Dirac fermions in borophene

TL;DR

This paper demonstrates that monolayer boron, specifically the eta 12 boron sheet, hosts Dirac fermions similar to graphene, confirmed by experiments and calculations, opening new avenues for high-speed electronic devices.

Contribution

It reveals that the eta 12 boron sheet can host Dirac cones due to its lattice structure, confirmed by experimental and theoretical methods, establishing borophene as a new Dirac material platform.

Findings

The eta 12 boron sheet hosts Dirac cones similar to graphene.

Periodic perturbations can split the Dirac cones.

Experimental and first-principles calculations confirm the electronic structure.

Abstract

Honeycomb structures of group IV elements can host massless Dirac fermions with non-trivial Berry phases. Their potential for electronic applications has attracted great interest and spurred a broad search for new Dirac materials especially in monolayer structures. We present a detailed investigation of the \beta 12 boron sheet, which is a borophene structure that can form spontaneously on a Ag(111) surface. Our tight-binding analysis revealed that the lattice of the \beta 12-sheet could be decomposed into two triangular sublattices in a way similar to that for a honeycomb lattice, thereby hosting Dirac cones. Furthermore, each Dirac cone could be split by introducing periodic perturbations representing overlayer-substrate interactions. These unusual electronic structures were confirmed by angle-resolved photoemission spectroscopy and validated by first-principles calculations. Our results suggest monolayer boron as a new platform for realizing novel high-speed low-dissipation devices.

Tables

Table 1. Table 1: The transfer of the MLWF effective Hamiltonian shown in a matrix form. Those having magnitude larger than 0.1 eV are shown. Those for the nearest neighbor site, t i ​ j subscript 𝑡 𝑖 𝑗 t_{ij} , are shown in the upper triangular part and others, t ~ i ​ j subscript ~ 𝑡 𝑖 𝑗 \tilde{t}_{ij} , are shown in the lower triangular part. Two values are assinged to t ~ a ​ c subscript ~ 𝑡 𝑎 𝑐 \tilde{t}_{ac} and t ~ c ​ c subscript ~ 𝑡 𝑐 𝑐 \tilde{t}_{cc} because there are two inequivalent pairs: The former value corresponds to those pairs with common x 𝑥 x -coordinate and the latter to those with different x 𝑥 x -coordinate.

$\tilde{t}\backslash t$	$a$	$b$	$c$	$d$	$e$
$a$	$0.500\text{ eV}$	$-2.04\text{ eV}$	$-1.79\text{ eV}$	$-$	$-2.12\text{ eV}$
$b$	$-$	$0.532\text{ eV}$	$-1.84\text{ eV}$	$-1.91\text{ eV}$	$-$
$c$	$0.308,0.389\text{ eV}$	$-0.130\text{ eV}$	$0.310,-0.126\text{ eV}$	$=t_{bc}$	$=t_{ac}$
$d$	$0.421\text{ eV}$	$-$	$={\tilde{t}}_{bc}$	$={\tilde{t}}_{bb}$	$=t_{ab}$
$e$	$-0.143\text{ eV}$	$0.421\text{ eV}$	$={\tilde{t}}_{ac}$	$-$	$={\tilde{t}}_{aa}$

Equations

$$\mathcal{H}=\sum _i{\epsilon }_i{c}_i^{†}{c}_i+\sum _{ij}^{\text{n.n.}}{t}_{ij}{c}_i^{†}{c}_j+\sum _{ij}^{\ne \text{n.n.}}{\tilde{t}}_{ij}{c}_i^{†}{c}_j,$$

$${\epsilon }_a={\epsilon }_d=0.196\text{ eV, }{\epsilon }_b={\epsilon }_e=-0.058\text{ eV, }{\epsilon }_c=-0.845\text{ eV.}$$

