Local Metallic and Structural Properties of the Strongly Correlated Metal LaNiO$_{3}$ using $^{8}$Li $\beta$-NMR

TL;DR

This study uses $eta$-NMR to investigate local metallic and structural properties of LaNiO$_{3}$, revealing two distinct environments and antiferromagnetic correlations without magnetic transition, challenging existing structural models.

Contribution

First $eta$-NMR investigation of LaNiO$_{3}$ revealing unexpected local environments and magnetic correlations, questioning the assumed crystal structure.

Findings

Identified two distinct local metallic environments in LaNiO$_{3}$.

Observed substantial antiferromagnetic correlations without magnetic transition.

Detected structural inconsistencies with common crystal models.

Abstract

We report $\beta$-detected NMR of ion-implanted $^{8}$Li in a single crystal and thin film of the strongly correlated metal LaNiO$_{3}$. In both samples, spin-lattice relaxation measurements reveal two distinct local metallic environments, as is evident from $T$-linear Korringa $1/T_{1}$ below 200 K with slopes comparable to other metals. A small, approximately temperature independent Knight shift of $\sim 74$ ppm is observed, yielding a normalized Korringa product characteristic of substantial antiferromagnetic correlations, but, we find no evidence for a magnetic transition from 4 to 310 K. Two distinct, equally abundant $^{8}$Li sites is inconsistent with the widely accepted rhombohedral structure of LaNiO$_{3}$, but cannot be simply explained by either of the common alternative orthorhombic or monoclinic distortions.

Tables

Table 1. Table 1: The total experimental asymmetry A 0 subscript 𝐴 0 A_{0} and the fractions in Eq. ( 1 ) from the global fit analysis of the SLR data in LNO.

Sample	$A_0$	$f_S$	$f_f$
Crystal	$0.085\left(3\right)$	$0.85\left(6\right)$	$0.50\left(2\right)$
Film	$0.090\left(3\right)$	$0.80\left(6\right)$	$0.50\left(3\right)$

Equations

$$\begin{array}{rl}R(t,t^{\prime })& =f_S[(1-f_f)e^{-{\lambda }_s(t-t^{\prime })}+f_f{e}^{-{\lambda }_f(t-t^{\prime })}]\\ & +(1-f_S)e^{-{\lambda }_b(t-t^{\prime })},\end{array}$$

$$K\text{mbox}\text{leavevmode}[ppm]=\left(\frac{{\nu }_{\mathrm{LNO}}-{\nu }_{\mathrm{MgO}}}{{\nu }_{\mathrm{MgO}}}\right)\times 10^6.$$

$$\frac{1}{T_1}=\frac{4k_{B}T}{\hslash }\sum _{\vec{q}}\frac{|A(\vec{q}){|}^2}{({\gamma }_e\hslash {)}^2}\frac{1}{\hslash {\omega }_0}{\chi }^{\prime \prime }(\vec{q},{\omega }_0),$$

$$\mathscr{K}=\frac{T_{1}T\mathcal{K}^{2}}{S}=$0.40(10)$,$$

$$\mathcal{K}=K+4\pi \left(N-\frac{1}{3}\right){\chi }_0(T)$$

Full text

I Introduction

On the path to understanding and controlling correlated electrons in solids, a great deal of effort has gone into studying how the Fermi liquid state can be destabilized to yield other more exotic ground states below a metal-insulator transition (MIT)Imada et al. (1998). The rare-earth ($R$) perovskite nickelates $R$NiO3 are a unique and important example that remain challenging despite intense scrutinyTorrance et al. (1992); Greenblatt (1997); Catalan (2008); Catalano et al. (2018). Besides the MIT, further interest in $R$NiO3 stems from their close relation to the high-$T_{c}$ cuprates, both as potential superconductorsAnisimov et al. (1999); Chaloupka and Khaliullin (2008), and for what the absence of nickelate superconductivity reveals about the cuprates. Among the $R$NiO3 series, LaNiO3 (LNO) is unique in avoiding the MIT, remaining a paramagnetic metal to low temperature, making it particularly interesting and potentially useful. Though metallic, LaNiO3 is highly correlatedSreedhar et al. (1992); Stemmer and Allen (2018), with a strongly enhanced magnetic responseZhou et al. (2014) and electronic heat capacitySreedhar et al. (1992). The origin of these properties and even the persistence of the metallic state itself remain open questionsShamblin et al. (2018); Li et al. (2015).

Recently, epitaxial strain and dimensional confinement have been used to modify the properties of $R$NiO3Catalano et al. (2018); Chakhalian et al. (2014), including causing an MIT in LaNiO3Boris et al. (2011). Advances in high pressure O2 crystal growth have also made high quality single crystals of LaNiO3 available for the first timeZhang et al. (2017); Guo et al. (2018), opening the prospect for refined studies of the bulk. Surprisingly, recent results on one crystal have cast doubt on LNO’s characterization as a nonmagnetic metal, instead concluding that, in sufficiently pure stoichiometric form, it is magnetically ordered below $T_{N}\approx$ $157\text{\,}\mathrm{K}$Guo et al. (2018). Other crystals show the propensity for slight substoichiometry and the ordering of oxygen vacancies into defect phases that may explain the observed magnetismWang et al. (2018).

