Charge-Induced Structural Stability and Electronic Property of Sb, Bi, and PbTe Monolayers

Abstract

Flat honeycomblike Sb and Bi monolayers have been fabricated epitaxially on Ag(111) and SiC(0001) substrates, respectively, although their freestanding structures are found to prefer a buckled form. Based on ab initio total energy calculations and phonon mode analysis, here we reveal that the charge (electron) can essentially induce the structural stability of planar antimonene and bismuthene. With increasing of the charge, the flat antimonene and bismuthene become more stable than the buckled form in energy, as the charge is larger than 0.22–0.24 electrons per atom. Meanwhile, the phonon modes can also be stable with increasing charge for flat monolayer. Similar behavior is also found in PbTe monolayers. The present results provide an excellent account for experimental observations and reveal the stabilization mechanism of the flat honeycomb-like Sb, Bi, and PbTe monolayers.

1. Introduction

Group-V elements can form stable 2D honeycomb-structured materials, except nitrogen [1,2,3,4,5,6,7,8,9,10]. The blue phosphorene has a buckled honeycomb structure, and theoretical calculations predict that an s-p band inversion can be induced at the Γ point in the band structure by applying tensile strain on monolayer phosphorene to about 17% [11,12,13]. Arsenene, antimonene and bismuthene can also form 2D honeycomb structures. The unique honeycomb structure of group-V elements is predicted to have excellent physical properties, including tunable electronic bandgap, nontrivial topological features and huge quantum spin Hall (QSH) gaps, which have attracted significant interest.

Antimonene and bismuthene are recent additions to the growing family of elemental two-dimensional (2D) materials. The energetic and dynamical stabilities of the buckled Sb and Bi monolayer ($P\overline{3}m1$ symmetry) are confirmed theoretically. Buckled antimonene is theoretically predicted to be a semiconductor [14] and is expected to be used in high-performance electronic and optoelectronic devices [15,16]. Zhao et al. also suggested that a large tensile strain (15%) applied on monolayer antimonene will lead to an s-p topological band inversion at the Γ point, transforming it from a normal insulator to a topological insulator with a QSH gap of up to 270 meV. So far, the epitaxial growth of buckled antimonene has been reported on Bi₂Te₃, Sb₂Te₃, PdTe₂, Ge(111), Ag(111), Pb(111), Cu(111), and Cu(110) surfaces [17,18,19,20,21,22]. On the other hand, the buckled bismuthene is determined to be dynamically and thermally stable with an indirect band gap. In particular, there is a peculiar Rashba spin-splitting that emerges in the valence band maximum (VBM) states.

The monolayer phases of antimony and bismuth also adopt flat honeycomb lattices ($P6/mmm$ symmetry). In 2018, the flat antimonene monolayer structure was successfully prepared on the Ag(111) surface [20] using the molecular beam epitaxy (MBE) method; the adsorption energy for this honeycomb structure is strong, which was predicted theoretically [23]. Meanwhile, as early as 2013, Huang et al. theoretically predicted that bismuthene can be fabricated on SiC(0001) surface with a flat honeycomb lattice [24]. In 2017, the flat bismuthene was grown on a SiC(0001) substrate successfully [25]. They detected a gap of 0.8 electron volts and conductive edge states consistent with theory by using scanning tunneling spectroscopy.

The flat Sb and Bi monolayers have been successfully synthesized in the laboratory [20,25]. However, the ground state of freestanding Sb and Bi monolayers tends to form a buckled structure [26,27,28]. Compared to flat monolayers, buckled structures are energetically more favorable and dynamically stable [26,27,28]. This fact indicates that there is a strong electronic interaction between the monolayer and the substrate, and charge transfer may play an important role in stabilizing the flat monolayer structure.

In this paper, we report a detailed ab initio study on the structural stability of the flat Sb, Bi, and PbTe monolayers with charge (electron). Total energy calculations show that the flat antimonene and bismuthene are energetically more stable than the buckled monolayer, as the charge is larger than 0.22–0.24 electrons per atom. Phonon spectrum simulations show that, with the increasing of charge, the buckled antimonene (or bismuthene) becomes dynamically unstable with the occurrence of imaginary frequency out-of-plane modes, while the flat antimonene (or bismuthene) becomes stable with the disappearance of imaginary frequency, showing the strong electron–lattice correlations in antimonene and bismuthene. We extend our horizon to the PbTe monolayer with the same number of valence electrons as the Sb(Bi) monolayer and find that it has a similar behavior.

