Tunability of Mg2si Bandgap by Formation of Mg2(Si,C) with an Anti-Fluorite Structure Examined by First-Principles Calculations

Tunability of Mg2Si Bandgap by Formation of Mg2(Si, C) with an Anti-Fluorite Structure Examined by First-Principles Calculations

Yoji Imai1,+, Atsushi Yamamoto1 and Ken-ichi Takarabe2

1National Institute of Advanced Industrial Science and Technology, AIST Tsukuba Central 2, Tsukuba 305-8568, Japan 2Okayama University of Science, Okayama 700-0005, Japan

We used first-principles calculations to investigate the effects of replacing Si atoms in Mg2Si with C atoms to tune its bandgap and enhance its thermoelectric performance. First-principles calculations suggest that the substitution of Si by C atoms in the Mg2Si lattice forms Mg8Si4¹zCz (z = 1, 2, and 3) with a sustained anti-fluorite structure, which results in an unexpected bandgap contraction. However, the bandgap of Mg8Si4(1¹x)C4x composed of a supercell structure of a rhombohedral Mg2Si primitive cell and Si-substitution with C has a wide bandgap when the C/Si ratio is sufficiently high. The formation enthalpies from Mg, Si, and C (diamond) are negative under pressures greater than ca. 15 GPa. [doi:10.2320/matertrans.M2019082]

(Received March 26, 2019; Accepted June 21, 2019; Published August 2, 2019)

Keywords: Mg2Si, carbon substitution, bandgap calculation, anti-ﬂuorite structure

1. Introduction in the Mg2Si lattice by transition metals to achieve a hoping chemical pressure effect.12) Magnesium silicide (Mg2Si) is a narrow-gap semi- In 2014, Balout et al. used a density functional theory conductor, which has attracted interest as a promising and (DFT) approach in a theoretical investigation of the effects eco-friendly thermoelectric semiconductor in the temperature of tensile and compressive strain on the electronic and 13) range of 600900 K at ambient pressure. Many studies have thermoelectric properties of Mg2Si. They stated that been performed on Mg2Si-based materials, as reviewed by isotropic tensile stress or negative pressure enhanced Gonçalves and Godart1) and Cheng et al.2) However, the thermoelectric performance. Substitution of Mg or Si with thermoelectric performance of these materials is insufficient heavier isovalent elements produces a negative chemical for practical applications and further improvements are pressure through volume expansion. However, Mg sub- required. The performance of thermoelectric devices depends stitution with heavier elements, such as Ca, Sr, and Ba, is not on a number of physical properties, and particularly on the expected to be possible because their half-silicides X2Si bandgap between electrons and holes, as theoretically (X = Ca, Sr) have the anti-cotunnite structure and the 3) suggested by Sofo and Mahan. Therefore, it is of vital solubility of these elements into the Mg2Si lattice is quite 4,5) importance to tune the bandgap of the materials to optimize limited. The solid solubilities of Ge and Sn into Mg2Si are device performance for given environmental conditions. It expected to be high because Mg2Ge and Mg2Sn have the has been claimed that the optimum bandgap of a direct same structure as that of Mg2Si. However, their indirect bandgap semiconductor should not be lower than 6kBT and bandgaps are 0.74 and 0.36 eV, respectively, which are may be higher, depending on the dominant scattering narrower than that of Mg2Si (0.77 eV) and a solid solution of mechanism, where T is the operating temperature of the Mg2SixGe1¹x or Mg2SixSn1¹x would have narrower bandgaps device and kB is the Boltzmann constant. than that of Mg2Si. To enhance the thermoelectric performance of Mg2Si in the To realize a negative pressure and increase the bandgap, relatively high temperature region, widening of the Mg2Si carbon (C) may have advantages over Ge and Sn as a solid bandgap is necessary. From this viewpoint, a number of solution element for Mg2Si. Carbon is more electronegative experimental and theoretical studies have been performed to than Si and causes a downward shift of the whole valence tune the bandgaps of alkaline-earth silicide semiconductors. band, which widens the bandgap. However, there are two These include investigations on Mg2Si, BaSi2, and Si- problems to be considered. clathrates, which are promising candidate materials for solar First, Mg2C with the anti-fluorite structure was unknown energy conversion. Examples include: (1) Mg substituted by until Kurakevych et al.’s work under high pressure.14) Their isovalent heavier elements, i.e., Ca, Sr, or Ba, in the Mg2Si successful synthesis of Mg2C with an anti-fluorite structure 4,5) 6) lattice; (2) Si substituted by isovalent Ge; (3) BaSi2 suggests the possibility of bandgap tuning of Mg2Si by 7,8) lattices substituted by isovalent light elements (Ca, Sr); (4) forming a solid solution of Mg2Si and Mg2C with a suitable 9) BaSi2 lattices substituted with alkali metals; (5) Si in BaSi2 bandgap for thermoelectric energy conversion. Second, C lattices substituted with isovalent carbon;10) (6) variation of incorporation might cause a positive chemical pressure effect the encapsulated alkali metal M and substitution of the Si through volume contraction, which would have the opposite atom with a group 13 element in silicon clathrates of the form effect to our expectations. 11) M8Si38A8 (A = Ga, Al, and In); and (7) substitution of Mg To determine which factors of electronegativity or volume contraction dominate, we performed 1st-principles calcu- +Corresponding author, E-mail: [email protected], imai-y@aist. lations of Mg8Si4(1¹x)C4x (x = 0, 1, 2, 3, and 4) with the anti- go.jp fluorite structure. During the calculations, we found that 1874 Y. Imai, A. Yamamoto and K. Takarabe substitution of Si with C atoms in the Mg2Si lattice decreased Initially, we considered that sufficient results for bandgap the bandgap to lower values than even that of Mg2Si. To widening of Mg2Si by substitution with C can be obtained better understand the discontinuity between the result for by considering the above structures. However, as stated in Mg8Si1C3 and that for Mg8C4, which has been confirmed to the introduction, the range of the calculations was not have a much wider bandgap than that of Mg8Si4, we extended sufficient by itself. We extended the range of the calculation the scope of this work from our original assumption to to include compounds with higher C/Si ratios. Details of include compounds with C/Si ratios higher than 3 for the construction method of the model compounds will be Mg8Si1C3. described later.

