High-symmetry Planes Research Articles

Observation of plaid-like spin splitting in a noncoplanar antiferromagnet.

Spatial, momentum and energy separation of electronic spins in condensed-matter systems guides the development of new devices in which spin-polarized current is generated and manipulated1-3. Recent attention on a set of previously overlooked symmetry operations in magnetic materials4 leads to the emergence of a new type of spin splitting, enabling giant and momentum-dependent spin polarization of energy bands on selected antiferromagnets5-10. Despite the ever-growing theoretical predictions, the direct spectroscopic proof of such spin splitting is still lacking. Here we provide solid spectroscopic and computational evidence for the existence of such materials. In the noncoplanar antiferromagnet manganese ditelluride (MnTe2), the in-plane components of spin are found to be antisymmetric about the high-symmetry planes of the Brillouin zone, comprising a plaid-like spin texture in the antiferromagnetic (AFM) ground state. Such an unconventional spin pattern, further found to diminish at the high-temperature paramagnetic state, originates from the intrinsic AFM order instead of spin-orbit coupling (SOC). Our finding demonstrates a new type of quadratic spin texture induced by time-reversal breaking, placing AFM spintronics on a firm basis and paving the way for studying exotic quantum phenomena in related materials.

A rod- and tessellation-based comparative analysis of polymorphic and structurally-invariant molecular crystals: application to sulfathiazole and 2-benzyl-5-benzylidenecyclopentanones.

A rationalization of the alternative crystal structures adopted by a given molecular compound or by a set of substitutionally related molecular compounds is provided by reference to the five known polymorphs of sulfathiazole and 16 substituted 2-benzyl-5-benzylidene cyclopentanones (BBCPs), respectively. Two-dimensional (2D) packing fractions (ϕ2D) take space-group symmetry into account, with a clear demarcation of closed-packed zones (CPZ) and molecular junction zones (JZ) in all Z' = 1 structures. Representation of the molecules as two linked rods allows a concise treatment of conformation and rapid visualization of crystal packing. Combined with calculations of intermolecular potential energies, the rod method provides insight into the stabilization mechanisms of alternative polymorphs. In sulfathiazole, the primary factor is to obtain satisfactory hydrogen bonding, with close packing a secondary consideration. In BBCP derivatives, by comparison, close packing is the primary mechanism of stabilization. Whereas the 2D structures arising in CPZ can be analysed as tessellations of molecular-based cells, a method based on 2D Dirichlet cells is required for the JZ. These are calculated from the centroids of the molecular envelopes in high-symmetry planes. It is shown that these centroid coordinates, when combined with space-group symmetry and unit cell coordinates, provide a concise parameterization of all structures containing JZ. It is anticipated that this parameterization may be exploited to predict such crystal structures from powder diffraction data.

Supercurrent-Induced Weyl Superconductivity

We show that Weyl superconductivity can be induced by finite supercurrent in noncentrosymmetric spin–orbit-coupled superconductors with line nodes. We introduce a three-dimensional tight-binding model of a tetragonal superconductor in a D+p-wave pairing state with a finite center-of-mass momentum, and elucidate that a line-nodal to point-nodal spectral transition occurs by applying an infinitesimal supercurrent. We also clarify that the higher-order effect in spin–orbit coupling is particularly important for this phenomenon. The point nodes are protected by topologically nontrivial Weyl charges, and therefore gapless arc states appear on the surface of the superconductor. Furthermore, both the positions and the Weyl charges of the point nodes depend on the direction of the current. In addition, a quantized Berry phase defined on high-symmetry planes characterizes the Weyl nodes when the in-plane supercurrent is considered. Our proposition paves a new way for controlling the superconducting gap structures by using an external field.

Elastic and magnetoelastic properties of TbMnO3 single crystal by nanosecond time resolved acoustics and first-principles calculations

Time resolved pump and probe acoustics and first-principles calculations were employed to assess elastic properties of the TbMnO3 perovskite manganite having orthorhombic symmetry. Measuring sound velocities of bulk longitudinal and shear acoustic waves propagating along at least two different directions in the high symmetry planes (100), (010) and (001), provided a powerful mean to selectively determine the six diagonal elastic constants C 11 = 227 GPa, C 22 = 349 GPa, C 33 = 274 GPa, C 44 = 71 GPa, C 55 = 57 GPa, C 66 = 62 GPa. Among the three remaining off-diagonal ones, C 23 = 103 GPa was determined with a bissectrice direction. Density functional theory calculations with colinear spin-polarized provided complementary insights on their optical, elastic and magnetoelastic properties.

