Charge-stripe order, antiferromagnetism, and spin dynamics in the cuprate-analog nickelate ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_8$

We report results of a muon spin rotation ($\mu \mathrm{SR}$) study of the cuprate-analog nickelate ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_8$, which undergoes a transition at 105 K to a low-temperature phase with charge-stripe and antiferromagnetic (AFM) order on square planar ${\mathrm{NiO}}_2$ layers. Zero-field $\mu \mathrm{SR}$ shows that the AFM transition is abrupt, commensurate, and has a quasi-two-dimensional character below $\sim 25$ K. Comparison of observed muon precession frequencies with Ni dipolar field calculations yields Ni moments $\lesssim 0.5{\mu }_B$. Dynamic muon spin relaxation above 105 K suggests critical slowing of Ni spin fluctuations, but is inconsistent with corresponding ${}^{139}\mathrm{La}$ NMR results. Critical slowing and an abrupt transition are also observed in the planar cuprate AFM ${\mathrm{La}}_2{\mathrm{CuO}}_{4+\delta }$, where they are taken as evidence for weakly interplanar-coupled two-dimensional AFM spin fluctuations, but our $\mu \mathrm{SR}$ data do not agree quantitatively with theoretical predictions for this scenario when applied to the nickelate.

Figure 1

Charge stripe [5] and proposed AFM moment configurations [23] on a nickel layer in ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_8$. Blue spheres: ${\mathrm{Ni}}^{1+}$ ions ($S=\frac{1}{2})$. Yellow spheres: ${\mathrm{Ni}}^{2+}$ ions ($S=0$). Arrows: AFM ordered moments. ${\mathrm{O}}^{2-}$ ions are not shown.

Figure 2

Unit cell of ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_8$ (space group $I4/mmm$, no. 139). Blue spheres: Ni; red spheres: O; green spheres: La. Small spheres: candidate muon sites (cf. Sec. 4, Table 1). Red: $\mu 1–\mu 3$; green: $\mu 4–\mu 5$; blue: $\mu 6–\mu 8$.

Figure 3

2D AFM structure of Fig. 1 [23] and projections of candidate ${\mu }^{+}$ stopping sites in ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_8$ (Fig. 2 and Table 1) onto the $ab$ plane in the $F4/mmm$ setting of the $3\sqrt{2}a\times \sqrt{2}a$ 2D stripe supercell. Large circles: blue: ${\mathrm{Ni}}^{1+}$ ($S=1/2$) ($\odot =$ up spins, $\otimes =$ down spins); yellow: ${\mathrm{Ni}}^{2+}$ ($S=0$); red: ${\mathrm{O}}^{2-}$. Small circles: ${\mu }^{+}$ sites from Table 1. Rectangle: supercell boundary.

Figure 4

Fourier-transform spectra from ZF-$\mu \mathrm{SR}$ asymmetry data (cf. Fig. 5) in ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_8, 2.2\text{K}⩽T⩽110$ K. The spectra are offset for clarity. Arrows: discrete frequencies at 2.2 K.

Figure 5

Representative ZF-$\mu \mathrm{SR}$ asymmetry spectra in ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_8, T⩽T_N=105$ K. Curves: fits of Eq. (1) to the data.

Figure 6

Temperature dependence of parameters in ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_8$ from fits of Eq. (1) to ZF-$\mu \mathrm{SR}$ asymmetry data (cf. Fig. 5). Numbers indicate peaks in Fig. 4. (a) ${\mu }^{+}$ precession frequencies. (b) Relative oscillation amplitudes of peaks 1 and 6 (Fig. 4). (c) Exponential damping rates of peaks 1 and 6.

Figure 7

Models for charge stripes in ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_8$. Numbering as in Ref. [5]. Blue spheres: ${\mathrm{Ni}}^{1+}$ ions ($S=\frac{1}{2}$); yellow spheres: ${\mathrm{Ni}}^{2+}$ ions ($S=0$); rectangles: $3\sqrt{2}a\times \sqrt{2}a\times c$ supercell outlines. The view is along the stripe ($b$) direction.

Figure 8

AFM moment configurations A–D in ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_8$ on charge-stripe model 1 [5]. Blue spheres: ${\mathrm{Ni}}^{1+}$ ions ($S=\frac{1}{2}$); yellow spheres: ${\mathrm{Ni}}^{2+}$ ions ($S=0$); rectangles: $3\sqrt{2}a\times \sqrt{2}a\times c$ supercell outlines; arrows: AFM ordered moments. The view is along the stripe ($b$) direction.

