Critical shell thickness for InAs-Al$_x$In$_{1-x}$As(P) core-shell nanowires

InAs nanowires with Al$_x$In$_{1-x}$P or Al$x$In$_{1-x}$As shells were grown on GaAs substrates by the Au-assisted vapour-liquid-solid (VLS) method in a gas source molecular beam epitaxy (GS-MBE) system. Core diameters and shell thicknesses were measured by transmission electron microscopy (TEM). These measurements were then related to selected area diffraction (SAD) patterns to verify either interface coherency or relaxation through misfit dislocations. A theoretical strain model is presented to determine the critical shell thickness for given core diameters. Zincblende stiffness parameters are transformed to their wurtzite counterparts via a well known tensor transformation. An energy criterion is then given to determine the shell thickness at which coherency is lost and dislocations become favourable.