Thermoelectric Properties of Mg2Si Thin Films by Computational

Article pubs.acs.org/JPCC

Thermoelectric Properties of Mg2Si Thin Films by Computational Approaches Hilal Balout,† Pascal Boulet,*,† and Marie-Christine Record‡ †

Aix-Marseille University and CNRS, MADIREL UMR 7246, Avenue Normandie-Niemen, 13397 Marseille Cedex 20, France Aix-Marseille University and CNRS, IM2NP UMR 7334, Avenue Normandie-Niemen, 13397 Marseille Cedex 20, France

‡

ABSTRACT: The semiclassical Boltzmann approach coupled with density functional theory calculations has been used to investigate the structural, electronic, and thermoelectric properties of Mg2Si thin films with 001, 111, and 110 orientations. The most stable slab is found to be that with the 110 orientation. The electronic band gap vanishes for the 001 and 111 thin films whereas for the 110 orientation the film is semiconducting with a band gap ranging from 0.27 to 0.36 eV depending on the number of atomic planes used to model the thin film. The energy gap decreases when the number of planes increases. As a consequence of the electronic band structure, the 110 semiconducting thin film exhibits the highest thermoelectric performance, especially the Seebeck coefficient (−350 μV K−1 at 600 K). The lower the number of atomic planes the larger the Seebeck coefficient. By comparing experimental data for the electrical conductivity σ to our calculated value of σ/τ at 600 K we have determined the electron relaxation time τ to be about 5 × 10−16 s. Using this value, and assuming a thermal conductivity of 2−3 W m−1 K−1 for the 110 Mg2Si thin film, we estimate that the figure of merit ZT at 600 K lies in the range of 0.4−0.6.

I. INTRODUCTION Thermoelectric (TE) materials are able to convert thermal energy directly into electricity through the Seebeck effect and conversely electricity to heat flow through the Peltier effect, making them attractive for energy harvesting and active cooling applications. The figure of merit ZT = (S2σ/κ)T, where S is the Seebeck coefficient, σ the electrical conductivity, κ the thermal conductivity, and T the absolute temperature, constitutes the main characteristic of TE materials for estimating their performance. In spite of continuous progress in materials design, the ZT value hardly reaches 1 for the best TE materials currently in use, so that the corresponding devices bear a thermoelectric efficiency that is less than 10% that of the Carnot efficiency. This TE efficacy can be improved by using the nanoscale engineering technique. The use of TE thin films should be an attractive route for a 3-fold reason. First, manufacturing thin films allows us to lower the consumption of the feedstock. Second, the quantum confinement of the electrons in the nanofilm should increase the Seebeck coefficient through the discretization of the electronic energy levels. Third, the nanosize of the thin film decreases the grain size and increases the number of interfaces, namely, the grain boundaries that should in turn reduce the thermal conductivity of the materials by decreasing the phonon mean-free path. In addition to these benefits, the rising interest in TE thin films is based on the ability of TE modules to be miniaturized and to be integrated into devices so as to reach higher power density.1 Silicides were proposed as potential thermoelectrics by Nikitin in 1958.2 The highest ZT value achieved to date is 0.7 for ptype higher manganese silicide (HMS−MnSi1.7) and 1.2 for n© 2014 American Chemical Society

type Mg2(Si,Sn) solid solution. Mg2Si is a compound of particular interest as it is constituted of lightweight, abundant, cheap, and environmentally harmless chemical elements. Moreover, Mg2Si should be viewed as utterly appealing materials for use in silicon integrated circuits and energy harvesting microsystems, since silicide technology exhibits a unique compatibility with today’s silicon-based microelectronics. Although it has been recognized that the fabrication of Mg2Si thin films remains a challenging task, due to the low condensation coefficient of Mg to Si and the high vapor pressure, volatility, and extreme reactivity and oxidability of magnesium, several authors have investigated the elaboration of these films. Mahan et al. have obtained polycrystalline Mg2Si thin films with (111) texture deposited on both the (111) and (100) silicon substrates by molecular beam epitaxy.3,4 Epitaxial Mg2Si(110) thin film on Si(111) substrate has been obtained by Wang et al. through thermally promoted solid-phase reaction between Mg film and Si substrate.5 Serikawa et al.6 have deposited Mg−Si thin films, containing Mg2Si intermetallic compound, on glass substrate by ion beam sputtering. Pulsed laser deposition has been used by Song et al. to prepare amorphous Mg 2Si, which became nanocrystalline after annealing.7 Galkin et al. have grown multilayered Mg2Si nanofilms using a preliminary deposited layer of Mg2Si nanoislands as precursor.8,9 Polycrystalline Mg2Si thin films on Si(100), Si(111), and silicon on insulator (SOI) substrates Received: June 25, 2014 Revised: July 21, 2014 Published: July 28, 2014 19635

