Introduction

Engineering techniques that use finite size effect to introduce tunable edge magnetism and energy gap are by far the most promising ways for enabling graphene1 to be used in electronics and spintronics.2,3 Examples of finize-sized graphene nanostructures include one-dimensional (1D) graphene nanoribbons (GNRs)4,5,6,7,8,9,10,11,12,13,14,15,16 and zero-dimensional (0D) graphene nanoflakes (GNFs) (also known as graphene quantum dots).17,18,19,20,21,22,23,24,25,26,27,28 It is well known that electronic and magnetic properties29 of GNRs and GNFs depend strongly on the atomic configuration of their edges, which are of either the armchair (AC) or zigzag (ZZ) types.8

Edge magnetism has been predicted theoretically10,11 and observed experimentally15,16 in ZZGNRs. The magnetism in ZZGNRs results from ferromagnetic (FM) coupling for each zigzag edge and antiferromagnetic (AFM) coupling between two parallel zigzag edges of ZZGNRs. The strong FM coupling along each zigzag edge has been predicted in theory11 and confirmed in experiments.16 However, the AFM coupling between two parallel zigzag edges in ZZGNRs is weak, which cannot be stabilized even at low temperature below 10 K14 and rapidly weakens (~w−2) as the ribbon-width w increases.13 Furthermore, the energy gap of GNRs depend on several factors, such as the edge type (armchair or zigzag) and the width of the nanoribbon,8 thus cannot be easily tuned. Such problem does not exist in GNFs due to the quantum confinement effect.30 The ability to control the energy gap has enabled GNFs to be used in promising applications in electronics.20 In addition, triangular ZZGNFs are theoretically predicted to have strong edge magnetism even in small systems.31,32 However, triangular ZZGNFs have large formation energy24 and have not been synthesized experimentally. Fortunately, hexagonal ZZGNFs exhibits significantly improved stability in ambient environment.24 Recent experiments33 have also demonstrated that edge magnetism can be observed in ZZGNFs when the edges are passivated by certain chemical groups. However, semi-empirical tight-binding model34,35 and first principle density functional theory (DFT) calculations25,26,27,28 for hexagonal ZZGNFs have been performed for small-sized systems but found no magnetism (NM). Thus the prospect of finding stable finite-sized graphene easily fabricated in experiments with both strong edge magnetism and tunable energy gap seems dim.

In this letter, we systematically investigate the electronic and magnetic properties of hexagonal ZZGNFs with the diameters in the range of 1–12 nm (from C24H12 to C3750H150). Using first-principles DFT calculations, we find that both strong edge magnetism and tunable energy gap can be realized simultaneously in large ZZGNFs stabilized at room temperature. We demonstrate that spin polarization plays a crucial role as the diameter of a ZZGNF increases beyond 4.5 nm (C486H54). A spin-unpolarized calculation shows that edge states become increasingly more localized as the size of a ZZGNF increases. These edge states form a half-filled pseudo-band and is thus unstable. Adding spin-polarization allows the edge states to spontaneously split into spin-polarized occupied and unoccupied states. This separation results in a magnetic configuration transition from an NM configuration to a strong inter-edge AFM configuration. It also opens a tunable band gap that can be easily controlled by quantum confinement effect. These properties make GNFs better candidate materials for nanoelectronics than GNRs.8 We also confirm that ZZGNFs passivated by different chemical groups all exhibit similar behavior. Such flexibility may facilitate future experimental synthesis of such ZZGNFs.

Results and discussion

We demonstrate the importance of spin polarization using C864H72 (6 nm) as an example (Fig. 2). From a spin-unpolarized calculation, we observe strong and localized edge states (Fig. 2e). These edge states contribute to high electron density along the edges of C864H72. Furthermore, these edge states become much stronger and more localized as the ZZGNF size increases.28 The presence of strong edge states makes the ZZGNF metallic at the nanoscale.35 The projected density of states (PDOS) of carbon edges of C864H72 plotted in Fig. 2c clearly show a considerably high density of states (DOS) near the Fermi level. This figure confirms that C864H72 is predicted to be metallic in a spin-unpolarized calculation. The metallic nature of the ZZGNF can be attributed to the presence of strong localized edge states.28

However, a spin-polarized calculation shows that half-filled metallic edge states are not stable, and can spontaneously split into two types of occupied and unoccupied states as shown in Fig. 2b, d. As a result, a magnetic configuration transition from a non-magnetic (NM) configuration to a magnetic configuration that exhibits intra-edge FM and inter-edge AFM characters can be observed in Fig. 2f. This transition can be interpreted as the consequence of Mott-type competition between the kinetic (hopping) energy and the intra-edge (on-site) electron–electron interaction energy as the system size increases. Lowering kinetic energy by increasing the system size tends to produce delocalized spin states across all edges, while reducing the electron–electron interaction energy as the system size tends to penalize simultaneous occupation of the same edge by spin up and spin down electrons. Both semi-local GGA-PBE and hybrid HSE06 calculations (the details are given in the Supplemental Material) indicate that for small systems, kinetic energy plays a more dominant role. This observation agrees with previous theoretical prediction of the NM configuration for hexagonal ZZGNFs.25,26,27,28,34,35 Only as the system size increases, the effective electron–electron interaction energy associated with the edge states starts to dominate and is ultimately responsible for this magnetic configuration transition.

