Electrical Properties of Materials
Electrical Properties of Materials MSE 310
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This 24 page Class Notes was uploaded by Keshaun Ferry on Saturday October 3, 2015. The Class Notes belongs to MSE 310 at Boise State University taught by William Knowlton in Fall. Since its upload, it has received 48 views. For similar materials see /class/217986/mse-310-boise-state-university in Material Science and Engineering at Boise State University.
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Date Created: 10/03/15
mum Kagfmpma ImperfectionsDefects in Solids CI Classi cation of Defects in Solids Point defects Zero dimensional defects 0 Vacancies o interstitials 0 foreign atoms Line defects One dimemional defects 0 dislocations Planar defecm Two dimemional defects 0 Interfaces 0 grain boundaries Bulk or Volume defects Three dimemiona defects 0 Precipitates 0 vacancy agglomerates Crystallinity 0 single poly amorphous l111 MSE 310 Elecincalpmparlwails meltnn ImperfectionsDefects in Solids CI Why would we want to study defects Defects affect plimaIily all propenies of mateIials 0 Electrical All defects especially point defects 0 Optical 11 defects especially point defects 0 Mechanical eg strength toughness hardness etc All defects especially dislocations 0 Magnetic 39 All defects 0 Kinetic e g diffusion All defects especially point defects MSEEIO Kagfmpma ImperfectionsDefects in Solids CI Point defects differ from 1D line or 2D defects in two important respects Point defects are dif cult to observe directly Thus Point defects are usually are detected and studied through their effect upon some physical property of the material They m material ay be present in appreciable concentrations even though the 39 in thermodynamic equilibrium 0 While dislocations and intelfacex raise the free energy of a material adding 39 uiyu39udeyecis 39 39 free energy to a minimum Value 0 This is because ofagain entropy caused by the many possible sets of places in the crystal in which thepoint defects can ex39 o This has to do with Con gurationa or Statistical Entropy which is determined using Statirtical ThermodynamicsMechanics MSE 310 ElectncalepnfMa s meltnn ImperfectionsDefects in Solids CI Question How many vacancies are needed to minimize the Free Energy a crystalline material n density ofvacancies in bulk crystal nW density ofatomslattjce sites in bulk crystal energy of formationactivation of vacancies 1a Boltzmann39s constant T temperature M35310 I fjfmwma ImperfectionsDefects in Solids ll llAll Vacancy hmaatbamnn for T We m 12m c 1 X in gtlt ID 5 E 1 X m E E U 1 gtlt ID 1 gtlt ID if K4 MEMO OISE ggggmwmws ImperfectionsDefects in Solids El Point defects essentially come in two avors 39Ve i e intrinsic point defecm 0 Intrinsic to the material No eign atom or impun39ty Extrinsic point defects Foreign atom o Impurity atom 0 Not native to the material El NativeIntrinsic point defects Vacancies Self interstitials Antisite defects Native complexespairs o Interstitialcy eg selfinterstitial pairs o Multiple vacancies M35310 Em mwmm s Vacanc1es amp lnterstltlals M1 CI Native Defects Vacancies and Interstitials wquot 41 Twodimensional represemalio o a vacancy and 39 1ch from Pearsall a d Vol 1 Sr39uclurt39 p 77 Copyright 1964 by ew York Reprinled by permission of John Wiley amp Sons Inc mm sum m Ehgmumge An mam M35310 Haifmwma b ImperfectlonsDefects 1n Sohds Alf Native complexespairs o Frenkel defect an interstitialVacancy pair not necessarily adjacent to one th ano er 0 Schog defect an anion eg As Vacancy and cation eg Ga Vacancy pair separated from one another Sdlo kydd39ect no Freukd defect Schutky and Frankel dd ects in an imic crystal M35310 Kmfmwma ImperfecuonsDefects 1n Sohds CI Example of a Native Point Defect Antisite defects in GaAs E Weber quot D M35310 Haifmwma b lmperfectwnsDefects 1n Sohds CI Extrinsic point defects Substitutional impurity