Magnetic phase transitions in pure zigzag graphone nanoribbons

(J. Phys. Lett. A 379 (2015) 753-760)

• Résumé de la publication 1 :

Dans cette publication, nous avons étudié les propriétés magnétiques et hystérétiques d’un nanoruban de graphone pur de type zigzag en utilisant la simulation Monte Carlo et le calcul de champ moyen. Dans ce travail, nous avons présenté une étude détaillée permettant de com- prendre le comportement magnétique et hystérétique d’un nanoruban de graphone pur avec des bords de type zigzag. Signalons que le magnétisme, dans cette dérivée de graphène, provient des électrons localisés sur les atomes de carbone qui ne sont pas hydrogénés.

Physics Letters A 379 (2015) 753–760

Contents lists available atScienceDirect

Physics

Letters

A

www.elsevier.com/locate/pla

Magnetic

phase

transitions

in

pure

zigzag

graphone

nanoribbons

L.B. Drissia,b,∗,S. Zriouelb

a_{LPHE,Modeling&Simulations,FacultyofScience,MohammedVUniversity,Rabat,Morocco} b_{CPM,CentreofPhysicsandMathematics,FacultyofScience,MohammedVUniversity,Rabat,Morocco}

a rt i c l e i nf o a b s t r a c t

Articlehistory:

Received 30 October 2014

Received in revised form 19 December 2014 Accepted 22 December 2014

Available online 30 December 2014 Communicated by V.A. Markel

Keywords:

Graphone Nanoribbons Triangular lattice Monte Carlo calculations Mean field theory Magnetic phases Critical temperature Hysteresis cycles

Magnetic properties and hysteresis loops of pure graphone nanoribbons (GONR) are studied using both Monte Carlo calculations and mean field theory. This study is relevant for understanding the magnetic behavior of pure GONR that exhibits magnetism due to the localized electrons on the carbon atoms without hydrogens. Magnetization and its corresponding susceptibility are given for various ribbon widths 3 ≤Nz≤100 and external magnetic field 0 <h≤10 kOe. The critical temperature Tc is deduced. It is

shown that temperature Tcreduces as a step function versus the ribbon widths Nzfor low values of h

up to 0.3 kOe. The effect of temperature, low/strong h andNzparameter on the hysteresis curves is also

examined. The findings of this work offer considerable promise for use of GONR in various nanoelectronic devices especially for high-energy-storage-capacitor applications that require square hysteresis loop behavior.

©2014 Elsevier B.V. All rights reserved.

1. Introduction

Since its first isolation in 2004, graphene has attracted a lot ofinterest forits fundamental studies[1,2] andits highpotential applications [3,4]. Graphene is a zero gap semiconductor which leads to the challenges of opening up and controlling the band gaptoadaptthismaterialtofuturehightech-electronicdevices.In order to overcome thislimitation severalapproaches are consid- ered using various processes. Quantum confinement of electrons by forming nanoribbons [5,6] is one waythat not only modifies theelectronicstructurebutalsointroduces magnetisminthisnon- magnetic material.

Graphene nanoribbons (GNRs) with different widths can be madeeitherbycutting exfoliatedgraphenesheetalonga straight

line[7,8]orbyepitaxialgraphenepattern[9,10].Theresultingone

dimensional systems can have either armchair or zigzag edges. Zigzag graphene nanoribbon (ZGNR) is a semi-conductor at its groundstate withband gapdependingonits widthW [11].Due tothezigzagedgestates,electronspinpolarization appearsspon- taneously.Indeed,ineachedge,localizedstatesareferromagneti- cally ordered[12]whilethemagneticmomentsonthetwo edges interact antiferromagnetically since they have opposite spin ori-

*Corresponding author at: LPHE, Modeling & Simulations, Faculty of Science, Mo- hammed V University, Rabat, Morocco.

E-mailaddress:ldrissi@fsr.ac.ma(L.B. Drissi).

entations[13].FM–AFM energy differencesper unit cell is afew meVandcouldpresentmetal stateatfinitetemperature[14].Dif- ferent applicationsin a numberof exceptional spintronicdevices havebeenproposedforone-dimensionalzigzaggraphenenanorib-

bons[15,16].

