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(A) Crystal structure of bulk bismuth. The principle lattice vectors a 1 = ( −1 2 , −1 2 √ 3 , 1 3 ), a 2 = ( 1 2 , −1 2 √ 3 , 1 3 ), and a 3 = (0, 1 √ 3 , 1 3 ) are expressed in the Cartesian coordinates. (B) The bulk Brillouin zone. The principle reciprocal space vectors ( b 1 , b 2 , and b 3 ) and high-symmetry points are noted. (C) Band structure of bismuth including spin-orbit coupling. The red and green circles represent positive and negative parity eigenvalues, respectively.
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We uncover the presence of a new topological crystalline insulator (TCI) state in bismuth, which is protected by a twofold rotational symmetry. In contrast to the recently discovered higher-order topological phase in bismuth, the present TCI phase hosts unpinned Dirac cone surface states that could be accessed directly through photoemi...
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... crystalizes in a rhombohedral structure (61). Its space group and point group are R-3m (#166) and D 3d , respectively. Each layer in this structure forms a buckled honeycomb lattice. The three principle lattice vectors ( ai=1,2,3) are tilted with respect to the Cartesian directions (Fig. 1A). In this construction, the out-of-planê z direction is [111], and the [1 ¯ 10], [10 ¯ 1], and [01 ¯ 1] directions lie in plane. Key pointgroup symmetries of bismuth are space inversion I; out-of-plane threefold rotation axis 3 [111] ; in-plane twofold rotational axes 2 [1 ¯ 10] , 2 [10 ¯ 1] , and 2 [01 ¯ 1] ; and mirror planes M [1 ¯ ...
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... [01 ¯ 1] directions lie in plane. Key pointgroup symmetries of bismuth are space inversion I; out-of-plane threefold rotation axis 3 [111] ; in-plane twofold rotational axes 2 [1 ¯ 10] , 2 [10 ¯ 1] , and 2 [01 ¯ 1] ; and mirror planes M [1 ¯ 10] , M [10 ¯ 1] , and M [01 ¯ 1] . The bulk Brillouin zone (BZ) has the shape of a truncated octahedron (Fig. 1B). The time-reversal invariant momentum (TRIM) points include one Γ, one T , three L, and three F symmetry points. Our band structure ( Table 1. Two possible topological states for the symmetry indicator Z 2,2,2,4 = {0, 0, 0, 2} ...
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... the preceding discussion in mind, we computed the surface band structure throughout the (1 ¯ 10) surface BZ (Fig. 2). As expected, we found two sets of surface Dirac cones, whose Dirac points are located at (±0.01 π, 0.269 π) of (ky , kz ) axes indicated in Fig. 1B. These are generic k points, which are related only by the twofold rotational symmetry 2 [1 ¯ 10] . The calculated surface states along the k paths passing through the Dirac points (DPs) (Fig. 2 E and F) directly show the presence of gapless Dirac surface states. Interestingly, these unpinned Dirac fermions are of type II (64) in that ...
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... Since the theoretical proposal and experimental observation of 2D [1][2][3][4][5][6] and 3D [7][8][9][10][11][12] topological insulators (TIs), these materials have been the subject of substantial research in condensed matter physics and related areas of science [11,[13][14][15][16][17]. Further investigations soon resulted in the discovery of another important class of topological materials named topological crystalline insulators (TCIs) [18][19][20][21][22]. Both TIs and TCIs host mid-gap boundary states related to a bulk topological invariant via the bulk-boundary correspondence [23,24]. ...
... By fitting the energy bands [Eqs. (22) and (23)] to the band structure obtained via DFT calculations, we obtain numerical values for the model parameters, shown in Table III. A range of ∼ 5.5% of the Γ − M line was used in the fitting performed with the LMFIT package for Python [81]. ...
... In Fig. 2-b), we have the spin-up conduction and valence bands [E 1,k and E 2,k in Eqs. (22) and (23), respectively] for the fitted parameters with the "orbital" composition (color code). Note that the conduction band is dominated by the |X⟩−i|Y ⟩ state (brownish red) at the Γ point, while the valence band is dominated by |X⟩ + i|Y ⟩ (green). ...
