Introduction

Normally, ferroelectric materials are compounds that consist of two or more different constituent elements1. By contrast, as the atoms in a unit cell of an elementary substance are identical, ordered electric dipole or even ferroelectric polarization seem difficult to form spontaneously. Ferroelectricity was predicted for the first time in single-element materials, namely, two-dimensional (2D) As, Sb and Bi2. A very recent experiment on high-quality Bi monolayers confirmed this prediction3. Moreover, coexistence of out-of-plane and in-plane ferroelectricity in ultrathin elemental group-Va (P, As, Sb, and Bi) nanotube arrays have also been predicted4,5. The origin of the ferroelectricity of elements monolayer is generally accepted to be charge transfer between sublattice, which produces lone-pair on the pz orbital and causes the buckled polar structure with in-plane polarization2. However, the mechanism of the formation of lone-pair and the origin of distinct buckling height between Group-Va elements is still unclear.

Our experiment also reported a surprising observation of abundant charged 180° head-to-head/tail-to-tail domain walls in high-quality two-dimensional single-element ferroelectric Bi monolayers3. The origin of this counterintuitive phenomenon remains unclear. Ferroelectric domain walls (DWs) are regions that separate ferroelectric domains with different electric polarization orientations. In most ferroelectric materials, the naturally occurring DWs are of neutral type, because charged domain walls (c-DWs) usually have much higher electrostatic energy cost hence are not stable6,7,8. The occasional appearance of c-DWs in practice is usually connected with extrinsic factors, such as lattice defects or additional charges introduced by defects9,10,11,12,13,14. In practice, most of them occurs in improper ferroelectrics or sliding ferroelectrics15,16,17,18. Interestingly, our experiment finds the popular appearance of 180° head-to-head/tail-to-tail c-DWs in Bi monolayers3, which is unique for such elemental ferroelectrics. Nevertheless, the fundamental question regarding this unusual phenomenon has not been answered yet, namely, why are 180° c-DWs favored in Bi monolayers?

In this work, we clarify the origin of this peculiar phenomenon. Based on combined first-principles calculations and non-contact atomic force microscopy (nc-AFM) measurement, we show that the coupling between lone-pair and empty antibonding orbitals activates ferroelectricity in Group-Va single-element monolayer, which also determines the formation of stable 180° c-DWs. The electrostatic energy of DWs, which is usually the dominant contribution, turns out to be inessential in Bi monolayers. Instead, the DW energetics here is dominated by local strain energy. Owing to its special character of lone-pair induced ferroelectricity, we find that the 180° c-DWs in fact have the lowest strain energy cost, hence they are intrinsically the most stable DW configuration in these elemental ferroelectrics (single layer Sb, Bi). This explains their popularity observed in experiment. Based on this understanding, we predict that a small compressive strain ~1% can change the most stable DW types from 180° c-DWs to 90° non-charged domain walls (n-DWs). Our calculated strain maps around DWs also match well with nc-AFM measurement results. Our work reveals a mechanism for the formation of intrinsic c-DWs, which may have important implications on DW design and device applications.

