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Structures and magnetic ordering of MnnFe (n=1–12) clusters
Solid State Communications 147 (2008) 53–56
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Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Structures and magnetic ordering of Mnn Fe (n = 1–12) clusters Bao-Ru Wang a,b,c , Jing Wang a , Qing-Ming Ma a , Ying Liu a,c,∗ a Department of Physics, Hebei Normal University, Shijiazhuang 050016, Hebei, China b College of Science, Hebei University of Science and Technology, Shijiazhuang 050018, Hebei, China c National Key Laboratory for Materials Simulation and Design, Beijing 100084, China
article
info
Article history: Received 25 January 2008 Received in revised form 1 April 2008 Accepted 14 April 2008 by P. Sheng Available online 24 April 2008 PACS: 61.46.Bc 31.15.Ew 36.40.Cg
a b s t r a c t The structures and magnetic ordering of Mnn Fe (n = 1–12) clusters have been investigated using all-electron density functional theory. The results indicate that the Mnn Fe clusters undergo a change in magnetic behavior from ferromagnetic ordering for the smallest size to ferrimagnetic ordering for intermediate sizes and beyond. Ferromagnetic ordering is clearly favored for n = 1, but ferromagnetic and ferrimagnetic states are nearly degenerate for n = 2, 3, and 4. A radical change occurs at n = 5 where the ferrimagnetic states completely prevail. The transition range of magnetic ordering from ferromagnetic to ferrimagnetic for Mnn clusters occurs early with the doping of one Fe atom. © 2008 Elsevier Ltd. All rights reserved.
Keywords: A. Manganese-iron clusters B. Density functional theory C. Magnetic ordering
1. Introduction There has been increasing effort in the study of magnetism of transition metal clusters (TMCs), which is motivated largely by the desire to explore fundamental understanding of the magnetic properties of materials with dimensions in nanometer length scales. The magnetic properties of Mnn clusters are unusual due to the electronic configuration 3d5 4s2 in Mn atoms. Therefore, the study of Mnn clusters has attracted considerable attention in both theoretical and experimental research areas [1–4]. Of particular interest, the transition range of the magnetic ordering of small Mn clusters from ferromagnetic to antiferromagnetic, is well studied in theory [5–8]. To date, most studies of cluster magnetism have involved TMCs composed of a single metal. The rich and varied magnetic behavior displayed in bulk binary alloys suggests that bimetallic TMCs may display interesting and potentially useful magnetic properties as well. Using a molecular beam deflection, it had been reported that small Com Mnn clusters were superparamagnets at temperatures substantially above the ordering temperatures of the corresponding Co1−x Mnx alloys, and that they possessed
∗ Corresponding author at: Department of Physics, Hebei Normal University, Shijiazhuang 050016, Hebei, China. Tel.: +86 311 86268649; fax: +86 311 86268314. E-mail address: [email protected] (Y. Liu). 0038-1098/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2008.04.020
mean per-atom moments that were substantially larger [9]. Using Stern–Gerlach deflection at low temperature (46.5 K), Yin et al. investigated the properties of Bim Mnn (m = 2–20, n = 0–7) clusters and found either ferromagnetic or ferrimagnetic coupling behavior among the manganese atoms, depending on size and composition [10]. In this note, we report the study of the magnetic ordering of the Mnn Fe (n = 1–12) clusters using all-electron density functional theory (DFT) by an extensive search for their lowestenergy configurations. A brief description of the theoretical method and computational details will be given in Section 2. In Section 3, the lowest-energy structures and some low-lying isomers for the Mnn Fe (n = 1–12) clusters will be shown, and the structural property, the stability and the magnetic ordering of the clusters will be discussed. The main computational results will be summarized in Section 4. 2. Computational details In order to search the lowest-energy structures of Mnn Fe (n = 1–12) clusters, the geometry optimization with constrained symmetry is firstly performed to find more low-lying metastable isomers among abundant initial structures, which are based on our previous studies [11–17]. In order to explore the full freedom in the potential energy surface and avoid possible saddle points, the geometry optimization is then performed without any symmetry
54
B.-R. Wang et al. / Solid State Communications 147 (2008) 53–56
Fig. 1. The various structures for Mnn Fe (n = 1–4) clusters, and the lowest-energy structures for Mnn Fe (n = 5–12) clusters. Light ball: Mn atom; dark ball: Fe atom. The spin orientation of Mn and Fe atoms and the Mn–Fe bond lengths are labeled.
restriction. The atomic charge and spin moments are obtained by the Mülliken population analysis.
3. Results and discussion
In our calculations, we chose a double-numerical basis with polarized functions (DNP), for which, in general, the most accurate results can be obtained. The spin-polarized BLYP functional (the Becke exchange functional [18] and the Lee–Yang–Parr correlation functional [19]) based on generalized gradient approximation (GGA) is employed to take into account the exchange and correlation effects of electrons. The charge density is converged to 1 × 10−6 e/Å in the self-consistent calculation, which corresponds to a total energy convergence of 1 × 10−5 Hartree. In the optimization process, the energy gradient and atomic displacement are converged to 1 × 10−4 Hartree/Bohr and 1 × 10−3 Å, respectively. All calculations are performed using allelectron DFT implemented in the DMol package [20,21], which is a comparatively accurate and efficient procedure for molecules.
