Nature of lattice distortions in the cubic double perovskite ${\mathrm{Ba}}_2{\mathrm{NaOsO}}_6$

We present detailed calculations of the electric field gradient (EFG) using a point charge approximation in ${\mathrm{Ba}}_2{\mathrm{NaOsO}}_6$, a Mott insulator with strong spin-orbit interaction. Recent ${}^{23}\mathrm{Na}$ nuclear magnetic resonance (NMR) measurements found that the onset of local point symmetry breaking, likely caused by the formation of quadrupolar order [Chen, Pereira, and Balents, Phys. Rev. B 82, 174440 (2010)], precedes the formation of long range magnetic order in this compound [Lu et al., Nat. Commun. 8, 14407 (2017); Liu et al., Physica B 536, 863 (2018)]. An extension of the static ${}^{23}\mathrm{Na}$ NMR measurements as a function of the orientation of a 15 T applied magnetic field at 8 K in the magnetically ordered phase is reported. Broken local cubic symmetry induces a nonspherical electronic charge distribution around the Na site and thus finite EFG, affecting the NMR spectral shape. We combine the spectral analysis as a function of the orientation of the magnetic field with calculations of the EFG to determine the exact microscopic nature of the lattice distortions present in low temperature phases of this material. We establish that orthorhombic distortions, constrained along the cubic axes of the perovskite reference unit cell, of oxygen octahedra surrounding Na nuclei are present in the magnetic phase. Other common types of distortions often observed in oxide structures are considered as well.

Figure 1

Schematic of the energy levels for a $I=3/2$ nucleus in both zero and a finite magnetic field $H$, in the presence of quadrupole interaction with the EFG generated by surrounding electronic charges, and the resulting NMR spectra. In principle, zero applied field and a finite EFG result in a single line at frequency ${\omega }_Q$ that can be observed in a nuclear quadrupole resonance (NQR) experiment. The frequency ${\omega }_Q$ is proportional to the product of nuclear quadrupole moment and the magnitude of the EFG. In a finite applied field and in the absence of quadrupole interaction, the spectrum consists of a single narrow line at frequency ${\omega }_0$. In the presence of a quadrupole interaction that acts as a perturbation to dominant Zeeman Hamiltonian in the depicted case, the central transition remains at frequency ${\omega }_0$. The satellite transitions appear at frequencies shifted by $\pm {\delta }_q$, which are proportional to the magnitude of the EFG [Eq. (2)]. For small values of the EFG, satellite transition cannot be resolved and only line broadening is observed. Strictly speaking, there is also a broadening due to the distribution of magnitude of the EFG itself, but this is manifested only on the satellites and not on the central transition.

Figure 2

Temperature evolution of ${}^{23}\mathrm{Na}$ in ${\mathrm{Ba}}_2{\mathrm{NaOsO}}_6$ spectra at 15 T with magnetic field applied parallel to the [001] crystallographic axis. At 20 K, a narrow single peak spectrum characterizes the high temperature paramagnetic (PM) state. In this state, the crystal structure of ${\mathrm{Ba}}_2{\mathrm{NaOsO}}_6$ is undistorted, as depicted. That is, point symmetry at the Na site is cubic and leads to a vanishing EFG. At intermediate temperatures, broader and more complex spectra reveal the appearance of finite EFG induced by the breaking of local cubic symmetry. In this case, the crystal structure of ${\mathrm{Ba}}_2{\mathrm{NaOsO}}_6$ is distorted so that point symmetry at the Na site is noncubic, inducing a finite EFG. At lower temperature, the splitting into two sets of triplet lines (labeled as I and II) reflects the existence of two distinct magnetic sites in the lattice. Zero of frequency is defined as ${\omega }_0={}^{23}\gamma H$. Splitting between quadrupolar satellites is denoted by ${\delta }_q$. Abbreviation: PM, paramagnetic; BLPS, broken local point symmetry; cFM, canted ferromagnetic.

