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1 Electoic Sulemetay Mateial (ESI o Joual o Mateial hemity A. Thi oual i The oyal Society o hemity 25 Suotig Iomatio hage tae ocee at the emicoducto/electolyte iteace o ola uel oductio: iight om imedace ectocoy. uca etoluzzi a*, Pila oez Vao b, Jua Atoio Jimeez Teada b, Jua iuet a,c a Gu de Dioitiu Fotovoltaic i Otoelectòic, Deatamet de Fíica, Uiveitat Jaume I, 27 atelló, Sai. b Deatameto de Electóica y Tecología de omutadoe, ITI-UG, Uiveidad de Gaada, 87 Gaada, Sai c Deatmet o hemity, Faculty o Sciece, Kig Abdulaziz Uiveity, Jeddah, Saudi Aabia betoluz@ui.e Steady tate euatio o the diect ad idiect chage tae model The claical dit diuio euatio that decibe the emicoducto ytem ae the cotiuity euatio o electo ad hole coductio cuet, the dilacemet cuet ad Poio euatio, a detailed i e. I teady tate (idicated by the ove ba, the dilacemet cuet i zeo ad thee euatio ca be witte a a ytem o 6 it ode dieetial euatio that gove the vaiatio o 6 vaiable: the electo cuet, the hole cuet, the deity o electo ad hole, the electotatic otetial ad the electic ield F : u F D u F D x x ( (2
2 2 U D G U x G U x F x F x N D N A Whee G ex( i the geeatio ate, beig the abotio coeiciet ad x U the hoto lux. i the bad to bad ecombiatio ate deied a. u ad u ae the electo ad hole mobility, eectively, D ad ae the electo ad hole diuio coeiciet, which ae lied to the mobility by the Eitei elatio: D u T. N ad N ae the total deity o ioized doo / D ad acceto, ad ae the dielectic emittivity o the vacuum ad the elative emittivityo the emicoducto. The total cuet deity i give by. A Fo the ytem eeted i Fig. o the mai text, we aume that at the emicoducto/metal iteace ( x, the maoity caie cocetatio (electo i cotat ad eual to it euilibium cocetatio: ( N ex( / T c Note that i ou eviou wo, 2 we had eglected the iluece o the electic ield ad the cocetatio at the cotact wa diectly modulated by the alied voltage a ( ex( V / T, whee T i the themal voltage. We tae a the eeece o the otetial, the electotatic otetial i x : ( We coide the metal to be a eect electo elective cotact (o hole cuet: ( At the emicoducto/electolyte iteace, the coductio ad valece bad edge ae ied ad the voltage do at the Schotty baie i lied to the electotatic otetial a: ( (. V ad theeoe ( aediectly modulated by the alied voltage a: V c ( V bi V c (3 (4 (5 (6 (7 (8 (9 ( whee Vbi i the built-i otetial. E ( ca be alo witte a:
3 3 Vc T ( e e Vbi V T ( Two bouday coditio o the cuet mut be imoed, deedig o the chage tae model o the emicoducto/electolyte iteace. We give the detail o uch bouday coditio heeate. a ouday coditio o the diect chage tae model. I the ollowig we model the iteace emicoducto/electolyte a a thi laye o emicoducto o legth, whee all caie deitie ae uoed to be uiom. 3 Fo diect chage tae o hole om the valece bad, the hole cuet lowig though the emicoducto/electolyte iteace ( x ollow the elatio: ( Whee Nv ex( / T. Fially, the electo cuet i aumed to be zeo at thi cotact ( The bouday coditio (7 (, (2 ad (3 allow obtaiig the comlete olutio to the ytem o euatio ( (6. (2 (3 b ouday coditio o the idiect chage tae model. Fo idiect hole tae om uace tate the ecombiatio cuet at the iteace emicoducto/electolyte mut be calculated. Oe ocedue to obtai uch cuet coit i olvig the cotiuity euatio couled to the mate euatio o the occuatio obability,, o the uace tate. We eglect diect chage tae o electo ad hole at the iteace, o that: ( (.The ecombiatio cuet ca be calculated by olvig the ollowig cotiuity euatio: x x ( N N N ( N (4 (5 whee i detemied by the mate euatio: ( ( ( (6 I additio, by uig the elatio imoed by the detailed balace (,
