Spin-lattice coupling of $R_{\mathrm{1-x}}$Lu$_{\mathrm{x}}$B$_{\mathrm{4}}$ revealing anomalous weak ferromagnetism ($R=$Sm, Gd, Tb, Dy, Ho)

$R$B$_{\mathrm{4}}$ ($R=$rare-earth elements) compounds exhibits antiferromagnetic ordering at low temperature and are classified as the Shastry-Sutherland lattice, which is a geometrically frustrated system. In previous study, it was reported that Y substitution in TbB$_{\mathrm{4}}$ single crystals causes anomalous WF (weak ferromagnetism) even though Y$^{\mathrm{3+}}$ is non-magnetic. The disturbance of a delicate equilibrium in a frustrated system can lead to new electronic and magnetic states. In this study, single crystals of $R_{\mathrm{1-x}}$Lu$_{\mathrm{x}}$B$_{\mathrm{4}}$ ($R=$Sm, Gd, Tb, Dy, Ho), (x$=$0 \textasciitilde 0.8) were synthesized. WF is also observed. TbB$_{\mathrm{4}}$ went through orthorhombic distortion below N\'{e}el temperature. To investigate the existence of orthorhombic distortion in TbLu$_{\mathrm{x}}$B$_{\mathrm{4}}$ (x$=$0.1, 0.35), high resolution single crystal x-ray diffraction was performed at 5 K. It was confirmed that the distortion was vanished with Lu substitution. Interestingly, lattice constant $a$ was increased with decreasing temperature below the T$_{\mathrm{C}}$. The strong correlation between spin-lattice coupling and WF will be discussed in detail.