ECE 340 Lecture 15 and 16: Diffusion of Carriers Class Outline:

Transcription

1 ECE 340 Lecure 5 ad 6: iffusio of Carriers Class Oulie: iffusio rocesses iffusio ad rif of Carriers

2 Thigs you should kow whe you leave Key Quesios Why do carriers diffuse? Wha haes whe we add a elecric field o our carrier gradie? How ca I visualize his from a bad diagram? Wha is he geeral effec of icludig recombiaio i our cosideraios? Wha is he relaioshi bewee diffusio ad mobiliy? M.. Gilber ECE 440 Lecure 5 ad 6 9/6/

3 iffusio rocesses Wha haes whe we have a coceraio discoiuiy?? Cosider a siuaio where we sray erfume i he corer of a room If here is o covecio or moio of air, he he sce sreads by diffusio. This is due o he radom moio of aricles. aricles move radomly uil hey collide wih a air molecule which chages i s direcio. If he moio is ruly radom, he a aricle siig i some volume has eual robabiliies of movig io or ou of he volume a some ime ierval. T 0 Should he same hig hae i a semicoducor if we have saial gradies of carriers? T 0 T 0 T 3 0 M.. Gilber ECE 440 Lecure 5 ad 6 9/6/

4 iffusio rocesses Le s shie ligh o a localized ar of a semicoducor ow le s moior he sysem 04 Assume hermal moio. Carriers move by ieracig wih he laice or imuriies. Thermal moio causes aricles o jum o a adjace comarme. Afer he mea-free ime ( c ), half of aricles will leave ad half will remai a cerai volume c c 6 c c rocess coiues uil uiform coceraio. We mus have a coceraio gradie for diffusio o sar. M.. Gilber ECE 440 Lecure 5 ad 6 9/6/

5 iffusio rocesses How do we describe his hysical rocess?? We wa o calculae he rae a which elecros diffuse i a simle oedimesioal eamle. Cosider a arbirary elecro disribuio λ λ λ e # of elecros movig from lef o righ i oe c. ivide he disribuio io icremeal disaces of he mea-free ah (λ). Evaluae () i he ceer of he segmes. Elecros o he lef of 0 have a 50% chace of movig lef or righ i a ime, c. Same is rue for elecros o he righ of 0. λ ( A) ( λa) M.. Gilber ECE 440 Lecure 5 ad 6 9/6/

6 iffusio rocesses So we have a flu of aricles The rae of elecro flow i he + direcio (er ui area): λ φ c ( ) Sice he mea-free ah is a small differeial legh, we ca wrie he elecro differece as: ( ) ( ) I he limi of small Δ, or small + λ φ ( ) λ c λ lim 0 c d( ) d φ mea-free ah bewee collisios ( ) ( + ) iffusio coefficie (cm /sec) M.. Gilber ECE 440 Lecure 5 ad 6 9/6/ λ λ

7 iffusio rocesses Bu we already eeced his efie he carrier flu for elecros ad holes: φ φ ( ) ( ) d d d ( ) ( ) d Ad he corresodig curre desiies associaed wih diffusio diff diff d( ) d d( ) d Carriers move ogeher, curres oosie direcios. M.. Gilber ECE 440 Lecure 5 ad 6 9/6/

8 iffusio ad rif of Carriers How do we hadle a coceraio gradie ad a elecric field? e- h+ E () () The oal curre mus be he sum of he elecro ad hole curres resulig from he drif ad diffusio rocesses ( ) µ ( ) E( ) ( ) µ ( ) E( ) + ( ) d d d ( ) d rif iffusio Where are he aricles ad curres flowig? φ (diff ad drif) φ (diff) ( ) + Elecros Holes φ (drif) (diff ad drif) (diff) (drif) ashed Arrows aricle Flow Solid Arrows Resulig Curres M.. Gilber ECE 440 Lecure 5 ad 6 9/6/

9 iffusio ad rif of Carriers A few era observaios φ (diff ad drif) φ (diff) φ (drif) (diff ad drif) (diff) (drif) ashed Arrows aricle Flow Solid Arrows Resulig Curres iffusio curres are i oosie direcios. rif curres are i he same direcio. Curres deed o: Relaive elecro ad hole coceraios. Magiude ad direcios of elecric field. Carrier gradies. ( ) ( ) E( ) ( ) µ ( ) E( ) ( ) d d d µ + iffusio curres ca be large eve if he carriers are i he mioriy by several orders of magiude. o rue for drif curres. ( ) d M.. Gilber ECE 440 Lecure 5 ad 6 9/6/

