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The Electronic and Thermoelectric Properties of Si1-xVx Alloys from First Principles

Amall Ahmed Ramanathan*, and Jamil Mahmoud Khalifeh

Department of Physics, University of Jordan, Amman 11942, Jordan
Correspondence to: Ramanathan AA, Tel: +962-6-5355000, E-mail: amallahmad@gmail.com
Received June 7, 2017; Revised August 20, 2017; Accepted August 21, 2017.
This is an open-access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract

The effect of temperature and vanadium metal concentration on the electronic and thermoelectric properties of Si in the diamond cubic structure has been investigated using a combination of density functional theory simulations and the semi classical Boltzmann’s theory. The BotzTrap code within the constant relaxation time approximation has been used to obtain the Seebeck coefficient and other transport properties of interest for alloys of the structure Si1xVx, where x is 0, 0.125, 0.25, 0.375, and 0.5. The thermoelectric properties have been extracted for a temperature range of 300 K to 1,000 K. The general trend with V atom substitution for Si causes the Seeback coefficient to increase and the thermal conductivity to decrease for the various alloys. The optimum values are for Si5V3 and Si4V4 alloys for charge carrier concentrations of 1021 cm−3 in the mid temperature range of 500~800 K. This is a very desirable effect for a promising thermoelectric and the figure of merit ZT approaches 0.2 at 600 K for the p-type Si5V3 alloy.

Keywords : Thermopower, Si, Thermal conductivity, Vanadium, Local density approximation
INTRODUCTION

The last decade has seen a renewed interest in thermoelectric (TE) materials. The stimulus to the interest in this class of materials is the recent progress in improving the efficiency of energy conversion by a variety of techniques. The direct conversion of heat energy to electricity using TE is a very appealing option, especially as currently there is a lot of heat energy wasted during the consumption of energy resources (Bell, 2008; Zhang & Zhao, 2015). Although, TEs offer many advantages like lack of dependence on heat source, lack of maintenance, etc. the major drawback of the low efficiency for energy conversion has kept this technology on hold.

A quantity ZT called figure of merit, measures the efficiency of conversion for TE materials and is given by the equation

$ZT=S2σT/κ.$

Where S is the Seebeck coefficient, σ is the electrical conductivity, T is the temperature in Kelvin and κ is the total thermal conductivity consisting of the electronic part κe and the lattice part κl. In order to improve the performance of TE materials, the power factor S2σ and κ need to be optimized to the maximum and minimum values respectively. This is easier said than done as the Wiedemann-Franz law requires the electronic part of thermal conductivity (κe) to be proportional to electrical conductivity (σ),

$κe=LσT.$

L is the Lorenz number. Also, there are limits to increasing S and σ simultaneously. Several different strategies have been adopted to enhance ZT to values greater than 1, namely, size effect (Heremans et al., 2013; Hicks & Dresselhaus, 1993), doping and carrier concentrations (Jaiyaraman et al., 2016; Pei et al., 2011; Yuli et al., 2016), Nano-structuring (Biswas et al., 2012; Hsu et al., 2004) etc.

Silicides of transition metals have been investigated as high potential TEs (Arita et al., 2001; Gu et al., 2002; Ivanenko et al., 2003) especially for the mid-temperature range of 300 K to 700 K. The ZT values attained for Rh-doped Ru2Si3 (Arita et al., 2001) is larger than the optimized n-type SiGe by 50%. This encourages us to investigate compounds of Si with the transition metal vanadium. Vanadium is as common as Cu and Zn and is the 22nd most abundant metal in the earth’s crust. The present study is an investigation of the TE properties of Si1xVx alloys formed by substitution of V atoms for Si in the conventional unit cell of Si in the diamond cubic structure of 8 atoms. The Si:V stoichiometric ratios in the alloys are 1:0, 7:1, 3:1, 5:3, and 1:1 with x=0, 0.125, 0.25, 0.375, and 0.5 respectively.

