Colossal magnetoresistance observed in Monte Carlo simulations of the one- and two-orbital models for manganites

The one- and two-orbital double-exchange models for manganites are studied using Monte Carlo computational techniques in the presence of a robust electron-phonon coupling (but neglecting the antiferromagnetic exchange $J_{\mathrm{AF}}$ between the localized spins). The focus in this effort is on the analysis of charge transport. Our results for the one-orbital case confirm and extend previous recent investigations that showed the presence of robust peaks in the resistivity versus temperature curves for this model. Quenched disorder substantially enhances the magnitude of the effect, while magnetic fields drastically reduce the resistivity. A simple picture for the origin of these results is presented. It is also shown that even for the case of just one electron, the resistance curves present metallic and insulating regions by varying the temperature, as it occurs at finite electronic density. Moreover, in the present study these investigations are extended to the more realistic two-orbital model for manganites. The transport results for this model show large peaks in the resistivity versus temperature curves, located at approximately the Curie temperature, and with associated large magnetoresistance factors. Overall, the magnitude and shape of the effects discussed here resemble experiments for materials such as ${\mathrm{La}}_{0.70}{\mathrm{Ca}}_{0.30}\mathrm{Mn}{\mathrm{O}}_3$, and they are in agreement with the current predominant theoretical view that competition between a metal and an insulator, enhanced by quenched disorder, is crucial to understanding the colossal magnetoresistance (CMR) phenomenon. However, it is argued that further work is still needed to fully grasp the experimentally observed CMR effect, since in several other Mn oxides an antiferromagnetic charge-ordered orbital-ordered state is the actual competitor of the ferromagnetic metal.

Figure 1

(Color online) Monte Carlo results obtained using a $4\times 4\times 4$ lattice. Shown are the resistivity and spin-spin correlations, the latter at the maximum allowed distance $\left(2\sqrt{3}\right)$, versus temperature, working at $\lambda =0.9$, $n=0.3$, and for the disorder strengths $\Delta$ indicated. The results shown are mainly for one configuration of quenched disorder, but as many as ten configurations were used in particular cases of temperatures and $\Delta$’s, and no substantial deviations were observed between disorder configurations.

Figure 2

(Color online) (a) Influence of the electron-phonon coupling $\lambda$ in the clean limit $\Delta =0$, and (b) effect of magnetic fields (values indicated) for $\lambda =0.9$ and $\Delta =0.6$, on the resistivity curves, using a $4\times 4\times 4$ lattice at density $n=0.3$.

Figure 3

(Color online) Influence of the disorder strength $\Delta$ (indicated) on the resistivity versus temperature curves, working at $\lambda =1.2$, $n=0.1$, and using an $8\times 8$ lattice.

Figure 4

(Color online) (a) Influence of the electron-phonon coupling $\lambda$ on the resistivity versus temperature curves, (b) influence of magnetic fields on the resistivity curve and on the spin-spin correlation at the maximum allowed distance $\left(4\sqrt{2}\right)$ on an $8\times 8$ lattice, and (c) magnetoresistance ratios versus temperature, calculated for two representative magnetic fields in the clean limit $\Delta =0$, at $n=0.1$, and using an $8\times 8$ lattice.

Figure 5

(Color online). Results mainly in the clean limit $\Delta =0$ illustrating a variety of issues discussed in the text. (a),(b), and (c) The first-order (discontinuous) character of the transition at $n=0.9$, $\lambda =1.4$, using an $8\times 8$ lattice. (a) The spin-spin correlations at the maximum distance. Also shown in red are results at $\Delta =0.4$ and averaged over five disorder configurations, showing the smearing of the transition with disorder; (b) is the resistivity versus $T$; and (c) is the Monte Carlo time evolution of the energy at $T=0.034$, showing the presence of two states. (d) The resistivity versus temperature at $n=0.7$ ($8\times 8$ lattice), at the $\lambda$’s indicated. (e) The same as (d) but at $n=0.5$ and using a $12\times 12$ cluster. (f) The same as (e) but for $n=0.2$.

Figure 6

(Color online) (a) Influence of quenched disorder on the size of the parameter-space region where the resistivity peak exists. In the plane $\lambda \text{−}\Delta$, $M$ $\left(I\right)$ denotes the region where the resistivity is metallic (insulating) at all temperatures, while $M\text{−}I$ is the area where the resistivity peak is present. The calculation was done on an $8\times 8$ cluster, with $n=0.1$; (b) and (c) illustrate the similarity of results obtained using a $6\times 6\times 6$ lattice (shown) as compared with the $4\times 4\times 4$ results discussed before. (b) Resistivity versus temperature at $n=0.9$ in the clean limit $\Delta =0$, for the $\lambda$’s indicated. (c) Resistivity $\rho$ versus $T$ at $n=0.8$ at the $\lambda$’s indicated and for $\Delta =0$.

