SPIN POLARIZED CHARGE CARRIER INJECTION,

Transcription

1 SPIN POARIZED CHARGE CARRIER INJECTION, TRANSPORT, AND DETECTION IN ORGANIC SEMICONDUCTORS A DISSERTATION SUBMITTED TO THE FACUTY OF THE GRADUATE SCHOO OF THE UNIVERSITY OF MINNESOTA BY MOHAMMAD YUNUS IN PARTIA FUFIMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHIOSOPHY PROF. P. PAU RUDEN, ADVISER APRI 0

2 Mohammad Yunus 0

3 ACKNOWEDGEMENTS Many individuals dsrv my thanks for thir hard work, patinc, and support all through th yars of rsarch rprsntd by this thsis. I thank my advisor, Prof. Paul Rudn, for his continuous guidanc and suprvision throughout th progrss of th work. My spcial rcognition gos to Dr. Darryl Smith, our rsarch collaborator at os Alamos National aboratory (Nw Mxico) for his insightful and fruitful commnts. I would lik to thank my fllow graduat studnts, Dr. Zahd Kausr, Dr. Dominic Schropfr, and Mr. Isaiah Stink for thir intraction on a daily basis. Accss to th facilitis of th Minnsota Suprcomputing Institut for Digital Simulation and Advancd Computation is gratfully acknowldgd. Finally, I would lik to xprss my apprciation to th National Scinc Foundation and th Dpartmnt of Elctrical and Computr Enginring (Univrsity of Minnsota) for financial support. i

4 ABSTRACT In this thsis w xplor spin polarizd charg carrir injction, transport, and dtction in organic smiconductors. Dvic structurs considrd hav on or mor frromagntic contacts to th organic smiconductor, and th condition for which charg carrir injction from frromagntic contacts is strongly spin polarizd is discussd. Spin injction into smiconductors can b gratly nhancd if th injction mchanism is spin slctiv, such as is th cas for tunnlling from frromagntic contacts. By contrast, if th carrir injction is by thrmionic mission or anothr procss that dos not dpnd on spin, th injction is only wakly spin polarizd. To discuss spin transport and spin dtction, w considr a unipolar organic spin valv consisting of an organic smiconductor layr sandwichd btwn two frromagntic contacts. Th polarizations of th magntic contacts can b paralll or anti-paralll. Spin and charg carrir transport in th organic smiconductor is dscribd by spin dpndnt transport quations in driftdiffusion approximation and th spin dtction procss is through magnto-rsistanc. W discuss th impact of various dgrs of spin rlaxation in organic smiconductors on th spatial variation of th spin currnt and its ffct on magnto-rsistanc. Th spatial profil of th spin currnt insid th organic smiconductor dpnds not only on th spin ii

5 diffusion lngth but also on th alignmnt of th contact polarizations. Howvr, th magnto-rsistanc dcrass strongly with dcrasing spin diffusion lngth. Elctron tunnlling from a frromagntic contact can hav significant spin dpndnc bcaus th spatial part of th lctron wav function is diffrnt for th majority and minority spin stats of th frromagntic contacts. Th tunnlling procss occurs from th frromagntic contact through an insulating layr into th organic smiconductor. Th insulating layr is modld first as an ohmic layr with spin dpndnt contact rsistancs. Th ffctivnss of spin dpndnt contact rsistancs on spin polarizd injction and magnto-rsistanc is xamind on th basis of a simpl analytical modl. W thn modl th insulating layr as a tunnl barrir with spin dpndnt rat quations. Both majority and minority spin lctrons of th frromagntic contact tunnl through th insulating layr into th localizd molcular stats of th organic smiconductor at th smiconductor/insulator intrfac. Tunnlling matrix lmnts and transition rats of th two spin typs ar calculatd using a Transfr Hamiltonian approach. Th transition rats ar thus spin dpndnt and usd in rat quations to calculat th injctd (xtractd) currnt for carrirs of ithr spin dirction. W xplor th various aspcts of th frromagntic contacts, th thicknss and barrir hight of th insulating layr, and th nrgy of th localizd molcular stats on spin injction and magnto-rsistanc. Consistnt with th xprimntal data, th spin iii

6 injction from frromagntic contacts can b ithr positiv or ngativ, and th magnto-rsistanc dcrass strongly with th applid bias across th dvic. iv

7 TABE OF CONTENTS Acknowldgmnts....i Abstract...ii Tabl of contnts......v ist of tabls..vii ist of figurs...viii ist of abbrviation.ix. Introduction. itratur rviw.... Objctiv of this work.6.3 Organization of th thsis....7 Spin Injction into organic smiconductors 9. Modl paramtrs...9. Spin and charg currnt insid FM contacts..0.3 Transport in organic smiconductors..4 Injction by thrmionic mission Numrical rsults for thrmionic mission 8.6 Injction by tunnlling....7 Numrical rsults for tunnl injction 5.8 Conclusions 8 3 Spin transport and its dtction in organic spin valvs Dvic structur Splitting of quasi Frmi lvls at th contact Transport in th smiconductor Rsults without spin rlaxation Finit spin rlaxation in th smiconductor Rsults with spin rlaxation Conclusions 49 4 Spin injction and xtraction by tunnlling Dvic structurs Tunnl injction/xtraction modl Rsults for tunnl injction/xtraction...60 v

8 4.4 Conclusions 73 5 Summary and suggstions for futur work Summary Suggstions for futur work Rfrncs Appndix A 86 Solution of spin diffusion quation vi

9 IST OF TABES Tabl. Transport paramtrs of frromagntic contacts and organic smiconductors..8 Tabl. Diffrnt combination of spin dpndnt contact rsistancs 8 vii

10 IST OF FIGURES Figur. Figur. Figur.3 Figur.4 Figur.5 Figur.6 Figur.7 Figur 3. Figur 3. Figur 3.3 Figur 3.4 Figur 3.5 Figur 3.6 Figur 3.7 Enrgy band diagram of a Schottky contact btwn a FM mtal contact and a smiconductor Calculatd charg, calculatd spin, and analytical spin currnt dnsitis with FM mtal contacts Calculatd charg and spin currnt dnsitis with SMO contacts.. Comparison of numrically calculatd spin currnt with analytical spin currnt Enrgy band diagram of a tunnl contact with spin dpndnt contact rsistancs Injctd charg and spin currnt dnsitis with spin dpndnt contact rsistancs....6 Spin polarization with diffrnt combinations of spin dpndnt contact rsistancs..7 A schmatic dvic structur consisting of an organic smiconductor sandwichd btwn two FM contacts...3 Calculatd charg and spin currnt dnsitis for P and AP contact magntizations Spin polarization for P and AP contact magntizations 36 Calculatd MR of an organic spin valv with spin dpndnt contact rsistancs Spin polarization for P and AP contact magntizations with bias dpndnt contact rsistancs...39 Calculatd MR of an organic spin valv with bias dpndnt contact rsistancs...40 Spatial dpndnc of spin polarization for P configuration...44 viii

