A two-step load-deflection procedure applicable to

A two-step load-deflection procedure applicable to extract the Young's modulus and the residual tensile stress of circularly shaped thin-film diaphragms Roman Beigelbeck, Michael Schneider, Johannes Schalko, Achim Bittner, and Ulrich Schmid Citation: Journal of Applied Physics 116, 114905 (2014); doi: 10.1063/1.4895835 View online: http://dx.doi.org/10.1063/1.4895835 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/116/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A resonant method for determining the residual stress and elastic modulus of a thin film Appl. Phys. Lett. 103, 031603 (2013); 10.1063/1.4813843 A method to determine the Young's modulus of thin-film elements assisted by dark-field electron holography Appl. Phys. Lett. 102, 051911 (2013); 10.1063/1.4790617 Effect of surface stress on the stiffness of micro/nanocantilevers: Nanowire elastic modulus measured by nanoscale tensile and vibrational techniques J. Appl. Phys. 113, 013508 (2013); 10.1063/1.4772649 Young’s modulus and density measurements of thin atomic layer deposited films using resonant nanomechanics J. Appl. Phys. 108, 044317 (2010); 10.1063/1.3474987 Residual stress distribution in the direction of the film normal in thin diamond films J. Appl. Phys. 86, 224 (1999); 10.1063/1.370720

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 52.2.249.46 On: Thu, 24 Sep 2015 18:25:37

JOURNAL OF APPLIED PHYSICS 116, 114905 (2014)

A two-step load-deflection procedure applicable to extract the Young’s modulus and the residual tensile stress of circularly shaped thin-film diaphragms Roman Beigelbeck,1,a) Michael Schneider,2 Johannes Schalko,2 Achim Bittner,2 and Ulrich Schmid2 1

Center for Integrated Sensor Systems, Danube University Krems, A-2700 Wiener Neustadt, Austria Institute of Sensor and Actuator Systems, Vienna University of Technology, A-1040 Vienna, Austria

2

(Received 28 June 2014; accepted 5 September 2014; published online 19 September 2014) We report on a novel two-step load-deflection (LD) formula and technique that enables an accurate extraction of the Young’s modulus and the residual tensile stress from LD measurements on circularly shaped thin-film diaphragms. This LD relationship is derived from an adaptation of Timoshenko’s plate bending theory, where the in-plane and out-of-plane deflections are approximated by series expansions. Utilizing the minimum total potential energy principle yields an infinitedimensional system of equations which is solved analytically resulting in a compact closed-form solution. In the appendant measurement procedure, the whole transverse bending characteristic of the diaphragm as a function of the radial coordinate is recorded for different pressure loads and introduced into the novel LD equation in order to determine the elastomechanical parameters of interest. The flexibility of this approach is demonstrated by ascertaining the Young’s modulus and the residual tensile stress of two disparate diaphragm materials made of either micromachined silicon or microfilC 2014 AIP Publishing LLC. tered buckypapers composed of carbon nanotube compounds. V [http://dx.doi.org/10.1063/1.4895835]

I. INTRODUCTION

The working principle of many mechanical microtransducers relies on static or dynamic out-of-plane deflections of implemented thin-film diaphragms.1,2 Designing such devices therefore requires accurate knowledge of the in-plane Young’s modulus and the internal residual stress of involved thin films.3–6 However, the values of these elastic coefficients typically depend on the fabrication process of the diaphragms and often considerably differ from those known for bulk matter.7–9 Even slight variations in the material composition, caused by minor changes in process parameters (e.g., deposition conditions such as temperature, pressure, or gas flow rate) or in subsequent processing steps (e.g., ion implantation or annealing), may influence the elastomechanical properties significantly.10–12 At present, it is impossible to predict these properties for an entire manufacturing sequence exclusively by theory. In order to still be able to fabricate diaphragms with well-defined elastic coefficients, a precise knowledge of the impact of the whole production process flow on Young’s modulus E and residual tensile stress r0 of the thin film is mandatory. Unfortunately, well-established measurement methods for testing elastic coefficients of bulk materials cannot be applied directly to thin films owing to difficulties associated with their small dimensions. Consequently, several custom methods optimized for thin film investigations have been developed during the past decades.6 The most prominent ones are based on static deflection measurements of thin-film specimens such as substrate curvature a)

Electronic mail: [email protected]

0021-8979/2014/116(11)/114905/13/$30.00

measurements,13,14 nanoindentation techniques,15–17 microtensile experiments,18–21 and bulge tests.22–24 Dynamic measurement principles utilizing resonance frequency determination of vibrating thin films are also in use.25–27 Other common methods rely on X-ray diffractometry28,29 and Raman spectroscopy.30,31 All these methods have their pros and cons. Examples for typical limitations are: Wafer curvature measurements may exhibit significant errors if the rigidity of the investigated thin film is similar to that of the substrate or if the structure undergoes large deformation.32 Standard bulge tests require precise acquisition of the center deflection as well as accurate knowledge of the specimen dimension and the type of the support of the thin film.33 X-ray diffraction techniques suffer from difficulties when applied to noncrystalline materials.34 Finally, Raman spectroscopy can only be applied to a limited class of materials.30 Moreover, stress information obtained by Raman spectroscopy or nanoindentation is highly localized making them moderately suited for not well-defined surfaces like those of carbon nanotube (CNT) buckypapers which comprise of highly irregularly arranged bundles of carbon nanotubes.30,35 In terms of dynamic measurement principles, a paper has been recently published where the resonance frequencies of the first two symmetric vibration modes of a circular thinfilm diaphragm were measured and compared to an analytically derived equation in order to determine E and r0.36 This method is simple and has versatile potential when applied to structures with a well-known mass density like the investigated epitaxial 3C-SiC diaphragms.36 However, it suffers from the same drawback as other resonant techniques because it requires knowledge of the mass density of the thin

116, 114905-1

C 2014 AIP Publishing LLC V

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 52.2.249.46 On: Thu, 24 Sep 2015 18:25:37

114905-2

Beigelbeck et al.

film which can differ considerably from bulk properties and is thus often also unknown.25–27 This paper focuses solely on bulge testing because it is the most versatile method. In the context of elastomechanical thin film probing, the bulge test was first reported in 1959 by Beams.37 Since then this method has evolved continuously38–42 and has been applied to a large variety of different thin-film materials.43–47 Nowadays, the bulge test, also called load-deflection (LD) method, is a very common and convenient procedure to extract Young’s modulus and residual tensile stress from measurements of the maximum outof-plane deflection w0 of laterally suspended diaphragms in response to a uniformly distributed transverse load p. This technique enables simultaneous determination of E and r0 by fitting a theoretically derived LD characteristic to experimentally obtained data. Evidently, proper knowledge of such a theoretical relationship, either in (semi)numerical48 or preferably analytical form, is necessary to achieve accurate values. The variety of utilized analytical methods spans approximate solutions of the nonlinear F€oppl–von Karman equations,3,49,50 Fourier expansions,51 plain-strain relationships,43,52 or energy methods.53,54 Depending on the shape of the diaphragm, several models have been developed and published. Particularly, rectangular probes have been extensively studied51,53,55,56 while circular diaphragms have been less frequently covered.54,57 Rectangular diaphragms with a high geometrical aspect ratio offer the advantage that the parameter extraction is less sensitive to uncertainties in the value of the Poisson’s ratio.51 However, depending on general conditions such as restrictions in the available fabrication technology or the envisaged sensor application, utilization of circular diaphragms can be beneficial. In addition, circular arrangements allow for a simple detection of distinctive anisotropic stress distributions where the bending behavior is not rotationally symmetric, which can be easily determined in the measurement. We report on a new analytical LD equation applicable to suspended circularly shaped diaphragms. Contrary to other closed-form relationships, which typically rely on one particular curvature characteristic (e.g., pure bending of ideal plates or membranes51,53,54), this novel formula contains a set of parameters that can be chosen to describe all practically relevant bending curves. This enlarges significantly the number of different diaphragm types which can be studied.