dx.doi.org/10.1021/jp506323r | J. Phys. Chem. C 2014, 118, 19635−19645

The Journal of Physical Chemistry C

Article

not interact with each other. In order to determine the size of the vacuum, the calculations of the potential energy curves have been performed with respect to the length of the simulation box perpendicular to the slab surface, which corresponds to the z axis for all of the films concerned by this study. The corresponding plots (not shown here) show that the interaction between the slab images vanishes when the vacuum size exceeds about four times the interplanar distance (d001 ≈ 1.6 Å, d110 ≈ 2.25 Å, and d111 ≈ 1.84 Å). A vacuum size of around 10 Å should therefore be sufficient. This size of the vacuum thickness is similar to that found by Liao et al. in their study of the 100 thin film.12 However, in our case, to ascertain the best possible quality of our results, the vacuum size has been set to 12 Å for all of the 001, 110, and 111 slabs. For completeness of information, the thicknesses of the various thin films are 1.11, 2.39, and 3.02 nm for the 001 film with N = 8, 16, and 20, respectively, 1.39, 2.48, and 2.85 nm for the 111 film with N=4, 7, and 8, respectively, and 1.58, 2.93, and 3.38 nm for the 110 film with N = 8, 14, and 16.

have been obtained by Kogut and Record by high-temperature treatment in vacuum using either conventional furnace10 or industrial rapid thermal processing.11 The crystalline orientations for the Mg2Si thin films were found to be (111), (200), and (220). Beside this abundant literature on the fabrication of Mg2Si thin films, neither experimental nor theoretical thermoelectric properties have been reported on these structures in literature. Only one theoretical study has been published on Mg2Si thin films. It concerns the calculation of its structural and electronic characteristics for the (100) crystalline orientation.12 The aim of the present work is to predict the thermoelectric properties of Mg2Si thin films, by quantum and semiclassical approaches, with various crystalline orientations. Since it has experimentally been found that the orientations encountered in the thin films are the (001), (111), and (110) ones, we have performed calculations on three types of films that model these particular orientations. For each film, several numbers of atomic planes have been considered.

II. CALCULATION DETAILS A. General Statements. The general approach followed in this study to ultimately obtain the thermoelectric coefficients consists of two steps. In a first step density functional theory (DFT)13,14 calculations using the Perdew−Burke−Ernzerhof (PBE) exchange-correlation functional15 have been performed to determine the equilibrium state of the model systems (bulk and surfaces) through the BFGS optimization procedure.16−19 The thresholds have been set to 10−9 Ry for the energy convergence and 10−5 Ry·bohr−1 for the energy gradient one. The k-point mesh used for optimizing the Mg2Si bulk structure has been set to 20 × 20 × 20 while that for the 001, 110, and 111 slab structures has been set to 8 × 8 × 1. Subsequently, a very thin k-point mesh (30 × 30 × 5) has been used to obtain a precise description of the band structure of the slab models for use in the transport calculations (see below). Ultrasoft pseudopotentials20 have been used throughout, and the kinetic energy cutoffs have been set to 30 and 320 Ry for the wave functions and charge density and potential, respectively. The Quantum ESPRESSO program package21 has been used to perform the DFT calculations. In the second step the Boltzmann transport equation has been solved using the constant relaxation time approximation to determine the thermoelectric coefficients, namely, the Seebeck coefficient and the electrical conductivity, from the calculated electronic band structures as described above. The BoltzTraP program22 has been used. B. Bulk Mg2Si Structure. Mg2Si crystallizes in the face centered cubic Bravais lattice with an antifluorite structure (space group Fm3̅m, number 225). The experimental cell parameter as determined from XRD experiment amounts to 6.35 Å.23 C. Surface Construction and Optimization. The construction of the 001, 110, and 111 slabs has been performed by cleaving the bulk Mg2Si structure perpendicularly with respect to these faces. The slabs consist of a number of atomic planes N (with N = 8, 16, and 20 for slab 001 and N = 8, 14, and 16 for slab 110). The number of atomic planes is different from one film to the next since it has been chosen to keep the Mg2Si stoichiometry in the considered film. In the case of the 111 slab for which atomic planes cannot be easily distinguished, N refers to the number of Mg2Si formula units present in the film, and N = 4, 7, and 8 for this slab. On-top vacuum has been chosen in such a way that the periodic images of the slabs do