11 Figur 3.8 Figur 3.9 Figur 3.0 Figur 3. Figur 4. Figur 4. Figur 4.3 Figur 4.4 Figur 4.5 Figur 4.6 Figur 4.7 Figur 4.8 Figur 4.9 Figur 4.0 Figur 4. Spatial dpndnc of spin polarization for AP configuration.. 45 Spin polarization of th injctd carrir dnsity Calculatd MR with diffrnt spin diffusion lngths Calculatd MR as a function of dvic thicknss. 48 Dvic structur for tunnl injction/xtraction modl 5 Calculatd tunnlling tim constants as function of th localizd molcular nrgy lvls Calculatd forward tunnlling rat Calculatd nt currnt dnsity for an applid fild of 00 V/cm. 64 Spin transmission of th tunnl barrir as a function of nrgy of th molcular stats Spin transmission of th tunnl barrir as a function of th barrir hight.. 66 Comparison of spin transmission and spin injction as a function of applid fild Comparison of spin transmission and spin injction as a function of dvic thicknss..69 Calculatd positiv MR Spin polarization for th cass shown in figur Calculatd ngativ MR ix

12 IST OF ABBREVIATIONS OED FET ED RTD FM Organic light mitting diod Fild ffct transistor ight mitting diod Rsonant tunnlling diod Frromagntic SMO a 0.7 Sr 0.3 MnO 3 MR Magnto-rsistanc P Paralll AP Anti paralll SU Spin up SD Spin down x

13 Chaptr Introduction Organic spintronics is a nascnt but rapidly growing fild whr organic smiconductors ar usd to conduct and control a spin polarizd currnt to incras dvic functionality. Crtain π-conjugatd polymrs and crystals of rlativly small hydrocarbon molculs known as organic smiconductors hav in rcnt yars bcom viabl matrials for lctronic, optolctronic, and photovoltaic dvics., Ths matrials hav procssing advantags ovr convntional smiconductors for lowcost, larg-ara, and flxibl dvic applications. Displays basd on organic light mitting diods (OEDs) ar alrady sing commrcial us. 3 Organic photovoltaic dvics ar also compting in xisting commrcial applications 4 and considrabl improvmnts hav alrady bn achivd in th fild of organic fild ffct transistors. 5 On th othr hand, spintronics is a st of idas that utiliz spin (instad of, or in addition to charg) as th physical information carrying quantity. Commrcial succss of mtal basd spintronic dvics has bn achivd with rcording hads and magntic mmoris that us th giant magnto-rsistanc and tunnling magnto-rsistanc ffcts in socalld spin valvs. 6,7 Basd on th succss of mtallic spintronics, intns rsarch fforts ar now dvotd to includ th spin dgr of frdom into th ralm of smiconductors. Ky rquirmnts for succss in this ffort includ th following: fficint injction of spin polarizd charg carrirs through on dvic trminal; fficint transport and sufficintly long spin rlaxation tims within th host smiconductor matrial; ffctiv control/manipulation of th spin polarizd carrirs in th structur (.g., by using a gat bias) to provid th dsird functionality; and ffctiv dtction of th spin polarizd carrirs at a scond dvic trminal. Coupling th spin dgr of frdom with lctronic

14 dvics could ultimatly incras considrably th functionality and prformancs of smiconductor dvics. 8, 9, 0 A numbr of spintronic dvic concpts, such as th spin polarizd fild ffct transistor (spin FET), spin polarizd light mitting diod,3 (spin ED), and spin dpndnt rsonant tunnlling diod 4 (spin RTD) hav bn discussd in th litratur, and som dgr of succss has bn ralizd with ach of ths dvics. In all of thm, th smiconductors ar inorganic matrials. Spintronic dvics basd on organic smiconductors hav also bn rportd rcntly. 5,6 Th prospct of introducing spintronics into organic smiconductor tchnology could mak th dvlopmnt of rsistiv mmoris and snsors basd on organic smiconductors possibl. 7. itratur rviw To dat, vry fw thortical works hav bn carrid out on organic spintronics. Th xprimntal situation is also not yt wll stablishd. Howvr, thr has bn considrabl progrss, both thortically and xprimntally, in undrstanding th lctron spin physics in inorganic smiconductors. In ordr to undrstand th spin dpndnt procsss in organic smiconductors, w will apply th rcnt progrss mad in inorganic smiconductors to organic matrials. Hnc, this litratur rviw will consist of works focusing on inorganic and organic smiconductors. An ssntial rquirmnt for spintronic dvic opration is fficint lctrical spin injction into smiconductors. Though frromagntic (FM) contacts ar spin polarizd, Schmidt t al. 8 rvald that th larg conductivity mismatch btwn FM contacts and smiconductors ffctivly supprsss spin injction. For th diffusiv transport rgim, thy showd that th spin injction cofficint γ is proportional to σ SC /σ FM, whr σ SC and σ FM ar th conductivitis of th smiconductor and th FM contact. Sinc σ SC <<σ FM, th spin injction cofficint is always much lss than unity. Howvr, E. I. Rashbha 9 showd that a spin slctiv tunnl contact at th FM/smiconductor intrfac can solv th conductivity mismatch problm and gratly nhanc spin injction from FM

15 contacts into smiconductors. Th magnitud of th tunnl contact rsistanc should b of th ordr of th rsistanc of th smiconductor. D.. Smith and R. N. Silvr, 0 and J. D. Albrcht and D.. Smith also showd that a tunnl contact can facilitat spin injction into smiconductors. Thy calculatd spin injction from FM contacts into smiconductors with diffrnt conductivitis. Thy showd that spin injction into smiconductors incrass with incrasing conductivity of th smiconductor for a fixd tunnl contact. Elctron tunnlling from a FM mtal contact is spin dpndnt bcaus th spatial part of th lctron wav function is diffrnt for majority and minority spin stats nar th Frmi surfac. In a dvic modl this procss can b ffctivly dscribd by contact rsistancs that ar diffrnt for majority and minority spin lctrons. Th ffcts of such spin injction modld by spin dpndnt contact rsistancs hav bn discussd in th litratur., 3, 4, 5 All ths calculations hav shown that spin dpndnt contact rsistancs can ffctivly injct spin polarizd charg carrirs into smiconductor matrials. Onc th mchanism of fficint spin injction into a smiconductor matrial had bn stablishd, spin injction and dtction from FM contacts wr succssfully dmonstratd in inorganic smiconductors. 6, 7, 8, 9, 30, 3 A FM mtal contact usually forms a Schottky contact with th smiconductor. Th xprimntal dvics us a prcis doping profil in th smiconductor nar th FM intrfac in ordr to achiv a thin dpltion layr. Tunnlling through th dpltion rgion provids spin injction into ths dvics. Th most studid matrial among th inorganic smiconductors has bn gallium arsnid (GaAs). GaAs provids som favorabl spin dpndnt optical proprtis that can b usd to dtct spin polarizd lctron injction. Spin dpndnt optical probs basd on slction ruls hav bn applid in dmonstrating lctrical spin injction, transport, and dtction xprimntally. 3 Unfortunatly, ths slction ruls ar not applicabl in organic smiconductors. This maks spin xprimnts in organic smiconductors difficult. Efforts to dtct spin injction in ths lattr matrials thrfor hav focusd on th masurmnt of th magnto-rsistanc of organic spin valvs. 3

