Resonant Structure of Second Harmonic Generation in Multilayer Graphene Polytypes

Abstract

Second harmonic generation (SHG) is a powerful optical tool for identifying non-centrosymmetric crystalline structures. Here, we analyze SHG in multilayer graphenes (MLG), with a focus on its dependence on the stacking order, encapsulation environment and biasing which break inversion symmetry in multilayers, as well as the SHG sensitivity to the electron-hole asymmetry in the MLG spectra and doping. In particular, we identify stacking-order-dependent resonant features in the SHG spectra for trilayers and tetralayers, suggesting that infra-red range SHG offers a non-invasive characterization method for distinguishing between MLG polytypes, as well as optical identification of crystallographic direction in MLG films.

Figures (4)

• Figure 1: (a) The Feynman diagram for the dominant process contributing to the SHG signal in MLG. (b) Top view of ABCB tetralayer lattice, with with red, gray, and blue color marking to the 'B', 'C', and 'A' layers, respectively (for a side view, see Fig. \ref{['fig:tetralayer_bands']}), and axes identifying a pair of armchair ($a$) and zigzag ($z$) axes. This sketch shows that the crystal has a threefold in-plane rotational symmetry $C_3$ and a mirror symmetry across the vertical planes with a projection aligned with armchair directions, such as shown by gray dashed line. (c) Dependence of a $2\omega$ emission intensity, $I^{2\omega}_{\rm out}$, on the orientation of the polarization axis in a single-polarizer SHG set-up (angle $\phi$ is counted from the armchair direction).
• Figure 2: Top row of panels shows a side view of tetralayer polytypes lattices with all hopping terms included in the implemented MLG Hamiltonians. We use the fine dashed lines to highlight the on-layer potentials, $\Delta_{t/b}$, induced by the encapsulation environment. Stacking diagrams for ABA and ABC graphene can be obtained by removing the top or bottom layer of the ABCB tetralayer. The bottom row of panels shows the band structure of each tetralayer polytype in the energy range relevant for optical processes in the infrared frequency range (nested bands are exposed by slicing along the $p_y=0$ line).
• Figure 3: The color maps show the incoming photon frequency ($\omega$) dependence of the magnitude, $\abs{\sigma^a_{aa}}$ (top panels), and phase, ${\rm Arg}(\sigma^a_{aa})$ (bottom panels), of the SHG tensor in ABCB graphene. Vertical axes of those maps are used to show the variation of the SHG tensor with doping density, $n$ (middle column), and with a potential, $\Delta_b$, induced by the substrate on the bottom layer of MLG. On the left, we show an orthographic projection of the band structure of ABCB graphene (bands are enumerated using encircled indices), where we illustrate transitions involved in three types of resonant SHG. Here, solid horizontal lines point towards real states near the band edges involved in single-resonance transition, marked as $\omega^n_m$ and $\Omega^n_m$; thin dashed lines mark off-resonant virtual states. For double-resonant transitions, $^nD^m_l$, arrows point directly at the corresponding bands, identified by band indices $n,m,l$.
• Figure 4: We plot the intensity of SHG $\abs{\sigma^a_{aa}}$ for all remaining tetralayer and trilayer polytypes of graphene with stacking order labeled on the panels. Also labeled are the choice of $\Delta_b$, the externally induced inversion symmetry breaking parameter representing a substrate. We induce $\Delta_b=18{\rm meV}$ on the stacking orders that poses intrinsic inversion symmetry so as to induce a second order optical response. Along side each SHG intensity we plot the associated band structure with numbered bands. We label the high intensity features by resonance type and leading resonant band numbers using the same notation adopted in Fig. \ref{['fig:ABCB']}.