The lowest-energy structures and some low-lying metastable isomers for Mnn Fe (n = 1–12) clusters are shown in Fig. 1, and the results are summarized in Table 1. For MnFe, the lowestenergy structure (C∞v ) with bond length 2.454 Å has ferromagnetic ordering and a total spin moment of 9µB . Our computed values are in good agreement with the early values (2.42 Å and 9µB ) obtained by Gutsev et al. [22]. For Mn2 Fe, an isosceles triangle (C2v ) is found to be the ground state, of which the ferromagnetic Mn–Mn coupling is the same as that of Mn2 [16], while the Mn–Fe couplings are ferrimagnetic. The C2v structure has a total spin moment of 8µB (considering spin orientation) and an average spin moment per Mn atom of 5.066µB (without considering spin orientation). The second isomer (Cs ) with a ferrimagnetic Mn–Mn coupling is 0.169 eV higher in energy than the ground state. The other two low-lying isomers
B.-R. Wang et al. / Solid State Communications 147 (2008) 53–56 Table 1 Calculated results for various structures of Mnn Fe (n = 1–12) clusters n
Symmetry
BE
HOMO
Gap
µs
QFe
1 2
C∞v C2v Cs C∞v D∞h Cs -1 Cs -2 C3v C2v -1 Cs -1 Cs -2 C2v -2 Cs -1 Cs -2 Cs -3 C4v Cs -4 C2v 1 C3v C2v -2 Cs C5v C2v -1 C2v -2 Cs -1 Cs -2 C2v -1 Cs C2v -2 C2v -3 C4v C4v -1 C2v C4v -2 C4v -3 C3v C4v C2 C2v C5v -1 C5v -2 D3h C2v C5v C2v D3h Ih Oh
−1.87 −2.94 −2.77 −2.63 −2.62 −4.68 −4.53 −6.74 −6.57 −6.53 −6.40 −6.31 −8.67 −8.59 −8.52 −8.30 −8.28 −10.92 −10.52 −10.32 −10.28 −10.17 −12.33 −12.30 −12.29 −12.23 −14.40 −14.16 −14.11 −13.66 −13.65 −16.22 −16.16 −15.57 −15.47 −15.26 −18.88 −18.21 −17.95 −20.65 −20.24 −20.03 −19.57 −22.32 −21.67 −21.09 −20.99 −20.47
A1.1 A1.1 A0 .1 B2.1 B2G.1 a A0 .1 a A0 .1 a A0 .1 B2.1 a A0 .1 A0 .1 A1.1 A0 .1 B2.1 A1.1 A1.1 A0 .1 E2.2 A1.1 A1.1 A00 .1 A0 .1 A1.1 A00 .1 A2.1 B2.1 B1.1 B1.1 A2.1 A1.1 E.2 E.2 A1.1 A.1 B2.1 E1.2 A1.1 E00 .2 B1.1 A1.1 A2.1 A10 .1 GU.3 T1U.3
0.388 0.068 0.253 0.514 0.304 0.497 0.391 0.529 0.468 0.604 0.365 0.268 0.501 0.426 0.411 0.230 0.343 0.338 0.319 0.320 0.341 0.103 0.489 0.373 0.210 0.526 0.247 0.202 0.260 0.357 0.158 0.256 0.556 0.197 0.331 0.071 0.172 0.290 0.224 0.210 0.139 0.154 0.237 0.225 0.056 0.213 0.084 0.256
9 8 2 2 8 1 3 12 22 12 2 2 3 7 3 9 3 10 0 10 5 22 3 9 7 9 0 6 4 20 2 1 15 27 29 7 2 6 4 1 5 5 4 38 16 6 46 52
−0.042 −0.168 −0.062 −0.140
3 4
5
6
7
8
9
10
11
12
0.024
−0.059 −0.069 −0.052 −0.029 −0.039 −0.037 0.048
−0.059 −0.010 −0.030 −0.022 −0.039 −0.040 −0.045 −0.012 −0.048 −0.010 −0.022 −0.028 −0.005 0.005
−0.051 −0.004 −0.014 0.002
−0.057 −0.100 0.010
−0.050 0.012
−0.011 −0.066 −0.065 −0.084 0.069
−0.059 −0.152 0.031
−0.013 −0.041 −0.028 0.018 0.111
The table lists the symmetry, binding energy BE (in eV), the HOMO state, the HOMO–LUMO gap, the total spin moment µs (in µB ) and the charge of Fe atom Q Fe (in e).
