Fall 2008 EE 410/510: Microfabrication and Semiconductor Processes M W 12:45 PM 2:20 PM EB 239 Engineering Bldg.

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1 Fall 008 EE 410/510: Mcrofabrcaton and Smconductor Procsss M W 1:45 PM :0 PM EB 39 Engnrng Bldg. Instructor: John D. Wllams, Ph.D. Assstant Profssor of Elctrcal and omputr Engnrng Assocat Drctor of th Nano and Mcro Dvcs ntr Unvrsty of Alabama n Huntsvll 406 Optcs Buldng Huntsvll, AL Phon: (56) Fa: (56) mal: wllams@ng.uah.du Tabls and harts takn from ambll, Scnc and Engnrng of Mcrolctronc Fabrcaton, Oford 001 And Wolf and Taubr, Introducton to Slcon Procssng for th VLSI Era, Vol. II

2 Fck s Laws of Dffuson Fck s 1 st law (, J D Accuratly dscrbs dffuson No convnnt masur of currnt dnsty Fck s nd law ombns frst law wth contnuty quaton J A( J J1) Ad Ad t Ylds concntraton ovr tm as a functon of scond drvatv of th concntraton gradnt through th dffuson constant t D Soluton rqurs knowldg of at last two boundary condtons

3 Undrstandng Atomstc Dffuson Physcal Mchansms of Dffuson To us Fck s scond law, w must assum that th crystal s sotropc Assumpton braks down whn th concntraton of th dopant s larg At larg concntratons, dffusvty bcoms a functon of concntraton and thrfor dpth. Intrsttal and substtuton dffuson Assum atoms ar corrctly rprsntd as mnma n parabolc potntal wlls. Ths atoms ar oscllat slghtly du to thrmal ctaton An nsrtd mpurty atom may thn st btwn lattc sts ntrsttally.

4 Undrstandng Atomstc Dffuson Intrsttal and substtuton dffuson Ths mpurts dffus rapdly du to th sharp localzd changs n potntal nrgy and do not contrbut to dopng Dffuson, howvr allows th mpurty to mov nto an mpty lattc st, thrby substtutng for ts potntal nto th lattc n plac of th matr matral Vacancs flld by substtuton rman wthn th lattc st untl suffcnt nrgy s provdd for th mpurty to mov to anothr mpty lattc st. Ths s achvd by charg rdstrbuton to mnmz th fr nrgy of th lattc Vacancs ar vry dlut n smconductors at typcal procss condtons Each of th possbl sts can b tratd as ndpndnt ntts. Th dffuson coffcnt thn bcoms th probablty of all possbl dffuson coffcnts, wghtd by th probablty of stnc D D + a 1 n n a D a + p n a D a+

5 Intrnsc arrr oncntratons Th ntrnsc carrr concntraton s n ( cm 3 ) n o T ( K) 3/ E whr n o 7.3*10 15 cm -3 for S and 4.*10 14 cm -3 for GaAs g /(k b T ) Th bandgap can b dtrmnd by E g E g0 αt ( K) β + T ( K) whr E g0, α, and β ar 1.17 V, V/K and 636 K for S and 1.5 V, V/K and 04K for GaAs

6 Undrstandng Atomstc Dffuson In havly dopd slcon, th bandgap s furthr rducd by th bandgap narrowng ffct Δ E g 7.1*10 10 V 3 n( cm ) T ( K) For havly dopd dffusons (>>n ) th lctron or hol concntraton s just th mpurty concntraton For low concntraton dffusons (<< n ), nn For substrats wth css fr lctrons, th D + trms n th dffuson constant ar nglctd For substrats wth css hols, th D - trms ar nglctd If chargd vacancs must b consdrd, th lctron or hol concntraton and thrfor th dffusvty, s a functon of poston and Fck s scond law must b solvd numrcally as t D

7 Undrstandng Atomstc Dffuson If vry dlut mpurty profls ar masurd bfor and aftr dffuson, thn dffuson coffcnts can b dtrmnd. Rptton of th procdur ovr svral tmpraturs provds D D o E a /( k T ) Whr E a s th actvaton nrgy of th ntrnsc dffusvty D o s a narly tmpratur ndpndnt trm that dpnds on vbratonal nrgy and gomtry of th lattc Donors (D) Accptors (A) As n S D P n S D Sb n S D B n S A Al n S A Ga n S A S n GaAs D S n GaAs D B n GaAs A Ga n GaAs I As n GaAs I D o E a D o + E a I s ntrsttal b D o E a D o + E a E

8 Analytc Soluton of Fck s nd Law: (onstant Sourc) In practc, dopant profls ara suffcntly compl and th assumpton that th coffcnt of dffuson s constant s qustonabl, thus numrcal solutons ar gnrally rqurd Howvr rough appromatons can b mad usng analytc solutons Solutons ar provdd for two thortcal condtons 1 st : Prdposton Dffuson: sourc concntraton ( s ) s fd for all tms, t > 0 D t ( z,0) 0 ( o, (, ( z, 0 s sfrc Fck s nd Law n 1-D Boundary ondtons z Dt, t > 0 Soluton

