Structures and Properties of Core-shell B@Mn - American

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C: Physical Processes in Nanomaterials and Nanostructures 8

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Structures and Properties of Core-shell B@Mn@Mg Cluster and Brief of Efficient Core-shell Cluster Algorithm and Global Minima Algorithm based on Sequential Micro-motion Press-expand Method Bai Fan, Guixian Ge, Bao-Lin Wang, Guanghou Wang, and Jianguo Wan J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b10166 • Publication Date (Web): 04 Apr 2019 Downloaded from http://pubs.acs.org on April 4, 2019

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The Journal of Physical Chemistry

Structures and Properties of Core-shell B@Mn8@Mg10 Cluster and Brief of Efficient Core-shell Cluster Algorithm and Global Minima Algorithm based on Sequential Micro-motion Press-expand Method

Bai Fan1, Gui-Xian Ge2, Bao-Lin Wang3, Guang-Hou Wang1 & Jian-guo Wan1,4,*

1

National Laboratory of Solid State Microstructures, and Department of Physics, Nanjing University, Nanjing 210093, China

2

Key Laboratory of Ecophysics and Department of Physics, College of Science, Shihezi University, Xinjiang 832003, China

3

College of Physical Science and Technology, Nanjing Normal University, Nanjing 210023, China

4

Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China * Corresponding author. Email: [email protected]

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Abstract We present a sequential micro-motion press-expand method. On this basis, we develop efficient core-shell cluster algorithm with given core-cluster structure from arbitrary initial shell-cluster structure, and further develop global minima algorithm from arbitrary initial structure. Many tests of stable clusters reveal that the structure of stable core-shell clusters is an optimal atomic distribution of the outer atoms according to the potential fields of the core-cluster. On this basis, we identify a new core-shell A@B8@C10 cluster. Applying a spin-polarized density functional theory (DFT) approach, we investigate the structural stability and magnetic properties of a B@Mn8@Mg10 cluster, which has high stability because of the strong p-d hybridization between B and Mn atoms and the strong s-d hybridization between Mn and Mg atoms. The most stable state is one in which the interface layer (Mn) atoms have an axial paramagnetic moment, while the Mn atoms have large local magnetic moments that are in reverse orientation to the magnetic moment orientations of the inner layer (B) atoms and the external layer (Mg) atoms. Using a B@Mn8@Mg10 cluster as the structural element, we design Mg2BN-1(Mn4Mg4)N (N = 2-8) nanochains when Mn atoms have an axial paramagnetic moment that remains in geometrical symmetry after DFT calculations. Moreover, the total number of magnetic moments of the nanochains increases linearly with the structural element N.

PACS number(s): 36.40.-c, 36.40.Mr, 61.46.Bc


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Introduction Nanoalloy clusters are an important research field because of their catalytic, optical, and magnetic properties, and thus their numerous practical applications1,2. Due to the difference in interatomic interaction strength, stable bimetallic clusters often have an evident core-shell structure3,4. Previous theoretical investigation has pointed out that an A@B12@C20 core-shell cluster has high stability and magnetic properties. Using genetic algorithm and density functional theory (DFT) calculations, it has been found that Sn@Mn12@Sn20 and Pb@Mn12@Pb20 clusters have the most stable structure5, and they have large total magnetic moments (28μB). A@B12@A20 (A=Sn, Pb; B=Mg, Zn, Cd) clusters have a large HOMO-LUMO gap (1.29 ~ 1.54eV). [Mn@Mn12@Au20]- clusters6 also have a large total magnetic moment (44μB). The magnetic moment of Co@Co12@Mn20 clusters7 is 113μB, which is mainly driven by Mn atoms. 4d transition-metal (TM) doped Mg8 clusters make up core-shell TM@Mg8. TcMg8 clusters8 have a large total magnetic moment (5μB) and also a large HOMO-LUMO gap (0.65eV), while the total magnetic moment of Fe@Mg8 is 4μB. Therefore, core-shell clusters often have high stability, making the evolutionary algorithm of core-shell clusters an important subject to study. From their geometrical structure, we may regard many clusters as core-shell clusters8,9. We designate the atoms in the center region of the core-shell cluster as the core-cluster, and we designate the atoms in the shell region of the core-shell cluster as the shell-cluster. For example, the core-cluster of an icosahedral 13-atom cluster is a single atom. The cage 3 ACS Paragon Plus Environment


cluster may be regarded as a non-core cluster. Whether for non-core clusters or for core-shell clusters, the interatomic distances of the optimized cluster fluctuate near its average bond length. For the icosahedral 13-atom pure cluster, all of the distances between the surface atoms and the center atom are almost equal. Considering the force field, the other 12 atoms are on one of the equipotential surfaces of the center atom. Applying molecular dynamic simulations to optimize the cluster structure, we find that the force will change dramatically when the interatomic distance fluctuates near the equilibrium distance. The small time step will consume a massive amount of computational time, while the large time step will lead to irrational structure disruption, especially for clusters under high compression. In order to avoid this case, we present a simple method for optimizing cluster structure, and we refer to it as a “sequential micro displacement press-expand method”. On this basis, we develop efficient core-shell cluster algorithm with given core-cluster structure from arbitrary initial shell-cluster structure and global minima algorithm from arbitrary initial structure. Applying our new algorithm, we obtained a new core-shell A@B8@C10 cluster. We further optimized the structures of B@Mn8@Mg10 clusters, and we calculated their magnetic properties using a spin-polarized density functional theory (DFT) approach. Using A@B8@C10 cluster as the structural element, we designed Mg2BN-1(Mn4Mg4)N (N=2-8) nanochains with Mn atoms that have an axial paramagnetic moment—such Mn atoms retain their geometrical symmetry after DFT 4 ACS Paragon Plus Environment

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calculations—and we investigated their magnetic moments.

