Electronic and magnetothermal properties of ferromagnetic clusters

Journal of Physics and Chemistry of Solids 66 (2005) 977–983 www.elsevier.com/locate/jpcs

Electronic and magnetothermal properties of ferromagnetic clusters Manickam Mahendran* Department of Physics, Thiagarajar College of Engineering, Madurai 625 015, India Received 29 June 2004; accepted 15 December 2004

Abstract The electronic structures and the magnetothermal properties of nickel clusters have been investigated. Their effective magnetic moments and specific heat capacities have been calculated assuming that the clusters undergo superparamagnetic relaxation. The average magnetic moments are computed adopting Friedel’s model of ferromagnetic clusters. The surface effect and the cluster size effect on the thermodynamic properties of these clusters have been analysed based on the mean field theory approximation. The specific heat capacity of Ni clusters for NZ300, where N is the number of atoms in the cluster, shows the peak value at TZ550 K and exhibits a steady increase with N. The effective potentials and energy eigen values of the clusters as a function of the number of atoms and radius of the cluster have also been calculated self-consistently using the local density approximation (LDA) of the density functional theory (DFT); this has been performed within the framework of the spherical jellium background model (SJBM). The results of this study have been compared with the Stern–Gerlach experimental data and other theoretical results already reported in literature q 2005 Published by Elsevier Ltd. PACS: 75.25CZ; 75.30Et; 75.30Pd; 75.50Cc; 81.05Ys Keywords: A. Magnetic materials; C. Ab initio calculations; D. Magnetic properties; D. Specific heat; D. Electronic structure

1. Introduction During the last few years, several investigations on the ferromagnetic transition metal clusters have increased tremendously been reported mainly because of their unique physical and chemical properties. The magnetic behaviour of metal clusters can be exploited in the manufacture of magnetic read heads used in the electronic gadgets. In addition, the fundamental science of these materials has not yet been fully understood [1–7]. A clear understanding of cluster magnetism would pave also the way for the emerging nanotechnology [1]. A large portion of the atoms in a cluster forming a surface is one of the important parameters to study the electronic structure and magnetic properties of transition metal clusters. It has been observed that the smaller coordination number produces large magnetic moments in ferromagnetic clusters [2]. The magnetic behaviour of the transition metal clusters such as * Tel.: C91 452 248 2240; fax: C91 452 248 3427. E-mail address: [email protected].

0022-3697/$ - see front matter q 2005 Published by Elsevier Ltd. doi:10.1016/j.jpcs.2004.12.008

Fe, Co and Ni has been measured using the Stern–Gerlach experiment. The average moment per atom of Ni cluster has been calculated using the tight-binding Hamiltonian model [8]. The effective magnetic moments of Fe and Co clusters are obtained as a function of the size, the temperature and the magnetic field. Khanna and Linderoth [9] have proposed the superparamagnetic model to investigate the effective magnetic moments of ferromagnetic clusters. This model paves the way for the experimental side to work on the metal clusters. The magnetic properties of Fe and Co clusters have been investigated that as the cluster size increases the effective magnetic moment decreases. Hence, the bulk values 2.2mB (Fe) and 1.72mB (Co) have been reached [6]. The cluster magnetism depends on the dimensionality of the system, the nearest neighbour distances and the atomic environment. Several theoretical studies reveal that the orientation of the magnetic moment of ferromagnetic clusters depends upon the atomic environment inside the cluster and vary from the deeper side to the surface of the cluster. The Stern–Gerlach Experiment showed some oscillations in the effective magnetic moments of ferromagnetic clusters such as those of Fe, Co, Ni as a function of the cluster size ranging from 50 to 750 atoms [4,10,11]. The effective magnetic

