Effects of charge doping on Mott insulator with strong spin-orbit coupling, Ba₂Na_1-xCa_xOsO₆

Introduction

Intricate interplay between strong electron correlations, and intertwined spin and orbital degrees of freedom leads to many diverse complex quantum phases of matter witczak2014correlated ; jackeli09mott ; PhysRevResearch.3.043016 ; doi:10.7566/JPSJ.90.062001 . Often, correlations of the spin and orbital degrees of freedom can be treated on distinct energy scales. However, this is not the case in systems containing $5d$ transition-metal ions, where spin-orbit coupling (SOC) and electron correlations are comparable in size Chen_PRB_2010 ; Chen:2011 ; svoboda2021orbital ; jackeli09mott ; MonteCarlo_2014magnetism ; cong2021monte ; PhysRevX.11.011013 ; PhysRevResearch.4.013062 . As a result, $5d$ compounds exhibit a wide range of exotic magnetic properties, structural distortions, and multipolar ordering witczak2014correlated ; Lu_NatureComm_2017 ; PhysRevB.97.054431 ; PhysRevMaterials.5.104410 ; paramekanti2020octupolar ; voleti2020multipolar ; churchill2021 ; PhysRevResearch.2.022063 ; PhysRevB.100.214113 ; maharaj2020octupolar . The underlying physical ground state, is controlled by the multiplet structure of the constituent ions, the nature of the chemical bonds in the crystal, and its symmetry. This complexity often leads to intricate quantum “hidden” orders, elusive to most standard experimental probes. Nevertheless, the structural, magnetic, and electronic properties can be finely tuned by altering the degeneracy of a multitude of ground states varying external perturbations, such as pressure, strain and doping PhysRevB.100.214113 ; voleti2021octupolar ; kesavan2020doping ; 2022polaron .

Results

Magnetic state - $\mu$SR magnetic volume fraction. First, we consider the evolution of the magnetic ground state of Ba₂Na_1-xCa_xOsO₆ as a function of the Na/Ca substitution $(0\leqslant x\leqslant 1)$ as probed by zero field muon spin relaxation (ZF-$\mu$SR) measurements. In the absence of an external field ($H=0\;\textrm{T}$), the spin $I=1/2$ muon implanted in the sample precesses around the spontaneous local magnetic field arising from the magnetically ordered state at the muon site. The muon precessions are reflected in damped oscillations of the muon asymmetry decay, probing the fraction of precessing muons, which in turn is proportional to the magnetic volume (see methods). Our ZF-muSR asymmetry measurements for the end members $x=0$ and $1$ are in agreement with those previously reported in Refs. steele2011low ; lovesey2020lone . In Fig. 1a, we plot the temperature evolution of the magnetic volume fraction $V_{mag}$ as a function of doping. We find that samples of all concentrations display a magnetic transition as the volume fraction approaches 100% in the low temperature limit. The transition temperature into a magnetically ordered state, $T_{N}^{\mu}$, defined to be at $V_{mag}=90\%$, grows monotonically from approximately $T=5\;\textrm{K}$ to 40 K as increasing doping induces a configuration change from the $5d^{1}$ to the $5d^{2}$ kesavan2020doping , as illustrated in Fig. 1d.

