Piezoreflectance Spectroscopy of Optical Transitions in van der Waals Layered Crystals

Summary

We present a reproducible strain-modulation piezoreflectance spectroscopy protocol that enables measurement of ΔR/R with lock-in detection for van der Waals crystals transferred onto piezoceramics, and results with characterization of direct optical transitions through ΔR/R spectra utilizing third-derivative fitting with the Aspnes formula.

Abstract

The goal of this protocol is to enable precise identification of direct optical transitions in van der Waals semiconductors by piezoreflectance spectroscopy. The method applies a small periodic strain modulation to a sample mounted on a piezoelectric transducer and detects the reflectance change with a lock-in amplifier, yielding derivative-like spectra in which a background signal is eliminated, and band-edge features become prominent. The workflow details assembly of a single-path, broadband optical setup (halogen source, monochromator, microscope optics, silicon photodiode), integration of a low-noise preamplifier and lock-in electronics, and safe high-voltage connection to the piezoceramic. The stepwise instructions cover exfoliation and transfer of flakes, application of ultrathin adhesive, silver-paste electrical contacts with room-temperature curing, and acquisition of the AC component (ΔR) and DC component (R) – reflectance referenced by chopper. Data processing includes baseline correction, computation of ΔR/R, and fitting with standard third-derivative line shapes to extract transition energies, broadenings, phases, and relative strengths. Representative settings and troubleshooting guidance optimize signal-to-noise ratio while preserving spectral accuracy. Compared with reflectance contrast and photoreflectance, this approach improves results for strain-responsive transitions and remains effective when intrinsic electric-field modulation is weak. The protocol supports micrometer-scale mapping and compatibility from thin layers to bulk, and it is encouraged to complement with photoluminescence or Raman measurements, providing a robust, generalizable route to optical characterization of layered semiconductors.

Introduction

Piezoreflectance spectroscopy (PzR) is a strain‑modulation variant of modulation spectroscopy that exploits well‑established light-matter interactions to extract with high precision excitonic and band-to-band transitions energies¹. In PzR, the sample is mounted on a piezoelectric transducer; an applied alternating voltage generates a small periodic strain (δa/a) that perturbs the crystal lattice and modulates the electronic band structure, primarily the band gap. This induces synchronous changes in the complex dielectric function and hence in the reflectance, ΔR, which are detected with phase‑sensitive (lock‑in) methods to yield derivative‑like line shapes with the slowly varying background strongly suppressed. The objective is to enable quantitative optical characterization of van der Waals (vdW) crystals and semiconductor microstructures - including superlattices, quantum wells, and heterojunctions - by revealing weak band‑edge features that are often invisible in conventional reflectance contrast (RC) and by providing a controlled handle on strain-exciton coupling relevant to advanced electronic and optoelectronic applications^2,3,4,5,6. The protocol is compatible with micrometer‑scale flakes and can be co‑registered with photoluminescence or Raman measurements for multimodal analysis under tunable strain. In addition, we are investigating an adhesive‑free configuration in which vdW crystals are exfoliated directly onto piezoceramics to measure micropiezoreflectance (µPzR) in micron-sized vdW flakes⁷.

Compared to reflectance contrast, usually applied to vdW crystals and heterostructures^8,9, piezoreflectance offers several practical and analytical advantages. Because the lock-in detected PzR signal is intrinsically normalized to ΔR/R, the spectral line shape is preserved even when absolute throughput drifts¹, provided the photon flux is sufficient for a good signal-to-noise ratio. The derivative nature of modulation spectra suppresses slowly varying backgrounds and sharpens optical transitions related to a critical point in the total optical density of states, which improves energy resolution and enables reliable extraction of even weak interband excitonic and band-to-band transitions^11,12. PzR measurements selectively probe only those transitions whose parameters (transition energy E, intensity I, or broadening Γ) respond to the applied strain; consequently, spectral regions that are strain-insensitive yield no signal, while truly responsive transitions stand out with high contrast^1,13,14. On the other hand, RC relies on subtracting two independently measured reflectance spectra (sample and reference usually coming from substrate), so any small mismatch or drift translates into a large, nonzero baseline across the entire spectral range; weak transitions can therefore be obscured by this background, a limitation explicitly demonstrated in materials where RC fails to resolve interband features that remain prominent in PzR (e.g., FePS₃ and NiPS₃), while strong transitions in MoS₂ remain visible by either method⁷.

