Graphene and few-layer graphite probed by second-harmonic generation: Theory and experiment

We have measured second-harmonic generation (SHG) from graphene and other graphitic films, from two layers to bulk graphite, at room temperature; all samples are mounted on a 300 nm oxide layer of a Si(001) substrate. With 800 nm, 150 fs fundamental pulses, the anisotropic response was recorded for combinations of $p$-, $s$-, and diagonally polarized fundamental and second-harmonic beams as the samples were rotated about their normal. Graphene samples display SHG signatures only slightly different from that of the bare substrate which shows SHG with fourfold rotational symmetry. All other layered systems show threefold symmetry, although the ratio of isotropic to anisotropic response varies with the number of layers. A model based on linear light propagation in layered media with interface dipole and bulk quadrupole SHG sources is presented for the analysis. We show that data from all layered samples can be understood in terms of well-known linear optical properties, the SHG response of the bare substrate and four independent, complex nonlinear dipole susceptibility tensor elements of the graphene/air interface.

Figure 1

(Color online) Diagram of interface structure including hypothetical vacuum interface in which the surface SHG occurs. A possible bulk contribution from the C-film is ignored.

Figure 2

(Color online) Magnitude (blue) and phase (red) of $c_{11}^{ps}$ vs number of layers. The contributions to the SHG from the top and bottom surfaces of the C-film add constructively for thin samples.

Figure 3

(Color online) Magnitude (blue) and phase (red) of $a_{31}^{pp}$ vs number of layers. The contributions to the SHG from the top and bottom surfaces of the C-film add destructively for thin samples.

Figure 4

(Color online) Normalized SHG signal as a function of rotation angle $\varphi$. Top two curves: $\left(pp\right)$ data for the bare substrate (black squares, lower curve) and graphene (red circles, upper curve), both fit to $A+B\text{cos}4\varphi$ as suggested by Eq. (19). Lower two curves: $\left(ps\right)$ data for a bilayer sample (blue circles, upper curve), and a $\sim 15$-layer sample (green squares, lower curve), both fit to $A{\text{cos}}^2\left(3\varphi \right)$ [as suggested by Eq. (16)]. The absolute angle is the same for the $\left(pp\right)$ curves but is otherwise arbitrary.

Figure 5

(Color online) For a $\sim 5$-layer sample, both $\left(pp\right)$ (black squares) and $\left(ds\right)$ (red circles) polarization combinations are shown. The absolute angle is arbitrary, but is the same in both plots. Both data sets are fit to $A+B\text{cos}\left(3\varphi +{\varphi }_0\right)$ and are normalized such that the isotropic component of the SHG signal from the substrate would be unity. The phase shift $\left({\varphi }_0\approx 144°\right)$ associated with $\left(ds\right)$ relative to $\left(pp\right)$ is evident.

Figure 6

(Color online) Fourier elements of $I_{2\omega }\left(\varphi \right)$ for various polarization combinations: (top-left and right: zeroth and third element for $\left(pp\right)$; second from top left and right: zeroth and third element for $\left(sp\right)$; third from top left: sixth element for $\left(ps\right)$; right: sixth element for $\left(ss\right)$; bottom left and right: zeroth and third element for $\left(ds\right)$ where the phase of the third Fourier element has been plotted as well). The $x$ axis is the number of layers of graphene in each graph. The SHG intensity has been normalized to the square of the incident intensity, and normalized such the isotropic contribution from the bare ${\text{SiO}}_2/\text{Si}$ substrate is unity. The curves are fits using the equations in Table .