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Inthiswork,wereportonanab-initiocomputationalstudyoftheelectronicandmagnetic propertiesoftransitionmetaladatomsonamonolayerofNbSe .WedemonstratethatCr,Mn,Fe andCopreferalltositabovetheNbatom,wherethedstatesexperienceasubstantial hybridization.Theinter-atomicexchangecouplingisshowntohaveanoscillatorynature accompaniedbyanexponentialdecay,inaccordancewithwhattheorypredictsforadamped Ruderman–Kittel–Kasuya–Yosidainteraction.Ourresultsindicatethatthequalitativefeaturesof themagneticcouplingforthefourinvestigatedadatomscanbeconnectedtothefinedetailsof theirFermisurface.Inparticular,theoscillationsoftheexchangeinFeandCoarefoundtobe relatedtoasinglenestingvector,connectinglargeelectronsandholepockets.Mostinterestingly, thisbehaviorisfoundtobeunaffectedbychangesinducedontheheightoftheimpurity,which makesthemagnetismrobusttoexternalperturbations.ConsideringthatNbSe isa superconductordowntoasinglelayer,ourresearchmightopenthepathforfurtherresearchinto theinterplaybetweenmagneticandsuperconductingcharacteristics,whichcouldleadtonovel superconductivityengineering. 1.Introduction anisotropy . Layered van der Waals (vdW) mater- ials have an intrinsic magnetocrystalline anisotropy, The recent years have seen a drastic increase in as a result of their reduced crystal symmetry, and the research on two-dimensional (2D) magnetism, hence are ideal, potential candidates for 2D magnet- motivated by a fascinating fundamental physics ism . In fact, several recent studies reported on as well as the perspective of various applications. the discovery of ferromagnetism in vdW materials According to the Mermin–Wagner theorem, long- down to the truly 2D limit [3–6]. Intrinsic magnet- range magnetic ordering in 2D systems should be ism is naturally the most attractive option in this suppressedatanyfinitetemperaturebythermalfluc- regard and was confirmed in CrI , Cr Ge Te and 3 2 2 6 tuations . However, this result is inferred from VSe ,albeitonlyforthelattertheorderingtemperat- simplemodelswherecontinuoussymmetrybreaking urewasfoundtobeaboveroomtemperature.Unfor- takesplaceanddoesnotholdwhentheHamiltonian tunately, structural degradation due to exposure to of the system admits a discrete symmetry, as e.g. airhasproventobedetrimentalfortheapplicationof in the presence of a sizeable magnetocrystalline thesematerialsindevices.Analternativestrategy ©2022TheAuthor(s). PublishedbyIOPPublishingLtd 2DMater.9(2022)045012 SSarkaretal to avoid this problem is to work with non-magnetic vector connecting large electron and hole pockets is 2D materials and induce magnetism via the func- found to essentially determine the oscillatory char- tionalizationwithdefects,dopingoradatoms[8–13]. acter ofthemagnetic interactions. Conversely, the Transition metal dichalcogenides (TMD) occupy a absence of this features for Cr and Mn gives rise to special place among 2D materials, due to their high acouplingofamuchshorterrange. versatility, tunability, ease of synthesis and manip- The rest of the paper is organized as follows. In ulation . Intrinsic as well as extrinsic magnet- section 2, the methodological aspects of our study ism has been predicted by theory and confirmed in arepresented.Then,section3iscenteredontheana- experiment . A substantial amount of research lysis of the interplay between geometry and mag- on extrinsic magnetism has been focused on insu- netism. The analysis of the long-range character lating systems, as e.g. MoS and MoSe , due to of the exchange coupling is reported in section 4. 2 2 the possibility of realizing localized impurity states Finally, the conclusions of this work are discussed in with large atomic-like magnetic moments [16–23]. section5. Metallic TMDs in 2D are also very interesting, due to the presence of fascinating collective phenom- ena such as superconductivity and charge density 2.Methodology waves (CDWs) . A 2D material that exhibits this type of physics is NbSe , whose structural and elec- 2.1.Structuraloptimization tronicpropertieshavebeenextensivelycharacterized The electronic structure was calculated using a pro- in recent years [25–32]. A peculiar feature of NbSe jected augmented wave method [45, 46] as imple- inthemonolayerlimitisthatitisclosetoamagnetic mented in the Vienna ab-initio simulation package instability, which may easily be triggered by external (VASP)[47–50].Thegeneralizedgradientapproxim- means. For example, long-range order has been ation  in the Perdew–Burke–Ernzerhof realiza- showntoariseunderasmallamountoftensilestrain, tion [51, 52] was used for the exchange-correlation albeit the precise amount of required strain and the functional. The system to investigate consisted of a type of magnetic pattern are still debated [33–36]. 