Deviations from Matthiessen's rule and resistivity saturation effects in Gd and Fe from first principles

According to earlier first-principles calculations, the spin-disorder contribution to the resistivity of rare-earth metals in the paramagnetic state is strongly underestimated if Matthiessen's rule is assumed to hold. To understand this discrepancy, the resistivity of paramagnetic Fe and Gd is evaluated by taking into account both spin and phonon disorder. Calculations are performed using the supercell approach within the linear muffin-tin orbital method. Phonon disorder is modeled by introducing random displacements of the atomic nuclei, and the results are compared with the case of fictitious Anderson disorder. In both cases, the resistivity shows a nonlinear dependence on the square of the disorder potential, which is interpreted as a resistivity saturation effect. This effect is much stronger in Gd than in Fe. The nonlinearity makes the phonon and spin-disorder contributions to the resistivity nonadditive, and the standard procedure of extracting the spin-disorder resistivity by extrapolation from high temperatures becomes ambiguous. An “apparent” spin-disorder resistivity obtained through such extrapolation is in much better agreement with experiment compared to the results obtained by considering only spin disorder. By analyzing the spectral function of the paramagnetic Gd in the presence of Anderson disorder, the resistivity saturation is explained by the collapse of a large area of the Fermi surface due to the disorder-induced mixing between the electron and hole sheets.

Figure 1

The area-resistance product as a function of the active disordered region length for two separate sets of calculations. The black circles (read using bottom and left axes) are calculations with collinear ferromagnetic $\alpha \text{-Fe}$ and a phonon mean-square displacement $\overline{u}=0.1572\AA$, and the gray circles (read using top and right axes) are calculations of Gd with random noncollinear spin disorder, current flowing parallel to the $c$ axis, and phonon mean-square displacement $\overline{u}=0.3629\AA$.

Figure 2

Electrical resistivity data taken from experiment. (a) Fe resistivity measurements compiled from Refs. [59, 60, 61]. The three lines correspond to fits to data from (1) Pallister [59]; (2) and (3) Cezairliyan et al. [61]. The temperature range, slopes, and intercepts of the fits are in Table 1. (b) High-temperature resistivity data for polycrystalline rare-earth metals compiled from Ref. [62].

Figure 3

Calculated resistivities for $\alpha$ and $\gamma$ Fe and hcp Gd. (a) $\rho \left({\overline{u}}^2\right)$ for ferromagnetic $\alpha$-Fe with atomic displacements. Filled circles: this work; open circles: data of Ref. [51]. (b) $\rho \left({\overline{u}}^2\right)$ for two phases of Fe with random vector spin disorder and atomic displacements. Open circles and solid line: $\alpha$-Fe, filled circles and dashed line: $\gamma$-Fe. (c) $\rho \left({\overline{u}}^2\right)$ for hcp Gd with random vector spin disorder and atomic displacements. Filled gray circles and gray fit line: $c$-axis transport, $m=7.72{\mu }_B$. Filled black circles and solid black fit line: in-plane transport, $m=7.72{\mu }_B$. Open circles and dashed fit line: in-plane transport, $m=7.45{\mu }_B$. (Inset) Enlarged plot of the linear region. (d) In-plane resistivity of Gd as a function of the squared Anderson disorder amplitude ${\Delta }^2$. Closed circles: with fictitious non-magnetic atomic potentials. Open circles: with random vector spin disorder ($m=7.72{\mu }_B$).

Figure 4

Average spin-projected local density of states of spin-disordered Gd ($m=7.72{\mu }_B$) for different amplitudes of lattice disorder $\overline{u}$: (a) no phonon disorder, (b) $\overline{u}=0.183$ Å, (c) $\overline{u}=0.257$Å, and (d) $\overline{u}=0.316$ Å.

Figure 5

The spectral function of paramagnetic Gd for different amplitudes $\Delta$ of Anderson disorder plotted along high-symmetry lines of the hexagonal Brillouin zone. (a) $\Delta =0$, (b) $\Delta =0.95$ eV, and (c) $\Delta =1.8$ eV.

Figure 6

The spectral function of paramagnetic Gd evaluated at the Fermi energy for the HLMK plane in the Brillouin zone.

Figure 7

The spectral function of paramagnetic Gd evaluated at the Fermi energy on the indicated planes of the Brillouin zone for different values of the Anderson disorder amplitude $\Delta$. (a), (d), and (g) $\Delta =0$. (b), (e), and (h) $\Delta =0.95$ eV. (c), (f), and (i) $\Delta =1.8$ eV.