Robust quantum spin Hall state and quantum anomalous Hall state in graphenelike BC₃ with adatoms

Based on the requirements above, we find that suitable adsorption of Tl atoms on BC₃ maintains the Dirac cone character and shifts Fermi level just to the Dirac cone simultaneously. These adsorption systems are the configurations with the stoichiometry of BC₃Tl, which corresponds to each BC₃ primitive cell adsorbing with two Tl atoms. Via traversing the total energies of all possible adsorption configurations for the (1 × 1), ($1\times \sqrt{3}$), ($\sqrt{3}\times \sqrt{3}$), and (2 × 2) BC₃ supercells, we obtain four low-energy structures, as shown in figures 2(a)–(d), in which Tl atoms tend to be adsorbed uniformly on each side of BC₃ sheet and fall on the hollow site of the B₂C₄ rings of BC₃. According to their space groups and sizes of the BC₃ supercells, the four low-energy structures are designated as C222@(1 × 1), Pmna@($1\times \sqrt{3}$), Pma2@($1\times \sqrt{3}$), and P-3m1@($\sqrt{3}\times \sqrt{3}$), as shown in figures 2(a)–(d), respectively. The binding energies between adatoms Tl and BC₃ in these structures are calculated by ${E}_{b}={E}_{{{\rm{BC}}}_{3}}+{E}_{{\rm{Tl}}}-{E}_{{{\rm{BC}}}_{3}{\rm{Tl}}}$, where ${E}_{{{\rm{BC}}}_{3}}$, E_Tl, and ${E}_{{{\rm{BC}}}_{3}{\rm{Tl}}}$ are the total energies of bare BC₃ sheet, isolated Tl atom, and adsorption system, respectively. The calculated binding energies E_b together with the structural parameters of these structures are listed in table 1. The values of E_b exceed the calculated cohesive energy E_c (∼1.95 eV) of the bulk Tl metal, which means that the clustering of adatoms Tl is completely suppressed.

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Table 1.  The calculated structural parameters (lattice vectors a and b, included angle γ between a and b, and thickness δ), binding energies (E_b) between adatoms Tl and BC₃ sheet, ratios between E_b and cohesive energy E_c (∼1.95 eV) of bulk Tl metal (E_b/E_c), nontrivial band gaps (E_g), and ${{\mathbb{Z}}}_{2}$ invariants for the Tl decorated BC₃ sheets (BC₃Tl) with the designations of C222@(1 × 1), Pmna@($1\times \sqrt{3}$), Pma2@($1\times \sqrt{3}$), and P-3m1@($\sqrt{3}\times \sqrt{3}$).

BC3Tl	a	b	γ	δ	Eb	Eb/Ec	Eg	${{\mathbb{Z}}}_{2}$
	(Å)	(Å)	(◦)	(Å)	(eV)		(meV)
C222@(1 × 1)	5.21	5.22	120.05	2.35	2.07	1.06	224	1
Pmna@($1\times \sqrt{3}$)	5.21	9.02	90	2.35	2.07	1.06	220	1
Pma2@($1\times \sqrt{3}$)	5.21	9.02	90	2.35	2.07	1.06	200	1
P-3m1@($\sqrt{3}\times \sqrt{3}$)	9.03	9.03	120	2.35	2.07	1.06	20	1

The four low-energy adsorption structures can be taken as stable monolayer BC₃Tl films with a fixed thickness of 2.35 Å, as listed in table 1. To verify whether these structures are more stable than other possible BC₃Tl films, we search more 2D BC₃Tl allotropes based on ab initio evolutionary algorithm. The searches are performed in the unit cells containing n units of BC₃Tl, where the integer n = 2, 3, 4, 5, 6, 7, and 8, which means that there are 10, 15, 20, 25, 30, 35, and 40 atoms, respectively, in the unit cells for each search. The initial structures are randomly generated using the plane group symmetry with a given initial thickness of 4.5 Å. In the searches with the unit cells containing 10, 20, and 30 atoms, the structures of C222@(1 × 1), Pmna/Pma2@($1\times \sqrt{3}$), and P-3m1@($\sqrt{3}\times \sqrt{3}$) are taken as their respective input seeds. Our search findings indicate that the structures with lower total enthalpies are found in the unit cells containing 10, 20, and 30 atoms. In all of the newly found BC₃Tl allotropes, only the four seed configurations of C222@(1 × 1), Pmna@($1\times \sqrt{3}$), Pma2@($1\times \sqrt{3}$), and P-3m1@($\sqrt{3}\times \sqrt{3}$) are the energetically stable structures, as shown in figures 2(e)–(g). Moreover, the phonon dispersion calculations demonstrate that all of these four structures locate at the local minimums in the energy landscapes due to the absence of any imaginary frequency in the phonon dispersion spectra, as shown in figure A1 in the appendix. These results imply that the four stable BC₃Tl adsorption structures may be synthesized directly on a suitable substrate in an epitaxial way.