atom Sha ow acceptors 0 Shallow donors InteIstitial impurity atom o Interstitial diffusers H Fe Cu Ni 0 etc 0 Part time interstitials Al C B etc ExtIirLsic point defect complexespairs A amp B foreign atom V vacancy I Selfimerxti al 0 A 0 Ge Sn P As Sb B Al etc o g A C B Al Ga 0 etc 0 AA or AB c N etc amp cAssCP c Sbs Fe Al etc Point Defect CI Extrinsic Defects Substitutional and Interstitial Flu m 42 Twodimensional schematic representations of substitutiona and inlerslitial impurin atoms Adapted from W G Moffatl G W Pearsall and J Wuiff The SIrncmre and Properties of Materials Vol I szcmre p 77 Copyright 1964 by John Wiley amp Sons York Reprinted by permission of John Wiley amp Sons Inc MSE 31D E Emffupumads ImperfectionsDefects in Solids CI Substitutional extrinsic point defects in GaAs Light points Si Dark points VGa sl iw l 5quot ml in m I mm lrl i E Weber 2 11 UC Berkeley amp LBN39L MsE 3m gmiwamms ImperfectlonsDefects 1n Sollds CI Substitutional extrinsic point defecm in GaAs Light point donor or acceptor Dark poinw VGa A teptor i EWeber quot 3 A ampLBN39L MSEZIU rmin wmws ImperfectlonsDefects 1n Sollds CI Point defecm Interrupt periodicity Cause strain sour 31m M The impurity atom is larger than the host atom HI HI HI H H H H H H 1H 5 IH I A It the crystal The impimty atom oeeupies an empty space between host atoms is smaller than the host atom Fig 144 Point defecm in the crystal structure The regions around i i i i A i i imam uarrru mum mm Dislocations in Materials 9m Fuse die fullnw g case scenar Consider ablock ofcrystalllne material on wlueb forces are applied Surn othe applied forces give rise to a Shear me on the block of crystalline material DA um ss e the shearforce is large enough to cause a ddsplacement 1 of the crystalllne material ddsplacement 1 imam uarrru mum mm oca Ions In Mater l The displacemmt occurs betwem two adjacent 100 slip planes B one rnanner in which be displacemmt occurs is that die bonds becween atoms on adjacent slip planes e I I Ifdisplacementis along atoms A B and c men I Amoves to B slte I B moves to C slt e etc Simultaneous aanslaaon of all atoms on slip plans must occur D It seerns as lbougln rnany bonds must be brokm simultaneously considerable energy would be required Question What shear force g would be required to cause unis type ofdisplacernenn D TryTo Answer This Quesu39ou Needro calculate the force requlredto slmultaneously break all the bonds along the slip plane mm Ehuxxd PNPvfM Dislocations In Materials mm D Compare stresses at which a material yields for I Theoretical results I Experimental results mm 1 I nwmlml mquotl I lllml39 mmm mm SIMIan m mm an mmle w umquot mum W mquot wquot ll l w Wle m u l W m w m hm Hum m m m m v w39 W hpl mm m m w Hm w l The theoretlcal stress ls much greaterthan the expenmental stress thtix gning nn39 Dislocations in Materials El In 1934 Taylnr Oman and leanyl lndepmdemly pnstulated The existence slalmss da crthatvmhddallnwtheblndcln the palms gure m sllp at muchlnm stress levels The defectthey pnsmmsdwas a hm ole n called a dulomnon By lntmdunng an em halfplane nfalnms mm the lawns they shnwe EIEI J Amm banamsmgs anLhz sllp plan cmlldbe xemlcudm m lmmsmu vlclmty fth bmmm edg arm halfplanz dislacahm hm J As dislacananlmz mvves39hr shp plane accurs consecutively llmslmmslmw an El Ma39nr cnmg tl as much less energyln break nne band at a me than all bands at nnce glass x39ul bmdbxeakag mssm E lt FIGUREZJ ll lmlmlsn mu Hummusle Illnan u n c c 1mm u Mum mm all n u um I rm lmmclnun J c u ndmunuilurx an wasquotle VII lvmml quotmum mm m Mama curl mm m X mm Dislocations in Materials 9m 3 Analog of me concggt of a dislocation I Simultaneous bondbreakmg movmg a large oor rug across the room by just puumg I Consecutive bond breakmg shake e gs up and own creating anpple streckme npples propagate from one end ofthe mg to the other and MM mm m rmve mg m m manner D This is shown below 33 y a RN 477 L Hanna 2 