Defect states and spin-orbital physics in doped vanadates Y${}_{1-x}$Ca${}_x$VO${}_3$

We present a model for typical charged defects in weakly doped Y${}_{1-x}$Ca${}_x$VO${}_3$ perovskites and study how they influence the magnetic and orbital order. Starting from a multiband Hubbard model, we show that the charge carriers introduced by doping are bound to the Ca defects with large binding energy of $\approx 1$ eV at small doping, and give rise to the in-gap absorption band observed in the optical spectroscopy. The central position of a generic Ca defect with eight equidistant vanadium neighbors implies a partly filled defect band and permits activated transport due to Coulomb disorder. We explore the effect of bound charge carriers on the dynamics of the $\{yz,zx\}$ orbital and spin degrees of freedom in the context of a spin-orbital $t$-$J$ model. After deriving the superexchange interactions around the doped hole, we show that the transition from $G$-type to $C$-type antiferromagnetic (AF) order is triggered by the kinetic energy of doped holes via the double-exchange mechanism. The defect states lead to local modification of orbital correlations within ferromagnetic chains along the $c$ axis; some of them contain hole defects, while the charge-orbital coupling suppresses locally $\{yz,zx\}$ orbital fluctuations in the others. Thereby, Ca defects provide a physical mechanism for spin-orbital dimerization along the ferromagnetic bonds, suggesting that, in the $C$-AF phase of weakly doped Y${}_{1-x}$Ca${}_x$VO${}_3$, dimerization increases with doping.

Figure 1

Schematic view of the lattice of vanadium sites with occupied $t_{2g}$ orbitals and a single hole $h$ introduced by doping a Ca defect $D$ (sphere in the center) in the $C$-type orbital structure of the low-temperature phase in weakly doped Y${}_{1-x}$Ca${}_x$VO${}_3$. For clarity, the Y and O ions and the orbital phases are not displayed. The hole occupies preferentially one of the V sites that form a cube around the Ca impurity due to the attractive Coulomb potential of the impurity. At the undoped V sites, only the topmost occupied $t_{2g}$ orbitals are shown. The $|c\rangle \equiv |xy\rangle$ orbitals occupied at each V ion and residing at lower energy are not shown. Orbital polarization [Eq. (3.12)] distorts the $|a\rangle \equiv |yz\rangle$ ($|b\rangle \equiv |xz\rangle$) $C$-type alternating orbital order in the neighborhood of the defect (dashed box) and favors occupation of one of the two ${\left\{\right|+\rangle }_i{,|-\rangle }_i\}$ rotated orbital states that minimize the orbital-defect interaction (see Sec. 3d).

Figure 2

Partial density of states $N\left(\omega \right)$ (in eV${}^{-1}$ units) for $t_{2g}$ states with $\{yz,zx\}$ symmetry of the Mott insulator YVO${}_3$ obtained by the unrestricted HF calculation. The multiplets represent the occupied lower Hubbard band (LHB) while the unoccupied upper Hubbard band consists of high-spin (HS) and low-spin (LS) multiplet states, respectively. The Mott-Hubbard gap ${\Delta }_{\mathrm{MH}}$ separates the LHB and the HS band. The dashed steplike curve indicates the integrated number of states $f\left(\omega \right)$ [Eq. (2.22)] normalized to one. Parameters: $U=4$, $J_H=0.6$, $t_c=t_a=t_b=0.2$ (all in eV).

Figure 3

Spectral weight of Hubbard subbands per site at hole doping $x$ for (a) the single-band Hubbard model and (c) Hubbard bands of doubly degenerate model with two spin and two orbital flavors. At doping concentration $x$, the number of states $(1-x)$ in each subband of the UHB is determined by the number of filled states in the LHB. Whereas in (a) a hole can be filled either by an up- or a down-spin electron, as shown for a representative electron configuration in (b), in the orbital degenerate case (c), there are four choices to fill up the hole in the atomic limit, i.e., $4x$ empty states in the LHB for a system with $x$ doped holes.

Figure 4

Partial density of states of a doped Mott insulator with degenerate $t_{2g}$ orbitals $\{a,b\}$ at $x=0.02$ doping concentration. Partially filled defect states $D$ are split off from the LHB by the defect potential $V_D=1$ eV. The vertical dashed line indicates the filling of the system at finite doping; $D^{*}$ marks defect states split from the highest LS excitation; further notations and parameters as in Fig. 2.

