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CHAPTER 2 RESEARCH METHODOLOGY 21 Thin Film Processing by Pulsed Laser Deposition Basic principles and advantages The current interest in the use of lasers either for scienti c investigations or for industrial applications is directly linked to the unique properties of laser light The high spatial coherence achieved with lasers permits extreme focusing and directional irradiation at high energy densities The monochromaticity of laser light together with its tenability opens up the possibility of highly selective narrowband excitation Controlled pulsed excitation offers high temporal resolution and often makes it possible to overcome competing dissipative mechanisms with the particular system under investigation The combination of all of these properties offers a wide and versatile range of quite different applications Material processing with laser takes advantages of Virtually all of the characteristics of laser light The light energy density and directionality achieved with lasers permits strongly localized heat and phototreatment of materials with a spatial resolution of better than its wavelength The monochromaticity of laser light allows for control of depth of heat treatment or selective nonthermal citation either within the surface of the material or within the molecules of the surrounding mediumsimply by changing the laser wavelength The advantages of laser processing also include reproduction of the target stoichiometry low contamination levels and insitu control of the film properties The controlled melting of the thin metallic overlayers deposited on various structures has been used in the formation of metal silicides and ohmic contacts for large band gap semiconductors and in modifying the surface properties of ceramics Conceptually and experimentally Hllsed Laser Deposition PLD is simple Figure 1 shows a schematic diagram of a typical experimental setup It consists of a multiple target holder and a substrate holder in a high vacuum chamber maintained by a turbomolecular pump The target consists of bulk material oriented at an angle of 450 with respect to the incident laser beam A highpower laser is used as an external energy source to vaporize materials and to deposit thin films A set of optical components is us ed to focus and raster the laser beam over the target mrface The evaporated material is deposited onto a substrate placed parallel to the target at distance of 35cm The substrate temperature can be varied from room temperature to 800 C The film quality depends various parameters such as substrate temperature laser energy density pulse repetition rate pressure in the chamber and substratetarget geometry The decoupling of the vacuum hardware and the evaporation power source makes this technique so exible that it is easily adaptable to different operational modes without the constrains imposed by the use of internally powered evaporation sources Film growth can be carried out in a reactive environment containing any kind of gas with or without plasma excitation It can also be operated in conjunction with other types of evaporation sources in a hybrid approach8 Elmnnl rrrrwrerb were yw Suhlnn m n my mower u lumxalmn um swvwmrlu mum usquot run n Iurbu wrtuhr yum Frgure 1 Schemauc dragmm of me laser physreal vapor deposmon technique Ref 8 r physreal phenomenon Theoreucal deserrpuorrs are mumdrserplmry and comme bom depends on laser charactensucs as well as me opueal wpologxcal and thermodyname propa ty of urga When the laser mdxauon rs absorbed by a sohd surface eleemeomagneue energy rs converted rst mm sledme exemauon and then rmo VHPrm HPm b Mm excxmuon plasma formauon and exfobauon Evapomuons results m a plume39 consxsung of a m m r n mm vaw m r n srzed sobd pamculates and monen globules The collmonal mean free pam msxde me dense pm 5 my 51m As a result mmmw an m 15 mam m We rapidly was mm m mm rm m Mg m farm a mu m wah maaynm aw cmamm my 5 m mam um wnh m chain a m mm m n canbe media W mm a myhndafmnlznal Th2 versatde 5 mama mm my ngm amends mm A Mm byEeechm 1991 155122 mm mm gwwnbyPLD A c mem H mm Emnlu m MHZ mm V wwumm WNW WWW l lumAWun Hum me WW WWW w M M youva Fxgure 2 Somatic xepxesmmmn af39he stages uf m mgzhmenmms mung shun pl sehgmpkulasenmenchmw hamhd mm In maquot m understand bass phyacal 91mph uf PLD mum dlxmg each uf m nmasecmd laser pulse lasaotazgn mummy an be amaaa mm mm was Shawn schzmmcallym glxe 2 Depmdmg mm type uf mum uf chains1 beam wah the target there are three separate regimes i interaction of the laser beam with the target materials resulting in evaporation of the surface layers ii interaction of the evaporated material with the incident laser beam resulting in isothermal plasma formation and expansion and iii anisotropic adiabatic expansion of the plasma and subsequent deposition The first two regimes start with the laser pulse and continue through laser pulse duration The third regime starts after the termination of the laser pulse 9 i Interaction of the laser beam with the target Intense heating of the surface layers by highpower nanosecond laser pulses results in melting and evaporation of the surface layers The heating rate melting and evaporation during pulsed laser irradiation depend on i laser parameterspulse energy density E pulse duration 139 wavelength A and shape of the laser pulse and ii materials properties re ectivity absorption coefficient heat capacity density thermal conductivity etc The free carrier hole collisions provide the mechanism for the absorption of the phonon energy in the bulk surface The heating and melting effects of pulsed laser irradiation on materials constitute a three dimensional heat ow problem In nanosecond laser processing the thermal diffusion distances are short and the dimensions of the laser beam large compared to the melt depth Hence the thermal gradients parallel to the interface are many orders of magnitude less than the thermal gradients perpendicular to the interface This makes the problem for heat ow during PLD one dimensional governed by 6t 61 6x p1 TCp ax x I 3amp1 mm 1 tl RTe 7 1 Where xrefers to the direction perpendicular to the plane of the sample and t refers to the time The subscripts il 2 refer to the solid and liquid phase respectively The terms p1 T and CpT are respectively the temperature dependent density and thermal heat capacity per unit mass of the target material RT and aT are the temperature dependent re ectivity and absorption coefficient of the material corresponding to laser wavelength The term 50 is the time dependent incident laser intensity striking the surface which depends on the intensity and shape of the laser pulse The term Ki refers to thermal conductivities of the solid and liquid phases at the interface The last term on the right hand side of the equation 1 is the heat generation term Accurate numerical solution to this equation by finite difference method gives the evaporation characteristics of the pulsed laser irradiated materials Using this method the thermal histories of the laser irradiated materials can be predicted Thus the effects of the variations in the pulse energy density E pulse