Strain energy and stability of Si-Ge compounds, alloys, and superlattices

James E. Bernard; Alex Zunger

doi:10.1103/PhysRevB.44.1663

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Strain energy and stability of Si-Ge compounds, alloys, and superlattices

James E. Bernard

Alex Zunger

2113 West Creighton Drive, Golden, Colorado 80401

Solar Energy Research Institute, Golden, Colorado 80401

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Phys. Rev. B 44, 1663 – Published 15 July, 1991

DOI: https://doi.org/10.1103/PhysRevB.44.1663

Abstract

First-principles total-energy pseudopotential calculations are carried out for Si, Ge, zinc-blende-structure SiGe, ( ${Si}_{2}$ $)_{p}$ /( ${Ge}_{2}$ $)_{p}$ superlattices in various layer orientations G and with various choices of substrate lattice parameter $a_{s}$ , and for the ${Si}_{0.5}$ ${Ge}_{0.5}$ random alloy. A subset of the results is used to construct an energy model, incorporating both strain (via an anharmonic valence force field) and chemical interactions (via a rapidly convergent cluster expansion) that closely reproduces the first-principles results, including those not used as input to the model. The model is applied to the study of larger superlattices than are amenable to first-principles treatment, revealing trends in (i) constituent strain energies, (ii) interfacial ‘‘strain-relief’’ relaxation energies, and (iii) interfacial chemical energies. The analysis reveals the major regularities in the dependence of superlattice stability on {p,G, $a_{s}$ }, and permits investigation of the nature of interactions at interfaces, including the substrate-film interface.

References (52)