III. RESULTS AND DISCUSSION A. Optimized Structure of Bulk Mg2Si. The bulk structure has been optimized using the PBE functional, and the calculated cell parameter amounts to 6.369 Å. To further ascertain the quality of the theoretical procedure, the elastic constants C11, C12, and C44 and the bulk modulus B0 of Mg2Si have been determined. The corresponding data (expressed in GPa) are the following: C11 = 115.1, C12 = 21.7, C44 = 46.1, and B0 = 52.8. The quality of our approach can be assessed by comparing these data with the experimental ones (GPa):24−26 C11 = 126, C12 = 26.0, C44 = 48.5, and B0 = 59. The optimized bulk structure has served to build the slab models. B. Structural Features and Surface Energy of the Slabs. The atom positions and x and y simulation box parameters, which are parallel to the slab surface, have been optimized for the slabs containing the various number of atomic planes (see section IIC). Some structural features of the slabs are given in Tables 1−3. In these tables we distinguish two types of bond distances, namely, the interplanar and intraplanar ones. Intraplanar distances correspond to those distances measured between two atoms belonging from the same atomic plane, whereas interplanar distances correspond to those distances measured between two atoms belonging from two adjacent atomic planes. These bond distances can be measured between atoms located either in innermost (bulk-like) or in outermost (nearby the surface) sections of the slab. The slabs are presented in Figure 1. Irrespective of the slab, the most affected bond distances are those between atoms located in the outermost planes. For the 001 slab (Table 1) the Mg−Si distances are most impacted by the relaxation. The intraplanar (interplanar) Mg− Si distances of the outermost planes decrease by 0.1 Å (slightly increase by 0.03 Å) when increasing the number of planes in the slab and are hence both departing from the bulk Mg−Si distances (2.76 Å). The Mg−Si distances for atoms located in the innermost planes are less affected (less than 1%) and tend toward the corresponding bulk value. The Mg−Mg and Si−Si distances are almost not affected by the relaxation. The degree of relaxation of the slab can be appreciated not only from the relaxation of the bond distances but also from that of the Si− Si−Si angles for silicon atoms located in the outermost planes of the slab. While the angles made by atoms belonging to the innermost planes do not vary much with N, those made from 19636



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The 110 slab presents the noticeable feature that the interplanar bond distances measured between atoms located in the outermost planes are very close to their corresponding bulk values (Table 3). The largest deviation (0.07 Å) is observed for the Mg−Mg distance. In addition, the bond distances are not much affected by the change in the number of planes for N > 8. All of the bond distances measured in the innermost region are close to the corresponding bulk value. The same observation applies to the Si−Si−Si angles, both for the innermost and the outermost planes. Overall, the structure of the slab is weakly perturbed when the bulk is cut through the 110 plane. The surface energy of the slabs has been calculated using the expression:

Table 1. Interatomic Distances (Å) between Atoms Located in the Inner and Outer Planes of the 001 Slaba

δ=

Eslab − nE Mg Si 2

2A slab

(1)

where Eslab is the total energy of the slab, EMg2Si is the total energy of the Mg2Si primitive cell which contains one Mg2Si formula unit, n is the number of Mg2Si chemical formula in the slab, and Aslab is the surface area of the slab. For each of the 001, 111, and 110 slabs, the values of δ have been calculated with respect to the number of planes in the slab. The evolution of δ is depicted in Figure 2 that shows that, first, the surface energy is already converged for 8, 4, and 8 planes for the 001, 111, and 110 slabs, respectively, and, second, the 110 slab is the most stable one with surface energy of about 49.3 meV/Å2. The 001 and 111 slabs are less stable by 50.7 and 66.7 meV/Å2, respectively, than the 110 one. The greatest stability is observed for the 110 slab even if a small reorganization of the surface atoms occurs as described previously in this section. Furthermore, in spite of the extensive reorganization of the surface atoms in the 111 slab the largest instability is observed. C. Electronic Properties. Density of States. The contribution of the inner planes of the slabs to the total density of states has been evaluated and compared to the total density of states (DOS) of bulk Mg2Si. A strong dependence on the total number of planes is observed (Figure 3). For the 001 slab (Figure 3a) the DOS resembles that of the bulk Mg2Si for N ≥ 16 only. For these slabs (N = 16, 20), the inner plane DOS does not vanish near the Fermi level and only

a

The atoms can be located either in the same plane (intraplanar distances) or in two adjacent planes (interplanar distances).