As a stress-modulated counterpart to conventional reflectance, PzR defines a distinct branch of modulation spectroscopy in which periodic uniaxial or coplanar stress isolates derivative-like features directly linked to deformation potentials and symmetry selection rules^{1,15,16,17,18,19,20,21,22,23,24}. Situated within six decades of modulation spectroscopy, piezomodulated reflectance has evolved from its 1960s first application in measurements of group-IV/III–V semiconductors into a versatile probe of deformation potentials, carrier character, and excitonic structure across bulk crystals, heterostructures, and vdW materials^{1,15,16,18,19,25,26,27}. PzR emerged in the early 1960s with landmark demonstrations of piezoreflectance in Ge¹⁵ and Si²⁸, as well as piezo-electroreflectance across Ge, GaAs, and Si. Foundational theory/experiment in the early 1970s consolidated the formalism and provided comprehensive reviews^1,25. A subsequent renaissance established PzR as a direct probe of deformation potentials in the elastic limit and as a vibronic-state-sensitive technique¹⁷, which catalyzed applications to strained heterostructures in the 1990s – resolving electron/hole character and spatial localization in epilayers, quantum wells, and superlattices under controlled uniaxial or coplanar stress geometries¹⁸. In the 2000s and beyond, the scope extended to semiconductor nanostructures and van der Waals (vdW) materials - capturing confined-state transitions in surface quantum dots¹⁹ and probing electronic band structures in TMDs and alloys^26,27. A unifying thread across this literature is the application of a well-controlled, periodically modulated stress: via piezoelectric transducers on single crystals, on evaporated metal films^21,22, or by low-frequency bending of ionic and metallic crystals (e.g., KI, KBr, Cu)²³, complemented by an apparatus enabling combined static and dynamic uniaxial stress²⁴. Building on this corpus, our method targets open opportunities for vdW crystals and heterostructures – probing excitonic transitions that can be weak or invisible in reflectance contrast, enabling stress coupling, and integrating PzR with PL measurements or Raman spectroscopy for multimodal, strain-resolved characterization^19,26,27. Therefore, readers can position PzR alongside and in combination with established optical probes when judging suitability for their systems.

Protocol

1. Assembly of the system for the piezoreflectance (ΔR/R) measurements with the lock-in technique

1. Build the optical path to perform piezoreflectance measurements in dark configuration using a broadband halogen light source, monochromator, exit slit with adjustable aperture, beam steering optics, sample stage with XYZ microcontrollers, micro-objective lens, and detector (e.g., silicon PIN photodiode detector). Route a preview of the beam to a CCD camera for alignment (Figure 1).

Figure 1: Schematic diagram of the experimental setup. Pink section only for chopper-referenced reflectance measurements with better signal-to-noise ratio. Please click here to view a larger version of this figure.

2. Integrate the electronic chain by adding a low-noise current preamplifier between the photodiode and the lock-in amplifier. Route the AC component and the DC component of the detected signal to the lock-in amplifier.
3. Connect a function/AC voltage generator to the piezoceramic element through insulated leads. Route the chopper reference output to the lock-in for reflectance measurements.
4. Develop a control and acquisition application (e.g., LabVIEW) that communicates with the monochromator and the lock-in amplifier, reads the detector signal, and writes the results to a text file. Configure instrument drivers and communication ports on the host computer. See Figure 2 for an application block diagram example.
5. Connect the lock-in output to a computer for data logging. After the optical alignment has been completed, the experiment can be paused with the system powered down.

Figure 2: Software workflow for monochromator lock-in control and data acquisition. Block diagram of the control program (e.g., LabVIEW). Please click here to view a larger version of this figure.

NOTE: The control/acquisition application is hardware dependent. Because valid setups vary in instrument models and wiring, a specific LabVIEW code example is not portable and is therefore not provided. For a representative screenshot, see Supplementary Figure 1.

2. Preparation of the sample

NOTE: See Figure 3 for necessary equipment.

1. Thin the bulk material to be characterized (e.g., a van der Waals crystal such as MoSe₂, MoS₂, WSe₂, WS₂) by repeated cleavage using adhesive tape (Figure 4A) until the preferred thickness is reached.
2. Transfer the thinned fragments from the tape to a PDMS stamp by laminating PDMS onto the tape (Figure 4B) and peeling the PDMS back after ~5 min. so that flakes remain on the PDMS (Figure 4C) for subsequent transfer to the piezoceramic.
3. Clean the piezoceramic surface by rinsing with acetone in a certified chemical fume hood. Dry the surface with clean, dry air or nitrogen.
   CAUTION: Acetone is highly flammable and volatile; avoid ignition sources, avoid inhalation and skin/eye contact, and collect waste in approved containers inside a fume hood. Wear appropriate PPE (lab coat, safety glasses, and chemical-resistant gloves).
4. Apply a very thin layer of an adhesive medium to the piezoceramic (Figure 4D) (e.g., colorless nail polish as a thin adhesive, or a superglue layer for thicker films). Handle volatile adhesives in a chemical fume hood and allow a minimal amount to remain before transfer. Proceed promptly to the next step of the protocol to prevent the thin layer of adhesive from drying.
   CAUTION: Many adhesives contain flammable solvents; handle inside a certified chemical fume hood and keep away from ignition sources.
5. Place the PDMS carrying the flake onto the wet adhesive on the piezoceramic, press gently to ensure contact (Figure 4E), wait ~30 s and remove (peel) the PDMS so that the material remains on the piezoceramic (Figure 4F). Verify adhesion by visual inspection (by eye); if needed or preferred, confirm under an optical microscope.
   NOTE: Avoid lateral sliding of the PDMS stamp during contact to prevent tearing/dragging of flakes; peel the PDMS back smoothly.

