Polar optical phonons and dielectric dispersion of GeSe crystals

Solid State Communications,Vol. 19, pp. 765-769, 1976.

Pergamon Press.

Printed in Great Britain

POLAR OPTICAL PHONONS AND DIELECTRIC DISPERSION OF GeSe CRYSTALS D.I. Siapkas, D.S. Kyriakos and N.A. Economou Department of Physics, University of Thessaloniki, Greece

(Received 16 January 1976 by M. Balkanski)

The polar optical phonons at k = 0 of GeSe have been investigated by polarized infrared reflection measurements in the range 20-4000 cm -1 . All the four, expected from group theory, infrared optically active phonons have been observed and identified with respect to their symmetry types. The phonon frequencies appear at 155 cm -l (B2u) for E II b and at 88 cm -~ (B3u), 173 cm -1 (B3~) and 181 cm -~ (Bau) for E II a in good agreement with an earlier determination from the optical absorption spectrum. Dielectric and optical dispersion properties due to the lattice vibrations and TO and LO frequencies have been derived from Kramers-Kronig integration and oscillator anal,y,,sis of the reflectivity spectra. The effective ionic charge is estimated as e /e "~ 0.3 by the Szigeti formula.

1. INTRODUCTION THE INFORMATION concerning the properties of germanium selenide is relatively scarce. In studying the optical properties of this compound in the vicinity of the absorption edge at this laboratory it was found that the absorption coefficient follows an exponential behaviour, which is known as Urbach's rule: By analysing this behaviour it was found that different phonons contributed to this effect, which we could not compare to the actual phonon spectrum of GeSe since the only data available where reststrahlen reflection spectra obtained by using unpolarized light, that were reported quite recently by Chamberlain et al. 2 Probably the reason for this lack of data is that single crystals of sufficient size and low carrier concentrations were not available. Therefore in order to check the previous results we decided to extend the optical measurements and to obtain the phonon spectrum of GeSe by using polarized reflectivity measurements in the far infrared.

2. SYMMETRY AND SELECTION RULES Crystalline GeSe has a layered structure which can be described as a deformed NaC1 lattice with the layers being perpendicular to the c axis. The unit cell is orthorhombic of space group D~6 (Pcmn) containing four molecules, and has lattice constants a = 4.40, b = 3.85 and c = 10.82. 3 In this structure the Ge atom is surrounded by six Se atoms, three at a short distance with the interatomic directions almost perpendicular to each other and three at a somewhat larger distance. The

o

(c

)

© ©

© Q I

0

O Ge

1

o

3A

OSe

Fig. 1. The D~6 [Pcmn] unit cell of GeSe as described by Ocazaki. 3 schematic projection of tile unit cell on (010) and the Ge-Se distances are shown in Fig. 1. The interatomic Ge-Se distances are 2.54 (1), 2.58 (2), 3.30 (2), 3.39 (1), where the numbers in parenthesis indicate the corresponding number of bonds. These six bonds correspond to the coordination expected in the NaC1 structures. The isomorphous point group is D2h and the factor group analysis of the P-point lattice modes is given in Table 1. Since there are eight atoms per unit cell this gives rise to 24 vibrational modes, which decompose into the following irreducible representations at the center of the Brillouin zone;

F = 4A~ + 2B,e + 4B2g + 2B3g + 2AL, + 4Blu + 2B2u + 4B3u.

(l)

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DIELECTRIC DISPERSION OF GeSe CRYSTALS

Vol. 19, No. 8

Table 1. Group-theoretical analysis of the zone-center lattice vibrations of GeSe. T i and aik are the vector and tensor components which transform according to the given irreducible representation Irreducible representations

Transformation properties

Selection rules

Ag

axx + ayy + azz Ty

Raman IR(E II b) Raman Inactive IR(E I[ a) Raman IR(E II c) Raman

B=u Big Au B3u

axy

B2e Bm Bsg

axz Tz ayz

Tx

Zone-center modes All

Acoustic

Optic

4 2 2 2 4 4 4 2

0

4

1

1

0 0 1 0 I 0

2 2 3 4 3 2

100

t

GeSe

~80

E# b

T=300°K ----

60

Exper Theor

4°f

20

0

:°°f

E//a f

7

60

40

20

0

11 h

100

200

300

400 W(cm-q

Fig. 2. Reststrahlen spectra of GeSe (at room temperature). The solid curves are the experimental data. The dashed curves are the theoretical single-phonon fit for E 1[ b and three-phonon fit for E II a calculated from the dispersion theory. Three of the modes are acoustic and are assigned to one of the four Blu, one of the two B2u and one of the four B3u representations. In this work the emphasis is on the active-infrared modes with a plane of polarization confined in the double layer. From Table 1 it is evident that the total number of infrared active fundamentals is four for transmission perpendicular to the plane under consideration, three for the case that the electrical field

vector E is parallel to the a axis (E II a) and one for the case E II b. 3. EXPERIMENTAL During the course o f growing single crystals of GeSe we have observed that crystals grown from a stoichiometric melt contained always yellow inclusions attributed to GeSe2. Therefore a vapor transport method of growth