Dual topological insulators (DTIs) are simultaneously protected by time-reversal and crystal symmetries, representing advantageous alternatives to conventional topological insulators. By combining ab initio calculations and the $\mathbf{k}\cdot\mathbf{p}$ approach, here we investigate the electronic band structure of a Na$_2$CdSn tri-atomic layer and derive a low-energy $4\times 4$ effective model consistent with all the symmetries of this material class. We obtain the effective Hamiltonian using the L\"owdin perturbation theory, the folding down technique, and the theory of invariants, and determine its parameters by fitting our analytical dispersion relations to those of ab initio calculations. We then calculate the bulk topological invariants of the system and show that the Na$_2$CdSn tri-atomic layer is a giant-gap (hundreds of meV) quasi-2D DTI characterized by both spin and mirror Chern numbers $-2$. In agreement with the bulk-boundary correspondence theorem, we find that a finite-width strip of Na$_2$CdSn possesses two pairs of counter-propagating helical edge states per interface. We obtain analytical expressions for the edge states energy dispersions and wave functions, which are shown to agree with our numerical calculations. Our work opens a new avenue for further studies of Na$_2$CdSn as a potential DTI candidate with room-temperature applications in areas of technological interest, such as nanoelectronics and spintronics.
... Bismuth was considered topologically trivial until it was identified as a high-order topological insulator [46]. A further analysis of Bismuth based on the symmetry indicators [23] found it to host a first-order topological phase with gapless surface states protected by a rotational symmetry [47]. Here, monolayer α-Sb, characterized by a spin Chern number C s = 2, exemplifies the distinction between Z 2 TIs and quantum spin Hall insulators. ...
For a time-reversal symmetric system, the quantum spin Hall phase is assumed to be the same as the Z 2 topological insulator phase in the existing literature. The spin Chern number C s is supposed to yield the same topological classification as the Z 2 invariant. Here, by investigating the topological electronic structures of monolayer α-phase group V elements, we uncover the presence of a topological phase in α-Sb, which can be characterized by a spin Chern number C s =2, even though it is Z 2 trivial. Although both being classified as trivial insulators by the existing topological classification schemes, we demonstrate the existence of a phase transition between α-As and Sb, which is induced by band inversions at two generic k points. In the absence of spin-orbit coupling (SOC), α-As is a trivial insulator, while α-Sb is a Dirac semimetal with four Dirac points (DPs) located away from the high-symmetry lines. Inclusion of the SOC gaps out the Dirac points and induces a nontrivial Berry curvature, endowing α-Sb with a high spin Chern number of C s =2. We further show that monolayer α-Sb exhibits either a gapless band structure or a gapless spin spectrum on its edges as expected from topological considerations.
... Bismuth based materials have been instrumental in the development of topological physics and bismuth itself has emerged as a higher-order topological insulator and first-order topological crystalline insulator [49,50]. Therefore, the superlattice can be realized with the metal layer being bismuth itself. ...
The conditions required by quantum matter to modify space-time geometry are explored within the framework of the general theory of relativity. The required characteristics for space-time modification in solid state structures, are met in either (a) massive photon Bose-Einstein condensate in a waveguide, or (b) the massive photons in superconductor's bulk, or (c) the Bose-Einstein condensate of acoustic phonons, or (d) a metal-insulator-topological insulator heterostructure.
... Bismuth exhibits a rich spectrum of topological phases in all its forms, from monolayer and bilayer to thin film and to bulk [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. Strong spin-orbit coupling (SOC) motivated early studies of the boundary states of bismuth in the context of spin splitting [17][18][19][20][21][22][23][24]. ...
... With the advent of topological materials, bismuth was one of the first proposed quantum spin Hall (QSH) insulators in 2D [1,2] and topological insulators in 3D [3], both of which have been confirmed experimentally [7][8][9][10]. More recently, it has been experimentally shown that bismuth exhibits exotic properties such as topological crystalline insulator states and higher-order topology [12][13][14]. ...