10 iffusio ad rif of Carriers Ca we relae he diffusio coefficie o he mobiliy? We ca by usig wha we kow abou drif, diffusio, ad bad bedig ( ) d d i k T b e I euilibrium, o curre flows. Ay flucuaio ha would begi a diffusio curre also ses u a elecric field which redisribues he carriers by drif. ( ) ( ) E( ) µ + ( ) d d M.. Gilber ECE 440 Lecure 5 ad 6 9/6/ 0 Solve for he elecric field E(): E( ) I s euilibrium, so we kow (): ( ) ( E E ) F k T b i dei d k T b E Assumig E is o-zero e i ( E E ) ( E ) µ ( E) 0 k T b F k b T i µ µ µ ( ) d( ) d kbt kbt

11 iffusio ad rif of Carriers These relaios are called he Eisei relaios µ µ kbt kbt The balace of drif ad diffusio curres creaes a buil-i elecric field o accomay ay gradie i he bads. Gradies i he bads ca occur a euilibrium whe: he bad ga varies. alloy coceraio varies. doa coceraios vary. E V de d dev d c de d i M.. Gilber ECE 440 Lecure 5 ad 6 9/6/

12 iffusio ad rif of Carriers Recall he revious eamle Assume ha: E c I is silico maiaied a 300 K. E f E i E g /4 a ± L ad E f E i E g /4 a 0. E i E f E v Choose he Fermi level as he referece eergy. V ( ) E c E ref V E -L 0 L V E de d dev d c dei d -L 0 L -L 0 L M.. Gilber ECE 440 Lecure 5 ad 6 9/6/

13 iffusio ad rif of Carriers Quesio: Is i i euilibrium? E c Eergy E i E f E f Maerial OS (E) F f (E) Maerial OS (E) F f (E) Assume wo maerials i iimae coac. I hermal euilibrium. o curre. o e eergy rasfer. Carriers movig from o mus be balaced by carriers movig from o. E v -L 0 L Rae ( E) f( E) ( E) [ f( E) ] - ( E ) f ( E ) ( E ) [ f ( E ) ] Rae - Rae - Rae - de F Therefore f (E) f (E) E f E 0 YES f d M.. Gilber ECE 440 Lecure 5 ad 6 9/6/

14 iffusio ad rif of Carriers Wha are he elecro ad hole curre desiies a ± L/: E c E i I is i euilibrium, so ad 0. E f E v Roughly skech ad iside he samle: -L 0 L i -L 0 L M.. Gilber ECE 440 Lecure 5 ad 6 9/6/

15 iffusio ad rif of Carriers Wha are he elecro diffusio curre a ± L/? If so, i wha direcio? There is a diffusio curre a boh L/ ad L/. A L/: A L/: d d d d > 0 < 0 diff diff Wha are he elecro drif curre a ± L/? If so, i wha direcio? A L/: drif E c E i E f E v -L 0 L drif vd µ E A L/: drif Wha is he diffusio coefficie? Use Eisei relaio kbt M.. Gilber ECE 440 Lecure 5 ad 6 9/6/ µ.9 cm /sec

16 iffusio ad Recombiaio So wha does his mea? Cosider his semicoducor: The hole curre desiy leavig he differeial area may be larger or smaller ha he curre desiy ha eers he area. This is a resul of recombiaio ad geeraio. e icrease i hole coceraio er ui ime, d/d, is differece bewee hole flu er ui volume eerig ad leavig, mius he recombiaio rae. M.. Gilber ECE 440 Lecure 5 ad 6 9/6/

17 iffusio ad Recombiaio How ca we elai his? The e icrease i hole coceraio er ui ime is he differece bewee he hole flu eerig ad leavig mius he recombiaio rae + ( ) ( + ) Rae of hole buildu. Icrease i hole coceraio i ΔA er ui ime. Recombiaio rae As Δ goes o zero, we ca wrie he chage i hole coceraio as a derivaive, jus like i diffusio (, ) Holes These relaios form he (, ) coiuiy euaios. Elecros M.. Gilber ECE 440 Lecure 5 ad 6 9/6/

18 M.. Gilber ECE 440 Lecure 5 ad 6 9/6/ iffusio ad Recombiaio Are here ay simlificaios? If he curre is carried maily by diffusio (small drif) we ca relace he curres i he coiuiy euaio diff diff We u his back io he coiuiy euaios iffusio euaio for elecros ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) d d E d d E + µ µ ( ) ( ),, iffusio euaio for holes Useful mahemaical euaio for may differe hysical siuaios

19 M.. Gilber ECE 440 Lecure 5 ad 6 9/6/ Seady Sae Carrier Ijecio To his oi, we bee assumig ha he erurbaio was removed Wha haes if we kee he erurbaio? The ime derivaives disaear L d d L d d Elecros Holes Where L L iffusio Legh