(C∞v , D∞h ) are linear chains. For Mn3 Fe, the most stable structure is a planar configuration (Cs -1), which has a total spin moment of 1µB and a mean spin moment per Mn atom of 4.354µB . A trigonal pyramid (Cs -2) is obtained as the second isomer. The Mn–Mn couplings of the structures (Cs -1, Cs -2) are ferrimagnetic. The ferromagnetic Mn–Mn couplings of Mn3 [16] are altered by the doping of one Fe atom. In the case of Mn4 Fe, the C3v structure is found to be the ground state with ferrimagnetic ordering. The total spin moment of the C3v state is 12µB and the mean spin moment per Mn atom is 4.333µB . It is worth mentioning that the C2v -1 structure has ferromagnetic ordering and a total spin moment of 22µB , which is obtained as the second isomer whose energy is 0.167 eV higher than that of the ground state. The other three structures (Cs -1, Cs -2, C2v -2) are a trigonal bipyramid, a distorted square pyramid and a planar configuration, respectively. For Mn5 Fe, the most stable isomer (Cs -1) is found to be a tetragonal bipyramid with a total spin moment of 3µB and a mean spin moment per Mn atom of 4.116µB . This is followed by four low-lying isomers (Cs -2, Cs -3, C4v , Cs -4). All the geometries
55
obtained here are ferrimagnetic. From n = 5 or more, the Mn–Mn couplings of all the geometries of Mnn Fe clusters are of ferrimagnetic ordering. The transition range of magnetic ordering from ferromagnetic to ferrimagnetic for the pure Mnn (n = 2–12) clusters [16] occurs early for the doping of one Fe atom. For Mn6 Fe, the C2v -1 structure is obtained as the lowest-energy structure, having a total spin moment of 10µB and a mean spin moment per Mn atom of 3.914µB . The C3v structure has energy 0.393 eV higher than that of the ground state and is found to be the second isomer. The C2v -2, Cs and C5v structures are obtained as the low-lying isomers as well. For Mn7 Fe, the C2v -1 structure has the lowest energy with a total spin moment of 3µB and an average per-atom spin moment of 4.105µB for Mn. The C2v -2 structure whose energy is only 0.022 eV higher than that of the ground state is considered to be the second isomer. Two other low-lying isomers (Cs -1, Cs -2) have been found, which are 0.034 eV, 0.097 eV higher in energy than the ground state. For Mn8 Fe, it is found that the total spin moment and the mean spin moment per Mn atom of the lowest-energy structure (C2v -1) are 0µB and 3.906µB . The other five low-lying isomers have Cs , C2v -2, C2v -3 and C4v symmetries, respectively, and in increasing order of binding energy, total spin moments of 6µB , 4µB , 20µB , and 2µB . For Mn9 Fe, the C4v -1 structure with Fe atom occupying the surface site is found to be the lowest-energy structure with a total spin moment of 1µB and a mean spin moment per Mn atom of 3.967µB . The second isomer with C2v symmetry is only 0.054 eV higher in energy than the ground state. For Mn10 Fe, the lowest-energy structure (C4v ) with Fe atom locating at the surface site has a total spin moment of 2µB and a mean per-atom spin moment of 3.642µB for Mn. The other two low-lying isomers (C2 , C2v ) are both with Fe atom locating at the central site. For Mn11 Fe, four low-lying isomers (C5v -1, C5v -2, D3h , C2v ) are obtained. The C5v -1 structure has a total spin moment of 1µB , which is found to be the ground state with a mean spin moment per Mn atom of 3.715µB . The Mn11 Fe as well as the Mn10 Fe is with Fe atom occupying the surface site in the lowest-energy structure. In the case of Mn12 Fe, the C5v structure with Fe atom occupying the surface site is obtained as the lowest-energy structure, the total spin moment and the average spin moment per Mn atom of which are 38µB and 3.912µB . The C2v structure is found to be the second isomer which is 0.652 eV higher in energy than the ground state. The structures with D3h , Ih and Oh symmetries, respectively, are all with Fe atoms locating at the central sites and they are much higher in energy than the ground state. It is favorable for the doped Fe atom to occupy the surface site. It can be seen from Table 1 that the charge transfers from the Mn atoms to the Fe atom except in the case of Mn11 Fe, in which the charge transfers from the Fe atom to the Mn atoms. In Fig. 2, the size dependence of the HOMO–LUMO gaps and the binding energies per atom (Eb ) of the Mnn Fe (n = 1–12) and MnN (N = 2–13) clusters are displayed separately. Fig. 2(b) illustrates the second derivative of the binding energies, ∆2 E(n) = Eb (n + 1) − 2Eb (n) + Eb (n − 1), for Mnn Fe (n = 1–12) clusters, as a function of cluster size. It can be seen that the stability of the pure MnN (N = 2–13) clusters is enhanced by the addition of one Fe atom, and the Mn6 Fe and Mn10 Fe have much higher stability compared to other clusters. The electronic structures of the pure MnN (N = 2–13) clusters have been greatly altered by the doping of one Fe atom. The size dependence of the average bond length and the µa of Mnn Fe (n = 1–12) clusters are shown in Fig. 3. Here µa is the sum of the absolute values of the spin moments divided by the number of atoms in each Mnn Fe (n = 1–12) cluster. From Fig. 3(a), it can be seen that µa decreases with the increasing cluster size, and exhibits the magnetic moment surface enhancement effect, which has been widely discussed by other authors [23– 26]. For Mn4 Fe, Mn7 Fe, Mn9 Fe and Mn12 Fe, the µa have larger values compared with those of their neighbors. At the same time, their corresponding average bond lengths are also larger,
B.-R. Wang et al. / Solid State Communications 147 (2008) 53–56
56
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Fig. 2. Size dependence of HOMO–LUMO gaps for Mnn Fe (n = 1–12) ( ) and MnN (N = 2–13) ( ) clusters (a); and the binding energies per atom (Eb ) for Mnn Fe (n = 1–12) ( ) and MnN (N = 2–13) ( ) clusters, and the second derivative of total binding energies (∆2 E) for Mnn Fe (n = 1–12) ( ) clusters (b). The data of the MnN (N = 2–13) clusters is from Ref. [16].
