Calculating spin transport properties from first principles: Spin currents

Local charge and spin currents are evaluated from the solutions of fully relativistic quantum mechanical scattering calculations for systems that include temperature-induced lattice and spin disorder as well as intrinsic alloy disorder. This makes it possible to determine material-specific spin transport parameters at finite temperatures. Illustrations are given for a number of important materials and parameters at 300 K. The spin-flip diffusion length $l_{\mathrm{sf}}$ of Pt is determined from the exponential decay of a spin current injected into a long length of thermally disordered Pt; we find $l_{\mathrm{sf}}^{\mathrm{Pt}}=5.3\pm 0.4\mathrm{nm}$. For the ferromagnetic substitutional disordered alloy permalloy (Py), we inject currents that are fully polarized parallel and antiparallel to the magnetization and calculate $l_{\mathrm{sf}}$ from the exponential decay of their difference; we find $l_{\mathrm{sf}}^{\mathrm{Py}}=2.8\pm 0.1\mathrm{nm}$. The transport polarization $\beta$ is found from the asymptotic polarization of a charge current in a long length of Py to be $\beta =0.75\pm 0.01$. The spin Hall angle ${\mathrm{\Theta }}_{\mathrm{sH}}$ is determined from the transverse spin current induced by the passage of a longitudinal charge current in thermally disordered Pt; our best estimate is ${\mathrm{\Theta }}_{\mathrm{sH}}^{\mathrm{Pt}}=4.5\pm 1\%$ corresponding to the experimental room-temperature bulk resistivity $\rho =10.8\mu \mathrm{\Omega }\mathrm{cm}$.

Figure 1

Calculated fractional spin conductance $G^{\uparrow \uparrow }/G^{\uparrow }$ for RT thermally disordered Pt sandwiched between the different ballistic leads: Pt (blue triangles), Au (red circles), and Cu (green crosses). $G^{\sigma {\sigma }^{\prime }}$ is ($e^2/h$ times) the transmission probability of a spin $\sigma$ from the left-hand lead into a spin ${\sigma }^{\prime }$ in the right-hand lead; $G^{\uparrow }=G^{\uparrow \uparrow }+G^{\uparrow \downarrow }$. The solid lines are the exponential fits to the calculated values giving rise to $l_{\mathrm{sf}}$. (Inset) The areal resistance of the $\mathrm{NM}\left|\mathrm{Pt}\right|\mathrm{NM}$ trilayer as a function of the length $L$ of Pt for all three ballistic leads. Solid lines are linear fits whose slopes yield identical resistivities in the three cases. Data for $L<4$ nm are excluded from the linear fit [28].

Figure 2

Example of a transverse supercell for a $\mathrm{Cu}\left|\mathrm{Py}\right|\mathrm{Cu}$ scattering geometry. Cu atomic layers form semi-infinite ballistic leads denoted $\mathcal{L}$ and $\mathcal{R}$. The scattering region $\mathcal{S}$ consists of a thickness $L$ of the substitutional disordered ${\mathrm{Ni}}_{80}{\mathrm{Fe}}_{20}$ alloy, permalloy, sandwiched between the leads. Each atomic layer in $\mathcal{L}\left|\mathcal{S}\right|\mathcal{R}$ contains $5\times 5$ atoms. The layers are parallel to the $xy$ plane and in the calculations this structure is repeated in the $x$ and $y$ directions so that an infinite periodic structure arises.

Figure 3

Illustration of a number of concepts defined in the text. Filled black circles represent atoms. Current flow is in the $z$ direction from left to right. The horizontal dashed lines indicate the (lateral) unit cell boundaries. The vertical dotted lines indicate layer boundaries in the $z$ direction. The gray area is a “wire” with assumed homogeneous current density that represents the general spatial distribution of the current between $Q$ and $P$, which can therefore be left unknown.

Figure 4

Majority (lhs) and minority (rhs) spin band structures of Cu when a repulsive constant potential of 1 Rydberg is added to the minority spin potential. The effect is to remove all minority spin states from the Fermi energy.

Figure 5

Natural logarithm of the spin current injected into RT Pt as a function of the coordinate $z$ in the transport direction. The inset shows the spin current on a linear scale. The current was extracted from the results of a scattering calculation for a two-terminal $\mathrm{Au}\uparrow |\mathrm{Pt}|\mathrm{Au}$ configuration using a $7\times 7$ lateral supercell. The red line is a weighted linear least squares fit; the error bar in the value $5.25\pm 0.05$ results from different “reasonable” weightings and cutoff values.

