Orbital polarons versus itinerant $e_g$ electrons in doped manganites

We study an effective one-dimensional (1-D) orbital $t\text{−}J$ model derived for strongly correlated $e_g$ electrons in doped manganites. The ferromagnetic spin order at half filling is supported by orbital superexchange $\propto J$ which stabilizes orbital order with alternating $x^2-y^2$ and $3z^2-r^2$ orbitals. In a doped system it competes with the kinetic energy $\propto t$. When a single hole is doped to a half-filled chain, its motion is hindered and a localized orbital polaron is formed. An increasing doping generates either separated polarons or phase separation into hole-rich and hole-poor regions, and eventually polarizes the orbitals and gives a metallic phase with occupied $3z^2-r^2$ orbitals. This crossover, investigated by exact diagonalization at zero temperature, is demonstrated both by the behavior of correlation functions and by spectral properties, showing that the orbital chain with Ising superexchange is more classical and thus radically different from the 1-D spin $t\text{−}J$ model. At finite temperature we derive and investigate an effective 1-D orbital model using a combination of exact diagonalization with classical Monte Carlo for spin correlations. A competition between the antiferromagnetic and ferromagnetic spin order was established at half filling, and localized polarons were found for antiferromagnetic interactions at low hole doping. Finally, we clarify that the Jahn-Teller alternating potential stabilizes the orbital order with staggered orbitals, inducing the ferromagnetic spin order, and enhancing the localized features in the excitation spectra. Implications of these findings for colossal magnetoresistance manganites are discussed.

Figure 1

(Color online) Competition between localized holes within an alternating orbital (AO) phase, with staggered localized $\mid x⟩$ (shadded vertical boxes) and itinerant $\mid z⟩$ (empty horizontal boxes) orbitals (a), and itinerant holes in ferro orbital (FO) $\mid z⟩$ phase (b), in a 1-D chain along the $c$ axis. A hole (shaded circle) is confined to a cluster of three sites in the AO state, while it is delocalized over the entire chain in the FO state, as indicated by dashed boxes. Part (c) shows a qualitative phase diagram of the 1-D orbital $t\text{−}J$ model (6), with a critical concentration $x_c$ separating the AO and FO phases, as obtained for $E_{\mathrm{JT}}=0$; and $\Delta =0$ (solid line), $\Delta =t$ (dashed line). Filled (empty) circles in (c) show the AO (FO) states found by exact diagonalization of an $N=14$ chain.

Figure 2

(Color online) Spectral functions $A(k,\omega )$ at half filling for an $N=14$ site chain: (a) orbital model, with hole excitations within an occupied $\mid z⟩∕\mid x⟩$ orbital shown by solid/dashed lines, respectively; (b) the spin $t\text{−}J$ model. Insets show the density of states $n\left(\omega \right)$. Parameters: $J=0.125t$, $E_{\mathrm{JT}}=0$, $h_s=0$. We assumed a finite broadening of the peaks by $\delta =0.1t$.

Figure 3

Ground state energy $E_0$ as a function of the number of itinerant electrons $N_z$ for 0 up to 5 holes (from top to bottom) added to a half-filled $N=14$ orbital chain, for $E_{\mathrm{JT}}=0$, and for (a) $J=0.25t$ and (b) $J=0.125t$.

Figure 4

Polarization of the orbital (filled circles, full lines) and spin (open circles, dashed lines) background $P\left(n\right)$ (8) at distance $n$ from a single hole doped to a half-filled $N=14$ chain, as obtained for the ground state of the orbital/spin $t\text{−}J$ model without staggered fields ($E_{\mathrm{JT}}=0$, $h_s=0$), and for (a) $J=0.25t$ $(N_z=7)$, and (b) $J=0.125t$ $(N_z=8)$.

Figure 5

Orbital correlations $R\left(n\right)$ (9) at distance $n$ from a single hole doped to a half-filled $N=14$ chain (filled circles, full lines), as obtained for the orbital $t\text{−}J$ model for $E_{\mathrm{JT}}=0$ and (a) $J=0.25t$ and (b) $J=0.125t$. Spin correlations obtained in the 1-D $t\text{−}J$ model $(h_Q=0)$ are shown for comparison by open circles and dashed lines.

Figure 6

Orbital correlations $R\left(n\right)$ (9) in a doped $N=14$ site chain (filled circles, full lines), as obtained for the orbital $t\text{−}J$ model at $J=0.125t$, $E_{\mathrm{JT}}=0$ for doping by (a) two holes (insulating phase), and (b) three holes (metallic phase). Spin correlations are as in Fig. 5.

Figure 7

Orbital and spin order (filled and empty circles) around a hole $Z$ (10) as obtained for an $N=14$ chain with increasing $J∕t$ doped by (a) one hole and (b) two holes. Other parameters are as in Fig. 2.

