Steady and Unsteady Thermo-Strucural Simulation of Thermally Actuated Micro- and Nano- Structures

Transcription

1 Louisiana Sae Universi LSU Digial Commons LSU Docoral Disseraions Graduae School 013 Sead and Unsead hermo-srucural Simulaion of hermall Acuaed Micro- and Nano- Srucures Elham Maghsoudi Louisiana Sae Universi and Agriculural and Mechanical College Follow his and addiional works a: hps://digialcommons.lsu.edu/gradschool_disseraions Par of he Mechanical Engineering Commons Recommended Ciaion Maghsoudi Elham "Sead and Unsead hermo-srucural Simulaion of hermall Acuaed Micro- and Nano-Srucures" (013). LSU Docoral Disseraions. 50. hps://digialcommons.lsu.edu/gradschool_disseraions/50 his Disseraion is brough o ou for free and open access b he Graduae School a LSU Digial Commons. I has been acceped for inclusion in LSU Docoral Disseraions b an auhoried graduae school edior of LSU Digial Commons. For more informaion please conacgraded@lsu.edu.

2 SEADY AND UNSEADY HERMO-SRUCURAL SIMULAION OF HERMALLY ACUAED MICRO- AND NANO-SRUCURES A Disseraion Submied o he Graduae Facul of he Louisiana Sae Universi and Agriculural and Mechanical College in parial fulfillmen of he requiremens for he degree of Docor of Philosoph in he Deparmen of Mechanical and Indusrial Engineering b Elham Maghsoudi B.S. Iran Universi of Science and echnolog 005 M.S. Swansea Universi 009 December 013

3 ACKNOWLEDGEMENS I would like o epress m sincere appreciaion o m advisor Dr. Michael James Marin for his suppor and guidance hroughou he research. I appreciae his encouragemen on aking courses aending conferences workshops rainings and inernships relaed o compuaional science which enhanced m knowledge and acceleraed he research. I also appreciae his paience and undersanding wih his eam. I would acknowledge suppor provided b a NASA and he Louisiana Space Gran Consorium hrough LEQSF(010)-DAR-4 Robus Nano-Mechanical Memor for Space Eploraion. I would hank Professor Michael Murph from Louisiana Sae Universi and Dr. Harish Manohara from NASA Je Propulsion Laboraor (JPL) California Insiue of echnolog for useful suggesions on hermal memor design and simulaion. I would also like o hank Professor Guoqiang Li from Louisiana Sae Universi and Professor Hosam Fah from Pennslvania Sae Universi for heir helpful commens on srucural simulaion. I would like o hank m commiee members Dr. Sumana Achara Dr. Dimiris Nikiopoulos Dr. Ingmar Schoegl and Dr. Marcio de Queiro for heir ime and useful commens. I would also like o hank he dean s represenaive Dr. Jerr rahan for monioring he general and final eaminaions. Special hanks o m parens Faraneh and Mohammad for encouragemen and inspiraion all he suppor and love. ii

4 ABLE OF CONENS ACKNOWLEDGEMENS... ii ABSRAC... v CHAPER 1: INRODUCION Acuaion Modes Applicaions Acuaion Phsics Disseraion Organiaion CHAPER : HREE DIMENSIONAL SEADY AND UNSEADY GOVERNING HERMO-SRUCURAL EQUAIONS Geomer and Boundar Condiions ransien Hea ransfer Equaion hermal Boundar Condiion Hea ransfer Coefficien ransien Srucural Equaion Srucural Boundar Condiions... 3 CHAPER 3: HREE DIMENSIONAL SIMULAION OF SEADY SAE HEA RANSFER IN MICRO- AND NANO-BRIDGES Geomer and Boundar Condiions hermo-srucural Formulaion Non-dimensional Parameers Scaling of Saisical Effecs Resuls and Discussion Conclusion CHAPER 4: HERMALLY ACUAED MICRO-SWICHES: AN APPLICAION O SEADY RESPONSE O CONSAN HEAING Geomer and Boundar Condiions Formulaion and Simulaion Mehod Resuls and Discussion Conclusion CHAPER 5: RANSIEN RESPONSE O CONSAN HEAING WIH APPLICAIONS O NANO-MECHANICAL MEMORY Spacecraf Sorage Memor Requiremens heor Simulaion iii

5 5.4 Resuls Conclusion CHAPER 6: RANSIEN RESPONSE IN NANO-BRIDGES Geomer and Boundar Condiions ransien hermo-srucural Formulaion Validaion of Unsead hermo-srucural Equaions emperaure Dependen hermal Properies Sensiivi Sud Resuls and Discussion Conclusion CHAPER 7: SUMMARY AND DISCUSSION hermal Posiioning hermal Acuaion Fuure Projecs REFERENCES VIA iv

6 ABSRAC his disseraion provides a hermo-srucural simulaion for nano-scale and micro-scale srucures wih pinned and fied boundar condiions which are eiher hermall posiioned buckled or acuaed. he sud begins wih simulaing a pinned-pinned beam in micro-scale and nano-scale. he sead sae hermo-srucural equaion is solved numericall using an implici Finie Difference mehod implemened in Malab o obain he hermal posiioning response which is he hermall sead sae cener displacemen b adding a consan ime-independen hea flu o he srucure. he resuls show he sead sae hermal displacemen of he ssem is a funcion of he geomer pressure maerial properies and consan hea flu in he free molecular model while his value is independen of pressure in he coninuum model. he hermal posiioning simulaion is used o improve he hermal efficienc of a hermal micro-swich b inroducing various heaing configuraions. he second hermal mode is hermal buckling which is used o inroduce a new hermal buckling sorage nano-memor. Using an unsead simulaion he power requiremens for hermal acuaions opimal geomer and wrie ime of he device for various maerials are invesigaed. he resuls show ha his memor consume a low power in he order of 1 nj per bi and has a daa sorage densi of bis/cm 3 which is accepable in comparison wih he curren memor devices. hermal buckling nano-memor is also radiaion-proeced making i a good alernaive for space eploraion compuer ssems operaing in high radiaion and elecromagneic environmens. In conras wih hermal posiioning and buckling hermal acuaion applies imedependen hea load leading o vibraion in he srucure. An implici Finie Difference mehod implemened in C++ was used o solve he coupled ransien hermo-srucural equaions wih v

7 consan hermal properies while an eplici approach was used o solve he variable properies hermo-srucural equaions. he response he cener displacemen in a doubl-clamped bridge is racked b ime and decomposed o he sead sae and vibraion ampliudes. he resuls show ha consan hermal properies assumpion is limied for small hea addiions lower han 1 mw. hermal acuaion resuls are applicable in simulaing he dnamic behavior of nano-scale devices used for swiching nano-manufacuring and measuremen. vi

8 CHAPER 1: INRODUCION 1.1 Acuaion Modes Nano-Elecro-Mechanical Ssems NEMS and Micro-Elecro-Mechanical Ssems MEMS devices are acuaed using several mehods including elecrosaic hermal mechanical and magneic acuaion. he response of hese ssems is he displacemen of he srucure. Figure 1.1 shows a micro-bridge and a micro-canilever schemaic. Boh srucures are acuaed using a disribued force along he srucure. he cener displacemen deermines he response of he bridge srucure. he ip displacemen deermines he response of he canilever srucure. (a) 1 (b) Figure. 1.1 he acuaion and response in NEMS and MEMS srucures (a) Micro-bridge (b) Micro-canilever If he acuaion force is creaed b heaing and creaing hermal sresses wihin he srucure hen he ssem is said o be hermall acuaed. he erm "hermal acuaion" is commonl used o describe hree relaed phenomenon. In he firs case a consan hea load is added o a ssem o change he saic deflecion of he ssem b changing he hermal sresses asmmericall o adjus he displacemen of he ssem. In he second case a consan hea load is added o a ssem

9 o change he saic deflecion of he ssem b hermall buckling he ssem. In he hird case a ime-dependen hea inpu is added o he ssem. his leads o ime-dependen hermal sresses and vibraion in he ssem. o avoid ambigui he firs case changing he hermal sresses asmmericall o adjus he displacemen of he ssem is referred o as "hermal posiioning" and he second case changing he hermal sresses asmmericall o buckle he ssem is called "hermal buckling". he erm "hermal acuaion" is lef for he hird case ime-dependen hea load and ime-dependen response. Figure 1. shows he acuaion modes caegories. Figure. 1. hermal posiioning hermal buckling and hermal acuaion he hermal response of he NEMS and MEMS devices deermines he mechanical response of hese ssems. he displacemen which is he mechanical response is deermined b hermal sress. An eample of his is hermal bimorph acuaion applied for a Parallel Dip-pen Nanolihograph (PDN) probe arra (Wang e al. 004). An adjusable dc volage source was used o suppl power o he acuaors. he silicon DPN probe hermal displacemen variaion verses acuaion power corresponds o a linear ssem. he maimum displacemen of 8 μm was achieved a an acuaion power of mw which is adequae o overcome he surface adhesion force.

10 1.1.1 hermal Posiioning As eplained in he inroducion hermal posiioning refers o changing he saic posiion of a micro- or nano-srucure. In his case he hermal acuaion is a sead consan hea load and no a funcion of ime. he mechanical response of he ssem which is defined as displacemen is deermined when he ssem reaches o hermall sead sae. his procedure can be applied o he phenomenon in which he sud of he response b ime is no imporan. For eample assume an open sae of a hermal-swich is called posiion 1 and a closed sae of i is called posiion. hermal posiioning can deermine he swich hermal power consumpion and he swich efficienc when i moves from posiion 1 o posiion. However i is no capable of discussing how long i akes for he swich o move from posiion 1 o posiion and an oher ime dependen erms such as opening ime lag and ec hermal Buckling hermal buckling occurs when he aial force due o hermal sress in he srucure becomes equal o he criical buckling force. A his ime he srucure is assumed buckled. he buckled srucure remains in is posiion wihou requiring adding eernal hea or force. hermal buckling is a famil of hermal posiioning as shown in Figure 1.1. I also refers o changing he saic posiion of a micro- or nano-srucure specificall from an un-buckled posiion o a buckled posiion. he hermal acuaion is a consan hea load. However he response is a ime dependen phenomenon. I has an applicaion in buckling beam memor sorages which will be discussed laer in chaper hermal Acuaion While appling a consan hea load leads o saic srucural behavior sinusoidal heaing leads o vibraion in he bridge. In hermal acuaion he hea load is no a consan. Insead i is a 3

11 harmonic funcion of ime. As a resul he srucure response of he hermal acuaion is imeindependen. In conras wih hermal posiioning hermal acuaion capures he ransien response of a hermal swich while i is changing posiion from open o close. Using he ime-dependen response he closing ime and opening ime (afer sop heaing he srucure) can be deermined. Figure 1.3 shows ime-dependen eciaion and response in a nano-scale resonaor. Figure. 1.3 ime-dependen eciaion and response in a hermall acuaed micro-srucure 1. Applicaions hermall posiioned and acuaed micro and nano-scale srucures have scienific and commercial applicaions including wireless communicaion ssems (Nguen 1999; Jensen e al. 007) nano-scale fabricaion (Piner e al. 1999; Fan e al. 000; Ding e al. 010) daa sorage (Veiger e al. 00) and opical ssems (Pal e al. 009). MEMS and NEMS (Micro- and Nano- Elecro-Mechanical Ssems) devices have widhs and lenghs on he order of microns and hicknesses on he order of 100 nanomeers. hese devices are eiher mechanicall or hermall acuaed. Laser beams hermall acuae devices in laboraor eperimens while elecrical heaers 4

12 are proposed for general applicaions. he mechanical response of hese ssems is he displacemen of he srucure. A varie of eperimens have been conduced in hermall acuaed devices wih a consan hea load. hese include he use of micro-scale elecrical heaers and laser which conrol he emperaure of phoonic crsal devices. A mehod using a laser beam was repored b Faraon e al. (Faraon e al. 007) o locall conrol he emperaure of he phoonic crsal b eernal heaing. In a relaed sud Faraon and Vuckovic (Faraon and Vuckovic 009) used his mehod b appling elecrical heaers o conrol he resonan frequenc of InAs quanum dos coupled o GaAs phoonic crsal resonaors enabling independen conrol of large ensembles of phoonic devices a high uning speeds. Zhu e al. (Zhu e al. 011) inegraed a novel elecrohermal nanoposiioner wih an elecrohermal acuaor in he same MEMS chip wihou he need for inclusion of era elecrical insulaion fabricaion process or assembling wo chips. he on-chip displacemen sensing enables a feedback conrol capabili. he also calibraed he nanoposiioner b suding he displacemen variaions b he acuaion volage. Wang e al. (Wang e al. 008) developed a wavelengh-selecive phoonic swich using hermo-opic effecs. he used a local micro-heaer o une he resonance wavelengh o a arge signal wavelengh Nano- and Micro-Bridge Swiches here are several hermall acuaed micro- and nano-devices such as swiches and resonaors. In hermal swiching an infrared laser beam was used b Hashimoo e al. (Hashimoo e al. 1994) o conrol he magneic acuaor. A spo-heaing infrared laser heas one of he saors in he swich and reduces is magneiaion. As a resul he force beween he wo saors changes and he oher saor aracs he armaure. Laser heaing can also pla a role as eernal eciaion sources in bridge resonaors (Svielski e al. 008). 5

13 Figure. 1.4 Magneosaic bisable microswich wih elecrohermal acuaors (Wu e al. 010) hermal-elecrosaic micro-swiches work based on displacemen variaion of he swich srucure wihou using a movable hermal conac liquid. Carmona e al. modeled hermal acuaion in a hermo-pneumaic micro-pump (Carmona e al. 003). hermal acuaion increases he air emperaure in his pump generaing he membrane deflecion. hermal-elecrosaic micro-swiches resul in a low acuaion volage bu consume a high power. here are various pes of hermal swiches such as bridges canilevers and laeral series of hese wo pes (Rebei 003). Various numerical and eperimenal sudies have been performed on differen designs of micro-swiches. Reid e al. simulaed a canilever micro-swich o invesigae he effecs of changes in he geomer on he operaional range of he swich (Reid and Sarman 003). Couu e al. modeled a canilever micro-swich pull-in volage collapse volage and conac force predicions analicall. he resuls were compared wih eperimenal resuls (Couu e al. 004). Dequenes e al. sudied he pull-in volage of several nanoube-based nano-swiches including he bridges and canilevers and proposed a coninuum model for he simulaion of carbon-nanoube-based NEMS swiches (Dequenes e al. 00). 6

14 Blond e al. designed and fabricaed a hermall acuaed silicon niride micro-swich bridge as shown in Figure Eperimenal measuremens showed ha he swich has a low acuaion volage. he swich also offered high isolaion and mechanical proecion (Blond e al. 001). Figure. 1.5 he hermall acuaed micro-swich designed b Blond e al. (001) Duong invesigaed he environmenal condiions effecs on he reliabili of five differen micro-swiches. All models are caegoried as bridge swiches bu wih differen geomerical designs. he acuaion volage and deflecion variaions versus emperaure variaions were sudied using modeling and eperimenal ools. he swich dilaion and pull-in volage variaions b emperaure and emperaure ccles were invesigaed (Duong e al. 005). 1.. Nano- and Micro-Bridge Memor he concep of mechanical bisabili of a doubl-clamped bridge was used o implemen nonvolaile elecro-mechanical memor (Nagami e al. 01) and volaile mechanical memor (Bade e al. 004) operaing based on he displacemen of he bridge. A poeniall simpler nonvolaile memor device is he buckled-beam nano-mechanical memor (Hälg e al. 1990; Roodenburg e al. 009; Charlo e al. 008). hese devices have been elecrosaicl acuaed. Afer he beam is buckled hrough applicaion of an elecrosaicl generaed force he bisable beam remains buckled afer he power is removed. hese devices have been successfull demonsraed in laboraor eperimens (Roodenburg e al. 009) bu have no been he subjec of eensive performance or reliabili analsis. 7

15 he buckling-beam concep was also used in fabricaion of oher devices such as snapping membranes which are used as emperaure indicaors. hese devices are hermall acuaed and buckle downward wih an increase in emperaure beond a criical value. Jus as in elecrosaicall acuaed devices he remain in he downward buckled sae as he emperaure decreases back o is iniial value (Ara e al. 006). 1.3 Acuaion Phsics Because hese devices are hermall acuaed he hea ransfer equaion in he ssem is a conrolling parameer. Because he lengh and he widh of he srucure are in he micro scale and he hickness is in he order of hundreds of nanomeer he coninuum conducive hea ransfer equaion can be used inside he device. Boh coninuum and free molecular regime were sudied for convecive hea ransfer depending on he air pressure. he dimensionless Knudsen number deermines which convecive hea ransfer coefficien mus be used in he simulaions as shown in Figure he Knudsen number is given b: Kn L L (1.1) where λ is he mean free pah of he gas and L is he lengh scale of he beam. he mean free pah is given b (Kennard 1983): 1 n d g (1.) where n is he densi of paricles per volume dg is he collision diameer of he gas molecule. he Knudsen number deermines if coninuum formulaions of he momenum and energ equaions are appropriae o use. As shown in Figure. 1.6 for Knudsen numbers more han 10 a free molecular hea ransfer coefficien is valid. However for Knudsen numbers less han 0.1 coninuum hea ransfer coefficien is valid. 8

16 Figure. 1.6 Knudsen number deermines he regime he ambien pressure will change he Knudsen number and as a resul he flow regime. Lee e al. sudied he hermal characerisics of a heaed microcanilever considering conjugae hea ransfer conducion in he srucure and convecion beween he srucure and air or helium for a range of pressures (Lee e al. 007). he showed for Kn>1 hermal ranspor from he canilever heaer is gas pressure dependen while for Kn<1 i remains consan. he coninuum breaks down for Kn numbers more han 1 affecs he convecion beween he cooling gas and he NEMS and MEMS devices which is an imporan parameer affecing he mechanical response of he ssem. A mehod was presened o opimie a recangular wo-beam micro-elecro-mechanical hermal acuaor using a simplified hea ransfer mechanism including convecion (Hicke e al. 00). he invesigaed buckling deflecion due o hermal sress a uniform emperaure and presened he measuremen of he deflecion as a funcion of acuaion volage. Phinne e al. (Phinne e al. 010) sudied he effecs of he pressure of he surrounding gas on he hermal performance of silicon bridges 10 μm wide.5 μm hick and 00 o 400 μm long. he measured he emperaure profile eperimenall in a nirogen amosphere. he numerical resuls obained using a FEM (Finie Elemen Mehod) were compared wih he eperimenal resuls. heir resuls show ha gas phase hea ransfer is an imporan parameer for devices of his sie a ambien pressure. As he pressure decreases below 5 orr he effecs of convecion become minimal. Hea ransfer in package MEMS was sudied numericall using he Direc Simulaion Mone Carlo mehod b Liu 9

