Physics of the Solid State, Vol. 45, No. 2, 2003, pp. 201–206. Translated from Fizika Tverdogo Tela, Vol. 45, No. 2, 2003, pp. 193–197. Original Russian Text Copyright © 2003 by KurganskiÏ, Pereslavtseva, Levitskaya. SEMICONDUCTORS AND DIELECTRICS Fermi Surface and Electrical Characteristics of Molybdenum Disilicide S. I. KurganskiÏ, N. S. Pereslavtseva, and E. V. Levitskaya Voronezh State University, Universitetskaya pl. 1, Voronezh, 394006 Russia e-mail: [email protected] Received April 9, 2002 Abstract—Semirelativistic self-consistent calculations of the electronic structure of MoSi2 are performed within the framework of the linearized augmented-plane-wave (APW) method in the local density functional approximation. The results of investigations of the band structure, the Fermi surface, and electrical character- istics (effective cyclotron masses, the conductivity anisotropy constant, the mean free path, and the coefficient γ of the heat capacity component linear in temperature) are reported. The Fermi surface consists of two sheets, namely, an electron sheet and a hole sheet. The extreme sectional areas of the Fermi surface agree well with the experimental data on the de Haas–van Alphen effect. The results of first-principles calculations need no addi- tional correction. © 2003 MAIK “Nauka/Interperiodica”. 1. INTRODUCTION tional approximation, specifically for the band structure [20, 21, 24], densities of states [17, 20–24], and the Molybdenum disilicide possesses high microhard- Fermi surface [22, 24]. However, none of the first-prin- ness [1], resistance to oxidation during heating both in ciples calculations of the Fermi surface [22, 24] agrees air and in an oxygen flow [1], high thermal stability [2], in full measure with the experiment [19]. It turned out temperature-independent high thermoelectric power that the results of direct calculations of the extreme sec- [3], low resistance [4–9], and good technological prop- tional areas of the Fermi surface [22, 24], which depend erties. This compound has long been in use in many on the direction of the magnetic field, considerably fields of science and engineering. Initially, molybde- exceed the experimental values [19]. This discrepancy num disilicide compounds were used as antirust coat- was explained as resulting from the use of the local den- ings. However, considerable advances made in micro- sity functional approximation. In order to achieve better and nanotechnologies substantially extended the field agreement between the calculation and the experiment of application of MoSi2 materials. In recent years, [19], the results of the direct calculations performed in molybdenum disilicide has been used in the fabrication [22, 24] were corrected by fitting to the experimental of diffusion barriers [10], protective coatings for inter- Fermi surface. In particular, Andersen et al. [24] dis- metallic compounds at elevated temperatures [11], placed the Mo d 2 2 states to the Fermi level by high-temperature thermocouples deposited on elec- x – y trodes [3], silicon integrated circuits [1], coatings for 0.41 eV and the Mo dxy states to the valence band bot- aircraft structures [12], and high-temperature turbine tom by 0.21 eV. As a result, the initially obtained density blades [13]. Such a wide spectrum of application of of states at the Fermi level (0.31 states/eV per unit cell) MoSi2 stems from the high stability of its properties. decreased by approximately 20% and almost reached The nature of these properties cannot be understood the experimental value (0.24 states/eV per unit cell) without detailed investigation of the electronic struc- [25]). Antonov et al. [22] applied a correction accord- ture. ing to the same scheme. However, the results of these At present, the electronic structure of MoSi has corrections are ambiguous: the corrected data in [24] 2 are in better agreement with the experiment [19] than been extensively investigated using experimental and those obtained in [22]. theoretical techniques. Among the experimental meth- ods are x-ray emission [14–16] and photoelectron [17, This paper reports on the results of direct theoretical 18] spectroscopies. The Fermi surface has been experi- investigations of the electronic structure of body-cen- mentally studied in terms of the de Haas–van Alphen tered tetragonal MoSi2. In the present work, we calcu- effect [19]. The electronic structure of MoSi2 has been lated the band structure, the Fermi surface, and electri- treated theoretically in the framework of the self-con- cal characteristics, such as the effective cyclotron sistent relativistic pseudopotential approach [20], the masses, the conductivity anisotropy constant, the mean linear muffin-tin orbital formalism [21–24], and the free path, and the coefficient γ of the heat capacity com- augmented-spherical-wave (ASW) method [17]. These ponent linear in temperature. Our results are in satisfac- calculations were performed in the local density func- tory agreement with the available calculated [20–22, 1063-7834/03/4502-0201 $24.00 © 2003 MAIK “Nauka/Interperiodica” 202 KURGANSKIÎ et al. of MoSi2 crystallizes in a body-centered tetragonal 17 structure (Fig. 1) (space group D4h –I4/mmm) with the unit cell parameters a = b = 0.3202 nm, c = 0.7852 nm, and c/a = 2.452 [31]. The calculation was performed in the basis set of 176 linearized augmented plane waves. Mo The basis function was expanded in spherical harmon- Si ics up to terms with l = 7. 