Spin-Flip Diffusion Length in $5d$ Transition Metal Elements: A First-Principles Benchmark

Little is known about the spin-flip diffusion length $l_{\mathrm{sf}}$, one of the most important material parameters in the field of spintronics. We use a density-functional-theory based scattering approach to determine values of $l_{\mathrm{sf}}$ that result from electron-phonon scattering as a function of temperature for all $5d$ transition metal elements. $l_{\mathrm{sf}}$ does not decrease monotonically with the atomic number $Z$ but is found to be inversely proportional to the density of states at the Fermi level. By using the same local current methodology to calculate the spin Hall angle ${\mathrm{\Theta }}_{\mathrm{sH}}$ that characterizes the efficiency of the spin Hall effect, we show that the products $\rho (T)l_{\mathrm{sf}}(T)$ and ${\mathrm{\Theta }}_{\mathrm{sH}}(T)l_{\mathrm{sf}}(T)$ are constant.

Figure 1

Natural logarithm of a spin current injected into RT bcc Ta as a function of the coordinate $z$ in the transport direction, $\mathcal{L}\to \mathcal{R}$. The upper inset sketches the transport geometry with a scattering region $\mathcal{S}$ with temperature dependent lattice disorder sandwiched between ideal semi-infinite ballistic leads, $\mathcal{L}$ and $\mathcal{R}$. A fully polarized spin current injected from the left lead $\mathcal{L}$ undergoes spin flipping leading to spin equilibration on a length scale given by the spin-flip diffusion (SFD) length $l_{\mathrm{sf}}$ as suggested by the red arrows. An unpolarized charge current injected from $\mathcal{L}$ undergoes spin dependent scattering leading to a transverse spin Hall current depicted by the purple arrow. The lower inset shows the spin current on a linear scale. The current was extracted from the results of a scattering calculation for a two-terminal $\mathrm{Ta}\uparrow |\mathrm{Ta}|\mathrm{Ta}$ configuration using a $7\times 7$ lateral supercell where $\mathrm{Ta}\uparrow$ indicates an artificial, fully polarized Ta lead. The red line is a weighted linear least squares fit; the error bar in the value $6.20\pm 0.06$ results from different “reasonable” weightings and cutoff values [17].

Figure 2

Black (left): spin-flip diffusion length $l_{\mathrm{sf}}$ for $5d$ transition metals calculated at room temperature (300 K), the error bars correspond to the spread of values for ten different configurations. For hcp metals, the $c$-axis values are shown. Orange (right): inverse of $\overline{g}({ϵ}_F)$, the density of states averaged over an energy window of $\pm k_{B}T$ about the Fermi energy ${ϵ}_F$. Inset: spin-orbit coupling parameter ${\xi }_d$ as a function of the square of the atomic number $Z$ for the $5d$ elements.

Figure 3

Product of the spin-flip diffusion length and resistivity for $5d$ metals calculated at different temperatures. For hcp metals, the $c$-axis values are shown. The error bars are estimated from the average spreads over 10 configurations for both $l_{\mathrm{sf}}$ and $\rho$.

Figure 4

Momentum relaxation time $\tau$ and spin-flip relaxation time ${\tau }_{\mathrm{sf}}$ in fs for all $5d$ elements at room temperature.

Figure 5

The spin Hall conductivity ${\sigma }_{\mathrm{sH}}$ ($={\mathrm{\Theta }}_{\mathrm{sH}}\times \sigma$) calculated at different temperatures compared to the room temperature spin Hall conductivity for all $5d$ elements (except Hf and Au) reported by Tanaka et al. [48] using a Green function method (GFM) and a phenomenological quasiparticle damping rate to account for disorder. The inset shows the product ${\mathrm{\Theta }}_{\mathrm{sH}}\times l_{\mathrm{sf}}$ as a function of temperature. Only one value is plotted for Au based on an extrapolation of $l_{\mathrm{sf}}$ to RT. For hcp metals, the $c$-axis values are shown.