Space charge. Lecture 8 09/11/2011. p-n junction with gradient. p-n junction with gradient. V. p-n junction. Space charge
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1 ecture 8 09/11/011 Sace charge. - jucto Sace charge th a gradet Out of equlbrum Sace charge -tye ad -tye regos Usually N >>N A thus q N x = N A /(N +N A x = N /(N +N A A ad x = The sace charge exteds towards the less doed rego N -N A -x x -tye -tye - jucto wth gradet The terface betwee the -tye ad -tye layers s o loger abrut - jucto wth gradet The bult- otetal s the esty of charges = qx wth the gradet (cm -4 d q x dx x Itegrate betwee ±/ where the electrc feld s zero (sace charge boudares kt kt l 3q 8q 3 deeds o the gradet The gradet the smaller, the lower
2 Characterstcs at equlbrum - jucto out of equlbrum Out of equlbrum meas that the jucto s olarzed (aled voltage Sace charge The, the exresso of as a fucto of s stll vald rovded that s relaced by - Fxed charges due to deleted murtes N A N q N AN No et curret (drft ad dffuso currets comesate thems elves htt:// N A N q N AN ( - jucto out of equlbrum - jucto out of equlbrum Forward as f : - f thus Reverse as r : + r thus N A N q N AN ( =0 > tye: 0 =N ad 0 = /N -tye: 0 =N A ad 0 = /N A ad are the cocetratos of electros ad holes at the sace charge boudares 8
3 At equlbrum the -tye ad -tye layers, we have: N v F Nve NN N N kt l kt l wth q Eg ktl v c ( ECEF/ kt ( EFE / kt A Eg ktl e e ktl NAN NAN e ca deduce ad 0 0 o =N ad 0 =N A 0 = 0 ex(-q /kt 0 = 0 ex(q /kt ( EF Ec / kt ce ( E E / kt Electro ad hole cocetrato at both sdes of the sace charge as a fucto of 9 0 = 0 ex(q /kt 0 = 0 ex(q /kt Polarsato (forward: >0, reverse <0 = ex(q( -/kt These relatos are stll vald whle the jucto s ased wth ad the cocetratos at the sace charge boudares e cosder the case of low jecto regmes. Thus 0 Smlarly, we have o 0 ex(q /kt ex(q( -/kt ad fally 0 ex(q/kt or (ex(q/kt-1 o the -sde at x=-x - 0 = 0 (ex(q/kt-1 o the -sde at x=x 10 Schockley s relatos Morty carrer cocetratos - ad -tye layers The carrers come from the - ad - layers e have exressed the carrer cocetratos at both sdes of the sace charge Currets: Rate equato: wth G R geerato rate recomato rate e have to descrbe the evoluto of the carrer cocetrato far from the jucto (whe gog back to the equlbrum 11 Overvew G R s gve by G R 0 Out of equlbrum oulato fetme (relaxato tme 1
4 Steady state regme (wth o electrc feld the - ad -tye regos d dt 1 dj q dx ( G R J = q(µ E - grad d dx ( 0 0 Itegrate wth = 0 ex(q/kt at x = - x ad = 0 at x = = 0 (ex(q/kt 1 ex[(x+x / ] wth s the dffuso legth of electros the -tye layer The electro curret at -x s the gve by d J( x q dx x ad the hole curret at x s q 0 q / kt e 1 13 d J x q q e q / kt 0 ( dx x 1 14 The total curret s J J ( x J ( x q q ( e 1 ( e 1 0 q / kt 0 q / kt J J ( e 1 wth J e e q / kt 0 0 s s Saturato curret The morty carrers recome wth the majorty carrers whle gog away the sace charge. 0 = ²/N A ad 0 = ²/N e J s N A e N wth N N c Ec Ev kt ve N N c Eg kt ve 15 16
5 J s a slco - jucto N P I N A = 5x10 16 cm -3, N = 1x10 16 cm -3, cm -3 =1 cm²/s, =10 cm²/s, = = 5x10-7 s J s = 8.6x10-1 A/cm the tycal sze of a devce s a few mcros whch leads to J s of the order of a few fa. There s o curret at reverse as =0, o curret (equlbrum I cotrast, at = 1 (forward as, J = x10 A/cm <0, J teds to J s >0, J radly creases Exermetal I- curves q k J J s e T B 1 dealty factor betwee 1 ad Overvew The assumto we made o the absece of currets the sace charge s qute rough. Actually, there s a cotrbuto to the curret due to recomato ad geerato rocesses The total curret wrtes J=J,dff + J,dff + J r,g 19 0
6 Geerato ad recomato currets the sace charge r >0 recomaso rate r <0 geerato rate e / kt 1 ( e 1 r ad J g, r e x x rdx Overvew 1 Reverse as: geerato curret ->>kt/e r the Forward as: recomato curret r max e / 1 ( e (1 e kt e 1 / kt J g e r maxmum for + mmum ad costat = = ex(e/kt Fally J r >>kt/e e eff r max wth e / e eff kt e N A N q N AN e / kt ( N vnc q Eg ktl NAN 1
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