7 Concluding remarks

 In this thesis work, I have presented $\mu$SR investigations of several many-body singlet ground state materials, and studied their response to vacancy/charge doping. Since the spin singlet is a most apparent quantum mechanical state of a spin system, this thesis work was intended to detect how quantum mechanics are realized in macroscopic spin systems. I also presented how quantum mechanical ground states are destroyed upon the perturbations due to doping.

The spin ladder materials Sr_n-1Cu_n+1O_2n, which alternate between a classical and quantum mechanical ground state were presented first. The experimental results supported the theoretical predictions about the ground state structures; the spin-gapped, non-magnetic ground state for the 2-leg ladder system, and a gap-less, ordered ground state for the 3-leg system.

The Haldane material Y₂BaNiO₅ was confirmed to take the non-magnetic ground state. With the Mg doping to the Cu site, this ground state was found to be stable against the vacancy doping on the spin chain. If positive charge is introduced to the chain, though, the non-magnetic ground state was strongly perturbed; on a macroscopic time scale, the system exhibited a spin-glass like signature, such as history dependent susceptibilities. On the other hand, on a microscopic time scale, however, the spins kept fluctuating down to milli-Kelvin regime in a unconventional way. If one calls a fluctuating spin ground state a `spin liquid', the charge doped Haldane system may be called as a `viscous' spin liquid.

The spin Peierls material CuGeO₃ exhibited a non-magnetic ground state. But, with a small amount of non-magnetic perturbation, a classical ground state was induced to appear; well defined Néel order was confirmed with a spontaneous muon spin precession in a Si-doped single crystal.

The experimental results presented in this thesis may be summarized as in the next table:

 

system pure 2cdoping

vacancy charge

3-leg ladder Néel/SG ? ?

2-leg ladder non-mag. ? ?

Haldane non-mag. non-mag. viscous liquid

spin Peierls non-mag. Néel order ?

In all of the singlet ground state materials investigated, the non-magnetic ground state was realized in the pure systems, and it induced muon spin relaxation in the fast fluctuation regime. But, when vacancy and/or charge is doped, the response to these perturbations seems to reflect the characters of the individual spin systems. For example, with a small amount of vacancy introduced to the chain, the spin Peierls material (Cu_1-xZn_x)GeO₃ exhibited bulk Néel order, while the Haldane material Y₂Ba(Ni_1-yMg_y)O₅ preserved the singlet ground state. The contrasting response to the vacancy doping may reflect the difference in the ground state structures; it seems that the ground state constituent of isolated singlet pairs are unstable against vacancy doping. To test this hypothesis, $\mu$SR investigations of the Zn-doped 2-leg ladder materials Sr(Cu_1-xZn_x)₂O₃ is underway.

The experimental results presented in this thesis may be summarized in the next figure:

  

Figure 62: Muon spin relaxation rate in the ground state $\lambda (T\!\rightarrow\! 0)$, as a function of low temperature Curie constant $C^{\rm LT}$. The open symbols correspond to relaxation in fast fluctuation regime; the closed symbols, to the static relaxation. The symbols with a cross inside indicates unconventionally dynamic relaxation. The dashed lines are the estimate of the paramagnetic relaxation rate due to doping induced unpaired moments. (see the second subsection of section 5.2.3.)

In this plot, the horizontal axis represents characteristic magnitude of perturbations to the system; if a singlet ground state is doped with vacancy and/or charge, the dopant creates unpaired spins which manifest themselves as the low temperature Curie term. The vertical axis reflects how the singlet ground state is perturbed upon doping; if majority of the moments remain singlet and muons detect the dipolar fields from the doping induced local moments, the plot should fall in the region of dashed lines. The pure/vacancy doped Haldane system Y₂Ba(Ni_1-yMg_y)O₅ seems to realize this situation. If the system shows bulk magnetic order, the relaxation becomes static (closed symbols) with a saturated relaxation rate, as the Zn doped CuGeO₃ exhibited.

One interesting question is why the pure CuGeO₃ and the 2-leg ladder system SrCu₂O₃ doesn't fall in the dashed-line region. There may be two explanations; (1) the native unpaired spins are already correlated and their fluctuation rate ($\nu$) becomes slow, and (2) muon itself works as an impurity to the system, and creates unpaired spins around it; namely, the dipolar field width (a) at the muon site becomes larger than that from the native unpaired spins observed in susceptibility. In the 2-leg ladder system, the enhanced dipolar field widths (a) estimated from the longitudinal field measurements (Table 4 in Chapter 4) seem to suggest a possibility of the latter case, namely, a muon-impurity effect.

Among all the results presented in this thesis, the most interesting one is the unconventional spin dynamics of the charge doped Haldane system. Since this problem involves two degrees of freedom, namely, the `spin' and the `charge', theoretical understanding may be challenging. But this system should contain interesting dynamical features inside.