Formulation for simulating the bi-layer attenuated phase shift contact hole diffraction in lithography

In order to evaluate the contact hole diffraction, a rigorous three dimensional (3D) mask model is established. The normal vector (NV) method is used to improve the convergence of the coupled-wave equation of the contact holes. The scattering-matrix approach (S-matrix algorithm) is applied to solve the linear equation system to avoid numerical instability. For the bi-layer attenuated phase shift contact hole, when the effective orders are above 1225, a more convergent result for the (0,0)th order is obtained. The diffraction efficiencies are 23.64%, 23.67%, 23.63% and 23.66%, when 1225, 1369, 1521 and 1681 orders, respectively, are retained in the computation. Then the change of polarization state as a function of mask and incident light properties is investigated by the mask models. When the linewidth of the contact hole is below 25nm, the mask is polarized predominately TM (transverse magnetic) polarization.