Magnetism behavior of $T^{\prime}$-type Eu₂CuO₄ revealed by muon spin rotation/relaxation measurements

Figure 1(a) shows the time spectra of ZF $\mu$SR for AS ECO at representative temperatures. A Gaussian-type slow decay due to the nuclear dipole fields is observed at 275 K. Upon cooling to 264 K, an exponential-type spectrum appears, indicating a slowdown of electron spin fluctuations. With further cooling, the spectra show faster decay at 235 K and, subsequently, another phase of slower decay at 164 K. This trend is not a typical ordering sequence; that is, the spin fluctuations slow down monotonically toward the freezing temperature. Furthermore, in the spectra at 164 K and 235 K, an oscillation component with a small amplitude is superimposed on the dominant exponential-type decay. Therefore, a partially ordered spin state coexists with a fluctuating spin state and/or spin-glass-like state at these temperatures. The small oscillation component disappears at 126 K, resulting in a simple exponential-type decay spectrum. A well-defined oscillation spectrum appears at even lower temperatures. Thus, the true magnetic order occurs at a much lower temperature than the previously reported temperature of 265 K Chattopadhyay1994 , and the ordering temperature is comparable to the $T_{\mathrm{N2}}$ of 110 K for $T^{\prime}$-type LCO Hord2010 . Here, we emphasize that the existence of the small oscillation component and the detailed thermal evolution of magnetism on the CuO₂ plane were revealed by the high-quality data with fine temperature steps.

Figure 1(b) shows the ZF $\mu$SR spectra of AN ECO. Similar to the results for AS ECO, exponential-type depolarization appears at $T_{\mathrm{N1}}=250$ K upon cooling (see the spectra at 200, 171, and 132 K). The spectrum at 171 K shows a slower time decay than that at 200 K. It transforms into a faster decay spectrum again at 132 K, following which oscillation spectra appear below $T_{\mathrm{N2}}=110$ K. Thus, the sequential development of magnetism with a low $T_{\mathrm{N2}}$ is the same in AS and AN ECO. One of the qualitative differences between the spectra of AS and AN ECO is the small oscillation component around 200 K, which is absent in the AS ECO spectra. Another difference is the considerably weaker depolarization in AN ECO around this temperature. These results indicate that the magnetism at higher temperatures is varied considerably by annealing.

For quantitative evaluation of the thermal evolution of magnetism, we analyzed the time spectra using the following functions:

where

The first (third) and second terms in Eq. 1 represent the fast (slow) decay and the small oscillation components, respectively. Similarly, the first and second terms in Eq. 2 represent the oscillation and decay components, respectively. $A_{0}$ is the initial asymmetry at $t=0$, and $A_{\mathrm{BG}}$ is the time-independent background. Both $A_{0}$ and $A_{\mathrm{BG}}$ were fixed throughout the analysis. $f_{\mathrm{se}}$ ($f_{\mathrm{osc}}$) is the fraction of the stretched exponential (oscillation) component. $\lambda_{\mathrm{se}}$ and $\beta$ are the relaxation rate and power of the stretched exponential component, respectively. $\lambda_{\mathrm{d1}}$ and $\lambda_{\mathrm{d2}}$ are the damping rates of the oscillations, and $r_{1}$ is the ratio of the damping components. $\omega_{1}$ and $\phi_{1}$ are the frequency and phase of the oscillation, respectively.

As shown in Fig. 2(a), AS and AN ECO exhibit similar temperature dependencies of $\lambda_{\mathrm{se}}$, i.e., with cooling, $\lambda_{\mathrm{se}}$ increases sharply at $T_{\mathrm{N1}}$, maintains a constant value until $\sim$150 K, and increases further below this temperature. However, there are apparent differences between the two samples; in AS ECO, the magnetism around $T_{\mathrm{N1}}$ develops in a rather broader temperature range, and the magnitude of $\lambda_{\mathrm{se}}$ below $T_{\mathrm{N1}}$ is much smaller. These results indicate the enhancement of spin fluctuations by annealing. Upon cooling, the small oscillation component in AS ECO disappears with a reduction in $f_{\mathrm{osc}}$, as shown in Fig. 2(b). This disappearance proceeds concomitantly with the development of $\lambda_{\mathrm{se}}$ toward $T_{\mathrm{N2}}$, suggesting an interrelation between the partially ordered and fluctuating spin states. In both AS and AN ECO, bulk long-range order is observed below $T_{\mathrm{N2}}$ of 110 K, which is comparable to $T_{\mathrm{N2}}$ for $T^{\prime}$-type LCO Hord2010 . Therefore, the static order with a low $T_{\mathrm{N2}}$ is a robust feature of $T^{\prime}$-type $R$CO. Furthermore, $\omega_{1}$ does not exist for 110 K $\lesssim T\lesssim$ 150 K owing to the disappearance of the small oscillation component. The extrapolation of $\omega_{1}$ at $150\>\mathrm{K}<T<T_{\mathrm{N1}}$ toward lower temperatures does not seem to connect smoothly with the values of $\omega_{1}$ below $T_{\mathrm{N2}}$. This discontinuity and the disappearance of the small oscillation component immediately above $T_{\mathrm{N2}}$ suggest that the partial order at high temperatures is not a direct precursor of the static magnetic order below $T_{\mathrm{N2}}$.