2. Computational Method

The calculations are carried out using the density functional theory as implemented in the Vienna ab initio simulation package (VASP) [29]. The exchange–correlation interactions are considered in the generalized gradient approximation (GGA) using the Perdew–Burke–Ernzerhof (PBE) functional [30]. The interactions between the valence electrons and ionic cores are described by the projector augmented wave (PAW) method [31], with valence electrons employed as 5$s^2$5$p^3$ for Sb and 6$s^2$6$p^3$ for Bi, respectively. A plane-wave basis set with a large energy cutoff of 600 eV is used [27]. The thickness of the vacuum region is set to 20 Å to avoid artificial interactions in the computational model. The Brillouin zone is sampled with a 24 × 24 × 1 Γ-centered Monkhorst-Pack k-point grid. Convergence criteria employed for both the electronic self-consistent relaxation and the ionic relaxation are set to 10⁻⁷ eV and 0.0001 eV/Å for energy and force, respectively. The electron doping is achieved by changing the total number of electrons of the unit cell, with a compensating jellium background of the opposite charge added. The amount of charge added to each atom is obtained by spherically integrated valence electrons around the atom [32]. To understand the dynamic stability, phonon band structure calculations are calculated based on a supercell approach using the phonopy code [33]. Due to the strong spin–orbit coupling (SOC) in antimonene and bismuthene [34], the SOC is used in our electronic band structure calculations. Moreover, there is a phase transformation from buckled toward flat structures using the solid-state nudge elastic band (SSNEB) [35,36].

3. Results and Discussion

3.1. Crystal Structures

Figure 1a,b show the buckled and flat structures of the Sb monolayer. The unit cells include two atoms (A and B) in $P\overline{3}m1$ (No. 164) symmetry with a trigonal lattice, occupying the 2d (0.3333, 0.6667, z) Wyckoff positions in Figure 1a. The lattice constant and the corresponding buckling height as a function of the charge (electron) per atom for flat (circle) and buckled (star) Sb monolayers are shown in Figure 1c. For the freestanding buckled Sb monolayer, the calculated lattice parameter a = 4.12 Å and the buckling height h = 1.64 Å. For the charge = 0.24 e/atom, the lattice constant increases to 4.68 Å and the buckling height decreases to 1.33 Å. For the charge = 0.31 e/atom, the lattice constant increases to 4.84 Å and the buckling height decreases to 1.24 Å. The unit cells include two atoms (A and B); $P6/mmm$ (No. 191) symmetry has a hexagonal lattice, occupying the 2d (0.3333, 0.6667, 0.5) Wyckoff positions in Figure 1b. For the freestanding flat Sb monolayer, the calculated lattice parameter is a = 5.03 Å. For the charge = 0.24 e/atom, the lattice constant increases to 5.25 Å. For the charge = 0.31 e/atom, the lattice constant increases to 5.31 Å.

In

Table 1. Calculated values of lattice constant, buckling height and stability with different charges are presented for the buckled ($P\overline{3}m1$) and flat ($P6/mmm$) Sb monolayers. Columns: charge in e/atom; lattice constants in (Å); buckling height of buckled structure in (Å).

System	Structure	Charge	a (Å)	Buckling (Å)	Stability
Sb	buckled	0	4.12	1.64	stable
		0.24	4.68	1.33	unstable
		0.31	4.84	1.24	unstable
Sb	flat	0	5.03		unstable
		0.24	5.25		stable
		0.31	5.31		stable
		Exp [20]	5.01
Bi	buckled	0	4.33	1.74	stable
		0.22	4.90	1.42	unstable
		0.31	5.09	1.30	unstable
Bi	flat	0	5.27		unstable
		0.22	5.46		stable
		0.29	5.53		stable
		Exp [25]	5.35
PbTe	buckled	0	4.33	1.69	stable
		0.25	5.02	1.33	unstable
		0.29	5.12	1.22	unstable
PbTe	flat	0	5.33		unstable
		0.25	5.54		stable
		0.30	5.59		stable

, calculated values of the lattice constant, buckling height, and stability are presented for buckled and flat Sb, Bi, and PbTe monolayers.