17 e al. (Liu e al. 007). he package was assumed as an enclosure wih a ho chip a he boom. he resuls showed if he boom emperaure was parl uniform a he cener he hea ransfer on he ho chip surface was enhanced comparing wih he uniform emperaure boom case. Marin and Houson (Marin and Houson 009) characeried high frequenc vibraing canilever and bridge srucures below he coninuum limi. A ne free molecular hea ransfer for he ssem was calculaed. Several researchers have sudied conducive hea ransfer effecs on he mechanical response of he ssem. Ilic e al. (Ilic e al. 010) invesigaed he energ ranspor mechanisms in silicon nano-canilevers eperimenall. A modulaed laser beam added hermal energ along he device laer. he hermal response of he device was obained for unsead hermal acuaors in he absence of convecion. Masropaolo and Cheung (Masropaolo and Cheung 008) sudied he hermo-mechanical behavior of silicon carbide clamped-clamped bridges resonaors. he simulaions were performed for differen elecrode lenghs widhs and spacings o invesigae he geomer effecs on he mechanical response of he ssem. In a relaed sud Masropaolo e al. (Masropaolo e al. 009) designed he elecro-hermall acuaed silicon carbide ring resonaors o achieve higher resonan frequenc compared o beam resonaors. he double elecrode configuraion was found o be he mos efficien design for acuaion wih a relaivel high average emperaure and he larges heaed area. 1.4 Disseraion Organiaion he hermo-srucural equaions are derived in chaper. his chaper begins wih he derivaion of unsead hea ransfer equaion for boh consan and variable hermal properies wih and wihou hermo-elasic erms. he unsead srucural equaion is derived using he equaion of moion for an infiniesimal elemen on he beam and Bernoulli beam assumpion. he srucural equaions boundar condiions are shown for boh pinned-pinned and fied-fied ends condiions. 10

18 Chapers 3 and 4 focus on hermal posiioning of micro- and nano-srucures. A hreedimensional pinned-pinned micro- and nano-bridge is simulaed in chaper 3. he sead response of he ssem wih sead acuaion (consan hea load) is discussed in his chaper o presen a universal scaling for he behavior of micro- and nano-scale bridge srucures over a range of dimensions (micro-scale o nano-scale) maerials (silicon silicon carbide and CVD diamond) ambien hea ransfer condiions (free molecular and coninuum approaches) and hea loads. he hea conducion equaion is solved numericall using a Finie Difference mehod implemened in Malab o obain he emperaure disribuion in he bridge. Using he nodal emperaure disribuion hermal sress due o he emperaure difference wih respec o he wall emperaure is calculaed. he srucural equaion is solved numericall o ge he displacemen along he beam. he algorihm and deails of ime domain and space domain discreiaion mehods are discussed in deails in chaper 3. he resuls are non-dimensionalied o provide insigh ino hermal posiioning across a range of srucure lengh scales and maerial properies. In addiion he coninuum level effecs are scaled wih he saisical mechanics effecs. Chaper 4 uses he Finie Difference mehod which is presened in chaper 3 o solve he hermo-srucural equaions o obain closing power consumpion and hermal efficienc of a hermall acuaed bridge micro-swich for various heaing configuraions. Ideall swich opening and closing imes should be calculaed hrough he full dnamic simulaion. However he sead sae approach wih simpler srucural and hea equaions is used in his sud o esimae he efficienc and power consumpion. hree heaing configuraions are used: disribued hea a he op surface concenraed hea a he cener of he op surface and concenraed hea a he sides of he op surface. A ime =0 a consan hea load q is applied o he op of he bridge unil he bridge reaches o a hermall sead sae condiion. he heaing procedure is also performed for 11

19 closed-swich models wih differen hermal boundar condiions. Simulaions are performed for wo differen maerials: silicon and silicon niride. Chaper 5 applies he wo echnologies of buckling beam and hermal eciaion o design a sorage memor. In his work a uni bridge of an arra of buckling-beam memor is simulaed using he Finie Difference mehod presened in chaper 3. he geomer and boundar condiions of he uni bridge is comparable wih he geomer presened in chaper 3. he hermal boundar condiions are idenical while he srucural boundar condiions are changed o fied-fied boundar condiions. he hea load is preserved sead and consan; however ransien response o consan heaing is sudied o esimae he power requiremens for hermal acuaions opimal geomer and wrie ime of he device for various maerials. In conrar wih consan hea load eciaion appling sinusoidal heaing leads o vibraion in he bridge. Chaper 6 discusses he harmonic response of he hermall acuaed doubl clamped nano-bridge o harmonic acuaion. he hea load is defined as a sinusoidal harmonic acuaion. An implici Finie Difference solver implemened in C++ was used o solve he hermo-srucural equaions wih consan hermal properies while an implici Finie Difference mehod was used for emperaure dependen hermal properies. he significance of emperaure dependen hermal properies wih he hea ampliude is sudied. he phase dela beween he eciaion and he response and he ampliude of he response are invesigaed b various acuaion frequencies a he pressure lower han amospheric pressure. 1

20 CHAPER : HREE DIMENSIONAL SEADY AND UNSEADY GOVERNING HERMO-SRUCURAL EQUAIONS his chaper discusses he derivaions of he governing equaion in deails for wo differen boundar condiions of a pinned-pinned bridge and a fied-fied bridge..1 Geomer and Boundar Condiions he sud in his disseraion is performed on a micro- and nano-bridge srucure wih pinnedpinned boundar condiions for sead hermo-srucural simulaion as will be used in chapers 3 and 4. he boundar condiions are changed o fied-fied for unsead hermo-srucural simulaion as will be discussed in chaper 6. Figure.1 shows he hermal boundar condiion in he bridge which is idenical in he disseraion. he hea ransfer includes conducion wihin he beam as well as convecion beween he beam and he quiescen gas are considered. he hea addiion is modeled as a consan hea load q" applied o he op surface of he beam for he sead case as will be discussed in chapers 3 and 4 while he hea addiion changes o harmonic hea load for he unsead sud as will be discussed in chaper 6. Figure..1 he geomer and boundar condiions in he model he cooling gas is air wih consan properies a ambien emperaure. Various maerials are used such as crsalline silicon silicon carbide and CVD diamond. he hermal properies of all maerials are assumed o be consan and defined a he wall emperaure. he wall emperaure w is fied a boh ends of he beam. 13

21 14. ransien Hea ransfer Equaion he full hree-dimensional ransien hea conducion equaion for consan hermal properies is given as follows (Incropera e al. 007): c k p (.1) where k is he hermal conducivi of he solid. his equaion is solved using an implici Finie Difference approach a each ime sep o obain he nodal emperaure disribuion when he hermal properies are assumed consan a he wall emperaure. his equaion is used in he sead sud in chapers 3 and 4. However he ransien hea ransfer equaion is modified in order o ake ino accoun he hermo-elasic erms and also emperaure dependenc of he hermal properies. Adding he hea equaion o he equaion of moion ields he following equaion (Landau and Lifshi 1959; Lifshi and Roukes 1999; Serra and Bonaldi 008): ) (1 1 j jj v p p c E k k k c (.) where νp is he poisson raio and cv is he specific hea per uni volume. εjj is he srain in he normal direcion in all hree direcions. Using Hookes law he srain componens are defined as follows (Lifshi and Roukes 1999): w E (.3) w p E (.4) where σ is he normal sress along he ais due o bending. he second erms in he righ hand side of Eqs. (.3) and (.4) represens he srain due o hea epansion.

22 he longiudinal normal srain of an elemen wihin he beam depends on is locaion on he cross secion and he radius of curvaure of he beam's longiudinal ais a ha poin as shown in Figure.: R c (.5) where ε is he srain a he bar middle surface: u 1 v (.6) where u is he aial displacemen and ν is he ransverse displacemen. Assuming pure ransverse moion in he direcion and making he usual Euler-Bernoulli assumpion ha he ransverse dimensions of he beam are sufficienl small compared wih he lengh and he radius of curvaure Rc ha an plane cross secion iniiall perpendicular o he ais of he beam remains plane and perpendicular o he neural surface during bending. As a resul ε is assumed negligible in comparison wih he oher erms. he curvaure equaion from calculus is: R c d d d 1 d 3 (.7) Figure.. he radius of curvaure in a deformed elemen 15

23 16 which for acual beams can be simplified because he slope d/d is small. he square is even smaller and can be negleced as a higher order erm. aking ino accoun his simplificaion Eq. (.7) becomes:. d d R c (.8) Combining Eqs. (.3) (.5) and (.8) he following epression is obained for he normal sress due o bending:. w E (.9) Subsiuing Eq. (.9) o Eq. (.4) he normal srains in and direcions become:. ) (1 w p p (.10) Subsiuing Eqs. (.5) and (.10) in Eq. (.) he finalied ransien hea ransfer equaion including hermo-elasic erms and emperaure dependen hermal properies is obained as follows:. ) ( c E k k k c c E p p p p p (.11) Equaion (.11) allows he incorporaion of hermal properies as emperaure dependen variables and akes ino accoun he hermo-elasic erms. his equaion is solved using an eplici Finie Difference approach o obain he nodal emperaure disribuion a each ime sep in chaper 6 for he sud of he unsead hermo-srucural behavior.

24 .3 hermal Boundar Condiion he hermal boundar condiions are idenical in his sud from chaper 3 o chaper 6. here are hree differen pes of boundar condiion in his model as shown in Figure.1. he Dirichle boundar condiion a boh ends is epressed as w. (.1) here are wo differen pes of Neumann boundar condiions which are applied o he heaed and unheaed surfaces. Equaion (.13) shows he boundar condiion for he heaed surface and Eq. (.14) shows he boundar condiion for he unheaed surfaces: k h a n q" (.13) k n h. a (.14).4 Hea ransfer Coefficien here are wo differen approaches o compuing he hea ransfer coefficien h. he seleced mehod depends on he Knudsen number. he Knudsen number is defined as Kn w W (.15) where λ is he mean free pah of he gas and W is he widh scale of he beam. For a dilue gas assumpion he molecular mean free pah is given b k b P a gas d g (.16) where kb is he Bolmann consan a is he ambien emperaure Pgas is he pressure of he gas and dg is he effecive diameer of he gas molecule (Lee e al. 007 ). he molecular collisions beween he beam and he gaseous medium decreases as Knudsen number approaches 1. he hea ransfer coefficien in a gas wih he Knudsen number more han 1 will be (Marin e al. 009): 17

25 1 h ni 1 3 kb a 8 m (.17) where σ is he hermal accommodaion coefficien γ is he specific hea raio of he gas m is he molecular weigh of he gas and ni is he number densi of he gas. For an ideal gas ni can be calculaed using Pgas ni m R a (.18) where R is he ideal gas consan. he coninuum heor is valid for a Knudsen number less han If he air is quiescen he hea ransfer from he srucure can be modeled as conducion ino an infinie medium. his can be reaed as an effecive value of he hea ransfer coefficien h: * h qsskgas (.19) WL where qss * is he sead-sae dimensionless conducion hea rae kgas is he conducivi of he gas and W and l are he widh and he lengh of he beam respecivel. For an infiniel hin recangle of lengh L widh W and consan emperaure in an infinie medium of consan emperaure qss * is esimaed as 0.93 (Incropera e al. 007). In he ransiion region where he Knudsen number is beween 0.01 and 1 i is compuaionall difficul o esimae he hea ransfer coefficien. In his sud he resuls in his region are assumed o fall beween he molecular and coninuum resuls. Previous researchers sudied he hea ransfer coefficien as a funcion of pressure (Lee e al. 007; Park e al. 007; Ramanan and Yang 009; Naraanaswam and Gu 011). A combined compuaional and eperimenal sud of he hea ransfer from a micro-heaer modeled a microcanilever beam b emploing he hermal resisance nework. A low pressures he convecive 18

26 hea ransfer coefficien is a funcion of pressure as epeced from free-molecular heor. As he pressure passes 40 kpa hea ransfer coefficien goes o a consan value of 150 W/mK (Lee e al. 007). his suggess ha a coninuum condiion has been reached. In order o validae Eq. (.19) hea ransfer coefficien is calculaed for he lengh of 100 μm and he widh of 10 μm. Eq. (.19) will give a value of 1890 W/m K suggesing i is a reliable order-of-magniude approimaion..5 ransien Srucural Equaion In order o derive he ransien srucural equaion equaion of moion for an infiniesimal elemen wih he lengh d on a Bernoulli beam shown in Figure.3 is derived. Figure..3 Infiniesimal elemen on a Bernoulli beam Figure.4 shows he forces acing on he elemen. Figure.4 (see a) shows he shear force variaion along he elemen. he equaion of moion in his case will be as follows: A V. (.0) Assuming he shear force does no change along he elemen as no eernal force is acing on he beam and onl he bending momen due o heaing he ssem changes along he elemen as shown in Figure.4 (see b) he balance of force and momenum gives: V M. (.1) 19

27 0 (a) (b) Figure..4 Eernal force acing on he elemen (a) Shear forces (b) Bending momen Combining Eqs. (.0) and (.1) he equaion of moion is obained as follows: M A (.) where M is he ne momen represening he momen due o he mechanical deflecion and he momen due o he hermal eciaion. aking ino accoun he compression aial force in he ssem Eq. (.) changes as follows (Jones 006): 0 M N A (.3) where N is he aial force in he beam. Bending momen and aial force can be obained using he aial sress given in Eq. (.3). Bending momen is calculaed b inegraing he aial sress over each plane: da E da E da E da E da E da M w w w (.4) where ε does no var over he bar cross-secional area because he origin is a he cenroid. As a resul he firs inegral erm in he las line is ero. he bending momen can be wrien as:

28 1 h w M EI da E EI M (.5) where Mh is he hermal bending momen and I is he momen of ineria:. 1 3 Wd dd da I (.6) he calculaed bending momen shown in Eq. (.5) includes he mechanical bending momen erm and hermal bending momen. he aial force N is calculaed using he aial sress given in Eq. (.3): da E da E da E da E da E da N w w w (.7) where he second inegral in he las line is ero because he firs momen of area is ero a he cenroidal ais. Using Eq. (.6) Eq. (.7) can be rewrien as follows: 1 1 h w N v u EA da E v u EA N (.8) where Nh is he hermal compression force. Subsiuing Eqs. (.5) and (.8) in Eq. (.3) and adding he damping erm he ransien srucural equaion becomes: 0 1 F M EI N v u EA A D h h (.9)

29 where he damping erm FD is he ne flow drag (Marin and Houson 008) in he free molecular regime. I is equivalen wih he consan damping erm Cf muliplied b vibraion veloci. FD is given b: 1 U C W d U c WP F f a w n n gas D (.30) where σn normal accommodaion coefficien and σ angenial accommodaion coefficien are assumed o be equal o 1. he hermal veloci c is defined in Eq. (.31) and U() is he veloci in he vibraion direcion which is defined as Eq. (.3). m k c a b (.31) ) ( U (.3) where m is he mass of he gas molecules in his case air wih m= kg. Oher erms kb and a are previousl defined as Bolmann consan and ambien emperaure. Subsiuing Eqs. (.30) in Eq. (.9) he finalied dnamic srucural equaion is obained: 0. 1 C dd E EI dd E v u EA A f w w (.33) he above equaion is used in chaper 6 o sud he dnamic behavior of a beam wih fied-fied boundar condiions. A sensiivi sud in chaper 6 will show ha he aial force and he srain a he bar middle surface can be ignored in comparison wih he bending momen. For pinned-pinned boundar condiions where he bending momen is ero a he suppors he sead srucural equaion can be wrien as follows: 0. M h EI (.34)

30 Equaion (.34) is used in chapers 3 and 4 o sud he sead hermo-srucural behavior of a bridge wih pinned-pinned boundar condiions..6 Srucural Boundar Condiions he bending momen is ero a he suppors for pinned-pinned boundar condiions which is used in chapers 3 and 4. In his case equaion (.34) is numericall solved using a Finie Difference mehod. he following boundar condiions is applied o he boh ends where he momen is ero a he boh ends: 0 L 0. (.35) In chaper 6 dnamic srucural equaion Eq. (.33) is solved for a fied-fied bridge o sud he unsead hermo-srucural behavior of he srucure. In his case slope ero and displacemen ero boundar condiions are applied a he boh ends: 0 L 0 0 L 0. (.36) 3

31 CHAPER 3: HREE DIMENSIONAL SIMULAION OF SEADY SAE HEA RANSFER IN MICRO- AND NANO- BRIDGES While appling sinusoidal heaing leads o vibraion in he bridge a consan hea load leads o saic srucural displacemen. his chaper simulaes a 3-dimensional pinned-pinned bridge in micro and nano-scales o obain a universal scaling for he behavior of micro- and nano-scale bridge srucures. A Finie Difference mehod implemened in Malab was used o solve he hermo-srucural equaions numericall. Simulaions are performed over a range of dimensions maerials ambien hea ransfer condiions and consan hea loads. his chaper performs his sud for hree differen maerials: silicon silicon carbide and CVD diamond. Boh free molecular and coninuum approaches are used o define he hea ransfer coefficien. he resuls are nondimensionalied o provide insigh ino hermal posiioning across a range of srucure lengh scales and maerial properies. In addiion he coninuum level effecs are scaled wih he saisical mechanics effecs. his chaper anales changing he hermal sresses asmmericall o adjus he displacemen of he ssem. o avoid ambigui his process will be referred o as "hermal posiioning" insead of "hermal acuaion." 3.1 Geomer and Boundar Condiions he sud uses wo geomeries: one pinned-pinned beam wih a lengh L of 100 microns a widh W of 10 microns and a hickness d of 3 microns and a second beam wih a lengh of 10 microns a widh of 1 micron and a hickness of 300 nanomeers. he hea ransfer and srucural equaions are solved numericall using a finie-difference mehod. Conducion wihin he beam as well as convecion beween he beam and he quiescen gas are considered. he hea addiion is modeled as a consan hea load q" applied o he op surface of he beam. his also corresponds 4

