Calculations of the Spin Structure of Trimer Cr3

Calculations of the wind Structure of Trimer Cr3Calculation of Magnetic Properties by Generalized Spin Hamiltonian and Generationof Global web Cr Trimer in molecule and on surfaceOleg V. Stepanyuk2, Oleg V. Farberovich11 Raymond and Bekerly Sackler Faculty of Exact Sciences,School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv 69978, Israel2 easy lay Planck Institute of Microstructure Physics, Halle, GermanyHere we present the results of the first-principles calculations of the straining structure of trimer Cr3with the use of a density- theatrical roleal scheme allowing for the non-collinear spin configurations in1. Using the results of these calculations we determine the Heisenberg-Dirac-Van Vleck (HDVV)Hamiltonian with an identical exchange couplings parameters linking the Cr ions with predominantspin density. The energy pattern was found from the effective HDVV Hamiltonian acting in therestricted spin space of the Cr ions by the application of the irreducible tensor op erators (ITO) technique. Comparison of the energy pattern with that obtained with the eolotropic exchangemodels conventionally used for the analysis of this system and with the results of non-collinearspin structure calculations show that our interwoven investigations provides a substantially description of thepattern of the spin levels and spin structures of the nano attractionized trimer Cr3. The results arediscussed in the view of a general line of spin frustration related to the orbital degeneracy ofthe antiferromagnetic ground state.PACS numbersI. INTRODUCTIONInformation technologies provide very interesting challengesand an extremely wide playground in which scientistsworking in materials science, chemistry, natural philosophy andnano-fabrication technologies may find stimuli for novelideas. Curiously, the nanometre scale is the molecularscale. So we may wonder whether, how or simply whichfunctional molecules bottom of the inning be regarded in some ways aspossible com ponents of nanodevices. The goal is ambitiousit is not just a matter to store information in a 3dmetaltrimer on a non-magnetic substrate, but we maythink to process information with a trimer and then tocommunicate information at the supramolecules containgfrom magnetic 3d-metal trimer on a surface.Spins are alternative complementary to charges as degreesof freedom to encode information. Recent examples,like for instance the discovery and application of Giantmagnetoresistance in Spintronics, take demonstratedthe effectual use of spins for information technologies.Moreover, spins are intrinsically quantum entities andthey have therefore been widely investigated in the surface areaof quantum-information processing. Molecular nanomagnetsare real examples of finite spin bonds (1D) or clusters(0D), and therefore they constitute a new benchmarkfor testing models of interacting quantum objects.New physics of molecular magnets feeds hopes of certainprospective applications, and such hope s pose theproblem of understanding, improving, or predicting desirablecharacteristics of these materials. The applicationswhich come into discussion are, for instance, magneticstorage (one molecule would store one bit, withmuch higher information storage density than favorablewith microdomains of present-day storage media or magneticnanoparticles of next future). In order to exploitthe quantum features for information processing, molecularspin clusters have to fulfil some underlying requirements.Magnetic transition metal nanostructures on nonmagneticsubstrates have attracted recently large attention cod to their novel and unusual magnetic properties2,3. The supported clusters inhabit both thereduction of the local coordination number, as in freeclusters, as well as the interactions with the electronicdegrees of freedom of the substrate, as in embedded clusters.The complex magnetic behavior is usually associatedwith the competition of several(prenominal) interactions, suchas inter atomic exchange and bonding interactions, andin some cases noncollinear effects, which merchantman give rise toseveral metastable states close in energy. The groundstate can therefore be easily tuned by external actiongiving rise to the switching between diametric states.In recent years, network has attracted the attentionof many physicists working in the area of quantummechanics 1, 2. This is due to the ongoing research inthe area of quantum information 3. Theoretical studiesare also important in the context of spin interactionsinside two structured reservoirs 9 such as genius magneticmolecule (SMM) and metal cluster on nonmagneticsurface. Cr is unique among the transition-metaladatoms, because its half-filled valence configuration(3d54s1) yields both a large magnetic indorsement and stronginteratomic bonding leading to magnetic frustration. Weapply our method to Cr trimers deposited on a Au(111)surface and the trinuclear hydroxo-bridged chromiumammine complex Cr3(NH3)10(OH)4 Br5 3H2O.Low-lying kindle states of a magnetic system are generallydescribed in ground of a general spin-Hamiltonian.For a magnetic system with many spin sites, this phenomenologicalHamiltonian is written as a sum of pairwisespin exchange interactions between adjacent spinsites in molecule and surface.In the present work we athletic field network betweenthe spin