- Email: [email protected]
Electronic structure calculations of thin films of RbF
Applied Surface Science 142 Ž1999. 43–47
Electronic structure calculations of thin films of RbF B. Stankiewicz ) , L. Jurczyszyn Institute of Experimental Physics, UniÕersity of Wrocław, pl. Maksa Borna 9, 50-204 Wrocław, Poland
Abstract Electronic structure of thin films of rubidium fluoride is investigated within a next-nearest-neighbour approximation of the LCAO approach, with overlap integrals included. For the bulk structure calculations, semi-empirical parameters are fitted in order to reproduce the experimental band gap width, electron affinity and ionicity of RbF. For the thin-film structure calculations, one center parameters are corrected to take into account the Madelung potential differences in the subsequent atomic layers of a considered slab. For very thin films Žtwo lattice constants or less., a strong dependence of the valence band width as well as the layer-density-of-states distributions on the slab thickness is found. The surface electronic structure of RbF is examined and it is shown—in accordance with previous estimations—that no surface states exist at the Ž001. surface. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Alkali halide; RbF; Next-nearest-neighbour; Electronic structure
1. Introduction Alkali halides are insulators with the main band gap of 7.5–13.6 eV and the lattice constant ranging ˚ which makes it possible to grow between 4–7.3 A, several epitaxial semiconductorralkali-halidersemiconductor systems, e.g., GerRbFrGaAs w1–3x. Electronic structure of the resulting semiconductorralkali-halide interfaces is known to resemble those of cleaved surfaces of the constituent crystals w4x. Since such ‘sandwich’ systems are important for specific device applications Že.g., in microelectronic circuits., it is interesting to study electronic properties of alkali-halide intralayers. In this paper we present a theoretical study of the electronic structure of thin films of RbF. The results obtained from numerical computations enabled us to )
Corresponding author. Tel.: q48-71-201343; Fax: q48-713287365; E-mail: [email protected]
examine the influence of the film thickness on the main energy gap width and on the density of states within the valence band. The role of surface effects on the electronic structure of RbF film is also discussed. 2. Model and method of calculation Electronic structure calculations have been performed for Ž001. oriented RbF slabs of thicknesses varying from 1 to 12 monolayers Ži.e., 6 lattice constants.. The slab model and Linear Combination of Atomic Orbitals ŽLCAO. method have been used within the sp 3 next-nearest-neighbour approximation, including the overlap integrals. The one-electron wave function has been constructed as a linear combination of 2p orbitals of fluoride and 5s orbitals of rubidium. Semi-empirical parameters, describing the electronic structure of the bulk RbF crystal, have been
0169-4332r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 9 - 4 3 3 2 Ž 9 8 . 0 0 6 3 0 - 8
B. Stankiewicz, L. Jurczyszynr Applied Surface Science 142 (1999) 43–47
44
Table 1 Semi-empirical matrix elements of Hamiltonian used for calculation of electronic structure of RbF bulk crystal Žin eV. On-site integrals EŽ2pF. EŽ5sRb.
y8.73713 3.36894
Type of interaction
Overlapping
Hopping
5s–5s 5s–2p 2p x –4p x 2p x –4p y 2p x –4p y
0.039436 y0.022316 0.001376 y0.004677 y0.006052
y0.059760 1.102665 y0.005989 0.066544 0.026350
for the bulk and for the films have been used to correct the values of on-site integrals. Parameters obtained in this way have been used to calculate the density of states and, subsequently, the highest occupied energy level, new values of both ionic charges in the successive molecular layers and again the one-center parameters. This procedure has been repeated until the highest occupied energy level stabilized.