4 4 ad itegatig e (4 ad (5, we obtai: (7 N ( (8 N ( Euatio (7 ad (8 ae the two lat bouday coditio that ae eeded to obtai the comlete olutio to the et o dieetial euatio ( (6 o idiect hole tae om the uace tate. I the cae that both the idiect chage tae model ad diect chage tae model ae coideed, the hole cuet deity at the emicoducto/electolyte iteace i give by the um o (2 ad (8 ad the electo cuet deity i give by (7. Seveal oit hould be emaed about e (7 ad (8. Fit, i the exeio o the ecombiatio cuet the it ight had ide tem coeod to a geealized Shocley ead Hall ecombiatio icludig chage tae ietic, while the ecod tem coeod to chage tae om thee uace tate. Secod, it hould be emaed that the total hoto-aodic cuet i give by: (9 ( ( ( N oeuetly, at high alied aodic otetial, the uace tate ae illed with hole ad. Theeoe, the maximum theoetical aodic cuet i. N Iluece o the ta aited ecombiatio o the aodic cuet. Fig. S Full dit diuio imulatio with two value o the hole taig ietic cotat ( idicated i the Figue. Highe hole taig ietic iduce highe electo-hole ecombiatio ad theeoe a lowe aodic cuet. Fo thi imulatio we choe - ad the othe aamete ae the oe idicated i Table S. 6 5
5 5 Value o the aamete ued o the imulatio. Table S.Paamete o the imulatio Paamete Value (m (m T 26 (mev = (cm 2 V N (cm -3 D N (cm -3 A 2 N (cm -3 2 N = (cm -3 E g c N v.2 (ev. (ev.4 (ev 9 G 5 (cm -3-3 (cm - 6 ( - ( - 8 (cm 3-2 (cm ( ( Note that,,, ad mut atiy the detailed balace. 2 I additio, we have choe the value o ad uch that the maximum achievable aodic cuet om the uace tate ad valece bad ae comaable. Accodig to euatio (2 ad (9, the atio betwee both cuet i N /(. Theeoe, ice the maximum value o mut have o the ode o -6 /. t i o the ode o 5 cm -3 (ee Figue 2b o the mai text, we
6 6 alculatio o the euivalet cicuit o Fig. 3 o the mai text. I the ollowig we deive the mall etubatio euatio that ae ued to obtai the E o Fig. 3 o the mai text. We ocu o the ocee that occu at the iteace ad eglect all the taot limitatio, which would give ie to tamiio lie behavio. I additio, the geometic caacitace, g, aociated to the dilacemet cuet i uually much malle tha the claical chemical caacitace o ee o taed caie. Thi caacitace i theeoe obeved at much highe euecie. Fo thi eao, we ca decoule the geometic caacitace om the et o the cicuit, eeeted with the imedace Z i Fig. S2. I the ollowig we calculate the imedace Z. el el Fig. S2 Geeal imliied E o both iteacial chage tae model dilayed i Fig. b ad c o the mai text. I thi aoximatio, the geometic caacitace ha bee decouled om the electoic taot a i e. 4 a Diect hole tae model. Fo diect hole tae, the cotiuity e (3 ad (4 ca be witte a: t t G U x x G U (2 (2 Sice we ca eglect all taot limitatio, all deitie ae homogeeou ad we ca itegate e (2 ad (2 ove the whole emicoducto laye. y alyig bouday coditio (9, (2 ad (3, we obtai the total extacted cuet: t t G U G U ( (22 (23 We ow ue the mall etubatio theoy, which coit i alyig a mall etubatio Vˆ to the teady tate voltage V. The caie deitie ad ca the be witte a: ad. Uig e (, we ca idetiy a:
7 7 Vˆ ˆ T (24 I additio we have: Uˆ ˆ Ate alyig the alace taom to e (22 ad (23, we obtai: iˆ ˆ ˆ i ˆ ombiig e (24, (26 ad (27, we obtai: Z el whee: ( cb ( ec ( ct i ( cb 2 T 2 T 2 T ( ec i ( ( ct (25 (26 (27 (28 (29 (3 (3 (32 Note that chage tae om the valece bad i otimum o aotuctued emicoducto ice i thi cae chage tae occu at all oit ad /. Fom e (28 we obtai Z el 3a o the mai text whee the exteal caacitace, (cb ad : g ( cb ( cb out ( cb g g ad coeuetly, om Fig. S2, we deduce the E o Fig. (cb out, i the aallel combiatio o (33 b Idiect hole tae model.
8 8 We aly the ame ocedue a the oe ued o the diect hole tae model. Howeve i thi cae we eglect the valece bad caacitace (i.e / t a well a the bul ecombiatio U. I thi cae, ate itegatig e (3 ad (4 ove the emicoducto laye ad alyig e (9, (4 ad (5 ad coideig the time deedet mate e, we get: t G ( N N G N ( N t ( Fom e (35 ad (36, we deduce: G /( N t ( ( ( G /( N ( Alyig the mall etubatio method ad the alace taom to e (34 ad (38, we obtai: iˆ ˆ ˆ( i ˆ( ˆ N (34 (35 (36 (37 (38 (39 (4 We deduce the total imedace Z el i ( cb ( td i ( Z el o idiect chage tae: ( (4 with: ( N ( ( cb ( ( td ( ( (42 (43 (44
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