---
author:
- 'Takuya Sugimoto$^1$, Daiki Ootsuki$^2$, Takashi Mizokawa$^{1,2}$'
title: ' Impact of Local Lattice Disorder on Spin and Orbital Orders in Ca$_{2-x}$Sr$_x$RuO$_4$ '
---

Introduction
============

The layered perovskite Ca$_{2-x}$Sr$_x$RuO$_4$ (CSRO) system has been attracting considerable interest due to the interesting evolution from the spin-triplet superconducting state in Sr$_2$RuO$_4$ to the Mott insulating state in Ca$_2$RuO$_4$ [@Maeno94; @Nakatsuji00a; @Nakatsuji00b; @Imada98]. The structural phase diagram of CSRO exhibits an interesting interplay between titling, rotation, and Jahn-Teller distortion of RuO$_6$ octahedron [@Friedt01]. The magnetic and electronic properties of CSRO strongly correlate with the structural distortions. The Mott transition at $x = 0.0$ (Ca$_2$RuO$_4$) is accompanied by the Ru 4$d$ orbital change due to the Jahn-Teller distortion [@Mizokawa01; @Mizokawa04]. For $x \leq 0.2$, CSRO is an antiferromagnetic insulator at low temperature due to the Jahn-Teller driven compression of RuO$_6$ octahedron along the $c$-axis. For $0.2 \leq x \leq 0.5$, the tilting of RuO$_6$ octahedron provides orthorhombic distortion, and the magnetic susceptibility shows a heavy Fermion behavior at low temperature. The orthorhombic distortion seems to suppress the ferromagnetic and/or small-$q$ antiferromagnetic fluctuation while it still enhances the mass renormalization towards $x= 0.2$. Although the mass enhancement around $x= 0.2$ is claimed to be explained by orbital selective Mott transition [@Anisimov02], existence of orbital selective Mott transition in multi-band Hubbard models depends on the details of parameters in the Hubbard Hamiltonians [@Liebsch03; @Koga04]. It is still controversial whether the orbital selective Mott transition of the multi-band Ru 4$d$ electrons is relevant for the electronic phase diagram of CSRO or not. As for the tiny Sr substitution in the Mott insulating state of Ca$_2$RuO$_4$, the transport properties are dramatically changed by the Sr doping [@Nakatsuji04].

Very recently, a systematic $\mu$SR study has revealed that static antiferromagnetic order exists at low temperature even in the Sr-rich region ($1.5 \leq x \leq 2.0$) of its phase diagram [@Carlo12]. This indicates that the local distortion of RuO$_6$ octahedron introduced by the Ca substitution plays important roles to stabilize the antiferromagnetic state. On the other hand, in the Ca-rich region ($0.0 \leq x \leq 0.5$), the Sr-substitution reduces the magnitude of the Jahn-Teller distortion, tilting, and rotation of the RuO$_6$ octahedron in Ca$_2$RuO$_4$ and, consequently, destroys the spin and orbital orders of Ca$_2$RuO$_4$. In order to gain deeper understandings of the phase diagram, it is important to study the Ru 4$d$ spin-orbital states using a realistic model in which the effects of spin-orbit interaction and lattice distortions are considered.

In this paper, we investigate the Sr or Ca doping effects on the electronic structure of CSRO system at both ends of its phase diagram (which are Sr$_2$RuO$_4$ and Ca$_2$RuO$_4$) by means of unrestricted Hartree-Fock (HF) calculation, which includes the spin-orbit interaction and lattice distortion induced by the chemical substitution.

