Two-Dimensional Honeycomb Monolayer of Nitrogen Group. Elements and the Related Nano-Structure: A First-Principle Study

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1 Two-Dimensional Honeycomb Monolayer of Nitrogen Group Elements and the Related Nano-Structure: A First-Principle Study Jason Lee, Wei-Liang Wang, Dao-Xin Yao * State Key Laboratory of Optoelectronic Materials and Technologies, School of Physics and Engineering, Sun Yat-Sen University, Guangzhou , China Corresponding author: yaodaox@mail.sysu.edu.cn Abstract In recent years, two-dimensional material attracts great attention, and new twodimensional materials are being predicted, discovered and studied. Based on the idea of octet stability, we predict a new family of two-dimensional material: honeycomb monolayer of nitrogen group elements in the periodic table. From our calculation we prove the existence of two-dimensional structure of all these elements, and their electronic structure has been intensively studied. The effect of spin-orbit coupling on the band structure has also been investigated. Based on analysis of projected density of states, we argue that only the s and p orbitals are sufficient to describe the electronic structure, and a tight-binding model with spin-orbit coupling using only s and p orbitals is constructed. We also shed light on the electronic structure of zigzag nano-ribbon, and found doubly-degenerate, spin-polarized ferromagnetic edge states. These materials may be synthesized and find their way in the electronic industry. Key Words: nitrogen group element, honeycomb, two-dimensional, nano-ribbon I. Introduction Various Two-dimensional materials are booming in recent years 1, 2. Successful exfoliation of graphene marked the beginning of study of two-dimensional materials 3,4,5. Particularly, black phosphorus has begun to be synthesized and studied 6,7,8. It was argued that black phosphorus has high carrier mobility, and has certain advantage over other two-dimensional materials 8. Recently, another allotrope of phosphorus call blue phosphorus has been predicted, and it has much larger band gap than black phosphorus 9. In addition to phosphorus, two-dimensional nitrogen atomic sheet has been grown epitaxially on GaAs(001) surface 10. Phosphorus and nitrogen belong to the nitrogen group (VA) elements in the periodic table. They have five valence electrons, and needs three additional electrons to achieve octet stability. In other words, if these elements form a two-dimensional honeycomb lattice so that each atom bonds to three neighboring sites, they achieve octet stability. Based on this idea, we argue that VA elements have stable form of two-dimensional honeycomb structure. In this work, we theoretically predict that VA elements in the periodic table, have two-dimensional stable structure. This is a generalization of previous study on blue phosphorus 11.

2 In this paper, we use first-principle calculation to explore the unknown phase of VA elements in the periodic table. Our calculation is based on Plane Augmented Wave (PAW) with Perdew-Burke-Ernzerh (PBE) of exchange-correlation 11 as implemented in the Vienna Ab initio Simulation Package (VASP) code 12. To ensure reliable result, we use large energy cut-off of 400eV for bulk and monolayer, but we use default cutoff energy for nano-ribbon. The systems are restricted to periodic boundary conditions in the study of bulk properties, while in the study of system with lower dimension (monolayer and nano-ribbon), a vacuum at least 15 Å thick is inserted to eliminate the interaction between different units. For optimization, ions are relaxed using conjugategradient algorithm until the total force is no greater than 0.01eV/ Å. Grimme s scheme for van de Waals (vdw) correction 13 is used for optimization of bulk structure, and results are compared with those obtained without vdw correction. Unfortunately, vdw correction is not included in case of bismuth because of lack of parameters. Some vdw scheme with higher precision 14 could have been used, but getting accurate vdw energy is not the main interest of this paper, and a simple time-saving scheme of Grimme is sufficient to prove our statements. Spin-orbit Coupling (SOC) is included only in the calculation of band structure of monolayer. II. Layered Structure and Monolayers The three-dimensional AB stacking structures of VA elements are obtained from our calculation. Their structures are similar, as shown in Figure 1. The optimized threedimensional structures of nitrogen (N), phosphorus (P), arsenic (As), antimony (Sb) and Bismuth (Bi) are listed in table 1. It is clear that the inclusion of vdw interaction does not change the structure parameters within a layer dramatically, even though the interlayer distance d is decreased by about 1Å for P and As. The calculated lattice constant a and bond length l bond for blue phosphorus are 3.32Å/3.28 Å (with/without vdw) and 2.27 Å respectively for bulk, and 3.28 Å and 2.27 Å for monolayer, in agreement with previous study by Z. Zhu et.al. 7. But their inter-layer distance is larger than our result. It is necessary to point out that, during the optimization process of three-dimensional structure of N, we found the tiny jittering of totally energy as the inter-layer distance changes. This may suggest that interaction between N layers are extraordinarily weak (if they have any) and the computer finds it difficult to optimize the structure to minimize the total energy. It means that these layers can detach from each other very easily, therefore the three-dimensional structure of N may not exists, and it is just a possibility. There is no monotonous increase of inter-layer distance d, but we do see the monotonous increases of lattice constant a, buckling z, bond length l bond and the nearest distance between atoms in different layers l with respect to increasing atomic number from P to Bi are clear. This is in agreement with the periodic law of elements. Interestingly, if we compare the bond length with twice the covalent radius, the bond length is a little bit longer.