Amorphous topological phases protected by continuous rotation symmetry

Protection of topological surface states by reflection symmetry breaks down when the boundary of the sample is misaligned with one of the high symmetry planes of the crystal. We demonstrate that this limitation is removed in amorphous topological materials, where the Hamiltonian is invariant on average under reflection over any axis due to continuous rotation symmetry. We show that the edge remains protected from localization in the topological phase, and the local disorder caused by the amorphous structure results in critical scaling of the transport in the system. In order to classify such phases we perform a systematic search over all the possible symmetry classes in two dimensions and construct the example models realizing each of the proposed topological phases. Finally, we compute the topological invariant of these phases as an integral along a meridian of the spherical Brillouin zone of an amorphous Hamiltonian.

Symmetry-enforced band nodes in 230 space groups

Crystallographic symmetries enforcing band touchings (BTs) in the Brillouin zone (BZ) have been utilized to classify and predict the topological semimetals. Though the early proposed topological semimetals contain isolated nodal points in the BZ, the proposed nodal line semimetals later could host various structures of several nodal lines/loops: nodal chains, nodal nets or Hopf-links, etc. In this work, using compatibility relations, we first list all possible high symmetry lines (HSLs) that can be nodal lines itself, high symmetry planes (HSPLs) that can host nodal loops, high symmetry planes (HSPLs) that are nodal surfaces for all 230 SGs, with spin-orbit coupling and time-reversal symmetry considered or not. We then show how to diagnose a nodal loop from the band crossing in an HSL, or nodal line/surface from irreducible representation (irrep) of an high-symmetry point (HSP), while the rest cases correspond to nodal points. Among our results, those essential cases, for which the nodal points/lines/loops/surfaces must exist, are highlighted since they are promising for the realizations of (nearly) ideal nodal point/line/loop/surface semimetals, as well as systems with flexible tunability owning fixed structure of topological nodal points/lines/loops/surfaces. Based on our results, SGs allowing Hopf-link structure with one straight nodal line threading a nodal loop, or two nesting nodal loops lying in two respective high symmetry planes, are highlighted, with the predicted materials being B$_5$Pb$_2$IO$_9$ in SG 34 and SrAl$_2$Au$_3$ in SG 62, respectively. Our exhaustive results could serve as a useful guide for efficiently predicting and designing materials or artificial systems owning exotic geometric nodal structures of energy bands simply based on structure symmetries.

Dual Topological Features of Weyl Semimetallic Phases in Tetradymite BiSbTe3 Supported by the National Natural Science Foundation of China (Grant Nos. 11604032, 11674040, and 51672270), and the Fundamental Research Funds for the Central Universities (Grant No. 106112016CDJZR308808).

Based on first-principles calculations and symmetry arguments, we reveal that the non-centrosymmetric ternary tetradymite BiSbTe3 possesses exotic dual topological features of Weyl semimetallic phases with Z 2 index (1:000). The results show that the helical Dirac-type surface states protected by the time-reversal symmetry are present in the vicinity of the Brillouin zone center, which is consistent with the experimental report. Furthermore, we show that four pairs of Weyl points reside exactly at the Fermi level, which are guaranteed to be located on high-symmetry planes due to mirror symmetries. The helical surface states and the projected Weyl nodes are well separated in the momentum space, facilitating their observations in experiments. This work not only uncovers a unique quantum phenomenon with dual topological features in the tetradymite family but also paves a fascinating avenue for exploring the coexistence of multi-topological states with wide applications.