Here we report results from $\beta$-detected NMR ($\beta$-NMR) measurements of implanted highly polarized 8Li+ ions in two very different samples of LNO – a high quality single crystal and an epitaxial thin film. Like muon spin rotation ($\mu$SR)Schenck (1985), $\beta$-NMR provides a sensitive local magnetic probe of solids, but, in contrast, due to the much longer lifetime, it is sensitive to the metallic state in close analogy with conventional NMRWalstedt (2008). Using a low energy beam of implanted nuclei, $\beta$-NMR has the additional capability that it can easily be applied to thin filmsMacFarlane (2015). In both samples, we find clear indication of conventional Korringa spin-lattice relaxation (SLR) below $200\text{\,}\mathrm{K}$, with no evidence of a magnetic transition from $4$ to $310\text{\,}\mathrm{K}$. We find two distinct, equally abundant, metallic local environments for the implanted probe, a robust feature that, so far, defies explanation. The quantitative similarity between the two samples strongly suggests all of these features are intrinsic.

II Experimental

In $\beta$-NMR, highly spin-polarized $\beta$-radioactive ions are implanted into the sample, and the NMR is detected by the subsequent $\beta$-decay. The $\beta$-decay asymmetry is proportional to the average longitudinal spin-polarization, with a proportionality constant $A_{0}$ that depends on the detection geometry and properties of the decayMorris (2014). The asymmetry is measured by combining $\beta$ count rates from two opposing scintillation detectors. All of the experiments were conducted using the $\beta$-NMR spectrometer at TRIUMF in Vancouver, CanadaMorris et al. (2004); Morris (2014). The 8Li+ probe nuclei (spin $I=2$, gyromagnetic ratio $\gamma/2\pi=$6.3016\text{,}\mathrm{MHz}\text{,}{\mathrm{T}}^{-1}, electric quadrupole moment $Q=+$32.6\text{\,}\mathrm{mb}, and radioactive lifetime $\tau=$1.21\text{,}\mathrm{s}$$) were spin-polarized in-flight using optical pumpingLevy et al. (2014) and subsequently ion-implanted into the LNO samples. The implantation energies were $2.9\text{\,}\mathrm{keV}$ and $27.9\text{\,}\mathrm{keV}$ corresponding to mean depths of $12.9\text{\,}\mathrm{nm}$ and $99.5\text{\,}\mathrm{nm}$ (details in Appendix A).

In an applied magnetic field $B_{0}=$ $6.55\text{\,}\mathrm{T}$ provided by a high homogeneity superconducting solenoid in persistence mode, two types of measurements were performed: relaxation and resonance. With a pulsed 8Li+ beam, SLR data were collected by monitoring the depolarization during and after the $4$ second long pulse. During the pulse, the polarization approaches a dynamic steady-state value, while afterwards, it relaxes to $\approx 0$. Since the probe nucleus is polarized prior to implantation, unlike conventional NMR, no radio frequency (RF) field is required to measure SLR. Resonance measurements used a continuous beam of 8Li+ with a transverse RF field $H_{1}$ stepped slowly in frequency through the 8Li Larmor frequency $\nu_{0}=\gamma B_{0}/(2\pi)\approx$ $41.27\text{\,}\mathrm{MHz}$. On resonance, the 8Li spin precesses rapidly due to the RF field, resulting in a loss of the time-averaged asymmetry. The resonance frequency was calibrated against a single crystal of MgO at $300\text{\,}\mathrm{K}$MacFarlane et al. (2014).

The film sample was deposited on a single crystal [LaAlO3]0.3[Sr2AlTaO6]0.7 (LSAT) substrate by pulsed laser deposition (PLD) and annealed in an O2 rich environment as described in Ref. Boris et al., 2011. Its thickness was determined by X-ray reflectivity to be $38(1)\text{\,}\mathrm{nm}$. The LSAT substrate is lattice matched to LNO, minimizing epitaxial strain. The LNO crystal was grown as detailed in Ref. Zhang et al., 2017. A $1.5\text{\,}\mathrm{mm}$ thick slice perpendicular to the pseudocubic [100] direction was cut from a cylindrical boule $\sim 5\text{\,}\mathrm{mm}$ in diameter. The surface was prepared by polishing with Al2O3 suspensions until a mirror-like surface was obtained. The samples were affixed to polished sapphire plates attached to a He cold-finger cryostat.

