Before the discussion of 2D layers, we firstly discuss the structure character of bulk group-V elements. There are five valence electrons in group-V elements and diverse valence states (–3, +3, and +5) are possible for them, resulting in a variety of structural allotropes for bulk of group-V elements at ambient conditions. Phosphorus, the most studied element of group-V, has several allotropes, which are named as white, red, blue, violet, and black phosphorus.^[48–51] In addition, amorphous phosphorus is also very popular.^[51,52] The discovery of black phosphorus, the most stable allotrope of phosphorus at ambient conditions, is dated back in 1914^[50,53] and it crystallizes in parallel puckered layer orthorhombic structure with space group Cmca as shown in Fig. 1(a). The 3s and 3p orbitals of phosphorus undergo sp³ hybridization iw three σ-covalent bonds along with an out-of-plane oriented lone pair of electrons. Two of these three covalent bonds lie parallel to the plane while the third one is almost perpendicular to the plane, forming a puckered layer. And, the out-of-plane lone pair electrons combine the puckered layers together, crystalizing in the orthorhombic structure (α phase). Phosphorus can also crystallize into a rhombohedral geometry, i.e., blue phosphorus (β phase) with space group .^[54] Blue phosphorus also adopts a layered structure with buckled layers consisting of knitted honeycomb rings as shown in Fig. 1(b), where the three σ-covalent bonds within the layer are equivalent, resulting in the hexagonal structure.

Fig. 1

	Figure Option ViewDownloadNew Window
	Fig. 1 Two typical stable phases of group-V elements: (a) orthorhombic α phase with space group Cmca, and (b) rhombohedral β phase with space group . Left and right panels show side and top views, respectively.

Although the most stable crystal structure of phosphorus is the α phase with orthorhombic space group, the other elements of group-V (As, Sb, and Bi) favor the β phase due to the weak s–p hybridization.^[55,56] The delocalized p–p coupling causes more and more metallic bonding between atoms from As, Sb to Bi and the hexagonal honeycomb structure becomes more stable. For example, the most common and stable allotrope of As and Sb has the rhombohedral layered structure with space group as shown in Fig. 1(b), consisting of two interpenetrating, face-centered-cubic lattices which are displaced in the [111] direction and slightly distorted in the [111] direction. While the α phase is a metastable phase, the phase transformation between β and α is observed for both As and Sb.^[57] For bulk Bi, the last element of group-V, only the rhombohedral structure (β phase) has been observed in experiment.

Generally, there are more crystal allotropes for 2D materials than the corresponding 3D ones due to the lack of periodic boundary constrain along one dimension. For 2D group-V layers, a lot of allotropes have been predicted by theoretical calculations,^[58–61] as shown in Fig. 2(a). Although several phases could be found under unique experimental conditions,^[48] the α and β phases are more stable than the others (Fig. 2(b)), which can be viewed as single layers cleaved from the bulk (Fig. 3(a)). For example, α-P (black phosphorene) ML has already been exfoliated from bulk α phase along [001] direction^[62] and β-P (blue phosphorene) ML can be grown along [0001] direction of bulk β phase by chemical vapor deposition (CVD).^[63] Both α and β group-V films have been experimentally obtained,^[64–68] although the experimental evidence of β-P ML is still under debate.^[69,70] Their dynamic stability has also been confirmed through phonon spectra as shown in Fig. 3(b), without any appreciable imaginary modes. In fact, most of group-V layers fabricated in experiment are α and β phases.^{[4,21,71–73]} These two phases are more popular and observed in experiment for all group-V elements.^[74] Thus, we only focus on these two phases in the following.

Fig. 2

	Figure Option ViewDownloadNew Window
	Fig. 2 (a) Top views of 9 group-V ML allotropes with five typical honeycomb structures (α, β, γ, δ, ε) and four non-honeycomb structures (ζ, η, θ, ι). (b) Comparison of stabilities in terms of energies for different allotropes of group-V MLs.[59]

Fig. 3

	Figure Option ViewDownloadNew Window
	Fig. 3 (a) Top view and side view of α and β phases with two atomic layers (green and blue). The buckling height h is marked for α phase. When h is non-zero, the inversion symmetry of the structure breaks. (b) Phonon dispersions of group-VA MLs in α and β phases.[59]

For the β phase, all group-V MLs have buckled honeycomb lattices with two atomic layers, the same as silicene, which are not in the same plane due to the special valence electrons (s²p³) and mixed sp³ hybridization. There is buckling difference for different group-V MLs, but the space group (symmetry) is the same for all of them. For the α phase, all group-V MLs take the puckered lattice with orthogonal structure and there is also buckling difference (Fig. 3(a)) for different group-V MLs, e.g., the buckling height (h) of α-P is zero and that of α-Bi varies from zero to nonzero. Importantly, this nonzero buckling breaks the inversion symmetry of the α phase, quite different from the buckling in the β phase, and largely correlates to the topological and ferroelectric properties of the α phase, which will be discussed below.

