Insulator-to-metal transition induced by disorder in a model for manganites

The physics of manganites appears to be dominated by phase competition among ferromagnetic metallic and charge-ordered antiferromagnetic insulating states. Previous investigations [Burgy et al., Phys. Rev. Lett. 87, 277202 (2001)] have shown that quenched disorder is important to smear the first-order transition between those competing states, and induce nanoscale inhomogeneities that produce the colossal magnetoresistance effect. Recent studies [Motome et al., Phys. Rev. Lett. 91, 167204 (2003)] have provided further evidence that disorder is crucial in the manganite context, unveiling an unexpected insulator-to-metal transition triggered by disorder in a one-orbital model with cooperative phonons. In this paper, a qualitative explanation for this effect is presented. It is argued that the transition occurs for disorder in the form of local random energies. Acting over an insulating states made out of a checkerboard arrangement of charge, with “effective” site energies positive and negative, this form of disorder can produce lattice sites with an effective energy near zero, favorable for the transport of charge. This explanation is based on Monte Carlo simulations and the study of simplified toy models, calculating the density-of-states, cluster conductances using the Landauer formalism, and other observables. A percolative picture emerges. The applicability of these ideas to real manganites is discussed.

Figure 1

Monte Carlo phase diagram of the $J_{\mathrm{H}}=\infty$ one-orbital model as given by the Hamiltonian in Eq. (7), for an $8\times 8$ lattice, and overall density $n=0.5$. Results at the values of disorder indicated are shown. The error bars are representative of the different critical temperatures obtained using different starting configurations, as discussed in the text. The number of quenched disorder configurations used for the averages varies from point to point, but the smoothness of the results indicates that there is no crucial dependence with that number.

Figure 2

(Color online) The criteria used for the determination of the Curie temperature $T_{\mathrm{c}}$: spin-spin correlation function at maximum distance for the cluster studied vs temperature, for $\lambda =0.6$ (a) and $\lambda =1.2$ (b). The $S\left(x_{\max }\right)=0.01$ line is 1% of its maximum value, the criteria used here to determine FM critical temperatures (indicated by the arrows). For $T_{\mathrm{CO}}$, $n(\pi ,\pi )$ vs temperature for $\lambda =1.0$ is shown in (c), for two strengths of disorder $\Delta$, and the approximate critical temperatures of the CO-state are indicated. The figures show results for one disorder configuration.

Figure 3

(Color online) Breakdown of the charge-ordered state with disorder: For small temperatures, $T=0.10$, the CO phase survives the disorder [(a), (c), and (e)]. However, at larger temperatures, $T=0.30$, the initial CO state melts with disorder [(b), (d), and (f)].

Figure 4

(Color online) Density-of-states obtained with Monte Carlo simulations at the values of $\lambda$ and temperature indicated. The lattice is $8\times 8$ in size, and averages over five disorder configurations are presented (the disorder was generated with a bimodal distribution in this example). ${10}^4$ thermalization steps were used, with ${10}^4$ measurements. Values of $\Delta$ are indicated. Similar results were also obtained using $4\times 4$ clusters (not shown).

Figure 5

The first five figures correspond to the DOS obtained with five different quenched-disorder configurations, at the same $\lambda$, lattice size, and temperature used in Fig. 4. A bimodal distribution of disorder was employed and results for $\Delta =0.4$ are shown. All five results are qualitatively similar. The last figure is the average over all five.

Figure 6

(Color online) Density-of-states (averaged over five disorder configurations) using a box distribution (instead of bimodal). ${10}^4$ thermalization steps were used, with ${10}^4$ sweeps for measurements.

Figure 7

(Color online) The “closing” and “re-opening” of the gap in the density-of-states as the disorder strength $\Delta$ is increased, using a bimodal distribution. Temperature, $\lambda$, and $\Delta$’s are indicated.

Figure 8

(Color online) Analog of Fig. 7 but for a box distribution. In this case, the gap that closes at intermediate $\Delta$ does not form again for larger values of the disorder strength.

Figure 9

Geometry used for the calculation of the conductance. The interacting region (cluster) is connected by ideal contacts to semi-infinite ideal leads. The configurations of classical localized spins and phonons are produced by Monte Carlo simulations.

Figure 10

(Color online) Conductance $(e^2∕h)$ vs temperature for the one-orbital model with cooperative phonons at $\lambda =0.7$ and using a $12\times 12$ cluster, in the geometry shown in Fig. 9.

Figure 11

(Color online) Logarithm (in base 10) of resistivity (inverse of conductivity) vs temperature for the same lattice and couplings used in Fig. 10. Note the similarity of the results with those obtained varying magnetic fields near AF-FM first-order transitions (see Ref. 9).

Figure 12

(Color online) Conductivity vs disorder strength for model Eq. (7), at $\lambda =0.7$ and $T=0.05$. 10,000 thermalization and 2000 measurement Monte Carlo sweeps were used. Results for two lattice sizes are shown, suggesting that finite-size effects could be small.