Although the crystals have a rather low average O deficiency (determined by TGA to be $\sim 0.015$/formula unit), it is possible that this could influence the measured properties. Indeed, large O vacancy concentrations can be purposely generated in LaNiO3 with the appearance of supercell defect phases that order magnetically, with composition LaNiO2.75 and LaNiO2.5Wang et al. (2018). If the O vacancy phases are uniformly distributed, they would amount to <5 mol %; however, in the case of nonuniform distribution, the concentration of ordered superlative phases would be sample-dependent and could exceed this upper bound.

III Results and Analysis

III.1 Spin-Lattice Relaxation

Representative SLR data are shown in Figure 1. At low temperatures in both the crystal and film, the relaxation is slowest, with the rate increasing monotonically with temperature. We identify a small amplitude fast relaxing component at early times as a background signal due to 8Li stopping outside the sample. It is easily distinguished from the sample signal with a nearly temperature independent rate faster than the sample at all temperatures. The remaining signal from LNO is also not comprised of a single relaxing component, but has two distinct SLR contributions with different rates. With this in mind, we require a model relaxation function $R(t)$, the analog of the magnetization recovery curve in NMR, to fit the data. The simplest form providing a good fit is a triexponential, which encapsulates the biexponential signal from LNO, as well as the background. Specifically, at time $t$ after an 8Li arriving at time $t^{\prime}$,

[TABLE]

where $\lambda_{i}\equiv 1/T_{1}^{i}$ ($i=s,f,b$) are the SLR rates, $f_{f}\in[0,1]$ is the fast relaxing fraction, $f_{S}$ is the fraction of 8Li in the sample, and the third term is the background. In order to fit the data, a global procedure was used wherein all spectra for each sample (i.e., at every temperature) were fit simultaneously using custom C++ code and the MINUIT minimization routinesJames and Roos (1975) provided by the ROOT frameworkBrun and Rademakers (1997). Best fits were obtained using temperature independent fractions: f_{f}=$$0.50(3) for both samples and $f_{S}=$ $0.85(6)$ for the crystal and $0.80(6)$ for the film (see Appendix B), leaving $\lambda_{i}$ as the only temperature dependent parameters. The fit quality is good in each case: $\chi^{2}$ = $0.94$ (crystal) and $1.05$ (film).

The slow and fast SLR rates extracted from the above analysis are shown as a function of temperature in Fig. 2(a) and (b). Consistent with the qualitative features in Figure 1, the rates are slowest at the lowest temperature and increase linearly up to $\sim 200\text{\,}\mathrm{K}$, above which there are sample dependent nonlinearities. Significantly, the fast and slow rates are quantitatively consistent between the two samples indicating the behavior is intrinsic. Over the linear range ($T<$ $200\text{\,}\mathrm{K}$), we fit $\lambda_{i}(T)$ to a line to obtain the slopes in Fig. 2. The two components are present in equal amplitudes in both samples with rates that differ by a factor of $\sim 4$ on average. Their origin is discussed in Section IV.

III.2 Resonances

The resonances (primarily in the crystal) are shown in Fig. 3. In contrast to the multiexponential SLR, the spectrum consists of a single broad line which is narrowest and most intense at the highest temperature. The spectra were fit to a single Lorentzian, and the resulting parameters are shown in Fig. 4. As the temperature is lowered, the peak broadens by a factor of \sim$$2, while its amplitude decreases by an order of magnitude. Remarkably, the resonance in the film at $300\text{\,}\mathrm{K}$ is comparable to the crystal. Aside from the breadth, the resonance is also slightly positively shifted from the calibration in MgO. We quantify its relative shift by

[TABLE]

From which we extract the Knight shift $\mathcal{K}$, due to the conduction band spin susceptibility, after accounting for demagnetization (see Appendix C). The resulting $\mathcal{K}\sim$ $74\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$, independent of temperature, is typical of 8Li in metals but is surprisingly small considering the rather large susceptibility. We discuss the shift further below.

IV Discussion

Like other atomic solutes, the implanted 8Li nucleus generally has some hyperfine coupling to the conduction band of the host, resulting in a phenomenology very similar to the conventional NMR of metalsWalstedt (2008). The predominant SLR mechanism is usually the Korringa process of spin exchange scattering with the conduction electrons, leading to a characteristic linear dependence of $1/T_{1}$ on temperatureMacFarlane (2015). One distinction from conventional NMR is that the lattice site of the implanted ion is not known a priori and is not independently accessible by X-ray diffraction. The site is important because it determines the sensitivity of $T_{1}$ to spin fluctuations at finite wavevector, such as when there is some tendency towards antiferromagnetism. The relaxation rate is related to the generalized frequency- and wavevector- dependent magnetic susceptibility $\chi=\chi^{\prime}-i\chi^{\prime\prime}$ via the Moriya expressionMoriya (1963),

[TABLE]

where the hyperfine form factor $A(\vec{q})$, the spatial Fourier transform of the local hyperfine coupling, acts as a filtering function on the magnetic fluctuations at wavevector $\vec{q}$ and the NMR frequency $\omega_{0}$. $\gamma_{e}$ is the electron gyromagnetic ratio. The resulting site dependence of $1/T_{1}$ is illustrated by the well known case of NMR in the cupratesWalstedt (2008).