Magnetic state - magnetization. We have also performed magnetization measurements to get a better insight into the nature of the magnetic transitions observed through ZF-$\mu$SR. In Fig. 1b, we plot magnetization curves as a function of an applied magnetic field for $T=2\;\textrm{K}$. The $x=0$ sample displays a non-linear field dependence with a characteristic S shape and a small hysteretic behavior consistent with a moderately weak ferromagnetic character due to the significant moment canting in the cAFM phase. This is in agreement with the magnetization behavior observed in Ba₂NaOsO₆ single crystals erickson2007ferromagnetism . This hysteretic behavior is rapidly suppressed with charge doping and is effectively undetectable for doping exceeding $x\sim 0.1$. Magnetic susceptibility measurements were performed as a function of temperature at $H=0.1\;\textrm{T}$ for all samples. The resulting magnetic susceptibility in the high temperature paramagnetic (PM) region fits well to a Curie-Weiss (CW) function plus a small temperature independent contribution. The resulting CW temperature ($\theta_{CW}$) and effective moment per formula ($\mu_{eff}$) are displayed in the inset to Fig. 1b. The values of the end members are in very good agreement with those previously reported for Ba₂Na_1-xCa_xOsO₆ at $x=0$ $(\mu_{\rm{eff}}=0.6\mu_{B})$ and $x=1$ $(\mu_{\rm{eff}}=1.6\mu_{B})$stitzer2002crystal ; erickson2007ferromagnetism ; yamamura2006structural ; thompson2014long . Furthermore, the extracted effective moments increase smoothly as the system evolves from the $5d^{1}$ to $5d^{2}$ configuration, while $\theta_{CW}$ becomes more negative. The extracted effective moment for both configurations is significantly suppressed from the theoretical value expected if SOC were negligible ($\mu_{\rm{eff}}=1.73\,\mu_{B}$ and $\mu_{\rm{eff}}=2.83\,\mu_{B}$ for $5d^{1}$ to $5d^{2}$ configurations, respectively) and is closer to the expected moments in the infinite SOC limit ($\mu_{\rm{eff}}=0\mu_{B}$ and $\mu_{\rm{eff}}=1.25\mu_{B}$ for $5d^{1}$ and $5d^{2}$ configurations, respectively) Chen_PRB_2010 ; PhysRevLett.118.217202 ; Chen:2011 . The experimentally determined values of $\mu_{\rm{eff}}$ being significantly reduced from those expected for negligible SOC limit indicates the presence of strong SOC (in agreement with predicted SOC coupling lambda $\sim 0.3\,\rm{eV}$), while their being larger than those in the limit of infinite SOC can be attributed to the hybridization of Os $d$ and oxygen $p$ orbitals, with extra moments coming from the $p$ orbitals, in agreement with predictions of Ref. PhysRevB.106.035127 . The most likely origin of the observed effective moment increase with doping $(5d^{1}\rightarrow 5d^{2})$ is the enhancement of the spin quantum number. This is because the spin $(S)$ is predicted to increase from $S=1/2$ to $S=1$ as doping changes from $x=0\rightarrow x=1$ PhysRevB.106.035127 .

Magnetic state - NMR. In order to obtain insight into the microscopic nature of the magnetically ordered state throughout the doping evolution, we performed detailed analysis of the ²³Na NMR spectra (see Methods). That is, we analyzed how the spectral shape changes in the magnetically ordered state, i.e. measured below $T_{N}$, as a function of the Ca content $x$. In Fig. 1c, we plot the absolute value of the Knight shift, deduced from first moment of the NMR spectra, and the linewidth of the spectra as a function of $x$. The magnitude of the Knight shift rapidly decreased on introduction of Ca dopants ($x>0$) and remains nearly constant for higher doping ($x\geqslant 0.1$). The absolute value of the shift is a measure of the local magnetic moment projected along the external magnetic field direction. In the magnetically ordered canted state, the shift is proportional to the projection of the non-compensated magnetic moment along the applied magnetic field, as was demonstrated by ²³Na NMR measurements on Ba₂NaOsO₆ single crystals Lu_NatureComm_2017 ; Liu_Physica_2018 . Therefore, the observed abrupt decrease of the shift upon the introduction of dopants indicates that the addition of charge effectively quenches the FM component, while further doping ($x\geqslant 0.1$) leads to a more uniform distribution of the projected moments. Furthermore upon the introduction of dopants ($x>0$), the linewidth, which reflects the distribution of the magnetic fields projected along the applied magnetic field, exhibits the same abrupt decrease as the NMR shift, in agreement with the suppression of the canted nature of the magnetic order upon charge doping. In the collinear AFM state, the spectral linewidth is qualitatively proportional to the size of the local ordered magnetic moment. Thus, the smooth increase of the linewidth observed for $x>0.1$ in Fig. 1c, indicates a progressive rise of the ordered magnetic moment as the charge concentration approaches $x=1$, i.e. the $5d^{2}$ configuration (full quantitative analysis of the linewidth is presented in the next section). We point out that one would naively expect that the linewidth monotonically increase with doping since it introduces inhomogeneity in the crystal, which is not what was observed here because the NMR linewidth reflects intrinsic inhomogeneities of the magnetic ground state. Both the magnetization and the low temperature NMR measurements are consistent with a picture where the canted magnetic state rapidly evolves into the collinear AFM phase upon the introduction of charge and that the ordered magnetic moment increases as a function of Ca doping.

