Modulation of electronic and mechanical properties of phosphorene through strain

We report a first-principles study on the elastic, vibrational, and electronic properties of the recently synthesized phosphorene. By calculating the Grüneisen parameters, we evaluate the frequency shift of the Raman and infrared active modes via symmetric biaxial strain. We also study a strain-induced semiconductor-metal transition, the gap size, and the effective mass of carriers in various strain configurations. Furthermore, we unfold the emergence of a peculiar Dirac-shaped dispersion for specific strain conditions including the zigzag-oriented tensile strain. The observed linear energy spectrum has distinct velocities corresponding to each of its linear branches and is limited to the $\Gamma \text{−}X$ direction in the first Brillouin zone.

Figure 1

(a) Schematic representation of the atomic structure of monolayer phosphorene from the top and side views. (b) The per-unit cell strain energy as a function of strain in uniform deformation regime. The shaded region indicates the plastic range. The inset shows the $4\times 4$ rectangular supercell used in the calculations. (c) The surface plot of (${\varepsilon }_x$, ${\varepsilon }_y$) and the corresponding per-unit cell strain energies. The points denote actual data and the background is the fitted formula. (d) The mesh plot of ${\Delta }_z$ is at the same data points as in panel (c).

Figure 2

(a) The phonon dispersion curve of the undeformed phosphorene. ZA marks the out-of-plane acoustic branch and LA (TA) denotes in-plane longitudinal (transverse) acoustic vibrations. (b) Frequencies of the Raman and infrared active modes at the $\Gamma$ point of phosphorene under strain. (c) Contour plots of Raman and infrared frequencies with strain. The eigenvector of the corresponding vibrational mode is depicted at the top of each panel.

Figure 3

(a) Modification of the electronic band structure under various strain configurations. (b) The surface plot of (${\varepsilon }_x$, ${\varepsilon }_y$) and the corresponding band gaps. The two dashed diagonal lines denote the symmetric (Sym.) and antisymmetric (A-Sym.) strain distributions. (c) Nature of band gaps for the same set of data presented in panel (b). Squares (circles) represent indirect (direct) band gaps. Panels (d) and (e) show the VBM and CBM in symmetric and antisymmetric strain distributions, respectively. Light (dark) gray regions correspond to direct (indirect) band gaps. Note that in panel (a) the Fermi energy is set to zero. In panels (d) and (e), the energies are referenced to vacuum level to further illustrate the modification of band offsets in strained structure.

Figure 4

(a) The band structures of phosphorene under various A-Sym. strain values and the emergence of a Dirac-shaped dispersion. Blue, cyan, green, and red denote ${\varepsilon }_y=-{\varepsilon }_x=0\%,6\%,8\%,\mathrm{and}11\%$, respectively. (b) Zoom of the band structure at ${\varepsilon }_y=-{\varepsilon }_x=11\%$ along the selected paths of the first Brillouin zone. Inset shows the equienergy contours for the conduction (top) and the valence (down) bands centered at $D$. Both rectangles span a length of $0.02\times \pi /a_1$ ($0.02\times \pi /a_2$) along the $k_x$ ($k_y$) direction of the first Brillouin zone. Panels (c) and (d) denote the orbital composition of the crystal wave functions close to the Dirac point for the topmost valence and the lowest conduction bands, respectively. (e) The mesh plot of the band gap at data points (${\varepsilon }_x$, ${\varepsilon }_y$). Bright cyan (yellow) region denotes Dirac-shaped dispersions with band gap smaller than 5 (20) meV. The cross marks denote selected strain distributions for which the existence of a Dirac-like dispersion is further validated by VASP. (f) Fermi velocity and $y$-directed effective mass for Dirac-shaped dispersions as functions of ${\varepsilon }_x$. Crosses and filled circles denote the results for conduction and valence energy bands, respectively.

Figure 5

(a) and (b) [(c) and (d)] depict the effective mass of electrons (holes) along the $x$ and $y$ directions, respectively. The corresponding locations of the VBM and CBM are denoted by circles (at $\Gamma$ point), crosses (along the $\Gamma \text{−}X$ high-symmetry line), and squares (along the $\Gamma \text{−}Y$ high-symmetry line).