- Email: [email protected]
Resonant Brillouin scattering in cadmium telluride
~Solid State C o m m u n i c a t i o n s , P r i n t e d in G r e a t Britain.
Vol.48,No. 7, pp. 581-584,
1983.
O O 3 8 - 1 0 9 8 / 8 3 $3.00 + .00 P e r g a m o n P r e s s Ltd.
RESONANT BRILLOUIN SCATTERING IN CADMIUM TELLURIDE
R. Sooryakumar + and M. Cardona, Max-Planck-Institut f~r Festk6rperforschung, Heisenbergstr. 7000 Stuttgart 80, Federal Republic of Germany,
~,
and J.C.
Merle,
Laboratoire de Spectroscopie et d'Optique du Corps Solide 5 Rue de l'Universit~, 67000 Strasbourg, France
(LA 232 du CNRS),
Received: July 20, 1983, by M. Cardona Resonant Brillouin scattering of exciton-polaritons from CdTe is reported. Selection rules are discussed and used to identify the scattering channel associated with each peak. The experimental Brillouin shifts are well reproduced by a model involving several exciton parameters which compare well with other available values.
INTRODUCTION
tion rules for exciton scattering before introducing the polariton concept. We then deduce the polariton and the corresponding parameters needed to fit the data. The excitonic dispersion can be obtained by the method of invariants. 6 The eight K=O exciton states formed with r 6 electrons and r 8 holes are
The original theQretical predictions of resonant Brillouin scattering (RBS) or exciton polaritons by Brenig et el. 1 have been tested successfully on a variety of semiconductors. 2 The measurements, which are now almost "classical", have been very useful in obtaining information on the exciton-polariton dispersion and exciton-phonon coupling mechanisms. We have studied the RBS of exciton-polaritons from high quality samples of CdTe in the [110] direction. Several transitions are followed through the resonance region. The spectra are more complicated than in the case of GaAs, 2 yet somewhat simpler than for CuBr. 3 We are faced with the delicate task of attributing correctly each peak to a given decay channel. To achieve this, scattering selection rules are deduced in more detail than previously established. W The polariton dispersion is then calculated and the parameters so obtained are compared with existing values. A full account of the selection rules together with further conclusions based on intensity calculations will be presented elsewhere. The standard RBS equipment using a double grating spectrometer as described in Ref. 2, has been used in these measurements. We studied freshly cleaved [110] surface samples of CdTe at 1OK in the backscattering geometry. The overall resolution of the system was 0.03 meV with incident and scattered beams polarized parallel to the [OO1] direction. The power of the incident beam was typically 4 mW. We also measured the normal reflectance during the course of the Brillouin scattering experiments. Our reflectance spectra are identical with those reported earlier except for a small shift in the absolute position of the peaks. Several Brillouin peaks are generally observed. They suggest a many branch polariton dispersion and the involvement of TA and LA phonons. The results are summarized in Fi~. 1.
decomposed into the dipole forbidden J=2 triplet state r3+r 4 at energy Eo, two transverse allowed J=l singlet states F5T at energy E=Eo+A and the longitudinal exciton at energy EL=Eo+~+~ ' . The dispersion involves terms quadratic in wavevector proportional to the effective exciton parameters ~i' Y2' and q which are defined by analogy to the 3 Luttinger parameters. In the perturbation approach 7 these are described by = mo
~i
2
W' 72 = Y2B'~' Y3 = Y3Bh a
in the notation used by Kane 7, where ~ is presently of the order of 0.7. K-linear terms, proportional to an effective constant C, are also present in the Hamiltonian. Following the magnetoreflectance studies of Dreybrodt et el. 5 such K-linear terms play an important role in the exciton dispersion of CdTe. For the scattering geometry chosen (KII[110] and [001] polarisation) four exciton branches couple with the incident light. 8 One of them is the longitudinal exciton, which for K~O becomes weakly dipole active due to the dispersive terms. In the present experiment we do not detect any effect which could be related to this exciton and therefore it will not be considered further. Furthermore, a recent measurement of the valence band parameters 9 and also the dependence of the RBS spectra on 13o_ larisation I0 shows that q2 and q3 have comparable values. In order to simplify the discussion we have used the approximation ~2--Y3. With these assumptions, the excitonic dispersion is given by the eigenvalues of the matrix
THEORY TO explain the Brillouin shifts we first discuss the excitonic dispersion as well as the selec+ ~resent address: Bell Laboratories, Holmdel, New Jersey 07733, U. S. A.
