Electronic correlations in Fe at Earth's inner core conditions: Effects of alloying with Ni

We have studied the body-centered cubic (bcc), face-centered cubic (fcc), and hexagonal close-packed (hcp) phases of Fe alloyed with 25 at.% of Ni at Earth's core conditions using an ab initio local density approximation + dynamical mean-field theory approach. The alloys have been modeled by ordered crystal structures based on the bcc, fcc, and hcp unit cells with the minimum possible cell size allowing for the proper composition. Our calculations demonstrate that the strength of electronic correlations on the Fe $3d$ shell is highly sensitive to the phase and local environment. In the bcc phase, the $3d$ electrons at the Fe site with Fe only nearest neighbors remain rather strongly correlated, even at extreme pressure-temperature conditions, with the local and uniform magnetic susceptibility exhibiting a Curie-Weiss-like temperature evolution and the quasiparticle lifetime Γ featuring a non-Fermi-liquid temperature dependence. In contrast, for the corresponding Fe site in the hcp phase, we predict a weakly correlated Fermi-liquid state with a temperature-independent local susceptibility and a quadratic temperature dependence of Γ. The iron sites with nickel atoms in the local environment exhibit behavior in the range between those two extreme cases, with the strength of correlations gradually increasing along the hcp-fcc-bcc sequence. Further, the intersite magnetic interactions in the bcc and hcp phases are also strongly affected by the presence of Ni nearest neighbors. The sensitivity to the local environment is related to modifications of the Fe partial density of states due to mixing with Ni $3d$ states.

Figure 1

The dependence of the inverse uniform magnetic susceptibility on the values of $U$, varying from 3 to 3.8 eV for different temperatures in the bcc phase of ${\mathrm{Fe}}_3\mathrm{Ni}$. The solid red lines correspond to the ${\mathrm{Fe}}_2$ type of Fe atoms; the dashed black lines correspond to the ${\mathrm{Fe}}_1$ type. Diamonds, circles, and squares are used for the temperatures 1400, 3866, and 5800 K, respectively. As one may see, nearly horizontal lines mean that the increasing or decreasing of $U$ by 10% does not quantitatively affect the results independent of the local environment around the Fe atoms (for both ${\mathrm{Fe}}_1$ and ${\mathrm{Fe}}_2$ types of atoms; the details on the atomic arrangement are described in Sec. 2).

Figure 2

The dependence of the quasiparticle lifetime on the value of $U$ for the temperature 5800 K in the ${\mathrm{Fe}}_3\mathrm{Ni}$ bcc phase. The solid red lines correspond to the ${\mathrm{Fe}}_2$ type of Fe atoms; the dashed black lines correspond to the ${\mathrm{Fe}}_1$ type. Circles and squares are used for $e_g$ and $t_{2g}$, respectively. As one may see, increasing or decreasing the value of $U$ by 10% does not quantitatively change the results independent of the local environment around the Fe atoms.

Figure 3

The dependence of the imaginary part of self-energy on the frequency for different values of $U$ for the bcc phase of ${\mathrm{Fe}}_3\mathrm{Ni}$: (a) $e_g$ and (b) $t_{2g}$ states. Temperature is equal to 5800 K. Black dashed, red solid, and blue dashed-dotted lines correspond to the $U$ values of 3, 3.4, and 3.8 eV, respectively. As one may see, all three curves are close to each other at low frequencies, meaning that the corresponding values for the mass enhancement and inverse quasiparticle lifetime exhibit a rather weak dependence on variations of $U$ within this range.

Figure 4

(a) The inverse uniform magnetic susceptibility in the paramagnetic state as a function of temperature for the hcp phase, with the dashed and solid lines corresponding to the ${\mathrm{Fe}}_1$ (six Ni and six Fe nearest neighbors) and the ${\mathrm{Fe}}_2$ (all nearest neighbors are Fe) types, respectively. The error bars show the stochastic CT-QMC error. (b) The same data for the bcc and fcc phases of the ${\mathrm{Fe}}_3\mathrm{Ni}$ alloy. Dashed red line corresponds to the ${\mathrm{Fe}}_1$ type of iron atoms, whose nearest neighbors are four Fe and four Ni atoms in the bcc structure. The solid red line shows the ${\mathrm{Fe}}_2$ type of atoms, which are surrounded exclusively by Fe atoms. The dashed-dotted black line shows the Fe atoms of the fcc phase. The error bars show the stochastic CT-QMC error.

Figure 5

The inverse uniform magnetic susceptibility of the paramagnetic state as a function of temperature for the Ni atom in the bcc, fcc, and hcp phases of the ${\mathrm{Fe}}_3\mathrm{Ni}$ alloy. The solid red line corresponds to the Ni atoms in the bcc structure of the ${\mathrm{Fe}}_3\mathrm{Ni}$ alloy; the dashed-dotted black line shows the Ni atoms in the fcc structure; and the blue dashed line corresponds to the Ni atoms in the hcp structure. The error bars show the stochastic CT-QMC error.