32 o he hea addiion of a hin resisive film. Figure. 3.1 shows he geomer and boundar condiions of he model. Figure. 3.1 he geomer and boundar condiions in he model he cooling gas is air wih consan properies a ambien emperaure. Simulaions are performed for hree differen maerials: crsalline silicon silicon carbide and CVD diamond. he hermal properies of all maerials are assumed o be consan and defined a he wall emperaure. he wall emperaure w is fied a boh ends of he beam. 3. hermo-srucural Formulaion 3..1 Hea ransfer Coefficien here are wo differen approaches o compuing he hea ransfer coefficien h. he appropriae mehod depends on he Knudsen number. he Knudsen number is defined as Kn w W (.15) where λ is he mean free pah of he gas and w is he widh scale of he beam. For a dilue gas assumpion he molecular mean free pah is given b k b P a gas d g (.16) where kb is he Bolmann consan a is he ambien emperaure Pgas is he pressure of he gas and dg is he effecive diameer of he gas molecule (Lee e al. 007 ). he molecular collisions beween he beam and he gaseous medium decreases as Knudsen number approaches 1. he hea ransfer coefficien in a gas wih he Knudsen number more han 1 will be (Marin e al. 009): 5

33 1 h ni 1 3 kb a 8 m (.17) where σ is he hermal accommodaion coefficien γ is he specific hea raio of he gas m is he molecular weigh of he gas and ni is he number densi of he gas. For an ideal gas ni can be calculaed using Pgas ni m R a (.18) where R is he ideal gas consan. he coninuum heor is valid for a Knudsen number less han If he air is quiescen he hea ransfer from he srucure can be modeled as conducion ino an infinie medium. his can be reaed as an effecive value of he hea ransfer coefficien h: * h qsskgas (.19) WL where qss * is he sead-sae dimensionless conducion hea rae kgas is he conducivi of he gas and w and l are he widh and he lengh of he beam respecivel. For an infiniel hin recangle of lengh L widh W and consan emperaure in an infinie medium of consan emperaure qss * is esimaed as 0.93 (Incropera e al. 007). In he ransiion region where he Knudsen number is beween 0.01 and 1 i is compuaionall difficul o esimae he hea ransfer coefficien. In his sud he resuls in his region are assumed o fall beween he molecular and coninuum resuls. Previous researchers sudied he hea ransfer coefficien as a funcion of pressure (Lee e al. 007; Park e al. 007; Ramanan and Yang 009; Naraanaswam and Gu 011). A combined compuaional and eperimenal sud of he hea ransfer from a micro-heaer modeled a microcanilever beam b emploing he hermal resisance nework. A low pressures he convecive 6

34 hea ransfer coefficien is a funcion of pressure as epeced from free-molecular heor. As he pressure passes 40 kpa hea ransfer coefficien goes o a consan value of 150 W/mK (Lee e al. 007). his suggess ha a coninuum condiion has been reached. In order o validae Eq. (.19) hea ransfer coefficien is calculaed for he lengh of 100 μm and he widh of 10 μm. Eq. (.19) will give a value of 1890 W/m K suggesing i is a reliable order-of-magniude approimaion. 3.. Governing Equaions In he curren work he full 3-dimensional sead hea conducion equaion Eq. (3.1) which is he sead sae form of Eq. (.1) is solved numericall o obain he emperaure disribuion: k 0 (3.1) where k is he hermal conducivi of he solid. here are hree differen pes of boundar condiion in his model (Figure. 3.1). he Dirichle boundar condiion a boh ends is epressed as w. (.1) here are wo differen pes of Neumann boundar condiions which are applied o he heaed and unheaed surfaces. Eq. (.13) shows he boundar condiion for he heaed surface and Eq. (.14) shows he boundar condiion for he unheaed surfaces: k h a n q" (.13) k n h. a (.14) he hea conducion equaion is solved numericall using a finie difference mehod (Incropera e al. 007). As a resul he emperaure is obained a each node. hese resuls are used o calculae he hermal momen due o hermal sresses as previousl shown in Eq. (.5): 7

35 M h E da. w (3.) In order o obain he displacemen disribuion along he beam Eq. (.34) is numericall solved using a finie difference mehod. EI M h 0. (.34) he momen and displacemen are ero a boh ends for pinned-pinned boundar condiion: 0 L 0. (.35) For small emperaure variaions he modulus of elasici E is a consan Discreiaion and Algorihm In order o solve he ssem of equaions numericall using a Finie Difference mehod he domain is discreied uniforml in he and direcions. he grid spacing is called δ δ and δ. he sud of hermal behavior of he ssem is performed b solving he ransien form of Eq. (3.1). he soluion is performed ieraivel unil he hermall sead sae soluion is obained. he emperaure difference beween wo consecuive ieraions is defined as a roo mean square error. As soon as his error becomes smaller han he olerance he ssem is assumed hermall sead sae. A cenral Finie Difference mehod is used for spaial implemenaion and uncondiionall sable full implici scheme is used for ime implemenaion (Paankar 1980). An eplici ime discreiaion scheme can be used alernaivel o avoid he complicaions of mari inversion mehods in solving he discreied ssem of equaions. However in his scheme he ime sep δ mus be small enough o saisf he sabili condiions of: c k (3.3) 8

36 c k c k. (3.4) (3.5) In his problem he grid spacing δ δ and δ are in he order of nanomeer. his requires he ime sep δ o be a mos in he order of 10-1 second o saisf he sabili condiion. his increases he compuaion ime o obain he sead sae resuls. However he implici scheme is uncondiionall sable and independen of ime sep. As a resul an implici scheme is seleced o reduce he compuaion ime and obain he resul faser Discreied Equaions. he number of boundaries deermines he number of discreied equaions are needed o solve his problem using Eqs (.1) o (.14). Equaion (3.6) shows he discreied ransien form of Eq. (.1) for he inerior nodes (all nodes ecep he boundaries): n i j k n1 n1 n1 1 Fo Fo Fo i j k Fo i1 j k i 1 j k n1 n1 n1 n1 Fo Fo i j1 k i j1 k i j k1 i j k1 (3.6) where i j and k deermine he locaion of discreied emperaure values in he and ais respecivel. he curren ime sep is deermined b n and he ne ime sep is deermined b n+1. Fo is non-dimensional Fourier number defined as following in each direcion: Fo Fo Fo k c k c k c. (3.7) (3.8) (3.9) 9

37 Equaion (.13) resuls in hree discreied equaions for he op heaed surface and lines which are used for he nodes on: he op heaed surface he fron op line and he back op line. Equaion (3.10) shows he discreied equaion for he nodes locaed on he op heaed surface: n i j p n1 n1 n1 1 Fo Fo Fo Bi Fo Fo Fo n1 n1 n1 Fo Bi Fo q i j1 p i j1 p i j p i j p1 i1 j p a i1 j p c node (3.10) where qnode is he hea added o each node. qnode is obained from he consan hea flu q added o he op surface: q node q" q". (3.11) Bi Bi and Bi are he dimensionless Bio numbers defined in he and direcions for he discreied equaions: h Bi k (3.1) Bi Bi h k h. k (3.13) (3.14) Unheaed boundar condiion equaion (Eq. (.14)) resuls in five discreied equaions which are applied o he nodes on: he unheaed back surface unheaed fron surface unheaed boom surface unheaed boom fron line and unheaed boom back line. Equaion (3.15) shows he discreied equaion for unheaed boom back line: n i l1 n1 n1 n1 1 Fo Fo Fo Bi Fo Bi Fo Fo Fo n1 i l 11 Fo n1 i l i l1 Bi Fo a i1 l1 Bi i1 l1 Fo. a (3.15) 30

38 Appling he Dirichle boundar condiion (Eq. (.1)) creaes 10 more discreied equaions which impose he nodes a he lef surface and lines as well as he righ surface and lines o be a he consan emperaure w. able 3.1 summaries he discreied equaions for hermal analsis where m is he number of nodes in he direcion l is number of nodes in he direcion and p is he number of nodes in he direcion Ssem of Equaions (Mari Consrucion and Soluion). In order o solve he ssem of equaions one mari and wo vecors are defined: Coefficien mari KK variable vecor A and consan vecor B. Coefficien mari KK is a M b M mari which includes he coefficiens muliplies in nodal emperaure values a ime n+1 (ijk n+1 ) where M is he oal number of nodes in he domain. he coefficiens are deermined b he discreied equaions shown in able 3.1. he vecor A includes he unknown variables nodal emperaure values a ime n+1 (ijk n+1 ). he vecor B includes he consans including he consan wall emperaure w and nodal hea added o he nodes a he op boundaries. he nodes corresponding o no hea addiion or convecive boundar condiions inerior nodes impose ero o he corresponding elemen in he mari B. he ssem of equaions should be se so KK A=B saisfies he discreied equaions. Since he mari KK and he vecor B are consan boh of hem are se ouside of he ransien loop (Figure. 3.). he purpose is o obain he vecor A a each ime sep which includes he nodal emperaure values a he ne ime sep n+1. An ieraive Gauss Seidel mehod (Kresig 005) is used o solve he ssem of equaions ieraivel. When he ieraive error reduces o he olerance he obained nodal emperaures ijk n+1 represen he emperaure a he ne ime sep. his means he previous nodal emperaure values can be subsiued wih he curren one o move o he ne ime sep calculaion. 31

39 3 able 3.1. hermal discreied equaions Node locaion Discreied Equaion Lef surface and lines w n k j 1 Righ surface and lines w n k j m Inerior nodes n k j i n k j i n k j i n k j i n k j i n k j i n k j i n k j i Fo Fo Fo Fo Fo Fo op heaed surface node a n p j i n p j i n p j i n p j i n p j i n p j i n p j i q c Fo Bi Fo Fo Fo Fo Bi Fo Fo Fo op fron heaed line node a a n p i n p i n p i n p i n p i n p i q c Fo Bi Fo Bi Fo Fo Fo Fo Bi Fo Bi Fo Fo Fo op back heaed line node a a n p l i n p l i n p l i n p l i n p l i n p l i q c Fo Bi Fo Bi Fo Fo Fo Fo Bi Fo Bi Fo Fo Fo Back unheaed surface a n k l i n k l i n k l i n k l i n k l i n k l i n k l i Fo Bi Fo Fo Fo Fo Bi Fo Fo Fo Fron unheaed surface a n k i n k i n k i n k i n k i n k i n k i Fo Bi Fo Fo Fo Fo Bi Fo Fo Fo

40 (able 3.1. coninued) Node locaion Discreied Equaion Boom unheaed surface n i j1 1 Fo Fo Fo Bi Fo n1 n1 n1 n1 Fo Fo Boom fron unheaed line i1 j1 Bi Fo n i11 Fo Bi a 1 Fo n1 i111 Fo a i1 j1 Fo n1 i111 Fo Fo i j11 n1 i1 Bi i j11 Fo Fo n1 i1 n1 i j1 Fo Bi Fo n1 i j Bi Fo n1 i11 a Boom back unheaed line n i l1 Fo Bi 1 Fo n1 i1 l1 Fo a Fo n1 i1 l1 Fo Fo n1 i l11 Bi Fo Fo Bi n1 i l Fo n1 i l1 Bi Fo a Srucural equaion Eq. (.34) is solved coupled wih he hermal equaions a each ime sep. he same approach as hermal analsis is used. he onl difference is ha his equaion is one dimensional in he direcion. he inegraion for obaining he hermal momen is performed using a rapeoidal mehod (Kresig 005). Ne he equaion is discreied using he same cenral Finie Difference mehod used in hermal analsis. Since he beam is assumed pinnedpinned ero displacemen is imposed o boh ends. he ssem of equaions is consruced using one mari and wo vecors and he ssem of equaions is solved using a Gauss Seidel mehod. Figure 3. shows he algorihm of he implemened code for solving hermal srucural equaion. 33

41 3.3 Non-dimensional Parameers Figure. 3. Algorihm flow char he resuls are non-dimensionalied using he Buckingham-Pi heorem (Whie 011). Eigh phsical variables are seleced as δ k h q" L W d and α. Here δ is he cener displacemen which is he maimum displacemen along he beam (δ=υ(=l/)). Four independen phsical variables were seleced as δ k l and α. As a resul five dimensionless groups are obained as follows: * k q" L (3.16) Bi h L k (3.17) * w * d W L d L (3.18) (3.19) 34

42 * k q" L (3.0) where δ * is he dimensionless cener displacemen. Bi is he Bio number. w * and d * are aspec raios Δ is he average emperaure difference and Δ * is he dimensionless emperaure difference. he dimensionless cener displacemen δ * is a funcion of dimensionless parameers Bi w * and d *. 3.4 Scaling of Saisical Effecs he displacemen creaed b hermal sresses can be scaled agains wo nano-mechanical effecs: quanum mechanical limis (Schwab and Roukes 005) and saisical mechanical effecs (Sowe e al. 1997). Because quanum mechanical effecs onl appear in conducion and mechanical moion a low emperaures he are ignored in his analsis. A room emperaure he saisical mechanics effecs are measurable. he individual aoms of he canilever are vibraing causing a ver small bu quanifiable displacemen which is defined as hermal noise (Sowe e al. 1997; Daskos e al. 003). In order o define he hermal noise he mechanical siffness ks of he ssem is defined as (Young and Budnas 00): k s 19EI 3 L. (3.1) he hermal displacemen is calculaed using he following formulaion: h kb k s (3.) where is he device emperaure (Sowe e al. 1997). he Displacemen Raio (DR) is defined as he cener displacemen creaed b he hermal sresses divided b he hermal displacemen. DR. h 35 (3.3)

43 he Displacemen Raio behavior will be a specific funcion of hea flu added o he op surface q". Dimensionless parameers are used o derive his funcion. he cener displacemen is a linear funcion of dimensionless cener displacemen (δ~δ*). he hermal displacemen changes wih he square roo of he emperaure as given in Eq. (3.). he average emperaure difference varies linearl b dimensionless emperaure difference (Δ=(-w)~Δ*). Subsiuing Eq. (3.16) and (3.) ino Eq. (3.3) he Displacemen Raio is a funcion of q" he geomer and maerial properies of he beam: DR k k k * q" L b s w q" l k * (3.4) where δ* and Δ* are funcions of Bio number. 3.5 Resuls and Discussion Convergence and Validaion he model was run for nodes in a quarer of he model. Grid independence was verified b repeaing he simulaions for and he resuls show ha is sufficien o obain accurae resuls. Figure 3.3 shows he convergence hisor in linear (see a) and logarihmic (see b) scales. he emperaure residuals deca o he convergence crieria which is Iniial resuls for an unheaed srucure were compared wih he analical soluion for a fin wih a recangular cross secion (Incropera 007) o validae he discreiaion. he analical soluion for he emperaure disribuion along a fin wih consan emperaure w a boh ends and ambien emperaure of a is given in Eq (3.5): 36

44 d hw d sinh L h W sinh kwd kwd ( ) ( w a ) a. hw d sinh kwd (3.5) (a) (b) Figure. 3.3 Convergence hisor (a) Linear scale (b) Logarihmic scale Figure 3.4 shows he analical soluion versus numerical soluion for an unheaed bridge wih he wall emperaure of 310 K and he ambien emperaure of 90 K. he hea ransfer coefficien is 411 W/m K a he pressure of 100 Pa. he resuls are in good agreemen for a bridge wih a lengh of 100 microns. Figure. 3.4 Numerical resuls verified wih analical fin resuls 37

45 Figure 3.5 shows he displacemen variaions along he ais for he free molecular model in silicon MEMS beam. he maimum displacemen occurs a he cener. his behavior agrees wih he emperaure disribuion behavior which shows displacemen variaions b oal hea correspond o a linear ssem. his is qualiaivel in agreemen wih eperimens b Wang e al. (Wang 004). he simulaion of nano-bridges shows a higher amoun of hea is required o deflec he bridges in he order of angsroms. Figure. 3.5 Displacemen variaions along he lengh of he bridge for differen hea flues in MEMS beam (Free molecular mehod - Bi=3.16e-6) 3.5. Dimensional Resuls Figure 3.6 (see a and b) show cener displacemen variaions b pressure for silicon in he MEMS and NEMS beams. Figure 3.6 (see c and d) show he same variaions for silicon carbide and CVD diamond in he NEMS beam. In all cases he cener displacemen increases as he hea load increases. I is clear ha cener displacemen is pressure-dependen in he free molecular approach due o he hea ransfer coefficien h. he hea ransfer coefficien is pressure-dependen in he free molecular case as eplained in secion.4. In nano-scale models cener displacemen behavior versus pressure was invesigaed for hree differen maerials. he coninuum and free molecular cases show a difference in cener displacemen of 10.5% 13.5% and 5.1% for silicon 38

46 silicon carbide and CVD diamond respecivel. he difference is less for CVD diamond because i is more conducive han he oher maerials. he percenage difference doubles in he microscale model. CVD diamond also shows cener displacemen of wo orders of magniude smaller han silicon and silicon carbide. (a) (c) (b) (d) Figure. 3.6 Cener displacemen variaion b pressure: (a) Silicon micro-scale (b) Silicon nano-scale (c) Silicon carbide nano-scale (d) CVD diamond nano-scale Non-dimensional Resuls Figure 3.7 shows dimensionless emperaure difference variaions b he Bio number. he plos collapse for differen maerials. Equaion (3.) and (.34) suggess ha dimensionless cener displacemen δ* can be obained from Δ*. his suggess ha dimensionless cener displacemen plos versus Bio number are epeced o collapse for differen maerials. 39

47 Figure. 3.7 Dimensionless emperaure difference versus he Bio number for differen maerials Figure 3.8 shows he dimensionless cener displacemen of he bridge versus he Bio number for hree differen maerials commonl used in micro-devices. he dimensionless cener displacemen plos collapse. I shows ha he cener displacemen changes as a funcion of maerial properies hea ransfer coefficien and he geomer. he resuls deermine he power required o posiion he wave guide. his also gives an esimae of he power required o acuae he nanomechanical bridge used as a swich. he micro-scale silicon resuls also collapse wih nano-scale resuls of oher maerials. I shows ha as long as w * and d * are idenical he dimensionless cener displacemen will be idenical. he resuls collapse for he micro- and nano-scales which shows ha he ssem behaves linearl. Figure. 3.8 Dimensionless displacemen versus he Bio number for various maerials 40