Fig. 3. Size dependence of µa (Fe ( ), Mn( ) atoms and the total ( ) clusters) for Mnn Fe (n = 1–12) clusters (a); and the average bond lengths (Mn–Fe ( ), Mn–Mn ( ) and the total bonds ( ) of the clusters) for Mnn Fe (n = 1–12) clusters (b).
which demonstrates the bond contraction effect discussed in Refs. [23,24].
[1] M.B. Knickelbein, Phys. Rev. B 70 (2004) 014424. [2] N.O. Jones, S.N. Khanna, T. Baruah, M.R. Pederson, Phys. Rev. B 70 (2004) 045416. [3] M.R. Pederson, F. Reuse, S.N. Khanna, Phys. Rev. B 58 (1998) 5632. [4] P. Bobadova-Parvanova, K.A. Jackson, S. Srinivas, M. Horoi, Phys. Rev. A 67 (2003) 061202. [5] R.C. Longo, E.G. Noya, L.J. Gallego, Phys. Rev. B 72 (2005) 174409. [6] M. Kabir, A. Mookerjee, D.G. Kanhere, Phys. Rev. B 73 (2006) 224439. [7] P. Bobadova-Parvanova, K.A. Jackson, S. Srinivas, M. Horoi, J. Chem. Phys. 122 (2005) 014310. [8] R.C. Longo, E.G. Noya, L.J. Gallego, J. Chem. Phys. 122 (2005) 226102. [9] M.B. Knickelbein, Phys. Rev. B 75 (2007) 014401. [10] S. Yin, X. Xu, R. Moro, W.A. de Heer, Phys. Rev. B 72 (2005) 174410. [11] Q.M. Ma, Z. Xie, J. Wang, Y. Liu, Y.C. Li, Phys. Rev. B 76 (2007) 035412. [12] J. Wang, Q.M. Ma, Z. Xie, Y. Liu, Y.C. Li, Phys. Rev. B 76 (2007) 035406. [13] Z. Xie, Q.M. Ma, Y. Liu, Y.C. Li, Phys. Lett. A 342 (2005) 459. [14] Q.M. Ma, Z. Xie, J. Wang, Y. Liu, Y.C. Li, Phys. Lett. A 358 (2006) 289. [15] Q.M. Ma, Z. Xie, J. Wang, Y. Liu, Y.C. Li, Solid. State. Commun. 142 (2007) 114. [16] Q.M. Ma, Z. Xie, Y. Liu, Y.C. Li, Chin. Phys. Lett. 24 (2007) 7. [17] N. Liu, Q.M. Ma, Z. Xie, Y. Liu, Y.C. Li, Chem. Phys. Lett. 436 (2007) 184. [18] A.D. Becke, J. Chem. Phys. 98 (1993) 5648. [19] C. Lee, W. Yang, R.G. Parr, Phys. Rev. B 37 (1988) 785. [20] B. Delley, J. Chem. Phys. 113 (2000) 7756. [21] B. Delley, J. Chem. Phys. 92 (1990) 508. [22] G.L. Gutsev, M.D. Mochena, P. Jena, C.W. Bauschlicher Jr., H. Partridge III, J. Chem. Phys. 121 (2004) 6785. [23] W.H. Zhong, C.Q. Sun, S. Li, H.L. Bai, E.Y. Jiang, Acta Materialia 53 (2005) 3207. [24] C.Q. Sun, H.L. Bai, S. Li, B.K. Tay, E.Y. Jiang, Acta Materialia 52 (2004) 501. [25] C.Q. Sun, T.P. Chen, B.K. Tay, S. Li, H. Huang, Y.B. Zhang, L.K. Pan, S.P. Lau, X.W. Sun, J. Phys. D: Appl. Phys. 34 (2001) 3470. [26] C.Q. Sun, L.K. Pan, H.L. Bai, Z.Q. Li, P. Wu, E.Y. Jiang, Acta. Materialia 51 (2003) 4631.
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4. Conclusions Using all-electron DFT-GGA calculation, the lowest-energy structures of the Mnn Fe (n = 1–12) clusters are obtained. The results indicate that the transition range of magnetic ordering from ferromagnetic to ferrimagnetic for Mnn Fe clusters occurs early compared to that of the pure MnN (N = 2–13) clusters. The MnFe has ferromagnetic ordering. The Mn–Mn coupling of Mn2 Fe is ferromagnetic similar to that of Mn2 . While the ferromagnetic Mn–Mn couplings of Mn3 are completely altered with the doping of one Fe atom. It is from n = 5 that the Mn–Mn couplings of all the states obtained here are ferrimagnetic. In addition, the stability, the surface enhancement and bond contraction effects of the magnetic moment for Mnn Fe(n = 1–12) clusters are also discussed. Acknowledgements We would like to thank Professor Chen Nan-Xian and Shen Jiang for helpful discussions. This work is supported by the 973 Project in China under Grant No. 2006CB605101, the National Natural Science Foundation of China (Grant No. 10574036).
