Numerical study of the lattice vacancy effects on the quantum transport of four-terminal graphene nanodevice

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1 Idia Joural of Pur & Applid Physics Vol. 52, Octobr 2014, pp Numrical study of th lattic vacacy ffcts o th quatum trasport of four-trmial graph aodvic A Jafari*, M Ghoraviss, M R Hathzadh & R Alipour 1 Plasma Physics Rsarch Ctr, Scic ad Rsarch Campus, Islamic Azad Uivrsity, Thra, Ira * jafari_82@yahoo.com Rcivd 21 Jauary 2014; rvisd 21 March 2014; accptd 20 August 2014 Th lctroic trasport proprtis of four-trmial graph aodvic hav b ivstigatd by mas of th Ladaur approach usig tight-bidig modl. Th ffcts of sigl vacacy ad divacacy o trasport proprtis of th lctro ijctd ito th systm hav b studid. Th xistc of sigl vacacy alog with th magtic fild crats additioal bad btw Ladau lvls, this ffct is appard by hacig th trasmissio. It is also foud that i th prsc of divacacy with small distac, th couplig btw two vacacis du "vacacy molculs", that th bodig btw ths vacacis ca b tud by th magtic fild. Th ffct of vacacis dcrass wh th siz of coductor rgio icrass. Th thortical rsults obtaid, ca b a bas for dvlopmt i dsigig graph aodvic. Kywords: Four-trmial graph aodvic, Trasport proprtis, Vacacy, Divacacy 1 Itroductio Rctly, graph shts wr succssfully isolatd ad dmostratd to b stabl udr ambit coditios 1. Th E-k rlatio is liar (E=±h v f ) for low rgis ar th six corrs of two-dimsioal hxagoal Brilloui zo, ladig to zro ffctiv mass for lctros ad hols 1,2 Graph has th idal proprtis to b a xcllt compot of itgratd circuits. It has a high carrir mobility, as wll as low ois, allowig it to b usd as th chal i a fildffct trasistor (FET). It may b a suitabl matrial for th costructio of quatum computrs usig lctroic circuits 3. Th uusual trasport proprtis of graph aris from its liar E-K rlatio from low rgis ar th six corrs of two-dimsioal hxagoal Brilloui zo 4. Bsid th high mobility ad miimum coductivity, graph displays aomalous quatum Hall ffct (QHE) diffrt from ordiary 2D coductors, with th squc shiftd by 1/2 with rspct to th stadard squc 5-7. Similar to th 2D Schrödigr lctros, 2D Dirac lctro stats i a prpdicular uiform magtic fild B also form highly dgratd discrt Ladau lvls (LLs). Wh th magtic flux is gttig largr, ths lvls form th Ladau sub-bads bcaus of th Harpr broadig 8. Most of th xprimtal ad thortical studis of graph hav focusd o th ffct of th rlativistic lctroic disprsio o diffrt phoma such as Ladau lvl structur or quatum Hall frromagtism. O of th most strikig faturs of graph is its vry larg lctro ma-fr path, which maks graph a ticig matrial for ovl lctroics applicatios. Ballistic trasport rquirs th sampl lgth to b shortr tha th ma fr path. Th ma fr path i graph is almost idpdt of tmpratur 9,10. Exprimtal obsrvd graph has a lot of dfcts du to imprfct cuttig. It is wll-kow that dfct is ubiquitous i graph ad its ffct o th lctroic structur has b studid xtsivly 11. O of th simplst dfcts is vacacy causd by th loss of o or svral arst atoms. Vacacis ar atural dfcts i graph lattic ad ca also b xtrally iducd by io bam irradiatio with high