ammiu mnummwmuun m n won5 vnrwmmu Mm llml lw mu mm mm Mum quotum I mmwnmmum m m m 1 mm mun r llvnm rn39rpm NM In Mm Ilurncquot sum In W m lemnm quotIrMrmum 2 quot1 Amquot mm mm va mm The planes on which dislocations move most readily are st chose ofgmate I Separauon I Atomic density mm mm W Dislocations in Materials Winn i B Each amyis on a slip plane and has a speci c slip direction indicating a slip system new am mlll r1 mi mull 7 mm DOIS en 0 E ninn i ns Dislocations in Materials Winn D Fundamental characuristics of dislocations I A dislocation is alamce line defect Tne 1mg ordxslacmmn de nes tlne boundary between slxpped and unslxpped ponions ofthe eiystal Disloeaaons ean tenninate at n e sii aee n bnundanes e g gmmbnundary inteifaee ete n AnnLher nislneatinn Disloeaaons ean neveitenninate Within the eiystal Consequently disloeaaons must ether form iisasir ems W Dislocations in Materials mm D Dislocations are described by two Vectors I 1 Burgers veetor I1 disloeatioh hhe veetor is the veetorthat designates the line ofbroken bonds that moves through the ervstal iisssir emf Dislocations in Materials Mar El Types ofDislocations I Edge I Serew Mixed I Perfeet Pure Whole in st mun 39 v sr ri tr n h pusiiri pawmm J Pan My emriiiriiimirvov whimmii rimh mu it irrir iirr KW Mir this4 hurt in h une 2x Sum murmin n n ullsplnrvml39nx ufnnr purl 0 H31 4 ma m olwn Nun maul n 2 ml r m 1 mmlM m n I rlauns m unm Ilklnm nmcnnlnin ng 1 mm m umwmm mm an An I wquot m 4x A m pm IlgrMB1leldislvnmunlnnu mlmk aul I Iudlul m mmr uraniyp lrd quu m DOIS mm mm Wuim mm DI oca ms nMater als D Dislocation Motion Edge Dislocations Cnns Muhammad a shy system Munscmsmam ummmm dnslacahmmt awner Thasmah xslmnwnasGlIDE n Nnncnnservauve Mvnanwumde m nmnal sup pm Massxsnm cansexved m u J mm m M mm M 0 quotmm m m o u a V mm valmvrlL a g 39r 4 n 0 o 5 A V 0 n t v 0 u 39 o B 039 u iisaain iinmi mm Dislocations In Materials M Kn Stress Field Aroun a 39 39 39 Dislocation Field D Disioeanons on same slip plane Disineaiinns nnne same signvnuiepei nne annLhEI a Disieeiannnienneineeen a Thsxesuhsmalarg saess enneenaiann inn leading edg athz pdew a Thscmleadmpnmauxe acuxe afma39znal Disineaiinn nfnppn le sign Win attract nne maths Dnvmg farce Stress eld B Disioeanons on dJSSlmllar slip planes Inlerammn Mnunnmay belmpeded BOI I mm emf Dislocations in Materials Winn D Dislocan39on snain Energy M n ag itude of stored energy in an elastically shamed region is always ofthe form Stran at any given polntls proportional to 1 Thus Elasne Stran Energy is proportional 0172 Mn 3m mm m D efects Dislocations Sem conductors 3353 Applying Thermo an Example 4pm W of Kinetics Diffusion El Fundamental Physics of Force and EneIgyWork Energy and Work 0 In general 0 The work is given by One can arguethatEqns 4 and 5 are really one in the same o Work or Enzrgy are scalar potzntmls e g voltage o Forcz is a Vectmfleld o If a patzntial is constant there is no FIELD o kais done by a Force o Combining equations 4 ands we have o Key Point Lfthefleld potential is not changing then no e work would be n Emugh mdamzrltals Let 5 apply this physics and thermal Knowhm meltm z 531395 Applying Thermo an Example m may of Kinetics Diffusion El Thermo amp Physics as applied to Diffusion Consider a forc forces acting on an atom producing atomic motion The applied force is given by the previous equation The motion of the atom will be often interrupted by other atoms and collisions occur Thus the velocity of the diffusing atom over a time period larger than the time between collisions is an average velocity The velocity is proportional to the applied force and can The constant of proportionality is called the mobility Consider o Flux of atoms A diffusing at an average v210ltyv through a homogeneous distribution ofB atoms o The ux ofA atoms through E is equal to the produ Number ofA atoms per unit volume r e comemattoh CA Average Velocity ofthe A atoms VA o This