Calculation of magnetic moments in Ho2C3 nanocrystals

Calculation of magnetic moments in Ho& nanocrystals S. A. Majetich,a) J. 0. Artman, and C. Tanaka Department of Physics, Carnegie Mellon University, Pittsburgh Pennsylvania 15213-3890 M. E. McHenry Department of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213-3890 A general approach to the computation of effective magnetic moments in rare-earth carbides is described, and details of this calculation for Ho3+ ions in Ho,C, are presented. This calculation is designed to explain the reduced magnetic moments, relative to free ion values, measured by dc SQUID magnetometry for Ho& nanocrystals. Crystal-field splittings of the rare-earth ion in a particular symmetry site are determined by the operator equivalent method. Using the eigenvalues and eigenfunctions of the crystal-field Hamiltonian, the effective magnetic moment is then determined. For Ho3+ ions in Ho&&, this method predicts a reduced magnetic moment, but the degree of reduction depends on the energy-level splittings and, therefore, the temperature. This magnetic moment is compared with previous experimental results, and the implications of the formal carbon charge, screening, and temperature are discussed. I. INTRODUCTION Carbon-coated gadolinium and holmium sesquicarbide nanocrystals have been prepared in a carbon arc, and the details of their synthesis, structure, and measured magnetic behavior have been reported previously.‘92 The gadolinium ions in Gd& have an 8S,,Z electronic ground state which is unaffected by the crystal field. However, the Ho3+ ions found in Ho& do not show the free ion behavior. For a ‘1s electronic ground state, the predicted effective moment, AJJ(J+-1P, is 10.61 ,un, but &n-7.5 ,&B was found experimentally.’ Here we present the details of a crystal-field calculation for Ho3+ ions in Ho.&, aimed at understanding the reduced magnetic moment, and in laying the groundwork for similar calculations for other rare-earth sesquicarbides, dicarbides, and rare-earth-containing fullerenes. II. CRYSTAL-FIELD CALCULATIONS The free ion energy levels are perturbed by a crystal field. The Hamiltonian for this perturbation, HcF, can be written in terms of operator equivalents, Oy, HcH=c BFOF, mGn. ,t,ll2 (1) The crystal-field intensity parameters, BF, are given by the expression B::‘=[-(e1/4rreol(r”)O.y,,N,,. (2) Here (Y) is the expectation value for the nth power of the f-electron radius and 0, is the Stevens’ factor, both of which are tabulated for various rare earth ions.3Z4Nnm is the coefficient for the Tesseral harmonic Zr, and Y,,~ is the lattice sum over the neighboring ions, which have fixed positions and charges for a particular structure. The operator equivalents 0: are functions of J, , J- , J,, and J. The eigenstates for the ion perturbed by the crystal field are generated from superpositions of free ion (JM) states. This method, reviewed by Hutchings,3 has recently been used to calculate ‘)Author to whom correspondence should be addressed. the energy levels of the rare-earth ions inside C,, molecules.’ Here we use this information to calculate the effective magnetic moment per ion ,ueff for HO&~. III. ENERGY LEVELS AND EIGENSTATES OF Ho3+ IN Ho&3 H02c3 has a body-centered-cubic structure (143d) with the Ho3+ ions in positions of Cs site symmetry. To model the crystal field experienced by a Ho3* ion in this solid, we included 11 neighboring carbon atoms and the 12 closest Ho3+ neighbors (Fig. 1). The atomic positions were taken from published x-ray and neutron data for bulklike Ho?C, powder,6 and the only free parameter was the charge on the carbon atoms. In many carbide molecules6 and solids,7 bonding occurs between the metal atom and GM dimer units, and the formal charge varies with the material.7 Previous results for Ho.& suggest that the holmium ions carry a 3f charge, and the carbon dimers have a reduced charge because they donate electrons to a conduction band.6 In our calculations the formal dimer charge was varied between 2- and 4-. The calculated lattice sums ynrn over the near-neighbor Ho3+ ions shown in Fig. 1 contributed only about 10% of the total, showing that the carbon dimers dominate the crystal field of a holmium ion in HO&~. Since the holmium ions have Cs site symmetry, only some of the operator equivalents contribute to the crystalfield Hamiltonian: HCF(C3)=B~O~+B400~$-B~O~+B~O~~B~O~+B660~ + B;iO;i+B&O;i+ B&O&. 