FEM analysis of microbeam bending experiments using ultra-micro indentation

COMPUTATIONAL MATERIALS SCIENCE Computational Materials Science 3 (1994) 169-176

ELSEVIER

FEM analysis of microbeam bending experiments using ultra-micro indentation M. Heinzelmann,

M. Petzold

Fraunhofer-lnstitut ffir Werkstoffmechanik, Aussenstelle Halle, Heideallee 19, D-06120 Halle, Germany

Received 27 March 1994; accepted 10 May 1994

Abstract

Bending experiments of cantilever microbeams in an ultra micro indenter can be used to determine Young's modulus E of micromechanical components. Using the Finite Element Method, it was investigated for anisotropically etched miocrobeams of monocrystalline silicon, whether elementary beam bending and shearing equations arc applicable for the evaluation of the experiments. It was found that the inclination and compliance of the beam support, and the indentation can lead to significant errors in E when the elementary equations are applied. Based on the FEM results, relations are derived that correct the elementary equations for these influences. In the experimental part of this study, Young's modulus E of monocrystalline silicon was determined from the microbeam experiments. The excellent agreement between the experimentally determined value of E with the well known literature value demonstrates the reliability of the microbeam testing technique.

1. Introduction

In the field of micromechanics, Young's modulus E is an especially important material property, because the operational principle of many micromechanical components is based on the elastic behavior of the material. Examples for such components are pressure sensors in which a m e m b r a n e is deflected by an applied pressure, or acceleration sensors, in which a seismic mass is deflected by an applied acceleration. Despite this importance of E, the often well known values for E obtained from macroscopic specimens, cannot simply be transferred to the microscopic scale, because the elastic behavior of microscopic components can differ significantly from the elastic behavior of macroscopic compo-

nents. Reasons for this difference can be found in (i) the increasing role of residual stresses in small components, (ii) the increasing ratio of surface to volume with decreasing component size, (iii) differences in the microstructure, e.g the grain size, of the material, and (iv) geometric nonlinearities, which occur frequently in microsized components [1,2]. Thus, Young's modulus has to be determined directly from specimens of a microscopic size, which requires the development of special testing techniques. However, the exerimental determination of mechanical properties from microsized specimens is generally problematic, as was pointed out in detail by Schweitz [3]. Nix [4] and Weihs et al. [5,6] were the first to use nanoindentation experiments to determine Young's modulus of thin

0927-0256/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0927-0256(94)00052-E

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M. Heinzelrnann, M. Petzold / Computational Materials Science 3 (1994) 169-176

films that were underetched to form thin cantilever beams. Ericson and Schweitz [7] determined the fracture strength and Young's modulus of heavily-doped monocrystalline silicon from bending cantilever microbeams in situ in a scanning electron microscope. Their experiments indicate a 30-40% reduction of Young's modulus in comparison to the well known value of undoped monocrystalline silicon, illustrating the tremendous difference that can exist between E in microscopic and macroscopic specimens. But although cantilever microbeam experiments have already been carried out to determine Young's modulus E, there is still some uncertainty regarding the evaluation of the experiments. The evaluations in Refs. [4-7] are based on elementary bending equations, although the assumptions underlying these equations may be violated for the microbeams, as was already noted in Refs. [5-7]. It is therefore the purpose of the present paper to analyse cantilever microbeam bending experiments with special attention on the applicability of elementary beam bending and shearing equations for the evaluation of the experiments. Under investigation is a special beam geometry that arises from anisotropic etching of undoped monocrystalline silicon. For this geometry, the applicability of the elementary equations has yet to be demonstrated, because contrary to their assumptions (i) the support is not perfectly stiff but somewhat compliant, (ii) the support is not vertical but inclined, and (iii) the local penetration of the indenter tip into the material is not accounted for in the equations. The analysis of the influences of the support is done using the Finite Element Method. The influence of the indentation is determined experimentally. Finally, experiments will be carried out to demonstrate the usefulness and reliability of the investigated testing technique.

