Single well or double well: First-principles study of 8$H$ and 3$C$ inclusions in the 4$H$ SiC polytype

Using first-principles calculations with a hybrid functional, we examined several fundamental issues for heterojunction structures formed by the same material but in different polytypes including the precise location of the boundary and the polarization. Particularly, we demonstrate that the inclusion of 8$H$ in 4$H$ SiC generates a single quantum well (QW) structure, rather than a double well consisting of two 3$C$ regions separated by a single hexagonal layer barrier. The level of the QW states for 8$H$ and 3$C$ inclusions are calculated to be 0.42 and 0.68 eV, respectively, below the conduction band minimum of 4$H$ SiC, in good agreement with the experiments.

Figure 1

Diagram of 8$H$ and 3$C$ inclusions in a 4$H$ SiC matrix. The single well and double well models of 8$H$ inclusion in 4$H$ SiC are shown. The large (red in color) and the small (blue) balls stand for the Si and the C atoms, respectively. The layers' numbers denote the Si-C bilayers consisting of Si atoms in one stack and the C atoms occupying the preceding stack.

Figure 2

The layer-by-layer PDOS and the wave function (shown as a green isosurface in the right panel) of the QW state of an 8$H$ inclusion in 4$H$. The layer number is as shown in Fig. 1. For comparison, the PDOS of cubic and hexagonal layers of 8$H$ and the density of states (DOS) of bulk 4$H$ are also shown.

Figure 3

The band diagram of the 4$H$/8$H$/4$H$ QW. The energies are aligned to the valence band maximum (VBM) position of bulk 4$H$ SiC. The red line shows the first electronic state in the QW. The black (dashed) lines show the average spontaneous potential in the QW. The electric field inside the QW is calculated to be around 0.35 MV/cm.

Figure 4

The layer-by-layer PDOS and the wave function of the QW state of a 3$C$ inclusion in 4$H$. The layer number is as shown in Fig. 1. For comparison, the DOS of bulk 3$C$ and 4$H$ are also shown.

Figure 5

(a) The average electrostatic potential of a SiC heteropolytype QW formed by a 3$C$ inclusion in 4$H$. (b) The average electrostatic potentials of an 8$H$ inclusion in 4$H$, in which the dotted, the solid, and the dashed lines are averaged over 2, 4, and 8 layers, respectively; please note the different scale. (c) Band diagram of a 3$C$ inclusion in 4$H$ SiC with a simulated one-dimensional (1D) wave function.