581
RESONANT
582
13>
12>
£ +26
SCATTERING
I1>
'-O.+ ~ 6
+~--6~
BRILLOUIN
~ +/-66
36
h - 3/-6~+ 36
where £ = ( ~ l + ~ 2 ) ~ 2 K 2 / 2 m o ; h =
~l-~2)~2K2/2mo
.
@ = C K / 2 ~ and 6--d/8. The wavefunctions I I> to j ~ are related to those defined in Ref. 6 by:
=
i//'2
-i/~
,7"J/2¢"~J~ 1+>-I I-> ) Iv
~/2~/\
I~"
For an arbitrary K, the exciton dispersion cannot be written down as a simple analytical expression. However, for large K, the quadratic and linear terms dominate over the exchange energy A and the exciton can then be classified simply into a light b~anch (branch 3) and a heavy branch. K-linear terms further split the heavy branch into two branches 1 and 2. In the large K limit, the exciton wavefunction being known, one can establish selection rules for the exciton scattering. This can be done by cons i d e r i n g t h e deformation p r o d u c e d by a given phonon in the long w a v e l e n g t h approximation. For the LA phonon it can be shown *0 that the only strong transitions are the intraband ones. An intraband transition relates two excitonic states having the same wavefunction. Note that due to the presence of Klinear terms ~n the matrix the eigenenergies associated with 1 and 2 are not symmetrical with respect to K. Therefore in terms of exciton transitions, an intrabranch backscattering process does not always connect two branches symmetric with respect to K=O. The TA phonon p o l a r i z e d along [O01] acts through both the deformation potential and the Fr6hlich mechanism (piezoelectric effect). In the resonance region, one can assume that the latter effect dominates. ~I Since the FrShlich interaction is diagonal, the TA scattering must also be an intraband process. The TA phonon polarized along [I[O] does not give any observable effect in agreement with selection rules. I0 We hence neglect this particular TA phonon in the following discussion. The above results are in agreement with those of Yamane and Cho. ~ The exciton-photon coupling will now be considered. This enables one to obtain the polariton dispersion curves as shown in Fig. 2. For large momentum, each polariton branch (except the highest one) is very close to a bare exciton and its properties are characterized by that exciton. In such a case, the selection rules established for the exciton transitions can be directly applied to the polariton. In Fig. 2 the polariton branches are labelled by Roman numbers. The dispersion for negative K-values can be obtained by reflection with respect to the K=O axis and corresponding primed values I', II'... will be used in this case. For large momenta, the exciton selection rules discussed earlier show that for both LA and T A phonons the polariton transitions III ~ III' (light-light), II + I' and I + II' (heavy-heavy) intraband processes must dominate. In the region very near E , both the exciton w a v e f u n c t i o n and the polariton~ are not simply described and thus clear selection rules cannot be deduced. DISCUSSION OF EXPERIMENTAL RESULTS The experimentally observed dispersion of the RBS peaks (Fig. i) is rich in structure. On the Stokes side we observe the following spectra. For low excitation energies E, two lines are observed which disappear with increasing E in a region ("gap") where no transitions are detected. Above the "gap" one follows two strong transitions whose intensities
IN C A D M I U M
Vol.
TELLURIDE
48, No.
