EE C245 – ME C218 Fall 2003 Lecture 6

EE C245 - ME C218Introduction to MEMS Design

Fall 2003

Roger Howe and Thara SrinivasanLecture 6 Mechanics of Materials

2EE C245 – ME C218 Fall 2003 Lecture 6

Today’s Lecture• Stress, strain, etc. � isotropic materials• Thin films: thermal stress, residual stress, and stress gradients,

internal dissipation• MEMS material properties and performance metrics

See also http://www.memsnet.org/material/

• Elastic constants for anisotropic materials (e.g. Si)

• Reading:Senturia, Chapter 8Yasumura, K. Y., et al, “Quality factors in micron- and submicron-thick

cantilevers,” Journal of Microelectromechanical Systems, 9, 2000, pp. 117-125.

Candler, R., et al, “Investigation of energy loss mechanisms in micromechanical resonators,” 12th Int. Conf. on Solid-StateSensors, Actuators, and Microsystems (Transducers ’03), Boston, Mass., June 8-12, 2003, pp. 360-363.

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Normal Stress = Force per unit Area

A

FF

σ = F / A units = [N/m2] = Pa

What happens?

L

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Strain = fractional change in length

FF

LL’

Define ε = (L’ – L)/L units: [ ]

Strain is found to be proportional to stress(for “small” stresses at “lower” temperatures):

ε = σ / E

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Poisson’s Ratio

FxFx W’W

εy = (W’ – W) / W

εy = - ν εx ν = Poisson’s Ratio

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Forces || to Surfaces � Shear Stress

F

FNote: compensating forces are appliedto the vertical faces to avoid a net torque!

θ

τ = F /A = shear stress G = shear modulus

A

θ = τ / G

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Normal and Shear Componentsfor an Isotropic Solid

• Isotropic = same in all directions• Add in off-axis strains from normal stresses in other

directions

εx = (1/E)[σx- ν (σy + σz)]

Poisson strains from off x-axis stress

εy = (1/E)[σy- ν (σx + σz)]

εz = (1/E)[σz- ν (σx + σy)]

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Important Special Case: Plane Stress• Common case: very thin film coating a thin, relatively

rigid substrate (e.g., a silicon wafer)

600 μm

Top surface is unstressed � σz = 0

z

εx = (1/E)[σx- ν (σy + 0)] εy = (1/E)[σy- ν (σx + 0)]

Symmetry in x-y plane � σx = σy = σ

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Plane Stress Case (cont.)• In-plane strain components εx = εy = ε

ε = (1/E)[σ- νσ] = σ (1- ν )/E

• Bi-axial modulus

E’ = E / (1- ν )

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Linear Thermal Expansion• As temperature increases, most solids expand in volume �

define the linear thermal expansion coefficient by

dTd x

Tεα =

• Why important?

Source of strain (due to mismatch between layers … thin film andsubstrate, substrate and package, package and board, etc.)

Can be exploited for thermal-expansion based actuators

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Linear Thermal Expansion: Varies with Temperature

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Thermal Stress• Assume film is deposited stress-free at a temperature Td

Si: αTs = 2.8 x 10-6 K-1

film: αTf

• At room temperature Tr , the film is under a thermal mismatch strain (a “built-in” strain) given by

)()( TT TfTsx Δ−−Δ−= ααε

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Residual Stress in Thin Films• What is measured in a thin film

(e.g., by the induced substrate curvature) �define as the residual stress

• Thermal mismatch stress: well understood• What is not explained by thermal mismatch �

define as the intrinsic stress

• Origins of intrinsic stress: many and varied!

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Measuring (Average) Residual Strain

• Mechanical leverage amplifies the tip deflection of the bottom structure �more sensitive measurement

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As-Deposited Stress in LPCVD Poly-Si and Poly-SiGe Films at Tr

Y.-C. Jeon, T.-J. King, and R. T. Howe,J. Electrochemical Society, 150, H1-H6 (2003)

αT’sarematched!

αT’saren’tmatched –SiGe: 4.7Si: 2.8

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Residual Stress in Poly-SiGe Alloys at Tr

Y.-C. Jeon, T.-J. King, and R. T. Howe,J. Electrochemical Society, 150, H1-H6 (2003)

Interpretation:

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Annealed “Stress-Free” Poly-SiGe Films

Eliminatedintrinsic stress

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Residual Strain GradientsIntrinsic strain can vary through the film thickness,resulting in warping of some structures

Extreme example:as-deposited poly-SiGe

Question:

Was this film tensile orcompressive?

Answer:

SEM: Carrie Low, BSAC

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Understanding the Difference between Average Strain and Strain Gradient

• Case 1. Average strain > 0 (tensile); top of film is more tensile than bottom

• Case 2. Average strain < 0 (compressive); bottom of film is more compressive than top

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Failure by Fracture (or Yielding)

(or Si at T > 900 oC)

(Si at T = 30 oC)

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Internal Dissipation (Losses)• All materials dissipate energy when undergoing cyclic

vibrations• Origins (see papers by Yasumura and Candler)

1. Anharmonic (non-quadratric) interatomic forces2. Mobile defects, especially at grain boundaries3. Surface films (thin oxides) and adsorbed species4. Thermoelastic damping (vibrational modes with volume change)

x(t)

t

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Example MEMS Material Properties

From Mark Spearing, MIT, Future of MEMSWorkshop, Cambridge, England, May 2003

Units:(m/s)2

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Young’s Modulus – Density Plot

Ashby, Mechanics ofMaterials, Pergamon,1992.

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Young’s Modulus and Useful Strength

From Mark Spearing, MIT, Future of MEMSWorkshop, Cambridge, England, May 2003

Stored mechanical energy

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What is a Material’s Useful Strength?

• Yield or fracture stress … ultimate limits• Practical limit: tolerable deviation from linearity of

stress-strain curve• Example: silicon (100) direction

E = Eo(1 + E1ε + E2 ε2)

Eo = 130 GPa, E1 = 0.65, E2 = -4.6result: εmax ˜ 2 x 10-3 (roughly 1/5 of fracture limit)

V. Kaajakari (VTT, Helsinki, Finland),Transducers ’03, Boston, paper 3E102.P

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Young’s Modulus vs. Strength

Ashby, Mechanics ofMaterials, Pergamon,1992.

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Single Crystal Silicon: a Cubic Material

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Silicon Elastic Constants

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Stiffness and Compliance Coefficients

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Young’s Modulus in (001) Plane

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Poisson’s Ratio in (001) Plane

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Design Implications• Young’s modulus variations are often ignored, but

can be a problem for structures in which mode-matching is critical (e.g., vibratory rate gyroscopes)

• M. E. McNie and V. Nayar, “Formation of suspended beams using SOI substrates, and application to the fabrication of a vibratory gyrometer,” PCT WO 00/16014A2, March 23, 2000.“As the skilled person will know, such silicon has anisotropic

properties … varying significantly in a cos 4θ mannerthrough the material. … For a perfectly formed circular ring the effect of the anisotropy is to cause a split in the frequencies of the drive and sense modes.”

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Solution: Mechanical Compensation

M. E. McNie and V. Nayar, “Formation of suspended beams using SOIsubstrates, and application to the fabrication of a vibratory gyrometer,”PCT WO 00/16014A2, March 23, 2000.

Manufacturability?

See K. Najafi, Univ. ofMichigan, ring gyrosusing isotropic poly-Si