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M. Mahendran / Journal of Physics and Chemistry of Solids 66 (2005) 977–983

moments of large ferromagnetic transition metals clusters have been predicted by assuming the statistical shell-by-shell growth of the cluster. The thermal properties of Fe, Co and Ni clusters containing 50–600 atoms are measured using the Stern–Gerlach experiment [4,7]. de Heer and co-worker [4,7] observed that the cluster lattice vibrations serve as a thermal bath for the spin system and therefore allow the clusters to have an internal thermometer, the magnetization. The cluster average magnetic moment of Ni clusters as a function of the number of atoms within the framework of Friedel’s model [12] has been reported for NZ5–26 atoms. The temperature dependence of the magnetic moments of these clusters has improved our knowledge on the stability of ferromagnetism and of the magnetic order against thermal excitations. A series of ionization potential of Fe, Co and Ni clusters has been measured using the laser photoionization method combined with the time-of-flight mass spectrometry [13–14]. Many novel properties of transition metal clusters have revealed that the evolution of the bulk electronic structure with increasing cluster size can be measured through the photo ionization experiment. Several theoretical and experimental investigations [15–23] have been made for small alkali metal clusters such as Li, K, Na, etc. The total energy, ionization potential, electron affinity and the structural relationship between the number of atoms and the energy eigen values have been obtained for these small metal clusters. The energy eigen values show discontinuities at certain cluster sizes pertaining to the mass spectra observation. The structure and magnetism of Fe, Co and Ni clusters, each with less than or equal to five metal atoms using the DFT scheme have been reported [21]. Duarte and Salahub [24] have studied the transition metal clusters using the embedded cluster approach within the framework of the Kohn–Sham DFT. As substantial work has not been done on the electronic structure of 3d transition metal clusters using LDA of Kohn–Sham DFT applied to the SJBM, our study focused on the medium sized Ni clusters containing 2–50 metal atoms. SJBM gives the external potential to arrive a solution to the Kohn–Sham equation easily [19]. The DFT computations have been performed to investigate the geometries and electronic properties of vanadium clusters within the linear combination of atomic orbitals. The atomic orbitals of Slater type have been used in the range from V2 to V8 of the vanadium clusters [25]. The dissociation energy, ionization energy and electron affinity have been studied. The magnetic properties and the electronic structure of vanadium clusters embedded in the bulk Fe are calculated using the spd-band Hubbard—like model [26]. The selfconsistent spin density distribution has also been computed using the unrestricted Hartree–Fock approximations and Gaussian orbitals method [26–29]. It has been observed that the cluster magnetic moments are dominated by the strong sensitivity of the itenerant 3d electrons to the local environment of the atoms [29]. The model Hamiltonian for the valence s, p, and d electrons has included the intra-

atomic Coulomb interaction in the unrestricted Hartree– Fock approximations. Using the Laser vaporization method the ionization energies of the 3d transition metal clusters have been investigated by Kurikawa et al [30]. Jena and co-workers [1–3] have calculated the magnetic properties and analyzed the electronic structure of Ni clusters using the molecular dynamics method. They have analysed the equilibrium geometries, energetics, electronic structure and ionization potential of Ni clusters for NZ21 atoms. The SJBM is chosen because it is an effective tool to study the electronic structure of transition metal clusters. Though, this is an old fashioned tool, it is versatile for evaluating the effective potential, the energy eigen values, the radius of the clusters and the electron density of Ni clusters self-consistently. For a long time the SJBM has been used to study the electronic structures of small alkali metal clusters such as those of Li, Na, K, etc. The LDA approach allows us to investigate large clusters having up to 50 metal atoms and to extract the electronic structure of ferromagnetic clusters satisfactorily [19]. The aim of the present work is to investigate the electronic structures (for NZ50 atoms) and magnetothermal properties (for NZ700 atoms) of Ni clusters and to extend the present theoretical understanding on metal clusters. It is believed that there is a strong effect of the exchange interaction in transition metal clusters to establish the thermal properties. The surface effect and the cluster size effect change the magnetothermal properties of Ni clusters, which give fruitful observable effects [4,31]. The specific heat capacities of metal clusters are expected to exhibit size effect and surface effect and are not carried out theoretically using the empirical relation cited in the references [12,31–33]. The following section discusses the computational methods and subsequently Section 3 contains the results and discussion and the conclusion is given in Section 4.

