Doping dependence of spin and orbital correlations in layered manganites

We investigate the interplay between spin and orbital correlations in monolayer and bilayer manganites using an effective spin-orbital $t\text{−}J$ model which treats explicitly the $e_g$ orbital degrees of freedom coupled to classical $t_{2g}$ spins. Using finite clusters with periodic boundary conditions, the orbital many-body problem is solved by exact diagonalization, either by optimizing spin configuration at zero temperature or by using classical Monte Carlo simulations for the spin subsystem at finite temperature. In undoped two-dimensional clusters, a complementary behavior of orbital and spin correlations is found—the ferromagnetic spin order coexists with alternating orbital order, while the antiferromagnetic spin order, triggered by $t_{2g}$ spin superexchange, coexists with ferro orbital order. With a finite crystal-field term, we introduce a realistic model for ${\mathrm{La}}_{1-x}{\mathrm{Sr}}_{1+x}\mathrm{Mn}{\mathrm{O}}_4$, describing a gradual change from predominantly out-of-plane $3z^2-r^2$ to in-plane $x^2-y^2$ orbital occupation under increasing doping. The present electronic model is sufficient to explain the stability of the CE phase in monolayer manganites at doping $x=0.5$ and also yields the $C$-type antiferromagnetic phase found in ${\mathrm{Nd}}_{1-x}{\mathrm{Sr}}_{1+x}\mathrm{Mn}{\mathrm{O}}_4$ at high doping. Also in bilayer manganites magnetic phases and the accompanying orbital order change with increasing doping. Here the model predicts $C$-AF and $G$-AF phases at high doping $x>0.75$, as found experimentally in ${\mathrm{La}}_{2-2x}{\mathrm{Sr}}_{1+2x}{\mathrm{Mn}}_2{\mathrm{O}}_7$.

Figure 1

(Color online) Schematic spin structures in the magnetic phases of doped manganites. The number of FM bonds decreases gradually from the FM to the $G$-AF phase, through the $A$-AF and $C$-AF phases, with eight and four FM bonds in a cube. For an $ab$ monolayer, the FM and $A$-AF phases are equivalent.

Figure 2

Phase diagram for the undoped layered manganites $(x=0)$, with the stability regions of FM, $G$-AF, and $C$-AF phases (see Fig. 1) in the $(J^{\prime },E_z)$ plane, as obtained at $T=0$ with a $4\times 4$ cluster. The core-spin superexchange $J^{\prime }$ and the crystal-field splitting $E_z$ are given both in units of $J$ and in units of $t$ for $J=0.125t$.

Figure 3

(Color online) Orbital correlations ${\mathcal{T}}_{\theta }\left(\vec{r}\right)$, Eq. (27), in the (01) direction in the FM phase, as obtained for the undoped $4\times 4$ cluster using $T_i^z$ operators defined for different bases of orthogonal orbitals: $\theta =0$—$\{\mid x⟩,\mid z⟩\}$; $\theta =\pi ∕2$—$\{(\mid x⟩\pm \mid z⟩)∕\sqrt{2}\}$, $\theta =2\pi ∕3$—$\{\mid \zeta ⟩,\mid \xi ⟩\}$. The latter choice involves directional orbitals within the $ab$ plane. Orbital correlations shown in the (11) direction for $\theta =\pi ∕2$ demonstrate an almost perfect AO order. Parameters: $E_z=J^{\prime }=0$. Spin correlations obtained for the spin model (i.e., the Heisenberg model for $s=1∕2$) on a $4\times 4$ cluster at $n=1$ are shown for comparison by squares and diamonds.

Figure 4

(Color online) Spin correlations $\mathcal{S}\left(\vec{r}\right)$, Eq. (26), in the undoped monolayer, as obtained in MC simulations with a $4\times 4$ cluster for $J^{\prime }=0$ and $E_z=0$ (FM correlations) and for $J^{\prime }=0$, $E_z=-0.5t$, as well as for $J^{\prime }=0.05t$, $E_z=0$ (both sets give AF correlations). Error bars are smaller than symbol sizes. Parameters: $J=0.125t$, $\beta t=100$.

Figure 5

(Color online) Spin correlations $\mathcal{S}\left(\vec{r}\right)$, Eq. (26), in the undoped monolayer as in Fig. 4, but for decreasing $J^{\prime }$ from $J^{\prime }=0$ (AF correlations) to $J^{\prime }=-0.04t$ (FM correlations). Parameters: $J=0.125t$, $E_z=-0.5t$, $\beta t=100$.

