Instabilities and insulator-metal transitions in half-doped manganites induced by magnetic-field and doping

We discuss the phase diagram of the two-orbital model of half-doped manganites by calculating self-consistently the Jahn-Teller (JT) distortion patterns, charge, orbital and magnetic order at zero temperature. We analyze the instabilities of these phases caused by electron or hole doping away from half-doping, or by the application of a magnetic-field. For the CE insulating phase of half-doped manganites, in the intermediate JT coupling regime, we show that there is a competition between canting of spins (which promotes mobile carriers) and polaronic self-trapping of carriers by JT defects. This results in a marked particle-hole asymmetry, with canting winning only on the electron doped side of half-doping. We also show that the CE phase undergoes a first-order transition to a ferromagnetic metallic phase when a magnetic-field is applied, with abrupt changes in the lattice distortion patterns. We discuss the factors that govern the intriguingly small scale of the transition fields. We argue that the ferromagnetic metallic phases involved have two types of charge carriers, localized and bandlike, leading to an effective two-fluid model.

Figure 1

(Color online) A depiction of the CE charge-ordered antiferromagnetic phase (CE-CO). The “bridge sites” (1,3), at the centers of the arms of the zig-zag chains which are ferromagnetically ordered, have alternate occupancies of the $3x^2-r^2$ and $3y^2-r^2$ orbitals. At the “corner sites” (2,4), there is no orbital order, unless JT interactions are present. $\delta$ is the charge disproportionation between the occupancies of the corner sites and the bridge sites.

Figure 2

Phase diagram of the 3D two-orbital model ($T=0$, $x=0.5$, $K∕t=10$). FM (resp. ${\mathrm{FM}}_d$): ferromagnetic metallic phase with no distortions (resp. small uniform distortions). FI-CO (resp. FM-CO): charge-ordered ferromagnetic insulating (resp. metallic) phase with distortions that favour occupancy of the $x^2-y^2$ orbitals (Fig. 3). $A_d$, ferromagnetic planes AF aligned with uniform distortions. $A$-CO, $A$ with charge order. CE-CO, ferromagnetic zig-zag chains AF ordered, charge and orbital ordered $(3x^2-r^2∕3y^2-r^2)$ (Fig. 1). $G$-CO, Néel AF phase with charge-order. Inc.; Possible incommensurate states that interpolate between CE and $G$. Dotted dashed lines come from analytical expressions derived in the strong-coupling limit (Sec. 2D). Solid (dashed) lines show first-order (second-order) phase transitions.

Figure 3

(Color online) A depiction of the ferromagnetic insulating (resp. metallic) charge-ordered phase [FI-CO (resp. FM-CO)] stable at strong JT coupling and small antiferromagnetic coupling (see Fig. 2). Alternate sites have charge disproportionation $\delta$ (given in Fig. 5 as a function of $\mathsf{g}∕t$). The sites with higher occupancies also have strong JT distortions (see Fig. 4), of such orientation as to promote the occupancy only of $(x^2-y^2)$ on these sites, leading to orbital order as well. Note that the lattice is rotated by 45° with respect to Fig. 1.

Figure 4

Amplitudes of distortions of the four inequivalent sites of the unit-cell as function of the coupling parameter $\mathsf{g}∕t$. The different panels represent the various phases found previously: $F,A,\mathrm{CE},G$. The last three phases do not exist for all values of $\mathsf{g}∕t$; the curves are then obtained by fixing the magnetic structure and optimizing with respect to the distortions. For the $G$-CO phase, the distortions are exactly given by $\mathsf{g}∕K$ since the electrons are completely localized.

Figure 5

Charge disproportionation defined by the valence of the two inequivalent Mn ions, ${\mathrm{Mn}}^{3.5+\delta }$ and ${\mathrm{Mn}}^{3.5-\delta }$ (see Figs. 1, 3), as a function of $\mathsf{g}∕t$ [or $E_{\mathrm{JT}}∕t$ (inset) defined by $2E_{\mathrm{JT}}=\mathsf{g}{Q}_{\max }$, where $Q_{\max }$ is the distortion of the site with the largest distortion] for the CE, $F$, and $A$ type-phases.

Figure 6

Charge gap vs $\mathsf{g}∕t$ for the CE, $F$, and $A$ type-phases. The CE phase is always insulating, while the $A$ and $F$ phases are insulating beyond ${\mathsf{g}}_{cA}∕t=5.1$, and ${\mathsf{g}}_{cF}∕t\sim 6.3$. The dotted line is the gap obtained with nonoptimized distortions [Eq. (11) in Sec. 3B].

Figure 7

Phase diagram when $\mathsf{g}∕t⪢1$ at $T=0$ $(x=0.5)$. The phases are the same as in Fig. 2. $J{S}^2={\tilde{t}}^2∕2E_{\mathrm{JT}}$. The CE-CO and $C$-CO phases are degenerate at zero field, but the latter wins at finite fields.

Figure 8

Energy change when a single JT defect is introduced in the FI-CO phase, vs $Q_d$. $Q-Q_d$ is the JT distortion on a defect site; all the other occupied sites having the same distortion $Q$. We see that there is an excitation with $Q_d\sim Q$ that softens when $\mathsf{g}∕t$ decreases. The excitation corresponds to a band particle-hole excitation with the removal of a lattice distortion of one site, while the $Q_d=0$ minimum is the polaron. The softening for ${\mathsf{g}}_c∕t\sim 6.8$ signals a phase transition with proliferation of mobile electrons and defects. Finite-size effects are small and shown for $\mathsf{g}∕t=6.7$ $(N=216,1000,1728)$.

