Elsevier

Calphad

Volume 74, September 2021, 102318
Calphad

Thermodynamic activity of solute in multicomponent alloy from first-principles: Excess Mg in Mg3(Sb1-xBix)2 as an example

https://doi.org/10.1016/j.calphad.2021.102318Get rights and content

Highlights

  • A first-principles thermodynamic model for estimating activity of solute in multicomponent alloys was developed.

  • Activity of Mg in Mg3(Sb1-xBix)2 alloy were estimated.

  • The solid solubility of Mg in Mg3(Sb1-xBix)2 alloy were evaluated and discussed.

  • The equilibrium phase diagram for Mg–Mg3Sb2–Mg3Bi2 system was constructed at different temperatures.

Abstract

Assessment of the thermodynamic activity and solid solubility for solute element in multi-component alloy is critical in thermodynamics, but an effective first-principles estimation of the parameter has been an unsolved issue for many years. By taking the excess Mg in Mg3(Sb1-xBix)2 alloy as an example, Mg atoms could occupy the regular lattice sites and the interstitial sites as well, which makes the estimation of thermodynamic activity of Mg solute not easy. In this work, we develop a model to estimate the activity and solid solubility of Mg in alloys such as Mg3(Sb1-xBix)2 by considering the formation energy contribution from both Mg interstitials and Mg vacancies, and the associated entropies to final Gibbs energy. Accurate assessment of the parameter by using first-principles thermodynamics approach is also realized. The estimated activity of Mg in Mg3Bi2 and Mg3Sb2 reproduce well the available experimental results. The solubility limits of Mg and equilibrium concentrations of Mg interstitials and Mg vacancies in Mg3(Sb1-xBix)2 are also estimated. The results could be used to understand Mg–Mg3Sb2–Mg3Bi2 phase diagram as well as the Mg effect on thermoelectric performance of Mg3(Sb1-xBix)2 alloy.

Introduction

In thermoelectrics, doping in host semiconductors is an important strategy for the optimization of thermoelectric properties due to lowering lattice thermal conductivity and tuning carrier concentration. The doping limit, i.e. the solid solubility of impurity or solute element, is largely determined by the thermodynamic states such as chemical potential and temperature as well as the composition of materials [1]. Recently, the thermodynamic phase diagram technique of solid solubility determination was widely applied to the design and optimization of thermoelectric materials. For example, the high filling limit of dopant in CoSb3-based filled skutterudite [2,3] was designed by understanding the dopant-Co-Sb phase diagram. The high thermoelectric performance in PbTe-based compounds [4,5] was further improved by optimizing the solubility of dopant based on the phase diagram. In half-Heusler compounds such as ZrNiSn [6] the phase diagram strategy was used to obtain high thermoelectric performance by precisely controlling Ni solubility. In addition, the phase diagram approach was also used for several other thermoelectric materials such as Zintl phase [7,8], diamond-like semiconductors Cu2HgGeTe4 [9], and AgSbTe2 [10], etc.

In thermodynamics, it is well-known that the phase diagram of alloy can be calculated with the aid of activity, and the solubility of solute element can be deduced based on phase equilibria information. The activity ai is the effective concentration of solute i in a mixed solution and is defined as ai=γix, where γi and x are the activity coefficient and concentration of the solute i, respectively. The activity coefficient can be measured by resorting to an experiment. However, the measurement of γi is not only time-consuming but also not easy in some situations such as dilute solutions and multi-component alloy. Therefore, the computational simulation approaches such as CALPHAD (CALculation of PHAse Diagrams) and first-principles calculations to estimate activity or activity coefficient had been developed in past decades [[11], [12], [13], [14]].