•
References
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Structures and magnetic ordering of Mnn Fe (n = 1–12) clusters Bao-Ru Wang a,b,c , Jing Wang a , Qing-Ming Ma a , Ying Liu a,c,∗ a Department of Physics, Hebei Normal University, Shijiazhuang 050016, Hebei, China b College of Science, Hebei University of Science and Technology, Shijiazhuang 050018, Hebei, China c National Key Laboratory for Materials Simulation and Design, Beijing 100084, China
article
info
Article history: Received 25 January 2008 Received in revised form 1 April 2008 Accepted 14 April 2008 by P. Sheng Available online 24 April 2008 PACS: 61.46.Bc 31.15.Ew 36.40.Cg
a b s t r a c t The structures and magnetic ordering of Mnn Fe (n = 1–12) clusters have been investigated using all-electron density functional theory. The results indicate that the Mnn Fe clusters undergo a change in magnetic behavior from ferromagnetic ordering for the smallest size to ferrimagnetic ordering for intermediate sizes and beyond. Ferromagnetic ordering is clearly favored for n = 1, but ferromagnetic and ferrimagnetic states are nearly degenerate for n = 2, 3, and 4. A radical change occurs at n = 5 where the ferrimagnetic states completely prevail. The transition range of magnetic ordering from ferromagnetic to ferrimagnetic for Mnn clusters occurs early with the doping of one Fe atom. © 2008 Elsevier Ltd. All rights reserved.
Keywords: A. Manganese-iron clusters B. Density functional theory C. Magnetic ordering
1. Introduction There has been increasing effort in the study of magnetism of transition metal clusters (TMCs), which is motivated largely by the desire to explore fundamental understanding of the magnetic properties of materials with dimensions in nanometer length scales. The magnetic properties of Mnn clusters are unusual due to the electronic configuration 3d5 4s2 in Mn atoms. Therefore, the study of Mnn clusters has attracted considerable attention in both theoretical and experimental research areas [1–4]. Of particular interest, the transition range of the magnetic ordering of small Mn clusters from ferromagnetic to antiferromagnetic, is well studied in theory [5–8]. To date, most studies of cluster magnetism have involved TMCs composed of a single metal. The rich and varied magnetic behavior displayed in bulk binary alloys suggests that bimetallic TMCs may display interesting and potentially useful magnetic properties as well. Using a molecular beam deflection, it had been reported that small Com Mnn clusters were superparamagnets at temperatures substantially above the ordering temperatures of the corresponding Co1−x Mnx alloys, and that they possessed
∗ Corresponding author at: Department of Physics, Hebei Normal University, Shijiazhuang 050016, Hebei, China. Tel.: +86 311 86268649; fax: +86 311 86268314. E-mail address: [email protected] (Y. Liu). 0038-1098/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2008.04.020
mean per-atom moments that were substantially larger [9]. Using Stern–Gerlach deflection at low temperature (46.5 K), Yin et al. investigated the properties of Bim Mnn (m = 2–20, n = 0–7) clusters and found either ferromagnetic or ferrimagnetic coupling behavior among the manganese atoms, depending on size and composition [10]. In this note, we report the study of the magnetic ordering of the Mnn Fe (n = 1–12) clusters using all-electron density functional theory (DFT) by an extensive search for their lowestenergy configurations. A brief description of the theoretical method and computational details will be given in Section 2. In Section 3, the lowest-energy structures and some low-lying isomers for the Mnn Fe (n = 1–12) clusters will be shown, and the structural property, the stability and the magnetic ordering of the clusters will be discussed. The main computational results will be summarized in Section 4. 2. Computational details In order to search the lowest-energy structures of Mnn Fe (n = 1–12) clusters, the geometry optimization with constrained symmetry is firstly performed to find more low-lying metastable isomers among abundant initial structures, which are based on our previous studies [11–17]. In order to explore the full freedom in the potential energy surface and avoid possible saddle points, the geometry optimization is then performed without any symmetry
54
B.-R. Wang et al. / Solid State Communications 147 (2008) 53–56
Fig. 1. The various structures for Mnn Fe (n = 1–4) clusters, and the lowest-energy structures for Mnn Fe (n = 5–12) clusters. Light ball: Mn atom; dark ball: Fe atom. The spin orientation of Mn and Fe atoms and the Mn–Fe bond lengths are labeled.
restriction. The atomic charge and spin moments are obtained by the Mülliken population analysis.
3. Results and discussion
In our calculations, we chose a double-numerical basis with polarized functions (DNP), for which, in general, the most accurate results can be obtained. The spin-polarized BLYP functional (the Becke exchange functional [18] and the Lee–Yang–Parr correlation functional [19]) based on generalized gradient approximation (GGA) is employed to take into account the exchange and correlation effects of electrons. The charge density is converged to 1 × 10−6 e/Å in the self-consistent calculation, which corresponds to a total energy convergence of 1 × 10−5 Hartree. In the optimization process, the energy gradient and atomic displacement are converged to 1 × 10−4 Hartree/Bohr and 1 × 10−3 Å, respectively. All calculations are performed using allelectron DFT implemented in the DMol package [20,21], which is a comparatively accurate and efficient procedure for molecules.
