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On the calculation of dielectric and optical properties of wide band gap semiconductors
HAtERlAlS SCIENCE ENGINEERING ELSEVIER
B
Materials Science and Engineering B46 (1997) 99- 100
On the calculation
of dielectric and optical properties of wide band gap semiconductors
Sergey K. Tikhonov Physico
&
Technical
Institute,
*, Sergey Yu. Davydov, Politekhnicheskaya
26, St. Petersburg
A.F. Ioffe 194021,
Russia
Abstract The bond orbital model has been used to calculate linear and quadric static permittivity for diamond and cubic modification of Sic, BN, AlN, Keywords:
dielectric susceptibility, electro-optical coefficient and and GaN. 0 1997 Published by Elsevier Science S.A.
Bond orbital model; Optical properties; Semiconductors
The well known bond orbital model (BOM) of Harrison [I -31 is a simple but a direct first principle method to calculate a wide range of semiconductor properties. The extended version of BOM, taking into account both electronic and ionic subsystems contributions and metallicity had been presented in Refs. [4,5] for the calculation of semiconductor dielectric and optical properties. Here we applied this extended version of BOM to describe the following characteristics for the cubic modifications of a number of wide band gap semiconductors: linear X, = X;’ + X’p” and quadric Xi, = X;f, + XI;o4” dielectric susceptibilities (here X” and Xion refers to electronic and ionic subsystems contributions), electrooptical coefficient r4,, high frequency E, and static ~0 permittivities, and parameter 11= a(ln X;‘)/a(ln d) (d is the bond length). Omitting the metallicity effect as a first step of our analytical form calculations we obtain the results listed in Table 1 (CX~and t?‘, stand for polarity or ionicity and covalency, respectively). The geometrical parameters were taken from Ref. [1], the energy parameters, from Ref. [3], and the experimental values, from Refs. [1,7-
91.
We chose for the parameter y, accounting the overlapping and field effects [ 11,the value 1.27, which is the mean of all the compounds considered here. Note that when used within the nearest-neighbor approach, the BOM can be applied to describe not only sphalerite structures but wurtzite structures in the ‘cubic approximation’ as well [l]. A comparison of the calculated E, and ~0 with * Corresponding author.
experimental data shows that while for covalent crystals (C, Sic) the calculated values of E, lie below the experimental figures, for the nitrides, which have a higher polarity, they lie above. The theoretical values of s,, are underestimated relative to the experimental values, but the agreement for the ionic compounds appears to be better than for the covalent ones. The published value of q for diamond [6] is 2.2, which is in good agreement with our value of 2. As follows from Ref. [4], the theory shows much better consistency with experimental data on q for low-polarity crystals. This warrants the conclusion that 1;7= 1.59 for Si C is a reliable value. As for the high-polarity compounds, here the disagreement between theory and experiment may be quite large. The value X2 = 2.88 x 10m8 cgs electrostatic units found for GaN [lo] is in satisfactory agreement with our result. Note that the theory [l 11 based on the model of Phillips-Van Vechten eves a value 22 times smaller than experiment. We have not been successful in finding any experimental information on r4r. The results of extended BOM calculations with the account of metallicity are presented in Table 2. Here a, is metallicity, g, and g,, are the metallicity correction factors for Xt’ and X$, respectively (without account of metallicity g, = g,, = 1). The parameter y was determined as this was done by Harrison in ch. 4 of Ref. [l]. The correction factors g, andg,, increase the values of susceptibility for the covalent crystals and reduce them for ionic crystals. The inclusion of metallicity significantly affects, in particular, the parameter 17 for crystals with a high bond polarity. It should be pointed out that taking into account the metallicity is identical to the inclusion of band effects.
0921-5107/97/$17.00 0 1997 Published by Elsevier Science S.A. All rights reserved PIISO921-5107(96)01941-l
100
S.K.
Tikhonor:
et al. /Materials
Science
Table 1 Coupling parameters and values of the dielectric and optical characteristics of wide band gap semiconductors, X;‘, X,, X$, X,4 and given in low8 cgs electrostatic units C
Sic
BN
0 1
0.26 0.97
0.34 0.94
AIN
and Engineering
aP
Theory x;’ ccc 4 Eo
0.59 0.81
C
Sic
BN
AlN
GaN
0.41 1.09 1.22 1.33 5.70
0.56 1.11 1.28 1.51 1.87 1.78 0.85
0.46 1.04 1.14 1.33 5.03 1.25 0.29
0.65 0.88 0.88 1.51 4.15 - 1.24 0.66
0.72 0.85 0.83 1.62 4.61 - 1.63 0.83
GaN O-60 0.80
0.31 4.90 0.31 4.90
0.38 5.83 0.43 6.36
0,32 5.02 0.38 5.83
0.38 5.72 0.73 10.12
0.40 5.03 0.80 11.05
l-;l, x14
-2 0
3.87 1.59 0.27
r41
-
0.10
2.18 1.31 0.29 0.23
- 4.36 0.09 2.31 1.30
-0.16 4.68 2.63 1.31
6.5 9.1
4.5 -
4.8 -
5.s 12.2
Experiment & 5.7 % 5.1
99-100
Table 2 Results of calculations of the dielectric and optical properties of wide band gap semiconductors, XT; given in 10m7 cgs electrostatic units
“m
%
B46 (1997)
g1 g14
Y 8,
References
In summary, we have calculated the dielectric and optical characteristics of wide band gap materials by the BOM, which was successfully employed in describing many properties of semiconductors. This method permits circumventing complicated computations that are usually difficult to avoid in calculations of dielectric and optical properties [12-141, and this is particularly useful in trying to predict and estimate the properties of new materials. The satisfactory agreement of our results with the available experimental data, regrettably very sparse, gives us grounds to hope that the values of physical characteristics predicted by us will likewise prove to be reasonable.