Vol. 19, No. 8

DIELECTRIC DISPERSION OF GeSe CRYSTALS

was used, which resulted in crystals suitable for optical studies. A detailed description of the method and the exact conditions of this growth is to be published soon The single crystals of GeSe were gray, fairly soft and with metallic lustre. They cleave easily along the (001) plane. Polarized reflectivity measurements have been performed using a sample with a (001) plane approximately 1 cm 2. Since the samples could be obtained in a form of thin platelets measurements with the electrical field vector along the c axis could not be performed at the present time. This third independent polarization E II c awaits experiments at oblique incidence or on pressed pellets, from which the oscillator frequencies and strengths of the B l u modes will be estimated. To study the two other directions E II a and E II b the crystals were oriented by back reflection Laue X-ray diagrams. The infrared reflectivity at the (001) plane was measured under nearly normal incidence in the spectral range 2 0 - 4 0 0 0 cm -~ , using a wire grid as a polarising medium. For the range 2 0 - 4 0 0 cm -~ a Beckman-RllC Michelson interferometer, model FS-720, was employed with a Hg arc source and a Golay cell detector, while to obtain spectra in the frequency range 2 0 0 - 4 0 0 0 cm -1 a Perkin-Elmer model 180 grating spectrometer with a globar source and a thermopile detector was used. The reference was a freshly prepared front surface A1 mirror. The error involved in the absolute reflectivity was in the order of 2% and for the frequency ~ 1 cm -l . The measurements were performed in the temperature range 3 0 - 5 5 0 ° K , mostly for qualitative comparison, the analysis confined only to the room temperature data. 4. RESULTS AND DISCUSSION Figure 2 displays the polarised reflectance spectra of the single crystal GeSe for E [[ b and E [1a. In each case the dashed line is a synthetic reflectivity spectrum obtained from an oscillator model and the solid line is the experimental data. As it is evident the one B2u mode expected from the theoretical analysis for the polarization E [[ b is clearly resolved at 154 cm -1 . For the polarization E [[ a (lower part of Fig. 2) all three Bau fundamental modes are clearly resolved. Two of these are lying close at 173 and 182 cm -1 respectively, their splitting becoming more pronounced as the temperature is lowered and one lying at 88 cm -1. An additional strong reststrahlen band, at ~ 250 cm -~, observed in the spectra of crystals grown from the melt is not evident in vapor grown crystals, therefore it is safe to assume that it is due to impurities, mainly GeSe2. The results of the optical behaviour of GeSe at the vicinity of the absorption edge seem to point that phase

767

transitions are to be expected. By varying the temperature in the range 3 0 - 5 5 0 ° K neither new peaks or pronounced frequency shifts have been observed in the phonon behaviour of GeSe, that can be attributed to phase transitions. The only effect that the rising of the temperature has on the spectra, is to smear out the splitting of the near lying two B3u fundamental modes, but even at 550°K, the two modes can be noticed. The optical and dielectric dispersion properties were deduced from the analysis of the reflectivity spectra using both Kramers-Kronig integration and the classical dispersion relation. An analytical description of the reflectance spectrum was obtained by fitting the data to a reflectance that was synthesized from superposition of dambed Lorentzian independent oscillators in which the oscillator frequencies, strengths and damping were taken as the adjustable parameters. This was done by writing the dispersion relation of the principal components of the complex dielectric tensor 4 as

sj o2

m

e((x)) = {l((,.o ) --/e2((.o ) = 6. --~ )_~ i=, ~ ] -- 002 - i%.~oj~o

(2)

with m = 3 for E [1a and m = 1 for E Jl b. eo. is the highfrequency dielectric constant, Si is the oscillator strength, coi the oscillator frequency (in wave number units), N the damping constant (dimensionless) of the flh mode and co the frequency of the radiation. The final reflectivity fits are the dashed lines of Fig. 2. The corresponding dielectric dispersions for EII b and E l[ a are plotted in Fig. 3. The synthesized reflectivity is related to the dielectric constant through the usual equations (n R

-

1) 2 + k 2

(n + 1)2 + k 2

el

= n 2-k

2

e2 = 2 n k

(3)

where n is the index of refraction and k is the extinction coefficient. The results for the optical constants, for both polarizations, E [1 b and E I[ a, are shown in Fig. 4. The fit is based on a free variation of all parameters of the dispersion relation [equation (2)]. The optimization criterion is the minimization of the r.m.s, deviation of the constructed reflectance from the experimental reflectance. Table 2 gives the values of the parameters obtained from the least-squares fit to the measured reflectivity. The frequencies of the LO-phonons have been obtained from the positions of the peaks of the energy-loss function -- Im (l/e) as synthesized from the oscillator fit parameters of Table 2. The least square fit gives an extrapolated value of 13.6 and 13.0 for the high frequency dielectric constant e~ for EII a and E II b respectively. Using these values of e~ and the sum rule

