Forbidden second harmonics in centrosymmetric bilayer crystals

Abstract

Optical spectroscopy based on second-order nonlinearity is a critical technique for characterizing two-dimensional (2D) crystals as well as bioimaging and quantum optics. It is generally believed that second-harmonic generation (SHG) in centrosymmetric crystals, such as graphene and other bilayer 2D crystals, is negligible without externally breaking the inversion symmetry. Here, we show that with a new homodyne detection technique, we can apparently circumvent this symmetry-imposed constraint and observe robust SHG in pristine centrosymmetric crystals, without any symmetry-breaking field. With its exceptional sensitivity, we resolve polarization-resolved SHG in bilayer hexagonal boron nitride (h-BN), bilayer 2H-WSe$_2$, and remarkably, Bernal-stacked bilayer graphene, allowing us to unambiguously identify the crystallographic orientation in these crystals via SHG for the first time. We also demonstrate that the new technique can be used to non-invasively detect uniaxial strain and optical geometric phase in these crystals. The observed SHG in our experiments is attributed to second-order nonlinearity in the quadrupole channel, which is controlled by the presence of the $C_2$ symmetry instead of the inversion symmetry. Our new technique expands the capability of nonlinear optical spectroscopy to encompass a large class of centrosymmetric materials that could never be measured before, and can be used for quantum sensing of moiré materials and twisted epitaxial films.

Figures (4)

• Figure 1: An ultrasensitive phase-resolved second-harmonic spectroscopy. (a) Illustration of CANS technique in comparison to traditional SHG spectroscopy. In CANS, coherent SHG waves from the reference and the sample interfere at the beam splitter (BS) to create a spectrum that contains the phase-resolved SHG amplitude from the sample. The upper-right panel shows the time domain pulses from the reference and the sample, at the fundamental (red) and second-harmonic (green) wavelengths. The lower-right panel shows the spectrum near $\lambda_0/2$, in a traditional SHG measurement, as well as in CANS measurements, when a phase difference of $\Delta\phi=0$ or $\pi$ is present between the reference SHG and sample SHG waves. (b) In a monolayer MoS2 flake, the armchair directions from Mo to S (AC+) are nonequivalent with the armchair directions from S to Mo (AC-). Scale bar is 5µm. (c) Polarization-dependent spectrum and SHG amplitude (square root of intensity) in a traditional SHG measurement. The six polarizations that yields maximum SHG signal are the armchair directions. However, the AC+ and AC- directions cannot be distinguished. (d) Spectrum and SHG amplitude in a CANS measurement. The phase information of the SHG wave is encoded into the interference pattern in the wavelength domain. The phase-resolved SHG amplitude extracted from CANS clearly differentiates AC+ and AC- directions. (e) SHG intensity map and crystallographic rotation map of a region with multiple overlapping MoS2 monolayer flakes. By using CANS, the crystallographic orientation of a 2D crystal can be determined unambiguously, respecting the underlying $C_3$ symmetry. Both maps are $100\times$100µm.
• Figure 2: Second harmonic generation from centrosymmetric crystals. (a) Crystal structure of bilayer 2H-TMDs, showing its inversion center. (b) While traditional SHG can hardly detect any SHG signal from a bilayer 2H-WS2, CANS can clearly detect an SHG interference pattern. See SI for background removal procedure for both data. The traditional SHG data is scaled by 10 times. (c) The complex-valued SHG amplitude extracted from the CANS measurement in (b). The SHG amplitude is represented by the radial distance (arbitrary units), and the SHG phase is represented by color (from $-\pi$ to $\pi$). The solid curve is a fitting curve to the data points (see SI for the fitting function). (d) Crystal structure of bilayer h-BN and its inversion center. (e-f) Same measurements as in (b-c) for bilayer h-BN. Traditional SHG in (e) is scaled by 5 times. The distortion seen in (f) is attributed to uniaxial strain in the crystal.
• Figure 3: Quardrupolar SHG and $C_2$ rotational symmetry. (a) In monolayer nonlinear crystals, SHG is generated by oscillating dipoles in the plane (e.g. $P_x^{2\omega}$ along the $x$ direction), excited by the fundamental. These dipoles radiate SHG waves in the $z$ direction. (b) In a bilayer crystal (2H-TMDs or h-BN), the SHG waves from the two oppositely rotated layers cancel, but a small residue remains. The total dipole density $P_{x}^{2\omega}$ vanishes, but the quadrupole density $Q_{zx}^{2\omega}$ is nonzero, which is can still radiates in the $z$ direction, giving rise to a detectable QSHG signal. (c) and (f) shows the crystal structure of monolayer graphene (MLG) and Bernal bilayer graphene (BLG), respectively. While both are centrosymmetric, MLG has a $C_2$ rotation axis while BLG does not. (d) MLG does not show any CANS signal apart from an polarization independent background. (g) BLG shows a clear CANS signal that can be attributed to QSHG in this $C_2$-broken crystal. (e) and (h) show the polarization-resolved complex-valued SHG amplitude extracted from (d) and (g), respectively.
• Figure 4: Strain and geometric phase sensing using CANS. (a) Uniaxial strain (including normal and shear strains) in a 2D crystal breaks the threefold rotational symmetry ($C_3$). (b) Complex-valued SHG amplitude and phase in a bilayer h-BN crystal with strain. The empty dots are data, solid curves are fitting that includes strain effects, and dashed curves are fitting without strain effects (see SI). (c) In a chiral system (e.g. a twisted crystal with twist angle $\theta_\mathrm{twist}$), a linearly polarized fundamental input light ($\omega$) becomes left circularly polarized (LCP) at the second harmonic frequency ($2\omega$). The Pancharatnum-Berry phase (geometric phase) is the optical phase of the $2\omega$ wave, which is related to the fundamental polarization $\theta$ by $\Delta\phi=3\theta$ when the $2\omega$ wave is purely LCP. (d) The SHG phase measured by CANS in a $\theta_{twist}=66^\circ$ pentalayer-pentalayer WS2 device, where the SHG is close to LCP. The fitting curve is $3\theta$ with an offset that represents the lattice orientation. (e) SHG chirality map of the $\theta_{twist}=66^\circ$ device. The dashed line highlights the twisted region. (f-h) Same measurements for a $\theta_{twist}=57^\circ$ device, where the system has opposite chirality and the SHG is right circularly polarized (RCP). The geometric phase is $\Delta\phi=-3\theta$ plus an offset in this case.