Appendix G Properties of Electronic & MEMS Semiconducting Materials Table G.1 Physical properties of common semiconductor materials used in microsensor and MEMS technology. Values are taken at 20°C where appropriate unless stated otherwise. Source: most of the values were taken from either the Handbook of Chemistry and Physics (CRC Press, Inc.) or MacMillans Chemical and Physical Data (James and Lord, MacMillans Press Ltd, 1992). These values are intended only as a guide and we recommend, wherever possible, validation against other sources Property Density, p m (kg/m 3 ) Melting point, Boiling point, r bp (°c) Electrical conductivity, a (10 3 S/cm) Energy band gap, E(eV) Thermal conductivity, k(W/m/K) Specific heat capacity, c p (J/K/kg) Temperature expansivity, a/(10~ 6 /K) Dielectric constant, s r Young's modulus, E m (GPa) Yield strength, Y (GPa) Breakdown field, (MV/cm) Si(c) 2330 1410 2355 4 x 10 –3 1.1 168 678 2.6 11.7 190 1 6.9 0.3 Material: Poly-Si Ge C 1 2320 5350 3510 937 3827 2830 4827 3 x 10 –5 ~10 –17 1.1 0.67 5.4 34 60 1000-2600 678 310 523 2-2.8 5.7 1 16.3 5.1 161 - 542 - - — 0.1 GaAs 5316 1238 N/A 1.35 370 5.7 12 - - 0.5 1 in diamond form and [111] Miller index. This page intentionally left blank Appendix H Properties of Electronic & MEMS Ceramic and Polymer Materials Table H.1 Physical properties of common ceramic and insulating materials used in microsensor and MEMS technology. Values are taken at 20 °C where appropriate unless stated otherwise. Source: most of the values were taken from either the Handbook of Chemistry and Physics (CRC Press, Inc.) or MacMillans Chemical and Physical Data (James and Lord, MacMillans Press Ltd, 1992). These values are intended only as a guide and we recommend, where possible, validation against other sources Property Density, p m (kg/m 3 ) Melting point, T mp (°C) Boiling point, Thermal conductivity, Specific heat capacity 1 , c p (J/K/kg) Temperature expansivity, a l (10 –6 /K) Dielectric constant, e r Energy band gap, Young's modulus, E m (GPa) Breakdown strength (MV/cm) A1 2 O 3 SiO 2 3965 2200 2045 1713 2980 2230 38 1.4 730 730 - — 85 3.8 18-23 9 57-85 39 10 Material: Quartz ||c Quartz J_c SiC Si 3 N 4 ZnO 2650 2650 3216 3100 5606 3070 1900 1975 - - - - 12 6.7 110 20 6 730 730 710 600-800 6.8 12.2 3.3 4.4 44 — — — - 3 5 3.35 72 72 440 304 - - - 2.3 - 1 Values at a temperature of 0°C. 466 APPENDIX H: PROPERTIES OF ELECTRONIC & MEMS CERAMIC MATERIALS Table H.2 Physical properties of common polymer and plastic materials used in microsensor and MEMS technology. Values are taken at 20 °C where appropriate unless stated otherwise. Source: most of the values were taken from either the Handbook of Chemistry and Physics (CRC Press, Inc.) or MacMillans Chemical and Physical Data (James and Lord, MacMillans Press Ltd, 1992). These values are intended only as a guide and we recommend, where possible, validation against other sources Material: Property Density, p m (kg/m 3 ) Maximum working point, ^max (°C) Thermal conductivity, k (W/m/K) Specific heat capacity, c p (J/K/kg) Temperature expansivity, a l (10 –5 /K) Dielectric constant, £r Young's modulus, E m (GPa) Tensile strength, Y m (MPa) Nylon Polyimide Polythene 2 1120-1170 1000-1600 926-941 100 - 71-93 0.25-0.27 0.15 0.33-0.42 1600-1900 1100 1900 28 - 14-16 3.7-5.5 - 2.3 1-4 ~3.1 0.4-1.3 50-90 69-104 8-24 PTFE 1 2100-2300 260 0.24-0.25 1050 10 2 0.4 10-31 PVC PVDF 1300-1400 1750-1780 70-74 150 0.16 0.1 840-1170 5-18 8-14 3.0-4.0 2.9 2.9 2.1 34-62 36-56 1 Trade name is Teflon. 