4×4 supercell of 1H-NbSe monolayer, including Functionalizationviadopingoradsorbedatomsand a total of 48 atoms, with a single magnetic adatom molecules has been also explored. Zhu et al have on top. The magnetic adatoms addressed in our shown in an experiment that superconductivity and study were Cr, Mn, Fe, and Co. A vacuum region ferromagnetism coexist in NbSe after adsorption of 14Å was considered between two monolayers in of hydrazine molecules . Liebhaber et al have the adjacent supercells along the z-direction to elim- used scanning tunneling microscopy and electronic inate any unphysical inter-layer interactions. The in- structure theory to show that Fe adatoms on NbSe plane lattice constant was fixed to the experimental lead to the formation of Yu–Shiba–Rusinov bound value of 3.45 Å corresponding to the NbSe unit state with the charge-density order . Similar cell . The internal positions of the atoms were analyses for thin films of NbSe were also recently optimized towards the minimum energy configur- published[39,40]. ation until the forces were found to be less than −3 −1 In a previous work , we investigated how 10 eVÅ . A gamma centered Monkhorst–Pack thepresenceofmagneticandnon-magneticadatoms meshof19×19×1k-pointsandaplanewaveenergy alterthelocalpatternsandenergyhierarchyofCDWs cutoff of 800eV were used in the calculations. Due in NbSe . In this work, instead, we intend to focus to the localized nature of the 3d states in trans- on the very nature of the exchange mechanism driv- ition metal (TM) adatoms, previous studies on 2D ing the formation of the magnetic order in NbSe in materials have emphasized the need of including an presence of selected magnetic adatoms, namely Cr, explicitcorrectionduetothelocalCoulombinterac- Mn,FeandCo.Bymeansofab-initioelectronicstruc- tion [13,18,19,41,54]. This term was then treated ture calculations based on density-functional the- in the Hartree–Fock approximation, via the DFT+U ory (DFT) , we analyze the magnetic landscape method[55,56].InVASP,weemployedtherotation- associated to different positions of adatoms on the allyinvariantformulationproposedbyLiechtenstein NbSe monolayer, showing the presence of several et al . The Coulomb interaction parameters U stableminimaatdifferentheights.Afterobtainingthe andJ forthe3dstatesoftheadatomswerechosenon ground state structures, we extract the inter-atomic the basis of previous studies [41, 54], as 4.0eV and exchange interactions between adatoms and show 0.9eVrespectively.Alltheotherstatesweretreatedin that they have a damped oscillatory nature, which standard DFT. Finally, we also investigated the effect is caused by the Ruderman–Kittel–Kasuya–Yosida ofvdWinteractionsonthecalculations,byusingthe (RKKY)-type coupling [43, 44]. Most importantly, DFT-D3 correction method by Grimme et al . the qualitative features of the magnetic coupling We found out that corrections to the DFT+U res- can be connected to the details of the Fermi sur- ults are rather small, as better illustrated in the face (FS). For Fe and Co, the presence of a nesting supplementarymaterial. 2 2DMater.9(2022)045012 SSarkaretal 2.2.Electronicandmagneticproperties 3.Interplaybetweenstructureand After performing the structural optimization, we magnetism investigatedthedetailsoftheelectronicandmagnetic properties by means of an all-electron theory, which We performed the structural relaxation of Cr, Mn, provides a more accurate solution, albeit at a higher Fe and Co adatoms on 1H-NbSe monolayer. We computational cost . To this aim, we employed focused on the 4×4 supercell, where the adatoms the full potential linear muffin-tin orbital method as are sufficiently distant to investigate the long range implementedintheRelativisticSpinPolarizedToolkit characterofthemagneticinteraction.Asillustratedin (RSPt) [60, 61]. The radii of the muffin-tin spheres figure1(a),weconsideredthreepossiblepositionsfor were set to 2.21÷2.34, 2.15÷2.27 and 2.09÷2.18 theadatoms,whicharethesitesontopofNbionsor a.u. for respectively Nb, Se and TM adatoms. The Seions,aswellasthehollow(H)site.Otherpossible intervals refer to the small variations when going positions, as e.g. the bridge site positions between from Co (smallest values) to Mn and Cr (largest val- Nb and Se atoms, can be ignored based on previ- ues). The basis functions were formally divided in ous studies on similar systems [18, 19, 41, 69, 70]. two energy sets, one for the valence states and one We first performed a detailed analysis of the ener- for the semi-core states. The former included 5s 5p getic landscape without including spin polarization. 