Method of calculation
=====================

We use the multiband $d$-$p$ model where full degeneracy of the Ru $4d$ orbitals and the O $2p$ orbitals are taken into account. The Hamiltonian is given by $$\begin{aligned}
\hat{\mathrsfs{H}} =& \hat{\mathrsfs{H}}_p + \hat{\mathrsfs{H}}_d + \hat{\mathrsfs{H}}_{pd}  \notag \\
 \hat{\mathrsfs{H}}_p =& \sum_{kl\sigma} \epsilon^p_k p^{\dagger}_{kl\sigma} p_{kl\sigma} + \sum_{kll'\sigma}V^{pp}_{kll'}p^{\dagger}_{kl\sigma} p_{kl'\sigma} + \text{h.c.} \notag \\
\hat{\mathrsfs{H}}_d =& \epsilon^0_d \sum_{i \alpha m \sigma} d^{\dagger}_{i \alpha m \sigma}d_{i \alpha m \sigma} + \sum_{i \alpha mm'\sigma \sigma'}h_{mm'\sigma \sigma'}d^{\dagger}_{i \alpha m \sigma}d_{i \alpha m' \sigma'}    \notag \\
&+ u\sum_{i \alpha m} d^{\dagger}_{i \alpha m \uparrow}d_{i \alpha m \uparrow} d^{\dagger}_{i \alpha m \downarrow}d_{i \alpha m \downarrow}  \notag \\
& +u'\sum_{i \alpha m m'} d^{\dagger}_{i \alpha m \uparrow}d_{i \alpha m \uparrow}  d^{\dagger}_{i \alpha m \downarrow}d_{i \alpha m \downarrow} \notag \\
&+ (u'-j)\sum_{i \alpha mm'\sigma}  d^{\dagger}_{i \alpha m \sigma}d_{i \alpha m \sigma}  d^{\dagger}_{i \alpha m' \sigma}d_{i \alpha m' \sigma}  \notag \\
&+ j \sum_{i \alpha mm'} d^{\dagger}_{i \alpha m \uparrow} d_{i \alpha m' \uparrow} d^{\dagger}_{i \alpha m' \downarrow}d_{i \alpha m \downarrow}  \notag \\
&+ j' \sum_{i \alpha mm'} d^{\dagger}_{i \alpha m \uparrow} d_{i \alpha m' \uparrow}  d^{\dagger}_{i \alpha m \downarrow}d_{i \alpha m' \downarrow} \notag \\
\hat{\mathrsfs{H}}_{pd} =& \sum_{kml\sigma}V^{pd}_{kml}d^{\dagger}_{km\sigma} p_{kl\sigma} + \text{h.c.} \notag \end{aligned}$$

Here, $d^{\dagger}_{i \alpha m \sigma}$ are creation operators for the Ru $4d$ electrons at site $\alpha$ of the $i^{\text{th}}$ unit cell and $d^{\dagger}_{km\sigma}$ and $p^{\dagger}_{kl\sigma}$ are creation operators for Bloch electrons which are constructed from the $m^{\text{th}}$ component of the $4d$ orbitals and from the $l^{\text{th}}$ component of the O $2p$ orbitals, respectively, with wave vector $\bm{k}$. The matrix $h_{mm'\sigma \sigma'}$ denotes the spin-orbit interaction and the effects of crystal field splitting. The magnitude of the spin-orbit interaction for the Ru $4d$ orbital is fixed as 0.15 eV. The transfer integrals between the O $2p$ orbitals $V^{pp}_{kll'}$ are given by Slater-Koster parameters $(pp\sigma)$ and $(pp\pi)$ which are fixed at $0.60$ eV and $-0.15$ eV respectively. The transfer integrals between the Ru $4d$ and O $2p$ orbitals $V^{pd}_{kml}$ are represented by $(pd\pi)$ and $(pd\sigma)$. They are fixed as $(pd\sigma) = -2.8$ eV and $(pd\pi) = 1.26$ eV for the longer in-plane Ru-O bond of Ca$_2$RuO$_4$ whereas $(pd\sigma) = -3.4$ eV and $(pd\pi) = 1.53$ eV for the shorter in-plane Ru-O bond of Sr$_2$RuO$_4$. The summary of the material-dependent parameters are shown in Table \[parameters\]. The tilting of the RuO$_6$ octahedron is included for Ca$_2$RuO$_4$. The distortion parameter $\delta_{\text{JT}}$ is defined as $\delta_{\text{JT}} = d_{\text{apical}}/d_{\text{in-plane}}$ which is the ratio between the apical and in-plane Ru-O bond distances. The distortion parameter $\delta_{\text{JT}}$ is utilized to express the elongation/compression of RuO$_6$ octahedron as in Fig. \[fig:deltaJT\](a). In the Sr-rich (Ca-rich) region, $\delta_{\text{JT}} = 1.07$ (0.95) for the host lattice and $\delta_{\text{JT}} = 0.95$ (1.07) for the locally distorted site. When the RuO$_6$ octahedron is distorted, the transfer integrals are scaled by Harrison’s rule. The intra-atomic Coulomb interactions between Ru $4d$ electrons are given by Kanamori parameters. They are fixed as $u = u'+ j + j' = 3.0$ eV and $j = j' = 0.5$ eV. The charge-transfer energy $\Delta$ (fixed as $-0.4$ eV) is defined by $\epsilon_d -\epsilon_p +4U$, where $\epsilon_d$ and $\epsilon_p$ are the energies of the bare Ru $4d$ and O $2p$ orbitals and $U [=u-(20/9)j]$ is the multiplet-averaged $d$-$d$ Coulomb interaction fixed at 1.89 eV.

