Calculations of the Spin Structure of Trimer Cr3

Calculations of the Spin Structure of Trimer Cr3 Calculation of Magnetic Properties by Generalized Spin Hamiltonian and Generation  of Global Entanglement: Cr Trimer in molecule and on surface Oleg V. Stepanyuk2, Oleg V. Farberovich1 1 Raymond and Bekerly Sackler Faculty of Exact Sciences,  School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv 69978, Israel 2 Max Planck Institute of Microstructure Physics, Halle, Germany Here we present the results of the first-principles calculations of the spin structure of trimer Cr3  with the use of a density-functional scheme allowing for the non-collinear spin configurations in  [1]. Using the results of these calculations we determine the Heisenberg-Dirac-Van Vleck (HDVV)  Hamiltonian with anisotropic exchange couplings parameters linking the Cr ions with predominant  spin density. The energy pattern was found from the effective HDVV Hamiltonian acting in the  restricted spin space of the Cr ions by the application of the irreducible tensor operators (ITO)   technique. Comparison of the energy pattern with that obtained with the anisotropic exchange  models conventionally used for the analysis of this system and with the results of non-collinear  spin structure calculations show that our complex investigations provides a good description of the  pattern of the spin levels and spin structures of the nanomagnetic trimer Cr3. The results are   discussed in the view of a general problem of spin frustration related to the orbital degeneracy of  the antiferromagnetic ground state.   PACS numbers: I. INTRODUCTION Information technologies provide very interesting challenges  and an extremely wide playground in which scientists  working in materials science, chemistry, physics and  nano-fabrication technologies may find stimuli for novel  ideas. Curiously, the nanometre scale is the molecular  scale. So we may wonder whether, how or simply which  functional molecules can be regarded in some ways as  possible components of nanodevices. The goal is ambitious: it is not just a matter to store information in a 3dmetal  trimer on a non-magnetic substrate, but we may  think to process information with a trimer and then to  communicate information at the supramolecules containg  from magnetic 3d-metal trimer on a surface. Spins are alternative complementary to charges as degrees  of freedom to encode information. Recent examples,  like for instance the discovery and application of Giant  magnetoresistance in Spintronics, have demonstrated  the efficient use of spins for information technologies. Moreover, spins are intrinsically quantum entities and  they have therefore been widely investigated in the field  of quantum-information processing. Molecular nanomagnets  are real examples of finite spin chains (1D) or clusters  (0D), and therefore they constitute a new benchmark  for testing models of interacting quantum objects. New physics of molecular magnets feeds hopes of certain  prospective applications, and such hopes pose the  problem of understanding, improving, or predicting desirable  characteristics of these materials. The applications  which come into discussion are, for instance, magnetic  storage (one molecule would store one bit, with  much higher information storage density than accessible  with microdomains of present-day storage media or magnetic  nanoparticles of next future). In order to exploit  the quantum features for information processing, molecular  spin clusters have to fulfil some basic requirements. Magnetic transition metal nanostructures on nonmagnetic  substrates have attracted recently large attention  due to their novel and unusual magnetic properties[2,3]. The supported clusters experience both the  reduction of the local coordination number, as in free  clusters, as well as the interactions with the electronic  degrees of freedom of the substrate, as in embedded clusters.   