Firstly, we discuss the α phase without buckling (h = 0 in Fig. 3(a)). In the inversion symmetric α phase, the structure has a non-symmorphic D_2h (7) space group with two sublattices (A and B in Fig. 4(a)). Based on the tight-binding model, two types of Dirac fermions have been predicted in the 1^st BZ as shown in Figs. 4(b) and 4(c).^[78] A pair of type-I Dirac points locate along Γ–X₂ and are mainly determined by the p_z orbital (Fig. 4(c)) between the two atomic layers, which are very sensitive to the interaction or distance between the layers. There are four type-II Dirac points protected by mirror symmetry, so that they carry opposite Chern numbers and appear in pairs (Fig. 4(b)). Type-II Dirac points are mainly determined by the p_x orbital (Fig. 4(d)) and sensitive to the in-plane strain. When a strain is applied, type I Dirac points only move along Γ–X₂ line while type II points are located at generic k-points. Thus, the α phase without buckling could hold six Dirac points in the 1^st BZ, like graphene, although their space groups are different. The main difference is that the Dirac points in α phase can move in BZ without changing symmetry while they are pinned at K points in graphene. Therefore, a QSH insulator with large topological gap can also be realized in α phase, although a strain or external field may be needed for the phase transition. For example, intrinsic α-P is a normal insulator with a bandgap around Γ point and a large strain (∼ 16 %) along the zigzag direction is needed for the emergence of type-II Dirac points. While, only 6 % strain is needed for the phase transition of α-As.^[79] Besides the strain modulation, application of electric field is also proposed to tune the bandgap and create the type-I Dirac points. In Fig. 5, an electric field (0.3 V/Å) perpendicular to the 2D plane is applied to close the bandgap of 4 ML α-P and causes the topological phase transition (0.55 V/Å for 3 ML).^[80] Adsorption of alkali atom on the surface of α-P has similar effect as electric field and Dirac states have been observed in experiment by surface doping of potassium.^[81] The structure parameters and electronic coupling between atoms of other group-V elements are different from those of α-P and the critical values for the topological phase transition are also different. Especially, the interlayer coupling of α-Bi is already weak enough and it is an intrinsic non-trivial 2D TI with large SOC gap^[78] (gapped type-I Dirac points as aforementioned). More importantly, the large-scale growth of α-Bi has been realized^[65] and the evidence of topological edge state along the edge of α-Bi has also been observed in experiment.^[73] Thus, α-Bi is a good elemental 2D material with intrinsic topological properties to investigate different kinds of quantum states controlled by thickness, gating, and substrate interaction. Very recently, α-Sb has been successfully grown on T_d-WTe₂ substrate^[74] and another experimental work claimed that 2 ML or 4 ML α-Sb becomes non-trivial due to the layer stacking.^[82]

Fig. 4

Figure Option ViewDownloadNew Window

Fig. 4 (a) Basis sites in a unit cell of centrosymmetric puckered α structure. (b) 2D rectangular BZ with Dirac points marked by red dots. A pair of type-I Dirac points are marked as D and D′ on Γ–X2, while two pairs of type-II Dirac points are marked (red dots) near F and F′ on Γ–X1. (c) Schematic energy dispersions around D and F points respectively illustrating type-I and type-II Dirac points. (d) Band structures of 2D phosphorene (α phase with h = 0) type lattice structure of group-V elements in the absence of SOC, the red and green dots size shows the projection on the pz (px) atomic orbitals.[78]

Fig. 5

Figure Option ViewDownloadNew Window

Fig. 5 (a) Band structure of phosphorene layers (4 L) with applied external electric field of (a) 0.3 V/Å and (b) 0.45 V/Å without including SOC. The red and blue lines show the top valence and bottom conduction states, respectively. (c) Inversion energy Δinv and (d) Z2 invariant of 2-, 3-, and 4-layer phosphorene as a function of electric field. The negative values of Δinv show non-trivial topological phases. It can be seen that the phase transition takes place for 3- and 4-layer phosphorene at a critical field of 0.55 V/Å and 0.3 V/Å, respectively.[80]