Acknowledgements
This work was supported by Arizona (USA) and INTAS grant 93-543.
University
[l] W.A. Harrison, Elecfronic Stmctnre and the Properties ofSolids, Freeman, San Francisco, 1980. [2] W.A. Harrison, Phys. Reo. B, 24 (1981) 5835. [3] W.A. Harrison, Plys. Rer. B, -77 (1983) 3592. [4] S.Yu. Davydov and E.I. Leonov, Sov. Ply. Solid State, 29 (1987) 1662. [g S.Yu. Davydov and E.I. Leonov, Sov. Phys. Solid State, 30 (1988) 768. [6] M. Kastner, Phys. Rev. B, 6 (1972) 2273. [7j I.K. Kikoin (ed.), Tables of Physical @mriries: A Handbook, Atomizdat, Moscow, 1976. [8] V.I. Gavrilenko, A.M. Grekhov, D.V. Korbutyak and V.G. Litovchenko, Optical Properties of Semiconductors: A Handbook, Naukova Dumka, Kiev, 1987 (in Russian). [9] IS. Grigor’ev and E.Z. Meilokhov (eds.), Physical Properties: A Handbook, Energoizdat, Moscow, 1991. IlO] J. Miragliotta, W.A. Bryden, T.J. Kistenmacher and D.K. Wickenden, in MC. Spencer et al. (eds.), Proc. 5th Co& Silicon Carbide afld Related Materials, Institute of Physics Conference Series, no. 137, Bristol, 1993, p. 537. [II] B.F. Levine, Phys. Rer. B, 7 (1973) 2600. 1121 D.J. Moss, J.E. Sipe and H.M. van Driel, Phys. Reu. B, 36 (1987) 9708. [13] Ya.0. Dovgii and I.V. Kityk, Sou. Phy.s. Solid Stare, 33 (1991) 238. [14] I.V. Kityk, Ser. Phys. Solid State, 33 (1991) 1026.
B
Materials Science and Engineering B46 (1997) 99- 100
On the calculation
of dielectric and optical properties of wide band gap semiconductors
Sergey K. Tikhonov Physico
&
Technical
Institute,
*, Sergey Yu. Davydov, Politekhnicheskaya
26, St. Petersburg
A.F. Ioffe 194021,
Russia
Abstract The bond orbital model has been used to calculate linear and quadric static permittivity for diamond and cubic modification of Sic, BN, AlN, Keywords:
dielectric susceptibility, electro-optical coefficient and and GaN. 0 1997 Published by Elsevier Science S.A.
Bond orbital model; Optical properties; Semiconductors
The well known bond orbital model (BOM) of Harrison [I -31 is a simple but a direct first principle method to calculate a wide range of semiconductor properties. The extended version of BOM, taking into account both electronic and ionic subsystems contributions and metallicity had been presented in Refs. [4,5] for the calculation of semiconductor dielectric and optical properties. Here we applied this extended version of BOM to describe the following characteristics for the cubic modifications of a number of wide band gap semiconductors: linear X, = X;’ + X’p” and quadric Xi, = X;f, + XI;o4” dielectric susceptibilities (here X” and Xion refers to electronic and ionic subsystems contributions), electrooptical coefficient r4,, high frequency E, and static ~0 permittivities, and parameter 11= a(ln X;‘)/a(ln d) (d is the bond length). Omitting the metallicity effect as a first step of our analytical form calculations we obtain the results listed in Table 1 (CX~and t?‘, stand for polarity or ionicity and covalency, respectively). The geometrical parameters were taken from Ref. [1], the energy parameters, from Ref. [3], and the experimental values, from Refs. [1,7-
91.
We chose for the parameter y, accounting the overlapping and field effects [ 11,the value 1.27, which is the mean of all the compounds considered here. Note that when used within the nearest-neighbor approach, the BOM can be applied to describe not only sphalerite structures but wurtzite structures in the ‘cubic approximation’ as well [l]. A comparison of the calculated E, and ~0 with * Corresponding author.
