Magnetoresistance effects in a spin-fermion model for multilayers

We consider a spin-fermion model consisting of free electrons coupled to classical spins where the latter are embedded in a quasi-one-dimensional superlattice structure consisting of spin blocks separated by spinless buffers. Using a spiral ansatz for the spins, we study the effect of the electron mediated Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction on the $T=0$ ground state of the system. We find that the RKKY interaction can lead to ferromagnetic, antiferromagnetic, or intermediate spiral phases for different system parameters. When the width is much larger than the length of the individual blocks, the spiral phases are suppressed, and the ground state oscillates between ferromagnetic and antiferromagnetic orders as the size of the buffer regions is varied. This is accompanied by a corresponding oscillation in the Drude weight reflecting an increased conductivity in the ferromagnetic state compared to the antiferromagnetic one. These results are reminiscent of classic giant magnetoresistance phenomena observed in a similar geometry of thin sandwiched magnetic and nonmagnetic layers. Our analysis provides a robust framework for understanding the role of the RKKY interaction on the ground-state order and corresponding transport properties of such systems, extending beyond the conventional perturbative regime.

Figure 1

The superlattice geometry. The blue spheres represent lattice sites in the spin blocks of length $L_1$ possessing classical Heisenberg spins ${\vec{S}}_i$ (represented by the red vectors), whereas the white spheres represent the intermediate buffer sites without spins. The combined spin and buffer blocks of length $L_c=L_0+L_1$ and width $L_y$ is repeated in the $x$ direction $N_c$ times. Periodic boundary conditions (PBCs) are imposed in both the $x$ and $y$ directions.

Figure 2

(a) Energy difference $\mathrm{\Delta }E$ between AF and FM states with increasing $L_0$ at $k_F=\pi /2, L_1=4$, and $J_H=1$. The period of oscillation is 2, and the envelope goes as $1/L_0$, consistent with the RKKY form $\sim \cos \left(2k_{F}r\right)/r=\cos \left(\pi L_0\right)/L_0$. The inset shows that the optimum $q=q^{★}$ value flips between FM and AF with changing chemical potential $\mu$ for the same parameters fixing $L_0=4$. (b) and (c) show the dispersion $E\left(q\right)$ (green circles, scaled) for $J_H=0.01$ and 0.1 at $k_F=\pi /10$. SP order at small $J_H$ gradually evolves towards FM/AF with increasing $J_H$. Orange squares show results from second-order perturbation for small $J_H$ (also scaled). In second order, $\delta E_2\left(q\right)=-{\sum }_{m,m^{\prime }}{J}_{m,m^{\prime }}{\vec{S}}_m\left(q\right)·{\vec{S}}_{m^{\prime }}\left(q\right)=-\mathrm{Re}\left[J\left(q\right)\right]$ (see Appendix pp1 for details), whose minimum coincides with that of $-\mathrm{Re}\left[J\right(q\left)\right]$ as the plots demonstrate.

Figure 3

(a) Optimum value $q^{★}$ over $L_y$ values $\sim [10,{10}^4]$ at $\mu =-0.4,L_0=25,L_1=4$, and $J_H=1.0$. We find that, in a large range $L_y\lesssim {10}^3$, the ground state consists of various SP orders, whereas, for large $L_y\gtrsim {10}^3$, these are suppressed and the system has an AF ground state at the chosen parameter values. (b) shows optimum $q=q^{★}$ vs $L_0$ for small $L_y=10$ where we see rapid oscillations without any uniform periodicity (see the text for more details). The number of cells $N_c$ is kept at 60.

Figure 4

(a) Energy difference $\mathrm{\Delta }E$ between FM and AF states (main) and optimum value $q^{★}$ (inset) vs $L_0$ for very wide superlattices ($L_y=20000$) at $\mu =-0.4,J_H=1$, and $L_1=4$. SP phases are suppressed in this regime, but in contrast to the results for $L_y\sim O\left(10\right)$, the ground state oscillates between FM and AF with a period approximately five sites. $\mathrm{\Delta }E=E_{\mathrm{FM}}-E_{\mathrm{AF}}$ shows corresponding oscillations with a magnitude that decays with $L_0$. (b) the Drude weight $D$ for the FM (orange) and AF (green) states for the same parameters. The FM state value is approximately two to three times larger throughout, leading to large oscillations in $D$ (red) as the ground state keeps flipping between FM and AF. The number of cells $N_c$ is kept at 60.