Assuming that graphene is a giant macromolecule allows the use of chemical reactions that create new derivatives such as graphane [17,18]. The use of this strategy tunes the gap energy

[19] and affects the magnetic properties of graphene. Graphane attain permanent magnetic moment through hydrogen vacancy domains [20] or by partial dehydrogenation giving rise to gra- phone[21].Graphoneisasemiconductorwithasmallindirectgap. Thisnewmaterial exhibitsferromagnetism groundstate originat- ingfrominteractionsbetween2p momentsattributedtoextended

p–p interactionsbetweenthelocalizedandunpairedelectrons on theunhydrogenatedcarbonatoms[22].Thepartial hydrogencov- eragehasastrikingeffectonphysicalpropertiesofgraphenethat canberestoredbyannealing[23].Removinghalfofthehydrogen atomsfromgraphoneleadstononmagneticmaterialaspzorbitals

oftwonearestunsaturatedCatomsform π-bondingthatquenches magnetism[21].

Although magnetic carbon nanostructures are of great inter- est asthey are light, stable, simpleto treat, andcheaperto pro- duce, they are not investigated extensively. The number of the- oretical studies existing are focusing on the role of topological

http://dx.doi.org/10.1016/j.physleta.2014.12.041

754 L.B. Drissi, S. Zriouel / Physics Letters A 379 (2015) 753–760

defects [24], carrier density [25] and edge states [26,27] to in- ducemagnetismingraphenenanoribbons.Recently,structuraland electronic propertiesof long nanoribbonsof graphone havebeen studied [28]. Unlike short samples with specific sizes that form carbonnanotubes [29], long graphone nanoribbons, dueto their

sp3 hybridization,spontaneously rollup to form spiralstructures withinteresting localization of frontier molecular orbitals. These rolledhydrocarbonstructuresarestablebeyondroomtemperature uptoatleastT=1000 K.

Thefeasibilityofengineeringmagneticgraphenenanostructure devicesisquestioned bymagnetic orderingat finitetemperature. This opens a new field in the research of the critical magnetic phenomenaatnanoscale.In[30]and[31],theinfluenceofthedis- orderon the magneticproperties ofnanoribbon are investigated. In[30],theeffectofboththenumberandthepositionsofs=3/2 substitutedmagnetic atomson themagnetic phase transitionsof mono-,bi- andtri-doped graphonenanoribbons indifferent con- figurationsare reported.It isshownthat the criticaltemperature

Tc increases with the number of dopants but for configurations

with fixed number of magnetic impurities, Tc is more sensitive

toedges. In[31],the magneticbehavior ofa mixeds=1/2 core and S=1 shell nanoribbon withan anti-ferromagnetic interface couplingshowsa number ofcharacteristic behaviorssuch asthe occurrenceofcompensationtemperatureandtheexistenceofsin- gleandtriplehysteresisloops.

Tounderstandmorethefundamentalbehaviorofthemagnetic properties of graphone nanostructures, we study in the present worktheGONRwidthdependenceofCurietemperatureintheab- sence and presence of external magnetic field. In particular, the influenceofthechangingwidthonthemagneticbehaviorofferro- magneticnanoribbon.Graphonenanoribbonsismodeledbytrian- gular latticewithperiodic boundaryconditionsinthe X -direction

andfreeboundary conditionsinthe Y -direction.Intheliterature, 2d Ising model phase transition on a triangular lattice is a long standingtopic [32] investigatedmainly forsystemswithperiodic boundary conditions using different methods as MC calculations

[33]andWang–Landausimulation[34].However,wearenotaware ofanywork reportingthe boundary effects onthe magnetic be- havior oftriangularlattice withfinitewidthsuch asnanoribbons evenifthesenanostructuresare verymuch requiredinengineer- ing magnetic devicessuch asgraphene derivatives andother 2D materials.UsingMonteCarlo(MC)simulationsandmeanfieldthe- ory (MFT),weshow theeffectsofexternalmagneticfield,aswell astemperatureandsizeontherelevantthermodynamicquantities such asmagnetization, susceptibility,hysteresis loopsand critical temperatureofthesystem.A detaileddescriptionofthemodeland theformalismaregiveninSection2.InSection 3we presentthe obtainedresultsanddiscuss, forweak andstrong magneticfield, theeffectofdifferentcitedparametersonthemagneticphasesand hysteresiscyclesthatchangefromsquarestoloopstoanhysteretic curves.Weendwithaconclusion.

2. Themodel

Inthepresentwork,weconsiderinfinitelylongone-dimensio- nal graphone nanoribbonswithzigzagedges. The systemisperi- odiconlyalong X -directionandtheunit cellusedinourcalcula- tionsisdelimitedbydashedlinesasshowninFig. 1.

