Effect of uniaxial strain on the excitonic properties of monolayer ${\mathrm{C}}_3\mathrm{N}$: A symmetry-based analysis

In recent years, the application of mechanical stress has become a widespread experimental method to tune the electronic and optical properties of two-dimensional (2D) materials. In this work, we investigate the impact of uniaxial tensile strain along zigzag and armchair directions on the excitonic properties of graphene-like ${\mathrm{C}}_3\mathrm{N}$, a single-layer indirect-gap material with relevant mechanical and optical properties. To do that, we develop a tight-binding Bethe-Salpeter equation framework based on a Wannier-function description of the frontier bands of the system, and use it to compute both dark and bright excitons of ${\mathrm{C}}_3\mathrm{N}$ for different applied strain configurations. Then, we use this model approach to classify excitons of pristine and strained ${\mathrm{C}}_3\mathrm{N}$ according to the crystal symmetry and to explain the appearance of bright excitons with intense optical anisotropy in strained ${\mathrm{C}}_3\mathrm{N}$, even at small strains. Finally, the effect of strain on the exciton dispersion at small center-of-mass momenta is discussed, with special focus on the implications for 2D linear-nonanalytic dispersions.

Figure 1

Left: Crystal structure of ${\mathrm{C}}_3\mathrm{N}$, where yellow (light-blue) spheres represent carbon (nitrogen) atoms. High-symmetry armchair and zigzag directions where strain is applied are also highlighted. Right: Hexagonal Brillouin zone of pristine ${\mathrm{C}}_3\mathrm{N}$, with the high-symmetry points considered in this work.

Figure 2

Example of maximally localized Wannier functions localized on carbon (a) and nitrogen atoms (b). The obtained Wannier functions exhibit a $2p_z$-like character, with comparable spatial spreads.

Figure 3

Absorption spectrum of monolayer ${\mathrm{C}}_3\mathrm{N}$ computed with the model described in Sec. 2. The continuous blue line (dashed red line) represents the spectrum computed with (without, independent particle IP) the electron-hole interaction in the BSE kernel. The green dotted line is the fully ab initio BSE spectrum obtained in Ref. [5], while the vertical dashed black line indicates the position of the quasiparticle direct band gap corresponding to the onset of independent particle (IP) absorption. These spectra were computed assuming light polarization along the zigzag direction, and analogous results were found for different polarizations. $\mathbf{k}$-resolved contributions to the exciton wave functions are shown for the first five lowest energy excitons, and for the resonances responsible for the higher-energy absorption peak at about 2.2 eV. All the spectra have been convoluted with a Lorentzian broadening of 10 meV. The labels $e_i$ indicate the excitation energies in ascending order.

Figure 4

(a) Absorption spectra of ML-${\mathrm{C}}_3\mathrm{N}$ under uniaxial strain along the zigzag direction: Continuous red lines correspond to light polarized along the zigzag ($X$) direction, while dashed blue lines to light polarized along the armchair ($Y$) direction. Spectra for different values of applied strain are rigidly shifted vertically to make the plot more readable. The black dashed lines highlight the splitting of the twofold degenerate exciton $e_{4,5}$ in pristine ${\mathrm{C}}_3\mathrm{N}$ into two separate excitons, here called $e_4$ and $e_5$. All the spectra are convoluted with a Lorentzian broadening of 10 meV. (b) Same as (a), for externally applied strain along the armchair direction.

Figure 5

Polar plot of the oscillator strengths $D_{\lambda }$ defined in Eq. (15) for excitons $e_4$ and $e_5$, in the case of zigzag strain (a) and armchair strain (b) of 2.0%, as a function of the in-plane light polarization direction, measured starting from the zigzag ($X$) axis. Concentric lines represent isovalues for the modulus of the exciton oscillator strength $|D_{\lambda }|$.

Figure 6

Functions $A\left(\mathbf{k}\right)$ defined in Eq. (16), computed for exciton $e_4$ (a) and $e_5$ (b) in the case of zigzag strain of 2.0%. (c) and (d) correspond to the same quantities shown in (a) and (b), but evaluated for ML-${\mathrm{C}}_3\mathrm{N}$ strained along the armchair direction.