states in the spin spectrum. In our model, aspin state interact with a continuum of the spin structureat breakup temperature 0 300 K, and entanglementproperties between the spin states in spin structure are realizeed. Using ball-shaped entanglement as a prize ofentanglement, we derive a pair of distributions that canbe interpreted as densities of entanglement in terms ofall the eigen valuate of the spin spectrum. This distributioncan be calculated in terms of the spectrum of spin excitationof cluster surface and supramolecule. With thesenew measures of entanglement we can study in detailentanglement between the spi n modes in spin structure.The method developed here, in terms of entanglementdistributions, can also be used when considering varioustypes of structured reservoirs ...II. THE conjectural APPROACHIn order to give a theoretical description of magneticdimer we exploit the irreducible tensor operator (ITO)technique ITO. Let us consider a spin cluster of arbitrarytopology formed from an arbitrary number of magneticsites, N, with local spins S1, S2,, SN which, ingeneral, can have different values. A successive spin couplingscheme is adoptedS1 + S2 = S2, S2 + S3 = S3, , SN1 + SN = S,where S represents the completed set of intermediate spinquantum numbers Sk, with k=1,2,,N-1.The eigenstates v of spin-Hamiltonian pass on be given by linear conspiracysof the basis states ( S)SM v =(S)SM(S)SM v (S)SM, (1)where the coefficients ( S)SM v can be evaluated oncethe spin-Hamiltonian of the system has been diagonalized.Since each term of spin-Hamiltonian can be rewrittenas a combination of the irreducible tensor operatorstechniqueITO.In ITO work focus on the main physicalinteractions which determine the spin-Hamiltonian andto rewrite them in terms of the ITOs. The exchangepart of the spin-Hamiltonian is to introducedHspin = H0 + HBQ + HAS + HAN. (2)The first term H0 is the Heisenberg-Dirac Hamiltonian,which represents the isotropic exchange interaction, HBQis the biquadratic exchange Hamiltonian, HAS is the antisymmetricexchange Hamiltonian,and HAN representsthe anisotropic exchange interaction. Conventionally,they can be expressed as follows ITOH0 = 2ifJif bSi bSf (3)HBQ = ifjif ( bSi bSf )2 (4)HAS =ifGif bSi bSf (5)HAN = 2if_J_ifbS_ibS_f (6)with = x, y, z We can add to the exchange Hamiltonianthe term due to the axial single-ion anisotropyHZF =iDi bSz(i)2 (7)where Jif and J_if are the parameters of the isotropic andanisotropic exchange iterations, jif are the coefficients ofthe biquadratic exchange iterations,and Gif=-Gfi is thevector of the antisymmetric excha nge. The terms of thespin-Hamiltonian above can be written in terms of theITOs.Both the HeisenbergDirac and biquadratic exchangeare isotropic interactions. In fact, the correspondingHamiltonians can be described by rank-0 tensor operatorsand thus have non zero ground substance elements onlywith states with the same total spin quantum numberS (S,M=0). The representative matrix can be decomposedinto blocks depending only on the value of Sand M. All anisotropic terms are described by rank-2tensor operators which have non zero matrix elementsbetween state with S=0,1,2 and their matrices cannot be decomposed into blocks depending only on totalspin S in account of the Smixing between spin stateswith different S. The single-ion anisotropy can be writtenin terms of rank-2 single site ITOs ITO. Finally,the antisymmetric exchange term is the sum of ITOs ofrank-1.The ITO technique has been used to design the MAGPACKsoftware ITO1, a package to calculate the energylevels, bulk magnetic properti es, and inelastic neutronscattering spectra of high nuclearity spin clusters thatallows studying efficiently properties of nanoscopic magnets.A. Calculation of the magnetic propertiesOnce we have the energy levels, we can evaluate differentthermodynamic properties of the system as magnetization,magnetic susceptibility, and magnetic specificheat. Because anisotropic interactions are not included,the magnetic properties of the anisotropic system do notdepend on the direction of the magnetic field. For thisreason one can consider the magnetic field directed alongarbitrary axis Z of the molecular coordinate frame thatis chosen as a spin quantization axis. In this case theenergies of the system will be _(Ms)+geMsHZ, where_(Ms) are the eigenvalues of the Hamiltonian containingmagnetic exchange and double exchange contributions(index runs over the energy levels with given totalspin protection Ms). Then the segmentation function in thepresence of the external magnetic field is given byZ(H Z) =Ms_exp_(Ms)/kTMsexpgeMsHZ/kT(8)Using this expression one can evaluate the magnetic susceptibility and magnetization M by standart thermodinamicaldefinitions =(MH)H0(9)M(H) = NkTlnZH(10)B. Entanglement in N-spin systemEntanglement has gained renewed interest with the developmentof quantum information science. The problemof measuring entanglement is a vast and lively field of researchin its own. In this parting we attempt to solve theproblem of measuring entanglement in the N-spin clusterand supramolecules systems. Based on the residualentanglement 9 (Phys. Rev. A 71, 044301 (2005)), wepresent the global entanglement for a N-spin state for thecollective measures of multiparticle entanglement. Thismeasures introduced by Meyer andWallach... The MeyerWallach(MW) measure