of two (E 11 c) and three (E I c) modes, which is in full agreement with the prediction of a group ... compounds are univalent TP+ ions, and the Ga and In ions are trivalent ions, each ... The interatomic bond within the (A3+B;-)I- fragment is mainly of covalent nature and that is why the above report [6] has suggested to write the.
N. 31. GASANLYet al. : Vibrational Spectra of Layer Single Crystais
367
phys. stat. sol. (b) 97, 367 (1980) Subject classification: 6 and 20.1; 22.3.1
Department of Physics, Azerbaidzhan State University, B a k u ( a ) and Institute of Spectroscopy, Troitsk ( b )
Vibrational Spectra of TIGaTez, TIInTez, and TlInSen Layer Single Crystals BY
N. M. GASANLY (a), A. F. GONCHAROV (b), B. M. DZHAVADOV (a), N. N. MELNIK(b), V. J. TAGIROV (a), and E. A. VINOGRADOV (b) Polarized IR-reflection and Raman scattering spectra are obtained for TlGaTe,, TlInTe,, and T1InSe, crystals (space group D:&. The IR-reflection spectra of all the crystals show the presence of two ( E 11 c) and three ( E Ic) modes, which is in full agreement with the prediction of a grouptheoretical analysis. All seven Raman active modes arc detected in TlInSe,. I n TlGaTe, and in TIInTc, only three of seven Raman active modes can be observed. Normal coordinates, effective force constants, and effective charges arc determined for the infra-red active lattice vibrations.
nOjryqeHbI IIOJIRpCI30BaHHbIe CIIeIcTpbI klH OTpaWeHHR M PaMaIIOBCIiOrO PaCCeRHHR HpHcTanJroB TlGaTe,, TlInTe, H TIInSe, (npocTpaHcTBeHHaR rpynna Dif,). Ann Bcex KpIlCTaJIJIOB B CneKTpaX klK OTpaXeHHR IIpHCYTCTBYIOT XBe ( E ( 1 C ) 51 TPH ( E 1C ) e~ MoabI, TO HaxoamcR B IIOJIHOM cornacm c n p e ~ c 1 c a 3 a ~ nTepeTI?Ko-rpyniioBoro aHaJIH3a. kl3 CeMH OXU4naeMbIX P2iMaH-aKTIIBIIbIX MOD B CIIeHTpaX 06HZipYWeHbI TPH (TlGaTc,, TlInTo,) H CeMb (TIInSe,) MOH. HaiiAeHbI HOpMaJIbHbIe K O O p ~ H I I a T L l , CHJIOBbIe klH-BKTHBHbIX IIOCTOFIHHbIe B3aPIMOnefiCTBHR G i 3@@eICTHBHbIeHOHHbIe 3apRDbI peIUeTO~HbIX~ o n e 6 a ~ ~ i i .
1. Introduction Semiconductive single crystals with a high anisotropy of physical properties find more and more applications in various devices in microelectronics, nonlinear optics, etc. TlSc-type compounds with layer and chain structures, including a number of other compounds such as TlGaTe,, TlInTe,, TlInSe, etc., stand out among anisotropic crystals. I R reflection spectra in the frequency range of 175 to 500 cm-l and Raman spectra of TlInSe, crystals are known [l,21.
2. Crystal Symmetry and a Group-Theoretical Analysis TlGaTe,, TlInTe,, and TlInSe, crystals crystallize in a tetragonal cell (space group D!:) having a symmetry centre [3,4]. A primitive cell of these ternary compounds contains two formula units (Fig. 1). I n this case all the thallium atoms in these compounds are univalent TP+ ions, and the Ga and I n ions are trivalent ions, each of them being in a tetrahedral environment consisting of Se or Te atoms [3 t o 61. An earlier work [6] has shown that fragments of a unit cell from (A3+B;-)l-, where A3+ is Ga, I n and B2- is Se, Te, form a chain extended along the e-axis coinciding with the crystallographic Oz-axis. Such negatively charged chains are bonded t o each other by Tll+ ions. The interatomic bond within the (A3+B;-)I- fragment is mainly of covalent nature and that is why the above report [6] has suggested t o write the formulas of these compounds as Tll+(A3+B;-)l-.