We set the $8\times 8$ supercell with periodic boundary conditions and put the Ru $4d$ and O $2p$ electrons on its each site. The local distortion is introduced at one Ru site of the supercell as shown in Figure \[fig:deltaJT\](b). The total number of electron in the supercell is 1792 (256 Ru $4d$ and 1536 O $2p$ electrons). The HF mean-field treatment is applied to the two-body part in $\hat{\mathrsfs{H}}_d$ by replacing its average values, for instance,

$$\begin{aligned}
u \sum_{i \alpha m} & d^{\dagger}_{i \alpha m \uparrow}d_{i \alpha m \uparrow} d^{\dagger}_{i \alpha m \downarrow} d_{i \alpha m \downarrow} \notag \\
\rightarrow &\notag \\
 & u\sum_{i \alpha m}\langle d^{\dagger}_{i \alpha m \uparrow}d_{i \alpha m \uparrow} \rangle d^{\dagger}_{i \alpha m \downarrow}d_{i \alpha m \downarrow}  \notag \\
 &+ u\sum_{i \alpha m} d^{\dagger}_{i \alpha m \uparrow}d_{i \alpha m \uparrow} \langle d^{\dagger}_{i \alpha m \downarrow}d_{i \alpha m \downarrow}\rangle  \notag \\
&-  u\sum_{i \alpha m} \langle d^{\dagger}_{i \alpha m \uparrow}d_{i \alpha m \uparrow} \rangle \langle d^{\dagger}_{i \alpha m \downarrow}d_{i \alpha m \downarrow}\rangle \notag\end{aligned}$$

In this Hartree-Fock calculation, we input the initial values of the order parameters such as $\langle d^{\dagger}d \rangle$ and diagonalize the mean-field Hamiltonian to get a set of eigen functions. Then the order parameters can be calculated using the obtained eigen functions. This self-consistency cycle is iterated until the successive difference of all the order parameters converge less than $10^{-4}$.

Results and Discussion
======================

We calculate the following four cases to see the doping effects. For the Ca-rich region, we calculate the orbital population of Ca$_2$RuO$_4$ and Ca$_{2-x}$Sr$_{x}$RuO$_4$ with a single-site distortion by the tiny Sr doping (let us label these two cases as case (i) and (ii), respectively). For the Sr-rich region, we calculate Sr$_2$RuO$_4$ and Ca$_{x}$Sr$_{2-x}$RuO$_4$ with a single-site distortion by the Ca doping (let us label these two cases as case (iii) and (iv), respectively). Since we set the $8\times8$ supercell, the ratio of one-site transposition is equal to $1/64 \simeq 0.0156$. In terms of the doping amount in this CSRO system, $1.56\%$ is equivalent to $x=0.0312$ in Ca$_{2-x}$Sr$_{x}$RuO$_4$ and Ca$_{x}$Sr$_{2-x}$RuO$_4$, respectively.