48 Figure 3.9 shows he changes of dimensionless cener displacemen of he bridge versus he Bio number in silicon showing he effecs of he geomer changes. As L increases while W and d are consan he displacemen increases. hese variaions are more significan a high Bio numbers. hese variaions occur for wo reasons. he hea flu added o he ssem b he heaing laser or elecrical heaers is consan. he firs reason is ha he oal hea power a he op given in Eq. (3.6) increases due o he increase in he op surface area: q q" LW. (3.6) he second reason is ha he displacemen varies relaivel wih he lengh of he bridge. he sud of widh variaions effecs on he dimensionless cener displacemen is performed. he lengh L and hickness d remain consan while onl widh W of he bridge decreases. he displacemen decreases due o he decrease in he oal laser hea power bu he displacemen variaion is negligible. Also he bridge hickness variaion effec on he displacemen is invesigaed a consan L and W. An decrease in he hickness resuls in an increase in he displacemen due o he volume reducion for he same oal hea hough he variaions are negligible. (a) (b) Figure. 3.9 Dimensionless displacemen versus he Bio number: (a) For differen maerials (b) For differen bridge lenghs 41

49 3.5.4 Saisical Effecs Figure 3.10 shows he hermal displacemen variaions b emperaure for silicon silicon carbide and CVD diamond in nano scale and silicon in micro scale (Eq. (3.)). he hermal noise of silicon carbide and CVD diamond is in he order of ens of angsroms. his value is much larger for silicon due o is smaller modulus of elasici. his suggess silicon carbide or CVD diamond ma be he preferred maerial for hese ssems. Because of he increased siffness he micro-scale silicon device generaes less noise in comparison wih nano-devices. As shown in Figure 3.10 CVD diamond shows he leas hermal noise while silicon shows he larges. However Figure 3.6 (see d) shows he cener displacemens in a CVD diamond nanodevice are so small ha an increase in he hea load will no make he resuls o be phsicall meaningful. In conras Figure 3.6 (see b) shows ha a significan increase in he hea load migh overcome he noise problem for silicon. Figure hermal displacemen versus emperaure variaions Figure 3.11 shows Displacemen Raio variaions b hea flu added o he op surface for hree differen maerials in nano-scale. Hea ransfer coefficien is calculaed based on free molecular approach a 0.1 Pa. he larges DR belongs o silicon carbide. As shown in Figure 3.10 silicon carbide has relaivel low hermal displacemen due o is large modulus of elasici while 4

50 i has he highes cener displacemen among all maerials as shown in Figure 3.6 (see c). Since emperaure increase variaions b hea flu corresponds o a linear ssem Eq. (3.4) shows ha Displacemen Raio increases b he square roo of he emperaure increase. Figure Displacemen Raio (DR) for hree differen maerials a Pair=0.1 Pa Figure 3.1 shows Displacemen Raio variaions b hea flu for silicon carbide a wo differen Bio numbers. As he hea flu increases he Bio number has a noiceable effec on he Displacemen Raio. 3.6 Conclusion Figure. 3.1 Displacemen Raio (DR) (Silicon carbide-free Molecular) he implici Finie Difference mehod is more suiable han he eplici one due o uncondiional sabili. Simulaions are performed b coupling hermo-srucural equaions a each ime sep. he simulaion resuls draw he following conclusions: 43

51 (1) Ambien cooling srongl influences he displacemen of hermall-posiioned nano-scale devices. he Bio number deermines he dimensionless displacemen of he bridge. () he displacemen of he ssem is a funcion of he geomer. he resuls show ha as he raio of he widh o he lengh of he bridge decreases for consan widhs he value of displacemen increases. (3) he displacemen behavior is also a funcion of pressure maerial properies and consan hea flu in free molecular model while his value is independen of pressure in he coninuum model. he displacemen increases as he consan hea flu a he op surface increases. his behavior shows ha he displacemen variaions b oal hea added o he srucure in his model corresponds o ha of a linear ssem. (4) As he dimensions of a geomer change b a consan facor he dimensionless displacemen versus he Bio number collapse in all cases for he same maerial proper. (5) hermal noise analsis and he displacemen variaions b pressure sugges silicon carbide as he mos appropriae maerial o fabricae nano-devices where posiioning accurac is a design requiremen. I shows he displacemens of he order of angsrom for an average hea load and hermal noises of ens of he order of he magniude. 44

52 CHAPER 4: HERMALLY ACUAED MICRO-SWICHES: AN APPLICAION O SEADY RESPONSE O CONSAN HEAING In 001 Blond e al. fabricaed a hermall acuaed silicon niride micro-swich bridge. he hea was added o he sides of he op surface of he swich. heir eperimenal measuremens showed ha he swich had low acuaion volage. he swich also offered high isolaion and mechanical proecion (Blond e al. 001). his chaper analses he power usage and opimiaion of he swich designed b Blond e al. (Maghsoudi and Marin 01b). hree heaing configuraions are used: disribued hea a he op surface concenraed hea a he cener of he op surface and concenraed hea a he sides of he op surface. he sud of various heaing configuraions gives he opporuni o compare he alernaive heaing configuraions for Blond s design and increase he efficienc. A ime =0 a consan hea load q ranging from 1.4 W o 6.5 W is applied o he op of he bridge unil he bridge reaches a hermall sead sae condiion. he heaing procedure is also performed for closed-swich models wih differen hermal boundar condiions. Simulaions are performed for wo differen maerials: silicon and silicon niride. 4.1 Geomer and Boundar Condiions he geomer is a 3-dimensional pinned-pinned beam as shown in Figure 3.1 in Chaper 3 wih a lengh L of 50 microns a widh W of 50 microns and a hickness d of 1 micron. Conducion wihin he beam as well as convecion beween he beam and he quiescen gas are considered. Hence here is convecion of hea from he wall o he gas and conducion of hea hrough he bridge o he wall. he hea losses hrough he conac are considered when he bridge is closed. he hin film resisor hea addiion is modeled as a consan hea load q applied o he op surface of he beam. his also corresponds o he hea addiion of a hin resisive film. Figure 45

53 4.1 (see a and b) show he geomer and boundar condiions for disribued hea addiion. Figure 4.1 shows he open-swich model (see a) and he close-swich model (see b). (a) (b) Figure. 4.1 he geomer and boundar condiions (a) Disribued hea in open swich (b) Disribued hea in closed swich here are wo alernaive heaing configuraions shown in Figure 4.. Figure 4. shows he geomer and boundar condiions for cener-heaing configuraion (see a) and shows he geomer and boundar condiions for side-heaing configuraion (see b). (a) 46 (b) Figure. 4. he geomer and boundar condiions for open swich wih (a) Concenraed hea a he op cener (b) Concenraed hea a he sides of he op surface

54 4. Formulaion and Simulaion Mehod Simulaion of hermal acuaion of micro-devices and he compuaion of he mechanical response of he ssem in his case he displacemen requires a hermo-srucural formulaion for his problem. he same hermo-srucural formulaion including governing equaions and boundar condiions which are defined in chaper 3 (secion 3.) is applied o his problem. he free molecular hea ransfer coefficien h is defined as Eq. (.17): 1 h ni 1 3 kb a 8 m (.17) where σ is he hermal accommodaion coefficien γ is he specific hea raio of he gas m is he molecular weigh of he gas and ni is he number densi of he gas. For an ideal gas ni can be calculaed using Pgas ni m R a (.18) where R is he ideal gas consan. he 3-dimensional ransien hea conducion equaion is he hermal governing equaion of he model: k c p (.1) where k is he hermal conducivi ρ is he densi and cp is he specific hea of he solid. is he emperaure disribuion in he solid Open Swich Simulaion he hermal governing equaion for boh he open and closed-swich models is he same. Equaion (.1) is used o obain he emperaure disribuion a each ime sep in he model. 47

55 However he hermal boundar condiions are differen for open and closed swich models as shown in Figure 4.1. here are hree differen pes of boundar condiions in he open-swich model as shown in Figure 4.1 (see a) and 4. (see a and b). he Dirichle boundar condiion as shown in Eq. (.1) is applied o he ends of he bridge when he swich is open. w. (.1) here are wo differen pes of Neumann boundar condiions which are applied o heaed and unheaed surfaces. Equaion (.13) shows he boundar condiion for he op heaed surface and Eq. (.14) shows he boundar condiion for unheaed surfaces. k h a n q" (.13) k n h. a (.14) Equaion (.14) is applied o he enire op surface for disribued heaing configuraion as shown in Figure 4.1 (see a). However i is onl applied o he heaed surfaces a he op for he cener heaing and side heaing configuraions as shown in Figure 4. (see a and b). A ime =0 a calculaed hea flu corresponding o a consan hea q is added o he ssem. In each ime sep he hea conducion equaion (Eq. (.1)) along wih open-swich boundar condiions (Eqs. (.1) o (.14)) is solved numericall using a finie difference mehod. he soluion gives he nodal emperaure disribuion in he bridge. Using he nodal emperaure values he hermal momen due o hermal sresses as previousl shown in Eq. (3.) is obained: M h E da. w (3.) 48

56 In order o obain he displacemen disribuion along he beam Eq. (.34) is numericall solved using a Finie Difference mehod: EI M h 0. (.34) he momen and displacemen are ero a boh ends for pinned-pinned boundar condiion: 0 L 0. (.35) Calculaions of he nodal emperaure values and displacemen disribuion a each ime sep coninue unil he ssem reaches hermal sead sae. he corresponding cener displacemen a his poin is called he hermall sead sae cener displacemen δ. he closing displacemen δc is defined as a sufficienl large displacemen o ouch he conac surface a he closing ime. In order o obain he closing displacemen he dnamic srucural equaion mus be simulaed. he dnamic srucural equaion is no solved in his work. Insead he sead sae srucural equaion is simulaed o obain hermall sead sae cener displacemen. he hermall sead sae cener displacemen δ is no necessaril he closing displacemen δc a which he swich is assumed closed. I can be larger or smaller depending on he value of he hea added o he ssem. he overall efficienc coefficien of he swich will be he displacemen per power added. his can be defined as η * : * q q L W (4.1) where q is he oal rae a which hea is added o he swich. 4.. Closed Swich Simulaion When he swich ouches he conac a reacion displacemen δr is creaed due o a reacion force. his means i is necessar o coninue heaing he device o keep he swich closed. As hea addiion o he ssem coninues b ime afer he swich is closed he value of reacion force 49

57 increases leading o he increase in he reacion displacemen. In his case δ increases o compensae he increase in δr in he opposie direcion and keep he swich closed. he reacion displacemen relaes wih δ and δc as: R c. (4.) In order o sud he required hea o keep he swich closed hermal boundar condiions are changed o closed-swich model s hermal boundar condiions as shown in Figure 4.1 (see b). Dirichle boundar condiion shown in Eq. (.1) is applied o he conac as well as he boh ends. he sud begins wih he conac emperaure c equal o he wall emperaure. In secion 4.4 he sud coninues for various conac emperaures. Neumann boundar condiions remain unchanged as shown in Eqs. (.13) and (.14). he procedure eplained in secion 4..1 is performed o obain he hermal sead sae displacemen δ. 4.3 Resuls and Discussion he analical soluion of a fin wih recangular cross secion validaes he resuls for he boundar condiions wihou he added hea flu. he model is run for nodes in a quarer of he model. Grid independence is verified b repeaing he simulaions for and he resuls show ha is sufficien o obain accurae resuls Maerials he maerial sud begins wih silicon which is a sandard maerial for nano- and microfabricaion. I coninues wih he mos common maerial for hermal swiches silicon niride (Blond e al. 001). Oher maerials used for micro-swiches such as i/gold and quar are no appropriae for hermal acuaion applicaions (Rebei 003). 50

58 Simulaions are performed for wo differen maerials: silicon and silicon niride. he hermal properies of maerials are assumed o be consan and defined a wall emperaure as shown in able 4.1. Figure 4.3 shows hermal sead sae displacemen variaions b heaing rae for silicon and silicon niride. he sead sae resuls show ha he hermal sead sae displacemen variaion a he cener of he bridge versus he oal hea changes a he op corresponds o a linear ssem. Furhermore for a consan heaing rae silicon niride shows overall efficienc coefficien η* of 5.6 imes larger han silicon s η*. Silicon niride is seleced for he res of he sud. able 4.1 Maerial properies k (W/K.m) c p (J/kg.K) ρ (kg/m 3 ) α (10-6 1/K) E (GPa) Silicon Silicon niride Figure. 4.3 hermal sead sae displacemen versus heaing rae for silicon and silicon niride 4.3. Open versus Closed Model Sead sae resuls are obained o compare he open swich behavior wih he closed swich. Figure 4.4 shows he hermal sead sae displacemen variaions wih he heaing rae for silicon niride open-swich and closed-swich models. For a specific δ a he cener he closed bridge 51

59 requires less heaing rae han he free bridge. he open silicon niride bridge shows η* of 66.1 nm/w while he closed silicon niride bridge shows η* of 89.1 nm/w. Figure. 4.4 hermal sead sae displacemen variaions b heaing rae (SiN) hese resuls show ha he closed bridge requires less power o remain in closed posiion han he power he open bridge requires reach he closed posiion. his means ha he swich is effecivel over-powered when in he closed posiion. Even if he heaers are sied perfecl o close he bridge here will be ecess heaing in he closed configuraion. When power is removed he swich will no open immediael. he swich will have o cool down b his era amoun before i can open Heaing Configuraions Efficienc In order o opimie he hea addiion wo addiional heaing configuraions are sudied. he firs approach is o add concenraed hea a he cener of he op surface which will be referred o he cener-heaing configuraion. Curren swiches add hea a he base of he bridge which will be referred o as he side-heaing configuraion (Blond e al. 001). he approach of adding hea uniforml along he op of he bridge will be referred o he disribued-heaing configuraion. Each of hese configuraions corresponds o a differen placemen of he heaers used in hermal acuaion. 5

60 Figures 4.5 shows he sead sae resuls for new models and disribued-heaing for open (see a) and closed (see b) swiches. he cener-heaing configuraion shows he larges hermal sead sae displacemen among all models for he same oal hea added o he op. able 4. summaried overall efficienc coefficien η* for he conac lengh of 14 microns and he heaing lengh of 40 microns for he side and cener-heaing configuraions. able 4. Overall Efficienc Coefficien η* for various heaing configuraions (SiN) Heaing Configuraion Open Swich η* (nm/w) Closed Swich η* (nm/w) Disribued Cener Side he cener- heaing configuraion shows he overall efficienc coefficien η* of nm/w for open-swich. his value decreases o 66.1 nm/w for disribued-heaing and 11.5 nm/w for side-heaing open-swich as shown in able 4.. I shows he cener-heaing configuraion is 8.8 imes and disribued-heaing configuraion is 5.7 imes more efficien han he side-heaing configuraion. (a) (b) Figure. 4.5 hermal sead sae displacemen for various heaing configuraions (a) Openswich model (b) Closed-swich model 53

61 As previousl discussed a closed swich shows larger overall efficienc coefficien han an open swich for disribued hea configuraion. he overall efficienc coefficien changes from 66.1 nm/w in open swich o 89.1 nm/w in closed swich. he cener-heaing configuraion follows he same behavior while he side-heaing configuraion does no. Cener-heaing he closed swich increases he overall efficienc coefficien from nm/w o 01.5 nm/w. However his value decreases from 11.5 nm/w o 11.4 nm/w for side-heaing he closed-swich. Sud of he emperaure disribuion in - plane clarifies he reason. he emperaure gradien along he ais deermines he magniude of he displacemen as shown in Eq (4.3): d d I Area dd a (4.3) where Eq. (4.3) is obained using Eq. (3.) and (.34). Figure 4.6 shows he mid plane cross secion along he ais. emperaure disribuions for various heaing configuraions illusraed in Figures 4.7 o 4.9 are ploed in his mid plane. he emperaure disribuion is ploed as emperaure difference -w. Figure. 4.6 Mid-plane cross secion along he ais Figure 4.8 shows he emperaure disribuion in mid plane for cener heaing configuraion for open-swich (see a) and closed-swich (see b) model. When he hea is added o he cener wall emperaure boundar condiion a he conac in closed bridge enhances he emperaure gradien 54

62 in he direcion. he increase in emperaure gradien along he ais leads o an increase in he hermal sead sae displacemen. As a resul overall efficienc coefficien increases (Eq. (4.1)). he same behavior is observed for disribued-heaing configuraion as shown in Figure 4.7. (a) (b) Figure. 4.7 emperaure disribuion in he mid plane (a) disribued heaing configuraion open-swich model (b) disribued heaing configuraion closed-swich model (a) (b) Figure. 4.8 emperaure disribuion in he mid plane (a) Cener-heaing configuraion openswich model (b) Cener-heaing configuraion closed-swich model 55

63 However his phenomenon does no occur for he side-heaing configuraion. In his case he swich is more efficien when open. his means he swich ma reach he closed posiion lose hea and re-open. his can onl be avoided b over-siing he heaers. Figures 4.9 shows he emperaure disribuion in mid plane for side-heaing configuraion for open-swich (see a) and closed-swich (see b) models respecivel. When he swich is closed wall emperaure boundar condiion a he conac does no enhance he emperaure gradien in he direcion (Figure. 4.9 (see b)). I means more energ is required o keep he swich closed when side-heaing configuraion is used. (a) (b) Figure 4.9 emperaure disribuion in he mid plane (a) Side-heaing configuraion openswich model (b) Side-heaing configuraion closed-swich model In cener-heaing and side-heaing configuraions he swich efficienc coefficien is a funcion of heaing lengh. As long as he heaing lengh affecs he emperaure gradien in he direcion i can change he efficienc coefficien. Figure 4.10 shows he efficienc coefficien variaions for cener-heaing and side-heaing configuraions and boh open and closed swich. he efficienc variaion is sudied wih he heaing lengh from 0 microns o 10 microns in boh cases. 56