Microscopic nature of the magnetic state. To investigate the direct effect of doping on the nature of the staggered magnetic moments, i.e. a two sublattice canted antiferromagnetic order observed in Ba₂NaOsO₆ single crystal Lu_NatureComm_2017 , we have simulated the powder spectrum corresponding to the two sublattice cAF order identified in Ref. Lu_NatureComm_2017 (Methods). The input parameters for the simulations include the electronic spin moment ($\vec{S}$) and the hyperfine coupling tensor ($\mathbb{A}$). For each Ca doping concentration, we fix the value of $\vec{S}$ to be that of the effective moment ($\vec{S}=\mu_{{\rm eff}}$) deduced from susceptibility measurements (the inset to Fig. 1b). Simulated spectra are then fitted to the observed ones with canting angle, and consequently staggered moment, as a fitting parameter. Here, the canting angle is that between the sub-lattice FM spin orientation and the [110] easy axis, while the staggered moment refers to the projection of the moments from the two sub-lattices, A and B, along the applied field direction. The simulation results are summarized in Fig. 2. We find that both, the canting angle and the staggered moment, change abruptly upon charge doping for $x>0.1$. For $x=0$, we obtain the best fit for a canting angle of $68\degree$, consistent with that reported for pure Ba₂NaOsO₆ single crystal Lu_NatureComm_2017 . For $x>0.1$, the canting angle rapidly approaches $90\degree$, the value associated with a collinear AFM state (see Fig. 2d). The deduced canting angles and staggered moments serve as input parameters to calculate the doping evolution of the Knight shift and linewidth associated with the local spin arrangement depicted in Fig. 2a. The calculated evolution of the shift and the linewidth is in an excellent agreement with observations, as shown in Fig. 2c. As $x$ increases, the Knight shift decreases as a direct consequence of the increase of the canting angle, as depicted in Fig. 2b. On the other hand, the linewidth decrease with increasing $x$ is associated with the weakening of the off-diagonal components of the hyperfine coupling tensor, for the canting angle $\sim 90\degree$. Furthermore in the inset to Fig. 2d we plot the Gaussian blur, which is used to account for the inhomogeneous magnetic broadening of the measured spectra, and its abrupt increase with doping approaching $x\rightarrow 1$ indicates that injected charge is inhomogeneously distributed.

Broken local point symmetry (BLPS). Next, we investigate the effect of charge doping on the BLPS phase Lu_NatureComm_2017 . The onset temperature of the visible broken local point symmetry is determined from the NMR Knight shift. We then proceed to analyze NMR powder spectral shapes at temperatures above the transition to the magnetically ordered state to determine the nature of the local crystal symmetry.

Nature of the broken symmetry. In Fig. 3a, we plot ²³Na NMR spectra of Ba₂Na_1-xCa_xOsO₆ for $x=0.125$ at $11\;\textrm{T}$. The cubic paramagnetic (PM) state is characterized by the narrow symmetric spectra, as expected in the highly symmetric cubic PM phase. The BLPS arises in the intermediate temperature range and is marked by asymmetric spectra with a more pronounced tail at lower frequencies. At low temperatures, in the magnetically ordered phase, the BLPS phase coexists with magnetism revealed by asymmetry of the NMR spectra. As elaborated in the Supplementary Note 4, a cubic local environment at the nuclear site must lead to a symmetric spectrum, while only non-cubic local symmetries, such as tetragonal or orthorhombic, can generate asymmetric lineshapes. We have performed detailed simulations of the ²³Na NMR powder pattern spectra in the presence of a quadrupolar interaction with the electric field gradient (EFG) and an anisotropic Knight shift $\mathbb{K}$ (see Methods & Supplementary Note 4), following the notation of Vachon08 . We find that above $T_{N}$, in the PM & BLPS phases, the resulting powder NMR spectra must reflect the symmetry of the $\mathbb{K}$ and EFG tensors. Therefore, the shape of the NMR spectra provides precise information about any deviation from the cubic symmetry. Indeed, our systematic analysis of the measured spectra demonstrates that the best fits can only be achieved by using orthorhombic distortions, in agreement with findings in Ba₂NaOsO₆ single crystals Lu_NatureComm_2017 .

