The structural characteristics of microengineered bridges

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1 351 The structural characteristics of microengineered bridges J S Burdess 1, A J Harris 2, D Wood 3 and R Pitcher 4 1 Department of Mechanical, Materials and Manufacturing Engineering, University of Newcastle, Newcastle upon Tyne, UK 2 Department of Electrical Engineering, University of Newcastle, Newcastle upon Tyne, UK 3 School of Engineering Science, University of Durham, UK 4 Druck Limited, Groby, Leicester, UK Abstract: The paper considers the measurement of Young's modulus and internal stress in microengineered bridge structures. Values determined from the results of static tests, obtained from the measurement of compliance using a nanoindenter, are compared with values derived from natural frequency measurements obtained from dynamic testing using a laser vibrometer. Both test procedures use mathematical models ` tted' to experimental data to estimate Young's modulus and internal stress. For the case of boron-doped silicon bridges the dynamic test method is shown to produce the superior estimates. Keywords: doping microbridge, tension, Young's modulus, nanoindenter, natural frequency, silicon, boron NOTATION 1 INTRODUCTION a b C E f h i J k l u x z non-dimensional position of the point of application of force bridge width compliance function Young's modulus function bridge thickness integer performance index non-dimensional stress parameter length of the bridge bridge displacement position on the bridge frequency equation factor Dirac delta function " error in a non-dimensional position on the bridge density stress! natural frequency The MS was received on 5 October 1998 and was accepted after revision for publication on 10 February *Corresponding author: Department of Mechanical, Materials and Manufacturing Engineering, University of Newcastle, Stephenson Building, Newcastle upon Tyne NE1 7RU, UK. When considering the static and dynamic behaviours of microengineered structures it is important to know the values of Young's modulus E of deposited or doped material and the internal stress caused by the fabrication process. It is well known [1, 2] that the di usion of boron into silicon to create an etch stop layer will introduce tensile stresses in the material of 60±80 MPa. Such stressses `sti en' the structure and can have a profound e ect on batch-to-batch repeatability of properties as sensor scale factor and resonant behaviour [3]. For the purposes of inspection and testing, at the wafer stage and onwards, it may be important to assess such a set of parameters, which can then be used as indicators of device quality and reproducibility. For this case an understanding of the requirements and limitations of the testing process is important. In this work, values for E and have been determined for boron-doped silicon and spun-on polyimide by considering the static and dynamic characteristics of a bridge structure. The methods described are based upon measurements of the bridge's natural frequencies using a laser vibrometer system [4] (dynamic method) and a measurement of its point compliance function using a Burkovich nanoindenter [5] (static method). It is important to compare the static and the dynamic C07998 ß IMechE 2000 Proc Instn Mech Engrs Vol 214 Part C

2 352 J S BURDESS, A J HARRIS, D WOOD AND R PITCHER methods of testing, as it is unclear, at this stage, which methodology o ers the most convenient and accurate approach. In this respect the work complements and extends the work presented in reference [6] where Young's modulus was determined from microcantilevers which are de ected by a nanoindenter. The work undertaken by Zhang et al. [7] has demonstrated the use of a frequency approach based upon two modes of vibration. However, for structures which are highly stressed, more measured natural frequencies are required in order to separate the e ects caused by elasticity and internal stress. Also an assessment of the uncertainty in the estimation of elasticity and stress is also an important requirement if either testing methodology is to be used as part of an inspection process. For example there is always an uncertainty in the precise nature of the structure's connection with its substrate and in the case of bridge-type structures this can be regarded as an error in the beam length. Likewise errors corresponding to thickness, sti ness and frequency uncertainties must be