2. The microbeam bending experiment As a typical example for the cantilever microbeam geometry, we have chosen the geometry of anisotropically etched microbeams of undoped monocrystalline silicon. The testing apparatus is

Fig. 1. Sketch of the anisotropically etched microbeam geometry.

an ultra micro hardness tester (Shimadzu DUH 202), in which the microbeams are loaded with a Vickers indenter. Using a Vickers indenter has the advantage that because the load is applied almost as a point load, the distance between beam support and load application can be measured with higher accuracy than it would be the case with a ball indenter, for example. Moreover, with a Vickers indenter the distance between beam support and load application stays constant during the course of the experiment. However, the local penetration of the indenter into the beam has to be accounted for in the evaluation of the experiments. We have chosen to analyse microbeams of monocrystalline silicon, because the elastic behavior of the material is already well known [8,9]. Therefore, it is possible to demonstrate the usefulness and reliability of the experimental technique by performing experiments and comparing the experimentally obtained value of E with the value found in the literature. The geometry of beam and support is depicted in Fig. 1. Characteristic features of this geometry are the trapezoidal cross-section with inclined sides and the inclination of the support. These inclinations result from the fabrication process of anisotropic etching that leaves the (liD-lattice planes with an angle of 35.26° to the vertical direction as side walls of the etched component. The dimensions of the investigated beam crosssection are a bottom beam width of 54 Ixm, a top beam width of 9 ~m, and a beam height of 31.8 ~m. The beam length, defined as the distance between the beam fixture at the top surface of

M. Hemzelmann, M. Petzold/ Computational Materials Science 3 (1994) 169-176 ...... .

: ~ .\

: ! i, '~-:7-!

.................

! . ....~::s:.%.................. ..?_~:=:..:~::::..!.........:~ ...... ................... . ~,

~'........................

Fig. 2. FEM mesh of beam and support.

the beam and the point of load application, was varied between 100 and 1100 ~m and could be determined with an accuracy of I Ixm. The high resolution of the testing machine made it possible to keep the beam deflections in the order of about 1-2 Ixm, such that geometric nonlinearities in the experiments could be excluded.

171

tioned influences of the support and the indentation, the actual beam and support geometry was modelled with finite elements in three dimensions. For this purpose, the FEM code ANSYS was used. A finite element model can be seen in Fig. 2. Note that because of symmetry, only one half of the geometry needs to be modelled, if all nodes in the plane of symmetry are fixed in the normal direction of the plane. Attention was paid to mesh the regions around the point of load application and around the beam fixture particularly fine. Depending on the actual beam length, the models consisted of approximately 8000 nodes having three degrees of freedom each. To facilitate the correlation of the calculated load versus deflection curves with the experiments presented in section 5, the FEM calculation were carried out with the known elastic constants of monocrystalline silicon.

3. FEM modelling of beam and support

In order to calculate the load vs. displacement behavior of the beams, including the aforemen-

4. FEM calculation of the influences of the support and the indentation on the load versus deflection behavior of the beams

4.1. Elementary cantilever beam equations

I

For a cantilever beam loaded by a point force F in a distance l from the fixed end (Fig. 3), the relationship between F and the deflection f at the point of load application is given as

Fl 3 3Ely ,

(1)

?

!/0

for long beams, and

/

Fl 3 f = - -

3EI~

with a = A f( Fig. 3. Sketch of the ideal cantilever beam geometry and the actual microbeam geometry. Differences to the ideal geometry result from the inclination of the support, the compliance of the support, which is described by the angle 0, and the local penetration of the indenter tip into the beam.

all +

-

(2)

-

GA

Sy )2 A) l y b ( y )

dA

(3)

for short beams. In Eqs. (1)-(3), z is the coordinate along the beam length, y the coordinate of the vertical direction, E Young's modulus, ly the axial moment of inertia, G the shear modulus, A

172

M. Heinzelmann, M. Petzold / Computational Materials Science 3 (1994) 169-176 L"

the cross-sectional area of the beam, Sy the static moment of inertia, and b(y) the beam width at the height y.

4.2. Influence of the inclination of the support

.... i

•

"S

-S

_

_ £,

I

-o

Let the beam length l be defined as the distance between the beam fixture at the top side of the beam and the point of load application. If ! is substituted into Eq. (2), Eq. (2) would result in erroneous values for E, because the distance between the beam fixture and the point of load application is shorter than l at any level below the top side of the beam (Fig. 3). With respect to the inclination of the support, Eq. (2) can therefore be corrected by assigning an effective length /eff to the beam which is by an amount of Al smaller than l, i.e. left = 1 + AI

with AI < 0

(4)

and using leff instead of l in Eq. (2). Obviously, Al should be independent of l for long beams. The magnitude of Al can be determined with FEM by calculating the load versus deflection behavior of a beam that is fixed in an inclined but rigid support and calculating the value of lcff with Eq. (2) that corresponds to this load versus deflection behavior. In Fig. 4(a), Al is plotted versus the beam length I. It can be seen that for long beams, Al takes on the constant value of -15.5 p.m. Only for very short beams does Al stay at somewhat smaller values. The error that arises in the evaluation of E with Eq. (2) when Al, i.e. the inclination of the beam, is neglected can be calculated as