Ultrathin bismuth exhibits rich physics including strong spin-orbit coupling, ferroelectricity, nontrivial topology, and light-induced structural dynamics. We use ab initio calculations to show that light can induce structural transitions to four transient phases in bismuth monolayers. These light-induced phases exhibit nontrivial topological character, which we illustrate using the recently introduced concept of spin bands and spin-resolved Wilson loops. Specifically, we find that the topology changes via the closing of the electron and spin band gaps during photoinduced structural phase transitions, leading to distinct edge states. Our study provides strategies to tailor electronic and spin topology via ultrafast control of photoexcited carriers and associated structural dynamics.
... Nevertheless, the corresponding symmetries also restrict the type of singularities that can be achieved. Bismuth (Bi) is an ideal material to explore the interplay of topological and correlated phases owing to its high spin-orbit coupling and long mean free path [20][21][22][23]. Although Bi has been studied in various contexts, such as unconventional superconductivity [24], electron fractionalization [25] and quantum hall ferromagnetism [26,27], its exact topological classification has been resolved only recently, as being on the verge of a phase transition between a high order topological insulator and a strong topological insulator [28,29]. ...
Strong singularities in the electronic density of states amplify correlation effects and play a key role in determining the ordering instabilities in various materials. Recently high order van Hove singularities (VHSs) with diverging power-law scaling have been classified in single-band electron models. We show that the 110 surface of Bismuth exhibits high order VHS with an usually high density of states divergence \sim (E)^{-0.7} ∼ ( E ) − 0.7 . Detailed mapping of the surface band structure using scanning tunneling microscopy and spectroscopy combined with first-principles calculations show that this singularity occurs in close proximity to Dirac bands located at the center of the surface Brillouin zone. The enhanced power-law divergence is shown to originate from the anisotropic flattening of the Dirac band just above the Dirac node. Such near-coexistence of massless Dirac electrons and ultra-massive saddle points enables to study the interplay of high order VHS and Dirac fermions.
... Bismuth exhibits a rich spectrum of topological phases in all its forms, from monolayer/bilayer to thin film and to bulk [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. Strong spin-orbit coupling (SOC) motivated early studies of the surface/edge states of 3D bismuth in the context of spin splitting [15][16][17][18][19][20][21][22]. ...
... With the advent of topological materials, bismuth was one of the first proposed quantum spin Hall (QSH) insulators in 2D [1,2] and topological insulators in 3D [3], both of which have been confirmed experimentally [6][7][8][9]. More recently, it has been experimentally shown that bismuth exhibits exotic properties such as topological crystalline insulator states and higher-order topology [10][11][12]. ...
... (2) The double-band inversion mechanism. In this mechanism, two sets of bands with opposite parities are inverted to induce higher-order topological energy gaps [182]. Experimental confirmation of this mechanism has been observed in only a few three-dimensional electronic materials, such as Bi [150], Td-WTe 2 [183], and Bi 4 Br 4 [184]. ...
The exploration of topologically nontrivial states in condensed matter systems, along with their novel transport properties, has garnered significant research interest. This review aims to provide a comprehensive overview of representative topological phases, starting from the initial proposal of the quantum Hall insulator. We begin with a concise introduction, followed by a detailed examination of first-order topological quantum phases, including gapped and gapless systems, encompassing relevant materials and associated phenomena in experiment. Subsequently, we delve into the realm of exotic higher-order topological quantum phases, examining both theoretical propositions and experimental findings. Moreover, we discuss the mechanisms underlying the emergence of higher-order topology, as well as the challenges involved in experimentally verifying materials exhibiting such properties. Finally, we outline future research directions. This review not only systematically surveys various types of topological quantum states, spanning from first-order to higher-order, but also proposes potential approaches for realizing higher-order topological phases, thereby offering guidance for the detection of related quantum phenomena in experiments.
... The band structure of Bi was initially classified as topologically trivial, with strong Z 2 TI index ν 0,GS = 0 8,36 . Now it is identified as a higher-order, topological crystalline insulator with strong Z 4 index κ 1,GS = 2 [37][38][39] . In contrast to this, the ground state of β-bismuthene is known to be a Z 2 QSH insulator 6,40-45 , supporting helical edge states 6,40,41 . ...
