Physics Laboratory III, National Technical University, Zografou Campus, Athens ... de Physique des Solides, UniversitÃ© Pierre et Marie Curie, 4, place Jussieu,.
J.
Physique
48
(1987)
239245
239
FÉVRIER 1987,
Classification Physics Abstracts 63.20R 78.30 
Anharmonic effects in Raman spectroscopy Y. S.
Raptis,
Mg2X (X
=
Si, Ge, Sn) compounds studied by
G. A. Kourouklis, E. Anastassakis, E. HaroPoniatowski
(*)
and M. Balkanski
(*)
Physics Laboratory III, National Technical University, Zografou Campus, Athens 15773, Greece 13(*) Laboratoire de Physique des Solides, Université Pierre et Marie Curie, 4, place Jussieu, 75230 Paris Cedex 05, France
(Requ
le 3 juillet 1986,
accept6
le 3 octobre
1986)
Le spectre Raman du premier ordre des composés Mg2X (X Si, Ge, Sn) a été mesuré en fonction de la température et de la pression. L’effet de la dilatation (effet implicite) sur le déplacement en fréquence induit par la température est calculé et comparé à l’effet de l’anharmonicité du réseau (effet explicite). Il est observé que l’effet explicite est plus important que l’implicite. Les valeurs de la fraction implicite (rapport de l’effet de dilatation à l’effet total) sont inférieures à 0,5 pour les trois composés et dépendent de la température. Ces valeurs sont consistantes avec le caractère covalent prédominant dans ces matériaux.
Résumé.
Abstract.
=
2014
Measurements of the firstorder Raman scattering of Mg2X (X Si, Ge, Sn) as a function of temperature and pressure are presented. The volume contribution (implicit effect) to the frequency shift with temperature is computed and compared to the latticeanharmonicity contribution (explicit effect). The latter 2014
=
contribution turns out to be larger than the former. The values of the implicit fraction (ratio of the volume to the total effect) for all three materials are found to be smaller than 0.5 and to depend on temperature. These findings are indicative of a predominantly covalent bonding in these materials.
1. Introduction.
Mg2X (X
=
Si, Ge, Sn)
and subtracted from the observed (total) shift. This allows us to obtain quantitative information about the anharmonic contribution (explicit effect) to the shift. Both contributions turn out to be important although the latter is more prominent. This is consistent with covalent bonding, as inferred from the values of the implicit fraction q (ratio of the volume to the total effect) [5]. The values of q are found to be smaller than 0.5 and to decrease at both high and low temperatures. Finally, the increase in the phonon linewidth with temperature is attributed to anharmonic interactions between the F2g mode and other phonons of the lattice. The results can be accounted for by (mainly) three and four phonon anharmonic interactions.
computed
are
smallgap
semiconductors
which
crystallize in the antifluorite structure. The primitive cell contains one Mg2X molecule with the two Mg atoms symmetrically located about the X atom. The point group is oh and there are two triplydegenerate longwavelength optical phonons with symmetries F2g (Raman active, IR inactive). The Raman spectra of all three materials have been studied extensively in the past [1, 2] including investigations under hydrostatic pressure (up to 0.8 GPa) [3] and uniaxial stress. [4] (for X Si, Sn only). =
In this work we report the results of firstorder Raman experiments for the three compounds under variable temperature T (4800 K) and hydrostatic pressure P (06.0 GPa). The Gruneisen mode parameters are obtained from the pressure measurements and are compared with the results from the uniaxial stress work [4]. The volume contribution (implicit effect) to the mode frequency shift with temperature is then
2.
Experimental details
and results.
The Raman spectra under hydrostatic pressure at T = 300 K were recorded with a double monochromator (SPEX) operating with holographically ruled gratings at a bandpass of  2.8 cm1. An RCA31034 cooled
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01987004802023900
240
photomultiplier operating in the photon counting mode used to detect the signal, which was then analysed with a LeCroy 3 500 multichannel analyser. The 488
was
and 514.5
nm
Ar+ laser lines
were
used for excitation
at a power level of  500 mW. The
hydrostatic pressure was generated by a gasketed diamond anvil cell [6] (DAC). The pressure was monitored using the Ruby fluorescence technique [7]. Typical spectra for various pressures are shown in figure 1. Raman spectra were difficult to record at pressures higher than 4.5 GPa for Mg2Si and 3.0 GPa for Mg2Ge. In the case of M92Sn it was not possible to observe meaningful spectra under pressure due to the very low signaltonoise ratio. After repeated loadings of the DAC, we were able to obtain a few spectra at pressures lower than 1.0 GPa, from which the slope (ðCù / ðP) T was determined with lesser, but acceptable, accuracy.
pressure phase is hexagonal (similar to that of N’2S’) and coexists with the antifluorite structure for a wide range of pressures. We were unable to experimentally observe Raman Spectra from the high pressure phase of Mg2Si.
Figure 2 shows the phonon frequency as a function of pressure. The solid line represents a linear fit to the experimental points, from which the slope ( all) / ap) T was obtained. The slopes and the room temperature values for the frequencies (at P 0) of the F2g phonons of the three materials are listed in table I. The values of the frequencies are in good agreement with those in the literature [1, 2]. No measurable changes in the phonon linewidths were observed as a function of pressure. =
Fig.
2.

