First-principles investigations of the atomic, electronic, and thermoelectric properties of equilibrium and strained Bi 2 Se 3 and Bi 2 Te 3 including van der Waals interactions

Bi${}_{2}$Se${}_{3}$ and Bi${}_{2}$Te${}_{3}$ are layered compounds of technological importance, being excellent thermoelectric materials as well as topological insulators. We report density functional theory calculations of the atomic, electronic, and thermoelectric properties of strained bulk and thin-film Bi${}_{2}$Se${}_{3}$ and Bi${}_{2}$Te${}_{3}$, focusing on an appropriate description of van der Waals (vdW) interactions. The calculations show that the van der Waals density functional (vdW-DF) with Cooper's exchange (vdW-DF${}^{\mathrm{C}09}$${}_{x}$) can reproduce closely the experimental interlayer distances in unstrained Bi${}_{2}$Se${}_{3}$ and Bi${}_{2}$Te${}_{3}$. Interestingly, we predict atomic structures that are in much better agreement with the experimentally determined structure from Nakajima than that obtained from Wyckoff, especially for Bi${}_{2}$Se${}_{3,}$ where the difference in atomic structures qualitatively changes the electronic band structure. The band structure obtained using the Nakajima structure and the vdW-DF${}^{\mathrm{C}09}$${}_{x}$ optimized structure are in much better agreement with previous reports of photoemission measurements, than that obtained using the Wyckoff structure. Using vdW-DF${}^{\mathrm{C}09}$${}_{x}$ to fully optimize atomic structures of bulk and thin-film Bi${}_{2}$Se${}_{3}$ and Bi${}_{2}$Te${}_{3}$ under different in-plane and uniaxial strains, we predict that the electronic bandgap of both the bulk materials and thin films decreases with tensile in-plane strain and increases with compressive in-plane strain. We also predict, using the semiclassical Boltzmann approach, that the magnitude of the $n$-type Seebeck coefficient of Bi${}_{2}$Te${}_{3}$ can be increased by the compressive in-plane strain while that of Bi${}_{2}$Se${}_{3}$ can be increased with tensile in-plane strain. Further, the in-plane power factor of $n$-doped Bi${}_{2}$Se${}_{3}$ can be increased with compressive uniaxial strain while that of $n$-doped Bi${}_{2}$Te${}_{3}$ can be increased by compressive in-plane strain. Strain engineering thus provides a direct method to control the electronic and thermoelectric properties in these thermoelectric topological insulator materials.