the atoms located in the outermost planes increase on average by 2.5°. Only the angles made by the silicon atoms in the innermost planes are very close the bulk value (60°). The structure of the 111 slab surface seems to be substantially modified with respect to the bulk structure when geometry relaxation occurs (Table 2). All of the bond distances (Si−Si, Mg−Mg, and Mg−Si) measured between atoms located in the outermost planes are significantly modified when N increases: Mg−Si and Si−Si decreases by 0.05 and 0.06 Å, respectively, and Mg−Mg increases by 0.08 Å. As N increases, all of these bond distances depart from their corresponding bulk value. By contrast, the bond distances measured in the innermost planes do not vary much with N and converge toward their bulk value when N increases. With regard to the Si−Si−Si angles, they remain close to 60° as N increases.


a

The atoms can be located either in the same plane (intraplanar distances) or in two adjacent planes (interplanar distances). 19637



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a

The atoms can be located either in the same plane (intraplanar distances) or in two adjacent planes (interplanar distances).

The DOS of the inner planes of the 110 slab (Figure 3c) resembles that of the bulk Mg2Si for N ≥ 14. Below this threshold, strong peaks appear both in the valence and conduction bands that can be attributed to the hybridization of the Si-3p orbitals with the Mg-3p ones. Whatever the number of planes, the slab shows semiconducting properties with a gap seems to increase as the number of planes decreases (see the total DOS in Figure 4c). Finally, although we have not particularly focused our attention to the search of surface states, it seems that these states, which should be located within the energy gap, do not exist in the present slabs. Electronic Band Structures. The electronic band structure of the bulk Mg2Si has been described at length in the literature.27−36 Mg2Si is a Γ−X indirect band gap semiconducting material with a calculated energy gap of 0.22 eV, very similar to that obtained by Liao et al.12 The experimental value is reported to be in the range of 0.66−0.78 eV.37−39 The evolution of the bands with respect to the number of atomic planes N for the 001 slab (Figure 5a) shows that for 16 and 20 atomic planes the band structures are essentially the same. The 001 slab is metallic, as indeed found by Liao et al.,12 with the Fermi energy located deep in the bands below the gap. As N increases the gap is gradually closing up at the Γ point. For N = 8, the slab is still metallic but the bands nearby the gap are more flattened, which could explain the appearance of large peaks in the DOS above the Fermi level (Figure 4a). Contrary to the bulk Mg2Si for which the gap is along the Γ−X line, the indirect gap of the 001 slab is along the Γ−L line. The band structure of the 111 slab (Figure 5b) shows that the slab is metallic and that the bands are much more interlaced than in the case of the 001 slab with no visible gap above the Fermi level. Comparing the band structure between the slabs with N = 4, 7, and 8, we observe that the electronic bands are similar to each other for N = 7 and N = 8 and are different from those for N = 4. This is in agreement with the observations made on the DOS (Figure 4). The gap of the 110 slab is a direct one located at the Γ point (Figure 5c). The 14- and 16-atomic planes slabs show identical electronic band structures. The 8-atomic plane slab has a

Figure 1. Structure of the 110 slab (left) with N = 14 atomic planes, 001 slab (middle) with N = 20 atomic planes, and 111 (right) slabs with N = 8 Mg2Si formula units. Color legend: brown for Si atoms; green for Mg atoms.

the slope of the inner plane DOS for states between −0.5 and 0 eV is similar to that of the bulk structure which shows a smooth decay. For the 8-planes slab the DOS shows a stair-like decay for the valence states near the Fermi level and a stair-like increase for the conduction states. In addition, the closure of the band gap is complete in this case. The total DOS of the 001 slabs with N = 8, 16, and 20 show that this slab has a metallic character (Figure 4a). We note that for N = 16 and 20 atomic planes a small hike appears at the Fermi level that can be explained from the features of the electronic band structure (see below). For the 111 slab (Figure 3b), the features of the inner plane DOS are similar to those of the 001 slab and as such do not deserve more comment. The total DOS of the 111 slab shows a metallic behavior of the slab (Figure 4b), and it is hence very different from the DOS of the bulk materials. 19638



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Figure 3. Contribution of the inner layers to the total density of states. The total density of states of the bulk Mg2Si is also depicted for comparison purpose: (a) 001 slab with N = 8, 16, and 20 atomic planes; (b) 111 slab with N = 4, 7, and 8 formula units; (c) 110 slab with N = 8, 14, and 16 atomic planes.