pressure for
Observed
MgzSi
function of represent linear fits.
phonon frequencies
and
Mg2Ge.
Lines
as a
The variable temperature measurements were recorded with a Coderg double monochromator (PHI) equipped with holographically ruled gratings and with variable bandpass (1 to 3 cm 1). An RCA 31034 cooled photomultiplier was used for detection operating in the
mode. The 488 and 514.5 nm Ar+ lines were used for excitation at power levels of about 200 mW. For the low temperature measurements (4300 K) a liquidhelium cryostat was used. The temperature was monitored with a calibrated Si diode (Lake. Shore Cryotronics, Inc.) and was controlled to an accuracy of ± 1 K. For the high temperature measurements the samples were placed in an evacuated oven [9]. The temperature was measured with a chromelalumel thermocouple attached to the holder close to the sample. An independent measurement of the temperature was performed by recording the Raman spectrum of a small Si chip placed near the sample. The temperature was deduced by comparison of the frequency and linewidth of the firstorder Raman spectrum of Si to published data [9]. The temperatures obtained by both methods agreed within 5 K, and this sets the limit of accuracy for all present high temperature
photon counting
Typical firstorder Raman spectra of Mg2Si and Fig. 1. Mg2Ge for three different pressures at 300 K. Excitation wavelength is 488 nm at a power of  500 mW. 
It should be noticed from figure 1 that the Raman intensity for Mg2Si and Mg2Ge decreases with increas
ing pressure and gradually vanishes at pressures higher than 6.0 GPa and 4.0 GPa respectively. This may be attributed to the onset of a structural phase transition which has been observed in radiographic studies [8]. In fact it has been established from the latter studies that this transition sets in at about 6.0 GPa for Mg2Si, 4.5 GPa for
M92Ge
and 3.0 GPa for
Mg2Sn.
The
high
241 Table I.

(F 2J of Mg2 X (X Si, Ge, Sn) at 300 K, their pressure derivatives, bulk compressibilities (Eq. (2)), thermal (Eq. (5)) and mode Grüneisen parameters for the same phonons.
Phonon frequencies
expansion coefficients
=
values. Figures 3 and 4 show the phonon frequency and full width at half maximum (FWHM) as a function of T. The solid lines in figure 3 are eyeguides while those in figure 4 represent least square fits (see Sect. 4). The data points of figure 4 have been corrected for instrumental broadening following standard techniques [10]. The samples used in this study are of the same origin as those in reference [1]. Cleaved (111) surfaces were used in the backward scattering geometry for both
types of
measurements. The incident light was linearly polarized and the scattered light was kept unpolarized, .
3. Elastic constants, thermal expansion coefficients, and mode Gruneisen parameters. In order to
thermal
expansion

the
analysis of the data it is compressibilities and the
coefficients
as a
function of T.
Corrected phonon linewidths (fullwidth at half Fig. 4. maximum) as a function of temperature. The solid lines represent best least square fits to equation (14). 
Observed phonon frequencies as a function of Fig. temperature. The solid lines are drawn only to assist the eye. 3.
proceed with
necessary to know the bulk
242
K
For the cubic system the volume compressibility is given in terms of the elastic stiffnesses Cij by the
where
D(n)
is the
Debye integral
relation
experimental points and the values of B obtained through equation (3) are very well described by the analytical expression [6]
The data of Cij for the three materials can be found in references [1113] for T 300 K. Using a linear extrapolation for the elastic stiffnesses for T > 300 K and equation (1) we find that K can be described accurately by the following analytical expression [6]
Temperature dependent
It is noted that by construction the slope of function (2) becomes zero as T  0 [6]. The values of the constants A, B, C and D are obtained from a least square fit of the experimental data to equation (2), and are listed in table I for all three materials. The experimental data along with equation (2) (solid lines) for the three materials are shown in figure 5. Equation (2) will be used later in order to derive the volume contribution to the phonon shift.
Fig. 5. Temperature dependence of the volume compressibility ( K ). The solid lines represent best least square fits of the calculated K values from equation (1) and experimental data for the elastic stiffnesses [1113] to the equation (2).
expression is plotted in figure 6 along with the experimental data and the values of 8 obtained from equation (3) for the three materials. The values of the constants E, F and Tl obtained from a least square fit of the experimental points and the values obtained from equation (3) to equation (5) are included in table I. It is also noted that equation (5) has the correct asymptotic behaviour, namely, .8 + 0 as T  0. This

