- Email: [email protected]
Electronic band structure of quaternary Be-chalcogenides, studied by ultraviolet ellipsometry and photoreflectance spectroscopy
Journal of Crystal Growth 214/215 (2000) 340}344
Electronic band structure of quaternary Be-chalcogenides, studied by ultraviolet ellipsometry and photore#ectance spectroscopy V. Wagner *, A. Fleszar, J. Geurts , G. Reuscher , M. Keim , A. Waag , G. Landwehr , K. Wilmers, N. Esser, W. Richter Physikalisches Institut der Universita( t Wu( rzburg, Am Hubland, 97074 Wu( rzburg, Germany Institut fu( r Theoretische Physik der Universita( t Wu( rzburg, Am Hubland 97074, Wu( rzburg, Germany Institut fu( r Festko( rperphysik, Technische Universita( t Berlin, Hardenbergstrabe 36, 10623 Berlin, Germany
Abstract Beryllium chalcogenides are a new class of II}VI materials and promising candidates for UV/VIS applications. In this paper we analyse the optical properties of the quaternary BeMgZnSe system lattice matched to GaAs. We investigate the compositional dependence of the fundamental and higher energy gaps (E , E , E ). The investigations are performed (a) experimentally by photore#ectance and by ellipsometry up to 9.5 eV and (b) theoretically by "rst principles band structure calculations within the virtual crystal approximation (VCA). The fundamental energy gap is found to vary from 2.7 eV for ZnSe to 3.7 eV for a (Be,Mg)-content of 70%. The VCA calculations predict the correct overall gap dependencies. Especially, a negative bowing of the E gap is predicted by the VCA model. This unusual behaviour may be explained by the absence of bond length redistribution dynamics in this lattice matched system. 2000 Elsevier Science B.V. All rights reserved. PACS: 78.20.!e; 78.40.Fy; 71.20.Nr; 81.05.Dz Keywords: BeMgZnSe; VUV-ellipsometry; Photore#ectance; Beryllium
1. Introduction Beryllium chalcogenides take a very special place among II}VI compounds because beryllium (atomic number Z"4, atomic mass A"9.01 u) has only one "lled electron shell and the lowest
* Corresponding author. Tel.: #49-931-888-5782; fax: #49931-888-5142. E-mail address: [email protected] (V. Wagner).
mass of all group-II elements. This leads to a strong in#uence on physical properties such as bond polarity [1], electronic band structure [2] and lattice dynamics [3]. A continuous variation of these physical properties can be achieved by ternary Be-chalcogenides, e.g. Be Zn Se [2] or Be V \V V Mg Se. However, in these compounds variation \V of the composition induces signi"cant changes of the lattice constant. Thus, the lattice matching condition for epitaxial growth on GaAs(1 0 0) can be ful"lled only for one composition. In contrast to
0022-0248/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 0 0 ) 0 0 1 0 4 - 4
V. Wagner et al. / Journal of Crystal Growth 214/215 (2000) 340}344
that quaternary Be-chalcogenides, consisting of Be Mg Zn Se, o!er the possibility of a conW X \W\X tinuous variation of the fundamental energy gap from 2.7 to about 4.5 eV, while the lattice parameter can be kept constant if the compositional ratio of Be, Mg and Zn is chosen appropriately [4]. The gradual replacement of Zn by Be and Mg should induce signi"cant e!ects on the electronic band structure, not only the fundamental gap E , but also higher gaps such as E and E . The gap dynamics, e.g. bowing parameters, are of fundamental interest and besides they are relevant for optoelectronic applications. In this paper we present a theoretical and experimental analysis of the compositional dependence of the E , E , and E gap. In contrast to former studies samples up to very high Be and Mg fractions are analysed in the present work.