evaluated. In this work the non-linear behaviour of bridge structures [8, 9], caused by the e ect of elasticity at `large' amplitudes of vibration, is avoided by keeping excitation amplitudes at resonance su ciently small. Temporal relaxation of internal stress [9] caused by ageing of the boron structures did not appear to be a problem, but this was not speci cally investigated. The procedures described rely only on the construction of a bridge structure to form a test piece and the method can be used to characterize other materials and composites formed from successive deposition of different material layers. In addition the bridge structure could be included on production wafers containing MEMS and used as test pieces to provide a quality assessment of a production process. de ection u and the point compliance C at the position ˆ x=l can be derived from the k2 u ˆ 12Fl 4, a Ebh3 p where k ˆ 12l 2 = Eh 2 and a ˆ x 0 =l: For this case the compliance at the point of application of the force is given by C a,, E ˆu a F ˆ l f k, a bh k 1 where the function f k, a is given in detail in the Appendix. When the factor k 1, i.e. the internal stress is large, the bridge behaves like a taut string and bending e ects can be neglected. For this situation the compliance is independent of E and equation (1) reduces to the simple form C a,, E ˆ l a 1 a bh 2 These theoretical values can now be compared with the experimental values of the compliance, C e a, measured at regular spaced intervals along the length of the bridge, to form an error function J s ˆ X C e a C a,, E Š 2 3 a The summation in equation (3) is taken over the set of measurement points. Estimates for and E are found by minimizing J s in a least-squares sense. 2 THEORY 2.1 Static method Figure 1 shows a bridge structure of length l, crosssection bh and density subjected to a point force F applied at a location distance x 0 from the end x ˆ 0. When the bridge has internal axial stress [10], the 2.2 Dynamic method By applying simple beam theory to a bridge structure having an internal stress [11], it can be shown that the natural frequencies! n of the bridge are given by the solutions to z 2 2 z2 1 2z 1 z 2 sinh z 1 sin z 2 cosh z 1 cos z 2 1 ˆ 0 4 Fig. 1 Bridge Proc Instn Mech Engrs Vol 214 Part C C07998 ß IMechE 2000

3 THE STRUCTURAL CHARACTERISTICS OF MICROENGINEERED BRIDGES 353 where the factors z 1 and z 2 are determined by! n and are the roots of z 4! 2 s! 2 z 2! 2 n b! 2 ˆ 0 5 b p In equation (5) the factors! s ˆ p = l 2 and! b ˆ Eh 2 = 12l 4 can be interpreted as frequencies which determine the taut string and beam behaviours respectively of the bridge. Thus the natural frequencies! n can be regarded as a function of the two unknown quantities! s and! b and, if! ne represent a set of measured values of! n, a mean square error function J! ˆ X! ne! n! s,! b Š 2 6 n can be de ned. This function is minimized to nd! s and! b, and and E are found from the de nitions of! s and! b once the dimensions of the bridge and the material density have been speci ed. 3 MEASUREMENTS Bridge structures were etched from the [110] surface of a silicon wafer of thickness 300 mm using KOH as the etchant and boron doping as an etch stop [12]. The bridges were oriented so that their lengths coincided with the [100] direction and were etched from a cavity, the four walls of which de ne the [111] planes. Two bridges were produced and these were 2.2 and 3.6 mm long (l ), 0.2 mm wide (b) and3mm thick (h). Figures 2 and 3 show scanning electron microscopy plan and edge photographs of the 2.2 mm beam and these were used to estimate values for its length, width and thickness. The errors associated with length, width and Fig. 2 Bridge structure thickness are of the order of 2, 2 and 5 per cent respectively. At a later date, bridge structures manufactured from polyimide were supplied by British Aerospace (Sowerby) and these had dimensions l ˆ 3:5 mm, b ˆ 0:5mm and h ˆ 19 mm: These bridges were produced from a spun-on Hitachi polyimide varnish-type PIQL-150 which was baked at 360 C: 3.1 Static tests Fig. 3 Bridge thickness These tests were only carried out using the silicon bridges, and the compliance function C e a was found by applying loads to each bridge using a Burkovich nanoindenter. The indenter is a precision piece of experimental equipment and is normally used to measure the load versus indentation characteristics of either a material or a surface lm. It is capable of measuring