Results

Lone-pair activate ferroelectricity

Group-Va elements have three p orbitals and the bonds between them are mainly determined by p orbitals hybridization. For example, the Bi monolayer is extended in the x–y plane, the bonds are mainly contributed by px and py orbitals and the pz electrons are left unpaired on one sublattice, which tend to bring in the buckling between the neighboring sublattices with the loss of centrosymmetry (space group Pmn21)2, as shown in Fig. 1a. Two non-equivalent atoms are labeled as M1 and M2, respectively. The centrosymmetric Bi monolayer has a lattice structure which consists of two flat atomic layers, as shown in the inset of Fig. 1f. In this case, Bi1 and Bi2 are equivalent. As shown in Fig. 1b, when Bi1 is in an asymmetric environment, more electrons locate at lower energy level than in a symmetric environment, indicating that the energy of the two degenerate pz orbitals splits with a lower energy level when the system goes through symmetry breaking. As a result, the unpaired pz electrons of one sublattice tend to move to the other sublattice, forming the lone-pair of electrons around Bi1. This corresponds to the position of the occupied pz level in Fig. 1e, which splits and shifts to a lower energy level than the pz level in Fig. 1f. In the meantime, the Crystal Orbital Hamiltonian Population (COHP)19 analysis shows the bonding character between Bi1 pz and Bi2 px/py, instead of the non-bonding character in centrosymmetric environment (Fig. 1c). This corresponds to the empty antibonding combination further interacts with the lone-pair and stabilize the buckling structure (asymmetric structure), as shown in Fig. 1e. The orbital interactions between Bi1 and the other Bi atom are also discussed in Supplementary Fig. 1, which show little impact on the buckling and ferroelectricity of Group-Va monolayer. The orbital interactions in Bi can be described in the following lone-pair model shown in Fig. 1d–f. The Bi1 px/py and Bi2 px/py interactions create filled bonding and empty antibonding combinations (Fig. 1d). At an asymmetric site, the spacing of Bi1 pz and Bi2 px/py orbitals decreases (Red double arrows in Fig. 1e) and it is favorable for the hybridization between these orbitals. As a result, the empty antibonding combination further interacts with the lone-pair formed by the charge transfer between Bi1 and Bi2 pz orbitals. The distorted Bi1 coordination environments stabilize the valence band maximum (VBM) and enlarge band gap, indicating the formation of stable buckling structure with lone-pair. In contrast, when Bi1 is at a high symmetry site without buckling, the distance between the Bi1 pz and Bi2 px/py orbitals is too large to have the hybridization (Red and purple double arrows in Fig. 1f). Thus, the interaction between the lone-pair of pz and px/py empty antibonding state is inhibited and increase the VBM in energy due to the suppressed coupling (Fig. 1f). This orbital interaction mechanism also can hold in the As and Sb monolayer. They have smaller buckling distortion than Bi monolayer due to the higher px/py antibonding energy (indicated in Fig. 1d and Supplementary Fig. 2), and thus weaker coupling between the empty antibonding state and pz lone-pair. In extreme case, the well-known structure of P preserves an inversion center without buckling (space group Pmna)20,21,22,23. As illustrated in Fig. 1d, the antibonding orbital energy level of P(px/py orbitals) is too high to couple with the lone-pair of pz orbital. This orbital interaction not only affects the ferroelectric phase transition, but also determines the formation of DWs in ferroelectric single-element monolayer, which will be discussed later.

Fig. 1: Orbital interactions of single-element monolayers.
figure 1

a Crystal structure of 2D ferroelectric single-element monolayers. Orange and purple spheres indicate the two non-equivalent atoms, labeled as M1 and M2, respectively. The partial charge density of pz is shown in both asymmetric and centrosymmetric environment. b Projected electronic density of states (DOS) for Bi1 pz and c crystal orbital Hamilton populations (COHP) for Bi1 pz and Bi2 px/py in asymmetric and centrosymmetric environment, respectively. Since the COHP of Bi1 pz–Bi2 px and Bi1 pz–Bi2 py are degenerate, only one of them is shown. The highest occupied state (Fermi level) is set to 0 eV. d The Bi1 px/py and Bi2 px/py interactions create filled bonding and empty antibonding combinations. The antibonding orbitals of P, As and Sb are also represented. e When the coordination environment of the Bi atom is asymmetric, the antibonding combination of px and py orbital is stabilized by the lone-pair of Bi1 pz coupling to form the upper valence band (VB). f When the coordination environment of Bi atom is centrosymmetric, this orbital interaction is suppressed due to destructive interference. In this case, the antibonding combination between Bi1 pz and Bi2 px/py mainly forms the upper VB without Bi1 pz orbital contribution. In e, f, the orange and purple balls represent Bi1 and Bi2, respectively. If adjacent orbitals are the same color, they represent symmetric wavefunctions, otherwise antisymmetric wavefunctions.

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Electrostatic energy vs. strain energy

There are two major contributions to DW energies: electrostatic energy and strain energy. Their interplay determines the most favorable type of DWs in a given material. Since the depolarization field and the bound charge at DW are both proportional to \({P}_{S}\), in conventional ferroelectrics, c-DWs almost always have much higher electrostatic energy than n-DWs. With reference to this huge electrostatic energy cost, the strain energy is usually negligible when performing the comparison.

A salient feature of ferroelectric Bi monolayers is that the ferroelectricity occurs in an elemental material. Compared to conventional ferroelectrics, the degree of charge transfer and polarization in 2D Bi is much weaker. This is evidenced by our calculated values of spontaneous polarization \({P}_{s}\) for 2D group-Va monolayers (including As, Sb, Bi monolayers), as listed in Table S1. In comparison, for the well-known compound ferroelectrics with the similar lattice structure, such as monolayer SnTe, GeTe and GeS, their values are at least an order of magnitude larger. Such small \({P}_{s}\) indicates that the bound charge appearing in a c-DW of Bi monolayers is also going to be small.