2. Calculation Details 2.2 Calculation method The calculation method was similar to that used in our 4) 2.1 Structures calculated previous study of the Ca2SiMg2Si system. Namely, Mg2Si has an anti-fluorite structure, belonging to the space calculations were performed using the CASTEP code 15) group No. 225. In the Mg2Si unit cell, Mg atoms occupy 8c developed by Payne et al.; this is a first-principles method sites [Wyckoff notation] at the eight («1/4, «1/4, «1/4) based on DFT, with a pseudopotential description of the positions, whereas Si atoms are located at the cube corners electroncore interactions and a plane-wave expansion of the and the three face centers, (4a sites). Therefore, the unit cell wavefunctions. The ultrasoft pseudopotential generated by contains four formula units, corresponding to the stoichiom- the Vanderbilt’s scheme16) was used, in which the Mg 2p state etry of Mg8Si4. However, it can be reduced to a primitive was explicitly treated as part of the valence. The Perdew 17) rhombohedral Mg2Si cell if we adopt a set of primitive Wang generalized gradient approximation (GGA) were vectors pointing from a corner site to any of the three kinds of used to approximate the DFT exchangecorrelation term. face-centered points of the cubic lattice, which is composed As for calculation method used here, it should be noted of Si atoms. The Mg2C considered here also has the same that typical local density approximation (LDA) (and GGA) structure as that of Mg2Si. functionals are known to face so-called ‘bandgap problem’. When some of the Si atoms in the Mg8Si4 cell are Several devises have been made to solve this problem such as substituted with C atoms, the symmetry of the system is (A), the so-called LDA+U method18) of including an on-site lowered from No. 225 to No. 221 when one or three of the Si Coulomb correlation to the effective Hamiltonian of the atoms are substituted, respectively, and to No. 123 when two LDA, (B), hybrid exchange-correlation functional usually of the Si atoms are substituted. In the former case, the cells of constructed as a linear combination of the HartreeFock exact Mg8Si3C1 and Mg8Si1C3 cannot be reduced further. In the exchange functional and any number of exchange-correlation 19,20) latter case, the cell of Mg8Si2C2 can be reduced to the cell of explicit density functionals and (C), the TranBlaha 21) Mg4Si1C1 with a tetragonal symmetry. First, we performed modified Becke-Johnson (mBJ) method. Among them, (A) calculations for Mg8Si4 and Mg8C4 with the assumed is effective to improve the bandgap of localized systems such symmetry of space group 225, Mg8Si3C1 and Mg8Si1C3 with as d-semiconductors and insulators but encounters many the assumed symmetry of the space group 221, and Mg4Si1C1 difficulties in describing the properties of systems with more with the assumed symmetry of the space group 123. The delocalized electrons such as metals.22) From this fact, it calculated structures are summarized in Table 1. We seems that its applicability to s- and p-semiconductors such performed several preliminary full-optimization calculations as Mg2Si is not good. Since (B) and (C) are likely to be without assuming the space groups of the compounds; promising in reducing the bandgap problem, the applicability however, there were no meaningful differences in these of those methods to the present work is kept for future results. studies. In the present study, we used somewhat classical

