3. Fundamentals of fabrication technology of air-gap based optical filters
3.6. Process control
3.6.2. Strain / Stress indicator structures
For the operation of the air-gap filters, a high position accuracy of the freestanding membranes is essential. Therefore, the strain / stress control and evaluation of the layers is important. Appropriate structures to determine the strain / stress are shown in the fol-lowing. The feedback obtained by the structures can be used to improve the device de-sign, the deposition process as well as the general fabrication process flow.
Generally, the total stress5 σtotal in a thin layer can be written in a polynomial form [Fan96]:
0
2 k
total k
k f
y σ ∞ σ t
=
=
∑
Eq. 3-9where y ∈ ( -tf / 2, tf / 2 ) represents the coordinate across the film thickness tf. The film central film plane is chosen as origin. In a first approximation, the higher order terms can be neglected and the total stress can be calculated from the mean stress σmean and the gradient stress σgrad:
2
total mean grad f
y σ ≈σ +σ t
Eq. 3-10
Since the deposition is often performed at higher temperatures, the mean stress can be caused by a mismatch of thermal expansion coefficients of neighbouring layers or be-tween layer and substrate. Reasons for the existence of the gradient stress can be diffu-sion through the layer or substrate, and interstitial as well as substitutional defects.
In general, the stress is called compressive for σmean < 0. Compressive stressed layers tend to expand if they are released. Contrarily, a tensile stress (σmean > 0) causes a con-traction of the layers after the release. Furthermore, the films can have a positive or negative stress gradient resulting in a bending of released structures. In Fig. 3-10, the different stress properties are visualised.
Generally, the stress of layers deposited on top of wafers can be determined by a meas-urement of the wafer bending (see for example [Seg80]). However, this measmeas-urement method requires complete wafers or large pieces of wafers, and it is essential that the substrates are initially flat. The obtained result is a global, average stress value. Alterna-tively, film stress can be determined by surface micromachined test structures. Many suggestions for stress evaluation structures, for example cantilevers, double clamped suspensions or rotatable indicators, can be found in the literature [Guc92, Elb97, Eri97, Win01]. The freestanding structures are fabricated simultaneously with the devices.
5 In the following, the considerations are focused on the stress of the structures. For elastic deformations,
Deflections or deformations evaluated by surface measurements are used for the deter-mination of the stress. Since the structures are very small and since they can be posi-tioned at different places distributed over the wafer, additional information about the film stress distribution can be obtained. In the following, some of the implemented test structures are explained. A complete overview can be found in chapter 10.2.
a) c)
compressive material 1 material 2 material 3
negative gradient
positive gradient
σmean<0 σgrad<0
σgrad>0 σmean>0
tensile
b) d)
Fig. 3-10: Assumed that 3 materials have different base units of extension, and a compound of these ma-terials is formed. Proposed that only the dimensions of the central layer can be changed (for example due to comparably large vertical extensions of the surrounding layers), the following stress is created in this layer: a) compressive stress in material 3 due to the embedding of two layers of material 2, b) tensile stress in material 1, c) negative gradient stress in material 2, d) positive gradient stress in material 2
Simple cantilevers are often applied for the stress determination. In [Fan96], a detailed evaluation of the bending of cantilevers is shown. In principle, the mean as well as the gradient stress can be detected by cantilevers. If the cantilevers are attached only at a single side, a mean stress causes a deviation of the cantilever at the position where the cantilever is released. If the film contains tensile stress, the cantilever will bend up-wards. Contrarily, the cantilever bends downwards for compressive stress. The gradient stress is visible by the curvature of the cantilever. Deformed cantilevers representing different mean and gradient stress cases can be seen in Fig. 3-11.
Possibly, an accurate stress measurement for negative stress values is restricted due to the bending of the cantilevers towards the substrate. If the mean stress is very small and can be neglected, the gradient stress can be calculated according to [Eri97] by:
2
2 1
grad
c c
d E
dt l
σ δ
= ν
− Eq. 3-11
where lc and tc are the length and thickness of the cantilever, respectively. δ is the de-flection of the cantilever tip due to the bending and can be determined using a white
light interferometer. E is the elastic module and ν the Poisson's ratio. If the cantilever has an additional tilt, i.e. a mean stress exist, the more accurate calculations of [Fan96]
have to be applied. The fabricated structures consisted of cantilever of different lengths.
To observe directional effects on the stress, some of the cantilevers were aligned in
<100> and others in <110> direction. In addition, test structure having stlike ar-ranged cantilevers were implemented.
σmean > 0, σgrad < 0 σmean > 0, σgrad > 0
σmean < 0, σgrad < 0 σmean < 0, σgrad > 0
Fig. 3-11: Determination of mean and gradient stress by evaluation of deformed cantilevers after release (according to [Fan96])
To achieve a more accurate determination of the mean stress, additional rotatable pointer structures as proposed by [Eri97] were fabricated. The structures consist of two symmetrically arranged parts (see Fig. 3-12). Each part has two supporting posts, which are connected by freestanding actuator suspensions.
The mean stress can be calculated by:
mean 1
a i
E d
σ l l δ
= ν
− Eq. 3-12
where E is the elastic module and ν the Poisson's ratio. la is the length of an actuator suspension, li corresponds to the length of an indicator and d is defined as distance
be-tween the hinge of the actuator beam and the centre of rotation. δ is the distance be-tween the two indicator tips and can be measured using a microscope (see Fig. 3-12).
la
li
d
Fig. 3-12: Labelled schematic sketch of the rotatable pointer structure
The mean stress causes an expansion or contraction of the actuator suspensions after they are released. If the actuator suspensions have compressive stress, the two indicator beams, both connected by narrow hinges to the actuators, will rotate counter clockwise after the release as shown in Fig. 3-13a). Without any mean stress, the ends of the two indicators will face each other (as seen in Fig. 3-13b)). Tensile stress causes a contrac-tion of the suspensions and, consequently, a clockwise rotacontrac-tion of the indicators (Fig.
3-13c)). Due to the rotation of both indicators, the sensitivity of the measurement is doubled. Since the magnitude of the stress is represented by the magnitude of the de-flection, a quantitative determination of the stress can be obtained.
The stress observation by rotatable pointer structures is restricted to gradient stress free films. In addition to the pointer test structures, double clamped suspensions are imple-mented for the evaluation of compressive stressed layers. Further theoretical details can be found in [Guc92]. For the semiconductor-based filters, the test structures were fabri-cated twice, using upper and lower layers to measure the stress of the p- and n-doped layers separately.
a) σmean < 0, σgrad = 0
b) σmean = 0, σgrad = 0
c) σmean > 0, σgrad = 0
Fig. 3-13: Indicators used to determine mean stress. Underetched rotatable pointer structures (after [Eri97, Chi00]) show a) compressive stress, b) no mean stress, c) tensile stress.