Revealing the vacuum level in an infinite solid by real-space potential unfolding

Although real materials are finite in size, electronic structure theory is built on the assumption of infinitely large solids, which led to a longstanding controversy: where is the vacuum level? Here, we introduce an analytic real-space potential-unfolding approach to uncover the vacuum level in infinitely large solids. First-principles calculations show that, in the absence of a physical surface, the bulk band structure, often measured with respect to an average bulk potential, is offset by a hereto unknown and orientation-dependent bulk quadrupole with respect to the vacuum level. By identifying intrinsic contributions of a bulk solid to its surface and interface properties, our theory eliminates the ambiguities surrounding the physical origin of the band alignment between matters.