Followingpreviouscustomarynotation[13,35],thefinitewidth

W of the ribbon is characterized by the number Nz of zigzag

chains ofcarbons that run along X . In Fig. 1 we plot the struc- ture of Nz=8 H-ZGONR-H with the edge carbon atoms all sat-

urated with non-magnetic atoms, namely H atoms to avoid the dangling bond states. In this model, we have two kinds of car- bonatoms: C1 andC2 formingthehexagonalstructure.Thebond

Fig. 1. Structural model of 8-zigzag graphone nanoribbon. Transparent atoms rep-

resent hydrogenated C1 atoms, black atoms are C2 atoms, and small blue atoms

denote atoms passivating the edges of graphone. (For interpretation of the refer- ences to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 2. (a) Representation of 8-zigzag graphone nanoribbon, (b) showing only mag-

netic nearest neighbors and ignoring non-magnetic atoms.

1.495 ˚A. In each hexagon, carbon atoms C1 are decorated with

hydrogenin sp3 hybridizationwhile C2 atomsthatremain unsat-

urated are sp2 hybridized. Soonly the unhydrogenated C2 atoms

carry a magneticmoment of about1μB astheir p-electrons are

localizedandunpaired.EachC2 atom,exceptthoseneartheedges,

hasthreenonmagneticfirstnearestneighborsC1andsixmagnetic

second nearest neighbors C2. Accordingto [21],the valenceelec-

tronsinp-statesaremoredelocalizedthanthoseind or f -states.

Therefore,theferromagnetic(FM)groundstateingraphoneisdue to interactions between 2p moments oflocalized electrons spins in C2 atoms. By ignoring non magnetic C1 atoms as they don’t

contribute in our calculations, we depict the corresponding C2

nanoribbon in Fig. 2.In this figure, ZGONR ismodeled by trian- gular latticewithperiodicboundaryconditionsinthe X -direction

andfreeboundaryconditionsintheY -direction.

Thesystemisdescribedbythehamiltonian:

H=J0Sz_iSz_j+h 

L.B. Drissi, S. Zriouel / Physics Letters A 379 (2015) 753–760 755

Fig. 3. Magnetization and susceptibility versus temperature for pure zigzag graphone

nanoribbons for three representative widths Nz=8, 20 and 40.

where J0isthecouplingbetweenmagneticnext-nearestneighbors

inthehexagonalstructureofZGONR,namely J0 istheinteraction

between the 2p S-moments attwo different sites i and j of C2

atoms and S_iz is the spin of the magnetic C2 atom atsite i. All

the spins are set to be ±1/2 forpure ZGONR. h is the external magneticfieldrangingas0≤h≤10 kOe.

3. Resultsanddiscussion

WeconsiderZGONRhavingdifferentwidthsW .Weconcentrate ontheeffectoftheparameterW onthevariationofCurietemper- atureforpureZGONRonthebasisofbothMonteCarlocalculations andmeanfieldtheory.Twocasesareinvestigated:ZGONRsinthe absenceandpresenceofexternalmagneticfield.

3.1. PureZGONRs

Monte Carlo calculations: By mean of MC simulations for the Isingmodeldescribedabove, westudyrectangularhydrogen- terminated one-dimensional zigzag graphone nanoribbons ofdif- ferentwidthsNz varyingfrom3upto100.Theperiodicboundary

conditionsareappliedinX -directionandfreeboundaryconditions are applied inY -direction. The groundstateelectronic configura- tionofpureZGONRischaracterizedbytheferromagneticarrange- ment ofspins.The MCsteps are5·105 steps perspin discarding thefirst 5·104 _{MonteCarlosimulations.We buildaprogram us-}

ingtheMetropolisalgorithm[36]tocalculatethethermalaverage ofmagnetization M andenergy E.Wesetthecoupling J0=1 and

theBoltzmann constantkB=1.Wecalculatealsothecorrespond-

ingmagneticsusceptibility

χ=1

T



M2− M2

andthespecificheatCV givenby

CV =

1

T2 

E2− E2.

Magnetization and susceptibility as functions of T for pure ZGONRwithspecificparameterwidthNz=8,20 and40 areplot-

tedinFig. 3.Fromtemperaturedependencesusceptibilitywe de-

duce theCurietemperature Tc=3.67,3.08 and2.66 respectively

forthethreevaluesofNz mentionedbefore.

For3≤Nz≤100,thatcorrespondsto varyingthewidthW in

the range from 5 ˚A up to 223 ˚A, we collect data from temper-

Fig. 4. Curie temperature versus ZGONR width parameter Nz≤100 (the lines guide the eye).

Fig. 5. ZGONR width parameter dependence of the Curie temperature for high values

of Nz≥100 (the lines guide the eye).