64 Figure 4.10 Efficienc coefficien variaion b he heaing lengh For he cener-heaing configuraion he efficienc increases as he heaing lengh Lh increases up o 50 microns. An increase in he heaing lengh over 50 microns leads o a decrease in he swich efficienc. For boh open and closed swiches 50 microns reurns he maimum pick of he efficienc coefficien in cener-heaing configuraion. he efficienc decas b increasing he heaing lengh furher han 10 microns and evenuall i becomes equal o he efficienc coefficien of disribued hea configuraion. he side heaing configuraion s efficienc coefficien increases linearl b he heaing lengh. he efficienc difference beween he open and closed swich is ignorable for he heaing lengh up o 10 micron. he efficienc is epeced o merge wih he disribued hea s efficienc coefficien as he heaing lengh increases up o 50 microns Conac Effecs In order o sud he effec of he conac lengh Lc on he efficienc coefficien conac lenghs of 7 14 and 1 microns are seleced. he resuls show he maimum difference of less han 3 nanomeers in he sead sae cener displacemen. able 4.3 shows he δ and η* variaions b increasing he conac lengh. he hermal sead sae cener displacemen increases from nm o 55.7 nm when he conac lengh increases from 7 o 14 microns. I has 4% decrease in he 57

65 swich hermal efficienc. he efficienc coefficien and sead sae hermal displacemen remain consan as he conac lengh increases o 1 microns. able 4.3 Conac lengh effecs on he efficienc for disribued hea (65 mw-sin) Lc (µm) δ (nm) η* (nm/w) Increase of conac lengh furher han 14 microns onl affecs he emperaure gradien in he direcion which does no affec he hermal sead sae cener displacemen. Anoher parameer ma affec he swich efficienc is he conac emperaure c. he resuls presened in Figures 4.3 o 4.10 are obained for he conac emperaure c equal o he wall emperaure w. he hermal conducance for a 0.1 microns gold conac is given as 0.1 mw/k which is a small value. A emperaure increase of 0 o 30 C in he conac is enough o sofen he conac area and lower he hardness of he conac maerial (Rebei 003). As a resul he emperaure increase in he conac mus no eceed 15 C. he conac emperaure sud is performed for c of and 305 K. he conac emperaure increase will decrease he emperaure gradien in he direcion leading o he lower efficienc. Figure 4.11 shows he emperaure difference disribuion in he mid-plane for conac emperaure of 95 K (see a) and 305 K (see b). he lower emperaure gradien is observed in c of 305 K. However he efficienc decrease is 1.5% for conac emperaure increase from 90 K o 305 K. 58

66 (a) (b) 4.4 Conclusion Figure 4.11 Mid-plane emperaure difference disribuion (a) c=95 (b) c=305 he following conclusions can be drawn from hese resuls: Sead sae resuls for disribued hea configuraion and cener-heaing configuraion show he closed swich operaes more efficienl han open swich. his will lead o a hermal lag in opening he swich. For a specific sead sae cener displacemen for disribued hea configuraion closed swich requires less hea a he op han open swich. Open silicon-niride-bridge shows he overall efficienc coefficien of 66.1 nm/w while closed silicon-niride-bridge shows he overall efficienc coefficien of 89.1 nm/w. Boh he conac lengh and conac emperaure variaions have negligible effecs on he sead sae cener displacemen of he bridge. he conac lengh decreases o half resuls in he maimum difference of less han 3 nanomeer in he sead sae cener displacemen. Hea addiion o he cener of he op surface of he bridge is he mos efficien wa o obain a larger cener displacemen per uni hea addiion. I is more hermall efficien han adding concenraed hea o he sides of he op surface b a facor of 17 in closed swich and 8.8 in open swich. 59

67 he qualiaive difference in swich behavior in he open and closed posiions will have implicaions for swich design. Swiches ha were side-heaed became less efficien when closed requiring addiional hea o remain closed. In micro-device design his effec has led o hermalracheing approaches o preven he swich from re-opening. hese resuls sugges ha his effec can be miigaed hrough heaer design offering a mechanicall simpler alernaive. 60

68 CHAPER 5: RANSIEN RESPONSE O CONSAN HEAING WIH APPLICAIONS O NANO-MECHANICAL MEMORY his chaper applies he wo echnologies of buckling beam and hermal eciaion o design a sorage memor. A uni bridge of an arra of buckling-beam memor (a hermall acuaed nanobridge) is simulaed using a Finie Difference mehod. Power requiremens for hermal acuaions opimal geomer and wrie ime of he device for various maerials are invesigaed. his pe of sorage memor can be applied o high radiaion environmens encounered in space eploraion. he suggesed memor is radiaion proeced agains high energ paricle collisions in high radiaion environmens such as Europa moon of Jupier. 5.1 Spacecraf Sorage Memor Requiremens he design of compuer ssems for spacecraf is complicaed b hree consrains beond hose found in erresrial ssems: limied available power a harsh environmen and he need for increased reliabili (Griffin and French 004). he power consrain is he mos severe for missions designed o go beond he reach of available solar power including missions o aseroids and eploraion of Saurn and Jupier s moons. Several of hese missions especiall eploraion of Io and Europa are likel fuure arges for NASA missions (Commiee on he planear science decadal surve 011). he environmenal consrains include eremes of emperaure far beond hose encounered b erresrial elecronics high radiaion levels and in some cases srong magneic fields. he criical role spacecraf elecronics pla in mission navigaion and in reurning scienific daa o earh requires hem o reach sandards of reliabili beond hose of an erresrial ssems. Spacecraf memor is affeced b all of hese consrains. he radiaion consrain is he mos difficul for hese ssems and is he larges barrier o qualificaion of memor ssems for 61

69 spacefligh (Nguen e al. 1999; Scheick e al. 000). In he highl radioacive environmen of space high energ proons and elecrons ma srike he srucure. hese paricles can be a resul of solar evens or he planear environmen (Forescue e al. 003). his can cause he single even upse for convenional semi-conducor devices. Curren missions o he Jovian ssem use a radiaion locker for all he elecronics o avoid scrambling he daa in convenional memories. In spie of his he life of he curren Juno mission is epeced o be limied b radiaion damage. NASA has recognied he need for new nano-echnolog based elecronic ssems for he fuure missions (Meador e al. 010). his has led o he consideraion of eoic memor ssems for spacecraf applicaions such as nano-ube based swiches (Rueckes e al. 000). Improving he performance of he memor is one of he significan seps in he design of a memor ssem. he high performance memories require a high memor densi less power consumpion and fas wrie/read and erase imes. A he presen ime here is no memor designed o saisf all menioned performance aspecs. For eample Phase Change random access Memories (PCM) have ver low power consumpion. he repored required energ for wriing and reading daa is in he order of 0.1 and 10 nj respecivel (Zhou e al. 009). However PCM do no wrie read and erase as fas as Dnamic Random Access Memories (DRAM) (Philip Wong e al. 010). Anoher eample is molecular memories which have memor densiies as high as 10 1 bis/cm (Chung e al. 010; Padavosi e al. 011). In he design of memories for spacecraf reliabili requiremens increase due o he long service life of he ssems he inabili o repair he ssem and he mission-criical role of spacecraf elecronics. An addiional challenge is funcioning in high radiaion environmens where high energ paricle collisions can scramble he daa. he performance of a memor ssem srongl depends on is operaional ssem. Mos of he recen memories such as volaile or non-volaile nano-crsal memories (Seimle e al. 003; De 6

70 Blauwe 00) and magneoresisive random access memories (Engel e al. 00) operae purel elecrosaicall. he concep of mechanical bisabili of a doubl clamped bridge was used o implemen non-volaile elecro-mechanical memor (Nagami e al. 010) and volaile mechanical memor (Bade e al. 004) operaing based on he displacemen of he bridge. A poeniall simpler non-volaile memor device is he buckled-beam nano-mechanical memor (Hälg e al. 1990; Roodenburg e al. 009; Charlo e al. 008). hese devices have been elecrosaisicall acuaed. Afer he beam is buckled hrough applicaion of an elecrosacill generaed force he bisable beam remains buckled afer he power is removed. hese devices have been successfull demonsraed in laboraor eperimens (Roodenburg e al. 009) bu have no been he subjec of eensive performance or reliabili analsis. 5. heor Buckling of he beam is one of he essenial conceps in he design of he suggesed memor. In his secion he heor of buckling and bisable beams including single maerials and membranes are briefl eplained. hen he operaion of he memor device is discussed heor of Buckling (Beer e al. 001) he sabili of he srucure is defined as is abili o suppor a given load wihou eperiencing a sudden change in is configuraion. Figure 5.1 shows wo rigid rods conneced b a pin and a orsional spring of consan ksp. Figure. 5.1 Sabili in wo rigid rods and spring ssem (Beer e al. 001) 63

71 As shown in Figure 5.1 (see a) he ssem is sable a he beginning while he connecion pin is slighl moved o he righ evenuall (see b). In his case here are wo forces acing on he beam: he couple formed b P and P of momen P(L/) sin(δθ) which ends o move he rod awa from he verical and he couple Msp eered b he spring which ends o bring he rod back ino is original verical posiion. Assuming he angle of deflecion of he spring is Δθ he momen of he couple Msp is M sp K sp. (5.1) If Msp is larger han he couple formed b P and P he ssem ends o reurn o is original equilibrium posiion as shown in Figure 5.1 (see a). In his case he ssem is called sable. However if he momen of he couple formed b P and P is larger han Msp he ssem ends o move awa from is original equilibrium posiion and is called unsable. he criical load Pcr is he value of he load for which he wo couples balance each oher as shown in Eq. (5.). P cr L sin K. sp For a small angle Δθ sin(δθ) Δθ (5.) P cr 4K L sp. (5.3) In summar he ssem is sable for P<Pcr and unsable for P>Pcr. Suppose designing a column of lengh L o suppor a given load P as shown in Figure 5. (see a). In his eample he column is pin-conneced a boh ends and he load P is a cenric aial load. If P>Pcr he slighes misalignmen or disurbance buckles he column o a curved shape (see b). 64

72 Figure. 5. Pinned-pinned column (a) unbuckled (b) buckled (Beer e al. 001) Using he equilibrium of he free bod AQ shown in Figure 5.3 he linear homogeneous differenial equaion of he second order wih consan coefficien is obained as follows: d d P EI 0. (5.4) he general soluion of Eq. (5.4) is Figure. 5.3 Free bod diagram Asin( P EI ) Bcos( P EI ). (5.5) he boundar condiion =0 =0 gives B=0 in Eq. (5.5). Subsiuing he second boundar condiion =L =0 will give Asin( P EI L) 0. (5.6) 65

73 Equaion (5.6) is saisfied eiher if A=0 or if he sin erm is ero. If A=0 Eq. (5.5) reduces o =0 represening he sraigh column shown in Figure 5. (see a). For he second condiion o be saisfied he following equaion imposes he sin erm o be ero: P EI L n (5.7) which gives he following for P: n P EI L. (5.8) he smalles of he values of P defined b Eq. (5.8) is ha corresponding o n=1. Subsiuing n=1 in Eq. (5.8) he criical load is given as follows for a pinned-pinned column: P cr EI L. (5.9) Equaion (5.9) gives he criical load onl for a pinned-pinned beam. he general form of he criical buckling load is given as follows: P cr EI Le (5.10) where Le is he effecive lengh column depending on he boundar condiions applied o Eqs. (5.4) and (5.5). Figure 5.4 shows he effecive lengh column for four differen column boundar condiions. In his sud he criical buckling load corresponding o fied-fied boundar condiions is used as shown in Figure 5.4 (see d). 66

74 Figure. 5.4 Effecive lengh column for differen boundar condiions (Beer e al. 001) 5.. Bisable Buckling Beams A hin buckled micromechanical bridge has been repored wih wo sable mechanical saes which could be swiched up and down b elecrosaic forces (Halg 1990 and Roodenburg e. al. 009). Figure 5.5 shows he schemaic of he buckling beam. he buckling beam is highl compressed so ha he free sanding par of i is sress-relieved buckles and become mechanicall bisable. As shown in Eq. (5.10) lower modulus of elasici provides a smaller criical buckling force. In order o make he beam bisable he buckling behavior mus remain wihin he elasic regime. SiO was used due o is low modulus of elasici E and he large elasic range (Halg 1990). he ransiion beween wo buckling saes was performed b elecrosaic acuaion. he elecrosaic forces are creaed b applicaion of a volage beween he bridge and an adjacen elecrode. he subsrae was used as an elecrode o pull he bridge downward. he laeral elecrodes where used o swich he bridge o he oher sable sae. A hin laer of Cr was spuered on he op of he bridge o provide a conducive pah o make he swiching b elecrosaic forces possible (Halg 1990). In a similar design a 600 nm laer of aluminum was used a he op of he bridge (Roodenburg e. al.). 67

75 Figure. 5.5 Single maerial bisable buckling beam (Halg 1990) he eperimenal analsis were performed for a range of buckling beams wih he lenghs of and 40 microns. he microphoographs of he eperimens showed ha he bridge is in he mechanicall sable up and down saes as shown in Figure 5.6. he previous sudies on bisable buckling beams showed ha hese srucures are reliable over housands of swiching he buckling sae ccles (Halg 1990 and Roodenburg e. al. 009). (a) (b) Figure. 5.6 Microphoograph of a sample bisable buckling beam in (a) up and (b) down saes (Halg 1990) I is repored ha for he single maerial buckling beam samples wih a small swiching volage (40 ±10 V) he bridges end o become monosable. For eample he alwas rela in he up sae. he possible reason is a ensile buil-in sress in he meal laer on he op of he bridge which induces a large sress gradien across he hickness of he bridge leading o a considerable bending momen. As a resuls using a sandwich srucure is srongl suggesed (Halg 1990). 68

76 Sandwich srucures such as membranes were fabricaed and esed as bisable buckling beams. Figure 5.7 shows a cross-secional diagram of a snapping membrane srucure (Ara e al. 006). he device consiss of a hin silicon (Si) membrane fied a boh ends wih a laer of hermall grown silicon dioide (SiO) wih a comparable hickness a he op. In he firs sep he emperaure increases up o he oidiing emperaure (1000 C) a which he oide flows plasicall leaving he silicon free of mechanical sresses. Figure. 5.7 Cross secion of snapping membrane srucure (Ara e al. 006) Afer he silicon is cooled he oide laer hardens ino an elasic maerial wih a low coefficien of hermal epansion ( K -1 ) which is much smaller han ha of silicon ( K -1 ). Cooling o he room emperaure induces large sresses a he silicon/oide inerface. he sress sae is biaial compression in he oide and biaial ension in he underling silicon. his ssem of sress will cause he membrane o buckle upward owards he oide laer. he upward buckled configuraion is sable and he defleced shape remains unchanged in he absence of an eernal mechanical force. However a second sable sae of mechanical equilibrium eiss. If sufficien force is applied in he downward direcion o he cener of he membrane he membrane buckles downward. he buckled downward shape is sable agains small mechanical perurbaions. A nearl fla equilibrium sae which is approimael midwa beween he upward- and downward buckled saes is mechanicall unsable. As a resul his ssem is called bisable. he ransiion from he upward sable sae o he downward sable sae occurs using differen mehods. In Ara s design i occurs b an increase in he ambien emperaure. A laer of aluminum is deposied underside of he membrane as shown in Figure 5.7 when i is in he upward- 69

77 buckled sae. he aluminum coefficien of hermal epansion ( K -1 ) is en imes ha of silicon. When he membrane is heaed he aluminum laer aemps o epand a a much faser rae han he silicon and oide laers. When a sufficien high emperaure has been reached a biaial sae of compression is induced in he aluminum couneracing he effec of he compression in he oide laer and disappears he upward-buckled sable equilibrium sae. As a resul he membrane snaps downward ino he single remaining sable sae. he upward- and downward-buckled saes eis in a range of emperaures ha includes room emperaure. Maoba repored a similar bisable membrane design as shown in Figure 5.8 (Maoba e al. 1994). he U-shaped bisable canilever membrane consiss of hree hin-film laered maerials: polcrsalline silicon silicon dioide and polcrsalline silicon. he ension band is made of silicon niride. his design uses resisive dissipaion o hea he upper and lower laers of he canilever for he ransiion from one sable sae o anoher. Figure. 5.8 Schemaic view o demonsrae he ransiion beween wo sable saes (Maoba e al. 1994) 5..3 Buckling Beam Memor Figure 5.9 illusraes he suggesed buckling-beam memor ssem which uses an arra of nanofabricaed beams and hermal buckling of hese devices. Each beam represens a single bi of 70

78 memor. he sorage densi is inversel proporional o he area occupied b each beam so he ideal arra would use he smalles beams possible. he daa is hen wrien b buckling he beams. A downward-buckled beam is a 1 and an upward-buckled beam is a 0. Beam (a) is in he iniial unbuckled sae. Beam (b) is being heaed eiher hrough a hin-film resisor or a heaed ip designed as a wriing device a a consan rae. Figure. 5.9 Nano-mechanical memor acuaion in a memor arra Afer a ime b he hermal sresses in he aial direcion will cause he compression force in he beam o eceed he force required for buckling as shown in Eq. (5.10) causing he beam o buckle down as shown in (c). he emperaure gradien in he beam a he momen of buckling is shown in Figure he op laers are a higher emperaure due o he hea addiion o he op of he beam. his causes he op laers o epand more creaing a downward bending momen in negaive direcion. If he hea is added o he boom of he beam he emperaure gradien will be reverse and he beam will buckle up as (d) which corresponds o a 0. According o he sabili of he buckled srucure beams (c) and (d) sa buckled wihou he need of a susaining hea (Ara e al. 006). Hence he device is a non-volaile memor. 71

79 Figure Cross secional emperaure disribuion (w-) a he ime of buckling he daa can be read and wrie and erased using a hermo-mechanical scanning-probe canilever (Veiger e al. 00). he scanning probe canilever was used o read and wrie he daa in a hermomechanical memor wih a oall differen operaion from he suggesed memor in his sud. Figure 5.11 shows he schemaics corresponding o wriing (see a) and reading (see b) process using scanning-probe canilever. he wriing process is a combinaion of appling a local force b he canilever/ip o he polmer laer and sofening i b local heaing. Iniiall he hea ransfer from he ip o he polmer hrough he small conac area is ver poor and improves as he conac area increases. his means he ip mus be heaed o a high emperaure abou 400 C o iniiae he sofening. Once sofening has commenced he ip is pressed ino he polmer which increases he hea ransfer o he polmer increases he volume of sofened polmer and hence increases he bi sie. he eac wriing process b Veiger is no used in he wriing procedure of he suggesed memor bu he idea can be used o add he required hea o buckle he bi wihou pressing he ip o he memor bi. 7