rgy. To rcogiz th vacacy dfcts usig X-ray missio spctra, th total dsity of stats for various sizs of vacacy dfcts hav b obtaid 12,13. I additio, th vacacy formatio rgy usig a tightbidig (TB) mthod ad ab iitio DFT-basd mthod has b studid rctly 14. Thortical works hav show that vacacis iduc th formatio of localizd stats ad wh particl-hol symmtry is brok, localizd stats bcom rsoacs clos to th Frmi lvl 16,17. I th followig, th ffcts of two kids of vacacis hav b aalyzd, sigl vacacy ad divacacy, i four-trmial graph aodvic wh a prpdicular magtic fild is applid. Basd o th critical rol that two-dimsioal four-trmial

2 JAFARI t al.: LATTICE VACANCY EFFECTS ON THE QUANTUM TRANSPORT OF GRAPHENE NANODEVICE 679 dvics hav playd i smicoductor aotchology, four-trmial graph aodvics should also play a importat rol i ay graph basd lctroic circuits. Du to its high lctroic quality, a four-trmial graph aodvic has also attractd th itrst of tchologists who s it as a way of costructig ballistic trasistors 29. I additio to th practical importac of ths four-trmial graph dvics, ths systms mak a usful framwork to study th ffcts of lattic dfcts o th lctro trasport i th dvic 18,19, Our rsults suggsts that sigl vacacy itroduc boudig ad ati-boudig stat, ad two clos vacacis may coupl to ach othr formig a "vacacy molcul" tad by th magtic fild. Furthrmor, wh a prpdicular magtic fild is applid to th four-trmial graph, w foud th vacacy stat btw Ladau lvls. Accordig to our rsults, currt i th xistc of vacacy i compariso with th pur cas, rducs, sic th vacacy stat cratd by dfct, producs additioal bad. 2 Thortical Modl Th systm is cosidrd as a ctral coductor rgio (C), coctd to four lads 31 L, R, B, T (s Fig. 1). Th lads ar formd by smi-ifiit prfct ribbos, simulatig four idal lads. Th coductor rgio cosists of Nc umbr of atoms. Thus, i ordr to accout for som gral faturs of th lctroic proprtis dpdc of th systm udr cosidratio o th vacacy positio, th sigl vacacy ad divacacy ar xamid i diffrt positios i th dvic i Fig. 1. Th coductor ad th lads ar dscribd by tightbidig with o -lctro pr atom which is succssfully usd to study th carbo-rlatd matrials. Th tight-bidig Hamiltoia of th systm 20 ca b writt as: i i + iφij + + i i i j i i i, j i H = ε a a + t a a + V a a (1) whr ε i is o-sit rgy ad t is th hoppig matrix lmt btw th arst ighbours. W oly cosidr th arst-ighbour itractio. Th o-sit rgis ar tak as zro 21 ad t=2.78 V. Th sum i i, j is rstrictd to th arst-ighbour atom, ad also a i (a i + ) is cratio (aihilatio) oprator for a lctro i th stat o sit i. A vacacy is simulatd by sttig th hoppigs to zro aroud th vacat sit ad its o-sit rgy quals to a larg valu or V/t. This has b show to b quivalt to actual vacacis whr bods ar xplicitly discoctd. Bcaus of th xistc of th magtic fild, a phas ϕ ij is addd i th hoppig lmt 22. I what follows, w show how to calculat th trasmissio of th systm. I th absc of thrmal ffcts ad th chargig trms, th trasmissio cofficit for lctros from th lft lad to th right lad (lft to right lad is idal 31 ) with rgy E is rlatd to Gr s fuctios usig Caroli s formula which provids high umrical accuracy 23,31,33 ad