Observation of Symmetry-Protected Dirac States in Nonsymmorphic α-Antimonene
Qiangsheng Lu, Kyle Y. Chen, Matthew Snyder, Jacob Cook, Duy Tung Nguyen, P. V. Sreenivasa Reddy, Tay-Rong Chang, Guang Bian
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Observation of Symmetry-Protected Dirac States inNonsymmorphic α -Antimonene Qiangsheng Lu, Kyle Y. Chen, Matthew Snyder, Jacob Cook, Duy Tung Nguyen, P. V. Sreenivasa Reddy, Tay-Rong Chang, and Guang Bian Department of Physics and Astronomy,University of Missouri, Columbia, Missouri 65211, USA Rock Bridge High School, Columbia, Missouri 65203, USA Department of Physics, National Cheng Kung University, Tainan 701, Taiwan (Dated: January 19, 2021)
Abstract
The discovery of graphene has stimulated enormous interest in two-dimensional (2D) electron gaswith linear band dispersion. However, to date, 2D Dirac semimetals are still very rare due to thefact that 2D Dirac states are generally fragile against perturbations such as spin-orbit couplings.Nonsymmorphic crystal symmetries can enforce the formation of Dirac nodes, providing a new routeto establishing symmetry-protected Dirac states in 2D materials. Here we report the symmetry-protected Dirac states in nonsymmorphic α -antimonene (Sb monolayer). The antimonene wassynthesized by the method of molecular beam epitaxy. 2D Dirac states with large anisotropy wereobserved by angle-resolved photoemission spectroscopy. The Dirac states in α -antimonene are spin-orbit coupled in contrast to the spinless Dirac states in graphene. The result extends the “graphene”physics into a new family of 2D materials where spin-orbit coupling is present. PACS numbers: 71.70.Ej, 73.20.At, 79.60.Dp, 73.21.Fg α -phase of antimonene (monolayer antimony). Bulk Sb is a group-V semimetalwith a small overlap between the valence and conduction bands. Despite small electron andhole pockets at the Fermi level, there exists an indirect negative energy gap traversing thewhole Brillouin zone and separating the valence and conduction bands [19]. This gappedband structure enables Sb, though semimetallic, to host the same Z topological invariants astopological insulators [3, 20]. The large spin-orbit coupling plays a key role in the generationof the nontrivial band topology. In the 2D limit, monolayer Sb, i.e. , antimonene, is knownto have two allotropic structural phases, namely, the black-phosphorus (BP)-like α -phaseand the hexagonal β -phase. The lattice of α -antimonene ( α -Sb for short) is nonsymmorphic,meaning that α -Sb can host symmetry-protected Dirac states. In this work, we synthesized α -Sb by the technique of molecular beam epitaxy (MBE) and detected the 2D Dirac statesby angle-resolved photoemission (ARPES) experiments. The results shed light on the searchof 2D Dirac materials in the vast territory of 2D nonsymmorphic crystals.In our experiment, α -Sb was grown on SnSe substrates under an ultrahigh vacuum envi-ronment. The SnSe crystals were cleaved in situ and provided an atomically flat surface forthe deposition of Sb. The crystallographic structure of α -Sb is shown in Figs 1(a) and 1(b).The surface unit cell is marked by a blue rectangular box. The in-plane lattice constants are4.49 Å and 4.30 Å in the x and y directions, respectively. The in-plane nearest-neighbor bondlength is 2.90 Å. α -Sb consists of two horizontal atomic sublayers. Each atomic sublayer isperfectly flat according to the first-principles lattice relaxations. For a single layer (1L) of α -Sb, the vertical spacing between the two atomic sublayers is 2.79 Å. For a two-layer (2L) α -Sb film, the vertical distance between the two atomic sublayers within each α -Sb layer is2.89 Å while the spacing between the two layers of α -Sb is 3.18 Å. The structure of one-and two-layer α -Sb belongs to the