Figure 6

(Color online) Electron density in the $\mid z⟩$ orbital perpendicular to an $ab$ plane of a monolayer manganite for increasing crystal-field splitting $E_z$ [see Eq. (11)], as obtained from MC simulations of undoped $(n=1)$ $\sqrt{8}\times \sqrt{8}$ clusters with various values of $J^{\prime }$. Parameters: $J=0.125t$, $\beta t=100$.

Figure 7

(Color online) Clusters used to investigate the CE phase at half doping $(x=0.5)$: (a) $\sqrt{8}\times \sqrt{8}$, (b) $4\times 4$, and (c) $2\times 8$. Dark (light) shading indicates possible realizations of staggered FM zigzag chains in the CE phase.

Figure 8

(Color online) MC snapshot obtained for doping $x=0.5$, with spin directions indicated by lines. Four $\sqrt{8}\times \sqrt{8}$ clusters are shown (heavy lines indicate the original cluster), and one clearly recognizes the magnetic order in the CE phase. Parameters: $J=0.125t$, $J^{\prime }=0.05t$, $E_z=V=0$, $\beta t=100$.

Figure 9

(Color online) Spin-spin correlation $\mathcal{S}\left(\vec{r}\right)$, Eq. (26), for the ideal CE phase and the MC data for an $\sqrt{8}\times \sqrt{8}$ cluster. There is one AF signal at (1,1). Parameters: $J=0.125t$, $J^{\prime }=0.05t$, $E_z=V=0$, $\beta t=100$.

Figure 10

(Color online) Finite-size extrapolation of the ground-state energy $E_0$ for the $C$-AF and CE phases on chains of different length $L⩽16$, as obtained at $T=0$ for different values of $J$: (a) $J=0.25t$, (b) $J=0.125t$, and (c) $J=0.05t$. The data for the $C$-AF phase alternate between the chains of length $L=4n$ and $L=4n+2$, with $n$ integer. Parameters: $J^{\prime }=E_z=V=0$.

Figure 11

(Color online) Orbital structure for the CE phase at $T=0$, with circles (crosses) for $3x^2-r^2$ $(3y^2-r^2)$ orbitals, as obtained for (a) $E_z=0$, (b) $E_z=0.2t$, (c) $E_z=0.5t$, and (d) $E_z=t$. The size of circles and crosses is proportional to the electron density in a given orbital. Parameters: $J=0.125t$, $V=0$.

Figure 12

(Color online) Possible orbital and spin structure for the CE phase, as suggested (a) in Fig. 1 of Ref. 52 and (b) from our calculations. The orientation of the rectangles indicates the type of occupied orbital—horizontal $z^2-x^2$ or vertical $y^2-z^2$.

Figure 13

Phase diagram at half doping $(x=0.5)$ for (a) $J=0.125t$ and (b) $J=-.25t$, as obtained from finite-size considerations at orbital degeneracy $(E_z=0)$. The black lines around the phase boundaries give their estimated numerical and finite-size errors. At $J∕t=1∕8$ the energies for the $C$-AF and CE phases differ only very little and we could not determine a phase boundary between them. For $J∕t=1∕4$ the CE phase has lower energy than the $C$-AF phase.

Figure 14

(Color online) MC results for the spin-spin correlations $\mathcal{S}\left(\vec{r}\right)$, Eq. (26), for a monolayer, as obtained in $\sqrt{8}\times \sqrt{8}$ cluster with two electrons (at high $x=0.75$ doping) for two parameter sets $J^{\prime }=0.05t$, $V=0$, $E_z=0$ [× for (10) and + for (11) direction] and $J^{\prime }=0.025t$, $V=t$, $E_z=0.25t$ [▵ for the (10) and ▿ for the (11) direction]. Error bars are smaller than symbol sizes. Parameters: $J=0.125t$, $\beta t=100$.

Figure 15

(Color online) Orbital electron densities for increasing doping $x=1-n$ as obtained in MC method for a $\sqrt{8}\times \sqrt{8}$ cluster: (a) percentage of density in $\mid z⟩$ orbitals $n_z∕n$ for a few selected values of the crystal-field parameter $E_z$ and for variable $E_z$ (see below) and (b) absolute densities $n_z$ and $n_x$ in the two orbitals from MC (symbols) compared with the densities found with the Kotliar-Ruckenstein (KR) slave-boson mean-field approximation Ref. 61 (lines) with variable $E_z=(\frac{1}{2}-x)t$. The charge distribution found at doping $x=0.5$ with the KR method ($n_z\simeq 0.14$, $n_x\simeq 0.36$) is indicated by diamonds. Remaining parameters in the MC calculations: $J=0.125t$, $J^{\prime }=0.025t$, $V=t$, $\beta t=100$.