3. Results fitted to reproduce, first of all, the experimental data, i.e., the band gap width w5x, the position of valence band edge with respect to the vacuum level w5x, and the ionic charge w6,7x. Agreement of the bulk electronic structure of RbF with the results obtained from ab initio calculations w8,9x has also been taken into account. It has been found that to reproduce—at the same time—the band gap width and the proper ionic charge, it is necessary to include overlap integrals. These have been assumed to be proportional to the appropriate overlap integrals of atomic orbitals of fluoride and rubidium w10x. The coefficients of proportionality appeared to be 0.078 for 5s–5s overlap integral, 2.751 for 2p–2p integrals and a geometric mean value of those for 2p–5s overlap integrals. The values of Hamiltonian matrix elements obtained in such a way are presented in Table 1. To calculate the thin-film electronic structure, one-center parameters in the successive molecular layers of the slab have been modified to include the changes Žwith respect to the bulk. of the ionic potential due to the lack of adequate number of neighbouring ions at both surfaces, hence the resulting changes in the charge density. Semi-empirical parameters for the films have been adjusted in the following way. First, the electrostatic potential energies, implied by the Coulomb interaction of 2p electrons of fluoride, as well as the 5s electrons of rubidium, with the surrounding ions, have been calculated in the bulk, using the ionic charges obtained for parameters presented in Table 1 Ži.e., 0.92e per ion., the classical Madelung potential and the dielectric constant in the low-frequency limit. Next, the respective electrostatic potential energies have been calculated for ions in the films, using the same data as in bulk. Differences in values obtained
Electronic properties of RbF film of thickness of one to twelve monolayers have been studied based on analysis of the density-of-states distributions and ionic charge distributions of particular molecular layers. The films of thickness of at least eight molecular layers reproduce already the main energy gap width, the valence band width and the density-of-states distribution of bulk RbF crystal. Thinner films Žfive to seven molecular layers. reproduce quite well the shape of the density-of-states distribution of the bulk crystal, but the valence band is slightly shrinked Žby about 0.05 eV, mainly from the bottom.. Electronic structure of slabs thinner than five molecular layers is considerably different from the bulk one. In none of the considered cases surface states have been found in the main band gap. Selected results are presented in Figs. 1–3. Fig. 1 shows, for comparison, the valence band density-ofstates distribution of the bulk RbF crystal. Fig. 2
Fig. 1. Density-of-states distribution within the valence band of the bulk RbF crystal.
B. Stankiewicz, L. Jurczyszynr Applied Surface Science 142 (1999) 43–47
45
Fig. 2. Layer density of states for successive molecular layers of RbF films of a different thickness: Ža. 1 molecular layer thick slab, Žb. one of molecular layers of 2 molecular layers thick slab, Žc. surface molecular layer Žc1. and central molecular layer Žc2. of 3 molecular layers thick slab, Žd. surface molecular layer Žd1. and central molecular layer Žd2. of 4 molecular layers thick slab.
presents layer density-of-states distributions calculated for very thin RbF films Žless than five molecular layers., when the electronic properties of a film differ from the bulk ones. The corresponding contributions to the valence band density of states from p x ,
p y and p z orbitals of fluoride are shown in Fig. 3 Žcontribution from s states of rubidium to the valence band is negligible.. In our model, slabs are assumed to be parallel to the Ž x, y . plane, so p x and p y orbitals are indistinguishable.
46
B. Stankiewicz, L. Jurczyszynr Applied Surface Science 142 (1999) 43–47
Fig. 2Ža. presents density-of-states distribution of the valence band of the system formed by one molecular layer only. In this case the width of the valence band is decreased by about one third as compared to the bulk crystal Žcf. Fig. 1.. The most prominent feature of this distribution is a high and narrow peak positioned f 0.1 eV below the upper valence band edge. As it follows from Fig. 3Ža2., the peak is built up from p z orbitals, while p x and p y contributions wcf. Fig. 3Ža1.x are much broader. Layer density-of-states distributions of slabs consisting of two wFig. 2Žb.x and three wFig. 2Žc1. and Žc2.x molecular layers are wider than that of Fig. 2Ža. and exhibit a number of sharp peaks. For the four molecular layers thick film wFig. 2Žd1. and Žd2.x, the layer density-of-states distributions are already quite similar to the bulk crystal one Žcf. Fig. 1.. It is interesting to note that this similarity refers not only to the central layers, but also to the surface layers. As it follows from the sequence of distributions presented in Fig. 3, the p z contribution becomes broader and broader with increasing slab thickness. Moreover, the peaks corresponding to p z states are not at the same energy position in the surface and subsurface layers wcf. Fig. 3Žb2. and Žb4. as well as Fig. 3Žc2. and Žc4.x. Gradually, the distribution of p z states becomes more and more similar to those of p x and p y states wcf. Fig. 3Žc.x.
4. Conclusions
Fig. 3. Layer density of states projected onto particular orbitals for Ža. p x orbital Ža1. and p z orbital Ža2. of 1 molecular layer thick slab, Žb. p x orbital Žb1. and p z orbital Žb2. of surface molecular layer and p x orbital Žb3. and p z orbital Žb4. of central molecular layer of 3 molecular layers thick slab, Žc. p x orbital Žc1. and p z orbital Žc2. of surface molecular layer and p x orbital Žc3. and p z orbital Žc4. of central molecular layer of 4 molecular layers thick slab.
