薄膜力学-5-应力与曲率

[
]
= C02 + 2C01ε − 2C11κ + C00ε 2 − 2C10εκ + C 20κ 2
where
Cmn = ∫− hs / 2
hs / 2 + h f
M ( z )z [ε
m
f
( z )] dz
n
κ=
∂U ∂U = =0 ∂κ ∂ε
C11C00 − C01C10 2 C20C00 − C10
EM 397: Thin Film Mechanics
V. Film stress and substrate curvature
Nanshu Lu The University of Texas at Austin Fall 2013
Stoney formula
Assume hf << hs so that the film contribute negligibly to the overall stiffness. f f The membrane force f can be related to a residual stress or an excess surface stress.
Multilayer films
Mulitlayer films are common in engineering applications, for example, to achieve multi-functionality and reliability. The layers (films) are deposited sequentially, with different material properties and stresses. Question 1: What is the curvature subjected to stresses in multilayer films? Question 2: Can the stress in each film be measured by the wafer curvature method?
σ
a Neutral plane: Bending moment:
1 (1 + α )αβ a = hs 1 + 2 1 + αβ
σ
α=
hf hs
β=
Ef Es
1 1+α 1 M = σh f hs + h f − a = σh f hs 2 1 + αβ 2
κ0 =
6σh f
Ehቤተ መጻሕፍቲ ባይዱ
2 s s
is the curvature predicted by Stoney formula.
The error is less than 1% if hs/hf > 300, assuming Es = Ef .
When the film is relatively thick, the stress in the film is reduced due to bending, and the substrate is also stressed in this case. Examples include thermostatic bimetals and some MEMS structures.
Measuring film stress
Stoney formula: Plane stress: Plane strain: Equi-biaxial: (circular wafer)
M s = Es Es Ms = 2 1 −ν s Es Ms = 1 −ν s
6f κ= M s hs2
The curvature is independent of material properties of the film.
2 s s
• • • •
Mechanical method Capacitance method X-ray diffraction Optical methods
– – – – Scanning laser method Multi-beam optical stress sensor Grid reflection method Coherent gradient sensor (CGS)
CGS: non-uniform film stress by full-field curvature measurements?
Thick film or thin wafer
• Relax the assumption hf << hs • Consider a bilayer composite beam
C11C10 − C01C20 ε= 2 C20C00 − C10
Same results can be obtained by an equilibrium approach.
Thin multilayer films
h f = ∑ hi << hs
C10 ≈ 0
6σ i hi κ ≈ ∑κi = ∑ M s hs2
Superlattices: periodic mulitlayer films
• Quantum well lasers • Optical waveguides • Thermoelectric devices
Compositionally graded films
• Continuous variation in composition, microstructure, and residual strain through the thickness of the film
2 f M h Average film stress: σ= = s s κ hf 6h f
Measurements of wafer curvature
d w( r ) R= = 2 κ dr 1
2 −1
M h 1 1 σ= − 6h f R R0
R
h / 2+ h
Minimize the total strain energy to determine κ, ε1, and ε2. Note that the curvature now depends on the lateral dimension R.
Nonlinear spherical bending
M s hs2 σi = 6hi
1 1 − R R i −1 i
Geometrically nonlinear deformation
• The linear solution assumes axi-symmetrical bending with uniform curvature (spherical bending) • When deflection is large compared to the substrate thickness, geometrically nonlinear effects must be considered. • Materials remain linear elastic, and the strain is still small, but with relatively large rotation.
1 M = fhs 2
M 6f κ= = 2 Es I s Es hs
Assumptions and limitations
• Both hs and hf are laterally uniform and small compared to the lateral dimension; • hf << hs • The substrate material is homogeneous, isotropic, and linearly elastic • The film material is transversely isotropic (could be inelastic) • The film stress is laterally uniform • Edge effect is negligible • Strains and rotations are infinitesimal (linear thin-plate theory)
C 20 ≈ 1 3 M s hs 12
C11 ≈
hs 2
∑ M i ε i hi
i
• The total curvature equals the sum of the curvatures induced by each layer individually. • Stress in each film can then be measured from the curvature change during deposition or etching.
2
1 2 2 Strain energy density: U (r , z ) = M (z )ε1 − zκ + 3 ε + κ r + ε ( z ) 2 f 2
Total strain energy:
W = 2π ∫0 ∫− hss / 2 f U (r , z )dzdr
Freund, JMPS 48, 1159-1174 (2000).
S=K
S = K [1 + (1 − ν s ) K 2 ]
The linear solution becomes invalid for S > 0.3.
Curvature:
6σh f M 1+α κ= = Es I s + E f I f Es hs2 1 + 4αβ + 6α 2 β + 4α 3 β + α 4 β 2