Figure 1a shows the variation of relative energy of NM, AFM, and FM magnetic configurations in ZZGNFs and ZZGNRs, respectively, with respect to system size. Our calculations show that AFM states are much more stable than NM and FM states in large ZZGNFs, and a magnetic configuration transition occurs as the diameter of the ZZGNF becomes larger than 4.5 nm (C486H54).31 We believe the FM coupling along each zigzag edge that belong to the same sublattice, are likely to be induced by intra-edge direct exchange interactions. The AFM coupling between two nearest-neighbor edges belonging to different sublattices are likely to be induced by inter-edge superexchange interactions facilitated by a carbon–carbon double bond (C=C) at the corner where two nearest-neighbor edges meet in ZZGNFs at the nanoscale. The local magnetic moment defined by Mi = \(\mid < \hat n_{i \uparrow } >\) − \(< \hat n_{i \downarrow } > \mid\), where \(< \hat n_{i\sigma } >\) is spin electron density with σ = ↑ (spin-up) or ↓ (spin-down)), at the carbon atom i. Figure 2b shows the local magnetic moment of carbon in the the middle of each zigzag edge in ZZGNFs (with the largest magnetic moment) increases with the system size, and converges to 0.3μB when the diameter is larger than than 6 nm (C864H72). Furthermore, there is no charge transfer (\(< \hat n_{i \uparrow } >\) + \(< \hat n_{i \downarrow } >\) ≈ 4) between carbon atoms sitting on different edges that belong to the same or different sublattices in ZZGNFs as the system size increases.

Fig. 1
figure 1

a Relative energy per edge atom (ΔE(AFM-NM) and ΔE(AFM-FM)) of NM, AFM, and FM coupling between different edges in ZZGNFs and ZZGNRs and (b) spin electron density \(< \hat n_{i\sigma } >\) (σ = ↑ (spin-up) or ↓ (spin-down)) at the carbon atom i in the middle of each zigzag edge in AFM ZZGNFs under the variation of the diameter size (ZZGNFs) or ribbon-width length size (ZZGNRs). The red and blue regions represent the stable NM (ΔE(AFM–NM) ≈ 0) and AFM (ΔE(AFM–NM) < 0) coupling between different edges in ZZGNFs, respectively. The critical temperature is estimated by the mean-field theory T = ΔE/kB, where kB is the Boltzmann constant

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Fig. 2
figure 2

Electronic structure of edge states in C864H72 in two different magnetic configurations (NM and AFM), including the schematic illustration of orbital diagram of superexchange interaction of edge states in the (a) NM and (b) AFM configurations, projected density of states (PDOS) of edges in the (c) NM and (d) AFM configurations, (e) local density of states (LDOS) of Fermi level (pink isosurfaces) in the NM configuration and (f) spin density isosurfaces in the AFM configuration. The red and blue isosuraces in (f) represent the spin-up and spin-down states, respectively. The red and blue lines in (d) represent the PDOS contributed by sublattice A (spin-up edges) and B (spin-down edges) atoms in graphene, respectively. The fermi level is marked by green dotted lines and set to zero

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Notice that the intra-edge direct exchange interaction via FM coupling along each zigzag edge in ZZGNFs is similar to that in ZZGNRs. However, the inter-edge superexchange interaction via AFM coupling between two nearest-neighbor edges through a C=C bond (Fig. 2a) in ZZGNFs can be stabilized at room temperature (298 K) and is much stronger than that via AFM coupling between two parallel edges though π-bonds in ZZGNRs as shown in Fig. 1a, where such AFM spin polarization weakens rapidly as the ribbon-width increases in ZZGNRs13 and cannot be stabilized even at ultra-low temperature (3 K).14 Our DFT calculations confirm that the energy difference associated with AFM and FM coupling between two parallel edges in large-scale 1D ZZGNRs is negligible compared to that reported in ZZGNFs.

