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Electronic structures of alkali metal (Li, Na, K, Rb, and Cs) monosilicides
Journal of Alloys and Compounds 478 (2009) 754–757
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Journal of Alloys and Compounds journal homepage: www.elsevier.com/locate/jallcom
Electronic structures of alkali metal (Li, Na, K, Rb, and Cs) monosilicides Yoji Imai ∗ , Akio Watanabe National Institute of Advanced Industrial Science and Technology (AIST), AIST Tsukuba Central 5, Higashi 1-1, Tsukuba, Ibaraki 305-8565, Japan
a r t i c l e
i n f o
Article history: Received 27 October 2008 Received in revised form 26 November 2008 Accepted 27 November 2008 Available online 7 December 2008 Keywords: Semiconductors Intermetallics Electronic band structure
a b s t r a c t The electronic structures of LiSi, NaSi, RbSi, and CsSi have been calculated using a first-principle pseudopotential method to clarify the possibility of band-gap broadening of BaSi2 by alloying with alkali metals. All of them are found to be semiconductors with the calculated indirect band gaps of 1.17 eV (LiSi) and 1.24 eV (NaSi) or the calculated direct band gaps of 1.32 eV (KSi), 1.38 eV (RbSi), and 1.66 eV (CsSi). The general tendency that the gap values of these ionic semiconductors are increased as their ionicities are increased and alkali metal incorporation into BaSi2 will be hopeful to broaden its band-gap. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Recently, BaSi2 has been attracting an attention for a good candidate material of solar cell because of its nearer band-gap (1.1–1.3 eV) to the suitable gap value (∼1.4 eV [1]) and its larger absorption coefficient compared to Si. It will be desirable to make this band-gap of BaSi2 approach to 1.4 eV by increasing the gap value. From the observed shift of optical absorption edge to shorter wavelength, Suemasu et al. suggested that increase in the gap value of BaSi2 would be expected by replacing some of the Ba atoms with Sr, an isoelectronic alkaline earth metal. However, the shift of the indirect absorption edge of Ba1−x Srx Si2 to shorter wavelength was approximately 0.1 eV when x = 0.52 and increase of x more than this value did not result in further shift of the absorption edge [2]. This is perhaps because Sr substitution into one of two Ba sites in BaSi2 causes increase in the gap value but that into another site causes decrease in the gap value, as predicted by a first-principle calculation [3]. Therefore, other methods should be developed in order to get wider gap value of BaSi2 . Correlation between the magnitude of the band-gap and the difference in electronegativity between anion and cation of ionic compounds, such as expressed in the form of van Vechten–Phillips relation, leads us to expect that band-gap will be wider if more electropositive elements than Ba are contained in the compounds. Alkali metal incorporation into BaSi2 is a possible method to bring about this effect. It is known that alkali metal monosilicides MSi (M = Na, K, Rb, Cs) are Zintl phases composed of isolated Si4 -tetrahedra, as seen in
∗ Corresponding author. Tel.: +81 29 861 4440; fax: +81 29 861 4440. E-mail address: [email protected] (Y. Imai). 0925-8388/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2008.11.149
BaSi2 . NaSi crystallizes in the monoclinic NaSi-type structure [4], while KSi, RbSi and CsSi in the cubic KGe-type structure [5].1 In addition, though corresponding LiSi has not been described in standard phase diagrams, a new Zintl phase of LiSi has been recently confirmed to exist as an equilibrium phase [6]. It belongs to the space group I 41 /a with the MgGa-type structure and is thermodynamically more stable than mixture of Li12 Si7 and Si, though the reaction of Li12 Si7 and Si to form LiSi are very low at ambient pressure and therefore it has not been prepared by the conventional method of cooling Li–Si melt. Properties of these alkali-monosilicides have not been determined yet and we do not know, at present, if they are semiconductors or not. The purpose of the present work is to clarify their electronic structures by first-principle calculations. This has been performed so as to find the possibility of band-gap tuning of BaSi2 from the viewpoint of solid-state chemistry, which suggests that increase in ionicity of the compounds will broaden the band-gap.2 We report the results of band calculation of these compounds and dis-
1 NaSi belongs to space group C 2/c. Silicon atoms form four isolated Si4 -tetrahedra but the arrangement of Na atom with respect to Si atoms is irregular, in contrast to the case of KSi, RbSi, and CsSi. The latter belong to the space group P43n ¯ . Silicon atoms form eight isolated Si4 -tetrahedra and alkali metal atoms are arranged around these tetrahedra. Each alkali metal atom is surrounded tetrahedrally by four Si4 . 2 From the analogy of the alkaline-earth metal silicides calculated before [7], the valence bands of alkali metal silicides would be composed mainly of Si 3s and Si 3p while the conduction bands would be mainly composed of the alkali metal s, p, (and d) states. Since the atomic orbitals of heavier alkali metal elements must have shallower energy levels, the energy gap between the valence and conduction band, Eg , would be broader in heavier elements. This tendency can be seen in the series of alkali metal halides; LiCl (Eg = 9.5 eV), NaCl (8.5 eV), KCl (8.5 eV), and CsCl (8.4 eV) [8].
