Skip to main content
SpringerLink
Log in

Shield-layers for reducing thermoelastic damping in resonating Silicon bars

  • Technical Paper
  • Published:
Microsystem Technologies Aims and scope Submit manuscript

Abstract

In this work it is theoretically shown that thermoelastic incompatibility between Silicon structures and their native-oxide layers induces thermoelastic damping. This damping dominates in structures that are packaged in vacuum and vibrate in pure axial motion. Analytic solutions of the thermoelastic response of axially loaded laminated bars are used to determine the material parameters which affect thermoelastic damping. The analysis suggests that thin shield-layers can significantly reduce thermoelastic damping which is associated with native-oxide layers in Silicon resonators.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

Abbreviations

A in :

eigenfunction amplitude

a :

bar thickness

B in :

eigenfunction amplitude

b 1, b 2 :

constants

C i :

specific heat

D :

stored elastic energy density per unit volume

ΔD :

mechanical energy dissipated per unit volume

E i :

Young’s modulus

\( \tilde G\left( {\gamma _{n} } \right) \) :

normalized variable

g i :

heat generation

\( \tilde H_{n} \) :

normalized variable

h :

thin layer thickness

i :

layer index

In:

Indium

K :

dimensionless ratio of the thermal conductivity

k i :

thermal conductivity

\( \tilde N\left( {\gamma _{n} } \right) \) :

normalized variable

n :

series index

Q :

quality factor

\( \tilde R_{n} \) :

normalized variable

r α :

dimensionless ratio of the thermal diffusivity

SiO2 :

Silicon native-oxide

SCS:

Single crystal Silicon

s P :

entropy produced per unit volume

T i :

temperature

\( \tilde T_{i} \) :

normalized temperature

T 0 :

ambient temperature

t :

time

\( \tilde t \) :

normalized time

Z i :

Zener coefficient

z :

vertical coordinate

\( \tilde z \) :

normalized vertical coordinate

ε xx :

time-harmonic uniaxial strain

ε 0 :

strain amplitude

\( \sigma _{{kk}}^{i} \) :

hydrostatic stress

ω :

harmonic frequency of the strain

α i :

thermal expansion coefficient

\( \alpha _{i}^{*} \) :

thermal diffusivity

ρ i :

density

β n :

eigenvalues

γ n :

normalized eigenvalue

δ :

thicknesses ratio

Ω2 :

normalized frequency

ϕ in :

eigenfunctions

χ :

normalized parameter

ψ :

volume-averaged specific damping capacity

ψ L :

local specific damping capacity

ψ c :

converged value of damping

ψ 0 :

normalized parameter

References

  • Ashby MF (1999) Materials selection in mechanical design. Butterworth-Heinemann, Oxford

    Google Scholar 

  • Bejan A (1988) Advanced engineering thermodynamics. Wiley, New York

    Google Scholar 

  • Bishop JE, Kinra VK (1993) Thermoelastic damping of a laminated beam in flexure and extension. J Reinf Plast Compos 12:210–226. doi:10.1177/073168449301200207

    Article  Google Scholar 

  • Cahill DG, Ford WK, Goodson KE, Mahan GD, Majumdar A, Maris HJ, Merlin R, Phillpot SR (2003) Nanoscale thermal transport. J Appl Phys 93:793–818. doi:10.1063/1.1524305

    Article  Google Scholar 

  • Campbell DS (1970) Mechanical properties of thin films. In: Maissel LI, Glang R (eds) Handbook of thin film technology, McGraw-Hill, New York

  • Chen G (1997) Size and interface effects on thermal conductivity of superlattices and periodic thin-film structures. J Heat Transf Trans ASME 119:220–229. doi:10.1115/1.2824212

    Article  Google Scholar 

  • Choi SH, Maruyama S (2005) Thermal boundary resistance at an epitaxially perfect interface of thin films. Int J Therm Sci 44:547–558. doi:10.1016/j.ijthermalsci.2004.12.006

    Article  Google Scholar 

  • Dean JA (1992) Lange’s handbook of chemistry, 14th edn. McGraw-Hill, New York

    Google Scholar 

  • Duwel A, Gorman J, Weinstein M, Borenstein J, Ward P (2003) Experimental study of thermoelastic damping in MEMS gyros. Sens Actuators A Phys 103:70–75. doi:10.1016/S0924-4247(02)00318-7