loads as small as a few micronewtons and displacements of the order of nanometres. A full description of the indenter used in these tests has been given in reference [5]. Using the indenter's X±Y table and viewing facilities the end of the bridge was located and a datum point for a set of local coordinates established. In the case of the shorter bridge the indenter was programmed to perform loading and unloading `load versus displacement' tests at intervals of ˆ 0:1 mm along the bridge length. For the longer bridge the data were collected at ˆ 0:2mm intervals. Figure 4 shows a typical load versus displacement plot taken on the 2.2 mm bridge for a nominal location 0.3 mm from the datum point. On loading, the plot is initially non-linear and this is because the ne point of the indenter penetrates into the bridge's surface. Thereafter the curve tends to converge towards a C07998 ß IMechE 2000 Proc Instn Mech Engrs Vol 214 Part C

4 354 J S BURDESS, A J HARRIS, D WOOD AND R PITCHER Fig. 4 Load versus displacement curve for the 2.2 mm bridge at the location x 0 ˆ 0:3mm straight line. During unloading, the curve is linear but does not follow exactly the same gradient as the loading curve. This is attributed to the perfect elastic recovery of the material during unloading as opposed to the inelastic conditions that occur during the loading phase. The indenter is thus programmed to t a `best' straight line to the unloading curve, and the compliance C e a at the measuring point is calculated from the gradient of this line. The compliance calculated from a best straight-line t of the loading curve is useful and is used to provide an estimate of the compliance error. This error is typically of the order of 8 per cent for measurement points located at the end regions of the bridge to about 2 per cent for points located in the central region. Due to the poor de nition of the end of the bridge it was very di cult to locate the datum point precisely and there is an o set error associated with the load point parameter a. To represent this unknown o set a small parameter " is introduced and a is written in the form a ˆ " i l 7 where i is the point number associated with the loading test and is the increment length. The error function (3) is modi ed to include " and the minimization of J s is achieved using the MATLAB [13] non-linear leastsquares routine `curve t' using ", and E as the unknown parameters. When the compliance function is de ned by equation (1) the procedure fails to predict values for E and. However, when equation (2) was used in the expression for J e, the `curve t' routine was successful and the procedure gave stress values of and MPa for the 2.2 and 3.6 mm beams respectively. Figure 5 shows the correlation between the estimated compliance (2) and the measured data associated with the 3.6 mm bridge. The errors in the above values are estimated by repeated application of the `curve t' routine assuming l, h, b and C e to be uniformly distributed random variables over their respective error ranges. The problem associated with the use of the compliance function given by equation (1) is due to the high value of stress within the bridge. Using the values of stress predicted from equation (2) and assuming that E 170 GPa (Young's modulus for undoped silicon), the factor k is of the order of 80 and is thus much greater than unity. In such circumstances the bridge behaves as an ideal taut string and the compliance is essentially independent of E. This independence of E was agged by MATLAB, which issued a warning message that the Jacobian matrix associated with the `curve t' routine was singular. Equation (1) shows that the elasticity of the bridge only becomes important if the curvature of the de ected shape is of the same order as the term k 2 u: The curvatures produced by a `small' point load are not likely to satisfy this requirement when k is large and this, together with the fact that the indentation test is very time consuming to perform, makes the static method an unattractive way of determining both and E. 3.2 Dynamic tests In these tests the bridges were mounted in the laser measurement system shown in Fig. 6. The details of this Proc Instn Mech Engrs Vol 214 Part C C07998 ß IMechE 2000