16 Organic smiconductors consist of π-conjugatd hydrocarbons in which th valnc stats ar formd primarily from bonding combinations of π-orbitals cntrd on carbon atoms and conduction stats ar formd primarily from th corrsponding antibonding combinations of th π-orbitals. Typically ths matrials ar highly disordrd and thir lctronic stats ar not lablld by a wav vctor. Conduction in most cass occurs by lctron or hol hopping rathr than by band transport, as dscribd by th Boltzmann Transport Equation. As a rsult, carrir mobilitis ar much smallr in magnitud and strongr functions of carrir dnsity and lctric fild strngth than in inorganic smiconductors. Thy ar gnrally undopd and ssntially fr of mobil charg carrirs in quilibrium. Elctrons and hols ar introducd into th smiconductor by injction from mtallic contacts. ow-work-function mtals (such as aluminum or calcium) can b usd to injct lctrons and high-work-function mtals (such as gold or platinum) can b usd to injct hols in dvics fabricatd from organic smiconductors. 33,34 Bcaus th principal lmnts making up most organic smiconductors (hydrogn and carbon) ar vry light, th spin orbit intraction in ths matrials is wak. 35 Th hyprfin intraction du to th spin-/ hydrogn nucli may play a significant rol in som of th magnto-rsistanc phnomna obsrvd with organic smiconductors, 36,37 and it may also provid th dominant rlaxation mchanism for spin polarizd carrir transport. Howvr, this may vary from matrial to matrial as lctron spin rsonanc xprimnts with crtain organic molculs hav yildd linwidths that ar quit narrow, 38 consistnt with th gnral notion that spin rlaxation tims in ths matrials ar rathr long. 39 Rcnt xprimnts in which hydrogn was rplacd by dutrium in crtain polymrs hav shd much light on this issu and hav shown that larg spin diffusion lngths ar indd attainabl. 40 In contrast to inorganic smiconductors whr th natural barrir formd by th dpltion rgion was usd to injct spin polarizd charg carrirs, a molcular monolayr grown by slf assmbly tchniqus to control th charg carrir injction 4,4 can b usd as th tunnl layr to injct spin polarizd charg carrirs into organic smiconductors. Th potntial for intgrating organic smiconductors with xtrmly spin polarizd colossal magntorsistanc manganats, for xampl a 0.7 Sr 0.3 MnO 3 (SMO), provids additional potntial for organic spintronics. 4

17 al. 43 Spin injction into organic smiconductors was first dmonstratd by V. Ddiu t Thy usd a dvic structur consisting an organic smiconductor sandwichd btwn two FM contacts; this dvic structur is rfrrd to as an organic spin valv. Thy masurd th magnto-rsistanc (MR) of an SMO/T 6 (sxithinyl)/smo structur. Hr MR rprsnts th chang of th rsistanc of th dvic structur as a rsult of applid magntic fild (i. th rsistanc of th structur with applid magntic fild minus th rsistanc of th structur without magntic fild). Th application of a magntic fild aligns th contact magntizations, and th rsultant MR is attributd to spin injction and transport in th T 6. Th vidnc of MR ffcts in organic spin valvs was vrifid by Z. H. Xiong t al. 44,45 Thy usd two diffrnt FM contacts (SMO and cobalt) in an organic (8-hydroxy-quinolin aluminum, known as Alq 3 ) spin valv. Sinc th corciv magntic fild is diffrnt for SMO and cobalt, th chang of th contact magntization will occur at diffrnt magntic filds. As a rsult paralll (P) and antiparalll (AP) alignmnts of th contact magntizations will form dpnding on th applid magntic fild. For a particular applid bias across th dvic, th rsistanc of th organic spin valv changs whn th alignmnt of th contacts changs from P to AP configuration, and a MR loop was found in thir xprimnts. A chang in th contact polarizations rsults in a chang of th rsistanc of th dvic structur. This xprimnt clarly stablishd spin injction and transport in organic smiconductors. Sinc th organic smiconductor layr thicknsss wr much largr than tunnl lngths, carrir transport in th organic smiconductor was assumd to b diffusiv, and th obsrvd MR was not attributd to tunnlling from on contact to th othr. Ths rsults wr followd by succssiv vidnc of MR ffcts in organic spin valvs. 46, 47, 48, 5, 40 All ths xprimnts hav shown that organic smiconductors ar suitabl for spintronic applications. A thortical modl of spin injction and transport in organic smiconductors was rportd by P. P. Rudn and D. Smith. 49 Thy showd that spin dpndnt contact rsistancs can ffctivly injct spin polarizd lctrons into organic smiconductors consistnt with th thortical modl in inorganic smiconductors., 3, 4, 5 Thy also showd that th voltag drop across a FM contact biasd to collct spin polarizd 5

18 lctrons from th smiconductor will b masurably diffrnt for P and AP alignmnt of th contact polarizations, and thus this diffrnc in voltag can b usd to dtct th spin polarizd currnt.. Objctiv of this work Organic smiconductors hav bn shown to b promising matrials for nxt gnration lctronic, optolctronic, and photovoltaic dvics. In addition, th MR phnomna discussd in th litratur dmonstratd that organic smiconductors ar also suitabl for spintronic applications. Th prospct of introducing spintronics into organic smiconductors could ultimatly incras th prformanc of som dvics alrady mad from organic smiconductors. For xampl, w rportd modl rsults that show how spin polarizd injction affcts th formation and distribution of (missiv) singlt xcitons. 50 Dvlopmnt of rsistiv mmoris and snsors ar prhaps also possibl using organic spintronics. 5, 6, 7 To charactriz ths dvics, a sound undrstanding of spin polarizd charg carrir injction, transport, and dtction in organic smiconductors is indispnsabl. Not much thortical work so far has bn don to undrstand spin injction and transport in organic smiconductors. Whn Z. H. Xiong t al. rportd th giant MR ffcts in organic valvs, th rsults promptd considrabl dbat in th scintific community. Th initial xprimntal rsults suffrd from a lack of rproducibility. 5 Som of th xprimntal rsults found for organic spin valvs ar puzzling. Th obsrvd magnto-rsistancs ar ithr positiv or ngativ and strongly dcras with th applid bias. Thrfor, to undrstand th xprimntal rsults, a comprhnsiv thortical study has bcom ssntial. This thsis srvs th abov purpos. In this thsis, w rport a comprhnsiv study of spin injction from FM contacts into organic smiconductors, transport of th injctd spin through th organic smiconductors, and dtction of th rsultant spin through an xtracting contact. First 6

