Effect of resonant impurity scattering of carriers on the Drude-peak broadening in uniaxially strained graphene

An explanation is proposed for the recently observed in optical spectra of monolayer graphene giant increase in the Drude-peak width under applied uniaxial strain. We argue that the underlying mechanism of this increase can be based on resonant scattering of carriers from inevitably present impurities such as adsorbed atoms that can be described by the Fano-Anderson model. We demonstrate that the often neglected scalar deformation potential plays the essential role in this process. The conditions necessary for the maximum effect of the giant Drude-peak broadening are determined. It is stressed that the effect is strongly enhanced when the Fermi level gets closer to the Dirac point. Our theoretical analysis provides guidelines for functionalizing graphene samples in a way that would allow to modulate efficiently the Drude-peak width by the applied strain.

Figure 1

The influence of uniaxial strain on the DOS, $D_{\varepsilon }\left(E\right)$. (a) Only the hopping term is taken into account. The triangular DOS of unstrained (strained) graphene is shown by the red (blue) lines. The unperturbed bandwidth is $W_0$ and the position of the Fermi level ${\mathcal{E}}_{\mathrm{F}}$ corresponds to the hole-doped sample. The tensile $\varepsilon >0$ strain results in the decrease of effective bandwidth $W_{\varepsilon }<W_0$. Since the number of carriers is fixed, the position of the Fermi level ${\mathcal{E}}_{\mathrm{F}}^{\varepsilon }>{\mathcal{E}}_{\mathrm{F}}$ shifts to the right in the hole-doped case. (b) Both the hopping term and deformation potential are included. The latter results in the shift of the Dirac point from ${\mathcal{E}}_{\mathrm{D}}$ to the position ${\mathcal{E}}_{\mathrm{D}}^{\varepsilon }={\mathcal{E}}_{\mathrm{D}}+\alpha \varepsilon$. The position of the Fermi level shifts to the value ${\mathcal{E}}_{\mathrm{F}}^{\varepsilon }$ as described by Eq. (19). For a doped strained graphene the work function $\mathcal{W}={\mathcal{W}}_D^{\varepsilon }+{\mathcal{W}}_F^{\varepsilon }$.

Figure 2

The real and imaginary parts of the self-energy function $\mathrm{\Sigma }\left(E\right)$ in vicinity of the Dirac point in the Lifshitz model. The model parameters are $V_{\text{L}}=6.3t_0, c=4\times {10}^{-4}$. The convex and concave curves correspond to $\mathrm{Re}\mathrm{\Sigma }\left(E\right)$ and $\mathrm{Im}\mathrm{\Sigma }\left(E\right)$, respectively. Blue lines and black lines were computed in the CPA approximation, while gray lines were obtained in the ATA approximation. The dashed and the solid lines correspond to the results for zero strain, while the dotted-dashed and dotted lines are for $\varepsilon =5\%$.

Figure 3

The real (upper red curves) and imaginary (lower green lines) parts of the self-energy $\mathrm{\Sigma }\left(E\right)$ for the Fano-Anderson model calculated in the CPA approximation with $E_0=-0.75t_0, t_{\mathrm{hyb}}=2t_0, c=4\times {10}^{-4}$ for two values of strain $\varepsilon =0$ and $\varepsilon =5\%$. The solid lines show results for zero strain, the dashed lines show results for $\varepsilon =5\%$ with the same deformation potential for host and impurity atoms ($\mathrm{\Delta }\alpha =0$), and the dotted lines show results for $\varepsilon =5\%$ with no impurity deformation potential (${\alpha }^{\mathrm{imp}}=0$ or $\mathrm{\Delta }\alpha =-2.5\mathrm{eV}$). Results obtained with ${\alpha }^{\mathrm{imp}}$ varying between these two limiting values span the shaded areas.

Figure 4

Imaginary part of the self-energy $\mathrm{Im}\mathrm{\Sigma }\left(E\right)$ calculated for the Lifshitz model (black solid line) and for the Fano-Anderson model (other curves). The impurity concentration $c=4\times {10}^{-4}$ is the same for all curves. The parameter $V_{\mathrm{L}}=6.3t_0$ for the Lifshitz model. The parameters for the Fano-Anderson model are $t_{\mathrm{hyb}}=2t_0, E_0=-0.75t_0$ (blue dashed line); $t_{\mathrm{hyb}}=t_0, E_0=-0.28t_0$ (green dotted-dashed line); $t_{\mathrm{hyb}}=0.5t_0, E_0=-0.17t_0$ (red dotted line). The zero strain is considered.

Figure 5

The Drude-peak width ${\mathrm{\Gamma }}_{\text{opt},\varepsilon }=-2\mathrm{Im}\mathrm{\Sigma }\left(E_{\mathrm{F}}\right)$, as a function of the Fermi energy $E_{\mathrm{F}}$, calculated for the strain $\varepsilon =0\%,1\%,2\%$ in the Fano-Anderson model. Inset: zoom-in the vicinity of $E_{\mathrm{F}}=-0.23\mathrm{eV}$ which is shown as the vertical line. The model parameters are the following: $c=4\times {10}^{-4}, E_0=-0.75t_0, t_{\mathrm{hyb}}=2t_0, \mathrm{\Delta }\alpha =0$.

Figure 6

The relative gain $\varkappa$ of the Drude width per $1\%$ of strain as a function of $E_{\mathrm{F}}$ for the Fano-Anderson model. The calculation is done assuming a constant strain independent $E_{\mathrm{F}}$. All curves are for $t_{\mathrm{hyb}}=2t_0$ and $c=5\times {10}^{-5}$. Four sets of the curves differ by the value of $E_0$: 1, $E_0=-1.25t_0$ (purple lines); 2, $E_0=-t_0$ (blue lines); 3, $E_0=-0.75t_0$ (green lines); 4, $E_0=-0.5t_0$ (orange lines). The dashed lines show the results with the same deformation potential for host and impurity atoms ($\mathrm{\Delta }\alpha =0$) and and the dotted lines are for no impurity deformation potential (${\alpha }^{\mathrm{imp}}=0$ or $\mathrm{\Delta }\alpha =-2.5\mathrm{eV}$). The areas between each pair of curves are shaded in same color tone as the lines. The dotted-dashed and double-dotted-dashed gray lines are obtained on the base of analytical estimates for the upper and lower bounds of the gain function for the cases ${\alpha }^{\mathrm{imp}}=\alpha$ and ${\alpha }^{\mathrm{imp}}=0$, respectively (see the main text for the explanation).

Figure 7

The relative gain $\varkappa$ of the Drude width per $1\%$ of strain as a function of the zero-strain Fermi energy $E_{\mathrm{F}}^0=W_0\sqrt{N_{\mathrm{c}}}$ for the Fano-Anderson model. The calculation is done assuming a constant strain-independent carrier imbalance $N_{\mathrm{c}}$. All parameters and notations are the same as in Fig. 6. The black dotted-dashed and double-dotted-dashed lines are obtained on the base of analytical estimates for the upper and lower bounds of the gain function for the cases ${\alpha }^{\mathrm{imp}}=\alpha$ and 0, respectively (see the main text for the explanation). The gray dotted-dashed and double-dotted-dashed gray lines are the same as in Fig. 6.