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by W.H. Müller , C. Liebold ICMS-workshop | Edinburgh | June 17-21, 2013

Are higher gradient theories of elasticity amenable to experiments? - A state-of-the-art review. by W.H. Müller , C. Liebold ICMS-workshop | Edinburgh | June 17-21, 2013. LKM. Technische Universität Berlin Institut für Mechanik - LKM Einsteinufer 5 D-10587 Berlin.

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by W.H. Müller , C. Liebold ICMS-workshop | Edinburgh | June 17-21, 2013

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  1. Are higher gradient theories of elasticityamenable to experiments? - A state-of-the-art review byW.H. Müller, C. Liebold ICMS-workshop | Edinburgh |June 17-21, 2013 LKM Technische Universität Berlin Institut für Mechanik - LKM Einsteinufer 5 D-10587 Berlin Supported by:

  2. Outline • Introduction and motivation: • Applications in micro-mechanics • The size effect (experiments from literature) • Overview on non-local continuum theories • Beam bending in couple stress theory • The AFM tool for deflection measurement • Performing the experiments: • Force calibration of the AFM • The beams • The Raman effect for strain measurement: • General remarks • Discussion of the penetration depth of the laser • Results for Si and SiN • Conclusions about the length scale parameter of Si and SiN • Improvements / discussion

  3. Outline • Introduction and motivation: • Applications in micro-mechanics • The size effect (experiments from literature) • Overview on non-local continuum theories • Beam bending in couple stress theory • The AFM tool for deflection measurement • Performing the experiments: • Force calibration of the AFM • The beams • The Raman effect for strain measurement: • General remarks • Discussion of the penetration depth of the laser • Results for Si and SiN • Conclusions about the length scale parameter of Si and SiN • Improvements / discussion

  4. Applications in micro-mechanics I A MEMS application: Micro 3D - gyroscope A MEMS application: Micro-sensors and black box in automotive technology

  5. Applications in micro-mechanics II • Objective:Stress and deflection measurements in micro beams subject to bending A MEMS application: An acceleration sensor in the automotive industry Raman peak position along the thickness of a deflected silicon beam; Srikar et al. (2003):

  6. Outline • Introduction and motivation: • Applications in micro-mechanics • The size effect (experiments from literature) • Overview on non-local continuum theories • Beam bending in couple stress theory • The AFM tool for deflection measurement • Performing the experiments: • Force calibration of the AFM • The beams • The Raman effect for strain measurement: • General remarks • Discussion of the penetration depth of the laser • Results for Si and SiN • Conclusions about the length scale parameter of Si and SiN • Improvements / discussion

  7. The size effect (experiments from literature) • The term “size effect”refers a stiffer (elastic) material behavior on the sub-micronscale. • Strain gradient theories can be used to model the size effect of micro-size structures, e.g., for simple beam bending or torsion of small wires. • A material length scale parameter called l is of importance in non-local theories, such as the couple stress theory used, e.g., by Yang et al. (2002). Left: Torsion of small copper wires performed by Chong et al. (2001) revealed:l = 3µm. Right: Simple beam bending experiments on epoxy by Gao and Park (2007) revealed:l =17.6µm. • Objective: Determination of the material length scale parameter l (additional stiffness parameter)of silicon and silicon nitride using atomic force microscopy and Raman spectroscopy.

  8. Outline • Introduction and motivation: • Applications in micro-mechanics • The size effect (experiments from literature) • Overview on non-local continuum theories • Beam bending in couple stress theory • The AFM tool for deflection measurement • Performing the experiments: • Force calibration of the AFM • The beams • The Raman effect for strain measurement: • General remarks • Discussion of the penetration depth of the laser • Results for Si and SiN • Conclusions about the length scale parameter of Si and SiN • Improvements / discussion

  9. Non-local theories (overview) local - non-local theory CS Couple Stress theory MSG Modified Strain Gradient theory SG Strain Gradient theory class. theory stored energy density: stress and strain measures: (isotropic) balance law: nr. of parameters: in use for finite element method by various authors published derivations: analytical displacement field:

  10. Outline • Introduction and motivation: • Applications in micro-mechanics • The size effect (experiments from literature) • Overview on non-local continuum theories • Beam bending in couple stress theory • The AFM tool for deflection measurement • Performing the experiments: • Force calibration of the AFM • The beams • The Raman effect for strain measurement: • General remarks • Discussion of the penetration depth of the laser • Results for Si and SiN • Conclusions about the length scale parameter of Si and SiN • Improvements / discussion

  11. Couple stress theory for beam bending I Derivation of the deflection curve, the strain and the bending rigidity. • The Euler-Bernoulli displacement field: • Components of the strain tensor: • Components of the stress tensor: • New: Components of the rotation gradient tensor:

  12. Couple stress theory for beam bending II • The principle of virtual work: • The differential equation: • The governing boundary conditions: • Integration of the differential equation: • Using the boundary conditions for a clamped beam the solution for deflection and strain reads:

  13. Couple stress theory for beam bending III • Derivation of the bending rigidity for x=L, ν=0: inverse quadratic function in h Figure: Simple beam bending experiments on epoxy by Gao and Park (2007) revealed: l=17,6µm. • Conclusion: - If l=0 the couple stress bending rigidity results in the conventional one. - The formula of the couple stress theory predicts a size effect.

