Origin of the magnetic and orbital ordering in $\alpha \text{−}{\mathrm{Sr}}_2\mathrm{Cr}{\mathrm{O}}_4$

Motivated by recent experimental progress in transition metal oxides with the ${\mathrm{K}}_2\mathrm{Ni}{\mathrm{F}}_4$ structure, we investigate the magnetic and orbital ordering in $\alpha \text{−}{\mathrm{Sr}}_2\mathrm{Cr}{\mathrm{O}}_4$. Using first-principles calculations, first we derive a three-orbital Hubbard model, which reproduces the ab initio band structure near the Fermi level. The unique reverse splitting of $t_{2g}$ orbitals in $\alpha \text{−}{\mathrm{Sr}}_2\mathrm{Cr}{\mathrm{O}}_4$, with the $3d^2$ electronic configuration for the ${\mathrm{Cr}}^{4+}$ oxidation state, opens up the possibility of orbital ordering in this material. Using real-space Hartree-Fock for multiorbital systems, we constructed the ground-state phase diagram for the two-dimensional compound $\alpha \text{−}{\mathrm{Sr}}_2\mathrm{Cr}{\mathrm{O}}_4$. We found stable ferromagnetic, antiferromagnetic, antiferro-orbital, and staggered orbital stripe ordering in robust regions of the phase diagram. Furthermore, using the density matrix renormalization group method for two-leg ladders with the realistic hopping parameters of $\alpha \text{−}{\mathrm{Sr}}_2\mathrm{Cr}{\mathrm{O}}_4$, we explore magnetic and orbital ordering for experimentally relevant interaction parameters. Again, we find a clear signature of antiferromagnetic spin ordering along with antiferro-orbital ordering at moderate to large Hubbard interaction strength. We also explore the orbital-resolved density of states with Lanczos, predicting insulating behavior for the compound $\alpha \text{−}{\mathrm{Sr}}_2\mathrm{Cr}{\mathrm{O}}_4$, in agreement with experiments. Finally, an intuitive understanding of the results is provided based on a hierarchy between orbitals, with $d_{xy}$ driving the spin order, while electronic repulsion and the effective one dimensionality of the movement within the $d_{xz}$ and $d_{yz}$ orbitals driving the orbital order.

Figure 1

(a) Schematic crystal structure of the canonical cell of ${\mathrm{Sr}}_2\mathrm{Cr}{\mathrm{O}}_4$ with the convention: green = Sr; blue = Cr; and red = O. (b) Schematic crystal structure of the primitive cell of ${\mathrm{Sr}}_2\mathrm{Cr}{\mathrm{O}}_4$. (b) Density of states near the Fermi level for the nonmagnetic state. Black = total; blue = Sr; red = Cr; and green = O. (c) Projected band structure of ${\mathrm{Sr}}_2\mathrm{Cr}{\mathrm{O}}_4$ for the nonmagnetic state. The Fermi level is shown with dashed lines. The weight of each chromium orbital is represented by the size of the circle. The Brillouin zone notation is $\mathrm{\Gamma }=(0,0,0), X=(0,0,\pi /2), X=(0,0,\pi /2), P=(\pi /4,\pi /4,\pi /4), N=(0,\pi /2,0)$, and $Z=(\pi /2,\pi /2,-\pi /2)$.

Figure 2

(a) The original DFT band dispersion for ${\mathrm{Sr}}_2\mathrm{Cr}{\mathrm{O}}_4$ is shown using red solid lines, while the Wannier interpolated band dispersion is presented using green dashed lines. (b) Schematic energy splitting of Cr's $3d$ orbitals with the $d^2$ configuration. (c) Sketch of nearest-neighbor hoppings along the $x$ and $y$ directions, as indicated. (d) Tight binding (TB) band structure for the three $t_{2\mathrm{g}}$ orbitals using the $3\times 3$ nearest-neighbor hopping matrices described in Sec. 3. Note that along $\mathrm{\Gamma }\text{−}P\text{−}X$, the $d_{yz}$ and $d_{xz}$ are identical.

Figure 3

$J_H/U$ vs $U/W$ phase diagram calculated using the Hartree-Fock method for a two-dimensional system. Lattice sizes $12\times 12$ and $16\times 16$ were used. Electronic density is $n=2/3$ i.e. two electrons per site. The notation PM, IC-SDW, FM, AFM, AFO, and SOS stands for paramagnetic, incommensurate spin-density wave, ferromagnetic, antiferromagnetic, antiferro-orbital, and staggered orbital stripe order, respectively. I and M stand for insulating and metallic, respectively.

Figure 4

(a) contains the energies of several states (indicated) varying $U/W$, at fixed $J_H/U=0.2$. In (b), (c), and (d) the real-space values of $\langle {\tau }_i^z\rangle$ are displayed color coded for the COS, SOS, and AFO states, respectively. See color explanation in text. In (e), the average orbital-resolved electronic occupation $\langle n_{\alpha }\rangle$ and average local spin moment $\langle {\mathbf{S}}^2\rangle$ is shown for various values of $U/W$.

