Researchers are using interlayer excitons to reveal how quantum capacitance alters electric fields within two-dimensional materials. The work demonstrates a breakdown of the standard geometric capacitor model in van der Waals heterostructures, observing an energy splitting that persists even when the applied electric field is canceled. This splitting, a result of incomplete cancellation between gate- and carrier-induced fields, provides a direct optical measure of the local chemical potential and a new method to probe quantum capacitance in operando. The results reveal the central role of quantum capacitance in interface electrostatics.

Interfacial Charge Transfer Disrupts Geometric Capacitor Model

Interlayer excitons now function as localized sensors revealing how quantum capacitance impacts electrostatic fields within two-dimensional van der Waals heterostructures, challenging established models of charge storage. Observations reveal a dipole-dependent energy splitting within these structures, persisting even when externally applied electric fields are nominally canceled. This splitting arises from incomplete compensation between fields induced by the gate voltage and those generated by the accumulating charge carriers.

This quantum capacitance, stemming from the finite density of states at the interface between materials, modifies the local chemical potential and alters the distribution of electric fields. Researchers fabricated devices by layering monolayer graphene onto bilayer molybdenum disulfide in a dual-gate architecture, then encapsulated the structures with hexagonal boron nitride and graphite electrodes to facilitate precise control and measurement.

The team found that the quantum capacitance becomes particularly important when comparable to the geometric capacitance, causing a shift in the chemical potential. “We show that the observed spectral asymmetry arises from a residual internal electric field governed by the quantum capacitance of graphene,” the paper reports, highlighting the direct link between the observed optical signal and the underlying quantum mechanical effects. The methodology employed uses reflection contrast to quantify changes in the heterostructure’s reflective properties, providing a means to assess the induced carrier density and electric field.

Within the geometric capacitor model, the change in the gate-induced external field and the carrier-induced internal field are expected to be equal and opposite, effectively canceling each other out in the top-gate configuration. However, the observed energy splitting demonstrates a breakdown of this symmetry, indicating that the internal electrostatics are governed by factors beyond simple geometric considerations.

van der Waals Heterostructures Enable Exciton-Based Electrostatic Probing

The experimental determination of a proportionality factor, denoted as 𝑝, revealed its value to be 0.48 e · nm, obtained from measurements of electric-field dependence. This value quantifies the relationship between induced carrier density and the resulting internal electric field within the heterostructure, providing a basis for accurately interpreting optical measurements. This refined understanding of the internal field allows for a more precise assessment of the Stark effect on interlayer excitons, separating it from the complicating influence of many-body screening.

The team’s approach moves beyond simply observing the exciton energy shift, instead focusing on isolating the contributions of both the external and internal electric fields. The observed asymmetry, therefore, is a fundamental property of the heterostructure’s electrostatic behavior, not a result of measurement error. This work establishes a method for probing quantum capacitance in operando by using the sensitivity of interlayer excitons to the local electrostatic environment.

This optical technique offers a distinct advantage over traditional transport measurements, providing micrometer-scale resolution without requiring complex device fabrication. The data that support the findings of this article are available from the authors upon reasonable request, facilitating further investigation and validation of these results.

Quantum Capacitance Modifies Internal Electric Fields in Bilayer MoS2

The standard geometric capacitor model falters when interfacial charge transfer occurs, altering internal electrostatics in two key ways: redistributed charges generate additional internal fields. Charge carriers transfer from bilayer MoS2 to graphene, creating an additional internal electric field and modifying excitonic interactions through many-body screening, resulting in shifts in the energies of interlayer exciton branches.

Reflection-contrast spectroscopy of interlayer excitons was used to probe these effects, revealing that the incomplete cancellation of fields originates from graphene’s finite density of states and its associated quantum capacitance. For a given gate voltage, quantum capacitance acts in series with geometric capacitance, reducing the transferred carrier density compared to the geometric capacitor model’s prediction. Consequently, the carrier-induced internal field does not fully compensate for the external gate field, leaving a residual electric field that causes a measurable Stark shift.

The dipole-dependent component of the energy splitting provides a measure of the local chemical potential, while the dipole-symmetric component tracks many-body screening at the interface. These results establish excitonic spectroscopy as a versatile probe of buried interfaces during device operation, highlighting the role of quantum capacitance in determining internal electric fields in van der Waals heterostructures.