  14. Outline • Introduction and motivation: • Applications in micro-mechanics • The size effect (experiments from literature) • Overview on non-local continuum theories • Beam bending in couple stress theory • The AFM tool for deflection measurement • Performing the experiments: • Force calibration of the AFM • The beams • The Raman effect for strain measurement: • General remarks • Discussion of the penetration depth of the laser • Results for Si and SiN • Conclusions about the length scale parameter of Si and SiN • Improvements / discussion

  15. The AFM tool I • The experiments for measuring w and ε: A Raman spectroscope integrated in an AFM used for the bending of µm beams The central equation of the AFM tool for determining the beam deflection reads: PSD LASER Raman laser Raman Laser AFM Laser observation variables: AFM laser

  16. The AFM tool II • 1st. The non-linear behavior of the PSD (Photo Sensitive Diodes): 2 1 AFM Laser cylindrical AFM probe Wheatstone bridge range of linearity 3 4 „signal=voltage“ non-linear behavior

  17. The AFM tool III • 2nd. Calibration of the tip deflection Δs: top view of piezosturning the feed table The Δx feed motion was produced by a calibrated screw thread driven by piezo-elements. Tapping the AFM probe against a rigid surface projected the Δx-input onto the voltage signal of the PSD. analysis points U [mV] U [mV] m 0 xmax x [nm] -single experiment - rescaled feed motion curves - slopes increase linearly! This gave rise to a quadratic relationship of the form:

  18. The AFM tool IV • 3rd. The AFM probe against a micro beam (separating both deflections): arig, nrig … values of the regression line of the rigid surface U [mV] a, n … values of the regression line with the micro beam in between x [nm] s…defl. of the tip wbeam … defl. of the beam • Conclusion: This approach compensated the non-linearity of the AFM tool.

  19. Outline • Introduction and motivation: • Applications in micro-mechanics • The size effect (experiments from literature) • Overview on non-local continuum theories • Beam bending in couple stress theory • The AFM tool for deflection measurement • Performing the experiments: • Force calibration of the AFM • The beams • The Raman effect for strain measurement: • General remarks • Discussion of the penetration depth of the laser • Results for Si and SiN • Conclusions about the length scale parameter of Si and SiN • Improvements / discussion

  20. Performing the experiments I • The force calibration of the AFM probe against a calibrated cantilever: • The AFM cantilever was calibrated at the PtB (Physikalisch Technische Bundesanstalt, Braunschweig, Germany) by a nano-compensational weight bridge for the force data in combination with a capacitive read out for the deflection data. • It revealed a spring constant of the cantilever of about: F=kPtBwPtB , kPtB=4.868 N/m Transfer of the spring constant

  21. Performing the experiments II • SEM analysis for determining the dimensions of the tested micro-beams: L=200±0.2 µm b=39±0.2 µm t=0.98±0.075 µm • Pictures via the optical microscope of some specimens: force application point SiN SiN SiN SiN SiN cSi

  22. Performing the experiments III • A video recording of the loading- unloading sequence via the optical microscope: (0) (1) (2) (3) (4)

  23. Outline • Introduction and motivation: • Applications in micro-mechanics • The size effect (experiments from literature) • Overview on non-local continuum theories • Beam bending in couple stress theory • The AFM tool for deflection measurement • Performing the experiments: • Force calibration of the AFM • The beams • The Raman effect for strain measurement: • General remarks • Discussion of the penetration depth of the laser • Results for Si and SiN • Conclusions about the length scale parameter of Si and SiN • Improvements / discussion

  24. The Raman effect for strain measurement I • The setup of our Raman spectrometer:

  25. The Raman effect for strain measurement II • The optical path of the laser beam: 1: live cam 2: beam splitter 3: polarizer (scattered) 4: aperture (slit) 5: notch filter 6: optical grating 7: CCD camera 8: filters for laser power 9: shutter 10: beam expander 11: polarizer (incident) 12: sample stage 13: optical lenses 14: shutter 15&16: laser source

  26. The Raman effect for strain measurement III • Inside the CCD camera (qualitatively): Si Raman peak @ ω 520 cm-1 A typical spectral distribution of silicon. I. DeWolf (1998)

  27. The Raman effect for strain measurement IV Some theoretical relationships • With the idea of effective force constants in the directions of the generalized coordinates of the phonons, we obtain the secular equation without presence of strain: Si Raman peak @ ω  98,5 THz 520 cm-1

  28. The Raman effect for strain measurement V Some theoretical relationships • Acc. to the so-called morphic effect, the strains εklenter the secular equation as follows: we can rewrite the secular equation: the secular equation under presence of strain • Consider pure tensionin x-direction:

  29. The Raman effect for strain measurement VI Some theoretical relationships • A separation of the three eigenvalues is possible by polarization of the incoming and scattered laser light (Raman selection rules), so that the strain-shift relationship reads: this holds for silicon in the present configuration Experimental procedure • Taking spectra in loaded and unloaded configuration at different points around the fixture: • numerical analysis for spectral position: Gauss / Lorentz mixture -reference- -loaded-

