often not suf®cient to represent the experimental results, whereas a reasonable ®t is obtained when A 4 ± 0 is included (obviously, the more parameters the better the ®t). This has been sometimes interpreted as evidence of fourth-order contributions to S [25,26]. However, when v 1 is not taken to be equal to v 2, and possibly even used as an adjustable parameter (while keeping v 2 ˆ v 2 v1 in the case of G ), many measured temperature dependences can be ®tted without having to invoke fourth-order terms [27]. For many tetrahedral semiconductors a reasonable ®t to G (T ) for the LO and TO phonons at G is obtained with thirdorder terms, of the type in Eq. 6a using v 1 ˆ 2v2 [27]. The existence of a dominant contribution to G for v 1 . 2v2 has

204

M. Cardona, T. Ruf / Solid State Communications 117 (2001) 201±212

Fig. 3. Temperature dependence of the Raman frequency of silicon [25,26]. The solid curve represents a ®t with the ansatz v1 ˆ 2v2 ˆ 2vR =3; including the thermal expansion effect [61]. The dotted asympotic line illustrates the construction described in the text to estimate the bare harmonic frequency v 0 from the linear extrapolation to T ! 0; (F. Widulle, to be published).

been con®rmed by ab initio calculations for Ge, Si [28], GaAs, and GaP [19], although for diamond [28], InP(TO), and AlAs (for both TO and LO) [19], v1 ˆ v2 seems to give a better approximation to the calculated G (T ). When trying to describe, in a similar way, the temperature dependence of D(T ), however, conceptual dif®culties remain: The frequencies v 1 and v 2 are now virtual frequencies which need not ful®ll v 1 1 v2 ˆ v: The various contributions to D(v ) can even reverse sign vs T because of the energy denominators in the second order perturbation expression for D(T ) (see, e.g. Fig. 6 of Ref. [19] for v (TO) in GaP). The contribution of the thermal expansion also has sign reversals at low T but, fortunately, the corresponding effects on D(T ) in this region are nearly negligible. While the ab initio calculation of D(T ) does not pose any more dif®culties than that of G (T ) (except for the term b of Fig. 1, see Refs. [8,9]), the experimental determination is more dif®cult since D(T ) is a frequency renormalization and, in principle, only the renormalized frequencies can be measured, not the bare ones. By postulating a given functional dependence for D(T ), of the type displayed in Eqs. (6a) and (6b), it is, however, possible to ®t the experimental data and to extract, from this ®t, the zero-point renormalization. A simple construction is based on the fact that Eq. (6a) becomes linear in T at high temperatures. If terms (6b) are negligible, the extrapolation of the linear high-temperature ®t to T ˆ 0 yields the unrenormalized phonon frequency. With this construction, a renormalization D…0† .

26 cm 21 is obtained for silicon (see Fig. 3). A similar construction for diamond yields D…0† . 220 cm21 [29]. One can ask how is it possible to obtain bare frequencies from experimental data. The answer is that it is not possible to do it exactly [30]. The estimate above was made on the basis of a crude assumption on the functional dependence of D(T ), namely that it is due only to third-order anharmonicity.

2. Stable isotopes in physics The constituent atoms of most semiconductors discussed here possess several stable isotopes which are present, with speci®c abundances, in the natural materials (Al, P, and As have only one stable isotope). The separation of the various isotopes of a given element is a costly process based on either mechanical (diffusion-ultracentrifugation) or electromagnetic techniques which were developed for separating uranium isotopes for nuclear warfare and as reactor fuel. In the past ten years, several institutions that ran isotope separators have made efforts to separate other elements and to ®nd scienti®c as well as industrial applications for their highly enriched stable isotopes. Most isotopes in the market come these days either from Oak Ridge National Laboratory or from a number of Russian centers, including the Kurchatov Institute. Macroscopic quantities of these materials, suf®cient to grow good size single crystals,

M. Cardona, T. Ruf / Solid State Communications 117 (2001) 201±212

205

3. Isotope effect on the self-energies of phonons in elemental semiconductors

Fig. 4. Intrinsic phonon linewidth of isotopic Ge vs. the average Ê . The ab mass, measured at 10 K, with a laser excitation of 6471 A initio linewidth reported in Ref. [42] for isotopically pure Ge with M ˆ 72:6 is indicated by the circle [42].

