Mechanically-tunable bandgap closing in 2D graphene phononic crystals

Phononic crystals (PnCs) are artificial structures with periodically varying material properties that create a phononic band structure analogous to Bloch waves in solids. PnCs offer broad control over band structure parameters through artificial patterning, enabling the realization of phenomena like quantum entanglement, topological states, and negative refraction. Applications include phononic shielding, wave guiding, and thermal management. Their low phonon propagation speed makes PnCs promising for quantum information technology based on manipulating mechanical motion at scales too small for photonic approaches. Most applications rely on phononic bandgaps, frequency ranges where phonon propagation is suppressed. Tension significantly affects phonon velocities. In conventional PnCs (e.g., silicon nitride), tension is fixed during fabrication, hindering coupling to external systems. Recent work demonstrated dynamic tension control in 2D materials using electrostatic pressure, enabling tunability of the bandgap center frequency. However, the band hierarchy remained unaffected—gapped systems stayed gapped. Precise control over bandgap size and coupling strength remains a challenge. This research demonstrates that applying uniaxial tension (unlike previous biaxial studies) alters the band hierarchy. A 2D phononic lattice in suspended graphene under biaxial tension exhibits an out-of-plane phonon bandgap (e.g., 48.8–56.4 MHz at 0.01 Nm⁻¹). This bandgap disappears when the uniaxial tension ratio (σxx/σyy) reaches 1.7, analogous to a metal-insulator transition (though not perfectly analogous, as the phononic chemical potential remains zero). This transition is relevant because out-of-plane modes in 2D materials are easily excited and detected. To control σxx/σyy, the researchers propose a device using electrostatic gating, showing that bandgap closing is achievable with experimentally feasible voltages (~8V). This system functions as a mechanical transistor with an on/off ratio of 10⁵ (100 dB suppression). Dynamic bandgap control enables tunable coupling between mechanical entities (e.g., qubits). The study also addresses fabrication challenges, finding that contaminant mass should be less than four times the graphene mass and tension variations less than 40% to maintain the bandgap and its tunability.

The introduction extensively reviews the existing literature on phononic crystals, highlighting their applications in various fields, including quantum information technology and thermal management. The authors cite several key papers demonstrating the use of phononic crystals for wave guiding, shielding, and entanglement. They also discuss the limitations of existing methods for controlling bandgaps, particularly in conventional rigid materials, and contrast them with the potential for dynamic control using 2D materials and electrostatic pressure. The literature review underscores the novelty of the presented work in achieving a mechanically tunable bandgap closing using uniaxial tension, a method not previously explored for achieving this level of control.

The research employed finite element method (FEM) simulations using COMSOL Multiphysics (Version 5.5) to model the behavior of the phononic crystal. The simulations used material parameters for monolayer graphene: Young's modulus E2D = 1.0 TPa, Poisson's ratio ν = 0.15, thickness h = 0.335 nm, and density ρ = 2260 kg m⁻³. The PnC design involved a honeycomb lattice of holes in a suspended graphene membrane. The simulations first calculated the tension distribution within the unit cell of the honeycomb lattice, revealing tension hotspots and relaxed regions. The phononic band structure was then calculated along high-symmetry lines for an infinite lattice using the first Brillouin zone. The simulations showed a bandgap for out-of-plane modes (48.8–56.4 MHz) under biaxial tension. The effect of uniaxial tension on the band structure was then investigated, revealing a bandgap closing at a specific uniaxial tension ratio. To simulate a realistic device, the researchers modeled acoustic transmission measurements in a finite-size PnC (9 µm × 28 µm). Transmission was calculated using an integral over the out-of-plane displacement of the graphene. The simulations investigated the effects of electrostatic pressure on the tension distribution in a finite-size device with a variable aspect ratio. Electrostatic pressure was modeled using the formula Pel = ε0(Vgate)²/(2d²), where ε0 is the vacuum permittivity, Vgate is the gate voltage, and d is the distance between the graphene and the gate electrode. The impact of fabrication-related challenges such as surface contamination (using PDMS as a model contaminant) and random tension variations was also simulated. These simulations involved adding random masses to the graphene membrane and generating random tension distributions using a superposition of randomized plane waves. The simulations used various parameters for contaminant mass and tension variations to determine the thresholds where the bandgap became significantly affected.

The key findings of the study are: 1. **Bandgap Closing under Uniaxial Tension:** Simulations revealed a clear phononic bandgap for out-of-plane modes under biaxial tension. This bandgap could be closed by applying uniaxial tension, with the bandgap completely closing when the tension ratio σxx/σyy reached 1.7. This behavior is analogous to a metal-insulator transition, though the analogy is not perfect due to the constant zero chemical potential. 2. **Realistic Device Design:** A realistic device design was proposed using suspended graphene and electrostatic gating. Simulations showed that a gate voltage of approximately 8 V was sufficient to close the bandgap, achieving a mechanically tunable system. 3. **Acoustic Transmission Measurements:** The bandgap closing was confirmed through simulations of acoustic transmission measurements in a finite-size device. These simulations showed a significant drop in transmission within the bandgap region, with an on/off ratio of approximately 10⁵ (100 dB suppression). 4. **Robustness to Fabrication Imperfections:** Simulations investigating the effects of surface contaminants (PDMS) and random tension variations showed that the bandgap closing remained observable even with a degree of disorder in realistic devices. A critical contaminant mass density (less than 4 times the graphene's mass) and a threshold for relative tension variations (approximately 40%) were identified, representing limits for maintaining bandgap functionality. 5. **Mechanical Transistor Analogy:** The tunable bandgap in the finite-size device closely resembled the behavior of a field-effect transistor, acting as a “mechanical transistor” for MHz-range phonons, offering a potential for novel phonon logic circuits. 6. **Tunable Coupling:** The ability to dynamically control the bandgap size enables the tuning of coupling strength between mechanical entities, suggesting applications such as controlling interactions between mechanical qubits.

The findings address the research question of achieving precise control over phononic bandgaps in 2D materials. The demonstration of bandgap closing under uniaxial tension represents a significant advance, providing a new degree of freedom in manipulating phononic systems. The analogy to a metal-insulator transition offers a powerful conceptual framework for understanding the observed behavior. The success in designing a realistic device that exhibits the predicted behavior is crucial for practical applications. The analysis of fabrication-related challenges provides valuable insights for experimental realization. The potential applications of this tunable phononic crystal in phonon logic and quantum information technology are substantial, opening new avenues for research and development in these fields. The high on/off ratio achieved in the simulated device suggests its suitability for practical implementations.

This research successfully demonstrated mechanically tunable bandgap closing in a 2D graphene phononic crystal. Using uniaxial tension engineering, the researchers achieved a transition between mechanically insulating and conductive states, analogous to a metal-insulator transition. A realistic device design using electrostatic gating showed bandgap closure at feasible voltages. Simulations, including the effects of fabrication imperfections, confirmed the practical feasibility of this approach. The resulting device functions as a mechanical transistor for MHz phonons, opening possibilities for phonon logic and tunable coupling between mechanical entities like qubits. Future research could focus on experimental validation of these findings and exploration of other complex condensed matter physics phenomena using this tunable system.

The study relies on FEM simulations, which may not perfectly capture all aspects of real-world device behavior. While the simulations accounted for some fabrication imperfections, other sources of disorder or material imperfections could affect the results. The analysis of contaminant effects focused on a specific contaminant (PDMS), and other materials might behave differently. The theoretical analogy to a metal-insulator transition is not complete, and a deeper theoretical exploration of this analogy might be beneficial. Experimental validation of the simulated results is needed to fully assess the practical potential of this technology.