Controllable Mechanical Dynamics in Golf-Tee Shaped Nanostructures

Abstract

We investigate the mechanical dynamics of golf-tee nanostructures, a common macroscopic geometry whose nanoscale implementation has received limited attention in the emerging field of hybrid nanophotonics. Using a theoretical analysis of the golf-tee geometry–characterized by its broad head, cylindrical support, tapered bottom sidewalls, and axisymmetric structure–we examine the mechanical characteristics of individual resonators and pairwise coupling between resonators. We demonstrate mechanical coupling of tens of MHz, providing improved control compared to conventional cylindrical pillars. The controllable mechanical dynamics of golf-tee structures offer an alternative approach to conventional cylindrical pillars, with enhanced tunability for mechanically configurable metasurfaces and potential for mechanically reconfigurable photonic applications.

1. Introduction

Pillar architectures are a common structural motif found in natural phenomena [1] and engineered systems [2,3], prompting extensive mechanical studies across various research fields [2,3,4,5,6,7,8]. In silicon photonics, pillar architectures have become as fundamental components for precise manipulation of optical fields at the nanoscale. They perform several critical functions: enabling the formation of photonic bandgaps and resonant modes essential for spectral control, [9,10] exhibiting tunable Mie resonances that enhance broadband light absorption [11], and supporting high-quality edge and bound states crucial for robust light confinement [12,13]. Furthermore, advanced nanowire designs utilizing whispering gallery mode resonances have improved infrared photodetection, [14] and complementary approaches, such as hybrid integration techniques, further boost silicon photonic device performance [15,16].

While conventional pillar structures are widely investigated and applied in silicon photonics, the golf-tee structure represents a distinctive architectural variant. Its unique hierarchical geometry enhances optical field localization and light–matter interaction. This design has been used to fabricate diamond mirrors for high-power applications [17], highlighting its potential to improve optical confinement and thermal robustness. These capabilities enable applications in silicon photonics, high-power optical systems, and optomechanical devices. The hierarchical design creates spatially varying electromagnetic field distributions: the planar top facet provides uniform field interaction, while the tapered bottom profile optimizes mode confinement and reduces substrate coupling losses. Moreover, the multi-region architecture of golf-tee structures offers exceptional potential for mechanically configurable metasurfaces, [18] where controlled deformation can selectively tune optical and mechanical properties through the distinct responses of each geometric region. Despite this promise, the golf-tee nanostructures remain underexplored in both nanophotonics and optomechanics.

In this study, we theoretically investigate the mechanical dynamics of golf-tee nanostructures, focusing on their unique mechanical properties and pairwise inter-resonator coupling. We show that these structures offer controllable mechanical dynamics that complement conventional cylindrical pillars, and we explore their potential as adaptive photonic platforms. This distinctive geometry–with a broad head and tapered bottom geometry–offers new opportunities to study mechanical properties in mechanically reconfigurable photonic applications.

2. Theoretical Framework and Design

The vibrational dynamics of golf-tee nanostructures can be described by the Euler–Bernoulli beam theory that accounts for non-uniform cross-sectional geometries with clamped base conditions. For such fixed-base structures with spatially varying geometry, the differential equation governing transverse vibrations is given by [19,20]

$$\begin{array}{c}\hfill {\displaystyle \frac{d^2}{d{x}^2}}\left[EI{\displaystyle \frac{d^{2}u}{d{x}^2}}\right]+\rho A{\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}=0\end{array}$$

subject to clamped boundary conditions at the base and free-end conditions at the top. Here, u represents the displacement field, E denotes the Young’s modulus, I is the areal moment of inertia, $\rho$ is the mass density, and A is the cross-sectional area. The spatially-varying cross-section of the golf-tee structures was modeled using mesh discretization in finite element analysis. For reference, the flexural eigenfrequency of a uniform cylindrical pillar is given by $f_n={\beta }_n^2/\left(2\pi h^2\right)\sqrt{EI/\rho A}$, [19] where the first two eigenvalues are ${\beta }_1$ = 1.875 and ${\beta }_2$ = 4.694, respectively [20].