MATERIALS AND METHODS

The electronic properties of Si1xVx alloys in the diamond cubic structure is calculated with the ABINIT software package (Gonze et al., 2005) using density functional theory (DFT) under the local density approximation (LDA). The Trouller-Martins pseudopotential with core corrections (Perdew & Wang, 1992) is used for Si and for V we have used the HGH pseudopotentials (Hartwigsen et al., 1998). As in our previous works (Ramanathan, 2013; Ramanathan & Khalifeh, 2017) a convergence criteria of less than 1×10−6 Ha for total energy differences in the self-consistent field is used for the optimization of cutoff energy, kpoint grid, lattice parameter and all structural calculations. The optimized Si lattice constant obtained for the structure is 10.18 Bohr which is close to experimental value of 10.23 Bohr and is used for the supercell to better compare with available experimental TE properties of pure Si as a test of the accuracy of our method. A sufficiently dense k-point grid of 21×21×11 with a cutoff energy of 14 Ha and 100 bands is used with the BoltzTraP code (Madsen & Singh, 2006) to extract the TE properties using a 1×1 supercell. The code solves the Boltzmann transport equation under the constant relaxation time (τ) approximation (RTA). Under this approximation the electrical/thermal conductivity distribution is given by the equations below, where, f is the electronic distribution at equilibrium, μ is the chemical potential, e is the charge of electron, T is the temperature, ɛ the electron energy and Ω is the unit cell volume of the crystal.

$σαβ=1Ω∫-∞+∞σαβ(ɛ)(-∂f∂ɛ) dɛ$$καβe=1e2TΩ∫-∞+∞σαβ(ɛ)(ɛ-μ)2 (-∂f∂ɛ) dɛ$

Taking into account the symmetry of the system BoltzTrap performs Fourier expansion of the band energies to obtain the transport coefficients. Under the constant RTA S and ZT are independent of τ and can be obtained as a function of the chemical potential or temperature. The electrical/thermal conductivities and also the power factor are dependent on τ and can be obtained only in terms of τ. So in order to obtain their actual values one has to depend on the experimental values of τ for the same or similar system. All the results except the Seebeck and ZT have been given in terms of τ, since no experimental values are available for the silicon-vanadium alloys under consideration.

RESULTS AND DISCUSSION

DFT calculations followed by BoltzTrap as implemented in the Abinit code is performed for the supercell of varying V atom concentrations in the diamond cubic conventional unit cell. The V atom size is comparable to that of Si and this permits the formation of substitution alloy. The V atoms in the alloys occupy inner positions on the body diagonals of the unit cell. The number of electrons at the Fermi level and the Fermi energy increases with each substitution of a Si atom with a V atom in the Si-V alloys. This is clearly seen in the total density of states (TDOS) plot, namely Fig. 1. The peak at the Fermi energy is maximum for Si4V4 and minimum for Si. The charge density color plot along the 101 plane is shown for the pure Si and the alloy Si5V3 in Fig. 2. The central point of the plane is 0 0 0 and the right hand scale shows the colors related to the regions of maximum and minimum charge density. The subsequent change in the charge density and the total DOS with each replacement of a Si with a V atom alters the transport properties and gives a more desirable trend for TE applications.

The two-dimensional total charge density plot along the 101 plane for (A) Si and (B) Si5V3 of the Si-V alloys is shown in Fig. 2.