However, there are few reports on the activity of solute element in multi-component alloy from first-principles calculations because of the complexity of multiple defect types and uncertain defect concentrations in the alloy. By taking the excess Mg in Mg3(Sb1-xBix)2 alloy as an example, Mg could occupy either the regular lattice sites or the interstitial sites to form Mg interstitials and Mg vacancies, which make the estimation of thermodynamic activity of solute very hard. Ab initio calculations of crystal structure and solute atom occupancy of multiple sites could be performed by cluster expansion method [15,16], which also can be used to predict the thermodynamic properties of alloy. The n-type Mg3(Sb1-xBix)2-based compounds recently have attracted much attention due to high thermoelectric performance near room temperature [[17], [18], [19], [20]]. It is known that almost all previous studies with Mg3Sb2-based hold on p-type materials [[21], [22], [23], [24], [25]] until the n-type Mg3·2Sb1·5Bi0·49Te0.01 was reported by Tamaki et al. [26]. It was believed that the extra Mg is necessary to compensate Mg vacancies and realize n-type thermoelectric behavior. Interestingly, Zhang et al. [27] reported that Mg3Sb1·48Bi0·48Te0.04 without excessive Mg prepared by arc melting method, had similar good n-type thermoelectric performance. Shuai et al. [28] and Imasato et al. [29] highlighted that the extra but a little bit excess Mg in Mg3+δSb1.5Bi0·49Te0.01 was enough to achieve good n-type performance. Therefore, it is particularly significant to know the exact solubility of Mg in Mg3(Sb1-xBix)2 for understanding the effect of excess Mg and designing thermoelectric materials with good performance. However, the exact solubility of Mg is difficult to measure experimentally due to the high volatility of Mg. Meanwhile, there is no computational reports on the Mg solubility in Mg3(Sb1-xBix)2 so far.

In this work, a computational model associated with defects to estimate the activity of Mg in Mg3(Sb1-xBix)2 is developed by using first-principles thermodynamics approach. With the aid of Mg activity, the phase diagram for Mg–Mg3Sb2–Mg3Bi2 is evaluated and the solubility limit of Mg and the concentration of Mg defects are deduced at equilibrium. Finally, the effect of excess Mg on thermoelectric properties of Mg3(Sb1-xBix)2 is discussed.

Section snippets

First-principles thermodynamic model of solute activity

Firstly, we describe the thermodynamic model for estimating the activity of solute Mg in Mg3(Sb1-xBix)2 by first-principles. To illustrate expediently the thermodynamic model, we take binary Mg3Sb2 system as an example. Mg3Sb2 compound has an inverse α-La2O3 type structure, space group is P3m1. The unit cell contains 3 Mg-sites and 2 Sb-sites, as shown in Fig. 1. Only the interstitial sites at (0 0 0.5) are considered because the configuration has the lowest energy for Mg interstitial [30].

Computational methods

Based on the above thermodynamic model, we tried to estimate the activity of Mg in Mg3(Sb1-xBix)2 alloy from first-principles calculations. The first-principles calculations were performed using the projector augmented wave (PAW) method [34] as implemented in the Vienna Ab-initio Simulation Package (VASP) [35] which based on the density functional theory (DFT) and the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA) [36] as the exchange correlation functional. For the

Formation energy of defects

Since the Gibbs free energy of the system is described as a function of defect formation energies, it is pertinent to calculate the formation energies of defects before discussing the activity of Mg in Mg3(Sb1-xBix)2. In our work, Monte Carlo (MC) simulations based on cluster expansion [15,40] and first-principles calculations were first carried out to understand the configurational arrangement of Sb and Bi atoms at Sb sublattice sites for Mg3(Sb1-xBix)2. Our results from MC simulations for a

Conclusions

We presented ab initio estimation of solute activity in multi-component alloy by combining thermodynamic analysis associated with defects. A first-principles thermodynamic model for estimating activity was presented and illustrated in Mg–Mg3(Sb1-xBix)2 system. The calculated activities of Mg in binary Mg3Bi2 and Mg3Sb2 were consistent with the available experimental results. The phase diagrams of Mg–Mg3Sb2–Mg3Bi2 system at equilibrium were constructed at different temperatures, which clearly

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

ZWQ and LWS thank National Key R&D Program of China (No. 2019YFA0704901) for support. This work is also supported by Guangdong Provincial Key Laboratory of Computational Science and Material Design (No.2019B030301001), Shenzhen Municipal Key-Lab program (ZDSYS20190902092905285), Guangdong Innovation Research Team Project (No.2017ZT07C062), Hunan Provincial Education Department research project (No. 17C1640) and the Centers for Mechanical Engineering Research and Education at MIT and SUSTech.

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