ON A SHOCKLEY-READ-HALL MODEL FOR SEMICONDUCTORS

Transcription

1 Theore.Al.Mech. TEOPM7,ol.40, No., , Belgrade 0 ON A SHOKEY-REA-HA MOE FOR SEMONUTORS ulka Kežević-Miljaović doi:0.98/tam0065k Mah. Subj. la.: 5A05; 5B0. Accordig o: Tib oural Abbreviaio () Mahemaical Review, he abbreviaio TEOPM7 ad for TEORSKA PRMENENA MEHANKA.

2 THEORETA AN APPE MEHANS Serie: Secial ue - Addre o Mechaic, ol. 40 (S), , Belgrade 0. ON A SHOKEY-REA-HA MOE FOR SEMONUTORS U:57.9; ulka Kežević-Miljaović Faculy of Mahemaic, Belgrade, Serbia Abac: The Shockley-Read-Hall model wa ioduced i 95 o decribe he aiic of recombiaio of hole ad eleco i emicoducor occurrig hrough he mechaim of aig ad we coider iiial-boudary value roblem wih iiial codiio. Key word: arial differeial euaio, iiial-boudary value roblem AMS ubjec claificaio: Primary 5A05. Secodary 5B0 NTROUTON The goverig euaio are give by µ U c c N () ( ( T )) c d ( ) ( ( U T )) c a ( N ) c b c a ( N ) c b c c c d ( N ) ( ) µ () () ε (4) Here deoe he deiy of eleco i he coducio bad, wherea i he deiy of hole i he valece bad, wih, beig ooie charge. The oiio deiy of occuied a i give by ; ad by c, c, c, c we deoe he rae coa. The uaiy U T i he o-called hermal volage. he followig, we coider a emicoducor cryal wih a coa (i ace) umber deiy of a N. he Poio euaio (4), (x; ) i he elecoaic oeial, a b c d ε he ermiiviy of he emicoducor, he elemeary charge, ad (x) he doig rofile. By addig euaio (),(),(), we obai he coiuiy euaio (5) 0 wih curre deiie

3 66 UKA KNEŽEĆ - MANOĆ ad ( U ) µ (6) T ( U ) µ (7) T Noe ha for he curre deiy we ue he imle oible model, he drif diffuio aaz, wih coa mobiliie µ, µ. Moreover, a here i o flux, here i o curre deiy.the ga bewee he valece ad he coducio bad (which i called he badga) i very large for emicoducor, which mea ha lo of eergy i eeded o afer eleco from he valece o he coducio bad. Thi roce i referred o a he geeraio of eleco-hole air (or air-geeraio roce), i.e., a eleco i creaed i he coducio bad ad a hole i he valece bad. The ivere roce i ermed recombiaio of eleco-hole air. We ow ioduce a recalig of,, ad i order o reder he euaio ()-() dimeiole:,, N, µ UT, x x,, µ, µ µ,,,,,, ad i a yical value for. Moreover, we recale ime o make ure N ha all coa are of order, ad e cc cd 0, cd cd, ca cb 0, N ad cb.give he calig aumio ε «, we fially obai R (8) R (9) ε R R (0) ε () where µ ( ) () ad µ ( ). () By R ad R we deoe he recombiaio-geeraio rae for ad, reecively: R ( 0 ( )) (4)

4 R 0 O a Shockley-read-hall model for emicoducor 67 ( ( ) ) Noe ha 0 hould hold from hyical oi of view. Moreover, boh ad are oegaive. (5) MAN RESUT We coider iiial-boudary value roblem wih iiial codiio x, 0 ( x), x,0 ( x), x,0 x (6), ad wih mixed irichle-neuma boudary codiio o Ω, i.e., le ( x, ) ( x), ( x, ) ( x), ( x, ) ( x), x Ω Ω (7) ad 0, Ω N : Ω \ Ω (8) ν ν ν where ν i he ouward ui ormal vecor alog ΩN. i allowed o imoe oly homogeou Neuma boudary codiio o all of Ω, i.e. we e Ω N Ø, ad he followig Theorem will hold. Theorem e, ( Ω) (ad o-egaive), 0, ( Ω). The, he oluio of (8)-() aifie, ( 0, ), ( Ω) H ( Ω) ad 0. loc ad le Proof: We will ue he reul from 5], which wa obaied for homogeou Neuma boudary codiio. We ca how by a aighforward comuaio d ( ) d µ µ ( ) ( ) R R µ µ µ µ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ] ( ) ( ) ( ) ( ) ( )

5 68 UKA KNEŽEĆ - MANOĆ 4 : (9) where he erm from (9) ca be rewrie a follow: ] )] ]( ] ε ] ) ( ε ) ( ε. We have ued arial iegraio, ad () o obai he la exreio. By alyig Holder ieualiy wih coefficie, r; ad uig he fac ha ', we obai he followig eimae ] r. W,, ρ ρ,where ε ρ. For ad eve, oe obai for (0) By rewriig he iegrad i he ecod iegral from (0) a () ad alyig he auchy-schwarz ieualiy, we have he followig eimae for (0): () For, he ame reaoig (wih. relaced by., reecively) yield a aalogou eimae. ollecig all he eimae, we fially obai: d d µ µ

6 O a Shockley-read-hall model for emicoducor 69 ] r () ] d d ] 4 (4) orollary Give he aumio of Theorem, coider euaio (8)- () wih homogeou Neuma boudary codiio. The Ω Ω, 0,, H loc. Ackowledgeme: Suored by Miiy of ciece, echology ad develome of Reublic of Serbia, rojec ON4400 REFERENES. Arold, A., Markowich, P., Tocai G., O arge Time Aym for rif-iffuio-poio Syem. 979, New York (i Eglih).. Arold, A., Markowich, P., Tocai, G., Uerreier, A., O covexsobolev ieualiie ad he rae of covergece o euilibrium for Fokker-Plack ye euaio. Arch. Raioal Mech. Aal., 9, 965, 5-6 (i Eglih). evillee,., Feller, K., Exoeial ecay oward Euilibrium via Eoy Mehod for Reacio-iffuio Euaio,. Mah. Aal. Al., 978, ( i Eglih) 4. Gajewki, H., O Exiece, Uiuee ad Aymoic Behaviour of Soluio of he Baic Euaio for arrier Traor i Semicoducor, ZAMM, 65():0-08, Markowich, P.A., Righofer,.A., Schmeier,., Semicoducor, Euaio, Sriger, New York,

7 70 UKA KNEŽEĆ - MANOĆ O SHOKEY-REA-HA MOEU ZA POUPROONKE ulka Keževic -Miljaovic Aak: Razmaamo SHOKEY-REA-HA model za olurovodike,i dokazuje e graiči roblem daim očeim ulovima. Ključe reči: arcijale diferecijale jedačie, graiči roblem AMS klaifikacija: Primara 5B00. Sekudara 5B0. 5A05. Submied o Aril 009, acceed o ue 0. O : 0.98/TAMS65K Mah.Subj.la.: 5A05; 5B0;