M. Ospelt, W. Bacsa, J. Henz, K. A. Mäder and H. von Känel, Superlatt. Microstruct. 5, 71 (1989).
E. Kasper, H. Kibbel, H. Jorke, H. Brugger, E. Friess and G. Abstreiter, Phys. Rev. B 38, 3599 (1988).
A. T. Fiory, J. C. Bean, R. Hull and S. Nakahara, Phys. Rev. B 31, 4063 (1985); T. P. Pearsall, J. Bevk, J. Bonar, M. P. Mannaerts and A. Ourmazd, ibid. 39, 3741 (1989).
S. J. Chang, K. L. Wang, R. C. Bowman and P. M. Adams, Appl. Phys. Lett. 54, 27 (1989).
E. Friess, H. Brugger, K. Eberl, G. Krötz and G. Abstreiter, Solid State Commun. 69, 899 (1989); K. Eberl, G. Krötz, R. Zachai, and G. Abstreiter,, J. Phys. (Paris) Colloq. 48, C5-S11 (1987); ibid. 48, C5-329 (1987).
D. Brasen, S. Nakahara and J. C. Bean, J. Appl. Phys. 58, 1860 (1985).
T. Narusawa and W. M. Gibson, Phys. Rev. Lett. 47, 1459 (1981).
S. Froyen, D. M. Wood and A. Zunger, Phys. Rev. B 36, 4547 (1987); Appl. Phys. Lett. 54, 2435 (1989).
S. Froyen, D. M. Wood and A. Zunger, Phys. Rev. B 37, 6893 (1988).
T. P. Pearsall, CRC Critical Reviews on Solid State and Materials Sciences (Chemical Rubber Company, Boca Raton, FL, 1989), Vol. 15, p. 551; J. Lumin. 44, 367 (1989).
J. L. Martins and A. Zunger, Phys. Rev. B 30, 6217 (1984).
J. L. Martins and A. Zunger, Phys. Rev. Lett. 56, 1400 (1986).
S. Ciraci and I. P. Batra, Phys. Rev. B 38, 1835 (1988).
B. Koiller and M. O. Robbins, Phys. Rev. B 40, 12554 (1989).
P. B. Littlewood, Phys. Rev. B 34, 1363 (1986).
S.-H. Wei and A. Zunger, Phys. Rev. Lett. 61, 1505 (1988).
R. G. Dandrea, J. E. Bernard, S.-H Wei and A. Zunger, Phys. Rev. Lett. 64, 36 (1990).
A. Ourmazd and J. C. Bean, Phys. Rev. Lett. 55, 765 (1985).
F. K. LeGoues, V. P. Kesan and S. S. Iyer, Phys. Rev. Lett. 64, 40 (1990).
For the rhombohedral structures we use the term ``bilayer'' to refer to pairs of (111) layers separated by a distance of a /4 sqrt 3 in the diamond structure, where a is the lattice constant. These bilayers are separated from each other by the larger distance a sqrt 3 /4.
There is also a third rhombohedral structure, RH3, which has half-integral p, and one mixed-bilayer and one pure-bilayer interface. This structure has energy intermediate between those of RH1 and RH2 and is not considered here.
J. Ihm, A. Zunger and M. L. Cohen, J. Phys. C 12, 4409 (1979).
W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).
J. E. Bernard, L. G. Ferreira, S.-H. Wei and A. Zunger, Phys. Rev. B 38, 6338 (1988).
For example, Ciraci and Batra (Ref. 13) partitioned the formation enthalpy into a bulklike strain term analogous to our Δ $E_{CS}$ and an interfacelike term attributed to the chemical process associated with the formation of the SibondGe bonds at the interface, and hence analogous to our $I_{CE}$ .
P. N. Keating, Phys. Rev. 145, 637 (1966).
R. M. Martin, Phys. Rev. B 1, 4005 (1970).
W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (Cambridge University Press, Cambridge, 1986), pp. 301-306.
A. Zunger, S.-H. Wei, L. G. Ferreira and J. E. Bernard, Phys. Rev. Lett. 65, 353 (1990); S.-H. Wei, L. G. Ferreira, J. E. Bernard and A. Zunger, Phys. Rev. B 42, 9622 (1990); S. H. Wei and A. Zunger, Appl. Phys. Lett. 56, 662 (1990); K. Hass, L. C. Davis and A. Zunger, Phys. Rev. B 42, 3757 (1990).
S. Froyen, Phys. Rev. B 39, 3168 (1989).
F. Birch, Phys. Rev. 71, 809 (1947); J. Geophys. Res. 57, 227 (1952).
D. M. Wood and A. Zunger, Phys. Rev. B 40, 4062 (1989).
R. G. Dandrea (private communication).
L. G. Ferreira, S.-H. Wei and A. Zunger, Phys. Rev. B 40, 3197 (1989); S.-H. Wei, L. G. Ferreira and A. Zunger, ibid. 40, 8240 (1990).
J. W. D. Connolly and A. R. Williams, Phys. Rev. B 27, 5169 (1983).
J. M. Sanchez, F. Ducastelle and D. Gratias, Physica A 128, 334 (1984).
In this section we use the term ``lattice'' in the less strict (but traditional in the literature on cluster expansions) sense of an array of empty sites, with no restriction to being equivalent under the operations of some translation group.
L. G. Ferreira (private communication).
Essentially full convergence for Si ${}_{0.5}{Ge}_{0.5}$ is reached with six interactions, including up to fourth-nearest-neighbor pairs and four-body second-nearest-neighbor interactions, though two of these (four-body first-nearest-neighbor and two-body third-nearest-neighbor) are smaller than the others and can be eliminated with little additional error.
S. de Gironcoli, S. Baroni, and P. Giannozzi, in Proceedings of the International Conference on the Physics of Semiconductors, Thessaloniki, 1990, edited by E. M. Anastussakis and J. D. Joannopoulos (World Scientific, Singapore, 1990), Vol. 2, p. 877.
In terms of the excess bond energy delta of the SibondGe bond relative to equivalent amounts of SibondSi and GebondGe bonds, Δ $E_{chem}$ ${}^{(α)}$ = $N_{α}$ δ /N, where $N_{alpha}$ is the number of SibondGe bonds per unit cell, and N is the number of atoms per unit cell. Since $N_{α}$ /N = - scrP ${bar}_{2, 1}$ (Table VI), the excess bond energy is just δ = - $D_{2, 1}$ $J_{2, 1}$ .
P. C. Kelires and J. Tersoff, Phys. Rev. Lett. 63, 1164 (1989).
A. Qteish and R. Resta, Phys. Rev. B 37, 1308 (1988); ibid. 37, 6983 (1988).
G. B. Stringfellow, J. Phys. Chem. Solids 34, 1749 (1973); G. B. Stringfellow and P. E. Greene, J. Electrochem. Soc. 117, 1075 (1970).
We have found, for example, that in our calculations with a 20-Ry cutoff energy, the formation enthalpy of the zinc-blende structure is about 3 meV/atom lower when two special k vec points are used than when ten are used. This is caused by the fact that Ge requires more dense k vec-point sampling to reach convergence than do ZB or Si.
K. A. Mäder, H. von Känel, and A. Baldereschi, Superlatt. Microstruct. (to be published).
Here we use the term ``lattice'' in the strict sense, meaning a particular set of points, all of which are equivalent under the operations of the translation group of the crystal, rather than the complete set of atomic sites.
The results of Ref. 46 were phrased in terms of the critical value of - 2 $J_{2, 1}$ required to produce epitaxial stability (hence instability of an abrupt Si/Ge interface) of ordered RH1 p =1 and RH2 p =1 phases, assumed as possible candidates for the phase of the interdiffused interface regions of a (001) Si/Ge SL.
Note that because the thin layers (q < 4) contain fractional unit cells of the ordered structures, we cannot claim that SQS-8 models a thin disordered region here, though we have shown that does so for thick layers.
D. J. Lockwood, K. Rajan, E. W. Fenton, J. M. Baribeau and M.W. Denhoff, Solid State Commun. 61, 465 (1987).
F. K. LeGoues, V. P. Kesan, S. S. Iyer, J. Tersoff and R. Tromp, Phys. Rev. Lett. 64, 2038 (1990).
E. Müller, H.-U. Nissen, K. A. Mäder, M. Ospelt, and H. von Känel (unpublished); see also E. Müller, H.-U. Nissen, M. Ospelt and H. von Känel, Phys. Rev. Lett. 63, 1819 (1989).