peak rapidly and then diminish gradually at higher energies. They are labelled (II + I')LA and T A in Fig. i. Two other peaks are comparatively m u c h weaker and are strongest around EL: ( I I I ÷ III')LA and TA (Fig. I). vial i ,o t is trir l . This also determineS the sound velocity ratio VLA/VTA which is then used in the higher energy region. For E>>E we make use of the selection rules given above an~ conclude that the dispersion of the four dominant transitions are caused by II+I' and IIl÷III' intrabranch TA and LA scattering. For the remaining part one notices that near the "gap" there exists a region for E=E~, where the polariton curve must be flat (no variatlon of energy with wavevector, see Fig. 2). The flatness of the dispersion curve became evident when it was noted ex-
I
(~.m)
(~rlImmIi~m~l, (N~I
1
xx
E L -
I
~xx ',; \~,
x~ ~\
_>
x ~
xx
E
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\ EO= 1-59565eV \ x x
o
x
xxx ,
~
¢n,rlh
"~x
%
~',
o
',,(111,g')
-0.5
" -
T A (001)
"---~x~.~
I
I
I
-0.8
-0.6
-0.4
AE
l~x$
-0.2
0
(meV)
Experimental and calculated Brillouin shifts for K i I[110] and incident and scattered polarisations parallel to [001]. The x's represent experimental data. The solid and dashed lines are c a l c u l a t e d shifts with the parameters given in the text. perimentally that for Stokes II~I ' LA phonon scattering (strong intraband), the final energy state of the polariton remained unaltered for variation in the incident energy by about 0.2 meV. Small changes in the excitation energy near Ef drastically changed the anti-Stokes Brillouin spectrum. This also reflects the singular behaviour of the polariton dispersion. The position of the flat branch approximately fixes E . This together with the Brillouin shifts at high°energy enables us to establish the heavy and light exciton masses.
7
Vol. 48, No. 7
RESONANT
BRILLOUIN
SCATTERING
The K-linear terms have little influence on the calculated shifts near E . However, just above the "gap" they modify greatl~ the II+I ' and somewhat the III÷III' curves. K-linear terms are also necessary to create a flat portion of the lower polar±ton dispersion. The analysis of the Brillouin shifts versus energy enables us to determine the following parameters (correction for the refractive index of air is included) E = 1595.65 ± 0.025 meV ~o YI = 2.44 ± O.12; ~2 = ~3 = 0.887 ± 0.045 -7 C m (2.06 ± 0.10). 10 meV cm; 6 b = 11.2 A 0.075 + O.O5 meV; A'= 0.65 ± 0.05 meV. This analysis is consistent with the following sound velocities v LA = 3.39.105 cm sac -I and VTA = 1.79. iO s cm sac -I which are within 5% of the velocities obtained at low temperatures by ultrasonic measurements. 12 E b is the background dielectric constant. The theoretical shifts calculated with the above values are reproduced in Fig. i where the agreement with the experimental results is seen to be very satisfactory. The accuracy for the fit values of Yl' ~ and C is good because a slight variation of one o~ these parameters changes appreciably the calculated shifts, at least in a specific excitation range. Although E is also determined with good accuracy for a give~ sample, EL, the longitudinal exciton energy, is found with less precision (±0.05 meV). This uncertainty arises because the strong IV+IV' LA scattering (see Fig. I) cannot be detected very near E where the Brillouin shift is very small. The influence of ~' is mainly on the I÷I' shifts at low energy a~id on the IV+IV' shift at high energy. How-
I
I
I
I
I
K II [110] Pol [001]
1.0
IN C A D M I U M
TELLURIDE
583
ever, an increase of 4' c a n easily be compensated by an identical decrease of ~ and a reduction of Cb. The parameters we obtain are in reasonable agreement with the values reported by Dreybrodt et al. 5 (~i = 2.17; ~2 = ~3 = 0.70; C = 1.92.10-' meV cm). An earlier measurement of RBS from CdTe 13 has been briefly reported. These limited results cannot be quantitatively compared with the present work. These have, however, been analysed in Ref. 2 where a value of the heavy exciton mass (~ -2~2)-| = 2.4 m has been reported. This value differs considerab~y from the value of 1.5 m reported here. However, the K-linear terms, whic~ strongly influence the dispersion, have been neglected in Ref. 2. This may explain the discrepancy. Several authors (Ref. 9 and other references therein) have determined the band parameters of CdTe by various methods and the following reliable values are now available: m* = O.O88 m ; y. = (5.3 ± 0.5); Y2 = (i .7 ± 0.3) ande73 = (2.0 ~ O.~) for the bands. With these parameters and using the results of Kane 7 we find for the excitons ~i = 2.62 ±0.30 and ~2 0.90 ± 0.20 which are in very reasonable agreement with our values. This indicates that the exciton dispersion seems to be well described by perturbation theory. Following Kane's theory, one also calculates an effective Rydberg R = 1.O5 R~ where R* is d e f i n e ~ for a non-degenerate valence ba~d of mas~ mo/Y~. The present situation for large m o m e n t u m is very similar to that encountered for CuBr in the [IIi] direction. I~ In both cases three exciton branches are present and the same kind of selection rules apply. In particular, the I+I ~ transitions are forbidden in the large K limit. 14 This selection rule is illustrated in Fig. I where the I+I' LA transitions observed at low energies do not reappear above the "gap". Note, however, that the RBS spectra are much less complicated in CdTe than in CuBr. 3 One probable reason is that the classification oF the excitons into light and heavy, which results in simple selection rules, is realized more easily in CdTe than in CuBr. This arises from the differences in the values of 4, y~, and C for the two compounds. The experiment~l results in CdTe seem closer to those for ZnSe. 15 However, the assignments of the Brillouin peaks are not fully consistent in the two cases. CONCLUSIONS
EL 0.5
>
E 0 Ill
'
ET
EF*
-05
""{i"/~ 0 0.5
l 1.0
J J I 1.5 2.0 2.5 WAVEVECTOR (10 6 cm -1)
Fis. 2: Exciton polar±ton dispersion in the [110] direction and for the [001] polarization. The curves are calculated with the parameters given in the text.