Figure 6

The inverse total uniform magnetic susceptibility of the paramagnetic state as a function of temperature for the Ni atom in the bcc, fcc, and hcp phases of the ${\mathrm{Fe}}_3\mathrm{Ni}$ alloy. The solid red, dashed-dotted black, and dashed blue lines correspond to the total magnetic susceptibility of the bcc, fcc, and hcp structures of the ${\mathrm{Fe}}_3\mathrm{Ni}$ alloy, respectively. The error bars show the stochastic CT-QMC error.

Figure 7

(a) The inverse local susceptibility of Fe atoms in the bcc, fcc, and hcp ${\mathrm{Fe}}_3\mathrm{Ni}$ phases vs temperature. The notation for iron sites (${\mathrm{Fe}}_1, {\mathrm{Fe}}_2$) is the same as in Fig. 1. (b) The average intersite nearest-neighbor exchange interaction as function of temperature. The positive/negative values correspond to ferromagnetic/antiferromagnetic interactions, respectively.

Figure 8

The ratio of the inverse quasiparticle lifetime Γ to temperature $T$ vs $T$ for the bcc, fcc, and hcp phases of the ${\mathrm{Fe}}_3\mathrm{Ni}$ alloy. The dashed black lines correspond to $e_g$ (circles) and $t_{2g}$ (squares) of the ${\mathrm{Fe}}_1$ type. The solid red lines correspond to $e_g$ and $t_{2g}$ of the ${\mathrm{Fe}}_2$ type, respectively.

Figure 9

The LDA + DMFT partial DOS of iron for the bcc, fcc, and hcp phases of the ${\mathrm{Fe}}_3\mathrm{Ni}$ alloy. The dashed black and solid red lines are the partial DOS of the ${\mathrm{Fe}}_1$ type and ${\mathrm{Fe}}_2$ types, respectively.

Figure 10

The LDA partial DOS of iron for the bcc, fcc, and hcp phases of the ${\mathrm{Fe}}_3\mathrm{Ni}$ alloy. The dashed black and solid red lines are the partial DOS of the ${\mathrm{Fe}}_1$ type and ${\mathrm{Fe}}_2$ types, respectively.

Figure 11

The dependence of the inverse uniform magnetic susceptibility on temperature for different values of $U$ in the bcc phase of ${\mathrm{Fe}}_3\mathrm{Ni}$. The solid and dashed lines correspond to the ${\mathrm{Fe}}_2$ and ${\mathrm{Fe}}_1$ types of Fe atoms, respectively. Red circles, green squares, and blue diamonds are used for the $U$ values 3, 3.4, and 3.8 eV, respectively. Black triangles are used for the $U$ value on Fe atoms equal to 3.4 eV and $U$ on Ni atoms equal to 2.75 eV. As one may see, the dependence of the susceptibility on temperature during the increasing or decreasing of $U$ by 10% does not quantitatively affect the results independent of the local environment around the Fe atoms. As one may also see, when $U$ on Fe is equal to 3.4 eV and $U$ on Ni is equal to 2.75 eV, the qualitative dependence remains the same; i.e., the linear behavior of the inverse magnetic susceptibility of the ${\mathrm{Fe}}_2$ type of atoms and nonlinear behavior on the ${\mathrm{Fe}}_1$ type remained unchanged.

Figure 12

The dependence of the inverse local magnetic susceptibility on temperature for values of $U$, varying from 3 to 3.8 eV in the bcc phase of ${\mathrm{Fe}}_3\mathrm{Ni}$. The solid and dashed lines correspond to the ${\mathrm{Fe}}_2$ and ${\mathrm{Fe}}_1$ types of Fe atoms, respectively. Red circles, green squares, and blue diamonds are used for the $U$ values 3, 3.4, and 3.8 eV, respectively. As one may see, the dependence of the susceptibility on temperature during the increasing or decreasing of $U$ by 10% does not quantitatively affect the results independent of the local environment around the Fe atoms.

Figure 13

The dependence of the inverse quasiparticle lifetime Γ to temperature $T$ vs $T$ for values of $U$, varying from 3 to 3.8 eV in the bcc phase of ${\mathrm{Fe}}_3\mathrm{Ni}$. The solid and dashed lines correspond to the ${\mathrm{Fe}}_2$ and ${\mathrm{Fe}}_1$ types of Fe atoms, respectively. Red circles, green squares, and blue diamonds are used for the $U$ values 3, 3.4, and 3.8 eV, respectively. As one may see, the dependence of Γ/$T$ on temperature during the increasing or decreasing of $U$ by 10% does not quantitatively affect the results independent of the local environment around the Fe atoms.