$$\mathcal{H}=-t_0\sum _{ij}^{\text{n.n.}}{c}_i^{†}{c}_j,$$

$$\varphi_{T_{1}}^{\alpha}\left(x+ma_{x},y+na_{y}\right)=\left\{\begin{array}[c]{c}\omega^{m}\left(\phi_{a}\left(x,y\right)+\omega^{2}\phi_{d}\left(x,y\right)\right)\text{ when }\alpha=1\\ \omega^{m}\left(\phi_{e}\left(x,y\right)+\omega^{2}\phi_{b}\left(x,y\right)\right)\text{ when }\alpha=2\end{array}\right.$$

$$\varphi_{\bar{T}_{1}}^{\alpha}\left(x+ma_{x},y+na_{y}\right)=\left\{\begin{array}[c]{c}\omega^{-m}\left(\phi_{a}\left(x,y\right)+\omega^{-2}\phi_{d}\left(x,y\right)\right)\text{ when }\alpha=1\\ \omega^{-m}\left(\phi_{e}\left(x,y\right)+\omega^{-2}\phi_{b}\left(x,y\right)\right)\text{ when }\alpha=2\end{array}\right.,$$

$$V_{n_x,n_y}\left(x,y\right)=V_0\cos \left(2\pi n_x\frac{x}{L_x}\right)\cos \left(2\pi n_y\frac{y}{L_y}\right),$$

$$q_y\simeq 2\frac{V_0}{t_0}+0.0628{\left(\frac{V_0}{t_0}\right)}^3,$$

$$q_x\simeq 1.23\frac{V_0}{t_0}$$

$$\left(\begin{array}[c]{c|cccc}&\varphi_{T_{1}}^{1}&\varphi_{T_{1}}^{2}&\varphi_{\bar{T}_{1}}^{1}&\varphi_{\bar{T}_{1}}^{2}\\ \hline\cr\varphi_{T_{1}}^{1}&A_{11}&A_{12}&C_{11}&C_{12}\\ \varphi_{T_{1}}^{2}&A_{21}&A_{22}&C_{21}&C_{22}\\ \varphi_{\bar{T}_{1}}^{1}&C_{11}^{\ast}&C_{21}^{\ast}&B_{11}&B_{12}\\ \varphi_{\bar{T}_{1}}^{2}&C_{121}^{\ast}&C_{22}^{\ast}&B_{21}&B_{22}\end{array}\right),$$

$$C=t_{0}\left(\begin{array}[c]{cc}1-3e^{-iq_{y}}-4\cos\left(q_{x}+\frac{2\pi}{3}\right)&-8\omega\sin^{2}\left(\frac{1}{2}q_{x}\right)\\ 8\omega^{2}\sin^{2}\left(\frac{1}{2}q_{x}\right)&1-3e^{-iq_{y}}-4\cos\left(q_{x}-\frac{2\pi}{3}\right)\end{array}\right),$$

$$\begin{array}[c]{c|c}\text{case}&\left\langle\varphi\right|\mathcal{H+}V_{n_{x},n_{y}}\left|\varphi\right\rangle\\ \hline\cr\mathrm{i}&\left(\begin{array}[c]{cc}6V_{0}J&C\\ C^{\dagger}&-3V_{0}J\end{array}\right)\\ \mathrm{ii}&\left(\begin{array}[c]{cc}6V_{0}J&C\\ C^{\dagger}&6V_{0}J\end{array}\right)\\ \mathrm{iii}&\left(\begin{array}[c]{cc}12V_{0}I&C\\ C^{\dagger}&-6V_{0}J\end{array}\right)\\ \mathrm{iv}&\left(\begin{array}[c]{cc}12V_{0}I&C\\ C^{\dagger}&12V_{0}I\end{array}\right)\end{array}.$$

$$\varepsilon=\left\{\begin{array}[c]{l}6\left(-V_{0}-t_{0}\sin\frac{q_{y}}{2}\right)\\ 6\left(-V_{0}+t_{0}\sin\frac{q_{y}}{2}\right)\\ 6\left(V_{0}-t_{0}\sin\frac{q_{y}}{2}\right)\\ 6\left(V_{0}+t_{0}\sin\frac{q_{y}}{2}\right)\end{array}\right.$$