Computational Methods I. Sequential micro displacement press-expand method We set the origin of the spherical coordinate system at the center of the core-cluster, and we select one atom of the shell-cluster as the operated atom and subject it to a tiny displacement while keeping the core-cluster stationary. The tiny displacement may be divided into radial displacement and transverse displacement. The radial displacement reflects that the operated atom approaches toward, or departs from, the center of the cluster, and thus reflects interactions between the operated atom and the atoms of the core-cluster. The transverse displacement is mainly due to the influence of interatomic interactions between the operated atom and the neighboring atoms of the shell-cluster, which are mainly distributed in the spherical shell neighborhood of the operated atom. Therefore, in order to achieve a tiny displacement, we give each atom of the clusters new coordinates by moving them with a tiny radial displacement; meanwhile, we search for the optimal position in the local spherical neighborhood that is near the new coordinates. The detailed illustration is as follows. We select one atom of the shell-cluster as the operated atom (indexed as A). The original coordinates of the A atom are (r, θ, φ), and the energy of the cluster under this situation is E0. The first step is to make the A atom move with a tiny radial displacement △r toward the cluster center (the origin of the coordinates), and thus the 5 ACS Paragon Plus Environment


new radial coordinate of A atom is r = r-△r. The second step is to search for the optimal position in the local spherical neighborhood at point (r-△r, θ, φ). For example, we adopt this local spherical neighborhood (r-△x, θ-△α ~ θ+△α, φ-△β ~ φ+△β), and we divide the mesh. We make the A atom move to each mesh point while keeping the other atoms stationary, and then we search for the optimal position by comparing the energy of the cluster. When the A atom is in the optimal position, the energy of the cluster is Et. We accomplish this operation when Et < EXr·E0. Otherwise, we return the A atom back to the original coordinates (r, θ, φ). We operate each atom of the cluster in the same manner by turns. We refer to this operation as the “press operation”, and we refer to △r and EXr in this press operation as the “press degree” and the “press energy restriction”. In like manner, we make the A atom move with a tiny radial displacement △r far away from the cluster center, and thus the new radial coordinate of the A atom is r+△r. Also, we also make the A atom move to each mesh point to search for the optimal position by comparing the energy of the cluster while keeping the other atoms stationary. We accomplish this operation when Et < EXa·E0. Otherwise, we replace the A atom back to the original coordinates (r, θ, φ). We refer to this operation as the “expand operation”, while we refer to △r and EXa in the expand operation as the “expand degree” and the “expand energy restriction”. This method causes the cluster to evolve in a slow continuous process, and prevents the irrational structure disruption of the cluster during the optimization process. Setting EXr < 1.0 (EXa < 1.0) means that the corresponding press (expand) 6 ACS Paragon Plus Environment

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operation will be accomplished, although the energy of the cluster will increase after one operation, which causes the cluster to evolve toward the expected core-shell structure from the arbitrary initial structure. On this basis, we further develop efficient core-shell cluster algorithm with given core-cluster structure from arbitrary initial shell-cluster structure and global minima algorithm from arbitrary initial structure.

II. Efficient core-shell cluster algorithm with given core-cluster structure from arbitrary initial shell-cluster structure based on the sequential micro displacement press-expand method. For optimized pure cluster structures with a high coordination number, the interatomic distances are often intensively distributed near its average bond length. If we set EXr < 1.0 and EXa < 1.0, the interatomic distances finally evolve near a certain distance (predesign-distance: RPRE), regardless of the initial structure, and this may efficiently achieve core-shell cluster evolution. RPRE should be set to a value that is close to the average bond length of the optimized stable cluster at given size. The predesign-distance (RPRE) and the energy restriction (EXr, EXa) are important parameters. Brief illustrations of the computational program are as follows. 1. We often set the predesign-distance according to the equipotential surface motif of the core-cluster, which may have two results: (a) for dimer or trimer core-clusters, we set the predesign-distance as approximately rAcos(rA, rB)+RA-B (B is the operated atom, while A is the atom of the core-cluster with the neighbors nearest to the B atom. rA and rB are the position vectors of the A atom and B atom, respectively, and RA-B is 7 ACS Paragon Plus Environment


the equilibrium distance of the empirical potential); (b) for icosahedron core-clusters, we set the predesign-distance as approximately rA + RA-B. 2. Mid-operation: we often set the operate-center at the center of the core-cluster. We apply the press operation or the expand operation to the atoms of the shell-cluster by turns, and we keep the core-cluster stationary in the meantime. We regard these actions as the mid-press operation and the mid-expand operation (RMPRE, EXMr, EXMa), respectively. Polar-operation: if the core-cluster is polyatomic, we select one atom of the core-cluster (A atom), and we set the operate-center on the A atom. We apply the press (expand) operation to the atoms of the shell-cluster within a certain distance near the A atom; the Rpolar is no more than the interaction distance in a tight-binding scheme as it approaches toward or departs from the A atom, respectively. We move the operate-center to each atom of the core-cluster by turns and regard these actions as the polar-press operation and the polar-expand operation (RPPRE, EXPr, EXPa), respectively. 3. We often divide the local spherical neighborhood with uneven mesh points, which are intensive in the center region and sparse at the edge. This contributes to the computational precision and scope of the local spherical neighborhood. For example, we apply the following mesh points: θ+(π/180)(I-MIT/2)|I-MIT/2|/MAT and φ+(π/180)(J-MJT/2)|J-MJT/2|/MBT, where MIT and MJT are the upper limits of I (0, …, MIT) and J (0, …, MJT), respectively, and both values are all even numbers greater than zero. 4. If the core-cluster is polyatomic, especially for large-size clusters, we take the 8 ACS Paragon Plus Environment