duration t and substrate temperature T on the maximum melt depths solidification velocities and surface temperatures can be computed Although the presence of a moving surface formed as a result of melting or evaporation and timedependent optical and material properties make it difficult the analytical solution simple energy balance considerations can be taken into account to assess the effects of interaction of the laser irradiation with the materials By using energy balance method the amount of material evaporated per pulse can be calculated The energy deposited by the laser beam on the target is equal to the energy needed to vaporize the surface layers plus losses due to the thermal conduction by the substrate and the absorption by the plasma The energy threshold Eth represents the minimum energy above which appreciable evaporation is observed Since the losses in plasma and substrate change with pulse energy density Eth varies with energy density too Thus the heat balance equation is given by 1REEm AXl AH CVAT 2 where Ax R AH Cv and AT are the evaporated thickness reflectivity latent heat volume heat capacity and the maximum temperature rise respectively This equation is valid for conditions where the thermal diffusion distance m is larger then the absorption length or attenuation distance of the laser beam in the target material lat In the above expression D is the thermal diffusivity of the target and t is the larger pulse duration In equation 2 the energy threshold depends on laser wavelength pulse duration plasma losses and the thermal and optical properties of the material ii Interaction of the laser beam with evaporated materials The interaction of the highpower laser beam with the bulk target materials leads to very high temperature gt2000K resulting in emission of positive ions and electrons from the free surface The emission of electrons and positive ions from a solid surface exhibits an exponential increase with temperature The thermionic emission of positive ions can be calculated by LangmuirSaha equation 139 no g g0exp IKT 3 where 1 and 139 represent positive and neutral ion uxes leaving the surface at 0 temperature T g and g0 are the statistical weights of the ionic and neutral states bis the electron work function and I is the ionization potential of the materials The fraction of ionized species increases with the temperature since Igt 1 Although the surface temperature of the target is close to boiling point higher temperatures can be induced in the evaporated plasma by the interaction of the laser beam with it The penetration and absorption of the laser beam by the plasma depend on the electronion density temperature and the laser wavelength The penetration or re ection of the incident laser beam depends on the plasma frequency vp 89gtlt103 712039s where ne is the electron concentration in the plasma The plasma frequency should be lower than the laser frequency for the laser energy to be transmitted or absorbed For example for XeC12 excimerlaser wavelength A 308 nm the laser frequency is 974XlO14 sec with critical electron density for re ection given by nel2X1022cm3 The material evaporated from the hot target is further heated by absorption of laser radiation Although the laser evaporation for deposition of thin lms occurs at much lower temperatures the plasma temperatures are f the order of 104K Different mechanisms become important in the ionization of the ionization of the laser generated species impact ionization photo ionization thermal ionization and electronic excitation The primary absorption mechanism for plasma is the electronion collisions The absorption occurs primarily by a process which involves absorption of a photon by free electron The absorption coefficient aP of the plasma is given by up 369 X10823n12T 395v3l eXp hvkT1 4 where Z ni and T are the average charge ion density and temperature of the plasma re ectively and h k and v are the Plank constant Boltzman constant and frequency of the laser hght respeeuvely The term leexpehvk739ln equauon 4 represents the losses due to sumulated emlsslon For KrF exclmer laser 1248nm the exponentlal term becomes unltv for Tltlt40000K and ean be apprommatelv by hvkT for Tgtgt40000K The absorptron term shows a T dependenee for low temperature Tltlt40000K for 1248nmTltlt40000K for 11 osnm and T for hlgh temperatures As rt ls seen the heatlng of the evaporated matenals depends on the eoneentratlon of the lomzed specles plasma temperature wavelength plasma temperature wavelength pulse durauon ete Also the pamcle denslty tn the plasma depends on the degree oflonlzatlon evaporatldn rate and the plasma expmslon velocmes um rARGE r msm CLOU Flgure 3 L the d d of a laser on a bulk target A unaffected bulktarget B evaporatedtargetmatenals C dense plasma absorblng the laser radlatlon and D expandlng plasma transparent to the laser beam From ref 9 Because of the high expansion velocities of the leading plasma edge the electron and ion densities decrease very rapidly with time This makes the plasma transparent to the laser beam for larger distances away from the target surface The inner edge of the plasma in a thin region close to the surface of the target it constantly absorbing laser radiation due to the constant augmentation of the plasma with evaporated particles A schematic diagram of the laser interaction with the plasma target is shown in gure 3 The diagram shows that four separate regions can be distinguished during the laser pulse These are l unaffected bulk target 2 evaporating target surface 3 area near the surface of the target absorbing the laser beam 4 rapidly expanding outer edge which is transparent to the laser beam A dynamic equilibrium exists between i the plasma absorption and ii the rapid transfer of thermal energy into kinetic energy These are two mechanisms controlling the isothermal temperature of the plasma The initial dimensions of the plasma are of the order of a millimeter in the transverse direction while they are less than 1 pm in the perpendicular direction During the isothermal regime and assuming an initial expansion velocity of 105106 cmsec the perpendicular dimension of the plasma is in the order of 10 lOOum at the end of a 30nsec laser pulse The rapid plasma expansion in vacuum results from the large density gradients The plasma absorbing the laser energy can be simulated as a HTHP gas which is initially confined in small dimensions and is suddenly allowed to expand in a vacuum The equations of gas dynamics governing the expansion of the plasma consist of the equation of continuity and the equation of motion The velocity density and the pressure profiles in the plasma are shown schematically in figure 4 It shows that the density is maximum while the minimum at the inner edge of the plasma The plasma density and the pressure gradients are monotonically decreasing vquot m mm a uf Expansmn the eeeelereuer rs very ugh whEn me Expansmn velunnes are luw When he zem resulting m are dungated plasma shape Velocil v xyzx Dcnsuy n x y z x Plexsurc p x y 7 r no0 x 7 4 0mm in x Kinsman Hm d a gure 4 SchEmanc preme shuwmg me danslty n pressure p and velamty v 333481115 m meplesmemx