Calculation of electronic energy and density of state of iron-disilicides using a total-energy pseudopotential method, CASTEP

Thin Solid Films 381 Ž2001. 176᎐182

Calculation of electronic energy and density of state of iron-disilicides using a total-energy pseudopotential method, CASTEP Yoji ImaiU , Masakazu Mukaida, Tatsuo Tsunoda Department of Inorganic Materials, National Institute of Materials and Chemical Research, Higashi 1-1, Tsukuba, Ibaraki 305-8565, Japan Received 26 April 1999; received in revised form 17 April 2000; accepted 17 April 2000

Abstract Electronic energies of ␣-, ␤-, and ␥-FeSi 2 were calculated so as to elucidate the possibility of the prediction of phase stability by a quantum-mechanical calculation using a total-energy pseudopotential code, CASTEP. It was properly predicted that the ␤-phase is more stable than ␣- and ␥-FeSi 2 . The effect of the non-stoichiometry of ␤-FeSi 2 and doping elements ŽMn, Cr, Co, and Ni. on the Fermi energy was also discussed. 䊚 2001 Elsevier Science B.V. All rights reserved. Keywords: FeSi 2 ; Electronic energy; Fermi level

1. Introduction Much effort has been devoted to develop effective materials for thermoelectric energy conversion w1x. Among them, ␤-FeSi 2 is expected because of its relatively high Seebeck coefficient, stability in high-temperature oxidizing atmosphere, abundance in natural resources, and relatively low toxicity to ecology w2x. It has been also attracting attention as an optoelectronic device integrated on well-established Si-technology w3x and photoconductivity w4x, due to its semiconducting nature. However, transformation to the metallic phase, usually referred to as ␣-FeSi 2 , at a higher temperature than 1210 K w5x and the low carrier mobility limits the field of application. Though substitution of constitutional elements has been tried to control the phase stability w6x or to control carrier concentration for obtaining a higher value of figure of merit w7x, the semiconducting phase becomes unstable when inappropriate doping is done. U

Corresponding author. Tel.: q81-298614547; fax: q81-298614709.

Relations between the crystal structure and the constitutional elements of various inorganic compounds are empirically studied and described on the maps w8,9x. However, they will not give a quantitative prediction for solid solubility of the added elements nor the electric properties such as an energy band gap. A quantum-mechanical approach by the first-principle calculation to the phase stability and the physical properties will be, if possible, more useful. In order to understand theoretically the phase stability, the structural energetics of the phases should be determined as a first step though entropy term might be important especially at higher temperatures. The cohesive energy of the specific phases is given by the sum of many terms such as Ž1. the absolute position of the mean electron level lying below its free atomic level; Ž2. the total one-electron band-structure energy measured relative to Ž1.; Ž3. the electrostatic potential energy of the ion lattice; Ž4. the repulsive contribution of the ion-core; and Ž5. exchange and correlation effects not included in the above terms. Fortunately, difference in the cohesive energy among the different structures is mainly determined by the term Ž2. in

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Y. Imai et al. r Thin Solid Films 381 (2001) 176᎐182