Another important difference of 8Li $\beta$-NMR from, e.g. transition metal NMR, is that, as a light isotope, the hyperfine coupling is quite weak, similar to the implanted muonSchenck (1985), dissolved hydrogenBarnes et al. (1997) or stable 7LiMacFarlane et al. (2000), leading to small Knight shifts on the order of $100\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$ and correspondingly long $T_{1}$. The small 8Li shifts make the demagnetization correction particularly importantXu et al. (2008). Unlike the muonBlundell and Cox (2001), the Korringa rate is usually in the accessible window for 8Li: $0.01\tau<T_{1}<100\tau$, making it a useful probe of metals, including correlated oxides such as Sr2RuO4Cortie et al. (2015).

In the present data, the linearity of $1/T_{1}$ for both fast and slow components below $200\text{\,}\mathrm{K}$ (Fig. 2) is clear microscopic evidence of a conventional metallic state, consistent with 139La NMR of bulk powderSakai et al. (2002). Recent powder neutron diffraction suggest that LNO is electronically inhomogeneous, possessing insulating pockets below $200\text{\,}\mathrm{K}$ Li et al. (2015). In contrast, we find no evidence for an insulating volume fraction.

$\beta$-NMR reports the average behavior over the sampled volume given by the implantation profile (Appendix A) and the millimetric beamspot. An insulating LNO phase would produce another SLR component with amplitude proportional to its volume fraction and a relaxation distinct in both magnitude and temperature dependence. For putative insulating regions, we expect strong local moment magnetism and low temperature magnetic order. In such regions, the relaxation would be fast and increasing with reduced temperature, peaking at the magnetic freezing, where it might even be so large as to “wipe out” the signal. Based on this, we rule out an insulating magnetic fraction greater than an estimated detection limit of \sim$$5\text{\,}\%.

The 8Li resonance spectrum characterizes the static, time-average magnetic properties. Surprisingly, we find a single broad line showing none of the quadrupolar fine structure observed in related insulating perovskitesMacFarlane et al. (2003); Karner et al. (2018a, b). A reduction of the electric field gradient (EFG) compared to isovalent LaAlO3 (LAO)Karner et al. (2018b) is reasonable in the negative charge transfer picture, as the extra $+1$ charge of Al3+ (vs. Ni2+) is spread over the oxide ligands. Metallic screening will further reduce the EFG, evidently resulting in an unresolved splitting of the NMR on the order of the linewidth or less.

The line is, in fact, very broad compared to both other metals and insulating perovskitesMacFarlane et al. (2003); Karner et al. (2019). The temperature dependence of the width [Fig. 4(b)] suggests a combination of a substantial temperature independent term and one that increases as temperature is decreased. Qualitatively, the former is consistent with quadrupolar broadening due to structural disorder and the latter to magnetic broadening from dilute magnetic defects. However, significantly, the widths in the two samples are comparable at $300\text{\,}\mathrm{K}$, the only temperature where we have a resonance in the film. One would not expect extrinsic disorder to be very similar in these vastly different samples, so this agreement is surprising and also points to an intrinsic origin for the width.

To understand these features, we must consider the structural details of perovskites. The nominal rhombohedral structure involves a rotation of the NiO6 octahedra about any of the equivalent $\langle 111\rangle$ directions of the cubic phase. As a result, the crystal is microtwinned - an intrinsic inhomogeneity that may contribute to the quadrupolar broadening. However, similar microtwinning is also a feature of rhombohedral LAO, where we find large well-resolved quadrupole splitting and substantially smaller linewidthKarner et al. (2019), as well as tetragonal SrTiO3MacFarlane et al. (2003). This suggests twinning alone cannot be responsible for broadening. We return to this point below.

There is also no evidence in the resonance spectra for two environments corresponding to the two relaxing components, but this is not surprising on quantitative grounds, since the observed shift is so small [Fig. 4(a)]. The relaxation rate is determined by the square of the hyperfine coupling [Eq. (3)], while the shift is only linear, so the factor of $4$ between the rates implies only a factor of $2$ in the shifts, yielding a magnetic splitting on the order of $50\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$ ($2\text{\,}\mathrm{kHz}$ at this $B_{0}$, significantly less than the linewidth). Moreover, though the relaxing components are practically equal in their initial (i.e., $t=0$) asymmetry, the resonance amplitude is determined instead by the time average asymmetryHossain et al. (2009), which, for the fast relaxing component, is suppressed to at most $27\text{\,}\%$ of the total. Finally, if there is some magnetic broadening, the larger coupling to the fast relaxing component would likely make its resonance broader and even smaller in relative amplitude.