5 THE STRUCTURAL CHARACTERISTICS OF MICROENGINEERED BRIDGES 355 Fig. 5 Compliance function for the 3.6 mm bridge system have been described fully in reference [4] and here it is su cient to say that the bridge and its substrate are waxed to a piezoelectric disc within the evacuated mount. An alternating voltage applied to the disc excited transverse vibrations into the bridge and these were measured by means of the laser vibrometer. By slowly changing the frequency of the applied voltage with a precision oscillator it was possible to identify the frequencies which produced a resonant response in the bridge. These frequencies could be measured to an accuracy of 10 Hz: The position of the laser spot of the surface of the bridge is generally unimportant although some adjustment was necessary in order to move the spot away from a nodal point. The measured natural frequencies of the 2.2 and 3.6 mm bridges are presented in Table 1. It should be noted here that the measured frequencies are restricted to 250 khz or less due to the limitations of the vibrometer controller. The values of! s and! b (and hence and E ), which minimize the frequency error function J!, are determined from an initial guess by application of the MATLAB [13] functions `fsolve', which determines the natural frequencies! n from equations (4) and (5) for given values of! s and! b, and `curve t', which provides adjusted values for! s and! b so that minimization of J! is achieved. For a given set of measured natural frequencies, such as those shown in Table 1, the `fsolve' and `curve t' routines work well and values of! s and! b are produced which yield frequency matches better than 100 Hz. When the data given in Table 1 are used in this procedure and when the density is taken to be 2330 kg=m 3, the MATLAB algorithms give ˆ 84 2 MPa and E ˆ GPa for the 2.2 mm bridge and ˆ 90 1 MPa and E ˆ GPa for the 3.6 mm beam. As before, the errors in and E are calculated by repeated application of the routines assuming that the Fig. 6 Vibrometer measurement system C07998 ß IMechE 2000 Proc Instn Mech Engrs Vol 214 Part C

6 356 J S BURDESS, A J HARRIS, D WOOD AND R PITCHER Table 1 Measured natural frequencies of silicon bridges Bridge Frequency (khz) for the following mode numbers length (mm) * * measured quantities are random variables uniformly distributed within their error range. The values for the stress compare well with the values obtained from the static tests and other published data [1, 2]. The estimated values of Young's modulus for boron-doped silicon are greater than the value of 166 GPa quoted for [100] undoped silicon [14]. The method of measuring natural frequencies as a means of determining internal stress and Young's modulus is clearly more successful than the method based upon measurements of static compliance. This can be explained by noting that at the higher natural frequencies the mode shape of vibration is approximately u /sin np : For this situation the bridge will only vibrate like a taut string if k= np l: For a given value of k this inequality shows that bridge elasticity becomes more important at the higher mode numbers. Thus, for a bridge with `high' internal stress, it is important to measure as many natural frequencies as possible. For example if the 2.2 mm bridge is considered, calculation of and E using the two lowest modes gives a reasonable estimate for stress, 86 MPa, but a low estimate for Young's modulus, E 134 GPa: As more modes are introduced into the calculation, the results converge to the values quoted above. Vibration tests similar to those described above were carried out on the polyimide bridge and the results shown in Table 2 were obtained. From these data, and taking the density to be 1379 kg=m 3, the internal stress and Young's modulus for the spun-on polyimide specimen were found to be ˆ 6:99 0:16 MPa and E ˆ 8:56 1:36 GPa respectively. An analysis of the errors in the evaluation of Young's modulus shows that, for the silicon and polyimide bridges, the major error Table 2 Mode number Measured natural frequencies of the polyimide bridge Frequency (khz) 1 11:57 0: :17 0: :51 0: :17 0: :65 0: :06 0:25 contribution comes from the estimation of the thickness h of the bridge. In order to calculate and E a value for the material density must be known and this is a disadvantage of this procedure. However, if E could be estimated from compliance tests on an annealed cantilever (in this case no internal stress is present), then can be treated as the unknown variable and found. 