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following steps in order to enhance computational effectiveness: before each mid-operation stage, we make the core-cluster once compress or expand slightly and linearly in certain geometric dimension, while keeping the atoms of the shell-cluster stationary. 5. In order to further increase the computational effectiveness, we could add a global optimal operation at the end of each complete cycle (after mid-operations and polar-operations). Each atom searches for its optimal position globally (△α = π/2, △β = π and EXG = 1.0) when the radial displacement of each operated atom is zero (△r = 0). The operate-center is the same as in the mid-operation. Using the Efficient core-shell cluster algorithm with Gupta potential, we quickly validate many magic structures of metallic clusters (such as 13, 19, 34, 38, 55, 79, 147, 309, 561). Apply this algorithm on plane and sphere, we also quickly validate some planar magic and spherical carbon clusters. These structures are equivalent to the geometric structures of clusters in previous investigations, and we also find some new clusters with high stability. Moreover, this algorithm also can be applied in carbon-metal cluster system. Therefore, this algorithm has high efficiency and reliability.

III. Global minima algorithm from arbitrary initial structure based on the sequential micro displacement press-expand method. 1. Polar-operation. The atoms of core-cluster often have high coordination number and near the cluster center. Therefore, in polar-operations, we often set operate-center 9 ACS Paragon Plus Environment


on the atoms with high coordination number, or (and) the atoms near the cluster center (especially in early stage). We take turns applying polar-press operation and polar-expand operation on the atoms nearby the atom on the operate-center. we often successively apply mid-operation up to certain cycles, which effectively keep the cluster structure in slow evolutionary process, and then move operate-center to other atoms. Because the interatomic interactions versus interatomic distance have high similarity, we could rewrite partial parameters (especially R0) of known empirical potential to obtain the press (expand) degree, or adopt analogous expression. Another adopted method in polar-operations is to unify press operation and expand operation to one operation, R → R - FT∙(R - RPRE), where R is the radial distance between the operated atom and operate-center, and FT is positive parameter to adjust amplitude degree. When R > RPRE, the operation produces press effect while the operation produces expand effect for R < RPRE. 2. Mid-operation. We often select the cluster center as the operate-center. When the cluster is in the linear compress (expand) process, the cluster could keep well geometric structure. Therefore, we apply the press degree and expand degree △L linearly versus R (the radial distance between the operate-center and the operated atom). We could successively apply mid-operation up to certain cycles, which enhances the strength of compress (expand) effect. In order to offset the shortage of fixing the operate-center on the cluster center, we could appropriately add (or apply) the mid-operations, the operate-center of which is cyclically moved to one of predesigned discrete positions or the atoms much nearby the cluster center (also could 10 ACS Paragon Plus Environment

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select partial inside atoms). In order to further consider the interaction between the operated atom and the atoms nearby it in mid-operations, we could appropriately add polar-operations. For example, we apply once polar-operation with much small amplitude of press (expand) degree after each mid-operation. For large size cluster, in order to further enhance efficiency, we could apply some polar-operations when the radial distance of operated atom(RB) is larger than that of operate-center (RA) by certain distance (for example, RB > RA+0.2 R0, where R0 is interatomic equilibrium distance). 3. Double-exchange operation. For n-size mixed cluster, we set two loop variable I (1, …, n) and J (1, …, n), which refer to the order of atoms. We directly exchange the positions of two different elemental atoms and conserve the exchange under the conditions of making the cluster more stable by comparing energy. We often successively apply double-exchange operation up to certain cycles. We combine double-exchange operation with mid-operations and polar-operations to apply global minima algorithm in mixed cluster system. 4. Steel-ball model. In order to enhance the efficiency, we make interatomic distance not less than minimum value Rs (for example, Rs ≈ 0.9 R0), which makes the atom like steel ball. In the computational program, we neglect the mesh points, which make the interatomic distance between the operated atom and other atoms less than Rs. 5. Moonie global optimization. In global minima algorithm, we present a new global optimal operation. We select one atom of the cluster, and then search for the optimization position in spherical neighborhood of each atom, which makes the 11 ACS Paragon Plus Environment


cluster like a pile of moonies. In the computational program, we often divide 10 equidistant concentric spherical surface on each atomic position when the radial distance from Rs to Rm (we often set Rm as the interatomic distance restriction in the tight-binding scheme). If the energy achieves the energy restriction, we move the atom to the optimization position, otherwise we keep the atom on original position. If we move each surface atom of the cluster away from the spherical location of the original position with the radius Rm, we cannot find other positions, which matches the condition of surface atom (for example, its coordination number no less than the minimum value). This could be used as standard of stable cluster. 6. Mold model. For the large size clusters, we could apply a mold, and the geometric size the mold fluctuates in the certain extent. Whether in the mold or out of the mold, the press (expand) degree in mid-operations is linear versus the radial distance between the operate-center and the operated atom. In the mold, we make press degree slightly larger (even could slightly smaller) than expand degree. However, outside of mold, we often make press degree much larger than expand degree, which makes the surface atoms of cluster effectively evolve to the surface characteristics of the mold. We adding auxiliary general mid-operations to achieve larger compress (expand) effect, which make the geometric size of the cluster fluctuate according to the mold size by adjusting the ratio between press degree and expand degree in the mid-operations. When the mold size fluctuates in certain extent, the geometric structure does not change, which can be used as the standard of stable cluster. We can use the minimum (maximum) of mold size to control the compress (expand) strength. 12 ACS Paragon Plus Environment