Figure 13

(Color online) Conductivity vs disorder strength $\Delta$, comparing results for the “real” phononic model, Eq. (7) at $\lambda =0.7$ against the “toy” model. In both cases a bimodal distribution is used, and results are shown using a $12\times 12$ lattice at $T=0.05$. In the calculation of the conductance $G$ for the real model, 10,000 sweeps for thermalization and 2000 for measurements were used. For the toy model the curve shows an average over 100 configurations of disorder, and $\alpha$ is 0.5. The different locations of the conductance peaks can be brought to the same value by modifying $\alpha$, an unnecessary task in our qualitative study.

Figure 14

(Color online) Density-of-states of the toy model, Eq. (15), for several values of the disorder strength $\Delta$. The lattice is $16\times 16$ and temperature $T=0.05$. At the Fermi level states are generated, as it occurs in the realistic model studied in previous sections.

Figure 15

(Color online) Conductivity vs disorder strength for the toy model, averaged over 100 disorder configurations at $T=0.05$. Results for several cluster sizes are shown. In 2D, conductivity and conductance are the same. The size effects appear to be mild, but the slight reduction of $G$ with lattice sizes in the central region could still lead to an insulating state in the bulk limit. More work is needed to fully understand this subtle issue.

Figure 16

(Color online) Conductivity (in $e^2∕hL$ units) vs disorder strength for the toy model in 3D, averaged over 100 disorder configurations. $L$ is the size length of a $L^3$ cube.

Figure 17

A cartoonish view of the generation of weight at the Fermi level by disorder. If in the clean limit the system is insulating (with no states available at the Fermi level), the mere broadening of these states by disorder certainly introduces states at $E_{\mathrm{F}}$ after some $\Delta$ strength similar to the gap is reached. However, it remains to be investigated whether the states at the Fermi level generated by disorder are truly metallic, or whether they are insulating.

Figure 18

Monte Carlo “snapshots” obtained using a $12\times 12$ cluster and temperature $T=0.05$ showing the electronic density at each site in tones of black and white. Results are shown for both the “real” and “toy” models [Eqs. (7, 15), respectively]. In the former, $\lambda =0.7$ is used. In the three cases, results for the clean limit $\Delta =0$, as well as intermediate and large values of $\Delta$ are presented. Both the real and toy models at $\Delta =0$ have densities approximately 0.7 and 0.3 in a checkerboard pattern, while at $\Delta =2$ the two clearly-distinguished densities are close to 0.95 and 0.05. A detailed discussion is in the text, but the reader can already visualize the localized nature of the charge in both limits of small and large $\Delta$, while the snapshots at intermediate disorder show a more disordered distribution of populated sites, leading to a metallic state. The similarity between the results for both models is also to be remarked.

Figure 19

An example showing that the sites with densities approximately 0.5 are correlated with those with small effective on-site energy, for the particular case $\Delta =0.8$, and $\alpha =0.5$, on a $24\times 24$ lattice. On the right panel, sites with small effective on-site energy are shown in black if $\Delta -\alpha =0.3$ and light grey if $\Delta -\alpha =-0.3$; the rest are in white. On the left panel, results for the toy model are shown. Here sites with densities between 0.50 and 0.75 are in black, between 0.25 and 0.50 in light grey, and the rest in white. The agreement between left and right is almost perfect. This two-color view helps in the understanding of percolation. In two dimensions the critical percolative fraction is 0.5, precisely the number of sites with small effective on-site energy. The system is then only at the verge of percolation. However, in 3D it would be fully percolated since the critical fraction is 0.25.

Figure 20

(Color online) A histogram of conductances observed in the study of the toy model, corresponding to $\Delta =0.9$, and at $T=0.05$, for many values of the disorder. The conductances can only take integer values; here they have been broadened for a better illustration. Size effects appear to be small. Several configurations are insulating $(G=0)$, while others conducting (although poorly).

Figure 21

(Color online) Density-of-states for the toy model where now the disorder is introduced through random hoppings, rather than random on-site energies. Here, the hopping disorder is again a bimodal function with values $\pm \tau$. The results are at $T=0.05$ and obtained on a $12\times 12$ cluster. The gap does not close under the influence of disorder in the hoppings.

Figure 22

Phase diagram of model Eq. (7) obtained in the limit where the localized spins are frozen in a ferromagnetic state at all temperatures. As in the case of nonfrozen spins, ${10}^4$ thermalization and ${10}^4$ measurement Monte Carlo sweeps were typically used at each temperature and $\lambda$. The lattice was $8\times 8$. The criteria used to determine the transition temperatures are the same as for the simulations with active spins. Results both with and without disorder are shown, with a behavior similar to that found in Fig. 1. The phase diagram at very low temperatures is difficult to obtain due to the rapid increase in CPU time required to remain ergodic (configurations are separated by large energy barriers).

Figure 23

(Color online) Density-of-states obtained using the model with frozen spins, leaving only the charge as an active degree-of-freedom. The clean-limit gap is still found to close with disorder, even in the absence of CO-FM competition. The lattice is $8\times 8$.