Multiple works suggest the possibility of classification of quantum spin Hall effect with magnetic flux tubes, which cause separation of spin and charge degrees of freedom and pumping of spin or Kramers-pair. However, the proof of principle demonstration of spin-charge separation is yet to be accomplished for realistic, ab initio band structures of spin-orbit-coupled materials, lacking spin-conservation law. In this work, we perform thought experiments with magnetic flux tubes on β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document}-bismuthene, and demonstrate spin-charge separation, and quantized pumping of spin for three insulating states, that can be accessed by tuning filling fractions. With a combined analysis of momentum-space topology and real-space response, we identify important role of bands supporting even integer invariants, which cannot be addressed with symmetry-based indicators. Our work sets a new standard for the computational diagnosis of two-dimensional, quantum spin-Hall materials by going beyond the Z2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}_{2}$$\end{document} paradigm and providing an avenue for precise determination of the bulk invariant through computation of quantized, real-space response.
... calized mode due to the geometry of the lattice [76,77]. In the case of the (110) surface, the dispersionless edge modes alongΓ-X can be similar to the one-dimensional topological hinge electronic states protected by threefold rotational and inversion symmetries in bismuth [78]. Similarly to the other systems, the phonon edge mode strongly depends on the surface termination [75]. ...
Shandite with Ni$_{3}$Pb$_{2}$S$_{2}$ chemical formula and R$\bar{3}$m symmetry, contains the kagome sublattice formed by the transition metal atoms. Recent experimental results confirmed the possibility of successfully synthesizing Pd$_{3}$Pb$_{2}Ch_{2}$ ($Ch$=S,Se) with the same structure. In this paper, we theoretically investigate the dynamical properties of such compounds. Furthermore, we study the possibility of realizing Pt$_{3}$Pb$_{2}Ch_{2}$ with the shandite structure. We show that the Pd$_{3}$Pb$_{2}${\it Ch}$_{2}$ and Pt$_{3}$Pb$_{2}$S$_{2}$ are stable with R$\bar{3}$m symmetry. In the case of Pt$_{3}$Pb$_{2}$S$_{2}$, there is a soft mode, which is the source of the structural phase transition from R$\bar{3}$m to R$\bar{3}$c symmetry, related to the distortion within the kagome sublattice. We discuss realized phonon nodal lines in the bulk phonon dispersions in upper frequency modes. We show that the shandite structure can host the phonon surface states, with strong dependence by the surface kind. Additionally, chiral phonons with circular motion of the Pb atoms around the equilibrium position are realized.
... However, the low damage threshold of 2D layered TQM, such as graphene, is unsuitable for high laser power detection [10]. As one of the TI materials, bismuth (Bi) has been proven to offer the potential of a broader band response than other TQM in the room temperature environment because of its bulk bandgap around 0.2 eV with the small electron effective mass, the long carrier mean free path, and the higher damage threshold [11][12][13][14][15][16][17][18]. Therefore, Bi has been widely used in the photoelectric field, such as broadband absorbers [19], photodetectors [20][21][22][23][24], and Hall sensors [25,26]. ...
Topological insulator bismuth has attracted considerable attention for the fabrication of room-temperature, wide bandwidth, and high-performance photodetectors due to the gapless edge state and insulating bulk state properties. However, both the photoelectric conversion and carrier transportation of the bismuth films are extremely affected by the surface morphology and grain boundaries to limit optoelectronic properties further. Here, we demonstrate a strategy of femtosecond laser treatment for upgrading the quality of bismuth films. After the treatment with proper laser parameters, the measurement of average surface roughness can be reduced from Ra = 44 nm to 6.9 nm, especially with accompany of the evident grain boundary elimination. Consequently, the photoresponsivity of the bismuth films increases approximately 2 times within an ultra-broad spectrum range from the visible to mid-infrared. This investigation suggests that the femtosecond laser treatment can help to benefit the performance of topological insulator ultra-broadband photodetectors.