Symmetry-enforced band crossings in trigonal materials: Accordion states and Weyl nodal lines

Nonsymmoprhic symmetries, such as screw rotations or glide reflections, can enforce band crossings within high-symmetry lines or planes of the Brillouin zone. When these band degeneracies are close to the Fermi energy, they can give rise to a number of unusual phenomena: e.g., anomalous magnetoelectric responses, transverse Hall currents, and exotic surface states. In this paper, we present a comprehensive classification of such nonsymmorphic band crossings in trigonal materials with strong spin-orbit coupling. We find that in trigonal systems there are two different types of nonsymmorphic band degeneracies: (i) Weyl points protected by screw rotations with an accordion-like dispersion, and (ii) Weyl nodal lines protected by glide reflections. We report a number of existing materials, where these band crossings are realized near the Fermi energy. This includes Cu2SrSnS4 and elemental tellurium (Te), which exhibit accordion Weyl points; and the tellurium-silicon clathrate Te16Si38, which shows Weyl nodal lines. The ab-initio band structures and surface states of these materials are studied in detail, and implications for experiments are briefly discussed.

4D texture of circular dichroism in soft-x-ray photoemission from tungsten

Photoemission-intensity distributions IRCP/LCP (EB, k) measured for right- and left-circularly polarized soft x-rays revealed a large circular dichroism in angular distribution (CDAD) in the 4D parameter space (EB binding energy, k momentum vector). Full-field k-imaging combined with time-of-flight energy recording at a high-brilliance soft x-ray beamline allowed mapping the CDAD in the bulk Brillouin zone of tungsten and the entire d-band complex within a few hours. CDAD-asymmetries are very high (up to 90%), persist throughout the whole photon-energy range (300–1300 eV) and show a pronounced dependence on momentum k and binding energy EB, visualized as movies or sequences of cuts through the 4D object. One-step photoemission calculations for the same photon energies show fair agreement with the measured results. In addition to the requirement of a ‘handed’ experimental geometry, known from previous experiments on adsorbates and surface states, we find an anti-symmetric behavior of the CDAD with respect to two bulk mirror planes. A new symmetry condition along the perpendicular momentum kz makes CDAD a valuable tool for an unambiguous identification of high-symmetry planes in direct transitions in the periodic zone scheme. Technically, the method provides a circular polarimeter for soft, tender and hard x-rays.

Characterization of Lifshitz transitions in topological nodal line semimetals

We introduce a two-band model of three-dimensional nodal line semimetals, the Fermi surface of which at half-filling may form various one-dimensional configurations of different topology. We study the symmetries and "drumhead" surface states of the model, and find that the transitions between different configurations, namely, the Lifshitz transitions, can be identified solely by the number of gap-closing points on some high-symmetry planes in the Brillouin zone. A global phase diagram of this model is also obtained accordingly. We then investigate the effect of some extra terms analogous to a two-dimensional Rashba-type spin-orbit coupling. The introduced extra terms open a gap for the nodal line semimetals and can be useful in engineering different topological insulating phases. We demonstrate that the behavior of surface Dirac cones in the resulting insulating system has a clear correspondence with the different configurations of the original nodal lines in the absence of the gap terms.

Symmetry-guaranteed nodal-line semimetals in an fcc lattice

We demonstrate theoretically that nodal-line semimetals (NLSs) can be realized in an fcc lattice with orbitals belonging to the same irreducible representation, such as ${{p}_{x},{p}_{y},{p}_{z}}$ or ${{d}_{xy},{d}_{yz},{d}_{zx}}$ orbitals on every lattice site. The three orbitals are divided into two subgroups in terms of the parity with respect to the mirror reflections on high-symmetry planes of the fcc lattice, which, with rotation symmetry, endows symmetry-guaranteed NL passing through $W$ points in the Brillouin zone. Depending on the parameters, there also appears an accidental NL around the $\mathrm{\ensuremath{\Gamma}}$ point. We notice that the symmetry-guaranteed NL addressed in the present work can be found in band structures of elemental solids taking the fcc structure, such as Cu, Ag, Au, In, Ga, etc., as well as opal, which is an fcc photonic crystal of ${\mathrm{SiO}}_{2}$ spheres. Furthermore, we clarify that the fcc lattice of Si spheres exhibits a NL in a frequency band where no other photonic band exists, which provides a unique platform to realize topological NLSs under intensive search, and can be explored for achieving slow light.