The lowest-energy structures and some low-lying metastable isomers for Mnn Fe (n = 1–12) clusters are shown in Fig. 1, and the results are summarized in Table 1. For MnFe, the lowestenergy structure (C∞v ) with bond length 2.454 Å has ferromagnetic ordering and a total spin moment of 9µB . Our computed values are in good agreement with the early values (2.42 Å and 9µB ) obtained by Gutsev et al. [22]. For Mn2 Fe, an isosceles triangle (C2v ) is found to be the ground state, of which the ferromagnetic Mn–Mn coupling is the same as that of Mn2 [16], while the Mn–Fe couplings are ferrimagnetic. The C2v structure has a total spin moment of 8µB (considering spin orientation) and an average spin moment per Mn atom of 5.066µB (without considering spin orientation). The second isomer (Cs ) with a ferrimagnetic Mn–Mn coupling is 0.169 eV higher in energy than the ground state. The other two low-lying isomers
B.-R. Wang et al. / Solid State Communications 147 (2008) 53–56 Table 1 Calculated results for various structures of Mnn Fe (n = 1–12) clusters n
Symmetry
BE
HOMO
Gap
µs
QFe
1 2
C∞v C2v Cs C∞v D∞h Cs -1 Cs -2 C3v C2v -1 Cs -1 Cs -2 C2v -2 Cs -1 Cs -2 Cs -3 C4v Cs -4 C2v 1 C3v C2v -2 Cs C5v C2v -1 C2v -2 Cs -1 Cs -2 C2v -1 Cs C2v -2 C2v -3 C4v C4v -1 C2v C4v -2 C4v -3 C3v C4v C2 C2v C5v -1 C5v -2 D3h C2v C5v C2v D3h Ih Oh
−1.87 −2.94 −2.77 −2.63 −2.62 −4.68 −4.53 −6.74 −6.57 −6.53 −6.40 −6.31 −8.67 −8.59 −8.52 −8.30 −8.28 −10.92 −10.52 −10.32 −10.28 −10.17 −12.33 −12.30 −12.29 −12.23 −14.40 −14.16 −14.11 −13.66 −13.65 −16.22 −16.16 −15.57 −15.47 −15.26 −18.88 −18.21 −17.95 −20.65 −20.24 −20.03 −19.57 −22.32 −21.67 −21.09 −20.99 −20.47
A1.1 A1.1 A0 .1 B2.1 B2G.1 a A0 .1 a A0 .1 a A0 .1 B2.1 a A0 .1 A0 .1 A1.1 A0 .1 B2.1 A1.1 A1.1 A0 .1 E2.2 A1.1 A1.1 A00 .1 A0 .1 A1.1 A00 .1 A2.1 B2.1 B1.1 B1.1 A2.1 A1.1 E.2 E.2 A1.1 A.1 B2.1 E1.2 A1.1 E00 .2 B1.1 A1.1 A2.1 A10 .1 GU.3 T1U.3
0.388 0.068 0.253 0.514 0.304 0.497 0.391 0.529 0.468 0.604 0.365 0.268 0.501 0.426 0.411 0.230 0.343 0.338 0.319 0.320 0.341 0.103 0.489 0.373 0.210 0.526 0.247 0.202 0.260 0.357 0.158 0.256 0.556 0.197 0.331 0.071 0.172 0.290 0.224 0.210 0.139 0.154 0.237 0.225 0.056 0.213 0.084 0.256
9 8 2 2 8 1 3 12 22 12 2 2 3 7 3 9 3 10 0 10 5 22 3 9 7 9 0 6 4 20 2 1 15 27 29 7 2 6 4 1 5 5 4 38 16 6 46 52
−0.042 −0.168 −0.062 −0.140
3 4
5
6
7
8
9
10
11
12
0.024
−0.059 −0.069 −0.052 −0.029 −0.039 −0.037 0.048
−0.059 −0.010 −0.030 −0.022 −0.039 −0.040 −0.045 −0.012 −0.048 −0.010 −0.022 −0.028 −0.005 0.005
−0.051 −0.004 −0.014 0.002
−0.057 −0.100 0.010
−0.050 0.012
−0.011 −0.066 −0.065 −0.084 0.069
−0.059 −0.152 0.031
−0.013 −0.041 −0.028 0.018 0.111
The table lists the symmetry, binding energy BE (in eV), the HOMO state, the HOMO–LUMO gap, the total spin moment µs (in µB ) and the charge of Fe atom Q Fe (in e).