80 (a) (b) Figure he scanning probe used in millipede design (a) Wriing process (b) Reading process (Veiger e al. 00) he hermal conducance beween he heaer plaform and he sorage subsrae changes according o he disance beween hem. his principle was used b Veiger o read he daa as shown in Figure 5.11 (see b). he medium beween a canilever and he sorage subsrae for eample air ranspors hea from one side o he oher. When he disance beween he heaer and sample is increased as he ip moves ino a bi indenaion he hea ranspor hrough air will be more efficien leading o a decrease in he heaer s emperaure and is resisance. hus changes in emperaure of he coninuousl heaed resisor are moniored while he canilever is scanned over daa bis providing a means of deecing he bis. Veiger increased he readback speed up o 1 Mb/sec b using an arra of 3 b 3 scanning canilever. In he suggesed memor he scanning probe can move over he arra of memor beams and repor 0 and 1 based on he heaers resisance variaions facing he buckled up o buckled down beams. While he Vieger hermo-mechanical ssem is no erasable he suggesed buckling beam memor is erasable b reposiioning all he bis buckled up as shown in d. As previousl eplained his posiion represens 0. 73

81 he required buckling energ and buckling emperaure can be esimaed heoreicall for a single bi of memor. he geomer and boundar condiions for complee and simplified geomeries are illusraed in Figure 5.1. A single bi of memor is a doubl clamped bridge as shown in Figure 5.1 wih fied-fied boundar condiions. he criical buckling load can be rewrien as follows using Eq. (5.10) (imoshenko and Goodier 1951): P cr EI L (5.11) where E is he modulus of elasici I is he momen of ineria L is he lengh of he bridge and k is he column effecive lengh facor equivalen wih Le/L as shown in Figure 5.4 which is equal o 0.5 for a bridge wih fied ends. (a) (b) Figure. 5.1 Geomer (a) Complee model (b) Simplified model he hermal force due o hermal sress is calculaed using he following formulaion (Bole and Weiner 1960): 74

82 F E h Area d d w (5.1) where α is he hermal epansion coefficien is he emperaure disribuion and w is he wall emperaure. In an ideal case where here are no losses o he surrounding maerial or he gas and he emperaure inside he bridge is uniform Eq. (5.1) simplifies o: F h A E c b w (5.13) where b is he oal required buckling emperaure and Ac is he cross secion of he beam. b can be found b equaing he formulaion for hermal sress inside he beam (Eq. (5.13)) wih he criical buckling load (Eq. (5.11)): d L 1 3 b w. (5.14) Using Eq. (5.14) he required buckling energ wih uniform volumeric heaing wih no conducion is obained as: Q v 3 c p W d L 3 (5.15) where ρ is he densi cp is he specific hea W is he widh d is he hickness and L is he lengh of he bridge as shown in Figure In order o invesigae he surface-heaing case wih conducion losses here is a need o perform he simulaions. Uninenional hea addiion o he beam srucure in he device can cause accidenal buckling or a bi flip which eiher scrambles he daa or erases i. In he highl radioacive environmen of space high energ proons and elecrons ma srike he srucure. hese paricles can be a resul of solar evens or he planear environmen (Forescue e al. 003). High energ paricle collision will add an insananeous high energ o he beam srucures. o avoid accidenal reseing of he 75

83 spacecraf memor he beam mus be sied so ha he energ of he paricles does no cause buckling. 5.3 Simulaion he single maerial sraeg is seleced in he simulaions. his means he buckling beam consiss of a hin maerial insead of laers of various maerials wih differen hermal epansion coefficiens (sandwich srucure) as described in secion 5... Apparenl he laers wih higher hermal epansion coefficien accelerae he buckling. his means single laer simulaion overesimaes he wrie ime of he device. Since his sud is limied o esimae he wrie ime and he power consumpion of he device no he memor operaion single maerial simulaion is accepable as he wors scenario case. Furhermore he maerial properies are assumed consan a he wall emperaure as shown laer in able 5.1. his assumpion does no seem feasible a he firs glance. For eample silicon conducivi drasicall decreases b he emperaure (Hull 1999). Bu he srucure reaches o he criical buckling force faser a a lower conducivi. As a resul hese simulaions overesimae he wrie ime of he device which is accepable as anoher wors seleced scenario. Figure 5.1 (see a) shows he complee geomer which mus ake ino accoun he mounings. here are wo mounings wih he lengh l and he widh w added o he beam. he mounings are fied and have he consan emperaure w a he boom and wo ends. However simplified geomer does no simulae hea ransfer in he mounings as shown in Figure 5.1 (see b). he beam is of lengh L widh W and deph d and is a an iniial emperaure w. Depending on how he device is packaged i ma be surrounded b gas molecules a a and have a convecive hea ransfer coefficien h. For hermal acuaion in boh models he hea load is assumed o be applied on he op of he beam a a rae of q (in Was/m ). 76

84 5.3.1 Buckling Simulaion As he emperaure of he beam increases he hermal sresses in he beam will increase unil he aial force reaches he force required for buckling as eplained in he previous secion and he beam will buckle. o capure his process he ransien hea conducion equaion inside he beam mus be solved which is given as Eq. (.1): k. c p (.1) he boundar condiions are based on he hermal condiions a he surface and ends of he beam. here are hree differen pes of boundar condiion in his model (Figure. 5.1): One Dirichle boundar condiion and wo pes of Neumann boundar condiions. he Dirichle boundar condiion is defined as Eq. (.1): w. (.1) he Neumann boundar condiion applied o he heaed surfaces is defined as Eq. (.13) and he Neumann boundar condiion applied o he unheaed surfaces is defined as Eq. (.14) as previousl eplained in secion.3 of chaper : k h a n q" (.13) k n h. a (.14) he hea ransfer coefficien h is calculaed based on he free molecular heor for an ideal gas. he formulaion is given in Eq. (.17) of secion.4 in chaper : 1 h ni 1 3 kb a 8 m (.17) 77

85 where σ is he hermal accommodaion coefficien γ is he specific hea raio of he gas m is he molecular weigh of he gas and ni is he number densi of he gas. For an ideal gas ni can be calculaed using Pgas ni m R a (.18) where R is he ideal gas consan. he ime-dependen emperaure field will be used o find he hermal sresses in he beam. When he aial load due o he hermal sress is equal o he criical buckling load he simulaed beam is assumed o buckle (Chiao and Lin 000). he aial load is calculaed using Eq. (5.1) where is he nodal emperaure disribuion. he formulaion is applicable o he boh models shown in Figure 5.1 because he mounings are fied a he boom. A Finie Difference solver (Secion 3..3 Chaper 3) ha allows he emperaure maerials ambien condiions hea load and he geomer o be changed is used o solve hese equaions. Using hese resuls a buckling ime b and a oal energ for buckling Q for an given geomer and ambien condiions are compued: Q b q (5.16) where q is he oal heaing rae added o he op: q W Lq". (5.17) 5.3. High Energ Paricle Collision As previousl eplained in secions 5.1 and 5. high energ proons and elecrons ma srike he srucure in he highl radioacive environmen of space. he high energ paricle bombardmen is no coninuous. For eample he inegral flu is given as 100 paricles per cm -s for elecron bombardmen and 10 paricles per cm -s for proon bombardmen in Europa orbi of 78

86 Jupier (Jun and Garre 005). his means he possibili of elecron collision in his environmen for a bi of 10 micron lengh and 1 micron widh is 6 elecrons per week. his value decreases o.4 proon per monh for proon bombardmen. Alhough he frequenc of collisions is low he spacecraf ma have o sore informaion such as he operaing ssem for ears. herefore he ssem mus be designed no o buckle afer high-energ collisions. In order o simulae high energ paricle collisions i is assumed ha onl one paricle his he op surface in he cener. he paricle is small enough o add an insananeous hea o he cenral node a he op surface. he amoun of his hea depends on he energ he paricle carries. he high emperaure is calculaed assuming he enire energ is ransferred o a node: hep c p Ehep w (5.18) where Ehep is he energ of he corresponding high energ paricle δ δ and δ are he space discreiaions in he and direcions respecivel. he same Finie Difference solver is used o invesigae he probabili of he buckling a he ime of collision. he rae of hea dissipaion is also calculaed using he same solver. 5.4 Resuls Numerical simulaions are performed o sud he effecs of various geomer dimensions maerial properies and heaing rae on he wrie ime of he device which is called buckling ime. Radiaion resisance of memor bis in high radiaion environmen is sudied for various maerials. he primaril simulaions are performed wih silicon because i is eas o micro-fabricae and has been inegraed wih IC circuis. he sud coninues wih four more maerials: silicon carbide PMMA parlene and kapon (able 5.1). 79

87 5.4.1 Geomer Opimiaion here are five inpu parameers and wo oupu parameers deermining he opimied geomer. he inpu parameers are he lengh L he widh W and he hickness d of he memor bi beam and he mounings lengh l and he widh w. he oupu parameers are he buckling ime and required energ. he objecive of he geomer opimiaion is o opimie inpu dimensions o minimie boh oupu parameers: buckling ime and required energ Beam Opimiaion. In order o invesigae how he buckling ime changes wih he mouning lengh and widh he buckling ime is obained for he complee model wih beam lengh of 0 μm widh of 1 μm and he hickness of 300 nm and various mounings widhs and lenghs. he resuls are compared wih hose of simplified model. Figure 5.13 (see a) shows he buckling ime for various mouning dimensions and he simplified model. he resuls are in agreemen for he mounings dimensions up o 5 μm. hese resuls show he single bi sud can be performed using he simplified model and also he mounings dimension has no effec on he wrie ime of he device. Figure (see b) shows he buckling ime variaions versus oal heaing rae a he op of he bridge for silicon using he simplified model. As he lengh of he beam increases he buckling ime decreases. his shows longer srucures buckle faser and reurn a smaller wriing ime. he buckling ime decreases as he oal hea added o he op increases. his means he wriing ime become faser b increasing he hea addiion o he op of he beam. Figure 5.14 shows he required energ variaion wih he oal heaing rae a he op. Calculaions show ha for a bridge wih he lengh of 0 microns widh of 1 micron and he hickness of 300 nm he leas oal energ of.06 nj is required o buckle he bridge. his value is smaller han he esimaed value of 3.96 nj calculaed using Eq. (5.15). Longer srucures will buckle a lower emperaures and will require less energ o acuae. If a raio of calculaed 80

88 surface-heaing energ divided b uniform volumeric heaing energ is defined his raio remains consan around 51% for all he beam lenghs wih he same hickness and widh. (a) (b) Figure Buckling ime variaions b oal hea added o he op (silicon) (a) Complee model for L=0 μm (b) Simplified model Alhough he required energ o buckle for a beam wih consan dimensions is epeced o be consan i slighl decreases as he oal heaing rae a he op increases. I shows i is more efficien o add more hea and make he buckling faser. he faser buckling does no increase he losses. 81

89 Figure Required energ versus oal heaing rae (silicon) Figure 5.15 shows he buckling ime variaions b hickness for silicon-bridge wih wo differen lenghs of 0 and 40 microns. As he hickness of he bridge increases he energ consumpion increases due o an increase in he momen of ineria (Eq. (5.11)) leading o an increase in he buckling ime. he graphs show ha he buckling ime for he same lengh decreases o he half as he oal heaing rae increases wice. he graph for he lengh of 0 microns and he oal heaing rae of 14 mw and he graph for he lengh of 40 microns and he oal heaing rae of 6 mw approimael collapse. hese resuls show he bucking ime changes inversel wih he beam lengh and oal heaing rae. Figure Buckling ime versus hickness for varing oal heaing raes (Silicon) 8

90 Figure 5.16 shows he buckling ime variaion b widh for he silicon bridge. An increase in he widh leads o an increase in he momen of ineria. As a resul he behavior of Figure 5.15 is epeced. he buckling ime increases as he widh increases. Figure Buckling ime versus widh (Silicon) Spacing Opimiaion. Wriing a high daa raes will require ha here be no cross-alk beween bis when he are wrien on in parallel. he mos ereme case is wriing on all of he bis bordering a bi ha is no being used. Figure shows an unheaed bi surrounded b heaed bis in an arra of 3 in 3 bis. he unheaed bi is a iniial emperaure w and supposed o sa unbuckled. Buckling of he unheaed bi due o hea conducion from he heaed bis will compromise he daa and mus be avoided. o invesigae if undesired buckling happens he maimum possible hea flu is applied o he unheaed beam s mounings a he sides. he resul shows ha he unheaed beam does no buckle and he cenerline emperaure is much smaller han he buckling emperaure. In a silicon beam he cenerline emperaure is 9.1 K while he buckling cenerline emperaure is 570 K. he sud is performed for a range of spacing beween 0.5 and 5 microns. he spacing of 0.5 microns is large enough for fabricaion and avoiding undesired buckling. 83

91 Figure An unheaed bi surrounded b heaed bis in an arra of memor 5.4. Maerial able 5.1 shows he lis of maerials used for he sud including hermal and mechanical properies a iniial emperaure 90 K. All of hese maerials are commonl used in microfabricaion making hem suiable candidaes for his device design. Afer obaining he iniial simulaions for silicon he sud coninues wih silicon carbide. Silicon carbide has a higher meling poin han silicon making i more appropriae for he harsh environmens. Plasic maerials are appropriae for mass fabricaion. wo arbirar plasic maerials PMMA and parlene are seleced for he preliminar plasic sud. he maerial sud erminaed wih kapon which has a low hermal conducivi and a high epansion coefficien like oher plasic maerials bu will las for a longer ime in high radiaion environmens. able 5.1 Maerial properies Kapon Parlene PMMA Silicon Silicon carbide k (W/m-K) c p (J/Kg-K) ρ (Kg/m 3 ) α (1/K) E (Pa) Meling poin (K) N/A

92 Figures 5.18 and 5.19 show he buckling ime and required energ o buckle for five maerials: silicon silicon carbide PMMA parlene and kapon. Silicon carbide buckles slighl faser han silicon because of is larger hermal epansion coefficien. PMMA parlene and kapon buckle much faser han silicon and silicon carbide because heir epansion coefficiens are an order of magniude larger han silicon and silicon carbide s ones. hese hree maerials also require he leas energ o buckle. Figure Buckling ime for various maerials Figure Required energ for various maerials 85

93 5.4.3 High Radiaion Collision he rapped elecron and proon energ specra were previousl obained using he radiaion bel models for he Europa orbi of Jupier. I shows he highes energ and he mos probable rapped elecrons and rapped proons carr 1000 Mev and 100 Mev respecivel (Jun e al. 005). Figure 5.0 shows he hea dissipaion of a bi wih he lengh of 0 microns widh of 1 micron and he hickness of 300 nm which is hi b he highes energ elecron and proon a he op face. If buckling occurs i mus be a he ver beginning of he collision. he silicon simulaion resuls show ha he buckling does no occur. In addiion he hea due o he collision dissipaes in less han 10 nsec which is much less han he quickes silicon bi buckling ime for he same dimensions 17 nsec. Figure. 5.0 Hea dissipaion in a bi of memor afer high energ paricle collision (silicon) Figure 5.1 shows he hea dissipaion of silicon silicon carbide PMMA parlene and kapon beams which are hi b he highes energ proon (see a) and elecron (see b) a he cener of he op surface respecivel. Because of he relaive infrequenc of hese collisions he collision is modeled as a discree hea addiion a he mos vulnerable poin of he ssem insead of a coninuous heaing. he hea dissipaion in silicon carbide is slighl differen from silicon while i is much slower in PMMA parlene and kapon because of he small hermal conducivi. In 86

94 elecron collision he force due o hermal sresses pass he required buckling force leading o undesired buckling in PMMA parlene and kapon beams. However in he proon collision case he hermal force does no eceed he required buckling force o buckle he PMMA parlene and kapon beam. (a) (b) Figure. 5.1 Hea dissipaion in a bi of memor afer (a) elecron collision (1000 Mev) (b) proon collision (100 Mev) able 5. shows he criical buckling force required o buckle a beam wih a lengh of 0 microns for silicon silicon carbide and kapon. he criical buckling force is calculaed using Eq. (5.11). he criical buckling force required o buckle a kapon beam is wo order of magniude 87

95 smaller han silicon and silicon carbide due o he smaller modulus of elasici as shown in able 5.. Figure 5. shows he hermal force deca in he beams due o high energ proon collision (see a) and high energ elecron collision (see b). he resuls show ha silicon and silicon carbide beams are radiaion resisan in boh high energ elecron and proon collisions. In elecron collision he force due o hermal sresses pass he required buckling force leading o an undesired buckling in kapon beam. However in he proon collision case he hermal force does no eceed he required buckling force o buckle he kapon beam. (a) (b) Figure. 5. Force deca in a bi of memor afer (a) proon collision (100 Mev) (b) elecron collision (1000 Mev) 88

96 able 5. Criical buckling force Kapon Silicon Silicon carbide Pcr (nn) Pressure Effecs and Packaging Previous sudies showed he pressure and dimension variaions in packaging will change he hea ransfer coefficien h (Masers e al. 005; Lee e al. 007). his has he poenial o change he wrie ime of he device. Previous researchers have sudied he hea ransfer in vacuum packaged ssems. his work has included modeling of gas phase conducion hea ransfer beween a subsrae and a canilever leg a consan emperaure in he free molecular flow regime (Masers e al. 005). A relaed work shows air pressure increase will raise he dissipaed power in a heaed microcanilever (Lee e al. 007). In boh cases he low pressure and small lengh scales showed ha free-molecular models for hea ransfer are appropriae. Pressure variaions appear in hea ransfer coefficien variaions as given in Eq. (.17) and Eq. (.18). Increasing he pressure leads o an increase in hea ransfer coefficien and dimensionless Bio number hl/k. In his sud all he resuls shown in Figures 5.13 o 5. are obained a consan pressure 5 kpa which ields Bio numbers of and for kapon parlene PMMA silicon and silicon carbide respecivel. Plasic maerials show larger Bio numbers in comparison wih silicon and silicon carbide. Figure 5.3 shows he buckling ime variaions b he pressure increase for a bi of kapon and silicon carbide. Kapon shows larger buckling ime as he pressure passes 100 Pa. However he buckling ime remains consan for silicon carbide. his behavior can be eplained using Bio numbers. Kapon has a large Bio number while silicon carbide has a Bio number 1000 imes 89

