1. Static Loading
1. Concentrated point load at center:
• Deflections: pyramidal along edges, conical near center where force is applied
• Simulates force - deflection behavior generated by AFM force curves
• Asymptotes to results of analytical calculations in small and large deflection limits
2. Uniform pressure across membrane:
• Reveals regions of high strain near edge centers – relevant to pressure sensor design
• Simulates laser interference measurements of membrane deflection under pressure
• Comparison with analytical calculations for limiting cases:
i. Small deflection limit: deflection (d) expected to scale linearly with
applied pressure (P) when d << h, the membrane thickness
ii. Large deflection limit: P ~ d3 when d >> h
Modeling Elastic Behavior in Freestanding Silicon Nanomembranes through Analytical and Finite Element Methods
Bryce Loveland, Alejandro Jimenez, Taylor Keesler and Gokul Gopalakrishnan
University of Wisconsin - Platteville
Background and Motivation
• Elastic constants of crystalline solids important for:
• Mechanical behavior
• Heat transport
• Electronic and optical properties
• Single and polycrystalline silicon ubiquitous in:
• Micro-electronic devices and integrated circuits
• Micro- and nano-electromechanical systems (MEMS/NEMS)
• Elastic parameters change from bulk to nanoscale materials
• Yet, elastic constants for silicon not well studied below 100 nm
• Our approach: computational and experimental investigation:
• Fabricate freestanding silicon sheets, 100 nm and thinner
• Measure behavior under different loading conditions:
concentrated static load, uniform pressure, vibration
• Model behavior using bulk elastic constants for silicon
• Modify effective elastic constants to fit modeled results to their
corresponding experimental values
• Computational techniques:
• Analytical calculations, where possible
• Finite Element Analysis (FEA), where needed
Vibrational Modes
Harmonic modes of a vibrating square membrane are given by:
𝑓𝑛 =
𝐶 𝑛
2𝜋𝑎2 𝐷 𝜌ℎ where 𝐷 =
𝐸ℎ3
12(1−ν2)
Mode #1 Mode #2
Mode #3 Mode #4
Mode #5 Comparison of Results:
Simulations relevant to reflection spectra from thermally excited nanomembranes
Special Case: Vibrational Modes of a 2D Phononic Crystal
• Short wavelength vibrational modes propagate heat and sound in a solid
• Modifying these modes by using periodic nanoscale boundaries provides a
better fundamental understanding of heat transport in nanoscale systems
• Also provides control over thermal properties through nanoscale engineering
Ongoing work:
a. Fabrication of 2D phononic crystals
b. Calculation of phonon dispersions
c. Comparison with Raman spectra
d. Demonstration of phononic bandgaps AFM and SEM images of 50 nm circular cavities on silicon
Mode #
Analytical
(Hz)
Finite Element
Model (Hz)
%
Difference
1 19847 19896 0.25
2 40482 40584 0.25
3 40482 40584 0.25
4 59702 59870 0.28
5 72589 72763 0.24
Challenges
• Polysilicon membrane:
100 nm x 0.2 mm
• High aspect ratio > 103
• Modeling internal stresses
• Highly non-linear response
• Managing memory
• Reducing computation time
Finite Element Analysis
• Analytical equations for mechanical behavior of plates and
membranes apply only in certain limiting cases:
• Example: very large or small deflection, no shear strength, etc.
• FEA works in all cases, given enough computing power, memory
• FEA: numerical technique to find approximate solutions to
differential equations with well-defined boundary conditions
• Structure “meshed” into small elements: simplifies calculation
• FEA software: ANSYS Workbench 14
Future Work
• Model the response for different thicknesses but fixed elastic modulus to determine size dependence of deflections, strains, and modal frequencies
• Vary the elastic modulus continuously for a given material and geometry to estimate the expected role of nanostructuring on mechanical response
• Incorporate uniform biaxial stresses in the membranes; if needed develop more sophisticated models for non-uniform stresses
• Perform analysis for other common polycrystalline membrane materials: stoichiometric and low-stress silicon nitride, amorphous silicon dioxide
• Develop models for single crystal materials (primarily silicon) to examine changes in behavior resulting from crystalline anisotropy
• Incorporate periodic boundary structures to study vibrational modes in two-dimensional phononic crystals (starting with a silicon-air system)