2. Experimental procedure The BeMgZnSe layers of 300 nm thickness were grown on GaAs(1 0 0) substrates in a Riber 2300 four-chamber MBE-system. After homoepitaxial bu!er growth the II}VI layers were grown by using elemental Be, Mg, Zn and Se e!usion cells. Six samples were fabricated corresponding to a nominal (Be,Mg)-fraction of 0%, 10%, 25%, 40%, 70% and 100%. Within the (Be,Mg)-fraction the compositional ratio Be : Mg was set to 67 : 33, as will be motivated in the next section. All samples, except the one with a 100% (Be,Mg)-fraction, showed during growth a clear RHEED-pattern. After growth a protective Se capping layer was deposited. The lattice match condition was checked by highresolution X-ray di!raction (HRXRD) to be ful"lled within *a/a(0.15%, except for the 70% and 100% (Be,Mg)-fraction sample, which could not be checked, since they were not stable in air ambient. The samples were transported in nitrogen atmosphere to the UHV chamber of the VUV ellipsometry chamber at the synchrotron BESSY-I in Berlin, which is described in detail elsewhere [2]. Prior to the ellipsometric measurements the Se cap was thermally desorped at 1903C. For the photoreflectance spectra a Jobin Yvon HR250 monochromator ( f"248 mm, grating 1200/mm) was
341
used, combined with a Xe-lamp and a silicon diode detector. A chopped (342 Hz) HeCd laser (j"325 nm) was used as a modulation source. The "rst-principles calculations were carried out selfconsistently by linear mixing of norm-conserving pseudopotentials [5] in the framework of the virtual crystal approximation (VCA) [6] and treating the valence electrons within the formalism of density functional theory [7,8] including the spin}orbit interaction. The wave functions were expanded in a planewave basis with kinetic energies up to 20 Ry.
3. Results and discussion For the discussion of quartenary Be Mg Zn Se system lattice matched to W X \W\X GaAs we use Vegard's law to calculate the allowed composition dependence parameterised with x"0}1. With literature data for the lattice constants at room temperature (a "5.667 As , 81 a "5.6533 As , a "5.152 As , a "5.904 As % 1 +1 for the zincblende structure) [9] we obtain the molar fractions of Be and Mg in dependence of parameter x to y(x)"y #(1!y !z )x and z(x)"z x with the constants y "(a !a )/ % 81 (a !a )"0.027; z "(a !a )/ 1 81 % 1 (a !a )"0.67. Since y is almost zero we +1 1 simplify the "rst relation to y(x)"(1!z )x and the composition variation for lattice matching to GaAs may be written as (Be Mg ) Zn Se. V \V Fig. 1 shows the theoretical prediction for the development of the band structure from x"0 (binary ZnSe) to 1 (Be Mg Se), based on the virtual crystal approximation. For all compositions we deal with direct semiconductors, whose fundamental gap is located at the !-point. Obviously, the energy of the E and E transitions increases. Its values shift from 2.0 to 2.7 eV and from 3.6 to 4.6 eV, respectively. In contrast, the E transition slightly decreases from 4.7 to 4.5 eV. All split-o! energies (* , * , * ) remain nearly unaltered at 0.4, 0.3 and 0.25 eV, respectively. This was expected, since the valence states are mainly located at the Se atoms, which are not modi"ed by compositional changes. In order to determine the quaternary composition of the samples, as a "rst step X-ray di!raction
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Fig. 1. Single-particle band structure of (Be Mg ) Zn Se V \V for (Be,Mg)-fraction x"0, 0.1, 0.25, 0.5, 0.7, 1.0, calculated within the framework of the virtual crystal approximation.
was applied to check their lattice matching to GaAs. Matching was con"rmed within *a/a( 0.15%. The determination of the remaining composition parameter, the (Be ,Mg )-fraction x,
is more di$cult, specially since beryllium is a very small atom and can hardly be detected with standard methods, e.g. electron probe microanalysis (EPMA). Instead, the calibrated MBE growth #ux ratios are used for the determination. Moreover, a "ne tuning was performed, based on the composition-induced shift of the E gap position. This gap shows only a very small bowing, both in our calculations and in experimental results for similar systems [2]. Therefore, we assumed a linear compositional dependence of the E gap position and corrected the nominal #ux-derived composition accordingly. The x-values after this correction amount to 0.18, 0.25, 0.32, 0.70. Fig. 2 shows the imaginary part of the dielectric function derived from the ellipsometry measurements for the di!erent compositions x. The spectra were evaluated in terms of an e!ective dielectric function 1e2, disregarding e!ects of the layer struc-
Fig. 2. Imaginary part of the e!ective dielectric function 1e2 of (Be ,Mg ) Zn /GaAs samples with various x values, V \V obtained from ellipsometry spectra at 300 K.