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The smaller the minimum of mold size is, the higher compress strength is. Moreover, in the early stage, if we apply relatively small amplitude of press (expand) degree outside of the mold, the outside atoms move into the mold in slow process. Several suggests in the computational program are useful, as follows. We use the coordination number and position of the mesh points to control the cluster evolutionary process. For example, in each operation, amount change of coordination number of the operated atom is not more than 2. We could make the atoms in preference to select the mesh points in certain area under the conditions by energy restriction and coordination number. Especially in Moonie global optimization, we often neglect the mesh points no more than the maximum radial distance, and make the atoms with high coordination number or the atoms near cluster center avoid to select the mesh points far away from their radial distance. We appropriately add moonie global optimizations on the atoms with radial distance near maximum and (or) with minimum coordination number to further enhance efficiency, and also could make the atoms in preference to select the mesh points near the cluster center. We often order the atoms from the near to the distant according to radial distance from the operate-center, and apply the operations sequentially according to the order, which could keep the structure characteristic of inside atoms in slower process in comparison with surface atoms. The operate-center also could be fixed at origin of coordinate system in mid-operations all the time, but this often consume more computational times in comparison with the operate-center in cluster center. We also could set the expand operation before press operation, but sometimes it also consumes 13 ACS Paragon Plus Environment


more computational times. In mold model, when the atoms are in the farside area (out of maximum size of the mold in mold model), local spherical neighborhood in the mid-operations could be set much smaller, which could save more computational times. Moreover, setting the energy restriction as 1.0 in partial operations, especially in mid-operations, will contribute to find new structures. Using the global minima algorithm without mold model with Gupta potential, we find some new stable structures of metallic Pdn clusters, and also quickly validate magic structures of Pdn clusters (such as 13, 19, 34, 38, 55). Using the spherical mold, we efficiently validate the magic structure of Pd55 and Pd147 clusters with the highest stability, which are equivalent to the structures of clusters in previous investigations3,4. Moreover, we quickly obtain Pd@Pd12@Pd20 cluster, structure of which is equivalent to the [Mn@Mn12@Au20]- cluster6. We also find the compress strength have close dependence on the structure of stable cluster. Therefore, the global minima algorithm also has high efficiency and reliability. The new methods and global minima algorithm mainly have two kinds of operators on the basis of sequential micro-motion press-expand method, and thus the computational program can be design artificially and flexibly. We also can fix geometric structure from partial atoms of the cluster, and embed the efficient core-shell cluster algorithm with given core-cluster structure into global minima algorithm. Moreover, by frst-principles calculations to calculate the cluster energy, the new algorithms will have larger application.


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IV. Computational process of A@B8@C10 cluster For metal clusters, the Gupta potential is conventional empirical potential, and the press degree and expand degree may be easily deduced from the Gupta potential. The predesign-distance (RPRE) may be adjusted easily, and thus we apply the Gupta potential to carry out the algorithm. The Gupta potential7 contains repulsive term Er(i) and attractive term Ea(i), and the energy of atom i can be expressed as follows: 𝐸 𝑖

∑

𝐸 𝑖

𝐴 𝑒

⁄

∑

𝐵 𝑒

⁄

(1)

where rij is the distance between atoms i and j, Rij is equilibrium distance, and the parameters Aij, Bij, Pij, Qij and Rij depend on the type of bond. For a dimer cluster, the expression of repulsive term Er(i) and attractive term Ea(i) are as follows: 𝐸 𝑖 𝐸 𝑖

⁄

𝐴 𝑒 𝐵 𝑒

(2) ⁄

(3)

If we set A atom as the reference frame, the absolute values of the repulsive and attractive forces of the B atom exerted by the A atom are as follows: 𝐹 𝑖

𝑒

⁄

(4)

𝐹 𝑖

𝑒

⁄

(5)

We separately consider the repulsive and attractive forces of the A atom. Also, we assume that the force does not change in the process of the tiny straight displacements. When given a tiny amount of energy △Er (△Ea), the B atom will overcome the repulsive (attractive) force exerted by the A atom, and it will approach (depart from) the tiny straight displacement △Lr(i) (△La(i)). Thus, the press degree and expand 15 ACS Paragon Plus Environment


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degree are △Lr(i) = △Er/Fr(i) and △La(i) = △Ea/Fa(i), respectively, and the concrete expressions of this may be rewritten as follows: ∆𝐿 𝑖

∆𝐸 ∙ 0.00005

𝑒

⁄

(6)

∆𝐿 𝑖

∆𝐸 ∙ 0.00005

𝑒

⁄

(7)

△Lr(i) and △La(i) will have one crossing point. If we set EXr < 1.0 and EXa < 1.0, while keeping the number of press operations and expand operations equal, rij will fluctuate near the predesigned distance RPRE at moment △Lr(i) = △La(i). Also, RPRE can be adjusted by △Er/△Ea. Like the Gupta potential of Pdn clusters, both the repulsive and attractive forces decrease sharply when rij > Rij, which leads to sharp increments of △Lr(i) and △La(i) when rij > Rij. Therefore, we set a cut-off distance Rmax (for example, Rmax = 1.2Rij) in equation (6) and equation (7) to maintain △Lr(i) and △La(i) at a small value. Moreover, we often set △Er and △Ea from 100 to 3000, which makes △Lr and △La reach the range of 0.001 ~ 0.1Å. When the radial displacement is about 0.05Å, the energy increment of the cluster does not exceed ǀ0.01·E0ǀ even though the energy increases after one operation, and thus we often set EXr = EXa = 0.99. Applying efficient core-shell cluster algorithm, we obtain a new core-shell cluster A@B8@C10 cluster (Figure 1). We assume one metal-like elemental atom (we index it as A), which has a small radius. In Table 1, we rewrite the Pd-Ag interatomic distance from 2.89Å to 2.2Å while keeping the other parameters unchanged, and we then use modified Pd-Ag parameters to reflect the Pd-A interaction. Both the interatomic distance restriction in the tight-binding scheme and the cut-off distance Rmax in 16 ACS Paragon Plus Environment

