spin Hall effect [@Kane2005; @Fu2007] to the bandgap engineering. [@Novoselov2007] As we will show later, the mass of Eq. (\[eq:mass\]) is indeed the most important quantity that governs the dynamical conductivity for photon energy of interest.

The wavefunction can be read from the following basic pattern of the first and second layers, $$\begin{aligned}
 |\Psi_{r{\bf p}}^s \rangle 
 =
 N_{r{\bf p}}^s
 \begin{pmatrix}
  \sin(k_r) & 0 & 0 & 0 & \cdots \cr
  0 & \sin(k_r) & 0 & 0 & \cdots \cr
  0 & 0 & \sin(2k_r) & 0 & \cdots \cr
  0 & 0 & 0 & \sin(2k_r) & \cdots \cr
  \vdots & \vdots & \vdots & \vdots & 
 \end{pmatrix}
 \begin{pmatrix}
  \frac{\Pi^*}{\varepsilon_{r{\bf p}}^s} \cr
  1 \cr
  1 \cr
  \frac{\Pi}{\varepsilon_{r{\bf p}}^s} \cr
  \vdots
 \end{pmatrix},
 \label{eq:wf}\end{aligned}$$ where $\Pi \equiv p_x + i p_y$ and $\Pi^* \equiv p_x - i p_y$. The probability amplitude of the $j$th layer having two components structure (pseudospin) is proportional to $\sin(jk_r)$. The diagonal components of the $2N \times 2N$ matrix show that the wavefunction is the symmetrical state for the case of a positive mass $m_r > 0$, while it is the antisymmetrical state for a negative mass $m_r <0$. Since degenerate symmetrical and antisymmetrical states can be 