Direct measurement of broken time-reversal symmetry in centrosymmetric and non-centrosymmetric atomically thin crystals with nonlinear Kerr rotation

TL;DR

This work introduces an all-optical, third-harmonic Kerr rotation method to directly detect broken time-reversal symmetry in both centrosymmetric bilayer and non-centrosymmetric monolayer WS$_2$ crystals. By employing elliptically polarized excitation, spin-selective bandgap shifts induced by optical Stark and Bloch–Siegert effects break TRS, imprinting TRS-odd components into the nonlinear susceptibility $\boldsymbol{\chi^{(3)}}$ and rotating the emitted TH polarization. The authors develop a semiconductor Bloch equation–based analytical model and validate it with polarization-resolved THG experiments near the TH resonance, extracting TRS-related parameters such as the gap differences $\Delta_{\text{gap}}^{\pm K}$ and dephasing time $T_2$. They demonstrate that in monolayers TRS breaking ties to spin–valley locking, while in bilayers it couples to spin–valley–layer locking, revealing distinct spin–valley–layer physics in atomically thin semiconductors. The approach enables TRS diagnostics independent of space-inversion symmetry and paves the way for ultrafast valleytronic device concepts in centrosymmetric and non-centrosymmetric 2D materials.

Abstract

Time-reversal symmetry, together with space-inversion symmetry, is one of the defining properties of crystals, underlying phenomena such as magnetism, topology and non-trivial spin textures. Transition metal dichalcogenides (TMDs) provide an excellent tunable model system to study the interplay between time-reversal and space-inversion symmetry, since both can be engineered on demand by tuning the number of layers and via all-optical bandgap modulation. In this work, we modulate and study time-reversal symmetry using third harmonic Kerr rotation in mono- and bilayer TMDs. By illuminating the samples with elliptically polarized light, we achieve spin-selective bandgap modulation and consequent breaking of time-reversal symmetry. The reduced symmetry modifies the nonlinear susceptibility tensor, causing a rotation of the emitted third harmonic polarization. With this method, we are able to probe broken time-reversal symmetry in both non-centrosymmetric (monolayer) and centrosymmetric (bilayer) crystals. Furthermore, we discuss how the detected third harmonic rotation angle directly links to the spin-valley locking in monolayer TMDs and to the spin-valley-layer locking in bilayer TMDs. Thus, our results define a powerful approach to study broken time-reversal symmetry in crystals regardless of space-inversion symmetry, and shed light on the spin, valley and layer coupling of atomically thin semiconductors.

Figures (5)

• Figure 1: Principle of TH Kerr rotation and mirror symmetries of a rotating sphere.a For a monolayer TMD, broken TRS is equivalent to an asymmetry between the $\pm$K valleys. TRS breaking introduces new $\boldsymbol{\chi^{(3)}}$ tensor elements, leading to a rotation of the TH output by the angle $\theta$. b A rotating sphere as analogue for spin-oriented states. A sphere rotating counter-clockwise (indicated by the black circular arrow and straight black arrow pointing upwards) coincides with its mirror image when mirroring along a horizontal plane (see bottom), but the rotation direction changes for the mirror image when mirroring along a vertical plane (see left, indicated by the red circular arrow and straight red arrow pointing downwards). To keep the rotating sphere invariant under vertical mirroring, the rotation direction needs to be reversed (antisymmetry operation). Figure adapted from Ref. Fecher2022.
• Figure 2: Probing TRS breaking in mono- and bilayer WS_2 via TH Kerr rotation.a Real-space schematic of a WS_2 monolayer with preserved SIS and TRS. b Stereographic projection of the $D_{3h} \equiv \bar{6}m2$ crystallographic point group. c Energy and spin of the $\pm$K valleys of a WS_2 monolayer with preserved TRS. Linear excitation leads to linear TH parallel to the input. d Real-space schematic of a WS_2 monolayer with broken SIS and TRS. Due to the spin-selective gap opening, one spin dominates. e Stereographic projection of the $\bar{6}m'2'$ magnetic point group. Red elements represent symmetry operations that are only allowed under antisymmetry. f Energy and spin of the $\pm$K valleys of a WS_2 monolayer with broken TRS. Elliptical excitation leads to the opening of a gap at the $+$K valley by $\mathbf{\Delta} E$ and a rotated TH output. g Real-space schematic of a WS_2 bilayer with preserved SIS and TRS. h Stereographic projection of the $D_{3d} \equiv \bar{3}m$ crystallographic point group. i Energy and spin of the $\pm$K valleys of a WS_2 bilayer with preserved TRS. Within one layer, the spin states differ for each valley because of spin-valley-layer locking. Linear excitation leads to linear TH parallel to the input. j Real-space schematic of a WS_2 bilayer with preserved SIS, but broken TRS. Due to spin-selective gap opening one of the spins is now dominant. k Stereographic projection of the $\bar{3}m'$ magnetic point group. l Energy and spin of the $\pm$K valleys of a WS_2 bilayer with broken TRS. Elliptical excitation leads to the opening of a gap by $\mathbf{\Delta} E$ in the top and bottom layers and thus to a rotated TH output.
• Figure 3: PL and TH rotation measurements for different ellipticities and power in monolayer WS_2.a Comparison of the total emitted TH intensity (black squares, left axis) as a function of FB wavelength and the emitted PL (pink line, right axis). b TH rotation angle as a function of the FB ellipticity angle and for an excitation power of 10mW (orange circles) and 15mW (pink squares). Solid lines are linear fits to the data. The inset shows the elliptical polarization pattern of the emitted TH for linear (violet triangles) and elliptical (blue circles) input polarization. c Power dependence of the TH rotation for -20° (blue circles) and -30° (green squares). Solid lines are linear fits to the data with a fixed intercept of 0° rotation at 0mW. d Wavelength dependence of the TH rotation angle for an input power of 15mW for ellipticity angles of -20° (blue circles) and -30° (green squares).
• Figure 4: Comparison of analytical model and experimental results for monolayer WS_2.a Wavelength-dependent TH rotation for fixed ellipticity of -20° and input power of 15mW (blue circles in Fig. \ref{['fig:FIG3_experimental_results_monolayer']}d). b Ellipticity-dependent TH rotation for a fixed input power of 15mW (pink squares in Fig. \ref{['fig:FIG3_experimental_results_monolayer']}b). c Power-dependent TH rotation for a fixed ellipticity of -20° (blue circles in Fig. \ref{['fig:FIG3_experimental_results_monolayer']}c). For the analytical calculations, we employ $T_2\space{=}\space 28\,\text{fs}$, $\Delta\space{=}\space$ 1.05eV and two values of the dipole element: $d\space{=}\space3\eangstrom$ (solid lines) and $d\space{=}\space3.5\eangstrom$ (dashed lines).
• Figure 5: PL and TH rotation measurements for different ellipticity angles and power values in bilayer WS_2.a Comparison of the total emitted TH intensity (black squares, left axis) and the emitted PL (pink line, right axis) as a function of the excitation wavelength. b Rotation of the emitted TH dependent on the ellipticity angle and for an excitation power of 10mW (orange circles) and 17mW (pink squares). Solid lines are linear fits to the data. c Power dependence of the TH rotation for -20° (blue circles) and -30° (green squares). Solid lines are linear fits to the data with a fixed intercept of 0° rotation at 0mW. d Wavelength dependence of the TH rotation for an input power of 15mW for ellipticity angles of -20° (blue circles) and -30° (green squares).