We have reported detailed EBS measurement for CdTe. For the assignment of each line we have b e n e fited from the existence of simple selection rules in regions where the experimental peaks are followed easily. Moreover, the experimentally found evidence for the existence of a flat p o r t i o n in the polar±ton dispersion - ~t energy E - imposes r a t h e r . f strong limitations on the pesszble values of some o f the parameters. The good agreement between the values of YI' Y2' and Y3 we have deduced and the v a lues calculated from known band parameters renders very improbably an error in the assignment we propose. CdTe, together with CuBr, are the best known examples where the K-linear terms are significant. It has been shown that such terms are necessary in the theory to reproduce the experimental results. The overall agreement between the calculated and experimental shifts places confidence in the values of the parameters deduced f r o m this m e a S U r e m e n t . These conclusions are further confirmed by a detailed discussion of the selection ruies and intensities which will be published elsewhere, t° ACKNOWLEDGEMENTS The excellent sam.plea of CdTe w e r e kindly supplied by R. Triboulet. Thanks are also due to P. Wurster, M. Siemers, and H. Hirt for technical assistance. Useful discussions with P.E. Simmonds are acknowledged. One of us (R.S.) would like to thank the Alexander yon Humboldt Foundation for support during the course of this work.
584
R E S O N A N T B R I L L O U I N S C A T T E R I N G IN C A D M I U M T E L L U R I D E
Vol. 48, No.
REFERENCES
i. 2.
3. 4.
5. 6. 7. 8.
BRENIG, W., ZEYHER, R., and BIRMAN, J.L., Phys. Rev. B 6, 4617 (1972). WEISBUCH, C. and ULBRICH, R.G., Light Scattering in Solids III, M. Cardona and G. GOntherodt, ede., Springer Verleg (1982), Chapt. 7. VU, D.P., OKA, Y., and CARDONA, M., Phys. Rev. B 24, 765 (1981). YAMANE, M. and CHO, K., Physica II7B & 118B, 377 (1983). (Proc. Int. Conf. of Semiconductors, Montpellier, France, 1 9 8 2 ) DREYBRODT, W., CHO, K., SUGA, S., WILLMAN, F., and NIJI, Y., Phys. Rev. B 2~I, 4692 (1980). CHO, K., Phys. Rev. B 14, 4463 (1976). KANE, E.O., Phys. Rev.-B i~i, 3850 (1975). H~NERLAGE, B., R~SSLER, U., V~, D.P., BIVAS, A.,
9. 10. 11.
12. 13.
14. 15.
and GRUN, J.B., Phys. Rev. B 22, 797 (1980). LE SI DANG, NEU, G., and ROMESTAIN, R., Solid State Commun. 44--, 1187 (1982). SOORYAKUMAR, R., MERLE, J.C., and CARDONA, M., to be published. CARDONA, M., Light Scattering in Solids II, M. Cardona and G. GOntherodt, eds., Springer Verlag, 1982, Chapt. 2. GREENOUGH, R.D. and PALMER, S.B., J. Phys. D, 587 (1973). NAKAMURA, A. and WEISBUCH, D., Abstract for the 35th meeting of the Phys. Soc. of Japan 2--, 211 (1980). CHO, K. and YAMANE, M., Solid State Commun. 4_~O, 121 (1981). SERMAGE, B. and FISHMAN, G., Phys. Rev. B 23, 5107 (1981).