19 w discuss spin injction from FM contacts into organic smiconductors. Tunnlling provids spin slctivity to injct spin polarizd carrirs into organic smiconductors. Th tunnlling procss occurs from th frromagntic contact through an insulating layr into th organic smiconductor. Th insulating layr is modld first by an ohmic layr with spin dpndnt contact rsistancs, and thn by a tunnl layr with spin dpndnt rat quations. Spin and charg carrir transport in th organic smiconductor is dscribd by spin dpndnt transport quations in drift-diffusion approximation and th dtction of th spin currnt is through magnto-rsistanc. W discuss th various aspcts of th frromagntic contacts, th insulating layr, and th spin rlaxation insid th organic smiconductor on spin injction and its ffcts on magnto-rsistanc..3 Organization of th thsis This thsis is structurd in fiv chaptrs. Each of chaptrs to 4 consists of an introduction followd by modl dscription, numrical rsults and discussion, and finally conclusions. Th introduction dals with th rlvant litratur and th motivation of th work. Th txt of th thsis is organizd in th following way: In chaptr w dscrib spin injction from FM contacts into organic smiconductors. W discuss spin injction into organic smiconductors if th carrir injction mchanism is dominatd by thrmionic mission and by tunnlling. Chaptr 3 xplains spin injction, transport of th spin through organic smiconductors, and dtction of th rsultant spin in organic spin valvs. Th tunnl layr is modld as an ohmic layr with spin dpndnt contact rsistancs. W xplor th various dgrs of spin rlaxation in th organic smiconductor on th dtction of spin. Spin polarizd charg carrir injction and xtraction by tunnlling through an insulating layr ar modld in chaptr 4. A dtaild dscription to calculat th 7

20 tunnlling tim constants for spin up and spin down lctrons is providd. W thn modl organic spin valv dvics using ths tim constants and discuss th MR of organic spin valvs. Finally, chaptr 5 prsnts th summary of th work and possibl futur dirctions. Th thsis is wrappd up by Appndix A, whr th solution of th spin diffusion quation is givn. 8

21 Chaptr Spin Injction into organic smiconductors Th most ssntial rquirmnt for a smiconductor basd spintronic tchnology is th fficint lctrical injction of spin polarizd charg carrirs from a FM contact into a smiconductor. In this chaptr, w will xplor that procss. W considr a FM mtal/smiconductor/non-magntic mtal dvic structur and will discuss th injction from th FM contact into th smiconductor. Initially, w will focus on th cas of a FM mtal in dirct contact with th organic smiconductor. atr, w will considr th prsnc of a tunnl contact rprsntd by spin dpndnt contact rsistancs.. Modl paramtrs In ordr to dvlop a modl for carrir injction from FM contacts into organic smiconductors, w nd to charactriz th FM contact and th organic smiconductor by suitabl paramtrs. Th (non-magntic) organic smiconductor is charactrizd by an nrgy gap, E g, and by bands of conduction and valnc stats with narrow nrgy width. For th purpos of charg carrir population, ths bands ar dscribd by qual ffctiv dnsitis of stats, N 0, which ar approximatly qual to th molcular dnsity of th matrial. (Nglcting th small spin-orbit coupling, qual ffctiv dnsitis of stats N 0 / may b attributd to spin up and spin down lctrons and hols). Th matrial is assumd to b undopd, hnc all mobil charg carrirs ar injctd from th contacts, and th larg ffctiv dnsity of stats nsurs that non-dgnrat statistics apply to ssntially all cass of intrst. Th charg carrir mobility is indpndnt of spin. It may b takn as fild indpndnt (as is rasonabl for som organic molcular crystals) or it may b takn to b fild dpndnt of th Pool-Frnkl form, μ(f) = 9

22 μ 0 xp( F /F 0 ) /, whr F is th lctric fild and μ 0 and F 0 ar matrial paramtrs. For convninc w will assum fild indpndnt mobility and formulat th problm in trms of lctron injction. In most cass, carrir injction into organic smiconductors is in fact du to hols. Th spin physics of lctrons and hols ar ssntially th sam in organic smiconductors. Hnc, modling lctron injction instad of hol injction dos not affct th rsults and conclusion rachd. Spin rlaxation insid th organic smiconductor is dscribd by a tim constant, τ s. Th FM mtal contacts ar dscribd by four paramtrs: th total conductivity, σ, a polarization cofficint, α, th spin diffusion lngth, Λ, and th quilibrium Schottky barrir hight, Φ B0. Th conductivitis of spin up and spin down lctrons ar rlatd to σ and α, as σ = α σ and σ = (-α)σ.. Spin and charg currnts insid FM contacts Th currnt dnsitis for th spin up (SU) and spin down (SD) lctrons in th FM contacts can b writtn as: j j = = σ σ dμ dx dx dμ (.a) (.b) whr μ ar th lctrochmical potntials (or quasi-frmi lvls) for th SU and SD, lctrons, and is th magnitud of th lctron charg. Evidntly, th charg currnt dnsity is givn by j = j j, and th spin currnt dnsity by = j j. Insid th bulk of th FM contact, th SU and SD lctrons ar in a quasi-quilibrium stat, μ = μ, and th diffrnc of th conductivitis of SU and SD lctrons giv ris to a nt spin currnt, (α-) j. Undr stady stat conditions, th charg currnt is constant j s 0

23 throughout th ntir structur. Th spin currnt tnds to dcras towards th intrfac to th non-magntic smiconductor and th lctrochmical potntials split. This is dscribd by 5 d ( μ μ ) dx μ μ = Λ (.) Th splitting of th lctrochmical potntials rsults in an incras in th ratio of th majority spin lctron dnsity to th minority spin lctron dnsity nar th intrfac, which mans that th majority and minority spin lctrons ar out of quasi-quilibrium nar th intrfac of th FM contact and th smiconductor. W dfin th splitting btwn th lctrochmical potntials as Δμ = μ μ. It can b asily drivd from quations (.) and (.) that th spin currnt at th injcting contact intrfac (x = 0 - as shown in figur.) is σ Δμ(0 ) j s (0 ) = (α ) j α ( α) (.3) Λ Sinc th currnt is ngativ, th magnitud of th spin currnt dcrass by an amount of α ( α) σδμ(0 ) / Λ at th intrfac rlativ to its bulk valu of (α-) j..3 Transport in th organic smiconductor Onc th carrirs ar injctd from th FM contact into th organic smiconductor, thir transport is govrnd by th tim and spin dpndnt continuity quations coupld with Poisson s quation.