have become affordable to the budgets of solid state physicists [31]. These crystals have made possible a number of basic experiments which were not possible before. We shall discuss next only experiments involving the physics of phonons. These experiments are based on the differences in isotopic masses. Isotopes have also different nuclear structures which can be used for doping semiconductors by nuclear transmutation [32]. Large 76Ge crystals are being used for neutrino physics in conjunction with double-beta-decay experiments [33]. 13C has been used to determine the width of a nuclear resonance around 3.5 MeV [34] and, as a by-product, this resonance is being used for determining the 13C concentration in arti®cial, isotopically tailored diamonds [35]. The 13.3 keV nuclear resonance of 73Ge has been suggested as a means of monochromatizing synchrotron radiation for MoÈssbauer spectroscopy experiments [36]. In this article, we shall concentrate on the applications of isotopically modi®ed semiconductor crystals in phonon physics. Some of these applications are straightforward. For instance, the phonon dispersion relations of crystals containing natural Cd cannot be measured by INS because of the strong absorption cross-section for slow neutrons of 113 Cd, an isotope with a natural abundance of 12% [37]. Phonon dispersion relations have, nevertheless, been determined by INS using crystals with less than 1% of 113 Cd, see Ref. [38] for CdTe, Ref. [39] for CdS and Ref. [40] for CdSe. These measurements are primarily concerned with the harmonic aspects of phonons. We shall discuss next the information on anharmonic effects and mass-disorder self-energies that can be obtained from semiconducting crystals with tailor-made isotopic composition.

We shall discuss here the effects of isotopic composition on the phonon self-energies of the four elemental semiconductors with diamond structure: diamond, silicon, germanium and gray tin (a zero-gap semiconductor). Experimental data are available for the four materials in single crystal form diamond [41], silicon [41a], germanium [42] and gray tin [43]. Two types of self-energies S (v ) will be considered: zero-point …T < 0† anharmonic self-energies and isotopic disorder contributions. In each case we shall discuss the real and the imaginary parts of S (v ). Most of the experimental data have been obtained by optical techniques and therefore they refer to optical phonons at the center of the BZ. A few data covering also phonons at general points of the BZ have been obtained by INS (see Ref. [22] for germanium) and by Raman scattering of strongly monochromatized, synchrotron generated X-rays for diamond [44]. Some information on phonons off the center of the BZ has also been obtained by means of two-phonon ir absorption for Ge [45] and by phonon-aided cathodoluminescence for diamond [46]. 3.1. Anharmonic self-energies of Raman phonons We discuss ®rst the imaginary part of S (v ) for the threefold degenerate optical phonons at the center of the BZ …k . 0† To lowest order in perturbation theory, the so-called ®rst Born approximation, the diagram of Fig. 1a leads to the zero-point imaginary part of the self-energy: 2S i …v† ˆ uV3 u2 r2 …v† ˆ G an =2

…8†

where V3 is the average anharmonic coupling constant and r 2(v ) the density of two-phonon states with the restrictions of k conservation and v ˆ v1 1 v2 : The three phonons attached to each of the two vertices of Fig. 1a contribute a factor ku 2l 3 to S i(v ). r 2(v ) scales like v 21, i.e. like M 1/2, when changing the average isotopic mass M. Hence, a dependence of G an / M 21 results from Eq. (8). This dependence has been con®rmed for four nearly isotopically pure samples of germanium ( 70Ge, 73Ge, 74Ge, 76Ge) [42], as illustrated in Fig. 4. In this work, great care was taken to avoid effects of the small laser penetration depth and surface conditions on the linewidth. Such care is not as important for diamond [41] because of the much larger penetration depth. For a-Sn, epitaxially grown samples were used [43] whose surfaces were of high quality. The anharmonic S r(v ) which can be obtained through a Hilbert±Kramers±Kronig transformation of Eq. (8) is:

S r …v† ˆ

2uV3 u2 Z1 v 0 r2 …v 0 † dv 0 2 02 p 0 v 2v

…9†

where we have assumed, for simplicity, that V3 is independent of frequency. The zero-point S r(v ) also varies like M 21 on the basis of Fig. 1a and b. Therefore, it is in

