Dominant phonon wave vectors and strain-induced splitting of the $2D$ Raman mode of graphene

The dominant phonon wave vectors ${\mathit{q}}^{*}$ probed by the $2D$ Raman mode of pristine and uniaxially strained graphene are determined via a combination of ab initio calculations and a full two-dimensional integration of the transition matrix. We show that ${\mathit{q}}^{*}$ are highly anisotropic and rotate about $\mathit{K}$ with the polarizer and analyzer condition relative to the lattice. The corresponding phonon-mediated electronic transitions show a finite component along $\mathit{K}$-$\mathit{\Gamma }$ that sensitively determines ${\mathit{q}}^{*}$. We invalidate the notion of “inner” and “outer” processes. The characteristic splitting of the $2D$ mode of graphene under uniaxial tensile strain and given polarizer and analyzer setting is correctly predicted only if the strain-induced distortion and red-shift of the in-plane transverse optical (iTO) phonon dispersion as well as the changes in the electronic band structure are taken into account.

Figure 1

The phonon-mediated electronic transitions yielding maximum intensity (black double-sided arrows) contributing to the $2D$ mode. These transitions were extracted from a full integration of the transition matrix. The underlying yellow-red contour plots of panels (a)–(c) are the product of the optical absorption and emission matrix elements at the polarizer and analyzer orientations (a) $x:x$, (b) $y:y$, and (c) $x:y$ ($x$, zigzag; $y$, armchair orientation of graphene), respectively. The equiexcitation energy contours corresponding to the laser energy are represented by green lines. (d) For the $x:x$ case, on adding a reciprocal lattice vector ${\mathbf{b}}_{\mathbf{2}}$ to ${\mathbf{k}}_{\mathbf{i}}^{*}$, from the seemingly outer dominant transition (black double-sided arrow connecting the black crosshairs) an equivalent inner electronic transition (blue double-sided arrow) is obtained.

Figure 2

The dominant phonon wave vectors ${\mathit{q}}^{*}$ (in blue-green) corresponding to polarizer and analyzer orientations: (a) $x:x$, (b) $y:y$, and (c) $x:y$. The underlying orange-hued contours depict the iTO phonon dispersion of Ref. 22 around $\mathit{K}$.

Figure 3

$2D$ Raman mode spectra calculated for increasing strain along the zigzag direction of graphene. Compare with Fig. 1a of Ref. 18. The inset highlights the orientation of graphene relative to strain and the polarization of incoming and outgoing light.

Figure 4

Experimental[18] [panels (a)–(c) (dotted points)] and calculated [panels (d)–(f) (solid lines)] polarized $2D$ Raman spectra for incoming polarization $P_{\mathrm{in}}$ and outgoing polarization $\psi$ relative to $P_{\mathrm{in}}$. Panels (a) and (d) and panels (b) and (e): Zigzag direction of strain. Panels (c) and (f): Armchair directions of strain. The dashed lines represent Lorentzian fits to the spectra. Panels (g)–(i): The orange-hued contours depict the iTO phonon dispersion and the blue-green regions are ${\mathit{q}}^{*}$ for graphene subject to $\varepsilon =1.5\%$ along the zigzag direction [panels (g) and (h)] and $\varepsilon =1.0\%$ along the armchair direction (i).

Figure 5

$2D$ Raman mode spectra obtained from (a) a full calculation considering the strained phonon and electronic bands for the zigzag orientation having $P_{\mathrm{in}}\parallel \varepsilon \parallel P_{\mathrm{out}}$ [compare with Fig. 4a]. (b) Calculation using an unstrained phonon dispersion plus strained electronic bands as input for the transition matrix calculation. (c) Calculation using a strained phonon dispersion and unstrained electronic bands. The corresponding orange-hued iTO phonon contours and blue-green regions for ${\mathit{q}}^{*}$ are given in panels (d)–(f).