Figure 5

Phase diagrams on $L_0-J_H$ plane for different $\mu$ values at $L_y=20000$ and $L_1=4$. Away from half-filling ($\mu \ne 0$), the systems show periodic oscillations between FM and AF with $L_0$ over essentially the whole range of $J_H\lesssim 4$. The basic period is approximately constant for a given $\mu$ but becomes smaller as it is reduced, reaching a minimum of two sites at $\mu =-2$, beyond which it increases again, displaying an approximate symmetry around this value. In contrast, $\mu =0$ (and $\mu =-4$, not shown) show no periodicity and persistent AF (FM for $\mu =-4$) regions forming narrow “fingers” running across the plots (see the text).

Figure 6

(a) shows the $\mathrm{\Delta }E$ vs $L_0$ plots (origins shifted for clarity, denoted by dotted black lines for each curve) at $\mu =-0.4$ (red squares), $-1.2$ (blue squares), and $-2.0$ (green squares). The solid lines of same color show fits to the form $f\left(L_0\right)\sim a\cos (k{L}_0+b)/L_0^l$. Plots of $L_y=20000$ at $\mu =-0.4$ (green circles, scaled by factor of 20, shifted slightly for clarity) and $L_1=10$ at $\mu =-1.2$ (magenta circles) show the that the basic period is independent of these parameters. Red dotted curves at $\mu =-2$ highlight the decaying amplitude of oscillations with increasing $L_0$. (b) shows the period (expressed as $2\pi /k_{\mathrm{eff}}$) extracted from the fits as a function of $\mu$. The number of cells $N_c=60$.

Figure 7

The fitting wave-vector $k_{\mathrm{eff}}$ as a function of $k_y$ for $L_1=4$ and $6,L_y=1000$, and $J_H=1$ for different values of $\mu$. “Conjugate” pairs, symmetrically located around $\mu =-2$ (see the text), have the same color but different symbols (circles and squares). Black lines show $2k_F$, where $k_F$ is defined by the usual relation $k_F={\cos }^{-1}[\cos \left(k_y\right)+\mu /2]$. The close fit shows that the RKKY form using the standard Fermi vector $k_F$ for the buffer regions is valid for $\mathrm{\Delta }E=E_{\mathrm{FM}}-E_{\mathrm{AF}}$ even outside the perturbative regime for $J_H$. The slight deviations for $L_1=6$ stem from the fitting function algorithm (fitting errors are not plotted for clarity) and is not indicative of a sudden change in the periodicity of the system. Effective period of the quasi-1D system seems to be dominated by the $k_y=0$ value.

Figure 8

Spin-resolved Drude weight $D_{\sigma }$ as a function of $\mu$ for (a) $L_0=L_1=4$, and (b) $L_0=L_1=10$, for a FM state, demonstrating finite-size oscillations and a concave central feature of width $\simeq J_H$ in both spin channels, leading to a higher value of $D$ for the down channel in this range. Bottom plots show spin-resolved density of states (DOS) ${\rho }_{\sigma }\left(\omega \right)$ at (c) $\mu =0$ and (d) $\mu =-1.2$, demonstrating: (i) only superficial resemblance with transport due to strong renormalization by current matrix elements in the transport formula (Appendix pp2), (ii) a mirror symmetry at $\mu =0$ (left) leading to same transport in both spin channels, which is no longer present away from half-filling (right) (see the text for more details).

Figure 9

The number density $\langle n\rangle$ for the FM state (filled circles) and the AF state (hollow squares of the same color, shifted slightly for visual clarity) as a function of $L_0$ at $\mu =-0.4,-1.2,-2.0,-2.8$, and $-3.6$ respectively. It is clear that the variation is marginal ($\lesssim 0.03$) for $L_0\gtrsim 5$. As a result, we choose to do our calculations keeping $\mu$ constant instead of $\langle n\rangle$.

Figure 10

The number density $\langle n_{\sigma }\rangle$ for the FM state as a function of $\mu$ and superlattice coordinate $x_i$. (a) plots the total density $\langle n_{\sigma }\rangle$ vs $\mu$, where $\langle n_{i,\uparrow }\rangle >\langle n_{i,\downarrow }\rangle$ throughout as expected. (b) shows $\langle n_{i,\sigma }\rangle$ at the center of the magnetic ($S$) and buffer ($B$) blocks where we find that the polarization in the density is largely limited to the $S$ blocks themselves. (c) and (d) show the variation of $\langle n_{i,\downarrow }\rangle$ and $\langle n_{i,\uparrow }\rangle$, respectively, with the superblock coordinate $x_i$ for two superblocks with expected oscillations between $S$ and $B$ blocks. Even here, the effect is almost local.