[c1] M. Ospelt, W. Bacsa, J. Henz, K. A. Mäder and H. von Känel, Superlatt. Microstruct. 5, 71 (1989).

[c2] E. Kasper, H. Kibbel, H. Jorke, H. Brugger, E. Friess and G. Abstreiter, Phys. Rev. B 38, 3599 (1988).

[c3] A. T. Fiory, J. C. Bean, R. Hull and S. Nakahara, Phys. Rev. B 31, 4063 (1985); T. P. Pearsall, J. Bevk, J. Bonar, M. P. Mannaerts and A. Ourmazd, ibid. 39, 3741 (1989).

[c4] S. J. Chang, K. L. Wang, R. C. Bowman and P. M. Adams, Appl. Phys. Lett. 54, 27 (1989).

[c5] E. Friess, H. Brugger, K. Eberl, G. Krötz and G. Abstreiter, Solid State Commun. 69, 899 (1989); K. Eberl, G. Krötz, R. Zachai, and G. Abstreiter,, J. Phys. (Paris) Colloq. 48, C5-S11 (1987); ibid. 48, C5-329 (1987).

[c6] D. Brasen, S. Nakahara and J. C. Bean, J. Appl. Phys. 58, 1860 (1985).

[c7] T. Narusawa and W. M. Gibson, Phys. Rev. Lett. 47, 1459 (1981).

[c8] S. Froyen, D. M. Wood and A. Zunger, Phys. Rev. B 36, 4547 (1987); Appl. Phys. Lett. 54, 2435 (1989).

