Tuning the magnetic states in AA-stacked bilayer zigzag graphene nanoribbons

Available reports mentioned that the magnetic ground state in the AA-stacked bilayer zigzag graphene nanoribbons is the non-magnetic state. As a consequence, it is impossible to exploit magnetism for future electronic devices. This paper aims to show how to generate magnetism in the AA-stacked bilayer zigzag graphene nanoribbons by employing ﬁrst-principles calculations. As we stacked different ribbon widths, the magnetic ground states appeared for all the thicknesses. In general, the G-type antiferromagnetic state, which is the antiferromagnetic alignment between both intraplane- and interplane-edge carbon atoms, is the ground state for all the thicknesses. We also found that the degenerate magnetic ground states and excited states may appear under certain thicknesses, thus yielding the richness of the magnetic state. As hole-electron doping was applied, a phase transition of magnetic ground state emerged for certain thicknesses, indicating that a new magnetic ground state in the AA-stacked bilayer zigzag graphene nanoribbons can be tuned by the doping.


Introduction
Investigations on the electronic and magnetic features in the low-dimensional materials have exhibited a number of significant impacts on the condensed matter physics after graphene has been discovered as a two-dimensional hexagonal carbon lattice [1][2][3]. Certainly, for being a highly prominent candidate for future devices in spintronics and nanoelectronics, graphene becomes a concern [4,5]. To exploit the electronic or magnetic properties of this material, recognition of its edge structures in the microscopic scale is deemed necessary. Previously, Fujita et al. showed that graphene nanoribbon (GNR), a one-dimensional structure of graphene, possesses extraordinary properties induced by the carbon edges [6,7]. Recently, several experiments have been realized to grow the GNR by some methods [8][9][10]. Later, theoretical explorations on the electronic and magnetic properties in the GNR are employed in such a way by applying the electric field [12][13][14] or doping [15][16][17] to seek applicable devices.
Based on the fundamental shape, the GNR is classified into two: the zigzag GNR (ZGNR) and armchair GNR (AGNR). It has been well-known that the ground state of ZGNR is an antiferromagnetic (AFM) state [6], while the AGNR has a nonmagnetic state [18]. Thus, the magnetic features in the ZGNR are more exploited than those in the AGNR. Arranging the spin orientations of carbon atoms at the edges in the ZGNR is necessary to determine the ground state. Even though the ground state of ZGNR is well-known as an AFM state, other configurations such as ferromagnetic (FM) state or spiral state can be useful for some electronic devices. As for an AFM ground state, it can be used to discuss the spin-waves excitation to predict the spin stiffness or critical temperature [19][20][21][22]. Those two properties are very useful to design spintronic devices, which will work at room temperature. At the same time, the FM state can induce a metallic state [23], while the spiral state can be utilized to construct a domain wall [24]. Note that the spiral state can be induced by incorporating a metal atom [25,26] or taking the electric field into account [27].
There are two types of stacked bilayer ZGNR, namely the AA-and AB-stacked bilayer ZGNRs. These bilayer structures have been successfully produced by available methods [28][29][30], possess the band gap by applying the electric field [31,32] or constructing the edge alignments [33], and are robust to the external perturbations [34]. For this, they are also prominent in the development of carbon-based spintronic devices, such as for the spin transport [35,36]. Due to the binding energy in graphene, the AB-stacked bilayer structure is more stable than the AA-stacked bilayer one [37,38]. However, from the magnetic perspective, the AA-stacked bilayer ZGNR is more energetically stable than the AB-stacked one. This is because the AA-stacked bilayer ZGNR displays the non-magnetic state, while the AB-stacked one shows the spin-polarized state [39]. For this reason, the consideration of magnetism in the AAstacked bilayer ZGNR is not completely explored.
Similar to the monolayer case, the magnetic ground state of bilayer ZGNR is determined by the alignments of magnetic carbon atoms at the edges. The ground state, therefore, will also be controlled by the interplane and intraplane edge orders, giving the different magnetic stability. In this paper, we investigated the magnetic ground state of AA-stacked bilayer ZGNR for five magnetic configurations as performed by Sawada et al. [40]. Previously, they showed that, of five magnetic configurations in the AB-stacked bilayer ZGNR for all thicknesses, there was only one magnetic ground state, namely, the C-type antiferromagnetic state, which had the interplane FM and intraplane AFM edge configurations. This suggests that the magnetic ground state is robust to the thickness.
As performing the same treatment above, on the contrary, we obtained the magnetic ground states, which were dependent upon certain conditions in the AA-stacked bilayer ZGNR. When we stacked two ZGNRs with different ribbon widths, we observed the degenerate magnetic ground states and excited states, which depended on the thickness. Also, there was only one state which always became a magnetic ground state for all thicknesses, namely, the G-type antiferromagnetic state which had the interplane AFM and intraplane AFM edge configurations. This magnetic ground state also emerged as hole-electron doping was considered, but then it was replaced by other magnetic ground states at certain conditions. This yielded phase transitions in the AA-stacked bilayer ZGNR under the doping. This means that controlling the magnetic state in the AA-stacked bilayer ZGNR can be realized by using the doping.

