Calculation method of spin accumulations and spin signals in nanostructures using spin resistors

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1 Clcultion method of pin ccumultion nd pin ignl in nnotructure uing pin reitor W. Svero Torre, A. Mrty*, P. Lczkowki, L. Vil, M. Jmet nd J-P. Attné Univerité Grenole Alpe, IAC-SPM, -38 Grenole, rnce CEA, IAC-SPM, -38 Grenole, rnce Atrct The undertnding nd clcultion of pin trnport re eentil element for the development of pintronic device. Here, we propoe imple method to clculte nlyticlly the pin ccumultion, pin current nd mgnetoreitnce in complex ytem. Thi cn e ued oth for CPP experiment in multilyer nd for multiterminl nnotructure mde of emiconductor, oxide, metl nd cron llotrope. I.- Introduction The development of pin-ed device relie on the injection nd mnipultion of pin current in nnocle ytem. Therefore, the clcultion nd the undertnding of pin trnport t the nnocle i eentil to optimize pintronic device. In thi context, everl pproche to clculte pin current nd pin ccumultion hve een propoed in the lt decde. The firt nlyticl clcultion of pin ccumultion w provided y vn Son et l. in the ce of ferromgnet/norml metl interfce. Vlet nd ert then derived the oltzmnn eqution to provide the pin ccumultion lndcpe in current perpendiculr to the plne CPP experiment. Thi generl model w rpidly extended to the ce of emiconductor 3, providing the min condition to optimize pin injection in uch mteril. Lter on, Tkhhi nd Mekw 4 ued imilr pproch to explin the pin trnport in lterl pin vlve, which w then refined y Hmrle et l. y including three dimenionl effect 5, nd ued y Kimur et l. to etimte the pin diffuion length y tking into ccount the pin ink in multi terminl device. 6 It i worth mention tht cn lo een clculted uing mtrix trnfer pproche 7,8 Spin-dependent diffuion eqution re differentil eqution tht cn e olved tep y tep uing oundry nd continuity eqution for chrge nd pin current t ech interfce. However, the clcultion ecome quickly hevy in complex tructure compriing everl element. Here, we propoe imple nd ytemtic method to derive mthemticl expreion of the mgnetoreitnce in complex ytem. It cn e ued oth for CPP experiment in multilyer, nd for multi-terminl

2 nnotructure mde of emiconductor, oxide, metl nd cron llotrope. In the following, we will explin thi method, nd illutrte it implicity y clculting the mgnetoreitnce of well-known ytem. Then, we will how how it cn e generlized nd pplied to ytem poeing more complex geometrie. II.- Method Thi method primrily ddree the pin trnport in multi-terminl pintronic tructure, typiclly compoed of everl nnowire connected through trnprent or nontrnprent interfce.,,4,9, Such ytem cn e decried y network of pin reitor connected y node, where ech node correpond to trnprent interfce. The concept of pin reitor i inpired from electric reitor concept, nd concern pin current trnport., Ech wire nd ech non-trnprent interfce of the nnotructure or ech lyer in the ce of CPP trnport correpond to pin reitor cf. ig. nd. A the ojective i to decrie the trnport propertie current ource i ued to inject n electricl current etween two node of the circuit, while meter detect the ge etween two node not necerily identicl to the former. An elementry wire i chrcterized y it mteril propertie electricl conductivity σ, pin polriztion β, nd pin diffuion length λ nd y it geometry contnt cro ection of re A nd length L. Within the wire, the flow of n electricl current I c nd of pin current I tke plce due to grdient of electrochemicl potentil ee cf. ig,. To provide n expreion of the pin current long the wire, the electrochemicl potentil μx will e expreed herefter in ge unit, nd the pin dependent prmeter will e denoted y the ucript ± for pin up nd - for pin down. Uing thi nottion, the up nd down current denitie long the pin reitor cn e expreed : ig.. Schemtic repreenttion of the pin ccumultion long the wire for n electricl current pplied long the x direction, nd ymol of the correponding pin reitor. c Sketch howing the electrochemicl potentil lndcpe in unit ner the interfce of two mteril poeing polrition β nd β β > β. The digonl lck line repreent the electric potentil in ech mteril, which exhiit drop t the interfce V produced y the pin ccumultion. The red nd lue curve repreent the electrochemicl potentil of the mjority nd minority pin popultion in ech mteril repectively.

