Properties Evaluation of Fe95-xSi5Nix Under the Earth's Inner-Core Conditions: Insight from the Ab-Initio Investigation

In this article we investigate under the same Earth's core conditions, the structural, electronic, and transport properties of Fe-Si-Ni ternary alloys based on Fe and 5% of Si with various concentrations 0%, 15%, 25%, and 40% of element Ni, by means of First-principles calculations. Based on Functional Density Theory (DFT). The Local Density Approximation (LDA) also has been adopted for the potential exchange correlation. We perform the calculation of electronic property at 360 GPa using the software Akai-KKR (machikaneyama), which used the Korringa-Kohn-Rostoker method along with coherent potential approximation (KKR-CPA). Afterward, we calculate the electrical resistivity of impurities formed on the Kubo-Greenwood formula with the vertex correction using SPR-KKR code, which is based on the relativistic polarized spin method. Then, we model the thermal conductivity by electrical resistivity for both varying in the range of 320–360 GPa and 4500-6000k of pressure and temperature, respectively; according to the conditions of the Earth’s inner core ICB using Wiedemann-Franz law. Hence, our results suggest that 85–115 µ Ω ·cm at 0 K and 320–360 GPa, then 225–285 µ Ω ·cm at 4500–6000 K and 360 GPa for electrical resistivity, and then 45–55 W·m − 1 ·K − 1 at 4500–6000 K and 360 GPa of thermal conductivity of Earth’s inner core. Lastly, the thermal and compositional convection is one of the major factors of global magnetic eld that is generated by geodynamo driven.


Introduction
Geophysics is a branch of the Earth Sciences that not only examines the internal core of our planet, but also all the seismic, electrical, thermal, and magnetic phenomena that occur on its surface and its surrounding. As matter of fact, these processes are measured by multiple experimental approaches by means of physically measurable parameters such as density, electrical resistivity…etc. For instance, electrical prospection is one of these methods based upon determining electrical resistivity and thermal conductivity under speci c conditions of high temperature and pressure, in order to understand the lithological formation as well as the mineralization of the Earth's core.
Multiple studies indicate that the internal structure of the Earth is made up of different chemical elements with different percentages. It was found that the most dominant element in Earth's core is iron with a slight change in percentages of lighter elements such as Ni, Si, S, N, O, C, H, and Mg (McDonougha and Suna. 1995). The Iron is abounded in nature in three structures, which are Face-Centered Cubic (FCC), Hexagonal-Close-Packed (HCP) (Mathon et al., 2004), and Body Centered Cubic (BCC) (Belonoshko and Ahuja, 2003;Luo et al., 2010). However, the HCP structure is the most stable one in the conditions of high temperature and pressure of iron in the Earth's core. Indeed, in order to better investigate the structure of the Earth's inner core, it is necessary to study the saturation of electrical resistivity and thermal conductivity of the structure of HCP of iron or iron doped by different alloys of elements existed in the heart of the Earth under high temperature and pressure. In the literature, several studies suggested that the saturation of the electrical resistivity of the outer and the inner cores was approximately equal to 1×10 -6 Ω.m. This saturation of electrical and thermal conductivity was studied by many researchers (Gomi et  where k is thermal conductivity in W/ (K.m), L = 2.45×10 −8 W.Ω/K 2 is the Lorenz number, T is temperature in K°, and ρ is electrical resistivity in Ωm.
In this present research article, we performed first-principles calculations based on Functional Density Theory (DFT) for ternary Fe 0.95-x Si 0.05 Ni x alloys by X=15%, 25%, 40% and binary Fe 0.95 Si 0.05 alloy under conditions of Earth's inner core. Our theoretical calculation are electronic properties and the electrical resistivity that were performed at high pressure. We also calculated the thermal conductivity by using the Wiedemann-Franz equation. Afterward, we discussed the implication of thermal conductivity of iron doped by silicon and nickel for the explanation of heat transport and its possible loss on Earth.