368
K. 31. GASAKLTet al.
''~3'4
Fig. 1. The projection of the unit cell of TlGaTe,, TIInTe,, and TIInSe, on the plane (001). The figures indicate the height above
0% 0 0
0
0n
.
(> i n , ~ a
.Je,Te
A group-theoretical analysis carried out by us with the use of a correlation method 171 gives the following set of irreducible representations for these compounds:
I' = 41,
+ 242, + B1, + 2B;, + 3E, + B1, + 3A2, + 4E,
The selection rules for IR-active and Raman-active modes are presented in Table 1. Thus seven optical modes must be observed in Raman spectra and five optical modes in I R spectra. Table 1 The results of group-theoretical analysis of vibrational spectra at the centre of the Brillouin zone
D:k
degree number of modes of the degeneracy optical acoustic
1
0 0 0 0 0 1 0
3
1
1 2 1 2 3 2
selection rules
IR
-
Raman
3, Experimental T h e present report deals with a n investigation of TIGaTe,, TlInTe,, and TlInSe, single crystals grown by a modified Bridgman method. I R reflection spectra were recorded in linearly polarized light in the spectral range from 20 t o 400 em-l with the use of a modified FIS-21 spectrometer having a resolution of 1 cni-l. Reflection nieasurements were performed a t room temperature from a natural cleavage plane containing the c-axis, a t E 1c and E 1 ) c polarizations. The IR-spectra were processed and the optical characteristics were determined by means of the well-known Lorentzian oscillator model with an additive dielectric function
is the high-frequency dielectric constant, S, is the oscillator strength, VTO) and where are the frequency and the damping constant of transverse optical modes.
?To,
369
Vibrational Spectra of TlGaTe,, TlInTe,, and TlInSe, Layer Single Crystals
~
200
-
150
100
50
0
V(cm-7)
Fig. 2
Fig. 3
Fig. 2 . Reflectivities of a ) TlGaTe,, b) TIInTe,, and c ) TIInSe, for (1) E 1c and ( 2 ) E )I c. The solid curves represent a least-squares fit t o the experimental reflectivity data
Fig. 3. Spectral dependences of Im (I/&)for (1) E 11 c and ( 2 )E TllnSe,
1c for a ) TIGaTe,, b) TlInTe,, and
Fig. 2 shows I R reflection spectra of TlGaTe,, TIInTe,, and TIInSe, single crystals, measured a t various polarizations of the incident radiation. At E 1c three reststrahlen bands are observed in the spectra for all the crystals, whereas a t E 1 1 c two reflection bands are present in the spectra. These results are in full agreement with the predictions of a group-theoretical analysis. Solid curves in Fig. 2 represent a least-squares fit t o the experimental reflection data. Dispersion parameters (Table 2) have been determined from the conditions of the best agreement between the experimental and calculated reflection spectra. The values of these parameters were used to calculate I m ( ~ / E ( Y ) )(Fig. 3). The last column of Table 2 gives the frequency values = niax. of longitudinal optical modes (~1’0)found from the condition I m (I/E(Y)) TlGaTe,, TlInTe,, and TlInSe, single crystals are opaque in the visible region ( E g= = 1.2 eV, 0.7 eV [8], and 1.44 eV [9], respectively). They are cleaved in two mutually perpendicular directions parallel to the c-axis, the cleavage planes being along the diagonal planes (110) of the tetragonal unit cells [lo]. Raman scattering spectra of TlInSe, were studied in the region of 10 to 200 c1n-l a t excitation with a continuous-type YAG:Nd3+ IR-laser (A = 1.064 pin) in rightangle scattering geometry. Fig. 4 depicts a non-polarized spectrum of a TIInSe, crystal a t a temperature of 100 K. All seven Raman-active modes have manifested 24
physica (IJ) g i l l
370
N. M. GASANLYet al. Table 2 Dispersion parameters for the calculated reflectivity of Fig. 2 crystal
parameters
TlGaTe,
E,
14.8
A2u
13.1
Ell
14.2
A2"
11.6
E,
10.2
TlInTe,
TIInSe,
Azu
8.0