3 J β β σ σ ; J Where μ± ccount for the electrochemicl potentil for pin up nd down popultion, nd where the pin polriztion i defined the ymmetry of the conductivitie, σ σ β. The verge potentil m σ σ nd the pin ccumultion re: ; m Then the chrge nd pin current denity cn e written follow: J J J σ m β σ V... J J J σ β βj... c Eq. how tht the chrge current depend on oth the verge electrochemicl potentil nd the pin ccumultion through the grdient of electric like potentil Vx m β. While electrochemicl potentil nd therefore the verge potentil nd pin ccumultion re continuou t trnprent interfce, thi electric like potentil i not continuou. At the interfce etween two mteril with different polriztion β nd β, potentil drop etween the mteril pper due to the polriztion chnge cf. ig c: β β V V Additionlly, Eq. how tht the totl pin current flowing long the wire i given y the um of two contriution. The firt contriution rie from the pure pin current, which i proportionl to the grdient of the pin ccumultion, while the econd one rie from the pin polrized current. It i worth noting tht c in the ce of non-mgnetic mteril the only contriution come from the pure pin current, ince there i no pin polriztion β. Unlike chrge current, pin current i not conervtive due to pin flip procee. Thi led to the differentil eqution governing the pin ccumultion:... 3 λ Where, λ i the pin diffuion length. With thi econd order liner differentil eqution, the pin ccumultion long the wire cn e expreed liner comintion of the pin ccumultion t the extremitie: L x x inh inh x λ λ L...4 L L inh inh λ λ Let u define the pin reitnce of the pin reitor : ρλ... 5 β A Where ρ i the uul electric reitivity of the wire. Uing Eq., 4 nd 5, the pin current t the extremitie of the pin reitor cn e written : ote tht in the literture "pin reitnce" ometime denote ρλ β A, ometime A ρλ, with eventully the ue of tr to ditinguih ferromgnet from nonferromgnet. or the ke of clrity we will in the following ue the generl definition of Eq. 5, with β in norml metl.

4 I I L L βi tnh inh...6 L βi c... 7 inh tnh Here, the dimenionle prmeter,, defined L / λ, chrcterie the pin-flip rtio within the pin reitor. Unlike electric reitor, where chrge current i contnt long the wire, in pin reitor, the pin current i not the me t the two extremitie. Eq. 6 nd 7 provide the generl expreion for the pin current when the pin reitor i connected on oth ide to other pin reitor, nd when it length i comprle with it pin diffuion length L~λ. However, uch reltion cn e implified when conidering one of the three following ce. When one extremity of the pin reitor, y L i unconnected, there i no chrge current flowing within the wire, therefore the pin current t thi unconnected extremity vnihe well the pin ccumultion grdient : μ. To tify thi condition, the olution of Eq. 3 hould e rewritten : L x coh x λ L coh λ Then uing Eq. nd 5 the pin current in tht ce i given y: tnh I When L>>λ i.e., Eq. 6 i reduced to: c I βi c ote tht the pin current t one extremity doe not depend on the pin ccumultion t the other extremity. In experiment, uch condition i required for the long contct proe. 3 When L<<λ, i.e. there i no pin flip in the wire nd Eq. 6 cn e written: β I r L βic where r-β i the electric reitnce of the wire. In the ce of non-trnprent interfce etween two wire for intnce ecue of the preence of tunnel rrier, 3,4,5,6 the interfce h to e conidered pin reitor connecting the two node, chrcteried y n electricl reitnce r, pin polriztion coefficient 7 nd pin flip rtio. The pin reitnce of the rrier i then defined : r ig. how n exmple of tructure compoed y everl mgnetic nd non mgnetic lyer. A decried ove, it cn e repreented y network of pin reitor connected y node numered red circle. The contct proe for the current ource nd the meter re not conidered node ince they re fr from the pin device nd uppoed to exhiit zero pin ccumultion.

5 ig. Schemtic repreenttion of multiterminl nnotructure nd it equivlent pin reitor repreenttion. The different color correpond to different mteril nd the red circle re node chrcterizing trnprent interfce. The lck rrow repreent the current nd ge pth long the tructure. To determine the pin ccumultion t ny node of thi ytem, we conider the conervtion of pin current t ech node. Hence, the ppliction of eqution 6 nd 7 to the ytem provide et of liner eqution tht cn e expreed imply uing mtrix formultion of the form: cur I c A Where i n-dimenionl vector correponding to the lit of pin ccumultion vlue t the n node. The mtrix A nd vector re detiled elow. In given proe configurtion, the pin ccumultion effect of the tructure cn e quntified through the evlution of the pin ccumultion reitnce ΣV /I c. I c i the chrge current pplied to the tructure nd ΣV i the um of the ge drop occurring t the node elonging to the loop including the meter. ote tht in non-locl proe configurtion the pin ccumultion reitnce i uully clled the pin ignl, where in locl meurement it correpond to the mgnetoreitnce. In mot experiment, thi pin ccumultion reitnce i meured function of n pplied field, it depend on the orienttion of the mgnetiztion in the ferromgnetic wire. In thi pproch, the pin ccumultion reitnce for mgnetic tte of the tructure i given y the iliner form : A cur...8 In Eq. 8, cur i n-dimenionl vector whoe element depend on the pth of the chrge current. It i th component cn e determined follow: -if there i no chrge current ping t the i th node, thi component i equl to -if chrge current flow from the reitor p to the reitor q through the i th node, the i th component of cur i given y β p - β q i the correponding vector long the ge pth, i.e., the prt of the circuit forming loop with the