AE=

3I(f_GAF(I+AI)

}

(5a)

which results in a relative error of

--E- -

I

I .: •

--

3

•

•

•

•

•

I

900

'CO0

'% •

09

2OC

!C,7, ,'(]0

2'2C

60t:, 70,2 qro

.p'r',

i 0 2'

-~

• ~.,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

,.(, • :C -

!

~ . . , , ..

~

i :'.z

6

g9

\

,.',0 40 ,:-j

10?Y

~ .,,,

.,?:',

.:Q,.

,.,O," ,j..

c ],)

,';?f5

gO';

~'7~:

' O0C

,

Fig. 4. Plot of the effective difference in the beam length A! due to the inclination of the support (a), and of the relative error in the evaluation of the experiments according to Fxl. (5b) (b).

This relative error is plotted in Fig. 4(b) versus the beam length I. It can be seen that for short beams, the negligence of the inclination of the support leads to significantly overestimated values of E. The overestimation becomes less severe with increasing beam length I.

4.3. Influence of the compliance of the support

FP

F(l+Al) 3

AE

•-

13 (1 + At) 3 +

3EIaAI - 1.

(5b)

The compliance of the support has the opposite effect on the load-deflection behavior of the beam than the inclination of the support, which resulted in an effectively shorter and therefore stiffer beam. To describe the influence of the compliance of the support quantitatively, the compliance of the support is characterized by an angle 0 by which the support is bent towards the vertical direction upon loading (Fig. 3). It can furthermore be assumed that for long beams with different lengths but the same cross-section, O is

M. Heinzelmann, M. Petzold / Computational Materials Science 3 (1994) 169-176

173

fact take on a constant value except for very short beams. The error that arises when evaluating an experiment and neglecting the influence of the compliance of the beam is

(

AE=

31 f •

O

F( l + A/) 3

o

GA

F ( l + Al)

)

O

F( l + A/) 3 -

(8a)

a F(l+Al)) 31 ( f + CFl 2 - GA .

.."

". . . . . . .

.......................

-- .

.

.

.

which results in a relative error of /

1

AE E

•

.

"

•

t

,

-"

?.."

?

'

7""

"

Fig. 5. Plot of the parameter C describing the effect of the compliance of the support (a), and the relative error in the evaluation of the experiments according to Eq. (8b) (b).

proportional to the bending moment M at the fixed side of the beam. Thus, the relation

0 = CM = CFI

(6)

with the constant proportionality factor C exists. For small angles 0, the additional deflection ~ f that is caused by 0 can be calculated as

A f = IO

(7a)

which leads with Eq. (6) to

A f = CFI 2.

(7b)

Similar to the determination of AI in the previous section, C can be calculated with two FEM runs, calculating the load-deflection behavior (i) for an inclined but rigidly fixed beam and (ii) for the rcal beam geometry. Performing both FEM runs with the same applied load, the difference in the two calculated beam deflections equals Af , and C can be determined from Eq. (7b). In Fig. 5a, the thus obtained values of C are plotted versus the beam length l, showing that C does in

(l + Al 3) (1 + AI 3) - 3CEIl 2 - 1.

(8b)

This relative error is plotted versus l in Fig. 5(b). For short beams, the negligence of the compliance of the support results in significant underestimations of E that become less severe with increasing beam length I. With AI and C, two parameters were found that correct the elementary cantilever beam equation, Eq. (2), with respect to the inclination and the compliance of the support. The corrected equation reads

E =

F(I + ~l)3 aF ) . 3 I ( f + CFI 2 - - ~ ( l + ~1)3

(9)

]

Substituting ~ l = - 1 5 , 5 g m and C = - 7 6 0 (Nm)-1 into Eq. (9), it can be seen how well Eq. (9) agrees with the FEM results. Since all FEM calculations were done with a Young's modulus of 169 GPa, a good agreement means that Eq. (9), too, leads to E = 169 GPa when the load F and deflection f of the FEM runs arc substituted into Eq. (9). Fig. 6 shows that a perfect agreement between Eq. (9) and the FEM calculations is achieved for all beam lengths except the two shortest, where E stays at lower values than 169 GPa. Thus, Eq. (9) is suitable for the evaluation of the microcantilever beam experiments once A/ and C have been determined with two FEM runs, as described above. Separate FEM calculations to evaluate E are only required for very short beam

M. Heinzelmann, M. Petzold / Computational Materials Science 3 (1994) 169-176

174

lengths, where Eq. (9) deviates from the FEM results.