Computational Details
In these rst-principles calculations, we relaxed the binary structure Fe 0.95 Si 0.05 and ternary structure Fe 0.95-x Si 0.05 Ni x with 15%, 25%, 40%. We calculated the electronic properties by using the AkaiKKR (MACHIKANEYAMA) package coded by Akai ( 1989). This calculation adopts Coherent Potential Approximation (CPA) (Butler, 1985), which employs the Local Density Approximation (LDA) based on Korringa-Kohn-Rostoker (KKR) and the Green function method with the Atomic Sphere Approximation (ASA) that was combined to exchange-correlation potential (Moruzzi et al., 1978). In addition, we employed the scalar relativistic approximation and the angular momentum quantum number l to calculate the wave functions up to 3, i.e., s, p, d, and f orbitals. Afterwards, we computed the electrical resistivity from the Kubo-Greenwood formula (Greenwood, 1958;Kubo, 1957)  In the current study, we focused on calculating the average value of the electrical resistivity by using the equation are the resistivities calculated perpendicular and parallel to the c-axis. Through the thermal conductivity of the Earth's core, the moving electrons mainly transported the heat in the metals. Thus, we can estimate this thermal conductivity of the Earth's core from the electrical resistivity by using the Wiedemann-Franz law indicated in the equation (1).

Structural properties
We know that Fe is the most dominant element in the inner core of the Earth based on the HCP structure, which is the most stable structure of 20 GPa. Therefore, in our work all the calculations are carried out with this last structure based on a space group of P6 3 /mmc (N°194) (Zidane et al., 2020). The unit cell of HCP structure of iron is containing two atoms that occupy the Wyckoff position 2(c) in the following positions Fe: (1/3, 2/3, 1/4); (2/3, 1/3, 3/4) (Takahashi et al., 1968).
We have calculated the volume at 320, 330, 340, 350, and 360 GPa for four structures doped by 0%, 15%, 25%, and 40% of Ni in the Fe 0.95-x Si 0.05 Ni x alloys of hcp structure (Figure 1). For all structures, we observe the decreasing of volumes related to the pressure. The decrement in the volume is almost linear in the interval (320-360) GPa. We also notice that the volume of Fe 0.95 Si 0.05 binary alloy is lower than the one of the other systems ternary alloys. This volume of systems Fe 0.95-x Si 0.05 Ni x is increasing with the increase of Ni concentration.

Electronic properties
We have performed the calculation for nonmagnetic HCP Fe-Si, and Fe-Si-Ni alloys of the Density of States (DOS), the electronic band structures, and Fermi Surface.
In Figure 2 we calculated the electronic Density of States (DOS) at about 360 GPa of pressure for nonmagnetic Fe 0.95-x Si 0.05 Ni x (x = 0, 15, 25, and 40%). The metallicity is con rmed for binary structure and all ternary structures. Nevertheless, we noticed a decrease in DOS at the Fermi level as the concentration of impurities increases at the expense of iron atoms with conserving the Si concentration xed at 5% and other concentration 0%, 15%, 25%, 40% of Ni.
The decrease in density at the Fermi level with the increasing of the dopant content of Ni in Fe-Si has been explained by the electronegativity of silicon and nickel impurities, which is higher than that of iron. Therefore, the multiplicity may continue to decrease, and the DOS will lower when the iron atoms are replaced by other atoms of higher electronegativity. Additionally, the surface under the DOS widens with the increase in light elemental impurities, which has been explained by the number of valence electrons that occupies this surface.
The electronic band structure proves the metallicity of all alloys for 360 GPa (Figure.  is the uncertainty of energy.
∆t: is the lifetime of the electron.
: is the reduced Planck constant.
Since the electrical resistivity is proportionally inversed to electronic density, we have explained the DOS by the reliant of disorder thermic. Thus, the band structure is justi ed by the escalation of the crossing bands at the Fermi level.
The cross-sections of the Fermi energy of the Bloch spectral function were presented for the same binary and ternary structures at 360 GPa based on Fe-Si ( Figure 4). These cross-sections have to be low at high percentage of impurities from Ni in Fe-Si. Moreover, it seems an acceptable display for small or null percentages of Ni. The saturation of the electrical resistivity of alloys and transition metals occurs at very high resistivity (Mooij, 1973;Bohnenkamp et al., 2002). This latter was produced when the interatomic distance t the mean of free path of conduction electrons. We call this condition the Mott-Ioffe-Regel criterion (Mott, 1972;Gurvitch, 1981), which also can be de ned from the Bloch Spectral Function (BSF) at the Fermi energy (Figure 4)