4d states for Nb, 4s 4p 4d states for Se, and 4s 4p 3d These data, which are discussed more extensively in states for adatoms. The latter included 4s 4p states the appendix, show that adatoms located on top of Nb, 3d states for Se, and 3s 3p states for adatoms. Nb can relax in two different configurations. In the Three kinetic energy tails were used for the valence first one, which we label as N , the adatom is loc- sp states, corresponding to the values of −0.1, −2.3 ated above the NbSe monolayer, as expected. In the and −1.5Ry. Only the first two tails were used for second one, which we label as N , the adatom sinks alltheotherbasisfunctions.TheDFT+Ucorrection slightly below the plane of the Se ions, pushing the was applied using the spin and orbital rotationally neighbors further away. The adatoms at the hollow invariant formulation described in [62, 63], using site and at the Se site relax to a single configuration a local basis that was constructed from the muffin- each, and for convenience we label them as H and S. tinheads.Thedoublecountingcorrectionwasbased Thefourpossibleconfigurationsobtainedarevisual- on the fully localized limit [56, 64]. The four-index izedinfigure1(c). U-matrix was constructed using the same Coulomb We can now proceed to investigate how mag- interaction parameters used in VASP and hence, in netism affects this scenario, by performing spin- absence of spin–orbit coupling, the two formalisms polarized calculations in DFT. The relative energy should be absolutely equivalent . The remaining ∆E, the height of the adatom above the Se plane, computational settings, including k-point mesh and h , as well as the magnetic moments of the impur- TM integrationmethod,werealsosettobeinaccordance ity and the nearest Nb atoms, respectively µ and TM with VASP, in order to ensure the consistency of our µ , are reported in table 1. For all impurities, h is Nb TM resultsacrossthemanuscript.Anexampleofthegood increasedwithrespecttothedatawithoutspinpolar- agreementbetweenthebandstructuresobtainedwith ization, which is consistent with an increased bond these two codes is reported in the SM. RSPt was also lengthassociatedto3dmagnetism.Thismechanism, usedtocalculatetheinter-atomicexchangeparamet- however,doesnotaffecttheN configuration,dueto ers,bymappingthemagneticexcitationsontothefol- the geometrical constraints on the adatom. As a res- lowingHeisenbergHamiltonian: ult, the N configuration is energetically penalized, by a correction that is proportional to the magnetic moment. Hence, all the adatoms prefer to occupy ⃗ ⃗ H = − J e ·e (1) ij i j the H position. As for the non magnetic calcula- i̸=j tions, the S configurations are always much higher HereJ istheexchangeinteractionbetweenthespins in energy. Focusing on the magnetic moments, we ij at sites i and j, while ⃗e and ⃗e are unit vectors along see an evident correlation between µ and h , for i j TM TM themagnetizationdirectionatthecorrespondingsite. alladatoms.IntheSconfiguration,thelargestheight TheJ ’swerecalculatedbyemployingthegeneralized corresponds to the largest moment, characterized by ij magnetic force theorem [66, 67], whose implement- a small hybridization with the NbSe monolayer. In ationinRSPtisdescribedin.Thelocalbasisfor the N configuration, the height and the moment the calculation of the J ’s was chosen to be equival- areslightlydecreased,whichindicatesalargerhybrid- ij ent to the one used in DFT+U, which was described ization. A further decrease is observed for the H above. Finally, the convergence of the inter-atomic configuration, which we already identified as the exchange parameters up to the precision used in our most favourable arrangement. Finally, the moment dataanalysisrequiredtheuseofaveryfinesampling is drastically quenched in the N configuration, due oftheBrillouinZone,extendinguptoameshconsist- to the strong overlap with states from Nb and Se. ingof70×70×1k-points. For Fe and Co, the hybridization is so strong that 3 2DMater.9(2022)045012 SSarkaretal Figure1.(a)Topviewofthe4×4supercellof1H-NbSe monolayer,showingthethreeconsideredsittingpositionsfortheTM atoms.Thedottedlineindicatesthesizeofthesupercell.(b)SideviewofapartoftheNbSe monolayer.ThetopmostlayerofSe atomsisshownwithadottedline,whichisusedasareferencefortheheightoftheTMatoms.(c)Schematicsshowingthefour possibleequilibriumpositionsidentifiedinDFTwithoutspinpolarization.IntheN configuration,theTMatomsitsatafinite positiveheightabovetheSelayer,whereasintheN configurationtheTMatommovesclosertotheNbatoms,makingthe heightnegative. Table1.Totalenergy ∆E(eV)relativetothegroundstate,heightoftheadatomh (Å)abovetheSeplane,totalmagneticmomentof TM theadatom µ (µ ),andtotalmagneticmomentoftheclosestNbatom µ (µ ).Thenegativesignof µ indicatesthatthismagnetic TM B Nb B Nb momentisanti-paralleltothemagneticmomentofthetransitionmetaladatom.DataforbothDFTandDFT+Uarereported.TheH, + − N ,N andSconfigurationsaredepictedinfigure1andexplainedinitscaption. DFT DFT+U ∆E h µ µ ∆E h µ µ TM TM Nb TM TM Nb Cr H GS 0.39 2.70 −0.21 0.24 1.47 4.20 −0.28 N 0.17 1.26 3.68 −0.20 GS 1.41 4.16 −0.24 N 0.50 −0.86 1.31 −0.10 1.67 −0.84 2.66 −0.46 S 1.48 2.21 4.27 −0.06 0.89 2.27 4.43 −0.07 Mn H GS 0.50 3.52 −0.18 0.08 1.18 4.40 −0.26 N 0.08 1.20 3.98 −0.15 GS 1.30 4.40 −0.15 N 0.61 −0.85 1.48 −0.30 1.91 −0.78 3.18 −0.56 S 1.88 2.29 4.66 0.02 1.27 2.38 4.89 0.04 Fe H GS 0.25 2.73 0.11 0.02 0.49 3.21 0.16 N 0.45 0.92 3.00 0.20 GS 1.03 3.34 0.28 N 0.18 −0.86 0.10 0.00 1.11 −0.64 2.34 −0.39 S 2.57 2.01 2.97 0.17 1.84 2.36 3.79 0.23 Co H GS −0.17 1.16 0.15 0.21 0.23 1.97 0.36 N 0.50 1.01 1.74 0.11 GS 1.12 1.94 0.20 N 0.29 −0.77 0.01 0.00 1.35 −0.87 0.02 0.00 S 2.65 1.98 1.92 0.13 2.30 2.07 2.22 0.22 the system does not even manage to form a signific- problem, we performed fully relaxed DFT+U cal- antlocalmoment.Amorequantitativeanalysisofthe culations,whoseresultsarereportedontherightside hybridization is provided in appendix B. Finally, we of table 1. As expected, the inclusion of an explicit also note that a small moment is induced on the Nb Coulomb interaction term increases the localiza- atoms.Forthenearestneighbors,thismomentisanti- tion of the TM-3d electrons, which in turn results ferromagnetically aligned for Cr and Mn, but ferro- in a weaker covalent bonding with the Nb-4d states. magneticallyalignedforFeandCo.Thisisconsistent Thus, the adatom moves farther from the mono- with the general behavior of these TMs in their bulk layer and µ increases, acquiring more atomic-like TM form[71,72]. character. This increased moment induces a larger Next, we consider that the physics of the 3d polarization of the Nb atoms too, i.e. a larger µ . Nb electrons on adatoms is usually not well cap- Although this may seem obvious at first, one should tured by standard DFT with local or semi-local also keep in mind that the increased localization of functionals[13, 18, 19, 41, 54]. To remedy this theTM-3dstatesalsoimpliesasmallerhopping,and 4 2DMater.9(2022)045012 SSarkaretal therefore a weaker exchange coupling with the Nb- of these systems. The second common feature for all 4d states. However, this effect does not seem to be the adatoms reported in figure 3 is that the inter- significantinthesesystems.Noticealsothatnomag- atomicexchangeinteractionoscillatesbetweenbeing netization is seen for Co in the N configuration, ferromagnetic(positivesign)andanti-ferromagnetic as the Coulomb interaction is not strong enough to (negative sign). This is another manifestation of the overcome the hybridization with the neighboring RKKY coupling and can also be connected to the orbitals,asbetterillustratedinappendixB.Concern- topology of the FS. Going more into the details of ing the energetic stability, in DFT+U all adatoms eachelement,wecannoticethatFeandCoarechar- prefer to arrange in the N configuration, in place acterizedbyoscillationsofsimilarperiodaswellasa of the H configuration. This is consistent with pre- similar scaling. For Cr, instead, the magnetic coup- vious studies on NbSe and MoS monolayers, des- ling seems to decay faster and the oscillations seem 2 2 pite the usage of supercells with different periodicit- tohaveaslightlyshorterperiod.Finally,thedecayof ies [19,41]. Finally, it is important to stress that our theexchangeinteractioninMnissofastthatwecan- analysisdidnotshowanyevidenceofmetastablespin notreallyresolvetheperiodoftheoscillationsforthe states,ase.g.thosereportedforadatomsongraphene inter-atomicdistancesunderconsideration. or MoS [19, 54]. This is due to the fact that NbSe Overall, figure 3 shows a non-trivial trend that 2 2 monolayer is metallic, which increases the hybridiz- cannot simply be interpreted in terms of a gradual ation of the 3d states of the impurity and therefore filling of the 3d orbitals. Nevertheless, a qualitative diminishes their atomic-like character. This is par- understanding can be gained by analyzing the basic ticularly important for DFT+U calculations, which features of the density of states (DOS) and FS. The may lead to a plethora of local minima for atomic- calculated FSs of all systems in their N configura- like systems as adatoms [73, 74]. Since the N and tionareshowninfigure4.Commonfeaturesinclude N configurations arise also without spin polariza- a triangular-shaped electron pocket at K or a lens- tion (see appendix A), they cannot be considered as shaped hole pocket at M, or both. The analysis of metastablespinstates. thebandstructureandthefatbands,includedinthe SM, illustrates that the electron pockets at K have 4.Long-rangeinter-atomicexchange major contributions from the TM-d states whereas coupling the hole pockets at M originate mainly from Nb-d 2 states. Besides these common features, some differ- Having clarified the relation between structural and ences are also evident for each adatom. The FS of magnetic configurations, we proceed to investigate the systems with Fe and Co have both electron and the long range behavior of the exchange interaction. hole pockets. Cr has only the electron pockets at K, We focus mainly on the ground state structure N , but no hole pockets at M, whereas the opposite hap- while additional data for the H configuration, which pens