7
Vol.48,No. 7, pp. 581-584,
1983.
O O 3 8 - 1 0 9 8 / 8 3 $3.00 + .00 P e r g a m o n P r e s s Ltd.
RESONANT BRILLOUIN SCATTERING IN CADMIUM TELLURIDE
R. Sooryakumar + and M. Cardona, Max-Planck-Institut f~r Festk6rperforschung, Heisenbergstr. 7000 Stuttgart 80, Federal Republic of Germany,
~,
and J.C.
Merle,
Laboratoire de Spectroscopie et d'Optique du Corps Solide 5 Rue de l'Universit~, 67000 Strasbourg, France
(LA 232 du CNRS),
Received: July 20, 1983, by M. Cardona Resonant Brillouin scattering of exciton-polaritons from CdTe is reported. Selection rules are discussed and used to identify the scattering channel associated with each peak. The experimental Brillouin shifts are well reproduced by a model involving several exciton parameters which compare well with other available values.
INTRODUCTION
tion rules for exciton scattering before introducing the polariton concept. We then deduce the polariton and the corresponding parameters needed to fit the data. The excitonic dispersion can be obtained by the method of invariants. 6 The eight K=O exciton states formed with r 6 electrons and r 8 holes are
The original theQretical predictions of resonant Brillouin scattering (RBS) or exciton polaritons by Brenig et el. 1 have been tested successfully on a variety of semiconductors. 2 The measurements, which are now almost "classical", have been very useful in obtaining information on the exciton-polariton dispersion and exciton-phonon coupling mechanisms. We have studied the RBS of exciton-polaritons from high quality samples of CdTe in the [110] direction. Several transitions are followed through the resonance region. The spectra are more complicated than in the case of GaAs, 2 yet somewhat simpler than for CuBr. 3 We are faced with the delicate task of attributing correctly each peak to a given decay channel. To achieve this, scattering selection rules are deduced in more detail than previously established. W The polariton dispersion is then calculated and the parameters so obtained are compared with existing values. A full account of the selection rules together with further conclusions based on intensity calculations will be presented elsewhere. The standard RBS equipment using a double grating spectrometer as described in Ref. 2, has been used in these measurements. We studied freshly cleaved [110] surface samples of CdTe at 1OK in the backscattering geometry. The overall resolution of the system was 0.03 meV with incident and scattered beams polarized parallel to the [OO1] direction. The power of the incident beam was typically 4 mW. We also measured the normal reflectance during the course of the Brillouin scattering experiments. Our reflectance spectra are identical with those reported earlier except for a small shift in the absolute position of the peaks. Several Brillouin peaks are generally observed. They suggest a many branch polariton dispersion and the involvement of TA and LA phonons. The results are summarized in Fi~. 1.
decomposed into the dipole forbidden J=2 triplet state r3+r 4 at energy Eo, two transverse allowed J=l singlet states F5T at energy E=Eo+A and the longitudinal exciton at energy EL=Eo+~+~ ' . The dispersion involves terms quadratic in wavevector proportional to the effective exciton parameters ~i' Y2' and q which are defined by analogy to the 3 Luttinger parameters. In the perturbation approach 7 these are described by = mo
~i
2
W' 72 = Y2B'~' Y3 = Y3Bh a
in the notation used by Kane 7, where ~ is presently of the order of 0.7. K-linear terms, proportional to an effective constant C, are also present in the Hamiltonian. Following the magnetoreflectance studies of Dreybrodt et el. 5 such K-linear terms play an important role in the exciton dispersion of CdTe. For the scattering geometry chosen (KII[110] and [001] polarisation) four exciton branches couple with the incident light. 8 One of them is the longitudinal exciton, which for K~O becomes weakly dipole active due to the dispersive terms. In the present experiment we do not detect any effect which could be related to this exciton and therefore it will not be considered further. Furthermore, a recent measurement of the valence band parameters 9 and also the dependence of the RBS spectra on 13o_ larisation I0 shows that q2 and q3 have comparable values. In order to simplify the discussion we have used the approximation ~2--Y3. With these assumptions, the excitonic dispersion is given by the eigenvalues of the matrix
THEORY TO explain the Brillouin shifts we first discuss the excitonic dispersion as well as the selec+ ~resent address: Bell Laboratories, Holmdel, New Jersey 07733, U. S. A.
