(c) and (d) show the EECs at various energies in the (c) absence and (d) presence of $E_z$ for $\lambda > |\vec{M}|$. ( $\vec{M} = 0.5\ \hat{y}$, $\lambda = 0.2$ in (a) and (b); $\vec{M} = 0.2 \hat{y}$ in (c) and (d). $E_z = 0.1$ in (b) and (d). ) []{data-label="gEZeec"}](EZeec.eps)

Fig. \[gEZeec\] compares the EECs in both the $\lambda < |\vec{M}|$ regime as well as the $\lambda > |\vec{M}|$ regimes in the presence and absence of the $E_z \tau_z$ term. The asymmetry between the upper and lower surfaces of the TI film due to the $E_z\tau_z$ term results in the states stemming from the Dirac cones corresponding to the two surfaces being shifted in opposite directions along the energy axis. The dispersion relations become ‘tilted’, and the EECs at a given value of energy becoming asymmetrical in $k$ space.

This asymmetry then results in a net out of plane spin accumulation after integrating over all the $k$ space points spanned by the EECs. Evidently, the spin accumulation increases with the magnitude of $\vec{M}$ and $E_z$. What is perhaps more interesting is the variation of $\langle \sigma_z (E) \rangle $, the out of plane spin accumulation integrated over the EECs 