1.5 3.2 3.2 4.9 15.9
5.2 5.4 2.5 2.6 1.7
192 88 44 175 27
202 97 52 205 38
1.5 4.4 5.3 3.2 25.8
4.4 8.4 2.6 8.2 4.9
159 80 41 152 23
168 90 46 169 37
1.6 3.8 4.8 3.7 17.4
7.4 5.3 1.9 2.3 2.5
196 98 54 179 28
212 114 61 218 40
themselves in it. It has been possible t o carry out polarization measurements for five of them (Fig. 5). The weak 16 and 200 cm-I lines have been assigned t o the 2Bzg modes missing according to a group-t heoretical analysis. TlGaTe, and TlInTe, crystals were opaque for the radiation wavelength of a PAG : Nd3f-laser, and therefore we were unable to measure polarization Ranian
--
200 v (cm-')
Fig. 4
roo
0
-
200
-30
0
vim-') Fig. 5
Fig. 4. Non-polarized Raman spectrum of a TlInSe, crystal a t T = 100 K (&,
=
1.064 pm)
Fig. 5. Polarization Raman spectra of TIInSe, crystal for various scattering geometry a t T = 300K (&, = 1.064 pm)
Vibrational Spectra of TIGaTe,, TlInTe,, and TIInSe, Layer Single Crystals
37 1
Pig. 6. Raman spectra of a) TlGaTe,, b) TlInTe?, c) TlIneS,, d) TlSe (Aexc = 5145
A)
Table 3 Frequencies (in cm-l) and assignment of the okserved lines in the Raman spectra
TlGaTe, TlInTe, TlInSe,*) TlSe
-
30 29
67 48 60 38
I65 138 174 141
-
98 93
135 127 184 159
-
-
-
200 204
16 -
*) I n [2] the lines 98 and 200 cm-l were wrongly assigned to the Bzg and B1, modes, respectively, because there it was supposed that the crystals were cleavable along the planes (100) and (010). 24 *
372
N. M. GASANLYe t al.
spectra by means of this laser. It was impossible to record polarization spectra in a backscattering geometry a t a n excitation with a 5145 A line of a n argon laser because of the high level of parasitically scattered light from the exciting line that gets into the spectrometer. That is why we nieasiired the Raman spectra of TlGaTe, and TlInTe, with the exciting radiation (A = 5145 A) incident a t the Brewster angle on the freshly cleaved crystal surface. These spectra are presented in Fig. 6 a and b a t various scattering geometries. I n order to assign properly the observed lines in the Raman spectra of TlGaTe, and TlInTe, we recorded the Raman scattering spectra of TlInSe, and TlSe crystals a t similar scattering geometries (Figs. 6 c and 6d). Only four of the seven expected lines have been found in this case: TlInSe, a t 184 cm-l (Alg) and 174,60, and 30 cm-l (Eg),and TlSe a t 159 em-l (Alg)and 141,38, and 29 cm-l (Eg).(The above frequencies in TlSe were assigned to symmetry types in [ll] on the basis of polarization measurements of Ranian spectra.) When the position of the Rpecimen is changed (turned
Fig. 7. Symmetrized displacements of atoms in TlGaTe,, TlInTe,, and TlInSe, crystals. The superscripts on the representations have no physical significance
Vibrational Spectra of TlGaTe,, TIInTe,, and TlInSe, Layer Single Crystals
373
through 90") the intensity of the A,, mode decreases, whereas that of the E, modes grows. Besides, two lines with frequencies of 93 and 204 cm-l appear in the TlSe spectrum. The 204 cm-l frequency line was attributed in [ l l ] to the B2g mode. It should be noted that the 93 cm-l line (Big) has been observed for the first time. A possible assignment of lines in TlGaTe, and TIInTe, spectra has been accomplished (Table3) similarly t o the assignment of lines in the TlInSe, and TlSe spectra. It should be emphasized that the lines of the TlInSe, and TlSe spectra whose intensities decrease severely whenthe specimen is turned through 90" (i.e. the Al, mode in which only selenium atoms vibrate (Fig. 7 ) ) are characterized by higher frequencies (184 and 159 cm-l) than the lines of a similar behaviour in the TIGaTe, and TlInTe, spectra (135 and 127 crn-l). This is likely to be due t o the fact that in the latter case Te atoms vibrate which are heavier than the Sc atoms.