Phase diagram. Both $T_{SB}$ and $T_{N}$ obtained from ²³Na NMR at 7 T and 11 T are displayed in the temperature versus doping phase diagram plotted in Fig. 4. The results show that the BLPS phase, a precursor to the magnetic state, is an intrinsic characteristic of these Mott-insulating metal oxides and persists to higher temperature, up to about 80 K, when approaching the $5d^{2}$ configuration. This orthorhombic local point distortion of the octahedra is a clear signature that these materials are intrinsically dominated by low temperature anisotropic spin-lattice interactions, an essential ingredient to be included in any microscopic quantum theory of Dirac-Mott insulators.

Discussions

Methods

Sample. The powder samples of Ba₂Na_1-xCa_xOsO₆ investigated here are the same as in Ref. kesavan2020doping . Powder x-ray diffraction (PXRD) measurements were performed to test the quality of the samples and analysed with Rietveld refinement. The compositional evolution of the lattice parameter is shown to follow Vegard’s Law, indicating a successful Na/Ca substitution kesavan2020doping .

Muon spin resonance. All $\mu$SR measurements were performed at the General Purpose Surface-Muon Instrument at the Paul Scherrer Institute in Switzerland. In $\mu$SR measurements, spin-polarized muons implant in the powder samples and precess around the local magnetic field with a frequency given by $\nu=\gamma_{\mu}\cdot$|B|/$2\pi$, where $\gamma_{\mu}=2\pi\cdot 135.5$ MHz/T. The muons decay with a characteristic lifetime of 2.2 $\mu$s, emitting a positron, preferentially along the direction of the muon spin. The positrons are detected and counted by a forward ($N_{F}(t)$) and backward detector ($N_{B}(t)$), as a function of time. The asymmetry function A(t) is given by

where $\alpha$ is a parameter determined experimentally from the geometry and efficiency of the $\mu$SR detectors. $A(t)$ is proportional to the muon spin polarization, and thus reveals information about the local magnetic field sensed by the muons. A typical zero-field $\mu$SR spectra below and above the magnetic transition temperature for Ba₂Na_1-xCa_xOsO₆ are presented in the Supplementary Material. The $\mu$SR spectra for all doping concentrations display damped oscillations at low temperatures (Supplementary Material 1), marking a transition to a state of long-range magnetic order. The spectra for the end elements, $x=0$ and $x=1$, are in agreement with those previously reported steele2011low ; thompson2014long . Each individual spectrum was fitted to a sum of precessing and relaxing asymmetries given by

The terms inside the brackets reflect the perpendicular component of the internal local field probed by the spin-polarized muons, the first term corresponds to the damped oscillatory muon precession about the local internal fields at frequencies $\nu_{i}$, while the second reflects a more incoherent precession with a local field distribution given by $\sigma$. The term outside the brackets reflects the longitudinal component characterized by the relaxation rate $T_{1}$.

Magnetization. Bulk magnetization measurements were performed using a superconducting quantum interference device (SQUID) magnetometer. Isothermal magnetization measurements as a function of applied field were performed at 2 K from -70 to 70 kOe. Zero-field and field-cooled magnetic susceptibility measurements were performed from 2 K to 400 K under an applied field of 1000 Oe.

NMR. NMR measurements were performed using high homogeneity superconducting magnets at Brown University, and the National High Magnetic Field Laboratory in Tallahassee, FL for magnetic fields exceeding 9 T. Temperature control was provided by ⁴He variable temperature inserts. ²³Na NMR data ware collected using state-of-the-art, laboratory-made NMR spectrometers from the sum of spin-echo Fourier transforms recorded at constant frequency intervals. Pulse sequences of the form ($\pi/2-\tau-\pi/2$) were used and none of the presented NMR observables depend on the duration of time interval $\tau$.

²³Na NMR. ²³Na NMR is a powerful local probe of both the electronic spin polarization (local magnetism), via the hyperfine coupling between electronic magnetic moments and the ²³Na nuclear spin $I=3/2$, and the charge distribution (orbital order and lattice symmetry), via the quadrupolar interaction between the electric field gradient (EFG) and the ²³Na quadrupole moment Q Lu_NatureComm_2017 ; PhysRevB.97.224103 . They affect both the first and second moments of the frequency distribution, i.e. the NMR spectra.