206

M. Cardona, T. Ruf / Solid State Communications 117 (2001) 201±212

Fig. 5. Raman spectra of several silicon crystals with different isotopic compositions. The vertical bars denote the VCA behavior. ªplº denotes a plasma line used for calibration. The isotopically mixed sample exhibits an additional frequency shift (real part of the phonon self-energy) of (11.15 cm 21. However, the additional line broadening ( , imaginary part of the self-energy) amounts to only 0.05 cm 21 (FWHM) [40].

principle possible to determine S r(v ) by ®tting the frequencies measured for several pure isotopes with the equation:

v ˆ AM 21=2 2 BM 21

…10†

The second term in the rhs of Eq. (10) represents S r. It is usually negative because in Eq. (9) the terms with v 0 . v dominate. Unfortunately, because of the small range of isotopic masses available, B can only be determined rather inaccurately: BM 21 has been found to be 4 ^ 1 cm 11 for silicon [40] and 16 ^ 4 cm21 for diamond [41]. These values of the anharmonic frequency renormalization are reasonably close to those obtained by linear extrapolation of v (T ) to T ˆ 0 (see Section 1 and Fig. 3). 3.2. Self-energies of Raman phonons due to isotopic disorder The isotopic disorder is characterized by the moments of the mass ¯uctuations: X  Mi 2 M m gm ˆ xi …11† M i where xi is the concentration of isotope i, Mi its mass and M the average mass. The ®rst Born approximation yields for the corresponding self-energies [42]:

S is;r …v† ˆ g2

v2 Z1 v 0 r1 …v 0 † dv 0 12 0 v2 2 v 02

2S is;i …v† ˆ g2

pv2 r …v† 24 1

…12a†

…12b†

Fig. 6. Raman spectra of diamond crystals with three different isotopic compositions. The vertical bars denote the VCA behavior. ªplº labels plasma lines used for calibration. The isotopically mixed sample exhibits an additional frequency shift (real part of the phonon self-energy) of 5 cm 21 and a strong additional line broadening of about 6 cm 21 FWHM [41].

where r 1(v ) is the density of one-phonon states normalized to six (six modes per unit cell). Fig. 5 indicates that S is,r is positive, a result which agrees with Eqs. (12a) and (12b) since r 1 …v† ± 0 only for v 0 , v . According to Fig. 5, S is;r ˆ 11:15 cm21 for 28Si0.5 30Si0.5 [40]. Similar values are found for 70Ge0.5 76Ge0.5 (1.06 cm 21) [42] while for 112 Sn0.5 124Sn0.5 S is;r ˆ 1:8 cm21 is found [43]. For diamond composed in equal parts of 12C and 13C, S is;r ˆ 5 cm21 is found (see Fig. 6). To the naked eye, the three Lorentzian spectra of Fig. 5 have the same width. A more careful analysis, however, indicates that the 0.5±0.5 sample has a slightly larger width (a contribution 2S is;i ˆ 0:025 cm21 ) than that which would correspond to isotopically pure 29Si, whereas 2S is;i ˆ 0:03 cm21 is found for 70Ge0.5 76Ge0.5 and 2S is;i ˆ 0:02 cm21 for 112Sn0.5 124Sn0.5. These very small values of 2S is,i, compared with S is;r ù 1:5 cm21 ; have been attributed to the vanishing of r 1(v ) at the Raman frequency, which is the maximum frequency of the phonon spectrum for Si, Ge, and a-Sn [47]. Isotopic ¯uctuations are static and therefore induce only elastic scattering, i.e. scattering with energy conservation. At the maximum frequency no states are available to scatter into and therefore 2S is,i, vanishes within the ®rst Born approximation. What is then the origin of the small but nevertheless nonnegligible isotopic broadening observed for Si, Ge, and a-Sn? It has been attributed [47] to the combined effect of anharmonicity and isotopic disorder. The anharmonicity broadened r 1(v ) becomes non-zero at the Raman frequency, a fact that leads to a non-zero 2S is,i, when using Eq. (12b). The small measured isotope broadening can be quantitatively explained in this manner.