Fig. 4.Wededucethat criticaltemperature Tc decreasesasastep

function ofthe ribbonwidth Nz,to minimize andstabilizein its

minimalvalue Tcmin forNz rangingfrom32 to100.Weexpectre-

covering thelimit ofgraphone infinitesheetfor largewidth that correspondstohighvaluesof Nz.

Indeed, when Nz varies in the large interval [100,4000], the

width W reaches the value of 8970 ˚A. The obtained results re- veal that the evolution of Tc has a different behavior compared

to the previouscase where Nz takesinteger values in theinter-

val [3,100]. As plotted in Fig. 5, the transition temperature Tc

increases with Nz until it stabilizes for Nz≥1300. The presence

of two regions with differentbehaviors suggestsa 1D→2D di- mensionalitycrossoverforNz around100.

To reproduce results in agreement with literature, we plot in

Fig. 6 the variation ofthe criticaltemperatureas functionof the

ribbonwidthvaryinginthetwointervals[3,100]and[100,4000] withlargestepsNz.NoticethatthestepfunctionfoundinFigs. 4

and5disappear leading toa decreaseofthe criticaltemperature infunctionofthesizeintheinterval[3,100]andanincreaseofTc

756 L.B. Drissi, S. Zriouel / Physics Letters A 379 (2015) 753–760

Fig. 6. Curie temperature versus ZGONR width parameter (a) for Nz≤100 and (b) for Nz≥100 for large steps (the lines guide the eye).

with[31,37].We deducethat theplateausfound inFigs. 4 and5

areadirectconsequenceofthevery smallstepsNz usedinour

numericalsimulation.

Meanfieldtheory: In order toanalyze theMC calculations of sizeeffectonTc,wehavecarriedoutananalyticcalculationswith

meanfield theory.For theHamiltonian inEq.(1), theGibbs free energypersite[38]isgivenby

¯

F0=F0+ H0− H00 (2)

where· · ·0=tr···_expexp_(−β.(−β._HH₀0₎) denotestheaveragevalueperformed

overtheeffectiveHamiltonianofthesystem H0,and F0 isitsas-

sociatedfreeenergy.Wehave

H0= − N  i=1 hi. Si, F0= − 1 βlog Z0 (3) withβ=_k1

BT,kB istheBoltzmannconstantandT isthetempera-

ture.

Fortheeffective fieldh parallel to−−−→O Z and Si= ±1₂,theparti-

tionfunctiongeneratedbytheaboveHamiltonianis

Z0= 

2.coshβ.h 2

N

whereN isthetotalnumberofC2 atoms.Therefore,Eq.(3)canbe

rewrittenas F0= − N β log  2.coshβ.h 2  (4) and H0= −J0  N1+ 3 2.N2  tanh2  β.h 2  (5) withN1 isthenumberofC2 atomsneartheedgethathaveonly

four C2 nearest neighbors while N2 is the number of C2 atoms

insideZGONRhavingsix nearestneighbors.Replacing Eqs.(4)and

(5)in Eq. (3),the minimization of the free energy (dF¯0/dh)=0

leadsto: N.β.h 4 = J0.β.tanh  β.h 2  .  N1+ 3 2.N2  . (6) Tc= J0.  2+ N2 (N1+N2) 

In theunit cell,when wevary thewidthparameter Nz,onlythe

number N2 vary while N1 remains constant. As the mean field

theoryisanapproximationthatbecomesexactonlyinthelimitof infinitesystemsize[39],we deducethatwhen N2→ ∞,thecrit-

ical temperatureofthesystem Tc→3 J0 whichagrees well with

previous MC result for large value of Nz having Tc=3.47. This

approach cannot reproduce the 1D limit as the mean field type approximationfailinonedimensionalsystemwherethequantum fluctuations dominate and the behavior is very exotic and non- standard[40].

3.2. Magneticfieldeffect

Inthissection,we runourMC calculationstostudytheinflu- enceofexternalmagneticfieldh onhydrogen-terminatedperiodic one-dimensional graphone nanoribbons of different widths 3≤

Nz≤100.Wevarythemagneticfieldinstepsofh=0.01 when

h≤1 kOe andh=0.5 forstrongh,equilibratingateachstatefor 5·105_{MonteCarlostepsperspin.Theresultsrevealthatwehave}

two main cases:(i) h takes very lowvalues 0≤h≤0.3 kOe and (ii)externalmagneticfieldisstrong.