J. Appl. Phys. 116, 114905 (2014)

Furthermore, our ansatz can be seen as an interesting alternative to other analytical state-of-the-art LD relationships, and it can even be used as benchmark for numerical models. The versatility of the presented approach is demonstrated by extracting the Young’s modulus and the residual tensile stress of two disparate diaphragm materials made of either micromachined monocrystalline silicon (Si) or microfiltered buckypapers composed of carbon nanotube compounds. Especially, silicon is well-suited for test purposes owing to its accurately defined mechanical properties. II. THEORY

In the standard version of the bulge test, the centerdeflection w0 of a freestanding, circumferentially supported diaphragm in response to a uniformly distributed (differential) pressure is measured for a wide range of different pressure values p. The measured data are then set in relation to an analytical LD equation p(w0) in order to receive E and r0. We applied an enhanced version of this principle to circularly shaped diaphragms. In contrast to the standard approach, the whole transverse bending characteristic w(r) as a function of the radial coordinate r is recorded for each pvalue and introduced into a novel two-step LD procedure. A. Rotationally symmetric model

The following LD analysis is based on the rotationally symmetric model shown in Fig. 1 of a single layer thin-film diaphragm. Importantly, the simplification to single layer structures imposes no restrictions on the presented method because all results can be straightforwardly extended to multilayer films.6 All quantities may be referred to the depicted cylindrical coordinate system. A well-established ansatz for the LD equation reads51 pðw0 Þ ¼ C1

r0 hw0 Ehw30 þ C  ; ð Þ 2 R2 R4

(1)

where w0 is the maximum deflection measured at the center of the diaphragm, p is the (differential) pressure load, 2R is the diaphragm diameter, h its thickness, r0 its residual tensile stress, E its flexural Young’s modulus, and  its in-plane Poisson’s ratio. Derivation of C1 and C2() requires known distributions of the radial u(r) and the transverse w(r)

FIG. 1. Cross section and top view of the utilized rotationally symmetric model featuring a linear isotropic, circumferentially supported, circular diaphragm with a diameter 2R, a thickness h, a Young’s modulus E, and a residual tensile stress r0. w(r) is the static transverse response of the diaphragm to a uniformly distributed (differential) pressure load p. Its maximum deflection, located at the center, is denoted by w0 ¼ w(0). The undeflected neutral plane coincides with z ¼ 0.

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 52.2.249.46 On: Thu, 24 Sep 2015 18:25:37

114905-3

Beigelbeck et al.

J. Appl. Phys. 116, 114905 (2014)

diaphragm displacement and therefore depends on the nature of the bending characteristic. Commonly used analytical models restrict w(r) to one distinctive bending behavior, e.g., pure plate "   2 #2 r (2) w ðr Þ ¼ w 0 1  R or pure membrane "

  2# r wðrÞ ¼ w0 1  R

(3)

bending. For example, Loy et al.54 approximated the radial displacement associated to Eq. (2) by means of a seventhorder polynomial and applied the minimum total potential energy principle to further obtain C1 and C2(). Such simplified approaches are no longer adequate if the measured transverse bending characteristics deviates significantly from Eqs. (2) and (3). To overcome this drawback, we approximate the transverse wð.; aÞ and the radial deflection uð.; bÞ using the functional series expansions ~ aÞ; wð.; aÞ ¼ w0 wð.;

~ aÞ ¼ wð.;

1 X

a2k fk ð.Þ ;

(4a)

k¼0

uð.; bÞ ¼

w20 u~ð.; bÞ; R

u~ð.; bÞ ¼

1 X

b2s gs ð.Þ ;

(4b)

b ¼ ½b0 ; b2 ; b4 ; …

(5)

are two, in general, infinite-dimensional tuples composed of expansion coefficients to be determined. Arguments (enclosed by brackets) on the left and right side of the semicolon indicate independent variables and parameters of interest, respectively. Tilde symbols signify non-dimensional, scaled ~ aÞ ¼ wð.; aÞ=w0  1 entails a0  1. quantities. Notably, wð.; The polynomial functions in Eqs. (4) are chosen to satisfy the boundary condition for a vanishing displacement at the supported edge (. ¼ 1) fk ð.Þ ¼ ð1  .2 Þ.2k ;

Importantly, the unknown coefficients C1 ðaÞ; C2 ða; Þ, and C3 ða; Þ depend on a and accordingly on the bending curvature of the diaphragm which is itself a function of the applied pressures, i.e., the coefficients depend implicitly on the applied pressure. The influence of C3 ða; Þ is marginal if w0  h, i.e., when the maximum deflection is much larger than the diaphragm thickness.

s¼0

where . ¼ r/R denotes the normalized radial coordinate and a ¼ ½a0 ; a2 ; a4 ; …;

~ FIG. 2. Four examples of normalized transverse deflections wð.Þ describable by Eqs. (4)–(6). Note that the tangent at the supported edge (. ¼ 1) is not necessarily horizontally. The shapes in case of pure plate and membrane ~ ~ ¼ 1  .2 , respectively. bending are given by wð.Þ ¼ ð1  .2 Þ2 and wð.Þ Characteristics in between are located in the so-called plate-to-membrane transition area.

gs ð.Þ ¼ .ð1  .2 Þ.2s

(6)

B. LD equation coefficients

The extensive auxiliary calculation of C1 ðaÞ; C2 ða; Þ, and C3 ða; Þ can be found in Appendix A in detail. In this section, we will just outline the basic approach. Starting from the generic Eqs. (4), we calculated the corresponding total potential energy of the diaphragm and applied the minimum total energy principle resulting in an infinitedimensional system of equations for the unknown coefficients b2s in Eq. (4b). This linear system (specified in Eq. (A10)) exhibits the remarkable property58 that if the number of terms in Eq. (4a) used to approximate the trans~ aÞ is limited to K verse deflection wð.; a ¼ ½a0 ; …; a2K ; |{z} 0…  ;

(8)

all zero

of wð1; aÞ  0 and uð1; bÞ  0. Moreover, the necessary precondition uð0; bÞ  0 is already fulfilled. Equations (4)–(6) enable analytical syntheses of virtually all practically relevant, sufficiently smooth curvatures of circumferentially supported diaphragms (Fig. 2). This can be seen as remarkable advantage compared to other approaches relying on circular diaphragms. Expansions (4)–(6) necessitate an improved version of Eq. (1) which we formulate as r0 hw0 pðw0 ; aÞ ¼ C1 ðaÞ R2 "



h þ C2 ða;  Þ þ C3 ða;  Þ w0

 2#

Ehw30 : R4

(7)

then an exact, closed-form solution for the b2s-coefficients in Eq. (4b) of the form bsol ¼ ½bsol0 ; …; bsol2S ; |{z} 0… 

(9)

all zero

exists, i.e., the number of terms required to describe the radial deflection uð.; bÞ is restricted to S ¼ 2K. Thus, solving the system for bsol can be accomplished by a finite number of matrix operations for any desired approximation order K < 1. This proves advantageous because, with respect to our envisaged measurement procedure, where the measured transverse bending characteristic wm (.) must be fitted to Eq. (4a), K is always limited from a practical view point. Once

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 52.2.249.46 On: Thu, 24 Sep 2015 18:25:37

114905-4

Beigelbeck et al.

