Fragility of the $A$-type AF and CE phases of manganites: Insulator-to-metal transition induced by quenched disorder

Using Monte Carlo simulations and the two $e_{\mathrm{g}}$-orbital model for manganites, the stability of the CE and $A$-type antiferromagnetic insulating states is analyzed when quenched disorder in the superexchange $J_{\mathrm{AF}}$ between the $t_{2\mathrm{g}}$ localized spins and in the on-site energies is introduced. At vanishing or small values of the electron-(Jahn-Teller) phonon coupling, the previously hinted “fragility” of these insulating states is studied in detail, focusing on their charge transport properties. This fragility is here found to induce a rapid transition from the insulator to a (poor) metallic state upon the introduction of disorder. A possible qualitative explanation is presented based on the close proximity in energy of ferromagnetic metallic phases, and also on percolative ideas valid at large disorder strength. The scenario is compared with previously discussed insulator-to-metal transitions in other contexts. This particularly severe effect of disorder must be present in other materials as well, in cases involving phases that arise as a compromise between very different tendencies, as it probably occurs with striped states in the cuprates.

Figure 1

(Color online) Schematic representation of the phase diagram corresponding to the model Eq. (1) in three dimensions (Ref. 39), and electronic density $n=0.5$. The influence of quenched disorder on the conductance will be mainly studied in this case in the $\lambda =0$ plane (i.e., without electron-phonon coupling), focusing on the $A$-type and CE phases.

Figure 2

(Color online) Results illustrating the appearance of a metal upon adding quenched disorder to an insulator. Shown is the numerically calculated conductance vs temperature $T$ for the $A$-type AF phase in three dimensions, with the antiferromagnetic spin arrangement along the $z$ axis, with and without disorder. The calculation was done on a $4^3$ lattice, working at $J_{\mathrm{AF}}=0.15$ and $\lambda =0$. For each temperature and $\Delta$, 5 000 thermalization and 15 000 measurement Monte Carlo steps through the whole lattice were carried out. The actual measurements of observables were separated by ten steps. The starting configuration corresponded to an $A$-type AF in the $z$-direction. The link disorder is bimodal, $J_{\mathrm{AF}}\pm \Delta$, and the average shown is over five quenched disorder configurations. For the error related to this average see Fig. 3a.

Figure 3

(Color online) (a) Dependence of the conductance on the particular quenched-disorder configuration used (labeled as 1–5, with “Av.” denoting the average). Although the results near the metal-insulator transition vary substantially from configuration to configuration, the qualitative behavior is similar in all of them. Shown is the conductance vs $T$ for the $A$-type AF phase (AF along the $z$-axis) for different disorder configurations, using $\Delta =0.1$. The calculation was done on a $4^3$ lattice at $J_{\mathrm{AF}}=0.15$ and $\lambda =0$, with the same number of Monte Carlo steps discussed in Fig. 2. The disorder is bimodal, $J_{\mathrm{AF}}\pm \Delta$, and the average shown is over the five disorder configurations. (b) Dependence of the conductance on the disorder strength $\Delta$, at low temperatures, showing the transition from an insulator to a metal upon disordering the system. Shown is the conductance vs $\Delta$ of the $A$-type AF phase (AF along the $z$ axis) at $T=0.02t$. The lattice, couplings, Monte Carlo steps, initial configuration, and type of disorder used are as in (a). The error bars mainly arise from the average over several quenched disorder configurations.

Figure 4

(Color online) Figures illustrating the expected dominance of quenched disorder over the spin configurations, at large $\Delta$. (a) Sign of ${\vec{S}}_i\bullet {\vec{S}}_j$ for nearest-neighbor sites $i,j$, indicated as red/thin (negative sign; antiferromagnetic) and blue/thick (positive sign; ferromagnetic) connections. The result corresponds to one particular quenched disorder and classical spin configurations, obtained after the Monte Carlo thermalization. (b) Value of the link coupling $J_{\mathrm{AF}}(i,j)$, indicated as colors red/thin $(J_{\mathrm{AF}}+\Delta )$ and blue/thick $(J_{\mathrm{AF}}-\Delta )$. In both left and right panels, the couplings were $J_{\mathrm{AF}}=0.15$ and $\Delta =0.1$. The correlation between the results in both panels is apparent, indicating that for this large value of the disorder strength the classical spins follow the disorder in the spin-spin correlations. (a) An example of a FM percolative path found in the study of the top panels. There are many of them (but only one shown).