Ca-rich region
--------------

The present HF calculation using the reasonable parameter set predicts that Ca$_2$RuO$_4$ is an antiferromagnetic insulator. The compression of the RuO$_6$ octahedra stabilizes the Ru 4$d$ $xy$ orbital compared to the Ru 4$d$ $yz/zx$ orbitals (Jahn-Teller type energy splitting). The combination of the Jahn-Teller type energy splitting and the Ru 4$d$ spin-orbit interaction can provide an interesting magnetic anisotropy. The antiferromagnetic insulating state with the in-plane ($x$ or $y$ axis) spin direction is lower in energy than that with the out-of-plane ($z$ axis) spin direction. The energy difference is 37 meV per Ru site for the present parameter set. The expectation value of $d^{\dagger}d$ for each orbital and spin component of the Ru $4d$ $t_{2g}$ states for Ca$_2$RuO$_4$ \[case (i)\] is shown in Table \[tab:orbpop2\] for the in-plane and out-of-plane antiferromagnetic states. In the out-of-plane case, the complex orbitals with type of $yz \pm izx$ are unoccupied \[namely, occupied by the two Ru 4$d$ $t_{2g}$ holes of the Ru$^{4+}$ ($d^4$) configuration\] while the $xy$ orbitals are almost fully occupied by the Ru 4$d$ $t_{2g}$ electrons. Consequently the orbital angular momentum along the $z$-axis is formed to align the spin moment along the $z$-axis. As for the in-plane case, the complex orbitals with type of $xy \pm izx$ or $xy \pm iyz$ are unoccupied to give orbital angular momentum in the $xy$-plane.

Assuming that one of the RuO$_6$ octahedra is elongated due to the tiny Sr doping in the case (ii), the orbital population of the elongated site would be affected due to the reverse of the Jahn-Teller energy splitting between the $xy$ and $yz/zx$ orbitals. Interestingly, the orbital population of the elongated site is dramatically changed for the in-plane antiferromagnetic state whereas the impact of the elongation is limited for the out-of-plane antiferromagnetic state. As shown in Table \[tab:orbpop2\], the orbital population of the elongated site is almost the same as that of the compressed sites for the out-of-plane antiferromagnetic state. On the other hand, the $xy$ orbitals accommodate more holes at the elongated site for the in-plane antiferromagnetic state. The impact of local reverse of the Jahn-Teller energy splitting strongly depends on the global spin direction due to the strong Ru 4$d$ spin-orbit coupling. Since the in-plane antiferromagnetic state is realized in Ca$_2$RuO$_4$, the present calculation indicates that the local elongation of the RuO$_6$ octahedron produces the Ru 4$d$ $xy$ hole which can affect the global spin and orbital orders of Ca$_2$RuO$_4$. The density of states (DOS) calculated for the cases (i) and (ii) (which are Ca$_2$RuO$_4$ and Ca$_{2-x}$Sr$_{x}$RuO$_4$ at $x=0.0312$) with the in-plane antiferromagnetic states are shown in Figs. \[fig:CaRich\](a) and (b), respectively. One can see that the Mott gap of Ca$_2$RuO$_4$ becomes narrower by the local elongation of RuO$_6$ octahedron ($\delta_{\text{JT}}=1.07$) induced by the tiny Sr doping. The reduction of the Mott gap is related to the local orbital change induced by the reverse of the Jahn-Teller energy splitting.