Based on these considerations, we conclude that the 8Li $\beta$-NMR is an unresolved composite of two lines, originating from two distinct local environments, whose spectrum is heavily weighted towards the slow component. The resonance shift ($\sim 74\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$) is then a weighted average of the two. To illustrate this, we decompose the spectrum at $300\text{\,}\mathrm{K}$, assuming equal linewidths and using the relative amplitudes from $\lambda$ as weightsHossain et al. (2009), to estimate the Knight shifts of the two components. These are shown in Fig. 4(a), as the horizontal lines for the slow (blue) and fast (green) component where the vertical arrows emphasize that they are lower limits, since the fast relaxing resonance may be wider (and hence smaller).

From this analysis, we estimate the Knight shift $\mathcal{K}\approx$ $71(10)\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$ for the major (slow) component at $300\text{\,}\mathrm{K}$. Combining this with the Korringa slope [Fig. 2(a)], we form the normalized Korringa product,Walstedt (2008)

[TABLE]

where for 8Li  $S=$ $1.20\text{\times}{10}^{-5}\text{\,}\mathrm{s}\text{\,}\mathrm{K}$. For an uncorrelated free electron metal, $\mathscr{K}\approx 1$. Our $\mathscr{K}$ is significantly less, indicating substantial antiferromagnetic correlationsMoriya (1963); Ueda and Moriya (1975). This is opposite to a recent Stoner enhancement interpretation of the susceptibility of powderZhou et al. (2014), but it is consistent with the occurrence of antiferromagnetism in the insulating nickelatesGarcía-Muñoz et al. (1995), including thin layers of LNOFrano et al. (2013). In contrast, 139La NMR finds $\mathscr{K}\approx 1.5$ (i.e. a relaxation rate slower than expected from $\mathscr{K}=1$) suggesting instead ferromagnetic correlationsSakai et al. (2002).

We now consider potential sources for this substantial discrepancy. One possibility is that, due to the different lattice sites, the different form factors in Eq. (3) blind the La to an important wavevector that implanted 8Li sees, reducing its $T_{1}$ and $\mathscr{K}$. The most obvious candidate is the reported AF ordering vector in LNO/LAO superlattices, $\vec{q}_{\mathrm{AF}}=(1/4,1/4,1/4)$Frano et al. (2013). However, for the La site $A(\vec{q}_{\mathrm{AF}})$ is nonzero, ruling out at least the most obvious explanation along these lines. A second possibility is that (unlike Li), La may have a substantial orbital shift which cannot easily be separated from the temperature independent Knight shift. Using the full 139La shift may thus result in a significantly overestimated $\mathscr{K}$.

While it is widely accepted that LNO remains a metal to low temperature, the occurrence of static antiferromagnetism (AF) was recently suggested from new data in a single crystal at $T_{N}\approx$ $157\text{\,}\mathrm{K}$.Guo et al. (2018) In contrast, we have no evidence of magnetic ordering at this temperature, either from the resonance line or $1/T_{1}$, in agreement with recent data suggesting that the AF is due to an oxygen deficient phase.Wang et al. (2018)

We turn now to the sample dependent deviation from the Korringa dependence of $1/T_{1}$ above $200\text{\,}\mathrm{K}$. This bifurcation coincides with a small change in the Ni-O-Ni angle from structural studies on powdersLi et al. (2015). Above about $300\text{\,}\mathrm{K}$, $\chi(T)$ for LNO crystals is well described by the phenomenological Curie-Weiss dependence, $\chi\propto(T+\theta)^{-1}$ with a large $\theta\approx$ $2000\text{\,}\mathrm{K}$, indicating a substantial departure from a simple T independent Pauli susceptibility. It is interesting that the Korringa dependence in Fig. 2 (the NMR hallmark of a metal) appears to break down close to the onset of this high $T$ decrease in $\chi$. From analysis of the RF conductivityShamblin et al. (2018), it was concluded that the carrier density decreases substantially above $200\text{\,}\mathrm{K}$ which would simultaneously diminish both Korringa slopes. The sample dependence is clearly more complex (Fig. 2). It is also not as simple as an additional relaxation, due to differing impurity content, for example. Rather, it has the opposite sense for the fast and slow components. This suggests the magnetic response depends on subtle details of the temperature evolution of the lattice which are in turn influenced by the epitaxial relation to the substrate.

The above features are consistent with a correlated metal without static magnetism. However, an essential feature of our data is the two component character of the relaxation indicating two metallic local environments for the 8Li. Based on the $\beta$-NMR of 8Li in simple metalsMorris et al. (2004); Parolin et al. (2007); Salman et al. (2007); Parolin et al. (2008, 2009); Hossain et al. (2009); Ofer et al. (2012), one possibility is simply that there are two inequivalent sites for the implanted 8Li within the unit cell. In elemental Ag and AuMorris et al. (2004); Parolin et al. (2008); Hossain et al. (2009), for example, there are two cubic 8Li sites that differ in their hyperfine coupling to the conduction band, and we find a $T$ dependent site change transition around $180\text{\,}\mathrm{K}$Parolin et al. (2008). In other perovskite oxidesMacFarlane et al. (2003); Karner et al. (2018a, b), we find a single interstitial 8Li+ site characterized by a large quadrupole splittingMacFarlane et al. (2003); Karner et al. (2018a, 2019), which we shall call the $P$ site, at the $3c$ Wyckoff position in the cubic ($Pm\bar{3}m$) phase. As illustrated in Fig. 5, $P$ is midway between two adjacent $A$-site (La3+) ions at the centre of a square Ni-O plaquette. In contrast to Au and AgMorris et al. (2004); Parolin et al. (2008); Hossain et al. (2009), the two components in LNO are equal in amplitude at all temperatures, and we find no hint of a site change. Equal population of two distinct lattice sites over such a wide range of $T$, while possible, seems unlikely.