Figure 3: Materials for exfoliation and transfer to the piezoceramic. Please click here to view a larger version of this figure.

Figure 4. Step-by-step preparation and transfer of exfoliated vdW flakes onto a piezoceramic actuator. (A) Thickness reduction of the bulk crystal by repeated cleavage using adhesive tape. (B) Lamination of a PDMS stamp onto the tape to pick up thinned fragments. (C) Peeling the PDMS stamp using tweezers; exfoliated flakes are visible on the PDMS by eye (depending on thickness). (D) Application of a thin adhesive layer onto the piezoceramic surface to promote flake adhesion and strain transfer. (E) Placement of the PDMS stamp carrying flakes onto the wet adhesive layer for transfer. (F) Final piezoceramic actuator with flakes adhered to after PDMS removal, ready for optical alignment and piezoreflectance measurements. Please click here to view a larger version of this figure.

3. Sample mounting and connection

1. Create separate electrodes to enable the application of an AC voltage across the piezoceramic.
   1. Place the piezoceramic with the transferred sample on the sample stage and secure it mechanically with silver paste, ensuring electric contact with the bottom electrodes of the piezoceramic.
   2. Form the top electrical contact using silver paste and copper wire. Allow the silver paste to fully dry (~10 h at room temperature) before applying any voltage to prevent contact damage.
      CAUTION: Silver paste is highly toxic to aquatic organisms, causing long-term adverse effects. May cause drowsiness or dizziness. Flammable liquid and vapor.
2. After applying the silver paste, cure at room temperature for approximately 10 h before proceeding. The prepared sample can be safely stored as long as the contacts are intact and conduct electricity.
3. Insert the mounted sample into the optical setup. Connect the electrodes with insulated wires (Figure 5) to the AC voltage generator and to the ground. Keep all exposed conductors insulated and protected, away from the optical path.
   ​CAUTION: High voltage may be present (e.g., up to 1500 V AC); use insulated leads, verify proper grounding, and do not touch live contacts. Enclose connections or use interlocks where available.

Figure 5: Piezoceramic disk actuator and sample-stage mounting for optical alignment. (A) Top view of the piezoceramic disk (Ø30 mm) with samples glued on its surface; leads are connected to the top (visible) and bottom (hidden under the piezoceramic disk) electrodes. (B) Actuator mounted on a manual X–Y translation stage with micrometer drives, enabling micrometric-precision positioning for optical alignment. Please click here to view a larger version of this figure.

4. Measurement of lock-in AC component proportional to ΔR and the DC component proportional to R

1. Turn on the halogen light source and set it to maximum output. Ensure that the optical chopper is not obstructing the monochromator entrance slit.
2. Power on the monochromator, preamplifier, lock-in amplifier, CCD camera, and control computer with the developed control and acquisition application. Wait at least 10 min until the system stabilizes. Among other things, stabilization of the signal depends on the heating due to turning on the halogen lamp.
3. Set the monochromator to zero wavelength so that the entire lamp spectrum passes through. Select the diffraction grating/groove density appropriate for the measurement.
4. Set the preferred width of the monochromator’s entrance/exit slit to achieve the desired spectral resolution.
   NOTE: The narrower the slit, the better the spectral resolution of the measurement; however, less light reaches the sample, and the detected signal becomes weaker. The parameters should be adjusted experimentally to achieve the optimal effect.
5. Route the beam to the CCD camera and align the spot in the selected region of the sample using the XYZ microcontrollers. Adjust the iris diaphragm aperture to set the spot diameter on the sample.
6. Switch the signal path from the CCD preview to the silicon PIN photodiode detector.
7. Configure the preamplifier: set the bias voltage, filter type/cutoff, input offset, gain mode, and sensitivity to achieve a stable signal without saturation. Document the chosen values. For sample settings, see Supplementary Figure 2.
8. Configure the lock-in amplifier to display X (in-phase). Set the reference frequency (e.g., 280 Hz) as intended for the AC voltage generator frequency and select the internal reference source. Then choose an appropriate time constant (typically ≥ 1 s; up to ~30 s for weak signals) and sensitivity. For sample settings, see Supplementary Figure 3A.
   NOTE: Optimize the time constant and sensitivity empirically based on material response and available light, starting from values previously successful for similar samples. The time interval (delay) between measurement points should be longer than the time constant set during the measurement.
9. Enable the AC voltage generator and set the desired amplitude (e.g., 100 V). Verify that the signal reaches the piezoceramic without overdriving the device. For sample settings, see Supplementary Figure 4B. The frequency of the generated voltage is referenced from the lock-in amplifier. Verify the output is disabled before changing leads; enable output only after securing all electrical connections.
10. Launch the acquisition application and select the measurement channel corresponding to the X lock-in output. Enter hardware-consistent parameters (time constant and sampling rate) and specify the output text file path for saving the data.
11. Define the spectral range (wavelength or photon energy) and the monochromator step size according to instrument limits.
    NOTE: Ideally, a sufficiently wide range is desired so that the spectrum baseline reaches zero outside the range of expected optical transitions. Remember that the spectral resolution of the measurements depends on the monochromator’s slit width.
12. Start the scan and allow it to continue uninterrupted until the end of the programmed range. Estimate the total duration as (number of steps) x (time constant) and plan accordingly. Do not interrupt the scan once it has started; continuous acquisition is required for consistent lock-in filtering.
13. After completion, turn off the AC voltage generator without disturbing the position of the sample. Proceed with the reflectance (R) measurement.