J. Appl. Phys. 116, 114905 (2014)

bsol is calculated, C1 ðaÞ; C2 ða; Þ, and C3 ða; Þ can be easily obtained by ordinary division (see Eqs. (A14)). K ¼ 3 is

reasonable for almost all practically relevant bending curvatures (see Appendix B) yielding

   8 35a22 þ 7ð7a4 þ 5ða6 þ 2ÞÞa2 þ 21a24 þ a4 ð34a6 þ 35Þ þ 3 5a26 þ 7a6 þ 35 ; C 1 ða Þ ¼ 7ð10a2 þ 5a4 þ 3a6 þ 30Þ

C2 ða;  Þ ¼

(10a)

 1  f546½275 21ð7 þ  Þ  168a2 þ 14ð11 þ  Þa22 315315ð1 þ  Þð30 þ 10a2 þ 5a4 þ 3a6 Þ    56a32 þ ð15 þ  Þa42 Þ þ 110a4 21ð9 þ  Þ þ 7ð67 þ 5 Þa2  ð265 þ  Þa22 þ ð103 þ 5 Þa32     þ 11a24 5ð353 þ 31 Þ  10ð181 þ  Þa2 þ ð1153 þ 47 Þa22 þ 22a34 5ð41 þ  Þ þ ð305 þ 11 Þa2    þ 3a44 ð479 þ 17 Þ þ 364a6 55 63ð5 þ  Þ þ 9ð103 þ 5 Þa2  3ð181 þ  Þa22 þ 7ð31 þ  Þa32   þ 33 15ð89 þ 7 Þ  2ð661 þ  Þa2 þ ð857 þ 27 Þa22 a4 þ3ð5005  7773a2 þ  ð121 þ 227a2 ÞÞa24    þ 4ð1736 þ 53 Þa34  þ 6a26 91 33ð371 þ 29 Þ þ 66ð171 þ  Þa2 þ 5ð1471 þ 41 Þa22 þ 182ð3795  99  6205a2 þ 155a2 Þa4 þ 22ð23873 þ 655 Þa24  þ 36a36 ½1001ð29 þ  Þ þ 35ð1461 þ 31 Þa2 þ 2ð24943 þ 633 Þa4  þ 90a46 ð4397 þ 107 Þg;

C3 ða;  Þ ¼

(10b)

    8 105 þ 385a22 þ 210a4 þ 637a24 þ 210a6 8 1470a4 a6 þ 897a26 þ 14a2 ð15 þ 65a4 þ 69a6 Þ þ 21ð1   2 Þð30 þ 10a2 þ 5a4 þ 3a6 Þ 21ð1   2 Þð30 þ 10a2 þ 5a4 þ 3a6 Þ

þ

105 ð1 þ a2 þ a4 þ a6 Þ2 : 21ð1   2 Þð30 þ 10a2 þ 5a4 þ 3a6 Þ

(10c)

These equations also imply the special cases for pure plate and pure membrane behavior listed in Table I. III. MEASUREMENT PROCEDURE

The measurement procedure comprises four steps, which can be straightforwardly implemented in LabVIEWTM facilitating autonomous measurement instrument control, data acquisition, data processing, and output of E and r0. Throughout this article, the index “m” denotes measured quantities. 1. A uniformly distributed (differential) pressure pm is subjected to the diaphragm and the responding transverse deflection curvature wm (.) is measured over the whole diameter distance. On this occasion, the symmetry condition (necessary for the validity of our rotationally

symmetric LD model) of the bending curvature regarding the diaphragm center should be checked. The maximum deflection follows from wm0 ¼ wm (0). 2. The first measurement step is repeated for a wide range of different pressures ensuring that the induced wm0-values span the linear- as well as the cubic-dominant domain of Eq. (7). In order to minimize the interdependence of the r0 and E terms in Eq. (7), deflections satisfying (h/wm0)2  1 should be preferred. For every pressure value, the expansion coefficients a are ascertained by fitting w(.) to wm (.) using a Kth-order truncation of Eq. (4a). Here, K must be chosen to achieve a good approximation with a minimum number of polynomial terms. As a rule of thumb, K ¼ 3 should be sufficient59 (see Appendix B). 3. Equations (A14) are solved for the selected K-value yielding expressions for C1 ðaÞ; C2 ða; Þ, and C3 ða; Þ.

TABLE I. Coefficients for pure plate and pure membrane bending in contrast to published relationships. The C2-value provided by Beams is only valid for  ¼ 0.25. C1 ðaÞ

C2 ða; Þ

C3 ða; Þ

[1, 1, 0…]

4

2ð239Þ 21ð1Þ 7 3ð1Þ

16 3ð1 2 Þ 4 3ð1Þ

3.56

a Pure plate Pure membrane

[1, 0, 0…]

4

Beams37 Pan et al.48

— —

4 4

2:67 1:0260:233

0 0

Loy et al.54

—

4

32ð264 562161 035103 523 2 Þ 3 864 861ð1 2 Þ

0

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 52.2.249.46 On: Thu, 24 Sep 2015 18:25:37

114905-5

Beigelbeck et al.

4. After inserting a into C1 ðaÞ; C2 ða; Þ, and C3 ða; Þ and subsequently in Eq. (7), a parameter fit of p(w0) to pm (wm0) extracts the values of E and r0, where the LD fit should be reasonably carried out on a double-logarithmic scale. IV. VERIFICATION BY COMPUTER NUMERICAL MODELING

Comprehensive and reliable validation of a new measurement method for E and r0 of thin films is generally very challenging because reference measurements using established methods are always (more or less) affected by uncertainties. Inaccurately known specimen dimensions, non-ideal boundary conditions induced by undercut, or even hardly quantifiable influences like variations in the material composition of the device under test may additionally deteriorate such reference measurements. To circumvent all these difficulties, we implemented an extensive three-dimensional finite element model (FEM) of a tensile-prestressed thin-film diaphragm. This model relies on the nonlinear theory for large deflections of thin flat plates by August F€oppl and Theodore von Karman60 and was utilized to “simulate” ideal measurement results. It incorporates all properties and assumptions of the analytical model summarized in Fig. 1, except that the limitation to rotationally symmetric analyses has been dropped.61 In order to heuristically validate our theoretically developed LD measurement procedure, we chose a sample diaphragm with a reasonable geometry and prescribed values for the Young’s modulus and residual tensile stress. Next, the computer numerical model was utilized to calculate the bending characteristics w(.) for different pressures. Then, these computational results were treated as measurement data wm (.), and the measurement procedure presented in Sec. III was applied to determine E and r0. Ideally,

FIG. 3. Computer numerical results for the normalized bending characteris~ m ð.Þ of a diaphragm (R ¼ 409.8 lm and h ¼ 1.61 lm) calculated for tics w p ¼ 5 kPa, E ¼ 169 GPa,  ¼ 0.064, and two different tensile prestresses and ~ their corresponding fits wð.Þ. Squares (FEM 1) and circles (FEM 2) label ~ m ð.Þ, respectively. It can be clearly seen r0 ¼ 0 MPa and r0 ¼ 100 MPa of w that the bending characteristic changes with the prestress. Equation (4a) with a ¼ ½1; 0:1170; 0:1588; 0:7028; 0… (fit 1) and a ¼ ½1; 0:1702; 0:3866; 0:9387; 0… (fit 2) enable perfect fits for FEM 1 and FEM 2, respectively. Normalized bending characteristics for pure plate and membrane behavior are indicated for comparison.