With no obvious second site, we now consider how site $P$ evolves as the crystal symmetry is lowered from the ideal cubic perovskite structure. In the rhombohedral phase (the nominal LNO structure), adjacent NiO6 octahedra rotate with an equal angle $\sim 9^{\circ}$ and alternating signGou et al. (2011). Despite this, there remains a single $P$ site, consistent with the 8Li spectrum in rhombohedral LAO.Karner et al. (2019) In the orthorhombic ($Pnma$) structure, the rotation pattern of the NiO6 octahedra differs ($a^{-}a^{-}c^{+}$ instead of $a^{-}a^{-}a^{-}$, in Glazer notationGlazer (1972)), resulting in three distinct $P$-derived Li sites and two distinct oxygen sites in the enlarged unit cell. With a further lowering of symmetry to monoclinic with the rocksalt alternating superlattice of long and short bond NiO6 octahedra, i.e., the bond disproportionated structure of insulating $R$NiO3, $P$ further diverges into 4 distinct sites (while there are two Ni and three O sites). Note that in all these structures there is a single La ($A$) site. Since these structures are all pseudocubic, the $P$ derived sites are all very similar, with comparable interstitial space to accommodate the implanted ion but slight differences in distances to the near neighbours and in Li-O-Ni angles that in the cubic phase are exactly $90^{\circ}$. Supercell density functional theory (DFT) calculations (Appendix D) confirm that energetically they are all very similar, so that one would expect a randomly implanted 8Li+ to occupy them with equal probability. While a distribution of similar (but distinct) sites in a lower symmetry structure constitutes an intrinsic microscopic source of inhomogeneity that could account for the resonance width, among these three structures, we have not found an obvious explanation for two equally abundant 8Li environments (detailed in Appendix E). Based on this, we suggest that the precise low temperature structure of LNO may be none of these commonly considered possibilities.

Interestingly, our DFT calculations predict that the ground state symmetry of LNO is rhombohedral $R\bar{3}$, where the $a^{-}a^{-}a^{-}$ rotation pattern of the NiO6 octahedra coexists with the breathing distortion. A similar observation was made earlier in Ref. Subedi, 2018. In this structure, there are two $P$ derived sites with energies as close as $20\text{\,}\mathrm{meV}$ when occupied by Li+. The distortions giving rise to these structural variants are still subtle and not easily distinguished even with high resolution diffractionLi et al. (2015); Shamblin et al. (2018). The two component character of the 8Li relaxation demonstrates that a local probe may provide important structural insight, particularly when there is substantial microscopic inhomogeneity as evident in the resonance width. It will be important to carry out structural refinement of the diffraction data based on the $R\bar{3}$ structure. In addition, other local probes (such as 17O or the very difficult 61Nivan der Klink and Brom (2010)) would provide further insight into the local structure and properties of LaNiO3.

V Conclusion

Using ion-implanted 8Li $\beta$-NMR, we studied the local electronic properties of LaNiO3. SLR measurements revealed two components with linearly temperature dependent $1/T_{1}$ below $200\text{\,}\mathrm{K}$, consistent with a Korringa mechanism, providing strong microscopic evidence of a conventional metallic state. The 8Li resonance spectrum comprised a single broad line, implying considerable static inhomogeneity at all temperatures. We find no evidence for either an insulating volume fraction or an antiferromagnetic ordering transition. However, the normalized Korringa product $\mathscr{K}=$0.40(10)$$ indicates substantial AF correlations. The two component SLR implies two 8Li environments, indicating the local crystal symmetry is inconsistent with the three commonly considered possibilities: the rhombohedral $R\bar{3}c$, the orthorhombic $Pnma$ and the monoclinic $P2_{1}/n$. Based on DFT calculations, we postulate that the local structure of LNO is $R\bar{3}$ where the rhombohedral unit cell is expanded by a breathing distortion of the NiO6 octahedra.

Acknowledgements.

We thank R. Abasalti, D.J. Arseneau, S. Daviel, B. Hitti and D. Vyas for technical assistance, and E. Benckiser, J. Chakhalian, Z. Salman and G. Sawatzky for helpful discussions. This work was supported by NSERC Canada.