fficicy: r a T = Tr( Γ G Γ G ) (2) whr m c c G r, a c ar rtardd ad advacd Gr s fuctios of th coductor ad Γ m, ar couplig matrics from th coductor to th lads. Th systms hav four lads, rsultig i a coductor Gr fuctio of form: r r r r r Gc = ( E+ iη ) I Hc L Σ R ΣT ΣB (3) 1 Fig. 1 Lattic cofiguratio of four trmial graphm dvic. Ctral rgio (boxd) is th coductor rgio, C, which is attachd to four lads L, T, B ad R whr I is th idtity matrix, Σ dots th slfrgy du to th couplig btw th coductor ad lad ;iη is a small imagiary trm addd to mak th Gr s fuctio (G) o-hrmitia. Wh thr ar mor tha two lads, th matrix algbra i Eq. (3) is somwhat mor complx as dscribd i th Rf. (23). Th couplig matrics ar xprssd as:

3 680 INDIAN J PURE & APPL PHYS, VOL 52, OCTOBER 2014 r a Γ = i[ Σ Σ ] (4) Th fuctio Γ is calld th broadig fuctio ad dscribs th couplig of th dvic to th lads, r a + whr Σ = ( Σ ).Th coductor rgio cosists of Nc atoms (whr Nc=32), makig all th matrics Nc Nc squar matrics.itgratig th trasmissio probability ovr th whol rgy rag ad for th xtral bias applid to th lctrods, o ca driv th tulig currt as th form: I( V, T ) = 2 / h T ( V, E)[ f ( E µ ) f ( E µ m )] de (5) whr T(V,E) is trasmissio probability pr at th rgy E, f(e) is Frmi-Dirac distributio, ad V is th bias voltag applid to th systm ad µ m (µ ) is th chmical pottial at (m) lad µ m = µ +V). Th coductac G (E) of th four-trmial graph ca b calculatd usig th Ladaur formula G (2 / h) T = (6) I ordr to dscrib th ffct of th vacacy o th coductac quatitativly, w itroduc th dcrasig rat of th avrag coductac, which is: E G E G E de 0 G E E G 0 G 0 ( E ) de E D = = [ ( ) ( )] (7) whr G 0 ( E ) is th coductac of a four-trmial graph without vacacy ad G(E) is th coductac of th four-trmial graph with vacacy. I smicoductig matrials, th ma fr path is vry importat quatity to assss trasport fficicy of th systm ad th corrspodig dvic prformacs, idd is a physical lgths i msoscopic trasport. W ot, howvr, that rct thortical ad xprimtal work hav dalt with ballistic trasport i msoscopic graph which shows uivrsal bhaviour i th rgim whr > W > L, whr W ad L ar th sampl width ad lgth. Hr, lastic ma-fr path is xtractd umrically from th lgth scalig aalysis of th quatum coductac by cosidrig that, accordig to th cosidrd rgims, T = N / (1 + L / ) (8) whr N is th umbr of activ coductio chals, L is th lgth of th ctral coductor rgio; is th ma fr path. I th prst papr, w hav calculatd all rsults by igorig th ffcts of tmpratur, spi-orbit itractio, lctro-lctro corrlatio, lctrophoo itractio, tc. I this modl, it is also assumd that th four sid-attachd lads hav gligibl rsistac. 3 Rsults ad Discussio Figur 2 shows th trasmissio cofficit of lctros from lad L to lad R i four-trmial dvic udr th ifluc of sigl vacacy. Th trasmissio cofficit through th pur dvic is show i Fig. 2 for compariso also. W fid that sigl vacacy affcts th bad structur ad as a rsult affcts th coductio chals ad dcrass th coductio i compariso with th cas that th systm is pur 30,31. Our rsults show that a vacacy xistig i th four-trmial graph will dtriorat th lctroic coductac i th coductio bad rgio du to dstructiv