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mid-operation are 1.2Rij. The origin of the spherical coordinate system is set at the center A atom, and the outer 10 Pd atoms are randomly generated in a cube of 10RPd-Pd length with a central A atom. The operate-center is fixed at the A atom (where rA = 0). The core-cluster is held stationary at all times and its structure does not change. Parameters in mid-operation are: MIT = MJT = 10, and MAT = MBT = 0.5. The setting △ErM = 500 and △EaM = 200, meaning that the predesigned distance RPRE = 1.9308 Å, while EXrM = EXaM = 0.9. Parameters in the global optimal operation are: MIT = 40, MJT = 60, MAT = 4, MBT = 5, △EG = 0, and EXG = 1.0. Each complete circle (nA) contains two steps. (1) Mid-operation: each mid-operation (nM = 1) contains 5 successive mid-press operations and then 5 successive mid-expand operations. (2) Global optimal operation: (nG = 1). We obtain an A@Pd8 cluster, the geometrical structure of which is equivalent to a TM@Mg8 cluster8. We apply A@Pd8 as the core-cluster without changing the parameters, and we keep RPRE fixed at 1.9308Å from the operate-center (A atom); we then obtain a new core-shell A@Pd8@Pd10 cluster.

V. DFT calculation details On the basis of the optimized structures obtained by molecular dynamic simulations, we further performed structural optimizations using the spin-polarized density functional theory (DFT) method. We use the Vienna ab initio simulation package (VASP)10, and implement the projector augmented wave (PAW) method11,12. 17 ACS Paragon Plus Environment


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The exchange-correlation functions are described by the Perdew, Burke, and Ernzerhof (PBE) function within the generalized gradient approximation (GGA)13. The plane-wave basis is set with an energy cut-off of 400 eV. The equilibrium geometries are obtained when the atomic forces are smaller than 0.01 eV/Å, and the total energy convergences are within 10−5 eV. All calculations are performed with a cubic box of 20 Å using a single k point (Г point) for the Brillouin-zone (BZ) integration. The accuracy of the DFT calculation is assessed by calculating dimers (B2, Mg2 and Mn2). Our calculated bond length, binding energy and total magnetic moment agree with the previous theoretical values listed in Table 2, indicating that our computational method in this work is reliable.

Results and discussion 1. Structure and electric properties of the B@Mn8@Mg10 cluster In order to investigate the stability of the B@Mn8@Mg10 cluster, we calculate its binding energy, which is defined as: Eb= E(B) + 8E(Mn) + 10E(Mg) - E(B@Mn8@Mg10)

(8)

Where E(B), E(Mn), E(Mg) and E(B@Mn8@Mg10) represent the energy of B, Mn, Mg and B@Mn8@Mg10, respectively. After DFT calculations, the B@Mn8@Mg10 cluster retains structural symmetry. In comparison with the B@Mn8@Mg10 cluster with the axial antiparallel magnetic moment of the Mn atoms, the B@Mn8@Mg10 cluster with the axial parallel magnetic 18 ACS Paragon Plus Environment

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moment of the Mn atoms is more stable (Figure 2), and the energy difference is large (≈0.52eV), as shown in Table 3. Their HOMO-LUMO gaps are 0.245eV and 0.312eV, respectively, and thus the cluster has electronic stability. Previous investigations have pointed out that the binding energy of an Mn8 cluster17 is about 1.3eV/atom, while the binding energy of an Mg10 cluster18 is 0.56eV/atom. The binding energy of a B@Mn8@Mg10 cluster with the axial parallel magnetic moment of the Mn atoms is 1.805eV/atom, indicative of the high stability of the B@Mn8@Mg10 cluster because of the B and Mg atoms. To further verify the geometrical stability of the B@Mn8@Mg10 cluster, we also calculated B0@Mn8@Mg0, B@Mn8@Mg0 and B0@Mn8@Mg10 clusters, the atom positions of which are equivalent to the corresponding atom positions of the B@Mn8@Mg10 cluster. The calculation results are listed in Table 3. After DFT calculations, the B0@Mn8@Mg0 cluster does not exist. Only a B@Mn8@Mg0 cluster with the axial parallel magnetic moment of the Mn atoms exists, and its HOMO-LUMO gap is small (0.024eV) while its binding energy is high (2.187eV). Therefore, we find that the B atom enhances the structural stability of the B@Mn8@Mg10 cluster. Interestingly, its total magnetic moment is large (30.9μB). Moreover, this B@Mn8 cluster is more stable than previous results17. B0@Mn8@Mg10 clusters with both the axial parallel and antiparallel magnetic moments of the Mn atoms exist, while their energy difference is much smaller (≈0.15eV). Therefore, Mg atoms enhance the energy difference of the B@Mn8@Mg10 cluster when the magnetic moment order is different, and this enhances the stability of 19 ACS Paragon Plus Environment