for Mn. A direct comparison between the sys- lays slightly above in energy, are shown in the SM. tems with Co and Fe suggests that the amplitude of Since we are interested in the long range coupling, it the J oscillations directly depends on the volume of ij is more convenient to discuss results obtained along the Fermi pockets. In fact, figure 4 shows that their the zigzag direction, a in figure 2(a), which ensures FSs are similar, but the volume of both electron and the maximum line density. Focusing on this direc- holepocketsissmallerforFethanforCo.Atthesame tion, we calculated the inter-atomic exchange inter- time, figure 3 shows that the J s of Fe and Co have ij actions J between adatoms at sites i and j, for Cr, oscillationsofsimilarperiod,butasmalleramplitude ij Mn, Fe and Co in the N configuration. The res- is observed for Fe, if compared to Co. In the case ults obtained from DFT+U calculations, as a func- of Cr, we see that the FS consists of small electron tionofthedistanceRbetweentheadatoms,areshown pockets at K, which originate from the Cr-d 2 states. in figure 3. For a better analysis of their asymptotic TheholepocketscomingfromNb-d statesaremiss- scaling, the J ’s have been multiplied times R . This ing,whichindicatesaweakcouplingbetweenCrand ij factorshouldaccountforthescalingexpectedforthe Nb states. As a result the J oscillations vanish very ij exchange coupling between two localized moments quicklyforincreasinginter-atomicdistance.ForMn, mediatedbya2Delectrongas,whichisageneraliza- weonlyhavetheholepocketsattheMpointscoming tionoftheRKKYinteraction[75,76].Previousstud- fromtheNb-d 2 states.TheMn-d 2 statesarefarfrom z z iesbasedonmodelHamiltonianshaveinfactshown theFermienergy,duetothelargeexchangesplitting. thattheinter-atomicexchangecouplingbetweenTM This is clearly visible in the TM-3d projected DOS, −2 adatoms on doped MoS [77, 78] scales as R at shown in figure 5. As a consequence of the limited largedistances.NbSe ismetallicfromtheoutsetand hoppinginvolvingtheTM-3dstates,theinter-atomic in principle one may expect a similar scaling even in exchangeinteractionhasaveryshortrange. absenceofdoping.However,theinspectionoffigure3 Amorecomprehensivetheoryforunderstanding reveals a more complex behavior, with a decay that thebehavioroftheRKKYcouplingbetweenadatoms is much faster than a quadratic scaling. As we will on metallic 2D materials can be formulated bor- see below, this is a consequence of the particular FS rowing from the seminal works by Rothetal , 5 2DMater.9(2022)045012 SSarkaretal Figure2.(a)Topviewofthe4×4NbSe supercellwithTMadatomsintheN configuration,includingthelatticevectorsa 2 1 anda aswellastheWigner–Seitz(WS)cell.J istheinter-atomicexchangeinteractionbetweenthecentralTMatomandits 2 02 secondneighboralongthea latticevectordirection.(b)CorrespondingBrillouinZone(BZ),includingthetworeciprocallattice vectorsb andb ,aswellashighsymmetrypoints.Thereciprocalspacedirectionalong Γ–K correspondstotherealspace 1 2 directionalonga . Figure3.Inter-atomicexchangeinteractionJ betweenadatomsatsitesiandjalongthezigzagdirection,asafunctionoftheir ij + 2 distanceR.DataforCr,Mn,FeandCointheN configurationinDFT+Uareshown.NotethattheJ ’saremultipliedbyR to ij takeintoaccounttheirexpectedasymptoticbehaviorandmakethelongrangeoscillationsmorevisible. 6 2DMater.9(2022)045012 SSarkaretal Figure4.FSofCr,MnFe,andCoonmonolayerNbSe intheN configuration,asobtainedinDFT+U.Twobasicfeaturesare visible:atriangular-shapedelectronpocketatKandalens-shapedholepocketatM.Thedominantspinandorbitalcharacterof theFermipocketsisalsoindicated. Figure5.ProjectedDOSfortheTM-3dstatesforselectedadatomsonNbSe asobtainedinDFT+U.Positiveandnegativesigns correspondtomajorityandminorityspins,respectively. 7 2DMater.9(2022)045012 SSarkaretal Figure6.Fittingoftheinter-atomicexchangeinteractionJ withanexponentiallydecayingsinewavefor(a)Coand(c)Fe. ij (b)PossiblecalipervectorsidentifiedintheFSofCo.InthecaseofFe,weobtainthesamevectors,albeitwithadifferentlength. (d)TableillustratingthelengthsofthepossiblecalipervectorsfoundforFeandCo. −1 Brunoetal[80–84]andPajdaetal.Theconcept and η =0.03 Å , providing a more quantitative Co of complex FS [44, 83] helps us understand why connection between J ’s and FS. The periods of the ij the long range exchange coupling decays faster than oscillationswereinsteadfoundtobeverysimilar,i.e. 2 −1 −1 R . As shown in figure 4, the contribution of the Q =0.110Å andQ =0.105Å .Thesevectors Fe Co TM states to the FS include at most one spin chan- canbetracedbacktothedetailsoftheFSandinpar- nel, which means that a gap characterizes the other ticulartothepossiblecalipervectors[44,79,87,88]. spin channel. This leads to an additional exponen- The latter have to be parallel to the Γ–K direction, tial decay that dominates over the quadratic scal- i.e. parallel to the real space direction a that we ing, as it happens