Quasicrystal Architected Nanomechanical Resonators via Data-Driven Design

Kawen Li; Hangjin Cho; Richard Norte; Dongil Shin

cond-mat.mes-hallApr 8, 2026

Quasicrystal Architected Nanomechanical Resonators via Data-Driven Design

Authors: Kawen Li, Hangjin Cho, Richard Norte, Dongil Shin

Organizations: Department of Precision and Microsystems Engineering, Delft University of Technology, Delft, The Netherlands · Department of Mechanical Engineering, Pohang University of Science and Technology, Pohang, Republic of Korea

Abstract

From butterfly wings to remnants of nuclear detonation, aperiodic order repeatedly emerges in nature, often exhibiting reduced sensitivity to boundaries and symmetry constraints. Inspired by this principle, a paradigm shift is introduced in nanomechanical resonator design from periodic to aperiodic structures, focusing on a special class: quasicrystals (QCs). Although soft clamping enabled by phononic stopbands has become a central strategy for achieving high- $Q_m$ nanomechanical resonators, its practical realization has been largely confined to periodic phononic crystals, where band structure engineering is well established. The potential of aperiodic architectures, however, has remained largely unexplored, owing to their intrinsic complexity and the lack of systematic approaches to identifying and exploiting stopband behavior. Here we demonstrate that soft clamping can be realized in quasicrystal architectures and that high- $Q_m$ nanomechanical resonators can be systematically predicted through a data-driven design framework. As a representative demonstration, the 12-fold QC-based resonator obtained numerically a quality factor $Q_m \sim 10^7$ and an effective mass of sub-nanograms at MHz frequencies, corresponding to an exceptional force sensitivity of $26.4$ ~aN/ $\sqrt{\text{Hz}}$ compared to previous 2D phononic crystals. These results establish QCs as a robust platform for next-generation nanomechanical resonators and open a new design regime beyond periodic order.

Figures & tables

Figure 1: Overview of the quasicrystal-enabled resonator design space (a) Quasicrystal tilings with 4-, 8-, and 12-fold symmetry. (b) Structural motifs inspired by phononic crystal unit cell designs. (c) Resulting quasicrystal nanomechanical resonator geometries enabled by aperiodic order.

Figure 2: Data-driven identification of stopbands for three representative design cases. Shaded regions denote the identified stopbands. Insets show representative mode shapes within these frequency intervals, highlighting clean stopbands (Case 1), isolated localized defect modes (Case 2), and modes with boundary leakage indicating a poorly formed stopband (Case 3).

Figure 3: Comparison of stopband behavior between periodic phononic crystal and quasicrystal nanomechanical resonator designs for Design 1. (a) Stopband evolution in phononic crystal-based designs as a function of lattice constant $a_{0}$ ( $r=0.26a_{0}$ ). (b) Stopband evolution in quasicrystal-based designs as a function of deflation order $n$ . (c) Representative quasicrystal geometries for $n=0,1,2$ , illustrating the geometric refinement and the symmetry-reduced sector with cyclic boundary conditions.

Figure 4: Optimized Designs 1 and 2. (a) Optimized mode shapes for the representative Designs 1 and 2 (see Fig. 6 for detailed design formulations), with insets highlighting the localized defect modes and the corresponding optimized design parameters. (b) Displacement profiles along the $x$ -axis through the membrane center, showing strong spatial confinement and suppression of motion toward the clamping boundary. (c) Simulated mechanical quality factors $Q_{m}$ for all modes within the identified stopband, with highlighted markers indicating the selected defect modes.

Designs	1	2
Frequency $f$ (MHz)	1.29	0.88
Effective mass $m_{\mathrm{eff}}$ (ng)	5.9	0.92
Dilution factor $D_{Q}$ ( $\times 10^{3}$ )	48.9	25.1
Quality factor $Q_{m}$ ( $\times 10^{6}$ )	118.3	60.5
$Qf$ product (THz)	153	53.3
Coherence time $\tau$ ( $\mu s$ )	3.0	1.54

Table 1: Numerically predicted performance of optimized Designs 1 and 2, shown in Fig. 4 with the mechanical quality factor

Q_{m}

as the primary optimization target, assuming room temperature (

T=300\,\mathrm{K}

Figure 5: Thermal force noise–limited sensitivity ( $\sqrt{S_{FF}}$ ) versus effective mass for representative nanomechanical resonator architectures. One-dimensional string-like resonators achieve exceptional sensitivity through extremely low effective mass, whereas two-dimensional pad-based resonators offer improved optical accessibility at the cost of increased mass. The QC-based designs presented in this work begin to occupy an intermediate regime, approaching the force sensitivity of one-dimensional resonators while retaining a fundamentally two-dimensional geometry. The orange circle denotes the best-performing design, while other circles indicate additional QC realizations (see Appendix G and Fig. G.1 ).

