Magnetism of two-dimensional triangular nanoflake-based kagome lattices

We report a new method for the design of kagome lattices using zigzag-edged triangular graphene nanoflakes (TGFs) linked with B, C, N or O atoms. Using spin-polarized density functional theory we show that the electronic and magnetic properties of the designed kagome lattices can be modulated by changing their size and the linking atoms. The antiferromagnetic coupling between the two directly linked TGFs becomes ferromagnetic coupling when B, C or N is used as the linking atoms, but not for O atom linking. All the designed structures are semiconductors which can be synthesized from graphene atomic sheets by using electron etching and block copolymer lithography techniques. This study is a good example of how mathematical models can be used to construct magnetic nanostructures involving only s, p elements.


Introduction
Spintronics that exploits the electron's spin degree of freedom has been a robust and challenging field of research over the last decade owing to its technological importance for new electronic devices. An ideal spintronics material should possess high spin polarization, long spin relaxation time, weak spin-orbit coupling and ferromagnetic (FM) ordering at room temperature. Past studies on spintronics materials have mainly focused on transition metal impurity-doped GaN and ZnO where the magnetism is due to d-electrons [1,2]. However, these materials have the disadvantages of clustering of impurity atoms [1], strong spin-orbit interaction and small spin relaxation time. Consequently, attention has now shifted to magnetic semiconductors where the magnetism is associated with s-, p-electrons. These materials have the advantage of low spin-orbit coupling, weak hyperfine interaction and no clustering. Among s, p elements, graphene-based structures are a natural choice. These include graphene nanoribbons (GNRs) and triangular graphene nanoflakes (TGFs), which can be fabricated by cutting graphene sheets using electron beam irradiation and etching techniques [3,4], or the atomic force microscopebased nanorobot [5]. In GNRs, although the zigzag edge states are FM, the opposite edges are coupled antiferromagnetically due to the anti-pattern rule, resulting in zero net magnetic moment [6][7][8]. However, in the TGFs the magnetic coupling between the zigzag edges is FM [9][10][11][12][13][14][15], and the non-equivalent sublattices A and B lead to a size-dependent magnetic moment of (N − 1) µ B , where N is the number of hexagons along one edge of the TGF. Compared to GNRs, the zigzag-edged TGFs therefore have some advantages in the synthesis of magnetic materials. In addition, unlike conventional magnetic materials where magnetism is due to d-or f-electrons, magnetism in the TGFs originates from the p-electrons, displaying weak spin-orbit coupling and low hyperfine interaction of the electron spins with the carbon nuclei, which are the main channels of relaxation and decoherence of electron spins [16]. Recent studies have focused on the assembling of TGFs by finding suitable linkers so as to promote FM ordering [17]. In this paper, we show that new graphene-based magnetic materials can be created by using an old mathematical model known as the kagome lattice.
A kagome lattice, composed of interlaced triangles in a two-dimensional (2D) pattern, is a well-known mathematical model widely used in the study of frustrated magnetism [18]. Although called a lattice, it is more closely related to tri-hexagonal tiling. A large number of research works in past years have been devoted to using this model to design and synthesize materials. For example, Ramirez et al [19] studied SrCr 8 Ga 4 O 19 where Cr ions form a kagome lattice with a high Curie-Weiss temperature of 515 K. Schweika et al [20] synthesized layered kagome materials consisting of Y 0.5 Ca 0.5 BaCo 4 O 7 , which exhibits a high Curie-Weiss temperature of 2200 K. Aidoudi et al [21] fabricated an inorganic-organic hybrid kagome 3 structure containing V ions, which displays antiferromagnetic (AFM) behavior. Currently, a new direction in synthesizing kagome lattices on the surface is by means of metal-organic coordination or supramolecular self-assembly of specific molecules. For instance, Mao et al [22] deposited Fe-phthalocyanine (FePc) molecules on highly ordered graphene monolayer (MG) supported on a Ru (0001) substrate that mimics a well-assembled kagome lattice. Wang et al [23] constructed a kagome lattice from Co ions and azide ligands showing antiferromagnetism. Colloidal kagome lattices using tri-block Janus spheres containing sulfate polystyrene with hydrophobic poles synthesized by Chen et al [24] showed no magnetism. Until recently, all existing methods for the synthesis of kagome lattices have been very complicated. Only transition metal ions have been used to induce magnetism, and the substrate surface has to be carefully chosen for assembling 2D kagome lattices. Thus, the question arises: can we find a simple route to achieve a 2D magnetic kagome lattice without using any metal ions? In this paper, a novel approach is provided for constructing a 2D periodic magnetic kagome lattice by using magnetic TGFs as the building blocks because of their excellent geometric shape and intrinsic magnetism. Using spin-polarized density functional theory (DFT) combined with Monte Carlo simulations, we have systematically studied the geometry, electronic structure and magnetic properties of TGF-based 2D kagome lattices.

