**3.1. 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 P63/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 Fe0.95-xSi0.05Nix 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 Fe0.95Si0.05 binary alloy is lower than the one of the other systems ternary alloys. This volume of systems Fe0.95-xSi0.05Nix is increasing with the increase of Ni concentration.

**3.2. 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 Fe0.95-xSi0.05Nix (x = 0, 15, 25, and 40%). The metallicity is confirmed 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 fixed 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. 3). The all-dispersion profiles are very similar. Concerning the energies of the bands, there are small changes with increasing doping concentration. At low or null concentration of doping Ni in Fe with the Si concentration fixed at 5%, the energy bands show sharp bands and similar to that of a perfectly ordered crystal. On the contrary, they become more smeared at high dopant concentration as we have in the DOS graph for the Fe-Si based alloys. In addition, the electronic energy bands are widening with the energy uncertainty that defines by the of energy over time as where:

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 justified 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 fit 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 defined from the Bloch Spectral Function (BSF) at the Fermi energy (Figure 4), because the width of the enlargement of the Fermi surface is varied inversely proportional at the mean free path. The inverse of the mesh parameter also is proportional at the edge of the first Brillouin zone (Butler and Stocks, 1984; Gomi et al., 2016; Zidane et al., 2020).

**3.3. 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 Fe0.95-xSi0.05Nix 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).

Generally speaking, the electrical resistivity of all the ternary Fe0.80Si0.05Ni0.15, Fe0.70Si0.05Ni0.25, and Fe0.55Si0.05Ni0.40, alloys are higher than the one of the Fe0.95Si0.05 binary alloy as shown in Figure 5. The increasing order of impurity resistivity is found as Ni (15%) < Ni (25%) < Ni (40%). For this light element Ni doped in Fe0.95Si0.05, we figured out a tiny decrease in electrical resistivity from 320 to 360 GPa for binary and ternary structures, which it can be the same for all pressure points in each concentration. At 0% concentration of Ni, i.e., the binary alloy Fe0.95Si0.05 has a resistivity in the range of 53–57 μΩ·cm. At 15%, 25%, and 40% concentration of Ni, i.e., the ternary alloys Fe0.80Si0.05Ni0.15, Fe0.70Si0.05Ni0.25, and Fe0.55Si0.05Ni0.40, and their resistivity are 88–92 μΩ·cm, 99–103 μΩ·cm, and 111–114 μΩ·cm, respectively.

The vibration modes harden along with increase in the pressure. In other term, when the phonons harder, this leads to a coupling decrease between the electron and phonons. This coupling is directly proportional to the electrical resistivity. In explanation, the electrical resistivity drops, because of the absence of structural, electronic, and/or topological transition, which was due to the increase in pressure. From phonons perspective, Lanzillo et al. (2014) discussed this outcome in term of the effect of pressure on resistivity. As a benchmark with theoretical results that were calculated at 0 K, we confirm that our findings are in a great agreement with others’ works such as 62-66 μΩ·cm of Fe0.85Ni0.10Si0.05, 100-102 μΩ·cm of Fe0.75Ni0.10Si0.15 and 115-118 μΩ·cm of Fe0.65Ni0.10Si0.25 including pressure (Zidane et al. 2020). Our values of electrical resistivity at 360 GPa are bigger than the one of 42 μΩ·cm of Fe0.85Ni0.10Si0.05, 78 μΩ·cm of Fe0.75Ni0.10Si0.15 and 95 μΩ·cm of Fe0.65Ni0.10Si0.25 at 120 GPa (Gomi et al. 2016). All the previous calculations of volume, density electronic, band structure, Bloch spectral function, and electrical resistivity were received at 0 K of temperature by using the Functional Density Theory (DFT).

In the context of electrical resistivity as a function of temperature, the electrical resistivity of hcp-Fe0.95-xSi0.05Nix (x = 0%, 15%, 25%, and 40%) alloys were calculated at fixed pressure 360 GPa and within range of temperature 4500-6000 K, which are the inner core conditions P-T of ICB. In figure 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 Fe0.95Si0.05 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 Fe0.95-xSi0.05Nix (x = 0%, 15%, 25%, and 40%), which is bigger than the measurement of hcp-Fe-5Ni-4Si and hcp-Fe-5Ni-8Si at 2000-4500 K and 140 GPa of CMB in a diamond-anvil cell (DAC) (Zhang et al. 2021). Regarding the hcp-Fe-4Si at 99 GPa and up to 1900 K are also by the diamond-anvil cell (Inoue et al., 2020). As a general agreement, the pressure and temperature of the CMB are less than that of ICB. Thus, this outcome makes a perfect sense because the electrical resistivity increases with temperature.

According to the literature, there are many studies found that the estimations of electrical resistivity by the method diamond-anvil cell, such as 102 μΩ·cm for CMB and 82 μΩ·cm for ICB of Fe77.5Si22.5 (Gomi et al., 2013), 104 μΩ·cm for CMB and 84.4 μΩ·cm for ICB of Fe67.5Ni10Si22.5 (Gomi et al., 2015), under the conditions of the core-mantle boundary (CMB; 135 GPa, 3750 K), and inner core boundary (ICB; 330 GPa, 4971 K) respectively. Gomi et al., 2016 concluded that the collapse of the large portion of the Fermi surface by using Matthiessen’s rule could violate at higher Si contents from 1 to ∼ 9 wt.% in the Fe-Si structure. Under the selected for the present study and the ones of the Earth’s inner core, which are 330–360 GPa of pressure and 5000–7000 K of temperature, the values of resistivity are between 0.7 and 1.5 ×10–6 Ωm (Pozzo and Alfè. 2016). Other authors used binary and ternary system based on iron with some element C, O, S, H, N, Ni at first principal calculations such as (de Koker et al., 2012; Xu et al., 2018; Pozzo et al., 2012); to that we would like to add the experiments of basic principles of Shock Compression in the condition of inner earth's core conducted by (Matassov, 1977).

**3.4. 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 fixed 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).**

Figure.7 indicates, on one hand, the thermal conductivity that increases with an increase of temperature for all our last structures in the range of 45-55 W·m−1·K−1. On the other hand, this latter increases with the decrease of concentration of Ni doped in Fe0.95Si0.05 system. Our result values of thermal conductivity at the Earth inner's core conditions are in good agreement with other studies such as Stacey and Anderson, 2001 who obtained 46 W·m−1·K−1 for CMB and 63 W·m−1·K−1 for ICB. Stacey and Loper, 2007 obtained 28.3 W·m−1·K−1 with slight variation in increase of depth for Fe-Ni-Si. In addition, Zhang et al., 2021 acquired the thermal conductivity in the range 2000-4500 k and 140 GPa under the conditions of Earth’s outer core at 40-75 W·m−1·K−1. In contrast, other investigators received higher values of thermal and electrical conductivity than ours (de Koker et al. 2012; Ohta et al. 2016; Pozzo et al. 2012, 2014).