Proc Plasma EECS 517
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This 10 page Class Notes was uploaded by Ophelia Ritchie on Thursday October 29, 2015. The Class Notes belongs to EECS 517 at University of Michigan taught by Staff in Fall. Since its upload, it has received 19 views. For similar materials see /class/231551/eecs-517-university-of-michigan in Engineering Computer Science at University of Michigan.
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Date Created: 10/29/15
EXAMPLE CYLINDRICAL GLOW DISCHARGE PARAMETERS What are the electron density electron temperature and selfsustaining electric eld in a low temperature plasma having the following properties j 10 cylindrical tube 15 cm diameter cm2 P3 Torr Tg400 K Mg40 AMU DION 100 T 3 3 km3 gtlt10 7 kion3 X 10 11 eXp As9eV e ELECTRON TEMPERATURE assume diffusion losses dominate 6n D NJr ate 0ne39kION39NG 2 0 N Ai R D D1 J TT ne 239405 A I TI I g 7 9654 X1018 gtlt 3T0rr K NG 724 gtlt1016C1If3 3 J 724 gtlt1016C1If3 L11 9 8V 3 X 10 expLTe 100cmzX 240512 1 TE j s 1522cm2 400115942e Solve for T6 and find Te 276 eV AXIAL ELECTRIC FIELD 2 2 1 anlekBTe Tg 1112 40g DERIVATION OF AMBIPOLAR DIFFUSION Start with continuity equations ne W 7V0Fe7V 7DeVneueEAne la 6N1 W V FI V DIVNI HIEANI 1b Assuming that V EA 0 and ne N1 then 6 De Vgnei ueEA Vne 2a 6N1 2 Tit DI V N1 M EA VNI 2 Multiply 2a by m multiply 2b by pa and add them together 2a m 2b Me with ne N1 to obtain ane DeHIDIHe l 2 2 at L Heui JV ne Dav me where the ambipolar diffusion coef cient is 1 De H1 DI He 1 I K He Using the Einstein relation D kq T u and assuming that ue gtgt m then Da D1 1Tr g DRIFT VELOCITY FROM BOLTZMANN S EQUATION From the twoterm spherical harmonic expansion HQ f0V f1V cos 9 for f0V21v2 sine d9 dV 1 0 0 Drift velocity for E E2 is VZ VX Vy Vd VZ vcose IOOOEGO V f1 V cos9vcos9 vaz sine d9 dV Since f0 is isotropic Ifo COSG dQ integrates to zero oo 7 a Vd I0 0 21W3 cosze sine d9 dV VdL 4 waf 1 3 3Ine 0 6V VmVVdViE HE 4n wafo 1 3 J where the mobility H i am 0 a V V V V dV 9 m If VmV constant then C 4 all 3 Cl P39mevmii0 3n6VVdV mevm where the brackets integrate to 1 DIFFUSION FLUX AND COEFFICIENT FROM BOLTZMANN S EQUATION From the twoterm spherical harmonic expansion at afo qEZ afo 6t V62 me av mel where J f0 2 1W2 Sine d9 no the average electron density Set 6 0 and E 0 only thermal motion 1 v v f1 f1 Vm 62 00 7E The diffusion ux in the z direction is FZ 11on n0 lt V cos 9 gt J J VZ f0 f1C0s6 dSV 0 0 Since Ifo COSG dQ integrates to zero then FZ IowLanf1COSGd3V JOOJWV cos9 Va f0cos9 vaz sine d9 dV 0 0 Vm 62 oo 7 2 J I lvz coszsine d9 dV 0 0 Vm 62 41W2 L263 lo 3 vm 62 dV n F ii winVZV Zfdvj Z 62 lo 3 Vm 0 Now returning to the momentum conservation equation we have 6nv VP T F anm 0 V1Tnvvm VnkT kT FZHVZ m Vm m VmVH DVH ANALYSIS OF THE CATHODE FALL OF A NORMAL GLOW cathode J01 jor current density of ions at cathode Ncmz j05 I joe current density ofe s at cathode Ncmz G e YqDI V0 Vo voltage drop across cathode fall I d thickness of cathode fall I cm I A x d rl Assume ne ltlt N1 and the N1 5 constant in the cathode fall Poisson s equation becomes iwm mimonmm 6x so so so soVI j1 q39Vdrift 39NI so that NI L qVI Integrating EX E o 6 Eo is electric eld at the cathode where EO gtgt E in positive d column The cathode fall voltage drop is v0 J EO o 2 X E0d 2V0 VX E X i V 7 E 7 0 2d 0 2 0 d Ions striking the cathode produce electrons by secondary emission with probability y The secondary electrons are accelerated back into the plasma producing ionizations in the dark space with Townsend coef cient