24 , s n n x n D F n x t n τ μ = (.4a), s n n x n D F n x t n τ μ = (.4b) ). ( 4 = n n x F ε π (.5) Hr n and n ar th SU and SD lctron concntrations and D is th lctron diffusivity which is rlatd to th mobility, μ, through th Einstin rlation, and ε is th static dilctric constant. Evidntly th charg ( = n n n ) and spin ( ) continuity quations bcom simply, = n n n s = x n D nf x t n μ (.6) s s s s s n x n D F n x t n τ μ = (.7) Equations (.4) and (.5) can b solvd numrically spcifying th boundary conditions. W discrtiz ths quations spatially using th Scharfttr-Gumml approach 53 and th rsulting first ordr diffrntial quations ar intgratd forward in tim. To find th stady stat solution at an applid voltag bias, a tim dpndnt potntial ramp that stops at th dsird voltag is applid to th right contact and th quations ar intgratd forward in tim starting from th thrmal quilibrium until stady stat is rachd. Th

25 position indpndnc of th charg currnt is usd to vrify that stady stat has bn rachd. Th quilibrium stat is calculatd following th approach dscribd in rf. 33. Th boundary conditions ar givn by spcifying th currnts for ach spin typ at th boundary. Th currnts at th boundary dpnd on th injction/xtraction mchanism btwn th contacts and th smiconductor. W will considr two kinds of injction mchanisms. First, w will considr currnt injction by thrmionic mission ovr th Schottky barrir that th smiconductor forms with th contact mtal. In this cas, th mtal is in dirct contact with th smiconductor. Scond, w will considr currnt injction by tunnlling into organic smiconductors. In this cas th mtal contact is sparatd from th smiconductor by a thin insulating layr..4 Injction by thrmionic mission A schmatic nrgy lvl diagram of th injcting contact for thrmionic mission is shown in figur. undr bias conditions. Also shown is th imag-charg inducd barrir lowring ffct. In this modl, th injctd currnt for ach spin dirction, j inj;, is th sum of a thrmionic mission currnt and an intrfac rcombination currnt (which is th tim rvrs procss of thrmionic mission). Th spin dpndnt currnts at th smiconductor intrfac (x = 0 as shown in figur.) ar givn by 54 j j n (0 ) ( ) A T xp( 0 = Φ kt ) (.8a) ; B; N 0 inj / ( n (0 ) ( 0 ) = A T xp Φ kt ) B (.8b) ; ; N 0 inj / 3

26 ΔΦ B μ μ Φ B0 E c 0 x Figur.. Schmatic nrgy band diagram of a Schottky contact btwn a FM mtal and a smiconductor undr bias such as to nabl injction of lctrons into th smiconductor. Hr A dnots th ffctiv Richardson constant, T th tmpratur, k Boltzmann s constant, and Φ B;, th spin dpndnt non-quilibrium barrir hight. Th barrir hight dpnds on spin bcaus th quasi-frmi lvls ar diffrnt for th two spin dirctions as shown in figur.. Th spin dpndnt lctron concntrations in th smiconductor at th intrfac ar dnotd by n, (0 ). Imag charg inducd Schottky barrir lowring is comparativly strong in th organic smiconductors du to thir small dilctric constants. Th ffct may b approximatd by lowring th quilibrium barrir hight by: ΔΦ B F( 0 ) / ε, whr F(0 ) is th lctric fild in th smiconductor. Imag charg inducd barrir lowring is includd in th dtrmination of th valu of Φ B;,. Th Schottky barrir, Φ, B and spin dpndnt barrirs ΦB;, ar rlatd to th spin 4

27 dpndnt lctrochmical potntials at th contact and th conduction band dg, E c, through μ (0 ) μ (0 ) Φ B = E c (0 ) (.9a) Φ = E (0 ) μ (0 ), B c (.9b) ; Φ = E (0 ) μ (0 ), B c (.9c) ; Th injctd charg currnt dnsity, j inj, is th sum of th SU and SD currnts givn by quations (.8a) and (.8b), which can b combind to: j inj (0 ) = AT n ) Φ B xp kt Δ (0 cosh kt (0 μ N 0 ) (.0) Hr n(0 ) is th total ( n n ) lctron concntration in th smiconductor at th intrfac. Similarly th injctd spin currnt dnsity, js,inj = j inj; - j inj;, can b xprssd as (0 ) (0 ) ns Φ B Δμ j, (0 ) = xp sinh, 0 s inj AT N kt kt (.) whr n s (0 ) is th lctron spin ( n n ) concntration at th intrfac. Th ky part of this modl is to dtrmin th splitting btwn th lctrochmical potntials at th contact i.. Δμ(0 ). W assum that thr is no spin scattring as th lctrons travrs th FM/smiconductor intrfac. Hnc, both th spin currnt and th charg currnt ar 5

28 continuous across th intrfac and th splitting of th quasi-frmi lvls at both sids of th intrfac is th sam: j( 0 ) = jinj (0 ) = j, j s ( 0 ) = js, inj (0 ) = js (0). Δμ( 0 ) = Δμ(0 ) = Δμ(0) (.a) (.b) (.c) Equations (.3), (.9a)-(.9c), (.0), (.), and (.a)-(.c) ar usd to dtrmin th spin dpndnt barrir hights Φ B;,, which ar usd to dtrmin th spin dpndnt particl currnts at th boundary (quations (.8a) and (.8b)). Th splitting btwn th lctrochmical potntials Δμ(0) is small du to th high conductivity of th FM mtal, and if Δμ(0)<<kT, th lctron spin dnsity, th charg currnt, and th spin currnt at th injcting contact can b approximatd by: Δμ(0) n s (0 ) = n(0 ), (.3a) kt j = AT n(0 ) Φ B xp, (.3b) N 0 kt j (0) = AT s n (0 ) (0) s Φ B Δμ xp, N 0 kt kt (.3c) 6