non-magnetic metallic compounds, as shown by the epoch-making work of Pettifor w10,11x who explained the transition-metal structure trend. Therefore, it is hopeful that the phase stability, at least at 0 K where entropy term has no effect, would be predicted by calculating the term Ž2.. The electronic structure in relation with the term Ž2. of ␤-FeSi 2 was calculated with augmented spherical wave ab initio band᎐structure method w12x or linear muffin᎐tin orbital method ŽLMTO. in conjunction with the local density approximation ŽLDA. to density᎐functional theory ŽDFT. w13x. That of ␣-FeSi 2 was calculated with LMTO w14x. However, it was difficult to compare directly these results because different methods were applied to the calculations. Recently, a method of ‘total energy pseudopotential calculation’ has been developed by Payne et al. w15x. It is based on DFT in describing the electron᎐electron interaction and on a pseudopotential description of the electron᎐core interaction, and has been publicized as a CAmbride Serial Total Energy Package ŽCASTEP.. It gives the sum of electronic energy of a large system, as well as its band structure. Transferability and robustness of the assumed pseudopotentials of each element seem to be confirmed by success in reproducing the physical properties such as lattice parameters of many compounds. Therefore, it can be expected to give the relative stability of different crystal structures. In the present paper, CASTEP is applied to the electronic energy calculation of iron-disilicides. Besides ␣- and ␤-FeSi 2 mentioned above, ␥-FeSi 2 with a calciumfluorite structure, which has been reported to be present in thin film w16x was also calculated and the relative stability and the density of states of these phases are discussed, in contrast to the case of nickeldisilicide. The effects of doping elements and nonstoichiometry of ␤-FeSi 2 will be also discussed. 2. Methods The CASTEP calculation of the above phases was performed through Cerius 2 graphical user interface.1 Some difficulties were found in the calculation. One of the difficulties in the calculation is the total energy dependence of the phases on the size of the assumed cell. For example, preliminary calculations showed that one-half of the total energy of two Fe atoms in the conventional body-centered cubic cell are approximately 0.03% smaller than that of one Fe atom in the primitive cell represented by the following translational vectors:

p 1 s Ž ar2.Ž i q j q k . , p 2 s Ž ar2.Ž yi q j q k . , p 3 s Ž ar2.Ž yi y j q k . , where a is the length of the side of the conventional unit cube, and i, j and k are orthogonal unit vectors parallel to the cube edges. This seemed to be caused by the different number of k-points sampled from the reciprocal space and the number of plane-wave expansion of wavefunctions used in the calculation. In principle, the number of k-points and the energy cutoff for plane-wave expansion of wavefunctions should be increased until the numerical error in the energy is reasonably small. The total energy should be systematically taken to complete convergence with respect to these. However, owing to the limitation of the computer used, it was not possible to carry out the above check for large primitive cells. Alternatively, structures that have resemblance to each other and include the same number of atoms were used for calculation in the present study. As for ␤- and ␥-FeSi 2 , the relation between them was clarified by Dusausoy et al. w17x. ␥-FeSi 2 belongs to the space group of Fm3m with cubic symmetry and ␤-FeSi 2 to Cmca with orthorhombic symmetry. However, the former can be regarded as belonging to Cmca when the lengths of the side Ž a, b and c . are selected as as 2 ⭈ a c , bs c s '2 ⭈ a c where a c is the side of the unit cube. If the unit cell of ␥-FeSi 2 is selected in such a way, ␤-FeSi 2 can be regarded as a form slightly distorted from ␥-FeSi 2 and both are described as Fe16 Si 32 . Another difficulty in the calculation is instability 2 in the self-consistent ŽSC. procedure for metallic systems with non-trivial symmetry. In CASTEP calculation for metallic systems, Gaussian-like smearing width of each energy level is introduced to eliminate discontinuous changes in energy when an energy band crosses a Fermi level during SC procedure. The symmetry of the structures was generally maintained by using an option to symmetrize the wavefunctions. However, pronounced instability was sometimes seen in the calculation during the charge density symmetrization process involved in each SC step when the value of smearing width was small, especially in case the structure is described using the conventional unit cells. This difficulty is known to be overcome by using a superlattice with a degraded symmetry. Thus, unit cells for the present calculation were generated by changing primitive translation vectors from those on the orthogonal coordinate, i, j and, 2

1

Cerius 2 is a trademark of Molecular Simulations Inc.

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The present calculation has been done using the CASTEP code, the user-interface of which has been provided by the Cerius 2 version 2.0. Improved stability in the SC procedure has been confirmed for the recent version of CASTEP.

