Since their discovery by Geim, 2D heterostructures have become a hotbed of research due to their novel structure. Stacking varying materials on top of each other allows for a vast range of possible materials and corresponding properties. However, this stacking typically results in an aperiodic, or incommensurate, material due to lattice mismatch or rotational misalignment. This incommensuration increases the potential for tunability of electronic and mechanical properties, leading, for example, to the famous discovery of unconventional superconductivity of twisted bilayer graphene at the magic angle.

 

Experimentation is vastly more expensive than numerical simulation, so a lot of work is needed on building analysis and algorithms for understanding and computing efficiently models of these incommensurate materials. In this work, we discuss configuration and momentum space methodologies to build algorithms to compute observables of incommensurate heterostuctures with mechanical relaxation. We exploit the ergodicity of the misalignment, and for appropriate materials (such as those with conic or parabolic bands) we use carefully selected perturbative expansions in momentum space.