The golf-tee nanostructures investigated consist of three distinct regions: a tapered base, cylindrical stem, and enlarged head section, as illustrated in Figure 1. All structural dimensions are implemented during fabrication. The stem radius r defines the cylindrical section, the base height $h_{\mathrm{B}}$ spans from the substrate to the stem junction, and the total height H encompasses the entire structure from substrate to head. The rotational symmetry inherent in golf-tee nanostructures introduces fundamental vibrational characteristics that govern their mechanical dynamics and inter-resonator interactions. This axial symmetry around the central vertical axis gives rise to mode degeneracies, where vibrational modes in orthogonal directions (x, y) emerge at identical eigenfrequencies. The inherent symmetry breaking arises from the bottom coupler configuration, which disrupts the perfect axisymmetric geometry by introducing controlled asymmetries at the base coupler interface. Through systematic variation of dimensional parameters across the structural parts, the degree of symmetry breaking can be controlled to obtain optimized mechanical response and adjustable coupling dynamics. The engineered asymmetry facilitates selective vibrational coupling along the coupler axis, enabling reconfigurable control over resonator-to-resonator mechanical interactions while providing enhanced mode selectivity for integrated photonic devices.

The baseline structural proportions are defined by ratios $r/h_{\mathrm{B}}=1$ and $h_{\mathrm{B}}/H=0.25$. The geometric parameters for the golf-tee nanostructures are summarized in

Table 1. Structural and material parameters for Si golf-tee nanostructures. Here, Y denotes the Young’s modulus, $\rho$ denotes the mass density, $w_{\mathrm{c}}$ is the ridge coupler width, and $t_{\mathrm{c}}$ is the coupler thickness.

r (nm)	${\mathit{h}}_{\mathbf{B}}$ (nm)	${\mathit{h}}_{\mathbf{B}}/\mathit{H}$	${\mathit{\theta }}_{\mathbf{B}}$ (Degrees)	${\mathit{\theta }}_{\mathbf{T}}$ (Degrees)	${\mathit{w}}_{\mathbf{c}}$ (nm)	${\mathit{t}}_{\mathbf{c}}$ (nm)	E (GPa)	$\mathit{\rho }$ (${\mathrm{kg/m}}^3$)
500	500	0.25	30	30	500	125	169 [21]	2329 [22]

. Unless otherwise specified for parametric analysis, all structural dimensions remain fixed at the tabulated values. The base region features a conical taper characterized by the base angle ${\theta }_{\mathrm{B}}$, measured from the vertical axis, governing the substrate-to-stem transition. The head section exhibits an inverted conical profile with a taper angle ${\theta }_{\mathrm{T}}$, forming the distinctive enlarged cap structure. Both taper angles are maintained at ${\theta }_{\mathrm{B}}$ = ${\theta }_{\mathrm{T}}$ = 30$°$ as the baseline configuration.

When two golf-tee structures are connected through bottom ridge coupler, their mechanical interactions follow a coupled oscillator framework. The coupling regions are thin ridge waveguides formed by etching the pillar bases, as shown in Figure 1b. The coupler width $w_{\mathrm{c}}$ equals the pillar radius r, while the coupler thickness $t_{\mathrm{c}}$ is set to 25% of the base height, with specific values provided in Table 1. The vibrational dynamics are governed by $m_i\ddot{u_i}+k_i{u}_i+\gamma (u_i-u_j)=0$ for resonators $i,j$ =$1,2$ ($i\ne j$), where $u_i$ are the displacement amplitudes, $m_i$ and $k_i$ represent the effective mass and spring constant of each resonator, and $\gamma$ characterizes the inter-pillar coupling spring constant. Solving these coupled equations, the mechanical coupling strength is quantified through:

$$\begin{array}{c}\hfill g={\displaystyle \frac{\gamma }{2\pi \sqrt{f_1{f}_2{m}_1{m}_2}}},\end{array}$$

which relates the coupling parameter $\gamma$ to the individual resonant frequencies $f_i$ and effective masses $m_i$ [23]. By examining temperature-dependent eigenfrequency shifts resulting from thermal modulation of Young’s modulus, the coupling strength between connected golf-tee resonators can be determined. The mechanical dynamics of these golf-tee nanostructures were investigated through finite element analysis using COMSOL software 6.0, building on established methodologies from previous studies of GaAs photonic nanowires [24,25].