Figure 3

The mean peak-to-peak splitting $\left({\delta }_q\right)$ between any two adjacent peaks of the triplets I and II as a function of the angle between [001] crystal axis and the applied magnetic field $\left(H\right)$. The red circles denote angular dependence of splitting for $H$ rotated in the $\left(1\overline{1}0\right)$ plane. The red solid line is the fit to $\left|\right(3{cos}^2\theta -1)/2|$, where $\theta$ denotes the angle between the principal axis of the EFG $\left(V_{ZZ}\right)$ and the applied magnetic field. The blue squares denote the angular dependence of splitting when the sample is rotated in the (010) plane. The blue dotted line is the fit to $\left|\right(3{sin}^2\theta -1-\eta {cos}^2\theta )/2|$ [Eq. (14)], where $\theta$ denotes the angle between the $\left(V_{ZZ}\right)$ and $H$, and $\eta$ is the asymmetric parameter as explained in the text.

Figure 4

Schematic of rotation of the applied field in the $\left(1\overline{1}0\right)$ plane. The red arrow denotes the applied field direction. $V_{ZZ}$ is parallel to $c, a$ and $b$ axis in $\left(a\right), \left(b\right)$, and $\left(c\right)$, respectively.

Figure 5

Schematic of rotation of the applied field in the (010) plane. The red arrow denotes the applied field direction. $V_{ZZ}$ is parallel to the $c, a$, and $b$ axis in (a), (b), and (c), respectively.

Figure 6

Crystal structure of ${\mathrm{Ba}}_2{\mathrm{NaOsO}}_6$ deduced from x-ray diffraction at room temperature [22, 23]. Solid lines show the unit cell. Oxygen, osmium, and sodium ions form face centered cubic structure, while barium ions arrange a simple cubic structure. This undistorted double-perovskite structure has $Fm\overline{3}m$ space group.

Figure 7

Schematic of the proposed lattice distortions. (a) One structurally distinct Na site in a noncubic environment is produced by elongation/compression of ${\mathrm{O}}^{2-}$ octahedra. (Model A) (b) Two structurally distinct Na sites are generated by elongation, or compression, of one ${\mathrm{O}}^{2-}$ octahedron along the [001] direction and its concurrent compression, or elongation, in the ($a,b$) plane. (Model B)

Figure 8

Schematic of the rotational lattice distortion generating single Na site. One structurally distinct Na site in noncubic environment is produced by in-plane rotation of ${\mathrm{O}}^{2-}$ octahedra, as depicted by shorter green arrows. (Model C)

Figure 9

Schematic of the rotational lattice distortion combined with elongation, or compression, of two oxygen ions along the [001] direction. One structurally distinct Na site in noncubic environment is produced by this lattice modification, consisting of in-plane rotation (green arrows) and distortion along the $c$ axis (orange arrows) of ${\mathrm{O}}^{2-}$ octahedra. (Model ${\mathrm{C}}_2$)

Figure 10

Schematic of the tilt-lattice distortion. One structurally distinct Na site in noncubic environment is produced by tilt of the $(a,c)$ plane. (Model D)

Figure 11

Schematic of the combined rotational and tilt lattice distortions. One structurally distinct Na site in noncubic environment is induced by concurrent $(a,b)$ plane rotation and $(a,c)$ plane tilt. (Model E)

Figure 12

Schematic of the proposed $\mathrm{GdFeO}{}_3$-type distortion. (a) The red solid line denotes distorted Na-O bond, which in its undistorted state points along the $c$ axis. The $\mathrm{NaO}{}_6$ octahedra is tilted by angle $\theta$ and $\varphi$ in spherical coordinates. (b) Schematic of the specific tilt of $\mathrm{NaO}{}_6$. The blue spheres/atoms represent $\mathrm{O}{}^{2-}$ ions in the undistorted state while the green ones represent distorted ones. We note that $\varphi$ is the angle in the $(a,b)$ plane but the four oxygen ions, originally in the $(a,b)$ plane, are no longer in that plane after the distortion. This corresponds to $a^{+}{a}^{+}{c}^{+}$ distortion in Glazer's notation. (Model F)

Figure 13

Tilting of essentially rigid oxygen octahedra. The schematic of the tilting of $\mathrm{NaO}{}_6$ octahedra with $\theta ={10}^{\circ }$ and $\varphi ={45}^{\circ }$, where $\theta$ and $\varphi$ are standard angles defining spherical coordinates. (Model ${\mathrm{F}}_2$)