Figure 6

Natural logarithm of the spin current density vs length for various $N\times N$ supercells ($N=3,5,7,10$) for a 35-nm-long Pt slab at 300 K. The color coordinated symbols and solid lines indicate the mean and a measure of the spread of the data from 20 different configurations, respectively. Inset: SDL obtained from the linear fit of $ln{j}_s\left(z\right)$ shown as a function of $1/N$. The numerical values of $l_{\mathrm{sf}}$ are given in Table 1.

Figure 7

Natural logarithm of the spin current density calculated for RT Pt with a $7\times 7$ lateral supercell and three different $Q\times Q$ samplings of the BZ: $Q=10$ (yellow squares), 16 (blue triangles), and 32 (red circles). The color coordinated symbols and solid lines indicate the mean and a measure of the spread of the data, respectively, for 20 different configurations for different Q samplings. The dashed black line indicates the linear fit determined for $ln{j}_s$ calculated with $Q=32$. Though the three curves initially overlap perfectly, for $z\ge 14\mathrm{nm}$ we see that noise in the mean and spread of $ln{j}_s$ for $Q=10$ is substantially larger than for the $Q=32$ data. (Inset) $l_{\mathrm{sf}}^{\mathrm{Pt}}$ as a function of the BZ sampling parameter $Q$.

Figure 8

Natural logarithm of the spin current injected into Pt as a function of the coordinate in the transport direction $z$ using different lead materials. The current was extracted from the results of a scattering calculation for a two-terminal $\mathrm{NM}\uparrow |\mathrm{Pt}|{\mathrm{NM}}^{\prime }$ configuration. The lattice constant of the Au leads was scaled to match to Pt. In the case of Cu, a lateral Cu $8\times 8$ supercell was matched to a $2\sqrt{13}\times 2\sqrt{13}$ lateral Pt supercell. Injection from permalloy was realized in a $\mathrm{Cu}\left|\mathrm{Py}\right|\mathrm{Pt}|\mathrm{Cu}$ configuration where lattice matching was realized using the same supercells as for $\mathrm{Cu}\left|\mathrm{Pt}\right|\mathrm{Cu}$. The straight lines are weighted linear least squares fits from which the interface region is omitted; the error bars result from different “reasonable” weightings and cutoff values.

Figure 9

A charge current passed through Py polarizes to $\beta$ in the middle of the scattering region. By fitting (solid black lines) the spin current calculated for an $N\times N$ supercell to Eq. (3), $\beta$ and $l_{\mathrm{sf}}^{\mathrm{Py}}$ are extracted for (a) $N=5$ (blue) and (b) $N=8$ (red). The dotted lines indicate the spread from 20 different configurations of disorder.

Figure 10

Spin currents ${\widehat{j}}_{sz}\left(z\right)$ obtained by injecting electrons into Py from a left Cu lead (red) and from injecting holes from a right Cu lead (green) for an $5\times 5$ lateral supercell. The black dashed line shows the fit to the averaged current discussed in the text. (Inset) Fractional nonequilibrium particle densities incident from left (red) and right (green) leads.

Figure 11

Fully polarized spin-up (blue) and spin-down (red) currents injected from the left ballistic $\mathrm{Cu}\uparrow$, respectively $\mathrm{Cu}\downarrow$ lead into 40 nm of Py with RT thermal lattice and spin disorder saturate to $\beta$ far from the lead. The difference of the two currents decays exponentially to zero. The supercell size is $8\times 8$ and the Brillouin zone $k$ sampling is $28\times 28$. (Inset) Natural logarithm of the difference is fit linearly (in yellow) to yield $l_{\mathrm{Py}}=2.83\pm 0.02$ nm.

Figure 12

Transverse spin currents driven by a charge current in the $z$ direction for a (111) oriented Pt slab embedded between (a) Pt and (b) Au leads. The error bars are the mean deviation of the currents for 20 configurations of disorder. The horizontal dotted and dashed lines indicate the extracted values of ${\mathrm{\Theta }}_{\mathrm{sH}}$. For both leads, $7\times 7$ supercells were used with a $22\times 22$ BZ sampling. (Inset) Integrated transverse spin currents as a function of $L_{\mathrm{Pt}}$ for a RT diffusive Pt scattering region embedded between ballistic Pt (green stars) (a) and Au (pink circles) (b) leads. The dotted and dashed lines indicate the weighted linear least squares fit with Pt (a) and Au (b) leads, respectively. The interface contributions in (a) are negligible compared to (b).

Figure 13

Band structure of Pt evaluated using Stuttgart LMTO code with an $spd$ (blue) and $spdf$ (red) basis. The dispersion about the Fermi energy and details of the Fermi surface are very sensitive to the choice of basis.

Figure 14

Spin Hall angle as a function of the energy in a rigid band approximation calculated for electrons incident from the left (red squares) and from the right (green stars). The sum of these contributions is shown as black circles. For reference, we show the Pt $d$ density of states in grey.