2 Medium density. Appendix I I1 THE COMPLEX RECIPROCITY RELATION The propagation of waves in a waveguide structure, such as the SAW substrate, with a thin film overlay can be accomplished using the technique of modal analysis. Any waveguide can support the propagation of an infinite number of solutions, or waveguide modes. As long as the set of modes is mathematically complete, any function can then be expressed as an infinite sum of these waveguide modes. Modal analysis also requires that the individual modes must be orthogonal, just as the sine and cosine functions in Fourier analysis are orthogonal. The proof of orthogonality requires the derivation of the complex reciprocity relation. The quasi-static and electromagnetic field equations needed derive the reciprocity. If an acoustic force vector F is included, the tensor equation given in Chapter 10 may now be written as follows T ij.j = pu i - F i , (I.1) and T ij = C ijkl u k,l + e ijk <t>,k (I.2) Another necessary equation is the time derivative of the electrical displacement equation D i = e ikl u k,l - £ ij <l> j (I.3) A second solution to the equations can be denoted by primed field quantities (T', u', f', . . .) and the complex conjugate of the field equations can be written (I.5) (I.6) 468 APPENDIX I: COMPLEX RECIPROCITY RELATION AND PERTURBATION ANALYSIS If Equations (I.l),(I.2) and (I.3) are multiplied by u*, u*j and —0 * added, then we have Tijju',* + Tijufj - Drf* ' (1.7) Similarly, Equations (I.4),(I.5) (I.6) can be multiplied by u i , u i,j and — <£,/ added to give (I.8) In a lossless material, the material constant matrices are real and symmetric and therefore satisfy the relations, P = P*, C ijkl =C klij , Sij = e*ij and e ijk - e* kij Under these conditions, Equations (I.7) and (I.8) can be added to get (I.9) Tijju'i* + T^UIJ + T^jiii } = +TUij - D 0J - D'fa \ (I.10) By combining various derivative terms, it is possible to rewrite the preceding equation as + - F'*Ui (I.11) In order to take into account an electrical forcing function, it can be assumed that some free charge distribution (q) is present. The equation D i,i = q simply describes Gauss's Law. This can be brought into the equation preceding by presenting equation in the following form: (I. 12) Figure Al Coordinate system for SAW waves showing the propagation vector. APPENDIX I: COMPLEX RECIPROCITY RELATION AND PERTURBATION ANALYSIS 469 Rearranging this would give ,i i ,i \ ' / ,i Adding Equations (I.12) and (I.13) and multiplying the result by —1 gives (I.14) T ,'.* T*,: \ I u i • •' — lijU i — 7 f jUj I j IpiiiUj The preceding equation is known as complex reciprocity relation. On the right-hand side of the Equation (I.14), each of the term inside the parentheses contains the product of a conjugated function with a nonconjugated function. Because the functional form of e ia>t is assumed in modal analysis, any term of the form A * B will have no time dependence (e-J wr gV^f = 1); therefore, the derivative term will be zero. The complex reciprocity relation for e l(at variations is then reduced to t'? + F'fui + 40'* + q*<t>\ (I.15) I2 PERTURBATION THEORY The perturbation analysis begins with the complex reciprocity relation with the added assumption that no sources (body force or charge density variations) are present within the substrate. The unperturbed