The enhanced stability of spin-polarized ZZGNFs can be understood by using the Heisenberg model. We consider each FM edge as one site and enumerate all possible magnetic configurations, and the Hamiltonian can be written as

$$\hat H = - {\sum} {J_{i,j}} \overrightarrow {M_i} \overrightarrow {M_j}$$
(1)

where Ji,j is the exchange parameter between two sites i and j, \(\overrightarrow {M_i}\) and \(\overrightarrow {M_j}\) are the corresponding spin magnetic moments. There are four different magnetic states in C864H72, there of which are AFM, AFM1, and AFM2 configurations and one is FM configuration as shown in Fig. 3. The total energies of magnetic configurations E(AFM), E(AFM1), E(AFM2), and E(FM) can be computed by DFT calculations, and the exchange parameters can be evaluated by solving the following least-squares-fitting problem36

$$\begin{array}{c}E({\mathrm{AFM}}) = (6J_1 - 6J_2 + 3J_3)M^2 + E_0\\ E({\mathrm{AFM1}}) = (2J_1 + 2J_2 - J_3)M^2 + E_0\\ E({\mathrm{AFM2}}) = ( - J_1 + 2J_2 + 3J_3)M^2 + E_0\\ E({\mathrm{FM}}) = ( - 6J_1 - 6J_2 - 3J_3)M^2 + E_0\end{array}$$
(2)

where J1, J2, and J3 are ortho-edge, meta-edge, and para-edge exchange interaction parameters, respectively, M is the spin magnetic moment at each edge, and E0 is the nonmagnetic reference total energy. The solution yields J1 = −0.038351 eV, J2 = 0.000954 eV, and J3 = 0.001633 eV for two nearest-neighbor edges of C864H72, which are 10 times stronger than the exchange interaction parameters between two parallel edges in ZZGNFs and ZZGNRs. Therefore, ZZGNFs at the nanoscale have strong edge magnetism at room temperature and can be directly used in nanospintronics, superior to that in ZZGNRs at the nanoscale.7,13

Fig. 3
figure 3

Spin density isosurfaces of hydrogen-passivated C864H72 in four different magnetic states, three types of antiferromagnetic ((a) AFM, (b) AFM1, and (c) AFM2) and one type of ferromagnetic ((d) FM) coupling at the inter edges. The red and blue isosurfaces represent the spin-up and spin-down states, respectively

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We perform ab initio molecular dynamics (AIMD) simulations on ZZGNFs and check the effect of temperature on electronic and magnetic properties of C486H54 in different AFM and FM configurations (the details are given in the Supplemental Material). We find that the AFM configuration of C486H54 can remain stable at room temperature of T = 300 K at least within 1.6 ps. Furthermore, the FM configuration of C486H54 rapidly transfers into the AFM configuration within 30.0 fs at room temperature of T = 300 K. Furthermore, after t = 1.5 ps, C486H54 is slightly bent,26 although it still keeps the AFM configuration.

We also check the effects of using different types of atoms (e.g., bare and fluorine) to passivate ZZGNFs, and how the shape (non-hexagonal) of ZZGNFs may alter their electronic and magnetic properties. We find that magnetic configuration transition (from NM to AFM) and semiconductor characteristics (The energy gaps of 0.54, 0.34, and 0.41 eV, respectively, for bare C864, fluorine-passivated C864F72 and non-hexagonal C839H71) of ZZGNFs are independent of the type of passivating atoms37 as plotted in Fig. 4. These properties suggest that it is relatively easy to create a chemical environment in which the synthesis of large scales ZZGNFs with tunable edge magnetism and energy gaps can be easily accommodated. The possibility of rapid synthesis makes ZZGNFs ideal candidates for electronic and spintronic devices.38

Fig. 4
figure 4

Spin density isosurfaces and total density of states (TDOS) of (a) bare (C864), (b) fluorine-passivated (C864F72), and (c) non-hexagonal (C839H71) ZZGNFs in the AFM configuration. The red and blue isosurfaces represent the spin-up and spin-down states, respectively. The energy differences (EAFM−ENM) between AFM and NM configurations of these ZZGNFs are shown above the figures. For hexagonal hydrogen-passivated C864H72, ΔE(AFM−NM) = −17.9 meV. The fermi level is marked by green dotted lines and set to zero