Figure 5

Partial density of states $N\left(\omega \right)$ of a $t_{2g}$ Mott insulator at $x=0.02$ doping concentration for different values for the defect potential: (a) $V_D=0.1$ eV, (b) $V_D=0.5$ eV, and (c) $V_D=1.0$ eV. The vertical dashed line indicates the chemical potential $\mu$, which is pinned by the defect states $D$. The binding energy $E_B$ of the defect states measured from the edge of the LHB grows with increasing $V_D$. An alternative measure is the defect energy $E_D$ ($E_D^{\prime }$) relative to the center of the LHB (HS band), respectively. Parameters: $U=4$, $J_H=0.6$, $t_c=t_a=t_b=0.2$ (all in eV).

Figure 6

Energy levels of the $t_{2g}$ orbital states $\gamma =a,b,c$ for a representative V${}^{3+}$ ion in YVO${}_3$ (left), and their modification in the vicinity of a Ca defect by the impurity potential $H_{\mathrm{imp}}$ (middle) and after adding the charge-orbital interaction $H_D$ (right). Two orbitals $\alpha =1$ and 2 are occupied at V${}^{3+}$ ions in the vicinity of the defect site $D$, as indicated by electron spins (arrows) that contribute to spin $S=1$. Occupied $t_{2g}$ orbitals $\alpha =2$ with intermediate energy are shown in the inset; the orbitals are linear combinations of $\left\{\right|a\rangle ,|b\rangle \}$ orbitals ${|\pm \rangle }_i$, defined in Eq. (3.11). Orbital phases and the $c$ orbitals occupied at each site are not shown for clarity. In the doped case one of the occupied orbitals $\alpha =2$ is replaced by a hole in the vicinity of the defect $D$ (see Fig. 1).

Figure 7

Schematic view of a section of an orbital chain parallel to the $c$ axis. A Ca defect $D$ in the vicinity of the chain is shown in the top part, close to the bond $\langle 12\rangle$, and marked by dark filled circles; more distant nonequivalent sites are labeled as $i=3,...,6$. The orbital chains correspond to (a) the $G$-AF phase with $a$ electrons in $|a\rangle$ states (dark semicircles) corresponding to $C$-AO order along the chain (here sites $i=1$ and $i=2$ are equivalent); (b) the $C$-AF phase with $G$-AO order represented by alternating $\left\{\right|a\rangle ,|b\rangle \}$ occupied orbital states along the chain (dark semicircles); (c) the $C$-AF phase with $G$-AO order of $\{a,b\}$ orbitals as in (b) and a hole $h$ at site $i=1$ near the defect $D$. The orbitals on two top sites $i=1,2$ belong to the V${}_8$ cube around Ca defect $D$ (see Fig. 6), and are modified toward the $|+\rangle$ orbital states by the increasing orbital polarization interaction $D$ [see Eq. (3.12)]. The hole in (c) is confined to sites $i=1,2$ by the trapping Coulomb potential $E_D$. Away from the defect $D$, orbital correlations in (a) are (almost) uniform and support AF spin coupling, while in (b) and (c) they generate instead FM interactions that alternate between strong ($s$) and weak ($w$) exchange bonds.

Figure 8

On-site orbital expectation values for the $G$-AF phase for increasing charge-orbital interaction $D$ [see Eq. (3.12)]: (a) orbital polarization $\langle {\tau }_i^x\rangle$ and (b) orbital order parameter $\langle {\tau }_i^z\rangle$. Labels of different curves indicate nonequivalent sites $i=1,3,4,5$ in a chain of $L_z=8$ sites [see Fig. 7a]. Parameters: $V_{ab}=0.2J$, $V_c=0.7J$.

Figure 9

On-site orbital expectation values obtained for the $C$-AF phase as a function of increasing charge-orbital interaction $D$ [see Eq. (3.12)], obtained for a chain of $L_z=8$ sites: (a) orbital polarization $\langle {\tau }_i^x\rangle$ and (b) orbital order parameter $\langle {\tau }_i^z\rangle$. Dotted line in (a) shows orbital polarization $\langle {\tau }_i^x\rangle$ for the free chain (at $h_{ab}=0$). Labels of different curves and parameters as in Fig. 8.

Figure 10

Orbital correlation functions $\langle {\vec{\tau }}_i·{\vec{\tau }}_{i+1}\rangle$ found for the orbital chain with $L_z=8$ along the $c$ axis in the $C$-AF phase shown in Fig. 7b: (a) self-consistent calculation with the MF term $\propto h_C$ [Eq. (4.3)] acting on the chain from the neighboring sites in the $ab$ planes; (b) free orbital chain ($h_C=0$). Parameters: $V_{ab}=0.2J$, $V_c=0.7J$, $t=6.25J$.

