Effects of the inter-layer couplings for the orthogonal dimer system SrCu$_2$(BO$_3$)$_2$ are discussed. The spin-gap $\Delta$ of the three-dimensional model is independent of the inter-layer couplings when they are small. Therefore at low temperatures ($T < \Delta$) thermodynamic properties are described well by the two-dimensional model. On the other hand at high temperatures the mean-field type scaling ansatz is useful to discuss the magnetic susceptibility for week inter-layer couplings. From fit of the magnetic susceptibility, the estimated coupling constants are J = 85 K for the nearest-neighbor couplings, $J^{ '} = 54$ K for the next-nearest-neighbor couplings, and $J^{ ''} = 8$ K for the inter-layer couplings. These parameters are consistent with the temperature dependence of the specific heat at low temperatures.
Categories: cond-mat.str-el