The complex magnetic behavior is usually associated  with the competition of several interactions, such  as interatomic exchange and bonding interactions, and  in some cases noncollinear effects, which can give rise to  several metastable states close in energy. The ground  state can therefore be easily tuned by external action  giving rise to the switching between different states. In recent years, entanglement has attracted the attention  of many physicists working in the area of quantum  mechanics [1, 2]. This is due to the ongoing research in  the area of quantum information [3]. Theoretical studies  are also important in the context of spin interactions  inside two structured reservoirs [9] such as single magnetic  molecule (SMM) and metal cluster on nonmagnetic  surface. Cr is unique among the transition-metal  adatoms, because its half-filled valence configuration  (3d54s1) yields both a large magnetic moment and strong  interatomic bonding leading to magnetic frustration. We  apply our method to Cr trimers deposited on a Au(111)  surface and the trinuclear hydroxo-bridged chromium  ammine complex [Cr3(NH3)10(OH)4]Br5  · 3H2O. Low-lying excited states of a magnetic system are generally  described in terms of a general spin-Hamiltonian. For a magnetic system with many spin sites, this phenomenological  Hamiltonian is written as a sum of pairwise  spin exchange interactions between adjacent spin  sites in molecule and surface. In the present work we study entanglement between  the spin states in the spin spectrum. In our model, a  spin state interact with a continuum of the spin structure  at interval temperature 0 – 300 K, and entanglement  properties between the spin states in spin structure are  considered. Using global entanglement as a measure of  entanglement, we derive a pair of distributions that can  be interpreted as densities of entanglement in terms of  all the eigenvalue of the spin spectrum. This distribution  can be calculated in terms of the spectrum of spin excitation  of cluster surface and supramolecule. With these  new measures of entanglement we can study in detail  entanglement between the spin modes in spin structure. The method developed here, in terms of entanglement  distributions, can also be used when considering various  types of structured reservoirs [..]. II. THE THEORETICAL APPROACH In order to give a theoretical description of magnetic  dimer we exploit the irreducible tensor operator (ITO)  technique [ITO]. Let us consider a spin cluster of arbitrary  topology formed from an arbitrary number of magnetic  sites, N, with local spins S1, S2,, SN which, in  general, can have different values. A successive spin coupling  scheme is adopted: S1 + S2 = SËÅ"2, SËÅ"2 + S3 = SËÅ"3, , SNËÅ"à ´Ã¢â€š ¬Ã¢â€š ¬Ã¢â€š ¬1 + SN = S, where ËÅ" S represents the complete set of intermediate spin  quantum numbers SËÅ"k, with k=1,2,,N-1.The eigenstates  | và ¢Ã… ¸Ã‚ © of spin-Hamiltonian will be given by linear combinations  of the basis states | ( ËÅ" S)SMà ¢Ã… ¸Ã‚ ©: | và ¢Ã… ¸Ã‚ © = ÃŽ £ (~S )SM à ¢Ã… ¸Ã‚ ¨(~S )SM | và ¢Ã… ¸Ã‚ © | (~S )SMà ¢Ã… ¸Ã‚ ©, (1) where the coefficients à ¢Ã… ¸Ã‚ ¨( ËÅ" S)SM | và ¢Ã… ¸Ã‚ © can be evaluated once  the spin-Hamiltonian of the system has been diagonalized. Since each term of spin-Hamiltonian can be rewritten  as a combination of the irreducible tensor operators  technique[ITO].In [ITO] work focus on the main physical  interactions which determine the spin-Hamiltonian and  to rewrite them in terms of the ITO’s. The exchange  part of the spin-Hamiltonian is to introduced: Hspin = H0 + HBQ + HAS + HAN. (2) The first term H0 is the Heisenberg-Dirac Hamiltonian,  which represents the isotropic exchange interaction, HBQ  is the biquadratic exchange Hamiltonian, HAS is the antisymmetric  exchange Hamiltonian,and HAN represents  the anisotropic exchange interaction. Conventionally,  they can be expressed as follows [ITO]: H0 = −2 ÃŽ £ i;f Jif bSi bSf (3) HBQ = − ÃŽ £ i;f jif ( bSi bSf )2 (4) HAS = ÃŽ £ i;f Gif [ bSi Ãâ€" bSf ] (5) HAN = −2 ÃŽ £ i;f ÃŽ £ _ J_ if bS_ i bS_ f (6) with ÃŽ ± = x, y, z We can add to the exchange Hamiltonian  the term due to the axial single-ion anisotropy: HZF = ÃŽ £ i Di bSz(i)2 (7) where Jif and J_   if are the parameters of the isotropic and  anisotropic exchange iterations, jif are the coefficients of  the biquadratic exchange iterations,and Gif=-Gfi is the  vector of the antisymmetric exchange. The terms of the  spin-Hamiltonian above can be written in terms of the  ITO’s. Both the Heisenberg–Dirac and biquadratic exchange  are isotropic interactions. In fact, the corresponding  Hamiltonians can be described by rank-0 tensor operators  and thus have non zero matrix elements only  with states with the same total spin