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

Abstract

Silicon can be heavily doped with phosphorus in a single atomic layer (a $δ$ 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 $δ$ layers using either boron or aluminum, making it feasible to create devices with interacting $δ$ layers with opposite polarity. Using Density Functional Theory, we calculate the electronic structure of a phosphorus-based $δ$ layer interacting with a boron or aluminum $δ$ layer, varying the distances between the $δ$ 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 $δ$ 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 $δ$ 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.

Figures (15)

• Figure 1: (a) The supercell used throughout this manuscript, indicating the $P$ and $B$$\delta$ layers and the distance separating them. The horizontal axis corresponds to the (100) direction. (b) The band structure of a pure silicon supercell. Because we are using a supercell, the more familiar Brillouin zone for a two-atom primitive cell is folded in on itself. The red line represents the Fermi level of the system. (c) The Brillouin zone for the supercell, labeling high-symmetry points with crystal reciprocal coordinates. The length of the box in the k$_x$ direction, associated with the (001) direction of the supercell is exaggerated for clarity. Given the significant length of the axis in real space, this dimension is essentially negligible in reciprocal space.
• Figure 2: The band structure and local density of states (LDOS) (i.e. the localized band structure) for boron and phosphorus $\delta$-doped layers separated by (a) 0.1 nm, and (b) 0.4 nm. As the separation distance increases, the $\delta$-layer potentials become more visible within the LDOS, decreasing the band gap of the overall structure. The red line represents the Fermi level of the system.
• Figure 3: The band structure and local density of states (LDOS) for boron and phosphorus $\delta$-doped layers separated by (a) 1 nm, (b) 2 nm, and (c) 10 nm. At these larger separation distances, the $\delta$-layer potentials become clearly distinct and, in the case of a 10 nm separation, overlapping in energy. The red line represents the Fermi level of the system.
• Figure 4: The band structure and local density of states (LDOS) for aluminum and phosphorus $\delta$-doped layers separated by (a) 0.1 nm, and (b) 0.4 nm. The aluminum atoms have less suppression of the $\delta$ doped layer induced bands, resulting in overlap even at these lower separation distances.
• Figure 5: The band structure and local density of states (LDOS) for aluminum and phosphorus $\delta$-doped layers separated by (a) 1 nm, (b) 2 nm and (c) 10 nm. The aluminum atoms have less suppression of the $\delta$ doped layer induced bands, resulting in overlap even at these lower separation distances.