J. Appl. Phys. 116, 114905 (2014)

prescribed and measured values must coincide. In real, their deviation can be interpreted as a measure for the quality of our method. Results of an exemplary benchmark for a diaphragm with R ¼ 409.8 lm and h ¼ 1.61 lm are collected in Figs. 3–5. Figure 3 contrasts our approach to popular bending models. We conclude that neither the plate nor the membrane approximation is able to reproduce the calculated shape of the deformed curvature correctly. While the plate model is inappropriate for the prestressed diaphragm (FEM 2), the membrane characteristic fits better, but the result is still unsatisfactory. Furthermore, it can be clearly seen that both popular models are ill-suited for diaphragms with no residual stress (FEM 1). In contrast, our model enables a perfect fit of the FEM results with just a three term approach of Eq. (4a), regardless of the prestress state. Figure 4 depicts three examples of LD characteristics pm (wm0) calculated by FEM and their associated fits p(w0) using Eq. (7). Figure 5 shows grid plots of prescribed and measured E- and r0-values depending on the Poisson’s ratios . The excellent agreement between measured and prescribed data strongly supports the presented two-step LD method. V. MATERIAL SELECTION AND SPECIMEN FABRICATION

With respect to sample measurements, two disparate types of specimens were fabricated involving either a micromachined or a microfiltered diaphragm. The former consists of monocrystalline silicon (Si) while the latter exhibits a buckypaper composed of single-wall CNTs, both were standardly available at our laboratory. Silicon is well-suited for test purposes owing to its accurately defined mechanical properties.62,63 However, silicon is an anisotropic crystalline material whose mechanical properties depend on orientation relative to the crystal lattice with an elastic anisotropy of about 30%.63,64 At first sight, this may violate the prerequisite of rotational symmetry utilized in our analytical model. Fortunately, computer numerical analyses of a circumferentially supported (1 0 0) silicon

FIG. 4. Double-logarithmic plots of FEM-calculated pm (wm0) (labeled by outlined symbols) and fitted p(w0) (indicted by lines) LD responses for a diaphragm featuring R ¼ 409.8 lm, h ¼ 1.61 lm, E ¼ 169 GPa, and  ¼ 0.064. Prescribed and measured prestresses are labeled by (E, r0)-pairs in units GPa and MPa for E and r0, respectively.

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 52.2.249.46 On: Thu, 24 Sep 2015 18:25:37

114905-6

Beigelbeck et al.

J. Appl. Phys. 116, 114905 (2014)

FIG. 6. FEM results for the transverse deflection w(.) of a (1 0 0) silicon diaphragm subjected to three different pressure values. Characteristic values of the diaphragm were chosen to R ¼ 409.8 lm, h ¼ 1.61 lm, and r0 ¼ 100 MPa. Coefficients of the stiffness tensor were taken after Hopcroft et al.63 Maximum and mean (over the radius) relative deviations between the crystal directions [1 1 0] and [1 0 0] are just 4.3% and 0.3%, respectively.14

FIG. 5. Prescribed E- and r0-values (dashed lines) and corresponding measurements (symbols, values in boxes) obtained for a diaphragm with R ¼ 409.8 lm and h ¼ 1.61 lm at different Poisson’s ratios. Theory and experiment show excellent agreement with maximum deviations between prescribed and measured E- and r0-values in the range of 1 GPa and 1 MPa, respectively. (a) Grid plot of (, E) at r0 ¼ 100 MPa for the prescribed Young’s moduli (130, 150, 170, 190) GPa. (b) Grid plot of (, E) at E ¼ 169 GPa for the prescribed residual stresses (50, 100, 150) MPa.

diaphragm using our computer numerical model revealed that the relative deviation from a radially symmetric bending curvature is only in the order of a few percent (Fig. 6). This is a consequence of the rotationally symmetric clamping condition in combination with the effective strain tensor components of a (1 0 0) silicon wafer, which both induce symmetry to the constitutive equations.14 Noteworthy, because of this symmetry, it is possible to define for (1 0 0) silicon wafers a symmetric biaxial modulus B(100)  180.3 GPa.63 Consequently, such diaphragms can be described by our LD model and therefore utilized for test measurements. Fabrication of the silicon diaphragms started from a silicon-on-insulator (SOI) wafer (supplier Ultrasil Corporation) featuring a silicon handle layer, a buried silicon dioxide (SiO2) layer, and a silicon device layer (for the actual thinfilm diaphragm) with thicknesses given by 350 lm, 1 lm, and 1.61 lm, respectively. In order to generate reference points for deflection measurements by white light interferometry, a thin-film dot grid (30 nm thick) was created on the almost translucent diaphragm using standard photolithography with subsequent sputter deposition of aluminum (Al) followed by lift-off patterning. In the same step, a coordinate

raster around the diaphragms as well as scribe marks, both on the device rim, were made. The influence of these metal dots on the mechanical properties of the specimens is negligible due to their small size and low thickness. Next, the photolithographic mask for deep reactive ion etching (DRIE) was patterned on the backside and plasma etching throughout the entire handle wafer was applied (stopping at the SiO2 layer right below the thin-film material). In the following steps, the resist on the backside and the SiO2 at the bottom of the etch trenches were removed by oxygen plasma stripping and diluted hydrofluoric acid (HF), respectively, releasing the circumferentially supported diaphragms (Fig. 7(a)). Finally, the different specimens were separated with a diamond scriber and the resist on the top side was removed by cleaning the samples in acetone. Figure 7(b) shows an optical micrograph of a sample specimen. Carbon nanotube buckypapers (CNTBPs) were selected as second test material for two reasons. First, to examine the applicability of our LD approach regarding further studies on how fabrication variations (e.g., different ultrasonic shaking procedures) influence the elastic parameters. Second, to demonstrate the flexibility of our LD approach because the bending characteristic of CNTBPs turned out to differ considerably from those of silicon. As source material for the CNTBP specimens, we took commercially available single-wall carbon nanotubes manufactured by Carbon Nanotube Incorporated. First step was to weigh the base material. For our geometry, a base unit of 2.4 mg was chosen resulting in a thickness of h  10 lm. Several specimens were fabricated from n times of this base unit. Next, the CNT material was dispersed in dimethylformamide (DMF). After ultrasonic shaking, the material was filled into a filtration ring where on the bottom a filter paper was placed. Subsequently, the CNT fabric was filtered out, desiccated for about one day, and then separated from the filter by etching the filter paper in sulfuric acid (H2SO4). Next, the freestanding, wet CNTBP was kept in deionized water for another day to remove the acid residuals. At least, the

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 52.2.249.46 On: Thu, 24 Sep 2015 18:25:37

114905-7

Beigelbeck et al.

J. Appl. Phys. 116, 114905 (2014)

FIG. 7. (a) Sketch of the specimen embodiment generated by means of the AutoCADTM mask file. The coordinate frame serves as reference for the determination of the bending characteristics. (b) Colorized optical micrograph of the diaphragm’s upper-right corner.

swollen fabric was clamped to an aluminum ring under water. In the course of drying, the CNTBP shrinks and tightens itself. The described fabrication flow is illustrated in Fig. 8. VI. MEASUREMENT SETUPS

Figure 9(a) shows a photograph of the measurement setup utilized to investigate the silicon diaphragms. Here, the supporting frame of the specimen was mounted on top of a small pressure chamber (to load the diaphragm) equipped with two ports. One port was connected to a syringe pump enabling chamber pressures 450 mbar below the ambient value while the other was linked to a JUMO pressure gauge displaying the current pressure state. In order to control and monitor the pressure in the chamber, the step motor of the syringe pump was driven by a computer running a software tool written in LabVIEWTM. The measurement setup for the CNTBPs (Fig. 9(b)) is very similar to the silicon one. Here, the ring with the CNTBP was mounted on top of a pressure cylinder (to

FIG. 8. Fabrication flow of the CNTBPs.

deflect the diaphragm) equipped with three ports. One port (inlet) was connected to a mass flow controller (MFC), another port (outlet) was supplied by a needle valve to vary the pressure at constant flow, and the remaining port was linked to a pressure gauge. Spatial transverse deflection measurements for both diaphragm types were carried out on basis of white light interferometry using a Micro System Analyzer (Polytec MSA-400).