quantum number  S (ΔS,ΔM=0). The representative matrix can be decomposed  into blocks depending only on the value of S  and M. All anisotropic terms are described by rank-2  tensor operators which have non zero matrix elements  between state with ΔS=0, ±1, ±2 and their matrices can  not be decomposed into blocks depending only on total  spin S in account of the S–mixing between spin states  with different S. The single-ion anisotropy can be written  in terms of rank-2 single site ITO’s [ITO]. Finally,  the antisymmetric exchange term is the sum of ITO’s of  rank-1. The ITO technique has been used to design the MAGPACK  software [ITO1], a package to calculate the energy  levels, bulk magnetic properties, and inelastic neutron  scattering spectra of high nuclearity spin clusters that  allows studying efficiently properties of nanoscopic magnets. A. Calculation of the magnetic properties Once we have the energy levels, we can evaluate different  thermodynamic properties of the system as magnetization,  magnetic susceptibility, and magnetic specific  heat. Because anisotropic interactions are not included,  the magnetic properties of the anisotropic system do not  depend on the direction of the magnetic field. For this  reason one can consider the magnetic field directed along  arbitrary axis Z of the molecular coordinate frame that  is chosen as a spin quantization axis. In this case the  energies of the system will be à Ã‚ µ_(Ms)+geÃŽ ²MsHZ, where  Ãƒ Ã‚ µ_(Ms) are the eigenvalues of the Hamiltonian containing  magnetic exchange and double exchange contributions  (index ÃŽ ¼ runs over the energy levels with given total  spin protection Ms). Then the partition function in the  presence of the external magnetic field is given by: Z(HZ) = ÃŽ £ Ms;_ exp[−à Ã‚ µ_(Ms)/kT] ÃŽ £ Ms exp[−geÃŽ ²MsHZ/kT] (8) Using this expression one can evaluate the magnetic susceptibility  Ãâ€¡ and magnetization M by standart thermodinamical  definitions: χ = ( ∂M ∂H ) H!0 (9) M(H) = NkT ∂lnZ ∂H (10) B. Entanglement in N-spin system Entanglement has gained renewed interest with the development  of quantum information science. The problem  of measuring entanglement is a vast and lively field of research  in its own. In this section we attempt to solve the  problem of measuring entanglement in the N-spin cluster  and supramolecules systems. Based on the residual  entanglement [9] (Phys. Rev. A 71, 044301 (2005)), we  present the global entanglement for a N-spin state for the  collective measures of multiparticle entanglement. This  measures introduced by Meyer andWallach[..]. The MeyerWallach  (MW) measure written in the Brennen form (G.K.Brennen,Quantum.Inf.Comp.,v.3,619 (2003)) is: Q(ψ) = 2(1 − 1 N ÃŽ £N k=1 Tr[Ï 2 k]) (11) where Ï k is the reduced density matrix for k-th qubit.   The problem of entanglement between a spin states in  N-spin systems is becoming more interesting when considering  clusters or molecules with a spectral gap in their  densities of states. For quantifying the distribution of  entanglement between the individual spin eigenvalues in  spin structure of N-spin system we use the density of entanglement. The density of entanglement ÃŽ µ(à Ã‚ µ_, à Ã‚ µ_, à Ã‚ µ)dà Ã‚ µ gives the entanglement between the spin eigenvalue à Ã‚ µ_ and spin eigenvalue à Ã‚ µ_ with in an energy interval à Ã‚ µ_ to à Ã‚ µ_ + dà Ã‚ µ_. One can show that entanglement distribution can be  written in terms of spectrum of spin exitation S(à Ã‚ µ_, à Ã‚ µ) = |c_|2 ÃŽ ´(à Ã‚ µ − à Ã‚ µ_) (12) and ÃŽ µ(à Ã‚ µ_, à Ã‚ µ_, à Ã‚ µ) = 2S(à Ã‚ µ_, à Ã‚ µ)S(à Ã‚ µ_, à Ã‚ µ) (13) where coefficient c_ = à ¢Ã… ¸Ã‚ ¨( ËÅ" S)SM | và ¢Ã… ¸Ã‚ © is eigenvector of the  spin-Hamiltonian of the cluster or supramolecule. Thus,  entanglement distributions can be derived from the excitation  spin spectrum Q(à Ã‚ µ) = 1− 2Δ2 Ï€2N ÃŽ £N _=1 |c_|2 (à Ã‚ µ − à Ã‚ µ_)2 + Δ2 ÃŽ £N _=_+1 |c_|2 (à Ã‚ µ − à Ã‚ µ_)2 + Δ2 (14) Though the very nature of entanglement is purely  quantum mechanical, we saw that it can persist for  macroscopic systems and will survive even in the thermodynamical  limit. In this section we discuss how it  behaves at finite temperature of thermal entanglement. The states in N-spin system describing a system in thermal  equilibrium states, are determined by the Generalized  spin-Hamiltonian and thermal density matrix Ï (T) = exp(−Hspin/kT) Z(HZ) (15) where Z(HZ) is the partition function of the N-spin system. The thermal entanglement is Q(à Ã‚ µ, T,HZ) = 1 − 2Δ2 Ï€2NZ(HZ)2 ÃŽ £N _=1 |c_|2 exp[−à Ã‚ µ_/kT] (à Ã‚ µ − à Ã‚ µ_)2 + Δ2 Ãâ€" (16) ÃŽ £N _=_+1 |c_|2 exp[−à Ã‚ µ_/kT] (à Ã‚ µ − à Ã‚ µ_)2 + Δ2 The demonstration of quantum entanglement, however,  can also be directly derived from experiments, without  requiring knowledge of the system state. This can be  done by using specific operators–the so-called entanglement  witnesses–whose expectation value is always positive  if the state Ï  is factorizable. It is quite remarkable  that some of these entanglement witnesses coincide  with well-known magnetic observables, such as energy  or magnetic susceptibility χ = dM/dB. In particular,  the magnetic susceptibility of N spins s, averaged over  three orthogonal spatial directions, is always larger than  a threshold value if their equilibrium state Ï  is factorizable: ÃŽ £ g χg > Ns/kBT [EW]. This should not be surprising,  since magnetic susceptibility is proportional to  the variance of the magnetization, and thus it may actually  quantify spin.spin correlation. The advantage in  the use of this criterion consists in the fact that it does  not require knowledge of the system Hamiltonian, provided  that this commutes with the Zeeman terms corresponding  to the three orthogonal orientations of the  magnetic field ÃŽ ± = x, y, z. As already mentioned, in  the case of the Cr3 trimer, the effective Hamiltonian includes,  besides the dominant Heisenberg interaction J ∠¼118 meV , smaller anisotropic terms G ∠¼ 1.1 meV and  D ∠¼ 0.18 meV , due to which the above commutation relations  are not fulfilled. This might, in principle, result in  differences between the magnetic susceptibility and the  entanglement witness WE (see Fig.). Apparently, the  difference is quite essential and therefore it is necessary to use a formula for global entanglement Q(ψ) in N-spin  system. 4 10−1 100 101 102 103 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 The calculated difference à ¯Ã¢â‚¬Å¡Ã‚ ½EW(T)−EWa(T)à ¯Ã¢â‚¬Å¡Ã‚ ½/EW(T)for Cr3 isosceles trimer T(K) à ¯Ã¢â‚¬Å¡Ã‚ ½EW(T)−EWa(T)à ¯Ã¢â‚¬Å¡Ã‚ ½/EW(T) FIG. 1: (Color online) The calculated difference j EW(T) à ´Ã¢â€š ¬Ã¢â€š ¬Ã¢â€š ¬ EWa(T) j =EW(T) for Cr3 isosceles trimer 0 100 200 300 400 0 2 4 6 0 0.2 0.4 0.6 0.8 1 1.2 Angle(Degrees) The calculated M(H) for Cr3 isosceles trimer H(T) M(à ¯Ã†â€™Ã‚ ¬B) FIG. 2: (Color online)Magnetization M(H) of the Cr3  isoscales trimer on metal surface as a function of angles from 0 to 360 degree C. Thermal global entanglement in static magnetic _eld 5 0 50 100 150 200 250 300 350 400 0 0.05 0.1 0.15 0.2 0.25 The calculated variation of M(H) vs angle (magnetization switching) Angle(Degrees) M(à ¯Ã†â€™Ã‚ ¬B) 0.1Ts 0.2Ts 0.5Ts 1.0Ts FIG. 3: (Color online)The calculated variation of M(H) vs  angle (magnetization switching) for Cr3 isoscales trimer   FIG. 4: (Color online)The calculated density of global entanglement  vs temperature and energy for Cr3 isoscales trimer 6 0 100 200 300 400 0 2 4 6 0 0.5 1 1.5 2 2.5 Angle(Degrees) The calculated M(H) for Cr3 molecular magnet H(T) M(à ¯Ã†â€™Ã‚ ¬B) FIG. 5: (Color online)Magnetization M(H) of the Cr3 molecular  magnet as a function of angles from 0 to 360 degree 0 50 100 150 200 250 300 350 400 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 The calculated variation of M(H) vs angle (magnetization switching) Angle(Degrees) M(à ¯Ã†â€™Ã‚ ¬B) 0.1Ts 0.2Ts 0.5Ts 1.0Ts FIG. 6: (Color online)The calculated variation of M(H) vs  angle (magnetization switching) for Cr3 molecular magnet 7 FIG. 7: (Color online)The calculated density of global entanglement  vs temperature and energy for Cr3 molecular magnet  FIG. 8: (Color online)The calculated entanglement for the  Cr3 isoscales trimer as a function of temperature and the  magnitude of the magnetic field Hpar. 8 FIG. 9: (Color online)The calculated entanglement for the  Cr3 isoscales trimer as a function of temperature and the  magnitude of the magnetic field Hper. FIG. 10: (Color online)The calculated entanglement for the  Cr3 isoscales trimer as a function of temperature and the magnitude  of the magnetic field Hav. 9 FIG. 11: (Color online)The calculated entanglement for the  Cr3 molecular magnet as a function of temperature and the  magnitude of the magnetic field Hav.