As mentioned in Section III.1, we found an anomalous ordering sequence above $T_{\rm N2}$ in AS ECO. Therefore, we performed LF $\mu$SR measurements to obtain further information on the spin states above $T_{\mathrm{N2}}$. Figures 3(a) and (b) show the field dependencies of the $\mu$SR spectra for AS ECO at 187 and 138 K, respectively. At 187 K, where the small oscillation component was observed, the spectra in the longer time region exhibited a parallel upward shift under the magnetic fields. This shift can be observed in the decoupling process of the interaction between muon spins and spontaneous local magnetic fields at the muon sites, providing the magnitude of the spontaneous magnetic fields. In AS ECO at 187 K, an external field of 20 mT is sufficient to shift the overall spectra over the entire time range, suggesting that the static magnetic fields at the muon sites are less than 20 mT in magnitude. Therefore, the partially and weakly ordered spin state exists at 187 K. At 138 K, where the small oscillation component was absent, the spectra slightly changed with an external field of 50 mT, and relaxation remained at 395 mT, as shown in Fig. 3(b). These results indicate that a dynamical nature indeed develops at a lower temperature of 138 K. This anomalous behavior is discussed in relation to the inhomogeneous spin state in Sec. IV.3.

The present $\mu$SR measurements demonstrate that a uniform static magnetic order exists only below $T_{\mathrm{N2}}$ (= 110 K) and a fluctuating spin state exists for $T_{\mathrm{N2}}<T<T_{\mathrm{N1}}$ in AS and AN ECO. These results are consistent with previous $\mu$SR studies for $T^{\prime}$-type LCO Luke1990 ; Hord2010 and contradictory to the absence of a fluctuating state with a higher Néel temperature ($T_{\mathrm{N2}}\approx T_{\mathrm{N1}}$ = 325 K) in $T$-type LCO Uemura1989 . Therefore, the sequential ordering process with a lower $T_{\mathrm{N2}}$ is a common feature of magnetism of $T^{\prime}$-type $R$CO. The variation of the magnetic order in the parent $R$CO with different structures most likely originates from differences in oxygen coordination. The spin direction can be perturbed by the contribution of the Cu 3$d_{3z^{2}-r^{2}}$ orbital to the spin correlations, causing the magnetic order to be more stabilized in $T$-type $R$CO having apical oxygen atoms.

The fact that AS $T^{\prime}$-type ECO ($a$ = 3.9049 Å) and LCO ($a$ = 4.0102 Å Hord2010 ) with different lattice constants $a$ have similar magnetic behaviors is in contrast to the variation of SC properties in AN $R$CO depending on the lattice constant $a$. In $T^{\prime}$-type $R_{2-x}$Ce_xCuO₄ with a larger $a$, the critical Ce concentration for the appearance of the SC phase is smallerNaito2002 ; Fujita2003 ; Krockenberger2008 ; Krockenberger2014 and the optimal SC transition temperature is higherNaito2002 ; Maple1990 . Thus, the appearance of non-doped superconductivity is not directly related to the lower $T_{\mathrm{N2}}$.

We note that the neutron diffraction measurements of $T^{\prime}$-type $R$CO showed magnetic Bragg peaks below approximately $T_{\mathrm{N1}}$ but no evidence of an anomaly around $T_{\mathrm{N2}}$ Chattopadhyay1994 . The different ordering temperatures determined using the $\mu$SR and neutron diffraction measurements originate from the different time scales to observe the spin dynamics, which are typically $\sim$10^-4 s to $\sim$10^-11 s and $\sim$10^-9 s to $\sim$10^-14 s, respectively. Therefore, a fluctuating spin state with a time scale of $\sim$10^-4 to $\sim$10^-9 s would be realized for $T_{\mathrm{N2}}<T\leq T_{\mathrm{N1}}$. (See Fig. 5.)