[c9] S. Froyen, D. M. Wood and A. Zunger, Phys. Rev. B 37, 6893 (1988).

[c10] T. P. Pearsall, CRC Critical Reviews on Solid State and Materials Sciences (Chemical Rubber Company, Boca Raton, FL, 1989), Vol. 15, p. 551; J. Lumin. 44, 367 (1989).

[c11] J. L. Martins and A. Zunger, Phys. Rev. B 30, 6217 (1984).

[c12] J. L. Martins and A. Zunger, Phys. Rev. Lett. 56, 1400 (1986).

[c13] S. Ciraci and I. P. Batra, Phys. Rev. B 38, 1835 (1988).

[c14] B. Koiller and M. O. Robbins, Phys. Rev. B 40, 12554 (1989).

[c15] P. B. Littlewood, Phys. Rev. B 34, 1363 (1986).

[c16] S.-H. Wei and A. Zunger, Phys. Rev. Lett. 61, 1505 (1988).

[c17] R. G. Dandrea, J. E. Bernard, S.-H Wei and A. Zunger, Phys. Rev. Lett. 64, 36 (1990).

[c18] A. Ourmazd and J. C. Bean, Phys. Rev. Lett. 55, 765 (1985).

[c19] F. K. LeGoues, V. P. Kesan and S. S. Iyer, Phys. Rev. Lett. 64, 40 (1990).

[c20] For the rhombohedral structures we use the term ``bilayer'' to refer to pairs of (111) layers separated by a distance of a /4 sqrt 3 in the diamond structure, where a is the lattice constant. These bilayers are separated from each other by the larger distance a sqrt 3 /4.

[c21] There is also a third rhombohedral structure, RH3, which has half-integral p, and one mixed-bilayer and one pure-bilayer interface. This structure has energy intermediate between those of RH1 and RH2 and is not considered here.

[c22] J. Ihm, A. Zunger and M. L. Cohen, J. Phys. C 12, 4409 (1979).

[c23] W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).

[c24] J. E. Bernard, L. G. Ferreira, S.-H. Wei and A. Zunger, Phys. Rev. B 38, 6338 (1988).

[c25] For example, Ciraci and Batra (Ref. 13) partitioned the formation enthalpy into a bulklike strain term analogous to our Δ $E_{CS}$ and an interfacelike term attributed to the chemical process associated with the formation of the SibondGe bonds at the interface, and hence analogous to our $I_{CE}$ .

[c26] P. N. Keating, Phys. Rev. 145, 637 (1966).

[c27] R. M. Martin, Phys. Rev. B 1, 4005 (1970).

[c28] W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (Cambridge University Press, Cambridge, 1986), pp. 301-306.

[c29] A. Zunger, S.-H. Wei, L. G. Ferreira and J. E. Bernard, Phys. Rev. Lett. 65, 353 (1990); S.-H. Wei, L. G. Ferreira, J. E. Bernard and A. Zunger, Phys. Rev. B 42, 9622 (1990); S. H. Wei and A. Zunger, Appl. Phys. Lett. 56, 662 (1990); K. Hass, L. C. Davis and A. Zunger, Phys. Rev. B 42, 3757 (1990).

[c30] S. Froyen, Phys. Rev. B 39, 3168 (1989).

[c31] F. Birch, Phys. Rev. 71, 809 (1947); J. Geophys. Res. 57, 227 (1952).

[c32] D. M. Wood and A. Zunger, Phys. Rev. B 40, 4062 (1989).

[c33] R. G. Dandrea (private communication).

[c34] L. G. Ferreira, S.-H. Wei and A. Zunger, Phys. Rev. B 40, 3197 (1989); S.-H. Wei, L. G. Ferreira and A. Zunger, ibid. 40, 8240 (1990).