ture of the samples. Of course, this approximation is only justi"ed if the light penetration depth is below the BeMgZnSe-layer thickness of 300 nm. As a consequence, at low energies up to the E transition Fabry}Perot interferences lead to arti"cial oscillatory structures in 1e2. These structures impede the quantitative assignment of the E gap shift, although the general trend of a (Be,Mg) induced increase is clearly observed. Therefore, photore#ectance spectra were used here to determine the E transitions accurately. For higher photon energies the light penetration depth is far below the layer thickness and the positions of higher gaps can be determined from the ellipsometric spectra. For higher x values the gap structures become less pronounced due to the alloying induced broadening. Nevertheless, by a derivative analysis the E and E gap values could be evaluated with an accuracy of $50 meV. Further structures of higher gaps are observed beyond 7 eV.
V. Wagner et al. / Journal of Crystal Growth 214/215 (2000) 340}344
However, they are much less pronounced and will not be evaluated quantitatively. Note that the x"1 sample exhibits a structure which is quite di!erent from the others. This is in accordance with the di!erent behaviour during growth (no RHEED pattern, minor quality). Therefore, it will be neglected in the following discussion. Fig. 3 shows a comparison of the experimental values with the theoretical predictions. The positions of the E , E #* , E , E #* , E and E #d gaps are plotted versus parameter x. In order to compensate the well-known underestimation of the optical gap energies by LDA calculations, energy o!sets were added as indicated in the "gure. These o!sets were chosen such that for x"0 for the E , E , and E gap the experimental and theory values coincide. The required o!sets increase with the gap energy, as it is known from literature [10]. They amount to 0.738, 1.063 and 1.607 eV, respectively.
343
We found a good overall agreement for the predicted shift direction and magnitude. For the E -gap the experimentally observed bowing is somewhat more pronounced than the theoretical prediction. This results in a deviation of 0.2 eV for x"0.70. A larger deviation is found for the E gap. It amounts to 0.4 eV for x"0.70. The experimentally derived bowing parameter amounts to b "!1 eV. The negative bowing of the E -gap, which occurs in the theoretical as well as in the experimental data is a very interesting aspect of this material system, because it is in contrast to the positive bowing behaviour, observed in most mixed crystal systems. One exception is ZnHgSe, which also shows b(0 [11]. Bernard and Zunger [6] discuss the bowing behaviour in mixed crystals by analysing ordered II}VI alloys. In their model, the parameter b is described as a sum of contributions b #b , where b and b describe order and dis' '' ' '' order contributions to the bowing parameter, respectively. While b is often small and may '' be neglected, the b part consists of three summands ' (b "b #b #b ): volume deformation (VD), ' 4" !# 1 chemical electronegativity di!erence (CE) and structural (bond length) redistribution (S). Generally, b is positive and overwhelms the other two 1 contributions. However, b correlates with the 1 atomic size mismatch of the alloyed atoms and just this mismatch should compensate for the considered quaternary lattice matched system. The chemical electronegativity contribution b is gen!# erally negative and scales with the electronegativity di!erence of the alloyed atoms (Pauling scale: Be: 1.57, Mg: 1.31, Zn: 1.65). Thus, specially Mg gives a large e!ect. The last term b concerning the 4" volume deformation is again expected to be small in the lattice-matched system. From this analysis it becomes clear, that a negative bowing is expected in systems such as (Be Mg ) Zn Se, whose V \V lattice constant is x independent.
4. Conclusions Fig. 3. Compositional dependence of electronic band gaps of (Be ,Mg ) Zn . Open circles: calculations from LDA. V \V Full circles: experimental results from ellipsometry and photore#ectance spectroscopy.