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DIELECTRIC DISPERSION OF GeSe CRYSTALS

Vol. 19, No. 8

Table 2. TO- and LO-phonon parameters, dielectric constants and effective charges o f GeSe from K - K analysis and least-square fit obtained from dispersion relation [equation (2)] Kramers-Kronig

Least-square fit

Polarization

eo

e~

coTO (cm -])

~OLO(cm -l)

~OTO(cm -1)

ooi,o (cm -l)

Si

7i

e*/e

E I[ b Ella

23.5 24.5

13.0 13.6

155 88 173 181

207 92 178 225

154 88 173 182

208 91 179 224

10.5 2.3 6.5 2.0

0.031 0.016 0.044 0.013

0.30 0.25 -

I

GeSe

300 it II

,,

GeSe

---n

K

i

--E

i

---E~

IL

2OO

E//b

E//b I

/

t I

100 co l

\k I l /

-

10(3 12 2OO jL

i

v

I ,: I

E#a

ii

E

E//a

8

4

100 -100

200

300 Wtcm

L

L

I)

_ ~

i

I00

300

200

W(crn -~)

Fig. 4. Dispersion of complex refractive index of crystalline GeSe.

J,

Fig. 3. Dispersion of el, the real dielectric responce and e2, the imaginary dielectric responce as synthesized from the oscillator-fit parameters of Table 2 for GeSe. m

e o - - e ~ = ~ Sj

(4)

Y

we determined the static dielectric constant eo = 24.5 for E II a and eo = 23.5 for E I1b. A Kramers-Kronig analysis of the reflectivity data was also carried out in order to select initial oscillator parameters for starting the oscillator-fit. The TO and LO-phonons frequencies as obtained from the K - K analysis are listed in Table 2 for comparison. The agreement seems to be excellent. As a quantitative criterion for the character of the chemical bonding of a crystal may serve the effective dynamic charge e*. We have calculated the effective dynamic charge for GeSe, a complex anisotropic crystal, by the Szigeti 5 formula, from the experimental reststrahlen strengths and an estimated reduced mass m,

assuming that the strongest mode for each polarization corresponds to pure rigid-sublattice displacements. The results are tabulated in Table 2. The estimated values e* ~ 0.3e for E II a and EII b, places GeSe among the predominantly covalent compounds like the A mBV group, where e*/e has values from 0.34 for InSb 6 to 0.68 for lnP. 7 However it is dangerous to draw conclusions about the binding character from the values of e*/e alone, although this conclusion is in agreement with Shiferl's suggestion 8 that the binding in GeSe should have a large covalent contribution. 5. CONCLUSION In conclusion, we have observed all the four, one

B2u and three B3u, polar optical phonons, predicted from the group theoretical analysis for the D ~ unit cell of GeSe. Analysis of the infrared reflectivity spectra, using both Kramers-Kronig intergration and summed

Vol. 19, No. 8

DIELECTRIC DISPERSION OF GeSe CRYSTALS

independent Lorentzian oscillators lead to values co co(LO) = 208 cm -1 or hv(LO) = 25.8 meV for the frequency, or the energy of the B2~ longitudinal lattice vibration, and ¢.o(LO)1 = 91 cm -x , to(LO)2 = 179 cm -l , 6o(LO)3 = 208 cm -1 or hv(LO)l = 11.3 meV, ;w(LO)2 = 22.2 meV, hv(LO)3 = 25.8 meV for the three B3u longitudinal lattice vibrations. These values are to be compared with the values 24 meV for E II b and hvl = 13 meV, hv2 = 22 meV for E II a found in analysing the behaviour of the absorption in the vicinity of the direct

76c~

energy gap. From this agreement is safe to assume that the observed phonons are responsible for the excitonphonon interaction responsible for the exponential dependence of the absorption on energy (Urbach's rule) reported. The phonon spectrum of GeSe seems to be a common feature for all I V - V I compounds with an orthorhombic structure e.g. GeSe, GeS, SnS, SnSe. Work to establish the exact values of the phonon frequencies in these compounds is in progress.

REFERENCES 1.

VLACHOS S.V., LAMBROS A.P. & ECONOMOU N.A. Solid State Commun. 19,759 (1976).

2.

CHAMBERLAIN J.M., SIRBEGOVIC S.S. & NICOLIC P.M., J. Phys. 7, L150 (1974).

3.

OKAZAKI A., J. Phys. Soc. Japan l3 ,1151(1958); KANNEWURF C.R., KELLY A. & CASHMAN R.J., Acta Cryst. 13,449 (1960).

4.

COCHRAN W. & COWLEY R.A., J. Phys. Chem. Solids 23,447 (1962).

5.

SZIGETI B., Trans. FaradaySoc. 45,155 (1949).

6.

HASS M. & HENVIS B.W., J. Phys. Chem. Solids 23, 1099 (1962).

7.

SPITZER W.G. & FAN H.Y.,Phys. Rev. 99, 1893 (1955).

8.

SCHIFERL D., Phys. Rev. B10, 3316 (1974).