Figure 5

(Color online) (a) Results illustrating the metal-to-insulator transition induced by disorder, focusing on the regime of the CE phase in a three-dimensional lattice. Shown are the conductance vs $T$ of the CE phase, with and without quenched disorder of strength $\Delta$. The calculation was done on a $4^3$ lattice, at $J_{\mathrm{AF}}=0.20$, $\lambda =0$, and using 5 000 for thermalization and 15 000 for measurement Monte Carlo steps (the latter actually evaluating observables at every ten steps to avoid self-correlations). The initial starting configuration had a CE phase with zigzag chains in the $xy$-plane. The disorder is bimodal, $J_{\mathrm{AF}}\pm \Delta$, and shown are averages over five disorder configurations. (b) Conductance vs $T$ for the FM phase, with and without disorder. In this case, where a metal exists at $\Delta =0$, the disorder simply decreases the conductance, as intuitively expected. The lattice, Monte Carlo steps, number of disorder configurations, and type of link disorder are as in (a). The superexchange coupling is $J_{\mathrm{AF}}$=$0.10$.

Figure 6

(Color online) Schematic phase diagram for model Eq. (1) in two dimensions, from Ref. 18. The effect of quenched disorder is studied in this section in the cases: (i) $\lambda =0$, $J_{\mathrm{AF}}=0.17$, corresponding to a CE phase without active phonons, and (ii) $\lambda =1$, $J_{\mathrm{AF}}=0.145$, corresponding to a CE phase stabilized including phonons. These two cases are indicated by dashed lines.

Figure 7

(Color online) Insulator-to-metal transition induced by quenched disorder in two dimensions. (a) Shown is the conductance vs $T$ of the two-dimensional CE phase, for different disorder strengths $\Delta$. The calculation was done on an $8\times 8$ lattice, at $J_{\mathrm{AF}}=0.17$, and $\lambda =0$. The number of Monte Carlo steps was 5 000 for thermalization and 15 000 for measurements, calculating observables every ten steps. The initial configuration was the perfect CE phase. The disorder is bimodal, $J_{\mathrm{AF}}\pm \Delta$, and the averages shown are over five quenched disorder configurations. (b) Same as (a) but for $J_{\mathrm{AF}}=0.145$ and $\lambda =1.0$.

Figure 8

(Color online) (a) Dependence of the conductance on disorder strength $\Delta$, at low temperature $T=0.01$, using results taken from Fig. 7a. The most dramatic evidence of fragility in the 2D insulating CE state is obtained in the absence of electron-phonon coupling. Inset: Dependence of the conductance on the disorder configuration chosen. The conductance is plotted for each of the 100 disorder configurations. This calculation was done for $J_{\mathrm{AF}}=0.17$ and $\lambda =0$ and for the values of the disorder strength shown. (b) Sign of ${\vec{S}}_i\bullet {\vec{S}}_j$ for nearest-neighbor sites $i,j$, indicated as red/thin (negative; AF) and blue/thick (positive; FM) connections. (c) Value of $J_{\mathrm{AF}}(i,j)$ indicated as two colors red/thin $(J_{\mathrm{AF}}+\Delta )$ and blue/thick $(J_{\mathrm{AF}}-\Delta )$. In both cases, $J_{\mathrm{AF}}=0.17$, $\Delta$=$0.1$, and the results were obtained for fixed classical spin and quenched disorder configurations. The correlation between the quantities in (b) and (c) is apparent, indicating that for this large value of $\Delta$ the system follows the disorder in the spin-spin correlations.

Figure 9

(Color online) Influence of a magnetic field on the transport properties of the 3D $A$-type AF state, as discussed in the text. The effect is particularly dramatic at small $\Delta$, including the clean limit $\Delta =0$. Shown are the conductances vs $T$ of the $A$-type AF phase, at different disorder strengths $\Delta$. Filled symbols (empty symbols) correspond to a magnetic field $B=0.1$ $(B=0.0)$. The calculations were done on a $4^3$ lattice, at $J_{\mathrm{AF}}=0.15$ and $\lambda =0$, and using 5 000 thermalization and 15 000 measurement Monte Carlo steps (calculating the conductance every ten steps). The starting configuration used was an $A$-type AF phase. The disorder is bimodal, $J_{\mathrm{AF}}\pm \Delta$, and shown are averages over five disorder configurations. The magnetic field is applied in the $z$ direction.

Figure 10

(Color online) Magnetoresistance (defined in Sec. 4) at $B=0.1$ vs $T$, for the $A$-type AF regime in 3D, and considering several disorder strengths $\Delta$ (indicated). The lattice, couplings, Monte Carlo steps, and other conventions are as in Fig. 9.

Figure 11

(Color online) (a) Influence of a magnetic field on the transport properties in the regime of the CE phase in three dimensions. Shown are the conductances vs $T$ of the CE-type phase, for different disorder strengths $\Delta$. Filled symbols (empty symbols) correspond to magnetic fields $B=0.1$ $(B=0.0)$. Note the large change at low temperatures, and small and vanishing disorder strength. Lattice, $\lambda$, Monte Carlo steps, number of quenched disorder configurations for averages, and other details are as in Fig. 9. The coupling $J_{\mathrm{AF}}$ is 0.2. (b) Magnetoresistance (defined in Sec. 4) corresponding to the data shown in (a).

