Electrical resistivity of FeSi alloy under high pressure

. The electrical resistivity of FeSi alloy under the influence of pressure is studied based on Matthiessen's rule and the saturation electrical resistivity model. Numerical calculations have been performed for FeSi up to a pressure of 100 GPa and compared with experimental data when possible. These results show that the electrical resistivity of the FeSi alloy diminishes gradually with pressure and saturates at high pressure. Moreover, in this paper, The impurity concentration effects on the electrical resistivity of FeSi alloy have also been investigated. Our numerical calculations, performed at 40 Gpa and 80 Gpa pressures, show that when the Si concentration is less than 20%, the electrical resistivity of the FeSi alloy increase rapidly, and they behave like linear functions of the concentration of Si. Our model gives electrical resistivity values that are in agreement with recent experimental measurements.


Introduction
To meet the growing need for energy efficiency in power electronics and electric machines as well as the recent demand for even more efficient electric motors and high-voltage transformers, a number of new soft magnetic materials are being investigated.Among them, a Fe-Si alloy with a high Si concentration is already being accepted as a commercially viable soft magnetic material, and is expected to be widely adopted by the industries [1,2].
On a larger scale, Earth's outer and inner cores are mainly composed of iron (∼85 wt %) alloyed with niken (∼5 wt %) and ∼8 to 10 wt % of light elements, respectively, such as Si, O, S, C, and H [3][4][5][6][7][8]. Convection of the liquid core generates Earth's magnetic field, which is controlled by the thermal conductivity of the core.Therefore, the light elements are important to the database of the thermal state, the structure of the core, as well as the generation and evolution of Earth's magnetic field [3,4,[9][10][11], where, Si was determined to account for 5 to 10% of the earth's core [4] and is one of the most promising materials for the main light element of the earth's core.Si is highly abundant cosmochemically, not highly volatile, and forms a continuous solid solution with iron over a wide compositional range.Therefore, the thermal conductivity and electrical conductivity properties of Fe-rich Si alloys have attracted intense interest in recent years [4,12].K) [20][21].For the Fe-8 wt% Si alloy, the conductivity results are ∼20 W.m -1 .K -1 at ∼132 GPa and 3000 K.However, the thermal conductivity of solid Fe-6,5 wt% Si at ∼99 GPa and 2000 K is ∼66 W.m -1 .K -1 [20].Thus, the studies on the thermal conductivity (electrical resistivity) of FeSi alloys at high pressure, especially at the Earth core pressure condition, are still controversial.There is no consensus among these past/previous results.
In this work, we study the effect of impurity concentration and pressure on the electrical resistivity of FeSi alloy.For such a goal, the influence of Si impurity concentration on the resistivity of the alloy is calculated based on Matthiessen's rule.To calculate the resistivity of alloys with a large doping concentration at high pressure, we use the suggestions of Côte and Meisel in which the electrical resistivity of the alloy system is not only depends on the impurity resistivity and temperature effects described by the Bloch-Grüneisen formula, but also on the effect of resistivity saturation in the estimates of core resistivity [15].This causes an increase in resistivity.
Consequently, alloy(s) has/have greater resistivity than metal(s).
Firstly, the dependence of the resistivity of the FeSi alloy on the concentration of Si will be studied based on Matthiessen's rule [14], which was introduced in 1864 [ The resistivity caused by phonon scatterring in a metal is calculated by the expression of Bloch-Grüneisen's law ur 0 , 0 , 11 with the note that we neglected residual resistivity caused by imperfections One of the remarkable points is that the experimental results of Gomi and his colleagues on FeSi and FeNi alloys [23][24] show that Matthiessen's rule is only suitable for dilute alloys.Therefore, in this research, to find out the dependence of the resistivities of alloys on pressure, we will apply the theory that allows us to calculate the resistivities of alloys based on the model of saturation resistivity that was considered by Côte and Meisel [13], and modeled by Gomi et al. [23][24] ( ) , where ( )


is the total resistivity of alloys, ( )  is the saturation resistivity, and is the Debye-Waller factor that can be determined from Debye or experiments.
For the resistivity of impurities, we use the value of the resistivity  of that is fitted to the respective experimental values [23] ( ) ( )   can be calculated as [15,25] ( ) ( ) ( ) To determine the influence of pressure on resistivity, we will apply the Vinet equation that shows the relationship between pressurevolume temperature [26] ( ) where ' 00 , BB is, respectively, the isothermal bulk compression module and the first order of its derivative.

Results and discussions
In the following, the expressions established in the previous section will be applied to numerically calculate FeSi alloys up to 100 GPa.For this purpose, the Debye temperature ( )  is assumed to be constant in the studied pressure range.The Debye-Waller coefficient is obtained experimentally in Gomi's work [23]: by the DAC method [23] (solid square) and shock compression measurement by Matassov [28] (solid diamond shape).Calculation result from the original principle [23] (dotted lines and solid circles) However, when the Si concentration is greater than 20%, previous experimental results show that the electrical resistivity of the alloy will no longer increase linearly but will decrease gradually.In this case, our calculation results do not explain the results of previous work.This is also consistent with the prediction of Gomi et al. [23,24] that we made section 2.
We now represent the pressure dependence of the electrical resistivity of FeSi alloys between 0 GPa and 100 GPa.As can be seen in Fig. 2, at pressures higher than 20 GPa, the electrical resistivity of FeSi alloy in the results of our theoretical calculations gradually decreases to saturation value in the high pressure region; his is in very good agreement with the ones of Ohta et al. [29].This phenomenon is explained by Gunnarsson et al. [30]: the electrical resistivity depends on the interaction distance between atoms, so this value will decrease with pressure.
The electrical resistivity tends to reach saturation value sat  when the ean free path of an electron is equivalent to the interatomic interaction distance at high temperature and pressure.

2
Materials and MethodsConduction in metals is hindered by the scattering of charge carriers due to the quantum vibrations of the lattice (phonons) and by their collisions with each other.Metals are generally considered good conductors; they allow current to flow through easily because, in their structure, there are many freely moving electrons.Metals will be alloys after adding some other metals and/or non-metals to get better properties.Nonmetals are poor conductors.When we add nonmetals to metals, the change in crystal structure will decrease their ability to conduct electricity.
14].Matthiessen et al.    discovered that most of the change in resistivity by temperature in a two-component alloy is equivalent to the resistivities of two conductive components that are completely independent and parallel to each other.When the concentration of a component decreases, the role of that component will become less important.From this result, they concluded that the addition of impurities concentrations will only add a constant amount to the total resistivity of pure metal, while the ingredient, depending on the temperature of resistivity, will not be affected.caused by phonon-phonon scatterring in pure iron.Here, Matthiessen's rule will be applied and improved by the idea that ( )

Fig. 1 .
Fig. 1.Effect of Si-doped concentration on the resistivity of FeSi alloy.Experimental data of Gomi et al.

Fig. 2 . 4 Conclusions
Fig. 2. Pressure dependence of the electrical resistivity of FeSi alloy.Experimental data of Gomi et al. by the DAC method [23]