structure that will be analysed is shown in Chapter 10, Figure 10.3. The semi-infinite piezoelectric slab extends to infinity in the ± x 1 and ± x 2 and — x 3 directions. The electromagnetic quantities extend out of the substrate into vacuum that extends to infinity in the +x 3 direction. The field quantities are independent of X2 and propagation is in the +x 1 direction. Under these conditions, all j = 2 terms in Equation (I.16) are zero and integration with respect to x 3 can be performed across the substrate resulting in / ° 9 / ^ -oo OX\ \ 9 / —T*i'j' — ' 4- > + The unprimed quantities represent the unperturbed field quantities in the bare substrate, whereas the primed quantities will represent the perturbed field quantities in the layered structure. The unperturbed quantities have the functional form e jkxl because the substrate is lossless, and the perturbed quantities vary as e jk'xl . If Ak is defined as Ak = k' - k, 470 APPENDIX I: COMPLEX RECIPROCITY RELATION AND PERTURBATION ANALYSIS then Equation (I.17) becomes or J I'' f ^(- ./-oo dxj In order to calculate the change in the propagation constant Ak, using Equation (I.19), it is necessary to know the perturbed field distributions. Without this knowledge, it is necessary to make several approximations to simplify the analysis. The first approximation can be made by replacing the perturbed quantities in the denominator with unperturbed quantities - because the perturbation is small. The denominator can then be simplified using the power flow equation for piezoelectric media that is given by 1 r 0 = -Re / (-TnuJ *• J -00 + Dfad* 3 (I.20) where P is the average power flow in the x\ direction per unit width. Because the real part of any complex variable x can be calculated using Re(jc) = (I.21) Equation (I.20) can be written as (I.22) The assumption that all quantities in the denominator are unperturbed quantities allows Equation (I. 10) to be written as (I.23) 4P The next assumption that greatly simplifies the calculation of Equation (I.23) is that small mechanical and electrical perturbations are independent and can therefore be calculated separately. For mechanical boundary perturbations, the electrical boundary conditions are assumed to be unperturbed, and for electrical boundary perturbations, the mechanical boundary conditions are considered unperturbed. When both types of boundary perturba- tions are present, the perturbations are calculated separately and added. When perturbation involves only mechanical effects, the last two terms in the numerator of Equation (I.23) are ignored. In addition, unperturbed situation includes a stress free surface at x 3 = 0 (i.e. T i3 = T 3i , = 0) and all field quantities disappear at x 3 = — oo; APPENDIX I: COMPLEX RECIPROCITY RELATION AND PERTURBATION ANALYSIS 471 therefore, only the second term evaluated at x 3 = 0 is nonzero. Because u* has the time dependence e jwt the equation reduces to 4P At this point, it is helpful to eliminate the stress terms by calculating T' i3 in terms of particle displacements. If the perturbed SAW propagation is in the Love wave layer structure, the surface at x = h is stress-free. If the film is isotropic, the Lame constants A. and /x, can be used to define the elastic constants as, c 11 = c 22 = c 33 = + ^ c 12 = c 21 = c 13 = c 31 = c 23 = c 32 = A. c 44 = c 55 = c 66 = U and all other c ij = 0 where the subscripts are Voigt index subscripts. Under