Table 1 Crystallographic data of the assumed structures for calculations of partially Si-substituted Mg8Si4 with C, Mg8Si4(1¹x)C4x. Tunability of Mg2Si Bandgap by Formation of Mg2(Si, C) with an Anti-Fluorite Structure Examined by First-Principles Calculations 1875 approximation method, expecting that qualitative guidance orbitals are present for Mg8Si4 and the other compounds would be given by the present calculations and keeping in calculated here; however, these bands are omitted from the mind to ensure consistency with our previous studies on numbering in this and succeeding figures for simplicity. 4,5,812,2326) Mg2Si and related materials. Mg2Si is an indirect semiconductor where the top of the Self-consistent iteration convergence in the present study valence band is located at the ! point in reciprocal space was assumed when the total energy difference between whereas the bottom of the conduction band is located at the successive cycles was less than 0.001 eV per atom. The X point. The gap width between them was calculated to be maximum error in the calculated energies was therefore 0.277 eV, which is approximately 36% of the observed value approximately «0.01 eV for Mg8Si4 and similar compounds. (0.77 eV), owing to the well-known tendency of DFT The kinetic cutoff energy for the plane-wave expansion calculations to underestimate bandgap values. (Ecutoff) was set to be 380 eV. To calculate the electronic The calculated BS of the optimized Mg8Si4 under energies, the MonkhorstPack (MP) k-point sampling atmospheric pressure is shown in Fig. 1(b). This diagram 27) scheme of reciprocal space was used with a spacing of suggests that Mg8Si4 is a direct semiconductor with a ! ¼ ! 0.5 nm¹1. We performed geometric optimization with the transition. However, this result is attributed to the fact that total energy minimization algorithm under the Broyden both ! (0, 0, 0) and X (1/2, 0, 1/2) of the reciprocal space FletcherGoldfarbShannon optimization procedure. The of Mg2Si correspond to ! (0, 0, 0) of the reciprocal space of band structures (BSs) along several high-symmetry lines in Mg8Si4. Thus, the indirectness of Mg2Si from ! to X becomes the Brillouin zone were calculated for the optimized unclear when the cells of Mg8Si4 are used for band-structure structures. The top of the valence bands was selected to be calculations. Hereafter, we decided to consider only the the zero energy level in drawing the BS diagrams. change of the bandgap values of the system and not the directness of the bandgap. 3. Results and Discussion The same can be said for Mg2C. Figure 2(a) shows the calculated BS of Mg2C with an anti-fluorite structure based 3.1 Band structures of Mg8Si4 and Mg8C4 on a rhombohedral primitive cell, whereas Fig. 2(b) shows As stated above, our initial target is substituted structures that of Mg8C4 based on a cubic unit cell. Here, the calculation of Mg8Si4 with the anti-fluorite structure and the composition was performed for the optimized structure under an applied of Mg8Si4(1¹x)C4x. Therefore, we first describe the results of pressure of 15 GPa because Mg2C is unstable under the terminal compounds of Mg2Si [or,Mg8Si4 (x = 0)] and atmospheric pressure. Mg2C[or,Mg8C4 (x = 1)]. Again, Mg2C is an indirect semiconductor with a The calculated BS of the optimized Mg2Si under calculated gap value of 0.874 eV, although it appeared to be atmospheric pressure (0 GPa) is shown in Fig. 1(a). Plotted a direct semiconductor if we use a cubic unit cell of Mg8C4. numerical values in this and succeeding band structure We could not compare the calculated bandgap with the diagrams are the numbers which express the order, counted observed one because no experimental values have been from the bottom of the valence band. Because the Mg2p reported. However, our calculated value is within the range of states are explicitly treated as a part of the valence band in the previous calculations (0.670.97 eV).28) pseudopotential used, 24 bands composed of Mg2p atomic The BSs of Mg2Si and Mg2C changed with applied pressure. Figure 3 shows the calculated BSs of Mg2Si at 30 GPa, calculated based on the rhombohedral primitive cell of Mg2Si, (a), and the cubic cell of Mg8Si4, (b), respectively. The top of the valence bands in Fig. 3(a), the 4th-band, is located at the !-point and the bottom of the conduction bands, the 5-th band, is located at the X-point; however, their electronic energy values are reversed in these two; the latter is 0.173 eV lower than the former. This is the so-called state of a ‘negative bandgap’ (or an inverted band structure). This situation is represented in Fig. 3(b) for a cubic unit cell. This energy inversion was calculated to occur at approximately 15 GPa. As for Mg2C, the changes of the bandgap under applied pressures were moderate, but slightly increased from 0.852 eV at 0 GPa to 0.880 eV at 30 GPa, as shown in Fig. 4.