Figure 11

Schematic view of a double exchange process. The hole hopping along the bond $\langle 12\rangle$ that belongs to the cube surrounding a Ca defect for the (a) $G$-AF phase and (b) $C$-AF phase. In the initial state $|i\rangle$, two spins $s=1/2$ due to electrons that occupy $|c\rangle$ and $|a\rangle$ orbitals form a high-spin $S=1$ state at $d^2$ site $i=1$. The $d^1$ site $i=2$ is occupied by the hole (filled circle) and has only a single $s=1/2$ spin due to the electron occupying the $|c\rangle$ orbital. In the $G$-AF phase shown in (a), a hopping process of the electron with $a$ orbital flavor $\propto t{a}_1^{†}{a}_2$ generates a finite canting angle $\theta$ for both spins with respect to their original orientation ($\theta =0$), which leads to a finite kinetic energy gain $\propto t\sin \theta$ by the transition to the final $|f\rangle$ high-spin $S=1$ state. In contrast, full hopping amplitude $t$ contributes along FM bonds in the $C$-AF phase [see (b)].

Figure 12

Total energy per site ${\mathcal{E}}_G$ calculated for the $G$-AF state with $C$-AO orbital order as a function of tilt angle $\sin \theta$, as obtained by exact diagonalization of an orbital chain of $L_z=8$ sites with $D=2J$ (solid line) and $D=0$ (dashed line). Parameters: $V_{ab}=0.2J$, $V_c=0.7J$, $t=6.25J$.

Figure 13

Orbital order parameter $\langle {\tau }_i^z\rangle$ in the $C$-AF phase around the charge defect neighboring with the bond $\langle 12\rangle$ shown in Fig. 7c, obtained for an embedded orbital chain with finite field $h_G$ for increasing orbital polarization interaction $D$ [see Eq. (3.12)]. Parameters: $V_{ab}=0.2J$, $V_c=0.7J$, and $t=6.25J$.

Figure 14

Alternation of the bond orbital correlations $\langle {\vec{\tau }}_i·{\vec{\tau }}_{i+1}\rangle$ as obtained for the $C$-AF phase in the vicinity of the charge defect [see Fig. 7c] as a function of orbital polarization interaction $D$ [Eq. (3.12)]. The correlation functions, which alternate between strong ($\langle 34\rangle$ and $\langle 56\rangle$, long-dashed lines) and weak ($\langle 23\rangle$, solid lines, and $\langle 45\rangle$, dashed lines) bonds, are obtained for an orbital chain with $L_z=8$ sites: (a) embedded with finite $h_C$ ($V_{ab}=0.2J$), and (b) free with $h_C=0$. Other parameters: $V_c=0.7J$ and $t=6.25J$.

Figure 15

Dimerization in the $C$-AF phase induced by charge defects as function of the orbital polarization interaction $D$ [Eq. (3.12)]: (a) orbital correlations $\langle {\vec{\tau }}_i·{\vec{\tau }}_{i+1}\rangle$ on strong ($i=3,5$, solid lines) and weak ($i=2,4$, dashed lines) bonds, and (b) magnetic exchange constants for strong ($J_{cs}$, solid line) and weak ($J_{cw}$, dashed line) FM bonds along the $c$ axis. Dotted lines indicate the average values in both cases. Parameters: $V_{ab}=0.2J$, $V_c=0.7J$, $t=6.25J$, $D=3J$.

Figure 16

Energies of two competing magnetic phases as a function of doping concentration $x$: $G$-AF phase (dashed line) and $C$-AF phase (solid line) obtained from Eqs. (6.8) and (6.9), respectively. Parameters: $V_{ab}=0.2J$, $V_c=0.7J$, $t=6.25J$, $E_C\left(0\right)=0.03J$.

Figure 17

Artist’s view of the virtual charge excitations $d_i^1{d}_j^2\to d_i^2{d}_j^1\to d_i^1{d}_j^2$ along a bond $\langle ij\rangle \in \mathcal{C}$ parallel to the $c$ axis that contribute to the AF superexchange between V${}^{4+}$ and V${}^{3+}$ ions in cubic vanadates. After a hopping of $a$ electron from state (a) to the excited state (b), the excitation energy ${\varepsilon }_1=2J_H$ arises. The excitation (b) may decay in two ways: (c) either the initial configuration is restored or (d) spin flips take place at both sites.

Figure 18

Artist’s view of the virtual charge excitations $d_i^1{d}_j^2\to d_i^2{d}_j^1\to d_i^1{d}_j^2$ along a bond $\langle ij\rangle \in \mathcal{C}$ along the $b$ axis, which contribute to the AF superexchange (a) between V${}^{4+}$ and V${}^{3+}$ ions in cubic vanadates. When the $a$ electron exchanges with a hole, the excited state (b) with energy ${\varepsilon }_1=2J_H$ arises: it leads to two ground-state configurations, either (c) without or (d) with spin flip. The hopping of the $c$ electron creates a double occupancy (e) with the excitation energy ${\varepsilon }_1=5J_H$; it gives again two ground-state configurations, either (f) without or (g) with spin flip.