these conditions and in the absence of external body forces, Equation (I.1) becomes T' 13,3 + J k'T' 11 = ~v 2 pu' 1 T' 23,3 + J k'T' 21 = ~v 2 pu' 2 (1.24) T' 33,3 + J k'T' 31 = ~v 2 pu' 3 where p is the density of the film. At this point, it is beneficial to describe the field quantities in the film as Taylor series about x 3 = h, the film surface. Any quantity / that is a function of x 3 is then approximated by f (x 3 ) = a 2 f 2! dxl (1.25) where only first two terms will be significant when h is small. Because the boundary is stress-free at x 3 = h and the stress tensor is symmetric in the absence of external body torques, it is also true that T' 13 (h) = T' 31 (h) = T' 23 (h) = T' 32 (h) = T' 33 (h) = 0 (I.26) 472 APPENDIX I: COMPLEX RECIPROCITY RELATION AND PERTURBATION ANALYSIS If the Equations (I.25) and (I.26) are substituted for the field quantities in Equation (I.17) and all terms containing x 3 = h are ignored (x 3 — h is small), Equation (I.24) becomes T' 23,3 (h) + jk'T' 2l (h) = T' 33,3 (h) = -a> 2 pu\(h) -a> 2 pu' 2 (h) -co 2 pu' 3 (h) (I.27) In order to simplify Equation (I.27), it is necessary to calculate T' 11 and T' 2l in terms of the particle displacements. Using Equations E.2, the elastic constants in terms of Lame constants and (I.26) the Voigt index notation described in [31], it can be shown that (I.28) T' 11 (h) = -jk'fr + 2n)u\(h) + Xu' 33 (h) T 33 (h) = 0 = jk'Xu\(h) + (A + 2fi)u' 33 (h) If the last two equations are solved simultaneously to eliminate u' 33 (h), the result is (I.29) The use of Equations (I.25) and (I.26) leads to the following derivation for the stress derivative terms T' i3 (0) = T' i3 (h) - hT' i33 (h) (I.30) Therefore, T' i3 (0) = -hT' i33 (h) (I.31) Substituting Equations (I.28) and (I.27) determines T i33 (h) and therefore T' i3 (0). The resul- tant equations are (I.32) and T' 33 (0) = ha> 2 pu' 3 (h) where v = CD/K = co/k' = the velocity of the wave. Equation (I.28) can be used to rewrite T' 33 (0) = ha) 2 pu' 3 (h) as A* T vhco ~4p~ p =-• v 2 . K(0) - 4) u' 2 (h)u 2 (0) + pu' 3 (h)u* 3 (0) _ (1.33) [...]... Structures: Dynamics and Control, John Wiley & Sons, New York, p 467, 1996 • Fiber Optic Smart Structures, E Udd, ed., John Wiley & Sons, New York, p 671, 1995 • Smart Material Structures: Modelling and Estimation and Control, H T Bauks, John Wiley & Sons, New York, p 467, 1996 • Smart Materials and Structures, M V Gandhi and B S Thompson, Chapman and Hall, London, p 309, 1992 (I) • Smart Materials and Structures,... Sensors International Conference on Solid-State Sensors and Actuators (akaTransducers) MEMS Annual 1987-now Biennial Biennial Annual 1984-now 1983-now 1981-now Annual 1989-now 488 APPENDIX L: WEBOGRAPHY Table L3 List of some 20 universities active in the fields of microsensors and MEMS University website is given because the activities tend to be spread across many departments or within research centres... p 362, 1991 (P) • Microsystems Technology and Microrobotics, S Fatikow and Rembold, Springer, New York, 408, 1997 (I) • Microtransducer CAD, A Nathan and H Baltes, Springer, New York, p 300, 1998 • Smart Electronics and MEMS, A Hariz, V K Varadan and O Reinhold, eds., Vol 3242, Proc SPIE, p 398, 1997 (P) (12) Smart materials and structures • Active Materials and Adaptive Structures, G Knowles, ed.,... G /it/Vc -r z ./)! i + _£:_^— t = Ze) —s- r l Ga(Rs + Zg) °^[ - i + eei ua(Ks + z,e)-i \+e/]\ (J.9) with