We presented an optical analysis of the quaternary (BeMg) Zn Se system lattice matched to V \V GaAs. We determined the direct transitions up to E versus the composition parameter x. The
344
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observed trends are interpreted by a band structure analysis obtained by "rst principles calculations within the virtual crystal approximation. Both theory and experiment yield a negative bowing of the E gap. This unusual behaviour is explained by the absence of bond length redistribution dynamics in this lattice-matched system. Acknowledgements The BeMgZnSe samples analysed here were grown within the Sonderforschungsbereich 410 of the Deutsche Forschungsgemeinschaft. The "rstprinciples calculations shown were supported by the Sonderforschungsbereich 410 too. References [1] W. Harrison, Electronic Structure of Solids, Freeman, San Francisco, 1980.
[2] K. Wilmers, T. Wethkamp, N. Esser, C. Cobet, W. Richter, V. Wagner, H. Lugauer, F. Fischer, T. Gerhard, M. Keim, M. Cardona, Phys. Rev. B 59 (1999) 10071. [3] V. Wagner, J.J. Liang, R. Kruse, S. Gundel, M. Keim, A. Waag, J. Geurts, Phys. Status Solidi B 215 (1999) 87. [4] A. Waag, F. Fischer, H.J. Lugauer, Th. Litz, T. Gerhardt, J. NuK rnberger, U. Lunz, U. Zehnder, W. Ossau, G. Landwehr, B. Roos, H. Richter, Mater. Sci. Eng. B 43 (1997) 65. [5] G.B. Bachelet, D.R. Hamann, M. SchluK ter, Phys. Rev. B 26 (1982) 4199. [6] J.E. Bernard, A. Zunger, Phys. Rev. B 36 (1987) 3199. [7] P. Hohenberg, W. Kohn, Phys. Rev. B 136 (1964) 864. [8] W. Kohn, L.J. Sham, Phys. Rev. A 140 (1965) 1133. [9] Landolt-BoK rnstein, in: O. Madelung, U. RoK ssler, M. Schultz (Eds.), Numerical Data and Functional Relationships in Science and Technology, New Series, Group III, Vol. 41, Subvol. B, Springer, Berlin, 1999. [10] D. Theis, Phys. Status. Solidi. B 79 (1977) 125. [11] N.O. Gavaleshko, W. Dobrowolski, M. Baj, L. Dmowski, T. Dietl, V.V. Khomyak, Physics of Narrow Gap Semiconductors, Elsevier, Amsterdam, 1978, p. 331.
Electronic band structure of quaternary Be-chalcogenides, studied by ultraviolet ellipsometry and photore#ectance spectroscopy V. Wagner *, A. Fleszar, J. Geurts , G. Reuscher , M. Keim , A. Waag , G. Landwehr , K. Wilmers, N. Esser, W. Richter Physikalisches Institut der Universita( t Wu( rzburg, Am Hubland, 97074 Wu( rzburg, Germany Institut fu( r Theoretische Physik der Universita( t Wu( rzburg, Am Hubland 97074, Wu( rzburg, Germany Institut fu( r Festko( rperphysik, Technische Universita( t Berlin, Hardenbergstrabe 36, 10623 Berlin, Germany
Abstract Beryllium chalcogenides are a new class of II}VI materials and promising candidates for UV/VIS applications. In this paper we analyse the optical properties of the quaternary BeMgZnSe system lattice matched to GaAs. We investigate the compositional dependence of the fundamental and higher energy gaps (E , E , E ). The investigations are performed (a) experimentally by photore#ectance and by ellipsometry up to 9.5 eV and (b) theoretically by "rst principles band structure calculations within the virtual crystal approximation (VCA). The fundamental energy gap is found to vary from 2.7 eV for ZnSe to 3.7 eV for a (Be,Mg)-content of 70%. The VCA calculations predict the correct overall gap dependencies. Especially, a negative bowing of the E gap is predicted by the VCA model. This unusual behaviour may be explained by the absence of bond length redistribution dynamics in this lattice matched system. 2000 Elsevier Science B.V. All rights reserved. PACS: 78.20.!e; 78.40.Fy; 71.20.Nr; 81.05.Dz Keywords: BeMgZnSe; VUV-ellipsometry; Photore#ectance; Beryllium
1. Introduction Beryllium chalcogenides take a very special place among II}VI compounds because beryllium (atomic number Z"4, atomic mass A"9.01 u) has only one "lled electron shell and the lowest
* Corresponding author. Tel.: #49-931-888-5782; fax: #49931-888-5142. E-mail address: [email protected] (V. Wagner).