[c35] J. W. D. Connolly and A. R. Williams, Phys. Rev. B 27, 5169 (1983).

[c36] J. M. Sanchez, F. Ducastelle and D. Gratias, Physica A 128, 334 (1984).

[c37] In this section we use the term ``lattice'' in the less strict (but traditional in the literature on cluster expansions) sense of an array of empty sites, with no restriction to being equivalent under the operations of some translation group.

[c38] L. G. Ferreira (private communication).

[c39] Essentially full convergence for Si ${}_{0.5}{Ge}_{0.5}$ is reached with six interactions, including up to fourth-nearest-neighbor pairs and four-body second-nearest-neighbor interactions, though two of these (four-body first-nearest-neighbor and two-body third-nearest-neighbor) are smaller than the others and can be eliminated with little additional error.

[c40] S. de Gironcoli, S. Baroni, and P. Giannozzi, in Proceedings of the International Conference on the Physics of Semiconductors, Thessaloniki, 1990, edited by E. M. Anastussakis and J. D. Joannopoulos (World Scientific, Singapore, 1990), Vol. 2, p. 877.

[c41] In terms of the excess bond energy delta of the SibondGe bond relative to equivalent amounts of SibondSi and GebondGe bonds, Δ $E_{chem}$ ${}^{(α)}$ = $N_{α}$ δ /N, where $N_{alpha}$ is the number of SibondGe bonds per unit cell, and N is the number of atoms per unit cell. Since $N_{α}$ /N = - scrP ${bar}_{2, 1}$ (Table VI), the excess bond energy is just δ = - $D_{2, 1}$ $J_{2, 1}$ .

[c42] P. C. Kelires and J. Tersoff, Phys. Rev. Lett. 63, 1164 (1989).

[c43] A. Qteish and R. Resta, Phys. Rev. B 37, 1308 (1988); ibid. 37, 6983 (1988).

[c44] G. B. Stringfellow, J. Phys. Chem. Solids 34, 1749 (1973); G. B. Stringfellow and P. E. Greene, J. Electrochem. Soc. 117, 1075 (1970).

[c45] We have found, for example, that in our calculations with a 20-Ry cutoff energy, the formation enthalpy of the zinc-blende structure is about 3 meV/atom lower when two special k vec points are used than when ten are used. This is caused by the fact that Ge requires more dense k vec-point sampling to reach convergence than do ZB or Si.

[c46] K. A. Mäder, H. von Känel, and A. Baldereschi, Superlatt. Microstruct. (to be published).

[c47] Here we use the term ``lattice'' in the strict sense, meaning a particular set of points, all of which are equivalent under the operations of the translation group of the crystal, rather than the complete set of atomic sites.

[c48] The results of Ref. 46 were phrased in terms of the critical value of - 2 $J_{2, 1}$ required to produce epitaxial stability (hence instability of an abrupt Si/Ge interface) of ordered RH1 p =1 and RH2 p =1 phases, assumed as possible candidates for the phase of the interdiffused interface regions of a (001) Si/Ge SL.

[c49] Note that because the thin layers (q < 4) contain fractional unit cells of the ordered structures, we cannot claim that SQS-8 models a thin disordered region here, though we have shown that does so for thick layers.

[c50] D. J. Lockwood, K. Rajan, E. W. Fenton, J. M. Baribeau and M.W. Denhoff, Solid State Commun. 61, 465 (1987).

[c51] F. K. LeGoues, V. P. Kesan, S. S. Iyer, J. Tersoff and R. Tromp, Phys. Rev. Lett. 64, 2038 (1990).

[c52] E. Müller, H.-U. Nissen, K. A. Mäder, M. Ospelt, and H. von Känel (unpublished); see also E. Müller, H.-U. Nissen, M. Ospelt and H. von Känel, Phys. Rev. Lett. 63, 1819 (1989).

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Strain energy and stability of Si-Ge compounds, alloys, and superlattices

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