$$\varphi=\left\{\begin{array}[c]{l}ie^{-q_{y}/2}\left(\varphi_{K_{1}}^{1}-\varphi_{K_{1}}^{2}\right)+\left(-\varphi_{\bar{K}_{1}}^{1}+\varphi_{\bar{K}_{1}}^{2}\right)\\ ie^{-q_{y}/2}\left(-\varphi_{K_{1}}^{1}+\varphi_{K_{1}}^{2}\right)+\left(-\varphi_{\bar{K}_{1}}^{1}+\varphi_{\bar{K}_{1}}^{2}\right)\\ ie^{-q_{y}/2}\left(-\varphi_{K_{1}}^{1}-\varphi_{K_{1}}^{2}\right)+\left(\varphi_{\bar{K}_{1}}^{1}+\varphi_{\bar{K}_{1}}^{2}\right)\\ ie^{-q_{y}/2}\left(\varphi_{K_{1}}^{1}+\varphi_{K_{1}}^{2}\right)+\left(\varphi_{\bar{K}_{1}}^{1}+\varphi_{\bar{K}_{1}}^{2}\right)\end{array}\right..$$

Full text

Nontrivial lattice structures of solids involving more than one atom per lattice site can host novel properties and behaviors; hence, the discovery and design of new structure-property combinations are at the forefront of materials science. A celebrated, but particularly simple, example is the two-dimensional honeycomb lattice with just two atoms per unit cell (Fig. 1a), such as grapheneGeim2007 ; Neto2009 , siliceneHoussa2015 ; Zhuang2015 ; Zhao2016 , germaneneLiLF2014 ; Derivaz2015 and staneneZhu2015 . These materials can host Dirac cones that give rise to rich physical propertiesNeto2009 ; Feng2013 ; Ezawa2012 ; XuY2013 . Recent theoretical investigations for new Dirac materials in simple two-dimensional structures have attracted great attentionMalko2012 ; ZhangLZ2015 ; WangJ2015 ; Miert2016 ; ZhouXF2014 , whereas experimental observations of Dirac cones beyond the honeycomb structure are still rare.

A promising route to realize novel two-dimensional materials is by tailoring or modifying the honeycomb lattice. An example is a monolayer boron sheet (i.e., borophene), which is realized by introducing periodic boron atoms in a honeycomb-like lattice. As boron has one less electron than carbon, the honeycomb structure is unstable but the introduction of additional boron atoms in the honeycomb lattice can stabilize the structures by balancing out the two- and multi-center bondsEvans2005 ; Tang2010 ; Tang2007 . Depending on the arrangements of the extra boron atoms, various monolayer-boron structures have been proposed, such as the $\alpha$-sheet, $\beta$-sheet, etcTang2010 ; Tang2007 ; WuX2012 ; Penev2012 ; LiuH2013 . Recently, several monolayer boron phases have been experimentally realized on Ag(111)Mannix2015 ; Feng2016 ; Zhang2016 ; Feng2016' . For example, Mannix et al. reported a stable striped phase and a metastable homogeneous phaseMannix2015 . The striped phase was proposed to be a complete triangular lattice with anisotropic, out-of-plane buckling. In another study, a similar striped phase with a different rotation angle has been observedFeng2016 . This phase, named a $\beta_{12}$-sheet (Fig. 1b), has an essentially flat structure and interacts weakly with the Ag(111) substrateFeng2016 ; Feng2016' ; Zhang2015 ; LiuY2013 . However, the experimental investigations on the electronic properties of monolayer boron are still rare.