Y. Imai, A. Watanabe / Journal of Alloys and Compounds 478 (2009) 754–757
Fig. 1. Band structure of NaSi. Broken lines in this and succeeding figures show the positions of the top of the valence band. Plotted numbers in this and succeeding figures express the order counted from the bottom of the valence band. Though Na 2s and 2p states are explicitly treated as a part of valence band in the pseudopotential used, 32 bands composed mainly of these atomic orbitals are omitted from the numbering in this figure.
cuss the possibility of the above-mentioned band-gap broadening of BaSi2 . 2. Calculation methods Calculations of the band structure (BS) along several high-symmetry lines in the Brillouin zone have been conducted using CASTEP (Cambridge Serial Total Energy Package) developed by Payne et al. [9], which is a first-principle pseudopotential method based on (1) the density-functional theory (DFT) in describing the electron–electron interaction, (2) a pseudopotential description of the electron–core interaction, and (3) a plane-wave expansion of the wavefunctions. The pseudopotential used is the ultrasoft pseudopotential generated by the scheme of Vanderbilt [10], which is bundled in the Cerius2 3 graphical User Interface. As for the method of approximation to the exchange–correlation term of the DFT, local density approximation (LDA) with generalized gradient correction [11] was used. The kinetic cutoff energy for the plane wave expansion of the wavefunctions was set at 380 eV, which had been confirmed beforehand to give well-converged band structures with respect to cutoff energy. The band structure calculations have been performed using observed crystallographic parameters [4–6]. We performed structural optimization processes before band structure calculations but that for LiSi did not converge well in contrast to other monosilicides.4 The energy zero of the BS diagrams given below is taken at the top of the valence band.
3
Cerius2 is a trademark of Accelrys, Inc. Preliminary optimization processes have shown that present calculations give quite good reproducibilities of the parameters of NaSi, KSi, RbSi, and CsSi. Optimized lattice parameters for NaSi are a = 1.2183 nm, b = 0.6573 nm, c = 0.1118 nm and ˇ = 119.03◦ , which can be compared with the observed parameters of a = 1.219 nm, b = 0.655 nm, c = 0.1118 nm, and ˇ = 119.0◦ [4]. Optimized lattice parameters for KSi, RbSi, and CsSi are a = 1.2617 nm, 1.3040 nm, and 1.3505 nm, which can be comparable to observed parameters of a = 1.262 nm, 1.304 nm, and 1.350 nm, respectively [5]. The errors are less than 0.3%. This was not the case for BaSi2 where 1.5% and 1.2% underestimations for a and c, respectively, was experienced in the previous study [3]. In contrast to those good reproducibilities, optimization process of LiSi 4
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Fig. 2. Band structure of KSi. Though K3s and 3p states are explicitly treated as a part of valence band in the pseudopotential used, 128 bands composed mainly of these atomic orbitals are omitted from the numbering in this figure.
3. Results and discussion At first, we present the calculated BS of NaSi near the Fermi level. Plotted in the BS figure are the numbers which express the order counted from the bottom of the valence band. The unit cell of NaSi contains 16 Na atoms and 16 Si atoms but can be reduced to Na8 Si8 which contains 40 valence electrons (1 electron from a Na atom and 4 electrons from a Si atom). The lowest 20 bands are fully occupied by these electrons and a gap is generated between the 20th and the 21st band. The top of the valence band is located at L (−1/2 0 1/2), whereas the bottom of the conduction band is located on the zone boundary plane vertical to c* axis. The latter is on the path from Z (0 0 1/2) to M (−1/2 −1/2 0) and on the path from Z (0 0 1/2) to L (−1/2 0 1/2). Therefore, the band-gap is indirect and the calculated gap value is 1.24 eV. Since there has been no observed gap value for NaSi to our knowledge, we cannot conclude that this value is reasonable. If we allow the tendency that about 60% of the measured gap is often obtained in the LDA calculation,5 NaSi must be a semiconductor with relatively wide gap value about 2 eV (Fig. 1). Because NaSi is too reactive with oxygen or moisture, it will be difficult to be used solely for practical applications. In addition, the estimated gap value above seems too wide for solar cell application.
did not converge well. Calculated Hellmann–Feynman force was as large as 12 GPa. This seems to be related to the fact that the form of tetrahedra contained in LiSi is far from regular tetrahedra. 5 For example, calculated gap value of BaSi2 is 0.72 eV in comparison with the observed value of 1.1–1.3 eV [3].