arreeuun paprmd mlartu me Erget surface The dansxty and plasma pressure are munutumcally decreasmg Guru the target surface whde me velamty mmeases hrreedy Ref 9 iii Adiahztic plasma Expans39nn and lm dqmsi nn Guru the target mm he plans because there is nu absurpuun uf laser energy Thus en 3642me Expansmn eeeurs with ememeayrrsrme relanun pven by TXYZIV sum 5 spa vulume a Dunng me emebeue repme me Lharmal Energy is cummed m kmen energy with the plasma achieving extremely high expansion velocities The temperature drop is slow because 1 the cooling due to expansion is balanced by energy regained from recombination processes by ions and 2 the plasma expands in one direction The initial dimensions of the plasma are much larger in the transverse direction The initial dimensions of the plasma are much larger in the transverse direction The initial dimensions of the plasma are much larger in the transverse directions y and 2 which are in the order of millimeter while in perpendicular direction x is in the order of 20 100um In the adiabatic expansion regime the velocity of the plasma increases in the direction of the smallest dimension References N L 4 V39 6 PulsedLaser Deposition ofThin Films edited by D B Chrisey and G K Hubler WileyInterscience New York pp 15 1994 Advances in Laser Ablation of Materials edited by R K Singh D H Lowndes D B Chrisey E Fogarassy and J Narayan Mater Res Soc 1998 Physical Vapor Deposition of Thin Films J E Mahan WileyInterscience New York pp 133 151 2000 R K Singh J Narayan A K Singh and J Krishnawmy Appl Phys Lett 54 22711989 J Narayan D Fathy O W Hollland B R Appleton R F Davis and P F Becher J Appl Physics 56 2913 1984 R K Singh K Jagannadham and J Narayan J Mat Res 3 1119 1988 7 J Narayan N Biunno R Singh 0 W Holland and O Auceiello Appl Phys Lett 51 1845 1987 8 R K Singh 0 W Holland and J Narayan J Appl Phys 68 l 233l990 9 R K Singh and J Narayan Phys Rev B 41 13 88431990 22 Characterization Methods of Thin Films Basic Principles and applications The properties of thin lms are determined by their chemical composition the content and type of impurities in the thin lm or on the surface crystal structure of the thin lm and on the surface and the types and density of structural defects In addition as the applications of thin lms extend to microelectronics optoelectronics magnets and other areas electrical optical and magnetic properties also have to be monitored and optimized In particular to TiN TaN thin lms as diffusion barriers in Cu metalization and interconnects electrical chemical and structural analysis should be monitored For TiN nanocrystalline hard coating TiNAlN alloys and superlattices hardness and structural information are the main properties need to be studied in detail While for GaN AlN and their superlattice growth optical and structural characteristics are the major requirement Most experimental procedures including techniques are introduced in the experimental part in each chapter In the following several important techniques which have been extensively used during this research are discussed in detail This study is focused on structural analysis Xray TEM and STEM techniques will be discussed in detail including basic principles and applications in materials science of thin lm Other analysis including mechanical and electrical property measurements will be described based on principles and de nitions The listed methods will be discussed in the following sections 1 Xray diffraction Structural XRD 2 Transmission Electron Microscopy Structural TEM 3 Scanning Transmission Electron Microscopy Structural STEM 4 Electron Energy Loss SpectroscopyElemental EELS 5 Nanoindentation Mechanical hardness 6 Electrical resistivity Electrical 221 X ray diffraction Structural XRD Xray diffraction is one of the most important nondestructive experimental techniques used to address issues related to the crystal structure of solids including lattice constant identi cation of unknown materials orientation of single crystals preferred orientations of thin lms defects stress etc Basic mechanism of Xray analysis is based on that when a parallel and monochromatic Xray beam with a wavelength A and angle of incidence 6 is diffracted by a set of planes oriented in specific directions there are sharp peaks corresponding to the spacing between the planes d when the conditions of the Bragg s law are satisfied 2dsin nl 1 These peaks are characteristic to the material and the crystal structure The crystal structure of a specific material determines the diffraction pattern and in particular the shape and size of the unit cell determine the relative intensities of these lines In the structural analysis by using Xray of known wavelength A and measuring 6 we can determine the spacing d of various planes in the crystal The essential features of an Xray spectrometer are shown in figure 1 It should be noted that l the incident beam normal to the re ecting plane and the diffracted beam are always coplanar and 2 the angle between the diffracted beam and the transmitted beam is always 26 X rays from the tube T are incident on a crystal C which may be set at any desired angle to the rnerdent beam by rotatron about an axls through 0 whlch ls the eenter of the speetrometer clrcle D ls a eounter whlch measures the rntensrty of the dlffracted erays rt ean also be rotated about 0 and set at any deslred angular posrtron Thus by measunng the peak posrtrons one ean determrne the shape and slze of the unrt eell and by measunng the rntensrtres of the dlffracted beams one ean determrne the posrtrons of atoms wrthtn the unrt eell Conversely lf the shape and slze of the unrt eell of the erystal are known we ean predet the posruons of all the posslble hnes of the lm Flgurel Schematlloerrayspectxometer Ref 1 In the deyrauon othe Bragg Law eertatn ldeal condluons are assumed for examplethe erystal ls perfeet and the rnerdent beam ls eomposed of stnetly parallel and monoehromaue radauon It should be noted that only rn mte erystal ean be eonsrdered as a perfeet erystal and that a small slze of an otherwrse perfeet erystal ean be eonsrdered as a erystal rmperfeeuon Thls eomes from the condluon that the waves lnvolved m a dlffracuon relnforce eaeh other and there ls a eontnbutron for the destrueuye rnterferenee from the planes deeper rnto erystal Destrueuye rnterferenee ls therefore just as mueh a consequence of the perroderty of atom arrangement as rt ls the coustructsve mterfa39mce As a result the wldth of the dlffractlon curve rucrease as the thlckness of the crystal decreases Schemauc represerrtauorr of the etlect of the ne parucle slze or drtlracuorr curves ls shown on gure 2 The followmg expresslon glves l l l m of therr L EL r l c small dlffractloncurves l ugllscossguhere B 792fmm gure 2 ls thewrdhre the dlfference between the two extrme angles at whlch the urterrsrty ls zero lxlmsltl spectral mm of the Xray source whrch ls proporuorral to tarts and becomes qulte uotrceable as 9 approaches 90 The eet ofthe mosalc structure of the crystal or lm can also m uence the broaderuug ofthe X ray hue Thus lfthe angle of mlsonentatlon between the blocks r t 411 u r r r but all angles between 9g and 