the cluster. Moreover, their total magnetic moments are 22μB and zero, respectively, which are relatively smaller in comparison with the B@Mn8@Mg10 cluster. To further understand the stability of the B@Mn8@Mg10 cluster, we calculate the LDOS of the B@Mn8@Mg10 cluster with the axial paramagnetic moment of the Mn atoms, and we find that the Mn atoms and the B, Mg atoms have strong hybridization, as shown in Figures 3a-3c. Figure 3a reveals a strong hybridization between the B and Mn atoms, especially the p-d hybridization between the p electrons of the B atom and the d electrons of the Mn atoms. Figure 3b reveals the strong hybridization between the Mn and Mg atoms, especially the s-d hybridization between the s electrons of the Mg atoms and the d electrons of the Mn atoms. In comparison with the B@Mn8@Mg10 cluster with the axial antiparallel magnetic moment of the Mn atoms, in the B@Mn8@Mg10 cluster with the axial paramagnetic moment of the Mn atoms, the average Mn-Mn bond length is larger, which is in agreement with previous theoretical investigations20, while the average Mg-Mn and B-Mn bond lengths are smaller. Therefore, we find that the Mn-Mn interaction is relatively weaker, while the Mg-Mn and B-Mn interactions are stronger in the B@Mn8@Mg10 cluster with the axial paramagnetic moment of the Mn atoms. Upon examining the charge transition, we find large charge transitions in the B@Mn8@Mg10 cluster with the axial paramagnetic moment of the Mn atoms, which leads to the strong hybridization and high stability of the cluster.

2. Magnetic properties 20 ACS Paragon Plus Environment

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Since both B and Mg atoms have weak local magnetic moments, we set their initial magnetic moments to zero, and we set the initial magnetic moment of each Mn atom as 5μB. We calculate the B@Mn8@Mg10 cluster with the axial paramagnetic and antiparallel magnetic moments of the Mn atoms, respectively. After DFT calculations, the magnetic moment orientation of each Mn atoms is found to be the same as the initial orientation. Whether for the B atom or for each Mg atom, the orientation is antiparallel with the magnetic moment orientation of the neighboring Mn atom. The B@Mn8@Mg10 cluster with the axial paramagnetic moment of the Mn atoms has a large total magnetic moment (23μB), which mainly comes from the Mn atoms. Mn atoms have a large local magnetic moment (2.689 ~ 3.028μB). The magnetic moment of each Mg atom is near zero, while the magnetic moment of the B atom is also very small, as shown in Table 4. We also calculate the B@Mn8@Mg10 cluster with the axial antiparallel magnetic moment of the Mn atoms, and we find that the magnetic moment orientation of each Mn atom is the same as the initial orientation. Whether for the B atom or for each Mg atom, the magnetic moment is near zero, and the magnetic moment orientation is antiparallel with the neighboring Mn atom. Although each Mn atom has a large local atomic magnetic moment, its total magnetic moment is 1μB. Previous theoretical investigations have pointed out that Mn atoms of the most stable Mnn clusters have antiparallel magnetic moments when n ≥ 7. Although Mn atoms have large local magnetic moments, the total magnetic moment is small16,19, e.g., less than 1.04μB/atom for Mnn (n = 5-8) clusters. Previous investigations of the 21 ACS Paragon Plus Environment


Page 22 of 35

core-shell cluster around the Mn atoms have revealed that the Mn-Mn bond length is closely related to the magnetic order of the Mn atoms, and that increments of Mn-Mn bond length achieve a large total magnetic moment5,6. However, this B@Mn8@Mg10 cluster achieves a paramagnetic moment of the Mn atoms with a smaller Mn-Mn bond length, as compared with previous investigations. From PDOS calculations (Figure 4), the magnetic moment of the Mn atoms mainly comes from the d electrons, and the magnetic moment of the B atoms mainly comes from the p electrons, while the magnetic moment of the Mg atoms mainly comes from the s electrons.

3. Mg2BN-1(Mn4Mg4)N nanochains assembled by the B@Mn8@Mg10 cluster Since the B@Mn8@Mg10 cluster has an evident symmetrical axis, we use the B@Mn8@Mg10

cluster

as

a

structural

element,

and

we

further

design

Mg2BN-1(Mn4Mg4)N nanochains, which have the axial paramagnetic moment of the Mn atoms. We perform DFT calculations in a box of 20×20×30 Å without changing the other parameters. After DFT calculations, we find that the nanochains retain geometrical symmetry, as shown in Figure 5. The most stable state occurs when the interface layer (Mn) atoms have an axial paramagnetic moment, while the orientation of the magnetic moment of the inner layer (B) atoms and external layer (Mg) atoms is the reverse of the orientation of the interface layer (Mn) atoms. Moreover, the Mn atoms have large local magnetic moments, while the magnetic moments of the B atoms and Mg atoms approach zero. Taking the 22 ACS Paragon Plus Environment

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example of the Mg2B5(Mn4Mg4)6 nanochain, we find the atomic magnetic moment of each Mn atom, as shown in Table 5. Interestingly, each Mn atom in the two terminals has a larger local magnetic moment than that of the internal Mn atoms. Moreover, the total magnetic moments of the nanochains increase linearly with the structural element N, as shown in Figure 5. These nanochains have potential value in nanoscale magnetic device applications.