    Article  Google Scholar 

  • Gad-el-Hak M (2002) The MEMS handbook. CRC Press, Boca Raton

    MATH  Google Scholar 

  • James AM, Lord MP (1992) Macmillan’s chemical and physical data. Macmillan, London

    Google Scholar 

  • Kaye GWC, Laby TH (1993) Tables of physical and chemical constants, 15th edn. Longman, London

    Google Scholar 

  • Kinra VK, Milligan KB (1994) A 2nd-law analysis of thermoelastic damping. J Appl Mech 61:71–76. doi:10.1115/1.2901424

    Article  Google Scholar 

  • Li FH, Balazs MK, Anderson S (2005) Effects of ambient and dissolved oxygen concentration in ultrapure water on initial growth of native oxide on a silicon (100) surface. J Electrochem Soc 152:G669–G673. doi:10.1149/1.1946487

    Article  Google Scholar 

  • Lide DR (2005–2006) Chemical rubber company handbook of chemistry and physics, 86th edn. CRC Press, Boca Raton

    Google Scholar 

  • Lifshitz R, Roukes ML (2000) Thermoelastic damping in micro- and nanomechanical systems. Phys Rev B 61:5600–5609. doi:10.1103/PhysRevB.61.5600

    Article  Google Scholar 

  • Lukes JR, Li DY, Liang XG, Tien CL (2000) Molecular dynamics study of solid thin-film thermal conductivity. J Heat Transf Trans ASME 122:536–543. doi:10.1115/1.1288405

    Article  Google Scholar 

  • Madou MJ (2002) Fundamentals of microfabrication, 2nd edn. CRC Press, Boca Raton

    Google Scholar 

  • Mahameed R, Elata D (2005) Shield layers for reducing thermoelastic damping in Silicon resonators. In: Proceedings of Eurosensors XIX, Barcelona, Spain

  • Maluf N (2000) An introduction to microelectromechanical systems engineering. Artech House, Boston

  • Nayfeh AH, Younis MI (2004) Modeling and simulations of thermoelastic damping in microplates. J Micromech Microeng 14:1711–1717. doi:10.1088/0960-1317/14/12/016

    Article  Google Scholar 

  • Nayfeh AH, Younis MI, Abdel-Rahman EM (2005) Reduced-order models for MEMS applications. Nonlinear Dynamics 41:211–236. doi:10.1007/s11071-005-2809-9

    Article  MATH  MathSciNet  Google Scholar 

  • Ozisik MN (1980) Heat conduction. Wiley, New York

    Google Scholar 

  • Roszhart TV (1990) The effect of thermoelastic internal friction on the Q of micromachined silicon resonators. In: Proceedings of solid-state sensor actuator workshop, Hilton Head, SC

  • Samsonov GV (1973) The oxide handbook. Plenum Press, New York

  • Vengallatore S (2005) Analysis of thermoelastic damping in laminated composite micromechanical beam resonators. J Micromech Microeng 15:2398–2404. doi:10.1088/0960-1317/15/12/023

    Article  Google Scholar 

  • Zener C (1937) Internal friction in solids: I. Theory of internal friction in reeds. Phys Rev 52:230–235. doi:10.1103/PhysRev.52.230

    Article  Google Scholar 

  • Zener C (1938) Internal friction in solids: II. General theory of thermoelastic internal friction. Phys Rev 53:90–99. doi:10.1103/PhysRev.53.90

    Article  Google Scholar 

  • Zener C (1939) Intercrystalline thermal currents as a source of internal friction. Phys Rev 56:343–349. doi:10.1103/PhysRev.56.343

    Article  Google Scholar 

Download references

Acknowledgments

The author Rashed Mahameed acknowledges the support of the Israel Ministry of Science and Technology.

Author information

Authors and Affiliations

  1. Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, 32000, Haifa, Israel

    Rashed Mahameed & David Elata

Authors
  1. Rashed Mahameed
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David Elata
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rashed Mahameed.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mahameed, R., Elata, D. Shield-layers for reducing thermoelastic damping in resonating Silicon bars. Microsyst Technol 15, 323–331 (2009). https://doi.org/10.1007/s00542-008-0667-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00542-008-0667-3

Keywords

Search

Navigation