2. Computational methods The magnetic properties of transition metal clusters are depending upon the size and the temperature, which favours the equal filling up of the spin states. The Hamiltonian model of the spin system is given by H ZK

1X J SS 2 ij ij i j

(1)

where the summation is over the nearest neighbour spins, i and j are the spin sites, Jij are the exchange interaction of the system, Si and Sj are the spins at the sites i and j respectively. The internal magnetic field at each spin site (j) is given by [34] the mean field equation X Hj Z Jðj K j 0 Þ! Szj 0O (2) j0

M. Mahendran / Journal of Physics and Chemistry of Solids 66 (2005) 977–983

where, J(n) is the spin–spin interaction with its nth neighbour. Szj 0 is the thermally averaged mean value of the spin component for the magnetic ions in the jth site. The spins of the atoms of the cluster interact with the nearest neighbour spins. The average spins of the system for a given Hj is given by [34] " #  mZ3=2 mZ3=2 X X !Szj 0O Z m expðHj m=TÞ expðHj m=TÞ mZK3=2

mZK3=2

(3) The average magnetic moment per atom of the single domain clusters having N atoms will have Nm moment, where m is the atomic magnetic moment. It is assumed that the bulk value of Ni clusters is 0.62mB. The effective magnetic moment per atom of the ferromagnetic clusters is given by the Brillouin function and is calculated using the relation meff Z mBJ ðxÞ;

x Z NmHj =KB T

(4)

where N is the number of atoms in the cluster and m is the magnetic moment of the atom in the cluster. The Eqs (2) and (4) are solved self-consistently. J has been treated as an adjustable parameter due to the inadequacy of experimental information [8,11,35,36]. To obtain the large magnetic moments to analyze the ferromagnetic transition metal clusters, the Friedel’s model is used. According to the model, the average magnetic moment, mavg of the cluster is computed using the relation [12] mavg Z ð1=NÞ

N X

meff i

(5)

iZ1

The surface effects through a geometrical dependent empirical relation proposed in the previous work [37] is included in the above expression, which is given by Neff Z ðN=2Þ½1 C ð1 K Ns =N

(6)

where Neff is the effective number of atoms in the surface of the cluster and Ns is the number of atoms residing on the surface of the cluster. We have incorporated the effective magnetic moment equation [37] in the specific heat capacity expression proposed by de Heer and co-worker [4,7] and it is used for the investigation of the magnetothermal properties of ferromagnetic transition metal clusters Cvmean Z Kð3NKB Tc Þ=ð2m2 Þðj=j C 1Þðvm2avg =vTÞ

(7)

where j is expressed as mmK1 B gJ and Tc is the Curie temperature. In the local density approximation applied to the spherical jellium background model of a system of electrons, the external potential, Vext(r) is [16] ( KN=r rO R Vext ðrÞ Z (8) 3 2 3 Kð3=2ÞðN=R Þ½R K ðr =3Þ r% R

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where N is the number of atoms forming the cluster. This external potential is due to the homogeneously positive charged sphere of radius R that is equal to N1/3rs where rs is the radius of the cluster. The positive charge density is [16]   4p 3 K1 rs rt ðrÞ Z QðR K rÞ (9) 3 with the unit step function Q (x) and rZjrj The electron density r(r) is given by rðrÞ Z

N X

jji ðrÞj2

(10)

iZ1

All the calculations have been carried out in units of a.u. (1 a.u. of energyZ27.2 eV). The Kohn–Sham equation has been solved self-consistently   1 2 K V C Veff ðr; rðrÞÞ ji ðrÞ Z2i ji ðrÞ (11) 2 The modified effective potential Veff is given by ð rðr 0 Þ K rt ðr 0 Þ 3 0 d r Veff ðr; rðrÞÞ Z Vext ðrÞ C jr K r 0 j 

1=3 3 K 3a rðrÞ 4p

(12)

The initial charge density computed from atomic wave functions in Eq. (10) is used in Eq. (12) to arrive at the initial Veff. This Veff is incorporated in Eq. (11) which is solved numerically to arrive at the eigen function Ji and eigen value 2i. The new Jis are used in Eq. (10) to compute r(r), which in turn used in Eq. (12) to get the new Veff. This computed Veff forms the new input for the system of Eq. (11) and the procedure is repeated until the iterated Ji, 2i and the electron density have converged. To compute r(r), the wave function values have been taken from Herman– Skillman [20].