The band structures excluding and including SOC for the four stable BC₃Tl structures based on the GGA-PBE calculations are plotted in the left and middle panels in figure 3. When the SOC is excluded, these structures are semimetals with Fermi level just at the Dirac cone. The atomic orbital projections on the bands exhibit that a fairly large part of the Dirac cone states are contributed by the p orbital electrons of the adatoms Tl, which hints strong SOC in these systems. As a result, when the SOC is included, the Dirac cones are prominently split, as shown in the middle panels in figure 3, and thus the large opened band gaps of 224, 220, and 200 meV are obtained for the C222@(1 × 1), Pmna@($1\times \sqrt{3}$), and Pma2@($1\times \sqrt{3}$) structures, respectively, as listed in table 1. However, the opened gap for the P-3m1@($\sqrt{3}\times \sqrt{3}$) structure is only 20 meV, which will be interpreted later. Due to three of the four stable BC₃Tl structures being noncentrosymmetric, the ${{\mathbb{Z}}}_{2}$ invariants of these structures are calculated by the evolution of Wannier charge centers during the time-reversal pumping process, which is a method independent of the presence of inversion symmetry [48, 49]. Meanwhile, the method suggested by Fu and Kane [50] is also used to check the ${{\mathbb{Z}}}_{2}$ of the centrosymmetric P-3m1@($\sqrt{3}\times \sqrt{3}$) structure, and consistent result is obtained. Our GGA-PBE calculations show that the ${{\mathbb{Z}}}_{2}$ of all the four stable BC₃Tl structures are +1, as listed in table 1, indicating that these structures are QSH insulators. Strikingly, for the C222@(1× 1), Pmna@($1\times \sqrt{3}$), and Pma2@($1\times \sqrt{3}$) structures, the nonzero ${{\mathbb{Z}}}_{2}$ combined with the large nontrivial gaps of 200–224 meV signifies that these structures are the 2D TIs that promise room-temperature dissipationless QSH states, similar to the functionalized germanene and stanene [13, 14].

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One of the prominent features of 2D TIs is the presence of the gapless edge state with spin-momentum locking. There are two methods to get the edge state spectrum. One is to compute the band structures of the nanoribbon systems directly, and the other is to calculate the edge Green's function for a semiinfinite system [51]. We use the later method and select the (01) edge to calculate the edge state spectrum. It is worth noting that the 2D Miller index of (01) mentioned here is based on the unit cells shown in figures 2(a)–(d). The GGA-PBE calculated edge state spectra for the four stable BC₃Tl structures are shown in the right panels in figure 3, in which the existence of helical Dirac edge states connecting the conduction and valence bands further indicates that these structures are indeed 2D TIs.

From ${{\mathbb{Z}}}_{2}$ invariants and edge states calculated by the GGA-PBE exchange-correlation functional, one can know that the topological properties of the four stable BC₃Tl structures remain unchanged with the changing of adsorption configurations, which is an important feature of the topological systems. However, the topological properties are not always remaining unchanged for every stable adsorption configuration. For instance, in adatoms decorated ($3n\times 3n$) graphene, since Dirac cone is folded back to the Γ point and adatoms decoration induces a coupling between two Dirac cones, a trivial band gap around the Dirac cone is opened, and thus the topological properties of the adsorption systems are destroyed [52]. This is the so-called intervalley scattering mentioned in literatures. A typical instance for the BC₃ sheet is the ($\sqrt{3}\times \sqrt{3}$) supercell. Due to the fold of BZ, there exists a pair of Dirac cones at the Γ point in the bare ($\sqrt{3}\times \sqrt{3}$) BC₃ sheet. After the decoration of six Tl atoms, the stable P-3m1@($\sqrt{3}\times \sqrt{3}$) BC₃Tl structure is gained, in which an accidental degeneracy between the valence and conduction bands occurs at the Γ point based on the GGA-PBE calculations, and the dispersion around the Dirac cone is no longer linear but become parabolic, signifying a strong intervalley scattering in this system, as shown in figure 3(d). When the SOC is included, a nontrivial gap of 20 meV is opened and thus the topological electronic properties are not destroyed by the intervalley scattering based on the GGA-PBE results. To further check the topological properties, the SCAN functional of meta-GGA [53], which is superior to most gradient corrected functionals [54], is used to recalculate the band structures and ${{\mathbb{Z}}}_{2}$ invariants. The meta-GGA-SCAN calculations provide the same topological properties as those of the GGA-PBE results for the C222@(1× 1), Pmna@($1\times \sqrt{3}$), and Pma2@($1\times \sqrt{3}$) BC₃Tl structures, as shown in figures A2(a)–(c) in the appendix. However, for the P-3m1@($\sqrt{3}\times \sqrt{3}$) structure, there is no degeneracy between the valence and conduction bands but a band gap of 39 meV around the Fermi level in the band structure without SOC based on the meta-GGA-SCAN calculations, as shown in figure A2(d). As a result, when the SOC is included, no band inversion occurs, the band gap only reduces to 13 meV, and consequently the calculated ${{\mathbb{Z}}}_{2}$ invariant is 0, which demonstrates a destruction of the topological electronic properties by the intervalley scattering.