Oe equal to (o>Ct + Ba)(Rs + Ze) (J.10) -iOt) /2G^ —s exp (Oti/2) exp (—/ (J-ll) (J.12) with #c equal to a)Ct(Rs + Ze) REFERENCES Cross, P S and Schmidt, R V (1977) "Coupled surface acoustic wave resonators," BellSys Tech Journal, 56, 144 7-1 482 Campbell, C (1998) SAW Devices for Mobile and Wireless... nature and are intended as exemplars for background or supplementary information - these are labelled by the letter 'I.' Other books are more advanced in nature and provide further technical detail to the reader than that given in our book - these are labeled by the letter 'A.' The list of books coming under the topics of microsensors, MEMS, and smart devices is intended to be comprehensive - that... Switzerland Germany Sweden France Spain www europractice com/MCI www tronics-mst.com www normic com www.csem.ch www.acreo.se www-dta cea.fr/homeJeti-uk htm www.cnm.es www.aml.co.uk www-imt unine ch www isit.fthg.de www.laas.fr www.mic.dtu.uk www mst-design co uk www nmrc ucc ie www sintef no/ecy MEMSOI NORMIC MC4 MAGFAB ACREO CEA-LETI CNM AML IMT ISIT LAAS MIC MST-DESIGN NMRC SINTEF UK Switzerland Germany... (I) • Designing Field-Effect 1990 (A) Transistors, E Oxner, ed., McGraw-Hill, New York, p 296, • Electrical and Thermal Characterisation of MESFETs, HEMTs, and HBTs, R Ahholt, Artech House, Boston, p 310, 1995 (A) • Electronic Devices and Circuit: Discrete and Integrated, M S Ghausi, Holt, Rinehart & Winston, New York, p 808, 1985 (I) • Metal-Semiconductor Contacts, E H Rhoderick and R H Williams, Oxford... http:/www wiley com http:/www vch-wiley com http :/iop.org/journals/ Sensor Technology Sensors Update Smart Materials and Structures Table L.2 List of some important conference series in Sensors and MEMS The list excludes many National conferences Title Frequency Duration European Conference on Solid-State Sensors and Actuators (aka Eurosensors) Hilton Head Solid-State Sensor and Actuator Workshop International... Conditioning, R Pallas-Areny and J G Webster, John Wiley & Sons, Chichester, 1991 • Transducers and Interfacing, B R Bannister and D G Whitehead, Van Nostrand, New York, 1986 (7) Microsensors and silicon sensors • Biosensors: Microelectrochemical Sensors, M Lambrechts and W Sansen, IOP Publis hing, Bristol, p 320, 1992 • Integrated Optics, Microstructures and Sensors, M Tabib-Azar, Kluwer Academic... (I.45) and the complex conjugate of the denominator, we obtain the variations in propagation velocity due to acoustic-electric interaction as equal to *—* I' / *—* * i I v ~" / — / ' — an / \~~ i w ~ \j ~ *J \ •" i w • ~~ f-* / \ /• T A f-" \ where Ay ^j / , n^l2 = -w(eair + £/, ) ^/ /SC and en7 = 4F /T^ ( 1-4 7) This page intentionally left blank Appendix J Coupled-mode Modeling of a SAW Device Material and . Polythene 2 112 0-1 170 100 0-1 600 92 6-9 41 100 - 7 1-9 3 0.2 5-0 .27 0.15 0.3 3-0 .42 160 0-1 900 1100 1900 28 - 1 4-1 6 3. 7-5 .5 - 2.3 1-4 ~3.1 0. 4-1 .3 5 0-9 0 6 9-1 04 8-2 4 PTFE 1 210 0-2 300 260 0.2 4-0 .25 1050 10 2 0.4 1 0-3 1 PVC . 8-2 4 PTFE 1 210 0-2 300 260 0.2 4-0 .25 1050 10 2 0.4 1 0-3 1 PVC PVDF 130 0-1 400 175 0-1 780 7 0-7 4 150 0.16 0.1 84 0-1 170 5-1 8 8-1 4 3. 0-4 .0 2.9 2.9 2.1 3 4-6 2 3 6-5 6 1 Trade name is Teflon. 2 Medium density. Appendix . ZnO 2650 2650 3216 3100 5606 3070 1900 1975 - - - - 12 6.7 110 20 6 730 730 710 60 0-8 00 6.8 12.2 3.3 4.4 44 — — — - 3 5 3.35 72 72 440 304 - - - 2.3 - 1 Values at a temperature of 0°C. 466 APPENDIX