5. Measurement of additional DC reflectance component (R) using chopper-referenced lock-in mode

1. Confirm that the AC voltage generator remains off.
2. Do not alter the beam position on the sample. If focus is uncertain, send the beam to the CCD preview, refocus and re-center the spot, and then return the signal path to the photodiode.
3. Start the optical chopper with its controller and set its frequency equal to the modulation frequency used previously (e.g., 280 Hz). Verify that the chopper reference output is connected to the lock-in reference input. For sample settings, see Supplementary Figure 4A.
4. Keep the preamplifier settings unchanged. In the lock-in amplifier, select the R (referenced reflectance) display, choose the chopper controller as the external reference source, and set input parameters identical to Section 4. Adjust the time constant (e.g., 0.1 s to 0.3 s typical) and the sensitivity to maintain a stable signal. For sample settings, see Supplementary Figure 3B.
   NOTE: Expect shorter time constants and a larger signal value for R compared to ΔR.
5. In the acquisition application, select the lock-in R channel and mirror the lock-in time-constant setting. Set a new path for the output text file. Name files to encode the sample, the location on the sample, and the acquisition parameters so that the data sets of Section 4 (ΔR) and Section 5 (R) can be matched as pairs.
6. Set the same spectral range and monochromator step size as used in Section 4 for the same spot on the sample. Start the scan and run it to completion.
7. After completion, turn off the chopper. If no further measurements are planned the same day, turn off the light source, monochromator, and other instruments.

6. Analysis of the piezoreflectance spectrum (ΔR/R)

1. Open the two text files collected in Section 4 and Section 5 in data-analysis software capable of plotting and data manipulation (e.g., OriginLab).
2. Label the datasets for clarity (e.g., PzR for ΔR from Section 4 and R for reflectance from Section 5). Plot the PzR signal versus the photon energy or wavelength on appropriate axes; prefer the photon energy (eV) on the x-axis.
3. If the PzR baseline deviates from zero outside expected transitions, perform a straight-line or baseline subtraction to zero the off-resonant regions. Document the operation parameters used.
4. Compute ΔR/R by dividing the baseline-corrected PzR data by the R data point-by-point. Save the resulting ΔR/R as a new dataset.
5. Plot ΔR/R versus photon energy to visualize resonances at optical transitions. Inspect the spectrum for features consistent with known transitions of the material.
6. Fit ΔR/R using the Aspnes line-shape formalism¹² (Equation 1) to extract transition energies and broadenings. Report the fit parameters and uncertainties.
   
   Where y₀ – baseline level (the flat line of the spectrum near the resonance), a, b – optional linear/quadratic baseline terms to absorb slow drift/curvature (use only if needed), θ – resonance phase, E_g – transition energy (peak position), Γ – linewidth/transition broadening, C – resonance amplitude, m – critical-point exponent, j – transition index.
   NOTE: For discrete excitonic features, use m = 2; for interband critical points, use m = 2.5 or m = 3. In our practice, at low temperature we usually set m = 2; for T > 100 K, we set m = 2.5. This parameter also depends on the studied material.
   1. Select the resonance window. Mark a fitting range that cleanly brackets the feature and contains short off-resonant segments on both sides.
   2. Enter strong initial guesses of the fit parameters. In the fit dialog, pre-fill the parameters you can read off, e.g., E_g – from the x-axis extremum; y0 – from the local baseline on the y-axis; C – from the peak-to-peak amplitude; m – per the rule above; a, b – set as zero, unlock only if needed for correcting fit line shape. Lock all parameters except for y₀.
   3. Iteratively unlock the fitting parameters (one at a time):
      Unlock C → fit
      Unlock θ → fit
      Unlock Γ → fit
      Unlock E_g → fit
      This will result in the final fine alignment. Only if residual slope/curvature remains, unlock a then b; otherwise, keep a = b = 0.
      ​NOTE: Bounds that stabilize convergence: Γ ∈ [5, 300] meV; θ ∈ [−π, π]; C > 0; E_g within ± 50 meV of the visual extremum; y₀ within ± 2× the off-resonant root mean square (RMS); keep ∣a∣ and ∣b∣ small (baseline only).
   4. Accept the fit when: residuals are structureless around zero inside the window; parameters change by <1% upon a small change of the window; and refitting from perturbed seeds returns the same solution.
7. Calculate the modulus for each direct transition using Equation 2.
   