The negligible annealing effect on $T_{\mathrm{N2}}$ in $T^{\prime}$-type ECO appears to be different from the annealing-induced suppression of AFM order reported for $T^{\prime}$-type LECO. As mentioned in the Introduction, reduction annealing plays two roles: electron dopingAsano2018 and the removal of chemical defect Radaelli1994c ; Schultz1996 ; Kang2007 . Because the $\delta$ values for the present ECO and LECO samples are comparable Asano2020 ; Takagi1989 ; Armitage2010 , the overall structural change due to annealing is expected to be the same for both samples.

Recent x-ray absorption near-edge structure measurements revealed a Mott insulating state in both AS LECO and NCO but a significant increase of electron density due to annealing ($n_{\rm AN}$) in LECO (0.4 electrons per Cu), which is much larger than that in NCO (0.05 electrons per Cu) Asano2020 . On this basis, a distinct electron-doping process was proposed: the self-doping of electrons and holes associated with the collapse of the charge-transfer gap for the former with a smaller gap size ($\Delta_{\rm CT}$) and electron doping into the upper Hubbard band for the latter with a larger $\Delta_{\rm CT}$. According to this scenario, the present ECO sample, which has a relatively large $\Delta_{\rm CT}$ Arima1991 ; Braden2005 , follows the $n_{\rm AN}=2\delta$ relation Asano2018 , as is the case for NCO. Therefore, the distinct effects of annealing on the long-range magnetic order in ECO and LECO Adachi2016 can be interpreted as due to different induced carrier densities. The suppression of the ordered state in LECO is attributed to the generated charges with a large $n_{\rm AN}$ value, while the negligible variation in $T_{\mathrm{N2}}$ and the small reduction of $\omega_{\rm 1}$ in ECO are likely to have originated from electron doping with a low $n_{\rm AN}$ value (= 2$\delta$ $\sim$0.04 electrons per Cu).

Next, we discuss the partially ordered spin state in AS ECO. Because this state disappears with annealing, its origin is argued from the viewpoint of the annealing effect. To exclude the possibility of electron doping aspect of annealing treatment, we performed additional ZF $\mu$SR measurements on AS Eu_2-xCe_xCuO₄ (ECCO) with $x$ = 0.07, 0.10, and 0.14. If the short-range magnetic order is related to the doping level, the small rotation component should vanish in AS ECCO with $x$ $\geq$ 0.04, the electron density in which is greater than that in AN ECO. Figure 4 shows normalized time spectra at $\sim$180 K. As indicated by arrows, the small oscillation component can be observed on the dominant exponential-type decay component for $x$ $\leq$ 0.10. Therefore, in contrast to long-range magnetic order, which is varied by the electron-doping effects of the annealing process, the partial order for 150 $<$ T $<$ $T_{\mathrm{N1}}$ is an intrinsic property of AS compounds independent of the doping level. Note that in $x$ = 0.14, no evidence of oscillation was detected at 181 K, because this temperature is higher than $T_{\mathrm{N1}}$.

To understand the origin of the partial order, the appearance of oscillation immediately below $T_{\mathrm{N1}}$ is noteworthy because it indicates a close connection between the partially ordered state and the dominant fluctuating spin state. In the slowly fluctuating spin state, chemical defect could become pinning centers for the fluctuating spins and induce local magnetic order Adachi2013 . The structural feature in AS samples containing chemical defects is qualitatively consistent with the appearance of the small oscillation component below $T_{\mathrm{N1}}$. Reduction annealing can remove such defectsRadaelli1994c ; Schultz1996 ; Kang2007 , resulting in the suppression of local magnetism and the enhancement of the dynamical spin nature. In AN ECO, the non-monotonic evolution of the muon spin relaxation suggests that a tiny amount of defects remains.

Finally, we discuss the disappearance of the small oscillation component and the anomalous enhancement in the dynamical spin nature of AS ECO for $T_{\mathrm{N2}}<T<150$ K with cooling. When the correlation length of the pristine fluctuating spin state develops toward static order, the partially and weakly ordered region contributes to the dominant state owing to the increase of magnetic energy. Consequently, the partial order could disappear immediately above $T_{\mathrm{N2}}$, where the correlation length of the fluctuating state diverges Endoh1989 ; Birgeneau1999 ; Motoyama2007 . Based on the present results, this thermal evolution of the magnetic behavior is schematically drawn in Fig. 5. The figure depicts the case in which $\sim$3$\%$ of pinning centers (open circle) exists on the CuO₂ plane. At high temperatures below $T_{\mathrm{N1}}$, static order is locally induced around the pinning center owing to the slowing down of spin fluctuations in the time scale of the $\mu$SR measurement. The ordered region shrinks upon cooling and vanishes immediately above $T_{\mathrm{N2}}$. The figures in the right column show the corresponding position dependence of the frequency of spin fluctuations along the red line near the pinning centers. All these states are observed as static order in the neutron scattering measurements.