In this letter, we present a combined theoretical and experimental investigations on the $\beta_{12}$ boron sheet. Our tight-binding analysis reveals that the lattice $\beta_{12}$-sheet can be decomposed into two triangular sublattices, analogous to the honeycomb lattice, and thus hosts Dirac cones. Moreover, each Dirac cone can be split by introducing periodic perturbations representing the moiré pattern observed by a scanning tunneling microscope. These intriguing electronic structures have been confirmed by angle-resolved photoemission spectroscopy (ARPES) measurements and first-principles calculations. Our results have experimentally confirmed the first monolayer Dirac materials beyond the honeycomb structure and have validated a novel approach to split the Dirac cones by periodic perturbations. Moreover, these results suggest monolayer boron as a promising material to realize high-speed, low-dissipation nanodevices.

In graphene, the $\pi$ bands near the Fermi level (EF) derived from the pz orbital form the Dirac cones at the K points (Fig. 1a)Neto2009 . The s, px and py orbitals are sp2 hybridized and contribute to the $\sigma$ bands which are far from EF. The $\beta_{12}$-sheet is also atomically flat, as is graphene, and, as confirmed later by our experiments and first-principles calculations, the bands near EF are also derived from the pz orbital. Interestingly, a simple tight-binding (TB) model, considering only the pz orbital for a freestanding $\beta_{12}$-sheet, shows the existence of Dirac cones centered at ($\pm 2\pi/3a$, [math]) in the first Brillouin zone (BZ), as illustrated in Fig. 1b.

For a detailed understanding of the electronic structure of the system, we present the wave function for each boron atom in Fig. 2a. Our TB analysis showed that the wave function at EF has a vanishing amplitude at site c, owing to phase cancellation at the six-fold coordinated boron atoms. The wave function originates instead from the atoms at sites a, b, d and e, which can be decomposed into two sublattices (Figs. 2a and 2b). As a result, the equivalent structure of the $\beta_{12}$-sheet is a honeycomb lattice, as shown in Fig. 2c. As with graphene, this honeycomb lattice gives rise to a Dirac cone at each $\rm\overline{K}$ point of the BZ. These Dirac cones are folded to ($\pm 2\pi/3a$, [math]), as illustrated in Fig. 2d, because the B atoms at site c alter the shape and size of the BZ. The band structure from our TB calculations is shown in Fig. 2e, where the two Dirac cones in the $\Gamma$-X axis are indicated by black arrows. For further confirmation, we also performed first-principles calculations for the freestanding $\beta_{12}$-sheet, and the Dirac cones at ($\pm 2\pi/3a$, [math]) were reproduced (Fig. 2f). The Dirac points were located at approximately 2 eV above the Fermi level, in agreement with previous reportsPenev2016 . The upward shift of the Dirac cones might originate from the electron deficiency of boron. It should be noted that the energy position of the Dirac points can be varied after being placed on a metal substrate to compensate for the electron deficiency (see Supplementary Materials for details). Similar Dirac cone states in a rectangular lattice have also recently been proposed in a graphene superlatticeJia2016 .

When the $\beta_{12}$-sheet is placed on a Ag(111) substrate, a long-range modulation arising from the lattice mismatch gives rise to a moiré pattern, as shown in Fig. S4. As the interaction of the boron layer and Ag(111) substrate is weak, the $\beta_{12}$-sheet remains largely intact and the moiré pattern can be explained by a modulated charge distribution on the surfaceFeng2016 . The long-range modulation yields an electronic perturbation; in our TB model, we simulate this effect by varying the on-site energy over a superlattice period of nax$\times$may, where ax$\times$ay is the original unit cell (Fig. 1b). The Dirac cones of the superlattice are folded onto the $\Gamma$ point when n is a multiple of three, and are further split into pairs in the $\Gamma$-Y direction when the sublattice symmetry is broken while retaining the inversion symmetry (Figs. 1c and 2g). The Dirac cones will split in the $\Gamma$-X direction when the inversion symmetry is also broken (see the Supplementary Materials for details). The splitting of the Dirac cones has also been confirmed by our first-principles calculations considering the periodic perturbation (Fig. S2b). From Fig. 2g, the split Dirac cones are non-concentric which is different from the Rashba-type splitting of the Dirac cones in grapheneVarykhalov2008 ; Marchenko2012 .