4. Normal Coordinates of Lattice Vibrations at the Zone Centre The optical frequencies a t the zone centre can be related to effective force constants and reduced masses of groups of ions involved in a particular normal vibration by the relation [12] 2 pc,w, = f c , . (1) The solution of this problem is facilitated by the fact that we have a t our disposal the data obtained as a result of a study of the spectra of optical phonons of three isostructural compounds practically with a n isotopic substitution of atoms (ions) in them. For these compounds it may be assumed in a first approximation that when a n atom is substituted by another one, only the masses of atoms but not the force constants of interaction inside the unit cell are changed. Therefore, it may be assumed that the ratio of frequencies of the same type of optical phonons of two crystals differing from one another by an atom is inversely proportional to the ratio of the square roots of the reduced inasses of dipoles. By going through the possible versions of relative displacements of groups of crystal lattice atoms onc can basically find the normal vibration coordinates meeting this relationship. It is most convenient to solve this problem with the use of symmetrized displaccments of atoms for various vibration types obtained by us by means of Melvin projection operators 1131. These symmetrized displacements are presented in Fig. 7 . Let us first consider the Az,-type vibrations. Fig. 7 shows that one can make up five types of independent linear combinations of Aiu, A:,, and A&:
+ +
+
1. A:, A;, A& acoustic vibration, 2. A&, A& - A&, = lAzu , 3. A&, - A;, ,A,, , 4. A,: - A&, , and 5 . A;, - A;, . Taking into account the investigation data of some reports [3 to 61 showing the differences in the bonding forces within the (A3fBg-)-1 chain fragments and between these chain fragments and the TI1+ions located between the chains, one can suppose that one of the normal vibrations is a shift of the (A3+B;-)l- ion with respect to the Tllf ion; then another normal coordinate may be represented by a vibration in (A3tB;-)1-. Then one should choose two of the four possible Azu symmetry vibrations, namely IA2, and 2A2n,for which the reduced masses of dipoles for TlInSe, are
~~
TO LO TO LO
TO LO TO LO TO LO
E,
~~~~
A2u
mode
(cm-l)
193 212 94 114 54 61
176 219 28 40
li
TIInSe,
~
159 168 80 90 41 46
152 169 23 37
TlInTe,
~~
~
192 202 88 97 44 52
175 205 27 38
TIGaTe,
1.23 1.26 1.23 1.27 1.32 1.33
1.17 1.29 1.22 1.08
v(T]InTe,)
1.06
1.22
1.09
1.06 0.83 0.83 0.91 0.93 0.93 0.88
0.87 0.82 0.85 0.97
p(T11nTe2) v(TlInTe,) p (TlInSe,) Y ( TlGaTe,)
1.09
v(~l~n~e,)
0.97
0.96
0.83
0.98
0.83
I .02 1.05 1.11 1.18 1.23 1.17
1.021 1.06 1.04 1.05
p(TIGaTe,) v(TIInSe,) p (TITnTe,) Y ( TlGaTe,)
Table 4 Frequencies of IR-active modes (in ern-I), their ratios, and square roots of the ratios of reduced masses
1.04
1.18
0.91
1.04
0.91
p (TlInSe,)
Vibrational Spectra of TlGaTe,, TIInTe,, and TIInSe, Layer Single Crystals
375
Table 5 The f o r e txmstamts Q€ interactions between atoms and effective ionic charges crystal
Azn
fl'
E"
( lo3 dyn/cm)
TlGaTe, 7.1 TIInTe, 6.1 TlInSe, 7.2
el:)/e
0.30 0.36 0.36
fk'
( lo5 dyn/cm)
1.1 1.2 1.4
e\;2)/e
fi' ey'/e ( lo3 dyn/cm)
(lo5 dyn/cm)
0.78 0.71 1.2
16.1 14.1 21.8
1.2 1.2 1.6
0.29 0.24 0.32
,(2)
1 le
0.44 0.48 0.70
