High-symmetry Directions Research Articles

Pressure-induced band gap enhancement and temperature-dependent thermoelectric characterization of semiconducting Transition Metal Chalcogenides LiMS (M = Cu, Ag)

The first principles study of Transition Metal Chalcogenides LiMS (M = Cu, Ag) has been conducted utilizing the concepts of Density Functional Theory (DFT). Both the compounds possess a cubic structure with space group F-43m. The mechanical stability of LiCuS and LiAgS has been predicted based on the values of elastic tensor. These Chalcogenides possess an interesting electronic band structure which reveals their semiconducting nature with a direct band gap. The electronic structure is further subjected to high pressure and the changes in band gap are deeply observed. For these materials, the changes in band gap in response to applied pressure suggest the possibility of their use in band gap engineering. The phonon curves are investigated over the Brillouin zone's high symmetry directions which reveal the lattice dynamical stability of the presently studied compounds. Moreover, the atoms present at different positions inside a crystal lattice exhibit vibrations of different intensities and in different directions which have been studied through the Eigenvector representations. The thermoelectric properties of LiCuS and LiAgS have been characterized using Boltzmann's theory over the range of temperature between 100 K and 1600 K. The calculated thermoelectric parameters reveal the high thermoelectric efficiency of these materials with excellent values of the figure of merit. The interesting properties of ternary Chalcogenides LiMS (M = Cu, Ag) make them promising candidates for thermoelectric applications.

Electronic structure investigation and anisotropic phonon anharmonicity in ternary ZrGeTe4 single crystals

Ternary two-dimensional (2D) transition metal chalcogenides have gained immense attention because of their ability to overcome the intrinsic limitations of their binary counterparts. Layered 2D materials are important for future electronic and photonic devices owing to their low structural symmetry and in-plane anisotropy with tunable bandgap. Herein, the electronic structure and detailed vibrational properties of bulk ZrGeTe4 layered single crystals were investigated using angle-resolved photoemission spectroscopy (ARPES) and Raman scattering (RS). The ARPES results revealed an anisotropic Fermi surface of different momentum along kx and ky from the zone center and an anisotropic band structure with varying band curvatures along the high-symmetry directions. Furthermore, the RS of ZrGeTe4 was investigated under different polarizations and varying temperatures. The polarized RS exhibited twofold and fourfold symmetry orientations in different configurations, revealing the anisotropic phonon dispersions for bulk ZrGeTe4. The observed softening of Raman modes was corroborated with the anharmonic phonon dispersion, which was further supported by our third-order force constant calculations of thermal transport using density functional theory. Low lattice thermal conductivity with increasing temperature is linked with enhanced phonon–phonon scattering, which is evident from the decreased phonon lifetime and peak linewidth. In addition to these fundamental aspects, the anisotropic nature and unique layered structure of such materials reveal their bright future for next-generation nanoelectronic applications.

Tilted Spins in Chains of Molecular Switches on Pb(100).

A complex based on a Ni(II) porphyrin exhibiting spin crossover on Ag(111) is studied on Pb(100) by scanning tunneling microscopy at 0.3 K. Strong molecular interactions between the phenyl and pentafluorophenyl moieties lead to the formation of molecular chains and cause a faceting of the substrate surface. The chains are located along double and multiple substrate steps that deviate from high-symmetry directions. Tunneling spectroscopy reveals spin-flip excitations of an S = 1 system. Measurements in high magnetic fields are used to identify a tilt of the complex and its hard anisotropy axis with respect to the surface normal. Electron injection into the substrate near the molecular rows induces a transition to a state with larger inelastic cross section, leaving the spin state unchanged.