Figure 6: Design formulation of 12-fold quasicrystal-based nanomechanical resonators. (a) Stampfli deflation process for generating a 12-fold dodecagonal quasicrystal lattice, illustrating the transition from the zeroth-order ( $n=0$ ) to the first-order ( $n=1$ ) structure and the associated characteristic length scales $L_{0}$ and $L_{1}$ . (b) Parameterized resonator designs implemented on the quasicrystal lattice using two representative phononic crystal–inspired motifs. Design 1 employs a low mass-contrast strategy based on hole patterns, while Design 2 adopts a high mass-contrast strategy using massive pads connected by thin strings. The geometric design parameters, including string widths ( $w_{i}$ ) and hole or fillet radii ( $r_{i}$ ), are defined with respect to individual quasicrystal tile types.

Appendix figures & tables17 assets

Supplementary material from the paper’s appendix.

Appendix

Figure A.1: Detailed deflation rules for 12-fold dodecagonal quasicrystal lattice with stampfli deflation rules. (a) Base unit cell structure ( $n=0$ ) used for our design with $L_{0}=1.5/(1+\sqrt{3}/2)$ [mm]. Major coordinates are marked in black for a cyclic symmetric slice. (b) Sub-structures used to deflate the base unit cell structure. By matching the marked coordinates in black, one can substitute the sub-structures into the base unit cell structure, giving a first-order ( $n=1$ ) structure.

Figure B.1: (a) Conventional bandstructure found for PnC structures, obtained via unit cell analysis with superimposed periodic boundary conditions. (b) Illustration of the 1D eigenfrequency array obtained for our QC-based designs, and possible definition of a stopband coming from data-driven perspectives. (c) Such 1D eigenfrequency array can be clustered based on the contrast in data density ( $C_{1}$ and $C_{2}$ ), the distance in between neighbouring clusters indicates candidate stopband. (d) Conceptual definition of the key parameters MinPts and Eps used in our DBSCAN algorithm. (e) An example of the k-th nearest neighbour distance density distribution ( $k=5$ ). 99th percentile of this distribution gives $\textit{Eps}\approx 0.05$ .

Figure B.2: Comparison of stopband behavior between periodic phononic crystal and quasicrystal nanomechanical resonator designs for Design 2. (a) Stopband evolution in phononic crystal-based designs as a function of lattice constant $a_{0}$ ( $r=0.26a_{0}$ ). (b) Stopband evolution in quasicrystal-based designs as a function of deflation order $n$ . (c) Representative quasicrystal geometries for $n=0,1,2$ , illustrating the geometric refinement and the symmetry-reduced sector with cyclic boundary conditions.

Figure B.3: Identified stopband boundaries within an optimization cycle of 100 design iterations for (a) Design 1 (b) Design 2

Figure B.4: Frequency-response analysis of Design 2 under harmonic excitation applied at the central defect.

Figure D.1: Simulation pipeline for the data-driven design of a quasicrystal-based nanomechanical resonator. A symmetry-reduced eigenfrequency analysis is used to identify stopbands, after which full-structure simulations are performed within the selected frequency windows to evaluate localized modes and their mechanical quality factors $Q_{m}$ .

Figure D.2: Mesh convergence analysis for Design 1 over different characteristic eigenmodes and mesh configurations. (a) Three common characteristic eigenmodes seen, color map indicates the magnitude of out-of-plane displacement. (From top to bottom) A well isolated defect mode, a less isolated defect mode, and a global mode. (b) Distribution of the energy dissipated per cycle due to bending loss, $W_{\rm bend}$ , in each individual eigenmode. Strong localization is observed close to the clamped boundary. (c) Effects of reducing mesh size on $Q_{m}$ performance for global and edge mesh configurations. The mesh size is defined relative to the minimum structural feature found in the design.

Figure D.3: Mesh convergence analysis for Design 2 over different characteristic eigenmodes and mesh configurations. (a) Three common characteristic eigenmodes seen, color map indicates the magnitude of out-of-plane displacement. (From top to bottom) A well isolated defect mode, a less isolated defect mode, and a global mode. (b) Distribution of the energy dissipated per cycle due to bending loss, $W_{\rm bend}$ , in each individual eigenmode. Strong localization is observed close to the fillet edges and clamped boundary for the global mode. (c) Effects of reducing mesh size on $Q_{m}$ performance for global, fillet and edge mesh configurations. The mesh size is defined relative to the minimum structural feature found in the design.

Figure E.1: (a) The actual model for optimization. (b) Posterior model constructed after a few initial samplings. (c) An exemplar acquisition function (Expected Improvements) used. (d) The next best sample point obtained by maximizing the acquisition function in (c).

