space points on the EECs where the spin accumulations already have large $x$ components.

The out of plane spin $z$ accumulations in the preceding figures are antisymetrically distributed in $k$ space on the EECs. This antisymmetry results there being no net out of plane spin accumulation in $k$ space after integrating over the entire Fermi surface. In order to break the antisymmetry, we now introduce a term $E_z \tau_z$ to the Hamiltonian Eq. \[tfHam0\] so that the Hamiltonian now reads $$H = (\vec{k}\times\vec{\sigma})\cdot\hat{z}\tau_z + \vec{M}\cdot\vec{\sigma} + \lambda \tau_x + E_z\tau_z. \label{tfHam1}$$

The $E_z\tau_z$ term introduces an asymmetry between the upper and lower surfaces. This asymmetry may physically result from the fact that in experimentally grown TI thin films the bottom surface of the film is in contact with the usually non-ferromagnetic substrate, and the upper surface either in contact with the vacuum (for $\vec{M}\cdot\vec{\sigma}$ being due to magnetic doping [@PRL102_156603; @Sci329_659; @NatPhy7_32] ) or with a FM layer (for $\vec{M}\cdot\vec{\sigma}$ being due to the proximity effect with a FM layer [@PRL104_146802; @PRL110_186807; @PRB88_081407] ) .

![ Panel (a) and (b) show the EECs at various energies in the (a) absence and (b) presence of $E_z$ for $\lambda < |\vec{M}|$. Panels 