Figure 9

Histograms of dipolar-field ${\mu }^{+}$ frequencies from AFM stripe models 1A–1D compared to 50-K $\mu \mathrm{SR}$ FT spectrum in ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_8, T=50$ K. Colors indicate ${\mu }^{+}$ site groups (Table 1): red: $\mu 1$–$\mu 3$; green: $\mu 4$–$\mu 5$; blue: $\mu 6$–$\mu 8$.

Figure 10

AFM Ni-ion moment configurations without stripes in ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_8, F4/mmm$ setting. $A$-AFM: FM intratrilayer, AFM intertrilayer. $C$-AFM: AFM intratrilayer, FM intertrilayer. $G$-AFM: AFM intratrilayer, AFM intertrilayer.

Figure 11

Histograms of dipolar-field ${\mu }^{+}$ frequencies from AFM configurations without stripes (Fig. 10) compared to 50-K $\mu \mathrm{SR}$ FT spectrum in ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_8$. See text for notation. Colors as in Fig. 9.

Figure 12

Representative zero-field $\mu \mathrm{SR}$ asymmetry relaxation in ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_8, T>T_N$. Legend: temperature values. Curves: fits of Eqs. (3) and (5) to the data.

Figure 13

Temperature dependence of zero-field dynamic relaxation parameters in ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_8, T>T_N$. (a) Stretched-exponential relaxation rate ${\lambda }^{*}$. Solid curve: fit of ${\lambda }^{*}\left(T\right)\propto T/(T-T_0)$ [32, 33] to the data, $T_0=90\left(1\right)$ K. Dashed curve: fit of Eq. (6) from the CO theory [10] to the data, $J=86\left(22\right)$ K. (b) Stretching power $K$.

Figure 14

Representative longitudinal-field $\mu \mathrm{SR}$ asymmetry relaxation in ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_8, T=200$ K. Legend: values of the longitudinal field $H_L$. Curves: fits of Eqs. (3) and (4) to the data.

Figure 15

Field dependence of longitudinal-field dynamic relaxation parameters in ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_8, T=200$ K. (a) Stretched-exponential relaxation rate ${\lambda }^{*}$. (b) Stretching power $K$.

Figure 16

Representative longitudinal-field $\mu \mathrm{SR}$ asymmetry relaxation in ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_8, T=110$ K. Legend: values of the longitudinal field $H_L$. Curves: fits of Eqs. (3) and (4) to the data.

Figure 17

Field dependence of longitudinal-field dynamic relaxation parameters in ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_8, T=110$ K. (a) Stretched-exponential relaxation rate ${\lambda }^{*}$. (b) Stretching power $K$.

Figure 18

AFM moment configurations A–D in ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_8$ on stripe model 2 [5]. See Fig. 8 for designations.

Figure 19

Histograms of dipolar-field ${\mu }^{+}$ frequencies from AFM stripe models 2A–2D compared to 50-K $\mu \mathrm{SR}$ FT spectrum in ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_8$. Colors as in Fig. 9.

Figure 20

AFM moment configurations A–D in ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_8$ on stripe model 3 [5]. See Fig. 8 for designations.

Figure 21

Histograms of dipolar-field ${\mu }^{+}$ frequencies from AFM stripe models 3A–3D compared to 50-K $\mu \mathrm{SR}$ FT spectrum in ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_8$. Colors as in Fig. 9.

Figure 22

AFM moment configurations A–D in ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_8$ on stripe model 4 [5]. See Fig. 8 for designations.

Figure 23

Histograms of dipolar-field ${\mu }^{+}$ frequencies from AFM stripe models 4A–4D compared to 50-K $\mu \mathrm{SR}$ FT spectrum in ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_8$. Colors as in Fig. 9.

Figure 24

Dashed curve: temperature dependence of transition temperature distribution $P(T_c$), normalized by $2\mathrm{\Delta }{T}_c$ for display. Solid curves: temperature dependencies of spontaneous NMR/NQR/$\mu \mathrm{SR}$ frequencies $f(T,T_c)$ for $T_c^{\min }$ (red), $T_{c0}$ (blue), and $T_c^{\max }$ (green). Dotted curve: spectral weight $w\left(T\right)$.

Figure 25

Representative zero-field frequency spectra from a uniform distribution of second-order magnetic phase transition temperatures. Spectra are normalized by 0.01 for display, and are offset so that each baseline is at the corresponding temperature $T/T_{c0}$ on the $P$ axis. For $T>T_c^{\min }$ spectral weight is transferred from nonzero frequencies to $f=0$.