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Y. Imai, A. Watanabe / Journal of Alloys and Compounds 478 (2009) 754–757
Fig. 3. Band structure of RbSi (a) and CsSi (b). Though Rb4s, Rb4p, Cs5s, and Cs5p states are explicitly treated as a part of valence band in the pseudopotential used, 128 bands composed mainly of these atomic orbitals are omitted from the numbering in these figures.
However, it may be hopeful to use Na as a doping element of BaSi2 because it will widen the gap of BaSi2 when doped. Next, we present the calculated BS of KSi in Fig. 2. This compound (K32 Si32 ) contains 160 valence electrons (one electron from a K atom and four electrons from a Si atom) and 80 bands are fully occupied by these electrons. A gap is generated between the 80th and the 81st band. The gap is direct, located at X (1/2 0 0) with the gap value of 1.32 eV, which is wider than that of NaSi. In Fig. 3(a) and (b), we show the calculated BSs of RbSi and CsSi near the Fermi level,6 respectively, both of which have the same crystal structure with KSi. As in the case of KSi, 80 bands are fully occupied by 160 valence electrons in the unit cells. Band-gap is generated between the 80th and the 81st band. In both cases, the gaps are direct. However, the gap is located at X (1/2 0 0) with the gap value of 1.38 eV in case of RbSi, whereas it is located at M (1/2 1/2 0) with the gap value of 1.66 eV in case of CsSi. From these calculated values, we can see a tendency that the gap values of this series of compounds are increased as their ionicities are increased.7 This is also the case for monosilicides with different crystal structure. In Fig. 4, we show the calculated BS of LiSi. The unit cell of LiSi with the MgGa-type structure is composed of 16 Li atoms and 16 Si atoms, but can be reduced to the primitive cell composed 8 Li atoms and 8 Si atoms. Therefore, 40 valence electrons are contained in the calculated cell and these electrons occupy 20 bands, again. It can be seen that the band structure of LiSi is somewhat peculiar in that some bands show deviation from parabolic dispersion. The origin of this behavior is not clear but this might mean the (local) minima or maxima of energy in the k-space do not exist around high-symmetry points in the Brillouin zone.8 Therefore, we cannot exclude the possibility that we could not succeed in obtaining real gap value. However, so far as present calculation, a gap is generated between the 20th and the 21st band and the top of the valence band is located between N and C or A and P.9 The 21st band has a local minimum energy there, which is 1.60 eV higher than the 20th
6
Bands below the 54th band are omitted from Fig. 3. The ionization energies of K, Rb, and Cs are 4.3 eV, 4.2 eV, and 3.9 eV, respectively. As is well-known, the heavier elements in the alkaline group are more positive than lighter elements. 8 This was experienced in the band diagram of BaSi2 where the valence band maximum of BaSi2 is located between (0 0 0) and Y (0 1/2 0) [12]. 9 Symbols to describe symmetry points for this structure are named rather arbitrarily. Please refer to the notation shown in Fig. 4. 7
Fig. 4. Band structure of LiSi. Though Li1s state is explicitly treated as a part of valence band in the pseudopotential used, eight bands composed mainly of this atomic orbital are omitted from the numbering in this figure.