M. Cardona, T. Ruf / Solid State Communications 117 (2001) 201±212

207

frequency of this peak, the denominator of Eq. (12a) is rather small, a fact that may render second-order perturbation theory inaccurate: Higher-order perturbation terms may be needed. The self-consistent method known as the coherent potential approximation (CPA) is often used [43,48,49]. It is also possible to add to the second-order terms of Eqs. (12a) and (12b) third-order terms involving the threefold scattering by an isotopic mass at a given site. In the case of a mixture of two isotopes, with concentrations x and (1 2 x), respectively, the corresponding second- and third-order contributions to the self-energy have the following dependence on x second order , g 2 , x…1 2 x†

…13a†

third order , g3 , x…1 2 x†…0:5 2 x†

…13b†

Whereas Eq. (13a) represents a parabola, symmetric with respect to x ˆ 0:5; Eq. (13b) contributes an antisymmetric term which displaces the parabola towards the larger average values of the mass. The corresponding asymmetry in S (x) has been observed for diamond [41,48] and silicon [60]. It is also reproduced by a CPA calculation which includes third and higher order terms of a single-site isotopic perturbation [62]. 3.3. Isotopic-disorder-induced Raman scattering Let us consider the spectrum of a Raman phonon as represented by Eq. (5). G (v ) is very small at the Raman frequency v R but, according to Eq. (12b), can become rather large at the maximum of r 1(v ). Around this maximum v M, the spectral function (Eq. (5)) becomes: I…v† . Fig. 7. Isotope-disorder contributions to phonon frequencies (in meV, 1 meV ˆ 8 cm21 ) determined in diamond for several phonons with zero and non-zero wavevectors. The open circles and squares are cathodoluminescence data; the open lozenges in (a) are Raman data; the solid lines are from a CPA calculation based on ab initio lattice dynamics [46].

Fig. 6 shows Raman spectra measured for diamond with the isotopic composition 12C12x 13Cx (x ˆ 0; 1, 0.47). Contrary to the case of the isostructural Si, Ge, and a-Sn, the linewidth for x ˆ 0:47 appears to be considerably broader …2S is;i ˆ 3 cm21 † than that of natural diamond. This can be accounted for by the fact that for diamond the highest phonon frequency does not occur exactly at k ˆ 0 but slightly off this point of the BZ. Under these conditions r 1(v ) is ®nite at the Raman frequency and a sizeable value of 2S is,i, obtains [41,48]. Eqs. (12a) and (12b) are based on second-order perturbation theory. In the materials under consideration the TO phonon bands are rather ¯at, a fact that results in a strong peak in r 1(v 0 ) slightly below the Raman frequency. At the

G…v† g2 v2 . r1 …v† 2 2p…v R 2 vM † 24…vR 2 vM †2

…14†

Eq. (14) indicates that the isotopic disorder, as represented by g2, should be observable as a band with the shape of r 1(v ), centered at the low frequency side of v R. It can be observed in natural crystals of Ge [50], disordered Si [60], a-Sn [43], and also in arti®cial 12C0.5 13C0.5 diamonds [41]. For the maximum possible values of g2 it reaches maxima of the scattering ef®ciency at vM ; .1023 of the maximum at v R. The perturbation expansion at v . vM involves very small energy denominators and therefore Eq. (14) can only give a rough approximation of the corresponding lineshape and intensity. Better quantitative results are obtained with the CPA [41,43,50]. 3.4. Dispersion of phonon self-energies: elemental semiconductors So far we have emphasized the self-energies of Raman phonons, although in Section 3.3 we have discussed the Raman spectra of the peak in the density of states of TO-phonons as re¯ected in the G (v ) of the Raman phonon. This peak corresponds mainly to TO-phonons at the edge of

208

M. Cardona, T. Ruf / Solid State Communications 117 (2001) 201±212

Fig. 8. Inelastic X-ray scattering spectra of natural and isotopically mixed diamond at the L-point of the BZ. An additional line broadening of 0.9 meV and a frequency shift of 0.9 meV (7 cm 21) due to the isotope disorder are clearly observed [44].