Dipole-Dependent Exciton Splitting Reveals Local Chemical Potential

The observed splitting is not merely a measurement artifact. It directly reflects the local chemical potential within the material, offering a new pathway for in operando characterization. This agreement holds true even with varying fixed bottom-gate voltages, confirming the robustness of the observed effect and eliminating dependence on specific device configurations. Data from Device 2 and Device 3 further corroborate these findings, demonstrating reproducibility across different heterostructure samples.

The analysis reveals that the observed splitting is distinct from many-body screening effects, which manifest as a shift in the average energy of interlayer excitons and are symmetric with respect to dipole orientation. In multilayer graphene regions, where the density of states is significantly higher, the chemical potential exhibits minimal shift during gate voltage sweeps, resulting in a correspondingly stable internal electric field.

This behavior contrasts sharply with bilayer graphene, where the chemical potential and resulting internal field are more sensitive to gate voltage modulation. The observed linear scaling between the measured energy splitting and the graphene chemical potential, as demonstrated in Device 1, simplifies data interpretation and allows for precise determination of the potential.

Graphene/MoS2 Stack Creates Carrier-Induced Electric Fields

The standard model for understanding electrostatic behavior in van der Waals heterostructures breaks down when quantum capacitance is considered, leading to a residual internal electric field even when external voltages are balanced. When the carrier distribution remains symmetric across the structure, the carrier-induced internal electric fields largely cancel, preserving the apparent validity of the geometric model. When this symmetry is broken by interfacial charge transfer, however, the geometric description breaks down in two ways.

First, the redistributed charges generate additional internal fields that modify the total electrostatic profile. Second, the finite density of states of the interfacial layer gives rise to quantum capacitance, which alters both the local chemical potential and the effective field distribution. Direct experimental access to this electrostatic reconstruction remains challenging.

The near-identical evolution of the energy splitting for two branches, coupled with the suppressed screened-exchange contribution expected when carriers reside primarily in graphene, supports the dominance of this residual Stark response. The team’s analysis indicates that the observed asymmetry is unlikely to be explained by dipole-dependent many-body renormalization, as the effects would not be consistent across the structure.

Reflection-Contrast Spectroscopy Maps Interlayer Exciton Energies

Reflection-contrast spectroscopy reveals that interlayer excitons respond to internal electric fields even when external fields are balanced, demonstrating a limitation of standard capacitor models in layered materials. Measurements performed at approximately 6 Kelvin show that the energies of interlayer excitons shift with carrier density, indicating the presence of an internal electric field despite a net zero external field imposed by gate voltages. This behavior suggests that charge transfer between layers generates an electric field that influences exciton behavior.

Analysis of reflection-contrast maps, focusing on the 1s interlayer exciton states with opposite dipole orientations (IX ↓ ଵୱ and IX ↑ ଵୱ), shows a linear Stark shift consistent with electric field interaction. Fitting the energy dispersion as a function of electric field yielded a dipole moment of approximately 0.48 e nm for both interlayer exciton branches, aligning with previously reported values.

However, further investigation into carrier density effects revealed a more subtle picture, as exciton energies continued to shift even with a maintained zero net external electric field. Sweeping the bottom-gate voltage while fixing the top gate at 0V, and vice versa with a -0.6V bottom gate, provided data for detailed analysis. During the bottom-gate sweep, the electric fields associated with each gate added constructively, creating a monotonic evolution similar to prior observations.

Examination of the 2s exciton states through second-derivative reflection-contrast mapping clearly resolved four distinct peaks corresponding to intralayer and interlayer excitons, allowing for energy extraction and comparison between 1s and 2s states for both interlayer exciton branches. The team’s analysis indicates that this internal field is not simply a result of dipole-dependent many-body renormalization effects, but a distinct phenomenon arising from charge redistribution. This finding challenges the traditional geometric capacitor model, which predicts carrier density based solely on applied voltage and capacitance, and suggests that quantum capacitance is important in interface electrostatics.

👉 More information
🗞 Excitonic Stark Response Induced by Quantum Capacitance in van der Waals Heterostructures
✍️ Xiangzhi Hong et al.
🧠 DOI: http://link.aps.org/doi/10.1103/3jqj-6ldy

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