band, but the 21st band has the lowest energy at D (−1/2 1/2 1/2). The indirect energy gap was calculated to be 1.17 eV. Calculated gap values of LiSi, NaSi, KSi, RbSi and CsSi have a tendency that the gap values are increased as their ionicities are increased, irrespective of their crystal structures. This is also valid when we take BaSi2 into the consideration, since the 2nd ionization energy of Ba is 10.0 eV, larger than the ionization energy of Li. As expected from the rule, calculated gap value of BaSi2 is narrower than that of LiSi. However, the 1st ionization energy of Ba is
Y. Imai, A. Watanabe / Journal of Alloys and Compounds 478 (2009) 754–757
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BaSi2 , which may cause ill effect on the efficiency of solar energy conversion. As for the case (2), there has been very limited information on the ternary compounds of Ba–Si–alkali metal system. However, even if new compounds have different crystal structures from BaSi2 or alkali metal monosilicides, their band gaps will be broader than that of BaSi2 if only they have a character of a Zintl phase. Energetic study on which will actually proceed is under way. 4. Summary
Fig. 5. Band structure of Ba4 Cs4 Si16 where all the BaI sites in the BaSi2 lattice are replaced by Cs and all the BaII sites are occupied by Ba along with the high-symmetry directions of a simple orthorhombic Brillouin zone. Symmetry points for the simple orthorhombic Bravais lattice in this figure are S (1/2 1/2 0), X (1/2 0 0), (0 0 0), Y (0 1/2 0), T (0 1/2 1/2), R (1/2 1/2 1/2), U (1/2 0 1/2), and Z (0 0 1/2). Though Ba5s, Ba5p, Cs5s and Cs5p states are explicitly treated as a part of valence band in the pseudopotential used, 32 bands composed mainly of these atomic orbitals are omitted from the numbering in this figure.
5.2 eV, which is smaller than the ionization energy of Li. Therefore, Li incorporation into BaSi2 may not work well to broaden the gap of BaSi2 . From the present calculations, Na, K, Rb, and Cs are the candidate incorporation elements into BaSi2 so as to broaden its gap value. Incorporation of these elements will proceed by either (1) substitution of Ba atoms, or (2) formation of ternary compounds of Ba–Si–alkali metal. As the preliminary study for the case (1) we have performed band calculation of Ba4 Cs4 Si16 , where one of crystallographic sites, BaII , are substituted by Cs.10 The calculated band diagram is shown in Fig. 5. As shown, the Fermi level is shifted to downward with respect to the top of the valence band, compared to the case of BaSi2 , and crosses the 40th and 41st band because of lack of electrons caused by substitution of Ba by Cs. However, the gap between the top of the valence band ( point of the 40th band) and the bottom of conduction band (on the –Y segment of the 41st band) is calculated to be 1.142 eV, much larger than that of BaSi2 .11 This calculation suggests that diluted and random substitution of BaI site by Cs (and other alkali metal) will generate the localized impurity states by the mechanism of Anderson localization between the broader gap. However, too much incorporation of alkali–metal will evoke metal/semiconductor transition of
10 In BaSi2 , there are two crystallographically inequivalent sites for Ba (BaI and BaII ) and three inequivalent sites for Si (SiIII , SiIV , and SiV ) [13]. The unit cell contains eight formula units and the stoichiometric description of the unit cell is Ba8 Si16 . Ba atoms are distributed over 4 BaI and 4 BaII and we conducted energy calculation in order to determine which sites can be more favorably substituted by Cs using geometrically optimization process. The results indicated that BaII sites would be more favorably substituted by Cs. 11 The calculated gap value for BaSi2 with the observed structure was 0.72 eV. This value was increased to 0.80 eV when the optimized structure was adopted [3] but still much smaller than the value of Ba4 Cs4 Si16 above calculated.
In summary, we have performed band calculations of LiSi with the MgGa-type structure, NaSi with the NaSi-type structure, and KSi, RbSi, and CsSi with the KGe-type structure. All of them are semiconductors with the calculated indirect band gaps of 1.17 eV, and 1.24 eV, and the direct band gaps of 1.32 eV, 1.38 eV, and 1.66 eV, respectively. Though these calculated gap values themselves may not be accurate due to the limitation of the local density approximation used, we can see the general tendency that the gap values of these ionic semiconductors are increased as their ionicities are increased. Therefore, alkali–metal incorporation into BaSi2 will be hopeful to widen the band-gap of BaSi2 so as to make the gap value of BaSi2 approximate to the suitable value of 1.4 eV. Acknowledgements The authors would like to express their sincere gratitude to Dr. Motoharu IMAI, National Institute for Materials Science (NIMS), Japan, for fruitful and instructive discussions. They are also grateful to Dr. Masaaki SUGIE of National Institute of Advanced Science and Technology (AIST), Japan, for his kind technical assistance. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
C.H. Henry, J. Appl. Phys. 51 (1980) 4494. T. Suemasu, K. Morita, M. Kobayashi, J. Crystal Growth 301/302 (2007) 680. Y. Imai, A. Watanabe, Thin Solid Films 515 (2007) 8219. J. Witte, H.G. Schnering, Z. Anorg. Allg. Chem. 327 (1964) 260. E. Busmann, Z. Anorg. Allg. Chem. 313 (1961) 90. J. Evers, G. Oehlinger, G. Sextl, Eur. J. Solid State Inorg. Chem. 34 (1997) 773. Y. Imai, A. Watanabe, M. Mukaida, J. Alloys Compd. 358 (2003) 257. C.A. Wert, R.M. Thomson, Physics of Solids, McGraw-Hill Book Company, 1964, p. 289. M.C. Payne, M.P. Teter, D.C. Allan, T.A. Arias, J.D. Joannopoulos, Rev. Mod. Phys. 64 (1992) 1045. D. Vanderbilt, Phys. Rev. B 41 (1990) 7892. J.P. Perdew, Y. Wang, Phys. Rev. B 45 (1992) 13244. D.B. Migas, V.L. Shaposhnikov, V.E. Borisenko, Phys. Stat. Sol. b 244 (2007) 2611. H. Schäfer, K.H. Janzon, A. Weiss, Angewandte Chemie, vol. 2, International Edition, 1963, p. 393.