Inflated nodes and surface states in superconducting half-Heusler compounds

Two topics of high current interest in the field of unconventional superconductivity are non-centrosymmetric superconductors and multiband superconductivity. Half-Heusler superconductors such as YPtBi exemplify both. In this paper, we study bulk and surface states in nodal superconducting phases of the half-Heusler compounds, belonging to the $A_1$ ($s+p$-like) and $T_2$ ($k_zk_x+ik_zk_y$-like) irreducible representations of the point group. These two phases preserve and break time-reversal symmetry, respectively. For the $A_1$ case, we find that flat surface bands persist in the multiband system. In addition, the system has dispersive surface bands with zero-energy crossings forming Fermi arcs, which are protected by mirror symmetries. For the $T_2$ case, there is an interesting coexistence of point and line nodes, known from the single-band case, with Bogoliubov Fermi surfaces (two-dimensional nodes). There are no flat-band surface states, as expected, but dispersive surface bands with Fermi arcs exist. If these arcs do not lie in high-symmetry planes, they are split by the antisymmetric spin-orbit coupling so that their number is doubled compared to the inversion-symmetric case.

Graph theory data for topological quantum chemistry.

Topological phases of noninteracting particles are distinguished by the global properties of their band structure and eigenfunctions in momentum space. On the other hand, group theory as conventionally applied to solid-state physics focuses only on properties that are local (at high-symmetry points, lines, and planes) in the Brillouin zone. To bridge this gap, we have previously [Bradlyn etal., Nature (London) 547, 298 (2017)NATUAS0028-083610.1038/nature23268] mapped the problem of constructing global band structures out of local data to a graph construction problem. In this paper, we provide the explicit data and formulate the necessary algorithms to produce all topologically distinct graphs. Furthermore, we show how to apply these algorithms to certain "elementary" band structures highlighted in the aforementioned reference, and thus we identified and tabulated all orbital types and lattices that can give rise to topologically disconnected band structures. Finally, we show how to use the newly developed bandrep program on the Bilbao Crystallographic Server to access the results of our computation.

Symmetry-Enforced Line Nodes in Unconventional Superconductors.

We classify line nodes in superconductors with strong spin-orbit interactions and time-reversal symmetry, where the latter may include nonprimitive translations in the magnetic Brillouin zone to account for coexistence with antiferromagnetic order. We find four possible combinations of irreducible representations of the order parameter on high-symmetry planes, two of which allow for line nodes in pseudospin-triplet pairs and two that exclude conventional fully gapped pseudospin-singlet pairs. We show that the former can only be realized in the presence of band-sticking degeneracies, and we verify their topological stability using arguments based on Clifford algebra extensions. Our classification exhausts all possible symmetry protected line nodes in the presence of spin-orbit coupling and a (generalized) time-reversal symmetry. Implications for existing nonsymmorphic and antiferromagnetic superconductors are discussed.

Nonlinear elastic response of cubic crystals to biaxial strain

The biaxial Poissons ratio in the nonlinear elastic theory is an important material parameter. We report here an accurate and efficient means of calculating the nonlinear elastic response of thin, homogeneous films to biaxial strain in arbitrary planes using a continuum-elasticity theory model. The general analytic expressions were derived for the elastic energy and the Poissons ratio under biaxial strain for cubic crystals. The biaxial Poisson’s ratio did not remain constants, but showed a linear relationship with strain when second- and third-order elastic constants are considered. The expressions were verified with simulated biaxial Poisson’s ratio and elastic energy of copper by density-functional theory calculations for three high symmetry planes.