581
RESONANT
582
13>
12>
£ +26
SCATTERING
I1>
'-O.+ ~ 6
+~--6~
BRILLOUIN
~ +/-66
36
h - 3/-6~+ 36
where £ = ( ~ l + ~ 2 ) ~ 2 K 2 / 2 m o ; h =
~l-~2)~2K2/2mo
.
@ = C K / 2 ~ and 6--d/8. The wavefunctions I I> to j ~ are related to those defined in Ref. 6 by:
=
i//'2
-i/~
,7"J/2¢"~J~ 1+>-I I-> ) Iv
~/2~/\
I~"
For an arbitrary K, the exciton dispersion cannot be written down as a simple analytical expression. However, for large K, the quadratic and linear terms dominate over the exchange energy A and the exciton can then be classified simply into a light b~anch (branch 3) and a heavy branch. K-linear terms further split the heavy branch into two branches 1 and 2. In the large K limit, the exciton wavefunction being known, one can establish selection rules for the exciton scattering. This can be done by cons i d e r i n g t h e deformation p r o d u c e d by a given phonon in the long w a v e l e n g t h approximation. For the LA phonon it can be shown *0 that the only strong transitions are the intraband ones. An intraband transition relates two excitonic states having the same wavefunction. Note that due to the presence of Klinear terms ~n the matrix the eigenenergies associated with 1 and 2 are not symmetrical with respect to K. Therefore in terms of exciton transitions, an intrabranch backscattering process does not always connect two branches symmetric with respect to K=O. The TA phonon p o l a r i z e d along [O01] acts through both the deformation potential and the Fr6hlich mechanism (piezoelectric effect). In the resonance region, one can assume that the latter effect dominates. ~I Since the FrShlich interaction is diagonal, the TA scattering must also be an intraband process. The TA phonon polarized along [I[O] does not give any observable effect in agreement with selection rules. I0 We hence neglect this particular TA phonon in the following discussion. The above results are in agreement with those of Yamane and Cho. ~ The exciton-photon coupling will now be considered. This enables one to obtain the polariton dispersion curves as shown in Fig. 2. For large momentum, each polariton branch (except the highest one) is very close to a bare exciton and its properties are characterized by that exciton. In such a case, the selection rules established for the exciton transitions can be directly applied to the polariton. In Fig. 2 the polariton branches are labelled by Roman numbers. The dispersion for negative K-values can be obtained by reflection with respect to the K=O axis and corresponding primed values I', II'... will be used in this case. For large momenta, the exciton selection rules discussed earlier show that for both LA and T A phonons the polariton transitions III ~ III' (light-light), II + I' and I + II' (heavy-heavy) intraband processes must dominate. In the region very near E , both the exciton w a v e f u n c t i o n and the polariton~ are not simply described and thus clear selection rules cannot be deduced. DISCUSSION OF EXPERIMENTAL RESULTS The experimentally observed dispersion of the RBS peaks (Fig. i) is rich in structure. On the Stokes side we observe the following spectra. For low excitation energies E, two lines are observed which disappear with increasing E in a region ("gap") where no transitions are detected. Above the "gap" one follows two strong transitions whose intensities
IN C A D M I U M
Vol.
TELLURIDE
48, No.
peak rapidly and then diminish gradually at higher energies. They are labelled (II + I')LA and T A in Fig. i. Two other peaks are comparatively m u c h weaker and are strongest around EL: ( I I I ÷ III')LA and TA (Fig. I). vial i ,o t is trir l . This also determineS the sound velocity ratio VLA/VTA which is then used in the higher energy region. For E>>E we make use of the selection rules given above an~ conclude that the dispersion of the four dominant transitions are caused by II+I' and IIl÷III' intrabranch TA and LA scattering. For the remaining part one notices that near the "gap" there exists a region for E=E~, where the polariton curve must be flat (no variatlon of energy with wavevector, see Fig. 2). The flatness of the dispersion curve became evident when it was noted ex-
I
(~.m)
(~rlImmIi~m~l, (N~I
1
xx
E L -
I
~xx ',; \~,
x~ ~\
_>
x ~
xx
E
i",x • cn.n"l ~,\
\ EO= 1-59565eV \ x x
o
x
xxx ,
~
¢n,rlh
"~x
%
~',
o
',,(111,g')
-0.5
" -
T A (001)
"---~x~.~
I
I
I
-0.8
-0.6
-0.4
AE
l~x$
-0.2
0
(meV)
Experimental and calculated Brillouin shifts for K i I[110] and incident and scattered polarisations parallel to [001]. The x's represent experimental data. The solid and dashed lines are c a l c u l a t e d shifts with the parameters given in the text. perimentally that for Stokes II~I ' LA phonon scattering (strong intraband), the final energy state of the polariton remained unaltered for variation in the incident energy by about 0.2 meV. Small changes in the excitation energy near Ef drastically changed the anti-Stokes Brillouin spectrum. This also reflects the singular behaviour of the polariton dispersion. The position of the flat branch approximately fixes E . This together with the Brillouin shifts at high°energy enables us to establish the heavy and light exciton masses.
