Impacts of Length Scale Parameter on Material Dependent Thermoelastic Damping in Micro/nanoplates Applying Modified Coupled Stress Theory

At a Glance

Metadata	Details
Publication Date	2022-06-21
Journal	Mechanika
Authors	R. Resmi, V.Suresh BABU, M.R. BAIJU
Institutions	University of Kerala, A P J Abdul Kalam Technological University
Analysis	Full AI Review Included

Technical Documentation & Analysis: MPCVD Diamond for High-Q Resonators

Reference Paper: Resmi, R., Suresh Babu, V., & Baiju, M. R. (2022). Impacts of Material Performance Indices and Length Scale Parameter on Thermoelastic Damping in Micro/Nanoplates Applying Modified Couple Stress Theory. Mechanika, 28(3), 171-189.

Executive Summary

This research provides critical validation for using high-performance materials, particularly diamond, in micro/nanoresonators where intrinsic energy dissipation (Thermoelastic Damping, TED) limits the achievable Quality Factor ($Q_{TED}$).

Core Finding: The study confirms that Diamond is a superior material for high-Q resonators, exhibiting the second-highest $Q_{TED}$ (after PolySi) among the five materials tested (PolySi, Diamond, Si, GaAs, SiC).
Size Effect Validation: The Modified Couple Stress Theory (MCST) is validated as essential for accurate modeling at the micro/nanoscale. For diamond plates, the difference between classical theory ($l=0$) and MCST ($l=1$ µm) reaches a maximum of 92.13% in $Q_{TED}$ prediction.
Material Dependency: $Q_{TED}$ is inversely related to the Thermoelastic Damping Index (TDI). Diamond’s high $Q_{TED}$ is attributed to its favorable mechanical and thermal properties (high Young’s Modulus and high thermal diffusivity).
Design Optimization: Critical Length ($L_c$), the dimension corresponding to peak energy dissipation, is identified as a key design parameter, dependent on the material’s thermal diffusion length ($l_r$).
6CCVD Value Proposition: 6CCVD specializes in providing the high-purity Single Crystal Diamond (SCD) required to achieve these theoretical maximum $Q_{TED}$ values, along with custom dimensions and ultra-low roughness polishing (Ra < 1 nm).

Technical Specifications

The following hard data points were extracted from the analysis, focusing on the material properties and performance indices relevant to CVD Diamond (SCD).

Parameter	Value	Unit	Context
SCD Young’s Modulus ($E$)	1050	GPa	High stiffness minimizes mechanical losses (Table 1)
SCD Poisson’s Ratio ($\nu$)	0.07	-	Low value indicates high rigidity (Table 1)
SCD Thermal Diffusivity ($\chi$)	9.31	cm²/s	Extremely high thermal transport property (Table 4)
SCD Thermal Diffusion Length ($l_r$)	3.69	nm	Key parameter governing Critical Length ($L_c$) (Table 4)
SCD TDI (Thermoelastic Damping Index)	2651	10^-6	Moderate TDI contributes to high $Q_{TED}$ (Table 4)
Maximum $Q_{TED}$ (Diamond, CC, $l=1$ µm)	8418.98	-	Clamped-Clamped (CC) boundary condition, MCST model (Table 2)
Max $Q_{TED}$ Difference (Diamond, 0 to 1 µm $l$)	92.13	%	Highest percentage difference observed between classical and MCST models (Table 3)
Plate Thickness ($h$) Analyzed	200	µm	Typical microplate dimension used in simulations
Critical Length ($L_c$) (Diamond, CC, $l=1$ µm)	1.04	µm	Length at which energy dissipation peaks (Table 5)

Key Methodologies

The study employed advanced non-classical continuum mechanics to accurately model size-dependent damping effects in micro/nanoplates.

Theoretical Basis: The analysis utilized the Modified Couple Stress Theory (MCST), which incorporates a single material length scale parameter ($l$) to account for size effects, overcoming the limitations of classical elasticity theories (Lifshitz and Roukes, LR).
Structural Model: Isotropic rectangular micro/nanoplates were modeled based on the Kirchhoff plate theory.
Governing Equations: Equations of motion and the coupled heat conduction equation were derived using the Hamilton principle.
$Q_{TED}$ Quantification: The thermoelastic damping limited quality factor ($Q_{TED}$) was derived using the complex frequency approach, providing a closed-form expression incorporating size effects and material performance indices (TDI, $l_r$).
Boundary Conditions and Modes: Simulations were conducted for two boundary types (Simply Supported (SS) and Clamped-Clamped (CC)) and two vibration modes (M-I(1,1) and M-II(2,1)).
Material Selection: Five distinct structural materials—PolySi, Diamond, Si, GaAs, and SiC—were selected to investigate the dependency of $Q_{TED}$ and Critical Length ($L_c$) on material properties.