3. Results and discussion 3.1. Magnetothermal properties of nickel clusters Large metal clusters require higher temperatures to study the specific heat capacity of transition metal clusters, which are computed using the superparamagnetic model [9]. Using the LSDA method [38] the average magnetic moments of Ni13 and Ni55 have been calculated as 0.62mB and 0.91mB respectively. The calculated effective magnetic moments of Ni clusters have been compared with the results obtained from various theories (Table 1). These moments have been compared with different cluster size ranges, NZ2–13, 19, 43, 56, 136, 164 and hence the bulk value 0.60mB has been reached. The magnetic moments for lower number of atoms are plotted in Fig. 1 and it has been noted that the effective

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Table 1 The computed magnetic moments of nickel clusters compared with various other theoretical calculations N

R1

N

R2

N

R3

N

R4

N

R5

N

R6

N

R7

N

R8

N

R9

N

R10

2 3 4 8 13 19 43 Blk

1.00 0.33 0.55 1.00 0.85 0.79 0.72 0.60

2 3 4 5 6 7 8 13

1.00 0.65 1.00 1.60 1.00 1.42 1.25 0.90

5 10 15 20 25 30 35 45

0.92 0.80 0.82 0.90 0.85 0.70 0.87 0.89

8 12 20 44 56 80 136 164

0.26 0.17 0.19 0.12 0.12 0.14 0.10 0.80

5 6 7 8 9 10 11 12

2.00 1.30 1.70 1.75 1.40 1.30 1.35 1.35

3 4 5 6 7 8 9 10

0.65 1.00 0.80 1.30 0.85 1.00 0.90 0.80

2. 3 4 5 6 8 13

1.00 0.67 1.50 1.60 1.00 1.00 0.62

5 10 20 30 40 50 60 70

1.00 0.80 0.42 0.70 0.50 0.80 0.70 0.60

5 6 7 8 9 10 11 12

1.80 1.50 1.55 1.60 1.30 1.25 1.20 1.15

5 10 15 20 25 30 35 40

1.8543 1.2239 1.1343 1.1658 1.0258 1.0657 0.8561 1.0683

R1: Ref. [36], local density approximation; R2: Ref. [32], tight-binding-molecular dynamics; R3: Ref. [31], tight-binding-Hamiltonian; R4: Ref. [46], Hund’s rule; R5: Ref. [33], Self-consistent tight-binding; R6: Ref. [1], molecular dynamics simulation; R7: Ref. [39], linear combination of atomic orbital; R8: Ref. [38], electronic shell model; R9: Ref. [12], Fridel’s model; R10: present, superparamagnetic model and mean field theory.

magnetic moments decrease with increasing the size of the clusters. It is found that in nickel clusters, there is a decrease of the effective magnetic moment per atom with magnetic field and cluster size. It has been observed [5] that the magnetic transition in Ni clusters occurred at higher temperature ranges than the Curie temperature (620 K). The convergence of the bulk saturation of the effective magnetic moments as a function of cluster size upto 700 atoms is shown in Fig. 2. In the bulk ferromagnetic transition metal clusters, the magnetic moment of the atom is independent of the temperature while the average magnetization decreases with temperature [6]. The thermal behaviour of the effective magnetic moments of Ni clusters shows a structural transition; such a transition is responsible for the decrease of the effective magnetic moment with temperature. We have observed that the clusters with NZ50–700 atoms have rich and enhanced effective magnetic moments compared with the bulk solid. The effective magnetic moment decreases with the temperature because the internal moment of the cluster increases with the temperature; this is because the thermal fluctuations reduce the tendency of forming spin