To confirm these intriguing properties of the $\beta_{12}$-sheet, we have performed high-resolution angle-resolved photoemission spectroscopy (ARPES) to directly measure its band structure. The sample was prepared by evaporating pure boron onto a Ag(111) substrate (see Supplementary Materials for details). From LEED measurements (Fig. S4a), we found that there is only one phase, the $\beta_{12}$-sheet. As the $\beta_{12}$-sheet has a rectangular structure, different from the hexagonal structure of Ag(111), there exist domains with three equivalent orientations related by 120*∘* rotations. A schematic drawing of the BZ of Ag(111) with the three domain orientations is shown in Fig. 3a, together with the measured Fermi surface. Because the coverage of boron was less than one monolayer in the experiments, there were some areas of bare Ag(111) surface. As a result, the Shockley surface state and bulk sp band of Ag(111) were clearly observed, as indicated by “SS” and “sp” in Fig. 3a. The band structure from the boron layer shows one Fermi pocket centered at the $\rm\overline{S}$ point of the $\beta_{12}$-sheet and a pair of Fermi pockets centered at the $\rm\overline{M}$ point of Ag(111), as indicated by the red and black arrows, respectively. The bands derived from the boron layer do not disperse with an increasing photon energy (Fig. S5), which is in agreement with its two-dimensional characteristic.

The pair of Fermi pockets centered at the $\rm\overline{M}$ point of Ag(111) is associated with Dirac cones, in agreement with the general picture based on our calculations. In Fig. 3b, we show constant energy contours (CECs) at different binding energies (EB). With increasing binding energies, the Fermi pockets first shrink in size and then become points at EB = -0.25 eV. Further increase of the binding energy leads to a pair of closed contours which touch each other at EB = -0.68 eV. The pair of closed contours merge into one contour at higher binding energies.

The band structure measured along typical cuts in the momentum space (the purple lines in Fig. 3a) is shown in Figs. 3c-3f. The measurements of cut 1 using p polarized light (Fig. 3d) reveal a Dirac cone as well as the bulk sp band of Ag(111). The Dirac point is located at approximately 0.25 eV below the Fermi level, in agreement with the evolution of the CECs in Fig. 3b. The linear dispersing bands extend to as deep as 2 eV. Within our experimental resolution, there is no obvious energy gap at the Dirac point; thus the quasiparticles are massless Dirac fermions. The Fermi velocities determined from Fig. 3d are approximately 6.1 eV$\cdot$Å and 7.0 eV$\cdot$Å for the left and right branches of the Dirac cone, respectively, which are close to the Fermi velocity of graphene ($\sim$6.6 eV$\cdot$Å). The slight difference of the Fermi velocity between the two branches originates from the anisotropy of the Dirac cones, in agreement with the CECs in Figs. 3(b) and S5. The neighboring bulk sp band of Ag(111) is clearly separated from the Dirac cone with no signs of hybridization, which indicates a weak interaction between the $\beta_{12}$-sheet and the Ag(111) substrate, in agreement with previous workZhang2015 . In Fig. 3e, we show the band structure along the $\rm\overline{K}$-$\rm\overline{M}$-$\rm\overline{K}$ direction; a pair of Dirac cones can be identified (indicated by the yellow dashed lines) although the one on the right side is only half-visible because of the limitation of our experimental configuration. The two cones touch each other at the $\rm\overline{M}$ point of Ag(111) at a binding energy of approximately 0.68 eV, which agrees with the evolution of the CECs discussed above. The band structure at the $\rm\overline{M}$ point of Ag(111) shows a “V” shape along the $\rm\overline{K}$-$\rm\overline{M}$-$\rm\overline{K}$ direction (Fig. 3e) and a “$\Lambda$” shape along the $\overline{\Gamma}$-$\overline{\rm M}$ direction (Fig. 3f); the bottom of the “V” and the top of the “$\Lambda$” are located at the same binding energy ($\sim$ 0.68 eV), which suggests a “saddle” point at the $\overline{\rm M}$ point of Ag(111). Within the first BZ of the $\beta_{12}$-sheet, we observed two pairs of Dirac cones in total, as schematically illustrated in Fig. 3g.

