4.2 Spin-ladder material Sr_n-1Cu_n+1O_2n

The spin-ladder material Sr_n-1Cu_n+1O_2n, which is synthesized under high pressure and high temperature [49,50], takes the structure shown in Fig.26a. The lattice structure is composed of (n+1)/2-leg spin-ladders, namely, strips of CuO₂ square lattice which have (n+1)/2- Cu²⁺ ions across their width. Each Cu²⁺ ion has spin S=1/2 with antiferromagnetic couplings within a ladder (strength: J), both in the `rung' and the `leg' directions. In the two directions, the difference of the coupling strengths are presumably small, because the 180$^{\circ}$ Cu-O-Cu bond lengths are almost equal for both directions [49].

  

Figure 26: (a) The `spin ladder' structure of Sr_n-1Cu_n+1O_2n. Oxygen ions locate at each corner of the squares. The figure shows the 3-leg ladder structure (n=5). (b) A magnified ladder-edge. The dotted lines represent the 90$^{\circ}$ Cu-O-Cu bonds which mediate the ferromagnetic inter-ladder interaction (-J').

In this material, the neighboring ladders are displaced by half the lattice constant, leading to a small inter-ladder interaction. Since the 90$^{\circ}$ Cu-O-Cu bonds mediate the inter-ladder interaction, the interaction may be ferromagnetic (-J'; see Fig.26b). The magnitude of the inter-ladder interaction has been theoretically estimated as J'/$J\approx 0.1-0.2$ [19,21]. The inter-ladder interaction brings about a geometrical frustration of the spins at the edge of the ladders, because of the triangular structure constituent of two ferromagnetic interactions (-J') and one antiferromagnetic interaction (J) [19,21].

Previous investigations of Sr_n-1Cu_n+1O_2n have measured magnetic susceptibility (Fig.27; [51]) and ⁶³Cu-NMR (Fig.28; [52]). In the 2-leg ladder system (n=3), the temperature dependence of the susceptibility and the T₁ relaxation rate are well described by thermal excitations over a gap, which may correspond to the spin gap between the non-magnetic ground state and magnetic excited states. The magnitude of the gap has been reported as 420 K (susceptibility: [51]) and 680 K (⁶³Cu-NMR: [52]).

The 3-leg ladder system (n=5), in contrast, has a finite susceptibility in the T$\rightarrow$0 limit, demonstrating that the ground state of this system can respond to the external magnetic field. Therefore, the ground state may exhibit magnetic order. In the 3-leg ladder system, the T₁ relaxation rate of ⁶³Cu nuclear moments was so large that it was hardly measurable with the conventional NMR technique. This result implies the existence of strong magnetic correlations in the 3-leg system [52].

As introduced in Chapter 2, continuous-beam muon spin relaxation ($\mu$SR) is a NMR-like local magnetic probe, but with a higher timing resolution ($\stackrel{<}{\sim}$1 ns) than typical NMR methods ($\sim$10 $\mu$s). Consequently, $\mu$SR is an adequate probe to study the 3-leg ladder system, in which the NMR relaxation rate was beyond its time resolution. Another advantage of $\mu$SR is its high sensitivity to small and/or dilute static moments. Using $\mu$SR one can best investigate the expected absence of static order in the 2-leg ladder system.

  

Figure: Susceptibility of the spin ladder materials Sr_n-1Cu_n+1O_2n. Cite from Ref. [51].

  

Figure: Temperature dependence of the ⁶³Cu-NMR T₁ relaxation rate (cite from [52]). The right axis is an estimated muon spin relaxation rate for the same relaxation mechanism. (See the discussion later.)

For our $\mu$SR measurements, polycrystalline specimens of the spin ladder cuprates (Sr_n-1Cu_n+1O_2n; n=3, 5) were prepared at the Institute for Chemical Research, Kyoto University, using a cubic anvil-type high pressure apparatus [53]. Powder X-ray analysis of our samples showed the stoichiometric ladder structure, except for small amounts ($\sim$10 Cu at.% ) of a CuO impurity phase [51]. Since CuO is an antiferromagnet ($T_{\rm N}$$\sim$230 K [54]), the impurity phase should not affect the muons which did not land within an impurity cluster. Therefore, in our $\mu$SR measurements $\sim$90% of the signal amplitude comes from the pure ladder structure.