Materials and Methods
For the investigation, a one-dimensional slab model of AAstacked hydrogen passivated bilayer ZGNR was constructed with the experimental lattice parameter of graphite (2.46 Å) along x-axis (periodic direction) as shown in figure 1(a). We constructed two structure models in which we fixed the 10 dimers at the bottom and varied the number of dimers at the top, as shown in figures 1(b-c). Here, the number of dimers was related to the ribbon width. To encounter the interaction in the non-periodic directions (y-and z-axes), we set 30 Å to generate a vacuum region. We also varied the thicknesses d to see the appearances of magnetic states. We carried out the first-principles calculations by using the OpenMX package [41] within the density functional theory. This package employed the localized basis orbitals [42] and norm-conserving pseudopotentials according to Troullier and Martins [43]. To obtain the converged results with good accuracy, we set the basis orbitals of s2p2 and s2p1 for the carbon and hydrogen atoms, respectively. Here, s and p are the valence and polarization orbitals generated numerically by the confinement method [42]. We also employed the generalized gradient approximation (GGA) in accordance with Perdew, Burke, and Ernzerhof [44] for treating the electron-electron interaction. In these self-consistent calculations, we also set a kpoint sampling of 62 × 1 × 1 and a cutoff energy of 250 Ry. Before running the calculations, we performed the optimized atomic positions on x and y planes at which the forces act on atoms were less than 0.0001 Hartree/Bohr and fixed the thickness d. Since the magnetic order induced by the edges has many possible configurations, we focused on the five collinear magnetic alignments, as shown in figure 2. These configurations are based on the previous work of Sawada et al. [40], who considered the interplane and intraplane edges. For the AFM configurations, the A-AFM was constructed by the interplane AFM and intraplane FM edges, the C-AFM was built by the interplane FM and intraplane AFM edges, while the G-AFM was established by the interplane AFM and intraplane AFM edges. Meanwhile, the ferrimagnetic (fm) state was constructed by a combination of FM/AFM at the interplane/intraplane edges, as shown in figure 2(e).