pairs [7]. When the cluster size is large, the effective magnetic moment gets saturated. The computed effective magnetic moments using the modified superparamagnetic model [37] have been compared with the Stern–Gerlach experimental results [4,7,10,11] for clusters having upto 700 atoms. The computed values of meff of Ni clusters in terms of mB are plotted for various temperatures for NZ45, 150, 220 and 575 (Fig. 3). From this figure, it is interesting to note that for a given cluster the meff decreases with increasing temperature. The meff reaches a constant value at TZ700 K for NZ45. The Ni magnetic moment is drastically reduced as compared to the bulk value. Also, the meff of Ni decreases initially with the increasing exchange interaction. Afterwards, with the increase of temperature, meff decreases gradually, when J decreases as reported by Vega et al [35]. The meff gradually decreases from 0.9352mB and 0.1548mB for NZ45 atom and NZ575 atom clusters respectively. Of course, there is no possibility of J having direct temperature dependence. It is seen that the meff decreases with the increase of cluster size in the Ni cluster.

Fig. 1. Computed values of the average magnetic moment of Ni Clusters at TZ200 K for various cluster sizes (upto 50 atoms). The meff s are given in terms of mB, the Bohr magneton. The lines are drawn to guide the eye.

Fig. 2. Computed values of the average magnetic moment of Ni Clusters at TZ200 K for various cluster sizes (50–700 atoms). The meff s are given in terms of mB, the Bohr magneton. The lines are drawn to guide the eye.

M. Mahendran / Journal of Physics and Chemistry of Solids 66 (2005) 977–983

Fig. 3. Computed values of the average magnetic moment of Ni Clusters for various cluster sizes as a function of temperature. The lines are drawn to guide the eye.

The results of the specific heat capacities are given in Fig. 4 for Ni clusters. The figure shows that the specific heat capacity of these clusters oscillates with the increase in temperature. The Ni cluster chosen for the specific heat calculation has an N value of 300 and the effect of temperature on this is studied upto 800 K. As the temperature is increased further, specific heat capacity also increase gradually. Some of the interesting oscillatory peaks of Ni clusters are shown in Fig. 4. After attaining a maximum temperature, near the Curie temperature, a gradual decrease in the specific heat capacity is noticed. The peaks confirm that the mean field theory approximation is suited for the estimation of specific heat capacities of magnetic materials. The peak value of specific heat capacity reaches at TZ350 K and TZ500 K confirming that more thermal fluctuations are produced in Ni clusters. More peaks may be found in the specific heat of Ni clusters, which are

Fig. 4. Computed values of the specific heat capacities of Ni Clusters for NZ300 atoms for various temperatures ranging from TZ100–800 K. The lines are drawn to guide the eye.

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due to the reduction in the spins at the surface of the clusters. The peak value of specific heat capacity of Ni clusters has been obtained at TZ500 K and the specific heat increases to nearly 7.5 cal molK1 KK1. The contraction of the interatomic distances at the surface of the cluster may reduce the effective magnetic moments [21]. The Friedel approximation gives the average transition behaviour from non-metallic to metallic region in the direction of increasing the temperature [12]. Greater accuracy cannot be achieved in the surface specific heat capacity at higher temperature. Therefore, there might be some variations in the positions and magnitude of the maxima as a function of the cluster size and is not expected to show fruitful results. This may be due to the fact that the maxima of effective magnetic moments and hence the density of states of Ni lies at higher energies than that of the bulk Ni clusters [39–41]. More experimental works on a variety of metal clusters with different sizes and surface areas are needed to develop the features of the magnetothermal properties of ferromagnetic clusters. An anomalous increase of the effective magnetic moments of Ni clusters with the increase of the temperature has been demonstrated experimentally [4,7]. The effective magnetic moments of Ni clusters are decreased with temperatures. 3.2. Electronic properties of nickel clusters The results of the electronic and structural properties of the Ni clusters as a function of N, the number of atoms forming a cluster and as a function of the radius of the cluster, rs are presented in this section. The enhancement of the structural relationships between the cluster size N and the energy eigen value, 2i, the effective potential, Veff and radius of the cluster, rs is due to the exchange-correlation effect. The electronic structure of transition metal clusters is associated within the spherical jellium of the LDA. Pietro Coronta [42] and Jones and Gunnarson [43] have studied the Ionization potential, the electron affinity, the electron density and the energy eigen values of transition metal clusters using the LDA-HF and the LSD methods. The Veff in the presence of the external potential Vext(r) increases gradually with the increasing rs. This gradual increase of the Veff may be due to the use of the converged electron density r(r). The ground state densities and the effective potential corresponding to the SJBM of the LDA are plotted. The effective potentials of Ni clusters as a function of the radius of the cluster for various cluster sizes (NZ10, 20, 30, 40 and 50) are plotted (Fig. 5). The potentials are smooth within the jellium spheres [37]. It is interesting to note that the effective potential increases customarily with increasing rs. The Veff is almost saturated when rs is increased continuously. Veff is customarily increased to saturation and then decreases. As rs increases the potential decreases. This may be due to the fact that it behaves like (1/r) for large r. The Effective potentials of Ni clusters as a function of the number of atoms, N is obtained in the self-consistent