7
Vol. 48, No. 7
RESONANT
BRILLOUIN
SCATTERING
The K-linear terms have little influence on the calculated shifts near E . However, just above the "gap" they modify greatl~ the II+I ' and somewhat the III÷III' curves. K-linear terms are also necessary to create a flat portion of the lower polar±ton dispersion. The analysis of the Brillouin shifts versus energy enables us to determine the following parameters (correction for the refractive index of air is included) E = 1595.65 ± 0.025 meV ~o YI = 2.44 ± O.12; ~2 = ~3 = 0.887 ± 0.045 -7 C m (2.06 ± 0.10). 10 meV cm; 6 b = 11.2 A 0.075 + O.O5 meV; A'= 0.65 ± 0.05 meV. This analysis is consistent with the following sound velocities v LA = 3.39.105 cm sac -I and VTA = 1.79. iO s cm sac -I which are within 5% of the velocities obtained at low temperatures by ultrasonic measurements. 12 E b is the background dielectric constant. The theoretical shifts calculated with the above values are reproduced in Fig. i where the agreement with the experimental results is seen to be very satisfactory. The accuracy for the fit values of Yl' ~ and C is good because a slight variation of one o~ these parameters changes appreciably the calculated shifts, at least in a specific excitation range. Although E is also determined with good accuracy for a give~ sample, EL, the longitudinal exciton energy, is found with less precision (±0.05 meV). This uncertainty arises because the strong IV+IV' LA scattering (see Fig. I) cannot be detected very near E where the Brillouin shift is very small. The influence of ~' is mainly on the I÷I' shifts at low energy a~id on the IV+IV' shift at high energy. How-
I
I
I
I
I
K II [110] Pol [001]
1.0
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TELLURIDE
583
ever, an increase of 4' c a n easily be compensated by an identical decrease of ~ and a reduction of Cb. The parameters we obtain are in reasonable agreement with the values reported by Dreybrodt et al. 5 (~i = 2.17; ~2 = ~3 = 0.70; C = 1.92.10-' meV cm). An earlier measurement of RBS from CdTe 13 has been briefly reported. These limited results cannot be quantitatively compared with the present work. These have, however, been analysed in Ref. 2 where a value of the heavy exciton mass (~ -2~2)-| = 2.4 m has been reported. This value differs considerab~y from the value of 1.5 m reported here. However, the K-linear terms, whic~ strongly influence the dispersion, have been neglected in Ref. 2. This may explain the discrepancy. Several authors (Ref. 9 and other references therein) have determined the band parameters of CdTe by various methods and the following reliable values are now available: m* = O.O88 m ; y. = (5.3 ± 0.5); Y2 = (i .7 ± 0.3) ande73 = (2.0 ~ O.~) for the bands. With these parameters and using the results of Kane 7 we find for the excitons ~i = 2.62 ±0.30 and ~2 0.90 ± 0.20 which are in very reasonable agreement with our values. This indicates that the exciton dispersion seems to be well described by perturbation theory. Following Kane's theory, one also calculates an effective Rydberg R = 1.O5 R~ where R* is d e f i n e ~ for a non-degenerate valence ba~d of mas~ mo/Y~. The present situation for large m o m e n t u m is very similar to that encountered for CuBr in the [IIi] direction. I~ In both cases three exciton branches are present and the same kind of selection rules apply. In particular, the I+I ~ transitions are forbidden in the large K limit. 14 This selection rule is illustrated in Fig. I where the I+I' LA transitions observed at low energies do not reappear above the "gap". Note, however, that the RBS spectra are much less complicated in CdTe than in CuBr. 3 One probable reason is that the classification oF the excitons into light and heavy, which results in simple selection rules, is realized more easily in CdTe than in CuBr. This arises from the differences in the values of 4, y~, and C for the two compounds. The experiment~l results in CdTe seem closer to those for ZnSe. 15 However, the assignments of the Brillouin peaks are not fully consistent in the two cases. CONCLUSIONS
EL 0.5
>
E 0 Ill
'
ET
EF*
-05
""{i"/~ 0 0.5
l 1.0
J J I 1.5 2.0 2.5 WAVEVECTOR (10 6 cm -1)
Fis. 2: Exciton polar±ton dispersion in the [110] direction and for the [001] polarization. The curves are calculated with the parameters given in the text.