3 For all cases, the bond length l bond s are much smaller than the nearest distance between atoms in different layers l s. From table 1 we can see, the inclusion of vdw correction mainly affects the inter-layer distance d while having minor effect on the lattice constant a, buckling z and bond length l bond. When vdw is included, all the l s are about twice the vdw radii.all these phenomena suggest that layers couple with each other by weak vdw interaction and thus can easily exfoliated to form monolayer 15. All the monolayers have silicone-like structures, and resembles that of the bulk. The structure parameters of monolayer only slightly differ from that of the bulk, which indicates the weak inter-layer vdw interaction. Moreover, it is clear that the structure parameters for nitrogen is much smaller than any other VA elements. As in the case of bulk structure, the bond lengths are a little bit longer than twice the covalent radii. III. Electronic structure of monolayer Electronic structures obtained from our first-principle calculation are very similar, as shown in Figure 2 (without SOC) and Figure 3 (with SOC). Three branches of bands are evident, as shown in three different colors in Figure 2 and Figure 3. For nitrogen monolayer, the band structure is similar to that of silicene 16, with the lower branch and the middle branch entangled. As atomic number increases, the lower branch gradually disentangled from the middle one, becomes lower in energy, and forms a graphene-like band structure. Meanwhile, the ranges of all these three bands shrink, suggesting the weakening interaction between orbitals as the covalent bonds become longer. As the atomic number increases, the effect of SOC on band structure increases. SOC mainly affects the middle branch and the upper branch of bands, and has no noticeable influence on the lower branch. As we will see in the projected density of states (PDOS), this is because the lower branch is made up of s orbit and while the middle and upper branch are made up of p orbitals. However, the overall effect of SOC is still minute. The calculated band gaps are listed in table 2. We obtained a smaller band gap (1.97eV) than previous study of blue phosphorus monolayer (over 2eV) 9. The band gap decreases from 3.70eV of nitrogen to just 0.55eV/0.42eV(without/with SOC) of Bi, which means they change from insulator to semiconductor. This makes sense in that the metal character increases as atomic number increases in VA elements. Without SOC, all these band structures have indirect band gaps except Bi. When SOC is included in the calculation, all band structures have indirect band gaps. As the atomic number increases, both the valence band maximum (VBM) and the conducting band minimum (CBM) shift towards Γ point in the BZ. In case of Bi, when SOC is not included, both the VBM and CBM rest in Γ point, making it a direct band gap. But strangely enough, when SOC is included, the VBM of bismuth shifts away from Γ point again, making it an indirect band gap. The calculated PDOS of VA element monolayers is presented in Figure 4. The evolution

