Unconventional dynamics in the charge doped systems

In the milli-Kelvin regime of the charge doped systems, we observed a Gaussian relaxation which showed much weaker LF decoupling than the Gaussian Kubo-Toyabe theory predicts. One idea to explain the unconventional Gaussian relaxation is to introduce dilution of the local field in a time-wise manner [106]. As has been presented in Chapter 3, conventional Kubo-Toyabe theory assumes that the local fields exist at all times, fluctuating with a time independent fluctuation rate ($\nu$). For the corresponding Gaussian Kubo-Toyabe function in the slow fluctuation regime, the zero-field relaxation rate ($\Delta$)reflects the Gaussian field width, and the decoupling longitudinal field is comparable to that width ($H_{\rm LF}\sim\Delta/\gamma_\mu$;see section 3.1). Now, suppose the local field is almost static during a certain fraction of time ($0<f\leq 1$), and fast fluctuating otherwise (Fig.44). If the muon spin relaxation occurs only during the slow fluctuation time, a Gaussian relaxation rate may be observed, but with a diluted relaxation rate as $\Delta\rightarrow f\cdot\Delta$. Even in this unconventional situation, the decoupling field $H_{\rm LF}$ will remain comparable to the instantaneous internal-field width $\Delta/\gamma_\mu$, because the decoupling happens as the competition between the external longitudinal field $H_{\rm LF}$ and the internal random fields $\Delta/\gamma_\mu$.Therefore, in this hypothetical situation, the decoupling may happen at much larger longitudinal fields than the zero-field relaxation rate suggests.

  

Figure 44: Hypothetical field dynamics which has slow fluctuations randomly appearing in the quickly fluctuating majority.

Taking into account the slow paramagnetic relaxation caused during the fast fluctuation time, the muon spin relaxation due to this unconventional field fluctuations may be expressed as [106]:

where $G^{\rm DGKT}$ is the dynamical Gaussian Kubo-Toyabe function (see section 3.1), which originates from the slow fluctuation time, and the stretched-exponential part from fast fluctuation time. We have analyzed our $\mu$SR spectra with this hypothetical relaxation function; Fig.45 shows the fits to the data from the Ca doped x=9.5% sample. The overall longitudinal field dependence is described by mainly adjusting the time-fraction parameter $f\sim 0.2$.

  

Figure 45: The same LF-$\mu$SR data with Fig.43. The solid lines are the fit with the hypothetical relaxation function (eq.44) to the LF=100, 200, 500, 1k and 2kG data.

Using the same hypothetical relaxation function (eq.44), we have analyzed the spectra from other Ca-doped systems (x=4.5, 14.9 and 30.5%). The fraction parameter (f) and the instantaneous Gaussian width ($\Delta$) are shown in Fig.46. It was found that the Gaussian width ($\Delta$) is almost independent of the Ca concentrations (x), while the fraction parameter (f) reflects the concentrations: a smaller Ca concentration results in a smaller fraction parameter (f), indicating more unconventional fluctuations (note that regular Markoffian spin fluctuations are described with f=1). The indifference of the Gaussian width ($\Delta$) to the charge concentrations suggests that the muon spin relaxation mechanism is common to all the charge doped systems, but how frequently the relaxation is caused ($\approx f$) is determined by the charge concentration.

  

Figure 46: (a) The fraction parameter (f) and (b) the instantaneous Gaussian width ($\Delta$)obtained from the analysis using the hypothetical relaxation function (eq.44).

One possible description of the relaxation mechanism is that the doped hole, which takes a localized state with hopping [103], occasionally comes close to the muon site and induces muon spin relaxation. When the charge is far away, the muon relaxation should be small and dynamic, because the majority of the spins on the chain may stay in the non-magnetic ground state.

Phenomenologically, the magnetic behavior of the charge doped Haldane material (Y_2-xCa_x)BaNiO₅ is very similar to that of the Kagomé-lattice compound (SrCr_zGa_12-zO₁₉), a geometrically frustrated antiferromagnet of Cr moments (S=3/2). The susceptibility measurements of the Kagomé-lattice system [107,108,109] have revealed the existence of a small portion of unpaired spins ($\sim 5$% of the total Cr ions for the z=8 sample), which exhibit a spin-glass-like history dependence below $T_{\rm g}\sim 3.5$ K. The unpaired moments observed in the susceptibilities are probably caused by Ga substitutions to the Cr sites, which are inevitable in this series of Kagomé compounds [109]. The ZF-$\mu$SR spectrum of the z=8 Kagomé material approaches a Gaussian shape as $T\rightarrow 0$ [106], while LF-$\mu$SR measurements at 100 mK suggest fast field fluctuations [95,106,110]. Neutron scattering measurements of the z=7.1 Kagomé compound [111] have also suggested persistent dynamics below the cusp temperature; a large fraction ($\sim 80\%$) of the scattering intensity was found to remain in the inelastic channel at $T/T_{\rm g}\sim 0.5$.

Theoretically, the S=1/2 Kagomé-lattice system may have a ground state composed of many singlet pairs [112], as expressed by the resonating valence bond (RVB) state. In this situation, the unpaired spins created by non-magnetic ion doping still have the ability to migrate spatially, because the surrounding spins have a large number of combinations for their singlet pairings. In the charge doped Haldane system, the doped charge may also move, with the surrounding spins in the singlet state. Considering these similarities, the persistent dynamics below the spin-glass like cusp temperature, as well as the hardly decoupled Gaussian relaxation of $\mu$SR spectra may be common signatures for migrating unpaired spins in singlet ground state materials.