Figure 9

(Color online) Canted CE phase with canting angle $\varphi$ $\left(\mathrm{CE}\text{−}M\text{−}C\right)$. The electrons can now hop from chain to chain. The original 1D zig-zag chains are marked with thicker lines.

Figure 10

(a) Energies of the canted phases as function of the canting angle, for different concentrations ($\mathsf{g}∕t=5$ and $Q=Q_{\mathrm{opt}}=0.35$). For $x<0.5$, there is a finite angle that minimizes the energy. For $x>0.5$, the angle is zero and the CE state is stable. (b) Comparison of the energies of the different phases. The canting angle is chosen such as to minimize the energy as in the top figure.

Figure 11

Band structure of the 1D zig-zag chains (solid lines) with Jahn-Teller distortions $(E_{\mathrm{JT}}∕t=0.875)$, given by Eqs. (8, 9, 10) (Refs. 28, 52, 54). At $x=1∕2$, only the lowest band is filled. Also shown is the splitting of the zero-energy states (dashed lines) when the core spins are tilted away by an angle $\varphi$ (where their dispersion becomes two-dimensional).

Figure 12

Energy gained by canting of the CE phase for a fixed canting angle of $\varphi =0.05$ as a function of doping away from half doping, showing the particle-hole asymmetry.

Figure 13

(Color online) Definition of the defect (a), that is an additional distortion of strength $Q_d$ on a single corner site [$(3z^2-r^2)$ orbital]. Energy levels vs $Q_d$ (b). At $Q_d=0$, there is no defect and the band structure is that of Fig. 11. At $Q_d>0$, bound states appear within the gap. Total excess energy of the system with $N_0+1$ electrons (c). This can be seen as the sum of the energy of the bound state $E_{\mathrm{bs}}$, the elastic energy $E_{\mathrm{elast}}$, and defines the scattering energy $E_{\mathrm{scatt}}$. $\mathsf{g}∕t=5$, $Q=Q_{\mathrm{opt}}=0.35$ $(E_{\mathrm{JT}}=0.875)$.

Figure 14

(Color online) Definition of the defect on a single bridge site with distortion decreased to $Q-Q_d$ ($Q_d=0$ corresponds to a fully distorted site as in the original structure). Energy levels vs $Q_d$ (b) and energies for $N_0-1$ electrons (c). $\mathsf{g}∕t=5$.

Figure 15

Phase diagram as a function of the model parameter $\mathsf{g}∕t$ for hole doping concentration $x$ close to $1∕2$ $(J_{\mathrm{AF}}{S}^2∕t=0.15)$. CE-$M\text{−}C$ denotes a metallic and canted CE phase. CE-$T$ refers to the CE phase with added carriers self-trapped in JT defects (Ref. 58). CE-$M$ is the CE phase with added holes that is metallic (no canting nor trapping of the holes). $C$ is a critical point ending a first-order transition line between two CE-$M\text{−}C$ states with differing canting angles that arise for small $\mathsf{g}∕t$.

Figure 16

First-order transition of the CE phase to a canted state when an external magnetic-field is applied to the system at $\mathsf{g}=0$. The threshold field that induces the first-order transition is given by $\mathrm{g}{\mu }_B{H}_c=0.010t$. The system is metallic beyond $H_c$.

Figure 17

Optimal canting angle vs external magnetic-field for different values of $\mathsf{g}∕t$ (from 0 up to 5) keeping the same distortions as at $H=0$ (CE phase) for all magnetic fields. There are first-order transitions between canted states. As we will show, there are other transitions, involving modified JT distortions, that preempt those shown in this figure (see Fig. 19).

Figure 18

Crossing of the energies of the distorted canted CE phases (solid lines, with $\mathsf{g}∕t=2,3,4,5$) with that of the undistorted highly-canted phase (solid line corresponding to $\mathsf{g}∕t=0$), or the fully ferromagnetic phase (dashed line) to which it merges for large fields. The cusps visible in the $\mathsf{g}∕t=2,3$ curves at higher fields correspond to the first-order transitions described in Fig. 17. But these are preempted by the transitions to the undistorted phase.

Figure 19

Magnetization or canting angle (a) and charge gap (b) vs external magnetic field for different values of $\mathsf{g}∕t=1–5$ $(J_{\mathrm{AF}}{S}^2∕t=0.15)$. There are clear first-order transitions to an undistorted canted metallic or fully ferromagnetic metallic phase.

Figure 20

Magnetization or canting angle (a) vs external magnetic-field for $\mathsf{g}∕t=6$ $(J_{\mathrm{AF}}{S}^2∕t=0.15)$. In the inset the energies of the undistorted canted phase (dotted line), the canted phase with distortions so as to favor $(x^2-y^2)$ orbital ordering (dashed line) and the CE canted phase (solid line) are given. The latter two cross for $\mathrm{g}{\mu }_{B}H∕t$ around 0.43, where the first-order transition takes place. Above this field the phase is fully ferromagnetic. (b): The resulting charge gap is shown as a function of $\mathrm{g}{\mu }_{B}H∕t$.

Figure 21

Energy for creating a single-site defect in the JT distortions with amplitude $Q_d$ for various canting angles in the FI phase. Such an excitation becomes soft at $\varphi =0.95$ and the defects proliferate. The transition corresponds to an instability towards a metallic phase with defects.