Theoretical procedure
A unit cell representing a 2D periodic structure has been generated by inserting two zigzagedged TGFs into the nearest triangles of kagome lattices and linking these with four different atoms M (M = B, C, N and O, respectively). A vacuum space of 12 Å in the non-periodic direction is used to avoid interaction between the two neighboring images. Here, the number N of the hexagons along one edge of the TGF N is used to define the size of a TGF N . To study the size dependence, we have changed the size of TGF N s from N = 2 to 4 and constructed different size 2D kagome lattices, labeled as TGF N -M-TGF N . The schematic structures of perfect and deformed kagome lattices with N = 3 are shown in figures 1 and 2, respectively. Calculations were performed by using spin-polarized DFT and the Vienna ab initio Simulation Package (VASP) code [25]. Generalized gradient approximation with the Perdew-Burke-Ernzerhof form was used to treat the exchange and correlation potentials [26]. The Brillouin zones are represented by the Monkhorst-Pack special k-point meshes of n × n × 1 with n = 7, 5 and 3 for the unit cells with size N of 2, 3 and 4, respectively. The energy cutoff was set to 400 eV. All the atoms were fully relaxed using a conjugate-gradient algorithm until the variations of energy and force between two steps were less than 0.01 meV and 0.004 eV Å −1 , respectively. In order to examine the magnetic behavior of these periodic frameworks at finite temperatures, we carried out Monte Carlo simulations for a (10 × 10) supercell system by using the Ising model. The simulations lasted for 1 × 10 5 loops in order to analyze the data, and in each loop all the magnetic moments of the TGFs and the linking atoms were allowed to change.

