5.2.2 $\mu$SR measurements

Fig. 40a shows the $\mu$SR spectra of the nominally pure (x=y=0) system. There was slow relaxation in zero-field (ZF, 2.8 K), but the relaxation did not disappear in an external longitudinal field (LF=100 G). This LF measurement proves that the relaxation is in the fast fluctuation regime; if the slow relaxation in zero-field were due to a static random field distribution, it should have been decoupled in a small LF$\sim$5 G (see section 3.4). In the nominally pure system, we have confirmed the absence of static order down to 100 mK.

The slow relaxation of the muon spin follows a square-root exponential function ($P_{\mu}(t)$$\approx$$\exp(-\sqrt{\lambda t})$; solid lines in Fig.40a), which is characteristic of dilute spin systems in a paramagnetic state (see section 3.2). As discussed later, the slow T₁ relaxation is most likely caused by native unpaired spins in the sample.

  

Figure 40: (a) $\mu$SR spectra of the nominally pure Y₂BaNiO₅. The solid line is the fit with the square-root exponential function. (b) LF=100 G $\mu$SR spectra are compared at T$\leq$100 mK. The solid lines are the fit with the stretched exponential functions.

In Fig. 40b, we compare $\mu$SR spectra from the pure, charge doped (Ca; x=4.5 and 9.5%) and vacancy doped (Mg; y=4.1%) systems in the milli-Kelvin regime. In the Ca doped, x=9.5% sample, there is fast muon spin relaxation, reflecting the spin-glass behavior in the susceptibility. In the Mg doped y=4.1% sample, muon spin relaxation is even slower than in the nominally pure system, which suggests an absence of static order in the Mg doped systems. This point was also confirmed with longitudinal field decoupling measurements (Fig.42).

In order to obtain the muon spin relaxation rate ($\lambda$), we analyzed the spectra with a phenomenological stretched exponential function, $\exp(-(\lambda t)^\beta)$, which describes paramagnetic relaxation with $\beta = 0.5\sim 1$ (see Chapter 3), as well as the slow fluctuation regime of dilute spin systems ($\beta\sim 1$) and the dense spin systems ($\beta\sim 2$). One problem of this universal relaxation function is that it often shows correlations between $\beta$ and $\lambda$, when the relaxation rate ($\lambda$) is small. Therefore, it is safer to fix $\beta$ for the analysis of the fast fluctuation regime, in order to obtain the appropriate temperature and/or field dependence of the relaxation rate ($\lambda$).

For the analysis of the nominally pure sample, we fixed as $\beta=0.5$, namely, to the square-root exponential function. The overall fit was good, as shown in Fig.40a. In the Mg-doped systems, the $\mu$SR spectra do not exhibit the fast front-end (Fig.40b), suggesting $\beta \gt 0.5$.We fixed $\beta$ to 0.72 (Mg 1.7%) and 0.77 (Mg 4.1%) which is the average of $\beta$, obtained from a preceding analysis without constraints on $\beta$. For the Ca-doped systems at T> 6 K, we adopted $\beta$=0.5. Below 6 K, we were able to obtain $\beta$ and $\lambda$ independently. It was found that $\beta$ of the Ca-doped systems approaches $1.5\sim 2$ in the milli-Kelvin regime. In Fig.41, relaxation rates ($\lambda$) for the LF=100 G measurements (and for the pure sample, the results of higher LF measurements as well) are shown, as a function of temperature.

  

Figure 41: Temperature dependence of muon spin relaxation rate ($\lambda$). For the doped samples, the results from LF=100 G measurements are shown; for the nominally pure sample, the data in higher LF's are shown as well. The solid lines (for the doped systems) and the dashed lines (for the pure system) are guides to the eyes.