Tilted stripes origin in La1.88Sr0.12CuO4 revealed by anisotropic next-nearest neighbor hopping

Spin- and charge- stripe order has been extensively studied in the superconducting cuprates, among which underdoped La2−xSrxCuO4 (LSCO) is an archetype with static spin stripes at low temperatures. An intriguing, but not completely understood, phenomenon in LSCO is that the stripes are tilted away from the high-symmetry Cu-Cu directions. Using high-resolution neutron scattering on LSCO with x = 0.12, we find two coexisting phases at low temperatures, one with static spin stripes and the other with fluctuating ones, both sharing the same tilt angle. Our numerical calculations using the doped Hubbard model elucidate the tilting’s origin, attributing it to anisotropic next-nearest neighbor hopping t′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${t}^{{\\prime} }$$\\end{document}, consistent with the material’s slight orthorhombicity. Our results underscore the model’s success in describing specific details of the ground state of this real material and highlight the role of t′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${t}^{{\\prime} }$$\\end{document} in the Hamiltonian, revealing the delicate interplay between stripes and superconductivity across theoretical and experimental contexts.

Exploring Structural, Electronic, Vibrational, and Thermophysical Properties of Fe–Pt, Fe3–Pt, and Fe–Pt3 Alloys: A Density Functional Theory Study

Utilizing density functional theory, the structural, electronic, vibrational, and thermophysical properties of L10 FePt, L12 Fe3Pt, and L12 FePt3 alloys are meticulously analyzed. Employing projected augmented wave pseudopotentials alongside the Perdew–Burke–Ernzerhof exchange‐correlation function, equilibrium lattice constants are computed, aligning closely with existing data, thus validating our approach. To ascertain the dynamical stability of these alloys, phonon frequencies and density of states across high symmetry directions of the Brillouin zone are computed, affirming their stability with positive phonon frequencies throughout. Furthermore, the electronic band structure, the total and projected density of states, electronic charge density, and Fermi surfaces of the alloys are delved. The thorough analysis of phonon dispersion curves, electronic band structures, and the density of states, charge densities, and Fermi surfaces provides conclusive insights into the properties and behavior of the alloys. In essence, comprehensive investigation offers valuable insights into the thermophysical properties of L10 FePt, L12 Fe3Pt, and L12 FePt3 alloys, spanning equilibrium lattice constants, phonon characteristics, and electronic properties. These findings significantly augment the understanding of the structural stability, phonon dynamics, and electronic behavior exhibited by these alloys.

Van Hove annihilation and nematic instability on a kagome lattice.

A nematic phase breaks the point-group symmetry of the crystal lattice and is known to emerge in correlated materials. Here we report the observation of an intra-unit-cell nematic order and associated Fermi surface deformation in the kagome metal ScV6Sn6. Using scanning tunnelling microscopy and scanning tunnelling spectroscopy, we reveal a stripe-like nematic order breaking the crystal rotational symmetry within the kagome lattice itself. Moreover, we identify a set of Van Hove singularities adhering to the kagome-layer electrons, which appear along one direction of the Brillouin zone and are annihilated along other high-symmetry directions, revealing rotational symmetry breaking. Via detailed spectroscopic maps, we further observe an elliptical deformation of the Fermi surface, which provides direct evidence for an electronically mediated nematic order. Our work not only bridges the gap between electronic nematicity and kagome physics but also sheds light on the potential mechanism for realizing symmetry-broken phases in correlated electron systems.

Symmetries and wave functions of photons confined in three-dimensional photonic band gap superlattices

We perform a computational study of confined photonic states that appear in a three-dimensional (3D) superlattice of coupled cavities, resulting from a superstructure of intentional defects. The states are isolated from the vacuum by a 3D photonic band gap, using a diamondlike inverse woodpile crystal structure, and they exhibit “Cartesian” hopping of photons in high-symmetry directions. We investigate the confinement dimensionality to verify which states are fully 3D-confined, using a recently developed scaling theory to analyze the influence of the structural parameters of the 3D crystal. We create confinement maps that trace the frequencies of 3D-confined bands for select combinations of key structural parameters, namely the pore radii of the underlying regular crystal and of the defect pores. We find that a certain minimum difference between the regular and defect pore radii is necessary for 3D-confined bands to appear, and that an increasing difference between the defect pore radii from the regular radii supports more 3D-confined bands. In our analysis, we find that their symmetries and spatial distributions are more varied than electronic orbitals known from solid-state physics. We surmise that this difference occurs since the confined photonic orbitals derive from global Bloch states governed by the underlying superlattice structure, whereas single-atom orbitals are localized. Based on this realization, we suggest that the extent symmetries of “photonic orbitals” could possibly translate to novel macroscopic behaviors of “photonic solid-state matter,” never before seen in the standard electronic solid-state systems. We also discover pairs of degenerate 3D-confined bands with p-like orbital shapes and mirror symmetries matching the symmetry of the superlattice. Finally, we investigate the enhancement of the local density of optical states for cavity quantum electrodynamics applications. We find that donorlike superlattices, i.e., where the defect pores are smaller than the regular pores, provide greater enhancement in the air region than acceptorlike structures with larger defect pores, and thus offer better prospects for doping with quantum dots and ultimately for 3D networks of single photons steered across strongly coupled cavities. Published by the American Physical Society 2024