As mentioned above, the atomic buckling in α phase is related to the inversion symmetry and quite important for topological properties. Except for α-P, the atomic buckling (h) of other group-V elements could be nonzero and different from each other due to different hybridization between two atomic layers or substrate effect. This buckling breaks the inversion symmetry and degeneracy of crossing points, and even causes the topological phase transition. This effect is most remarkable in α-Bi. Because of the negligible s–p hybridization and the formation of lone pair,^[43] the spontaneous buckling height of α-Bi can vary in a wide range (0.0–0.5 Å). The different buckling height results in different p_z interaction and thus changes the topological gap. The non-trivial QSH effect (phase transition) is determined by atomic buckling (h), as shown in Fig. 6.^[73] The α-Bi is topologically non-trivial at low buckling heights (e.g., h < 0.1 Å ) and transforms to trivial insulator when h is large enough. Normally, atomic buckling is sensitive to charge ordering or charge transfer between substrate and epitaxial film. Thus, this buckling-dependent QSH effect provides a unique way to control the topological character at atomic level and very helpful for the design of topological device.

Fig. 6

Figure Option ViewDownloadNew Window

Fig. 6 (a) Side and top views of α-Bi ML with buckling. The upper and lower atomic layers are presented in blue and green colors, respectively. The buckling height h is defined by the distance between the atoms within the same ML. (b) The h-dependent energy gap (Δ) at Dirac point of α-Bi ML. (c) Band structures of non-buckled (h = 0) α-Bi without and with SOC as well as buckled α-Bi with SOC at different buckling heights (h = 0.1 Å and 0.5 Å).[73]

Bismuth ML in β phase has attracted attention immediately after the discovery of graphene due to the similar honeycomb structure. Bismuth is the heaviest group-V element and actually, it is also the heaviest element in periodic table without radioactivity. So, it possesses large SOC effect and is the most promising candidate to achieve 2D QSH phase. In 2006, β-Bi ML had been predicted to be topologically non-trivial with protected edge states^[83] and the non-trivial gap is large enough for experimental measurement. Several experimental works provide evidences of the topological edge states in β-Bi ML as shown in Fig. 7, where the dI/dV mapping shows the conducting states at the edges^[84] and the STS profile crossing the edge shows a Dirac-point-like signal.^[3,4]

Fig. 7

Figure Option ViewDownloadNew Window

Fig. 7 (a) Crystal structure of β-Bi. Red and blue colors label the upper and lower atomic layers of the topmost bilayer, respectively. Here, bilayer means two (upper and lower) atomic layers. (b) Topography around the hexagonal pit-like defect, differential conductance is shown with color plot, showing high conductance at alternative edges.[89] (c) STS of the step edge (blue) and the inner terrace (red) of Bi/Bi2Te3 island averaged over the area as marked in the inset of (c). The STS, the bands aligned Dirac point “DP” as indicated by the blue dashed line. The orange dashed lines mark the band gap of the topmost Bi layer.[3] (d) Differential conductivity dI/dV at different distances from the edge. A large gap of ∼ 0.8 eV is observed in bulk bismuthene (black curve). Inset shows the STS measurement locations at uphill substrate step causing the boundary.[4]

The β-Sb and β-As in ground state are trivial insulators. However, they can sustain large tensile strain (e.g., 18.4 % for As^[85] and 18 % for Sb^[86]) that assists the topological phase transition. It was proposed that the topological phase transition from normal insulator to 2D QSH insulator occurs under biaxial strain of 11.7 % and 14.5 % for Sb and As, respectively.^[85,86] This large strain may be achieved by some substrate as buffer layer during epitaxial growth. As the second heaviest element of group-V, β-Sb could become 2D TI by other ways. It was predicted that β-Sb becomes 2D TI when the thickness increases between 4 ML and 7 ML, where the surface band splitting provides a robust QSH gap at room temperature.^[87] With further increased film thickness, a crossover from the 2D TI to the 3D TI occurs.^[87] On the other hand, coupling with substrate could also cause the topological phase transition in β-Sb and most recently, an experimental evidence of nontrivial topological edge states has been observed in epitaxially grown 2D β-Sb islands on copper Cu (111) substrate,^[72] where the topological gap lies ∼ 1.5 eV above the Fermi level (Fig. 8). Similarly, β-P was also predicted to be a large indirect bandgap semiconductor ( > 2 eV),^[58] and it has been proposed to be topologically non-trivial under 16 % tensile strain.^[88]