e.g. in strong ferromagnets [44, used to calculate the J ’s, see figure 2(b). As illus- ij 85]. This also explains why Mn stands out as hav- trated in figure 6(b) for Co, we can identify a total ing an exchange coupling that has a much shorter of seven possible caliper vectors that are parallel to range and no marked oscillations, given that its Γ–K . Their precise values for Fe and Co are repor- FS shows states that have no dominant TM con- ted in figure 6(d). It is evident that the caliper vec- tribution, for either spin. Determining the pre- tors that are in better agreement with the fitting are cise factor governing the decay is complicated, but those labeled as Q , connecting the electron pockets insightful information can be obtained by fitting the at the K or K points to the hole pockets at the M −1 −1 inter-atomic exchange via an expression as J(R) ∼ pointsandhavingavalueof0.133Å and0.117Å −2 Asin(Q×R + ϕ)R exp(−ηR). Here, A and Q are forrespectivelyFeandCo.Thesecalipervectorshave respectivelyamplitudeandperiodoftheoscillations, also important nesting properties, connecting large ϕ is a phase factor and η describes the exponential parallel regions from the electron and hole pockets. decay. In figure 6, we illustrate the results of the fit- The small discrepancy between the identified caliper ting for Fe and Co, which are the two elements for vectors and those obtained from the fitting is attrib- whichtheRKKYnatureofthecouplingismoreevid- utedtothenumericaluncertaintiescharacterizingthe ent. Although in principle one could have a super- latter. position of more oscillatory functions , the data When comparing the properties of the exchange in figure 6 suggest that a single mode is sufficient coupling for Fe and Co adatoms, we should keep in to describe the asymptotic behavior of Fe and Co mindthatthedifferencebetweenthesetwosystemsis adatoms on NbSe . The coefficients determining the notlimitedtotheFS,butinvolvesalsothelocalcharge −1 exponential decay were found to be η =0.04 Å andmagneticmoment,whichinturnaffectthelocal Fe 8 2DMater.9(2022)045012 SSarkaretal Figure7.(a)SystemswheretheheightoftheCoadatomabovetheSeplaneismodifiedof ±0.20Åwithrespecttoitsequilibrium value;norelaxationofthedegreesoffreedomofNbSe isperformedafterfixingtheheight.(b)Comparisonofthebandstructure obtainedfordifferentheightsoftheCoatomalongtheimportanthigh-symmetrypathsintheBZ.(c)Comparisonofthe inter-atomicexchangeinteractionsJ sfordifferentheightsoftheCoatom. ij hybridization. To avoid these multiple changes, we caliper vector Q remains unchanged is that it con- performed additional calculations for the Co system nects two parallel regions of electron and hole pock- only, where the height of the impurity is varied with ets that move toward the same direction when these respecttoitsGSvalue,asshowninfigure7(a).Relat- pockets are both increased or decreased in volume. ingthechangesobtainedfortheJ stothoseobtained Therefore, we can conclude that this caliper vector ij in the FS can help us clarify if the previously iden- determines the long-range behavior of the magnetic tified caliper vectors are really those responsible for couplingforFeandCoadatomsonNbSe . the long-range behavior of the RKKY coupling. The Forsakeofcompleteness,weinvestigatedhowthe inter-atomic exchange interactions calculated in the previous discussion is affected by the precise choice systems where the Co adatom height is modified of of the Coulomb interaction parameters. To this aim, ±0.20Å with respect to h =1.12 Å are reported in we performed selected calculations for Co adatoms GS intheN configuration,whoseresultsareillustrated figure 7(c). It is clear that the changes induced by a different height are minimal and concern mainly the in the SM. Despite some quantitative differences in themagneticcouplingatshortdistances,theasymp- amplitudeofthewaveandnotitsperiod.Thecorres- totic behavior of the long-range oscillations remains pondingbandstructuresalongtherelevanthighsym- unchanged. This reflects the fact that the degree of metrydirectionsareshowninfigure7(b).Weseethat localization of the 3d states of the adatoms, which increasing the height leaves the electron pocket at K substantially unvaried, while the hole pocket around is directly affected by U, does not induce relevant M becomes smaller, due to the band moving down- changesinthecharacteristicsoftheFS.Overall,these resultsshowthatourconclusionsaresolidanddonot ward.Decreasingtheheight,instead,leadstoalarger electron pocket at K and a larger hole pocket at M, dependonthedetailsofourcalculations. with their corresponding bands moving downward Finally, we also performed the analysis of the inter-atomic exchange interaction of the adatoms in andupward,respectively.FromtheFS,wecanextract thefullsetofcalipervectorsandtheirvariationswith H configuration, corresponding to the first excited