Conclusion We present a sequential micro-motion press-expand method, and on this basis, we develop efficient core-shell cluster algorithm with given core-cluster structure from arbitrary initial shell-cluster structure and further develop global minima algorithm from arbitrary initial structure. The computational program can be design artificially and flexibly, because new algorithms only have two kinds of operators based on sequential micro-motion press-expand method. Many tests of stable clusters reveal that the structure of stable core-shell clusters is an optimal atomic distribution of the outer atoms according to the potential fields of the core-cluster. On this basis, we obtain a new core-shell A@B8@C10 cluster. We further investigate the structure and magnetic properties of the B@Mn8@Mg10 cluster by DFT calculations. We find that this cluster has high stability because of the strong p-d hybridization between the B and Mn atoms and the strong s-d hybridization between the Mn and Mg atoms. The most stable state occurs when the interface layer (Mn) atoms have an axial paramagnetic moment. Whether for B atoms or for each Mg atom, 23 ACS Paragon Plus Environment


Page 24 of 35

the magnetic moment orientation is antiparallel with the neighboring Mn atom. The total magnetic moment of the B@Mn8@Mg10 cluster is 23μB, and the local atomic magnetic moments of the Mn atoms are 2.689 ~ 3.028μB. Using

the

B@Mn8@Mg10

cluster

as

structural

element,

we

design

Mg2BN-1(Mn4Mg4)N (N = 2-8) nanochains when Mn atoms have an axial paramagnetic moment, which retains geometrical symmetry after DFT calculations. Mn atoms also have large local magnetic moments. The most stable state occurs when the magnetic moment orientations of the inner layer (B) atoms and the external layer (Mg) atoms are in reverse orientation with the interface layer (Mn) atoms. Moreover, the total magnetic moments of the nanochains increase linearly with the structural element N.

Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos. 11464038, 51472113). We are also grateful to the High Performance Computing Center of Nanjing University for doing the numerical calculations in this work.

Conflict of interest: The authors declare no competing financial interest.

Reference (1) W. Y. Li, F. Y. Chen, Structural, electronic and optical properties of 7-atom Ag-Cu 24 ACS Paragon Plus Environment

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nanoclusters from density functional theory. European Physical Journal D. 2014, 68, 91. (2) D. A. Kilimis, D. G. Papageorgiou, Structural and electronic properties of small bimetallic Ag-Cu clusters. European Physical Journal D. 2010, 56, 189. (3) F. Baletto, C. Mottet, R. Ferrando, Growth simulations of silver shells on copper and palladium nanoclusters. Physical Review B. 2002, 66, 155420. (4) R. Ferrando, J. Jellinek, R. L. Johnston, Nanoalloys: From Theory to Applications of Alloy Clusters and Nanoparticles. Chemical Reviews. 2008,108, 845. (5) X. M. Huang, J. J. Zhao, Y. Su, Z. F. Chen, R. B. King, Design of Three-shell Icosahedral Matryoshka Clusters A@B12@A20 (A=Sn, Pb; B=Mg, Zn, Cd, Mn). Scientific Reports. 2014, 4, 6915. (6) J. L. Wang, J. Bai, J. Jellinek, X.C. Zeng, Gold-Coated Transition-Metal Anion [Mn13@Au20]- with Ultrahigh Magnetic Moment. Journal of the American Chemical Society. 2007, 129, 4111. (7) Q. Jing, H. B. Cao, G. X. Ge, Y. X. Wang, H. X. Yan, Z. Y. Zhang, Y. H. Liu, Giant magnetic moment of the core-shell Co13@Mn20 clusters: First-principles calculations. Journal of Computational Chemistry. 2011, 32, 2474. (8) G. X. Ge, Y. Han, J. G. Wan, J. J. Zhao, G. H. Wang, First-principles prediction of magnetic superatoms in 4d-transition-metal-doped magnesium clusters. Journal of Chemical Physics. 2013, 139, 174309. (9) B. Fan, G. X. Ge, C. H. Jiang, G. H. Wang, J. G. Wan, Structure and magnetic properties of icosahedral PdxAg13-x (x = 0-13) clusters. Scientific Reports. 2017, 25 ACS Paragon Plus Environment


7, 9539. (10) G. Kresse, J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Physical Review B. 1996, 54, 11169. (11) G. Kresse, D. Joubert, From ultrasoft pseudopotentials to the projector augmented-wave method. Physical Review B. 1999, 59, 1758. (12) P. E. Blöchl, Projector augmented-wave method. Physical Review B. 1994, 50, 17953. (13) J. P. Perdew, K. Burke, M. Ernzerhof, Generalized Gradient Approximation Made Simple. Physical Review Letters. 1996, 77, 3865. (14) I. A. Howard, A. K. Ray, A correlation study of boron dimers and trimers. Zeitschrift Für Physik D. 1997, 42, 299. (15) Y. W. Mu, Y. Han, J. L. Wang, J. G. Wan, G. H. Wang, Structures and magnetic properties of Pdn clusters (n = 3-19) doped by Mn atoms. Physical Review A. 2011, 84, 053201. (16) P. Bobadova-Parvanova, K. A. Jackson, S. Srinivas, and M. Horo, Emergence of antiferromagnetic ordering in Mn clusters. Physical Review A. 2003, 67, 061202(R). (17) C. J. Feng, B. Z. Mi, Configurations and magnetic properties of Mn–B binary clusters. Journal of Magnetism and Magnetic Materials. 2016, 405, 117 (18) J. Jellinek, P. H. Acioli, Magnesium Clusters: Structural and Electronic Properties and the Size-Induced Nonmetal-to-Metal Transition†. Journal of Physical Chemistry A. 2002, 107, 10919. 26 ACS Paragon Plus Environment

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(19) M. B. Knichkelbein, Magnetic ordering in manganese clusters. Physical Reviews B. 2004, 70, 014424. (20) A. K. Singh, T. M. Briere, V. Kumar, Y. Kawazoe, Magnetism in Transition-Metal-Doped Silicon Nanotubes. Physical Review Letters. 2003, 91, 146802.