LOflD(mN) 1.0

--

--

--

i +

_.c

4.4. Influence of the load application

/

In the investigated experiments, the load is applied through a vickers indenter, and the local penetration of the indenter tip into the beam also contributes to the measured deflection in an experiment. Thus, a way has to be found to distinguish between the deflection corresponding to the bending and shear deformation of the beam and the support, from which E can be evaluated using Eq. (9), and the deflection caused by the local penetration of the indenter into the material. This distinction can be done either by including the indentation in the FEM calculations and modelling the indentation as a contact problem or by determining the penetration of the indenter tip experimentally. In this paper, the experimental determination of the indentation effect was favoured, because the FEM modelling of the indentation would lead to nonlinear and thereby extremely time-consuming FEM calculations, and because the proper modelling of the contact problem would furthermore require the determination of some yet unknown material and geometric properties, e.g. the coefficient of friction between the indenter tip and the material or the indenter tip radius [10]. The experimental determination was done by measuring a load vs. penetration depth curve of an indentation into the support, i.e. into the bulk material (Fig. 7(a). Note that this curve corresponds to the local

!!!t •

.......

!)9

I')0 110

f

I I

a.8

L-'-".

--

A . 8t~lB

DEPTH ( pm )

/

-~C

•

-2~

~"

-30

c

-4C

j

i

/ / -60

1 ,".O,

2 '."(:

3351

403

".'>7:0

633

~ 73'~"

803

Fig. 7. Load vs. penetration depth curve for an indentation into the bulk material (a), and the relative error in the evaluation of the experiments according to Eq. (10b) (b).

indentation into the beam only if the slope of the beam at the point of the indentation is negligible. It will be shown in the following chapter that this condition is fulfilled in our experiments, where the total beam deflection stays in the range of 1-2 Ixm compared to a total beam length of several hundred micrometers. The error that arises when evaluating E from Eq. (2) and neglecting the influence of the indentation can be roughly assessed by approximating the load versus penetration depth curve of Fig. 7(a) with a linear curve having a constant indentation depth to load ratio of t~in d = 0.07 I~m/mN

F(l+Al) 3 AE=

I+

Ol

+

)

F(l + AI) 3

i O r'

. •

O0

,

•

200

,

•

300

,

L

4~.0

-,

533 60C (um,

|

;'03

,

,

800

i

900

',u-- i

as

I r i

l

i

:t

30

,~--"

,

,

96,SI I 3130

Fig. 6. Comparison of the results of eq. (9) with the elastic modulus of 169 GPa that was used in the FEM calculations.

(

3I f + Cfl --fi.d

o

G'A f ( l + Al)

) (10a)

M. Heinzelmann, M. Petzold / Computational Materials Science 3 (1994) 169-176

0o0::.....

where find is the local penetration of the indenter tip at the applied load F. Eq. (10a) leads to a relative error of AE

i; !ii

I I:L

( l + AI) 3 -

E

175

(10b)

.

(l + A / ) 3 _ 38indE 1

This relative error is plotted versus the beam length l in Fig. 7(b). For short beams, the negligence of the indentation effect leads to significantly underestimated values of E. Only for long beams the indentation effect can be neglected.

5. E x p e r i m e n t s

with monocrystalline

( 11 )

For cubic materials, s u is characterized by only three independent coefficients, which are in the case of monocrystalline silicon [8,9]

[ .....

Z.SBPI

modulus E with respect to the direction 1 in a cubic material is sl) - 2(sll - s 1 2 -

5344)

12l 2 + 121.~ 2 , + 1212 3 ,

(13) where l i are the indices of the direction 1 [8,11]. For monocrystalline silicon, Eqs. (12) and (13) lead to a Young's modulus of 169 GPa with respect to the [110] lattice direction, i.e. with respect to the direction of the bending stresses. A typical load vs. deflection curve obtained in an experiment is shown in Fig. 8. From this curve, the load vs. penetration depth curve of the indentation into the bulk material (see Fig. 7(a)) has to be subtracted, as was discussed in the preceeding chapter, and the resulting load vs. deflection curve can be evaluated either by the FEM results or by Eq. (9). The experiments were conducted with five different microbeams, and for each beam, the distance between beam support and the point of load application was varied between 100 ~ m and 1100 Ixm. The experiments yielded a mean value of Young's modulus of (14a)

with a standard deviation of

sl2 = - 0 . 2 1 4 X 10 - i t Pa - l s44 = 1.26× 10 -~' P a

.......