3. Results

The dependence of resonance eigenfrequencies $f_{\mathrm{res}}$ on the base angle ${\theta }_{\mathrm{B}}$ is shown in Figure 2 for the first and second flexural, radial, and longitudinal modes. For frequencies in kHz–MHz range, all four mechanical modes exhibit significant variations of $f_{\mathrm{res}}$ within the range of base angle deviations commonly encountered in practical etching processes. The golf-tee-like bottom profiles commonly arise unintentionally during conventional pillar fabrication due to etching imperfections. In silicon pillar dry processing using deep reactive ion etching, sidewall profiles are influenced by mask edge geometry, with angular features from the lithographic pattern being replicated in the final structures. Achieving vertical sidewalls requires optimized ion bombardment parameters that promote directional removal and minimize lateral erosion. The reported base angles typically deviate by 5–10$°$, and in some cases by over 20$°$ from vertical (${\theta }_{\mathrm{B}}$ = 90$°$) [26,27,28].

The first flexural mode shows considerable variation (detailed in the inset) with relative frequency shifts of 6%, 14%, and 34% for ${\theta }_{\mathrm{B}}$ deviations of 5$°$, 10$°$, and 20$°$, respectively, from the ideal vertical configuration. When evaluating frequency changes relative to the vertical stem configuration rather than instantaneous gradients, distinct frequency shift patterns emerge across different vibrational modes: at 10$°$ deviation, breathing modes (radial and longitudinal) show approximately 5% frequency reduction, while flexural modes exhibit around 10% change. At 20$°$ deviation, breathing modes demonstrate approximately 20% frequency reduction, while the second flexural mode shows around 33% change. Despite these varying degrees of frequency shift among different modes, the fundamental flexural mode remains of primary interest due to its favorable noise performance and quality factor characteristics. Interestingly, a degeneracy point occurs at approximately ${\theta }_{\mathrm{B}}\sim {50}^{\circ }$, where the resonance frequencies of the second flexural and the longitudinal modes become identical. For all four modes, the increase in the resonance frequency with increasing ${\theta }_{\mathrm{B}}$ indicates an enhanced effective stiffness of the structure.

For the hierarchical geometry, the proportional dimensions of the golf-tee structure also influence its vibrational characteristics. Figure 3a shows the dependence of eigenfrequency shifts on the normalized height ratio $h_{\mathrm{B}}/H$ for various vibrational modes. As the normalized height ratio increases from 0.25 to 0.5, all vibrational modes exhibit enhanced frequency shifts, reflecting a stiffening effect of the stem structure. The first flexural mode shows a strong dependence on the base height, with $\Delta f/f_{\mathrm{c}}$ increasing from approximately 0.0 to 0.65, indicating that the base height ratio significantly influences the structure’s flexural stiffness.

Controlling the normalized radius ratio ($r/h_{\mathrm{B}}$) in the fabrication has distinct effects on different vibrational modes, as shown in Figure 3b. With increasing $r/h_{\mathrm{B}}$ above unity for a given height of 500 $\mathrm{nm}$, the first flexural mode shows negligible frequency shifts. In contrast, both radial and longitudinal modes show decreasing frequency shifts as the radial dimension increases, reflecting enhanced structural stiffness from the larger stem cross-sectional area. These observations confirm that manipulating the base geometry provides systematic frequency control for golf tee resonators. Vertical scaling primarily controls the manipulation of the flexural eigenfrequencies, while radial scaling decouples the radial and longitudinal mode frequencies from the flexural characteristics. Overall, strategic base geometry manipulation enables systematic frequency control in golf-tee resonators, with vertical scaling serving as the primary mechanism for adjusting flexural resonance characteristics and radial scaling providing independent control over radial and longitudinal mode behaviors.

