Grüneisen parameter of the $G$ mode of strained monolayer graphene

We present a detailed analysis of the effects of uniaxial and biaxial strain on the frequencies of the $G$ mode of monolayer graphene, using first principles calculations. Our results allow us to explain discrepancies in the experimentally determined values of the Grüneisen parameter. The direction and strength of the applied strain, Poisson's ratio of the substrate, and the intrinsic strain in different experimental setups turn out to be important. A reliable determination of the Grüneisen parameter is a prerequisite of strain engineering.

Figure 1

Ideal graphene lattice, decomposed into the sublattices $A$ and $B$. (a) Rectangular four-atom unit cell, and (b) rhombohedral two-atom unit cell. The dashed lines indicate the cell vectors ${\mathit{a}}_{{1}}$ and ${\mathit{a}}_{{2}}$. The solid blue arrow represents strain in the direction given by the angle θ.

Figure 2

Phonon band structure of unstrained graphene and graphene under ɛ $=$5% tensile strain along the zigzag direction. The paths in the Brillouin zone are indicated in the inset.

Figure 3

Atomic displacements for the two degenerate $G$ modes, (a) and (b), and the two split $G$ modes, (c) and (d). The displacements are parallel to the strain axis for $G^{-}$, while they are perpendicular to the strain axis for $G^{+}$.

Figure 4

(a) Frequency shift $\Delta {\omega }_{G^{\pm }}$ with respect to the frequency of the equilibrium structure as a function of uniaxial tensile strain (<1.5%), together with data from Ref. 10. (b) Variation of the phonon frequencies ${\omega }_{G^{\pm }}$ of the $G^{+}$ and $G^{-}$ modes under strain in the θ $=$0° (zigzag), 10°, 20°, and 30° (armchair) directions, respectively. The inset shows the derivative of $\Delta {\omega }_{G^{\pm }}/\varepsilon$ as a function of the strain direction θ.

Figure 5

Slopes $\Delta {\omega }_{G^{-}}/\varepsilon$ and $\Delta {\omega }_{G^{+}}/\varepsilon$ as functions of Poission's ratio $v$ of the substrate.

Figure 6

(a) Poisson's ratio $v$ as a function of uniaxial strain in the θ $=$0° (zigzag), 10°, 20°, and 30° (armchair) directions, respectively. The inset shows the derivative $\partial \nu /\partial \varepsilon$. (b) Grüneisen parameter for the $E_{2g}$ mode for strain up to 20% in different directions.

Figure 7

(a) Frequency shift $\Delta {\omega }_G$ with respect to the frequency of the equilibrium structure under biaxial strain (−0.2% to 0.2%), together with data from Ref. 13. (b) $E_{2g}$ mode frequency and Grüneisen parameter under biaxial strain (−16% to 20%).