experimental data shows that while for covalent crystals (C, Sic) the calculated values of E, lie below the experimental figures, for the nitrides, which have a higher polarity, they lie above. The theoretical values of s,, are underestimated relative to the experimental values, but the agreement for the ionic compounds appears to be better than for the covalent ones. The published value of q for diamond [6] is 2.2, which is in good agreement with our value of 2. As follows from Ref. [4], the theory shows much better consistency with experimental data on q for low-polarity crystals. This warrants the conclusion that 1;7= 1.59 for Si C is a reliable value. As for the high-polarity compounds, here the disagreement between theory and experiment may be quite large. The value X2 = 2.88 x 10m8 cgs electrostatic units found for GaN [lo] is in satisfactory agreement with our result. Note that the theory [l 11 based on the model of Phillips-Van Vechten eves a value 22 times smaller than experiment. We have not been successful in finding any experimental information on r4r. The results of extended BOM calculations with the account of metallicity are presented in Table 2. Here a, is metallicity, g, and g,, are the metallicity correction factors for Xt’ and X$, respectively (without account of metallicity g, = g,, = 1). The parameter y was determined as this was done by Harrison in ch. 4 of Ref. [l]. The correction factors g, andg,, increase the values of susceptibility for the covalent crystals and reduce them for ionic crystals. The inclusion of metallicity significantly affects, in particular, the parameter 17 for crystals with a high bond polarity. It should be pointed out that taking into account the metallicity is identical to the inclusion of band effects.
0921-5107/97/$17.00 0 1997 Published by Elsevier Science S.A. All rights reserved PIISO921-5107(96)01941-l
100
S.K.
Tikhonor:
et al. /Materials
Science
Table 1 Coupling parameters and values of the dielectric and optical characteristics of wide band gap semiconductors, X;‘, X,, X$, X,4 and given in low8 cgs electrostatic units C
Sic
BN
0 1
0.26 0.97
0.34 0.94
AIN
and Engineering
aP
Theory x;’ ccc 4 Eo
0.59 0.81
C
Sic
BN
AlN
GaN
0.41 1.09 1.22 1.33 5.70
0.56 1.11 1.28 1.51 1.87 1.78 0.85
0.46 1.04 1.14 1.33 5.03 1.25 0.29
0.65 0.88 0.88 1.51 4.15 - 1.24 0.66
0.72 0.85 0.83 1.62 4.61 - 1.63 0.83
GaN O-60 0.80
0.31 4.90 0.31 4.90
0.38 5.83 0.43 6.36
0,32 5.02 0.38 5.83
0.38 5.72 0.73 10.12
0.40 5.03 0.80 11.05
l-;l, x14
-2 0
3.87 1.59 0.27
r41
-
0.10
2.18 1.31 0.29 0.23
- 4.36 0.09 2.31 1.30
-0.16 4.68 2.63 1.31
6.5 9.1
4.5 -
4.8 -
5.s 12.2
Experiment & 5.7 % 5.1
99-100
Table 2 Results of calculations of the dielectric and optical properties of wide band gap semiconductors, XT; given in 10m7 cgs electrostatic units
“m
%
B46 (1997)
g1 g14
Y 8,
References
In summary, we have calculated the dielectric and optical characteristics of wide band gap materials by the BOM, which was successfully employed in describing many properties of semiconductors. This method permits circumventing complicated computations that are usually difficult to avoid in calculations of dielectric and optical properties [12-141, and this is particularly useful in trying to predict and estimate the properties of new materials. The satisfactory agreement of our results with the available experimental data, regrettably very sparse, gives us grounds to hope that the values of physical characteristics predicted by us will likewise prove to be reasonable.
Acknowledgements
This work was supported by Arizona (USA) and INTAS grant 93-543.
University
[l] W.A. Harrison, Elecfronic Stmctnre and the Properties ofSolids, Freeman, San Francisco, 1980. [2] W.A. Harrison, Phys. Reo. B, 24 (1981) 5835. [3] W.A. Harrison, Plys. Rer. B, -77 (1983) 3592. [4] S.Yu. Davydov and E.I. Leonov, Sov. Ply. Solid State, 29 (1987) 1662. [g S.Yu. Davydov and E.I. Leonov, Sov. Phys. Solid State, 30 (1988) 768. [6] M. Kastner, Phys. Rev. B, 6 (1972) 2273. [7j I.K. Kikoin (ed.), Tables of Physical @mriries: A Handbook, Atomizdat, Moscow, 1976. [8] V.I. Gavrilenko, A.M. Grekhov, D.V. Korbutyak and V.G. Litovchenko, Optical Properties of Semiconductors: A Handbook, Naukova Dumka, Kiev, 1987 (in Russian). [9] IS. Grigor’ev and E.Z. Meilokhov (eds.), Physical Properties: A Handbook, Energoizdat, Moscow, 1991. IlO] J. Miragliotta, W.A. Bryden, T.J. Kistenmacher and D.K. Wickenden, in MC. Spencer et al. (eds.), Proc. 5th Co& Silicon Carbide afld Related Materials, Institute of Physics Conference Series, no. 137, Bristol, 1993, p. 537. [II] B.F. Levine, Phys. Rer. B, 7 (1973) 2600. 1121 D.J. Moss, J.E. Sipe and H.M. van Driel, Phys. Reu. B, 36 (1987) 9708. [13] Ya.0. Dovgii and I.V. Kityk, Sou. Phy.s. Solid Stare, 33 (1991) 238. [14] I.V. Kityk, Ser. Phys. Solid State, 33 (1991) 1026.