written in the Brennen form(G.K.Brennen,Quantum.Inf.Comp.,v.3,619 (2003)) isQ() = 2(1 1NNk=1Tr2k) (11)where k is the reduced density matrix for k-th qubit.The problem of entanglement between a spin states inN-spin s ystems is becoming more interesting when consideringclusters or molecules with a unearthly gap in theirdensities of states. For quantifying the distribution ofentanglement between the individual spin eigenvalues inspin structure of N-spin system we use the density of entanglement.The density of entanglement (_, _, )dgives the entanglement between the spin eigenvalue _and spin eigenvalue _ with in an energy interval _ to_ + d_.One can show that entanglement distribution can bewritten in terms of spectrum of spin exitationS(_, ) = c_2 ( _) (12)and(_, _, ) = 2S(_, )S(_, ) (13)where coefficient c_ = ( S)SM v is eigenvector of thespin-Hamiltonian of the cluster or supramolecule. Thus,entanglement distributions can be derived from the excitationspin spectrumQ() = 1 222NN_=1c_2( _)2 + 2N_=_+1c_2( _)2 + 2(14)Though the very nature of entanglement is purelyquantum mechanical, we proverb that it can persist formacroscopic systems and will survive even in the thermodynamicallimit. In th is section we discuss how itbehaves at finite temperature of thermal entanglement.The states in N-spin system describing a system in thermalequilibrium states, are determined by the Generalizedspin-Hamiltonian and thermal density matrix(T) =exp(Hspin/kT)Z(HZ)(15)where Z(HZ) is the partition function of the N-spin system.The thermal entanglement isQ(, T,HZ) = 1 222NZ(HZ)2N_=1c_2 exp_/kT( _)2 + 2(16)N_=_+1c_2 exp_/kT( _)2 + 2The demonstration of quantum entanglement, however,can also be directly derived from experiments, withoutrequiring knowledge of the system state. This can bedone by using specific operatorsthe so-called entanglementwitnesseswhose expectation value is always positiveif the state is factorizable. It is instead remarkablethat some of these entanglement witnesses coincidewith well-known magnetic observables, such as energyor magnetic susceptibility = dM/dB. In particular,the magnetic susceptibility of N spins s, averaged overthree orthogonal spatial directions, is always larger thana threshold value if their equilibrium state is factorizableg g Ns/kBT EW. This should not be surprising,since magnetic susceptibility is proportional tothe variance of the magnetization, and thus it may actuallyquantify spin.spin correlation. The profit inthe use of this criterion consists in the fact that it doesnot require knowledge of the system Hamiltonian, providedthat this commutes with the Zeeman terms correspondingto the three orthogonal orientations of themagnetic field = x, y, z. As already mentioned, inthe case of the Cr3 trimer, the effective Hamiltonian includes,besides the dominant Heisenberg interaction J 118 meV , smaller anisotropic terms G 1.1 meV andD 0.18 meV , due to which the above commutation relationsare not fulfilled. This might, in principle, result indifferences between the magnetic susceptibility and theentanglement witness WE (see Fig.). Apparently, thedifference is quite essential and therefore it is necessary to use a formul a for global entanglement Q() in N-spinsystem.4101 100 101 102 10300.0050.010.0150.020.0250.030.0350.040.045The calculated difference EW(T)EWa(T)/EW(T)for Cr3 isosceles trimerT(K)EW(T)EWa(T)/EW(T)FIG. 1 (Color online) The calculated difference j EW(T) EWa(T) j =EW(T) for Cr3 isosceles trimer0100200300four hundred024600.20.40.60.811.2Angle(Degrees)The calculated M(H) for Cr3 isosceles trimerH(T)M(B)FIG. 2 (Color online)Magnetization M(H) of the Cr3isoscales trimer on metal surface as a function of angles from 0 to 360 degreeC. Thermal global entanglement in static magnetic_eld50 50 100 150 200 250 300 350 cd00.050.10.150.20.25The calculated variation of M(H) vs angle (magnetization switching)Angle(Degrees)M(B)0.1Ts0.2Ts0.5Ts1.0TsFIG. 3 (Color online)The calculated variation of M(H) vsangle (magnetization switching) for Cr3 isoscales trimerFIG. 4 (Color online)The calculated density of global entanglementvs temperature and energy for Cr3 isoscales trimer60100200300400024600.511.522.5 Angle(Degrees)The calculated M(H) for Cr3 molecular magnetH(T)M(B)FIG. 5 (Color online)Magnetization M(H) of the Cr3 molecularmagnet as a function of angles from 0 to 360 degree0 50 100 150 200 250 300 350 40000.050.10.150.20.250.30.350.40.450.5The calculated variation of M(H) vs angle (magnetization switching)Angle(Degrees)M(B)0.1Ts0.2Ts0.5Ts1.0TsFIG. 6 (Color online)The calculated variation of M(H) vsangle (magnetization switching) for Cr3 molecular magnet7FIG. 7 (Color online)The calculated density of global entanglementvs temperature and energy for Cr3 molecular magnetFIG. 8 (Color online)The calculated entanglement for theCr3 isoscales trimer as a function of temperature and theorder of magnitude of the magnetic field Hpar.8FIG. 9 (Color online)The calculated entanglement for theCr3 isoscales trimer as a function of temperature and themagnitude of the magnetic field Hper.FIG. 10 (Color online)The calculated entanglement for theCr3 isoscales trimer as a function of temperature a nd the magnitudeof the magnetic field Hav.9FIG. 11 (Color online)The calculated entanglement for theCr3 molecular magnet as a function of temperature and themagnitude of the magnetic field Hav.