mass of all group-II elements. This leads to a strong in#uence on physical properties such as bond polarity [1], electronic band structure [2] and lattice dynamics [3]. A continuous variation of these physical properties can be achieved by ternary Be-chalcogenides, e.g. Be Zn Se [2] or Be V \V V Mg Se. However, in these compounds variation \V of the composition induces signi"cant changes of the lattice constant. Thus, the lattice matching condition for epitaxial growth on GaAs(1 0 0) can be ful"lled only for one composition. In contrast to
0022-0248/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 0 0 ) 0 0 1 0 4 - 4
V. Wagner et al. / Journal of Crystal Growth 214/215 (2000) 340}344
that quaternary Be-chalcogenides, consisting of Be Mg Zn Se, o!er the possibility of a conW X \W\X tinuous variation of the fundamental energy gap from 2.7 to about 4.5 eV, while the lattice parameter can be kept constant if the compositional ratio of Be, Mg and Zn is chosen appropriately [4]. The gradual replacement of Zn by Be and Mg should induce signi"cant e!ects on the electronic band structure, not only the fundamental gap E , but also higher gaps such as E and E . The gap dynamics, e.g. bowing parameters, are of fundamental interest and besides they are relevant for optoelectronic applications. In this paper we present a theoretical and experimental analysis of the compositional dependence of the E , E , and E gap. In contrast to former studies samples up to very high Be and Mg fractions are analysed in the present work.
2. Experimental procedure The BeMgZnSe layers of 300 nm thickness were grown on GaAs(1 0 0) substrates in a Riber 2300 four-chamber MBE-system. After homoepitaxial bu!er growth the II}VI layers were grown by using elemental Be, Mg, Zn and Se e!usion cells. Six samples were fabricated corresponding to a nominal (Be,Mg)-fraction of 0%, 10%, 25%, 40%, 70% and 100%. Within the (Be,Mg)-fraction the compositional ratio Be : Mg was set to 67 : 33, as will be motivated in the next section. All samples, except the one with a 100% (Be,Mg)-fraction, showed during growth a clear RHEED-pattern. After growth a protective Se capping layer was deposited. The lattice match condition was checked by highresolution X-ray di!raction (HRXRD) to be ful"lled within *a/a(0.15%, except for the 70% and 100% (Be,Mg)-fraction sample, which could not be checked, since they were not stable in air ambient. The samples were transported in nitrogen atmosphere to the UHV chamber of the VUV ellipsometry chamber at the synchrotron BESSY-I in Berlin, which is described in detail elsewhere [2]. Prior to the ellipsometric measurements the Se cap was thermally desorped at 1903C. For the photoreflectance spectra a Jobin Yvon HR250 monochromator ( f"248 mm, grating 1200/mm) was
341
used, combined with a Xe-lamp and a silicon diode detector. A chopped (342 Hz) HeCd laser (j"325 nm) was used as a modulation source. The "rst-principles calculations were carried out selfconsistently by linear mixing of norm-conserving pseudopotentials [5] in the framework of the virtual crystal approximation (VCA) [6] and treating the valence electrons within the formalism of density functional theory [7,8] including the spin}orbit interaction. The wave functions were expanded in a planewave basis with kinetic energies up to 20 Ry.