9 Estmaton of Dffuson profls Dos of prdposton profls vars wth th tm of dffuson Dos can b obtand usng Q T ( ( z, dz (0, 0 π masurd n mpurts pr unty ara (cm - ) Th dpth of th profl s typcally lss than 1 μm Dos of cm - wll produc a larg volum concntraton (>10 19 cm -3 ) Snc th surfac concntraton ( s ) s fd for a prdposton dffuson, th total dos ncrass as th squar root of tm Dt

10 Analytc Soluton of Fck s nd Law: (onstant Dos) st approach: Drv Dffuson: Intal amount of mpurty (Q T ) s ntroducd nto th lattc D t ( z,0) 0 ( o, 0 z (, 0 0 ( z, dz Q ( z, T QT πdt z /(4D z K 0 Q T constant t > 0 Fck s nd Law n 1-D Boundary ondtons Soluton

11 Analytc Soluton of Fck s nd Law: (onstant Dos) Wth dos s constant, surfac concntraton must dcras wth tm: s (0, πdt At 0, d/dz s zro for all t K 0. On classc ral world ampl of ths two solutons s a prdposton surfac followd by drv n dffuson Rcall that th boundary condton for drv n was that th ntal mpurty concntraton was zro vrywhr cpt at th surfac Thus drv n s a good appromaton for dffuson provdd that Dt prdp << Dt drvn Boron (B) s dffusng nto S that as a unform composton of phosphorus (P), B. Also assum that S >> B A dpth wll st at whch S B Snc B s p-typ and P s n-typ, a p-n juncton wll st at ths dpth, j : Q T j 4Dt ln B Q T πdt Drv n j 1 B Dtfrc Prdp S

12 Dffuson of Varous Dopants n S Onln Thrmal Dffuson alculator: alculators/dffalc.html

13 orrctons to Smpl Thory Substtutonal mpurts ar almost compltly onzd at room tmpratur Thus an lctrc fld always st wthn th substrat Total currnt du to th fld ffcts both drft and dffuson componnts Rcallng Ohm s Law: D z Whr μ s th moblty, E s th lctrc fld, and th Enstn rlatonshp btwn moblty and dffusvty as bn nvokd. Assumng that th carrr concntraton s compltly dtrmnd by th dopng profl, thn th fld can b calculatd drctly kbt 1 E η q d dz Whr η s th scrnng factor varyng from 0 to 1. J + μe d J D( 1+ η) dz

14 orrctons for Dopng undr Ods and Ntrds For ( dopng >> n, Sub ), th profls own lctrc fld wll nhanc movmnt of th mpurty Not that th quaton s dntcal to Fck s frst law wth th slght modfcaton of th scrnng factor multplr omparson of nrt, odzng, and ntrdzng dopant dffuson prmnts has provdd th followng conclusons: Dffuson of mpurts dpnds drctly on th concntraton of mpurts Odzd smconductors produc a hgh concntraton of css ntrsttals at th od smconductor ntrfac Intrsttal concntraton dcays wth dpth du to rcombnaton Nar surfac, ths ntrsttals ncras th dffusvty of B and P Thrfor t s blvd that B and P mpurts dffus prmarly ntrsttally Arsnc s dffusvty s found to dcras undr odzd condtons Ecss ntrsttal concntraton s pctd to dcras local vacancy concntraton, thrfor, arsnc s prmarly blvd to dffus through vacancy mchansms (at last n odzd systms) Ths rsults hav bn confrmd by usng ntrd slcon surfacs whch ar domnatd prmarly by vacancs and NOT ntrsttals. Dopant dffusvts undr odzng condtons dt D D + ΔD D + α dt Whr th ponnt n has bn found prmntally to rang from and th α trm s (+) for odaton and (-) for ntrdaton o n

15 Hgh oncntraton Dopng At hgh concntratons, fld nhancmnt s vdnt Arsnc Phosphorous n D D + ( D ) n DAs n ( D ) As Ths lads to mamum carrr concntratons of Ph n D tal D + D

4 Prob Analyss of Dffusd Profls 4 prob Rsstanc masurmnt Sht carrr concntraton can also b combnd wth a masurmnt of juncton dpth to provd a complt dscrpton of th dffusd profl (z) s

16 4 Prob Analyss of Dffusd Profls 4 prob Rsstanc masurmnt Sht carrr concntraton can also b combnd wth a masurmnt of juncton dpth to provd a complt dscrpton of th dffusd profl (z) s th carrr concntraton μ() s th concntraton dpndnt moblty 1 V1 V R 4 + I34 I π 1 Rsq ln() R Rsq 3 41 [ q μ( ) dz] 1 V + I ( z) 34 1 V + I 41 3 Rcall from bfor: Q T ( ( z, dz (0, 0 π Dt

17 Hall ffct Analsyss of Dffusd Profls B w v V B v E B qv F z h z y r r r s j h j X j j R q qv B I D D qw I v j j μ Hall Voltag ntgratd carrr concntraton Lorntz Forc Hall moblty (for ptay consdratons)

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