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 52.2.249.46 On: Thu, 24 Sep 2015 18:25:37

114905-8

Beigelbeck et al.

J. Appl. Phys. 116, 114905 (2014)

~ FIG. 11. Measured normalized transverse deflections wð.Þ of the diaphragm ID 14-11 (R ¼ 409.8 lm and h ¼ 1.61 lm) for increasing pressures. The shift from plate towards membrane behavior can be clearly seen. Pure plate and membrane characteristics are indicated as reference. Each characteristic contains 700 measured points.

uncertainties in the measured diaphragm deflections and the thin-film dimensions:65

We applied the measurement procedure described in Sec. III to both diaphragm types, whereas special attention had to be taken on data processing because results obtained by bulge tests are, in general, highly sensitive to

1. In order to single out spurious influences on the measured diaphragm deflection owing to clamping forces (imposed through the mount) and the tilt of the specimen, the displacement of the frame next to the diaphragm support was measured for each imposed pressure value. From these data, a coil correction algorithm based on a 10th-order polynomial approximation was derived and pointwise applied yielding corrected wm (.) values. Additionally, the zero deflection (i.e., wm (.)  0) of the diaphragm at p ¼ 0 was checked. The measurement run for each specimen was repeated several times to minimize random measurement uncertainties. 2. Since the LD response of the diaphragm governed by Eq. (7) varies in the cubic term with w30 and R4, accurate measurements of the center deflection and the diameter are necessary.33,66 A rough error estimate is deduced in Appendix C. For the determination of the diaphragm

FIG. 10. Measured normalized transverse deflections in [1 1 0]- and [1 0 0]direction at p ¼ 50.2 kPa for the specimen with the ID 14-11 (Table II) featuring R ¼ 409.8 lm and h ¼ 1.61 lm. Both are virtually identical which confirms Fig. 6. Each characteristic consists of 1400 measured points.

FIG. 12. Double-logarithmic plots of measured pm (wm0) and Eq. (7)-fitted p(w0) LD characteristics for different specimen geometries identified by their reference number. Measurements and fits are labeled by symbols and dashed lines, respectively. Determined E- and r0-values are collected in Table II.

FIG. 9. (a) Setup for the silicon measurements. The detail views emphasize the MSA objective and the specimen in mounting position, respectively. (b) Arrangement for the CNTBP measurements.

VII. EXPERIMENTS, RESULTS, AND DISCUSSION

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 52.2.249.46 On: Thu, 24 Sep 2015 18:25:37

114905-9

Beigelbeck et al.

J. Appl. Phys. 116, 114905 (2014)

TABLE II. Measured Young’s moduli E and residual stresses r0 for different geometries of silicon specimens. The last two columns were derived by an intricate dual-parameter fit using the computer numerical diaphragm model with isotropic elastic behavior. By contrast, the equivalent Young’s modulus for circularly supported diaphragms deduced from the biaxial modulus of monocrystalline silicon amounts Elit ¼ 169 GPa at  lit ¼ 0.064. Generally, the agreement between theory63 and our measurements can be rated as excellent. The relative deviation in E between the solution of Loy et al.54 and the fit obtained by Eq. (7) is about 11%. Loy et al.54 fit

Geometry

Eq. (7) fit

Isotropic FEM fit

Specimen identifier

06-03 13-04 13-06 13-07 14-11

R (lm)

h (lm)

E (GPa)

r0 (MPa)

E (GPa)

r0 (MPa)

E (GPa)

r0 (MPa)

252.4 6 1 514.9 6 1 718.3 6 1 823.4 6 1 409.8 6 1

1.61 1.61 1.61 1.61 1.61

186.1 6 2.6 188.2 6 1.3 186.0 6 0.9 186.5 6 0.8 187.6 6 1.6

57.2 6 0.4 94.0 6 0.4 92.8 6 0.3 90.4 6 0.2 81.0 6 0.4

167.0 6 2.6 168.8 6 1.3 166.9 6 0.9 167.4 6 0.8 168.3 6 1.6

55.6 6 0.4 91.3 6 0.4 90.1 6 0.3 87.8 6 0.2 78.8 6 0.4

168.6 6 2.6 169.3 6 1.3 168.2 6 0.9 168.0 6 0.8 168.4 6 1.6

60.4 6 0.4 88.6 6 0.4 90.1 6 0.3 90.4 6 0.2 88.8 6 0.4

dimension, we utilized a white light interferometer (Polytec MSA-400) and a thin-film thickness analyzer (Filmetrics F20) combined with an optical microscope (Nikon Optiphot 66). A. Silicon diaphragms

We investigated (1 0 0) silicon diaphragms with diameters between 400 lm and 1600 lm, where the wafers’ primary flat was aligned to [1 1 0]. For diameters 400–800 lm, the utilized MSA objective was able to capture the entire diaphragm surface at once yielding the three-dimensional bending curvature. Above, a one-fourth of the diaphragm starting from the center towards to the clamp was recorded. From these data, deflection curves along the crystal directions [1 1 0], ½ 1 1 0, and [1 0 0] can be directly extracted. By crystal symmetry, the results in [1 1 0]- and ½1 1 0-direction must be identical, which was also confirmed by our measurements. Figure 10 contrasts deflection measurements in [1 1 0]- and [1 0 0]-direction. It can be seen that the deviation between both normalized transverse deflections is negligible, i.e., the computationally deduced prediction of Fig. 6 is fulfilled. Notably, the curvature observed near the clamp reveals a strong plate component in the bending characteristics, which is the characteristic for silicon diaphragms in such geometry and pressure ranges.

~ FIG. 13. Measured normalized transverse deflection wð.Þ for the CNTBPs. This characteristic is independent of the applied pressure and can be closely approximated by Eq. (11).

Figure 11 illustrates the change of the bending curvature with the applied pressure of a representative silicon specimen. From low to high pressures, the diaphragm undergoes a transition from plate towards dominant membrane behavior. This characteristic is fully taken into account by our LD relationship Eq. (7) yielding the LD response shown in Fig. 12. In Table II, measured E and r0 for different diaphragm geometries are summarized and compared to values derived from an extensive intricate dual-parameter fit using the computer numerical diaphragm model and by taking the anisotropic elastic behavior into account.

B. CNTBP diaphragms

The diameters of all investigated CNTBPs exceeded the aperture angle of the MSA objective. In order to still gather the bending curvature over the complete diameter range, we shifted the radial coordinate by varying the sample position with micrometer screws. Three CNTBP specimens were investigated, where multiple measurements along their radial coordinate revealed that the circular CNTBPs deform perfect rotationally symmetric which is not immediately obvious due to their material composition involving highly irregular arranged bundles of carbon nanotubes.35 Figure 13 illustrates the measured

FIG. 14. Measured pm (wm0) and fitted p(w0) LD data (indicated by symbols and dashed lines, respectively) for three representative CNTBP samples (ID 130907-4, ID 140606-1, and ID 220506-1) with R ¼ 12 mm, h ¼ 10 lm, and   0.27.35 Determined Young’s moduli and residual tensile stresses of all three samples are collected in 3.