respecttotheheight,asreportedintable2.Aswecan state. These data, presented in the SM, show a more see, the smallest absolute and relative variations are complex behavior, where one cannot resolve well obtainedforthevectorQ ,whichisexactlytheonewe defined oscillatory asymptotics. This behavior can hadidentifiedabove.Alltheothervectorsexperience be traced back to a higher complexity of the FS, a change of about 10%, which does not correspond indicating the presence of two overlapping modes. to what observed in figure 7(c). The reason why the The exponential decay is also present and seems to 9 2DMater.9(2022)045012 SSarkaretal −1 Table2.PossiblecalipervectorsQ(Å )calculatedwhentheheighth(Å)oftheCoadatomonNbSe ischanged.Theabsolute variationiscalculatedusingtherangeofthevalues,whiletherelativevariationisobtainedfromitbydividingbytheGSvalue.No furtherstructuralrelaxationisperformedwithrespecttotheGSconfiguration.Theprecisevisualizationofthecalipervectorsis providedinfigure6(b). h Variation Co 0.92 1.12 1.32 Absolute Relative Q 0.081 0.074 0.072 0.009 12.2% Q 0.206 0.184 0.186 0.020 10.9% Q 0.116 0.117 0.118 0.002 1.7% Q 0.142 0.160 0.160 0.018 11.3% Q 0.079 0.089 0.086 0.010 11.2% Q 0.178 0.193 0.202 0.024 12.4% Q 0.270 0.255 0.249 0.021 8.2% be more marked than the one observed for the N difficulties, having an exchange interaction that is configuration. smallerandextremelyshortranged.Nevertheless,this limitation may become an advantage for controlling the spin interaction at the atomic scale and tailor 5.Conclusions more complex systems with well defined magnetic properties. Since monolayer NbSe is a supercon- In this work, we have investigated the nature of ductor,themanipulationofthemagneticcouplingis the magnetic coupling between adatoms deposited alsoconnectedtothestudyofthecoexistenceofmag- on a monolayer of NbSe . We have shown that the neticorderandsuperconductivity.Asrecentlyshown, adatoms may occupy several stable configurations, thelocalmagneticorderleavesaclearsignatureonthe whose energetic hierarchy depends on the method Yu–Shiba–Rusinov states that, in presence of strong usedforthecomputations.InDFT+U,whichwecon- spin-orbit coupling, may support low-dimensional sider as the method of choice among those we used, topological superconductivity with Majorana bound all adatoms prefer to occupy a position just on top states. of the Nb atoms and above the Se plane, which we namedasN configuration.Themagneticmoments Dataavailabilitystatement obtained for each adatom are found to be slightly lower than their corresponding ionic values, which The data that support the findings of this study are reflects the presence of a finite hybridization with availableuponreasonablerequestfromtheauthors. the metallic substrate. The calculated inter-atomic exchangecouplingsshowthatpairsofadatomsinter- actwitheachotherviaRKKYinteraction,accompan- Acknowledgments ied by a further exponential decay that is due to the absence of one of the spin channels from the FS, in We thank Prof. Yunkyu Bang and Prof. Peter Wahl the case of Cr, Fe and Co. For these elements the for useful discussions on this study. The computa- exchangeisfoundtooscillatebetweenferromagnetic tional work was enabled by resources provided by and anti-ferromagnetic character, and the period of the Swedish National Infrastructure for Computing the oscillation is found to be determined by the cal- (SNIC)attheNationalSupercomputerCentre(NSC) ipervectorconnectingelectronpocketsatKandhole of Linköping University (Sweden) and at the High pockets at M. We have also shown that this vector Performance Computing Centre North (HPC2N), is rather insensitive to changes of the height of the partially funded by the Swedish Research Council adatom.Thisimpliesthatonecaninprinciplemanip- through Grant Agreement No. 2018-05973. Finan- ulate the magnetic character of pair of atoms on top cial support from the National Research Found- of NbSe by moving them in-plane, without signi- ation (NRF), funded by the Ministry of Science ficant effects arising from the out-of-plane modi- of Korea, is acknowledged by F C (Grant Nos. fications. This seems to offer a better realization of 2017R1D1A1B03033465, 2020R1C1C1005900 and themagneticinteractioncontrolbetweenadatomsby 2022R1I1A1A01071974), as well as by S S and IDM means of external bias voltage, recently proposed by (Grant No. 2020R1A2C101217411). I D M is also Badrtdinovetalforphosphorene.Furthermore, supported by the appointment to the JRG program considering that NbSe is metallic and that adatoms at the APCTP through the Science and Technology have a single spin configuration (without low and Promotion Fund and Lottery Fund of the Korean high spins), probing their magnetic interactions via Government, as well as by the Korean Local Gov- inelasticelectrontunnelingspectroscopyseemsmuch ernments, Gyeongsangbuk-do Province and Pohang more feasible. Cr, Fe and Co adatoms seem easier to City.YOKacknowledgesthefinancialsupportfrom be measured, as they experience a stronger exchange theSwedishResearchCouncilVR(ProjectNo.2019- interaction. Mn adatoms may instead involve more 03569)andGöranGustafssonFoundation. 