3.2 Mg8Si4(1¹x)C4x with anti-ﬂuorite structure where one to three Si atoms in the Mg8Si4 lattice are substituted with C Next, we calculated the band structures of Mg8Si4(1¹x)C4x Fig. 1 Band structure (BS) of Mg2Si at 0 GPa; (a) Calculated BS for the with the anti-ﬂuorite structures where Si atoms are partially rhombohedral primitive cell of Mg Si. (b) Calculated BS for the cubic cell 2 substituted with C atoms. The calculated BSs of the of Mg8Si4. Plotted numerical values in these and succeeding band structure diagrams are the numbers that express the order, counted from optimized structures of Mg8Si3C1,Mg8Si1C3, and Mg4Si1C1 the bottom of the valence band. at ambient pressure (0 GPa) are shown in Fig. 5(a), (b), and 1876 Y. Imai, A. Yamamoto and K. Takarabe

Fig. 2 Band structure (BS) of Mg2C; (a) Calculated BS for the rhombohedral primitive cell of Mg2C. (b) Calculated BS for the cubic cell of Mg8C4.

Fig. 3 Band structure (BS) of Mg2Si at 30 GPa. (a) Calculated BS for the rhombohedral primitive cell of Mg2Si. (b) Calculated BS for the cubic cell of Mg8Si4.

(c), respectively, where the former two belong to the space One possible reason for this result is that Si substitution group 221 whereas the latter belongs to the space group 123. with C causes a volumetric contraction of Mg2Si, which Both Mg8Si3C1 and Mg8Si1C3 are zero-gap semiconduc- decreases the bandgap value. Thus, the volume effect caused tors where the 16-th band (i.e., the highest valence band) and by substitution might exceed the chemical effect caused by the 17-th band (i.e., the lowest conduction band) are in the difference of the electronegativity between C and Si. To contact at the only ! points of their own reciprocal lattice investigate whether predicted bandgap contraction can be [Fig. 5(a) and (b)]. Conversely, the energy range of the explained by the volumetric contraction caused by Si highest valence band, the 8-th band, and the lowest substitution of Mg2Si with C, we decided to investigate the conduction band, the 9-th band, of Mg4Si1C1 overlapped, effects of the volume change on the bandgap values of these which indicates that Mg4Si1C1 has a negative energy gap. compounds. The result that Si-substitution with C atoms causes a bandgap The bandgap dependences of those compounds on their contraction of Mg2Si is contrary to our presumptions prior to unit-cell volumes are shown in Fig. 6, along with those of calculations. Mg2Si and Mg2C, as stated in the previous section. Tunability of Mg2Si Bandgap by Formation of Mg2(Si, C) with an Anti-Fluorite Structure Examined by First-Principles Calculations 1877

Fig. 4 Variation of band structure (BS) of Mg8C4 from 0 to 30 GPa.

