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Summary

SEM and stochastic simulation of stressed low-K dielectric lines
SEM and stochastic simulation of stressed low-K dielectric lines used in nanoscale integrated circuits.

Modeling and Simulation are vital for metrology. Models help define measurable quantities and establish their significance and application. One of the simplest and well-known examples is the Elongation (Young) Modulus that is defined by a simple tensile experiment. Euler-Bernoulli beam theory allows the results of simple tensile experiments to be applied to bending of beams, and Finite Element simulations extend its application to stiffness and small deformation of arbitrary structures and assemblies. In the example of tensile experiments, the modelling path from experiment to measured parameter is very short. In other examples, the path can be quite long. The focus of Modeling and Simulation in the Nanomechanical Properties Group is more complex applications, optimal and robust design of experiments and measurement requiring complex deconvolution of confounding effects. These goals are accomplished through development, application and integration of analytic formulas and in-house software, combined with application and extensions of commercial software such as Finite Element packages. Here is some recent and ongoing work:

Description

Fracture Paths in Ferroelectric Domains

Cross-correlation Electron Backscatter Diffraction was used to map stresses in and around ferroelectric domains in single-crystal BaTiO3, a ubiquitous dielectric in electronic devices. Building on the usual stress maps, virtual surfaces experiencing maximum principal stress were evaluated, indicating possible fracture driving forces and initial facture paths. See Related Publication 1.

ferroelectric domains in BaTiO3
Analysis of isolated ferroelectric domains in BaTiO3. (a) optical micrograph with visible (darker) isolated domains. (b) EBSD domain orientation mapping with cross-correlation EBSD reference points indicated. (c) Maximum stress from cross-correlation EBSD, planes on which it acts and integrated plane-tangents to indicating directions of possible crack opening and extension.

Dynamical Electron Simulation for Nanoscale Shape Metrology

POC Measured Pattern
POC Measured Pattern
Simulated Electron Wave-Function
Simulated Electron Wave-Function

 

Extending ongoing work in dynamical simulation of electron backscatter diffraction, simulations were developed and applied for shape metrology using electrons. Similar in concept to Grazing Incidence Small Angle X-ray Scattering, Reflective Small Angle Electron Scattering and Electron Reflectometry

Reflection and evanescent wave at surface of crystalline Si
Reflection and evanescent wave at surface of crystalline Si

use electrons to measure nanoscale geometry with an eye towards meeting the needs of integrated circuit manufacturers with subsequent advantages and disadvantages, the greatest advantage being the immense practicality of a compact cheap electron source instead of a cost-prohibitive cyclotron X-ray source. Electron shape metrology has been demonstrated through a proof-of-concept experiment using a TEM and subsequent dynamical electron simulation. Building on the existing RSAES patent, a patent for ER was awarded (related publication 2). Soon, atomic-scale applications will enable better Electron Back Scattering Strain metrology. Python-based simulation software is planned to be publicly available.

Classical Electron Simulation in 3D Nanoscale Devices

Electron simulation plots
Electron simulation plots

Towards high-resolution strain mapping in 3D nanoscale devices, cross-correlation Electron Backscatter Diffraction was performed on 3D nanoscale strained SiGe lines on a Si substrate. In 3D structures, it was found that a significant number of electrons could undergo long-range classical scattering resulting in a convoluted Electron Backscatter Diffraction pattern. Adapting a classical electron simulation, appropriate weighting factors were found to deconvolute the measured patterns and yield correct strain measurements. (related publication 3)

 

Statistical Mechanics of Nanoparticle Size-Distribution

Lattice gas model
Lattice gas model

To synthesize nanometer-scale materials such as colloids with high precision and reproducibility, researchers and manufacturers must struggle against the materials' tendency to coarsen and coalesce to reduce their surface energy and increase their entropy. However, nanoparticle dispersions with narrow size distributions can be thermodynamically stabilized, even with respect to bulk solids, if their excess energy density due to interfacial (capillary) forces is designed to be small and increasing with decreasing particle size. Conversely, measurements of stable size distributions can be used to extract excess surface energies enabling quantitative comparison of their curvature-dependent behavior with electronic structure and molecular dynamics simulations. To perform this accurately requires detailed evaluation of the entropy for the multicomponent system, for example via a lattice gas model. (related publication 4)

Thermodynamic Stabilization of particle size distribution
Thermodynamic Stabilization of particle size distribution

 

Nanoscale Mechanical Stochasticity

Micro-Tensile Specimen with nanoscale line edge roughness
Micro-Tensile Specimen with nanoscale line edge roughness

Nanoscale mechanical experiments suffer from lack of control and loss of repeatability, often requiring a statistical approach that adds inherent uncertainty to measurement uncertainty. Smaller effective volumes mean larger fluctuations in effective properties. Even more alarming, in non-linear systems, fluctuations move the means of distributions. In the multiscale finite element / stochastic differential equation simulation depicted in the Summary, it was found that stochastic fluctuations shift the mean buckling stress by 5-15%. In fracture simulations, it was found from detailed modeling of sample geometry using image-processed micrographs, that line-edge roughness can overwhelm intended experimental design, leading to greatly broadened strength-distributions. (related publication 5)

 

Related Publications

  1. Howell, Jane A., Mark D. Vaudin, Lawrence H. Friedman, and Robert F. Cook. 2020. “Microscale Mapping of Structure and Stress in Barium Titanate.” Journal of Research of the National Institute of Standards and Technology 125 (April): 125013. https://www.nist.gov/publications/microscale-mapping-structure-and-stress-barium-titanate.
  2. Friedman, Lawrence H., and Wen-Li Wu. 2019. “Electron Reflectometry for Measuring Nanostructures on Opaque Substrates.” Applied Physics Letters 115 (2): 023105. https://www.nist.gov/publications/electron-reflectometry-measuring-nanostructures-opaque-substrate. Friedman, Lawrence H., and Wen-Li Wu. 2019. United States Patent: 10424458 - Electron reflectometer and process for performing shape metrology. 10424458, issued September 24, 2019. https://federallabs.org/technology/electron-reflectometer-and-process-for-performing-shape-metrology. Friedman, Lawrence H., Wen-Li Wu, Wei-En Fu, and Yunsan Chien. 2017. “Reflective Small Angle Electron Scattering to Characterize Nanostructures on Opaque Substrates.” Applied Physics Letters 111 (12): 123106. https://www.nist.gov/publications/reflective-small-angle-electron-scattering-characterize-nanostructures-opaque-substrate.
  3. Osborn, William, Lawrence H. Friedman, and Mark Vaudin. “Strain Measurement of 3D Structured Nanodevices by EBSD.” Ultramicroscopy, (August 20 2017). https://www.nist.gov/publications/strain-measurement-3d-structured-nanodevices-ebsd.
  4. Nelson, Andrew and Lawrence H. Friedman. “Thermodynamic stabilization of nanoparticle size distributions: interfacial energy measurement and limits on monodispersity” presented at Fall 2020 Virtual Meeting & Expo of The American Chemical Society (August 17, 2020).
  5. Friedman, Lawrence H., Igor Levin, and Robert F. Cook. “Stochastic Behavior of Nanoscale Dielectric Wall Buckling.” Journal of Applied Physics 119 (11): 114305 (2016). https://www.nist.gov/publications/stochastic-behavior-nanoscale-dielectric-wall-buckling. DelRio, Frank; Boyce, Brad; Benzing, Jake; Friedman, L; Cook, Robert “Shoulder fillet effects in strength distributions of MEMS components.” in review.
Created September 3, 2020, Updated September 9, 2020