Effects of charge doping on Mott insulator with strong spin-orbit coupling, ${\mathrm{Ba}}_2{\mathrm{Na}}_{1-x}{\mathrm{Ca}}_x{\mathrm{OsO}}_6$

The effects of doping on the electronic evolution of the Mott insulating state have been extensively studied in efforts to understand mechanisms of emergent quantum phases of materials. The study of these effects becomes ever more intriguing in the presence of entanglement between spin and orbital degrees of freedom. Here, we present a comprehensive investigation of charge doping in the double perovskite ${\mathrm{Ba}}_2{\mathrm{NaOsO}}_6$, a complex Mott insulator where such entanglement plays an important role. We establish that the insulating magnetic ground state evolves from canted antiferromagnet (cAFM) [Lu et al., Nat. Commun. 8, 14407 (2017)] to Néel order for dopant levels exceeding $\approx 10\%$. Furthermore, we determine that a broken local point symmetry (BLPS) phase, precursor to the magnetically ordered state, occupies an extended portion of the ($H-T$) phase diagram with increased doping. This finding reveals that the breaking of the local cubic symmetry is driven by a multipolar order, most likely of the antiferro-quadrupolar type [Khaliullin et al., Phys. Rev. Res. 3, 033163 (2021); Churchill and Kee, Phys. Rev. B 105, 014438 (2022)]. Future dynamical measurements will be instrumental in determination of the precise nature of the identified multipolar order.

Figure 1

Representative zero-field $\mu \mathrm{SR}$ data. (a) Temperature evolution of muon asymmetry with decreasing temperature for x = 0. (b) Asymmetry for all doping concentrations at low temperatures ($T\in [1.6\mathrm{K},5\mathrm{K}]$), well below the magnetic ordering. In this case, the vertical axis are offset for clarity. Solid lines are results of the asymmetry fit to Eq. (2). Blue colors denote a canted antiferromagnetic (cAFM), red an antiferromagnetic (AFM), and green paramagnetic (PM) phase.

Figure 2

Representative temperature dependence of zero-field cooled magnetic susceptibility. (a) High-temperature range of magnetic susceptibility. (b) Low-temperature range of magnetic susceptibility with linear Curie-Weiss fittings in the PM state.

Figure 3

Magnetic state evolution as a function of charge doping ($x$) in ${\mathrm{Ba}}_2{\mathrm{Na}}_{1-\mathrm{x}}{\mathrm{Ca}}_{\mathrm{x}}{\mathrm{OsO}}_6$. (a) Magnetic volume fraction extracted from $\mu \mathrm{SR}$ asymmetry for all doping levels. The magnetic transition temperature is defined as the 90% filling of the magnetic volume and increases monotonically with increasing Ca doping. (b) Magnetization as a function of applied magnetic field at 2 K. The results of high-temperature ($T\gtrsim 50\text{K}$) Curie-Weiss fittings for magnetic susceptibility measurements in the PM state are shown in the inset. (c) ${}^{23}\mathrm{Na}$ NMR spectral linewidth (top) and absolute value of Knight shift (bottom) as a function of doping concentration at various temperatures. (d) Magnetic phase diagram. Markers denote magnetic transition to the canted AFM and collinear AFM state for zero-field $\mu \mathrm{SR}$ and high-field NMR measurements. Solid line serves as a guide to the eye. Typical error bars are on the order of a few percent and not shown for clarity in panels (a)–(c).