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Fig. 1. Ža. Transformation of the primitive translation vectors of the conventional orthogonal lattice to the triclinic lattice used in the present calculation. Žb. Two basis atoms in the triclinic lattice are required to describe one basis atom in the orthogonal primitive cell.

k, to those on the triclinic coordinate, p 1 , p 2 and p 3 , as shown in Fig. 1a and given by the following equations p 1 s b j q ck, p 2 s ck q ai, p 3 s ai q b j,

Fig. 2. Ža. ␣-, ␤-, and ␥-FeSi 2 on the conventional orthogonal lattice Ž ␣-FeSi 2 , ␤-Fe16 Si 32 , and ␥-Fe 4 Si 8 .. Žb. ␣-, ␤-, and ␥-Fe 32 Si 64 used in the calculation.Bigger open circles represent silicon atoms and smaller closed circles indicate iron atoms.

Ž1.

where a, b and c are the lengths of sides of the conventional orthogonal unit. It should be noted that two basis atoms in the triclinic lattice thus formed are required to describe one basis atom in the orthogonal primitive cell, as shown by Fig. 1b. Therefore, this way for selection of primitive translation vectors requires that the superlattice of the cell should be used for calculation. Though the volume of the cell to be calculated is doubled, this way of building up a superlattice drastically improved the stability of self-consistent procedures. It can be shown that one atom, the fractional coordinate of which is Ž u,¨ ,w . in the orthogonal primitive cell corresponds to the two atoms, the fractional coordinates of which are Žy0.5⭈ u q 0.5⭈ ¨ q 0.5⭈ w, 0.5⭈ u y 0.5⭈ ¨ q 0.5⭈ w, 0.5⭈ u q 0.5⭈ ¨ y 0.5⭈ w . and Žy0.5⭈ u q 0.5⭈ ¨ q 0.5⭈ w q 0.5, 0.5⭈ u y 0.5⭈ ¨ q 0.5⭈ w q 0.5, 0.5⭈ u q 0.5⭈ ¨ y 0.5⭈ w q 0.5.. From the above considerations, all the calculations were done for Fe 32 Si 64 . For the ␤- and ␥-phase, Fe16 Si 32 cells belonging to the space group of Fm3m are transformed by Eq. Ž1. to make F32 Si 64 cells. For the ␣phase, a conventional Fe1 Si 2 cell belonging to the space group of P4r mmm is transformed by Eq. Ž1. to

make a Fe 2 Si 4 cell. After that: Ži. one of the primitive translation vectors of the Fe 2 Si 4 cell was doubled; Žii. next the other primitive vector was doubled; and Žiii. the last primitive translational vector was quadruplicated. The ␣-F32 Si 64 cell could be thus obtained. The atomic arrangements of each phase, thus given, are schematically presented in Fig. 2b, in comparison with those on the conventional primitive lattices ŽFig. 2a.. The total energies of the phases were calculated for varied volumes of unit cells. This was done assuming the unchanged ratios of the lengths of the unit cell edges parallel to each reference axis and the unchanged fixed interaxial angles, given by Table 1, and the fixed fractional coordinate of each atom calculated by the above equation using the values from the data of real crystals w18,19x. The total energy calculation conditions are as follows. The values of initial and final Gaussian-like smearing width of each energy level were set at 4 eV and 0.1 eV, respectively. Finite basis-set correction to the calculated total energy was performed according to the scheme proposed by Francis and Payne w20x. The correction term was evaluated numerically using three

Table 1 Assumed values of the ratio of a, b, and c Žthe lengths of the unit cell edges parallel to each reference axis. and the values of ␣ , ␤, and ␥ Žthe interaxial angles. in the calculated structures of ␣-, ␤-, and ␥-Fe 32 Si 64