The buckled and flat Bi monolayers have the same $P\overline{3}m1$ and $P6/mmm$ symmetry as shown in Figure 1a,b, respectively. The lattice constant and the corresponding buckling height as a function of the charge (electron) per atom for flat and buckled Bi monolayers are shown in Figure S1. For the buckled Bi monolayer, as the charge increases from 0 to the 0.31 e/atom, the lattice constant increases from 4.33 Å to 5.09 Å and the buckling height decreases from 1.74 Å to 1.30 Å. For the flat Bi monolayer, as the charge increases from 0 to the 0.29 e/atom, the lattice constant for the flat Bi monolayer increases from 5.27 Å to 5.53 Å. Experimentally, the lattice parameters of the Sb and Bi monolayer on Ag(111) and SiC(0001) surface are 5.01 and 5.35 Å, respectively [20,25].

We also study the buckled and flat PbTe monolayers, which have different symmetry from Sb monolayers. The unit cells include two atoms (A and B) in $P3m1$ (No. 156) symmetry with a trigonal lattice, occupying the 1a (0.0, 0.0, z) and 1b (0.3333, 0.6667, −z) Wyckoff positions in Figure 1a. The lattice constant and the corresponding buckling height as a function of the charge (electron) per atom for flat and buckled PbTe monolayers are shown in Figure 1d. For the freestanding buckled PbTe monolayer, the calculated lattice parameter a = 4.33 Å and the buckling height h = 1.69 Å. For the charge = 0.25 e/atom, the lattice constant increases to 5.02 Å and the buckling height decreases to 1.29 Å. For the charge = 0.29 e/atom, the lattice constant increases to 5.12 Å and the buckling height decreases to 1.22 Å. The unit cells include two atoms (A and B) and $P\overline{6}m2$ (No. 187) symmetry with a hexagonal lattice, occupying the 1b (0.0, 0.0, 0.5) and 1d (0.3333, 0.6667, 0.5) Wyckoff positions in Figure 1b. For the freestanding flat PbTe monolayer, the calculated lattice parameter is a = 5.33 Å. For the charge = 0.24 e/atom, the lattice constant increases to 5.54 Å. For the charge = 0.31 e/atom, the lattice constant increases to 5.59 Å.

3.2. Structural Stability

We next investigate the charge effect on the structural stability by electron doping in the Sb, Bi, and PbTe monolayers. In Figure 2a, the energy as a function of the charge is plotted for the buckled and flat Sb monolayer. It is shown that, without charge (e = 0), the buckled structure is more stable than the flat one with an energy gain of 524 meV per Sb atom. Meanwhile, as the charge increases, the relative stability changes, and the flat structure becomes more stable when the charge is larger than 0.24 electron per Sb atom.

In Figure 2b, the energy as a function of the charge is plotted for the buckled and flat Bi monolayer. A similar behavior to that of the Sb monolayer is shown, and the flat structure is energetically more stable than the buckled one, as the charge is larger than 0.22 electron per atom [see Figure 2b].

In Figure 2c, the energy as a function of the charge is plotted for the buckled and flat PbTe monolayer. It shows a similar behavior to those described above, and the flat structure is energetically more stable than the buckled one, as the charge is larger than 0.22 electron per atom [see Figure 2c]. These results suggest that the charge can effectively change the relative energetic stability and make the flat structure more stable than the buckled structure, as the charge is larger than 0.22–0.24 electron per atom.

In order to understand the kinetic stability of these flat phases, we examine the phase transformation from buckled to flat structures using the solid-state nudge elastic band (SSNEB) [35,36] with different charges. The enthalpy results along the pathways for the transformation from the buckled to flat Sb, Bi, and PbTe monolayers with different charges are shown in Figure 2d–f, respectively. For the Sb monolayer, as the charge increases, the flat phases are more favorable than the buckled phases, and the energy barrier for the transition gradually decreases [see Figure 2d], showing the stability of the flat Sb monolayer with charge from both the kinetic and energetic views. Similar behavior is also found for the transformation of the buckled to flat Bi (PbTe) monolayers, which confirms the stability of the Bi (PbTe) flat structure with charge [see Figure 2e,f].

