Theoretical $2D$ Raman band of strained graphene

We study the 2$D$ Raman band of in-plane uniaxially strained graphene within a nonorthogonal tight-binding model. At nonzero strain, the obtained 2$D$ band splits into two subbands at strain angles $0^{\circ }$ and ${30}^{\circ }$ or into three subbands at intermediate angles. The evolution of the 2$D$ subbands is calculated systematically in the range of the accessible strains from $-1\%$ to $3\%$ and for the commonly used laser photon energy from 1.5 to $3.0$ eV. The strain rate and dispersion rate of the 2$D$ subbands are derived and tabulated. In particular, these two quantities show large variations up to $50\%$. The results on the 2$D$ subbands can be used for detecting and monitoring strain in graphene for nanoelectronics applications.

Figure 1

(a) The hexagonal atomic structure of graphene. The atoms are depicted by solid circles and the solid lines are the interatomic bonds. The strain direction is defined by the angle $\theta$ relative to the $x$ axis, which is chosen along a zigzag of carbon bonds. The limiting cases of $Z$ strain and $A$ strain are also shown. (b) Schematic representation of a double-resonant Raman-scattering process, which contributes to the 2$D$ band. An electron is scattered between the conelike conduction bands at the $K$ and $K^{\prime }$ points of the hexagonal Brillouin zone of graphene by a $K$-point phonon with a wave vector, denoted by an arrow. The electron is scattered back by a phonon with an opposite wave vector. The three scattering paths $K{K}^{\prime }$ are denoted by 1, 2, and 3. Similarly, holes can be scattered by phonons between conelike valence bands at the $K$ and $K^{\prime }$ points.

Figure 2

Calculated 2$D$ band at $\theta =0^{\circ }$ ($Z$ strain), ${10}^{\circ }$, ${20}^{\circ }$, and ${30}^{\circ }$ ($A$ strain), at $E_L=2.0$ eV, and for parallel and perpendicular light polarization (solid and dashed lines, respectively) in comparison with that at zero strain (dotted line). In most cases, two Raman features are clearly seen. They correspond to the experimentally observed $2D^{-}$ and $2D^{+}$ bands.

Figure 3

Scaled Raman shift of subbands $2D_1$ and $2D_3$ (symbols) as a function of the strain $\epsilon$ at $\theta =0^{\circ }$ and ${30}^{\circ }$, and at different values of $E_L$.

Figure 4

Calculated Raman spectra at $Z$ strain ($\epsilon =1.9\%$) and $A$ strain ($\epsilon =1.3\%$), at $E_L=2.33$ eV and parallel or perpendicular incident light polarization. The scattered light is polarized at an angle of $0^{\circ }$, ${45}^{\circ }$, or ${90}^{\circ }$ relative to the incident light polarization. The NTB red shift of the 2D band is scaled by $0.8$ to match the data from Ref. 12, where experiment and ab initio results are in agreement, as discussed in the text. The obtained spectra agree qualitatively with those reported in Fig. 3 of Ref. 10.

Figure 5

Scaled Raman shift of subbands $2D_1$ and $2D_3$ as a function of $E_L$ at different $\epsilon$ and at $\theta =0^{\circ }$ and ${30}^{\circ }$. The triangles are the shifts of the 2$D$ band of unstrained graphene. The lines are guides to the eye.

Figure 6

Scaled Raman shift of the 2$D$ subbands (symbols) as a function of $\theta$ at different values of $\epsilon$ and $E_L$. The three branches at a given strain magnitude are the shifts of subbands $2D_1$, $2D_2$, and $2D_3$ in order of increasing (decreasing) shift at $\epsilon <0$ ($\epsilon >0$). The horizontal lines are drawn at the Raman shift of unstrained graphene.