6 meter. It component cn e determined follow: - If the i th node doe not elong to the ge pth, the i th component of i - If the ge pth goe from the reitor p to the reitor q through the i th node, the i th component of i given y β p -β q. ote tht cur nd depend on the mgnetic tte of the tructure: β p i poitive rep. negtive if the mgnetiztion of the ferromgnetic element p point towrd the direction correponding to the up rep. down pin. In ddition, when the pin reitor correpond to tunnel rrier, the pin polriztion β i uully leled with the letter. A i nxn mtrix chrcterizing the tructure of the pin reitor lttice. The mtrix element ij re contructed from eqution. 6 nd 7 in the following wy: I.- Digonl element ii : The ii element i derived y conidering the i th node. It i equl to the um of the following term: tnh for ech pin reitor connected to the i th node t one end nd to nother node t hi other end e.g., 4 nd 5 in fig. for ech pin reitor connected to the i th node t one end nd uch L>>λ. Thi i uully the ce for long electricl contct. tnh for ech pin reitor connected only to the to the i th node, the other end of the reitor eing unconnected. II.- on digonl element ij : The mtrix eing ymmetric y contruction, only hlf of the non digonl element hve to e clculted. The ij element i equl to : when the node i inh nd j re connected through ingle reitor e.g., the node nd in ig when the node i nd j re not directly connected e.g., the node nd 3 in ig. inlly, the pin ignl mplitude Δ, which i the vrition of the pin ignl etween two tte of mgnetiztion of the tructure I nd II, i given y: III.- Appliction I In the following we will how how our method to clculte the pin ignl mplitude Δ cn e pplied to everl well-known ytem nd more complex one II.-/ junction.- Junction with trnprent interfce

7 Thi tructure, firt tudied y Johnon nd Silee 8 nd Vn Son et l., provide the implet ytem where pin ccumultion crete reitnce contriution. It cn e repreented y two pin reitor, which re connected to one node t one extremity nd to the electricl contct t the other ee cf. ig 3..- Junction with inerted tunnel rrier // The inertion of tunnel rrier t / interfce i imple ytem tht i extenively ued to incree the pin ccumultion ner the / interfce, 3 leding to higher mgnetoreitnce nd pin trnfer torque, 9 nd thu llowing to develop competitive memory nd logic ppliction., ig.3 Schemtic repreenttion of trnprent / junction nd it equivlent pin reitor repreenttion. According to the mtrix contruction method explined previouly, there i only one node the mtrix A i clr whoe vlue i given y: A A the current flow nd ge proe follow the me pth,, nd cur re equl: β. Therefore, cur the pin ccumultion reitnce in thi tructure i imply given y: β cur A β β Thi i in greement with the reult otined y Vn Son et l. ig.4 Schemtic repreenttion of // junction nd it equivlent pin reitor repreenttion The tunnel rrier i repreented pin reitor, chrcterized y the pin flip prmeter, the polriztion nd y the rrier pin reitnce cf. ig. 4,. A there re two node, A i x mtrix tht cn e contructed uing the rule propoed previouly: tnh A inh inh tnh

8 A the current nd ge follow the me pth, the vector nd cur re lo equl nd given y: cur β where β- nd re the chnge of polriztion t the / nd / interfce, repectively. Therefore, fter n lgeric reorgniztion, the pin ccumultion reitnce of thi tructure cn e written : β tnh β β tnh tnh Thi expreion cn e implified in the ce of rrier without pin flip. It tke the form: r r β r β ig.5 Schemtic repreenttion of // junction nd it equivlent pin reitor repreenttion To clculte the pin ignl in thi tructure, let u tke into ccount the pin reitor repreenttion howed in ig. 5. A the tructure poee two node, A i x mtrix with element given y: tnh A inh inh tnh where r i the electricl reitnce of the rrier. c.-doule junction // In thi prt, the method i pplied to the ce of doule junction with trnprent / interfce. Thi ytem, commonly known pin vlve,, i the implet multilyer device exhiiting Gint Mgneto eitnce GM. 3 The current nd ge proe follow the me pth, therefore when oth mgnetiztion re prllel the ocited vector re given y: cur β β Where, when they re ntiprllel: cur β β The pin ignl mplitude i AP P, where nd re P AP the pin ccumultion reitnce in the prllel nd ntiprllel tte, nd re clculted uing Eq. 8.