The TE properties extracted using the BoltzTrap code for a charge carrier concentration of 1021 cm−3 is displayed in the subsequent graphs. The Seebeck characteristics at a temperature of 300 K for the different alloys as a function of the chemical potential are displayed in Fig. 3. We see that the maximum values are for Si5V3 in blue color. The bulk Si Seebeck coefficient obtained is 392 μV/K which compares well with the experimental value of 435 μV/K (Lasance, 2006). The effect of temperature on the Seebeck coefficient and the thermal conductivities for the different alloys is shown in Fig. 4A and B respectively. The optimum values of the transport properties at each temperature is extracted and displayed as a function of temperature. The thermal conductivities are given in terms of τ. In Fig. 4A the modulus of the Seebeck coefficient is plotted since we have both positive and negative values for the different alloys. We see an increase with temperature of the Seebeck coefficient for all the alloys. The maximum values are for Si5V3 followed by Si4V4 with appealing TE properties in the mid temperature range. This is very encouraging as the mid temperature is where most of the industrial processes operate. The size and sign of the of the Seebeck coefficient are related to the electron distribution asymmetry around the Fermi level. The sign of S determines if the material is n-type or p-type TE at a given temperature and whether the charge carriers are electrons or electron holes. Our results show that negative Seebeck coefficient predominates for Si, Si7V and Si6V2 whereas Si5V3 has positive values throughout the temperature range. On the other hand Si4V4 has positive and negative values, and for temperatures greater than 800 K negative S predominates. The thermal conductivity is considerably lowered by alloying with V and we find that the lowest value is for Si5V3 as shown in Fig. 2B. Si4V4, Si6V2 and Si7V show slightly higher values of the thermal conductivities but, pure Si on the other hand, has a very sharp increase and the maximum values is for Si. The calculated value of thermal conductivity for pure Si alloy is 164 W/mK at 300 K using a typical relaxation time τ of 10−14 s and this compares well with the available technical data (http://periodictable.com/Elements/014/data.html) of 150 W/mK. With low thermal conductivities and high Seebeck coefficients for Si5V3 and Si4V4 the recipe for optimizing the figure of merit ZT will be complete provided the electrical conductivities for these two alloys are also comparatively high.

The electrical conductivities σ in terms of τ and the figure of merit ZT are plotted as a function of temperature for the different alloys in Fig. 5A and B respectively. From the Figure it is clear that the electrical conductivities are not affected by the temperature and are almost constant. The maximum values are for Si followed by Si7V. Although, Si5V3 and Si4V4 have lower values of σ the best combination of S, σ and κ are for Si5V3 and Si4V4. The highest value of ZT is for Si5V3 followed by Si4V4. This is due to the combination of high Seebeck and low thermal conductivity for these alloys. This is seen in Fig. 5B with the optimum values in the mid temperature range reaching 0.2 for Si5V3 and around 0.1 for Si4V4. The figure of merit is almost constant with temperature for pure Si, Si7V and Si6V2 and is comparatively lower than the other alloys.

CONCLUSIONS

The TE properties of the Si1xVx alloys in the diamond cubic structure have been investigated using DFT-LDA pseudopotentials and the BoltzTrap code. The study has provided useful insight regards the best stoichiometric ratio of Si:V that enhances the TE performance of these alloys. We find very promising TE properties for Si5V3 and Si4V4 which show appreciable ZT values. This leads to the conclusion that a 37.5% vanadium Si5V3 alloy is the best choice for p-type TE operating in the mid temperature range 500 K~800 K. As a first study, the findings will help establish a data base for future investigations and proof of possibility of high TE efficiency with this class of materials. Further experimental and theoretical investigations to verify the result and establish the best Si:V ratio lying between Si5V3 and Si4V4 are required.

ACKNOWLEDGMENTS

This work has been carried out in the Theoretical Physics Laboratory, Physics Department, at The University of Jordan.

Figures
 Fig. 1. The total density of states (TDOS) for the different alloys.
 Fig. 2. Two dimensional charge density plot along the 101 plane of the Unit cell for (A) Si and (B) Si5V3 alloy.
 Fig. 3. The Seebeck coefficient as a function of the chemical potential.
 Fig. 4. Seebeck S (A) and thermal conductivity κ (B) as a function of temperature.
 Fig. 5. Electrical conductivity (A) and figure of merit ZT (B) as a function of temperature.
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