Layer dependence of third-harmonic generation in thick multilayer graphene

We demonstrate experimental evidence for the existence of an optimal thickness that yields the maximum third-harmonic (TH) signal in thick multilayer graphene (MLG). We measure the layer-dependent TH signal from MLG on a quartz substrate for $N$ graphene layers ranging from $\sim 3$ to 50. Theoretical predictions of the layer-dependent TH signal are compared with experiment, including those measured using atomic force and optical microscopy. We find the optimal TH signal for $N\approx 24$, in good agreement with two theoretical models: one in which the graphene layers are assumed isolated but contribute coherently, and one in which the MLG is assumed as one continuous thin film. Our measurements reveal that for MLG, the stacking property of the linear and nonlinear conductivities are found to be remarkably preserved up to the very thick layers $\left(N\sim 50\right)$, i.e., ${\sigma }_{N,g}^{\left(i\right)}=N{\sigma }_g^{\left(i\right)}$, where $i=1$ and 3, respectively, and ${\sigma }_g^{\left(i\right)}$ is the corresponding monolayer conductivity.

Figure 1

(a) Transmission, reflection, and absorption coefficients for MLG on quartz substrate vs $N$ for the interface (blue curves) and slab (red curves) models. Interface model (left inset): MLG is treated as a source of interface current via the surface conductivity ${\sigma }_{N,g}^{\left(1\right)}=N{\sigma }_g^{\left(1\right)}$. TH reflected and transmitted fields $E_{3\omega ,R}$ and $E_{3\omega ,T}$ are generated via the third-order conductivity ${\sigma }_{N,g}^{\left(3\right)}=N{\sigma }_g^{\left(3\right)}$. Slab model (right inset): The MLG is treated as a single slab with nonlinearity ${\chi }^{\left(3\right)}$. Four TH waves are generated: $E_{3\omega ,R},E_{3\omega ,T},E_{3\omega ,M}$, and $E_{3\omega ,M^{\prime }}$, denoting the reflected, transmitted, and internal down- and up-going waves, respectively. (b) Reflected THG signal predicted by the interface and slab models for MLG. Inset: Slab model prediction for a large range of $N$.

Figure 2

Experimental setup. A Ti:sapphire (TS) laser beam passes through dichroic mirror (DC) and is focused on the MLG (G) sample via a microscope objective (MO). The TH signal is reflected by DC into a Pellin-Broca prism (PB), reflected by mirror (M), focused by a lens (L) into the monochromator (MC), and directed into a photomultiplier tube/photon-counting system (PMT/PC). Left inset: Cubic-power dependence of TH signal; right inset: TH spectrum.

Figure 3

(a) Optical microscopy of a typical MLG region with a large variation in layer thickness. (b) TH mapping of the same region. (c) AFM image of a $13\times 20\mu {\text{m}}^2$ region. (d) A comparison of TH and AFM line scans of the AFM image indicated in (c).

Figure 4

Normalized layer-dependent TH generation in MLG. Circles: Experimental data. Blue and red curves: Theoretical predictions using the interface and slab models, respectively.