To assess the dynamical stability of the buckled and flat Sb monolayer with charge, we calculate the phonon band structures along the high symmetric directions in the first Brillouin zone. The obtained phonon band structures and partial density of states (PDOS) for the buckled and flat Sb monolayer are plotted in Figure 3 with a series of charges of 0, 0.19, 0.24, and 0.31 e/atom.

For buckled Sb monolayer, when charge = 0 as shown in Figure 3a, throughout the entire Brillouin zone (BZ), no imaginary frequencies are observed, confirming the dynamic stability of the buckled Sb monolayer; on the other hand, as shown in Figure 3e, for the flat Sb monolayer, there are some imaginary frequencies distributed in the BZ, confirming the dynamic instability of the freestanding flat Sb monolayer. The imaginary frequency at the M point is −48.2 cm⁻¹. The PDOS shows that all the imaginary frequencies of the freestanding flat Sb monolayer are related to the out-of-plane mode (z direction). These results suggest that the freestanding flat Sb monolayer is indeed unstable without charge.

We next discuss the charge effect on the dynamic stability of the Sb monolayer. As shown in Figure 3b, for the buckled Sb monolayer with 0.19 e/atom, no imaginary frequencies are observed throughout the entire BZ, confirming its dynamical stability. However, the lowest phonon band is remarkable softened along the Γ-M direction. Further increasing the charge to the energy exchange point (charge = 0.24 e/atom), as shown in Figure 3c, the softened modes become imaginary around the M point. The imaginary frequency at the M point is −13.2 cm⁻¹. We find that these imaginary phonon modes correspond to the Sb-Sb bond stretching modes. And PDOS also shows that the imaginary frequencies are related to both out-of-plane (z direction) and in-plane mode (x and y direction). With a large charge of 0.31 e/atom, as shown in Figure 3d, the imaginary frequencies around M point become larger, showing the strong dynamical instability.

On the other hand, as shown in Figure 3f, for the flat Sb monolayer with 0.19 charge per atom, the imaginary frequencies in the entire BZ are remarkable attenuated, and there are some small imaginary frequencies near the Γ point related to the out-of-plane mode (z-direction). When we increase the charge to 0.24 charge/atom as shown in Figure 3g, the phonons exhibit obvious hardening and all imaginary mode disappear, confirming the dynamic stability of the flat Sb monolayer. When the charge is 0.31 e/atom, as shown in Figure 3h, no imaginary frequencies are observed, confirming its stability. We can see that the charge can essentially induce the dynamical stability of the flat Sb monolayer, relative to the out-of-plane mode. The phonon band structures and density of states (DOS) projected along the x, y, and z Cartesian coordinates of the flat antimonene with several different electron concentrations are shown in Figure S2.

We also study the charge effect on the dynamical stabilities of the Bi and PbTe monolayers. The detailed phonon band structures are shown in Figure 4 and Figure S3, respectively. Calculated phonon band structures of buckled bismuthene monolayers with 0, 0.10, 0.22 and 0.29 electrons per atom are shown in Figure 4a–d, respectively. With increasing of the charge, the phonon modes around the M point in buckled form are softened as shown in Figure 4d. On the other hand, the calculated phonon band structures of the flat bismuthene monolayers with 0, 0.09, 0.22, and 0.28 electrons per atom are shown in Figure 4e–h, respectively. With the increasing of the charge, the imaginary frequency modes in the flat structure can be completely suppressed, confirming the dynamic stability of the flat Bi monolayer as shown in Figure 4f–h.

For the flat PbTe monolayers, the calculated phonon band structures with 0, 0.18, 0.25, and 0.30 electrons per atom are shown in Figure S3, respectively. With the increasing of the charge, throughout the entire Brillouin zone, no imaginary frequencies are observed, confirming the dynamic stability of the flat PbTe monolayer. The Bi and PbTe monolayers show similar behavior to that described above for the Sb system. We can see that the charge plays a key role in stabilizing the flat lattice and inducing a structure phase transition from the buckled to flat lattice.