   ​​NOTE: The index j labels each individual critical point/transition included in the model (e.g., j = 1 for the A exciton, j = 2 for the B exciton, j = 3 for a higher-energy interband feature, etc.). Each j has its own fitted parameters (E_j, Γ_j, m_j, C_j, θ_j); only the modulus |C_j∣ enters Equation 2, so Δρ_j is phase-independent.
   1. Compute Δρ_j(E) for each fitted transition with parameters (E_j, Γ_j, m_j, C_j) taken from the Aspnes (TDFF) fit (Section 6.6). Plot Δρ_j(E) (and, if needed, the total Δρ(E) = ∑_jΔρ_j(E)) to compare the transition strengths between samples or conditions.

Results

Lock-in piezoreflectance (PzR) yields derivative-like signatures at direct optical transitions across a broad set of van der Waals (vdW) crystals and semiconductor microstructures. Figure 6 shows typical raw output from a single spot in WS₂ acquired with the workflow in Sections 4–6: strain-modulated AC reflectance ΔR, reference DC component proportional to reflectance R (Figure 6A), and the baseline-corrected ratio ΔR/R (Figure 6B). With a piezoceramic disk (thickness TH = 1 mm, OD = 30 mm) driven at 280 Hz and 100 V, ΔR/R peak magnitudes on the order of 10⁻⁵–10⁻⁴ are routinely obtained using lock-in time constants of 3 s to 30 s, allowing clear visualization of weak excitonic resonances that are often ambiguous or invisible in reflectance contrast (RC).

Figure 6: Representative outputs of the piezoreflectance workflow: strain-modulated reflectance and calculated PzR. (A) Strain-modulated AC reflectance ΔR (red, left axis) and chopper-referenced DC reflectance R (blue, right axis) from a single spot, measured at room temperature using a piezoceramic disk (TH = 1 mm, OD = 30 mm) driven at 280 Hz and 100 V. (B) Baseline-corrected piezoreflectance spectra (PzR) obtained by calculating ΔR/R. Please click here to view a larger version of this figure.

The ΔR/R baseline is typically flat (<1 x 10⁻⁵) in non-resonant regions when the spectral range extends beyond all strong features (Sections 4.11, 6.3). Residual curvature usually indicates an insufficient spectral window or slow lamp performance drift; both are corrected by extending the scan and/or subtracting a linear baseline from ΔR prior to forming ΔR/R (Section 6.3–6.4).

To quantify optical transitions in WS₂, the ΔR/R spectrum in Figure 7A (black) was fit with the Aspnes third-derivative functional form (Section 6.6, Equation 1). The resulting fit (cyan) reproduces all features; Figure 7B decomposes the model into four components (colored curves) with the dashed line showing their sum. The gray lines at the bottom of Figure 7A depict the phase-independent moduli Δρi(E) for direct transitions computed from Equation 2 using the parameters returned by the Aspnes fit, providing a convenient measure of the relative strength of each direct transition. The transition energies represented by peaks 1-4 are, respectively, 1.829 eV, 1.991 eV, 2.291 eV, and 2.421 eV. The energies obtained for the A and B transitions are consistent with prior modulation-spectroscopy reports on WS²²⁹.

Figure 7: Aspnes-model fitting of WS₂ piezoreflectance and decomposition into critical-point components. (A) Room-temperature PzR spectrum of bulk WS₂ (black; plotted as 10⁴ ΔR/R vs. photon energy) with best fit (cyan) obtained using the Aspnes third-derivative functional form. Thin gray curves show the decomposition into the individual modulus components of ΔR/R used in the fit. (B) Component line-shapes from the Aspnes model corresponding to panel A: Fit Peak 1 (green), Fit Peak 2 (red), Fit Peak 3 (blue), and Fit Peak 4 (magenta). The dashed black line represents the cumulative fit (sum of all components). Please click here to view a larger version of this figure.

Dependence on measurement parameters
Here, we assess how the measurement parameters specified in the protocol influence the experimental outcome. The parameter choices are not universal: each optical setup must be calibrated first on well-characterized reference materials such as common transition-metal dichalcogenides (TMDs, namely WS₂, WSe₂, MoS₂, and MoSe₂), whose excitonic transitions lie in the measurement window for the silicon diode utilized as a detector and provide known resonance energies. The absolute magnitude of the detected ΔR is mainly determined by fixed optical factors – the halogen-lamp output, transmission losses through lenses/beam splitters/mirrors, and the illumination spot size defined by the aperture, objective, and monochromator slit. After alignment, the electronic settings determine the signal-to-noise ratio, in particular the preamplifier/amplifier gain and bandwidth, the modulation frequency and alternating voltage amplitude applied to the piezoceramic and the lock-in time constant. When preparing a new setup or starting measurements for a new material family, these parameters need to be optimized with signal-to-noise ratio (SNR) as the primary figure of merit, ensuring the possibility of observing resonances in measured spectra.