To break the TRS for seeking possible QAH state, the 3d–5d transition metal atoms are used to be adsorbed on BC₃ sheet. By traversing all of the 3d–5d transition metal atoms on the (1 × 1), ($\sqrt{3}\times \sqrt{3}$), and (2 × 2) BC₃ supercells, we find that the (2 × 2) BC₃ with one adatom of technetium (Tc), rhenium (Re), or ruthenium (Ru) and the ($\sqrt{3}\times \sqrt{3}$) BC₃ with one adatom of osmium (Os) are possible structures for realizing QAH effects. In these four adsorption structures, the transition metal adatoms are apt to be adsorbed on the hollow site of the C₆ rings of BC₃. The structural parameters and binding energies E_b for the four structures are listed in table 2, in which the cohesive energies E_c of the bulk counterparts of adatoms are also shown. Evidently, the E_b values are smaller than those of the E_c, although they are much higher than the E_b between Tl and BC₃. To verify whether the adatoms tend to cluster or not, we use a (4 × 2) BC₃ to adsorb two Tc, Re, or Ru atoms and a ($2\sqrt{3}\times \sqrt{3}$) BC₃ to adsorb two Os atoms on the same side. Via calculating the total energies of the adsorption configurations with different adatom–adatom distances, we find that the transition metal adatoms tend to be adsorbed on BC₃ discretely and thus the clustering of adatoms is effectively suppressed, which can be interpreted by the strong Coulomb repulsion between adatoms due to the electron transfer from adatoms to BC₃ sheet. In addition, there is no imaginary frequency in the phonon dispersion spectra of these four adsorption structures, and thus these structures locate at the local minimums in the energy landscapes, as shown in figure A3 in the appendix.

Table 2.  The calculated structural parameters (lattice vectors a and adsorption height h), binding energies (E_b) between adatoms and BC₃ sheet, cohesive energies (E_c) of adatoms, ratios between E_b and E_c (E_b/E_c), energy differences (ΔE) of ferromagnetic state relative to antiferromagnetic state, ferromagnetic moments (μ), energies (E_MCA), nontrivial band gaps (E_g), and Chern numbers (C) for the (2 × 2) BC₃ with one adatom of Tc, Re, or Ru and the ($\sqrt{3}\times \sqrt{3}$) BC₃ with one adatom of Os.

	a	h	Eb	Ec	Eb/Ec	Δ E	μ	EMCA	Eg	C
	(Å)	(Å)	(eV)	(eV)		(meV)	(μB)	(meV)	(meV)
Tc-(2 × 2)BC3	10.34	1.22	3.73	6.94	0.54	−24	1	−1.1	38	1
Re-(2 × 2)BC3	10.35	1.21	2.72	7.83	0.35	−27	1	−5.8	50	1
Ru-(2 × 2)BC3	10.35	1.32	3.67	7.03	0.52	−70	2	2.9	47	−1
Os-($\sqrt{3}\times \sqrt{3}$)BC3	8.97	1.36	3.10	8.27	0.37	−132	2	9.8	70	−2

