Lecture 2a. Crystal Growth (cont d) ECE723

Transcription

1 Lctur 2a rystal Grwth (cnt d) 1

2 Distributin f Dpants As a crystal is pulld frm th mlt, th dping cncntratin incrpratd int th crystal (slid) is usually diffrnt frm th dping cncntratin f th mlt (liquid) at th intrfac. Th rati f ths tw cncntratins is dfind as th quilibrium sgrgatin cfficint,. s = (1) l Whr s and l ar, rspctivly, th quilibrium cncntratins f th dpants in th slid and liquid nar th intrfac. 2

3 Equilibrium grgatin fficints fr Dpants in ilicn Dpant Typ Dpant Typ B 8 x 10-1 p As 3 x 10-1 n Al 2 x 10-3 p b 2.3 x 10-2 n Ga 8 x 10-3 p T 2 x 10-4 n In 4 x 10-4 p Li 1 x 10-2 n O 1.25 n u 4 x x 10-2 n Au 2.5 x P 0.35 n 3

4 Z rystal Grwth nsidr a crystal bing grwn frm a mlt having an initial wight M with an initial dping cncntratin in th mlt (i.., th wight f th dpant pr 1 g f mlt). At a givn pint f grwth whn th crystal wight M has bn grwn, th amunt f dpant rmaining in th mlt (by wight) is. Fr incrmntal amunt f th crystal with wight dm, th crrspnding rductin f th dpant (-d) frm th mlt is s dm, whr s is th dping cncntratin in th crystal (by wight). d = sdm (2) 4

5 Z rystal Grwth (2) Nw, th rmaining wight f th mlt is M -M, and th dping cncntratin in th liquid (by wight), l, is givn by = (3) l M M mbining Eqs. 2 and 3 and substitut t Eq. 1 d = dm M M (4) 5

6 Z rystal Grwth (3) Givn th initial wight f th dpant, M, w can intgrat Eq. 4: M d = M 0 dm M M (5) lving Eq. 5 and cmbining with Eq. 3 givs s 1 = M M 1 (6) 6

7 urvs fr grwth frm th mlt As crystal grwth prgrsss, th cmpsitin initially at will incras cntinually fr <1 and dcras cntinually fr >1. Whn =1, a unifrm impurity distributin can b btaind. urvs fr grwth frm th mlt shwing th dping cncntratin in a slid as a functin f th fractin slidifid (M/M ). 7

8 Effctiv grgatin fficints Whil th crystal is grwing, dpants ar cnstantly bing rjctd int th mlt ( >1). If th rjctin rat is highr than th rat at which th dpant can b transprtd away by diffusin r stirring, thn a cncntratin gradint will dvlp at th intrfac. Th sgrgatin cfficint givn by q 1 is = s / l (0). Dping distributin nar th slid-mlt intrfac. 8

9 Effctiv grgatin fficints Th ffctiv sgrgatin cfficint, which is th rati f s and th impurity cncntratin far away frm th intrfac: s = vδ / D l + (1 ) whr v = crystal grwth vlcity δ = mlt width, and D = dpant diffusin (7) 9

10 FZ rystal Grwth T valuat th dping distributin f a flat-zn prcss, cnsidr a simplifid mdl shwn hr. Th initial unifrm dping cncntratin in th rd is (by wight). L is th lngth f th mltn zn at a distanc x alng th rd, A is th crss-sctinal ara f rd, ρ d is th spcific dnsity f silicn, and is th amunt f dpant prsnt in th mltn zn. Flat-zn prcss. (a) chmatic stup. (b) impl mdl fr dping valuatin. 10

11 FZ rystal Grwth (2) As th rd travrss t a distanc dx, th amunt f dpant addd t its advancing nd is ρ d Adx, whr th amunt f dpant rmvd frm it at th rtrating nd is (dx/l), whr is th ffctiv sgrgatin cfficint. Thus, d d = ρd Adx dx L = ρd A dx L (8) 11

12 12 FZ rystal Grwth (3) that whr = ρ d AL is th amunt f dpant in th zn whn it was first frmd at th frnt nd f th rd. Frm q 8a, w btain r = d x L A dx ) / ( 0 ρ (8a) ) / ( ) / ( / L A L A d d L x = ρ ρ (9) ( ) [ ] L x d AL / 1 1 = ρ (9a)

13 FZ rystal Grwth (4) inc s, ( th dping cncntratin in th crystal at th rtrating nd) is givn by s = ( dx/l), thn s [ ( ) ] x / L 1 1 = (10) If it is dsirabl t dp th rd rathr than t purify it, cnsidr th cas in which all th dpants ar intrducd in t first zn ( = l Aρ d L), th initial cncntratin is nglibly small. Equatin 9 givs: = * x L (11) inc s = (/Aρ d L), w btain th fllwing quatin frm 11. s = l x / L (12) 13

14 Prblms 1. A silicn ingt which cntain brn atms/cm 3, is t b grwn by th zchralsi tchniqu. What cncntratin f brn atms shuld b in th mlt t giv th rquird cncntratin in th ingt? If th initial lad f th silicn in th crucibl is 60g, hw many grams f brn (atmic wight 10.8) shuld b addd? Th dnsity f th mltn silicn is 2.53g/cm Plt th distributin f arsnic at distancs f 10, 20, 30, 40, and 45 cm frm th sd in a silicn ingt 50 cm lng that has bn pulld frm a mlt with an initial dping cncntratin f cm In silicn, th lattic cnstant is 5.43 angstrm. Assum a hard sphr mdl. (a) alculat th radius f th silicn atm. (b) Dtrmin th dnsity f silicn atms in atms/cm -3. (c) Us Avgadr cnstant t find th dnsity f silicn. 4. W us th flat zn-prcss t purify a silicn ingt that cntains a unifrm gallium cncntratin f 5x cm -3. On pass is mad with mltn zn 2 cm lng. Ovr what distanc is th rsulting gallium cncntratin blw 5x cm