The orbital contribution of the boron bands can be probed by switching the linear polarization of the incident light. The s polarized light primarily probes the in-plane px and py orbitals, while the p polarized light probes both the in-plane (px and py) and out-of-plane (pz) orbitals. The band structures along cut 1 measured with s and p polarized light are shown in Figs. 3c and 3d, respectively. The Dirac cone was not observed with the s polarized light, leaving only the bulk sp bands of Ag(111). This means that the Dirac cones originate from the pz orbital of boron. For further confirmation, we performed first-principles calculations for the B/Ag(111) system. The relaxed atomic structure shown in Fig. 3h agrees with previous workFeng2016 . The partial density of states (PDOS) of the boron and silver atoms is shown in Figs. 3i and 3j. Near EF, the density of states (DOS) is mainly derived from the pz orbital of boron; the contributions from the px and py orbitals of boron are essentially negligible (Fig. 3i). Likewise, the contributions from Ag orbitals are much smaller compared with those from the pz orbital of boron (Fig. 3j). We conclude that the Dirac cones are predominantly derived from the pz orbital of boron, with little hybridization with the Ag substrate states. This observation validates our TB analysis in terms of the boron pz orbital only.

Although the contributions from the Ag atoms is much smaller than the pz orbital of boron, there are still considerable contributions from the 5s, 4d${}_{z^{2}}$, 4dxz and 4dyz orbitals of Ag atoms over much of the valence band range. These orbitals have large out-of-plane components and could potentially hybridize with the pz orbital of boron, which can explain the origin of the weak interaction between the $\beta_{12}$-sheet and the Ag(111) substrate. This interaction might energetically shift the bands of the free-standing $\beta_{12}$-sheet, moving the Dirac points below the Fermi level. Another important consequence of this interaction is the appearance of the moiré pattern. From Fig. S4b, the period of the moiré pattern is approximately 5.5ax, which is approximately two times the period of the perturbation in our TB model (Fig. 1c). This observation validates our qualitative explanation for the splitting of the Dirac cones. Alternatively, the splitting of the Dirac cones could be interpreted in terms of a uniaxial strain in the $\beta_{12}$-sheet associated with the moiré pattern. The strain in the lattice could break the equivalence of bonds, inducing a splitting of the $\pi$ bandsJia2016 . The net results would be similar to those caused by a modulation of the on-site energy. On the other hand, owing to the existence of the moiré pattern, the pair of Dirac cones centered at ($\pm 2\pi/3a$, [math]) of the $\beta_{12}$-sheet are folded to the $\rm\overline{M}$ point of Ag(111), in agreement with our experiments. As a further test of our explanation, first-principles calculations of B/Ag(111) also reveal the same pair of Dirac cones, as shown in Fig. 3k. The calculated Fermi velocity is approximately 3.5 eV$\cdot$Å, which is in the same order of magnitude as the experimental value. The difference between the theoretical and experimental results might originate from the many-body interactions, which have already been extensively studied in grapheneElias2011 ; Hwang2012 .

All of our results support or confirm the existence of gapless Dirac cones in the $\beta_{12}$ boron sheet grown on Ag(111). These Dirac cones are split into pairs owing to the interaction of the boron layer with the substrate. An important implication of our analysis and discussion of the underlying physics is that Dirac cone features can arise in lattices with large unit cells; such systems tend to exhibit multiple motifs and are conducive to atomic scale engineering of the structure. Our work suggests opportunities and strategies in connection with the realization of Dirac and possibly other exotic phases; it might also stimulate further investigation of the novel properties of monolayer boron, such as superconductivityPenev2016 , topological order, and high-speed electronic transport and switching.