Temperature dependent data for the volume thermal expansion coefficient f3 can be found in references [14] and [15] for Mg2Si and Mg2Ge respectively, in the temperature range 300530 K. For M92Sn only the temperature value of fl could be found in the literature [16], while for higher temperatures provision has been taken so that the f3 ( T) curve has the same behaviour as those of Mg2Si and Mg2Ge. The values of f3 for lower temperatures were calculated using existing Debye temperatures [17] and the expression [6, 18] room
6. Temperature dependence of the volume expansion coefficient B. Crosses represent experimental points and full circles represent values calculated from the Debye model (Eq. (3)) and Debye temperatures. The solid lines represent best least square fits of all points to equation (5).
Fig.

The isothermal mode Grdneisen parameters y T are calculated directly from the slopes of figure 2 and the definition
where V is the volume of the crystal and w the phonon at P 0. The results for y T at 300 K are listed in table I together with the values of y T obtained previously either from hydrostatic pressure [3] or
frequency
=
243
uniaxial stress [4] measurements. The values are in reasonable agreement. Due to the K ( T) and w ( T) factors in equation (6) the parameters y T depend on T in
general.
Measurements of
(see Table I) for 09to ( ap ) T T
exhibits a small decrease of about 7 % from its value at 300 K. Assuming that this tendency persists in a linear manner over the whole temperature region (0835 K), the values of all calculated quantities will also vary accordingly by no more than 20 %. However, because of the lack of experimental observations of such a tendency for the whole temperature region and, further, since the introduced mean error of  10 % will not affect the general character of the results, we chose to handle the slope ( aCd / ap) T as temperature independent in the subsequent analysis. Assuming this is the case for all three materials we calculate the percentage change of y T between 4 K and 600 K. It turns out that the change in y T is 6 % and 10 % for M92Si and M92Ge respectively. This variation is comparable to that exhibited by the parameter y T for the various Ramanactive optical phonons of LaF3 [6]. The variation of y T in Mg2Sn is much higher ( ~ 30 % ) . This is connected to the fact that the volume compressibility in Mg2Sn is more sensitive to T than in the other two materials (Fig. 5).
Mg2Si
at 25 K showed that this
slope
Temperature dependence of the frequency shift and Iinewi4th. 4.
observed frequency shift with temperature 4Cd total = Cd (T)  Cd ( 0 ) shown in figure 3 consists of two terms [5], the implicit, or volumedriven term 4Cdvol’ and the explicit, or amplitudedriven term The
4Cd expl’
so
that
Volume (dashed lines) and explicit (solid lines) contributions to the total frequency shift of figure 3 as a function of temperature. The solid lines in the explicit contribution represent the best least square fits of the points
Fig.
to
7.

equation (9).
where n2
and n3 are the phonon occupation numbers at frequencies wo/2 and wo/3 and c, d are constants. The c and d terms in equation (9) represent anharmonic contributions to Awexpl (T) due to the decaying of one optical phonon with harmonic frequency wo to two and three phonons respectively, of frequencies wo/2 and (t)o/3. The values of wo, c, d resulting from the fitting are tabulated in table II. Notice that according to the present formalism we have wo w (0). As shown in figure 7 both contributions to the total frequency shift, i.e., 4(JJvol and 4Cù expl’ are comparable in size and of the same sign (both contributions tend to lower the frequency as T increases). It is interesting to compare these results with those for fluoritetype compounds such as CaF2, SrF2 and BaF2. In the latter =
The volume contribution is due to thermal expansion of the crystal and can be calculated directly from the
equation
Figure 7 shows the values of åCdvol ( T) (dashed lines), obtained from equation (8) using the analytic expressions for f3 (T) (Eq. (5)) and K ( T) (Eq. (2)) and assuming, as before, a Tindependent
( a Cd / a P ) T. The
is now estimated from of the measured values of equation (7) by from 3 and the calculated values of figure Aw total ( T) from 7. The results are shown by dots figure åCdvol ( T) in figure 7. The solid lines represent best least square fits of such data to the following equation [19, 9]
explicit
term
åCd expl ( T)
use
/
1 1