Figure 16

(Color online) Kinetic (approximately parabolic shape) and superexchange (weakly increasing) energies for increasing doping $x=1-n$, as obtained for $\sqrt{8}\times \sqrt{8}$ clusters with various parameter sets: 엯 and ◻, for $J^{\prime }=0$ and $E_z=V=0$; × and +, for $J^{\prime }=0.05t$ and $E_z=V=0$; ▵ and ▿, for $J^{\prime }=0.025t$, $V=t$, and $E_z$ given by Eq. (35). Parameters: $J=0.125t$, $\beta t=100$.

Figure 17

Phase diagram of the undoped bilayer system as obtained in the $(E_z,J^{\prime })$ plane with a $\sqrt{8}\times \sqrt{8}\times 2$ cluster at $T=0$. The phases found between FM and $G$-AF order for two $ab$ planes with interlayer coupling along the $c$ axis are $C$-AF and $A$-AF phases as in Fig. 1, $C^{\prime }$-AF: FM along the $c$ axis and AF within the $ab$ planes, $A^{\prime }$-AF: FM along the $a$ and $c$ axes, and AF along the $b$ axis. The OO which accompanies the FM and $G$-AF phase at $E_z=0$ is also indicated and shown in Figs. 18a, 18b, 18c. Units $J^{\prime }∕t$ and $E_z∕t$ correspond to $J=0.125t$.

Figure 18

(Color online) Orbital correlations ${\mathcal{T}}_{\theta }\left(\vec{r}\right)$ (27) in the (01) direction within one $ab$ plane and between the two different planes of the bilayer, as obtained for the undoped $\sqrt{8}\times \sqrt{8}\times 2$ clusters at $E_z=0$ and $T=0$ in the (a) $G$-AF phase, (b) FM phase, and (c) $A$-AF phase. In each case $T_i^z$ operators are defined by different bases of orthogonal orbitals: $\theta =0$—$\{\mid x⟩,\mid z⟩\}$ and $\theta =\pi ∕2$—$\{\mid x⟩+\mid z⟩,\mid x⟩-\mid z⟩\}$. Parameters: (a) $J^{\prime }=0.05t$; (b) and (c) $J^{\prime }=-0.02t$.

Figure 19

Phase diagram of the undoped $2\times 2\times 2$ cluster as obtained in the $(E_z,J^{\prime })$ plane at $T=0$. The phases and parameters are the same as in the bilayer system (Fig. 17), with the directions of FM bonds in $A$-AF, $A^{\prime }$-AF, and $C^{\prime }$-AF phases distinguished by the symmetry-breaking crystal field $\propto E_z$ [see Eq. (11)]. The $C$-AF phase almost vanishes in this case.

Figure 20

(Color online) Ground-state energy $E$ of various magnetic phases for increasing $t_{2g}$ superexchange $J^{\prime }$ in the half-doped bilayer $\sqrt{8}\times \sqrt{8}\times 2$ cluster, as obtained at $T=0$ for (a) $V=0$ and (b) $V=t$. Different phases with coexisting AF and FM bonds are defined as follows: $C$-AF, FM in the $a$ direction; $A$-AF, AF in the $c$ directions, FM in the $ab$ planes; ${\mathrm{CE}}_1$, FM zigzag chains in the $ab$ planes with FM interlayer coupling; ${\mathrm{CE}}_2$, FM zigzag chains in the $ab$ planes with AF interlayer coupling. Parameters: $J=0.125t$, $E_z=0$.

Figure 21

(Color online) Spin correlations (26) as obtained from MC simulations with a $\sqrt{8}\times \sqrt{8}\times 2$ cluster in the highly doped regime for increasing coordinate $r$ in the $ab$ plane: (a) $G$-AF phase for one electron $(x=15∕16)$ and (b) $C$-AF phase for three electrons. Different symbols and lines indicate different types of neighbors: × and dashed (blue) line, within the $ab$ planes along either the (01) or (10) direction; 엯 and solid (red) line, within the $ab$ planes along the (11) direction; ◻ and dashed (green) line, between the two planes along the $a$ or $b$ direction; ▵ and solid (black) line, between the two planes and along the (11) direction in the $ab$ plane. Parameters: $J=0.125t$, $J^{\prime }=0.05t$, $V=E_z=0$, $\beta t=100$.