97 smaller han Kapon. Previous sudies show ha he cener displacemen variaion for a heaed bridge is significan a high Bio numbers (Maghsoudi and Marin 01a). 5.5 Conclusion Figure. 5.3 Air pressure effecs on wriing ime variaions (logarihmic scale) he energ required for wriing daa per bi is in he order of 1 nj which shows low power consumpion of he device. he daa sorage densi is a rade-off. In order o balance hese consrains he lengh of 0 microns and he smalles possible hickness o fabricae is suggesed. his design suggess a memor densi in he order of 10 6 bis/cm and bis/cm 3 which are accepable in comparison wih he curren memor devices and can be improved b decreasing he widh and he hickness of he beam. Decreasing he hickness and he widh is desired for boh energ consumpion and buckling ime. However decreasing he lengh has inverse effecs. he buckling ime changes inversel wih he oal heaing rae and he lengh of he beam. he simulaion resuls show ha he wrie ime of less han 10 nsec/bi is achievable (100 MB/sec). Because his echnolog allows reading and wriing in parallel he wriing process can be acceleraed using muliple reading/wriing heads. 90

98 he resuls obained of simulaion a bi of memor including mounings are in agreemen wih he resuls of simplified model. he simplified model is a good approimaion of he complee geomer. PMMA parlene and kapon bis consume he leas power and buckle faser han silicon and silicon carbide bis. However elecron collision causes undesired buckling in PMMA parlene and kapon bis. High energ paricle collision will no lead o undesired buckling in he silicon and silicon carbide memor bis. he hea due o he collision dissipaes in less han 10 nsec. As a resul eiher silicon or silicon carbide is he mos appropriae maerial o fabricae he device for Jovian moon or oher ereme environmens. However PMMA parlene and kapon are more appropriae alernaives for oher missions where radiaion environmen is no ereme. he pressure variaion and packaging affec he wrie ime of plasic maerials while do no change he wrie ime in silicon and silicon carbide bis significanl. Alhough he simulaions repored accepable wriing ime and power consumpion for he suggesed buckling memor and confirmed a radiaion proeced design for harsh environmens here are a few weaknesses associaed wih he simulaions: 1. he hermal properies are assumed consan as he wors scenario case. Increasing he emperaure will drasicall decrease he hermal conducivi and increase he hermal epansion which accelerae he buckling resuling in smaller buckling ime.. he single maerial sraeg is seleced while i has he endenc o become mono-sable evenuall. Sandwich maerials repored as bisable membranes are more appropriae approach for he simulaions. 3. he wriing ime was no aken he probe speed ino accoun. his becomes ver imporan especiall because wo arras of wriing probes are required: one a he op and anoher a he boom of he arra of buckling memor. 91

99 he simulaions can be significanl improved b using he emperaure dependen hermal properies solver for hea conducion equaion. However his will show a smaller wriing ime while he curren wriing ime falls wih a reasonable range. Fabricaion and preliminar ess on a single bi of buckling memor can provide necessar informaion for he coninuaion of he simulaions and deermine he necessi of using a sophisicaed srucural formulaion. 9

100 CHAPER 6: RANSIEN RESPONSE IN NANO-BRIDGES As previousl discussed in secion 1.3 of he inroducion appling sinusoidal heaing leads o vibraion in he bridge called hermal acuaion. his chaper simulaes a hree-dimensional doubl clamped bridge wih he sep funcion and harmonic hea addiion. he simulaion is onl performed in nano-scale. Silicon is he seleced maerial for he sud. Onl he free molecular approach is used o define he hea ransfer and damping coefficiens. his chaper conains analsis of he sep funcion response which deermines he direcion of deflecion he dominan frequencies in he ssem and seling ime. he harmonic eciaion sud includes compuing he phase dela beween he eciaion and he response he sead sae ampliude and he vibraion ampliude for a range of frequencies. I also discusses he changes of he vibraion ampliude and he sead sae ampliude for a range of pressures below he amospheric pressure. he ransien hea ransfer and srucural equaions are solved using a Finie Difference mehod implemened in C++. he implici approach is used for consan hermal properies while he eplici approach is used for emperaure dependen hermal properies. 6.1 Geomer and Boundar Condiions he geomer is a nano-scale bridge as inroduced in Chaper. Figure.1 shows he geomer which is a beam wih a lengh of 10 microns a widh of 1 micron and a hickness of 300 nanomeers. Conducion wihin he beam and convecion beween he beam and he quiescen gas are considered. In conrar wih chaper he is doubl clamped. For sep funcion eciaion a consan hea flu q is added o he op of he bridge uniforml. However for harmonic eciaion a ime-dependen hea load is added o he ssem. If he ssem consiss of a resisor R wih curren I() = Io sin(ω) passing hrough i hen he power can be 93

101 defined as q = R[Io sin(ωa)]^. his means ha he ime-dependen harmonic hea load q" applied o he op surface of he beam can be epressed as follows: q A h sin A 1 cos( ) a h (6.1) where Ah is he hea load ampliude is he ime and ωa is he acuaion frequenc and ω is he frequenc of he ssem: a (6.) he eciaion frequenc and he response frequenc will be equal o ω because he sin erm raised o power is equivalen wih (1-cos(ω)) as shown in Eq. (6.1). his is equivalen wih an unbalanced harmonic eciaion. All oher hermal boundar condiions ecep he hea load are preserved as hermal boundar condiions eplained in chaper (Figure..1). Figure..1 he geomer and boundar condiions in he model he cooling gas is air wih consan properies a ambien emperaure. Simulaions are performed for crsalline silicon. he hermal properies of crsalline silicon are assumed o be consan and defined a he wall emperaure in he implici approach. Anoher approach updaes he silicon hermal properies b emperaure a each ime sep. he wall emperaure w is fied a boh ends of he beam. 94

102 95 6. ransien hermo-srucural Formulaion 6..1 ransien Hea ransfer Equaion For he consan hermal properies approach Eq. (.1) is used: c k p (.1) where k is he hermal conducivi of he solid. his equaion is solved using an implici Finie Difference approach a each ime sep o obain he nodal emperaure disribuion when he hermal properies are assumed consan a he wall emperaure. However for he emperaure dependen hermal properies approach Eq. (.11) is used which akes ino accoun he hermo-elasic erms and also emperaure dependenc of he hermal properies (Landau and Lifshi 1959; Lifshi and Roukes 1999; Serra and Bonaldi 008): ) ( c E k k k c c E p p p p p (.11) where νp is he Poisson raio and cp is he specific hea. he hermal boundar condiions are he same as chaper from Eq. (.1) o Eq. (.14). here are hree differen pes of boundar condiion in his model (Figure.1). he Dirichle boundar condiion a boh ends is epressed as: w. (.1) here are wo differen pes of Neumann boundar condiions which are applied o he heaed and unheaed surfaces. Equaion (.13) shows he boundar condiion for he heaed surface and Eq. (.14) shows he boundar condiion for he unheaed surfaces:

103 k h a n q" (.13) k n h. a (.14) he hea conducion equaion is solved numericall using a Finie Difference mehod (Incropera e al. 007). As a resul he emperaure is obained a each node. hese resuls are used o calculae he hermal sress due o emperaure differences in he beam and he hermal momen and force. Onl he free molecular approach is seleced o compue he hea ransfer coefficien h. he formulaion is presened in secion.4: 1 h ni 1 3 kb a 8 m (.17) where σ is he hermal accommodaion coefficien γ is he specific hea raio of he gas m is he molecular weigh of he gas and ni is he number densi of he gas. For an ideal gas ni can be calculaed using Pgas ni m R a (.18) where R is he ideal gas consan. 6.. ransien Srucural Equaion In his chaper he ransien srucural equaion which akes ino accoun he unseadiness is used. he derivaion of he equaion is eplained in chaper. he primar form of he equaion is given as follows: A u 1 v EA N h EI M h F D 0 (.9) 96

104 97 where he damping erm FD is he ne flow drag (Marin and Houson 008) in he free molecular regime. I is equivalen wih he consan damping erm Cf muliplied b vibraion veloci. FD is given b: 1 U C W d U c WP F f a w n n gas D (.30) where σn he normal accommodaion coefficien and σ he angenial accommodaion coefficien are assumed o be equal o 1. he hermal veloci c is defined in Eq. (.31) and U() is he veloci in he vibraion direcion which is defined as Eq. (.3): m k c a b (.31) ) ( U (.3) where m is he mass of he gas molecules in his case air wih m= kg. Oher erms kb and a are previousl defined in chaper as Bolmann consan and ambien emperaure. Subsiuing Eqs. (.30) in Eq. (.9) he finalied dnamic srucural equaion is obained: 0. 1 C dd E EI dd E v u EA A f w w (.33) Equaion (.33) is solved numericall using a Finie Difference mehod o obain he displacemen along he bridge. he presence of he fourh-order derivaive of he displacemen wih respec o in Eq. (.33) requires wo addiional poins oher han boh ends a he boundaries. o fulfill his requiremen wo ghos poins are creaed a boh ends. he fied-fied boundar condiion requires ero displacemen and slope ero a boh ends. Beginning he firs ime sep he following boundar condiion is applied o boh ends a all he ime seps:

105 0 L 0 0 L 0. (.36) he discreiaion algorihm and soluion mehod will be as discussed is secion 3..3 of chaper 3. In his case he slope ero boundar condiion is applied using he ghos poin. Figure 6.1 shows he compuaional poins a he lef end. he ghos poin is labeled as g and he end poin is numbered 1 he nodes numbers increase owards he oher end poin in he righ. Figure. 6.1 Compuaional nodes a he boundar Equaion (6.3) shows he discreied form of Eq. (.36) a each ime sep. 1 0 g 0 (6.3) where νg is he displacemen a he ghos poin ν1 is he displacemen a he end poin and ν is he displacemen a he firs node ne o he end poin. his simpl means ha a each ime sep he governing ransien srucural equaion Eq. (.33) is solved a all he nodes ecep he wo ends and he ghos poins. A he end of each ime sep he slope ero boundar condiion is saisfied b equaing νg wih ν. 6.3 Validaion of Unsead hermo-srucural Equaions Alhough here is no se of analical resuls o validae he coupled hermo-srucural solver he unsead hermal solver and unsead ransien srucural solver are validaed independenl o ensure he agreemen wih analical soluion. he unsead hermal solver is validaed for consan hermal properies. Laer in his chaper he consan properies hermal solver shows he same resuls as emperaure dependen hermal properies solver a low emperaures. 98

106 In dnamic srucural equaion he hea added o he ssem is se o ero which means no hermal bending momen or eernal force acs on he srucure. hen he beam is given he iniial displacemen corresponding o he firs mode shape for fied-fied boundar condiion. he resuls are validaed using free vibraion analical soluion Unsead hermal Equaion Validaion In Chaper 3 he sead hermal equaion was validaed wih analical soluion of uniform cross secional recangular fins. he resuls showed good agreemen. In order o validae he unsead conducion equaion he analical resul of a long recangular bar was used. Figure 6. shows a long recangular bar wih hea convecion a he op boom fron and back sides. Figure. 6. Long recangular bar wih convecion a four sides (Yener e. al. 008) he analical soluion for he emperaure disribuion b ime in he cross secion of he bar is given as (Yener e. al. 008): w a a W d w a a W w a a d (6.4) where w is he iniial emperaure since he bar is a wall emperaure a ime =0 and a is he ambien emperaure. he firs and second erm in he righ hand side can be obained using he following formulaion: a Fo n Cne cos n 0.5W w a W n1 (6.5) where Fo is he Fourier coefficien: 99

107 k c p Fo 0.5 W (6.6) he coefficien Cn is: C n 4sin sin n n n (6.7) and he discree values of ξn (eigenvalues) are posiive roos of he ranscendenal equaion: sin n n Bi w h 0.5W. k (6.8) he firs four roos of his equaion are abulaed (Incropera e al. 007). he second erm in he righ hand side of Eq. (6.4) can be calculaed b subsiuing W b d in Eqs. (6.5) o (6.8). In order o validae he numerical soluion of consan hermal properies conducion equaion Eq. (.1) wih analical soluion Eq. (6.4) a larger lengh of 5 micron was used which is 5 imes larger han he widh of 1 micron and 83 imes larger han he hickness. Ambien emperaure a of 30 K was seleced. Cener emperaure was racked b ime and ploed versus he analical soluion for various number of divisions along he ais. he resuls are in agreemen for 89 divisions along ais as shown in Figure 6.3. Figure. 6.3 Cener emperaure (analical versus numerical soluion) 100

108 6.3. Unsead Srucural Equaion Validaion In he dnamic srucural equaion as shown in Eq (9) if he hea is se o ero he dnamic srucural equaion for no eernal force will be as follows: 4 EI C. A 4 f (6.9) he beam wih no eernal force acing on i is given an iniial displacemen corresponding o he firs mode shape for fied-fied boundar condiion wih A0 of 1 nanomeer as shown in Figure 6.4. he iniial displacemen is calculaed using he following formulaion over half of he beam (Marin and Houson 009): A0 1.00cos 4.73 L cosh 4.73 L 0.985sin 4.73 L sinh 4.73 L. (6.10) Figure. 6.4 Iniial displacemen he beam vibraes wih air damping coefficien. If he damping coefficien was ero he beam would vibrae wih he naural frequenc of he bridge ωn which is given b (imoshenko e al. 1974): 101

109 k n n L EI m (6.11) where kn is he mode consan which is equal o for he bridge and m is he mass per uni lengh (ρwd). Including damping coefficien he ssem vibraes wih damping frequenc ωd which is given as follows for ξ<1: d n 1 (6.1) where ξ is he corresponding non-dimensional damping coefficien which is a funcion of he air pressure and ambien emperaure. he quali facor Q is he raio of he sored energ of he ssem o he dissipaion energ: U Q U d i (6.13) where Ui is he sored vibraional energ in he ssem and Ud is he energ dissipaion per period (Blom e al. 199). For his model he quali facor Q is equal o fluidic quali facor QF because he losses are dominaed b gas losses. If he flow is free-molecular he following equaion holds (Marin and Houson 008): Q F kn k P m gas d L E 1 (6.14) where km is given b: k m m kb a 1 w. n n a d W (6.15) he damping coefficien ξ is defined using he following formulaion: 10

110 1 Q F. (6.16) A 500 kpa he corresponding damping coefficien is ξ= ): he analical soluion for he ssem of equaion is given as follows (imoshenko e al. v ) b0 n v b0 cos( ) sin( ) d d d 0 ( n e (6.17) where b0 is he iniial cener displacemen and v0 is he iniial displacemen derivaive wih respec o ime. he analical deca b ime is given as follows (imoshenko e al. 1974): v( ) A e C f / A 0. (6.18) he numerical resuls were ploed versus analical soluion and he analical deca as shown in Figures 6.5 o 6.7. he doed plo refers o Eq. (6.18) and he dashed plo refers o Eq. (6.17). As he discreiaion along he beam becomes finer agreemen beween he numerical and analical soluion is achieved. Figure. 6.5 Numerical versus analical soluion for 7 compuaion nodes along he beam (NnodX=7) 103

111 Figure. 6.6 Numerical versus analical soluion for 15 compuaion nodes along he beam (NnodX=15) Figure. 6.7 Numerical versus analical soluion for 1 compuaion nodes along he beam (NnodX=1) he accurac is defined as he difference beween he analical and numerical soluion a he las ime sep. Figure 6.8 shows he accurac variaions b he number of divisions along he beam. Since he displacemen is in he order of nano in he validaion sud for 7 divisions along he beam he accurac is wihin he firs significan digi. As he number of divisions increases he accurac of up o fourh significan digi is achieved. his shows ha he analical and numerical soluion are in agreemen for 17 and 1 nodes along he beam. 104

112 Figure. 6.8 Accurac versus number of divisions along he ais 6.4 emperaure Dependen hermal Properies he sudies in chapers 3 4 and 5 were performed assuming consan hermal properies a he wall emperaure. However as he srucure is heaed he emperaure raise makes significan changes in he hermal properies such as hermal conducivi k specific hea cp and hermal epansion coefficien α. Figures 6.9 o 6.11 show crsalline silicon hermal properies b emperaure variaions (Hull 1999). Figure. 6.9 hermal conducivi of crsalline silicon b emperaure 105

113 Figure hermal epansion coefficien of crsalline silicon b emperaure Figure Specific hea of crsalline silicon b emperaure Using he abulaed hermal properies versus emperaure (Hull 1999) hermal properies deviaion b emperaure is calculaed as shown in able 6.1. emperaure raise Deviaion in hermal conducivi Deviaion in hermal epansion coefficien Deviaion in specific hea able 6.1 Maerial properies 40 K 100 K 300 K (from 90 K o 330 K) (from 300 K o 400 K) (from 300 K o 600 K) 15% 3% 59% 11% 4% 47% 4% 10% 19% 106

114 he hermal conducivi variaions due o he emperaure raise is significan. he emperaure raise of 100 K decreases he hermal conducivi over 3%. he hermal epansion coefficien variaions due o he emperaure increase is less significan han hermal conducivi bu is no negligible for higher emperaure raise. he specific hea shows he leas variaions due o he emperaure increase. able 6.1 shows he crsalline silicon hermal properies are highl emperaure dependen. As he emperaure increases beond 100 K he consan hermal properies assumpion will be limied in predicing he hermo-srucural behavior of he ssem. In order o ake ino accoun he emperaure dependenc of hermal properies hree subrouines are added o he eplici compuaional code which updae he hermal properies a each ime sep using he provided abulaed daa for crsalline silicon hermal properies versus emperaure (Hull 1999). As a resul he hermal conducivi hermal epansion coefficien and specific hea are updaed a each ime sep based on he corresponding discreied emperaure domain a ha ime sep. 6.5 Sensiivi Sud erm sensiivi sud over unsead conducion equaion for variable hermal properies Eq. (. 11) is performed. he eplici emperaure dependen hermal properies solver is run wih and wihou hermo-elasic erms. he resuls show ha he hermo-elasic erms creae a difference up o he hird significan digi in he emperaure disribuion. his is obained for he oal emperaure raise of 100 K. However Eq. (.11) is solved as full in he res of he sud. erm sensiivi sud is also performed over he ransien dnamic srucural equaion as shown in Eqs (.9) and (.33). he equaion is solved once wih he aial force erm and anoher ime wihou he aial force erm for a consan disribued hea load of 50 mw added o he op of he beam. Figure 6.1 shows he cener displacemen versus ime in boh models. he small difference shows he aial force erm does no have significan impac on he overall resuls. 107

