/8/4 C 340 eure 6 ad 7: iffusio of Carriers Class Oulie: iffusio roesses iffusio ad rif of Carriers Thigs you should kow whe you leave Key Quesios Why do arriers use? Wha haes whe we add a eleri field o our arrier gradie? How a I visualize his from a ad diagram? Wha is he geeral effe of iludig reomiaio i our osideraios? Wha is he relaioshi ewee usio ad moiliy? M.. Giler C 340 eure 6 ad 7 iffusio roesses Wha haes whe we have a oeraio disoiuiy?? Cosider a siuaio where we sray erfume i he orer of a room If here is o oveio or moio of air he he se sreads y usio. This is due o he radom moio of ariles. ariles move radomly uil hey ollide wih a air moleule whih hages i s direio. If he moio is ruly radom he a arile siig i some volume has eual roailiies of movig io or ou of he volume a some ime ierval. Should he same hig hae i a semioduor if we have saial gradies of arriers? M.. Giler C 340 eure 6 ad 7 T 0 T 0 T 0 T 3 0 iffusio roesses e s shie ligh o a loalized ar of a semioduor 04 0 56 30 56 9 5 5 5 56 56 ow le s moior he sysem Assume hermal moio. Carriers move y ieraig wih he laie or imuriies. Thermal moio auses ariles o jum o a adjae omarme. Afer he mea-free ime ( half of ariles will leave ad half will remai a erai volume. 384 384 04 8 8 6 M.. Giler C 340 eure 6 ad 7 3 roess oiues uil uiform oeraio. We mus have a oeraio gradie for usio o sar.
/8/4 iffusio roesses How do we desrie his hysial roess?? We wa o alulae he rae a whih eleros use i a simle oedimesioal eamle. Cosider a arirary elero disriuio e # of eleros movig from lef o righ i oe. M.. Giler C 340 eure 6 ad 7 ivide he disriuio io iremeal disaes of he mea-free ah (. valuae ( i he eer of he segmes. leros o he lef of 0 have a 50% hae of movig lef or righ i a ime. Same is rue for eleros o he righ of 0. ( A ( A iffusio roesses So we have a flu of ariles The rae of elero flow i he + direio (er ui area: ( Sie he mea-free ah is a small ereial legh we a wrie he elero eree as: ( + I he limi of small or small mea-free ah ewee ollisios lim 0 d ( + d iffusio oeffiie (m /se M.. Giler C 340 eure 6 ad 7 iffusio roesses Bu we already eeed his iffusio ad rif of Carriers How do we hadle a oeraio gradie ad a eleri field? efie he arrier flu for eleros ad holes: d d d d Ad he orresodig urre desiies assoiaed wih usio d d d d e- The oal urre mus e he sum of he elero ad hole urres resulig from he drif ad usio roesses h+ ( d + d ( d d rif iffusio Where are he ariles ad urres flowig? ( ad drif ( ( ad drif ( + leros (drif (drif Carriers move ogeher urres oosie direios. M.. Giler C 340 eure 6 ad 7 ashed Arrows arile Flow!!Solid Arrows Resulig Curres!!! M.. Giler C 340 eure 6 ad 7
/8/4 iffusio ad rif of Carriers A few era oservaios iffusio ad rif of Carriers Ca we relae he usio oeffiie o he moiliy? ( ad drif ( (drif We a y usig wha we kow aou drif usio ad ad edig ( ad drif ( ashed Arrows arile Flow!!Solid Arrows Resulig Curres!!! iffusio urres are i oosie direios. rif urres are i he same direio. Curres deed o: Relaive elero ad hole oeraios. Magiude ad direios of eleri field. Carrier gradies. d M.. Giler C 340 eure 6 ad 7 (drif d + iffusio urres a e large eve d if he arriers are i he mioriy y several orders of magiude. d o rue for drif urres. d i e d k T I euilirium o urre flows. Ay fluuaio ha would egi a usio urre also ses u a eleri field whih redisriues he arriers y drif. d + 0 M.. Giler C 340 eure 6 ad 7 d Solve for he eleri field (: I s euilirium so we kow (: ( F i k T di d Assumig is o-zero ( F i ie ( ( 0 k T d d iffusio ad rif of Carriers These relaios are alled he isei relaios The alae of drif ad usio urres reaes a uil-i eleri field o aomay ay gradie i he ads. Gradies i he ads a our a euilirium whe: he ad ga varies. alloy oeraio varies. doa oeraios vary. V M.. Giler C 340 eure 6 ad 7 d dv di d d d iffusio ad rif of Carriers Reall he revious eamle Assume ha: I is silio maiaied a 300 K. i g /4 a ± ad i g /4 a 0. Choose he Fermi level as he referee eergy. ( V V ref M.. Giler C 340 eure 6 ad 7 i V d dv di d d d 3
/8/4 iffusio ad rif of Carriers iffusio ad rif of Carriers Quesio: Is i i euilirium? ergy i Wha are he elero ad hole urre desiies a ± /: i Maerial OS ( F f ( Maerial OS ( F f ( Assume wo maerials i iimae oa. I hermal euilirium. o urre. o e eergy rasfer. Carriers movig from o mus e alaed y arriers movig from o. Rae ( f( ( [ f( ] - Rae ( f ( ( [ f ( - ] Rae - Rae - I is i euilirium so ad 0. Roughly skeh ad iside he samle: i d F Therefore f ( f ( 0 f d M.. Giler C 340 eure 6 ad 7 YS M.. Giler C 340 eure 6 ad 7 iffusio ad rif of Carriers iffusio ad Reomiaio Wha are he elero usio urre a ± /? If so i wha direio? There is a usio urre a oh / ad /. A /: d > 0 d d A /: < 0 d Wha are he elero drif urre a ± /? If so i wha direio? A /: A /: drif drif Wha is he usio oeffiie? Use isei relaio M.. Giler C 340 eure 6 ad 7 i drif vd.9 m /se So wha does his mea? Cosider his semioduor: The hole urre desiy leavig he ereial area may e larger or smaller ha he urre desiy ha eers he area. This is a resul of reomiaio ad geeraio. e irease i hole oeraio er ui ime d/d is eree ewee hole flu er ui volume eerig ad leavig mius he reomiaio rae. M.. Giler C 340 eure 6 ad 7 4
/8/4 5 M.. Giler C 340 eure 6 ad 7 iffusio ad Reomiaio How a we elai his? The e irease i hole oeraio er ui ime is he eree ewee he hole flu eerig ad leavig mius he reomiaio rae + + ( Rae of hole uildu. Irease i hole oeraio i A er ui ime. Reomiaio rae As goes o zero we a wrie he hage i hole oeraio as a derivaive jus like i usio leros These relaios form he oiuiy euaios. M.. Giler C 340 eure 6 ad 7 iffusio ad Reomiaio Are here ay simlifiaios? If he urre is arried maily y usio (small drif we a relae he urres i he oiuiy euaio We u his ak io he oiuiy euaios iffusio euaio for eleros d d d d + iffusio euaio for holes Useful mahemaial euaio for may ere hysial siuaios M.. Giler C 340 eure 6 ad 7 Seady Sae Carrier Ijeio To his oi we ee assumig ha he eruraio was removed Wha haes if we kee he eruraio? The ime derivaives disaear d d d d leros Where iffusio egh