calculate using Eq. \[c2oe\] . Fig. \[gspinZpoln\] shows the out of plane spin $z$ accumulation generated at various $k$ space points on the EECs of a TI thin film with $\lambda > |\vec{M}|$ (panels (a) and (b) ), and $\lambda > |\vec{M}|$ (panels (c) and (d) ) for electrical fields applied in the $x$ ( panels (a) and (c) ) direction perpendicular to the magnetization, and the $y$ direction ( panels (c) and (d) ) parallel to the magnetization. The sign of the resulting spin $z$ accumulation can be understood in terms of how the applied electric field changes the direction of the SOI field experienced by the charge carriers. We noted in our earlier discussion in Sect. \[spinhalf\] that each point on the EECs may be associated with the Fermi circle of either the $+(\vec{k}\times\vec{\sigma})\cdot\hat{z}$ Dirac cone, or the $-(\vec{k}\times\vec{\sigma})\cdot\hat{z}$ cone. This is also indicated on the left panels of Fig. \[gspinZpoln\] where the two Fermi circles are indicated by dotted circles of different colors. Consider now the $k$ space region denoted in the inset of panel (b). The inset shows the spin accumulations on two points in $k$ space with the red (blue) arrows denoting the spin accumulation 