Derivation of the Metal-Semiconductor Junction Current
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1 .4.4. Derivio of e Mel-Seiouor uio Curre Derivio of e iffuio urre We r fro e epreio for e ol urre e iegre i over e wi of e epleio regio: q( µ + D (.4.11 wi be rewrie b uig -/ uliplig bo ie of e equio wi ep(- /, ielig: (.4.1 ep( qd ( + ep( qd ep( Iegrio of bo ie of e equio over e epleio regio iel: qd ep( ep( qd N ep( ep( 1 ep( (.4.1 Were e followig vlue were ue for e elero ei e poeil: ( f( N ep(- / - i + N N ep(- / ep( i / + i -. Te iegrl i e eoior be olve uig e poeil obie fro e full epleio pproiio oluio, or: qn ε ( (.4.14 o be wrie : qn qn ( ( i ε ε (.4.15
2 were e eo er i roppe ie e lier er i oi if <<. Uig i pproiio oe olve e iegrl : (.4.16 ep( ( for ( i >. Ti iel e fil epreio for e urre ue o iffuio: i qdn q( i N (.4.17 ep( [ep( 1] ε Ti epreio iie e urre epe epoeill o e pplie volge,, e brrier eig,. Te prefor be ueroo pill if oe rewrie er fuio of e eleri fiel e el-eiouor ierfe, : q( i N ε (.4.18 ielig: qµ N ep( [ep( 1] (.4.19 o e prefor equl e rif urre e el-eiouor ierfe, wi for zero pplie volge el ble e iffuio urre derivio of e erioi eiio urre Te erioi eiio eor ue elero, wi ve eerg lrger e op of e brrier will ro e brrier, provie e ove owr e brrier. Te ul pe of e brrier i ereb igore. Te urre be epree : rig lef ( + q qv (.4. For o-egeerel ope eril, e ei of elero bewee + i give b: (uig (.4.7 uig F, < - 4π ( / (.4.1 F, g( F( ep[ ( ] Auig prboli ouio b (wi o effeive, e rrier eerg,, be rele o i veloi, v, b: ee lo S.M. Sze "Pi of Seiouor Devie", Wile So, eo eiio, p. 55
3 v, vv, v (.4. Cobiig (.4.1 wi (.4. iel: ( F, v ( ep[ ]ep[ ]4πv v (.4. we replig v b v + v + v z 4π v v b v v v z e urre beoe: v r l qv ( q( ep[ v ep[ ] v π ep[ v ] v ep[ ( vz ep[ ] v ( F. F. ] v ]ep[ z ] (.4.4 uig v ep[ ] v vz ep[ ] v z π (.4.5 Te veloi v o i obie b eig e kiei eerg equl o e poeil ro e -pe regio: v (.4.6 o q o v o i e iil veloi of elero i e qui-eurl -pe regio, eee o ro e brrier. Uig 1 i [ ( F, ] q (.4.7 wi i vli for el-eiouor juio, oe obi: MS A T (.4.8 ep( [ep( 1] 4π q k were A i e Riro o i e Sok brrier eig wi equl e ifferee bewee e Feri level i e el, F,M e ouio b ege,, evlue e ierfe bewee e el e eiouor. Te -1 er i e o
4 ou for e urre flowig fro rig o lef. Te urre flow fro rig o lef i iepee of e pplie volge ie e brrier i iepee of e b beig 4 i e eiouor equl o. Terefore i be evlue volge. For e ol urre u be zero, ielig e -1 er. Te epreio for e urre ue o erioi eiio lo be wrie fuio of e verge veloi wi wi e elero e ierfe ppro e brrier. Ti veloi i referre o e Riro veloi give b: v R π (.4.9 So e urre ei beoe: qv N R (.4. ep( [ep( 1].4.4..Derivio of e uelig urre To erive e uel urre, we r fro e ie iepee Sröiger equio: wi be rewrie Ψ ( ( Ψ Ψ Ψ ( (.4. Ψ Auig ( - i iepee of poiio i eio bewee + i equio be olve ielig: Ψ ( + Ψ( ep( k wi k [ ( ] (.4. Te iu ig i oe ie we ue e prile ove fro lef o rig. For lowl vrig poeil e pliue of e wve fuio be rele o e wve fuio : Ψ Ψ [ ( ] ( ( ep (.4.4 Ti eo ue e effeive of e rrier i e e o bo ie of e brrier. 4 igorig e Sok brrier lowerig ue o ige rge
5 Ti equio i referre o e WK pproiio 5. Fro i e uelig probbili, Θ, be lule for rigulr brrier for wi (- q (1- Ψ Ψ Θ Ψ Ψ ( ( ep q (1 ( ( (.4.5 Te uelig probbili e beoe 6 : were e eleri fiel equl /. 4 Θ ep q / (.4.6 Te uelig urre i obie fro e prou of e rrier rge, veloi ei. Te veloi equl e Riro veloi, e veloi wi wi o verge e rrier ppro e brrier wile e rrier ei equl e ei of vilble elero uliplie wi e uelig probbili, ielig: qvr Θ (.4.7 Te uelig urre erefore epe epoeill o e brrier eig o e / power. 5 Ne fer Wiger, Krer rilloui 6 Uig 1
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