ECE 44 Power Semicoductor Devices ad Itegrated circuits Srig, 6 Uiversity of Illiois at Chicago Lecture - 4
ecombiatio, geeratio, ad cotiuity equatio 1. Geeratio thermal, electrical, otical. ecombiatio rate ad equilibrium 3. Differet tyes of recombiatio rocess 4. Cotiuity equatio derivatio 5. Simle examles of the alicatio of cotiuity equatio
Origi of geeratio Carriers are geerated thermally by the way of collisios with crystal lattice. Otical excitatio ca also trasort electros from valece bod to coductio bad ad geerate free carriers. High exteral electric field ca also break the covalet bods ad create free carriers. Electro desity Hole desity Geerally, + + Excess electro cocetratio Excess hole cocetratio Whe excess carriers are reset, i
Origi of recombiatio The cause of recombiatio is geeratio! Nature does ot like oequilibrium. So recombiatio occurs to reduce the excess carrier cocetratios to zero. So, o average G ; G ad beig the geeratio ad recombiatio rate resectively. d G d G Questio: Does the recombiatio rate deed o carrier cocetratio?
Thermal ad otical geeratio ad recombiatio
ecombiatio rate ecombiatio rate is roortioal to both electro ad hole cocetratios At thermal equilibrium, αi α Let s say excess carriers have bee created i a semicoductor, ad Therefore, et geeratio rate, d G α d Now, d ( ) d + () t () t α () t () t i α d Therefore, α α[ ( + )( + ) ] [( ) ] α + + (Assumtio, )
ecombiatio rate ad lifetime >> + We ivoke low ijectio level aroximatio i.e. ( ) d d α( + ) t () [ α ( + )] t τ t. e. e Therefore, α ( + ) Where carrier lifetime is defied as: τ α 1 ( + ) Exoetial decay
ecombiatio tyes: direct, idirect, Auger Direct recombiatio occurs betwee bad to bad without the eed for ay itermediate levels. Domiat i direct badga semicoductors like GaAs. Idirect recombiatio occurs through a itermediate tra cetre. Domiat i idirect badga semicoductors like silico. C d ( ) i ; τ τ
Auger recombiatio Auger recombiatio occurs through 3 article iteractios. It is imortat oly at very high carrier cocetratios. Eergy associated with a iitial geeratio/recombiatio rocess is used by aother third article. Auger C A ( ) ( ) + C A
Other geeratio tyes Imact ioizatio: Auger geeratio triggered by electric field heated carrier. Zeer tuelig or field ioizatio: Direct tuelig of electro from valece bad to coductio bad i the resece of a strig electric field. Imact ioizatio Zeer tuelig Iteractio with eergetic articles like α-articles.
Cotiuity equatio Electro curret desity: Hole curret desity: Carrier dyamics: J J qµ E + qd qµ E qd d d G d dx d dx We still eed a book-keeig equatio for comlete descritio. That is eeded to solve roblems like the followig
Cotiuity equatio rate of icrease of electros i V rate of electro geeratio i V rate of electro recombiatio i V et flow of electro leavig V i uit time ( V ) t G V V r v F. ds e Dividig by V ad aroximatig i the limit of small V t G r r. F e
Cotiuity equatio I terms of curret desity e J q G t r r. 1 + electro curret cotiuity equatio h J q G t r r. 1 hole curret cotiuity equatio If we use reviously derived curret desity equatios, for 1-dimesioal case: x D x E E x G t + + + µ x D x E E x G t + + + µ
Comlexity of cotiuity equatio Cotiuity equatios are couled, oliear set of secod order artial differetial equatios. They are hard to solve aalytically excet uder simlifyig hysical coditios or aroximatios. Noliearity arises from the deedece of electric field, mobility, or diffusio coefficiet o electro or hole cocetratio. Couled ature rimarily arises from the electric field goverig equatio i.e. Poisso's equatio: r. E q ε ( + ) N D N A Couled ature also arises because of deedece of recombiatio rate (carrier lifetime) o both electro ad hole cocetratios. ; τ τ