Appendix A Beam Energy and Implantation Depth

The energy of the 8Li+ beam determines the mean implantation depth of the probe into the sample. The $\beta$-NMR spectrometer is located on an isolated high voltage platform and the ion beam can be decelerated (i.e., the desired implantation depth can be chosen) by varying the biasMorris et al. (2004); Morris (2014). We modeled the implantation using the SRIM Monte Carlo simulation packageZiegler et al. (2010), and results for the beam energies used here are shown in Figure 6. For the $38\text{\,}\mathrm{nm}$ film, the implantation energy was chosen to be $2.9\text{\,}\mathrm{keV}$ (mean depth $13\text{\,}\mathrm{nm}$) to minimize the amount of 8Li in the LSAT substrate. In contrast, no such concerns were warranted for the crystal and the full beam energy $27.9\text{\,}\mathrm{keV}$ (mean depth $100\text{\,}\mathrm{nm}$) was used.

Appendix B Decomposition of the Spin-Lattice Relaxation

Here we present a more detailed view of the analysis of the SLR data. The phenomenological fitting function based on Eq. (1) consists of three exponentials with independent rates whose contributions are illustrated in Figure 7 at $300\text{\,}\mathrm{K}$, where the relaxation is fastest. The background component has the largest relaxation rate but by far the smallest amplitude. In contrast, the slow and fast sample components have equal amplitudes, but rates that differ by a factor of $\sim 4$. Note that though small, the background signal is essential for a good fit. It is comparable in both amplitude and rate to similar signals that are clearly evident in samples with much slower relaxation, such as nonmagnetic insulators, e.g. Fig. 5 in Ref. Karner et al., 2018a. Table 1 lists the (temperature independent) values of the full asymmetry $A_{0}$, $f_{S}$, and $f_{f}$ for the two samples, with statistical errors from the global fits.

Appendix C Demagnetization Correction

The raw shift with respect to MgO is proportional to the static average internal field within the sample. To extract the Knight shift $\mathcal{K}$, it is necessary to account for the contribution from demagnetizationXu et al. (2008), which depends on the shape and uniform magnetization of the sample. The crystal is approximately cylindrical, so we use the results of Ref. Joseph and Schlömann, 1965, noting that the demagnetizing field is inhomogeneous in such a non-ellipsoid, and that the sampled volume corresponds to the central region of the face of the crystal to estimate the relevant demagnetization factor $N\approx$ $0.739$. Using this, we compute the Knight shift $\mathcal{K}$ as,

[TABLE]

where $\chi_{0}(T)$ is the CGS volume susceptibility from Ref. Zhang et al., 2017. Fig. 8 illustrates the effect of the demagnetization correction by comparing the raw and corrected shifts. In our temperature range, the susceptibility of LaNiO3 is almost constant, making the correction an approximately temperature independent positive offset of $77(2)\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$. We note that for Li the orbital (chemical) shift $K_{\mathrm{orb}}$ is very small, at most $5\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$Kartha et al. (2000), and comparable to (or less than) the uncertainty in the measurements. We have, therefore, not attempted to account for it.

Appendix D Density Functional Calculations

Determination of the LNO ground state symmetry was performed using the projector augmented wave method (PAW)Blöchl (1994); Kresse and Joubert (1999) as implemented in the Vienna ab initio simulation package VASPKresse and Furthmüller (1996, 1996); Paier et al. (2005). The generalized gradient approximation (GGA) PBEsolPerdew et al. (2008) was employed to account for exchange and correlation effects, while a Hubbard-like on-site repulsion $U$ and Hund’s exchange coupling $J$ was introduced for the Ni-$3d$ ($U=$ $2.4$ or $6\text{\,}\mathrm{eV}$ and $J=$ $1\text{\,}\mathrm{eV}$) and La-$4f$ ($U=$ $14$ and $J=$ $1\text{\,}\mathrm{eV}$) electrons, following the scheme of Liechtenstein et alLiechtenstein et al. (1995). The kinetic energy cut-off was set to $550\text{\,}\mathrm{eV}$ and a well-converged density of the $k$-vector mesh, corresponding to 6$\times$6$\times$6 in a simple cubic perovskite unit cell, was chosen. We compared the total energies of the monoclinic $P2_{1}/n$ unit cell of LNO and of the rhombohedral $R\bar{3}$ unit cell, both featuring two structurally inequivalent Ni sites but differing in the octahedra’s rotation pattern. It is consistently found that the rhombohedral $R\bar{3}$ unit cell has a lower energy irrespective of the magnetic order imposed [corresponding to either $\vec{q}_{\text{AF}}=(1/2,0,0)$ or $(1/4,1/4,1/4)$ ordering vector] and the $U$ value in the GGA+$U$ scheme chosen ($U=$ $6\text{\,}\mathrm{eV}$ for a robust insulating state and $U=$ $2.4\text{\,}\mathrm{eV}$ for a vanishing charge gap maintaining bond disproportionation). The energy difference between the two structures, however, is only on the order of a few meV per formula unit.