6CCVD Solutions & Capabilities

The research confirms that diamond is a critical material for next-generation, high-performance MEMS/NEMS resonators, filters, and sensors where intrinsic damping must be minimized. 6CCVD is uniquely positioned to supply the necessary high-quality CVD diamond materials and customization services required to replicate and advance this research.

Applicable Materials

To achieve the high theoretical $Q_{TED}$ values demonstrated in this study, researchers require diamond with exceptional purity and crystal quality.

Optical Grade Single Crystal Diamond (SCD): Recommended for replicating the high-Q performance demonstrated in the paper. SCD offers the highest Young’s Modulus (1050 GPa) and thermal diffusivity (9.31 cm²/s), which are the primary factors maximizing $Q_{TED}$ and minimizing TED.
Polycrystalline Diamond (PCD): Suitable for large-scale resonator arrays or applications where plates up to 125 mm in diameter are required, offering robust mechanical properties at scale.

Customization Potential

The study focuses on microplates (200 µm thick, 10 µm x 5 µm). 6CCVD provides the necessary material flexibility and precision engineering services for MEMS/NEMS fabrication.

Research Requirement	6CCVD Capability	Relevance to High-Q Design
Precise Thickness Control	SCD/PCD Thickness: 0.1 µm up to 500 µm	Enables precise tuning of resonant frequency and optimization of the plate thickness ($h$) relative to the critical length ($L_c$).
Custom Dimensions	Plates/Wafers up to 125 mm (PCD)	Supports both small-scale research (10 µm x 5 µm) and industrial scaling of resonator arrays.
Ultra-Smooth Surfaces	Polishing: Ra < 1 nm (SCD), Ra < 5 nm (Inch-size PCD)	Minimizes extrinsic energy dissipation mechanisms (e.g., surface scattering losses) that limit the practical $Q$ factor in real-world devices.
Electrode Integration	Custom Metalization: Au, Pt, Pd, Ti, W, Cu	Essential for integrating electrical functionalities (actuation and sensing) in MEMS/NEMS devices, as referenced in the paper’s introduction.

Engineering Support

The paper highlights the complexity of modeling size-dependent TED using MCST and the necessity of optimizing material length scale parameters ($l$) and thermal diffusion length ($l_r$).

6CCVD’s in-house PhD team offers expert consultation on material selection and design optimization for high-Q MEMS/NEMS projects, including:

Material Selection: Assisting engineers in selecting the optimal diamond grade (SCD vs. PCD) based on required $Q_{TED}$ and dimensional constraints.
Modeling Support: Providing material parameters (E, $\nu$, $\chi$, etc.) with high precision for accurate non-classical modeling (MCST/Strain Gradient Theory) of thermoelastic damping in micro/nanoplates.
Global Logistics: Ensuring reliable global shipping (DDU default, DDP available) of sensitive diamond wafers and plates.

Call to Action: For custom specifications or material consultation regarding high-Q diamond resonators, visit 6ccvd.com or contact our engineering team directly.

View Original Abstract

Among the different energy dissipation mechanisms, thermoelastic damping plays a vital role and need tobe alleviated in resonators inorder to enhance its performance parameters by improving its thermoelastic dampinglimited qualityfactor, QTED. The maximum energy dissipation is also interrelated with critical length (???????? ) of theplates and by optimizing the dimensions the peaking of energy dissipation can be diminished. As the size of thedevices is scaled down, classical continuum theories are not able to explain the size effect related mechanicalbehavior at micron or submicron levels and as a result non-classical continuum theories are pioneered with theinception of internal length scale parameters. In this paper, analysis of isotropic rectangular micro-plates based onKirchhoff model applying Modified Coupled Stress Theory is used toanalyzethe size-dependent thermoelasticdamping and its impact on quality factor and critical dimensions.Hamilton principle is adapted to derive thegoverning equations of motion and the coupled heat conduction equation is employed to formulate the thermoelasticdamping limited quality factor of the plates. Five different structural materials (PolySi, Diamond,Si, GaAs andSiC)are used for optimizing QTED which depends on the materialperformance index parameters. ThermoelasticDamping Index [TDI] and thermal diffusion length, lT. According to this work, the maximum QTED is attained forPolySi with the lowest TDI and Lcmax is obtained for SiC which is having the lowest lT. The impact of lengthscaleparameters (l), vibration modes, boundary conditions (Clamped-Clamped and Simply Supported), and operatingtemperatures on QTED and Lcare also investigated. It is concluded that QTED is further maximized by selecting lowtemperatures and higher internal length scale parameters (l).The prior knowledge of QTED and Lchelp the designers tocome out with high performance low loss resonators.

Tech Support

Original Source

DOI: https://doi.org/10.5755/j02.mech.25841