Fig. 8. Typical load vs. deflection curve of an experiment.

/~ = 163.7 GPa sl, = I).768 X 10 - l ' Pa - l ,

J

DEPTh(I'~)

I E

silicon

In Section 4, the elementary beam bending and shearing equations have been corrected with respect to the influence of the inclination and compliance of the support and the influence of the indentation. This chapter now presents experimental results with monocrystalline silicon microbeams. It is the purpose of the experiments to demonstrate the reliability of the microbeam testing method by comparing the experimentally obtained value of E with the well known value from the literature. The elastic stress-strain behavior of anisotropic materials can generally be described by the 6 × 6 matrix s u that relates the strain vector Ei to the stress vector ~ according to E i = sii%..

:/2 L .....

A E = 9.8 G p a (12)

Instead of s u, the elastic behavior of anisotropic materials can also be described by direction-dependent Young's moduli, shear moduli, and Poisson's ratios, which can be calculated from s,j. The corresponding equation to calculate Young's

(14b)

which corresponds to 6.0% of the mean value. This scatter can surely be regarded as being satisfactorily low. The experimentally obtained mean value agrees very well with the well known value of 169 GPa. It could thus be shown that when the influence of the inclination and compliance of the support

176

M. Heinzelmann, M. Petzold / Computational Materials Science 3 (1994) 169-176

and the influence of the indentation effect are taken into account, the cantilever microbeam experiment is a reliable merthod for the determination of Young's modulus.

demonstrated that taking the effects of the inclined and compliant support and the effect of the indentation into account, the cantilever microbeam experiment is a reliable method to determine Young's modulus of micromechanical components.

6. Summary and conclusions Bending experiments of anisotropically etched cantilever microbeams of monocrystalline silicon have been investigated with the Finite Element Method. It has been shown that the inclination and the compliance of the support as well as the local penetration of the indenter tip into the beam can lead to significant errors when the experiments are evaluated with elementary beam bending and shear equations. These errors are such that the inclination of the support leads to overestimations of Young's modulus, whereas the compliance of the support and the indentation effect result in underestimations of Young's modulus. All three effects are most pronounced for short beams. Analytical relations have been derived that correct the elementary beam bending and shear equations for the influence of the inclination and the compliance of the support. The influence of the local penetration of the indenter into the beam has been determined experimentally. The experimental determination of Young's modulus using the corrected beam bending and shear equation yielded a very good agreement with the well known value for monocrystalline silicon from the literature. Thus, it could be

Acknowledgements Financial support of the Ministerium f'tir Wissenschaft und Forschung des Landes SachsenAnhalt is gratefully acknowledged.

References [1] E. Sommer and J.M. Olaf, to be published in Materialpriifung (1994). [2] E. Arzt, J. Sanchez and W.D. Nix, proceedings to the VDI Tagung Werkstoffe der Mikrotechnik, Karlsruhe (1991). [3] J.A. Schweitz, Mater. Res. Soc. Bulletin (1992) 34. [4] W.D. Nix, Metallurgical Transactions A 20A (1989) 2217. [5] T.P. Weihs, S. Hong, J.C. Bravman and W.D. Nix, J. Mater. Res. 5 (1988), 931. [6] T.P. Weihs, S. Hong, J.C. Bravman and W.D. Nix, Mater. Res. Symp. Proc. 130 (1989) 87. [7] F. Ericson and J.A. Schweitz, J. Appl. Phys. 68 (1990) 5840. [8] W.A. Brantley, J. Appl. Phys. 44 (1973) 534. [9] J.F. Nye, chap. 7 in Physical Properties of Crystals (Clarendon Press, Oxford, 1957). [10] J.M. Olaf, Ph.D. dissertation, Universit~it Freiburg, (1992). [11] Sirotin and Shoskolskaya, Fundamentals of Crystal Physics (in Russian), (Moscow, 1975).