Quasiperiodic Van der Waals Heterostructures of Graphene and Hexagonal Boron Nitride

Advancement toward opening a bandgap at the Dirac point induced by symmetry breaking paved the way to realize 2D heterostructures with graphene and hexagonal boron nitride (h‐BN). An alternate arrangement of graphene and h‐BN layers in a 3D stacking can tune the bandgaps of these composites depending on the position of B and N atoms with respect to C atoms of graphene. Herein, a unique possibility of arranging graphene and h‐BN atomic layers in a quasiperiodic Fibonacci sequence to study the possibilities of controlling the electronic properties of these heterostructures is explored. Density functional theory calculations combined with van der Waals corrections reveal that these quasiperiodic heterostructures are more stable than normal periodic stacking of monolayers of graphene and h‐BN. Moreover, for certain arrangements of atomic layers, sizeable bandgaps can be obtained.

DOI: 10.1002/pssb.202100423 Advancement toward opening a bandgap at the Dirac point induced by symmetry breaking paved the way to realize 2D heterostructures with graphene and hexagonal boron nitride (h-BN). An alternate arrangement of graphene and h-BN layers in a 3D stacking can tune the bandgaps of these composites depending on the position of B and N atoms with respect to C atoms of graphene. Herein, a unique possibility of arranging graphene and h-BN atomic layers in a quasiperiodic Fibonacci sequence to study the possibilities of controlling the electronic properties of these heterostructures is explored. Density functional theory calculations combined with van der Waals corrections reveal that these quasiperiodic heterostructures are more stable than normal periodic stacking of monolayers of graphene and h-BN. Moreover, for certain arrangements of atomic layers, sizeable bandgaps can be obtained.
High mobility and a large spin lifetime have been realized in graphene spintronics [25] by using layered h-BN as a substrate. Recently, the importance of Rashba spin-orbit coupling and spin relaxation properties has been highlighted [26] theoretically for graphene/h-BN heterostructures.
So far, the normal periodic stacking of graphene and h-BN has been considered as models of 3D hybrid heterostructures. Recent experiments have shown that the vertical graphene heterostructures with h-BN or molybdenum disulfide (MoS 2 ) can work as field-effect tunneling transistors. [15,27] Tuning of bandgap is also possible using BLG h-BN heterostructures. [28] Bipolar doping of double-layer graphene with hydrogenated BN has also been studied, where it has been shown that by tuning the interlayer distance, the interfacial dipole and screening charge distribution can be significantly affected. [29] Low-loss plasmon damping has also been observed in graphene h-BN heterostructures, which can be key to nanophotonic and nano-optoelectronic devices. [30,31] In these heterostuctures, it is also possible to tune the optical conductivity of graphene in the visible range and obtain optical excitations that are not present in the individual building blocks. [32][33][34] The stacking has a strong impact on the spatial distribution and the character of the generated electron-hole pairs. [34][35][36] Bandgaps of these 3D heterostructures can also be modified using an external electric field. [37] Ferromagnetic junctions with these kinds of heterostructures are also gaining interest in the field of spintronics devices. [38] However, there are not any studies on quasiperiodic 3D stacking of graphene and h-BN monolayers.
Therefore, in this article, we aim to explore the possibilities of breaking sublattice degeneracies in quasiperiodic heterostructures. In this article, we have considered a quasiperiodic 3D stacking of 2D graphene and h-BN monolayers in a Fibonacci sequence. The Fibonacci quasicrystal has exotic spectral features such as critical eigenfunctions, a spectrum with zero measure, and unique scaling behavior. [39][40][41] Binary layered structures have been synthesized in a Fibonacci sequence in one direction keeping pure translation periodicity in other directions. SiO 2 and TiO 2 dielectric slabs arranged in Fibonacci order have been grown [42] to verify the theoretical prediction [43] of localization of light in Fibonacci dielectric multilayers. Metallic Fibonacci multilayers were grown, and superconductivity was experimentally studied by Kartkut et al. using MoV superlattices. [44] In this work, we have studied structural and electronic properties of 3D Fibonacci heterostructures of graphene and h-BN to explore the possibilities of bandgap opening and stability of quasiperiodically arranged atomic layers using first-principles electronic structure calculations.

