Figure 4. Stable charge neutrality point and 2D conductance in e-beam irradiated topological materials.

(a) Sheet resistance R_xx versus temperature of Bi₂Te₃ crystal at the charge neutrality point (red line) exhibits a plateau at low temperatures—a thumbprint of 2D surface conduction, see right panels in e and f). We note that in Bi₂Te₃, CNP and Dirac point do not coincide, with the Dirac point situated within a valley in the bulk valence bands⁵, and is expected to reflect that²³. Magnetic field breaks time reversal symmetry and gaps Dirac bands, resulting in localizing behaviour shown as dash. Inset: optical image of the crystal showing van der Pauw contact configuration used. (b) Annealing protocol with time steps Δt=30 min implemented to tune Bi₂Te₃ crystal with dose 1 C cm⁻² back to stable CNP. Inset: magnetoresistance at 1.9 K after each annealing step, with colours matched to indicate different annealing temperatures. (c,d) ARPES spectra of a Bi₂Te₃ crystal irradiated with electron dose of 1.7 C cm⁻² taken along Γ−K direction in the Brillouin zone. (c) Before annealing, the irradiated sample is n-type, with Dirac point at E_DP∼−290(10) meV relative to the Fermi level E_F. (d) After annealing at 120 °C, Dirac point upshifts to a binding energy of E_DP∼−160(10) meV. The same shift is seen for the scans along Γ−M. (e) WAL low-field quantum interference correction to the linear-in-field magnetotransport ((f), also see text) at CNP in a Bi₂Te₃ crystal at 1.9 K with its characteristic low-field cusp. The 2D character of WAL is evident in its scaling with transverse field H_‖=Hcosθ, where θ is the tilt angle of the field measured from sample's c axis. A fit to 2D localization (HLN) theory²⁶ (solid line) confirms that the contribution is only from two surfaces and yields a dephasing field . (f) Linear magnetoresistance at CNP shows 2D scaling with H_‖. (g) WAL contribution at CNP in a Bi₂Se₃:Ca(0.09%) crystal at 1.9 K also scales with H_‖. At high fields outside the cusp, the scaling is seen to fail for θ≳60°. A fit to HLN theory (solid line) again confirms the contribution only from two surfaces and yields a smaller dephasing field B_ϕ∼0.004 T (corresponding to dephasing length l_ϕ∼220 nm). (h) Linear magnetoresistance at CNP also scales with H_‖.