Fig. 5 Band structure of Mg8Si3C1 (a); Mg8Si1C3 (b); and Mg4Si1C1 (c); with the anti-ﬂuorite structures where one to three Si atoms in the Mg8Si4 lattice are substituted with C atoms.

As described in the previous section, the bandgap of result suggests that the bandgap value would not generally Mg8Si4 (Mg2Si) decreased with decreasing cell-volume (or have a monotonous dependence on the applied pressure but a as the applied pressure increased from 0 to 30 GPa) in the maximum value at a certain pressure; however, this was not assumed pressure range for calculations. Conversely, the confirmed within the range of calculated pressures in the pressure dependence of the bandgap of Mg8C4 (Mg2C) was present calculations. small, although the bandgap value slightly increased with Next, the gaps have zero or negative values and do not decreasing cell-volume (increasing applied pressure). necessarily increase with changes to the cell volume. This Calculations of Mg8Si4(1¹x)C4x where Si atoms in the result conflicts with our presumption that Si substitution by C Mg8Si4 lattice were partially substituted with C atoms, atoms would increase the gap value. revealed the following. First, the pressure dependence of the Third, the bandgap value of Mg8Si1C3, marked by the gap values become more moderate than that of Mg8Si4. The symbol (©), was persistent with the variation of cell-volume. bandgap of Mg8Si3C1 has a maximum value of 0 eV at 0 GPa, This result suggests that the bandgap contraction was not and decreases at both positive and negative pressures. This only caused by the volumetric effect but also by the Brillouin 1878 Y. Imai, A. Yamamoto and K. Takarabe