29 W can liminat th lctrochmical potntials from quations (.3) and (.3) and xprss th spin currnt as a function of th charg currnt: j s (0) = (α ) j σ kt α ( α) Λ j (.4) For lctron injction w hav j < 0. Equation (.4) is valid providd that th currnt injction is dominatd by thrmionic mission and its tim rvrs, intrfac rcombination. From quation (.4), w can draw two conclusions. First, whn α tnds to on, th spin currnt approachs th charg currnt. Scond, if th ratio σ/λ for th contact matrial is sufficintly small, th spin currnt is qual to (α -)j, which is th spin currnt in th bulk of th FM contact. Hnc, in ordr to achiv significant spin injction w nd ithr strongly polarizd contacts ( α ) or a contact matrial that has small σ/λ. For convntional FM mtal contacts, whr th spin polarization is not clos to on and th σ/λ ratio is larg, quation (.4), can b furthr approximatd by j s (0) (α ) j = j σ kt α ( α) Λ (.5) Hr j s (0)/j is th spin polarization, SP, of th injctd currnt. 7

30 .5 Numrical rsults for thrmionic mission W considrd an organic smiconductor, 00 nm thick, sandwichd btwn a FM contact and a non-magntic mtal contact. Th FM contact forms a Schottky barrir of 0.3 V with th smiconductor. A contact mad from colossal magnto-rsistanc manganats, SMO, is also considrd. SMO acts narly as half mtal, whr α. Th transport paramtrs adoptd for a FM mtal, SMO, and th organic smiconductor ar listd in tabl.. Tabl. Transport paramtrs σ (S/cm) α Λ (cm) FM mtal CMR half-mtal Organic Smiconductor E g (V) μ (cm /Vs) N 0 (cm -3 ) First, w considr spin injction from a FM mtal contact into an organic smiconductor. Figur. shows th calculatd injctd charg and spin currnt dnsitis as a function of th voltag applid. Th charg currnt dnsity is constant throughout th structur as rquird by charg consrvation, but th spin currnt dnsity varis du to spin rlaxation. Th spin currnt dnsity shown in figur. is valuatd at 8

31 th lctron injcting contact. Th spin currnt dnsity is much smallr than th charg currnt dnsity du to th high lctrical conductivity and short spin diffusion lngth of th contact mtal. Th dottd lin shows th approximat spin currnt dnsity obtaind from th charg currnt dnsity via Eq. (.5). Th dottd lin ssntially coincids with th dashd lin (numrically calculatd rsult), indicating that th spin currnt is to a good approximation proportional to th squar of th charg currnt. This parabolic dpndnc of spin currnt on charg currnt can b xplaind by th larg conductivity mismatch btwn th mtallic contact and th organic smiconductor. 0 3 Currnt dnsity (A/cm ) Applid bias (V) Figur.. Calculatd charg (solid), calculatd spin (dashd), and approximat spin (dottd Eq..5) currnt dnsitis as a function of th applid voltag across th dvic structur with a FM mtal injcting contact. 9

32 At low bias, charg carrir injction is low, implying that th conductivity of th smiconductor is low. As th bias voltag incrass, th charg carrir injction incrass, which incrass th smiconductor conductivity and rducs th conductivity mismatch btwn th smiconductor and th FM mtal, and spin currnt incrass with th charg currnt. Nxt, w considr spin injction from SMO. Figur.3 displays th calculatd injctd charg and spin currnt dnsitis from SMO contacts into an organic smiconductor for diffrnt contact polarizations, α. Again th spin currnt dnsity shown in figur.3 is takn at th lctron injcting contact. Evidntly, spin injction is gratly nhancd only whn th contact is narly half-mtallic. Two paramtrs assist high spin injction. On is a low conductivity of th contact, which for half-mtals lik SMO is almost four ordrs magnitud smallr than that of convntional FM mtals. Th othr is th high contact polarization of th half-mtallic contact, which is clos to unity. From figur.3, w can s that, as w incras α, th spin currnt incrass and tnds towards th charg currnt. It is also vidnt that α nds to approach unity vry closly in ordr to b ffctiv in nabling strong spin injction. 0

33 0 3 Currnt dnsity (A/cm ) α = α = α = α = Charg currnt Spin currnt Bias (V) Figur.3. Calculatd charg (solid) and spin (dashd) currnt dnsitis as a function of th applid voltag across th dvic structur with SMO as th injcting contact. Contact polarization, α is usd as a paramtr. W nxt xplor how th spin currnt prdictd by quation (.4) matchs with th calculatd spin currnt for SMO injcting contact. W plot th numrically calculatd spin currnt and q. (.4) as a function of th injctd charg currnt in figur.4. Th simpl analytical rsults (dottd lins) nicly match th calculatd spin currnts (dashd lins) for all valus of α. This rsult indicats that th analytical xprssion givn by Eq. (.4) is a good approximation of th injctd spin currnt for any FM contacts providd that th currnt injction is dominatd by thrmionic mission.

34 0 3 0 Spin currnt (A/cm ) α = α = α = α = Charg currnt (A/cm ) Figur.4. Numrically calculatd spin currnt (dashd) and analytical spin currnt (dottd) projctd from th charg currnt (Eq..4) as a function of th injctd charg currnt for SMO injcting contact. Contact polarization, α is usd as a paramtr..6 Injction by tunnlling To modl injction by tunnlling, w may nvision a thin insulating tunnl layr btwn th FM contact and th smiconductor as shown in figur.5. Elctron tunnlling from a FM contact through this layr is spin slctiv bcaus th spatial part of lctron wav function is diffrnt for majority and minority spin dirction. Th

35 tunnlling procss through th insulating layr can b dscribd by spin dpndnt contact rsistancs, r and r, for th SU and SD lctrons rspctivly. W assum that th tunnl layr is mor transparnt for th majority spin lctrons such that r < r. μ μ Φ B E c -δ 0 x Figur.5. Schmatic nrgy band diagram for a contact btwn a frromagntic mtal and a smiconductor through a thin insulating tunnl layr undr bias. If thr is no spin scattring in th insulating layr, th currnt for ach spin dirction is continuous and is rlatd to th spin dpndnt lctrochmical potntial chang across th tunnl layr via Ohm s law, giving ris to th following boundary conditions: μ ( 0) μ ( δ ) = r j ( δ ) (.6a) μ ( 0) μ ( δ ) = r j ( δ ) (.6b) 3

36 j δ ) = j (0) (.6c) s ( s Equations.6(a)-(c) can b usd to calculat th diffrnc of th quasi-frmi lvls at th insulator/smiconductor intrfac, and it is givn by: Δμ ( 0) = Δμ( δ ) ( r r ) j ( r r ) js (0) (.7) Hr w allow for a possibl additional discontinuity Δμ(-δ) in th quasi-frmi lvls at th contact/insulator intrfac as it occurs for xampl if carrir injction is limitd by thrmionic mission in th absnc of th tunnl barrir. It is convnint to combin th ffcts of th FM mtal and th tunnl contact and to xprss th polarization ffct in th smiconductor in trms of Δμ(0) Δμ ( 0) = ( r r ) j ( r r ) js (0) (.8) whr th ffctiv rsistancs ar dfind by: (α ) Λ r r = r r r r α( α) σ (.9a) Λ r r = r r r r α ( α) σ (.9b) bcaus Λ/σ tnds to b vry small for convntional frromagntic mtals on th scal of th contact rsistancs. Onc th diffrnc of th quasi-frmi lvls ar dfind in th 4