at jo j01j0e j01Yj01 1Yj01 Withjor qNIVI V1 1an 6E znz i 2 so J Ezm 5wa d 80 1 dq 01 dq 10 d The total current density is then 2 2 s E 1 48 V Jo1 1011Y70H 0 Y 0M 0 d3 De ne je electron current at edge of dark space Dark Cathode Space Negative Glow 7 Je J0e W01 Je gtgtl I 4 J01 4 J1 Sinceje gtgtj1 then Jo J0eJOlJeleJe1 JOIJOJOemJeJOe BUtJ0eYJO1Y00J0e Uel0e JOeN Y1 Je The electron current from the cathode through the dark space is ampli ed by electron impact ionization 3 dx j OLX 0c First Townsend Coef cient d 1 JOe exp Locum d J39Oe exp 00 aC xlx d With Joe E J39e J39Oe exp IOONXMXJ 1 d h1wa 1 e on where EX EOE E E0 From empirical data B 0LApexp BF 1 7 V B 7 7 Cm To cm Torr p pressure Torr A 1 d Bp So ln l7j JO Ap exp EO1de The integral can be solved analytically 2 101 Altpdgt B s j v 2V0 pdB x 74L 7 where SX J e ydy xe X E1 E1 Exponential Integral o We can now solve for V0 in terms of d A second relationship between V0 and d is 2 480V0 H11 Y JO d3 We could in principle solve for two of jo V0 and d as a function of the third Up to now there is nothing unique to the normal glow De ne two functions lnl 1 2A c c 1 B1n1YL Z 80AB2P2P H11Y 13 13 where 1 S 0210 K0210 Note Since HI vi then pul is a constant 1 If no is the mobility at pressure p0 ul uo pl Plot all points satisfying relationship I Abnormal C1V0 Minimum V 0 jo I Normal Glow C 2 J0 Minimum occurs at C1 VNORMAL 60 C2 jn 067 which yields 3B 1 Vnormal X1n l E function of gas metal but not pressure ABZH0P01Y pz Jn592x10 1n1i Y EECS 517 Fall 2008 Units and Best Practice Units prove to be a confusing aspect of this course The units which are commonly in use in the eld are the quotstandardquot for this course Unfortunately the units are quotmixedquot that is a mixture of cgs and mks Some useful conversion factors are listed below Some best practices you should follow are I ALWAYS perform a units analysis and perform a quotsanityquot check to determine that your answer is reasonable In most cases quotunreasonablequot answers are a result of unit problems For example if your answer is that the argon ion density in a plasma etching reactor is 105 ionscm3 your answer is unreasonable and you probably have a units problem You know your answer is unreasonable since if the density is really 1050 argon ionscm3 the mass of 10 cm3 of the plasma would be equal to twice the mass of the earth 2 Never ever be confused by expressing temperature in Energy Units or Vice versa Temperature in Energy Units ALWAYS Means T eV E kT eV 3 Unless speci ed otherwise you nal answers in homework problems should be expressed in the following units Electron energies or temperatures EV Atomic or molecular energies or temperatures K or eV Lengths cm Electron atomic or molecular masses AMU or g Electron atomic or molecular speeds cms Cross sections cm2 or A2 Mobilities cmzVs Diffusion coef cients cmzs Rates coef cients 1st 2nd 3rd order s39l cm3s cm6s Electric elds Vcm391 Normalized Electric Fields Vcm392 or Td Densities cm393 Power W Power deposition speci c Wcm393 Current density Acm392 Unitsl EECS 517 Fall 2008 Useful Conversion Factors k 138 X 103916 ergK 138 X 103923 JK leV 16 X 103912 ergs 16 X 103919 JE 115942 K q e 16 X 103919 C coulomb 48 X 103910 esu 1V 1 JC 107 ergC so 885 x 1012 Fm 0r CZm J 885 x 1014 Fcm or CZcm J me electron mass 0911 X 103927 g 0911 X 103930 kg EN 1 Td Townsend 103917 Vcm2 103921 Vm2 0354 VcmTorr at T 273 K 1A2 103916 cm2 103920 m2 latm 760 Torr 1013 bar Pg 0er P T Gas Dens1ty N E 9654 X 1018 TK cm393 1 m3 106 cm3 Units2
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