Considering the long-range interaction between DWs, neighbor DWs can be seen as two parallel capacitors to estimate the electrostatic energy around a 180° c-DW (Supplementary Fig. 3). The electrostatic energy, also known as depolarization energy when free charges are absent, is approximated by the energy stored between neighbor DWs, namely:

$${{{{\mathcal{E}}}}}_{{{{static}}}}=\frac{{L}_{\parallel }h}{2}{\rho }_{b}U,$$
(1)

where \({L}_{\parallel }\) is the lattice parameter parallel to the wall, h is the thickness of the monolayer (defined as maximum distance between two atoms along the out-of-plane direction plus two van der Waals radiuses, more discussions about definition of thickness see Supplementary Fig. 4), \({\rho }_{b}\) is the 2D bound charge density at DW and U is the potential difference between neighbor DWs. The calculated polarization, potential difference and bound charge profiles for 2D compound ferroelectrics SnTe and GeS are shown in Supplementary Figs. 5–7. The total electrostatic energy value for c-DW is large. The values are ~86.31 and ~185.48 meV/Å for SnTe and GeS, respectively. These values are much larger than their strain energies (~10 meV/Å for SnTe and 100 meV/Å for GeS) (Supplementary Fig. 8). This dominance of electrostatic energy cost indicates that in these conventional compound ferroelectrics, n-DWs are intrinsically favored than c-DWs24,25,26. Now, let’s turn to the results for the elemental ferroelectrics Bi and Sb in Supplementary Figs. 6 and 7. One observes that the bound charge density and potential difference for c-DWs here are dramatically smaller. The electrostatic energies values are estimated to be only 3.34 meV/Å and 3.20 meV/Å for Bi and Sb monolayers, respectively, which are two order of magnitude smaller than the compounds. As we discussed above, such weak electrostatic (depolarization) energy in monolayer Bi and Sb is a result of the weak polarization (which is tied to its elemental character).

The cost of strain energy always exists for a DW. As discussed above, for compound ferroelectrics (like SnTe and GeS), the strain energy is usually subdominant compared to electrostatic energy. Here, we shall investigate whether this is the case also in Bi monolayers. We compute and compare the strain energies for four typical DW configurations for Bi monolayers, including the 90° and 180° n-DWs and c-DWs, as illustrated in Fig. 2a. For each DW, we estimate its strain energy \({{{{\mathcal{E}}}}}_{{{{strain}}}}\) by the discrete sum of the energy difference in each unit cell (labeled by an integer index i), namely:

$${{{{\mathcal{E}}}}}_{{{{strain}}}}=\mathop{\sum}_{i}{{{{\mathcal{E}}}}}_{{{{strain}}}}\left(i\right)=\mathop{\sum}_{i}\left[{{{{\mathcal{E}}}}}_{s}\left(i\right)-{{{{\mathcal{E}}}}}_{0}\right]/L,$$
(2)

where \({{{{\mathcal{E}}}}}_{s}(i)\) is the energy of cell \(i\) (shown in the red dashed frame in Fig. 2a) near the DW, \({{{{\mathcal{E}}}}}_{0}\) is energy for an unstrained cell (or equivalently, a cell far away from the DW) and L is the supercell lattice parameter parallel to the DW. The energy difference between \({{{{\mathcal{E}}}}}_{s}(i)\) and \({{{{\mathcal{E}}}}}_{0}\) aroused by the lattice and internal strain between a cell near DW and a perfect cell. More details about strain energy calculation see Supplementary Fig. 9. The calculated distribution of \({{{{\mathcal{E}}}}}_{{{{strain}}}}(i)\) for the four types of DW is plotted in Fig. 2b. From the result, one observes that 90° DWs generally have a higher strain energy than 180° DWs. This is due to the lattice structure. 2D ferroelectric Bi has a rectangular lattice with \(a \, > \, b\) (\(a\approx 1.03{b}\)), and the electric polarization is along the \(a\) axis. It follows that for 90° DWs, the angle of polarization on both sides of DW is deviated by 90° and there must be a sizable lattice constant mismatch27, which leads to large strain energy. In comparison, for 180° DWs, there is no lattice constant mismatch. The strain energies for 180° DWs are mainly of internal strain contribution (i.e., the displacement of atomic positions in a cell). Table 1 verifies that 90° DWs have a larger structural distortion than 180° DWs around DWs.