Supercell calculations were performed using both the PAW method via VASP and the linearized augmented plane wave method (LAPW) implemented in the WIEN2k codeBlaha et al. (2001). For the monoclinic $P2_{1}/n$ structure we used a $2a\times 2a\times 2a$ supercell, where $a$ is the pseudocubic lattice constant, and imposed the $\vec{q}_{\text{AF}}=(1/2,0,0)$ antiferromagnetic order onto Ni spins. For the rhombohedral $R\bar{3}$ structure we used a face-centered $4a\times 4a\times 4a$ supercell and imposed the $\vec{q}_{\text{AF}}=(1/4,1/4,1/4)$ antiferromagnetic order onto Ni spins. A single Li+ ion was introduced into the various inequivalent $P$ sites (with a compensating uniform background charge) and the system was allowed to fully relax its ionic positions, with the volume fixed to the equilibrium volume of LaNiO3 in PBEsol. The resulting total energies corresponding to different Li+ positions are found to differ by no more than $140\text{\,}\mathrm{meV}$ per formula unit in the $P2_{1}/n$ structure and $20\text{\,}\mathrm{meV}$ per formula in the $R\bar{3}$ structure.

Appendix E Detailed Properties of the Interstitial $P$ Site

In an attempt to explain the two 8Li environments, here we present a detailed account of the interstitial $P$ site in the relevant pseudocubic LNO structures. Adopting the approach used successfully in NMR of the cupratesWalstedt (2008), we focus on the hyperfine coupling to the nearest Ni atomic moments. A transferred hyperfine coupling will result from unpaired $3d$ ($e_{g}$) orbital spin density being mixed into the mostly vacant Li $2s$ orbital, where it has a Fermi contact coupling to the 8Li nucleus. This mixing can be due to direct overlap, or it can be mediated by the oxygen neighbours. Symmetry considerations, analogous to the Goodenough-Kanamori rules for exchange couplingGoodenough (1955, 1958); Kanamori (1959), apply to these overlaps, and we expect the coupling to the $d_{x^{2}-y^{2}}$ orbital to be zero (both direct and through the oxygen) in the limit of a $90^{\circ}$ Li-O-Ni angle. However, for all the pseudocubic distorted structures, the angles are not precisely $90^{\circ}$, so this coupling will be nonzero, but relatively small. The other degenerate $e_{g}$ orbital, $d_{z^{2}}$ should also have a nonzero coupling. This dependence of the hyperfine coupling on angle is clearly demonstrated in the conventional NMR of transition metal oxide Li battery materialsGrey and Dupré (2004).

The Knight shift is just proportional to the spin susceptibility, $K=A\chi_{s}$. Assuming the macroscopic susceptibility $\chi$ is predominantly due to the Ni spins, we estimate the hyperfine coupling $A$ for the slow relaxing site as $0.78$ kG/$\mu_{B}$, a value that is relatively small, e.g., compared to 8Li in simple metalsParolin et al. (2008) or for substitutional Li in YBCOBobroff et al. (1999).

In the nominal rhombohedral structure of LNO, the $P$ site has two distinct Li-O-Ni angles of $\sim 84\text{\,}\mathrm{\SIUnitSymbolDegree}$ and $\sim 95\text{\,}\mathrm{\SIUnitSymbolDegree}$, and two distances to the coordinating oxygens. However, all the $P$ sites are equivalent, so this structure is inconsistent with our two component relaxation. Neither of the conventional lower symmetry distorted structures provide a simple doubling of the $P$ site into two equal populations of $P$-derived sites, but each inequivalent $P$ site will have a different EFG and hyperfine coupling. Although the multiplicity does not match the 1:1 ratio in our data, based on the above considerations, we calculated the distribution of Li-O-Ni angles for all the $P$-derived sites in both lower symmetry structures ($Pnma$ and $P2_{1}/n$) to see if it clustered approximately into two categories with equal weights. We considered both the ideal structures and ones relaxed around the interstitial Li+, as estimated by DFT, but we found no indication of an appropriately symmetric bimodal distribution.

The DFT calculations, outlined in Appendix D, find that the ground state structure is none of the three discussed above, but rather has symmetry $R\bar{3}$. In this structure, the $a^{-}a^{-}a^{-}$ rotation of the NiO6 octahedra of the rhombohedral unit cell ($R\bar{3}c$) is preserved; however, the symmetry is lowered by bond disproportionation. Resulting in two Li $P$ sites (and two Ni sites), consistent with our SLR data. However, these two sites must have a different hyperfine coupling to account for the ratio of $\sim 4$ in SLR rates. The distribution of Li–O–Ni angles for both $P$ sites is peaked at $90\text{\,}\mathrm{\SIUnitSymbolDegree}$, with one much narrower than the other. Given the sensitivity of the hyperfine coupling to anglePan et al. (2002), this structure is a good candidate for explaining the two component nature of our SLR data.