Electronic transport in ferromagnetic alloys and the Slater-Pauling curve

Experimental measurements of the residual resistivity $\rho \left(x\right)$ of the binary-alloy system ${\text{Fe}}_{1-x}{\text{Cr}}_x$ have shown anomalous concentration dependence which deviates significantly from Nordheim’s rule. In the low $\left(x<10\%\right)$ Cr concentration regime the resistivity has been found to increase linearly with $x$ until $\approx 10\%$ Cr where the resistivity reaches a plateau persisting to $\approx 20\%$ Cr. In this paper we present ab initio calculations of $\rho \left(x\right)$ which explain this anomalous behavior and which are based on the Korringa-Kohn-Rostoker method in conjunction with the Kubo-Greenwood formalism. Furthermore we are able to show that the effects of short-range ordering or clustering have little effect via our use of the nonlocal coherent-potential approximation. For the interpretation of the results we study the alloy electronic structure by calculating the Bloch spectral function particularly in the vicinity of the Fermi energy. From the analysis of our results we infer that a similar behavior of the resistivity should also be obtained for iron-rich ${\text{Fe}}_{1-x}{\text{V}}_x$ alloys—an inference confirmed by further explicit resistivity calculations. Both of these alloy systems belong to the same branch of the famous Slater-Pauling plot, and we postulate that other alloy systems from this branch should show a similar behavior. Our calculations show that the appearance of the plateau in the resistivity can be attributed to the dominant contribution of minority-spin electrons to the conductivity which is nearly unaffected by the increase in Cr/V concentration $x$, and we remark that this minority-spin electron feature is also responsible for the simple linear variation in the average moment in the Slater-Pauling plot for these materials.

Figure 1

(Color online) Spin-projected coherent-potential approximation (CPA) density of states of disordered bcc ${\text{Fe}}_{0.8}{\text{Cr}}_{0.2}$. The solid (black) line shows the total DOS, the dashed (red) line shows the $\text{Fe}d$ states and the dotted (blue) line shows the $\text{Cr}d$ states. The Fermi level is located at the zero of the energy axis.

Figure 2

(Color online) The residual resistivity of ${\text{Fe}}_{1-x}{\text{Cr}}_x$ and ${\text{Fe}}_{1-x}{\text{V}}_x$ as a function of the Cr/V concentration $x$. The asterisks (red line) show the experimental data for ${\text{Fe}}_{1-x}{\text{Cr}}_x$ of Ref. 7 at 4.2K. The triangles and diamonds (black lines) show our CPA results for ${\text{Fe}}_{1-x}{\text{Cr}}_x$ and ${\text{Fe}}_{1-x}{\text{V}}_x$, respectively.

Figure 3

Total and spin-projected BSF of ${\text{Fe}}_{1-x}{\text{Cr}}_x$ at the Fermi energy in the (001) plane for different Cr concentrations (top: 4% Cr; middle: 12% Cr; bottom: 20% Cr). The black regions correspond to values $>50\text{arb}\text{.}$ units. For a better resolution the cusps of the BSF have been cut.

Figure 4

BSF along the $\Gamma \text{−}X$ direction for the minority-spin subsystem in ${\text{Fe}}_{1-x}{\text{Cr}}_x$ for three different Cr concentrations (4%, 12%, and 20% Cr). The cusps of the BSF have been cut at 300 arb. units.

Figure 5

Total and spin-projected BSF of ${\text{Fe}}_{0.8}{\text{V}}_{0.2}$ at the Fermi energy in the (001) plane. The black regions correspond to values $>50\text{arb}\text{.}$ units. For a better resolution the cusps of the BSF have been cut.

Figure 6

BSF along the $\Gamma \text{−}X$ direction for the minority-spin subsystem in ${\text{Fe}}_{1-x}{\text{V}}_x$ for three different V concentrations (4%, 12%, and 20% V). The cusps of the BSF have been cut at 300 arb. units.

Figure 7

Projected majority component of the ${\text{Fe}}_{1-x}{\text{Cr}}_x$ BSF at 4% Cr. The left plot shows the total BSF whereas the middle and the right plots show the $s+p$- and $d$-projected BSFs, respectively.

Figure 8

(Color online) Spin-projected CPA density of states of ${\text{Fe}}_{0.96}{\text{Cr}}_{0.04}$. The solid (black) line shows the total DOS, the dashed (red) line shows the $\text{Fe}d$ states, and the dotted (blue) line shows the $\text{Cr}d$ states. The Fermi level is located at the zero of the energy axis.

Figure 9

(Color online) Spin-resolved conductivities ${\sigma }_{\uparrow }$ and ${\sigma }_{\downarrow }$ for ${\text{Fe}}_{1-x}{\text{Cr}}_x$.

Figure 10

(Color online) Cluster-resolved NL-CPA density of states of disordered ${\text{Fe}}_{1-x}{\text{Cr}}_x$ (top: 4% Cr; bottom: 20% Cr). The solid (black) line shows the total DOS, the dashed (red) line shows the contribution of the FeFe cluster, the (blue) dotted line of the FeCr cluster, the (green) dashed-dotted line of the CrFe cluster, and the (orange) dashed-dotted-dotted line of the CrCr cluster.

Figure 11

(Color online) The resistivity of ${\text{Fe}}_{1-x}{\text{Cr}}_x$ and ${\text{Fe}}_{1-x}{\text{V}}_x$ as a function of the concentration $x$. ${\text{Fe}}_{1-x}{\text{Cr}}_x$: the asterisks (red line) show the experimental data of Ref. 7 and the triangles (black line) show our CPA results. The circles (yellow line) show the NL-CPA results for the disordered alloys; the diamonds (green line) show results when SRO is included and the squares (blue line) when short-range clustering is included. ${\text{Fe}}_{1-x}{\text{V}}_x$: the diamonds (black line) show our CPA results and the crosses (magenta line) show results when SRO is included.