the BZ. As already mentioned, some experimental information is available for phonons at general points of the BZ, concerning both, anharmonic [51] and isotope disorder contributions [22,45,46]. We shall discuss here recent work performed on 12C0.53 13 C0.47 diamond by means of cathodoluminescence [46] and X-ray Raman scattering [44]. The cathodoluminescence work is based on the fact that the energy gap of diamond is indirect, involving valence states at the BZ-center and conduction states close to the X point [k ˆ …2p=a†…0:76; 0; 0†; 0.76X for short]. Upon carrier excitation across the gap (with an electron beam), light emission becomes possible through phonon-aided electron-hole recombination. The phonons involved have k ˆ 0.76X. It is thus possible to observe exciton luminescence peaks corresponding to the following phonons: LO(0.76X) at 1284 cm 21, TO(0.76X) at 1095 cm 21 and TA(0.76X) at 674 cm 21. The disorder-induced frequency shifts of these phonons are shown in Fig. 7, together with Raman and other optical data for the Raman phonon (labeled LO(G )). The latter shows a maximum of S is,r equal to 15 cm 21. The LO and TO phonons at k ˆ 0.76X show self-energies S is,r of opposite signs. This fact can be attributed to the existence of a strong band in r 1(v ) between LO(0.76X) and TO(0.76X). According to Eq. (12a), this band contributes a positive term to the S is,r of the LO(0.76X) phonon and a negative one for TO(0.76X). The corresponding effect for the TA(0.76X) phonon is small and negative, as expected on the basis of the same equation. We show in Fig. 8 Raman spectra obtained with monochromatized X-rays for natural diamond (nearly pure 12C) and for 12C0.53 13C0.47 diamond. These measurements were performed with the synchrotron radiation emitted by the European Synchrotron Radiation Source (ESRF). It is possible to see in Fig. 8, even with the naked eye, an increase in the width of the LO(L) phonon for the isotopically disordered diamond. This increase corresponds to 2S is;i ˆ 3:5 cm21 ; slightly larger than that shown in Fig. 6 for the G-phonons.

4. Phonon self-energies in zincblende-type semiconductors

Fig. 9. TO Raman spectra of ZnSe with several isotopic compositions, measured at 2 K. The isotope compositions are given in the form [Zn]/[Se]. ªdisº stands for 64-68-Zn and 76-80-Se in 50/50 mixtures, respectively. Note that the linewidth of natural ZnSe is largely due to isotope disorder. The importance of the partial density of phonon states shows up in the different widths observed for the samples with only Zn or Se disorder [52].

The investigations described in Section 3 were con®ned to the four elemental semiconductors with diamond structure. Their zincblende-type counterparts exhibit a much larger variety which includes III±V, II±VI and I±VII compounds, in which both, the cation and the anion can be varied. It is thus possible to choose the material so as to exhibit a variety of interesting self-energy effects. The splitting of the Raman phonons into LO and TO components near the center of the BZ also adds an additional degree of freedom to the family under consideration. For speci®c materials (e.g. GaP, CuCl, CuBr)

M. Cardona, T. Ruf / Solid State Communications 117 (2001) 201±212

209

Fig. 10. Raman spectra of the TO phonon of: (a) 69GaP; (b) nat GaP ( 69Ga0.6 71Ga0.4P), (c) natGaP( 69Ga0.4 71Ga0.6P), and (d) 71GaP at T ˆ 6 K: The solid lines are ®ts to the experimental data (symbols) using the indicated two-phonon density of states r 2(v ), (dashed lines). The vertical lines at v TO represent the d -function-peaks of the harmonic TO phonons prior to renormalization [59].

the optical phonons (either TO or LO) fall onto singularities in the two-phonon density of states r 2(v ) (see Eqs. (8) and (9)). In these cases, strong anharmonic coupling between the Raman phonon and the two-phonon states (e.g. a Fermi resonance) takes place, leading to complicated non-Lorentzian lineshapes which depend strongly on isotopic composition. 4.1. The simple case of non-singular S (v ) nearly independent of frequency: natural ZnSe The LO±TO splitting of the Raman phonons implies different values of both contributions, anharmonic and mass-disorder induced for each of the split phonons. For most of these materials, the LO(G ) phonons have the highest frequency of all phonon modes and, therefore, S is,r is positive and large, whereas 2S is,i, is very small (see Section 3.2). On the other hand, the TO(G ) frequency lies in the

middle of the continuum of optical phonons, a fact that leads to large values of 2S is,i. This is illustrated in Fig. 9 for the TO phonons of ZnSe. In this ®gure, the Raman spectra of several nearly isotopically pure ZnSe crystals are shown. Their FWHM ( ˆ 22S is,i) is small ( < 0.4 cm 21). The TO spectra of the four isotopically disordered samples, also shown in Fig. 9, are considerably broader, the broadest being that of the crystal grown out of elements with the natural isotopic abundance (FWHM ˆ 1.8 cm 21). In order to interpret these results one should use two different mass variance parameters g2, one for the cation and another for the anion [52]. While in the case of natural ZnSe one can approximately assume that S is,i is independent of v , leading to a symmetric Lorentzian shape, for the dis80 and 68dis, the spectra shown in Fig. 9 are asymmetric. This asymmetry can be explained on the basis of a frequency dependence of S is (v ) [52], of the type discussed in the following section.