Two practical dependencies are illustrated in the figures. First, the strain-drive scaling (Figure 8): with τ = 1 s, the ΔR line shape in WS₂ is preserved while the amplitude increases monotonically as the AC voltage is increased from 100 to 1200 V (Figure 8A). The peak magnitude at selected energies scales linearly with the voltage amplitude (Figure 8B), confirming the operation in the elastic regime of the transducer and the coherent strain coupling at the resonances. Second, the lock-in time constant (Figure 9A): increasing τ reduces high-frequency noise and reveals weaker features, as seen by comparing MoS₂ spectra at 3 s, 10 s, and 30 s measured with the same AC amplitude of 100 V. The trade-off is a greater susceptibility to baseline drift and longer scan durations; in practice, τ ≈ 3-10 s offers a robust compromise that preserves derivative-like line shapes while delivering adequate signal-to-noise ratio. However, for some materials, it can become impossible to obtain satisfying PzR spectra without setting the time constant to 30 s. These calibrations define an operating window that can be reused across samples and revisited after any optical realignment.

Figure 8: Drive-voltage dependence of ΔR: amplitude scaling and elastic-regime verification. (A) Room temperature spectra acquired with lock-in time constant τ = 1 s as the AC drive is increased from 100 V to 1200 V (light to dark blue). The ΔR line shape is preserved while the amplitude grows monotonically. (B) Peak magnitudes at selected energies (1.29 eV, 1.58 eV, 1.88 eV, 1.93 eV) vs. applied voltage with linear fits (red), confirming elastic-regime transducer operation and coherent strain coupling at the resonances. Please click here to view a larger version of this figure.

Figure 9: Measurement optimization and cross-validation: lock-in time constant effects and comparison to photoreflectance. (A) Room temperature MoS₂ PzR spectra acquired at 100 V with lock-in time constants τ = 3 s, 10 s, and 30 s (blue to red to gray). (B) Room temperature results obtained for bulk WS₂: PzR data (black) with Aspnes fit (cyan) compared to photoreflectance (PR, orange). Please click here to view a larger version of this figure.

Comparison with photoreflectance
Photoreflectance (PR) is a pump-probe variant of modulation spectroscopy in which a spectrally scanned probe beam measures the relative reflectance change ΔR/R induced by a periodically modulated pump. Moreover, PR is contactless electroreflectance: the chopped pump creates excess carriers that modulate built-in electric fields (typically in surface/depletion regions), thereby perturbing the complex dielectric function and, as a result, the sample reflectance. Because the detection is phase-sensitive (lock-in at the pump modulation frequency), PR yields derivative-like line shapes with high sensitivity to interband critical points while suppressing a large, slowly varying background^12,13,30.

For this section, in addition to the PzR signal, the PR signal was measured using the lock-in technique, which also allows extracting: (i) weak AC signals proportional to ΔR from the background and (ii) the DC component (proportional to reflectance, R). Both components were recorded with the same Si PIN photodiode and lock-in detection used for PzR. The relative changes in the reflection spectrum (ΔR/R) were evoked by modulation of the built-in surface electric field of the investigated sample, generated by illuminating the sample with a CW 405 nm laser that was mechanically chopped at 280 Hz. All measurements were performed in a dark configuration³⁰.

Figure 9B presents the results for WS₂, a representative of the material class called transition-metal dichalcogenides with the general formula MX2 (where M = Mo, W, and X = S, Se, Te). In WS₂, PR produces clear derivative-like features at the A/B excitons, and the transition energies agree with those obtained from PzR (cyan Aspnes fits over the black PzR traces). This reflects that both modulation spectroscopies probe the same optical critical points in WS₂. The match between PR features and the Aspnes fits to PzR confirms that the PzR response is governed by the same excitonic dielectric-function derivative, validating PzR as a promising alternative to PR for extracting excitonic transition energies in TMDs.

Piezoceramic Actuator Selection
To generate a measurable piezoreflectance signal, the primary actuator-selection criterion is the achievable strain amplitude at the sample. In practice, we therefore prioritize piezoceramic elements with a large piezoelectric charge/strain coefficient (piezo modulus) d_ij, which quantifies either the induced charge density per applied stress or, the mechanical strain per applied electric field. For a thickness-poled disk actuator driven by a voltage (U) across its thickness (TH), the in-plane deformation is conveniently estimated in the small-signal regime by the radial (effective) coefficient d₃₁: the disk outer diameter changes by ΔOD ≈ d₃₁(OD/TH)U, corresponding to an average in-plane strain ε ≈ ΔOD/OD ≈ d₃₁U/TH. Because thin disks satisfy OD >> TH, this geometry is efficient for producing in-plane strain at modest voltages. When the sample is placed on the top electrode, a thickness-driven disk typically produces an approximately biaxial in-plane (radial) strain (Figure 10), which is sufficient for lock-in detection of ΔR/R as the optical probe measures a spot-averaged modulation. The strain profile becomes important primarily when symmetry is crucial to the scientific question (e.g., uniaxial strain to deliberately break in-plane rotational symmetry or split anisotropic features), whereas for routine enhancement/derivative detection of weak optical transitions, the repeatable strain amplitude is usually the dominant requirement. For completeness, note that the lowest mechanical resonance of thin disks is commonly the radial extensional mode, with a characteristic scaling f_s ≈ N_p/OD, where N_p is the frequency coefficient of the planar oscillation of a round disk; while resonant driving can increase strain, we generally operate well below resonance for linear response and stable phase, and practical resonance matching is often constrained by available actuator geometries/materials. Finally, electrode design can matter for reliability and mounting: thin-film electrodes deposited by PVD (e.g., sputtering) are typically ~1µm thick; we use silver electrodes but encourage testing alternative metallizations where needed. For an overview of available actuator shapes (disks, plates, shear elements), polarization directions, and electrode options, the PI Ceramic catalog provides a useful reference³¹.