95 3 Another effect of the mosatc structure ls to rucrease the urterrsrty of the rellected beam wrth respect to that theoreucally calculated for arr ldeally perfect crystal The diffracted beam is rather strong compared to the sum of all the rays scattered in the same direction because of the reinforcement strengthening which occurs but extremely weak compared to the incident beam If the scattering atoms are not arranged in a regular periodic way then the xrays will have a random phase relationship to one another and neither constructive nor destructive interference takes place under such conditions Then the intensity of a beam scattered in a particular direction is simply the sum of intensities of all X rays scattered in that direction If there are N scattered rays each with amplitude A and therefore with intensity A in arbitrary units then the intensity of the scattered beam is NA2 However if the rays are scattered by atoms of a crystal in a direction satisfying the Bragg Law then they are all in phase and the amplitude of the scattered beam is N times the amplitude A of each scattered Xray or NA Therefore the intensity of the scattered beam is therefore N2A2 or N times as large as if the reinforcement had not occurred This explained why the Xray intensity of a crystal is much higher than that of an amorphous solid As stated earlier the intensities of the diffracted beams are determined by the positions of the atoms in the unit cell To establish an exact relationship between intensity and atom positions is a complex problem because of many variables involved since the Xrays are scattered by electrons by atoms and by all the atoms in the unit cell When a monochromatic beam of X rays strikes an atom two scattering processes occur Tightly bound electrons are set into oscillation and radiate X rays of same wavelength as that of the incident beam coherent scattering More loosely bound electrons scatter part of the incident beam and slightly increase their wavelength with an amount depending on the scattering angle incoherent angle Since the intensity of the coherent scattering is MWVFFM dueeunh where fsm EM and the values fur f fur vannus atnms and vannus values nf HA Vh w qur The Enhermdy scattered ramaunn 39nm all the atnms undergnes remfnrcemem nunstxumve interference m mam dxrecunns and thus prnducmg mf acted beams Fxgm33 The atnrm seattehhg factnr nfcnpper Ref 1 There are six factan affeetmg the relatwe mtehsmes hf the dxf 39amnn lmes 1 pnlanzatmn factnr 2 stxucture factnr 3 mumphuty factnr 4 Lnrenz factnr 5 h mu v F M r Thus dif 39acnnn hhe 5 men by 2 lcn52211 1 F 7 z pm29cns9 0 1h equatan 2 F15 the stxucture factnr accnuntmg fur the resulting enhmhuunh hf a umt eeu fur a speci c hkl re echnn and 1t 5 a mmehsmhless whmh is the man nf the amplitude scattered by the unit cell to the amplitude scattered by one electron in the same direction The multiplicity factor p accounts for the relative proportion of the planes contributing for same re ection and can be de ned as the number of different planes in a form having same spacing Example p6 for 100planes in a cubic crystal and p8 for the lllplanes for a tetragonal crystal p4 for 100 planes and p2 for 001 planes The expression in the numerator in the brackets of Equation 2 is so called polarization factor and it arises from the fact that the incident Xray beam is not polarized The term in denominator is the so called Lorenz factor and it is a sum of three factors 1 the value of maximum intensity Imax depends on the angular range of crystal rotation over which the energy diffracted in the direction 26 is appreciable and depends as lsin6 and therefore Imax is large at low scattering angles and small at large scattering angles 2 the number of microcrystallites favorably oriented for re ection is proportional to cos 6 and is quite small for high angles of re ections and 3 geometrical factor giving the length of any diffraction line as proportional to Rsin2 6 where R is the radius of the camera and hence the relative intensity per unit length of the line is proportional to lsin2 GB The whole term in brackets in Equation 2 is called Lorentz polarization factor and it is plotted in Figure 4 The overall effect of these geometrical factors is the reduction of re ection intensities at intermediate angles compared to those in forward or backward directions The absorption factor resulting in a decreased intensity of a diffraction beam due to absorption of the incident beam cancel with the temperature factor due to the thermal vibration of atoms since the two factors depend on the scattering angle in opposite ways 100 FrgureA Lmentzrpnlanzatmn factnrys Bragg angle Ref 1 The yrhynhes rnechanrcal electrrcal chnnrcal etc Df a slngle phase aggregate are detnrnrned by twh facthrs l the yrhynhes hfthe slngle crystal hfthe rnaterral and z the way ln whrch the slngle crystals are put thgethn th fm39m chrnyhsrte rnass Thus the r Ml l n VF N n deterrnrne the yrhynhes hfthernaterrals The are Df grams ln a yhlycrystalhne rnaterral has a pmmunced effect an many Df rts ymyerhes manly rncrease ln strength and hardness wrth the reducth hfthe gran nze ery mffrachhn can we smrquamtatlve rnfhrrnanhn ahhut the gran srze almngth m the mfhachhn hne ls reyealed by the hne bmadmmg The marneter Df the crystal pamde trs Lhm measured as 0 mschse where B ls the llnebmadmmg The crystal pelfectllm DI the effect hfthe sham bath umem and nhnunrfhrrn alsh deterrnrnes the dl iractmn hne yhsrhhn and shane as shhwn ln gure 5 lf the gmm DI lm is gwen umfm39m tensile strain at nght angles tn the re eenng planes merr spaungs became larger than an and me cmnespnndmg mf 39acunn lme sh s tn 1nwer angles but dues nut change its shape Hnwever nnnumfm39m sham nn me up tensmn nne une bn nm nmpressmn eause lme bmademng HqHU FxgurES Effect nflamce sham nnthe lme mm and pnsmnn Ref 2 And nally the presence nr absence nfaprefmed nnentahnn m the lm ean be revealed Sam the intensity msmbuunn nf lmes parallel tn the substrate surface In suen ease nn1y grams whmh ean nnmbute tn the hkl re emnn are these wnnse th planes are parallel tn the substrate surface mne texture is suen that here are Very few suen grams une then would indicate that the corresponding lm planes are preferentially oriented parallel or nearly parallel to the lm surface References l B D Cullity Elements of X Ray Diffraction AddisonWesley Publishing Co Inc 1982 2 L Azaroff R Kaplow N Kato R Weiss A Wilson R Young Xray Diffraction Mc GtawHill Inc 1974 103 222 Transmission Electron Microscopy Structural TEM Transmission electron microscopy TEM is a method used to obtain structural and morphological information from specimens that are thin enough to transmit electrons This technique can help revolutionizing our understanding of materials by completing the processingstructureproperties down to the atomic levels TEM has increasing applications not only in materials science research but also in the semiconductor device technology biotechnology and other material or microstructure related eld For example In device fabrication TEM has been extensively used to provide i1formation about the geometry of patterned lms the uniformity