37 form of quation (.8), w can calculat th injctd charg and spin currnts using th modl dvlopd in sction.4, i.. by using quations (.9)-(.)..7 Numrical rsults for tunnl injction Th spin currnt and th charg currnt in th prsnc of spin dpndnt contact rsistancs ar plottd in figur.6. Th dvic paramtrs ar th sam as discussd in th cas of thrmionic mission. Th valus of th spin slctiv contact rsistancs ar: r = Ω cm and r = 0 - Ω cm, such that r / r = /. In th plot, w xclud th voltag drop across th contact rsistanc, as it is ngligibl compard to th ovrall applid bias. At thrmal quilibrium, lctron spins ar polarizd in th FM contact but unpolarizd in th smiconductor. In th absnc of contact rsistancs, th lctrons in th mtal and in th smiconductor ar in good thrmal contact, and thrfor th lctrons in th smiconductor stay clos to local thrmal quilibrium. In ordr to achiv ffctiv spin injction, this quasi-quilibration must b supprssd. Th spin dpndnt contact rsistanc braks th quasi-quilibration btwn th organic smiconductor and th mtallic contact and gratly nhancs spin injction from th mtallic contact into th smiconductor as vidnt in figur.6. 5

38 0 Currnt dnsity (A/cm ) Bias voltag across th smiconductor (V) Figur.6. Injctd charg (solid) and spin (dashd) currnt dnsitis from a FM mtal contact into th organic smiconductor in th prsnc of spin slctiv contact rsistancs. Th valus of th contact rsistancs ar r = Ω cm and r = 0 - Ω cm. Nxt w xplor how th magnitud and th ratio of th spin dpndnt contact rsistancs affct spin injction. W can control th spin polarization, j s (0)/j, of th injctd currnt by controlling th spin dpndnt contact rsistancs, r and r. At a givn lvl of charg injction, spin injction can b incrasd ithr by incrasing th contact rsistancs (whil kping th ratio r / r constant) or by dcrasing th ratio r / r (whil kping r constant). Figur.7 shows th rlativ SP for diffrnt combinations of contact rsistancs. Again, th bias voltag is takn only across th 6

39 organic smiconductor in ordr to indicat a givn lvl of charg injction. Charg injction is th sam in all cass at a givn bias across th smiconductor. From figur.7, w can conclud that spin injction can b incrasd ithr by incrasing th contact rsistancs (whil kping r / r constant) or by dcrasing th ratio of r / r (whil kping r constant). Dcrasing th ratio of r / r has a mor pronouncd ffct on SP ovr incrasing th magnitud of th rsistancs. Morovr, incrasing th contact rsistancs incrass th voltag drop across th contacts. For a fixd ratio r / r, incrasing th contact rsistancs incrass th spin polarization significantly whn th bias voltag is low. For larg biass, incrasing th contact rsistancs with fixd r / r ratio has ngligibl ffct on spin polarization..0 SP, j s (0)/j A B C D E F r /r = 0. r /r = r /r = Bias voltag across th smiconductor (V) Figur.7. SP, j s (0)/j, for diffrnt combinations of spin slctiv contact rsistancs. Th valus of th contact rsistancs ar giv in tabl.. 7

40 Tabl.. Spin slctiv contact rsistancs for th diffrnt plots shown in figur.7 Plot r (Ω cm ) r (Ω cm ) r / r A B C D E F Conclusions W may summariz ths rsults as follows: thrmionic mission cannot injct spin polarizd charg carrirs from a convntional frromagntic mtal contact into organic smiconductors, and ffcts such as imag charg inducd barrir lowring do not altr that conclusion. Th physical rason for infficint spin injction is as follows. At thrmal quilibrium, th FM contact is spin polarizd but th smiconductor is unpolarizd. For a FM mtal/smiconductor junction, th lctrons in th mtal and in th smiconductor ar in good contact, and thrfor th lctrons in th smiconductor stay clos to local thrmal quilibrium. As a rsult, th charg carrirs in th smiconductor ar only wakly spin polarizd, but spin polarization incrass slowly with 8

41 incrasing bias, i.. whn th smiconductor is drivn furthr out of quilibrium. In ordr to driv th smiconductor far out of quilibrium, w nd to injct a high charg currnt dnsity; within practical valus for th injctd charg currnt dnsity, th spin currnt dnsity rmains ordrs magnitud smallr than th charg currnt dnsity. On th othr hand an SMO contact may nhanc spin injction. Howvr, spin injction from th SMO contact is apprciabl only if th contact is indd vry clos to half mtallic. In othr words spin polarization of th contact should b %, which is practically impossibl at a rasonabl tmpratur. Spin injction is gratly nhancd if thr is a spin dpndnt contact rsistanc btwn th mtallic contact and th smiconductor. An insulating tunnl barrir with a spin polarizd FM contact has spin dpndnt intrfac rsistanc bcaus of th diffrnc in Frmi wav vctors for th two spin typs in th contact matrial. A FM insulator tunnl barrir can also provid spin dpndnt contact rsistanc. 55 Th spin dpndnt contact rsistanc braks th quasi-quilibration btwn th organic smiconductor and th FM contact and significantly nhancs spin injction. 9

42 Chaptr 3 Spin transport and its dtction in organic spin valvs A thory of spin injction was dvlopd in th prvious chaptr. W showd that a spin dpndnt contact rsistanc (tunnl barrirs) at th injcting contact can gratly nhanc spin injction. Th rsulting spin polarization is, in principl, dtctabl through th xtracting contact via th MR of a spin valv. In this chaptr w will xplor spin transport through th organic smiconductor and dtction of th spin polarization. 3. Dvic structurs Th dvic structurs w will considr in this chaptr consist of an organic smiconductor of thicknss sandwichd btwn two FM contacts a dvic commonly known as an organic spin valv. Fig. shows such a schmatic dvic structur. In this spin valv gomtry nvisiond, th lft lctrod is th injcting contact and right lctrod is th xtracting contact. Th polarization of th lft lctrod is always in th up-dirction. For paralll alignmnt of contact magntization, th polarization of th right lctrod is in th up-dirction, whras for anti-paralll alignmnt, th polarization of th right lctrod is in th down-dirction. Th spin 30