Ratio of the lengths of primitive vectors Ž a:b:c . Interaxial angles Ž⬚. ␣ ␤ ␥

␣-Fe32 Si64

␤-Fe32 Si64

␥-Fe32 Si64

1.0:1.0:1.3126

1.0:1.140:1.138

1.0:1.225:1.225

70.84 70.84 38.316

52.085 64.079 63.836

48.189 65.906 65.906

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different kinetic cutoff energies for plane-wave expansion of wavefunctions of 294, 297 and 300 eV. Because of the very large cells to be calculated and the limitations imposed by the ability of computational resources, the spacing in the reciprocal space to generate k-points by Monkhorst᎐Pack scheme w21x was set at 0.05 Ay1 , resulting in sampling of only four k-points from the irreducible part of the Brillouin zone. The same structural and calculation conditions were also used for the following calculations. First, the effect of non-stoichiometry on the generation of localized energy levels in the energy gap of ␤-FeSi 2 were studied by introducing a vacancy at the Fe- or Si-site. Thus, Fe 31 Si 64 and Fe 32 Si 63 cells were considered. The effect of Coulomb interactions between repeated images of the charged defects must be considered when energetics of the isolated vacancy is the main focus. However, the contribution to the interaction energy can be obtained by regarding the defects as point charges in a uniform matrix when the system considered is large enough even in the case of ionic crystals. For example, cells up to 48 or 32 ions in the case of Li 2 O or MgO, respectively, were considered to be large enough for the above model w22x, and it can be expected that the density of state ŽDOS. of the structures with defects can be correctly produced by the above simple treatment. The positional relaxation of the surrounding atoms was not considered, neither, because no definite information on the relaxation is available. Doping effects of Mn, Cr, Co, and Ni to ␤-FeSi 2 were also calculated by substitution of these atoms into an Fe site without the positional relaxation of the surrounding atoms. As for the method of approximation to the exchange-correlation term of DFT, local density approximation ŽLDA. was used. Spin-polarization of iron atoms was not considered because preliminary calculations for ␣-FeSi 2 gave precise lattice parameters. Though it is generally admitted w23x that the LDA gives a few percent smaller interatomic distances for 3d magnetic metals if spin-polarization is disregarded, the effect of magneto-volume effects can be ignored in the present case for Si-rich phases. 3. Results and discussion 3.1. Order of electronic energy of iron-disilicides and comparison to the case of NiSi 2 The calculated total energies vs. the volumes of the unit cell for ␣-, ␤- and ␥-FeSi 2 are shown in Fig. 3. The energy of the phase has its minimum when the cell volume is approximately 1200 = 10y3 nmy3 , which is very close to their experimental equilibrium value shown by the arrows in the figure.

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Fig. 3. Calculated variation of the total energies of ␣-, ␤-, and ␥-Fe 32 Si 64 . The arrows indicate the equilibrium volume of ␣- and ˚3 and 1204 A ˚3, respectively.. ␤-Fe 32 Si 64 Ž1193 A

Among these phases, ␤-FeSi 2 is correctly the lowest in energy; following that is the ␣-phase. This may sound strange because ␥-FeSi 2 exists in an ultrathin configuration on SiŽ111. w16x, whereas the ␣-phase with stoichiometric composition does not. However, phase stability diagrams of bulk phases shows that the ␣-phase is stable at higher temperature, where the entropy term becomes important in free energy, and ␥-FeSi 2 is not. In addition, Jedrecy et al. reported w24x the tetragonal symmetry of FeSi 2 epitaxial grains on SiŽ111.. The deviation of chemical composition from the stoichiometric FeSi 2 to Si-rich side was, also, reported to bring about the formation of the ␣-phase w25x. Thus, it would be reasonable to consider that the ␣-phase is energetically more stable than ␥-FeSi 2 and ␥-FeSi 2 can appear only in the form of ultrathin film, where the strain-induced energy difference between these phases are predominant. The reason for different total energies is clearly demonstrated by Fig. 4 which presents the densities of states ŽDOS. of these phases. The phase, the DOS at the position of Fermi level Ž EF . which is high compared to other phases, cannot be energetically stable because the total energy of that structure is larger than those of the phases whose DOS values at EF are low. As Christensen pointed out w13x, EF of ␥-FeSi 2 falls right in a sharp peak of DOS, whereas distortion of the crystal lowers the energy Ža Jahn᎐Teller-type effect. to make semiconducting ␤-FeSi 2 . DOS of the ␣-phase slopes gently at EF and the total energy of the ␣-phase is lower than ␥-FeSi 2 . The gap of DOS of ␤-FeSi 2 in the present calculation is approximately 0.7 eV and underestimated compared with the experimental value Žapprox. 0.85 eV

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Fig. 5. Calculated variation of the total energies of Ni 32 Si 64 of CaF2 structure and of the hypothetical ␤-FeSi 2 structure.

