First principles band structure of interacting phosphorus and boron/aluminum $\delta$-doped layers in silicon

Silicon can be heavily doped with phosphorus in a single atomic layer (a $\delta$ layer), significantly altering the electronic structure of the conduction bands within the material. Recent progress has also made it possible to further dope silicon with acceptor-based $\delta$ layers using either boron or aluminum, making it feasible to create devices with interacting $\delta$ layers with opposite polarity. Using Density Functional Theory, we calculate the electronic structure of a phosphorus-based $\delta$ layer interacting with a boron or aluminum $\delta$ layer, varying the distances between the $\delta$ layers. At separations 1 nm and smaller, the dopant potentials overlap and largely cancel each other out, leading to an electronic structure closely mimicking intrinsic silicon. At separations greater than 1 nm, the two $\delta$ layers behave independently of one another, with an equivalent electronic structure to a p-n diode with an intrinsic layer taking the place of the depletion region. One mechanism for charge transfer between $\delta$ layers at larger distances could be tunneling, where we see a tunneling probability exceeding what would be seen for a standard silicon 1.1 eV triangular barrier, indicating that the interaction between delta layers may enhance tunneling compared to a traditional junction.