Page 28 of 35

Table 1. Parameters of Gupta potential7. Aij (eV)

Bij (eV)

Pij

Qij

Rij (Å)

Pd-Pd

0.1715

1.7019

11.0

3.794

2.75

Pd-Ag

0.1607

1.5597

10.895

3.492

2.82

Table 2. Comparison of our calculated values of bond length (Å), magnetic moment (μB), and binding energy (eV/atom). average bond length

magnetic moment

Ours

Ref

Ours

Ref

Ours

Ref

B2

1.617

1.61a

2.0

-

1.869

1.33a

Mg2

3.499

3.51b

0

-

0.069

0.07b

Mn2

2.580

2.586c

10.0

10d

0.473

0.449c

binding energy

a

From Ref. 14, bFrom Ref. 8, cFrom Ref. 15, dFrom Ref. 16.

Table 3. Average bond length (Å), total magnetic moment M (μB), binding energy Eb (eV/atom) and HOMO-LUMO gap (eV) of B@Mn8@Mg10, B@Mn8@Mg0 and B0@Mn8@Mg10 clusters (ferro and anti represent Mn atoms with axial parallel and antiparallel magnetic moments, respectively). Average bond length B-Mn

Mn-Mn Mn-Mg Mg-Mg

Eb

M

gap

B@Mn8@Mg10 (ferro)

2.130

2.576

2.720

3.404

1.805

23

0.245

B@Mn8@Mg10 (anti)

2.134

2.564

2.758

3.646

1.778

1.0

0.312

B@Mn8@Mg0 (ferro)

2.145

2.593

-

-

2.187

30.9

0.024

B0@Mn8@Mg10 (ferr)

-

2.494

2.687

3.403

1.462

22

0.287

B0@Mn8@Mg10 (anti)

-

2.446

2.694

3.659

1.453

0

0.231


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Table 4. Local atomic magnetic moment M (μB), atomic charge Q (e) of B@Mn8@Mg10 clusters with axial paramagnetic moment of Fe/Mn atoms. N is the serial number of the atom. If Q is positive, it means a decrease in atomic charge; if it is negative, it means an increment in atomic charge. N

M(μB)

Q(e)

N

M(μB)

Q(e)

1-B

-0.249

-0.93

2-Mg

-0.025

0.84

12-Mn

3.025

-0.72

3-Mg

-0.026

0.91

13-Mn

3.027

-0.70

4-Mg

-0.001

0.71

14-Mn

3.026

-0.71

5-Mg

-0.044

0.79

15-Mn

3.028

-0.73

6-Mg

-0.044

0.79

16-Mn

2.690

-1.12

7-Mg

-0.001

0.71

17-Mn

2.885

-0.83

8-Mg

-0.042

0.80

18-Mn

2.887

-0.83

9-Mg

-0.042

0.80

19-Mn

2.689

-1.11

10-Mg

-0.042

0.80

11-Mg

-0.042

0.80

Table 5. Local atomic magnetic moment of Mn atoms (M) of Mg2B5(Mn4Mg4)6 clusters with the axial paramagnetic moment of Mn atoms. N is the number of layers (Mn atoms) from top to bottom along the z axis. N

M(μB)

N

M(μB)

3.142 N=1

N=2

3.142

N

2.269 2.269

N=3

M(μB) 1.531

N=5

1.530

3.142

2.270

1.530

3.142

2.269

1.530

1.531

2.267

3.143

1.530

2.267

N=4

N=6

3.143

1.531

2.267

3.143

1.530

2.268

3.143



Captions: Figure 1. Geometrical structure of A@B8@C10 cluster, where red, blue and origin balls represent center, shell and shell atoms, respectively.

Figure 2. Structure of B@Mn8@Mg10 cluster with the axial paramagnetic moment of the Mn atoms, where red, violet and origin balls represent B, Mn and Mg atoms, respectively. The arrow nearby the atom represents the magnetic moment orientation of its atom. The numerical value of each ball is the serial number of the atom.

Figure 3. LDOS of B@Mn8@Mg10 cluster with the axial paramagnetic moment of the Mn atoms. The black vertical dotted line refers to the Fermi level.

Figure 4. PDOS of B@Mn8@Mg10 cluster with the axial paramagnetic moment of the Mn atoms. The black dashed line refers to the Fermi level.

Figure 5. The total magnetic moments (μB) of Mg2BN-1(Mn4Mg4)N nanochains versus structure element N, and the structure of the Mg2B5(Mn4Mg4)6 nanochain with the axial paramagnetic moment of the Mn atoms. The red, violet and origin balls represent B, Mn and Mg atoms, respectively.


Page 30 of 35

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Fig. 1. Geometrical structure of A@B8@C10 cluster, where red, blue and origin balls represent center, shell and shell atoms, respectively. 99x87mm (200 x 200 DPI)

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Fig. 2. Structure of B@Mn8@Mg10 cluster with axial paramagnetic moment of Mn atoms, where red, violet and origin balls represent B, Mn and Mg atoms, respectively. The arrow nearby the atom represents magnetic moment orientation of its atom. The numerical value on each ball is the serial number of the atom. 200x162mm (72 x 72 DPI)


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Fig. 3. LDOS of B@Mn8@Mg10 cluster with axial paramagnetic moment of Mn atoms. Black vertical dotted line refers to the Fermi level. 99x133mm (600 x 600 DPI)



Fig. 4. PDOS of B@Mn8@Mg10 cluster with axial paramagnetic moment of Mn atoms. Black dashed line refers to the Fermi level. 99x80mm (600 x 600 DPI)


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Fig. 5. The total magnetic moments (μB) of Mg2BN-1(Mn4Mg4)N nanochains versus structure elements N, and structure of Mg2B5(Mn4Mg4)6 nanochain with axial paramagnetic moment of Mn atoms. Red, violet and origin balls represent B, Mn and Mg atoms, respectively. 69x50mm (600 x 600 DPI)


Structures and Properties of Core-shell B@Mn - American

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