Sr-rich region
--------------

The present HF calculation using the reasonable parameter set predicts that Sr$_2$RuO$_4$ is a paramagnetic or ferromagnetic metal. For the present parameter set, the ferromagnetic metallic state is slightly lower in energy than the paramagnetic metallic state, and the energy difference is about 0.4 meV per Ru site. In the HF approximation, stability of ferromagnetic or antiferromagnetic states tends to be overestimated. Therefore, the present calculation indicates that the parameter set is reasonable to analyze the paramagnetic metallic state of Sr$_2$RuO$_4$. The expectation value of $d^{\dagger}d$ for each orbital and spin component of the Ru $4d$ $t_{2g}$ states for Sr$_2$RuO$_4$ \[case (iii)\] is shown in Table \[tab:orbpop\] for the paramagnetic metallic state and the ferromagnetic metallic state. In the paramagnetic state, the spin-up and spin-down states have the same population for the $xy$, $yx$, and $zx$ orbitals, respectively. In the ferromagnetic state, the spin polarization for the $yz/zx$ orbitals is larger than that of the $xy$ orbital, indicating that the $yz/zx$ states stabilized by the elongation of the RuO$_6$ octahedra are playing important roles to provide the ferromagnetic interaction. As for the effect of tiny Ca doping to Sr$_2$RuO$_4$ \[case (iv)\], it is assumed that one of the RuO$_6$ octahedra is compressed due to the Ca doping. The local compression of the RuO$_6$ octahedron changes the stability of the paramagnetic state and the nature of the magnetic state. The paramagnetic metallic state becomes unstable and only the ferromagnetic and/or antiferromagnetic states are obtained as stable solutions. The $xy$ orbitals are more occupied at the compressed site to induce antiferromagnetic superexchange interaction. The global antiferromagnetic state is stabilized by the local orbital change. The spin direction is intermediate between the in-plane and out-of-plane directions probably due to the competition between the Ru 4$d$ spin-orbit interaction and the local Jahn-Teller splitting of the Ru 4$d$ levels. In the stable magnetic state, the in-plane components of the Ru 4$d$ spins are ferromagnetically aligned whereas the out-of-plane components are antiferromagnetically arranged. The DOS calculated for the cases (iii) and (iv) (which are Sr$_2$RuO$_4$ and Ca$_{x}$Sr$_{2-x}$RuO$_4$ at $x=0.0312$) are shown in Figs. \[fig:SrRich\](a) and (b), respectively. The local compression of RuO$_6$ octahedron by the tiny Ca doping induces the small band gap at the Fermi level, which can be assigned to the magnetic ordering.

Conclusion
==========

Investigating the electronic structure of CSRO by unrestricted HF calculation, we find the effects of local distortions on spin and orbital orders of the Ru $4d$ $t_{2g}$ states. As for the Ca-rich region, we find that the antiferromagnetism survives when the single-site distortion in Ca$_2$RuO$_4$ does carry on. The orbital state is locally disturbed for the in-plane antiferromagnetic state resulting in the reduction of the Mott gap. As for the Sr-rich region, on the other hand, the single-site distortion in Sr$_2$RuO$_4$ changes the entire system to the ferromagnetic/antiferromagnetic state from the paramagnetic state. The present study captures the trend of effects on Sr$_2$RuO$_4$ by the tiny amount of Ca doping, which could explain the recent $\mu$SR study [@Carlo12].

Acknowledgements {#acknowledgements .unnumbered}
================

The authors would like to thank Profs. Y. J. Uemura and N. L. Saini for fruitful discussions.

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Tables:\
\

                   ($pd\sigma$) for in-plane Ru-O bond   tilting angle   $\delta_{\text{JT}}\;\;\;$
  --------------- ------------------------------------- --------------- ----------------------------
   Sr$_2$RuO$_4$                $-3.4$ eV                   0 deg.                  1.07
   Ca$_2$RuO$_4$                $-2.8$ eV                  12.5 deg.                0.95

  : Parameter sets for Sr$_2$RuO$_4$ and Ca$_2$RuO$_4$.[]{data-label="parameters"}

\
\

        case (i) Ca$_2$RuO$_4$                                                                                                        
  ----------------------------------- -------------- ---------------- -------------- ---------------- -------------- ---------------- --
          in-plane ($xy$) AFM          $xy\uparrow$   $xy\downarrow$   $yz\uparrow$   $yz\downarrow$   $zx\uparrow$   $zx\downarrow$  
                                           0.95            0.95            0.53            0.76            0.52            0.77       
        out-of-plane ($z$) AFM         $xy\uparrow$   $xy\downarrow$   $yz\uparrow$   $yz\downarrow$   $zx\uparrow$   $zx\downarrow$  
                                           0.98            0.98            0.98            0.28            0.99            0.26       
   case (ii) Ca$_{2-x}$Sr$_x$RuO$_4$                                                                                                  
          in-plane ($xy$) AFM          $xy\uparrow$   $xy\downarrow$   $yz\uparrow$   $yz\downarrow$   $zx\uparrow$   $zx\downarrow$  
       compressed site (Ca site)           0.97            0.96            0.64            0.64            0.64            0.64       
          in-plane ($xy$) AFM          $xy\uparrow$   $xy\downarrow$   $yz\uparrow$   $yz\downarrow$   $zx\uparrow$   $zx\downarrow$  
       elongated site (Sr site)            0.66            0.66            0.64            0.64            0.96            0.96       
        out-of-plane ($z$) AFM         $xy\uparrow$   $xy\downarrow$   $yz\uparrow$   $yz\downarrow$   $zx\uparrow$   $zx\downarrow$  
       compressed site (Ca site)           0.98            0.98            0.98            0.28            0.99            0.26       
        out-of-plane ($z$) AFM         $xy\uparrow$   $xy\downarrow$   $yz\uparrow$   $yz\downarrow$   $zx\uparrow$   $zx\downarrow$  
       elongated site (Sr site)            0.99            0.98            0.99            0.28            0.99            0.26       