The ground states of the four stable adsorption structure are found to be ferromagnetic with the magnetic moments of 1 μ_B for the (2 × 2) BC₃ with one adatom of Tc or Re and 2 μ_B for the (2 × 2) BC₃ with one adatom of Ru and the ($\sqrt{3}\times \sqrt{3}$) BC₃ with one adatom of Os, which are about 24–132 meV lower than their corresponding antiferromagnetic states, as listed in table 2. For the (2 × 2) BC₃ with one adatom of Tc or Re, the GGA-PBE band structures excluding SOC for the ferromagnetic states are plotted in the left panels in figure 4. Owing to internal magnetization, the majority-spin (red line) and minority-spin (blue line) bands are split away from each other, and thus only the majority-spin Dirac cone bands are left around the Fermi level, giving rise to the formation of the half semimetals. Similar minority-spin Dirac cone bands are about 0.3 eV above the Fermi level. The spin densities (${\rho }_{\uparrow }-{\rho }_{\downarrow }$) shown in the insets exhibit that the ferromagnetic moments are mainly distributed around the adatoms. The atomic orbital projections on the bands demonstrate that a large part of the Dirac cone states around the Fermi level are contributed by the d orbital electrons of the adatoms, implying strong SOC in these adsorption systems. Therefore, when the SOC is included, the large nontrivial band gap of about 38 (58) meV is opened at the K point for the (2 × 2) BC₃ with one adatom of Tc (Re), as shown in the right panels in figure 4. In addition, for the (2 × 2) BC₃ adsorbed with one Re, due to the drastic up-move of conduction band at the K point, the conduction band minimum shifts to the M point, which results in an indirect band gap of about 50 meV in this system, as shown in the right panel in figure 4(b).

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The GGA-PBE band structures excluding SOC and including SOC for the (2 × 2) BC₃ with one adatom of Ru and the ($\sqrt{3}\times \sqrt{3}$) BC₃ with one adatom of Os are plotted in figures 5(a) and (b), respectively. Different from the bands of the (2 × 2) BC₃ adsorbed with one Tc or Re, the Dirac cone states at the K point in the band excluding SOC for the (2 × 2) BC₃ with one adatom of Ru are far away from Fermi level. However, a majority-spin semimetal state at the Γ point is present around the Fermi level, which is mainly provided by the d orbital electrons of the Ru adatom. Thus, when the SOC is included, a nontrivial band gap of about 47 meV is opened in this system, as shown in the right panel in figure 5(a). For the ($\sqrt{3}\times \sqrt{3}$) BC₃ adsorbed with one Os, a majority-spin degeneracy point exists at the K point nearby 7 meV below Fermi level in the band excluding SOC based on the GGA-PBE results. Meanwhile, the valence band along the MK line and the conduction band around the K point are both crossed some by Fermi level, which makes the system be an ordinary halfmetal. Fortunately, because of the strong SOC of Os adatom, the band structure including SOC gives an insulating state with the conduction band minimum at the M point, the valence band maximum at the middle point of the MK line, and a nontrivial gap of about 70 meV, as shown in the right panel in figure 5(b).

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To identify the topological properties of the SOC induced insulating states, we calculate the Berry curvatures and Chern numbers for the four stable adsorption structures. The Berry curvatures Ω(k) are calculated using the Kubo formula [55, 56]

$$\begin{eqnarray}{\rm{\Omega }}({\bf{k}}) & = & \displaystyle \sum _{n}{f}_{n}{{\rm{\Omega }}}_{n}({\bf{k}}),\\ {{\rm{\Omega }}}_{n}({\bf{k}}) & = & -\displaystyle \sum _{m\ne n}2{\rm{Im}}\displaystyle \frac{\langle {\psi }_{n{\bf{k}}}| {v}_{x}| {\psi }_{m{\bf{k}}}\rangle \langle {\psi }_{m{\bf{k}}}| {v}_{y}| {\psi }_{n{\bf{k}}}\rangle }{{({\varepsilon }_{m{\bf{k}}}-{\varepsilon }_{n{\bf{k}}})}^{2}},\end{eqnarray} \tag{ 1 }$$

where the summation is over all occupied states, ε_nk and ψ_nk are the eigenvalue and Bloch function of band n at wave vector k, f_n is the Fermi–Dirac distribution function of band n, and v_x and v_y are the velocity operators. Through the gauge transfer between Bloch functions and real space Wannier functions, the Berry curvatures Ω(k) are computed directly from the formula (32) of [57]. Via integration of the Ω(k) over the first BZ, the Chern numbers (C) and anomalous Hall conductivities (σ_xy) are calculated by the formulas $C=\tfrac{1}{2\pi }{\int }_{{\rm{BZ}}}{{\rm{d}}}^{2}k{\rm{\Omega }}$ and ${\sigma }_{{xy}}=C\tfrac{{e}^{2}}{h}$.