of thickness and coverage in addition to surface morphology and topography including size and shape and presence of compounds Especially recently the submicron technology has been introduced into device fabrication the resolution of SEM and other normal surface techniques can not fulfill the requirements of detailed atomic structure study TEM has become one technique which can probe detail structural and defect information Additionally TEM can also give crosssectional view of interface regions such as the interfacial reactions perfection of devices and diffusion study The combination of TEM with other analytical techniques such as STEM EELS and EDX TEM has become a powerful technique which can combine imaging and chemical compositional study down to a single atomor a column of atoms strictly speaking For example JEOL 2010FEG NCSU has point to point resolution of 018 nmTEM combined with STEM with resolution of 012nm and EELS analysis 104 In this section major parameters of TEM are listed Several important imaging and diffraction techniques are discussed and related examples for different techniques are presented Resolution and Aberration The main parameter in a transmission electron microscope is its resolution Dr Duscher had a question in midterm of 2002 spring TEM course Xerox Microscopy the patent author claimed to be able to image atoms after copying something with a copy machine 20 times with a magni cation of 2 2201048576 Is it possible The answer is obviously no Usually the resolution of a Xerox copy machine is about 01um Although the magni cation is increasing after several copies the resolution has been decided by the rst copy of the machine This question clari ed the difference between magni cation and resolution According to classical Raleigh criterion without any aberration of lenses the smallest distance that can be resolved is given approximately by 6 0611 where is the wavelength of the radiation u the refractive index of the viewing medium and B is the semiangle of collection of the magnifying lens Due to the coherency of electron beam in TEM and the short wavelength of the accelerated electrons l is in the order of hundredths of an A the resolution limit is down to 1A for lMeV electrons 100087A and 21510393 rad Spherical aberration along with the chromatic aberration and the stigmatism are the main electromagnetic lens defects that limit the resolution of the electron microscope shown in gure la The spherical aberration is a lem defect arising from the non 105 paraxiality of the electron beam causing electrons leaving the point P at higher angles with respect to the optic axis of the microscope to cross to focus before the image plane while electrons left the objective lens closer to the optic axis are focused on the image plane Thus instead of a point a disk with a radius rs is formed where rsCS 32 3 is the angular aperture of the lens Chromatic aberration arises from the nonchromaticity of the electron beam ie the electrons leaving the electron gun are with slight difference of the energy E 3eV This causes the faster electrons to be less strongly refracted from the objective lens than the lower energy electrons as shown in figure lb The higher energy electrons thus are brought in focus beyond the image plane which they pass at distance 15CCBEE where Cc is the chromatic aberration coefficient Astigmatism arises from the asymmetric magnetic field and it occurs when the lens exhibits different focal lengths depending on the plane of the ray paths Thus in figure the rays traveling plane A are focused at PA while those in plane B focus at point PB This leads to the fact that a point on the object is imaged as a disk with radius rABfAY where fA is the maximum difference in focal length arising from astigmatism If all the astigmatisms are corrected and the sample is thin enough that chromatic aberration is negligible then the spherical aberration error limits the resolution The resolution is given by the combination of Rayleigh criterion and the aberration error ri091cs3 4 2 This equation gives the practical resolution of the microscope Here r is the spherical aberration error and Cs is the spherical aberration l is the wavelength of electron 106 m a my 22mg 5 mm a mvS my 45 mg 1 Mug 1 amnn v w gm 1 1 gure 1 Objemlve augman a Spherical b Ehrumahc c asngnansm Ref 1 Ith a eld and Ith anncIIs Depth uf eld D schanancally Shawn m gum 5 the value uf meme specimen pusmuns un bum sxdes uf the themencal amen plane 1 uf the ubjemve Ian mum quot1 uh in a disc of radius raD2 where a is the objective angular aperture In order to keep the resolution from diminishing the diameter of this disk should be smaller than or equal to the resolution distance d so that aDd The larger the depth DNSOOA the better chance that the specimen is focused on its whole thickness Depth of focus D39 is de ned similarly in the nal image plane as depth of eld in the object plane Now d dM and a39aM where M is the nal magni cation Thus depth of focus D39DM2 Magni cation of the order of 105 would give depth of focus 500m which means that the uorescent screen position is not critical Image and Diffraction Modes The objective lens takes the electrons emerging from the exit surface of the specimen disperses them to create a diffraction pattern DP in the back focal plane and recombines them to form an image in the image plane To see the diffraction pattern you have to adjust the imaging system lenses so that the back focal plane for the intermediate lens Then the diffraction pattern is projected onto the viewing screen see gure 2A For imaging mode you readjust the intermediate lens so that its object plane is the image plane of the objective lens Then an image is projected onto the viewing screen as shown in 2B 108 Sncchncn kemme Obicuiw lens n wnnr Obicclwc upnnulc Wed buck l39ucul puma 39 Rumuvn npnnlne Change slmnglh Immmudmlc lcm I l mjcclor lens Flnlll Hung Flgure 2 The two basl operatlons of the TEM lnnaglng system lnvolve A Projectlng 1n eaen ease the lntel39medAate lens selects either he baek focal plane or me lnnage plane of me objectlve lens as its objects Ref 4 Image mode and image contrast The image contrast in TEM arises because of the scattering of the incident beam by the specimen As the electron beam transverse the specimen it changes both its amplitude and phase These changes give rise to image contrast There is a fundamental distinction between amplitude contrast and phase contrast which in many cases can both contribute to the image The amplitude contrast can be described in terms of massthickness contrast and diffraction contrast Williams and Carter 1996 Diffraction contrast is most widely used to identify defects and distinguish between different types of crystal defects This type of contrast arises due to coherent elastic scattering at special Bragg angles This intensity in a diffracted beam depends strongly upon the deviation parameter s the crystal defects distort the diffracting planes and therefore the diffraction contrast from regions close to the defect would depend upon the properties of the defects such as strain field To get good interpretable diffraction contrast the