Fig. 4. Density of states of ␣-, ␤-, and ␥-Fe 32 Si 64 . The arrows indicate the Fermi levels.

w26x., as is often observed when the calculation is done under a local density approximation. As described above, the presence of semiconducting ␤-FeSi 2 is closely related to the position of EF of the ␥-FeSi 2 ŽCaF2 structure .. This is in contrast to CoSi 2 and NiSi 2 , both of which have a CaF2 structure. To confirm this, the total energy and DOS of NiSi 2 were also calculated, assuming the hypothetical phase of ␤-FeSi 2 type has the same fractional coordinates as ␤-FeSi 2 . The results were shown in Figs. 5 and 6, respectively. As clearly demonstrated, EF of NiSi 2 with a CaF2 structure rises above a peak of the DOS, and distortion of the crystal observed in ␤-FeSi 2 will not lower the energy. Thus, NiSi 2 has the structure of CaF2 and shows metallic properties.

non-stoichiometric ␤-FeSi 2 were calculated. As there seems to be no definite information on the kind and site of defects in ␤-FeSi 2 , a site vacancy of Fe or Si was assumed. The calculated DOS of ␤-FeSi 2 with a Si vacancy and that with an Fe vacancy near the gap are shown by Fig. 7a,c, respectively, with that of perfect ␤-FeSi 2 ŽFig. 7b.. As shown by the figures, EF of Si-deficient ␤-FeSi 2 ŽFe 32 Si 63 . exists a bit higher than the top of the valence band, though some defect levels are formed between the gap. On the contrary, EF of Fe-deficient ␤-FeSi 2 ŽFe 31 Si 64 . exists between the defect levels formed between the gap. Thus, p-type conduction can be expected for Si-deficient ␤-FeSi 2 . The reason for n-type conduction for Fe-deficient ␤-FeSi 2 is not clearly demonstrated from the present calculation, however, more precise calculations using a large number of sampling points in k-space may be necessary to obtain a more precise estimation of the energy levels between the

3.2. Effect of non-stoichiometry and doping elements on density of states From the DOS shown by Fig. 4, ␤-FeSi 2 is an intrinsic semiconductor. However, it is known that non-intentionally doped ␤-FeSi 2 is p-type and the reason for this is not known. Also, n-type conduction is recently observed in ␤-FeSi 2 thin films with a Si excess w27x. Therefore, defects in ␤-FeSi 2 seem to play an important role on this nature. To elucidate the effect of defects on the type of conduction, DOS values of

Fig. 6. Density of states of Ni 32 Si 64 of CaF2 structure and hypothetical ␤-FeSi 2 structure. The arrows indicate the Fermi levels.

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gap. We may also have to take into consideration the trace amounts of coexisting metallic ␣-phase in Si-rich composition range w25x. Replacement of Fe in ␤-FeSi 2 by Cr or Mn gives p-type conduction and that by Co or Ni produces n-type conduction w28x. The effects of these elements on DOS calculations are shown in Fig. 8. As shown by the figure, EF of Cr- or Mn-doped ␤-FeSi 2 ŽFig. 8a or b. is higher than the top of the valence band of undoped ␤-FeSi 2 ŽFig. 8c. and that of Co- or Ni-doped ␤-FeSi 2 ŽFig. 8d or e. is lower than the bottom of the conduction band of undoped ␤-FeSi 2 . Thus, p- and n-type conduction can be expected for Cr- or Mn-doped ␤-FeSi 2 and for Co- or Ni-doped ␤-FeSi 2 , respectively; this agrees with experimental observations. As for positions of impurity levels between the bandgap, the present estimations are different from the values using the so-called ‘hydrogen-atom model’, as given by the following term: Ž m 0rm ) . ⭈ Ž ␧y2 . ⭈ 13.6 Ž eV. where m 0 is the free electron mass, mU is the effective mass of electron or hole, and ␧ is a static dielectric constant of the semiconductor matrix. If we assume the following values of mU K 1.0⭈ m 0 w29x and ␧ s 61.6 w30x, the above terms mean the generation of impurity levels at approximately 4 meV higher than the top of the valence band or lower than the bottom of the conduction band. The energy levels of impurity bands calculated in the present study, thus,