Photoinduced self-trapped hole center inTiO2crystals

Electron paramagnetic resonance is used to identify the intrinsic self-trapped hole center in bulk ${\text{TiO}}_{2}$ crystals having the rutile structure. Two additional extrinsic trapped hole centers associated with a defect on a neighboring ${\text{Ti}}^{4+}$ site are also observed. For each center, the unpaired spin (i.e., the hole) is localized in a nonbonding $p$ orbital on an oxygen ion. The three hole centers are photoinduced with 442 nm laser light at 4 K. After the laser light is removed, the majority of the hole centers disappear immediately at 4 K with the remainder disappearing when the temperature is raised above approximately 10 K. Angular-dependence data, taken in the three high-symmetry planes, provide spin-Hamiltonian parameters for the self-trapped hole center and one of the two defect-related trapped hole centers. The principal values of the $g$ matrix for the self-trapped hole center are 2.0040, 2.0129, and 2.0277 and the corresponding principal axes are along the [110], [001], and $[1\overline{1}0]$ directions in the crystal. Although the principal values of the $g$ matrix for the extrinsic trapped hole center are similar (2.0036, 2.0182, and 2.0307), the directions of two of its principal axes are not along high-symmetry directions in the crystal because of the neighboring perturbation.

Grain boundary self-diffusion in Ni: Effect of boundary inclination

We examined the influence of the boundary plane on grain-boundary diffusion in Ni through a series of molecular dynamics simulations. A series of 〈010〉 ∑5 tilt boundaries, including several high symmetry and low symmetry boundary planes, were considered. The self-diffusion coefficient is a strong function of boundary inclination at low temperature but is almost independent of inclination at high temperature. At all temperatures, the self-diffusion coefficients are low when at least one of the two grains has a normal with low Miller indices. The grain boundary self-diffusion coefficient is an Arrhenius function of temperature. The logarithm of the pre-exponential factor in the Arrhenius expression was shown to be nearly proportional to the activation energy for diffusion. The activation energy for self-diffusion in a (103) symmetric tilt boundary is much higher than in boundaries with other inclinations. We discuss the origin of the boundary plane density–diffusion coefficient correlation.

Paramagnetic Properties of Noncentrosymmetric Superconductors: Application toCePt3Si

In a noncentrosymmetric crystal, the Zeeman interaction of the band electrons with an external magnetic field is highly anisotropic in the momentum space, vanishing along some high-symmetry planes. One of the consequences is that the paramagnetic susceptibility in superconductors without inversion symmetry, such as CePt3Si, shows an unusual temperature dependence.

Theoretical Study of Orientation Dependence of Piezoelectric Effects in Wurtzite Strained GaInN/GaN Heterostructures and Quantum Wells

We calculated the crystal orientation dependence of piezoelectric fields in wurtzite strained Ga0.9In0.1N/GaN heterostructures. The highest longitudinal piezoelectric field of 0.7 MV/cm can be generated in (0001)-oriented biaxial-strained Ga0.9In0.1N layer coherently grown on GaN. On the contrary, no longitudinal piezoelectric field is induced in strained layers grown along orientations at an off angle of 39° or 90° from (0001). The high symmetry planes with these angles are, for instance, (1124) and (1012) for 39°, and (1120) and (1010) for 90°. We also calculated the crystal orientation dependence of the transition probability in a 3-nm strained Ga0.9In0.1N/GaN quantum well, which indicated that the transition probability with these non-(0001) orientations becomes 2.3 times larger than that with the (0001) orientation. We conclude that high-performance strained nitride-based optical devices can be obtained by control of the crystal orientation.

Crystal structure and domain-wall orientations of antiferroelectric

The crystal structure of antiferroelectric with ordered complex perovskite structure was reexamined by TEM (transmission electron microscope). Its domain-wall orientations were investigated by the spontaneous strain method. From the CBED (convergent-beam electron diffraction) patterns, its point group was determined to be orthorhombic mmm. During transition from the high-temperature cubic to low-temperature orthorhombic phase, the prototype cubic cell is monoclinically distorted. The monoclinic cell was found to be contracted along the direction of shifts. The reduced symmetry of the orthorhombic phase transformed from the prototypic phase allows the existence of ten species of domain walls, i.e. six W-walls and four -walls. The breaking of reflection symmetry during transition occurred at (1 0 0), (0 1 0), (0 1 1), and planes of the cubic prototype. These high-symmetry planes correspond to the W-walls in the ferroelastic phase. Taking into account the domain geometry, the and planes are the domain-boundaries and the others can be both and ones. The orientations of -walls calculated from the strain components were approximately , and planes at room temperature. On approaching the transition these -walls tend to orient themselves toward the direction of or domain W-walls.