Contents lists available at ScienceDirect
Journal of Alloys and Compounds journal homepage: www.elsevier.com/locate/jallcom
Electronic structures of alkali metal (Li, Na, K, Rb, and Cs) monosilicides Yoji Imai ∗ , Akio Watanabe National Institute of Advanced Industrial Science and Technology (AIST), AIST Tsukuba Central 5, Higashi 1-1, Tsukuba, Ibaraki 305-8565, Japan
a r t i c l e
i n f o
Article history: Received 27 October 2008 Received in revised form 26 November 2008 Accepted 27 November 2008 Available online 7 December 2008 Keywords: Semiconductors Intermetallics Electronic band structure
a b s t r a c t The electronic structures of LiSi, NaSi, RbSi, and CsSi have been calculated using a first-principle pseudopotential method to clarify the possibility of band-gap broadening of BaSi2 by alloying with alkali metals. All of them are found to be semiconductors with the calculated indirect band gaps of 1.17 eV (LiSi) and 1.24 eV (NaSi) or the calculated direct band gaps of 1.32 eV (KSi), 1.38 eV (RbSi), and 1.66 eV (CsSi). The general tendency that the gap values of these ionic semiconductors are increased as their ionicities are increased and alkali metal incorporation into BaSi2 will be hopeful to broaden its band-gap. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Recently, BaSi2 has been attracting an attention for a good candidate material of solar cell because of its nearer band-gap (1.1–1.3 eV) to the suitable gap value (∼1.4 eV [1]) and its larger absorption coefficient compared to Si. It will be desirable to make this band-gap of BaSi2 approach to 1.4 eV by increasing the gap value. From the observed shift of optical absorption edge to shorter wavelength, Suemasu et al. suggested that increase in the gap value of BaSi2 would be expected by replacing some of the Ba atoms with Sr, an isoelectronic alkaline earth metal. However, the shift of the indirect absorption edge of Ba1−x Srx Si2 to shorter wavelength was approximately 0.1 eV when x = 0.52 and increase of x more than this value did not result in further shift of the absorption edge [2]. This is perhaps because Sr substitution into one of two Ba sites in BaSi2 causes increase in the gap value but that into another site causes decrease in the gap value, as predicted by a first-principle calculation [3]. Therefore, other methods should be developed in order to get wider gap value of BaSi2 . Correlation between the magnitude of the band-gap and the difference in electronegativity between anion and cation of ionic compounds, such as expressed in the form of van Vechten–Phillips relation, leads us to expect that band-gap will be wider if more electropositive elements than Ba are contained in the compounds. Alkali metal incorporation into BaSi2 is a possible method to bring about this effect. It is known that alkali metal monosilicides MSi (M = Na, K, Rb, Cs) are Zintl phases composed of isolated Si4 -tetrahedra, as seen in
∗ Corresponding author. Tel.: +81 29 861 4440; fax: +81 29 861 4440. E-mail address: [email protected] (Y. Imai). 0925-8388/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2008.11.149
BaSi2 . NaSi crystallizes in the monoclinic NaSi-type structure [4], while KSi, RbSi and CsSi in the cubic KGe-type structure [5].1 In addition, though corresponding LiSi has not been described in standard phase diagrams, a new Zintl phase of LiSi has been recently confirmed to exist as an equilibrium phase [6]. It belongs to the space group I 41 /a with the MgGa-type structure and is thermodynamically more stable than mixture of Li12 Si7 and Si, though the reaction of Li12 Si7 and Si to form LiSi are very low at ambient pressure and therefore it has not been prepared by the conventional method of cooling Li–Si melt. Properties of these alkali-monosilicides have not been determined yet and we do not know, at present, if they are semiconductors or not. The purpose of the present work is to clarify their electronic structures by first-principle calculations. This has been performed so as to find the possibility of band-gap tuning of BaSi2 from the viewpoint of solid-state chemistry, which suggests that increase in ionicity of the compounds will broaden the band-gap.2 We report the results of band calculation of these compounds and dis-
1 NaSi belongs to space group C 2/c. Silicon atoms form four isolated Si4 -tetrahedra but the arrangement of Na atom with respect to Si atoms is irregular, in contrast to the case of KSi, RbSi, and CsSi. The latter belong to the space group P43n ¯ . Silicon atoms form eight isolated Si4 -tetrahedra and alkali metal atoms are arranged around these tetrahedra. Each alkali metal atom is surrounded tetrahedrally by four Si4 . 2 From the analogy of the alkaline-earth metal silicides calculated before [7], the valence bands of alkali metal silicides would be composed mainly of Si 3s and Si 3p while the conduction bands would be mainly composed of the alkali metal s, p, (and d) states. Since the atomic orbitals of heavier alkali metal elements must have shallower energy levels, the energy gap between the valence and conduction band, Eg , would be broader in heavier elements. This tendency can be seen in the series of alkali metal halides; LiCl (Eg = 9.5 eV), NaCl (8.5 eV), KCl (8.5 eV), and CsCl (8.4 eV) [8].