Results and Discussions
To start the discussion, we considered the experimental thickness d = 3.35 Å for all the ribbon widths N. Figure 3 displays the energy dispersions for the magnetic configurations of bilayer N-ZGNR. We observed that the FM, C-AFM, and fm states exhibited the metallic state while the A-AFM and G-AFM states showed the insulating state. It was also found that all the configurations became the magnetic states where the A-AFM and G-AFM states together became the magnetic ground states, in opposite to the AB-stacked bilayer ZGNR [40]. Furthermore, we found that all the AFM states in the AB-stacked structure always exhibited the insulating state while the AFM-state, in the case of C-AFM state, in the AAstacked structure may exhibit the metallic state, as shown in figure 3.
For discussing other thicknesses d, we fixed the atomic positions on x and y planes and only changed d. As we varied d, we found interesting tendencies, as shown in figure 4. First of all, the G-AFM state became the magnetic ground state for all d and N,   Regarding the magnetic moment, the clear difference between small d and large d can be found in table 1. At the small d, even the localized edge state occurred, the bonding between edge carbon atoms along the interplane-edge direction were sufficiently strong, thus resisting the spin polarization. Consequently, this led to the small magnetic moments at the edges. Contrarily, as we increased d, the bonding between layers became weak; thus in this condition the localized edge states overcame the bonding between edge carbon atoms along the interplane-edge direction, appearing the spin polarization. In this moment the large magnetic moments at the edges occurred then. It can be assumed that the magnetism in the AA-stacked bilayer ZGNR can be strengthened by increasing d, thus reducing the bonding between edge carbon atoms along the interplane-edge direction.
For the general notes, we assumed that the G-AFM ground state was independent with the N and d in the AA-stacked bilayer ZGNR, which differed from the AB-stacked structure [40]. This meant that the AFM coupling (FM coupling) at the interplane edge atoms dominated the magnetism in the AAstacked (AB-stacked) bilayer ZGNR. This result is also consistent with the previous works of Lee et al. by different approaches [45,46]. Note that the G-AFM ground state also emerges in the AB-stacked bilayer ZGNR for the non-collinear structure [47].  Next, we considered the carrier-doped bilayer ZGNR by introducing hole-electron doping. To do so, we applied the Fermi level shift approach as we gave a homogeneous background charge as if the system remained neutral [48,49]. The realization of this doping can be performed experimentally by using the chemical doping [48]. We specified the number of carrier doping per cell as x (e/cell). Here, hole (electron) doping was expressed by the positive (negative) e values. We only considered d = 2.35 Å , 3.35 Å and 4.35 Å for N = 6 and 14. We then computed the total energy difference ∆E with respect to the ground state as a function of x. At d = 2.35 Å as shown in figure 5, the G-AFM state became the ground state for all the doping and ribbon widths, indicating that, in this case, the G-AFM state was robust to the doping and ribbon widths. Meanwhile, the excited states became degenerate under N. For N = 6 except for the A-AFM state, the FM, C-AFM, and fm states became degenerate, as displayed in figure 5(a). Contrarily, all the excited states became degenerate for N = 14, as displayed in figure 5  In the bilayer ZGNR, the physical properties were determined by the magnetic couplings at the intraplane-and interplane-edge atoms. At d = 2.35 Å in figure 5, since the G-AFM state was only the magnetic ground state for all the doping and N, the AFM couplings at both the intraplane and interplane-edge atoms maintain as | | increased. Thus, we expected that both the intraplane and interplane AFM couplings remained unchanged under the carrier doping caused by the lattice defects or impurities in the AA-stacked bilayer N-ZGNR. Moreover, although the FM, A-AFM, C-AFM, and fm states were the different magnetic excited states, they had the same FM coupling at the interplane-edge atoms except for the A-AFM state. This implies that the interplane FM coupling will be never found as the most stable state under the doping.

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At d = 3.35Å , in the case of N = 6 in figure 6 (a) it can be seen that at | | ≤ 0.06 e/cell, the AFM coupling held at the interplane-edge atoms while either the FM coupling (A-AFM state) or the AFM coupling (G-AFM state) at the intraplaneedge atoms was maintained. Except for the fm state, the AFM coupling was then replaced by the FM coupling at the interplane-edge atoms while either the FM coupling (FM state) or the AFM coupling (C-AFM state) at the intraplane-edge atoms wa sstill maintained as | | increased. This indicates that the stability of the interplane FM ordering only works at high x, which is the same result as in the AB-stacked case [50]. As for N = 14 in figure 6 (b), a similar behavior occurred as in the case of N = 6 but for all the doping, i.e., the A-AFM and G-AFM states were the magnetic ground states. This indicates that the interplane AFM ordering is robust under the doping.
At d = 4.35 Å it can also be seen that the magnetic ground state was still dominated by the interplane AFM ordering, but no degenerate magnetic state appeared for all N in both the ground state and excited state. From these results, we expected that the degenerate magnetic states sensitively depended on the thickness. As the thickness increased, the bonding between interplane-edge carbon atoms was weakened; thus the magnetism was maintained. However, a consequence corresponds to the unavailable degenerate magnetic states for both the magnetic ground state and excited states.

Conclusion
We investigated the magnetic ground states of AA-stacked bilayer ZGNR and found the degenerate magnetic ground states, dependent upon the ribbon width and thickness. We observed that the magnetic ground state was degenerate for certain thickness and became non-degenerate as the thickness increased. In general, the G-AFM state also became the magnetic ground state, which is robust to the ribbon width and thickness.
As hole-electron doping was introduced, the magnetic phase transitions generally occurred for all N at the high d, for example at d = 4.35 Å as we plotted the total energy difference with respect to the doping. However, no degeneracy appeared in both the magnetic ground state and excited state. When the thickness decreased, no magnetic phase transition appeared except for N = 6 at d = 3.35 Å . On the contrary, the degenerate magnetic states appeared. Unlike the AB-stacked bilayer ZGNR where there was no degenerate magnetic state, the degeneration in the AA-stacked form appeared due to the bonding between interplane-edge carbon atoms. In addition, this bonding governed the state in ZGNR to become magnetic or non-magnetic.