4 CONCLUSIONS The paper has considered the measurement of internal stress and Young's modulus of boron-doped silicon by considering the static and dynamic behaviours of a bridge structure. The procedure based upon the static response of the beam uses compliance measurements derived from nanoindenter tests to predict the internal stress. The values of internal stress are predicted to be and MPa for the samples tested and these values compare well with published data. Young's modulus is not predicted by this method because the high value of the internal stress causes the bridge to behave like a taut string rather than a exurally sti beam. The method using the dynamic response of the bridge utilizes measurements of natural frequency as the determining quantity and for the structures tested gives values of 84 2 and 90 1 MPa for internal stress and and GPa for Young's modulus. These values compare well with other published data. The dynamic method overcomes the problem of high stress by measuring the response of the higher modes of vibration where bending e ects start to become signi cant. The measurement of as many natural frequencies as possible is shown to be important for estimation of Young's modulus. For the polyimide bridge the dynamic method gives the internal stress and Young's modulus to be 6:99 0:16 MPa and 8:56 1:36 GPa respectively. The measurement of the thickness of the bridge is the principal source of error associated with the estimation of Young's modulus. Vibration tests were much simpler to perform than the static tests and are shown to produce superior results. A disadvantage of the dynamic test is that a value for material density is needed. Proc Instn Mech Engrs Vol 214 Part C C07998 ß IMechE 2000

7 THE STRUCTURAL CHARACTERISTICS OF MICROENGINEERED BRIDGES 357 ACKNOWLEDGEMENT The authors would like to express their thanks to the Engineering and Physical Sciences Research Council for providing the funding for this work under the Mechanical Characterization of Micromechanical Structures GR/J43356 Contract. REFERENCES 1 Bourouina, T., Vauge, C. and Hekki, H. Variational method for tensile stress evaluation and application to heavily boron-doped square shaped diaphragms. Sensors and Actuators A, June 1995, 49(1±2), 21±27. 2 Ding, X., Ko, W. H. and Mansour, J. M. Residual stress and mechanical properties of boron-doped p** plus-silicon lms. Sensors and Actuators A, April 1990, 23(1±3), 866± Burdess, J. S., Harris, A. J., Cruickshank, J. J., Wood, D. and Cooper, G. Silicon membrane gyroscope with electrostatic actuation and sensing. In Micromechanical and Microfabrication Process Technology, Proceedings of the SPIE, Vol. 2642, 1995, pp. 74±83 (SPIE, Bellingham, Washington). 4 Burdess, J. S., Harris, A. J., Wood, D., Pitcher, R. J. and Glennie, D. A system for the mechanical characterisation of microstructures. J. Microelectromech. Systems, December 1997, 6(4). 5 Page, T. F., Oliver, W. C. and McHargue, C. J. The deformation behaviour of ceramic crystals subjected to very low load (nano) indentations. J. Mater. Res., February 1992, 7(2), 450± Weighs, T. P., Hong, S., Bravman, J. C. and Nix, W. D. Mechanical de ection of cantilever microbeams: a new technique for testing the mechanical properties of thin lms. J. Mater. Res., September±October 1988, 3(5), 931± Zhang, L. M., Uttamchandi, D. and Culshaw, B. Measurement of the mechanical properties of silicon microresonators. Sensors and Actuators A, 1991, 29, 79±84. 8 Zook, J. D., Burns, D. W., Guckel, H., Sniegowski, J. J., Engelstad, R. L. and Feng, Z. Characteristics of polysilicon resonant microbeams. Sensors and Actuators A, 1992, 35, 51±59. 9 Pember, A., Smith, J. and Kemhadjian, H. Long term stability of silicon bridge oscillators fabricated using the boron etch stop. Sensors and Actuators A, 1995, 46±47, 51± Roark, R. R. Formulas for Stress and Strain (McGraw-Hill, New York). 11 Rao, S. S. Mechanical Vibrations, 1986 (Addison-Wesley, Reading, Massachusetts). 12 Marden, M. Fundamentals of Microfabrication, 1997 (CRC Press, Boca Raton, Florida). 13 MATLAB, Optimization Toolbox, User's Guide, 1996 (The Math Works Incorporated). 14 Auld, B. A. Acoustic Fields and Waves in Solids, Vol. 2, 1973 (John Wiley, New York). APPENDIX The compliance function f k, a for a bridge structure with internal stress is given by f k, a ˆ where C 4 C a3 C 3 C a4 cosh ka 1Š C 3 C a3 C 2 C a4 sinh ka kaš C 2 3 C 2C 4 C 2 ˆ sinh k, C 3 ˆ cosh k 1, C 4 ˆ sinh k k C a3 ˆ cosh k 1 a Š 1 C a4 ˆ sinh k 1 a Š k 1 a C07998 ß IMechE 2000 Proc Instn Mech Engrs Vol 214 Part C

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