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 52.2.249.46 On: Thu, 24 Sep 2015 18:25:37

114905-10

Beigelbeck et al.

J. Appl. Phys. 116, 114905 (2014)

TABLE III. Measured Young’s moduli E and residual stresses r0 of three CNTBP specimens. The last two columns were obtained by an intricate dualparameter fit using the computer numerical diaphragm model with isotropic elastic behavior. The relative deviation in E between Loy’s approximation and the fit obtained by Eq. (7) is about 16%. Loy et al.54 fit

Geometry

Eq. (7) fit

FEM fit

Specimen identifier

130907-4 140606-1 220506-1

R (mm)

h (lm)

E (GPa)

r0 (MPa)

E (GPa)

r0 (MPa)

E (GPa)

r0 (MPa)

12 6 0.2 12 6 0.2 12 6 0.2

10 6 1 10 6 1 10 6 1

3.55 6 0.5 3.57 6 0.5 3.56 6 0.5

1.14 6 0.15 0.47 6 0.06 0.45 6 0.06

3.06 6 0.5 3.08 6 0.5 3.06 6 0.5

1.11 6 0.15 0.46 6 0.06 0.44 6 0.06

3.06 6 0.5 3.08 6 0.5 3.06 6 0.5

1.11 6 0.15 0.46 6 0.06 0.44 6 0.06

~ m ð.Þ, which is almost indenormalized bending curvature w pendent of the applied pressure. Consequently, for all measured CNTBP diaphragms, an approximation close to ~ wð.Þ ¼ 1  0:929.2 þ 0:124.4

(11)

holds. From the bending curvature in the vicinity of the clamping, we conclude that investigated CNTBPs behave like ideal membranes. Measured LD characteristics are compared in Fig. 14 and the associated E- and r0-values are summarized in Table III.

supported by the Austrian Science Fund (FWF, research grant P25212-N30). The Center for Integrated Sensor Systems gratefully acknowledges partial financial support by the European Regional Development Fund (EFRE) and the province of Lower Austria. APPENDIX A: DERIVATION OF THE LD CONSTANTS

The work input to the diaphragm imposed by the external pressure is governed by67 ~ WðaÞ ¼ 2pR2 pw0 WðaÞ ;

VIII. CONCLUSION

We presented a novel two-step load-deflection technique enabling an easy but still accurate extraction of the Young’s modulus and the residual tensile stress from bulge test measurements on circularly shaped diaphragms. This relationship was obtained by analytically solving an infinite-dimensional system of equations arising from the application of the minimum total potential energy principle. Compared to previously published approaches, our ansatz allows accurate consideration of any practically relevant bending curve of circular diaphragms ensuring applicability to a large variety of different diaphragm types. The versatility of this new load-deflection formula was first proven theoretically by computer numerical modeling and then by extracting the Young’s modulus and the residual tensile stress of two disparate thin-film diaphragm materials made of either micromachined silicon or carbon nanotube buckypapers. This novel method turned out to be very flexible and potentially very accurate for a large variety of circular diaphragms. Basically, its accuracy is determined by uncertainties related to the bulge-test measurements itself and not by the loaddeflection model. ACKNOWLEDGMENTS

The authors are grateful to Sophia Ewert and Dr. Artur Jachimowicz, both from the Institute of Sensor and Actuator Systems of the Vienna University of Technology, for their assistance during the fabrication of the test specimens. Special thanks go to Professor Dr. Bernhard Jakoby from the Institute for Microelectronics and Microsensors of the Johannes Kepler University Linz as well as Dr. Ephraim Suhir from the Physical Sciences and Engineering Research Division of Bell Labs for many stimulating discussions regarding nonlinear modeling in structural mechanics. This work has been financially

(A1a)

ð1

~ ðaÞ ¼ wð.; ~ aÞ. d. : W

(A1b)

0

Using Timoshenko’s plate bending theory for large deflections,67 we calculate the radial w20 ~e . ð.; a; bÞ ; R2

(A2a)

~e . ð.; a; bÞ ¼ u~0 þ ðw ~ 0 Þ2 =2 ;

(A2b)

e. ð.; a; bÞ ¼

and the transverse strain et ð.; bÞ ¼

w20 ~e t ð.; bÞ ; R2

~e t ð.; bÞ ¼ u~=. ;

(A3a) (A3b)

where prime symbols denote derivatives regarding .. Subsequently, the strain energy due to stretching of the middle surface V1 ða; bÞ, the strain energy caused by the internal tensile stress V2 ða; bÞ, and the strain energy owing to pure bending V3 ðaÞ are obtained by V1 ða; bÞ ¼ 2pr0 hw20 V~1 ða; bÞ ;

(A4a)

ð1

V~1 ða; bÞ ¼ ½~e . þ ~e t  . d. ;

(A4b)

0

V2 ða; bÞ ¼

pEh w40 ~ V ða; bÞ ; ð1   2 Þ R 2 2

ð1 ~ V 2 ða; bÞ ¼ ½~e 2. þ ~e 2t þ 2~e .~e t  . d. ;

(A4c)

(A4d)

0

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 52.2.249.46 On: Thu, 24 Sep 2015 18:25:37

114905-11

Beigelbeck et al.

J. Appl. Phys. 116, 114905 (2014)

pEh3 w20 ~ V 3 ðaÞ ; 12ð1   2 Þ R2 " # ð1 ðw0 Þ2 2w0 w00 2 00 ~ Þ þ 2 þ V~3 ðaÞ ¼ ðw . d. : . . V3 ðaÞ ¼

(A4e)

and applying the minimum total potential energy principle68 regarding the coefficients bi yields

(A4f)

@V ða; bÞ ¼ 0 8 i 2 ½0; 1Þ ; @b2i

(A8)

0

After substituting Eqs. (4), (A2), and (A3) to Eqs. (A1) and (A4), an extensive auxiliary calculation yields for the scaled strain energies and the work input 1 X 1 X

1 V~1 ðaÞ ¼ 2 V~2 ða; bÞ ¼

a2k a2l /kl ;

(A5a)

k¼0 l¼0

1 X 1 X b2s b2t w‹ st s¼0 t¼0

þ

1 X 1 X 1 X

which simplifies to @V2 ða; bÞ ¼0 @b2i

1 X

k¼0 l¼0 s¼0

V~3 ðaÞ ¼

1 X 1 X

a2k a2l vkl ;

1 X

Ais b2s ¼ Bi

(A10)

with (A5c) ‹ Ais ¼ w‹ is þ wsi ¼

a2k gk

8 fi; sg 2 ½0; 1Þ

s¼0

(A5b)

k¼0 l¼0

~ ðaÞ ¼ W

(A9)

because V1 ðaÞ; V3 ðaÞ, and WðaÞ are independent of b. Interchanging derivation and integration, applying product and chain rule, and evaluating the integrals results in an infinite-dimensional, symmetrical (Ais ¼ Asi), linear system of equations for the expansion coefficients

a2k a2l b2s w› kls

1 X 1 X 1 X 1 1X þ a2k a2l a2m a2n wfi klmn ; 4 k¼0 l¼0 m¼0 n¼0

8 i 2 ½0; 1Þ

(A5d)

8ði þ 1Þðs þ 1Þ ; ði þ s þ 1Þði þ s þ 2Þði þ s þ 3Þ (A11a)

k¼0

Bi ¼ 

with the abbreviations

1 X 1 X

a2k a2l w› kli :

(A11b)

k¼0 l¼0

ð1

/kl ¼ .fk0 fl0 d. ;

(A6a)

0

w‹ st

¼

ð1 

.g0s g0t þ

 gs gt þ 2g0s gt d. ; .