materials it has been established from similar experiments at 300 K that Awexpl is smaller than Awvol and of the opposite sign, i. e. , explicit term tends ’ to increase the frequency with increasing T [20]. A different way of looking at the relative importance of the two contributions is to calculate the socalled implicit fraction [5], that is the ratio
where
y p, y v are the Grüneisen parameters at constant
pressure and constant volume,
respectively, given by
244
Table II.

Values for the various coefficients appearing in equations (9, 14).
of the factor K / J3. For the entire range 150800 K the fraction q remains between 0.3 and 0.6 indicating comparable contributions from volume expansion and anharmonicity, the latter being slightly larger. For T 150 K however ( n  0) , the volume contribution diminishes and the anharmonic effects dominate. 1 indicates predominantly To the extent that q covalent bonding the Tdependence of q in the region 0800 K reflects the way that such bonding depends on
dependence
and
The values of q obtained at 300 K are smaller than 1 (see Table II). Since y T > 0, this means that" v > 0 and therefore (aw / aT) v 0, i. e. , the explicit contribution tends to decrease the frequency with increasing T. According to a general scheme suggested in reference [5], the fact that q 0.5 is indicative of predominantly covalent bonding. This is consistent with the results of references [1214] and [21, 22] where analogous conclusions are reached. On the other hand, the above results are contrary to those of references [23, 24] where it was concluded that the bonding is largely ionic. In the case of the fluorites mentioned earlier, it has been found that q > 1 which is consistent with the ionic character of their bonding [20]. Since all three parameters y T yP and yV are temperature dependent, the implicit fraction q is also temperature dependent, according to its definition (Eq. (10)). It would be interesting therefore to study the temperature dependence of q in more detail. First we write q in the following form, ,
The analytic forms of K ( T) and 8 ( T) are already known. The slope (aid / aT) v can be computed from equation (9) while ( a w / aP ) T is assumed temperature independent. The results of the calculation, that is the functions n n ( T ) , are shown in figure 8 for the three materials. Following an alternative approach, the same calculation can be based on equation (10) and the numerical values for (aid / ap ) T and (aid / aT) p derived directly from the experimental data. The results of both approaches differ by 10 % at most. The behaviour of n ( T) is similar for the three materials. For T 200 K the implicit fraction falls rapidly and tends to zero due to the dominant variation of the K / {3 factor in equation (13). The latter factor diverges as T + 0, while (aid / aT) v approaches zero much slower and from negative values. Between 200 K and 400 K, n ( T) exhibits a broad maximum and then falls slowly for T > 400 K. In this high temperature region the slope (aid / aT) v dominates over the T
temperature. Since, according
to
Phillips’ ionicity
theory [25], the covalent and/or ionic behaviour of a material is critically connected to its energy band structure, the Tdependence of q inferred from the present data reflects in a subtle way the temperature dependence of the energy gaps. The whole subject of the temperature influence on the balance between covalent and ionic bonding seems to be an interesting problem worthwhile of independent study. An expression similar to equation (8) describes the Tdependent broadening of the phonon bands shown in
figure 4,
=
Fig. 8. Temperature dependence according to equation (13). 
of the
implicit fraction 17
245
where a, b are nonnegative constants and ro represents
broadening due to lattice imperfections of [18]. Fitting equation (14) to the data leads to the solid line of figure 4 and the values of To, a, b listed in
Acknowledgments
any residual
table II. On the basis of the results for c, d, a and b of table II, it is concluded that whereas threephonon mechanisms prevail in the anharmonic shift of the phonon frequencies, contributions from fourphonon mechanisms to the shift and the broadening are also present and non
This work has been supported by the FrenchGreek Collaboration Program in Research and Technology. The pressure measurements at 25 K were performed at the Max Planck Institute for Solid State Physics, Stuttgart. One of us (G. A. K.) acknowledges valuable discussions with Drs. Hochheiner of MPI and Jayaraman of AT & T Bell Laboratories during his stay at MPI.
negligible.
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