13) Here B~i refers to the imaginary part of Bc, and the crystal axes were chosen to minimize the contributions from the Bzi and B~i terms. Using the free ion basis set, IJM), with J=8, the 17X17 matrix was diagonalized to determine the energy eigenvalues Ei and the eigenfunctions )~i): lW=C QfIJM). (41 6307 0 1994 American Institute of Physics J. Appt. Phys. 76 (IO), 15 November 1994 0021-8979/94/76(10)/6307/3/$6.00 Downloaded 23 Jan 2001 to 128.2.132.154. Redistribution subject to AIP copyright, see http://ojps.aip.org/japo/japcpyrts.html. 2.5 FIG. 1. (a) Ho’+ ion with 11 nearest-neighbor Ho3’ ions. (b) Ho3’ ion with 12 nearest-neighbor carbide ions. From group theory, the J =8 electronic ground state for a free Ho3’ ion should split into five rl?t states and six ar2 states when perturbed by a C, crystal field. As shown in Table I, the formal calculation did indeed result in six doublets and five singlet states. While the energy-level splittings varied with the formal charge on the carbon dimer, the relative ordering was the same (Fig. 2), and the makeup of the energy eigenstates was not significantly affected. The lowest doublet con- 3.0 4.0 3.5 FIG. 2. Energy levels of Ho3+ as a function of carbon diier charge magnitude. Energies are relative to that of a free Ho3+ ion. A more detailed description of the energy eigenstates for the dimer charge of -2.0 is found in Table I. sisted predominantly of M = +8 and -8 states, and the next lowest doublet was mainly M = +7 and -7 states. For all formal charge values there is significant population of excited states at 300 K. IV. MAGNETIC MOMENT OF Ho3+ IN Ho2C3 In the limit of small HIT, ity is given by the paramagnetic susceptibil- TABLE I. Ho&$ eigenvectors for C$ Eigenvector (energy) (91,J=0.988~+-8)+(+0.142-0.056i~t5), [.&,lk=O.OO K] lW,,)=O.98617~)+(~0.149-O.O63i)l+-4), [E3,,/k=18.39 K] ~~,)=(-0.505-0.479i)~6)f(-0.117-0.040i)~3)~0.002i10)+(-0.112-0.051i)~-3)+0.696~-6), [&/k=72.87 K] ~~~=0.697~6)+(0.109-0.049i)(3)+(0.0~-0.010~)~0)+(-0.113+0.040i)~-3)f0.506-0.479i)~-6}, [&/k=72.96 K] ~~,,s)=~0.003~~4)+(0.022t0.04i)~t1)+(T0.164-0.079i))52)+0.971~~5)+(0~142-0.056~)~~8), [.E,,s/k=145.56 K] ~~~)=(-0.136-0.057i)17)+(0.002-0.002i)~~+~.880~4)+(-0.010~+0.011i)~2)+(0.154-0.081i)~1)c(0.078-0.011i)~-1)f(0.032+0.0071)\2~ -t(-0.326+0.234i)~-4)+(0.006+0.001i)~-5)+(-0.035+0.057i)~-7), [Eg/k=217.00 K] ~~,,)=(-0.001+0.001i)~5)+(0.035-0.008i)~2)+(-0.169-0.089i)~-1)+0.967~-4)+(0.149-0.063i)~-7), [EJk=217.00 K] ~~,~)=(-0.112-0.048i)16)+0.681(3)+(0.076-0.194i)~0)+(0.501+0.461i)~-3)+(0.115+0.041i)~-6), [E,,/k=275.36 K] ~\u~~=(0.112-0.047i))6)+(-0.464+0.503i)~3)C(-0.169+0.073i)~0)+0.684~-3)+(0.110-0.050i)~-6), [EJk=283.13 K] ~‘Y~3)=(O.OO6-O.OO9i)~8)+(-O.l~-O.O76i).l8l-O.O82~)~-1)+(-0.011-0.038i)~-4)+(-0.004-0.003i)~-7), @&k=326.61 ~~,,)=(0.003-0.003i)~7)+(-0.011+0.038i)]4)+(0.001+0.001i)~2)+(-0.181-0.082i)~1)+0.962j-2)+(0.165-0.0761)~-5) +(-0.004-O.O03i)(-8), [E&=326.61 K] ~~,,)=(0.001-0.001i)~8)+(0.001+0.002i)~5)+(-0.179-0.085i)[2)+0.961~-1)+(0.172-0.082i)~-4)+(0.006-0.014i)~-7), [E&k=355.84 K] ~~~a)=~O.OO6+O.Ol4i)~7)+(O.OO1+O.OOl~)~5)+(-O.l72-O.O8li)~4)+(-O.Ol4+O.OO2i)~2)+O.959~l)+(O.O55-O.O38i)~-l)+(O.l79-O.O85i)~-2) +(0.007-0.011i)~-4)+(0.001-0.0Wi)(-5)+(-0.001i~~-7)+(-0.001-0.001i)~-8), [IZ,&k=355.84 K] ~~~~)=(0.005+0.019i~6)+(-0.177-0.085i)13)+0.960[0)+(0.177-0.0851)~-3)+(0.005-0.0191)~-6), [E,,/k=365.73 K] 6308 J. Appl. Phys., Vol. 76, No. 10, 15 November 1994 K] Majetich et al. Downloaded 23 Jan 2001 to 128.2.132.154. Redistribution subject to AIP copyright, see http://ojps.aip.org/japo/japcpyrts.html. Earlier SQUID magnetometry measurements’ yielded a temperature-averaged value of ,~+~=7.5 ,u~, There the data were scaled as’a function of HIT, and results from all temperatures were fit simultaneously.” Experiments are underway to study the degree of temperature dependence in order to place limits on the carbon dimer charge. Refinements in the crystal-field calculations to incorporate screening of the conduction electrons will also have an impact on the energylevel splittings, and therefore on the temperature dependence of the magnetic