3. Results and discussion For the discussion of quartenary Be Mg Zn Se system lattice matched to W X \W\X GaAs we use Vegard's law to calculate the allowed composition dependence parameterised with x"0}1. With literature data for the lattice constants at room temperature (a "5.667 As , 81 a "5.6533 As , a "5.152 As , a "5.904 As % 1 +1 for the zincblende structure) [9] we obtain the molar fractions of Be and Mg in dependence of parameter x to y(x)"y #(1!y !z )x and z(x)"z x with the constants y "(a !a )/ % 81 (a !a )"0.027; z "(a !a )/ 1 81 % 1 (a !a )"0.67. Since y is almost zero we +1 1 simplify the "rst relation to y(x)"(1!z )x and the composition variation for lattice matching to GaAs may be written as (Be Mg ) Zn Se. V \V Fig. 1 shows the theoretical prediction for the development of the band structure from x"0 (binary ZnSe) to 1 (Be Mg Se), based on the virtual crystal approximation. For all compositions we deal with direct semiconductors, whose fundamental gap is located at the !-point. Obviously, the energy of the E and E transitions increases. Its values shift from 2.0 to 2.7 eV and from 3.6 to 4.6 eV, respectively. In contrast, the E transition slightly decreases from 4.7 to 4.5 eV. All split-o! energies (* , * , * ) remain nearly unaltered at 0.4, 0.3 and 0.25 eV, respectively. This was expected, since the valence states are mainly located at the Se atoms, which are not modi"ed by compositional changes. In order to determine the quaternary composition of the samples, as a "rst step X-ray di!raction
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Fig. 1. Single-particle band structure of (Be Mg ) Zn Se V \V for (Be,Mg)-fraction x"0, 0.1, 0.25, 0.5, 0.7, 1.0, calculated within the framework of the virtual crystal approximation.
was applied to check their lattice matching to GaAs. Matching was con"rmed within *a/a( 0.15%. The determination of the remaining composition parameter, the (Be ,Mg )-fraction x,
is more di$cult, specially since beryllium is a very small atom and can hardly be detected with standard methods, e.g. electron probe microanalysis (EPMA). Instead, the calibrated MBE growth #ux ratios are used for the determination. Moreover, a "ne tuning was performed, based on the composition-induced shift of the E gap position. This gap shows only a very small bowing, both in our calculations and in experimental results for similar systems [2]. Therefore, we assumed a linear compositional dependence of the E gap position and corrected the nominal #ux-derived composition accordingly. The x-values after this correction amount to 0.18, 0.25, 0.32, 0.70. Fig. 2 shows the imaginary part of the dielectric function derived from the ellipsometry measurements for the di!erent compositions x. The spectra were evaluated in terms of an e!ective dielectric function 1e2, disregarding e!ects of the layer struc-
Fig. 2. Imaginary part of the e!ective dielectric function 1e2 of (Be ,Mg ) Zn /GaAs samples with various x values, V \V obtained from ellipsometry spectra at 300 K.
ture of the samples. Of course, this approximation is only justi"ed if the light penetration depth is below the BeMgZnSe-layer thickness of 300 nm. As a consequence, at low energies up to the E transition Fabry}Perot interferences lead to arti"cial oscillatory structures in 1e2. These structures impede the quantitative assignment of the E gap shift, although the general trend of a (Be,Mg) induced increase is clearly observed. Therefore, photore#ectance spectra were used here to determine the E transitions accurately. For higher photon energies the light penetration depth is far below the layer thickness and the positions of higher gaps can be determined from the ellipsometric spectra. For higher x values the gap structures become less pronounced due to the alloying induced broadening. Nevertheless, by a derivative analysis the E and E gap values could be evaluated with an accuracy of $50 meV. Further structures of higher gaps are observed beyond 7 eV.
V. Wagner et al. / Journal of Crystal Growth 214/215 (2000) 340}344
However, they are much less pronounced and will not be evaluated quantitatively. Note that the x"1 sample exhibits a structure which is quite di!erent from the others. This is in accordance with the di!erent behaviour during growth (no RHEED pattern, minor quality). Therefore, it will be neglected in the following discussion. Fig. 3 shows a comparison of the experimental values with the theoretical predictions. The positions of the E , E #* , E , E #* , E and E #d gaps are plotted versus parameter x. In order to compensate the well-known underestimation of the optical gap energies by LDA calculations, energy o!sets were added as indicated in the "gure. These o!sets were chosen such that for x"0 for the E , E , and E gap the experimental and theory values coincide. The required o!sets increase with the gap energy, as it is known from literature [10]. They amount to 0.738, 1.063 and 1.607 eV, respectively.
