Fig. 4: Magnetoresistance in HfTe₅ under different strains.

a \({\rho }_{{xx}}\) as a function of perpendicular magnetic field (B) measured under different strains of B1. Inset: Schematic of the measurement setup. b Oscillatory part of the resistivity (\({\Delta \rho }_{{xx}}\)) after background subtraction, plotted vs. 1/ B in the field range from 0.25 T to 1 T. \({\Delta \rho }_{{xx}}\) is vertically shifted for clarity for strains \({\epsilon }_{1}\) and \({\epsilon }_{2}\). The bright solid green lines represent the LK fit (See Methods). The arrows mark the positions of integer Landau levels. c \({\rho }_{{xx}}\) vs. B measured at T ~ 1.5 K at different angles of B2 at \(\epsilon \sim 4.5\%\). d Angular dependence of the oscillation frequency extracted by the fast Fourier transform (FFT) for samples F2 and B2 (\(\epsilon \sim 4.5\%\)). The solid lines represent the fitting of a 3D ellipsoid model \(F\left(\theta \right)={F}_{\perp }{F}_{//}/\sqrt{{({F}_{//}\cos \theta )}^{2}+{({F}_{\perp }\sin \theta )}^{2}}\) (black) and a 2D model \(F\left(\theta \right)={F}_{\perp }/\cos \theta\) (red), where \({F}_{\perp }\) (\({F}_{//}\)) is the frequency under a perpendicular (parallel) magnetic field. Inset shows the configuration between the sample and the magnetic field. The X error bars represent the instrumental error when controlling the gear baser rotator. The frequency error bars are caused by the uncertainty of the Fast Fourier Transform (FFT) peak positions. e \({\rho }_{{xx}}\) as a function of parallel magnetic field (B) at different strains of B1, measured with B // I (for this we rotate the sample and aligned the crystal a-axis to (B)). The red dashed line is a guide for the linear magnetic field dependence of \({\rho }_{{xx}}\) vs. B measured under strain \({\epsilon }_{2}\). Inset: Schematic of the measurement setup. f Longitudinal magnetoresistance (LMR) vs. B (up to 2 T) for B1 at different strains and B2 at \(\epsilon \sim 4.5\%\). The bright solid lines represent the fittings with different equations, \({LMR}\left({{{{{\boldsymbol{B}}}}}}\right)=\eta {{{{{{\bf{B}}}}}}}^{2}\) for \({\epsilon }_{0}\), \({LMR}\left({{{{{\bf{B}}}}}}\right)=\frac{1}{\sigma \left(0\right)+{\eta }_{1}{{{{{{\bf{B}}}}}}}^{2}}-\frac{1}{\sigma (0)}+{\eta }_{2}{{{{{{\bf{B}}}}}}}^{2}\) for \({\epsilon }_{1}\), and \({LMR}\left({{{{{\bf{B}}}}}}\right)={k|}{{{{{\bf{B}}}}}}|\) for B1 at \({\epsilon }_{2}\) and B2 at \(\epsilon \sim 4.5\%\). Where \({\eta }_{1}\), \({\eta }_{2}\), σ(0), and k are fitting parameters. Inset: Zoomed-in plot of LMR vs. B (up to 3 T) for strain \({\epsilon }_{1}\).