For TIInTe, we have ,ul = 131.66 and ,u2= 79.19, whereas for TlGaTe, ,ul = 125.46 and pz = 54.76. Table 4 gives frequency values of Azu symmetry vibrations for all three crystals, frequency ratios of these phonons) and the square roots of the corresponding reduced mass ratios. As seen from this table, there is a good agreement in these ratios for the lAnu and 2A2u vibrations. The two remaining linear combinations of symmetrized displacements fail t o show such a good agreement. Thus, for Azu symmetry vibrations we have two normal coordinates in accordance with a group-theoretical analysis. Based on the knowledge of the reduced masses of vibrating dipoles, one can determine with equation (1) the force constants of the interactions between atoms. Table 5 lists the force constants of the interaction between the ions Tll+ and (A3+B;-)l-, fy, and between 2B2- and A3i, fi, calculated from experimental data. A similar treatment gives the following sets of normal vibrations with El, symmetry.
2. E:
+ E! + E: + Eg - Ei
3. Et
i:EZ f Ei Ifi E:
1. E:
G
EE
3
l3:
4. E: - EZ sz EZ - E!
+ E: + E: + E: - E:
acoustic vibration , low-frequency vibration,
medium-frequency vibration, high-frequency vibration.
The reduced inasses of dipoles for the E, vibrations for TlInSe, are:
Et ,
p3 - 4 ___ + 2nz(Se) m(1n)
I-'
= 66.48.
I n a similar way one can also calculate the masses for the two other compounds. Table 4 gives the frequencies and the square roots of the frequencies and the square roots of the reduced mass ratios for the E, vibrations. It is evident that a rather good agreement is observed between the frequency ratio and the inverse ratios of the square roots of reduced masses of dipoles. The minor difference between the ratios is likely to be associated to the fact that the polarizabilities and hence the effective ionic charges in various compounds ares omewhat different. It is not surprising in a s much as the I n and Ga as well as the Se and Te ions are of different sizes. The difference between the values of the effective ionic charges must lead to a difference in the force constants of interaction between the atoms. Table 5 gives the values of calculated force constants of interatomic binding for the E, vibrations.
376
N. &I.GASANLYet al.
I n a similar way one can also find the nornial coordinates of the vibrations of atoms active in Raman spectra. I n the case of Al, and B1, vibrations the symmetrized atomic displacements (Fig. 7 ) represent normal coordinates. I n the other cases it is somewhat difficult to determine the mode of the normal coordinates of the atoniic vibrations because of the scantiness of Raman scattering spectra for TlGaTe, and TlInTe, crystals. Here, one can confine oneself only to a qualitative explanation. For the BZg vibrations two normal coordinates are possible, lB2, and ?-Bzg. I n this case, it is most likely that % ' z, is the high-frequency vibration involving only anions, whereas lBZg is the low-frequency vibration in which only trivalent ions of I n and Ga vibrate. Similarly, one can suppose that the low-frequency lE,-vibration is due to the relag. tive displacements of ion chains -TI-In(Ga)-T1-, i.e. lEg= E3, - Ei = E%- Eb Then the medium-frequency ?-E, vibration results from the vigrations of T1 and In(Ga) ions: 2E,3 Ei Ei = l3: E$. The high-frequency 3E, vibration is due to vibrations of anions Se(Te).