Fig. 2: Electrostatic energy and strain energy of DWs.
figure 2

a Schematic representation of the relaxed structure of four typical DW models in Bi. Purple and orange spheres indicate the two non-equivalent atoms. The centers of the DWs are presented by black dashed lines and the red frame is defined as the length of per cell i near the DW, which is used to calculate the distribution of \({{{{\mathcal{E}}}}}_{{{{strain}}}}\). The four angles θ and side a, b of the cell i are presented. b The strain energy distribution of four typical DW models in Bi. The black dashed line indicates the center of DWs. c The electrostatic energy and strain energy of four typical DW models in Bi.

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Table 1 The structural distortion of cell i nearest to DW, as shown in Fig. 2a
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In Fig. 2c, we compare the total energy, including both electrostatic and strain energies, for the four types of DWs in Bi monolayers. One can see that, first, the 180° c-DW has the lowest energy. Second, the result shows that electrostatic energy is negligible compared to strain energy for c-DWs in this system. Therefore, we have a rather rare situation where the dominating role of electrostatic energy is replaced by strain energy for a c-DW. This switch of dominant factor in energetics is origin of the unusual popularity of c-DWs in Bi monolayers. In addition, our calculation shows that monolayer Sb shares the similar behavior (Supplementary Fig. 10), so 180° c-DWs are the intrinsic stable DW type for this family of elemental ferroelectrics. With the decreased importance of electrostatic energy cost, this opens the new possibility that in these elemental ferroelectrics, n-DWs may no longer win the energy competition against c-DWs, and the strain energy may play a more important role in determining the winner.

Lone-pair induces stable 180° c-DW

Let us return to the behavior of lower strain energy than the 180° n-DW for 180° c-DW in Bi monolayer as noticed before. It is related to the lone-pair model mentioned above. When Bi is at a high symmetry site without buckling, there is no charge transfer between two pz orbitals and the lone-pair will be suppressed. As a result, the interaction between the lone-pair of pz and px/py empty antibonding state is inhibited, as depicted in Fig. 1c. Thus, the pz orbital could not be stabilized by the coupling with antibonding combination of px/py orbital. Therefore, the local symmetric environment is an important factor in determining the energy level. It is noticed that 180° c-DW and 180° n-DW have different local symmetric environments around DW. As shown in Fig. 3a, for 180° c-DW, Bi1 and Bi2 are in the symmetric environment, and COHP result in Fig. 3b shows that the interaction between Bi1 px/py and Bi2 pz orbital has non-bonding characteristic. Therefore, for 180° c-DW, only one interaction (Bi1 px/py–Bi2 pz) shows non-bonding characteristic. Similarly, for 180° n-DW, there are four atoms (Bi1, Bi1’, Bi2, Bi2’) in the symmetric environment, as shown in Fig. 3c. COHP result in Fig. 3d shows that the interaction of both Bi1 px/py–Bi2 pz and Bi1’ px/py–Bi2’ pz pairs have non-bonding characteristic. Thus, there are more non-bonding interaction for 180° n-DW than 180° c-DW and result in large strain energy at DW. Besides, the strain energy is also related to the local structural distortion around DWs. As illustrated in Table 1, the cell near 180° n-DW have greater lattice distortion compared to 180° c-DW. Therefore, the more symmetrical environment and larger lattice distortion of 180° n-DW makes its strain energy higher than 180° c-DW. As for 90° DWs, no Bi atoms in symmetric environment for 90° n-DW, while both Bi1(Bi2) and Bi1’(Bi2’) in symmetric environment for 90° c-DW (shown in Supplementary Fig. 11), resulting in larger strain energy in charged 90° than non-charged one. Consequently, due to more lone-pair formation and slight local structural distortion, the 180° c-DW is indeed the intrinsically stable DW type in this system. This explains the experimental observation of the popularity of these DWs (Fig. 3e).

Fig. 3: Orbital interactions of 180° DW.
figure 3

The local structure of a 180° c-DW and c 180° n-DW around the DW. The black dashed box is a subcell formed from four atoms at DW. COHP for Bi1 pz and Bi2 px/py of b 180° c-DW, for Bi1 (Bi1’) pz and Bi2 (Bi2’) px/py of d 180° n-DW. Since the COHP of Bi1 (Bi1’) pz–Bi2 (Bi2’) px and Bi1 (Bi1’) pz–Bi2 (Bi2’) py are not degenerate (Supplementary Fig. 14), the average of them is shown. The highest occupied state (Fermi level) is set to 0 eV. e Top: STM image of typical Bi monolayer. Bottom: zoom-in nc-AFM image and perspective view of 180° c-DW structure. Purple and orange spheres indicate the two non-equivalent atoms.