is given by MSEEIO Em mpg Applying Thermo an Example 4pm W of Kinetics Diffusion El TheIIno amp Physics as applied to Dif ision cont By combining the last two equations we obtain Substituting the Force Field Eqn 7 eld gradient o This last equation is a generalme of Flak 15 Low of ammo o That is the ux ofA atoms through a homogeneous 39 39bution ofB atoms is due to the gradient of some potential field o NOTE This potential eld can be any ofthe ones shown in the previous table Because the gradient ofa potential eld follows the superposition principle the more general form of Fick s 15t Law is39 5 2 V7 12 o where the sum indicates the superposition of potential field gradi ents anlton 3 531395 Applying Thermo an Example p may of Kinetics Diffusion El Thermo amp Physics as applied to Diffusion cont 7 Derivation of Fickx 15 Law In the following slides Fisk s 15t law in which the concentration gradient is obtained from the chemical potential is explicitly derived Ker goint Diffusion in solids is based on this delineation ermodynamlc relation where EA is a ee energy of the A atoms in B Examples ofFree Energy E o Gibbs G 677 R N o Helmhotz F FT V N o Enthalpy H Hs R o Omega potential or Grand Potential 0 127 V 11 The chemical potential of atoms A is given by the th 39 meltm 4 533 Applying Thermo an Example 4pm W s of Kinetics Diffusion El TheImo amp Physics as applied to Dif ision cont Substituting eqn 13 into eqn 11 e ux with respect to the chemical potential gradient is obtained Assuming onedimensional diffusion equation 14 simpli es to The chemical potential ofthe A atoms may be wrinen as a function of the chemical activity ie aA ofA in a distribution of B o where nA is the octiwty ofA among B kis Boltzmann s constant and m is the chemical potential ofA in the pure state he a may be described as the amount that the chemical potential of A deviates from the ideal or pure state i e ideality The ideality can be interpreted as the absence ofAA interaction upon adding an extra A atom to the system Thus the entlwlpy change AH ofthe system is zero Knowhm 5 i321 Applying Thermwn Example W of Kinetics Diffusion El Thermo amp Physics as applied to Diffusion cont The mathematical description of the activity is given by o where 7A is the octiwty coc iciznt and CAis the canczntmtlan ofthe A atoms Case 1 Henry 5 Law a range of CA much smaller than the concentration ofB atoms CB Kg becomes constant 0 The chance ofintel39action between A andB atoms is small since A is so dilute in B o The primary interaction ofA is with B o This phenomenon is known as Henry a law o Mathematically as CA approaches zero the activity coef cient of A u is given by Case 11 Raoult s law CA gtgt CE ie B rmherthan A atoms follow Henry s law activity coe cient ofA is 1 o The A atoms have a small probability ofinteracting with B atoms o The primary interaction ofA atoms is with other A atoms o Hence the solution ofA is effectivelypwe meltm a MSEEIO Em mpg Applying Thermo an Example 4pm W of Kinetics Diffusion El TheImo amp Physics as applied to Dif ision cont In this case and the acti ty coef cient is given by 0 This condition is known as Raault 5 law o Raoult s law is predominant for most diffusion processes in si since c3 gtgt can m For either Henry s or Moult s law Fisk s 1quot and 2 law may still be derived from the chemical potential This eventually can be seen by combining equations 15 16 and 17 into the following form o since the chemical patentml of a pure substance is constant its derivative is zero 0 Furthermore k and Tare constant o Under these circumstances andtaking the derivative ofthe natural logarithm equation 20 becomes39 0 For Henry 5 and Raoul 5 law his a constant or one respectively Winn o In either case 7Amay be taken out ofthe differential 7 531395 Applying Thermo an Example 4mm may of Kinetics Diffusion El Thermo amp Physics as applied to Diffusion cont Sincethe ratios ofboth 7A and CA factor to 1 