specimen has to be tilted into twobeam conditions Massthickness contrast arises from incoherent Rutherford elastic scattering of electrons any scattered beam can contribute to the formation of the image Two beam conditiom bright eld and dark eld Twobeam conditions bright field and dark field are specific imaging conditions when the specimen is tilted so that only one diffracted beam is strong The electrons in the strongly excited hkl beam have been diffracted by a specific set of hkl planes and so the area that appears bright in the dark field DF mode is the area where the hkl planes are at Bragg condition Therefore the DF image contains specific orientation information Bright field image is obtained when only the directly transmitted beam is used for the formation of the image However when the image is created using only diffracted beam 110 it is ealled dark eld image The Specimen can be tilted to set up several dlffefem two beam conditions The DF images can be formed from each strongly diffracted beam a ee tilting me Specimen and each will give a dlffefem image Bright eld BF and DF images show almost complanentary contnst under twobeam conditions To further L L A H 39 39 39 39 quot awayfromcxact Bragg condition where 54 towards positive value of s lds way lde inleepmable images under kinematical diffraction conditions are obtained Pammeur s is ealled deviation from Bragg Condition B mm m mm m lumuw mm m lm nu mudml kc lu M mullm m mung hludum phm i l 1 n pmmn in r lhllmrlul l l Wm WM l f y dmm m o o o o o o o o 5342 ddled 9w 52m O 0 0 RD 9 mm lyeam mam mquot Ullmuml hum I L07 0 o o o o r Figural 39 39 39 w produce A a BF image formed from me direct beam B a daplaeedapenme DF image formed with a specific offaxis scattered beam and C a CDF image where the incident beam is tilted so that the scattered beam remains on aixs Ref 4 Phase contrast and HRTEM Phase contrast is formed when the transmitted and diffracted beams recombine as shown in gure thus preserving their amplitudes and phases and this way lattice images or even structure images can be formed see Figure 4 If details in the structure are to be seen which can give atomic structure atomic positions and defects in that structure to obtain higher and higher resolution information has to be included from higher and higher angles in reciprocal space ie images should be taken with as many beams as possibleHRTEM One of the approaches to model the image formation in atomic resolution mode or high resolution mode is to consider the transmission electron microscope as a linear system which is the information theory approach For a linear system 058397 Sg 3 06S1 3Sr Where the expression on the left arrow is the input signals and that on the right side out signal Or in other words linear combination of input signals is transmitted by the system as a linear combination of output signals Then analogy from the information theory as shown in figure 5 first input signal is the object phase shift ie phase shift in the electron wave function cause by the specimen in TEM Second the action of the transmission system TEM is to take this input signal in the real space and consider it in Fourier space ie consider the input spectrum as object phase shift spectrum in TEM The way input spectrum is achieved from input signal is by Fourier Transform Third then the output spectrum is achieved from image spectrum by multiplying it by transfer function or the 112 socalled phase contrast transfer function in TEM Fourth nally be output signal is ukeu by inverse Fourier Transform achieving mus die image eouuasi This seheme from he information theory can be used to explain me mixes in TEM A mush 0g uiuuuuio sPtciMEu 0mm wr LENS 545 mm MANE was um i Figure 4 Phase contrast imaging from a periodic object The diffracted and transmitted beams recombine atthe image plane Ref 2 Input signal Object phase shift Input spectrum Object phase shift spectrum Output spectrum Image contrast spectrum Output signal Image contrast Figure 5 Schematic representation of TEM as a transmission system Diffraction Mode and Selected Area Diffraction The most common diffraction mode is the socalled parallel beam mode with angular dispersion 1039410395 rad while for speci c applications in convergent mode the angular aperture is of the order of 10392 rad The main features of the electron diffraction are 1 very small wavelength of electrons compared to lattice parameters resulting in small diffraction angles Thus the Bragg equation becomes 26nldhk1 and the diffraction angle 261 and 2 There is a very strong interaction with the matter so the kinematic approximation does not apply In transmission electron diffraction the specimen has to be very thin With a thin single crystal the resulting diffraction domains are in the form of fine rods normal to the specimen For a hkl re ection to be active the re ection sphere has to intersect the corresponding diffraction domain The observed re ections are those relative to the lattice planes hkl which have the incident direction uvw as a zone axis 114 and therefore the re ection indices are those obeying the following expression for the zero Laue zone hukvlw0 This is an equation of the reciprocal lattice plane uvw passing through the origin and containing lattice point hkl Selected Area Electron Diffraction Depending on the adjustment of the post objective lenses intermediate lens diffracted lens on one of the planes the nal viewing screen or lm displays either the electron diffraction pattern or the electron image of the object The selected aperture inserted into the image plane limits an image selection area of diameter ds dsm Choosing a given aperture results in selecting a specific area of the specimen The active area of SAD is of the order of dslum so with magnification of the objective lens m100 the image selection area ds 100 pm The diffraction length L is not directly expressed The diffraction constant or camera constant KL which describes the magnification of the diffraction pattern can be determined from standard specimen knowing that LdR as shown in Figure Here R is the distance between the transmitted beam and the diffracted spot corresponding to interplanar distance d the diffraction mode can be utilized in sub modes like selectedarea electron diffraction microdiffraction or nanodiffraction convergent beam electron diffraction etc 115 References 1 D B Williams Practical Analytical Electron Microscopy in Materials Science Philips Electronic Instruments Inc Electron Optics Publishing Group New Jersey 1984 J W Eddington Practical Electron Microscopy in Materials Science V Nostrand Reinhold NY 1976 J Spence Experimental High Resolution Electron Microscopy Oxford Univerisity 1988 D B Williams and C B Carter Transmission Electron Microscopy Plenum Press New York 1996 116 223 Scanning Transmission Electron Microscopy and Electron Energy Loss Spectroscopy Structural and elemental STEM amp EELS STEM with Z contrast Z contrast techniques in scanning transmission electron microscopy STEM can provide strong compositional sensitivity at atomic level Transmitted beam is used to get bright eld STEM image Highangle Rutherfordscattered