42 layer group ( pman ). The lattice is nonsymmorphic,because it is invariant under a glide mirror reflection operation. The glide mirror is parallelto the x - y plane and lies in the middle between the two atomic sublayers of 1L α -Sb. Theglide mirror reflection is composed of a mirror reflection and an in-plane translation by (0.5 a ,0.5 b ), where a = 4 . Å and b = 4 . Å are the lattice constants in the x and y directions,respectively. For 2L α -Sb, the glide mirror sits in the middle between the two α -Sb layers.Figures 1(c-e) show the STM image of two α -Sb/SnSe samples with atomic resolution. Thefirst sample consists of mainly 1L α -Sb islands, see Fig. 1(c). A line-mode reconstruction(Moiré pattern) can be seen on the α -Sb surface, as marked by the green dashed lines. Theheight profile is taken along the blue arrow (shown in Fig. 1(d)) indicates that the height ofthe 1L α -Sb island on the SnSe surface is 6.5 Å. The second sample possesses both 1L and2L domains as shown in Fig. 1(e).The glide mirror symmetry of the lattice leads to band degeneracy at high-symmetrypoints ¯X = ( π, and ¯X = (0 , π ) . A detailed analysis of the location of Dirac points canbe found in the previous work [21]. We performed first-principles calculations for the bandstructure of 1L and 2L α -Sb films. The ABINIT package [22, 23] and a plane-wave basis setwere employed in the calculations. The energy cut is 400 eV. Relativistic pseudopotentialfunctions constructed by Hartwigsen, Goedecker, and Hutter (HGH) were used [24]. TheSOC of the system is varied from 0 to 300% by linearly scaling the relativistic parts of theHamiltonian [25]. The calculated band structures of 1L and 2L α -Sb are shown in Figs. 2(a)and 2(b). The Brillouin zone of α -Sb is plotted in Fig. 2(c). Both 1L and 2L α -Sb have asemiconducting behavior, which can be seen in the calculated density of states. There existband crossings at points ¯X and ¯X . The band degeneracy occurs for every band at thesetwo high-symmetry points. Each band splits into two branches as it disperses away from ¯X , . Therefore, the band crossings at ¯X , create 2D Dirac states. The Dirac points at ¯X and ¯X in the top valence band are marked by ‘D1’ and ‘D2’, respectively. The location ofDirac points is entirely determined by the underlying nonsymmorphic lattice symmetry.The Dirac bands at ¯X , can be described by an effective k · p model constructed aroundeach Dirac point. For D1 at ¯X , the matrix representations of the symmetry operations are T = − iσ y ⊗ τ K (time reversal), f M z = σ z ⊗ τ y (glide mirror reflection), P = σ ⊗ τ x (spaceinversion), and M x = − iσ x ⊗ τ x (mirror reflection with respect to a plane parallel to the y - z plane), where K is the complex conjugation operator, σ j and τ j ( j = x, y, z ) are the Paulimatrices representing spin and orbital degrees of freedom, respectively, σ and τ are the × identity matrices. These matrices of symmetry operators can be found in the standardreference [26]. Subjected to these symmetry constraints, the effective model in the vicinityof D1 expanded to linear order in the wave vector k ′ takes the form of H ( k ′ ) = v x k ′ x (cos θ σ x ⊗ τ z + sin θ σ ⊗ τ y ) + v y k ′ y σ y ⊗ τ z , (1)where the energy and the wave vector k ′ = ( k ′ x , k ′ y ) are measured from D1, the modelparameters v x and v y are Fermi velocity in the x and y directions, respectively, and θ isa real parameter that depends on the microscopic details. The dispersion around D1 is E = ± q v x k ′ x + v y k ′ y , which indeed corresponds to a linear spin-orbit coupled Dirac cone.According to the first-principles bands, v x = 2 . × m/s and v y = 2 . × m/s . TheDirac cone exhibits a minor anisotropy. The effective model for D2 at ¯X can be describedin a similar way. With T = − iσ y ⊗ τ K , f M z = σ z ⊗ τ y , P = σ ⊗ τ x , and M x = − iσ x ⊗ τ ,the effective Hamiltonian can be written as H ( k ′ ) = v x k ′ x σ y ⊗ τ z + v y k ′ y (cos θ σ x ⊗ τ z + sin θ σ ⊗ τ y ) , (2)where v x = 1 . × m/s and v y = 5 . × m/s , according to the first-principlescalculation. A huge anisotropy is found in the Dirac bands at D2. To find the SOC effecton the Dirac states, we calculated the the band structure with various strength of SOC andextract the Fermi velocity at D1 and D2, see Figs. 2(d) and 2(e). Without SOC, the twobands of the Dirac cone become degenerate in the direction of ¯X - ¯M - ¯X . This can be seenin Eqns. (1) and (2): the terms depending on the spin matrices σ x,y,z vanish in the absenceof SOC, leaving only one term with σ , which induces the band splitting