  30. The Raman effect for strain measurement VII Experimental application to silicon Raman @ 190 µN Raman @ 81 µN Euler Bernoulli-Theorie FEM @ 190 µN F x=0 strain εxx 

  31. Outline • Introduction and motivation: • Applications in micro-mechanics • The size effect (experiments from literature) • Overview on non-local continuum theories • Beam bending in couple stress theory • The AFM tool for deflection measurement • Performing the experiments: • Force calibration of the AFM • The beams • The Raman effect for strain measurement: • General remarks • Discussion of the penetration depth of the laser • Results for Si and SiN • Conclusions about the length scale parameter of Si and SiN • Improvements / discussion

  32. The problem of penetration depth of the laser I • Theoretical penetration depth of an extended laser beam in silicon (De Wolf*, 1996): with Aspnes* (1983): I0 F D I ε dp x • Specification for the choice of d: • 0.1 chosen a = absorption coefficient measured for Si at different wavelengths *De Wolf, I. (1996). Micro-Raman spectroscopy to study local mechanical stress in silicon integrated circuits. Semicond. Sci. Technol. 11, pp.139–154. *Aspnes, D. E., Studna, A. A. (1983). Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV. Am. Phys. Society. 27(2), pp. 985-1009.

  33. The problem of penetration depth of the laser II • Because the used beam thicknesses are 7000 nm, 3000 nm and 1000 nm a correction factor ffor the strains can be introduced as follows: εmax I0 calculated by conventional theory ε=0 x 0 dp increase of uncertainties f exp.

  34. Outline • Introduction and motivation: • Applications in micro-mechanics • The size effect (experiments from literature) • Overview on non-local continuum theories • Beam bending in couple stress theory • The AFM tool for deflection measurement • Performing the experiments: • Force calibration of the AFM • The beams • The Raman effect for strain measurement: • General remarks • Discussion of the penetration depth of the laser • Results for Si and SiN • Conclusions about the length scale parameter of Si and SiN • Improvements / discussion

  35. Results I • Corrected strain values for silicon micro beams measured using the Raman spectroscope: t=7µm, L=175µm measurement at different points around the fixture corr factor: f=1.19 measurement at different points along the beam length t=3µm, L=90µm t=3µm, L=130µm corr corr factor: f=1.71 factor: f=1.78

  36. Results II • No strain values for silicon nitride, due to a thin metallic layer. • The corresponding Raman signal is smeared out.

  37. Results III • The deflections of silicon micro beams measured via the AFM tool: t=1µm, L=130µm t=7µm, L=175µm t=1µm, L=50µm t=3µm, L=130µm

  38. Results IV • The deflections of silicon nitride micro beams measured via the AFM tool: t=0,8µm, L=170µm t=0,8µm, L=120µm t=0,8µm, L=80µm t=0,2µm, L=29µm t=0,6µm, L=80µm t=0,6µm, L=170µm … otherwise large bending angle

  39. Outline • Introduction and motivation: • Applications in micro-mechanics • The size effect (experiments from literature) • Overview on non-local continuum theories • Beam bending in couple stress theory • The AFM tool for deflection measurement • Performing the experiments: • Force calibration of the AFM • The beams • The Raman effect for strain measurement: • General remarks • Discussion of the penetration depth of the laser • Results for Si and SiN • Conclusions about the length scale parameter of Si and SiN • Improvements / discussion

  40. Conclusions about the length scale parameter I • The bending rigidities of silicon micro beams measured via the AFM tool: Bending rigidity vs. thickness of Epoxy 1 2 3 4 5 6 7 8 • Conclusion: The material length scale parameter of single crystalline silicon is smaller than l < 200 nm.

  41. Conclusions about the length scale parameter II • The bending rigidities of silicon nitride micro beams measured via the AFM tool: Bending rigidity vs. thickness of Silicon Nitride 0,1 0,2 0,3 0,6 0,8 1 • Conclusion: The material length scale parameter of silicon nitride is smaller than l < 50 nm.

  42. Outline • Introduction and motivation: • Applications in micro-mechanics • The size effect (experiments from literature) • Overview on non-local continuum theories • Beam bending in couple stress theory • The AFM tool for deflection measurement • Performing the experiments: • Force calibration of the AFM • The beams • The Raman effect for strain measurement: • General remarks • Discussion of the penetration depth of the laser • Results for Si and SiN • Conclusions about the length scale parameter of Si and SiN • Improvements / discussion

  43. Improvements / Discussion • Raman / Silicon: • A minimization of the penetration depth of the Raman laser can be achieved either by an UV laser source in combination with a UV spectrometer or by an independently moveable AFM-TERS probe. • For detecting a size effect even smaller micro beam geometries (particularly smaller thicknesses) are needed. • Raman / Silicon-Nitride: • For detecting a Raman signal the metallic layer should be removed (e.g., by etching). • Size Effect: • In order to detect a more pronounced size effect materials with a more complex micro-structure should be tested.

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