The Knight shift. The Knight shift is defined as

where $\omega_{0}=\;^{23}\gamma\cdot H_{0}$, where ${}^{23}\gamma$ = 11.2625 MHz/T and $H_{0}$ is the externally applied magnetic field, and $\omega^{i}$ is obtained from the first moment and/or peak position of the NMR spectral lines. This equation forms the general definition of the NMR shift $K$ in terms of the observed NMR frequency, $\omega^{i}\equiv\gamma H_{0}(1+K)$, where $K\equiv H_{loc}/H_{0}$ is the shift. Therefore, $K$ is a measure of the relative strength of the component of the local magnetic field ($H_{loc}$) parallel to the applied magnetic field, $H_{0}$. In the more general case where the shift varies as a function of the orientation of $H_{0}$ (anisotropic shift), the scalar $K$ is promoted to a second-rank tensor $\mathbb{K}$ and the expression for the observed NMR frequency becomes Abragam1961

where $\hat{\mathbf{h}}=\mathbf{H}_{0}/H_{0}$ is a unit vector in the direction of the applied magnetic field.

Calculation of NMR spectra below $T_{N}$. In the magnetically ordered phase, we model the local magnetic field at a Na site as $H_{loc}=\hat{\bf{h}}\cdot\sum_{i}\mathbb{A}_{i}\cdot\vec{S}_{i}$, where $\hat{\bf{h}}$ is a unit vector in the applied field direction, $\mathbb{A}_{i}$ is the hyperfine coupling tensor with the $i^{{\rm th}}$ nearest-neighbour Os atom, and $\vec{S}_{i}$ is its local spin moments. We note that contributions from dipolar effects are relatively small and are thus neglected. By performing a full lattice sum, we calculate the local fields projected in the direction of the applied field at the Na sites for the entire single crystal. The corresponding NMR spectra is a histogram of these local fields. The powder spectrum is then obtained by rotating the $\hat{\bf{h}}$ over the solid angle and integrating the results (see Supplementary Note 4). The diagonal components of the hyperfine tensor are optimized based on the hyperfine coupling values obtained from the NMR Knight shift and magnetic susceptibility measurements (see Supplementary Figure 4), while the symmetry of the hyperfine tensor is assumed to be the same as that found in the single crystal Ba₂NaOsO₆ Lu_NatureComm_2017 . The quadrupolar broadening effects are accounted for utilizing EFG parameters determined from the intermediate temperature powder spectrum simulation (see Supplementary Note 4).

Transition temperature. The transition temperature $(T_{N})$ from the PM to low-temperature magnetically ordered state was determined by examining the magnetic volume fraction $V_{mag}$ from zero-field $\mu$SR, and in finite fields from the temperature dependence of the NMR observables. That is, $T_{N}^{\mu}$ is delineated as a temperature at which $V_{mag}$ exceeds 90%. From NMR analysis, $T_{N}$ is defined as the onset temperature of a significant drop in the Knight shift associated with the formation of the AF ordering. Such $T_{N}$ is consistent with the one determined from the peak of the NMR relaxation rate, induced by critical fluctuations when approaching a transition into a long-range ordered magnetic state as a function of temperature.

ACKNOWLEDGEMENTS
We are thankful to D. Fiore Mosca, P. Santini, Brad Marston, and A. Paramekanti for enlightening discussions. This work was supported in part by U.S. National Science Foundation (NSF) grant No. DMR-1905532 (V.F.M.), the NSF Graduate Research Fellowship under Grant No. 1644760 (E.G.), NSF Materials Research Science and Engineering Center (MRSEC) Grant No. DMR-2011876 (P.M.T. and P.M.W.), and University of Bologna (S.S. and P.C.F.). Part of this work is based on experiments performed at the Swiss Muon Source S$\mu$S, Paul Scherrer Institute, Villigen, Switzerland. The study at the NHMFL was supported by the National Science Foundation under Cooperative Agreement no. DMR-1644779 and the State of Florida.

AUTHOR CONTRIBUTIONS
E.G., R.C., P.C.F., A.T. , A. P. R. and G.A. obtained NMR data and performed the data analysis. S.S. lead the muon spin spectroscopy experiment for which E.G. and R.C. contributed equally to data acquisition and interpretations. P.M.T. and P.M.W. grew the samples and performed the magnetic susceptibility measurement. S.S., V.F.M. and C.F. ideated the project. S.S. and V.F.M. supervised the experiments, lead data analysis and interpretations and writing of the manuscript. R.C., G.A., and V.F.M. were involved in the simulations of the NMR data. C.F. performed first principle calculations and was involved in data interpretations. A.P.R. enabled all measurements above 10 T at the NHMFL. All authors have read the paper and approved it. All authors discussed the results and commented on and edited the manuscript.