Phonon dispersion, defining the frequency of lattice vibrations, is crucial for understanding thermal and mechanical properties of materials. Traditional techniques like inelastic neutron or X-ray scattering require large-scale facilities and are unsuitable for nanoscale crystals. Picosecond acoustic measurements offer high temporal resolution to study interfaces in thin-film heterostructures, but frequency ranges have been limited. Two-dimensional (2D) van der Waals (vdW) materials, known for their unique properties and atomically clean interfaces, provide ideal platforms for studying strain wave behavior at THz frequencies. This study introduces a method leveraging picosecond acoustics to measure full phonon dispersion in nanoscale 2D vdW materials. The approach utilizes a heterostructure, generating broadband coherent acoustic phonons, and observing their propagation and reflections within a precisely defined nanoscale region to extract phonon dispersion information. This technique offers advantages over traditional methods by enabling measurements in nanoscale systems and utilizing readily accessible experimental setups.

Previous studies have explored coherent lattice vibrations in 2D vdW materials using various techniques, such as Raman spectroscopy and time-resolved optical measurements. However, measuring the complete phonon dispersion relation in nanoscale systems has remained challenging. Existing methods, such as inelastic neutron or X-ray scattering, require large-scale facilities and struggle with the small scattering cross-sections of nanoscale samples. Picosecond acoustic measurements have been employed to investigate interfaces and nanostructures, but they've been limited to lower GHz frequencies due to light diffraction. This study bridges this gap by utilizing the unique properties of 2D vdW heterostructures to access the THz frequency range.

The researchers created a van der Waals heterostructure consisting of a few layers of black phosphorus (BP) encapsulated between top and bottom hexagonal boron nitride (hBN) layers on a quartz substrate. Two samples were prepared with varying top hBN layer thicknesses (12 nm and 29 nm). A pump-probe experiment using 60 fs pulses from a Ti:sapphire laser (760 nm) was employed. The pump pulse excited carriers in the BP layer, generating a photoacoustic strain pulse due to inhomogeneous photoexcitation. This pulse propagated into the hBN layers, reflected at the hBN/air interface, and produced echoes that were detected in the BP layer using a probe pulse (sensitive to BP optical transitions). The time-resolved transient differential transmission (ΔT/T) was measured to capture the strain pulse and its echoes. Sliding window Fourier transform (SWFT) analysis determined the frequency-dependent time-of-flight (TOF) and group velocity dispersion. A 1D linear chain model (LCM) was developed to simulate the out-of-plane longitudinal acoustic (LA) mode of hBN, and Density Functional Theory (DFT) calculations using the local density approximation (LDA) and optB86b-vdW functional were performed for comparison. Ab initio simulations, including DFT band structure calculations and molecular dynamics (MD) simulations using a machine-learned force field (MLFF), were used to investigate the microscopic origin of the photoinduced strain pulse generation and propagation. The MLFF simulations provided a real-time atomistic-scale description of acoustic pulse propagation and were compared to the experimental ΔT/T signals.

The experimental results showed that the strain pulse generated in the BP layer extended over a wide spectral range (0-3 THz). The frequency-dependent TOF measurements allowed for the determination of the hBN phonon group velocity dispersion (GVD). The LCM model accurately predicted the strain wave arrival times, confirming its effectiveness in describing the out-of-plane LA phonon dispersion of bulk hBN. DFT calculations (LDA functional) showed good agreement with the LCM and experimental GVD. The Fourier transform (FT) spectrum of the strain wave exhibited pronounced frequency combs due to multiple echoes, which further confirmed the phonon dispersion and allowed for an independent determination of the group velocity. First-principles calculations revealed the microscopic origin of photoinduced strain pulse generation: photoexcitation created charge depletion in the inner BP layers, inducing lattice relaxation and strain pulse generation. The MD simulations based on MLFF accurately reproduced the experimental strain wave propagation and echoes, confirming the relation between the differential transmission signal and the time-transient atomic displacements in the BP layer. The echo times showed a linear correlation with the top hBN layer thickness, highlighting the method's potential for nondestructive inspection of nanoscale vdW heterostructures.

The study successfully demonstrates a novel method for measuring full phonon dispersion in nanoscale 2D vdW materials using picosecond acoustics. The combination of experimental measurements, LCM modeling, DFT calculations, and ab initio simulations provided a comprehensive understanding of the photoacoustic strain pulse generation and propagation. The method's ability to access THz frequencies and its applicability to nanoscale systems represents a significant advancement over traditional techniques. The good agreement between experimental data, LCM simulations, and DFT calculations validates the accuracy and reliability of the proposed approach. The frequency combs observed in the FT spectrum provide an independent way to determine phonon dispersion, further strengthening the findings. The detailed atomistic simulations illuminated the microscopic mechanisms underpinning the experimental observations.

This work presents a new method for measuring the complete phonon dispersion along the stacking direction in nanoscale van der Waals heterostructures. The use of picosecond acoustics, coupled with detailed theoretical modeling, offers a powerful tool for characterizing these materials. Future work could extend this method to other 2D vdW materials and investigate the effects of different heterostructure configurations on phonon dispersion. Exploring the use of this technique for characterizing other nanoscale systems, like semiconductor quantum wells, is also a promising direction.

The study focuses on a specific vdW heterostructure (hBN/BP/hBN). While the LCM model provides a simplified representation of the hBN lattice, further refinement might be needed to capture more complex interactions. The analysis assumes primarily out-of-plane LA mode propagation. Other phonon modes might contribute, particularly at higher frequencies. The photoexcitation density was moderate in the simulation, and investigation across a range of excitation densities could provide a broader understanding of the photoacoustic response.