Magnetic and transport signatures of Rashba spin-orbit coupling on the ferromagnetic Kondo lattice model in two dimensions

Motivated by emergent phenomena at oxide surfaces and interfaces, particularly those involving transition metal oxides with perovskite crystal structure such as ${\mathrm{LaTiO}}_3/{\mathrm{SrTiO}}_3$, we examine the ferromagnetic Kondo lattice model (FKLM) in the presence of a Rashba spin-orbit coupling (RSOC). Using numerical techniques, under the assumption that the electrons on localized orbitals may be treated as classical continuum spins, we compute various charge, spin, and transport properties on square clusters at zero temperature. We find that the main effect of the RSOC is the destruction of the ferromagnetic state present in the FKLM at low electron fillings, with the consequent suppression of conductivity. In addition, near half filling the RSOC leads to a departure of the antiferromagnetic state of the FKLM with a consequent reduction to the intrinsic tendency to electronic phase separation. The interplay between phase separation on one side, and magnetic and transport properties on the other, is carefully analyzed as a function of the RSOC/hopping ratio.

Figure 1

Energy relative to the energy of the FM state, per site, vs electron density for various values of ${\lambda }_{SO}/t_0$. Symbols are PMC results, lines correspond to trial spiral order. The inset is a zoom of the $[0.75,0.9]$ density region for the PMC results only. Results for the $8\times 8$ cluster with open BC.

Figure 2

Energy relative to the energy of the FM state vs electron density, per site for various values of ${\lambda }_{SO}/t_0$. (a) ${\lambda }_{SO}/t_0\ge 1$ $({\lambda }_{SO}=0$ added for completeness), $8\times 8$ cluster. Lines and open symbols were obtained for open BC; stars $\left({\lambda }_{SO}/t_0=1\right)$ and filled squares $\left({\lambda }_{SO}/t_0=2\right)$ for mixed BC. (b) Comparison of results obtained for the $8\times 8$ (lines and open symbols) and $12\times 12$ (filled symbols) clusters with open BC in the range $0.6<\nu <0.9$. For ${\lambda }_{SO}/t_0=1$, results for the $12\times 12$ cluster with mixed BC (circles) are also included.

Figure 3

(a) Magnetic phase diagram in the electron density-${\lambda }_{SO}/t_0$ plane determined from the modulus of the peak of the magnetic structure factor, where dark blue corresponds to FM order $\left(|{\mathbf{k}}_{\mathrm{peak}}|=0\right)$ and dark red to AFM order $\left(|{\mathbf{k}}_{\mathrm{peak}}|=1\right)$. (b) Intensity of $\chi \left({\mathbf{k}}_{\mathrm{peak}}\right)$ in density-${\lambda }_{SO}/t_0$ plane. Results for the $8\times 8$ cluster with open BC.

Figure 4

(a) Magnetic phase diagram in the electron density-${\lambda }_{SO}/t_0$ plane determined from the distance of the peak of $\chi \left(\mathbf{k}\right)$ to the diagonal line in momentum space (see text), with dark blue (red) corresponding to diagonal (off-diagonal) $\mathbf{k}$, obtained for the $8\times 8$ cluster with open BC. Right: $\chi \left(\mathbf{k}\right)$ for $\nu =0.97$, (top) and $\nu =0.69$ (bottom), $12\times 12$ cluster, ${\lambda }_{SO}/t_0=0.5$, with open BC. The color bar code is the same as in Fig. 3

Figure 5

Kinetic energy per site along $x$ direction for (a) the spin-conserving hopping, and (c) the spin flipping or RSOC terms of the Hamiltonian as a function of electron density, for various values of ${\lambda }_{SO}/t_0$. Drude weight for (b) the spin-conserving hopping and (d) the spin flipping or RSOC as a function of electron density, for various values of ${\lambda }_{SO}/t_0$. Symbols and colors are the same as in Fig. 1. Results for the $8\times 8$ cluster with periodic BC and strips with twisted BC.

Figure 6

(a) Total kinetic energy along the $x$ direction, and (b) total Drude peak for various values of ${\lambda }_{SO}/t_0$. Lines are guides to the eye. Results for the $8\times 8$ cluster averaged over periodic BC and strips with twisted BC.

Figure 7

Drude weight $D$ and maximum value of the magnetic structure factor $\chi$, for (a) ${\lambda }_{SO}/t_0=0.5$, (b) 1.0, and (c) 1.5 at electron density $\nu =0.5$. The hopping and SO contribution to $D$ are plotted with circles (red) and diamonds (green) respectively. Up triangles (black) and stars (blue) correspond to the total $D$ and ${\chi }_{\max }$ respectively. $\chi$ has been normalized in this plot in such a way that ${\chi }_{\max }\le 0.25$. Vertical dotted and dashed lines correspond to changes of the momentum of the peak of $\chi$. Results for the $8\times 8$ strip BC.

Figure 8

Left: density-chemical potential stability diagram for various values of ${\lambda }_{SO}/t_0$. Right: snapshots of PMC simulations of FM and AFM bonds (see text) for ${\lambda }_{SO}=0$ (top) and ${\lambda }_{SO}/t_0=1$ (bottom) at $\nu =0.875$. Results for the $8\times 8$ cluster with open (left) and periodic (right) BC.

Figure 9

Spectral functions $A^{(\pm )}(\mathbf{k},\omega )$ for ${\lambda }_{SO}/t_0=0$ (a), 0.75 (b), 4.0 (c), and $\infty$ (d), at density $\nu =0.5$ (bonding band). The PES is plotted with full lines (black) and the IPES with dashed lines (red). The Fermi level is indicated with vertical dotted lines. The Lorentzian width employed is $\epsilon =0.1$. Results obtained for the $8\times 8$ with periodic BC. Lines with dots in (a) and (d) correspond to the $32\times 32$ obtained with fixed FM and AFM order of localized spins respectively.

Figure 10

Energy dispersion for ${\lambda }_{SO}=1$, $t_0=0$, at $\nu =0.5$, obtained from $A^{(\pm )}(\mathbf{k},\omega )$. Results for the $32\times 32$ cluster with AFM fixed localized spin order, periodic BC.