Figure 4

Doping evolution of the staggered moment in the low-temperature magnetic state. (a) Schematic of the spin model used to fit the NMR observables. Different colors of the arrows denote different spin environments at the Os sites. The two planes with distinctly oriented moments from sublattice A and B are shown in different shades. (b) Schematic of the canted spin arrangement by angle $\varphi$ with respect to the [110] direction in the XY plane. (c) Simulated and measured NMR spectra linewidth and Knight shift at $T=1.4$ K and $H=11$ T as a function of Ca doping $x$ in ${\mathrm{Ba}}_2{\mathrm{Na}}_{1-\mathrm{x}}{\mathrm{Ca}}_{\mathrm{x}}{\mathrm{OsO}}_6$. (d) Simulated evolution of the staggered moment, defined as the projection of moments from two sublattices, A and B, along the applied field, as a function of doping in the magnetic state. The Gaussian blur, used to properly account for magnetic broadening, of simulated spectra is shown in the inset. The blur increases abruptly for $x=0.9$. This might be related to the increased inhomogeneity of the local magnetic field environment at the Na nuclei site. Details of this simulation can be found in Sec. 2d and Appendix pp4. (e) Simulated evolution of the canted angle, defined as the angle between the sublattice FM spin orientation and the [110] easy axis, as a function of doping in the magnetically ordered state. Typical error bars are on the order of a few percent and not shown for clarity.

Figure 5

NMR spectral evidence of broken local point symmetry. [(a)–(c)] Temperature evolution of ${}^{23}\mathrm{Na}$ spectra for (a) $x=0$, (b) $x=0.125$, and (c) $x=0.9$. [(d)–(f)] ${}^{23}\mathrm{Na}$ powder NMR spectrum simulation results at $H=11$ T for (d) x = 0, (e) $x=0.125$, and (f) $x=0.9$ in the BLPS phase.

Figure 6

NMR shift evidence of broken local point symmetry. Representative NMR Knight shift as a function of temperature at 11 T and 7 T. Arrows indicate the corresponding structural $T_{SB}$ and magnetic $T_N$ transition temperatures. $x=0.9$ (Ca = 90%) data is vertically offset for presentation clarity. Typical error bars are on the order of a few percent and not shown for clarity.

Figure 7

Phase diagram of ${\mathrm{Ba}}_2{\mathrm{Na}}_{1-\mathrm{x}}{\mathrm{Ca}}_{\mathrm{x}}{\mathrm{OsO}}_6$. Solid markers denote the magnetic transition to the canted AFM and collinear AFM state for zero-field $\mu \mathrm{SR}$ and high-field NMR measurements. Open markers denote structural transitions into the BLPS phase. Solid lines serve as a guide to the eye.

Figure 8

Curie-Weiss fittings of magnetic susceptibility. The red lines denote the results of Curie-Weiss (CW) fittings for (a) x = 0, (b) x = 0.25, (c) x = 0.5, (d) x = 0.9, and (e) x = 1.0 in the PM state. The extracted CW temperature (${\theta }_{\mathrm{CW}}$) and effective moment per formula unit (${\mu }_{\mathrm{eff}}$) are displayed in the inset to Fig. 3.

Figure 9

Hyperfine coupling constant determination. (a) ${}^{23}\mathrm{Na}$ Clogston-Jaccarino plot with temperature as implicit parameter for each point. Solid lines denote linear fit to the data, with a fitting range from the highest temperature available to the magnetic transition temperature $T_N$ for each doping concentration. (b) Hyperfine coupling constant extracted from linear fits shown in the top figure.

Figure 10

Schematic graph showing the staggered spin model pattern relative to an arbitrary field direction. The red arrow H represents the direction of external magnetic field. ${\mathrm{S}}_A$ and ${\mathrm{S}}_B$ represent the spin moment orientations on the two sublattices. The angles $\theta$ and $\varphi$ refer to the angles ${\theta }_Q$ and ${\varphi }_Q$, which are the polar angles of the field direction relative to the EFG (and crystal) coordinate.

Figure 11

${}^{23}\mathrm{Na}$ powder NMR spectra simulation results for ${\mathrm{Ba}}_2{\mathrm{Na}}_{(1-x)}{\mathrm{Ca}}_x{\mathrm{OsO}}_6$. Spectra shown at $H=11\mathrm{T}$ in the low-temperature magnetic phase for (a) x = 0, (b) x = 0.125, (c) x = 0.25, (d) x = 0.375, (e) x = 0.50, and (f) x = 0.90. Green lines represent measured NMR spectra, blue lines represent simulated histogram and orange lines represent fitted spectra. The simulation fit parameters of these NMR spectra are displayed in Table 2.