Structural Information
It has been shown theoretically that a graphene layer deposited on an h-BN substrate induces a bandgap opening in graphene. [45] This has been followed by several works, where a graphene monolayer or a bilayer sandwiched between the h-BN layer is studied. In these structures, the graphene layer is protected by h-BN apart from forming C─N, C─B out-of-plane dipoles. Formation of this dipole and its position with respect to two C-sublattices define the π À π Ã band degeneracy breaking at the Dirac points. A similar approach, like opening up a bandgap in BLG with applied electric field, is also seen to tune the bandgap in these kinds of systems. [46,47] In our proposed superlattices, h-BN and graphene layers occur in a Fibonacci sequence, S j , following a recurrence relation: S jþ1 ¼{S j , S jÀ1 }, S 0 ¼{B}, and S 1 ¼{A}. A and B correspond to graphene and h-BN single layers, respectively. With this approach, the first few sequences of Fibonacci will be S 2 ¼ {AB}, S 3 ¼ {ABA}, and S 4 ¼ {ABAAB}. Figure 1 shows a schematic representation of this quasiperiodic sequence. We followed the Bernal stacking for the arrangement of the individual layers. Being limited by the computational feasibility, we have considered finite-sized Fibonacci heterostructures with 13, 21, 34, and 55 layers with the number of BN layers (N BN ) and the number of graphene layers (N C ) to be (5,8), (8,13), (13,21), and (21, 34), respectively. By swapping the composition, reverse structures are also obtained with the number of N BN and N C layers as (8,5), (13,8), (21,13), and (34,21), respectively. The swapping has a twofold impact: 1) the concentration of C and h-BN layers changes and 2) the interface ordering also varies, although the number of interfaces remains the same.
To achieve the golden ratio (two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. It is an irrational number with a value of φ ¼ 1þ reveal under which conditions the sublattice degeneracy can be broken and also the effect of increasing the separation between graphene layers on its electronic properties. Also, one may get information on how BLG is affected when sandwiched between two h-BN layers. A careful inspection of the Fibonacci-ordered superlattices reveals that they contain the building blocks of monolayer and BLG sandwiched between h-BN layers. Therefore, the studies of these periodic blocks might shed some light on the understanding of the stability and electronic properties of Fibonacci superlattices. Now, we discuss the geometrical aspects of these heterostructures. In Table 1, we have shown the c=a ratios of the structures for forward and reverse (in parentheses) directions (c ¼ length of a superlattice in the growth direction z, a ¼ in-plane lattice parameter) for both LDA and PBE-vdW methods. (For details of the used methods, see Section 4.) The values of the lattice parameters are very similar for both LDA and PBE-vdW methods. The LDA lattice parameter for the forward and reverse structure is 2.46 and 2.47 Å, respectively. The lattice parameter within PBE-vdW for the forward and reverse structure is 2.48 and 2.49 Å, respectively. From Table 1, it can be seen clearly that the LDA lattice constants and c=a ratios of the multilayered structures are respectively smaller and larger as compared to the PBE-vdW values. But overall, the volumes of the supercells become smaller in PBE-vdW case. The reduced c=a ratios for PBE-vdW calculations indicate that the inclusion of vdW interactions enhances the bonding between the layers, especially for the reverse direction. Figure 2 shows the calculated interplanar separations in the Fib21 superlattice for both forward and reverse directions. In LDA calculations, the difference between forward and reverse structures is not prominent. On the contrary, significant differences are observed for calculations, including vdW corrections. Surprisingly, no CÀC vdW bonds are present between the adjacent layers in the reverse direction, whereas a number of such bonds are present in the forward direction. For the latter case, the calculated CÀC vdW bonds are 3.2 Å or more, giving rise to the peak structures in the figure.
Let us consider the regularly periodic multilayers with five structures having h-BN graphene ratios 1:1, 1:2, 1:3, 1:4, and 1:5. For the ratio 1:1, we have considered two structures. The first structure consists of one graphene and one h-BN layer in the unit cell. This means that the sublattice A (say) of graphene will always have B atom (in our case) from top and bottom h-BNlayers, whereas sublattice B will have nothing as we have considered the AB stacking. The other structure consists of four layers, where in the alternative h-BN layers, the positions of N and B atoms are altered. This means that sublattice A of a graphene layer will have C─B and C─N vdW bonds both coming from the top and bottom layers. We have found that in the former case, there is a bandgap opening, while it is still metallic for the latter case. So, a B─C─B (or N─C─N) situation opens up a bandgap, while B─C─N does not. These features are also reflected in other structures. For the reverse structures with higher h-BN content, we have considered h-BN layers with alternative B and N atom positions. That means in the perpendicular direction and on sublattice A, we will find C─B─N for 1:2, C─B─N─B for 1:3, C─B─N─B─N for 1:4, and C─B─N─B─N─B for 1:5. So, only for 1:3 and 1:5 ratios, we will have the B─C─B situation, and in the rest of the cases, B─C─N will be found. Following the observation in structures with 1:1 ratios, we can expect a gap opening in structures with 1:3 and 1:5 ratios. This is precisely seen in our calculations of Fibonacci superlattices having some of the abovementioned building blocks. A small gap opens up resembling the calculations for the building blocks. The increased number of h-BN layers between two graphene layers seems to have no effect as the gaps in 1:3 and 1:5 are the same. Hence, one may conclude that the similarity between the results of building block and multilayer calculations suggests that the properties of the graphene layer can only be affected by the local environment (neighboring h-BN layers) and they are decoupled from each other. Now we consider the possible building blocks of the forward Fibonacci structures, i.e., BLG sandwiched between two h-BN layers, as depicted in Figure 3. Following the Bernal stacking   Figure 3a-c). The formation energy analysis indicates that the B─C─C─B structure is energetically most probable among the others. However, from the literature, it is known that the B─C─C─N sequence can only open up a bandgap. [28] One thing to note here is that in B─C─C─N configurations, the layers are in AB stacking and the two C atoms belong to two different sublattices of two different graphene layers. These two situations of single-layer graphene and BLG are building blocks for our Fibonacci superlattices. As we have AB stacking in the Fibonacci stacking and have always similar h-BN layers (i.e., all h-BN layers have B atom as sublattice A and N atom as sublattice B), single-layer graphene will always have the B─C─B situation, while a BLG will have only B─C─C─B configuration, which are semiconducting and metallic blocks, respectively. So, as a whole, we got metallic situations for the forward structures, no matter how many layers we have considered. Table 2 shows the formation energies of these multilayered structures for both forward and reverse (in parentheses) directions. From the data presented in the table, we can conclude that it is possible to stabilize these quasiperiodic structures from 2D graphene and h-BN layers. To check the stability of these quasiperiodically stacked heterostructures as compared to the corresponding normal multilayer periodic stacking, we have also compared the formation energies of these two kinds of systems.