in ¯X - ¯Γ and ¯X - ¯Γ directions. Consequently, the bands form a nodal line at the boundary of the Brillouin zone,since the bands are degenerate along ¯X - ¯M - ¯X , see Fig. 2d. In this sense, the nodal-lineband structure in the absence of SOC also arises from the nonsymmorphic symmetry of thelattice. However, this band degeneracy is not robust against spin-orbit coupling. Turningon SOC, the nodal line is gapped everywhere except ¯X and ¯X . The Fermi velocity v y atD1 and v x at D2 grows from zero as SOC increases, leading to the formation of Dirac cones.In other words, SOC transforms the system from a nodal-line system into a Dirac fermionstate. It is worth noting that v x at D2 remains highly suppressed even at artificially enlargedSOC, leading to an anisotropic Dirac cone at D2.The ARPES result taken from the 1L α -Sb sample is shown in Fig. 3. The photon energyis 21.2 eV. There are three prominent features on the Fermi surface, namely, one electronpocket at ¯Γ and two hole pockets near ¯X , see Fig. 3(a). The band spectrum taken alongthe ¯X - ¯Γ - ¯X and ¯X - ¯Γ - ¯X directions are plotted in Figs 3(b) and 3(c). The calculated banddispersion is overlaid on the ARPES spectrum for comparison. The theoretical bands agreewith the ARPES spectrum, especially both showing the band crossings at ¯X and ¯X . Wenote that the MBE sample is slightly electron-doped due to the charge transfer between thefilm and the substrate, and the Fermi level of the calculated bands is shifted to march theARPES spectrum. Figures 3(d) and 3(e) show the band dispersion around ¯X and ¯X in aperpendicular direction. In the ¯M - ¯X - ¯M direction, the two subbands dispersing away fromthe Dirac nodes are nearly degenerate, which leads to a huge anisotropy in the Dirac bandcontours. The anisotropy of Dirac bands is less prominent in the ¯M - ¯X - ¯M direction. This isconsistent with the calculated band structure. We note that the minor discrepancy betweenthe ARPES spectrum and first-principles bands can be attributed to the substrate effectson the MBE samples. The ARPES results taken from the 2L α -Sb sample is shown in Fig.4.According to the STM characterization, the sample possesses 1L and 2L domains, thereforewe can see contributions from 1L and 2L to the total ARPES spectrum. From a comparisonwith the calculated bands, we can identify the spectrum from 2L α -Sb films. The ARPESresults again show Dirac points (band degeneracy) at ¯X and ¯X . The Dirac cone centeredat ¯X exhibits a large anisotropy in the band dispersion. The ARPES results along withthe first-principles band simulations unambiguously demonstrate the existence of 2D Diracstates in the nonsymmorphic α -Sb films.In summary, Our ARPES measurements and first-principles calculations showed that α -Sb hosts Dirac-fermion states at the high-symmetry momentum points ¯X and ¯X . Theband degeneracy at the Dirac points is protected by the nonsymmorphic symmetry of thelattice. The lattice symmetry guarantees that the Dirac states in α -Sb are robust evenin the presence of strong SOC. 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FIG. 1: (a) Side view of α -Sb/SnSe lattice structure. (b) Top views of α -Sb lattice structure. Theunit cell is indicated by the blue rectangular box. The structure belongs to the
42 layer group pman . (c) STM image of α -Sb grown on SnSe substrate. (d) The height profile taken along theblue arrow in (c). (e) STM image of an α -Sb sample with 1L and 2L domains. FIG. 2: (a) The band structure and density of states of 1L α -Sb. (b) The band structure anddensity of states of 2L α -Sb. (c) The Brillouin zone of α -Sb. (d) The band structure of 1L α -Sbwithout SOC. (e) Fermi velocity of the bands at D1 and D2 of 1L α -Sb with various strength ofSOC. FIG. 3: (a) Fermi surface taken from the 1L α -Sb sample by ARPES. (b, c) ARPES spectrumtaken along ¯X - ¯Γ - ¯X and ¯X - ¯Γ - ¯X directions. (d, e) ARPES spectrum taken along ¯M - ¯X - ¯M and ¯M - ¯X - ¯M directions. FIG. 4: (a) Fermi surface taken from the 2L α -Sb sample by ARPES. (b, c) ARPES spectrumtaken along ¯X - ¯Γ - ¯X and ¯X - ¯Γ - ¯X directions. (d, e) ARPES spectrum taken along ¯M - ¯X - ¯M and ¯M - ¯X - ¯M¯M