3.3. Electronic Properties

The electronic band structures of freestanding flat antimonene monolayers without or with the SOC effect are plotted in Figure 5a,b, respectively. As shown in Figure 5a without SOC, around the Fermi level, the conduction band and valence band are reversed beyond the crossing at the K-point [26] and then generate two different band intersections along the K-Γ and K-M directions, respectively, showing metallic behavior. However, as shown in Figure 5b with SOC, the intersections along the K-Γ and K-M directions are broken due to the strong SOC effect [34]. The gaps induced by SOC are 0.18 and 0.33 eV at the points along the M-K direction and the K-Γ direction, respectively. The electronic band structures of flat antimonene monolayers with 0.24 charge per atom without or with the spin–orbit coupling are plotted in Figure 5c,d, respectively. Our results show that the charge changes the Fermi level of the system, and the gaps induced by SOC are 0.17 and 0.31 eV at the points along the M-K direction and the K-Γ direction, respectively.

The electronic band structures of freestanding flat bismuthene monolayers without or with the SOC effect are plotted in Figure 6a,b. Our results show that the large gaps opened by SOC are 0.51 and 0.87 eV at the points along the M-K direction and the K-Γ direction, respectively. The electronic band structures of flat bismuthene monolayers with 0.22 charge per atom without or with the spin–orbit coupling are plotted in Figure 6c,d. Our results show that the charge changes the Fermi level of the system, and the large gaps opened by SOC are 0.47 and 0.81 eV at the points along the M-K direction and the K-Γ direction, respectively.

The electronic band structures of freestanding buckled antimonene monolayers without and with the SOC effect are plotted in Figure S4a and Figure S4b, respectively. And electronic band structures of buckled antimonene monolayers with 0.24 charge per atom without and with the SOC are plotted in Figure S4c and Figure S4d, respectively. The calculated results show that the buckled Sb monolayer tends to transition from a semiconductor to a metallic state with the increasing of the charge. The electronic band structures of freestanding buckled bismuthene monolayers without and with the SOC effect are plotted in Figure S5a and Figure S5b, respectively. And the electronic band structures of buckled bismuthene monolayers with 0.22 charge per atom without and with the SOC are plotted in Figure S5c and Figure S5d, respectively. These calculated results show that the buckled Bi monolayer exhibits electronic properties similar to those of the Sb monolayer.

To understand the bonding nature of electrons in both antimonene and bismuthene, the electron localization function (ELF) maps are illustrated in Figures S6 and S7. The ELF maps can give a clear and quantitative description of the basic chemical bond (high ELF values 1 > ELF > 0.5 indicate the formation of covalent bonds). We find that the freestanding buckled Sb monolayer exhibits covalent bonding between the Sb atoms (see Figure S6). However, with 0.24 charge per atom, its electronic properties are affected, and the chemical bonding begins to transition toward metallic bonding. In contrast, the flat Sb freestanding monolayer tends to exhibit metallic bonding in both states with 0 and 0.24 charges per atom. Also, the Bi monolayer exhibits similar chemical bonding behavior as shown in Figure S7.

3.4. Bi/4H-SiC(0001) System

We also calculate the spherically integrated valence electron density around different Bi atoms as a function of the integration radius. The electron number of Bi atoms in the flat bismuthene monolayer (black line) and on the 4H-SiC(0001) surface (red line) are shown in Figure 7. The configuration of the Bi/4H-SiC(0001) system is consistent with that in [25], and the bismuthene monolayer in this system is flat. In the Bi/4H-SiC(0001) system, the closest distances of Bi-Bi and Si-Si are the same value 3.09 Å respectively. Therefore, we use 1.545 Å as the radius of integration (half of the distance between Bi and Bi or Si and Si). We find that the flat bismuthene monolayer receives 0.22 charge per atom from the 4H-SiC(0001) system, which is consistent with the calculation results above.

4. Conclusions

In conclusion, we have performed a detailed ab initio study on the structural stability of flat antimonene and bismuthene with charge (electron). We found that the charge can essentially induce the stability of flat antimonene and bismuthen in both energetics and dynamics. With the increasing of the charge, the imaginary frequency modes in the flat structure can be completely suppressed, while the phonon modes around the M point in the buckled form become softened, showing a strong electron–lattice correlation in antimonene and bismuthene. Meanwhile, total energy calculations show that the flat antimonene and bismuthene are energetically more stable than the buckled monolayer when the charge is larger than 0.22–0.24 electrons per atom, respectively. These results provide an excellent account for experimental observations and reveal the stabilization mechanism for the flat honeycomb-like Sb and Bi monolayer self-assembly on the Ag(111) and SiC(0001) surfaces, respectively.