4 of orbital character of band structure can be seen from PDOS. For nitrogen monolayer, the lower branch mainly comes from 2s orbital while the upper one mainly from 2p orbitals, and in the middle branch, 2s and 2p orbital are highly mixed, making roughly the same contribution to this branch of bands. For phosphorus, the s component of the lower branch increases with decreasing p character, but the middle and upper branch takes a totally opposite trend, which is the p character increases as s character decreases. This trend persists all the way to bismuth. This can explain why SOC can influence the middle and upper branch of bands while having no noticeable effect on the lower branch: SOC have nonzero contribution to p orbitals, but it vanishes for s orbital because the angular moment of s orbital is zero. Moreover, from arsenic on, d character begins to creep up, but its contribution is still minute for all cases. Another interesting trend is that, the gap between the lower branch and the middle branch enlarges as atomic number increases, which is an indication of increasing relativistic effect. Based on the analysis of PDOS, it is clear that a tight-binding (TB) model with just s and p orbital of both atoms is sufficient to describe the system. Recently Zolymoi et. al. have constructed a TB model for silicane and germanane with just s and p orbital 17. In view of certain similarity between the systems we study and the system in Zolymoi s paper (silicane and germanane), we can construct a similar TB model, with minor modification. H = H 0 + H 1 + H 2 + H SO H 0 = [ε s a (R i )a(r i ) + ε p b (R i ) b(r i )] i H 1 = [h s (i, j) + h p (i, j) + h sp (i, j) + h. c. ] i,j H 2 = [h s (i, j) + h p (i, j) + h sp (i, j) + h. c. ] i,j H SO = iλ I ν ij σ z αβ b α (R i ) b β (R j ) i,j αβ + iλ R μ ij (e ij σ) z b α (R i ) b β (R j ) i,j αβ h s (i, j) = γ ss a (R i )a(r j ) h p (i, j) = γ ppσ [e ij b (R i )][e ij b(r j )] + γ ppπ {b (R i ) b(r j ) [e ij b (R i )][e ij b(r j )]} h sp (i, j) = γ sp a (R i ) (e ij b(r j ))

5 h s (i, j) = γ ss a (R i )a(r j ) h p (i, j) = γ ppσ [e ij b (R i )][e ij b(r j )] + γ ppπ {b (R i ) b(r j ) [e ij b (R i )][e ij b(r j )]} h sp (i, j) = γ sp a (R i ) (e ij b(r j )) e ij = R i R j R i R j, ν ij = (d z i d j ) d i d j Here, a (R i ) and a(r i ) are the creation and annihilation operators of the s electron in lattice site R i, and b (R i ) = (b x (R i ), b y (R i ), b z (R i )) and b(r i ) = (b x (R i ), b y (R i ), b y (R i )) are the creation and annihilation operators of the p electron in lattice site R i. ε s and ε p in H 0 are the onsite energies of the s and p orbital. H 1 contains all the nearest neighbor hoppings and H 2 contains all the next nearest neighbor hoppings. Both H 1 and H 2 can be divided into three parts: hopping between s orbitals (h s and h s ), hopping between p orbitals (h p and h p ), and hopping between s orbital and p orbital (h sp and h sp ). γ ss, γ ss, γ ppσ and so on are the corresponding hopping integrals. The first term in H SO is intrinsic SOC and the second term is Rashba SOC 18. α and β in H SO are spin indexes. μ ij = ±1 for A(B) sites. d i and d j are the two nearest bonds connecting the next nearest neighbors. All these parameters can be obtained by fitting TB band structure to the result from firstprinciple calculation. IV. Zigzag Nano-ribbon We have calculated blue phosphorus nano-ribbon with 20 phosphorus atoms wide, as shown in Figure 5. Before calculation of the electronic structure, the outermost three atoms on each side are fully relaxed. From our optimization, we observe only minute atom displacements. Compared with the band structure of monolayer, the correspondence of the branches of bands are obvious, as shown in Figure 6. In particular, the lower branch mimics the band structure of zigzag graphene nano-ribbon. Any bands of nano-ribbon that have this correspondence can be obtained by projection. And the bulk band gap is 2.05eV, slightly larger than that of the monolayer. It is interesting to notice that there are two bands that do not correspond to any of the bands in the monolayer. These two bands are doubly degenerate, and spin polarized. There is a gap between these two bands, which is about 0.28eV, and the Fermi level lies inside the gap. In other words, this system is ferromagnetic, and demonstrates a