Electrical resistivity
There are several experiments with dilute alloys that measured their electrical resistivity conducted by Norbury in 1920. This author noticed a horizontal increase in term of distance between host metal in the periodic table and the positions of the impurity element. We know in the Earth's core that the values of pressure and temperature proportionally increase with depth. As a matter of fact, in the inner core of the Earth, the values of pressure vary from 320 to 360 GPa and the temperature from 4500 to 6000 k. By using the Kubo-Greenwood method, we performed the calculations of electrical resistivity based on alloy Fe 0.95-x Si 0.05 Ni x doped with some percentage 0%, 15%, 25%, and 40% of Ni alloys at high-pressure values, which are 320, 330, 340, 350, and 360 GPa of HCP structure ( Figure 5). Similarly, analysis for the HCP morphology can be found in (Zidane et al., 2020). In the context of electrical resistivity as a function of temperature, the electrical resistivity of hcp-Fe 0.95x Si 0.05 Ni x (x = 0%, 15%, 25%, and 40%) alloys were calculated at xed pressure 360 GPa and within range of temperature 4500-6000 K, which are the inner core conditions P-T of ICB. In gure 6, we observed for the all-last alloys an increase between 225 μΩ·cm and 285 μΩ·cm of the electrical resistivity as a function of temperature. Consequently, the electrical resistivities get closer to each other, and then converge to a single point at very high temperature; because these structures are depressed under very high conditions of pressure-temperature. Furthermore, we also notice an increase in the electrical resistivity with increasing of Ni percentage doped in Fe 0.95 Si 0.05 for all the points of temperatures as 4500 k, 4750 k, 5000 k, 5250 k, 5500 k, 5750 k, and 6000 k.
In the conditions of the Earth's inner core, we modeled the electrical resistivity for HCP structure for Fe 0.95-  (Matassov, 1977).

Thermal conductivity
In this part of our reach, we converted the electrical resistivity (ρ) to thermal conductivity (k) by using the Wiedemann-Franz law (Equation 1) at pressure and temperature condition of Earth's core. Hence, we carried out all our calculation the thermal conductivity at xed pressure 360 GPa and in the range 4500-6000 k of temperature for 0%, 15%, 25% and 40% concentration of Ni in alloys based of Fe+5at%Si ( Figure.7).

Conclusion
In this scienti c article, we used the ab initio method of functional density theory to calculating the structural, electronic, and transport proprieties of hcp-Fe0.95-xSi0.05Nix with 0%, 15%, 25%, and 40% of Ni alloys in the ranges 320-360 GPa and 4500-6000 k as the conditions of pressure and temperature as the Earth's inner core. We calculated the electrical resistivity of all structures by using the Kubo-greenwood formula (Butler 1985;Oshita et al. 2009). Afterward, we determined the thermal conductivity by applying Weidman Franz law in the ICB conditions. Consequently, our ndings of the electrical resistivity and the thermal conductivity calculations of Fe-Si-Ni alloys are consistent with the ones found by our predecessors like (Stacey and Anderson, 2001;Stacey and Loper, 2007;and Zhang et al., 2021). At the end, we would like to join the consensus of the geodynamo that has been driven by thermal convection in the Erath's inner core that the values of thermal conductivity should be low. As well as said by Korte [1], "these physical properties are not exactly known because we cannot just go down to the Earth's core and directly measure them." Nevertheless, she mentioned, "they have to be inferred." Throughout this work, we a rm this inference and corroborate previous threshold of the thermal conductivity of the Earth's inner core, which remains at lower values. [1] Monika Korte (0000-0003-2970-9075) -ORCID | Connecting Research and Researchers