We have found that surface states do not appear in the main energy gap Žneither at the top of the valence band, nor at the bottom of the conduction band. in any of the considered RbF films, i.e., for slabs from 1 to 12 molecular layers thick. Moreover, for slabs of thickness of 5 molecular layers or more, the density-of-states distributions for surface layers do not really differ from the bulk ones. As follows from the obtained results, even very thin RbF slabs Ž4 molecular layers thick. reproduce already the basic electronic properties of the RbF crystal, i.e., the width of the main energy gap as well as the density-of-states distribution within the valence band are very similar to those of the bulk RbF crystal.
B. Stankiewicz, L. Jurczyszynr Applied Surface Science 142 (1999) 43–47
Acknowledgements This work has been supported by the University of Wrocław within the grant No. 2016rWrIFDr98. References w1x R. Klauser, M. Kubota, Y. Murata, M. Oshima, Y.Y. Maruo, T. Kawamura, T. Miyahara, Phys. Rev. B 40 Ž1989. 3301. w2x R. Klauser, M. Oshima, H. Sugahara, Y. Murata, Surf. Sci. 242 Ž1991. 319.
47
w3x R. Klauser, M. Oshima, H. Sugahara, Y. Murata, H. Kato, Phys. Rev. B 43 Ž1991. 4879. w4x B. Stankiewicz, Thin Solid Films 280 Ž1996. 178. w5x R.T. Poole, J.G. Jenkin, J. Liesgang, R.C.G. Leckey, Phys. Rev. B 11 Ž1974. 5179. w6x A.K. Koh, J. Phys. Chem. Solids 50 Ž1989. 39. w7x L. Bosi, Phys. Status Solidi Žb. 168 Ž1991. K81. w8x A.B. Kunz, Phys. Rev. B 26 Ž1982. 2056. w9x W.Y. Ching, F. Gan, M.-Z. Huang, Phys. Rev. B 52 Ž1995. 1596. w10x E. Clementi, C. Roetti, Atomic Data and Nuclear Data Tables, Academic Press, New York, 1974.
Electronic structure calculations of thin films of RbF B. Stankiewicz ) , L. Jurczyszyn Institute of Experimental Physics, UniÕersity of Wrocław, pl. Maksa Borna 9, 50-204 Wrocław, Poland
Abstract Electronic structure of thin films of rubidium fluoride is investigated within a next-nearest-neighbour approximation of the LCAO approach, with overlap integrals included. For the bulk structure calculations, semi-empirical parameters are fitted in order to reproduce the experimental band gap width, electron affinity and ionicity of RbF. For the thin-film structure calculations, one center parameters are corrected to take into account the Madelung potential differences in the subsequent atomic layers of a considered slab. For very thin films Žtwo lattice constants or less., a strong dependence of the valence band width as well as the layer-density-of-states distributions on the slab thickness is found. The surface electronic structure of RbF is examined and it is shown—in accordance with previous estimations—that no surface states exist at the Ž001. surface. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Alkali halide; RbF; Next-nearest-neighbour; Electronic structure
1. Introduction Alkali halides are insulators with the main band gap of 7.5–13.6 eV and the lattice constant ranging ˚ which makes it possible to grow between 4–7.3 A, several epitaxial semiconductorralkali-halidersemiconductor systems, e.g., GerRbFrGaAs w1–3x. Electronic structure of the resulting semiconductorralkali-halide interfaces is known to resemble those of cleaved surfaces of the constituent crystals w4x. Since such ‘sandwich’ systems are important for specific device applications Že.g., in microelectronic circuits., it is interesting to study electronic properties of alkali-halide intralayers. In this paper we present a theoretical study of the electronic structure of thin films of RbF. The results obtained from numerical computations enabled us to )
Corresponding author. Tel.: q48-71-201343; Fax: q48-713287365; E-mail: [email protected]
examine the influence of the film thickness on the main energy gap width and on the density of states within the valence band. The role of surface effects on the electronic structure of RbF film is also discussed. 2. Model and method of calculation Electronic structure calculations have been performed for Ž001. oriented RbF slabs of thicknesses varying from 1 to 12 monolayers Ži.e., 6 lattice constants.. The slab model and Linear Combination of Atomic Orbitals ŽLCAO. method have been used within the sp 3 next-nearest-neighbour approximation, including the overlap integrals. The one-electron wave function has been constructed as a linear combination of 2p orbitals of fluoride and 5s orbitals of rubidium. Semi-empirical parameters, describing the electronic structure of the bulk RbF crystal, have been
0169-4332r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 9 - 4 3 3 2 Ž 9 8 . 0 0 6 3 0 - 8
B. Stankiewicz, L. Jurczyszynr Applied Surface Science 142 (1999) 43–47
44
Table 1 Semi-empirical matrix elements of Hamiltonian used for calculation of electronic structure of RbF bulk crystal Žin eV. On-site integrals EŽ2pF. EŽ5sRb.