6 magnetic moment about 2.02μ B. This is reasonable, because the band below Fermi level is doubly degenerate, and contains two electrons from each unit cell. This differs from the antiferromagnetic edge state of graphene nano-ribbon 19,20. Analysis of charge density reveals the existence of edge states. We found that the charge of this band is mainly distributed on both of the edges of the nano-ribbon, as shown in Figure 7. V. Conclusion Using first-principle calculation, we demonstrate the existence of two-dimensional honeycomb monolayer of VA elements, and the corresponding structure parameter is given. From the three-dimensional structure, it is clear that the systems have AB stacking layered structure (with some uncertainty about N), and each layer is similar to silicene. The geometry of monolayer is very similar to that of the bulk, indicating that layers are bound together by weak van de Waals interaction. Bond length, lattice constant and buckling increases monotonously as atomic number increases from nitrogen to bismuth, obeying the periodic law of elements. Electronic structure shows a clear trend that obeys the periodic law of elements, as atomic number goes from nitrogen to bismuth. Three branches of bands are obvious in the band structure, and they are similar for all VA elements with minor variations. Spinorbit coupling (SOC) plays certain role in the band structures, and its effect becomes greater as atomic number increases, but its influence is still limited. SOC influences p orbitals, which can be seen from minor changes of bands that have much p characters. There is no noticeable effect of SOC on s orbitals. The band gap decreases monotonously, from 3.7eV for nitrogen to only 0.55eV(without SOC)/0.43eV(with SOC), which means it goes from insulator to semiconductor. From projected density of states, it is clear the s and p character of bands changes monotonously. When atomic number is small, s and p orbitals are mixed, such as for cases of nitrogen and phosphorus. As atomic number increases, the s character shifts downwards in energy, and becomes the dominant component of lower branch of bands; on the other hand, the p character shifts upwards in energy, becoming the dominant component of the middle branch and upper branch of bands. In all these cases, the effect of d orbital in the making of band structure is negligible. Based on this analysis, we have constructed a tight-binding (TB) Hamiltonian using only one s orbital and three p orbitals from each atom in the unit cell. SOC is also included in this TB Hamiltonian. Lastly, the electronic structure of blue phosphorus zigzag nano-ribbon has also been studied. We found an obvious correspondence of band structure to that of monolayer, and exhibits spin-polarized ferromagnetic edge states. All these finding may contribute to the study of future use these materials.

7 Acknowledgments: This project is supported by National Basic Research Program of China (2012CB821400), NSFC , NSFC , Specialized Research Fund for the Doctoral Program of Higher Education ( ), and NCET

8 Table 1 Structure Parameter for Monolayer of VA Elements Element N P As Sb Bi Bulk a (without z vdw) d l l bond Bulk (with vdw) a z d l l bond Monolayer a z l bond Covalent radius vdw radius Unit in Å Table 2 Band Gaps of Monolayer of VA Elements Element N P As Sb Bi Band gap/ev Without Gap type Indirect Indirect Indirect Indirect Direct SOC VBM Not Γ Not Γ Γ Γ Γ CBM Not Γ Not Γ Not Γ Not Γ Γ Band gap/ev With SOC Gap type Indirect Indirect Indirect Indirect Indirect VBM Not Γ Not Γ Γ Γ Not Γ CBM Not Γ Not Γ Not Γ Not Γ Γ Change of band gap duo to SOC/eV

9 Figure 1 (a) The top view and (b) the side view of three-dimensional structure of VA elements. Red arrows are the lattice vectors in the plane. Green rhombus is the unit cell. d is the inter-layer distance.

10 (a) (b) (c) (d) (e) (f) Figure 2 (a) The first Brillouin Zone and the high symmetric points of two-dimensional honeycomb lattice. (b)-(f)band structure evolution of 2-dimensional monolayer of VA elements as it goes from nitrogen to bismuth. Different colors refer to different branches of bands. Fermi energy is set to zero denoted by green dash line.

11 (a) (b) (c) (d) (e) Figure 3 The same as Figure 2, but with SOC. (f)

12 Figure 4 The density of states of monolayers of (a) nigrogen, (b) phosphorus, (c) arsenic, (d) antimony and (e) bismuth. Red, blue and purple solid lines denote the projection into s orbital, p orbital and d orbital, respectively. Fermi energy is set to zero.

13 Figure 5 Schematic drawing of blue phosphorus zigzag nano-ribbon with 20 atoms wide. The system is periodic in x direction. Green rectangle represents the atoms in the unit cell. Vacua of 15 Å are inserted on both sides of the nano-ribbon. (a) (b) (c) Figure 6 Band structure of zigzag blue phosphorus nano-ribbon (b) compared with that of monolayer (a). (c) Same as (b), but the energy scale is different. Red and blue solid lines in (b) and (c) refer to spin-up and spin-down, respectively.

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