then Einstein s relation states that the al39 iuivigl D of an atom is proportional to is mobility where the constant of proportionality is kT Mathematically amp in terms ofA atoms this is written as o Mm units square of the distance per unit time per unit energy 0 kTunits en o Thus DA units square ofthe distance per unit time Invoking Einstein s relation with respect to equation 23 Fisk s 15t law is obtained Fick s 1Sl law Note It is the concentration gradient that drives the ux of atoms from one area to another Fick i39 1n law Steady State D m ion c z 0 c at every point does not change wrt time meltm 3353 Applying Thermo an Example 4pm W of Kinetics Diffusion El Theimo amp Physics as applied to Dif ision cont NONrSTEADY STATE DIFFUSION It has been found that as the concentration ofA atoms in B changes with time the concentration changes with position Known as Fisk s 2 law it has the following form Assume that DA is concentration independent For diffusion in SCs the concentration of dopant atoms is Very small thus assumption may be comfortably made Substituting equation 24 into equation 25 Fisk s 2 d law s given y a FiCk S 2nd law Knowhm 9 MS 3m Eu Pmps ufMat39ls El Diffusion Solving Fick s 2 quot1 Law Solve for Cxt Infinite Solutions Need Boundary Conditions Two Primary Boundary Conditions o Fixed Surface Concentration Infinitz Source Solution Compiimentary Error Functi o Redistnbution ofa constant totai numb Finite Sburcz 39 Saluttari Gaussian Function KIM m i u gt4 Applying Thermo an Example m 0f Kinetics Diffusion on er of diffusing atoms ff jfps Applying Thermo an Example 4pm 531395 Applying Thermo an Example W W of Kinetics Diffusion may of Kinetics Diffusion El Diffusion Solving Fick s 2 dLaW o Fixed Surface Concentration In nltz Some motion Solution Complimentary Error El Diffusion Solving Fick s 2 quot1 Law 1quot of2 Primary Boundary Conditions 0 Fixed Surface Concentration Infinite Source Solution Complimen Error unc on Boundary Conditions Fort0cc at USxSw Forigtocc atx0 551 atxoo x TH H rquotmunrrirrrrrrmritrrrrrrrwrirrrrri H r erivlihHiwnumm r r M mm my min in win mum rirrmi r mmr mm m rr meitm 533 Applying Thermo an Example 4pm W s of Kinetics Diffusion El Diffusion Solving Fick s 2 quot1 Law 2 of2 Primary Boundary Conditions 0 Redistribution ofa constant total number of diffusing atoms Flle Some Solution Gaussian Function Boundary Conditions Applying Thermo an Example W s of Kinetics Diffusion CI Diffusion Thermally Activated Processes Temperature Plays a significant role in diffusion Temperature is not the driving force Remember DRIVING FORCE GRADIENT of a FIELD VARIA LE Remember Driving force for diffusion is a difference in the chemical potential 1 eqn 13 mimiwnm is Awe er VH O HOWEVER Temperature increases the activity of a diffusing species Dm m ivity or Dm ilxion Comient Em is the activation energy for diffusion kb Tis the thermal energy Do the preexponential factor contains a number of physical constants and properties including o entrapy affarmatxan ofthe defect o anemptfrequeney for jumps into available 0 lattice mutant 0 crystal structure dependence meitm Knowhm MSEEIO Em m 5 Applying Thermo an Example 50 5 553105 Applying Thermo an Examp e iiitin All iiiiiiquot of Kinetics Diffuswn l l39nm of Kinetics Diffuswn El Diffusion Thermally ActivatedProcesses Di iuivity or Di iuion Coe cient El Diffusion Mechanisms Processes Reac ons We will use Si as an example ofa system with Various Wmionm ch 39 Two types ofdiffusion mechanism 0 Direct diffusion mechanisms diffusion without the aid of fects point de Interstitial diffusion war mm man 500 n 0 Indirect diffusion mechanisms diffusion with the aid of point defects KnowHm meitm 3353 Applying Thermo an Example 4pm W of Kinetics Diffusion El Diffusion Mechanisms Processes Reactions Indirect diffusion mechanisms diffusion with the aid of point defecm anmm 17
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