Figure F.1: Demonstration of in-plane strain engineering techniques with optimized Design 2. (a) Distribution of radial zones in enabling radial direction variation of design features. Shows the radial direction design feature profile for three separate designs, from no strain engineering to most extreme strain engineering accomplished. (b) Von Mises stress field near the defect for design with most strain engineering applied. (c) Frequency spectrum (simulated) for three designs, stopband regions are highlighted. (d) Soft-clamped mode shape of the most strain engineered design. (e) Averaged von Mises stress in regions A-F near the defect for three designs, labels in (b). (f) Averaged von Mises stress across all regions A-F for three designs. (g) Quality factor (

Q_{m}

) for three designs.

Figure F.2: (a) Construction of our optimized Design 2 with the most extreme in-plane strain engineering implemented for different sizes: 3mm, 6mm, 11.2mm (from left to right). (b) Averaged von Mises stress in regions A-F (defined in Fig. F.1 b) for three designs with different sizes.

Figure F.3: (a) Design parameters after introducing absolute drifts caused by fabrication variability. Note that the defect region in the PnC-based design also employs a fillet-based geometry, and the corresponding drift is applied accordingly (not shown in the figure). (b) Resonance frequency variation in both QC- and PnC-based nanomechanical resonators with respect to the fractional drift in the design parameters.

Designs	3	4	5	6
Frequency $f$ (MHz)	1.52	1.50	0.83	0.92
Effective mass $m_{\mathrm{eff}}$ (ng)	2.8	1.11	21.7	3.1
Quality factor $Q_{m}$ ( $\times 10^{6}$ )	63.1	41.1	140	99.7
$Qf$ product (THz)	96	61.7	117	91.3

Appendix

Table G.1: Simulated performance of non-12 fold Designs assuming room temperature (

T=300\,\mathrm{K}

). Designs are defined in Fig. G.1 (b)

Figure G.1: Design and optimization results with alternative quasicrystal geometries. (a) Deflation rule for the 4-fold and 8-fold quasicrystal geometries. (b) Quasicrystal-based resonator designs using the same design motif (Fig. 6 ) with 8-fold and 4-fold quasicrystals.(c) Stopband in 8-fold low-mass-contrast design. (d) High- $Q_{m}$ soft-clamped mode shape in 4-fold and 8-fold designs.

Figure H.1: Eigenfrequency analysis validation: comparison between symmetry-reduced and full-model simulations. (a) Mode spectra of the two models. (b) Relative frequency differences obtained using nearest-neighbor matching. (c) Frequency-count spectra using a 20 kHz bin size.

Parameters
Thickness $t$ (nm)	35
Device dimension (mm)	3
Young’s Modulus (GPa)	270
Density (kg $/m^{3}$ )	3200
Pre-stress (GPa)	1.27

Appendix

Table I.1: Design parameters used in the simulation of the QC-based design and the representative PnC-based reference based on [ 19 ] .

Designs	1	2	PnC
Frequency $f$ (MHz)	1.29	0.88	0.78
Stopband Width (MHz)	0.12	0.34	0.1
Effective mass $m_{\mathrm{eff}}$ (ng)	5.9	0.92	14.2
Dilution Factor $D_{Q}$ ( $\times 10^{3}$ )	48.9	25.1	68.7

Appendix

Table I.2: Performance of the QC-based designs (Designs 1 and 2) compared with the PnC-based design.

Materials Science

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Physical reservoir computing uses nonlinear dynamics and a trained linear readout to process information. Nanoelectromechanical (NEMS) resonators combine geometric Duffing nonlinearity with fading memory, but most electromechanical implementations use a single resonance mode. Here, we demonstrate reservoir computing with two interacting modes of a single NEMS resonator measured through one readout port. We introduce dynamical diversity through complementary modal drive settings: the same input sequence is replayed under different allocations of drive amplitude between the modes, and the responses are concatenated into a single feature matrix. This multiplexing expands the representation available to the readout without additional devices or training of internal parameters. On NARMA-2, it reduces variance-normalized test error more than 28-fold relative to single-mode operation and more than threefold relative to the best individual two-mode setting. Linear memory-capacity measurements show that accessible recall spans only a few symbols at the tested symbol duration. Its rapid decline with delay, consistent with mechanical dissipation, accompanies rising NARMA error and the eventual loss of multiplexing gains at higher orders. We also use electrical feedthrough as an internal reference for assessing the computational contribution of the NEMS response. Separate linear readouts are trained on feedthrough features and features derived from the measured NEMS response, using the same recordings and matched drive settings and processing. On challenging nonlinear mapping tasks, the multiplexed NEMS features yield substantially lower errors than the feedthrough features. These results demonstrate how dynamical diversity through variations in modal drive amplitudes expands the computational capability of a single multimode NEMS resonator.

Abstract

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