Strong spin-orbit coupling and Zeeman spin splitting in angle dependent magnetoresistance of Bi₂Te₃

$lϕ∼T−12$ indicates the 2D nature of the carriers. In thin TI film, both the surfaces are coupled with a gap opening at the Dirac point, and the otherwise bulk states become quasi-2D due to confinement in one direction^{4,34,39,59–61} and they become strongly coupled with the surfaces through scattering allowing the system to behave like a single phase-coherent channel.^{18,23,34,35,61}

We assume parabolic quasi-2D CB with an effective mass of 0.178 times⁶² the rest mass of an electron, the bottom of CB being about 0.3 eV above the Dirac point,⁶³ and a linear 2D Dirac cone with the Fermi velocity of 4.0 × 10⁵ m/s.^45,63

The film thickness $d<(ℏ/(4eB)2)$ should hold for HLN equation to be valid in 2D. Here, d = 4 nm and $ℏ/(4eB)2=4.3 nm$ for B = 9 T. So for high field 2D HLN equation is less valid, which may be a reason for very little deviation in fitting near 9 T.

This is an upper limit on $τSOZ$ obtained from fitting, as lowering $τSOZ$ further is insensitive to fitting due to the $η(x)$ function. So actual value of $τSOZ$ could be lower than our estimate.

Although the EEI theory of Lee-Ramakrishnan is not derived under strong SOC, we have not considered any further theory of EEI in strong SOC system for better fitting of perpendicular field MR data, as original HLN theory alone can explain it.

The free space permittivity is used for calculation, as the film thickness is comparable to the mean in-plane distance between electrons (order of $kF−1$⁠) and most of the electric field lines responsible for EEI pass through the surrounding media.⁶⁴ 

To fit the angular dependency of the data with HLN equation, $σ(B,θ=90°)$ data-point was replaced by $σ(B=0)$⁠; as the change in MR for perpendicular field is not due to WAL effect but due to the Zeeman effect, which is suppressed for θ < 80°.²³