Phonon softening and direct to indirect band gap crossover in strained single-layer MoSe${}_2$

Motivated by recent experimental observations of Tongay et al. [Nano Lett. 12, 5576 (2012)] we show how the electronic properties and Raman characteristics of single layer MoSe${}_2$ are affected by elastic biaxial strain. We found that with increasing strain: (1) the $E^{\prime }$ and $E^{\prime \prime }$ Raman peaks ($E_{2g}$ and $E_{1g}$ in bulk) exhibit significant redshifts (up to $\sim$30 cm${}^{-1}$), (2) the position of the $A_1^{\prime }$ peak remains at $\sim$180 cm${}^{-1}$ ($A_{1g}$ in bulk) and does not change considerably with further strain, (3) the dispersion of low energy flexural phonons crosses over from quadratic to linear, and (4) the electronic band structure undergoes a direct to indirect band gap crossover under $\sim$3$\%$ biaxial tensile strain. Thus the application of strain appears to be a promising approach for a rapid and reversible tuning of the electronic, vibrational, and optical properties of single layer MoSe${}_2$ and similar MX${}_2$ dichalcogenides.

Figure 1

(a) Phonon dispersion for single layer MoSe${}_2$ and displacement directions of Raman-active modes. Inset: Side view of atomic structure of MoSe${}_2$. (b) Electronic band structure and partial density of states calculated within GGA and $G_0{W}_0$. The energies are relative to the Fermi level (i.e. $E_{\mathrm{F}}=0$). Inset: Comparison of the band edges calculated by GGA, GGA+SO, and LDA.

Figure 2

(a) Phonon dispersions for 1$\%$ and 7$\%$ strained single layer MoSe${}_2$. (b) Evolution of the Raman-active modes as a function of applied strain.

Figure 3

Evolution of the electronic band dispersion of single layer MoSe${}_2$ as a function of strain. Fermi level shown by red line is set to zero in the energy spectrum.

Figure 4

Evolution of (a) direct and indirect energy band gaps, (b) electron and hole effective mass, normalized to unstrained value ($m_e^{*}/m_{e0}^{*}$ and $m_h^{*}/m_{h0}^{*}$), under biaxial tensile strain. (c) Energy band gaps calculated by using PBE+SO and PBE+$G_0{W}_0$ are shown by solid and dashed lines, respectively. Red, green, and blue colors are used for transitions $K\to K$, $K\to \Gamma K$, and $\Gamma \to K$, respectively.