Activity Coefficients of Electrons and Holes in Semiconductors

Dilu te-solut ion t ranspor t equa t ions wi th cons tan t act ivi ty coeff icients are c o m m o n l y used to mode] s emiconduc tors. These equa t ions are cons is ten t wi th a Bo l t zmann dis t r ibut ion and are inval id in regions where the Species concent ra t ion is close to the respec t ive site concentra t ion. A more r igorous t r ea tmen t of t ranspor t in a s emiconduc to r requi res ac t iv i ty coeff icients wh ich are funct ions of concentra t ion. Express ions are p resen ted for act ivi ty coeff icients of elect rons and holes in s emiconduc to r s for wh ich conduct ionand va lence-band energy levels are g iven by the respec t ive b a n d e d g e energy levels. These act ivi ty coeff ic ients are funct ions of concen t ra t ion and are t he rmodynamica l l y consistent. The use of act ivi ty coeff icients in macroscop ic t ranspor t re la t ionships al lows a descr ip t ion of e lectron t ranspor t in a m a n n e r cons is ten t wi th the Fermi-Dirac distr ibut ion. The concen t ra t ions of holes and e lect rons in a semiconduc tor are g iven by the Fermi-Dirac d is t r ibut ion (1, 2). A Bo l t zmann d is t r ibu t ion is f r equen t ly used as an approxima t ion to this d i s t r ibu t ion in s ta t i s t ica l -mechanical analyses of s e m i c o n d u c t i n g systems. Di lu te-solut ion t ranspor t equa t ions wi th cons tan t act ivi ty coefficients, cons i s ten t w i th a Bo l t zmann dis t r ibut ion, are also used in character iz ing the behav io r of s e m i c o n d u c t i n g systems. These a p p r o x i m a t e me thods are popula r because of their relat ive ma thema t i ca l s implici ty , but are inval id w h e n elect ron or hole concen t ra t ions are close to the respec t ive site concen t ra t ions in any region. Calcula t ion of ind iv idual ionic act ivi ty coeff icients for e lec t rons and holes has been p roposed as a means of ident i fy ing the regions in wh ich these approx ima t ions are justif ied. Rosenbe rg (3), Pan i sh and Casey, Jr. (4, 5), and H w a n g and Brews (6) have p resen ted act ivi ty coeff icients for e lec t rons and holes tha t are funct ions of potent ia l as wel l as concent ra t ion . Ha rvey (7) d iscusses the separat ion of the act ivi ty coeff ic ient into parts due to chemica l and electr ical effects. L a n d s b e r g and Guy (8) p resen t an activity coeff icient based upon an Eins te in re la t ion that inc ludes wi th in it the nonideal i t ies associa ted wi th the act ivi ty coeff ic ient (9). Act iv i ty coeff icients are de r ived here that are func t ions of concent ra t ion . This der iva t ion is i n d e p e n d e n t of the E ins te in relation. These act ivi ty coeff icients are t h e r m o d y n a m i c a l l y cons i s ten t and can be used to check the val id i ty of the B o l t z m a n n func t ion as an approx imat ion to the Fermi-Dirac distr ibut ion. These coeff icients can also be used in the appl ica t ion of mac roscop ic transport equa t ions to s e m i c o n d u c t i n g sys tems in a way that is genera l ly valid. Theoretical Development The e l ec t rochemica l potent ia l of a g iven species can arbi trari ly be separa ted into t e rms r ep resen t ing a re fe rence state, a chemica l contr ibut ion , and an electr ical contr ibut ion (10) ~ = ~o + R T in (c~) + z~F~P [1] whe re 4) is a potent ia l wh ich character izes the electr ical state of the phase and can be def ined in a n u m b e r of ways. The potent ia l used here is the e lect ros ta t ic potential wh ich is ob ta ined th rough in tegra t ion of Po i sson ' s equa t ion (11). Equa t i on [1] can be v i ewed as the def ining equa t ion for the act ivi ty coefficient, f~. U n d e r the a s sumpt ion of a di lute solution, the flux of an ind iv idua l species wi th in the s emiconduc to r is d r iven by a grad ien t of e l ec t rochemica l potent ia l N i = -ciuiVp. i [2a]