Controlling Goos-H\"anchen shifts in phosphorene via barrier and well

We study the effect of the double potentials (barrier, well) on the Goos-H\"anchen (GH) shifts in phosphorene. We determine the solutions of the energy spectrum associated with the five regions that make up our system. By studying the phase shifts, we find that the GH shifts are highly sensitive to the incident energy, the $y$ directional wave vector, the potential heights and widths. To validate our findings, we perform a numerical analysis of the GH shifts as a function of the transmission probability under various conditions. In particular, we observe a consistent pattern in which a positive peak in the GH shift is always followed by a negative valley, a behavior evident at all potential height values. Notably, the energies at which the GH shift changes sign coincide exactly with the points at which transmission drops to zero. In particular, the transmission resonances that occur just before and just after the transmission gap region are strongly correlated with the points at which the GH shift changes sign. This study advances our understanding of how the double potential influences the GH shift behaviors in phosphorene. The ability to fine-tune the GH shifts by changing system parameters suggests potential applications in optical and electronic devices using this two-dimensional material.