itrfrc of lctroic wavs. Du to th spcific lctroic structur of graph, missig o carbo atom crats vacacy stat at th zro (i.. Dirac) rgy. This vacacy stat is quasi-localizd ad modifis th coductio proprtis of th dvic. If w mak a sigl vacacy at a vacat sit i a graph sht, a vacacy boud stat appars at E = 0 ad its wav fuctio has a obodig charactr, whr th amplituds of th vacat Fig. 2 Trasmissio cofficit of lctro ijctd to th four-trmial graph aodvic i pur cas i th prsc of a sigl vacacy ad divacacy

4 JAFARI t al.: LATTICE VACANCY EFFECTS ON THE QUANTUM TRANSPORT OF GRAPHENE NANODEVICE 681 sits ar zro.th fi paks i pur curvs, kow as Va Hov sigularitis 25 (VHSs), corrspod to xtrm poits i th rgy bads ad th w paks i th trasmissio curv with vacacy corrspod to quasi-localizd stats. Th quasi-localizd stats ca b foud i th scaig tulig spctroscopy (STS) imags or low bias scaig tulig microscopy (STM) imags 26. Th ijctd lctro will b rflctd wh its rgy is qual to th rgy lvl of th quasi-localizd stats. Th ist of graph shows th trasmissio cofficits i th viciity of th Frmi rgy i th prsc of two ar ad far vacacis Figur 2 shows th trasmissio cofficit of lctros from lad L to R i four-trmial graph aodvic udr th ifluc of divacacy. It is assumd that itr-vacacy distac (d) is small (d a, whr a is th distac btw two ighbour carbo atoms). Th divacacy crats two paks i th trasmissio cofficit at rgis abov ad blow th Frmi rgy. Cratig a vacacy i o sit is similar to lowrig th lctroic dsity of stats ad icrasig th lctroic dsity o all th ighbour sits aroud th vacacy sit. Th paks apparig symmtrically aroud Frmi rgy ar du to th boudig ad ati-boudig stats 26. I Fig. 3, magtic fild ad divacacy ar applid to th systm simultaously. Th ffct of th magtic fild o trasmissio is two fold: (1) th magtic fild lads to o-disprsiv magtic lvls, with a larg dgracy; (2) th rgy lvl spacig is modifid givig ris to a pilig up of rgy lvls as o mov away from th Dirac poit. Th xistc of divacacy alog with th magtic fild crats additioal bad btw Ladau lvls. Th stats i th ctr of th Ladau lvls ar xtdd ad thir wav fuctios ar dlocalizd. Not that our discussio is valid for wak magtic filds. W fid that two clos vacacis may coupl to ach othr, formig a vacacy molcul tud by th magtic fild. Th couplig btw th wav fuctio of two vacacis is vrifid to b dtrmid by th distac btw thm ad th magtic fild. If th two sigl vacacis ar itroducd far from ach othr i th lattic, thy caot b coupld ach othr, th w assum that itr-vacacy distac (d=a) is small.th bodig btw ths vacacis ca b tud by magtic fild; this ffct is appard by cratig a pak i Frmi rgy wh th magtic fild ad divacacy ar applid to th systm simultaously (s th ist of Fig. 3). I ordr to ivstigat th bhaviour of chags i currt voltag from lad L to lad R, ad also ivstigatig th ffct of vacacy o currt, w apply xtral pottial to th systm, so that lad L lis i th pottial of V/2 ad lad R lis i th pottial of-v/2. Th currt is ot rliabl wh th applid bias is so big i th msoscopic systm hc th ist of Fig. 4 shows th chags i currt voltag i [-1V,1V].Th small oscillatios i th curv ar du to Va Hov sigularity ad origiatd from th currt quatizatio i this msoscopic systm (s th ist of Fig. 4). It is obvious that th currt dsity icrass liarly, at low voltags, ad th lowrs to a limitd valu. Dcras of currt mas that a gativ rsistac ffct appars i Fig. 3 Trasmissio cofficit of lctro ijctd to th fourtrmial dvic i th prsc of magtic fild ad divacacy Fig. 4 Currt-voltag curvs of four-trmial dvic i pur cas ad i th prsc of a sigl vacacy ad divacacy