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We remark that magnetic configuration transition and the associated tunable electronic structures in ZZGNFs, especially energy gaps, can also be understood in terms of the Hubbard model.31 From our first principle calculations, we find that choosing the parameters t = 2.5 eV and U = 2.1 eV in the Hubbard model can well reproduce the size-dependent energy gaps (the details are given in the Supplemental Material). In Fig. 5, we plot how the HOMO-LUMO energy gap Eg changes with respect to the size of ZZGNFs and ACGNFs in two different magnetic configurations (NM and AFM). Our DFT calculations and mean-field Hubbard model show similar results, i.e., the energy gap Eg of ZZGNF decreases as its size increases. In particular, we find that the energy gap of NM ZZGNFs decreases more rapidly with respect to the system size than that of AFM ZZGNFs, due to the presence of edge states whose electron density near the edges of ZZGNFs as shown in Fig. 2e. This observation is consistent with previous results obtained from tight-binding models34,35 and DFT calculations.27,28 However, AFM semiconducting ZZGNFs show similar energy gap scaling compared to that of NM ACGNFs at the nanoscale.28 Therefore, edge states should have little effect on the energy gaps of AFM ZZGNFs and the quantum confinement effect30 is the only factor to control the energy gaps in ZZGNFs and ACGNFs (Fig. 5a). In detail, NM ZZGNFs exhibits metallic characters (Eg is smaller than the thermal fluctuation (25 meV) at room temperature) when the diameter is larger than 7 nm (C1350H90), but AFM ZZGNFs with the diameter of 12 nm (C3750H150) still behaves as a semiconductor with a sizable energy gap Eg = 0.23 eV, similar to the case of NM ACGNFs.28 Therefore, ZZGNFs at the nanoscale can be directly used in nanoelectronics.

Fig. 5
figure 5

Energy gap Eg (eV) of ZZGNFs and ACGNFs in two different magnetic configurations (NM and AFM) as a function of the number of carbon atoms, computed with two different methods: (a) DFT calculations and (b) Hubbard model (t = 2.5 eV and U = 2.1 eV)

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In summary, using large-scale first principle calculations, we demonstrate that the electronic and magnetic properties of hexagonal zigzag that graphene nanoflakes (ZZGNFs) can be significantly affected by the system size. We found that the zigzag edge states in ZZGNFs become much stronger and more localized as the system size increases. The presence of these edge states induce strong electron–electron interactions along the edges of ZZGNFs, resulting in a magnetic configuration transition from nonmagnetic to intra-edge FM and inter-edge AFM when the diameter is larger than 4.5 nm. On the other hand, such strong and localized edge states are also responsible for making ZZGNFs semiconducting with a tunable energy gap. The energy gap can be controlled by merely adjusting the system size. Therefore, ZZGNFs with strong edge magnetism, tunable energy gaps and room-temperature stability may be promising candidates for practical electronic and spintronic applications.

Methods

We use the Kohn–Sham DFT-based electronic structure analysis tools implemented in the Spanish Initiative for Electronic Simulations with Thousands of Atoms (SIESTA)39 software package. We use the generalized gradient approximation of Perdew, Burke, and Ernzerhof (GGA–PBE)40 exchange correlation functional with collinear spin polarization, and the double zeta plus polarization orbital basis set (DZP) to describe the valence electrons within the framework of a linear combination of numerical atomic orbitals (LCAO).41 Because semi-local GGA–PBE calculations are less reliable in predicting the electronic structures of ZZGNFs, the screened hybrid HSE0642 calculations implemented in HONPAS43,44,45 (Hefei Order-N Packages for Ab Initio Simulations based on SIESTA) are also used to compute the electronic and magnetic properties of ZZGNFs. All atomic coordinates are fully relaxed using the conjugate gradient (CG) algorithm until the energy and force convergence criteria of 10−4 eV and 0.02 eV/Å, respectively, are reached.

For initial magnetic moment setting of spin-polarized DFT calculations in ZZGNFs, we set all the carbon atoms with initial magnetic moments of 1μB for the FM configuration and only set the edged carbon atoms with initial magnetic moments of 1 or −1μB, and then optimize the structures and magnetic moments of ZZGNFs.

Due to the large number of atoms contained in hexagonal hydrogen-passivated ZZGNFs (\({\mathrm {C}}_{6n^2}\)H6n, n = 1–25), we use the recently developed Pole EXpansion and Selected Inversion (PEXSI) method46,47,48 to accelerate the eigenvalue problem in the Kohn–Sham DFT calculations. The PEXSI technique can efficiently utilize the sparsity of the Hamiltonian and overlap matrices generated in SIESTA and overcome the cubic scaling limit for solving Kohn–Sham DFT, and scales at most as quadratic scaling even for metallic systems, such as graphene. Furthermore, the PEXSI method is highly scalable49 and can scale up to 100,000 processors on high performance machines.

We perform AIMD simulations on ZZGNFs to check the effect of temperature on electronic and magnetic properties of ZZGNFs. The simulations are performed for about 1.6 ps with a time step of 2.0 fs at room temperature of T = 300 K controlled by a Nose–Hoover thermostat.50,51