Fig. 8. Comparison of density of states of doped ␤-FeSi 2 . Ža. Fe 31Cr 1 Si 64 , Žb. Fe 31 Mn 1 Si 64 , Žc. undoped ␤-Fe 32 Si 64 , Žd. Fe 31Co 1 Si 64 , and Že. Fe 31 Ni 1 Si 64 . The arrows indicate the Fermi levels.

seem to be separated too much from the top of the valence band or the bottom of the conduction band. However, experimental observation on the dependence of the gap energy of Co-doped ␤-FeSi 2 shows that approximately 20% reduction of the gap energy is brought about by a replacement of 15% of the iron atom by cobalt w31x. Therefore, this may be better explained by the present calculation than the hydrogen-atom model. The present calculation is done using only four kpoints in the Brillouin zone. Calculations using a larger number of sampling points in k-space are expected to produce a more precise estimation of the energy level. 4. Conclusion

Fig. 7. Comparison of density of states of ␤-FeSi 2 with Si vacancy Ža., and Fe vacancy Žc.. Perfect ␤-FeSi 2 is shown in Žb.. The arrows indicate the Fermi levels.

In summary, the following results are presented using the CASTEP code. ␤-FeSi 2 is the lowest in electronic energy followed by ␣-FeSi 2 . ␥-FeSi 2 is energetically the most unstable among the iron-disilicides considered because the Fermi level of ␥-FeSi 2 falls right in a sharp peak of the density of states ŽDOS. curve. This is not the case of nickel-silicides where the Fermi level of CaF2 structure rises above a peak of the DOS and distortion of the crystal observed in ␤-FeSi 2 will not lower the energy. The DOS of non-stoichiometric ␤-FeSi 2 suggested

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the p-type conduction of Si-deficient phases, however, n-type conduction by Fe-deficient phases were not clearly shown by the present calculation. p- and n-type conduction can be expected for Mn- or Cr-doped ␤FeSi 2 and for Co- or Ni-doped ␤-FeSi 2 , respectively. Acknowledgements The authors would like to express their sincere gratitude to Dr Masaaki SUGIE, National Institute of Materials and Chemical Research, for his kind technical assistance during calculations. References w1x International Conference on Thermoelectrics, Dresden, 1997 and Nagoya, 1998 Žpublished many papers related to thermoelectric energy conversion.. w2x U. Birkholz, E. Gross, in: M. Rowe ŽEd.., Thermoelectrics, CRC press, 1994, p. 287 Žfor review of FeSi 2 .. w3x J. Derrien, J. Chevrien, V. Le Thanh, J.E. Mahan, Appl. Surface. Sci. 56 Ž1992. 382 Žand references therein .. w4x E. Arushanov, E. Bucher, Ch. Kloc, O. Kulikova, L. Kulyk, A. Siminel, Phys. Rev. B52 Ž1995. 20. w5x O. Kubaschewski, Iron-Binary Phase Diagrams, SpringerVerlag, Berlin, 1982, p. 136. w6x H. Takizawa, P.F. Mo, T. Endo, M. Shimada, J. Mat. Sci. 30 Ž1995. 4199. w7x T. Tsunoda, unpublished data. w8x C. Calandra, O. Bisi, G. Ottaviani, Surf. Sci. Rep. 4 Ž1985. 271. w9x D.G. Pettifor, Mat. Sci. Technol. 4 Ž1988. 675. w10x D.G. Pettifor, J. Phys. C3 Ž1970. 367. w11x D.G. Pettifor, CALPHAD 1 Ž1977. 305. w12x R. Eppenga, J. Appl. Phys. 68 Ž1990. 3027. w13x N.E. Christensen, Phys. Rev. B42 Ž1990. 71489.

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