Figure 5

(a) Schematic representation of a two-leg ladder with three-orbitals at each site. [(b) and (c)] DMRG phase diagram at fixed $J_H/U=0.2$. (b) contains the spin ordering, where PM stands for paramagnetic phase, IC for incommensurate ordering, and AFM for antiferromagnetic staggered spin ordering. (c) contains the orbital ordering, where RFO stands for rung ferromagnetic orbital ordering, and AFO for antiferro-orbital ordering along both the leg and rung of the ladder.

Figure 6

(a) Real-space spin correlation $S\left(r\right)=\langle {\mathbf{S}}_1·{\mathbf{S}}_j\rangle$ (with $r=|1-j|$), and (b) spin structure factor $S(q_x,\pi )$, for different values of $U/W$, at a fixed Hund coupling $J_H/U=0.2$, and using a $L=2\times 8$ cluster. (c) Schematic of a two-leg ladder, showing the stabilized real-space spin arrangement at large $U$ with the “snake” counting of ladder sites index used in (a) to calculate the distance $r$.

Figure 7

(a) Site-average electron occupancy $n_{\gamma }$ for the three orbitals {$\gamma =1,2,3$} vs $U/W$. (Inset) Site-averaged charge fluctuations $\delta N$ vs $U/W$. (b) Site-average spin structure factor $S(\pi ,\pi )$ vs $U/W$. (Inset) Site-averaged expectation value of the total spin squared vs $U/W$. These results were obtained using DMRG with cluster size $L=2\times 8$ at fixed $J_H/U=0.2$.

Figure 8

(a) Electronic charge occupancy $\langle n_{\gamma ,i}\rangle$ for the three orbitals $\{\gamma =1,2,3\}$ vs site index $i$ at $U/W=6.0$. [(b) and (c)] The orbital-ordering structure factors (b) $T(q_x,\pi )$ (c) $T(q_x,0)$ at $U/W=6.0$ and $J_H/U=0.2$. In (d), there is the schematic representation of the electronic occupancies for the three orbitals $d_{xy}$ (green circles,) $d_{xz}$ (red circles), and $d_{yz}$ (blue circles) at each site of a two-leg ladder system. The colored circles with up or down arrows represent occupied orbitals with the corresponding orientation of the spins, while circles with no arrows denote empty orbitals. The size of the circles does not denote electronic density; it just represents the different orbitals.

Figure 9

Orbital-resolved density of state (DOS) vs $\omega -\mu$ for different values of interaction strengths (a) $U/W=0.5$, (b) 2.0, and (c) 6.0, at fixed $J_H/U=0.2$, using the Lanczos diagonalization method for a small $L=2\times 2$ cluster.

Figure 10

(a) Sketch representing the $d_{xy}$ orbitals, which have the largest hopping amplitude along the $x$ and $y$ directions in the $t_{2g}$ sector. This orbital is always occupied by one electron, thus it develops staggered spin ordering. This AFM order in the $d_{xy}$ sector fixes the AFM order in the rest of the orbitals due to the robust Hund coupling. (b) Schematic representation of the $d_{xz}$ (blue) and $d_{yz}$ (red) orbitals. The AFM spin order is fixed as explained in (a), and here we aim to understand the orbital order. Circles in dark color represent occupied orbitals while light color are empty orbitals. The hopping amplitudes for $d_{xz}$ orbitals along the $x$ direction ($t_{1,1}^x=-0.193$) dominate over the $y$ direction ($t_{1,1}^y=-0.039$). Reciprocally, the hopping amplitudes for the $d_{yz}$ orbitals along $y$ ($t_{2,2}^y=-0.193$) dominate over the $x$ direction hopping ($t_{2,2}^x=-0.039$). Those dominant hoppings are represented by colored dashed lines.

Figure 11

Spin-resolved charge occupancy of the $d_{xz}$ and $d_{yz}$ orbitals vs site index $i$, using DMRG applied to a two-leg ladder of size $L=2\times 8$. (a) shows that $\langle n_{i,1,\uparrow }\rangle >\langle n_{i,1,\downarrow }\rangle$ for the $d_{xz}$ orbitals. (b) shows that $\langle n_{i,2,\uparrow }\rangle <\langle n_{i,2,\downarrow }\rangle$ for the $d_{yz}$ orbitals, in agreement with the qualitative description presented in Sec. 6.

Figure 12

(a) Sketch of FM and G-AFM spin configurations in the 2D square lattice, considered in the DFT calculations. Spin-up and spin-down are indicated by arrows. [(b) and (c)] Cr-projected local DOS corresponding to the spin-up and spin-down Cr atoms in one plane with a G-AFM type magnetic configuration, respectively. The Fermi level is indicated with dashed lines.