Fig. 8

Figure Option ViewDownloadNew Window

Fig. 8 Band structure of freestanding (β phase) antimonene ML with and without SOC, represented by red and blue lines, respectively. (b) STM topography of a hexagonal honeycomb shaped antimonene island. The lower panel shows three dI/dV curves measured at the edges and center of the island marked by blue, green, and red arrows in the upper panel. (c) Plot of spatially resolved data of 35 dI/dV curves along the black dashed arrow in panel (b). The three dI/dV curves marked by arrows correspond to those shown in panel (d) with peaks at 1.68 eV (green), 1.85 eV (blue), and 1.95 eV (red). (d) The intensity plot of the dI/dV curves shown in panel (c).[72]

In general, crystal defects are really important for semiconductors, which could introduce additional defect states in bandgap affecting the transport properties. For topological materials, these defect states in bulk could introduce additional conducting states overcoming the advantage of the protected boundary states within bandgap. In comparison with compound systems, elemental systems are better with less defect problems, which is very important for the application of topological materials. This is one reason that researchers struggle to realize topological character in elemental systems. Recently, due to rising interest in QSH effect in group-V elemental MLs, 2D honeycomb structures of group-V binary compounds have also gained much attention,^[90] e.g., SbAs, BiP, BiAs, BiSb, AsP, and SbP. Because all elements are group-V elements with isovalent electrons, there is no additional state appearing in the bandgap and the gap size can be tuned by applying a lattice strain. They all exhibit metal–indirect gap semiconductor–direct gap semiconductor–TI transitions under strain.^[91] This broadens the potential candidates of 2D TIs in hexagonal β phase of group-V elements.

Chemical functionalization is an important tool to control electronic properties of materials and oxidation process is easy for group-V elements. This could change the topological character and help the emergence of new topological states. For example, β-P ML (blue phosphorus) is a trivial insulator with large bandgap. This gap size decreases largely after oxidation (P₂O₂) due to the saturation of the lone pair electrons. And, a moderate strain can easily tune the three bands around the Fermi energy, resulting in tunable quantum phase transition between pseudospin-1 fermions and 2D double Weyl fermions (Figs. 9(a)–9(c)).^[92,93] β-As ML also becomes a topological insulator exhibiting the quantum spin Hall effect when surface functionalized with H, F, OH, and CH₃ chemical group.^[94,95]

Fig. 9

Figure Option ViewDownloadNew Window

Fig. 9 (a) Electronic band structure of blue phosphorus oxide (BPO) without strain. A and B label the non-degenerate state at the conduction band bottom and the doubly-degenerate state at the valence band top, respectively. (b) Energies of A and B states versus strain. The crossing of the two corresponds to the quantum phase transition between a semiconductor phase and a symmetry protected semimetal (SSM) phase. (c) Energy dispersion around the band crossing point at Fermi level for critical strain +0.6 % (upper panel) and (lower panel) +4 % strain. The low-energy quasiparticles around the crossing point are described by 2D pseudospin-1 fermions and 2D double Weyl fermions.[92] Band structures of the (d) BiF and (e) SbF monolayers with (red) and without (gray) SOC. The bands around the Fermi level consist of the px and py orbitals. The size of the symbols is proportional to the population of the px and py orbitals.[96]

Hydrogenation or halogenation is another effective method to introduce topological phase transition in group-V MLs. Here, we firstly discuss the functionalized β phases. After functionalization, Bi₂X₂ or Sb₂X₂ (X = H, F, Cl, Br) were predicted to be topologically non-trivial insulators and stable above room temperature (Figs. 9(d) and 9(e)).^[96] These features make Bi₂X₂ (or Sb₂X₂) an ideal platform for the realization of many striking quantum phenomena at room temperature. Similarly, halogen functionalized As₂X₂ also opens up a large non-trivial gap, signifying its potential for future applications as high-speed devices. Furthermore, if functionalized 2D TI is grown on the magnetic substrate, the quantum anomalous Hall insulator may be achieved, e.g., Sb₂H₂ on antiferromagnetic substrate LaFeO₃ (111).^[97] For the α phase of group-V, it is predicted that halogen-decorating on one side of MLs would generate a Dirac nodal line around the Fermi energy without SOC or hourglass fermions protected by nonsymmorphic symmetry when considering the SOC.^[98]