Figure 8

(Color online) Spectral functions $A(k,\omega )$ for $N=14$ chains doped with one hole for $J∕t=0.125$: (a) orbital $t\text{−}J$ model (1) with $N_z=8$ and $N_x=5$—solid and dashed lines for $\mid z⟩$ and $\mid x⟩$ excitations; (b) spin $t\text{−}J$ model (12) with $N_{\uparrow }=6$ and $N_{\downarrow }=7$—solid and dashed lines for hole and electron excitations. Insets show the densities of states $n\left(\omega \right)$. Other parameters and peak broadening as in Fig. 2.

Figure 9

(Color online) Spectral functions $A(k,\omega )$ for $N=14$ chains doped with three holes for $J∕t=0.125$: (a) orbital $t\text{−}J$ model (1) with $N_z=11$ and $N_x=0$, (b) spin $t\text{−}J$ model (12) with $N_{\uparrow }=5$ and $N_{\downarrow }=6$. The meaning of solid (dashed) lines, insets and other parameters are the same as in Fig. 8.

Figure 10

(Color online) Spectral functions $A(k,\omega )$ for an $N=14$ orbital chain doped with three holes for $J∕t=0.25$. Solid and dashed lines for $\mid z⟩$ and $\mid x⟩$ excitations. Inset, other parameters and peak broadening as in Fig. 2.

Figure 11

Evolution of orbital correlations with increasing hole doping, as obtained for $J=0.125t$ and $E_{\mathrm{JT}}=0.25t$: (a) hole-orbital correlations $P\left(n\right)$, and (b) hole-orbital-orbital correlations $R\left(n\right)$. Filled circles, empty circles, filled triangles, and empty triangles for one, two, three, and four holes doped to an $N=14$ orbital chain.

Figure 12

(Color online) Spectral functions $A(k,\omega )$ for an insulating $N=14$ orbital chain for $J∕t=0.125$ and $E_{\mathrm{JT}}=0.25t$, obtained for increasing doping by (a) one hole, and (b) three holes. Solid and dashed lines for $\mid z⟩$ and $\mid x⟩$ excitations. Insets and peak broadening are as in Fig. 2.

Figure 13

(Color online) Spectral functions $A(k,\omega )$ for an $N=14$ spin $t\text{−}J$ chain (12) at doping by one hole, as obtained for $J∕t=0.125$ with a staggered field $h_s=2J$. Solid and dashed lines indicate hole and electron excitations. The inset shows the density of states $n\left(\omega \right)$.

Figure 14

Orbital correlations $⟨T_i^z{T}_{i+1}^z⟩$ (7) at half filling, as obtained at low ($\beta t=100$, open circles), and at intermediate ($\beta t=50$, filled circles) temperature for $N=12$ site orbital chain (20), with $J=0.125t$, and (a) $J^{\prime }=E_{\mathrm{JT}}=0$, (b) $J^{\prime }=0.02t$, $E_{\mathrm{JT}}=0$, and (c) $J^{\prime }=0.02t$, $E_{\mathrm{JT}}=0.25t$. Statistical errors are smaller than the symbol sizes.

Figure 15

(Color online) Spin structure factor $S\left(k\right)$ as obtained for a half-filled $N=12$ orbital chain (20), and for increasing hole doping up to five holes: (a) at low temperature $\beta =100$, and (b) at intermediate temperature $\beta =50$ (in units of $t^{-1}$). Statistical errors are smaller than the symbol sizes. Parameters: $J=0.125t$, $J^{\prime }=0$, $E_{\mathrm{JT}}=0$.

Figure 16

(Color online) Spin structure factor $S\left(k\right)$ for an $N=12$ orbital chain (20) as in Fig. 15, but for $J^{\prime }=0.02t$.

Figure 17

(Color online) Spectral functions $A(k,\omega )$ for a half-filled $N=12$ orbital chain (20) at temperature $\beta t=100$. Solid and dashed lines for $\mid z⟩$ and $\mid x⟩$ excitations. The inset shows the density of states $n\left(\omega \right)$. Parameters: $J=0.125t$, $J^{\prime }=0.02t$, $E_{\mathrm{JT}}=0$.

Figure 18

(Color online) Spectral functions $A(k,\omega )$ as obtained for an $N=12$ orbital chain (20) at temperature $\beta t=100$ doped by (a) two holes and (b) three holes. Solid and dashed lines, inset and parameters as in Fig. 17.

Figure 19

(Color online) Spin structure factor $S\left(k\right)$ as in Fig. 15, but for up to four holes. Parameters: $J=0.125t$, $J^{\prime }=0.02t$, $E_{\mathrm{JT}}=0.25t$.

Figure 20

(Color online) Spectral functions $A(k,\omega )$ for an $N=12$ orbital chain (20), as obtained at temperature $\beta t=100$ for (a) three holes and (b) five holes. Solid/dashed lines and inset are as in Fig. 17. Parameters: $J=0.125t$, $J^{\prime }=0.02t$, $E_{\mathrm{JT}}=0.25t$.