  : Expectation values of $<d^{\dagger}_{m\sigma}d_{m\sigma}>$ with $m=xy, yz, zx$ and $\sigma =\uparrow, \downarrow$ in the Ru $4d$ $t_{2g}$ states for case (i) and case (ii).[]{data-label="tab:orbpop2"}

\
\

             case (iii) Sr$_2$RuO$_4$                                                                                                            
  ---------------------------------------------- -------------- ---------------- -------------- ---------------- -------------- ---------------- --
                        PM                        $xy\uparrow$   $xy\downarrow$   $yz\uparrow$   $yz\downarrow$   $zx\uparrow$   $zx\downarrow$  
                                                      0.71            0.71            0.88            0.88            0.88            0.88       
                        FM                        $xy\uparrow$   $xy\downarrow$   $yz\uparrow$   $yz\downarrow$   $zx\uparrow$   $zx\downarrow$  
                                                      0.67            0.77            0.80            0.94            0.80            0.94       
   case (iv) Ca$_x$Sr$_{2-x}$RuO$_4$ (Ca-doped)                                                                                                  
                       AFM                        $xy\uparrow$   $xy\downarrow$   $yz\uparrow$   $yz\downarrow$   $zx\uparrow$   $zx\downarrow$  
             elongated site (Sr site)                 0.60            0.85            0.85            0.88            0.86            0.87       
                       AFM                        $xy\uparrow$   $xy\downarrow$   $yz\uparrow$   $yz\downarrow$   $zx\uparrow$   $zx\downarrow$  
            compressed site (Ca site)                 0.79            0.94            0.61            0.89            0.61            0.89       
                        FM                        $xy\uparrow$   $xy\downarrow$   $yz\uparrow$   $yz\downarrow$   $zx\uparrow$   $zx\downarrow$  
             elongated site (Sr site)                 0.67            0.78            0.80            0.94            0.80            0.94       
                        FM                        $xy\uparrow$   $xy\downarrow$   $yz\uparrow$   $yz\downarrow$   $zx\uparrow$   $zx\downarrow$  
            compressed site (Ca site)                 0.88            0.91            0.68            0.81            0.68            0.81       

  : Expectation values of $<d^{\dagger}_{m\sigma}d_{m\sigma}>$ with $m=xy, yz, zx$ and $\sigma =\uparrow, \downarrow$ in the Ru $4d$ $t_{2g}$ states for case (iii) and case (iv).[]{data-label="tab:orbpop"}

Figure captions:\
Figure 1:\
(Color online) Schematic diagram of (a) RuO$_6$ octahedron with the definition of $\delta_{\text{JT}}$ and (b) the supercell with a single-site distortion.\
\
Figure 2:\
(Color online) DOS (calculated with the Ru 4$d$ spin-orbit interaction) of (a) Ca$_2$RuO$_4$ and (b) Ca$_{2-x}$Sr$_{x}$RuO$_4$ at $x=0.0312$.\
\
Figure 3:\
(Color online) DOS (calculated with the Ru 4$d$ spin-orbit interaction) of (a) Sr$_2$RuO$_4$ and (b) Ca$_{x}$Sr$_{2-x}$RuO$_4$ at $x=0.0312$.

![image](Sugimoto_Fig1.eps){width="90.00000%"} \[fig:deltaJT\]

![image](Sugimoto_Fig2.eps){width="90.00000%"} \[fig:CaRich\]

![image](Sugimoto_Fig3.eps){width="90.00000%"} \[fig:SrRich\]
