Temperature dependent electronic transport in concentrated solid solutions of the $3d$-transition metals Ni, Fe, Co and Cr from first principles

An approach previously developed for the calculation of transport coefficients via the Mott relations is applied to the calculation of finite temperature transport properties of disordered alloys—electrical resistivity and the electronic part of thermal conductivity. The coherent-potential approximation is used to treat chemical disorder as well as other sources of electron scattering, i.e., temperature induced magnetic moment fluctuations and lattice vibrations via the alloy analogy model. This approach, which treats all forms of disorder on an equal first-principles footing, is applied to the calculation of transport properties of a series of fcc concentrated solid solutions of the $3d$-transition metals Ni, Fe, Co, and Cr. For the nonmagnetic alloys ${\mathrm{Ni}}_{0.8}{\mathrm{Cr}}_{0.2}$ and ${\mathrm{Ni}}_{0.33}{\mathrm{Co}}_{0.33}{\mathrm{Cr}}_{0.3}$, the combined effects of chemical disorder and electron-lattice vibrations scattering result in a monotonic increase in the resistivity as a function of temperature from an already large, $T=0$, residual resistivity. For magnetic ${\mathrm{Ni}}_{0.5}{\mathrm{Co}}_{0.5},{\mathrm{Ni}}_{0.5}{\mathrm{Fe}}_{0.5}$, and ${\mathrm{Ni}}_{0.33}{\mathrm{Fe}}_{0.33}{\mathrm{Co}}_{0.33}$, the residual resistivity of which is small, additional electron scattering from temperature induced magnetic moment fluctuations results in a further rapid increase of the resistivity as a function of temperature. The electronic part of the thermal conductivity in nonmagnetic ${\mathrm{Ni}}_{0.8}{\mathrm{Cr}}_{0.2}$ and ${\mathrm{Ni}}_{0.33}{\mathrm{Co}}_{0.33}{\mathrm{Cr}}_{0.33}$ monotonically increases with temperature. This behavior is a result of the competition between a reduction in the conductivity due to electron-lattice vibrations scattering and temperature induced increase in the number of carriers. In the magnetic alloys, electron scattering from magnetic fluctuations leads to an initial rapid decrease in thermal conductivity until this is overcome by an increasing number of carriers at temperatures slightly below the Curie temperature. Similar to the resistivity above $T_C$, the electronic parts of the thermal conductivities are close to each other in all alloys studied.

Figure 1

Projected density of states of magnetic NiCo (left) and nonmagnetic ${\mathrm{Ni}}_{0.8}{\mathrm{Cr}}_{0.2}$ (right) shown by dashed blue lines for Co/Cr and a solid red line for Ni atoms. The Fermi energy corresponds to zero on the vertical axis. The centers of the $d$ bands are shown by horizontal lines, where splitting is denoted by Δ.

Figure 2

Static electrical conductivity as a function of energy calculated in three alloys. The Fermi energy corresponds to zero.

Figure 3

Electrical conductivity in ${\mathrm{Ni}}_{0.5}{\mathrm{Co}}_{0.5}$ as a function of energy calculated with magnetic moment fluctuation at temperature $T=300$ K (blue line with triangles) and with perfect magnetic ordering (red line with circles). The derivative of the Fermi distribution function $\left(-\partial f/\partial E\right)$ is shown by black solid $(T=300$ K) and dashed $(T=50$ K) lines in relative units.

Figure 4

(a) Electrical resistivity in Ni as a function of temperature calculated using CFM results for the magnetization (red line with filled circles) and experimental magnetization (green down triangles); the experimental resistivity is shown by cyan triangles [28] and blue squares [75]. Both experimental (green down triangles) and calculated (red circles) magnetization dependences of T/${\mathrm{T}}_{\mathrm{C}}$ are shown in the inset. (b) Thermal conductivity, κ, as a function of temperature calculated using Eq. (7) and theoretical results for magnetization (the WF law result is shown by a red line with filled circles; the result with correction to the WF law is shown by empty diamonds). The lattice contribution to κ is shown as a blue line with open circles. Total κ calculated using theoretical magnetization is shown as a green line with down triangles, whereas that calculated using experimental magnetization is shown as a gray line with filled squares; experiment [29] is shown as cyan triangles. The experimental $T_C$ is indicated by a vertical line. All the calculated dependencies include scattering on both lattice vibrations and magnetic moment fluctuations.

Figure 5

(a) Electrical resistivity and (b) thermal conductivity in ${\mathrm{Ni}}_{0.5}{\mathrm{Co}}_{0.5}$. For notations see Fig. 4. In addition to the contribution from electron scattering on lattice vibrations and magnetic moment fluctuations, the result presented by a red line with filled circles contains a contribution from scattering on chemical disorder. The calculated $T_C$ is indicated by a vertical line.

Figure 6

(a) Electrical resistivity and (b) thermal conductivity in ${\mathrm{Ni}}_{0.5}{\mathrm{Fe}}_{0.5}$. For notations see Fig. 4. In addition to the contribution from electron scattering on lattice vibrations and magnetic moment fluctuations, the result represented by a red line with filled circles contains a contribution from scattering on chemical disorder. The calculated $T_C$ is indicated by a vertical line.

Figure 7

(a) Electrical resistivity and (b) thermal conductivity in ${\mathrm{Ni}}_{0.33}{\mathrm{Co}}_{0.33}{\mathrm{Fe}}_{0.33}$. For notations see Fig. 4. In addition to the contribution from electron scattering on lattice vibrations and magnetic moment fluctuations, the result represented by a red line with filled circles contains a contribution from scattering on chemical disorder. The total thermal conductivity calculated as a sum of lattice contribution and the electronic contribution obtained from experimental resistivity [cyan triangles in Fig. 7] through the WF law is shown by gray color squares. The calculated, $T_C$, and experimental, ${T_C}^{\exp }$, Curie temperatures are indicated by solid and dashed vertical lines, respectively.

Figure 8

(a) Electrical resistivity and (b) the electronic part of the thermal conductivity in ${\mathrm{Ni}}_{0.8}{\mathrm{Cr}}_{0.2}$. For notations see Fig. 4. The calculations have been done in the nonmagnetic state.

Figure 9

(a) Electrical resistivity and (b) the electronic part of the thermal conductivity in ${\mathrm{Ni}}_{0.33}{\mathrm{Co}}_{0.33}{\mathrm{Cr}}_{0.33}$. For notations see Fig. 4. The calculations have been done in the nonmagnetic state. The experimental resistivity obtained in the current paper for the compound ${\mathrm{Ni}}_{0.35}{\mathrm{Co}}_{0.35}{\mathrm{Cr}}_{0.3}$ is shown as cyan color triangles. The electronic part of the thermal conductivity calculated from the experimental resistivity [cyan triangles in Fig. 9] through the WF law is shown as gray color squares.