Energetics
Our calculations revealed that formation energies of the quasiperiodic structures are quite lower than the corresponding normal multilayer structure (see Table 2). From the analysis of energetics, we also found that the reverse structures are more stable than the forward structures, only visible after the inclusion of vdW interactions.

Electronic Structure
In Table 2, we have tabulated the bandgaps of Fib13, Fib21, and Fib34 reverse quasiperiodic structures as obtained using the PBE-vdW method. It is seen that for Fib21 reverse structure, we find the maximum value of the bandgap. An oscillation in the values of the bandgaps is observed. This is related to the quasiperiodic behavior of the heterostructures. A similar behavior was reported previously for quasiperiodic carbon-BN nanoribbons by Pedreira et al. [48] Figure 4a shows the band structure plots for the reverse quasiperiodic structure of Fib21 where a small bandgap opening at the Dirac point can be observed (see close-up view of Figure 4a. To find out the nature of orbital charateristics of the valence band maximum (VBM) and conduction band minimum (CBM) at Dirac point for reverse Fib21 structure, we have calculated the partial charge densities for VBM and CBM at the Dirac point. Figure 4b shows a schematic diagram of band level splitting in the Fib21 reverse structure along with the partial charge density plots for VBM and CBM at Dirac point.

Conclusion
In summary, we have studied 3D heterostructures of 2D layers of graphene and h-BN stacked in a quasiperiodic Fibonacci sequence by vdW corrected density functional theory. We demonstrate that this quasiperiodic arrangement is more stable than a normal 1:1 periodic arrangement of graphene and h-BN layers. The effect of adding vdW correction in the stability and interplanar separations is shown. Finally, we show that for certain arrangements of Fibonacci stacking, it is possible to open up bandgaps of the order of tens of meV at the Dirac point. We hope that our prediction will encourage the experimentalists to grow these quasiperiodic heterostructures.  code. [49] An energy cutoff of 500 eV was used to expand the wave functions. Generalized gradient approximation as proposed by Perdew, Burke, and Ernzerhof (PBE) [50,51] has been used for the exchange-correlation functional. To describe the electronion interactions, we have used the projector augmented wave methods (PAW). [52,53] We have also included vdW interactions within the DFT-D2 formalism of Grimme (PBE -vdW) [54] to incorporate the interlayer interactions. Recent papers [55,56] have shown that the choice of vdW correction method is not significantly important for describing the properties of graphene and h-BN composites. Moreover, for comparison, we have also performed calculations with local density approximation (LDA) as it is known that in many cases, LDA can give rise to a reasonable agreement with the experimental results for vdW bonded systems. [57] The structures have been optimized using the conjugate gradient method with forces calculated from the Hellmann-Feynman theorem. We have used the force tolerance of 0.01eV Å À1 along with the energy tolerance of 10 À5 eV for our calculations. For geometry optimizations, a 15 Â 15 Â 2 Γ-centered k-point mesh has been used for the full BZ integration. For density of states (DOS) calculations, we have used 45 Â 45 Â 2 Γ-centered k-point mesh.

Methods
The formation energy E f of graphene/h-BN hybrid system is defined as follows where EðGraphene þ BNÞ is the total energy of the hybrid graphene/h-BN system, EðGrapheneÞ is the total energy of a single-layer graphene, and EðBNÞ is the total energy of a single-layer h-BN. m and n represent the number of graphene and h-BN layers, respectively. www.advancedsciencenews.com www.pss-b.com