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Strain-determined DW type

Since the DW energetics in this system is dominated by the strain contribution, one can expect that applied strain may have a strong influence in determining the most stable DW type. For example, applying uniaxial strain that decreases the difference between lattice parameters \(a\) and \(b\) of a unit cell can reduce the lattice parameter mismatch across a 90° DW hence reduce its strain energy cost. In comparison, the energy of a 180° DW is less susceptible to this strain as there is no lattice parameter mismatch across DW. These contrasting behaviors suggest that there might be possibility that the most stable DW type may be affected under applied strain.

In Fig. 4a, we plot the strain energy variation for 90° DWs under an applied compressive strain \(\eta\) normal to the wall. One observes that the strain energy decreases with the strain. The decrease is quite significant. For the 90° n-DW, 1% strain can decrease the energy by ~25%. In the inset, we show the unit cell distortion near the wall. One indeed finds that the applied strain tends to reduce the lattice anisotropy, thereby reduce the strain energy cost for a 90° DW. On the other hand, such strain increases the polarization (green line in Fig. 4a) because of giant negative piezoelectricity of this single-element monolayer28, resulting in increased electrostatic energy of c-DW as discussed above. Therefore, the relative stability of different DW configuration can be determined by in-plane strain.

Fig. 4: Strain application to DWs.
figure 4

a Polarization of periodic unit cell and total strain energy of 90° DW as a function of applied compressive strain η. The inset is a schematic of lattice mismatches under compressive strain. b The formation energy \({E}_{{DW}}^{f}\) of four typical DW models in Bi as a function of compressive lattice strain η. The dashed red line represents the critical point of DW type conversion. The inset shows how the applied compressive strain η acts on a DW. nc-AFM image of c 90° n-DW and d 180° c-DW of Bi. The orange box is defined as the size of the unit cell away from DWs, with the lattice constants labeled (in units of Å).

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In Fig. 4b, we plot the total DW formation energy \({E}_{{DW}}^{f}\) versus applied strain for the four types of DWs in Fig. 2a. One can see that the formation energies for 90° DWs decrease, whereas the energies for 180° DWs increase. Above ~1% strain, the energy for 90° n-DW becomes less than 180° c-DW, so under applied strain, it can replace 180° c-DW to be the most stable intrinsic DW type. Our calculation shows that monolayer Sb also exhibits the similar behavior (Supplementary Fig. 12). It is very interesting that a small strain can determine the stable intrinsic DW type between charged and neutral ones in these elemental ferroelectrics.

This assumption can be directly probed by nc-AFM measurement. In experiment, we have prepared ferroelectric monolayer Bi samples following the method in ref. 3. In the Fig. 4c, d, we show the nc-AFM images of two DWs in the samples. The strain distribution around the DWs can be derived from nc-AFM image by the Lawler-Fujita algorithm29. The obtained results are superposed (with gray colormap) on the nc-AFM images in Supplementary Fig. 13. One can see that the lattice parameter of a cell far away form 90° n-DW (4.85 Å) is reduced by ~1% compared to 180° c-DW (4.93 Å), as shown in Fig. 4c, d. This confirms the above assumption that 90° n-DW prefers to appear when applying uniaxial strain to reduce the lattice mismatch. Therefore, we suppose that strain is the factor leading to the emergence of different types of DW. On the other hand, for both DW types, the local strains are concentrated around the DW. It is clear that the strain around 90° DW is larger than the 180° DW (Supplementary Fig. 13), consistent with our analysis above.

Discussion

We have clarified the orbital origin of ferroelectricity in Group-Va monolayer and the surprising popularity of c-DWs in ferroelectric Bi monolayers. We find that lone-pair activity is important for atomic layer buckling and thus the occurrence of ferroelectricity in Group-Va elemental monolayer. The weak polarization of elemental ferroelectrics and the special lattice structure of Bi monolayers lead to a rather rare scenario where the electrostatic energy is greatly suppressed for c-DWs in this system, and so the local strain energy can replace it to become the dominant factor to determine the stable DW type.

The overall decreased energy scale associated with DWs opens the possibility to determine intrinsic DW types by a small applied strain. In practice, strains on 2D materials can be readily applied by using a stretchable substrate30,31. The DW types can be directly probed by nc-AFM or scanning tunneling microscopy. The switch of popular DW type between charged and neutral walls may also affect the electric conductance of the sample, resulting novel electronics.