InFig. 7,wedisplay themagnetization M andthecorrespond-

ing susceptibility χ versustemperaturefordifferentvaluesofex- ternal magnetic field h=1,. . . ,10 kOe for Nz=3 ZGONR. All

themagnetization curvesdecreasefromtheirsaturationvalue 0.5 andgotozerowhenT increases.Asmagneticfield increases,the slope of magnetization curvesbecomes lesspronounced. Forthe temperaturedependenceofthesusceptibility,thecurvespeakfor different values of T that increases as h increases. These peaks thatcharacterizethephasetransitionfromferromagneticphaseto paramagnetic one become lesspronounced for highvalues of h.

Thisisduetothestrongeffectofh onthealignmentofspins.No- ticethatthesamebehaviorforM and χ versus T isobtainedfor differentribbonwidths3≤Nz≤100.

Thesamecalculationsareperformedasafunctionoflowmag- netic field for width Nz up to 100.The temperaturedependence

ofthesusceptibilityisreportedinFig. 8forthecaseNz=6.Com-

pared to previous results for strong h, the behavior is the same asthecurvespeakfordifferentvaluesof T thatincreaseswithh.

However,themaximumof χ islargerthanthatfoundforstrongh

andthetransitiontemperatureismuchlower.

Notice that themost importantinfluence on the criticaltem- perature Tc seems to be the size [41], which indicates that the

L.B. Drissi, S. Zriouel / Physics Letters A 379 (2015) 753–760 757

Fig. 7. Magnetization and susceptibility versus temperature of 3-zigzag graphone nanoribbon for various values of external magnetic field h.

Fig. 8. Susceptibility versus temperature for 6-zigzag graphone nanoribbon at low

external magnetic field h =0.03, 0.15, 0.2 and 0.28 with a zoom on susceptibility

peaks.

a designedtemperaturerange[42].Todeterminetheevolutionof the criticaltemperatureinfunctionofthe width,we plotthees- timated transition temperatures as function of Nz≤10 for low

magnetic field inFigs. 9 and 10 andforstrong magneticfield in

Fig. 11.

When 0<h≤0.3 kOe, the critical temperature Tc decreases

asa stepfunctionof ZGONRwidth Nz.At fixedwidth Nz,Tc in-

creasesashincreasesandstabilizesatits approximatevalue 4.08 for h=0.3 kOe as shown in Fig. 9. When h vary in the range

[0.35,0.8],thetemperature Tc increasesasastepfunctionofthe

ribbonwidthNz tomaximizeandsaturateasdepictedinFig. 10.

Forstrongexternalmagneticfield1≤h≤4,thetransitiontem- perature increases slowly and similarly with the size Nz. When

h≥5, the variation becomes slow for Nz≥6 and stabilizes for

highervaluesofthewidth.

ItisworthnotingthattheZGONsisinsensitivetoh when0<

h≤0.3 kOe and the behavior of the critical temperature versus the width Nz isthe sameasthe oneobtained forzeromagnetic

field.

Inordertoinvestigatetheinfluenceofthetemperatureonthe hysteresis behavior ofthe Nz ferromagneticGONR in thecaseof

low external magnetic field, a series ofhysteresis curves atvari- ous valuesofTisplottedinFig. 12forthecaseof Nz=6 GONR

having Tc=3.88.Thecurvesshow hystereticbehaviorwithloops

becomingnarrowerasthetemperatureincreases.Wecanalsosee

when thetemperature approachesits criticalvalue Tc. Moreover,

all themagnetization curves aresymmetric forboth positiveand negativevaluesoftheexternalmagneticfield.Thispropertycomes directly from the symmetry of the Hamiltonian with respect to changes h→ −h and M→ −M. At T=0.8,the hysteresis is al- mostsquareindicatingthatthenormalorientationisaneasyaxis. In this case, the remanent magnetization is equal to the satura- tion magnetization that is an obviousconsequence of thesquare hysteresis loop.Thesquare behaviorofcurvesisduetomorethe temperatureislowthecloserwegettothegroundstate,therefore the system becomes ordered and encounters the thermal equi- librium. At higher T , the shapes of the ferromagnetic hysteresis loops change from square to loop. The slanted curves demagne- tize and remagnetize quickly. It is worth noting that all curves saturate quicklyand when increasing temperature, the remanent magnetization decreases while the saturation magnetization in- creases. For T ≥2.2, we find that the slanted curves disappear. This behavior can be explained by the fact that higher tempera- ture disorders the system and namely, the state of the nanorib-

Dans le document Contributions à l’étude Monte Carlo des propriétésmagnétiques des nanomatériaux type graphyne et graphone (Page 110-119)