electrons are collected by high angle annular dark eld detector to obtain Zcontrast dark eld images Probe size is critical in obtaining atomic resolution Zcontrast image STEM mimics the parallel beam in a TEM by scanning strictly parallel to the optic axis at all times It is a key feature of STEM that scanning beam must not change direction as the beam is scanned Fig l is an illustration of how the STEM achieves the parallel beam Two pairs of scan coils are used to pivot the beam about the front focal plane of the upper objective C3 pole piece The C3 lens then ensures that all electrons emerging from the pivot point are brought parallel to the optic axis and an image of the Cl lens crossover is formed in the specimen plane Now if the objective lens is symmetrical then a stationary diffraction pattern is formed on the back focal plane One big advantage of forming images this way is that no lenses is used in an SEM So defects in the imaging lenses do not affect your image resolution which is controlled by beam only Hence chromatic aberration which can limit TEM image is absent in STEM images which is advantageous if you are dealing with a thick specimen Similar to TEM techniques STEM can form its own bright eld BF and dark eld DF images 117 Flg 1 Scannlng me conva39gmt probe for STEM lrnage formauon uslng two palrs of scan colls belween me C2 lens usually smelled off and me upper oblecuye poleplece The probe rernalns pamllel to the opuc alrls as l scans 5 BF STEM images dlffa39mt 39om that for a aw beam TEM lrnage 1n the TEM a poruon of the eleclrons emeglng 39om an area of the speclrnen ls selec1ed and the dlslrllluuon ls prolec1ed mm a screen wnlle ln STEM the beam ls scanned on the speclmm by adlusung the scan colls mese sarne colls are used to scan me CRT synchronously The elecuon demecwr acts as the l on the CRT slnce ll lakes up w 2043 scan llnes w bulld up an lrnage on the CRT the whole process of me creang a STEM lrnage ls much slowa39 than TEM lrnaglng 1n TEM ln orda39 to form a BF lrnage an apmure ls lnsened lnw me plane of the TEM dif 39actlon paneln and only allowed me direct elecuons mrougn lc into me lrnaglng 113 qualiuns and Newton s M lorTransiem CnnLimiation Constrainnd heo E quotplea Conclusions di gsexutdilz i39reimg ieimit Clt9gtimitimiuieiiii gtim C T Kelley NC State University timkelleancsu du Joint with Liqun Qi Li Zhi Liao Corey Winton Todd Coffey P Iovd iv August 2007 Nonlineal qualiuns and Newton s Method P quotlarTransit m Continuation Constrain Conclusions Nonlinear Equations and Newton39s Method Implementation Pseudo Transient Continuation IJtc What39s wrong with Newton Integration to Steady State and lJtc Flow Through a Nozzle Constrained lJtc Utc Theory Convergence Dynamics Examples Bound Constrained Optimization Inverse Singular Value Problem Model Calibration Kelley Baumx rms A Nonlinear Equations and Newton s Method PE lorTransiem Cmninn c Cons c T xeory E mple Conclusions Cdlllalbmamrs NCSU Students Scott Pope Ji Reese Corey Winton NCSU Alums Todd Coffey Katie Fowler Chris Kees Hong Kong Liqun Qi Li Zhi Liao NCSU MAE Scott McRae Jeff McMuIIan UNC Casey Miller Matt Farthing Elsewhere David Keyes Stacy Howington VVVVVV c T Kelley lymmwy maniax Farmsth Manama Nonlinear Equations and Newton39s M rTran n Cmninnalion Constraii in mm w mm Conclusions Newtoms method Problem solve Fu O F R a RN is Lipschitz continuously differentiable Newton39s method u uc s The step is s iFuc 1FuC F uc is the Jacobian matrix and Newton39s M n Cmninnalion Constraii liminmvgnmxi Conclusions lnexact formulation llFUc5 FUcll S ncllFUcll 7 O for direct solvers l analytic Jacobians If Fu 0 F u is nonsingular and HG is close to u MM 7 ll 0ncllUc 7 ll M 7 Wilz Meditation exercise What39s 7 Nonlinear Equations and Newton39s Method PseudorTransiem Cmninnalion Constrain liminmvgnmxi Bum viihat if 1 few mde m Armijo Rule Find the least integer m 2 0 such that llFUc 2 m5ll S 1 a2 mllFUcll gt m O is Newton39s method gt Make it fancy by replacing 2 rgt Oz 10 4 is standard Nonlinear Equations and Newton39s Method PseudorTr n 39 mm39nnallo Constrai immi mn Conclusions If F is smooth and you get 5 with a direct solve or GM RES then either zgt BAD the iteration is unbounded i e lim sup Hunll oo gt BAD the derivatives tend to singularity lim sup HF un 1H 00 or D GOOD the iteration converges to a solution u in the terminal phase In 0 and llUn1 ll 0nnllUn ll llUn Wig Bottom line you get an answer or an easy to detect failure Lx rqu quot 311quotan Nonlinear Equations and Newton s Method PseudorTransienl Con rmation Constrai Conclusions Why Kimmy lgt Stagnation at singularity of F really happens t steady flow A shocks in CFD lgt Non physical results gt fires go out tgt negative concentrations gt Nonsmooth nonlinearities D are not uncommon flux limiters constitutive laws gt globalization is harder Igt finite diff directional derivatives may be wrong lJtc is one way to fix some of these things Nonlinear Equations and Newton s Method PseudorTransienl Con rmation Constrai Conclusions naming Think about a PDE du E FU L 0 U07 and its solution ut Fu contains D the nonlinearity D boundary conditions and D spatial derivatives We want the steady state solution u IimtH00 ut Nonlinear Equations and Newton s Method PseudorTransienl Con rmation Constrai Conclusions What an wrong If He is separated from u by gt complex features like shocks igt stiff transient behavior or D unstable equlibria the Newton Armijo iteration can gt stagnate at a singular Jacobian or gt find a solution of Fu O that is not the one you want mum m mm mm m l i Conclusions A Qua lldea One way to guarantee that you get u is gt Find a high quality temporal integration code gt Set the error tolerances to very small values gt Integrate the PDE to steady state tgt Continue in time until ut isn39t changing much gt Then apply Newton to make sure you have it right Problem you may not live to see the results Nonlinear EqualiunS and Newton s Method PseudorTransienl Con rmation Consirai ami quotmt Conclusions Integrate du E to steady state in a stable way with increasing time steps Equation for lJtc Newton step 6671 FLICD s 7Fuc 7Hm H 5211 F Uc 5 FUcii S nciiFUcH quotmm i ami MM m quotum i ii Implicit Euler for y 7Fy Un1 Ur 5FUn1 un1 is the solution of Cu ui un6Fu 0 Since G u I 6F u a single Newton iterate from uc un is u uc 7 I 6F uc 1uC Lln 6FuC uc 76 1 FucirlFiucii since uc 7 un 0 Nonlinear Equations and Newton s Method PseudorTransienl Con rmation Consirai ami MM lgt Low accuracy PECE integration v Trivial predictor Igt Backward Euler corrector one Newton iteration b 1st order Rosenbrock method High order possible Luo K Liao Tam 06 D Begin with small time step 6 Resolve transients Igt Grow the time step near u Turn into Newton Is this stable semlorTransienl Continuation c Const inn ra 1 I wear Conclusions Time Grow the time step with switched evolution relaxation SER 5n min50llFUollllFUnll7hex lf 6max 00 then 6 6W1HFun1HHFunH Alternative with no theory SER B 6n 6n71llun 7 unilll Nonlinear EqualiunS and Newton s Method PseudorTransienl Con rmation L lc Consxr mm mm mm Estimate local truncation error by T uxtn 2 and approximate u by 2 in