+
+
5. Effective Ionic Charges I n crystals with a fraction of an ionic bond between the atoms the dipole vibrations split u p into longitudinal and transverse components due t o long-range Coulomb forces [la]. The frequencies of transverse and longitudinal phonons in uniaxial crystals are determined by the expressions [14]
where woo,(j)is the frequency of atomic vibrations without taking into account the long-range Coulomb forces, pa(j) the reduced mass of the j-th dipole vibration, N f the nuniber of ion pairs (dipoles) per 1 em3 involved in the j-th vibration, L, the Lorentz factor, ELthe high-frequency dielectric constant of the crystal (a indicates the direction with respect to the optical c-axis), and e , ( j ) the niicroscopic effective ionic charge introduced in [15]. From equations ( 2 ) and (3) we obtain
For the vibrations involving the univalent T1 ion, i.e. for lAzu and IE, vibrations we find from expression ( 5 )the values of effective charges el/)/e and e(:)/e, respectively. The values of these charges are given in Table 5 . Owing to the electrical neutrality of the crystal the (A3+B;-)I- ion has also the same charges. The A3+ and B2- ions take part in the relative displacements in the 2A2, and 3E, vibrations. Substituting the experimental frequencies, E&, N,, and p a ( j ) in (5) we find the value of the effective charge ek2) of A3f ions (Table5). The effective charge values ei3) of the B2- ions will be obtained on the basis of the electrical neutrality of a unit cell:
2 i e ~ ) ( i )= i ieiwi
+i e t w .
Vibrational Spectra of TlGaTe,, TIInTe,, and TlInSc, Layer Single Crystals
377
Thus, we have succeeded not only in singling out the normal coordinates of the atoniic vibrations of crystals with TlSe structure, but also in determining some crystallocheniical constants essential for calculating dynamic characteristics of crystal lattices. Acknozcledgement
The authors wish to thank Dr. G. N. Zhizhin for helpful discussions and encouragements.
References V. I. TACIROV, and B. AT. DZHAVADOV, Tezisy dokladov [1] A. KH. KHUSEIN,N. M. GASANLY, pervoi respublikanskoi konferentsii aspirantov Azerbaidzhana, Baku 1978 (p. 46). [2] N. M. GASANLY, A. F. GONCHAROV, B. M. DZHAVADOV, N. N. MELNIK,V. I. TAGIROV, and E. A. VINOGRADOV, phys. stat. sol. (b) 99,11139 (1979). [3] J. A. A. KETELAAR, W. H. T’HART,M. MOEBEL,and D. POLDER,Z. Krist. 101h, 396 (1939). [4] H. HAHNand B. WELLMANN, Naturwisseiischaften 54, 42 (1967). [5] L. N. MAN,R. &I. IwAnlov, and S. A. SENILETOV, Kristallografiya 21, 628 (1976). L6] D. MULLER,G. EULENBERGER, and H. HAHN,Z. anorg. allg. Chem. 398, 207 (1973). 171 W. G. FATELEY, N. T. RICDEVITT, and F. F. BENTLEY, Appl. Spcctroscopy 25,155 (1971). [8] G. D. GUSEINOV, G. B. ABDULLAEV, S. &I. BIDZINOVA, F. 31. SEIDOV, M. Z. IsninILov, and A. M. PASIIAYEV, Phys. Letters A 33, 421 (1970). [9] V. I. TAGIROV, A. E. BAKHYSHOY, M. A. SOBEIKII,A. M. AKHMEDOV, and V. &I. Sa~nfaxov, IZV.YUZOV, Fiz. No. 11, 131 (1978). [lo] F. M. GASHIMZdDE and Af. A. ~IZAlKETDINOVA4, F i L . tverd. Tela 10, 2665 (1968). [Ill K. R.ALLAKHVERUIEV, E. A. VINOGRADOV, X. N. NELNIK, Af. A. NIZAA~ETDINOVA, E. YU. SALAEV, and R. M. SARDARLY, phys. stat. sol. (b) 87, K115 (1978). [I21 H. POULET and J.-P. MATHIEU,Spectres de Vibration et Symetrie des Cristaux, Gordon & Breach, Paris/London/New York 1970. [13] M. A. MELVIN, Rev. mod. Phys. 2S, 18 (1956). [14] M. BALKANSKI, Proc. Internat. Conf. 11-VI Semicond. Compounds, Rhode Island 1967 (p. 1007). [15] B. SZIGETI,Trans. Faraday Soc. 46, 155 (1949). (Received June 5, 197’9)