210

M. Cardona, T. Ruf / Solid State Communications 117 (2001) 201±212

Fig. 11. (a) Experimental, and (b) theoretical Raman spectra of the anomalous TO structure for several isotopically modi®ed CuCl samples. The spectra of samples with the same copper mass are grouped together. Within these groups the reduced mass m decreases from the bottom to the top. The TO(g ) line does not shift according to changes in m but with changes of the copper mass. The broad TO(b ) line shifts according to changes in m . The calculated Raman spectra were obtained in the Fermi resonance scenario (Eq. (5) with frequency dependent self-energy). The peak amplitudes are normalized with respect to TO(g ) [23].

4.2. Effects of singularities in the two-phonon density of states

Fig. 12. Fermi resonance model of the TO anomaly in CuCl. (a) The solid line represents the imaginary part of the phonon self-energy G (v )/2. The dashed line is the Kramers±Kronig related real part ~ j††: (b) D(v ). The straight dotted line represents …v 2 v TO …0; Spectrum of natCu natCl simulated with Eqs. (5), (8) and (9), using uV3 u2 ˆ 70 cm22 ; the spectral intensity is given in arbitrary units [23].

It was noticed early that the TO phonons of ªnaturalº GaP are anomalously broad [53]. This anomaly was attributed to the existence of a singularity in the density of states r 2(v ), corresponding to one TA and one LA phonon at the edge of the BZ, nearly degenerate with the TO(G ) frequency [53]. This anomaly could be removed by application of hydrostatic pressure [53]. In order to clarify the nature of the phenomenon, Raman measurements at 6 K for several isotopic compositions of the Ga (phosphorus has only one stable isotope) have been recently performed [24]. The results are shown in Fig. 10, together with the calculated r 2 (v ) which displays a square root singularity close to v TO. 2S i(v ) has a shape similar to that of r 2(v ). Replacement of S (v ) and its Hilbert transform S r(v ) into Eq. (5) leads to the complex lineshapes drawn through the experimental points in Fig. 10. Note that for 71GaP the singularity has almost been removed from the region of the main TO spectrum and a nearly Lorentzian lineshape (except for the weak shoulder labeled B) is recovered. Recently, a very detailed analysis of the pressure dependence of the TO Raman phonon has been performed for 69GaP [54]. It con®rms the picture just mentioned: Upon application of

M. Cardona, T. Ruf / Solid State Communications 117 (2001) 201±212

pressure the TO(G ) mode moves considerably up in frequency while the two-phonon TA 1 LA singularity moves much less because of the negative GruÈneisen parameter of the TA phonons. Consequently, TO(G ) moves away from the singularity in r 2(v ) and sharpens. The most spectacular effects related to a singularity in r 2(v ) have been observed for the TO(G ) phonons of CuCl, a material isoelectronic with GaP. The effects on the TO phonons of CuCl are similar to those observed for GaP, except stronger. They are illustrated in Fig. 11 for various isotopic combinations [23]. One may naively guess that the sharp peak between 170 and 175 cm 21, labeled TO (g ), represents the TO (G ) phonon. This assumption is, however, contradicted by the shift observed for this peak when the isotopic masses are varied: Instead of behaving like m 21/2 (where m is the reduced mass given 21 21 21=2 by m 21 ˆ MCu 1 MCl ) it varies like MCu : However, the 21 broad peak centered at ,155 cm , labeled TO (b ), contains most of the spectral strength and shifts like m 21/2. It thus seems to be more closely related to the TO(G ) mode. These anomalies are easily understood when considering the singular dependence of S (v ) (Fig. 12), similar to that found for GaP. Eq. (5) leads to a sharp peak for v 2 v 0 2 S r …v† ˆ 0; provided S i(v ) is small: such is the case for the TO(g ) peak. Note in Fig. 12 that when isotope masses are varied, the corresponding change in the bare v (TO) frequency changes the zero of v 2 v 0 2 S r …v† very little, whereas 21=2 the shift in the broad two-phonon band, related to MCu ; 21=2 changes the zero in a way nearly proportional to MCu ; as observed experimentally. In the rhs of Fig. 11 we display the spectral functions calculated for the various isotopic compositions using Eq. (5) and the r 2(v ) derived from lattice dynamical calculations. The model proposed [55] explains the details of the observed TO spectra and therefore other possible models can be discarded [56]. Recently [57] an anomaly similar to that just discussed for the TO(G ) phonon of CuCl has been found for the LO(G ) phonon of CuBr. The shift of this anomaly from TO(G ) to LO(G ) is easy to understand. The increase in the anion mass when replacing chlorine by bromine lowers the TO(G ) mode and brings the higher LO(G ) mode into near degeneracy with the TA 1 LA singularity.