Figure 7

(Color online) (a)–(c) Results obtained in the one electron limit. (a) Resistivity $\rho$ versus $T$ at $\Delta =0$ for an $8\times 8$ lattice, showing that even the $\lambda =0$ results present a peak. The inset contains the spin-spin correlations at the maximum distance; (b), (c) the $\lambda =0$ spatially resolved nearest-neighbor spin-spin correlations $NN\left(i\right)={\sum }_{⟨ij⟩}{\vec{S}}_i\bullet {\vec{S}}_j$, where the sum is over the four neighbors $j$ of site $i$. The results were obtained at $T=0.004$ and $T=0.072$, respectively, namely before and after the resistivity peak. Dark colors denote large values of $NN\left(i\right)$, namely regions where the spins are aligned ferromagnetically. (d)–(i) Density-of-states at $\lambda =1.5$, $\Delta =0$, using an $8\times 8$ lattice, at the various temperatures indicated. The red lines (vertical) indicate the location of the chemical potential such that $n=0.1$ in all the panels. (j) Natural logarithm of the resistivity $\rho$ versus temperature plotted on the same scale with ${\sigma }_n$ (defined in text) and the inverse of the density of states at the chemical potential, $1∕N(\omega =\mu )$, using the same parameters as in (d)–(i). Both ${\log }_{10}\left(\rho \right)$ and $1∕N(\omega =\mu )$ are normalized to coincide with the maximum of ${\sigma }_n$. Results shown correspond to averages over several independent Monte Carlo runs.

Figure 8

(Color online) Possible explanation for the existence of the resistivity peak, using a $4\times 4\times 4$ and the couplings and densities indicated. Results shown are for the artificial “left-right” system described in the text, where the left (right) of the lattice has a relatively large (small) $\lambda$. (a) Various quantities (see middle inset) versus temperature. The growth of ${\sigma }_n$ (left) and $n$ (left) with decreasing temperature is indicative of localization of charge in the large $\lambda$ region (left). The trends in these quantities are very similar to the resistivity in its insulating range. At the Curie temperature (see spin-spin correlations), the localization features remain the same but now the mobile carriers can conduct much better than in a paramagnetic spin background. (b)–(e) Electronic density (proportional to the diameter of the spheres) at the temperatures indicated.

Figure 9

(Color online) Clean limit ($\Delta =0$ results) (a): Resistivity versus $T$ for various values of $\lambda$, using the two-orbital model for manganites. The simulation was performed on an $8\times 8$ lattice with 20 electrons $(n\approx 0.3)$. (b) Spin-spin correlation at the maximum distance versus $T$ for various values of $\lambda$. Lattice, density, and MC steps are as in (a). (c) Resistivity versus $T$ for a $12\times 12$ lattice, considering 2 orbitals per site, 44 electrons $(n\sim 1∕3)$, $J_{\mathrm{AF}}=0.0$, and the values of $\lambda$ indicated. (d) Spin-spin correlation at the largest possible distance $\left(6\sqrt{2}\right)$ on a $12\times 12$ lattice, in the clean limit, with the same convention and parameters as in (a).

Figure 10

(Color online) (a) Resistivity versus $T$ for several $\lambda$’s (indicated). Results were obtained with (closed symbols) and without (open symbols) a magnetic field $H=0.1$. The simulation was performed on an $8\times 8$ lattice, with 20 electrons $(n\approx 0.3)$. (b) Magnetoresistance (defined as $((\rho \left(0\right)-\rho \left(H\right))∕\rho \left(H\right))\times 100$ versus $T$ for the same parameters as in (a). (c) Resistivity versus $T$ for various values of $\lambda$ and with and without quenched disorder, as indicated. The enhancement of the resistivity peak with increasing $\Delta$ is shown. The lattice size and various parameters are as in Figs. 9c, 9d.

Figure 11

(Color online) Results for the two-orbital model at $n=0.1$ and the $\lambda$’s indicated. (a) Resistivity versus temperature using an $8\times 8$ lattice. Note how sharp is the low temperature transition from low to high resistance. (b) Spin-spin correlation at the maximum distance.