Y. Imai, A. Watanabe / Journal of Alloys and Compounds 478 (2009) 754–757
Fig. 1. Band structure of NaSi. Broken lines in this and succeeding figures show the positions of the top of the valence band. Plotted numbers in this and succeeding figures express the order counted from the bottom of the valence band. Though Na 2s and 2p states are explicitly treated as a part of valence band in the pseudopotential used, 32 bands composed mainly of these atomic orbitals are omitted from the numbering in this figure.
cuss the possibility of the above-mentioned band-gap broadening of BaSi2 . 2. Calculation methods Calculations of the band structure (BS) along several high-symmetry lines in the Brillouin zone have been conducted using CASTEP (Cambridge Serial Total Energy Package) developed by Payne et al. [9], which is a first-principle pseudopotential method based on (1) the density-functional theory (DFT) in describing the electron–electron interaction, (2) a pseudopotential description of the electron–core interaction, and (3) a plane-wave expansion of the wavefunctions. The pseudopotential used is the ultrasoft pseudopotential generated by the scheme of Vanderbilt [10], which is bundled in the Cerius2 3 graphical User Interface. As for the method of approximation to the exchange–correlation term of the DFT, local density approximation (LDA) with generalized gradient correction [11] was used. The kinetic cutoff energy for the plane wave expansion of the wavefunctions was set at 380 eV, which had been confirmed beforehand to give well-converged band structures with respect to cutoff energy. The band structure calculations have been performed using observed crystallographic parameters [4–6]. We performed structural optimization processes before band structure calculations but that for LiSi did not converge well in contrast to other monosilicides.4 The energy zero of the BS diagrams given below is taken at the top of the valence band.
3
Cerius2 is a trademark of Accelrys, Inc. Preliminary optimization processes have shown that present calculations give quite good reproducibilities of the parameters of NaSi, KSi, RbSi, and CsSi. Optimized lattice parameters for NaSi are a = 1.2183 nm, b = 0.6573 nm, c = 0.1118 nm and ˇ = 119.03◦ , which can be compared with the observed parameters of a = 1.219 nm, b = 0.655 nm, c = 0.1118 nm, and ˇ = 119.0◦ [4]. Optimized lattice parameters for KSi, RbSi, and CsSi are a = 1.2617 nm, 1.3040 nm, and 1.3505 nm, which can be comparable to observed parameters of a = 1.262 nm, 1.304 nm, and 1.350 nm, respectively [5]. The errors are less than 0.3%. This was not the case for BaSi2 where 1.5% and 1.2% underestimations for a and c, respectively, was experienced in the previous study [3]. In contrast to those good reproducibilities, optimization process of LiSi 4
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Fig. 2. Band structure of KSi. Though K3s and 3p states are explicitly treated as a part of valence band in the pseudopotential used, 128 bands composed mainly of these atomic orbitals are omitted from the numbering in this figure.