(C∞v , D∞h ) are linear chains. For Mn3 Fe, the most stable structure is a planar configuration (Cs -1), which has a total spin moment of 1µB and a mean spin moment per Mn atom of 4.354µB . A trigonal pyramid (Cs -2) is obtained as the second isomer. The Mn–Mn couplings of the structures (Cs -1, Cs -2) are ferrimagnetic. The ferromagnetic Mn–Mn couplings of Mn3 [16] are altered by the doping of one Fe atom. In the case of Mn4 Fe, the C3v structure is found to be the ground state with ferrimagnetic ordering. The total spin moment of the C3v state is 12µB and the mean spin moment per Mn atom is 4.333µB . It is worth mentioning that the C2v -1 structure has ferromagnetic ordering and a total spin moment of 22µB , which is obtained as the second isomer whose energy is 0.167 eV higher than that of the ground state. The other three structures (Cs -1, Cs -2, C2v -2) are a trigonal bipyramid, a distorted square pyramid and a planar configuration, respectively. For Mn5 Fe, the most stable isomer (Cs -1) is found to be a tetragonal bipyramid with a total spin moment of 3µB and a mean spin moment per Mn atom of 4.116µB . This is followed by four low-lying isomers (Cs -2, Cs -3, C4v , Cs -4). All the geometries
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obtained here are ferrimagnetic. From n = 5 or more, the Mn–Mn couplings of all the geometries of Mnn Fe clusters are of ferrimagnetic ordering. The transition range of magnetic ordering from ferromagnetic to ferrimagnetic for the pure Mnn (n = 2–12) clusters [16] occurs early for the doping of one Fe atom. For Mn6 Fe, the C2v -1 structure is obtained as the lowest-energy structure, having a total spin moment of 10µB and a mean spin moment per Mn atom of 3.914µB . The C3v structure has energy 0.393 eV higher than that of the ground state and is found to be the second isomer. The C2v -2, Cs and C5v structures are obtained as the low-lying isomers as well. For Mn7 Fe, the C2v -1 structure has the lowest energy with a total spin moment of 3µB and an average per-atom spin moment of 4.105µB for Mn. The C2v -2 structure whose energy is only 0.022 eV higher than that of the ground state is considered to be the second isomer. Two other low-lying isomers (Cs -1, Cs -2) have been found, which are 0.034 eV, 0.097 eV higher in energy than the ground state. For Mn8 Fe, it is found that the total spin moment and the mean spin moment per Mn atom of the lowest-energy structure (C2v -1) are 0µB and 3.906µB . The other five low-lying isomers have Cs , C2v -2, C2v -3 and C4v symmetries, respectively, and in increasing order of binding energy, total spin moments of 6µB , 4µB , 20µB , and 2µB . For Mn9 Fe, the C4v -1 structure with Fe atom occupying the surface site is found to be the lowest-energy structure with a total spin moment of 1µB and a mean spin moment per Mn atom of 3.967µB . The second isomer with C2v symmetry is only 0.054 eV higher in energy than the ground state. For Mn10 Fe, the lowest-energy structure (C4v ) with Fe atom locating at the surface site has a total spin moment of 2µB and a mean per-atom spin moment of 3.642µB for Mn. The other two low-lying isomers (C2 , C2v ) are both with Fe atom locating at the central site. For Mn11 Fe, four low-lying isomers (C5v -1, C5v -2, D3h , C2v ) are obtained. The C5v -1 structure has a total spin moment of 1µB , which is found to be the ground state with a mean spin moment per Mn atom of 3.715µB . The Mn11 Fe as well as the Mn10 Fe is with Fe atom occupying the surface site in the lowest-energy structure. In the case of Mn12 Fe, the C5v structure with Fe atom occupying the surface site is obtained as the lowest-energy structure, the total spin moment and the average spin moment per Mn atom of which are 38µB and 3.912µB . The C2v structure is found to be the second isomer which is 0.652 eV higher in energy than the ground state. The structures with D3h , Ih and Oh symmetries, respectively, are all with Fe atoms locating at the central sites and they are much higher in energy than the ground state. It is favorable for the doped Fe atom to occupy the surface site. It can be seen from Table 1 that the charge transfers from the Mn atoms to the Fe atom except in the case of Mn11 Fe, in which the charge transfers from the Fe atom to the Mn atoms. In Fig. 2, the size dependence of the HOMO–LUMO gaps and the binding energies per atom (Eb ) of the Mnn Fe (n = 1–12) and MnN (N = 2–13) clusters are displayed separately. Fig. 2(b) illustrates the second derivative of the binding energies, ∆2 E(n) = Eb (n + 1) − 2Eb (n) + Eb (n − 1), for Mnn Fe (n = 1–12) clusters, as a function of cluster size. It can be seen that the stability of the pure MnN (N = 2–13) clusters is enhanced by the addition of one Fe atom, and the Mn6 Fe and Mn10 Fe have much higher stability compared to other clusters. The electronic structures of the pure MnN (N = 2–13) clusters have been greatly altered by the doping of one Fe atom. The size dependence of the average bond length and the µa of Mnn Fe (n = 1–12) clusters are shown in Fig. 3. Here µa is the sum of the absolute values of the spin moments divided by the number of atoms in each Mnn Fe (n = 1–12) cluster. From Fig. 3(a), it can be seen that µa decreases with the increasing cluster size, and exhibits the magnetic moment surface enhancement effect, which has been widely discussed by other authors [23– 26]. For Mn4 Fe, Mn7 Fe, Mn9 Fe and Mn12 Fe, the µa have larger values compared with those of their neighbors. At the same time, their corresponding average bond lengths are also larger,
B.-R. Wang et al. / Solid State Communications 147 (2008) 53–56
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Fig. 2. Size dependence of HOMO–LUMO gaps for Mnn Fe (n = 1–12) ( ) and MnN (N = 2–13) ( ) clusters (a); and the binding energies per atom (Eb ) for Mnn Fe (n = 1–12) ( ) and MnN (N = 2–13) ( ) clusters, and the second derivative of total binding energies (∆2 E) for Mnn Fe (n = 1–12) ( ) clusters (b). The data of the MnN (N = 2–13) clusters is from Ref. [16].