43 dpndnt contact rsistancs ar such that for spin up polarization of th contact, r < r, and vic vrsa. -δ 0 δ r l < r l r r > r r Figur 3.. A schmatic dvic structur consisting of an organic smiconductor sandwichd btwn two FM contacts with anti-paralll contact magntizations. Th ffcts of th tunnl barrirs ar dscribd by spin dpndnt contact rsistancs. 3. Splitting of quasi-frmi lvls at th contacts Th spin dpndnt contact rsistancs incras th splitting btwn th quasi- Frmi lvls at th smiconductor/tunnl barrir intrfac as discussd in sction.6. Th splitting at th injcting and xtracting contacts ar givn by 56 Δμ ( 0) = ( r r ) j ( r r ) js (0) l l l l (3.a) Δμ ( ) = ( r r ) j ( r r ) js ( ) r r r r (3.b) Hr l and r idntify th lft and right contacts, rspctivly. 3

44 3.3 Transport in th smiconductor Undr stady stat conditions, th carrir transport in th organic smiconductor is govrnd by th spin dpndnt continuity quations, d kt dn n n μ n F = 0, dx dx τ s (3.a) d kt dn n n μ n F = 0, dx dx τ s (3.b) whr th SU and SD currnts ar xprssd in drift-diffusion approximation. Hr n, ar th spin dpndnt lctron dnsitis, F is th lctric fild, k is Boltzmann s constant, and T is th tmpratur. Th carrir mobility and spin rlaxation tim ar dnotd by μ and τ s, rspctivly. Non-dgnrat carrir statistics is assumd for th organic smiconductor. Th continuity quations ar to b solvd togthr with Poisson s quation, df dx = π ( n n ) / ε 4 (3.3) whr ε is th prmittivity of th smiconductor. It is instructiv first to considr a cas in which an ssntially analytical solution to th transport problm may b obtaind. W mak th following approximations: (i) τ s 3

45 is long compard to th transit tim, i.. th last trms in quations (3.a) and (3.b) ar nglctd; (ii) th ffct of th injctd spac charg is nglctd, i.. th lctric fild is constant throughout th organic smiconductor; and (iii) ohmic boundary conditions apply, i.. μ and μ ar continuous at x = 0 and x =. Th rsulting charg currnt and spin currnt in th organic smiconductor can thn b xprssd as, V j = F( V ) xp kt Δμ(0) Δμ( ) cosh cosh kt kt V Δμ(0) Δμ( ) j s = F( V ) xp sinh sinh kt kt kt (3.4a) (3.4b) V is th voltag droppd across th smiconductor, and F(V ) is givn by, F( V ) = V V Φ b B; μ N 0 xp kt V Vb xp kt r (3.5) Hr V b is th built-in potntial, (Φ B;r - Φ B;l ) and Φ B;r(l) ar th barrir hights of th right (lft) contact. W assum τ s >> /(μkt/), hnc th injctd spin currnt dnsity is constant throughout th smiconductor. From Eqs. (3.4a) and (3.4b), w s that if V >> kt, th charg and spin currnts dpnd only on th diffrnc of th quasi-frmi lvls at th lft (injcting) lctrod. Howvr, in th low bias rgim, both th injcting and xtracting contacts control th currnts in th smiconductor. Equations (3.) and (3.4) 33

46 can b solvd slf-consistntly for a particular bias, V. Th total voltag applid to th dvic is thn obtaind from, V = V ΔV l ΔV r (3.6a) Δ V = / 4)[( r r ) j ( r r ) j ] (3.6b) l, r ( l, r l, r l, r l, r s Spin dpndnt contact rsistancs gratly nhanc spin injction and th rsulting spin polarization can b xtractd via th MR of th organic spin valv. Th MR is dfind as th diffrnc of th rsistancs btwn anti-paralll (AP) and paralll (P) alignmnts of th contact magntizations. Th P and AP configurations ar xprssd through th trms involving r r r, which hav th sam sign as r r in P r l l configuration, but opposit sign in AP configuration. Thus, th currnts and total applid voltags for th two contact alignmnts ar obtaind as a function of V. Finally, th MR is dfind as, VAP jp MR = 00% VP j AP (3.7) Hr V AP and V P dnot th applid biass for th P and AP configurations, and j P and j AP ar th corrsponding charg currnt dnsitis. 34

47 3.4 Rsults without spin rlaxation W us th calculation outlind abov for rlativly thin organic smiconductor layrs whr spac charg ffcts du to th injctd carrirs ar small undr low bias. Th thicknss is takn to b 00nm, and th mobility is 0 - cm /Vs. Th spin rlaxation tim thus nds to b much gratr than 400ns to b consistnt with our approximations abov. For simplicity, w tak V b = 0, and choos a barrir hight of 0.V. All calculations ar for room tmpratur conditions. 4 Currnt dnsity (A/cm ) 3 P P AP AP Applid bias (V) Figur 3.. Charg currnt dnsity (solid curvs) and spin currnt dnsity (dashd curvs) for P and AP contact magntizations as a function of applid bias. Th ratios of for both lft and right contacts ar ½ and r = 0 Ωcm r. = l r r / r 35

48 Figur 3. shows th calculatd charg and spin currnts vrsus total applid voltag for P and AP alignmnts. Th ratios r / r for both lft and right contacts ar ½ and r = 0 Ωcm = l r r. W also considr cass in which th contact rsistanc valus of th injcting and xtracting contacts ar diffrnt. Th currnt polarizations for thr cass ar dpictd in figur P j s /j AP Applid bias (V) Figur 3.3. Spin polarization, j s /j, for P and AP contact alignmnts and diffrnt ffctiv contact rsistancs. Th ratios of r / r for both lft and right contacts ar ½ in all cass. Th magnituds of th contact rsistancs ar r = 0 Ωcm r (solid lins), = l r r 0 Ωcm = l = 0r r (dashd lins), and 0r = 0 Ωcm = l r r (dottd lins). 36

49 Hr th rsults for ffctiv contact rsistancs ar compard with thos obtaind for rducd ffctiv rsistancs on th injcting or th collcting sid. It can b sn that in th non-symmtric situations vn in th AP alignmnt cas th currnt polarization xtrapolats to a non-zro valu at vanishing bias. If th ffctiv contact rsistanc of th xtracting contact is largr than that of th injcting contact, it tnds to dtrmin th spin polarization at low bias, which in th sign convntion adoptd hr is ngativ. Figur 3.4 shows th calculatd MR valus for thr sts of ffctiv contact rsistancs. It is vidnt that th MR may dcras with incrasing bias if th ffctiv contact rsistancs of th injcting contact ar smallr than thos of th xtracting contact. 0 5 MR (%) Applid bias (V) Figur 3.4. MR plottd as a function of th applid bias for thr cass of diffrnt ffctiv contact rsistancs as shown in figur