3. Results and discussion At first, we present the calculated BS of NaSi near the Fermi level. Plotted in the BS figure are the numbers which express the order counted from the bottom of the valence band. The unit cell of NaSi contains 16 Na atoms and 16 Si atoms but can be reduced to Na8 Si8 which contains 40 valence electrons (1 electron from a Na atom and 4 electrons from a Si atom). The lowest 20 bands are fully occupied by these electrons and a gap is generated between the 20th and the 21st band. The top of the valence band is located at L (−1/2 0 1/2), whereas the bottom of the conduction band is located on the zone boundary plane vertical to c* axis. The latter is on the path from Z (0 0 1/2) to M (−1/2 −1/2 0) and on the path from Z (0 0 1/2) to L (−1/2 0 1/2). Therefore, the band-gap is indirect and the calculated gap value is 1.24 eV. Since there has been no observed gap value for NaSi to our knowledge, we cannot conclude that this value is reasonable. If we allow the tendency that about 60% of the measured gap is often obtained in the LDA calculation,5 NaSi must be a semiconductor with relatively wide gap value about 2 eV (Fig. 1). Because NaSi is too reactive with oxygen or moisture, it will be difficult to be used solely for practical applications. In addition, the estimated gap value above seems too wide for solar cell application.
did not converge well. Calculated Hellmann–Feynman force was as large as 12 GPa. This seems to be related to the fact that the form of tetrahedra contained in LiSi is far from regular tetrahedra. 5 For example, calculated gap value of BaSi2 is 0.72 eV in comparison with the observed value of 1.1–1.3 eV [3].
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Fig. 3. Band structure of RbSi (a) and CsSi (b). Though Rb4s, Rb4p, Cs5s, and Cs5p states are explicitly treated as a part of valence band in the pseudopotential used, 128 bands composed mainly of these atomic orbitals are omitted from the numbering in these figures.
However, it may be hopeful to use Na as a doping element of BaSi2 because it will widen the gap of BaSi2 when doped. Next, we present the calculated BS of KSi in Fig. 2. This compound (K32 Si32 ) contains 160 valence electrons (one electron from a K atom and four electrons from a Si atom) and 80 bands are fully occupied by these electrons. A gap is generated between the 80th and the 81st band. The gap is direct, located at X (1/2 0 0) with the gap value of 1.32 eV, which is wider than that of NaSi. In Fig. 3(a) and (b), we show the calculated BSs of RbSi and CsSi near the Fermi level,6 respectively, both of which have the same crystal structure with KSi. As in the case of KSi, 80 bands are fully occupied by 160 valence electrons in the unit cells. Band-gap is generated between the 80th and the 81st band. In both cases, the gaps are direct. However, the gap is located at X (1/2 0 0) with the gap value of 1.38 eV in case of RbSi, whereas it is located at M (1/2 1/2 0) with the gap value of 1.66 eV in case of CsSi. From these calculated values, we can see a tendency that the gap values of this series of compounds are increased as their ionicities are increased.7 This is also the case for monosilicides with different crystal structure. In Fig. 4, we show the calculated BS of LiSi. The unit cell of LiSi with the MgGa-type structure is composed of 16 Li atoms and 16 Si atoms, but can be reduced to the primitive cell composed 8 Li atoms and 8 Si atoms. Therefore, 40 valence electrons are contained in the calculated cell and these electrons occupy 20 bands, again. It can be seen that the band structure of LiSi is somewhat peculiar in that some bands show deviation from parabolic dispersion. The origin of this behavior is not clear but this might mean the (local) minima or maxima of energy in the k-space do not exist around high-symmetry points in the Brillouin zone.8 Therefore, we cannot exclude the possibility that we could not succeed in obtaining real gap value. However, so far as present calculation, a gap is generated between the 20th and the 21st band and the top of the valence band is located between N and C or A and P.9 The 21st band has a local minimum energy there, which is 1.60 eV higher than the 20th
6
Bands below the 54th band are omitted from Fig. 3. The ionization energies of K, Rb, and Cs are 4.3 eV, 4.2 eV, and 3.9 eV, respectively. As is well-known, the heavier elements in the alkaline group are more positive than lighter elements. 8 This was experienced in the band diagram of BaSi2 where the valence band maximum of BaSi2 is located between (0 0 0) and Y (0 1/2 0) [12]. 9 Symbols to describe symmetry points for this structure are named rather arbitrarily. Please refer to the notation shown in Fig. 4. 7
Fig. 4. Band structure of LiSi. Though Li1s state is explicitly treated as a part of valence band in the pseudopotential used, eight bands composed mainly of this atomic orbital are omitted from the numbering in this figure.