y8.73713 3.36894
Type of interaction
Overlapping
Hopping
5s–5s 5s–2p 2p x –4p x 2p x –4p y 2p x –4p y
0.039436 y0.022316 0.001376 y0.004677 y0.006052
y0.059760 1.102665 y0.005989 0.066544 0.026350
for the bulk and for the films have been used to correct the values of on-site integrals. Parameters obtained in this way have been used to calculate the density of states and, subsequently, the highest occupied energy level, new values of both ionic charges in the successive molecular layers and again the one-center parameters. This procedure has been repeated until the highest occupied energy level stabilized.
3. Results fitted to reproduce, first of all, the experimental data, i.e., the band gap width w5x, the position of valence band edge with respect to the vacuum level w5x, and the ionic charge w6,7x. Agreement of the bulk electronic structure of RbF with the results obtained from ab initio calculations w8,9x has also been taken into account. It has been found that to reproduce—at the same time—the band gap width and the proper ionic charge, it is necessary to include overlap integrals. These have been assumed to be proportional to the appropriate overlap integrals of atomic orbitals of fluoride and rubidium w10x. The coefficients of proportionality appeared to be 0.078 for 5s–5s overlap integral, 2.751 for 2p–2p integrals and a geometric mean value of those for 2p–5s overlap integrals. The values of Hamiltonian matrix elements obtained in such a way are presented in Table 1. To calculate the thin-film electronic structure, one-center parameters in the successive molecular layers of the slab have been modified to include the changes Žwith respect to the bulk. of the ionic potential due to the lack of adequate number of neighbouring ions at both surfaces, hence the resulting changes in the charge density. Semi-empirical parameters for the films have been adjusted in the following way. First, the electrostatic potential energies, implied by the Coulomb interaction of 2p electrons of fluoride, as well as the 5s electrons of rubidium, with the surrounding ions, have been calculated in the bulk, using the ionic charges obtained for parameters presented in Table 1 Ži.e., 0.92e per ion., the classical Madelung potential and the dielectric constant in the low-frequency limit. Next, the respective electrostatic potential energies have been calculated for ions in the films, using the same data as in bulk. Differences in values obtained
Electronic properties of RbF film of thickness of one to twelve monolayers have been studied based on analysis of the density-of-states distributions and ionic charge distributions of particular molecular layers. The films of thickness of at least eight molecular layers reproduce already the main energy gap width, the valence band width and the density-of-states distribution of bulk RbF crystal. Thinner films Žfive to seven molecular layers. reproduce quite well the shape of the density-of-states distribution of the bulk crystal, but the valence band is slightly shrinked Žby about 0.05 eV, mainly from the bottom.. Electronic structure of slabs thinner than five molecular layers is considerably different from the bulk one. In none of the considered cases surface states have been found in the main band gap. Selected results are presented in Figs. 1–3. Fig. 1 shows, for comparison, the valence band density-ofstates distribution of the bulk RbF crystal. Fig. 2
Fig. 1. Density-of-states distribution within the valence band of the bulk RbF crystal.
B. Stankiewicz, L. Jurczyszynr Applied Surface Science 142 (1999) 43–47
45
Fig. 2. Layer density of states for successive molecular layers of RbF films of a different thickness: Ža. 1 molecular layer thick slab, Žb. one of molecular layers of 2 molecular layers thick slab, Žc. surface molecular layer Žc1. and central molecular layer Žc2. of 3 molecular layers thick slab, Žd. surface molecular layer Žd1. and central molecular layer Žd2. of 4 molecular layers thick slab.
presents layer density-of-states distributions calculated for very thin RbF films Žless than five molecular layers., when the electronic properties of a film differ from the bulk ones. The corresponding contributions to the valence band density of states from p x ,
p y and p z orbitals of fluoride are shown in Fig. 3 Žcontribution from s states of rubidium to the valence band is negligible.. In our model, slabs are assumed to be parallel to the Ž x, y . plane, so p x and p y orbitals are indistinguishable.