Figure 12

(Color online) Magnetization vs applied magnetic field, for various disorder strengths $\Delta$ at $T=0.01$. Inset: The magnetic susceptibility $dM∕dB{\mid }_{B\to 0}$ vs disorder strength $\Delta$. Parameters and other details are as in Fig. 11.

Figure 13

(Color online) (a) Monte Carlo “summed snapshot” showing the “clustered spins” stabilized by a small amount of quenched disorder at $B=0$, for the case of an $8\times 8$ lattice, $J_{\mathrm{AF}}=0.17$, $\lambda =0$, and $\Delta =0.02$. In (a), the presence of FM clusters is clear to the eye: quenched disorder with a small value of $\Delta$ transforms the original CE state into a state with local ferromagnetism (but not global). The conductance of this state is zero. (b) Same as in (a) but for $B=0.02$. A very small magnetic field is sufficient to align the preformed FM clusters and simultaneously increase the conductance (to a value $C=3$, in this case).

Figure 14

(Color online) (a) Conductance vs $T$ of the two-dimensional CE phase for different disorder strengths $\Delta$. Full (empty) symbols correspond to a magnetic field $B=0.1$ $(B=0)$. The calculation was done on an $8\times 8$ lattice at $J_{\mathrm{AF}}=0.17$, $\lambda =0$, and using 5 000 thermalization and 15 000 measurement Monte Carlo steps, and the CE phase as initial configuration. The disorder is bimodal, $J_{\mathrm{AF}}\pm \Delta$, and the average shown is over five quenched disorder configurations. (b) Magnetoresistance at $B=0.1$ vs $T$ for the 2D CE phase, at different disorder strengths $\Delta$, as indicated. The calculation was done using the same lattice, couplings, and other details as in (a).

Figure 15

(Color online) (a) Resistance vs temperature for a $12\times 12$ lattice in the clean limit $\Delta =0$, $\lambda =0$, and considering $J_{\mathrm{AF}}=0.15$ (squares), $J_{\mathrm{AF}}=0.16$ (circles), and $J_{\mathrm{AF}}=0.17$ (triangles). The figure illustrates the metallic character of the $J_{\mathrm{AF}}=0.15$ state (with a FM ground state), and the insulating properties of the $J_{\mathrm{AF}}=0.16$ and 0.17 samples (with a CE ground state). The Monte Carlo simulation was carried out starting with a random state at each temperature, using a warming up of 10 000 steps, followed by an additional 10 000 steps to measure properties. (b) Same as (a) at $J_{\mathrm{AF}}=0.17$, but now considering the on-site disorder strengths ${\Delta }_{\mathrm{P}}$ indicated, and averaging over three different disorder configurations. The resistance of the clean sample is reduced several orders of magnitude by the addition of the on-site potential disorder, similarly as it occurs for link disorder. (c) Same as (a), but on a $4^3$ lattice and with $J_{\mathrm{AF}}=0.20$.

Figure 16

(Color online) (a) Effect of quenched disorder on the two-dimensional CE phase, running the simulation with the TPEM. Conductance vs $T$ of the CE phase on a $20\times 20$ lattice calculated with the TPEM method using $M=30$ moments, and the TPEM parameters ${ϵ}_{\mathrm{pr}}={10}^{-6}$ and ${ϵ}_{\mathrm{tr}}={10}^{-6}$. The coupling was $J_{\mathrm{AF}}=0.17$, density $n=0.5$, and 2000 for thermalization and 1000 for measurement Monte Carlo steps were carried out. The CE phase shows zero conductance at low temperature in the clean limit. This transforms into a disordered metal with finite conductance when link quenched disorder of strength $\Delta =0.05$ is added. (b) Effect of disorder on the three-dimensional CE phase, studied with TPEM. Details are the same as in (a) but on an $8^3$ lattice and at $J_{\mathrm{AF}}=0.20$.

Figure 17

(Color online) Results of Monte Carlo simulations that summarize the main conclusions of this paper (see text for discussion). Shown are the signs of the classical spin correlation ${\vec{S}}_i\bullet {\vec{S}}_j$ considering nearest-neighbor sites $i,j$, indicated as red/thin (negative; AF) and blue/thick (positive; FM) connections. The lattice is $8\times 8$, $J_{\mathrm{AF}}=0.17$, and the disorder strength is (a) $\Delta =0.01$, (b) $\Delta =0.02$, (c) $\Delta =0.08$. In (a), the disorder is too weak to change the CE spin pattern, which is clear in the result. However, in (b) the CE phase is replaced by long ferromagnetic (metallic) chains, which in this case are vertical. These chains provide conductance channels in the vertical direction. An equal number of horizontally conducting systems are found when analyzing many quenched disorder realizations, thus restoring rotational invariance after averaging. In (c), the state is more reminiscent of the high disorder regime, with shorter chains. At large $\Delta$, the spins follow the disorder as explained already in Sec. 3C. Of the three configurations, the largest conductance is obtained in (b).