As both α and β phases of group-V MLs have one lone pair of electrons, functionalization by monovalence chemical groups is normally preferred to maintain chemical stability with tunable electronic structures. Although chemical functionalization is a good way to change the electronic structure of 2D materials including the topological properties of 2D group-V MLs, this functionalization is permanent with fixed change of electronic structure. It is hard to reversibly tune the electronic structure by chemical reaction which mostly evolves irreversible bond formation. It is more desirable for device application to change or switch the electronic character between two or more topological states reversibly.

By scanning tunneling microscope and transport measurement, the 2D ferroelectricity has been confirmed in group-IV monochalcogenides, SnTe ML, which has puckered atomic structure.^[100] The key condition of this ferroelectricity is the resonance bonding between cations and anions composing the puckered structure.^[106] After that, ferroelectricity has been identified in a lot of 2D systems by replacing elements with isoconvalent state as SnTe, e.g., MX (M = Ge, Sn; X = S, Se).^[45] The α phase of 2D group-V elemental MLs has similar structure and the same valence electrons with 2D group-IV monochalcogenides. Thus, they are also possible to present ferroelectricity in puckered α phase.

Ferroelectricity has been mostly observed in compounds of different elements rather than elemental materials. The appearance of FE in elemental materials does not contradict the requirement of broken inversion symmetry. The major obstacle is that most nonmetallic elemental materials are stabilized in nonpolar structures of high symmetry, which prevents the electric polarization. This obstacle could be overcome in 2D system. As mentioned above, the atomic buckling in α phase breaks the inversion symmetry and very recently, Xiao et al. predicted the existence of FE in buckled α-As, α-Sb, and α-Bi elemental MLs for the first time, as shown in Figs. 10(a)–10(c).^[43] The spontaneous polarization in these buckled MLs is along the in-plane x-axis with a predicted high Curie temperature, which makes them promising candidates for ferroelectric devices. Further analysis of the free-energy contour and phonon spectrum confirmed that beyond the two FE phases (A and A′), there also exist anti-ferroelectric (AFE) metastable phases (C and C′) (Figs. 10(d) and 10(e)).

Fig. 10

Figure Option ViewDownloadNew Window

Fig. 10 (a) Top view of α phase group-V elemental ML. The unit cell is indicated with black dashed rectangle lines. (b) Free energy contour for α-As ML versus the buckling heights (hU, hL). The phases A, A′, and B are marked. (c) Side views of the two energy-degenerate distorted noncentrosymmetric structures (A and A′ phases) and undistorted centrosymmetric structure (phase B). The height differences between upper (red) and lower (blue) atomic layers are marked as hU and hL, respectively. (d) Side view of α-Bi ML in antiferroelectric (AFE) configuration. The atomic sites shifted up and down are colored in red and green, respectively. (e) Free energy contour of Bi MLs versus buckling heights (h(R), h(G)). For Bi, besides the FE phases A and A′, there also exist metastable AFE phases C and C′.[43]

Compared with the 2D FE group-IV monochalcogenide materials, besides the fundamental distinction (compound vs. elemental), the mechanisms for the ferroelectric polarization of 2D group-V MLs are also different. The group-IV monochalcogenide materials break the centrosymmetry via in-plane distortion from the cubic structure as well as the occupation of lattice sites with different kinds of elements. In comparison, the group-V MLs break centrosymmetry by the out-of-plane atomic-layer buckling, whereas the in-plane distortion does not break the symmetry. As a result, the transformation between two degenerate FE structures of group-IV monochalcogenide materials requires the breaking and reforming bonds between atoms. This is the reason that the resonance bonding between atoms is important and required for the generation of ferroelectronic polarization in group-IV monochalcogenide compounds.^[107] Whereas for the group-V monolayers, there is no significant bonding and bond-breaking character in the transformation between two degenerate FE structures. Moreover, the puckered structure of the elemental MLs permits the presence of an AFE phase (for α-Bi), which is absent in the compound materials. The coexistence of FE and AFE phases in the same materials is important. As the symmetry and geometrical structure of AFE phase are different from those of FE phase, its electronic structures could be largely changed in topological nature. This interplay between geometrical phase transition and electronic polarization allows an extraordinary control of the electronic structure (e.g., electronic topological properties) via external field. In addition, the electronic polarization is sensitive to buckling height h and interaction with the substrate, giving us more freedom to engineer the electronic structures and phase transition of the van der Waals heterostructure devices based on group-V elemental MLs.