Results and discussion
We first studied the magnetic coupling between the two zigzag-edged isolated TGF N s (N = 2, 3 and 4). In agreement with previous calculations [17] we found that the two TGF N s are coupled antiferromagnetically when they are directly connected, regardless of the size N of TGF N s. Stimulated by the pioneering work of Rao and Jena [1] and Tono et al [27] on tuning the magnetic coupling in clusters by using 2p electrons of N and O atoms, respectively, we introduced a single atom, M (M = B, C, N and O), between the two TGF N s to mediate their magnetic coupling. We refer to this as the 0D structure. It is interesting to see that the AFM coupling between the two TGF N s turns into FM when they are connected by a single B, C or N atom, while the AFM coupling configuration still exists when an O atom is introduced between them. For instance, the FM configuration is found to be lower in energy by 0.20, 0.18, and 0.15 eV than the AFM one for the systems composed of two TGF 3 s linked by a single B, C or N atom, respectively. However, the AFM configuration is found to be lower in energy by 0.01 eV than the FM one when one O atom is used as the linker. This indicates that the magnetic coupling between the two TGFs can be changed from AFM to FM by introducing B, C or N atoms. The calculated results for the 0D structures are given in table 1. From their geometries we can see the difference: the centers of two TGFs and the kagome site remain in a straight line when they are linked by a B atom, i.e. the angle θ between the two B-C bonds is 180 • . However, when C, N and O atoms are used as the linkers, the corresponding angles change to 146.8 • , 131.2 • and 132.5 • , respectively. From the spin density isosurfaces we can see the similarity: for all the systems the magnetic moment mainly comes from the 2p orbitals of C atoms at the sublattice A sites of the TGFs. The spin density on the C atoms at the sublattice B sites is small and has the opposite spin direction, which results in a net magnetic moment of 5.0, 6.0 and 5.0 µ B for the systems containing B, C and N atoms, respectively. This can be understood from Lieb's Based on the above study of two isolated TGF N s linked by B, C, N and O atoms, we then construct a kagome lattice by forming a 2D periodic structure with these 0D structures as building blocks. The initial schematic structure of the TGF N s-based kagome lattice is plotted in figure 1, where each unit cell of the kagome lattice contains the two TGF N s and three linking M (M = B, C, N and O) atoms. Four different spin coupling configurations are considered to determine the preferred magnetic state. For each structure of TGF N -M-TGF N with N = 2, 3 and 4, the total energies corresponding to the four spin configurations were calculated. The energy difference E between the FM configuration, as shown in figure 1(a), and the AFM configurations, as shown in figures 1(b)-(d), is defined as E = E AFM -E FM . Geometry optimizations were carried out for all the above configurations. The optimized geometries can be divided into two kinds: one is the TGF N s-based perfect kagome lattice, as shown in figure 1. The other is the deformed kagome lattice, as shown in figure 2. We note that the results for the 0D systems are also valid in their corresponding 2D structures. We find that the centers of two TGF   Table 2. The calculated angle θ between the linking atom and its nearest two C atoms, the lattice parameter L, the magnetic moment M per unit cell and the energy differences E between the FM (as shown in figure 1(a)) and the AFM configurations ( E = E AFM -E FM ) (as shown in figures 1(b)-(d)) for the 2D periodic kagome structures. All the constructed 2D periodic TGF 3 -M-TGF 3 with M = B, C and N structures are found to have FM ground states. Among them, the TGF 3 -C-TGF 3 has the largest energy difference E C between the FM and AFM states, indicating that the C atom is more effective in mediating the magnetic coupling between the TGFs, as compared to B, N and O atoms, and the system clearly prefers FM coupling as the AFM states lead to higher total energy, as shown in table 2, We note that the spin direction of the linking atom M (M = B and N) at the kagome site in the coupling configuration, as shown in figure 1(b), flips to the same direction as its nearest neighbors (TGF N s) upon geometry optimization, namely the AFM configuration of figure 1(b) turns into FM coupling, as shown in figure 1(a). We note that the difference between E (c) M and E (d) M for all the constructed FM 2D periodic structures decreases with an increase in size N . This is because the coupling between the TGF N s becomes weaker when their size gets larger and the spin direction of the linking atom plays a relatively weaker role in mediating the coupling. Therefore, the configurations (c) and (d) become energetically nearly degenerate when N = 4. For the assembled 2D TGF N -O-TGF N (N = 2, 3 and 4) structures, the AFM configuration, as shown in figure 1(c), is found to be lower in energy than the FM one (see figure 1(a)). Taking N = 3 as an example, the energy of the AFM ground state is lower than that of the FM state by 10 meV; this energy difference is small and in view of the accuracy of DFT methods, one may argue that the AFM and FM states are nearly degenerate.  The PDOS shows that the split bands are dominated by the 2p orbitals from both the TGF 3 s and kagome sites. However, the contribution from kagome sites of C atoms to the bands near the Fermi level is larger than that from the B and N atoms due to the different numbers of unpaired valence electrons. The TGF 3 -C-TGF 3 has the largest band gap of 0.93 eV, while the TGF 3 -B-TGF 3 and TGF 3 -N-TGF 3 have smaller band gaps of 0.32 and 0.39 eV, respectively. Therefore, the different band structures of the 2D TGF N -M-TGF N for M = B, C and N could be used to identify their geometric structures.
We now carry out Monte Carlo simulations to investigate the magnetic behavior at finite temperatures. In order to calculate the energies for different possible magnetic configurations, we use the Ising model Hamiltonian, H = − i, j J i j s i s j , where i and j represent two nearest neighbor magnetic sites, namely the linking atoms and the TGF N s. In each case, the exchange parameter J can be derived from the energy difference between the FM and AFM states. The J of the 2D periodic structures of TGF 3 -B-TGF 3 , TGF 3 -C-TGF 3 and TGF 3 -N-TGF 3 are calculated to be 34, 23 and 30 meV, respectively. We plot the variation of average magnetic moment per unit cell of the TGF 3 -M-TGF 3 (M = B, C and N) with respect to temperature in figure 5. We see that the magnetic moment remains in the high-spin state in the lowtemperature range for all the structures and then drops to near zero at the Curie temperature. The corresponding Curie temperatures of the 2D TGF 3 -M-TGF 3 structures with M = B, C and N are found to be about 400, 460 and 340 K, respectively. These values indicate that the ferromagnetism that we observed in these TGFs-based kagome lattices can be detected at room temperature. Thus, the designer 2D FM kagome lattices may have advantages over the wellknown dilute magnetic semiconductors in the applications of spintronic materials.

Conclusions
We present a first-principles study of the geometry and electronic and magnetic properties of TGF N -based kagome lattices. We show that a magnetic kagome lattice can be achieved by using zigzag-edged TGFs as building blocks and B, C and N atoms as linkers at the kagome sites. B, C and N atoms play an important role in mediating ferromagnetism. Further, their FM coupling strength can be tuned by changing the size of TGF N s and the linking atoms. The long-range ferromagnetism together with the ideal Curie temperature calculated by the MC simulations, and the homogeneous porosity makes the TGF N -based magnetic kagome structures promising for many potential applications. We hope that our work will stimulate experimental studies.