115 Figure. 6.1 Cener displacemen versus ime including and no including aial load erm In order o check he order of magniude of he force and momen erms is Eq. (.9) or (.33) hese erms are calculaed using he hermo-srucural solver for he full equaion. he hermo-srucural equaion is solved including all erms presened in Eq. (.9) or (.33). A consan disribued hea load of 50 mw is added o he op of he beam. Figure 6.13 shows he logarihmic scale of absolue values of he oal aial load and he hermal momen erm. We used he absolue values so we can appl he log funcion. Because he values are ver small he were muliplied o 10 0 while saving. he resuls show ha he smalles values of he hermal momen erm are order of magniudes larger han he larges values of he aial load erm and he larges values of he hermal momen erm are 4 o 5 order of magniudes larger han he larges values of he aial load erm. 108

116 Figure Logarihmic scale of absolue values of aial and hermal momen erms Figure 6.14 plos he hermal momen erm B divided b he aial load erm A in logarihmic scale. Noe ha erms B and A are shown in Figure his figure shows clearl ha he hermal momen erm is o 5 order of magniudes larger han he aial load. As a resul he aial load erm can be ignored in comparison wih he hermal momen erm as previousl done b oher researchers (Lifshi and Roukes 1999). Figure 6.14 Logarihmic scale of relaive order of he hermal momen o he aial load erm Scaling analsis was used as a mahemaical ool o esimae he order of magniude of erm A he aial load erm in comparison wih erm B he hermal momen erm. Using Eqs (.8) and (.9) erm A can be scaled as follows: 109

117 N u EdW L 0 1 u0 L u E dw L 0. (6.19) M h Using Eqs. (.5) and (.9) erm B can be scaled as follows: d 1 E W (6.0) L where in Eqs (6.19) and (6.0) E is he modulus of elasici d is he hickness W is he widh and L is he lengh of he beam. u0 is he maimum deflecion which is in he order of nanomeer Δ is he maimum emperaure difference and α is he hermal epansion coefficien. In order o compare he order of magniudes of erms A and B Eq. (6.19) is divided b Eq. (6.0). Afer canceling he same erms A/B becomes: A B u0 1 u0 u L L d 0 u0. d (6.1) Since u0 is in he order of nanomeer and L is 10 micromeer l>>u0. As a resul u0/l>>(u0/l). Eq (6.1) can be rewrien as follows: A B u0 u0 L u0. d d If he values of each scaled erm is subsiued in Eq. (6.): (6.) A B (6.3) 110

118 Equaion (6.3) shows for he curren dimensions erm A can be ignored in comparison wih erm B. Using he resuls presened in Figures 6.1 o 6.14 and scaling analsis he simulaions used in his sud do no ake ino accoun he aial load erm in Eq. (6.33). 6.6 Resuls and Discussion Sep Funcion Eciaion In his secion sep funcion eciaion is applied o he ransien hermo-srucural equaion for boh he consan hermal properies solver and he variable hermal properies solver. A range of air pressures leading o a range of damping coefficiens are used. Appling a consan hea flu o he ssem leads o vibraion in he ssem bu he ssem evenuall converges o a consan cener displacemen. he sud begins wih higher pressure indicaing a higher damping coefficiens and coninues wih lower pressures Variable Properies Solver a 5 MPa Air Pressure. A consan disribued hea load of 50 mw is added a ime =0 o he op of he beam wih he lengh of 10 micron widh of 1 micron and he hickness of 300 nm. he beam sars vibraing in he air wih he pressure of 5 MPa corresponding o he dimensionless damping consan of ξ=0.6. Figure 6.15 shows he response he negaive cener displacemen b ime. he response is ploed for hree ime duraions as shown in green red and blue. As shown in he blue plo he vibraion is evenuall damped o a consan cener displacemen value. he response plo shows he beam deflecs downwards b adding he hea o he op of he srucure. Adding he hea o he op of he srucure creaes a emperaure gradien along he hickness of he beam. he larger epansion of he op laers due o he higher emperaure increase will creae a downward bending momen. 111

119 Figure Response o consan hea load of 50 mw a 5 MPa (Variable hermal properies) In order o rack he emperaure variaions b ime he emperaure a he op cener of he beam was moniored b ime. Figure 6.16 shows he op cener emperaure variaions b ime corresponding o green red and blue response plos shown in Figure he resuls show ha he response is hermall sead sae a he end of each se of simulaions. his means he emperaure disribuion becomes idenical a he las ime sep of he firs run. Figure op cener emperaure b ime for a consan hea load of 50 mw a 5 MPa (Variable hermal properies) In order o sud he relaion beween he emperaure disribuions along he beam wih he corresponding defleced shape emperaure disribuion and displacemen along he beam are 11

120 ploed a he las ime sep of each response plo shown in Figures 6.17 and Figure 6.17 shows he op cener-line emperaure disribuion along he ais. Figure 6.18 shows he corresponding defleced shape of he beam where poin 1 represens he resuls a he las ime sep of he green plo poin represens he resuls a he las ime sep of he red plo and poin 3 represens he resuls a he las ime sep of he blue plo shown in Figures 6.15 and An idenical emperaure disribuion along he beam is obained because as shown in Figure 6.16 he hermal equaion becomes sead sae a he corresponding 3 poins. However due o he dnamic effecs he displacemen along he ais is no idenical in he las ime sep of each response simulaion. Figure op cener emperaure disribuion along he beam for a consan hea load of 50 mw a 5 MPa (Variable hermal properies) Figure 6.17 shows a sudden change in emperaure a he edges. However here is no a significan emperaure gradien along he beam oher han he edges. his creaes a large bending momen close o he edges. As a resul as shown in Figure 6.18 a sudden change appears in he defleced shape close o he edges. 113

121 Figure defleced shape for a consan hea load of 50 mw a 5 MPa (Variable hermal properies) A Fas Fourier ransformaion (FF) code wrien in MALAB is applied o he converged soluion shown in Figure 6.15 in blue. his gives he dominan frequencies in he ssem. Figure 6.19 shows he dominan frequencies for he ssem. his shows ha he ssem responds wih all he modes for he sep funcion eciaion. Figure Dominan response frequencies for a consan hea load of 50 mw a 5 MPa (Variable hermal properies) Variable Properies Solver a MPa Air Pressure. his secion presens he hermal and srucural response of he beam wih he same dimensions and hea load as secion he air pressure is decreased o MPa corresponding o he dimensionless damping 114

122 consan of Figure 6.0 shows he response b ime for hree ime duraions. In comparison wih 5 MPa response he seling ime is longer for MPa due o a smaller damping coefficien. Figure. 6.0 Response o consan hea load of 50 mw a MPa (Variable hermal properies) Figure 6.1 shows he op cener emperaure variaions b ime. he ssem is hermall sead sae a he end of each response plo. he behavior is comparable wih wha presened in secion Figure. 6.1 op cener emperaure b ime for a consan hea load of 50 mw a MPa (Variable hermal properies) Sharp changes appear in he emperaure disribuion along he beam leading o sharp changes in he defleced shapes a he edges as shown in Figures 6. and

123 Figure. 6. op cener emperaure disribuion along he beam for a consan hea load of 50 mw a MPa (Variable hermal properies) Figure. 6.3 defleced shape for a consan hea load of 50 mw a MPa (Variable hermal properies) Figure 6.4 shows he dominan frequencies corresponding o he response shown in Figure 6.0. his shows ha he ssem responds wih all he modes for sep eciaion. he resuls obained in secion are comparable wih he resuls obained in secion

124 Figure. 6.4 Dominan response frequencies for a consan hea load of 50 mw a MPa (Variable hermal properies) Consan versus Variable Properies Solver a 1 MPa Air Pressure. his secion presens he hermal and srucural response of he beam wih he same dimensions as secions and for a hea load of 50 mw added o he op of he beam a ime =0. he air pressure is reduced o 1 MPa corresponding o he dimensionless damping consan of he sud was performed for he boh consan and variable properies solvers. Figure 6.5 shows he response for he boh cases. Due o a much smaller damping coefficien i akes a longer ime for he response o be damped o a consan value. As a resuls four ime duraions were used o sud he behavior. he consan hermal properies solver is damped o a smaller cener displacemen due o a smaller hermal epansion coefficien. he seling ime is comparable beween he consan and variable hermal properies. 117

125 (a) (b) Figure. 6.5 Response o consan hea load of 50 mw a 1 MPa (a) Consan hermal properies (b) Variable hermal properies Figure 6.6 shows he op cener emperaure variaions b ime for he boh solvers. he emperaure is no full sead a he end of he firs run as shown in he dashed line. (a) (b) Figure. 6.6 op cener emperaure b ime for a consan hea load of 50 mw a 1 MPa (a) Consan hermal properies (b) Variable hermal properies Figure 6.7 shows he emperaure disribuion a he las ime sep of each response along he beam. Oher han poin 1 which is he las ime sep of he firs response plo he oher emperaure disribuions are idenical. he reason is ha he firs response plo ends a he ver beginning of ransiion o sead behavior as shown in Figure

126 (a) (b) Figure. 6.7 op cener emperaure disribuion along he beam for a consan hea load of 50 mw a 1 MPa (a) Consan hermal properies (b) Variable hermal properies Figure 6.8 shows he defleced shapes a he four ime poins. Each poin represens he resuls a he las ime sep of each response shown in Figure 6.5. he defleced shape becomes approimael idenical a poins 3 and 4. he reason is ha he dnamic effecs become minimal a hese poins as shown in Figure 6.5. As previousl eplained in secions and he defleced shape is a funcion of emperaure disribuion. A emperaure disribuion uniforml disribued along he beam and wih a sudden change a he edges leads o large bending momen a he ends and creaes sharp displacemen slope a he ends as shown is secions and Figure 6.8 (see a) shows he sharp slope a he ends in he defleced shape for consan properies however Figure 6.8 (see b) clearl shows he ero slope a he ends for he defleced shape for variable properies. his rend is observed because he epansion coefficien is parabolic disribued along he beam for he variable properies solver while i is uniform along he beam for consan properies. hese resuls sugges hermal properies as anoher erm plaing a significan role in he defleced shape. 119

127 (a) (b) Figure. 6.8 defleced shape for a consan hea load of 50 mw a 1 MPa (a) Consan hermal properies (b) Variable hermal properies Figure 6.9 shows he dominan frequencies in he boh cases. Good agreemen is obained beween he eplici consan properies solver and he implici variable properies solver. Boh consan and variable solvers indicae he firs and second dominan frequencies as eac same values. However he consan properies solver does no indicae an dominan frequenc larger han H. he implici solver for consan properies equaion is uncondiionall sable. As a resul he ime sep seleced for implici solver is wo order of magniude larger han he ime sep seleced for he variable properies solver o make he compuaion faser. As a resul he implici consan properies solver is limied in indicaing he large frequencies in he ssem. Figure. 6.9 Dominan frequencies for a consan hea load of 50 mw a 1 MPa for variable versus consan hermal properies 10

128 Seling ime. he seling ime is invesigaed for a range of pressures from 0.7 MPa o 5 MPa. As shown in he previous secion and Eq. (.30) damping coefficien is a funcion of air pressure. Damping coefficien increases as he air pressure increases. his decreases he seling ime. Alhough iniiall he hea ampliude does no seem o affec he seling ime a sud is performed o invesigae he effec of hea ampliude over he veloci of he movemen and he seling ime of he vibraion. Figure 6.30 shows he seling ime variaions b pressure. As he pressure decreases he seling ime increases. Figure Seling ime b air pressure Figure 6.31 shows he seling ime b he hea addiion o he ssem for he air pressure of 1 MPa. Alhough no variaion was epeced b increasing he hea addiion he seling ime increases. 11

129 Figure Seling ime b hea addiion o he ssem Figure 6.3 shows he cener veloci variaion b ime for a range of hea addiion o he ssem. Increase in he hea addiion increases he cener veloci which indirecl affecs he damping of he ssem. Figure. 6.3 Cener veloci b ime for a range of hea addiion 6.6. Harmonic Eciaion Afer performing he mesh independence sud as will be eplained in secion he resuls are obained for 100 nodes 11 nodes 9 nodes respecivel in he and direcions number of ime seps are seleced in each vibraion period. he ime sep is defined as follows: (6.4)

130 In order o quanif he resuls for simplici anoher erm is defined as he frequenc raio r. Frequenc raio is he response and eciaion frequenc ω divided b he naural frequenc: r n (6.5) where ωn is he naural frequenc of he bridge. Figure 6.33 shows he eciaion (q ) versus response (ν(=l/)) variaions b ime for he hea load ampliude of Ah=600 MW/m and frequenc raio of.4. he cener displacemen (ν(=l/)) is measured in he opposie direcion of he ais as shown in Figure.1 for he hea added o he op of he bridge. he ais is no quanified since he eciaion and response have differen unis. his Figure shows he boh eciaion and response changes b ime wih he same frequenc equal o ω. However here is a lag beween he eciaion and he response. his lag is called phase dela φ if i is in degrees or ime dela τ if i is in second. he response sars from ero and i akes ime unil i becomes harmonic. he resuls also show ha i akes a couple of periods unil he vibraion becomes seadil harmonic. Figure Response versus eciaion (Ah=600 MW/m and r=.4) 13

131 he firs sep o quanif he resuls is o fi he ransien hermo-srucural response wih a funcion given as follows: cos s v (6.6) where δs is he sead sae displacemen δv is he vibraion displacemen and φ is he phase dela beween he eciaion and he response. In order o obain a reasonable fi δs and δv mus be deermined when he response becomes harmonic (sead sae) hermal versus Mechanical Eciaion. Prior o sud of he vibraion and sead sae displacemen variaions due o he changes in he hea and he pressure he response is compared wih a spring-damper ssem o sud he hermall ecied ssem versus a mechanicall ecied ssem. Alhough he sud arges he pressures lower han amospheric pressure geing he soluion converged is faser for higher pressures as i will be discussed laer. he resuls were obained using a consan hermal properies solver a 1 MPa. he phase dela beween he response and he mechanical eciaion for forced vibraion of a single degree-of freedom mass-spring-damper is given b (Kell 01): 1 r an 1 r (6.7) where ξ is he damping coefficien and r is he frequenc raio as shown in Eqs (6.16) and (6.5). Figure 6.34 shows phase dela variaions b he frequenc raio for he hermo-srucural simulaion for oal hea ampliude Ah of 600 MW/m and he phase dela in he mechanical eciaion in a mass-spring-damper ssem (Eq. (6.7)). he hermo-srucural simulaion phase dela increases b he frequenc. he resuls also show ha hermal acuaion reurns larger phase dela han he phase dela in he mechanical acuaion in he mass-spring-damper ssem. he reason is when he srucure is hermall acuaed an addiional dela is creaed due o he ime ha i akes he hea o ravel and cause he hermal sresses. 14

132 Figure Phase-dela variaions b he frequenc raio Mesh-Independence. he model is iniiall run for nodes in full model for a fied ime sep of 1 micro-sec. he objecive is o fi he mesh domain for a fied ime sep. Afer obaining he fied domain he run will be repeaed for a range of ime seps o obain he ime-sep-independen resuls. Figure 6.35 show he resuls for various divisions in he direcion. he displacemen onl racked a he cener of he beam b ime (ν(=l/)). Figure Response b ime for various divisions in direcions he resuls show ha he mesh independence is obained as he number of divisions in direcion is increased o 101. he mesh independenc is accurae o he hird significan digi for his number of divisions. 15

133 Figure 6.36 shows he maimum cener displacemen (wihin 000 nsec) versus he number of divisions in direcions. he resuls oall approach he asmpoe value which shows meshindependen resuls are obained. Figure Maimum cener displacemen versus number of divisions in direcion Response Variaion b Hea. Chaper showed sead sae cener displacemen variaions b oal hea correspond o a linear ssem. he same sud mus be performed for he sead sae ampliude δs and he vibraion ampliude δv. Figure 6.37 shows he vibraion ampliude variaions b oal hea ampliude for frequenc raio of 1 a 100 kpa for boh he consan hermal properies solver and he emperaure dependen hermal properies solver. he response for consan hermal properies solver corresponds o a linear behavior wih oal hea variaions while he emperaure dependen hermal properies solver reurns a nonlinear behavior of he response wih he hea addiion. In general increasing he oal hea ampliude Ah has more significan effecs on increasing he vibraion ampliude δv raher han he sead sae ampliude δs. he variaion beween sead sae ampliude and hea is no shown because a he pressures lower han amospheric pressure his variable is ver small in he order of angsroms. 16

134 Figure Vibraion ampliude variaions b hea (r=1) Response Variaion b Pressure. Decreasing he pressure affecs he vibraion and sead sae ampliudes b decreasing he damping coefficien ξ (Eqs. (6.14) and (6.16)) and he hea ransfer coefficien. I significanl increases he vibraion and sead sae ampliudes. Figure 6.38 shows he cener displacemen versus ime for four pressures of 1 MPa (see a) 500 kpa (see b) 50 kpa (see c) and 100 kpa (see d). hese resuls are obained using he consan hermal properies solver a a low hea ampliude of 1 mw where he behavior is sill linear. As i shown in Figure 6.38 (see d) as he pressures drops o he amospheric pressure he convergence ime becomes significanl higher. Figure 6.39 shows he ampliudes variaions b pressure for oal hea ampliude of 50 mw. he higher rae of hea addiion requires using he eplici emperaure dependen hermal properies solver. he vibraion ampliude decrease b increasing he pressure. he rae of decrease is significanl lower for he sead sae ampliude δs. As previousl eplained in secion sead sae displacemen is ver small in he order of an angsrom. 17

135 (a) (c) (b) (d) Figure Response b ime for a range of pressures Figure Ampliudes variaions b Pressure As shown in Figure 6.38 obaining he sead harmonic response becomes challenging as he pressure decreases. Alhough he sead harmonic response was no obained for pressures 18

136 below 70 kpa he firs 11 ccles as shown in Figure 6.40 (see a and b) show as he pressure decreases below.5 kpa he sead and vibraion ampliudes remain unchanged. I is he poin where hea ransfer does no affec he response of he srucure. (a) Figure.6.40 Response b ime a low pressures (b) Non-Resonan Response. As he frequenc changes o non-resonan frequencies he convergence ime significanl increases. Figure 6.41 shows he cener displacemen variaions b ime a frequenc raio of 1. and air pressure of 100 kpa. he response behavior shows various frequencies oher ha he eciaion frequenc a he beginning. However afer 6000 nsec he response is damped o onl one frequenc which is he eciaion frequenc. Figure Non-resonan response b ime 19