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Fig. 5. The Effective potentials, Veff (in eV) as a function of the radius of the clusters of Ni upto NZ50 atoms are obtained in the self-consistent spherical jellium model within the framework of DFT. The lines are drawn to guide the eye.

spherical jellium model (Fig. 6). As the number of atom increases, there is a deep decrease in the effective potentials in the negative side. The quantity 2i is calculated self-consistently while iterating the electron density using Eqs (10)–(12). The 2i as a function of the N for Ni clusters is presented in Fig. 7. For smaller number of atoms (NZ2–5) the clusters have shellclosing configuration, and hence the shell structure effects are systematically averaged out. It is clear from Fig. 7 that as the N in a cluster increases the 2 i values show discontinuity. The electronic energy eigen values obtained from the DFT scheme have discontinuities whenever shells are completely filled. The solutions to the Kohn–Sham equation formed from the wave function Ji(r) includes the principle quantum number n and the angular quantum number l. We have assumed the spherical symmetry for the states with s are n and l values; therefore each shell has a 2(2lC1) degeneracy [24,31,32]. The spherical average electron density is used in the calculation of the open shell clusters. The measured ionization energies of

Fig. 7. The Energy eigen values, 2i calculated for jellium spheres as a function of the number of atoms, for Ni clusters. The lines are drawn to guide the eye.

Fe, Co and Ni clusters reported by Jarrold [44] have discontinuities as the number of atoms in the cluster increased (up to 100 atoms). The overall decreasing trend in the energy eigen value has been observed, but they have converged slowly in all the three cases. These interpretations given by us are in agreement with Perdew and Alex Zunger [45] where they have calculated the energy eigen values using the LSD method for these clusters.

4. Conclusion The occurrences of maxima and minima in the specific heat capacities of Ni clusters with increasing temperature have been observed using the superparamagnetic model and Friedel’s approximation. These maxima occurred at TZ550 K for Ni clusters. The effective magnetic moments of Ni clusters decrease with the increase in size, magnetic field and the temperature of the clusters. The experimental bulk values of 0.60mB is reached when the cluster size is increased to 700 atoms. The self-consistent electronic structure calculation of transition metal Ni clusters has been substantiated. The LDA applied to the SJBM explains the average trends of the effective potentials, the energy eigen values and the cluster radii in the electronic structure calculation. The effective potential obtained by assuming the jellium sphere is reflected in the electronic properties.

Acknowledgements

Fig. 6. The Effective potentials of Ni clusters as a function of the number of atoms, N is obtained in the self-consistent spherical jellium model. The line is drawn to guide the eye.

The author thanks Prof. K. Iyakutti, Madurai Kamaraj University and Prof. K. Tsuruta, Okayama University for their useful suggestions and discussions. The management and the Principal of Thiagarajar College of Engineering are duly acknowledged for their support and assistance to carry out this work. The author also thanks the referees’ constructive comments.

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