211

discovered for the TO(G ) phonons of CuCl may also be found. It is hoped that high resolution INS and Raman scattering with monochromatic X-rays (of the synchrotron radiation variety), will add in the future information concerning general points of the BZ. Isotopic substitution has also been useful for investigating the phonon dispersion relations of Cd-based semiconductors [38±40]. In the case of wurtzite-type materials, isotopic substitution has helped to determine phonon eigenvectors that are not ®xed by symmetry [58]. For silicon carbide, it has been used to determine the eigenvectors of the zincblende modi®cation along the [111] direction [59]. Acknowledgements We thankfully acknowledge the collaboration of many colleagues at the Max-Planck-Institut and worldwide, who have participated in the research mentioned above. Representative among them we mention: T.R. Anthony, A. Debernardi, E.E. Haller, J. Kuhl, R. Lauck, C.T. Lin, V.I. Ozhogin, A. Goebel, F. Widulle and J. Zegenhagen. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

5. Conclusions

[14] [15]

We have shown that the availability of crystals with different isotopic compositions has been very helpful for unraveling the mechanisms which determine anharmonic and disorder-induced phonon frequency renormalizations. They have also helped to clarify anomalous lineshapes related to singularities in the corresponding self-energies. Most of the work has been performed by Raman spectroscopy, with emphasis on Raman phonons. The detailed information so obtained is still lacking for most of the other phonon states, for which anomalies similar to that

[16] [17] [18] [19] [20] [21] [22] [23] [24]

A. Einstein, Ann. Phys. (Leipzig) 22 (1907) 180. P. Debye, Ann. Phys. (Leipzig) 39 (1912) 789. C.T. Walker, G.A. Slack, Am. J. Phys. 38 (1970) 1380. S. Baroni, S. de Gironcoli, A. Dal Corso, Rev. Mod. Phys. (2000) (in press). E. GruÈneisen, Handb. der Physik 10 (1926) 22. M. Born, K. Huang, The Dynamical Theory of Crystal Lattices, Oxford University Press, Oxford, 1954. M. Cardona, Raman scattering in high-Tc superconductors, in: W.H. Weber, R. Merlin (Eds.), Raman Scattering in Materials Science, Springer, Heidelberg, 2000, p. 151. A. Debernardi, Solid State Commun. 113 (2000) 1. G. Lang, et al., Phys. Rev. B 59 (1999) 6182. L. Bergman, et al., J. Appl. Phys. 71 (1997) 2157. T. Ruf, et al., Solid State Commun. 115 (2000) 243. A. Debernardi, M. Cardona, Phys. Rev. B 54 (1996) 11305. A. Kazimirov, J. Zegenhagen, M. Cardona, Science 282 (1998) 930. R.A. Cowley, Rep. Prog. Phys. 31 (1968) 123. T.H.K. Barron, M.L. Klein, Perturbation theory of anharmonic crystals, in: G.K. Horton, A.A. Maradudin (Eds.), Dynamical Properties of Solids, Vol. 1, North Holland, Amsterdam, 1974, p. 391. R.A. Cowley, J. Phys. (Paris) 26 (1965) 659. P.G. Klemens, Phys. Rev. 148 (1966) 845. A.A. Maradudin, S. Califano, Phys. Rev. B 48 (1993) 12628. A. Debernardi, Phys. Rev. B 57 (1998) 12847. G. Lang, et al., Phys. Rev. B 59 (1999) 6182. C.Z. Wang, K.M. Ho, Phys. Rev. B 42 (1990) 11276. A. GoÈbel, et al., Phys. Rev. B 58 (1998) 10510. A. GoÈbel, et al., Phys. Rev. B 56 (1997) 210. F. Widulle, et al., Phys. Rev. Lett. 82 (1999) 5281.