We have reported detailed EBS measurement for CdTe. For the assignment of each line we have b e n e fited from the existence of simple selection rules in regions where the experimental peaks are followed easily. Moreover, the experimentally found evidence for the existence of a flat p o r t i o n in the polar±ton dispersion - ~t energy E - imposes r a t h e r . f strong limitations on the pesszble values of some o f the parameters. The good agreement between the values of YI' Y2' and Y3 we have deduced and the v a lues calculated from known band parameters renders very improbably an error in the assignment we propose. CdTe, together with CuBr, are the best known examples where the K-linear terms are significant. It has been shown that such terms are necessary in the theory to reproduce the experimental results. The overall agreement between the calculated and experimental shifts places confidence in the values of the parameters deduced f r o m this m e a S U r e m e n t . These conclusions are further confirmed by a detailed discussion of the selection ruies and intensities which will be published elsewhere, t° ACKNOWLEDGEMENTS The excellent sam.plea of CdTe w e r e kindly supplied by R. Triboulet. Thanks are also due to P. Wurster, M. Siemers, and H. Hirt for technical assistance. Useful discussions with P.E. Simmonds are acknowledged. One of us (R.S.) would like to thank the Alexander yon Humboldt Foundation for support during the course of this work.
584
R E S O N A N T B R I L L O U I N S C A T T E R I N G IN C A D M I U M T E L L U R I D E
Vol. 48, No.
REFERENCES
i. 2.
3. 4.
5. 6. 7. 8.
BRENIG, W., ZEYHER, R., and BIRMAN, J.L., Phys. Rev. B 6, 4617 (1972). WEISBUCH, C. and ULBRICH, R.G., Light Scattering in Solids III, M. Cardona and G. GOntherodt, ede., Springer Verleg (1982), Chapt. 7. VU, D.P., OKA, Y., and CARDONA, M., Phys. Rev. B 24, 765 (1981). YAMANE, M. and CHO, K., Physica II7B & 118B, 377 (1983). (Proc. Int. Conf. of Semiconductors, Montpellier, France, 1 9 8 2 ) DREYBRODT, W., CHO, K., SUGA, S., WILLMAN, F., and NIJI, Y., Phys. Rev. B 2~I, 4692 (1980). CHO, K., Phys. Rev. B 14, 4463 (1976). KANE, E.O., Phys. Rev.-B i~i, 3850 (1975). H~NERLAGE, B., R~SSLER, U., V~, D.P., BIVAS, A.,
9. 10. 11.
12. 13.
14. 15.
and GRUN, J.B., Phys. Rev. B 22, 797 (1980). LE SI DANG, NEU, G., and ROMESTAIN, R., Solid State Commun. 44--, 1187 (1982). SOORYAKUMAR, R., MERLE, J.C., and CARDONA, M., to be published. CARDONA, M., Light Scattering in Solids II, M. Cardona and G. GOntherodt, eds., Springer Verlag, 1982, Chapt. 2. GREENOUGH, R.D. and PALMER, S.B., J. Phys. D, 587 (1973). NAKAMURA, A. and WEISBUCH, D., Abstract for the 35th meeting of the Phys. Soc. of Japan 2--, 211 (1980). CHO, K. and YAMANE, M., Solid State Commun. 4_~O, 121 (1981). SERMAGE, B. and FISHMAN, G., Phys. Rev. B 23, 5107 (1981).
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