Figure 10: Thin piezoceramic disk actuator geometry and voltage-induced in-plane deformation. (A) Schematic of a thin, thickness-poled piezoceramic disk (outer diameter OD, thickness TH, with OD ≫ TH) equipped with electrodes on the top and bottom faces and driven by an applied voltage U across the thickness. On the left, the orthogonal coordinate system (1-2-3) used to describe poled piezoelectric ceramics is shown; the polarization vector P (right side of panel A) is parallel to the 3 (Z) axis. (B) Illustration of the dominant in-plane (radial) deformation produced under thickness excitation, yielding an approximately biaxial lateral strain at the top surface relevant for strain-modulated optical measurements. Please click here to view a larger version of this figure.

Supplementary Figure 1: User interface of the control and acquisition software. Representative screenshot of the UI of the control/acquisition application supporting experimental setup.Please click here to download this file.

Supplementary Figure 2: Preamplifier configuration for piezoreflectance measurements. Sample settings.Please click here to download this file.

Supplementary Figure 3: Lock-in amplifier configurations for AC (ΔR) and DC (R) reflectance detection. Sample settings (A) of the lock-in amplifier for the measurement of lock-in AC component proportional to ΔR and DC component proportional to R, and (B) of the chopper-referenced lock-in amplifier for measurement of the additional DC reflectance component (R).Please click here to download this file.

Supplementary Figure 4: Instrument settings for AC drive and optical chopping. Sample settings (A) of the optical chopper controller and (B) of the custom-made AC voltage generator.Please click here to download this file.

Discussion

Here we present piezoreflectance (PzR) spectroscopy for van der Waals crystals, a protocol that uses periodic strain from a piezoceramic transducer to produce derivative-like reflectance spectra and isolate critical points with high contrast. Compared with reflectance contrast (RC), the method sharpens weak or overlapping resonances and extends to cases where RC provides ambiguous contrast, while retaining a single broadband optical path and compatibility from monolayers to bulk; the primary tradeoffs are the lack of absolute strain calibration and longer acquisition times.

All measurements here were performed in the dark configuration³⁰, as specified in the Protocol, because placing the monochromator upstream of the sample minimizes stray light and minimizes noise signal in ΔR. The bright configuration can also be utilized in piezoreflectance measurement, but the dark path should rather be considered the default. In general, three steps dominate outcome quality:

Sample mounting: The adhesive should be applied as thin as possible - sufficient to glue the flake but not to overflow its edges - then the PDMS should be peeled immediately to avoid meniscus ridges. The over‑thick adhesive increases the optical background and mechanically decouples the flake, degrading the accuracy of the line‑shape in the derivative‑like PzR features.

Electrical contacts and curing: After painting the silver paste, a full room‑temperature cure (~10 h) should be allowed; attempts to accelerate curing (e.g., warm bakes) increase contact brittleness and reduce durability, leading to intermittent lock-in signals and baseline wander. Prolonged operation at high drive voltages (≈1000 V) should be avoided: extended bias can irreversibly reorient ferroelectric domains in the piezoceramic (depolarizing can occur if the field exceeds a threshold - coercive field, especially in the opposite direction of poling) and can char or lift the silver paste, necessitating complete re‑contacting.

Optical alignment and modulation settings: Optics should be chosen properly so that the entire illumination spot lies within the flake area, because collecting light from the surrounding piezoceramic introduces background at the reference frequency. A 50x objective is typically sufficient; use 100x when a smaller spot is necessary. Finally, baseline should be handled manually by design: select a linear or non-linear baseline by eye using off‑resonant regions; this improves with practice and should be kept minimal to avoid distorting amplitudes.

The setup tolerates a limited set of changes. To extend the spectral range, swap the detector (e.g., to InGaAs for sub‑1.2 eV or to a proper photomultiplier for >3 eV); do not substitute the light source, which can compromise stability and baseline quality. If ΔR is noisy, first increase the lock‑in time constant. The increase in applied AC voltage tends to amplify the background and promote non-linear baselines. If the baseline drifts across long scans, allow a non-linear baseline subtraction and, if unavoidable, include a small quadratic term in the fit. In Aspnes' analysis, enabling non‑zero background parameters (a, b) is acceptable when justified by the data. As a practical success criterion, a visible resonance that can be fit with a high R² and produces well‑defined moduli is sufficient for reliable parameter extraction. Quick‑reference order of operations:

• Noise present then increase τ (before increasing V).

• Need a smaller spot, then use a higher‑NA / 100x objective; confirm the spot lies fully within the flake.

• Baseline curvature then permit a mild quadratic background; revisit alignment and τ.

• Contact instability, then remake silver contacts; avoid high‑V stress.

