Transport properties of doped permalloy via ab initio calculations: Effect of host disorder

Transport properties of permalloy doped with V, Co, Pt, and Au are explored via ab initio calculations. For this purpose, the Kubo-Bastin formula is evaluated within the fully relativistic Korringa-Kohn-Rostoker Green function formalism. Finite-temperature effects are treated by means of the alloy analogy model. It is shown that the fact that the host is disordered and not crystalline has a profound effect on how the conductivities characterizing the anomalous Hall effect and the spin Hall effect depend on the dopant concentration. Several relationships between quantities characterizing charge and spin transport are highlighted. The rate of decrease of the longitudinal charge conductivity with increasing doping depends on the dopant type, following the sequence Co-Au-Pt-V. The dependence of the anomalous Hall and spin Hall conductivities on the dopant concentration is found to be nonmonotonic. Introducing a finite temperature changes the overall trends significantly. The theoretical results are compared with available experimental data.

Figure 1

Theoretical longitudinal conductivity ${\sigma }_{xx}$ (left), relative anisotropic magnetoresistance $({\rho }_{zz}-{\rho }_{xx})/{\rho }_{\text{aver}}$ (middle), and the difference of resistivities ${\rho }_{zz}-{\rho }_{xx}$ (right) for Py doped with V, Co, Pt, and Au, for $T=0\mathrm{K}$. Calculated values are shown by markers, the lines are guides for an eye. The dopant type is indicated in the legend.

Figure 2

Average resistivity ${\rho }_{\text{aver}}$, difference between resistivities ${\rho }_{zz}-{\rho }_{xx}$, and anomalous magnetoresistance $({\rho }_{zz}-{\rho }_{xx})/{\rho }_{\text{aver}}$ for Py doped with V, Co, Pt, and Au calculated for $T=300\mathrm{K}$, compared to experiment. Experimental data are from Nagura et al. [4], Yin et al. [7], and Hrabec et al. [9]. For the average resistivity ${\rho }_{\text{aver}}$, theoretical results obtained when only the spin fluctuations or only the atomic vibrations are accounted for, are also shown.

Figure 3

Dependence of the resistivity ${\rho }_{\text{aver}}$ of undoped Py on the temperature as provided by our calculation (solid line), by calculations of Starikov et al. [27] (dashed line), and by experiment [28] (squares).

Figure 4

Bloch spectral function $A_B(\mathit{k},E)$ of undoped Py projected on majority-spin states and on minority-spin states. Three instances where the majority-spin band crosses the Fermi level are marked by the numbers 1, 2, and 3.

Figure 5

Majority-spin Bloch spectral function at the Fermi level $A_B(\mathit{k},E_F)$ for $\mathit{k}$ in the vicinity of the point marked as 1 in Fig. 4. The $\mathit{k}$ vector lies on the $X\text{−}\mathrm{\Delta }$ path, its values are in units of $2\pi /a_0$. The concentration of the dopants is 1% (left) and 10% (right).

Figure 6

Lower panel: Atomic phase shifts of majority-spin $d$ electrons for scattering at Fe and Ni as host atoms and at V, Co, Pt, and Au as impurity atoms in Py. Upper panel: Total atomic scattering cross section of majority-spin electrons for the same types.

Figure 7

Theoretical off-diagonal conductivities ${\sigma }_{xy}$ (corresponding to AHE) and ${\sigma }_{xy}^z$ (corresponding to SHE) for Py doped with V, Co, Pt, and Au, for $T=0\mathrm{K}$. Calculated values are shown by markers, the lines are guides for an eye. The dopant types are indicated in the legend.

Figure 8

Theoretical dependence of the AHE conductivity ${\sigma }_{xy}$ (two left panels) and of the SHE conductivity ${\sigma }_{xy}^z$ (two right panels) on the longitudinal conductivity ${\sigma }_{xx}$ if the concentration of the dopants is varied (for $T=0\mathrm{K})$. The Py host is treated either within the CPA or within the VCA, respectively, as indicated at the top of the panels. The dopant type is specified in the legend.

Figure 9

Theoretical off-diagonal conductivities ${\sigma }_{xy}$ (corresponding to AHE) and ${\sigma }_{xy}^z$ (corresponding to SHE) for Py doped with V, Co, Pt, and Au, for $T=300\mathrm{K}$.

Figure 10

Temperature dependence of ${\sigma }_{xy}$ and ${\sigma }_{xy}^z$ for Py doped with 6% of Co and with 6% of Au, calculated for the vertex corrections either included or omitted.

Figure 11

Theoretical temperature dependence of ${\sigma }_{xy}$ and ${\sigma }_{xy}^z$ for undoped Py (diamonds) and for Py doped with 6% of V (asterisks), Co (crossed circles), Pt (triangles), and Au (squares).

Figure 12

Temperature dependence of the longitudinal conductivity ${\sigma }_{xx}$ and of off-diagonal conductivities ${\sigma }_{xy}$ and ${\sigma }_{xy}^z$ for Py doped with 6% of V (upper panels) and with 6% of Pt (lower panels). Data are shown separately for calculations where both spin fluctuations and atomic vibrations are accounted for (full lines with asterisks), where only the spin fluctuations are considered (dotted lines with crosses), and where only the atomic vibrations are taken into account (dashed lines with diamonds).

Figure 13

Left: Experimental anomalous Hall resistivity ${\rho }_{xy}$ for Pt-doped Py measured by Hrabec et al. [9] at $T=300\mathrm{K}$ (triangles) together with our theoretical data for $T=0\mathrm{K}$ (dashed line) and for $T=300\mathrm{K}$ (full line). Right: Temperature dependence of the AHE resistivity ${\rho }_{xy}$ and the SHE resistivity ${\rho }_{xy}^z$ for undoped Py obtained from our calculations (lines) and from the experiment of Omori et al. [42] (markers). Note that the calculations concern bulk Py while the experiment [42] was done for a thin film.

Figure 14

Left: Dependence of the SHE conductivity ${\sigma }_{xy}^z$ on the AHE conductivity ${\sigma }_{xy}$ for Py doped by V, Co, Pt, and Au (of various concentrations) calculated for $T=0\mathrm{K}$ if all contributions to ${\sigma }_{xy}$ and ${\sigma }_{xy}^z$ are taken into account. Middle: As the left panel but considering only the contributions due to the vertex corrections. The straight line corresponds to the apparent ${\sigma }_{xy}\text{(VC)}={\sigma }_{xy}^z\text{(VC)}$ relation. Right: The temperature dependence of the contributions to ${\sigma }_{xy}$ (lines) and to ${\sigma }_{xy}^z$ (markers) due to the vertex corrections for Py doped by 6% of V, Co, Pt, and Au.

Figure 15

Reduced magnetization $M\left(T\right)/M\left(0\right)$ used as input for the alloy analogy model. Data are shown for undoped Py and for Py doped by 6% or 20% of V, Co, Pt, and Au.