Methods

DFT calculations

The density functional theory (DFT) calculations were performed using the Vienna Ab initio Simulation Package (VASP)32 with the projector augmented wave (PAW) method33. The generalized gradient approximation (GGA) with Perdew, Burke, and Ernzerhof (PBE) realization was used for the exchange-correlation functional34. A vacuum space of 20 Å perpendicular to the monolayer is employed to avoid the periodicity of the images. We construct 24 × 1 × 1 supercells for 180° c-DWs, 1 × 24 × 1 supercells for 180° n-DWs and 14 × 1 × 1 supercells for 90° DWs, respectively, which are comparable with the average experimental domain sizes. For total energy calculations, Monkhorst-Pack k-point grids35 with one k-point in the direction perpendicular to the DW was obtained. In the direction parallel to the DW we sampled 10 k-points in the 180° DW supercells and 8 k-points in the 90° DW supercells. A sufficiently high kinetic-energy cut-off above 400 eV and the force and energy convergence criteria were set to 10−3 eV/Å and 10−6 eV, respectively. The van der Waals (vdW) correction was included by the Grimme-D3 approach36. We also considered Spin-orbit coupling (SOC) effect in all electronic structure calculation for single-element monolayer. For the strained DWs calculations, first, apply uniaxial strain to the unit cell and fully relax. Then, construct the supercells for four types of DW models and fix lattice to relax the structures. These structures are the DW models after applied strain.

Polarization profiles calculations

The polarization of the subcell α can be written as7:

$${P}_{i,\alpha }=\frac{1}{A}\mathop{\sum}_{j,a\in \alpha }{Z}_{{ij}}^{*\,a}{d}_{j}^{a},$$
(3)

Here A is the unit cell area, \({d}_{j}^{a}\) is the displacement of atom α with respect to its position in the direction j, \({Z}_{{ij}}^{*{a}}\) is the Born effective charge tensor of atom α and the sum runs over atoms in each unit cell.

Bound charge density calculations

Bound charge density ρb arises from the polarization profile and the ρb of unit cell i can be described by \({\rho }_{b}=-[(\overrightarrow{{{{{\bf{P}}}}}_{{{{\boldsymbol{i}}}}{{{\boldsymbol{+}}}}{{{\mathbf{1}}}}}}-\overrightarrow{{{{{\bf{P}}}}}_{{{{\boldsymbol{i}}}}}})+(\overrightarrow{{{{{\bf{P}}}}}_{{{{\boldsymbol{i}}}}}}-\overrightarrow{{{{{\bf{P}}}}}_{{{{\boldsymbol{i}}}}{{{\boldsymbol{-}}}}{{{\mathbf{1}}}}}})]\cdot \overrightarrow{{{{\bf{n}}}}}/2\), where \({{{{\bf{P}}}}}_{{{{\boldsymbol{i}}}}}\) and \({{{{\bf{P}}}}}_{{{{\boldsymbol{i}}}}{{{\boldsymbol{+}}}}{{{\mathbf{1}}}}}\), \({{{{\bf{P}}}}}_{{{{\boldsymbol{i}}}}{{{\boldsymbol{-}}}}{{{\mathbf{1}}}}}\) are the spontaneous polarization for each unit cell \(i\) and its neighborhoods, respectively, and \(\overrightarrow{{{{\bf{n}}}}}\) is the unit vector along polarization direction.

Formation energy calculations

The different DWs formation energy \({E}_{{DW}}^{f}\) can be calculated by:

$${E}_{{DW}}^{f}=\frac{\left({E}_{N}-N{E}_{0}\right)}{2L},$$
(4)

where L is the supercell lattice parameter parallel to the DW, E0 is the energy per unit cell, EN is the total energy of the supercell with two DWs and N is the number of unit contained.

Experiment measurement

Monolayer Bi (α-phase) was prepared by room-temperature deposition of Bi on the surface of highly oriented pyrolytic graphite (HOPG) substrate. Prior the deposition, HOPG was exfoliated in the ambient condition and loaded into the high-vacuum chamber to degas the surface contaminations. Nc-AFM measurement was carried out at 4.3 K with a CO-terminated tungsten tip glued to a qPlus sensor. During the measurement, qPlus sensor has a resonance frequency of 27 kHz and a quality factor of about 50,000. Oscillations amplitude of 40–50 pm was chosen to perform all the AFM topography scans. The data have been processed and analyzed using MATLAB and WSxM software37.