Figure 2. Evolution of the surface energy (meV/Å2) of the slabs, as calculated from eq 1, with respect to the number of layers for slabs 001 (a) and 110 (c) and with respect to the number of Mg2Si formula units for slab 111 (b).

Thermoelectric Properties. The calculation of the thermoelectric properties has been achieved using the constant relaxation time approximation. In this approximation we admit that the relaxation time τ is independent of the energy level of the electron and k-point. This is certainly a weak approximation to use for thin film. However, we have not found in the literature data that report the evolution of τ in Mg2Si thin films. Therefore, in the following only the σ/τ and PF/τ values are presented and discussed. Nonetheless, Ogawa et al.40 have very recently reported electrical conductivity data for asdeposited and annealed Mg2Si thin films with the 110 orientation. We have used these data to estimate the relaxation time at intermediate temperature of 600 K and hence the power factor and the figure of merit ZT at this temperature.

markedly different band structure, especially on the L−Γ line where some degeneracy removals can be seen. The electronic band gap tends to increase slightly as the number of atomic planes in the thin film decreases (0.274, 0.3, and 0.362 eV for N = 16, 14, and 8, respectively). In contrast to the bulk case, the Fermi level is located nearby the conduction band, depicting an n-type conduction of the thin film. In conclusion, both the 001 and the 111 Mg2Si thin films are metallic and only the 110 film is semiconducting with band gap varying with the number of atomic planes. Considering our purpose, which is the study of electronic transport properties of Mg2Si films, we can here and now assume that the 110 film is the most interesting one for thermoelectric concern. 19639



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Figure 4. Total density of states of the slabs and bulk Mg2Si: (a) 001 slab with 8, 16, and 20 atomic planes and bulk Mg2Si; (b) 111 slab with 4, 7, and 8 Mg2Si formula units; (c) 110 slab with 8, 14, and 16 atomic planes.

εF the larger the Seebeck coefficient. This feature occurs for semiconductors with flat bands located near the Fermi level. The Seebeck coefficient has been calculated for all of the Mg2Si thin films and plotted with respect to the energy in Figure 7. As expected from expression 2, the Seebeck coefficient of semiconducting bulk Mg2Si near the band edges is by far larger than that of the 001 (Figure 7a) and 111 (Figure 7b) thin films which are metallic. At low chemical potentials and temperature (300 K) the oscillations observed for S result from the peaks in the DOS visible for the low-energy states (Figure 4a,b). As the temperature increases (600 and 900 K), the oscillations are smoothed out. By contrast, the Seebeck coefficient for the semiconducting 110 thin film (Figure 7c) shows improvements with respect to the bulk Mg2Si. Interestingly, the film with smallest thickness (8-atomic planes) bears the highest Seebeck coefficient. The explanation for these observations can be as follows. The band structure of the 8atomic-planes film (left panel in Figure 5c) reveals a very flat band on the L−Γ line above the Fermi level at about 0.25 eV. This flat band appears neither in the bulk Mg2Si band structure (left panel in Figure 5a) nor in the band structure of the 14atomic-planes and 16-atomic-planes Mg2Si thin films (middle and right panels in Figure 5c, respectively). The power factor S2σ (PF) is depicted with respect to the energy in Figure 8. The 001 and 111 metallic films (Figure 8a,b) show generally much lower PF than the bulk Mg2Si around E = εF where moderate doping of the materials is amenable. An exception lies for the 001 thin film at 600 and 900 K (top and middle panels in Figure 8a) for which a slight increase of PF occurs for energies where the PF of the bulk Mg2Si drops to zero. Still, the PF of the thin film remains very small in this region. The 110 thin film shows interesting

The procedure we have followed and the corresponding results are described at the end on this section. The electrical conductivity of all of the thin films evolves in a very similar way when plotted with respect to the Fermi energy (Figure 6). For each film, and for comparison purpose, the electrical conductivity of the bulk Mg2Si is also displayed. This type of plot allows us to evaluate the accessible range of doping for tuning the property. A reasonable doping range (n, p < 5 × 1020 cm−3) is obtained for ΔE ≈ 0.13, 0.26, and 0.39 eV at 300, 600, and 900 K, respectively (ΔE = 5kT).41 These statements assume the validity of the rigid band approximation. The most noticeable feature is in the range of 0−0.5 eV where the electrical conductivity of the bulk Mg2Si tends to zero and corresponding to its energy band gap; in this region, the electrical conductivity of the films is larger than that of the bulk. For the 110 orientation (Figure 6c), the moderately n-doped films (1 × 1019

Thermoelectric Properties of Mg2Si Thin Films by Computational

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