Phonons of single quintuple Bi${}_2$Te${}_3$ and Bi${}_2$Se${}_3$ films and bulk materials

Phonons of single quintuple films of Bi${}_2$Te${}_3$ and Bi${}_2$Se${}_3$ and corresponding bulk materials are calculated in detail by MedeA (a trademark of Materials Design) and Vienna ab initio simulation package (VASP). The calculated results with and without spin-orbit couplings are compared, and the important roles that the spin-orbit coupling plays in these materials are discussed. A symmetry breaking caused by the anharmonic potentials around Bi atoms in the single quintuple films is identified and discussed. The observed Raman intensity features in Bi${}_2$Te${}_3$ and Bi${}_2$Se${}_3$ quintuple films are explained.

Figure 1

Structures of Bi${}_2$Te${}_3$ (or Bi${}_2$Se${}_3$) bulk and thin film (top) and their corresponding Brillouin zone (bottom), where (a) is for bulk and (b) is for film. The black balls are Bi atoms, and the gray balls are Te (or Se) atoms. In the above figures, $a_1,a_2,a_3$ are the translational vectors in the real space, and $b_1,b_2,b_3$ are the reciprocal lattice vectors derived from them. The gray lines show the Brillouin zone. Here $L$ is $(0\frac{1}{2}0)$ and $Z$ is $(\frac{1}{2}\frac{1}{2}\frac{1}{2})$ in the corresponding reciprocal coordinate systems.

Figure 2

The irreducible representations of vibrational modes of bulk Bi${}_2$Se${}_3$ or Bi${}_2$Te${}_3$ taken from Ref. 27. From this figure, we can see that the $A$ modes are mainly perpendicular to the films and and the $E$ modes are mainly within the films.

Figure 3

Phonon dispersions of bulk Bi${}_2$Se${}_3$ (a) with SO and (b) without SO. Here $L$ stands for $(0\frac{1}{2}0)$ and $Z$ stands for $(\frac{1}{2}\frac{1}{2}\frac{1}{2})$ in the trigonal coordinate system.

Figure 4

Phonon dispersion of Bi${}_2$Te${}_3$ film with anharmonic interactions (a) with SO and (b) without SO. Here $L$ stands for $(0\frac{1}{2}0)$ and $Z$ stands for $(\frac{1}{2}\frac{1}{2}\frac{1}{2})$ in the hexagonal coordinate system.

Figure 5

Phonon dispersion of Bi${}_2$Se${}_3$ film with anharmonic interactions (a) with SO and (b) without SO. Here $L$ stands for $(0\frac{1}{2}0)$ and $Z$ stands for $(\frac{1}{2}\frac{1}{2}\frac{1}{2})$ in the hexagonal coordinate system.

Figure 6

Phonon dispersion of two bulk Bi${}_2$Te${}_3$ structures with different pressures. The VASP energies are −18.709 566 eV for (a) and −18.639 107 eV for (b). Electronic pressures are 53.000 MPa for (a) and 2.066 GPa for (b). Here $L$ stands for $(0\frac{1}{2}0)$ and $Z$ stands for $(\frac{1}{2}\frac{1}{2}\frac{1}{2})$ in the trigonal coordinate system.

Figure 7

Phonon dispersion of (a) Bi${}_2$Te${}_3$ and (b) Bi${}_2$Se${}_3$ film with harmonic interactions and SO. Here $L$ stands for $(0\frac{1}{2}0)$ and $Z$ stands for $(\frac{1}{2}\frac{1}{2}\frac{1}{2})$ in the hexagonal coordinate system.