vectors. The crystallographic data of the former representa- tion, including the fractional coordinates of each atom before geometrical optimization, are shown in Table 2. In the last column of Table 2, the values of the Wyckoff position, not regulated, are described for convenience. These values are generated by the operation of 2 © 2 © 1, 2 © 2 © 2, or 3 © 2 © 2 supercell formation of Mg2Si but slightly varied during the optimization procedure. The calculated BSs of these compounds at 0 GPa are shown in Fig. 7. In Fig. 7(a), a negative bandgap is found in the BS of Mg8Si1C3, which is different from that of Mg8Si1C3 belonging to the space group No. 221, as previously stated. From an energetic viewpoint, the structure belonging to the Space Group No. 65 was slightly more negative (³0.16 eV per formula unit) than the structure belonging to the Space Group No. 221; however, further calculations for this Fig. 6 Bandgap dependences of Mg8Si4 ( ), Mg8C4 ( ), Mg8Si3C1 ( ), structure have not been conducted because the predicted © Mg4Si1C1 ( ), and Mg8Si1C3 ( ) on their unit-cell volume. Numerical negative bandgap was beyond the scope of the present study. values attached to these symbols indicate the assumed applied pressures in As shown in Figs. 7(b) and (c), a wider bandgap than that the unit of GPa during the geometrical optimization procedures. of Mg8Si4 can be expected for Mg16SiC7 and Mg24Si1C11. Thus, the bandgaps of Mg8Si4(1¹x)C4x were larger than that of zone folding owing to the superlattice formation by the Mg2Si and smaller than that of Mg2C when a high value of ordered substitution of atoms. It has been predicted that the x/(1 ¹ x)(0¯ x ¯ 1) can be attained. In fact, volume (or, band structure of nitrogen (N)-doped graphene changes applied pressure) dependences of the bandgap of those markedly from a gapless state to states with a definite gap compounds, as shown in Fig. 8, indicate that the blank region value, depending on the calculated cell-size of the assumed of the bandgap range in Fig. 6 can be completed with those supercell size (or, the concentration of doped N atoms).29) compounds and bandgap tuning may be possible by varying The observed persistency of the contact of the top of the the composition and applied pressure during the synthesis valence band and the bottom of the conduction band at only procedure. the ! point suggested a limiting current in a one-dimensional Calculated standard formation enthalpies of those com- situation, so-called Luttinger-liquid behavior. pounds are shown in Fig. 9. Here, the reference states of Mg, To conclude the present section, our calculations suggest Si, and C are metallic Mg with a hexagonal closed-packed that the bandgap of Mg2Si cannot be widened by substituting structure, semiconducting Si with a diamond structure, and Si atoms with C atoms without altering the atomic diamond for carbon. The interlayer attraction energy of arrangement. A region of the bandgap = 0.30.8 eV in graphite is hard to evaluate by DFT calculations; hence, Fig. 6 is blank. Therefore, there is a need for compounds graphite, which is an equilibrium phase under ambient to fill this space. To search for possible compounds with conditions, was not selected as the reference state of carbon. wider bandgaps than that of Mg2Si but narrower than that As shown in Fig. 9, Mg8Si4 with an anti-fluorite structure of Mg2C, we decided to relax our constraints and search for is more stable than a mixture of Mg, Si, and C throughout the possible compounds with similar atomic arrangements to calculated pressure range. However, Mg16Si1C7,Mg24Si1C11, Mg2Si. We considered, as an example, a structure in which and Mg8C4 are stable at pressures greater than ca. 15 GPa. the primitive units of Mg2Si were the same and only the The results for Mg8C4 agreed well previous calculations by stacking order was different. Kurakevych et al.14) In the following section, we describe the BSs of those Thus, the bandgaps of the compounds in which the basic compounds where all but one Si atom in the superlattice of structural units of Mg2Si are stacked and all but one of the Mg2Si were substituted with carbon. other Si atoms are substituted with C atoms are wider than that of Mg2Si, if the C/Si ratio is greater than 7, and narrower 3.3 Mg8Si4(1¹x)C4x composed of a superlattice of the than that of Mg2C, as expected. These are stable at a high rhombohedral Mg2Si where all but one Si atom are pressure of approximately 15 GPa or more and bandgap substituted with C atoms tuning is possible through formation of a Mg2SiMg2C solid We constructed 2 © 2 © 1, 2 © 2 © 2, and 3 © 2 © 2 solution. Notably, further investigations are necessary to superlattices of the rhombohedral primitive cell of Mg2Si determine the following possibilities: and all but one of the Si atoms were replaced by carbon (1) SiC is quite stable. To create Mg2(Si, C) while avoiding atoms. The cell compositions thus prepared were Mg8Si1C3, generation of SiC, it is necessary to devise a non- Mg16Si1C7, and Mg24Si1C11, respectively. Those compounds equilibrium process. are found to belong to the space groups 65, 225, and 12, (2) An orthorhombic phase with the anti-cotunnite (Pnma, respectively. The unit cells, which express their symmetry Z = 4) structure and a hexagonal phase with Ni2In-type more directly are Mg16Si2C6,Mg64Si4C28, and Mg48Si2C22, (P63/mmc, Z = 2) structure, both of which have been and can be reduced to Mg8Si1C3,Mg16Si1C7, and known to be stable at higher pressures and to be Mg24Si1C11, respectively, by selecting the proper primitive metallic; however, Ca2Si and Sr2Si with the former Tunability of Mg2Si Bandgap by Formation of Mg2(Si, C) with an Anti-Fluorite Structure Examined by First-Principles Calculations 1879

Table 2 Crystallographic data of the assumed structures for calculations of stacked Mg2Si where all but one of the Si atoms are substituted with C.

Fig. 7 Band structures of Mg8Si1C3,Mg16Si1C7, and Mg24Si1C11 at 0 GPa.

structure are known to be direct semiconductors. These 4. Summary properties might aﬀect the phase stability calculated here. In summary, we have calculated the band structures of (3) There is a possibility that a carbon atom enters into the Mg8Si3C1,Mg8Si1C3, and Mg4Si1C1 with anti-ﬂuorite Mg2Si crystal to form an interstitial compound. structures where one to three Si atoms in the Mg8Si4 lattice are substituted with C atoms and Mg8Si4(1¹x)C4x composed of 1880 Y. Imai, A. Yamamoto and K. Takarabe

Mg2Si if the C/Si ratio is suﬃciently high (more than 7 or so). These are stable phases at a high pressure of approximately 15 GPa or more.

Acknowledgements

The authors would like to express their sincere gratitude to Dr. Naomi Hirayama, the Institute for Solid State Physics, the University of Tokyo, for her informative suggestions on the progress of the DFT calculations.

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