46
B. Stankiewicz, L. Jurczyszynr Applied Surface Science 142 (1999) 43–47
Fig. 2Ža. presents density-of-states distribution of the valence band of the system formed by one molecular layer only. In this case the width of the valence band is decreased by about one third as compared to the bulk crystal Žcf. Fig. 1.. The most prominent feature of this distribution is a high and narrow peak positioned f 0.1 eV below the upper valence band edge. As it follows from Fig. 3Ža2., the peak is built up from p z orbitals, while p x and p y contributions wcf. Fig. 3Ža1.x are much broader. Layer density-of-states distributions of slabs consisting of two wFig. 2Žb.x and three wFig. 2Žc1. and Žc2.x molecular layers are wider than that of Fig. 2Ža. and exhibit a number of sharp peaks. For the four molecular layers thick film wFig. 2Žd1. and Žd2.x, the layer density-of-states distributions are already quite similar to the bulk crystal one Žcf. Fig. 1.. It is interesting to note that this similarity refers not only to the central layers, but also to the surface layers. As it follows from the sequence of distributions presented in Fig. 3, the p z contribution becomes broader and broader with increasing slab thickness. Moreover, the peaks corresponding to p z states are not at the same energy position in the surface and subsurface layers wcf. Fig. 3Žb2. and Žb4. as well as Fig. 3Žc2. and Žc4.x. Gradually, the distribution of p z states becomes more and more similar to those of p x and p y states wcf. Fig. 3Žc.x.
4. Conclusions
Fig. 3. Layer density of states projected onto particular orbitals for Ža. p x orbital Ža1. and p z orbital Ža2. of 1 molecular layer thick slab, Žb. p x orbital Žb1. and p z orbital Žb2. of surface molecular layer and p x orbital Žb3. and p z orbital Žb4. of central molecular layer of 3 molecular layers thick slab, Žc. p x orbital Žc1. and p z orbital Žc2. of surface molecular layer and p x orbital Žc3. and p z orbital Žc4. of central molecular layer of 4 molecular layers thick slab.
We have found that surface states do not appear in the main energy gap Žneither at the top of the valence band, nor at the bottom of the conduction band. in any of the considered RbF films, i.e., for slabs from 1 to 12 molecular layers thick. Moreover, for slabs of thickness of 5 molecular layers or more, the density-of-states distributions for surface layers do not really differ from the bulk ones. As follows from the obtained results, even very thin RbF slabs Ž4 molecular layers thick. reproduce already the basic electronic properties of the RbF crystal, i.e., the width of the main energy gap as well as the density-of-states distribution within the valence band are very similar to those of the bulk RbF crystal.
B. Stankiewicz, L. Jurczyszynr Applied Surface Science 142 (1999) 43–47
Acknowledgements This work has been supported by the University of Wrocław within the grant No. 2016rWrIFDr98. References w1x R. Klauser, M. Kubota, Y. Murata, M. Oshima, Y.Y. Maruo, T. Kawamura, T. Miyahara, Phys. Rev. B 40 Ž1989. 3301. w2x R. Klauser, M. Oshima, H. Sugahara, Y. Murata, Surf. Sci. 242 Ž1991. 319.
47
w3x R. Klauser, M. Oshima, H. Sugahara, Y. Murata, H. Kato, Phys. Rev. B 43 Ž1991. 4879. w4x B. Stankiewicz, Thin Solid Films 280 Ž1996. 178. w5x R.T. Poole, J.G. Jenkin, J. Liesgang, R.C.G. Leckey, Phys. Rev. B 11 Ž1974. 5179. w6x A.K. Koh, J. Phys. Chem. Solids 50 Ž1989. 39. w7x L. Bosi, Phys. Status Solidi Žb. 168 Ž1991. K81. w8x A.B. Kunz, Phys. Rev. B 26 Ž1982. 2056. w9x W.Y. Ching, F. Gan, M.-Z. Huang, Phys. Rev. B 52 Ž1995. 1596. w10x E. Clementi, C. Roetti, Atomic Data and Nuclear Data Tables, Academic Press, New York, 1974.