212

M. Cardona, T. Ruf / Solid State Communications 117 (2001) 201±212

[25] M. Balkanski, R.F. Wallis, E. Haro, Phys. Rev. B 28 (1983) 1928. [26] E. Haro, et al., Phys. Rev. B 34 (1986) 5358. [27] J. MeneÂndez, M. Cardona, Phys. Rev. B 29 (1984) 2051. [28] A. Debernardi, S. Baroni, E. Molinari, Phys. Rev. Lett. 75 (1995) 1819. [29] H. Herchen, M.A. Cappelli, Phys. Rev. B 43 (1991) 11740. [30] P.B. Allen, Philos. Mag. 70 (1994) 527. [31] E.E. Haller, Proceedings of the 25th International Conference on Physics of Semiconductors, Osaka, Springer, Heidelberg, 2000 (in press). [32] M. SchnoÈller, IEEE Trans. Electron Devices (1974) 313. [33] H.V. Klapdor-Kleingrothaus, Nucl. Phys. B 28A (1992) 207. [34] R. Moreh, et al., Phys. Rev. C 48 (1993) 2625. [35] O. Beck, et al., J. Appl. Phys. 83 (1998) 5484. [36] A.I. Chumakov, et al., Phys. Rev. Lett. 71 (1993) 2489. [37] J. Emsley, The Elements, Clarendon Press, Oxford, 1991. [38] J.M. Rowe, et al., Phys. Rev. B 10 (1974) 671. [39] A. Debernardi, et al., Solid State Commun. 103 (1997) 297. [40] F. Widulle, et al., Physica B 263±264 (1999) 448. [41] J. Spitzer, et al., Solid State Commun. 88 (1993) 509. [41a] F. Widulle, et al., Physica B 263±264 (1999) 381. [42] J.M. Zhang, et al., Phys. Rev. B 57 (1998) 1348. [43] D. Wang, et al., Phys. Rev. B 56 (1997) 13167. [44] T. Ruf, et al., Proceedings of the 25th ICPS, Osaka, Springer, Heidelberg, 2000 (in press).

[45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57]

[58] [59] [60]

[61] [62]

H.D. Fuchs, Phys. Rev. B 45 (1992) 4065. T. Ruf, et al., Solid State Commun. 105 (1998) 311. H.D. Fuchs, et al., Phys. Rev. B 44 (1991) 8633. K.C. Hass, et al., Phys. Rev. B 45 (1992) 7171. R.J. Elliott, in: M.F. Thorpe (Ed.), Excitations in Disordered Systems, Plenum, New York, 1982, p. 3. H.D. Fuchs, et al., Phys. Rev. Lett. 70 (1993) 1715. C.Z. Wang, C.T. Chan, K.M. Ho, Phys. Rev. B 40 (1989) 3390. A. GoÈbel, et al., Phys. Rev. B 59 (1999) 2749. B.A. Weinstein, Solid State Commun. 20 (1976) 999. S. Ves et al., Proc. Ninth Int. Conf. on High Pressure in Semiconductors, Sapporo 2000 (Phys. Status Solidi B (2001)). M. Krautzman, et al., Phys. Rev. Lett. 33 (1974) 528. M. Cardona, et al., Phys. Rev. Lett. 84 (2000) 4511. J. Serrano, Proceedings of the 25th International Conference on Physics of Superconductors, Osaka, Springer, Heidelberg, 2000 (in press). J.M. Zhang, et al., Phys. Rev. B 56 (1997) 14399. F. Widulle, et al., Phys. Rev. Lett. 82 (1999) 3089. J. Widulle, et al., Proceedings of the 25th International Conference on Physics of Superconductors, Osaka, Springer, Heidelberg, 2000 (in press). P. Pavone, S. Baroni, Solid State Commun. 90 (1994) 295. F. Widulle, to be published.