5 682 INDIAN J PURE & APPL PHYS, VOL 52, OCTOBER 2014 Fig. 6 Ma fr path(m) for a four-trmial dvic i pur cas ad i th prsc of a sigl vacacy ad divacacy Fig. 5 Dcrasig rat of th avrag coductac (btw ±0.5 V) du to a sigl vacacy(blu curv) ad divacacy (rd curv) currt voltag charactristics. Th raso for this ffct is th rducd ovrlap btw th rgy bad of th two lads (L ad R) as th bias is icrasd. Accordig to th Fig. 4, currt i th prsc of vacacy i compariso with th pur cas, dcrass, sic th vacacy stat producs additioal bad, i this way th ijctd lctro has mor chals for trasport ad its itsity dcrass i compariso with th pur cas (by lss umbr of chals). It is obvious that th currt icrass liarly, at low voltags i th prsc of vacacy. Basd o Eq. (7), th dcrasig rat of th avrag coductac (btw ± 0.5 V) as a fuctio of th ctral coductor rgio siz (Nc) is show i Fig. 5. Th divacacy affcts dvic much mor strogly tha sigl vacacy. Morovr, th ffct of vacacis dcrass wh th siz of ctral rgio icrass. This is bcaus thr ar mor atoms i th crosssctio of a widr dvic, so lctros ar asir to go aroud th dfct. To udrstad coductac scalig i such systms, a aalysis of trasport lgth scals is prformd. Figur 6. shows th ma fr path as a fuctio of th rgy E of th ijctd lctros. Log ma-fr paths for lctros ar th Frmi rgy hav idd b ifrrd from rgular Coulomb oscillatios ad cohrt tulig at low tmpraturs 27,28. Th strog hacmt of aroud th zro rgy implis a rducd umbr of scattrig procsss. It is wll kow that scattrig of lctros, th charg carrirs at graph dvic will dcras thir ma fr path. Hr, magtic fild ad divacacy ar applid to th systm simultaously. I th prsc of a magtic fild, th stats of graph ar dscribd i trms of Ladau lvls. At low rgis, wh th Dirac frmio dscriptio is valid, th rgy lvls ar giv by: 1 E δ l ± ( ) =± 2 B (9) 3ta ad dfid lb = / B, δ = ad =1, 2, 2 Notic that th cyclotro rgy is much largr tha th Zma rgy, thus, w disrgard th Zma rgy. Both vacacy ad th magtic fild ca affct th trasport proprtis i th systm. Thr ar two importat lgth scals: vacacy causs lctro scattrig, ladig to a lctro ma-fr path ( ) ad th magtic fild itroducs a magtic lgth ( B ). Elctroic stats i a Ladau lvl (LL) ca b viwd as cyclotro motios of a lctro aroud orbits of radius ( B ) ctrd at diffrt locatio. This should hav som cosqucs o th trasport gaps ad could b at th origi of larg sampl-to sampl low-tmpratur coductac fluctuatios. 4 Coclusios Basd o Ladaur approach ad usig tightbidig modl, th ifluc of two kids of vacacy, sigl vacacy ad divacacy i th prsc of prpdicular magtic fild o th trasport proprtis of a four-trmial graph aodvic, has b studid. Our rsults show that sigl vacacy