The GGA-PBE results of Berry curvatures Ω(k) (in au) for the whole valence bands along high symmetry lines for the (2 × 2) BC₃ with one adatom of Tc, Re, or Ru and the ($\sqrt{3}\times \sqrt{3}$) BC₃ with one adatom of Os are plotted by the green solid circle lines in figures 4 and 5, which exhibits that the most nonzero Berry curvatures Ω(k) only occur nearby the SOC induced nontrivial gaps. The corresponding distributions of the Ω(k) in the momentum spaces are shown in the left panels in figure 6. For the (2 × 2) BC₃ adsorbed with one Tc or Re, the nonzero Ω(k) are localized around the K and K' points with the same sign, while the nonzero Ω(k) for the (2 × 2) BC₃ adsorbed with one Ru are located around the Γ point, as shown in the left panels in figures 6(a)–(c). For the ($\sqrt{3}\times \sqrt{3}$) BC₃ adsorbed with one Os, there exists conspicuous distribution of Ω(k) around the whole KK' line with the maximum value nearby the K and K' points based on the GGA-PBE calculations, as shown in the left panel in figure 6(d). By integrating the Ω(k) over the first BZ, the anomalous Hall conductivities σ_xy as a function of the energy around the Fermi level are obtained for the four adsorption structures, as shown in the right panels in figure 6. One can find that the quantized Hall conductance platforms appear around the Fermi level and the widths of the platforms for the four adsorption structures accord completely with their global band gaps obtained from the GGA-PBE first-principles calculations. The Chern number of all occupied bands for the (2 × 2) BC₃ with one adatom of Tc or Re is the integer value of 1, and the Chern numbers for the (2 × 2) BC₃ with one adatom of Ru and the ($\sqrt{3}\times \sqrt{3}$) BC₃ with one adatom of Os are −1 and −2, respectively, as listed in table 2. These nonzero Chern numbers indicate that the four adsorption structures are topologically nontrivial and thus two chiral edge channels appear on each side of their samples. In addition, for further verification of the topological properties, the SCAN functional of meta-GGA is also used to recalculate the band structures and Chern numbers of the four adsorption structures. The meta-GGA-SCAN calculations provide the same topological properties as those of the GGA-PBE results for the three adsorption structures of the (2 × 2) BC₃ with one adatom of Tc, Re, or Ru, as shown in figures A4(a)–(c) in the appendix. But, for the ($\sqrt{3}\times \sqrt{3}$) BC₃ with one adatom of Os, the band structure excluding SOC have a band gap of 268 meV around the Fermi level based on the meta-GGA-SCAN results, which is in sharp contrast to the halfmetal state obtained from the GGA-PBE calculations, as shown in figure A4(d). Thus, when the SOC is included, there is no band inversion occurring, the band gap only becomes 172 meV, and consequently the calculated Chern number is 0, which can also be explained by the strong intervalley scattering in the ($\sqrt{3}\times \sqrt{3}$) BC₃ supercell. These results means that the realization of the robust QAH effects are only feasible in the systems of the (2 × 2) BC₃ with one adatom of Tc, Re, or Ru.

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In order to confirm the easy magnetization orientation, we also calculate the magnetocrystalline anisotropy of the four adsorption structures, which is the total energy difference between two magnetic states where the magnetization is aligned along the [100] or [001] direction, i.e., ${E}_{{\rm{MCA}}}={E}_{[100]}-{E}_{[001]}$. Our calculations indicate that the energy of the out-of-plane spin orientation for the (2 × 2) BC₃ adsorbed with one Tc or Re is about 1.1 or 5.8 meV lower than that of the in-plane spin orientation, while the ground states of the (2 × 2) BC₃ with one Ru adatom and the ($\sqrt{3}\times \sqrt{3}$) BC₃ with one Os adatom have the in-plane spin orientations, as listed in table 2. Thus, the (2 × 2) BC₃ adsorbed with one Tc or Re are the suitable systems to observe the QAH states in experiments. Furthermore, due to the existence of the largest ratio of E_b/E_c in the four adsorption structures, the (2 × 2) BC₃ adsorbed with one Tc is the best system to realize the QAH effect. In addition, it is known that the QAH effects can be effectively manipulated through changing of the easy magnetization axis by external electric fields [22]. For the (2× 2) BC₃ adsorbed with one Ru, a suitable external electric field is suggested to tune the QAH effects for possible experimental observation.