Spin-caloric transport properties of cobalt nanostructures: Spin disorder effects from first principles

The fundamental aspects of spin-dependent transport processes and their interplay with temperature gradients, as given by the spin Seebeck coefficient, are still largely unexplored and a multitude of contributing factors must be considered. We used density functional theory together with a Monte-Carlo-based statistical method to simulate simple nanostructures, such as Co nanowires and films embedded in a Cu host or in vacuum, and investigated the influence of spin disorder scattering on electron transport at elevated temperatures. While we show that the spin-dependent scattering of electrons due to temperature-induced disorder of the local magnetic moments contributes significantly to the resistance, thermoelectric, and spin-caloric transport coefficients, we also conclude that the actual magnitude of these effects cannot be predicted, quantitatively or qualitatively, without such detailed calculations.

Figure 1

General setup of our model system with atoms placed on the fcc crystal lattice. The direction of current is parallel with the $z$ axis ([001] direction). Cu and Co atoms are shown as spheres and arrows, respectively. The transmission probability is evaluated for pairs of atomic layers (indicated by vertical lines) placed far enough from the Cu/Co interface.

Figure 2

Top panel: Cobalt [blue (dark) spheres] nanostructures sandwiched between Cu leads [yellow and red (bright) spheres] used in this study: (c) thin Co layer “TL” consisting of 8 monolayers; (a), (f) monoatomic wires “W11(Cu)” and “W11(Va)” having 1 Co atom in all layers; (b), (e) biatomic wires “W22(Cu)” and “W22(Va)” having 2 Co atoms in all layers; (d) wire “W54(Va)” with alternating 5 and 4 Co atoms. The (Cu) and (Va) indicate the type of embedding (copper and vacuum, respectively) of the Co nanostructure. Periodic boundaries are indicated by solid lines. Crystal structures were plotted using VESTA [40]. Bottom panel: Setup of the nanowires geometry for the (g) W11, (h) W22, and (i) W54 nanowires: Co sites (red filled circle), their first nearest neighbor (green thick line circle), second nearest neighbor (blue medium thick line circle), and embedding sites (black thin line circle) are shown in the two adjacent layers (large and small circles, respectively). The $3\times 3$ and $4\times 4$ supercell is indicated by solid and dashed lines, respectively.

Figure 3

(a) The electron density of states (DOS) for majority and minority spin channel (upper and reverse scale lower part of the graphs, respectively) averaged over four Cu layers adjacent to the scattering region (circles) and all Co atoms in the system (squares). Data corresponding to the spin-ordered and spin-disordered case (calculated at MC simulation $T\approx 1100$ K) is shown as open and filled symbols, respectively. (b) Transmission probability $\Gamma$ divided by the number of atoms ($N_{\mathrm{at}}$) in the cross section of the scattering region. Open and filled symbols correspond to the spin-ordered and average spin-disordered case, respectively. In the spin-disordered case, upper and reverse scale lower parts of the graphs depict ${\Gamma }^{\uparrow }={\Gamma }^{\uparrow \uparrow }+({\Gamma }^{\uparrow \downarrow }+{\Gamma }^{\downarrow \uparrow })/2$ and ${\Gamma }^{\downarrow }={\Gamma }^{\downarrow \downarrow }+({\Gamma }^{\uparrow \downarrow }+{\Gamma }^{\downarrow \uparrow })/2$, respectively. The MC simulation temperature of the spin-disordered data (left to right) corresponds to 1100, 300, 400, and 300 K. The color coding of data sets in (b) is consistent with the system labels in Fig. 2.

Figure 4

Absolute value of the leading exchange coupling parameters $J_{ij}$ in logarithmic scale as a function of the Co intersite distance $r_{ij}$. The color coding of data sets is consistent with the system labels in Fig. 2.

Figure 5

(a) Monte Carlo (MC) site averages of the Co atoms magnetic moment $m$ (solid line) and magnetic susceptibility $\chi$ (dashed line). (b) Spatial correlation $C_N$ of the magnetic moment orientation between $N$th nearest neighbor layers in the $z$ direction. Thick-to-thin lines corresponds to first to seventh nearest neighbors, respectively. Vertical dotted line in (a) indicates temperature for which $C_3$ drops under 0.12 [dotted line in (b)]. (c) Electrical conductance of the ordered magnetic configuration; the temperature dependence enters via Fermi function smearing. Triangles pointing up/down correspond to $\uparrow /\downarrow$ spin, respectively. (d) Electrical conductance of the spin-disordered magnetic configurations calculated by MC method. Triangles pointing up/down and diamonds correspond to $\uparrow \uparrow /\downarrow \downarrow$ and average of $\uparrow \downarrow$ and $\downarrow \uparrow$ of spin matrix elements, respectively. (e) Polarization of the electrical conductance. (f) Total electrical resistance. Position of the vertical dotted line is equivalent to (a). (g) Charge and (h) spin Seebeck coefficient. In (e)–(h), dashed and solid lines correspond to the spin-ordered and spin-disordered data, respectively. The color coding of data sets in the individual columns corresponds to the system labels in Fig. 2.