Let bsol ¼ ½bsol0 ; bsol2 ; … denote its solution and insert it into Eq. (A7), then the minimum total potential energy principle68 with respect to w0 postulates @V ða; bsol Þ ¼ 0: @w0

(A6b)

(A12)

0

ð1

0 0 0 w› kls ¼ fk fl ð.gs þ gs Þ d. ;

(A6c)

0

ð1 ¼ .fk0 fl0 fm0 fn0 d. ; wfi klmn

(A6d)

0

vkl ¼

ð1 

.fk00 fl00

 fk0 fl0 0 00 þ 2fk fl d. ; þ .

(A6e)

0

(A6f) C 1 ða Þ ¼ 2

0

All integrals involve ordinary power functions and can be straightforwardly evaluated in closed-form. Introducing the total potential energy functional of the diaphragm as superposition Vða; bÞ ¼ V1 ðaÞ þ V2 ða; bÞ þ V3 ðaÞ  WðaÞ

V1 ðaÞ þ 2V2 ða; bsol Þ þ V3 ðaÞ ¼

w0 @W ðaÞ : 2 @w0

(A13)

Rearrangement and equating the coefficients on both sides of the equation yields the final result

ð1

gk ¼ .fk d. :

Notably, although relating u(.) to w(.) through the scaling factor w20 =R in Eqs. (4) seems unconventional, it offers the major advantage that the linear system (A10) and therefore its solution (i.e., each bsol2i-value) is independent of w0. As a consequence, the scaled energy expressions in Eq. (A5) can be treated as constant regarding @/@w0 and Eq. (A12) reduces to

(A7)

C2 ða;  Þ ¼

V~1 ðaÞ ; ~ ðaÞ W

2 V~2 ða; bsol Þ ; ~ ðaÞ ð1   2 Þ W

C3 ða;  Þ ¼

1 V~3 ðaÞ : ~ ðaÞ 12ð1   2 Þ W

(A14a)

(A14b)

(A14c)

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 52.2.249.46 On: Thu, 24 Sep 2015 18:25:37

114905-12

Beigelbeck et al.

J. Appl. Phys. 116, 114905 (2014)

APPENDIX B: APPROXIMATION ORDER OF THE TRANSVERSE DEFLECTION

In Sec. III, we stated that K ¼ 3 is sufficient to approximate a huge class of practically relevant normalized transverse deflections. The validity of this assumption can be either checked by comparing measured and fitted plots optically or by the plausibility consideration below. For K ¼ 3, Eq. (4a) reduces to ~ aÞ ¼ ð1  .2 Þð1 þ a2 .2 þ a4 .4 þ a6 .6 Þ : wð.;

(B1)

Here, the first factor (1  .2) describes an ideal membrane bending behavior while the sum enclosed by the brackets on the right is a sixth-order polynomial approximation. Equation (B1) already satisfies the vanishing deflection at ~ aÞ  0) as well as the normalized the support (i.e., wð1; ~ aÞ  1) and the horizontal slope value at the center (i.e., wð0; ~ 0 ð0; aÞ  0). Consequently, all three paat the center (i.e., w rameters (a2, a4, and a6) are left as degrees of freedom to closely fit theory and measured LD data and thus allow for description of a very large class of different bending shapes in the range 0 . 1. From an empirical point of view, we measured hundreds of different (pressure loads, membrane geometries, membrane types, etc.) transverse bending curva~ m ð.Þ and evaluated for each characteristic the maxitures w mum absolute and the mean (averaged over the radius) ~ aÞ. ~ m ð.Þ and fit wð.; square error between measurement w The achieved values were always below 102 and 105, respectively. APPENDIX C: INFLUENCE OF GEOMETRY AND DEFLECTION UNCERTAINTIES ON THE MEASURE PARAMETERS

Utilizing the first order Taylor series expansion  n Dx Dx Dx ;  1 1þn 1þ x x x

(C3)

in Eqs. (C2) and neglecting the second order error products yields worst case estimates for the maximum relative error DR Dw0 Dr0 Dh þ (C4a) þ 2 R w ; r0 h 0 DR DE Dh þ 3 Dw0 : (C4b) þ 4 w0 E h R As one can see, variations in the radius influence the measured parameters the most. For example, a relative error of 1% in h yields relative errors in the Young’s modulus and residual stress of 1%. On the other hand, a relative of 1% in R causes relative errors in E and r0 of 4% and 2%, respectively. 1

In order to minimize the interdependence of the linear and cubic term in Eq. (7) through E and to separate the influence of E in the determination of r0, the imposed center deflection in the measurement should be much larger than the diaphragm thickness. In this case, the worst case estimation of the errors in the residual stress Dr0 and the Young’s modulus DE induced by measurement uncertainties of the diaphragm geometry becomes easier because the influence of the h3-term (i.e., the C3 ða; Þ-term) in Eq. (7) can be neglected. Under this prerequisite and the assumption that h, R, and w0 are burden by small uncertainties Dh, DR, and Dw0, respectively, Eq. (7) reads    Dh Dw0 w0 1 þ r0 h 1 þ w0 h pðw0 ; aÞ ¼ C1 ðaÞ  2 DR R2 1 þ R    3 Dh 3 Dw0 Eh 1 þ w0 1 þ h w0 þC2 ða;  Þ :  4 DR R4 1 þ R

Applying this equation to measured LD data would result in wrong values for the residual stress and Young’s modulus    Dh Dw0 1þ 1þ w0 h r00 ¼ r0 ; (C2a)  2 DR 1þ R   3 Dh Dw0 1þ 1þ w0 h : (C2b) E0 ¼ E  4 DR 1þ R



(C1)

M. Elwenspoek and H. Jansen, Silicon Micromachining (Cambridge University Press, 1999). 2 W. Menz, J. Mohr, and O. Paul, Microsystem Technology, 1st ed. (Wiley & Sons, 2008). 3 J. Voorthuyzen and P. Bergveld, Sens. Actuators 6, 201 (1984). 4 O. Paul and H. Baltes, J. Micromech. Microeng. 9, 19 (1999). 5 H. Lim, B. Schulkin, M. Pulickal, S. Liu, R. Petrova, G. Thomas, S. Wagner, K. Sidhu, and J. Federici, Sens. Actuators, A 119, 332 (2005). 6 O. Brand, G. Fedder, C. Hierold, J. Korvink, and O. Tabata, Reliability of MEMS: Testing of Materials and Devices (Wiley Press, 2013). 7 V. Ziebart, “Mechanical properties of CMOS thin films,” Ph.D. dissertation (ETH Zurich, 1999). 8 K. Danaie, A. Bosseboeuf, C. Clerc, C. Gousset, and G. Julie, Sens. Actuators, A 99, 78 (2002). 9 D. R. Lide, CRC Handbook of Chemistry and Physics, 86th ed. (CRC Press, 2005). 10 D. Maier-Schneider, J. Maibach, E. Obermeier, and D. Schneider, J. Micromech. Microeng. 5, 121 (1995). 11 A. Ababneh, U. Schmid, J. Hernando, J. Sanchez-Rojas, and H. Seidel, Mater. Sci. Eng., B 172, 253 (2010). 12 R. Sah, L. Kirste, M. Baeumler, P. Hiesinger, V. Cimalla, V. Lebedev, H. Baumann, and H.-E. Zschau, J. Vac. Sci. Technol. A 28, 394 (2010). 13 C. Klein, J. Appl. Phys. 88, 5487 (2000). 14 G. Janssen, M. Abdalla, F. van Keulen, R. Pujada, and B. van Venrooy, Thin Solid Films 517, 1858 (2009). 15 H. Espinosa, B. Prorok, and M. Fischer, J. Mech. Phys. Solids 51, 47 (2003). 16 X. Chen and J. Vlassak, J. Mater. Res. 16, 2974 (2001). 17 W. Oliver and G. Pharr, J. Mater. Res. 19, 3 (2004). 18 R. Keller, J. Phelps, and D. Read, Mater. Sci. Eng., A 214, 42 (1996). 19 H. Huang and F. Spaepen, Acta Mater. 48, 3261 (2000). 20 J. Gaspar, M. Schmidt, J. Held, and O. Paul, J. Microelectromech. Syst. 18, 1062 (2009).

