Fig. 2: Basic physical properties of Ba₂Co_1−xZn_xGe₂O₇ (BCZGO).

a Energy level splitting of 3d electrons at Co²⁺. The free-ion state splits into three levels in the T_d crystal field environment. The lowest four-fold degenerate ⁴A₁ state can be described by S = 3/2 with easy-plane anisotropy (\(D{\hat{S}}_{z}^{2}\)) due to the S₄ crystal field and spin-orbit coupling (SOC). b Schematic magnetic structure of Ba₂CoGe₂O₇ (BCGO). Red arrows represent spin orientations in the ordered phase. The spin canting is exaggerated from the actual angle for visibility (see Supplementary Note 1). c Temperature dependence of electric permittivity ε_a along the a-axis of BCZGO. Zero levels are also shifted. Inset shows x dependence of antiferromagnetic transition temperature T_N. The geometrical percolation threshold x_c = 0.41 of the square-lattice is shown as a red triangle. In x = 0.5, no sign of long-range order was observed down to 0.4 K. The mean-field line of T_N(x)/T_N(0) = 1 − x is shown by a green dotted line.