9 It led to 8β e e which i equivlent to the expreion otined in ef nd 7..- Lterl pin vlve Let u conider lterl pin vlve, 9,8 with inerted tunnel junction t the interfce, in order to clculte the nlyticl expreion of the pin ignl cf. ig. 6. ig. 7 Schemtic repreenttion of lterl pin vlve with inerted tunnel junction nd the equivlent pin reitor repreenttion ollowing the propoed contruction rule, The A mtrix i: tnh inh A inh tnh inh tnh inh tnh inh tnh inh tnh Here, the current nd the ge follow different pth, nd therefore the vector cur nd re different. When the mgnetiztion of the electrode re prllel, thee vector re given y: β ; cur β A the current nd ge pth involve only two node ee ig 5, their correponding vector poe only two non vnihing element. Here gin, the pin ignl mplitude i P AP determined from: where the vrition of the mgnetic tte i tken into ccount y chnging the pin polrition from β-β in the lyer nd from Ɣ-Ɣ in the rrier. The complete nlyticl expreion cn e otined y pplying Eq. 8 giving rther lrge expreion:

10 4 h ch e h ch e ch h β If we retrict our nlyi to the ce where there i no pin flip t the interfce i.e., the pin ignl mplitude i given y: e e r r β 4 ow, y conidering the limit ce, we cn eily otin the following reult: Exp when r >> nd h r when << r << h β when r << << The expreion otined in the implified ce re in greement with thoe otined y Tkhhi et Mekw 4, nd hve een ued in experimentl ytem to chrcterize the pin-dependent propertie in non mgnetic mteril 5,9,4, More complex geometrie Up to now we hve tudied imple nd well known ytem, in order to check the enefit of thi pproch, nd to underline the fcility of the propoed method to derive mthemticl expreion. In the following, we will how how it cn e ued in more complex ce, conidering n exmple tructure with four ferromgnetic wire with inerted tunnel junction. ig. 7 Schemtic repreenttion of complex geometry compoed y everl ferromgnetic wire nd tunnel junction. Sketch howing the pin reitor repreenttion.

11 In tht ce ll the ferromgnetic element nd ll the tunnel junction re uppoed to e identicl. The node hving imilr urrounding cn e gthered into three group; {,7}, {3,5} nd {,4,6,8}. Symmetrie of thi geometry thu imply tht the 8x8 mtrix A cn e written uing only five independent coefficient: 3 A With: 33 3 tnh inh tnh tnh tnh ; ; inh 3 ; tnh Then, to write nd cur, one h to conider the current nd ge pth howed in ig. 7. If the four ferromgnetic electrode poe prllel mgnetiztion, cn e written : β β, cur cur nd β β ; When conidering other mgnetic configurtion of the electrode, cur nd cn contructed y replcing β y -β nd Ɣ y Ɣ for the electrode with revered mgnetiztion. ote tht in thi rticle we focued on the clcultion of the pin ccumultion reitnce given y Eq. 8, thi reitnce i uully the mot importnt prmeter in experiment. ow, let u note S the vector whoe i th component i the pin ccumultion t the i th node. The knowledge of otined y uing: S I C A cur S i, nd the whole pin ccumultion lndcpe cn then e clculted uing Eq. 4. IV.- Concluion The propoed method llow thu to otin n nlyticl expreion of the pin ccumultion nd of the pin ignl in ny complex tructure. Still, there re three limittion to thi method. The firt one i tht of the Vlet-ert model: It pplie only to ytem with colliner mgnetiztion i.e., pin cn only e up or down. The econd one i common to ll exiting nlyticl reolution of the Vlet-ert differentil eqution: the clcultion i one-dimenionl, in the ene tht the electrochemicl potentil depend only of the poition long the length of the wire. The lt limittion come from our will to del with pin ccumultion lone: in thi model the chrge current mut e known prior to clcultion. Thi i poile if the impoed current follow unique pth, in the ene tht there i no cloed loop in the circuit. ote

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