in71 7 in71 in72 6n71 6n72 6n71 6n72 Adjust step so that 739 75 c T Kelley Baumx fmami mmnmqmm Nonlinear Equations and Newton39s Method Pseudo Transient Continuation lltc W Constrained Li tc quot I H Wtc Theory 7 39 V H E w Examples mm I m Conclusions Rimmmsw ssil mp1s m s gt Direction L 500 Flow quotmil Rmm swm m m mm om C T Kelley Emit Nonlinear Equations and Newton s Method PseudorTransienl Con rmation Constrai Conclusions unknowns density velocity energy V pv O Vpvvpl0 v pe pm 0 Ideal gas law p p y 71e 7 lvl22 where 39y is the ratio of specific heats Use nonsmooth slope limiter to get second order accuracy Nonlinear Equations and Newton39s Method Pseudo Transient Continuation lltc Constrained Li tc Wtc Theory Examples Conclusions Firm WT m N J C Q Newton Density C T Kelley Pt o pszctfmxdl Rmm swm m on iamvm om Nonlinear Equations and Newton39s Method Pseudo Transient Continuation lltc Constrained Ll tc Wtc Theory Examples Conclusions Firm WT h N Newton ogreative residual 0 50 100 150 200 Iterations C T Kelley Pt o tsxo39mxtt Rmmt ewmotmm on tmm tom Nonlinear Equations and Newton39s Method Pseudo Transient Continuation lltc Constrained Li tc Wtc Theory Examples Conclusions Fikw 7 mm gm Mom Amy ail an Ifai W m W1 to 3 L Di Residual Update and Timestep for SER 8 7 g 4 Yes 3 2 1 li O O 3 O A O 15 O 1 O C T Kelley Fm quottrail Rm by swmmu ojmnma w om Nonlinear Equations and Newton39s Method Pseudo Transient Continuation lltc Constrained Li tc Wtc Theory Examples Conclusions Fiona 7 no PTC Density C T Kelley Pm quot rs Rmm swm m m mm om Nonlinear Equations and Newton s Method PseudorTransienl Continuation l lc Consiraiii int Conclusions t if But If if not sinnimtir Typical Euler equation approach Igt Discretize with 2nd order scheme with slope limiter Slope limiters can be nonsmooth but Lipschitz continuous rgt Use Jacobian of a smooth 1st order scheme Modified method u uc i s where ii 521 Jc 5 FUcii S nciiFUciii and JC is the Jacobian of the smooth low order discretization Folwer K and Liao Qi K results explain this Nonlinear Equations and P mlorTr Constrained L lc l tc Theory quotrile Conclusions F K 39 An umm mrnm 7Fu u0 Ho 6 Q ut E Q Fu E Tu tangent to Q Examples D Q has interior bound constrained optimization rgt Q smooth manifold inverse singular value problem Problem lJtc will drift away from Q Lx Emmet quot f wlmnm Nonlinear EqualiunS and P morTr I Constrained m w Theory rile Conclusions u PM 7 61 Huc 1Fuc where 1gt 73 is map to nearest R V a Q HPum 1 for u E Q lgt Huc makes Newton like method fast Lx rqu quot f mmnm Equations and Newton s Method IorTranSiem Cnnunnanion Coner Nonlinear P u 1 Theory Exnnnies Conclusions Chimera Liao Qi K 2006 F Lipschitz no smoothness assumptions 73L c 7 671 i HUC71FUC7 where H is an approximate Jacobian Theory H bounded other assumptions imply un a u and iur un1 LIL 061 m where LIA1 un 7 Hun 1Fun which is as fast as the underlying method C Kelley nomisz Pkan rqu quotmmmzt Wham these gammy Nonlinear Equations and Method P m rTr I l Newton s mninnalio Consirai Conclusions one VVVVVV ut a u 60 is sufficiently small lipum 1 or Lip const of 73 1 u is dynamically stable Hu is uniformly well conditioned near ut l t 2 O u uc 7 Huc 1FuC is rapidly locally convergent near u Nonlinear Equations and Newton s ethod rTran m Con rmation Constraii i heory quotrile Conclusions A dynamics 1 A worrdl du 7 dt 7 implies ut a u if F Vf and 7Fu u0 HQ gt f is real analytic gt the Lojasiewicz inequality llVfUll 2 CW ful holds or D f has bounded level sets and finitely many critical points But none of this implies that u is dynamically stable Nonlinear Equations and Newton s Method P mlrTr I 39 mnjnnauo Constrai mglmt Fixing TTE SElm If the underlying problem is minimization of f and D you reduce 6 until f is reduced gt 60 is sufficiently small and D u is the unique root of F Then either 6 a O or you converge to u Nonlinear EqualiunS and Newton s Method PseudoyTransit m lelimlalioll Collsiraill w Theory Examples Conclusions E wm C6Wl h ll l l Qp tll muzmm c T Kelley Baumx Emami mmnliqujl qualiuns and Newton s M dorTransiem Cnnu39nna COIISS ion raiund l tc xeory Examples Conclusions PM 7 n cwmm1wcwltogtmr eff Identify binding constraints mu i w u L and Vfu lt 0 or u U and Vfu gt 0 with an over estimate to get fast convergence a binding set B u U 7 u g a and Vfui lt 7w or u 7 L a and Wu gt Nonlinear EqualiunS and Newton s Method P m rTr ml nnalion Consxrai I c w Theory nples Conclusions App m mn 91 Resdmaadj nggg am Set 0u and V2fuj i B u 6 i6 B u Then it all works Nonlinear Equations and Newton s Method rTran m Con rmation Constraii in Theory quotpies Conclusions Piwbiiem i Limear Aw Chu 92 Find 6 6 RN so that the M x N matrix N 3a Bo chBk k1 has prescribed singular values 0 V1 Data Frobenius orthogonal Bi v1 07V1 Nonlinear Equations and Newton s Method rTran m Con rmation Constrail w Theory nples Conclusions Pormtwllaitfmn Least squares problem min m V 2 MW VWF where N RU V UXVT 7 B0 7 Z lt UXVT Bk gtF Bk k1 Manifold constraints U is orthogonal M x M and V is orthogonal N x N EqualiunS and Newton sudorTransit m Cmm39nnat Coner Conclusions Digimmmfm Fomm gmiom U QV Projected gradinet 6 RMXM 63 RNXM U and V orthogonal 1 RUVVXTU747UXVTRUVYU gU V RUVTUXV747VXTUTRUV V 0 I alt gtimmgiguvy c T Kelley Baumx Emust mmnmmv Nonlineal qualiuns and Newton s Method P udoyTransiem Cmninnalion Constrain Conclusions Higham 86 04 Projection of square matrix onto orthogonal matrices A a Up where A UpHp is the polar decomposition Compute Up via the SVD A UXV Up UVT Projection of onto Q is Nonlineal qualiunS and Newton s Method P udoyTransiem Continuation Constrain Given u E Q let PTu Pu be the projection onto the tangent space to Q at u Let H I 7 PTu l PTuFuPTu Locally very locally superlinearly convergent if Q is OK near u Nonlinear Equations and Newtun39s Method Pseudo Transient Con rmation Wit M V ained Li tc 5 am Wtc Theory 39 I I 5W Examples Conclusions m H Ma Ea 39L a c 43 Liz h kallkar UJVUFQJQ QJUWMWHQW italng f Qi yrg quot iquot i m m D I I mum m Kquot iterations Kelley Pk g psxdmxdl Rmm wahm m mmvm romm qualiuns and Newton s M ulorTransit m Cmm39nnat COIISS ion railind inc xeory E mples Conclusions i722 it non Example H Parameter ID Find 6 and k for simple harmonic oscillator W l CW l kW 0 WO W07 WO 07 by sampling output of odelSs with rtol at ol10 6 and comparing to exact solution c k 1 at 100 equally spaced points u ckT M 100 N2 Ru Wexadt 7 Wu Nonlinear Equations and Newton s Method P m rTr I mninnalion Constrai I i c in Theory v t PIGS iurxim wmm Conclusions Results rm i Bounds u uo 1010 in all cases Interior 07 0 g u zero residual Exterior 27 0 g u non zero residual Exterior 17 0 g u zero residual degenerate Nonlinear EqualiunS and Newton s 2 IorTranSit m Cnnu nnanion m A m Consxra39 j I V r n Examples mm mm m Conclusions gt Emust mmnm Nonlinear Equations and Newton s Method P quotlorTr I mninnalio Constrai Conclusions gt lJtc computes steady state solutions gt Can succeed when traditional methods fail b It is not a general nonlinear solver rgt Theory and practice for many problems b ODEs DAEs gt Nonsmooth F gt Tempting idea for optimization Lx rqu quot fimmmzv