It has also been predicted that 2D group-V binary compounds composed of both light and heavy group-V elements present ferroelectricity in α or γ phase with notable large and strain tunable spontaneous polarization, e.g., BiN,^[107] SbN, BiP.^[108] SbP, and SbAs.^[109] Such large polarization is caused by the immense electronegativity difference of the light and heavy group-V elements and their displacement in the polarization plane. The mechanism for the ferroelectric transition is almost the same as that of 2D group-IV monochalcogenide compounds requiring the breaking and reforming bonds between atoms. In addition, ferroelectricity is also predicted in halogen-intercalated α-P bilayer without buckling, where the ferroelectric transition is determined by the shift of the halogen atoms covalently bonded to up-layer or down-layer.^[110] This concept may be extended to other van der Waals bilayers, but it is really challenging in experiment to precisely functionalize the target material to control these quantum orders.

In hexagonal honeycomb β phase, the out-of-plane atomic-layer buckling does not break the inversion symmetry and the group-V elemental MLs remain centrosymmetric without ferroelectricity no matter whether they are planar or buckled 2D layers. One possible way to break centrosymmetry is to fabricate AB binary monolayer with a bipartite corrugated honeycomb lattice, where the A and B atoms are different elements. Binary group-V compounds show a spontaneous FE polarization in freestanding buckled β phase and their polarization is switched by changing the relative position of the two atoms along out-of-plane direction. The presence of a diatomic basis that breaks the inversion symmetry even in the planar geometry leads to the emergence of Zeeman-like spin-split bands with coupled spin-valley physics analogous to the MoS₂ case.^[111] Meanwhile, it is mainly responsible for the onset of the FE phase when the honeycomb lattice buckles, allowing for an electrically controllable Rashba-like spin-texture around the valleys, whose chirality is locked to the polarization direction and therefore fully reversible upon FE switching.^[42] These spin split bands can be effectively detected by spin-resolved spectroscopic techniques and binary β phase compounds allow for the engineering of 2D spin field-effect-transistors.^[112]

Chemical functionalization of polar chemical group is another direct and possible way to transform centrosymmetric β phase into polar structure. It is proposed that ferroelectricity can be introduced into β-As ML by passivation of CH₂OCH₃ ligand.^[90] The fully relaxed structure of CH₂OCH₃ functionalized β-As is shown in the inset of Fig. 11(a), where arsenene is flat without any out-of-plane buckling and remains centrosymmetry. When CH₂OCH₃ is adsorbed on β-As ML, it becomes polarized with local dipole. The in-plane electronic polarization is achieved by ligand molecules alignment and a ferroelectric transition can be driven by a ligand rotation mechanism (Fig. 11(b)). Actually, this concept can date back to an early work^[114] and is expected to be a more general phenomenon in 2D-layered materials. Similar to β-As passivated by simple ligands as mentioned before, CH₂OCH₃-functionalized β-As becomes a topologically nontrivial insulator with QSH effect. The electronic structure and topological character are mainly determined by the bonding between β-As and ligand, not the relative position or rotation of CH₂OCH₃ sitting on top of the As atoms. Thus, the topological phase and nontrivial band gap are robust and well preserved when ligand CH₂OCH₃ molecules are rotated (Fig. 11(c)) without interplay between the topological character and ferroelectricity. On the other hand, the total energies of multiple rotated configurations with different polarizations are quite close to each other, which means that the ligand molecules may not be orderly arranged as shown in Fig. 11(a). However, this controllable in-plane ferroelectricity maybe has wide applications in memory or logical device designs and the storage density can be quite high as the information bit can be directly related to one ligand arrangement.

Fig. 11

Figure Option ViewDownloadNew Window

Fig. 11 (a) Energy variation and kinetic barrier for CH2OCH3 rotations in functionalized arsenene; the insets show the structural models. (b) Variation of the polarization along the x and y directions. (c) Electronic band structures of the symmetric ligand configuration with the rotation angle of 0° (left) and the the asymmetric ligand configuration with the rotation angle of 120° (right). The topological order is preserved under the ligand rotation, although there is a band splitting due to an out-of-plane symmetry breaking.[113]