Dilution effect in correlated electron systems with orbital degeneracy

Theory of dilution effect in orbital ordered system is presented. The $e_g$ orbital model without spin degree of freedom and the spin-orbital coupled model in a three-dimensional simple-cubic lattice are analyzed by the Monte Carlo simulation and the cluster expansion method. In the $e_g$ orbital model without spin degree of freedom, reduction in the orbital ordering temperature due to dilution is steeper than that in the dilute magnet. This is attributed to a modification of the orbital wave function around vacant sites. In the spin-orbital coupled model, it is found that magnetic structure is changed from the A-type antiferromagnetic order into the ferromagnetic one. Orbital-dependent exchange interaction and a sign change in this interaction around vacant sites bring about this phenomenon. Present results explain the recent experiments in transition-metal compounds with orbital dilution.

Figure 1

(Color online) (a) Schematic of the degenerate PS configurations of the type (i) degeneracy and (b) that of the type (ii) one.

Figure 2

(Color online) (a) System size dependence of the orbital correlation function $M_{\text{OO}}\left(x=0\right)$ and (b) that of the orbital angle function $M_{\text{ang}}\left(x=0\right)$. The minimum energy $E_{\text{min}}$ in the WL method is taken to be $0.95E_{\text{GS}}$ in (a) and $0.98E_{\text{GS}}$ in (b).

Figure 3

(Color online) Scaling plot of the correlation length $\xi \left(x=0\right)$ for the staggered orbital correlation. Numerical data are obtained by the Metropolis algorithm.

Figure 4

(Color online) Impurity concentration dependence of the staggered orbital correlation function $M_{\text{OO}}\left(x\right)$. System size is taken to be $L=18$. Numerical data are obtained by the Metropolis algorithm.

Figure 5

(Color online) (a) System size dependence of the orbital correlation function $M_{\text{OO}}\left(x\right)$ at $x=0.1$ and (b) that at $x=0.2$.

Figure 6

(Color online) (a) System size dependence of the correlation length $\xi \left(x\right)/L$ at $x=0.15$ and (b) that at $x=0.2$. The inset of (a) is the scaling plot for $\xi \left(x\right)$ at $x=0.1$. The OO temperature and the critical exponent at $x=0.15$ are obtained to be $T_{\text{OO}}\left(x\right)=0.248\pm 0.003$ and $\nu =0.755\pm 0.085$, respectively.

Figure 7

(Color online) Impurity concentration $x$ dependence of the OO temperature $T_{\text{OO}}\left(x\right)$. Filled circles are obtained by the MC method. Results from the quantum and classical CE methods are shown by broken lines. For comparison, $x$ dependence of the Néel temperature $T_N\left(x\right)$ in the 3D $XY$ model obtained by the MC method and that in 3D Heisenberg model by the classical CE one are presented by filled triangles and dotted line, respectively. Thick arrow indicates the percolation threshold in a 3D simple-cubic lattice.

Figure 8

(Color online) (a) A snapshot in the MC simulation for the PS configuration at $x=0.1$ and (b) that at $x=0.3$. Filled circles indicate impurities.

Figure 9

(Color online) (a) A schematic PS configuration without impurity and (b) that with an impurity. The filled circle represents an impurity.

Figure 10

(Color online) (a) Total energy $E$ and (b) the A-type AFM correlation function $M_{\text{A-AFM}}\left(x=0\right)$ calculated for several values of the minimum energy $E_{\text{min}}$ in the WL method. System size is chosen to be $L=10$.

Figure 11

(Color online) Temperature dependence of the orbital correlation function $M_{\text{OO}}\left(x=0\right)$, the PS angle function $M_{\text{ang}}\left(x=0\right)$, the A-type AFM one $M_{\text{A-AFM}}\left(x=0\right)$, and the FM one $M_{\text{FM}}\left(x=0\right)$. System size is chosen to be $L=8$.

Figure 12

(Color online) (a) System size dependence of the orbital correlation function $M_{\text{OO}}\left(x=0\right)$ and (b) that of the A-type AFM correlation function $M_{\text{A-AFM}}\left(x=0\right)$.

Figure 13

(Color online) (a) Impurity concentration $x$ dependence of the A-type AFM correlation function $M_{\text{A-AFM}}\left(x\right)$ and (b) that of the FM correlation function $M_{\text{FM}}\left(x\right)$. System size is chosen to be $L=8$.

Figure 14

(Color online) (a) Contour map of $J_i^z$ defined in Eq. (29) and (b) a snapshot of the PS configuration around an impurity in the $xy$ plane obtained in the MC method. The filled circle represents an impurity. Temperature is chosen to be $T/J_2=0.3$.

Figure 15

(Color online) A schematic PS configuration around an impurity at site $i_0$. The filled circle represents an impurity.

Figure 16

(Color online) Impurity concentration dependence of the A-type AFM transition temperature, $T_N\left(x\right)$, and the FM one, $T_c\left(x\right)$, calculated in the realistic parameter values for ${\text{LaMn}}_{1-x}{\text{Ga}}_x{\text{O}}_3$. The parameters and the system size are chosen to be $\left(J_1/J_2,g/J_2\right)=\left(2.5,5\right)$ and $L=8$, respectively.