6 JAFARI t al.: LATTICE VACANCY EFFECTS ON THE QUANTUM TRANSPORT OF GRAPHENE NANODEVICE 683 xistig i th four-trmial graph will dtriorat th lctroic coductac i th coductio bad rgio du to dstructiv itrfrc of lctroic wavs. W fid that divacacy crats boudig ad ati-boudig stats aroud Frmi rgy du to th icrasig lctroic dsity o ighbour sits. I th othr had, xistc of two sigl vacacis with small distac btw thm, i th prsc of magtic fild, producs vacacy molculs by couplig btw thir wav fuctio. From th I-V curv, w foud th currt quatizatio i th systm, also th currt i th systm with vacacy dcrass i compariso with th pur cas, bcaus of th cratio of vacacy bad. It shows that th ffct of vacacis dcrass wh th siz of ctral rgio icrass. Th ma fr path pattrs ar foud to strogly dpd o th dfct. This fatur could mak such a quasi-2d carbo-basd juctio a possibl cadidat for ao-lctroic dvics. Rfrcs 1 Zhag Y, Ta Y-W, Stormr H L & Kim P, Natur (Lodo), 438 (2005) Novoslov K S, Gim A K, Morozov S V, Jiag D, Katslso M I, Grigoriva I V, Duboos S V & Firsov A A, Natur (Lodo), 438 (2005) Hirayama Y, Saku T, Tarucha S & Horikoshi Y, Appl Phys Ltt, 58 (1991) Schwab M G & Kim K, Natur, 490 (2012) Shg D N, Shg L & Wg Z Y, Phys Rv B, 73 (2006) Sahoo S & Das S, Idia J of Pur & Appl Phys, 47 (09) (2009) Sahoo S, Idia J of Pur & Appl Phys, 49 (06) (2011) Wakabayashi K, Fujita M, Ajiki H & Sigrist M, Phys Rv B, 59 (1999) Klos J W & Shylau A A, Phys Rv B, 80 (2009) Lhrbir Aur li, Blas X & Niqut Ya-Michl, Phys Rv Ltt, 101 (2008) Hjort M & Stafstrom S, Phys Rv B, 61 (2000) Prira V M, Guia F, Satos J M B Lops dos, Prs N M R & Nto A H Castro, Phys Rv Ltt, 96 (2006) Prira V M, Satos J M B Lops dos & Nto A H Castro, Phys Rv B, 77 (2008) Hashimoto A, Suaga K, Glotr A, Urita K & Iijima S, Natur, 430 (2004) Novoslov K S, Jiag Z, Zhag Y, Morozov S V, Stormr H L, Zitlr U, Maa J C, Bobigr G S, Kim P & Gim A K, Scic, 315 (2007) Prs N M R, Guia F & Nto A H Castro, Phys Rv B, 73 (2006) Prira V M, Satos J M B Lops dos, Nto A H Castro, Phys Rv B, 77 (2008) Brow D, Bad Y B & Avishai Y, Phys Rv B, 53 (1996) Gol N, Chug S J, Satos M B, Suzuki K, Miyashita S & Hirayama Y, Physica E, 21 (2004) Datta S, Elctroic Trasport Proprtis of Msoscopic Systms, Cambridg: Cambridg Uivrsity Prss, (1995). 21 Chico L, Phys Rv B, 54 (1996) Johso M H & Lippma B A, Phys Rv, 76(1949) 828, Rabi I I, Z Phys, 49 (1928) Lopz-Sacho M P, Lopz-Sacho J M & Rubio J, J Phys F Mt Phys, 14 (1984) Kobayashi Y, Fukui K-I, Eoki T & Kusakab K, Phys Rv B, 73 (2006) Odom T W, Huag J L, Kim P & Libr C M, J Phys Chm B, 104 (2000) Orllaa P A, Domiguz-Adam F, Gómz I & Guvara M L Ladró d, Phys Rv B, 67 (2003) Crsti A, Nmc N, Bil B, Niblr G, Triozo F, Cuibrti G & Roch S, Nao Rs, 1 (2008) llr Markus Mu, Brauigr Matthias & Trauzttl Bjor, P R L, 103 (2009) Liag Ggchiau, Nophytou Nophytos, Mark S, Ludstorm & Nikoov Dmitri, J Appl Phys, 102 (2007) Hicks J, Tjda A, Talb-Ibrahimi, Nvius A, Wag M S, Shpprd F, Palmr K, Brtra J, Fèvr F L, Kuc P, Hr J d, Brgr W A, Corad C & Nat E H, Phys, 9 ( 2013) Rittr C, Pachco M, Orllaa P & Latg A, J Appl Phys, 106 (2009) Maiti Satau K, Solid Stat Commuicatios, 150 (2010) Saha Kamal K, Lu Wchag, Brholc J & Muir Vict, Physical Rviw B, 81(12) (2010)