3. RESULTS AND DISCUSSION 3.1. Band Structure and the Fermi Surface Figure 2 represents the band structure of MoSi2. According to the calculation, the valence band is equal to 13.4 eV. The energy bands at the valence band bot- tom are attributed to the Si s electron states. The region Fig. 1. Crystal structure of MoSi2. from –0.3 to –4.7 eV is determined primarily by the Mo d states with an admixture of Si p states. This is in good agreement with the available calculated data on the density of states [17, 20–24]. The density of states at 10 the Fermi level (EF) is relatively low [N(NF) = 0.28 states/eV per unit cell], which indicates that the compound is stable. This value is less than that deter- 5 mined prior to the fitting of the results in [24] (0.31 states/eV per unit cell) but is closer to the experi- mental value obtained in [25] (0.24 states/eV per unit cell). The Fermi level is crossed by two energy bands, 0 namely, by the seventh and eighth energy bands counted off from the valence band bottom. The seventh Γ band passes through the Fermi level EF near the point Ð5 of the Brillouin zone and is empty in this vicinity. The eighth band crosses the Fermi level in the vicinity of the Z point. In this part of the Brillouin zone, the eighth Ð10 band is characterized by occupied states. These bands form the Fermi surface of MoSi2 (Fig. 3). The intersec- tion of the seventh band with the Fermi level generates a hole sheet on the Fermi surface (Fig. 3a). This sheet Ð15 Za b Z É cdXÉ N PX has the shape of a cylinder that is slightly widened both in the upper and lower half and is then narrowed in the Γ Fig. 2. Band structure of MoSi2. bases along the Z direction. The eighth band forms an electron sheet on the Fermi surface (Fig. 3b) having the shape of a Z-centered rosette. A comparison showed 24] and experimental [4, 19, 25, 26] data. However, no that the results of our calculations of the Fermi surface additional correction, such as that performed in [22, are in good agreement with the theoretical investiga- 24], was needed in our case. tions performed in [22, 24] and the experimental data on the de Haas–van Alphen effect [19]. Nonetheless, there are some differences. As follows from our calcu- 2. COMPUTATIONAL TECHNIQUE lation, the seventh and eighth bands are degenerate in Γ The electronic structure of an MoSi2 crystal was cal- the Z direction. Consequently, the electron and hole culated in the local density functional approximation sheets of the Fermi level in the ΓZ direction touch each using an exchange correlation potential within the other at a point. However, according to [22, 24], the approach proposed in [27]. The energy spectrum was electron and hole sheets are separated by a gap. This calculated by the self-consistent linearized augmented- difference stems from the fact that our calculation plane-wave (APW) method in terms of the formalism ignores the spin–orbit splitting, which relieves the used in [28] within the scalar relativistic approximation degeneracy of these bands. As was shown in [20, 21], [29], which includes all relativistic effects, except for the spin–orbit splitting of energy levels in MoSi2 is a spin–orbit splitting. The linearized APW method was small quantity; as a result, there arises a gap of described in detail in [30]. The low-temperature phase 0.007 bohr–1 between the electron and hole sheets of PHYSICS OF THE SOLID STATE Vol. 45 No. 2 2003 FERMI SURFACE AND ELECTRICAL CHARACTERISTICS OF MOLYBDENUM DISILICIDE 203 the Fermi surface [20]. This value is less than 2% of the (a) characteristic size of the Fermi surface. Therefore, the [001] error introduced by this neglect is small compared to the contributions made by other factors (for example, the potentials used in the calculations) and does not change the topology of the calculated Fermi surface. A2 It was expedient to analyze quantitatively the topol- A1 ogy of the Fermi surface and to compare our results with the experimental data [19] on the de Haas–van [010] Alphen effect. For this purpose, we calculated the area [100] of the extreme section of the Fermi surface as a function of the direction of the magnetic field (Fig.