Measuring Valley Polarization in Two-Dimensional Materials with Second-Harmonic Spectroscopy

A population imbalance at different valleys of an electronic system lowers its effective rotational symmetry. We introduce a technique to measure such imbalance (a valley polarization), which exploits the unique fingerprints of this symmetry reduction in the polarization-dependent second-harmonic generation (SHG). We present the principle and detection scheme in the context of hexagonal two-dimensional crystals, which include graphene-based systems and the family of transition metal dichalcogenides, and provide a direct experimental demonstration using a molybdenum diselenide monolayer with 2H polytype at room temperature. We deliberately use the simplest possible setup, where a single pulsed laser beam simultaneously controls the valley imbalance and tracks the SHG process. We further developed a model of the transient population dynamics, which analytically describes the valley-induced SHG rotation in very good agreement with the experimental data. In addition to providing the first experimental demonstration of the effect, this work establishes a conceptually simple, compact, and transferable way of measuring instantaneous valley polarization, with direct applicability in the nascent field of valleytronics.