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 52.2.249.46 On: Thu, 24 Sep 2015 18:25:37

114905-13 21

Beigelbeck et al.

J. Gaspar, A. Gualdino, B. Lemke, O. Paul, V. Chu, and J. Conde, J. Appl. Phys. 112, 024906 (2012). 22 E. Bromley, J. Randall, D. Flanders, and R. Mountain, J. Vac. Sci. Technol. B 1, 1364 (1983). 23 M. Allen, “Measurement of mechanical properties and adhesion of thin polyimide films,” Master’s thesis (Massachusetts Institute of Technology, 1986). 24 M. Mehregany, “Application of micromachined structures to the study of mechanical properties and adhesion of thin films,” Master’s thesis (Massachusetts Institute of Technology, 1986). 25 W.-H. Chuang, T. Luger, R. Fettig, and R. Ghodssi, J. Microelectromech. Syst. 13, 870 (2004). 26 P. Gkotsis, “Development of mechanical reliability testing techniques with application to thin films and piezo MEMS components,” Ph.D. dissertation (Cranfield University, 2010). 27 B. Ilic, S. Krylov, and H. Craighead, J. Appl. Phys. 108, 044317 (2010). 28 M. Birkholz, Thin Film Analysis by X-Ray Scattering, 1st ed. (Wiley, 2006). 29 U. Welzel, J. Ligot, P. Lamparter, A. Vermeulen, and E. Mittemeijer, J. Appl. Crystallogr. 38, 1 (2005). 30 I. De Wolf, Semicond. Sci. Technol. 11, 139 (1996). 31 S. Yang, R. Miyagawa, H. Miyake, K. Hiramatsu, and H. Harima, Appl. Phys. Express 4, 031001 (2011). 32 K.-S. Chen and K.-S. Ou, J. Micromech. Microeng. 12, 917 (2002). 33 M. Small and W. Nix, J. Mater. Res. 7, 1553 (1992). 34 W. Nix, Metall. Mater. Trans. A 20, 2217 (1989). 35 Y. Ji, “Multiscale investigations on structural properties and mechanical applications of carbon nanotube sheets,” Ph.D. dissertation (University of Akron, 2008). 36 S. Ma, S. Wang, F. Iacopi, and H. Huang, Appl. Phys. Lett. 103, 031603 (2013). 37 J. Beams, in Structure and Properties of Thin Films, edited by C. Neugebauer, J. Newkirk, and D. Vermilyea (Wiley, New York, 1959), pp. 183–192. 38 M. Allen, M. Mehregany, R. Howe, and S. Senturia, Appl. Phys. Lett. 51, 241 (1987). 39 J. Vlassak, “New experimental techniques and analysis methods for the study of the mechanical properties of materials in small volumes,” Ph.D. dissertation (Stanford University, 1994). 40 R. Vinci and J. Vlassak, Annu. Rev. Mater. Sci. 26, 431 (1996). 41 X. an Fu, J. Dunning, C. Zorman, and M. Mehregany, Thin Solid Films 492, 195 (2005). 42 F. Yang and J. Li, Micro and Nano Mechanical Testing of Materials and Devices, 1st ed. (Springer Press, 2008). 43 V. Ziebart, O. Paul, U. M€ unch, J. Schwizer, and H. Baltes, J. Microelectromech. Syst. 7, 320 (1998). 44 Y. Xiang, T. Tsui, and J. Vlassak, J. Mater. Res. 21, 1607 (2006).

J. Appl. Phys. 116, 114905 (2014) 45

J. Gaspar, O. Paul, V. Chu, and J. Conde, J. Micromech. Microeng. 20, 035022 (2010). 46 A. Wright, J. Appl. Phys. 82, 2833 (1997). 47 U. Schmid, M. Eickhoff, C. Richter, G. Kr€ otz, and D. Schmitt-Landsiedel, Sens. Actuators, A 94, 87 (2001). 48 J. Pan, P. Lin, F. Maseeh, and S. Senturia, in IEEE Solid-State Sensors and Actuators Workshop 1990 (Hilton Head Island, South Carolina, USA, 1990), pp. 70–73. 49 E. Saibel and I. Tadjbakhsh, Z. Angew. Math. Phys. (ZAMP) 11, 496 (1960). 50 M. Sheplak and J. Dugundji, J. Appl. Mech. 65, 107 (1998). 51 O. Tabata, K. Kawahata, S. Sugiyama, and I. Igarashi, Sens. Actuators 20, 135 (1989). 52 Y. Xiang, X. Chen, and J. Vlassak, J. Mater. Res. 20, 2360 (2005). 53 D. Maier-Schneider, J. Maibach, and E. Obermeier, J. Microelectromech. Syst. 4, 238 (1995). 54 C. Loy, S. Pradhan, T. Ng, and K. Lam, J. Micromech. Microeng. 9, 341 (1999). 55 D. Maier-Schneider, J. Maibach, and E. Obermeier, J. Micromech. Microeng. 2, 173 (1992). 56 M. Mehregany, R. Howe, and S. Senturia, J. Appl. Phys. 62, 3579 (1987). 57 E. Cianci, A. Coppa, and V. Foglietti, Microelectron. Eng. 84, 1296 (2007). 58 C. Aliprantis and K. Border, Infinite Dimensional Analysis, 3rd ed. (Springer Press, Berlin, 2007). 59 K 4 is only required to synthesize highly sophisticated bending characteristics. In such cases, the number of terms to express C1 ðaÞ; C2 ða; Þ, and C3 ða; Þ boosts tremendously making application of software tools like MathematicaTM inevitable. 60 L. Landau, E. Lifshitz, A. Kosevich, and L. Pitajewski, Theory of Elasticity, 3rd ed., Course of Theoretical Physics Vol. 7 (Pergamon Press, 1986). 61 This generalization allows studying the bending curvature and the LD response of diaphragms exhibiting anisotropic elastic material properties (e.g., monocrystalline silicon), which will be needed in Sec. VII. 62 J. Wortman and R. Evans, J. Appl. Phys. 36, 153 (1965). 63 M. Hopcroft, W. Nix, and T. Kenny, J. Microelectromech. Syst. 19, 229 (2010). 64 B. Choubey, E. J. Boyd, I. Armstrong, and D. Uttamchandani, J. Microelectromech. Syst. 21, 1252 (2012). 65 J. Mitchell, C. Zorman, T. Kicher, S. Roy, and M. Mehregany, J. Aerospace Eng. 16, 46 (2003). 66 V. Srikar and M. Spearing, Exp. Mech. 43, 238 (2003). 67 S. Timoshenko, Theory of Plates and Shells, 2nd ed. (McGraw-Hill Higher Education, 1969). 68 J. N. Reddy, Energy Principles and Variational Methods in Applied Mechanics, 2nd ed. (Wiley & Sons, 2002).

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 52.2.249.46 On: Thu, 24 Sep 2015 18:25:37