Fig. 3. Size dependence of µa (Fe ( ), Mn( ) atoms and the total ( ) clusters) for Mnn Fe (n = 1–12) clusters (a); and the average bond lengths (Mn–Fe ( ), Mn–Mn ( ) and the total bonds ( ) of the clusters) for Mnn Fe (n = 1–12) clusters (b).
which demonstrates the bond contraction effect discussed in Refs. [23,24].
[1] M.B. Knickelbein, Phys. Rev. B 70 (2004) 014424. [2] N.O. Jones, S.N. Khanna, T. Baruah, M.R. Pederson, Phys. Rev. B 70 (2004) 045416. [3] M.R. Pederson, F. Reuse, S.N. Khanna, Phys. Rev. B 58 (1998) 5632. [4] P. Bobadova-Parvanova, K.A. Jackson, S. Srinivas, M. Horoi, Phys. Rev. A 67 (2003) 061202. [5] R.C. Longo, E.G. Noya, L.J. Gallego, Phys. Rev. B 72 (2005) 174409. [6] M. Kabir, A. Mookerjee, D.G. Kanhere, Phys. Rev. B 73 (2006) 224439. [7] P. Bobadova-Parvanova, K.A. Jackson, S. Srinivas, M. Horoi, J. Chem. Phys. 122 (2005) 014310. [8] R.C. Longo, E.G. Noya, L.J. Gallego, J. Chem. Phys. 122 (2005) 226102. [9] M.B. Knickelbein, Phys. Rev. B 75 (2007) 014401. [10] S. Yin, X. Xu, R. Moro, W.A. de Heer, Phys. Rev. B 72 (2005) 174410. [11] Q.M. Ma, Z. Xie, J. Wang, Y. Liu, Y.C. Li, Phys. Rev. B 76 (2007) 035412. [12] J. Wang, Q.M. Ma, Z. Xie, Y. Liu, Y.C. Li, Phys. Rev. B 76 (2007) 035406. [13] Z. Xie, Q.M. Ma, Y. Liu, Y.C. Li, Phys. Lett. A 342 (2005) 459. [14] Q.M. Ma, Z. Xie, J. Wang, Y. Liu, Y.C. Li, Phys. Lett. A 358 (2006) 289. [15] Q.M. Ma, Z. Xie, J. Wang, Y. Liu, Y.C. Li, Solid. State. Commun. 142 (2007) 114. [16] Q.M. Ma, Z. Xie, Y. Liu, Y.C. Li, Chin. Phys. Lett. 24 (2007) 7. [17] N. Liu, Q.M. Ma, Z. Xie, Y. Liu, Y.C. Li, Chem. Phys. Lett. 436 (2007) 184. [18] A.D. Becke, J. Chem. Phys. 98 (1993) 5648. [19] C. Lee, W. Yang, R.G. Parr, Phys. Rev. B 37 (1988) 785. [20] B. Delley, J. Chem. Phys. 113 (2000) 7756. [21] B. Delley, J. Chem. Phys. 92 (1990) 508. [22] G.L. Gutsev, M.D. Mochena, P. Jena, C.W. Bauschlicher Jr., H. Partridge III, J. Chem. Phys. 121 (2004) 6785. [23] W.H. Zhong, C.Q. Sun, S. Li, H.L. Bai, E.Y. Jiang, Acta Materialia 53 (2005) 3207. [24] C.Q. Sun, H.L. Bai, S. Li, B.K. Tay, E.Y. Jiang, Acta Materialia 52 (2004) 501. [25] C.Q. Sun, T.P. Chen, B.K. Tay, S. Li, H. Huang, Y.B. Zhang, L.K. Pan, S.P. Lau, X.W. Sun, J. Phys. D: Appl. Phys. 34 (2001) 3470. [26] C.Q. Sun, L.K. Pan, H.L. Bai, Z.Q. Li, P. Wu, E.Y. Jiang, Acta. Materialia 51 (2003) 4631.
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4. Conclusions Using all-electron DFT-GGA calculation, the lowest-energy structures of the Mnn Fe (n = 1–12) clusters are obtained. The results indicate that the transition range of magnetic ordering from ferromagnetic to ferrimagnetic for Mnn Fe clusters occurs early compared to that of the pure MnN (N = 2–13) clusters. The MnFe has ferromagnetic ordering. The Mn–Mn coupling of Mn2 Fe is ferromagnetic similar to that of Mn2 . While the ferromagnetic Mn–Mn couplings of Mn3 are completely altered with the doping of one Fe atom. It is from n = 5 that the Mn–Mn couplings of all the states obtained here are ferrimagnetic. In addition, the stability, the surface enhancement and bond contraction effects of the magnetic moment for Mnn Fe(n = 1–12) clusters are also discussed. Acknowledgements We would like to thank Professor Chen Nan-Xian and Shen Jiang for helpful discussions. This work is supported by the 973 Project in China under Grant No. 2006CB605101, the National Natural Science Foundation of China (Grant No. 10574036).
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References