moment. . M CONCLUSIONS 01,’ 60 100~ I 200 150 Temperature I 250 c 300 (K) FIG. 3. Effective magnetic moment for Ho3’ with a carbon dimer charge or -2.0 as a function of temperature, with assumptions concerning offdiagonal elements as noted in the text. Xpm=N&g2JJ(J+ 1)/3kT=N&3kT, (5) for a free rare-earth ion with ground-state angular momentum J. Here p is the magnetic moment of the free ion. In the full quantum-mechanical treatment,* x depends on the statistical averaging over the energy eigenstates. While there are several contributions,g the effective magnetic moment determined from the magnetization curve measured with a SQUID magnetometer is related to the low-frequency term Crystal-field calculations were performed to determine the ground-state energy-level splittings for Ho3+ ions in Ho2C,. The energy eigenstates were used to calculate the low-frequency paramagnetic susceptibility and the effective magnetic moment. In comparison with SQUID magnetometry measurements, this calculation underestimates the reduction of the magnetic moment. However, information concerning important fitting parameters may be obtained by detailed temperature-dependent measurements of peff. Note added in proof A marked increase in the magnetic moment with increasing temperature has been observed experimentally, thought the error bars for fits of the hightemperature data are quite large. When the data taken at different temperatures are scaled and fit as a group, the lowtemperature contributions dominated in the determination that ,L&=r=7.5 /.&B. ACKNOWLEDGMENTS ,. 2 2 J M (JMlM2 exp( -EJlkT)IJM) (6) F z (JMI expt-EJlkTlIJM) ’ For Ho3+ ions in the Ho2C3 crystal field, we treated only the ground electronic state multiplet, which has J=8. The next lowest electronic state in the free ion, ‘17, lies over 5000 cm-’ higher in energy and is not appreciably populated in the experimental measurements.l* With the crystal-field eigenfunctions and eigenvalues, the expression for x is 2 C i C i =N,u&I3kT. (JM\a,*,aiMM’ ed-EiJkT)IJM) M C (JMla&aiu M exp(-EilkT)IJM) ’ (7) The effective moment per Ho3+ ion, Fan, differs from the free ion moment because of the removal of the free ion degeneracy, and because the crystal-field splittings lead to a temperature-dependent moment. Figure 3 shows the temperature dependence of the moment for the case where the carbon dimer charge equals -2.0. For larger carbon dimer charge values, the temperature-dependent drop-off is less pronounced. M.E.M. and S.A.M. would like to thank the National Science Foundation for support through NY1 Awards No. DMR-9258450 and No. DMR-9258308, respectively. Support from the Carnegie Mellon University SURG program and the assistance of E. M. Brunsman, C.-S. Niu, P. Sivaramakrishnan, and the CMU Buckyball Project members have also been valuable. ‘S. A. Majetich, J. 0. Artman, M. E. McHe~y, N. T. Nuhfer, and S. W. Staley, Phys. Rev. B 48, 16845 (1993). “E. M. Brunsman, M. E. McHenry, S. A. Majetich, J. 0. Artman, M. De Graef, S. W. Staley, R. Sutton, E. Bortz, S. Kirkpatrick, K. Midelfort, J. Williams, and B. Bnmett, J. Appl. Phys. 75, 5879 (1994). ‘M. T. Hutchings, in Solid StatePhysics, edited by F. Seitz and D. Turnbull (Academic, New York, 1964), pp. 227-276. 4W. E. Wallace, S. G. Sankar, and V. U. S. Rao, Structure and Bonding 33, 1 (1977). ‘M. E. McHenry and S. G. Sankar (unpublished). 6M. Atoji and Y. Tsunoda, J. Chem. Phys. 54, 3510 (1971). ‘G. J. Miller, J. K. Burdett, C. Schwartz, and A. Simon, Inorganic Chemistry 25, 4437 (1986). ‘5. H. Van Vleck, Phys. Rev. 29,727 (1927); 30,31 (1927); 31,587 (1928). 9S. V. Vonsovskii, Magnetism (Wiley, New York, 1974), Vol. 1, pp. 120126.. lo G H. Dieke, Spectra and Energy Levels of Rare Earth Ions in Crystals &iley, New York, 1968). “In this sample, electron diffraction has shown the presence of cubic (Y-HO&. No evidence of Ho&,, has been observed. In comparing the experimental and calculated values, a factor of l/3 should be introduced in the susceptibility formula to account for random orientations of the individual crystals. J. Appl. Phys., Vol. 76, No. 10, 15 November 1994 Majetich et al. 6309 Downloaded 23 Jan 2001 to 128.2.132.154. Redistribution subject to AIP copyright, see http://ojps.aip.org/japo/japcpyrts.html. View publication stats