Figure 7

(a) DFT-PBE band structure of ML-${\mathrm{C}}_3\mathrm{N}$ in the pristine case (solid black lines) and with a zigzag strain of 2.0% (dashed red lines); (b) same as (a), but for a 2.0% strain along the armchair direction. (c) Energy difference ${\epsilon }_c\left(\mathbf{k}\right)-{\epsilon }_v\left(\mathbf{k}\right)$ between the lowest unoccupied conduction $c$ and the highest occupied valence $v$ for $\mathbf{k}$ along the path $M\text{−}\mathrm{\Gamma }\text{−}{M}^{\prime }$ for the unstrained case and for a 2.0% zigzag strain; (d) same as (c) for an armchair strain.

Figure 8

Excitonic wave functions ${\mathrm{\Psi }}_{\lambda }^{\alpha ,\beta }\left(\mathbf{R}\right)$, as defined in Eq. (18), computed respectively for the excitons $e_4$ (left) and $e_5$ (right), assuming a 2.0% strain along the zigzag direction (top) and the armchair direction (bottom). In all cases, we assume the hole to be fixed on the carbon atom marked by the black dot and positioned on the vertical dashed line, which represents the symmetry $y=0$ axis.

Figure 9

(a) Low-energy tail of the absorption spectrum in ML-${\mathrm{C}}_3\mathrm{N}$ under uniaxial strain along zigzag direction. (b) Same as (a), but considering an externally applied strain along the armchair direction. All the spectra were convoluted with a Lorentzian broadening of 10 meV. Curves corresponding to different strain values are rigidly shifted vertically to make the plot more readable.

Figure 10

Functions $A\left(\mathbf{k}\right)$ for excitons $e_1, e_2$, and $e_3$ are shown in panels (a), (c), and (e); the applied strain is fixed to $1.5\%$ ($5.0\%)$ in the case of excitons $e_1$ and $e_2 \left(e_3\right)$, and it is always assumed along the armchair direction. Real-space representations ${\mathrm{\Psi }}^{\alpha ,\beta }\left(\mathbf{R}\right)$ for the excitons $e_1, e_2$, and $e_3$ are displayed in panels (b), (d), and (f), computed for the same set of strain configurations.

Figure 11

Oscillator strength $D_{\lambda }$, defined in Eq. (15), computed for exciton $e_3$ as a function of the light polarization direction (measured with respect to the zigzag $x$ axis), in the case of zigzag (a) and armchair (b) strain equal to 5.0%. The circles in both figures represent isovalues for the modulus of the exciton oscillator strength $|D_{\lambda }|$.

Figure 12

Exciton dispersion in pristine ${\mathrm{C}}_3\mathrm{N}$ for $\mathbf{Q}$ in proximity of $\mathrm{\Gamma }$ and along the path $M\text{−}\mathrm{\Gamma }\text{−}{K}^{\prime }$. Red dots denote the excitonic bands departing from the bright excitons at $\mathrm{\Gamma }$, transforming as $E_1$; the linear band dispersion of the highest band is also highlighted by dashed red lines, representing the obtained linear fits along the two high-symmetry directions. For completeness, $\left|\mathrm{\Gamma }M\right|=0.746 {\AA }^{-1}$ while $|\mathrm{\Gamma }{K}^{\prime }|=0.861 {\AA }^{-1}$.

Figure 13

Exciton dispersion in ${\mathrm{C}}_3\mathrm{N}$ for $\mathbf{Q}$ along the path $M\text{−}\mathrm{\Gamma }\text{−}{K}^{\prime }$, in the case of zigzag (a) and armchair strain (b). In both cases, the strain is fixed to 1.0%. The red (blue) dots denote the excitonic band dispersing from the $B_2 \left(B_1\right)$ exciton at $\mathrm{\Gamma }$. The insets show the excitonic branches departing from the bright excitons $B_1$ and $B_2$ at $\mathrm{\Gamma }$. For completeness, $\left|\mathrm{\Gamma }M\right|=0.748 {\AA }^{-1}$ and $|\mathrm{\Gamma }{K}^{\prime }|=0.858 {\AA }^{-1}$, in the case of zigzag strain, while $\left|\mathrm{\Gamma }M\right|=0.739 {\AA }^{-1}$ and $|\mathrm{\Gamma }{K}^{\prime }|=0.863 {\AA }^{-1}$ in the case of armchair strain.