10 2DMater.9(2022)045012 SSarkaretal TableA1.Totalenergy ∆E(eV)relativetothegroundstateandheightoftheadatomh (Å)abovetheSplane,asobtainedinDFT TM withoutspinpolarization. ∆E h TM + − + − H N N S H N N S Cr GS 1.26 0.17 4.78 −0.03 1.07 −0.87 1.87 Mn 0.06 1.24 GS 4.68 −0.22 1.00 −0.88 1.85 Fe 0.08 1.22 GS 4.26 −0.28 0.93 −0.86 1.87 Co GS 0.85 0.28 3.40 −0.25 0.89 −0.76 1.91 AppendixA AppendixB Here we focus on the energetic landscape obtained Inthissectionweprovideamorequantitativeanalysis in DFT without considering any spin polarization. of the hybridization of the TM-3d states with their In table A1, we illustrate the energy and equilib- chemical environment. To this aim, we will employ rium height of all the investigated adatoms posi- the local hybridization function ∆(E), which can be tioned on the NbSe monolayer. The positions are obtained directly from the local Green’s function, labeled as explained in the main text and illustrated projected on a given set of local orbitals . Recent infigure1(c).WefirstnoticethattheenergyoftheS workshaveshownthatthehybridizationfunctioncan configurationismuchhigherthanthoseoftheother provide significant insight into the physical proper- configurations, for all adatoms, and is also accom- ties of various compounds [90–92]. For sake of sim- paniedbythelargestheight.Thissuggestthatalarge plicity we will limit our analysis to Mn impurities, energy gain results from the strong covalent interac- as the other impurities show a similar behavior. The tionbetweentheTMandNbatoms,whichispresent trace of the imaginary part of ∆(E) for the Mn-3d for all configurations except S. As a matter of fact, states, as obtained in spin-polarized DFT, is illus- onecanseeastrongcorrelationbetweentheequilib- tratedinthebottompaneloffigureB1.The4differ- rium height and the relative energy with respect to entcurvescorrespondtothefourpossibleconfigura- thegroundstate.Asdiscussedinthemaintext,there tions discussed in the text. The other three panels of aretwopossibleequilibriumheightsforadatomsloc- figure B1 show the total DOS, as well as the projec- ated on top of the Nb site. These two stable con- ted densities of the 4p states of the nearest Se atoms figurations arise from the competition between two and the 4d states of the nearest Nb atoms. Focus- possiblemicroscopicinteractions.Oneofthemisthe ing on ∆(E), we observe that the smallest hybrid- aforementionedcovalentinteractionbetweentheTM ization is found for the S configuration (blue dot- atom and the Nb atoms. The other one is due to ted line), which is consistent with having the largest the steric effects between the TM atom and the top- distances from Nb and Se atoms. A slightly larger most Se layer. The competition between these two hybridization is observed for the N configuration types of interaction results in a potential well with (red dashed line), mainly determined by the Se-4p twominimawithrespecttothevariationoftheheight states.Amorestructuredhybridizationcharacterizes of the TM atom along the z-direction, perpendicu- the H configuration (black line), which is also the larly to the monolayer. The location of the two min- groundstateinspin-polarizedDFT.Thelargestpeak ima is on either side of the top Se layer, separated by is found at −1.5eV and can find correspondence in the potential barrier resulting from the steric effects. theSe-4pstatesinthesameenergyrange.Finally,the Therefore, we obtain two different geometries at the largest hybridization is found for the N configura- + − Nb site, which were labeled as N and N in the tion (green line) and the most important contribu- main text. As expected from the large difference in tionisvisiblearoundtheFermienergy.Inthisenergy height, the N configuration is more favourable in range, there is a clear correspondence with Nb-4d energy than the N configuration, when spin polar- states,Se-4pstates,aswellasotherstatesarisingfrom ization is not considered. Table A1 shows that the differentelectronicshellsandvisibleinthetotalDOS. H configuration and the N configuration are very The strength of the hybridization is also found to be close in both height and energy. In fact, Cr and proportional to the difference observed between the Co prefer the H structure, while Mn and Fe prefer calculatedMnmagneticmoment,reportedintable1, theN structure. and its expected ionic value. 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2D Materials – IOP Publishing
Published: Oct 1, 2022
Keywords: surface adatoms; magnetic interactions; Heisenberg model; first-principles calculations; transition metal dichalcogenides
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