343
We found a good overall agreement for the predicted shift direction and magnitude. For the E -gap the experimentally observed bowing is somewhat more pronounced than the theoretical prediction. This results in a deviation of 0.2 eV for x"0.70. A larger deviation is found for the E gap. It amounts to 0.4 eV for x"0.70. The experimentally derived bowing parameter amounts to b "!1 eV. The negative bowing of the E -gap, which occurs in the theoretical as well as in the experimental data is a very interesting aspect of this material system, because it is in contrast to the positive bowing behaviour, observed in most mixed crystal systems. One exception is ZnHgSe, which also shows b(0 [11]. Bernard and Zunger [6] discuss the bowing behaviour in mixed crystals by analysing ordered II}VI alloys. In their model, the parameter b is described as a sum of contributions b #b , where b and b describe order and dis' '' ' '' order contributions to the bowing parameter, respectively. While b is often small and may '' be neglected, the b part consists of three summands ' (b "b #b #b ): volume deformation (VD), ' 4" !# 1 chemical electronegativity di!erence (CE) and structural (bond length) redistribution (S). Generally, b is positive and overwhelms the other two 1 contributions. However, b correlates with the 1 atomic size mismatch of the alloyed atoms and just this mismatch should compensate for the considered quaternary lattice matched system. The chemical electronegativity contribution b is gen!# erally negative and scales with the electronegativity di!erence of the alloyed atoms (Pauling scale: Be: 1.57, Mg: 1.31, Zn: 1.65). Thus, specially Mg gives a large e!ect. The last term b concerning the 4" volume deformation is again expected to be small in the lattice-matched system. From this analysis it becomes clear, that a negative bowing is expected in systems such as (Be Mg ) Zn Se, whose V \V lattice constant is x independent.
4. Conclusions Fig. 3. Compositional dependence of electronic band gaps of (Be ,Mg ) Zn . Open circles: calculations from LDA. V \V Full circles: experimental results from ellipsometry and photore#ectance spectroscopy.
We presented an optical analysis of the quaternary (BeMg) Zn Se system lattice matched to V \V GaAs. We determined the direct transitions up to E versus the composition parameter x. The
344
V. Wagner et al. / Journal of Crystal Growth 214/215 (2000) 340}344
observed trends are interpreted by a band structure analysis obtained by "rst principles calculations within the virtual crystal approximation. Both theory and experiment yield a negative bowing of the E gap. This unusual behaviour is explained by the absence of bond length redistribution dynamics in this lattice-matched system. Acknowledgements The BeMgZnSe samples analysed here were grown within the Sonderforschungsbereich 410 of the Deutsche Forschungsgemeinschaft. The "rstprinciples calculations shown were supported by the Sonderforschungsbereich 410 too. References [1] W. Harrison, Electronic Structure of Solids, Freeman, San Francisco, 1980.
[2] K. Wilmers, T. Wethkamp, N. Esser, C. Cobet, W. Richter, V. Wagner, H. Lugauer, F. Fischer, T. Gerhard, M. Keim, M. Cardona, Phys. Rev. B 59 (1999) 10071. [3] V. Wagner, J.J. Liang, R. Kruse, S. Gundel, M. Keim, A. Waag, J. Geurts, Phys. Status Solidi B 215 (1999) 87. [4] A. Waag, F. Fischer, H.J. Lugauer, Th. Litz, T. Gerhardt, J. NuK rnberger, U. Lunz, U. Zehnder, W. Ossau, G. Landwehr, B. Roos, H. Richter, Mater. Sci. Eng. B 43 (1997) 65. [5] G.B. Bachelet, D.R. Hamann, M. SchluK ter, Phys. Rev. B 26 (1982) 4199. [6] J.E. Bernard, A. Zunger, Phys. Rev. B 36 (1987) 3199. [7] P. Hohenberg, W. Kohn, Phys. Rev. B 136 (1964) 864. [8] W. Kohn, L.J. Sham, Phys. Rev. A 140 (1965) 1133. [9] Landolt-BoK rnstein, in: O. Madelung, U. RoK ssler, M. Schultz (Eds.), Numerical Data and Functional Relationships in Science and Technology, New Series, Group III, Vol. 41, Subvol. B, Springer, Berlin, 1999. [10] D. Theis, Phys. Status. Solidi. B 79 (1977) 125. [11] N.O. Gavaleshko, W. Dobrowolski, M. Baj, L. Dmowski, T. Dietl, V.V. Khomyak, Physics of Narrow Gap Semiconductors, Elsevier, Amsterdam, 1978, p. 331.