Figure 4 presents the distance-dependent coupling strength ($g/2\pi$) for golf-tee-shaped resonators as a function of normalized center-to-center separation ($d_{cc}/D$), where $d_{cc}$ is the center-to-center distance between resonators and D is the stem diameter of the pillar. A decay tendency in inter-resonator coupling is observed with increasing distance, ranging from tens of MHz to sub-MHz levels. Direct comparison of strain-induced coupling with conventional GaAs pillar structures [29,30] is limited by our hierarchical Si golf-tee geometry, where the heavy head mass and conical base design create distinct stress field distributions. This structural configuration amplifies stress localization at the base contact area, reminiscent of strain-enhancement effects observed in conical nanowire geometries [31], resulting in modified coupling dynamics relative to conventional coupled pillars, with coupling strength further influenced by bottom geometry parameters as demonstrated previously [24]. The demonstrated strain-mediated coupling, combined with the golf-tee architecture’s inherent advantages in base geometry control and optical coupling efficiency, makes these platforms especially promising for scalable optomechanical networks and mechanically configurable metasurfaces.

4. Discussion

We investigated golf-tee-shaped Si nanostructures inspired by the morphology of everyday sporting equipment, establishing their potential as promising platforms for advanced optomechanical coupling networks. These hierarchical structures represent a transformative alternative to conventional cylindrical pillar designs; they offer enhanced geometric control and adjustable inter-resonator coupling, while providing excellent optical coupling efficiency for photonic applications. Manipulation of the base geometry enables systematic frequency tuning of the golf-tee resonators. Vertical scaling primarily controls the flexural mode, while radial scaling independently tunes the radial and longitudinal modes for mode-selective frequency adjustment.

The hierarchical geometry offers promising opportunities for mechanically configurable metasurfaces, where controlled strain can exploit the differential mechanical responses of the base, stem, and head regions to achieve selective mode tuning and dynamic optical property modulation. These structures allow for strain-mediated control with distance-dependent coupling strengths through geometric parameter optimization. All structural modifications are implemented during fabrication through controlled variation of process parameters. Additionally, beyond the coupler-induced asymmetry, the engineered radial/height asymmetries induce slightly different resonance frequencies between structures, enabling selective actuation while maintaining strain-mediated coupling via a bottom coupler. Previous theoretical work predicted decreased inter-resonator coupling strength with larger pillar base angles [24]. While experimental studies have demonstrated similar coupling mechanisms in conventional pillar resonator arrays [30], providing precedent for the strain-mediated interactions in golf-tee structures, our results still demonstrate tens of MHz coupling strengths suitable for practical applications. Achieving precise control over these geometric parameters for engineered golf-tee structures presents certain challenge as the etching processes responsible for such profiles exhibit inherent variability and high sensitivity to process conditions. Nevertheless, the axisymmetric nature of golf-tee structures facilitates their integration into structured photonic networks with engineered mechanical coupling and coherent array dynamics. These findings position golf-tee nanostructures as an enabling framework for scalable optomechanical systems, where the unique combination of hierarchical geometry and strain-mediated coupling paves the way for dynamically reconfigurable metamaterials and adaptive photonic networks.

5. Conclusions

This study demonstrates that golf-tee-shaped Si nanostructures offer enhanced control over mechanical dynamics compared to conventional pillar designs. The hierarchical geometry enables independent control of different vibrational modes through strategic geometric parameter control: height scaling produces relative frequency shifts of 10–30% in flexural modes for typical base angles of 10–20$°$, while radial dimension control independently adjusts radial and longitudinal modes. Two base-coupled resonators demonstrate strain-mediated coupling with strengths ranging from tens of MHz to sub-MHz levels across varying separation distances. This approach facilitates the construction of reconfigurable optomechanical networks with tailored coupling architectures. Golf-tee nanostructures thus provide a versatile platform for advanced optomechanical applications, offering selective mode control and tunable mechanical coupling for reconfigurable photonic devices and integrated optical systems.