band, but the 21st band has the lowest energy at D (−1/2 1/2 1/2). The indirect energy gap was calculated to be 1.17 eV. Calculated gap values of LiSi, NaSi, KSi, RbSi and CsSi have a tendency that the gap values are increased as their ionicities are increased, irrespective of their crystal structures. This is also valid when we take BaSi2 into the consideration, since the 2nd ionization energy of Ba is 10.0 eV, larger than the ionization energy of Li. As expected from the rule, calculated gap value of BaSi2 is narrower than that of LiSi. However, the 1st ionization energy of Ba is
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BaSi2 , which may cause ill effect on the efficiency of solar energy conversion. As for the case (2), there has been very limited information on the ternary compounds of Ba–Si–alkali metal system. However, even if new compounds have different crystal structures from BaSi2 or alkali metal monosilicides, their band gaps will be broader than that of BaSi2 if only they have a character of a Zintl phase. Energetic study on which will actually proceed is under way. 4. Summary
Fig. 5. Band structure of Ba4 Cs4 Si16 where all the BaI sites in the BaSi2 lattice are replaced by Cs and all the BaII sites are occupied by Ba along with the high-symmetry directions of a simple orthorhombic Brillouin zone. Symmetry points for the simple orthorhombic Bravais lattice in this figure are S (1/2 1/2 0), X (1/2 0 0), (0 0 0), Y (0 1/2 0), T (0 1/2 1/2), R (1/2 1/2 1/2), U (1/2 0 1/2), and Z (0 0 1/2). Though Ba5s, Ba5p, Cs5s and Cs5p states are explicitly treated as a part of valence band in the pseudopotential used, 32 bands composed mainly of these atomic orbitals are omitted from the numbering in this figure.
5.2 eV, which is smaller than the ionization energy of Li. Therefore, Li incorporation into BaSi2 may not work well to broaden the gap of BaSi2 . From the present calculations, Na, K, Rb, and Cs are the candidate incorporation elements into BaSi2 so as to broaden its gap value. Incorporation of these elements will proceed by either (1) substitution of Ba atoms, or (2) formation of ternary compounds of Ba–Si–alkali metal. As the preliminary study for the case (1) we have performed band calculation of Ba4 Cs4 Si16 , where one of crystallographic sites, BaII , are substituted by Cs.10 The calculated band diagram is shown in Fig. 5. As shown, the Fermi level is shifted to downward with respect to the top of the valence band, compared to the case of BaSi2 , and crosses the 40th and 41st band because of lack of electrons caused by substitution of Ba by Cs. However, the gap between the top of the valence band ( point of the 40th band) and the bottom of conduction band (on the –Y segment of the 41st band) is calculated to be 1.142 eV, much larger than that of BaSi2 .11 This calculation suggests that diluted and random substitution of BaI site by Cs (and other alkali metal) will generate the localized impurity states by the mechanism of Anderson localization between the broader gap. However, too much incorporation of alkali–metal will evoke metal/semiconductor transition of
10 In BaSi2 , there are two crystallographically inequivalent sites for Ba (BaI and BaII ) and three inequivalent sites for Si (SiIII , SiIV , and SiV ) [13]. The unit cell contains eight formula units and the stoichiometric description of the unit cell is Ba8 Si16 . Ba atoms are distributed over 4 BaI and 4 BaII and we conducted energy calculation in order to determine which sites can be more favorably substituted by Cs using geometrically optimization process. The results indicated that BaII sites would be more favorably substituted by Cs. 11 The calculated gap value for BaSi2 with the observed structure was 0.72 eV. This value was increased to 0.80 eV when the optimized structure was adopted [3] but still much smaller than the value of Ba4 Cs4 Si16 above calculated.
In summary, we have performed band calculations of LiSi with the MgGa-type structure, NaSi with the NaSi-type structure, and KSi, RbSi, and CsSi with the KGe-type structure. All of them are semiconductors with the calculated indirect band gaps of 1.17 eV, and 1.24 eV, and the direct band gaps of 1.32 eV, 1.38 eV, and 1.66 eV, respectively. Though these calculated gap values themselves may not be accurate due to the limitation of the local density approximation used, we can see the general tendency that the gap values of these ionic semiconductors are increased as their ionicities are increased. Therefore, alkali–metal incorporation into BaSi2 will be hopeful to widen the band-gap of BaSi2 so as to make the gap value of BaSi2 approximate to the suitable value of 1.4 eV. Acknowledgements The authors would like to express their sincere gratitude to Dr. Motoharu IMAI, National Institute for Materials Science (NIMS), Japan, for fruitful and instructive discussions. They are also grateful to Dr. Masaaki SUGIE of National Institute of Advanced Science and Technology (AIST), Japan, for his kind technical assistance. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
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