This protocol is broadly applicable, from few-layer structures to bulk crystals, but several constraints should be recognized. First, we do not perform quantitative strain calibration. Instead, effective modulation is inferred pragmatically from the emergence of resonant derivative-like features in PzR spectra, which confirms strain transfer without assigning absolute values. Consequently, deformation‑potential constants are compared qualitatively rather than absolutely. Second, while the method works across sample thickness, very thick crystals can partially screen or relax the periodic strain, reducing modulation at the surface. On the other hand, ultrathin vdW flakes require careful handling to avoid mechanical damage and to ensure the illumination spot remains fully on the flake. Third, the spectral window of the Si detector constrains routine operation to ~1.1–3.2 eV, with reduced signal and increased noise near edges. Fourth, there is a throughput-precision trade-off: depending on the sensitivity of the material and the temperature of the acquisition, the typical uncertainty in the extracted transition energies is on the order of tens to hundreds of meV, and the total acquisition times range from dozens of minutes to several hours, following the step‑size and time‑constant choices in the Sections 4.8, 5.4. Finally, it is necessary to ensure minimal impact of environmental factors such as temperature drift, surface condition, or mechanical vibrations. Careful alignment, stable curing of contacts, and conservative drive voltages can usually mitigate these effects in standard, room temperature conditions.

Relative to reflectance contrast (RC), PzR emphasizes derivative‑like line shapes that enhance weak or overlapping features and provide robust critical‑point discrimination. As shown above, PzR spectroscopy can reveal resonances in materials where RC yields weak or ambiguous contrast (e.g., centrosymmetric layered phosphorous chalcogenides), while for systems with strong, isolated transitions (e.g., some TMDs such as WS₂ under our conditions), RC can suffice, and PzR serves as a complementary confirmation rather than a necessity^7,29. Thus, selection between RC and PzR should be guided by signal strength, feature crowding, and desired sensitivity to subtle modulations, with PzR favored when derivative enhancement is advantageous.

This method provides a practical route to study vdW crystals and related 2D materials, including transition‑metal dichalcogenides (TMDs), as well as their alloys and heterostructures. By lowering alignment complexity (single optical path), maintaining compatibility from monolayers to bulk, and producing derivative‑like spectra that sharpen weak features, PzR supports probing direct optical transitions, tracking thickness-dependent shifts, and extracting energies and broadenings of direct optical transitions parameters via Aspnes fits. The approach integrates naturally with steady-state PL and Raman for multimodal insight, provided that spot isolation and mechanical coupling are maintained on small flakes. These capabilities make the protocol a versatile addition to optical spectroscopy toolkits for strain‑sensitive electronic structure studies in modern layered materials^2,5,7.

Materials

List of materials used in this article

Name	Company	Catalog Number	Comments
Acetone (analytical grade)	Sigma-Aldrich / Merck	650501
CCD Camera	e.g. Hamamatsu Photonics / Princeton Instruments / Thorlabs	N/A
Computer with control and data acquisition software (e.g., LabVIEW)	National Instruments / Custom setup	N/A
DSP Lock-In Amplifier	Stanford Research Systems (SRS)	SR810
EKE Halogen Lamp, 150 W 21 V (GX5.3 base)	OSRAM	93638	Used as the replacement bulb for the Thorlabs OSL1-EC Fiber Illuminator.
Gel-Film PDMS-based viscoelastic film, 4.0″×4.0″ sheet	Gel-Pak	WF-40×40-0060-X4
High Intensity Fiber Light Source (halogen lamp) - 150 W	Thorlabs	OSL1-EC	The model is listed as discontinued (superseded by OSL2) as of 07 Jan 2014.
High Voltage Rectangular Signal Generator	Custom-made in our laboratory	HVG-3
Low-Noise Current Preamplifier	Stanford Research Systems (SRS)	SR570
M Plan APO NIR 50× Objective Lens	Mitutoyo Corporation	378-825-17	Other objectives with suitable magnification and spectral transmission can also be used depending on the optical setup.
Monochromator / Imaging Spectrograph, focal length 320 mm - Zolix Omni-λ300i	Zolix Instruments Co., Ltd.	Omni-λ300i
Optical Chopper Controller	Stanford Research Systems (SRS)	SR540
Optomechanical and optical components (lenses, mirrors, irises, optical chopper, beam splitters, optical fibers, and mounts)	Thorlabs / custom-made	Various
Piezoelectric Ceramic Disk (Ø30 × 1 mm, PZT-5O, silver electrodes)	He-Shuai Ltd.	N/A
PVC Surface Protection Low-Adhesion Cleanroom Tape (blue)	Cleanroom Supply	N/A
Silicon PIN Photodiode	e.g. Hamamatsu Photonics / Thorlabs	N/A	Other types of photodetectors can be employed depending on the spectral range of interest.
Silver conductive varnish (“Leitlack”)	Busch GmbH & Co. KG	5900
Transparent nail polish	Revlon, Inc.	N/A	Used as an adhesive layer for fixing vdW flakes. Any clear nail polish of similar composition can be used as an alternative.
Tungsten Disulfide (WS2), CAS 12138-09-9	HQ Graphene	N/A	sample material