Computational systematic study of pressure driven changes in electronic, optical, elastic, mechanical, thermodynamic and thermoelectric properties of CaZrO3 for optoelectronic and thermoelectric applications

Calcium Zirconate Oxide (CaZrO3) was studied in this work for its electronic, optical, structural, thermoelectric and thermodynamic properties in the consistent external pressure from 0 to 100 GPa. The correlation functionals utilized have been Generalized Gradient Approximations (GGA) and Heyd-Scuseria-Ernzerhof (HSE03). The structure maintains its cubical structure for a maximum of 100 GPa, while the lattice parameters gradually decrease when exposed to external pressure. Band structure are seen an indirect band at 0–40 GPa whereas direct band at 60–100 GPa (4.737–4.629 eV). To further comprehend the band gap, decrease with the estimated partial densities of states (PDOS) for bulk CaZrO3 for Ca, Zr and O are also estimated. Optical properties such as absorption I(ω), complex optical conductivity σ (ω), complex dielectric function ε(ω), loss function L(ω), reflectivity R(ω) and complex refractive index n(ω) varies significantly with external pressure ranges from 0 to 100 GPa. Mechanical stability is only observed at pressures up to 0–100 GPa becomes stable according to elastic constants. According to Pugh's ratio, the compound is ductile at 0–100 GPa. Whereas Poisson ratio shows metallic bonding up to 100 GPa, our predicted findings show anisotropic nature. According to the thermoelectric properties, these phases could potentially beneficial for thermoelectric devices. Thermodynamic parameters with respect to both temperature and pressure, including heat capacity, thermal expansion, and Debye temperature, from 0 to 1000 K and 0–100 GPa. CZO has phonon density of states and dispersion are calculated along high symmetry directions. This structure is dynamically stable and all frequencies are in the positive phonon spectrum.

PH-NODE: A DFPT and finite displacement supercell based python code for searching nodes in topological phononic materials

Exploring the topological physics of phonons is fundamentally important for understanding various practical applications. Here, we present a density-functional perturbation theory and finite displacement supercell based Python 3 software package called PH-NODE for efficiently computing phonon nodes present in real material through a first-principles approach. The present version of the code is interfaced with the WIEN2k, Elk, and ABINIT packages. In order to benchmark the code, six different types of materials are considered, which include (i) FeSi, a well-known double-Weyl point; (ii) LiCaAs, a half-Heusler single-type-I Weyl topological phonon (TP); and (iii) ScZn, coexisting nodal-line and nodal-ring TPs; (iv) TiS, six pairs of bulk Weyl nodes; (v) CdTe, type-II Weyl phonons; (vi) CsTe, coexisting TP and quadratic contact TP. In FeSi, the node points are found at Γ(0,0,0) and R(0.5,0.5,0.5) high symmetric points. Also, there are 21 energy values at which the node points are situated, corresponding to the full Brillouin zone. For LiCaAs, the previously reported literature claims that there is a node point along the W-X high symmetry direction between the highest longitudinal acoustic and the lowest transverse optical branch, while in our DFT calculations, a gap of 0.17 meV is found. Furthermore, ScZn hosts six nodal-ring TPs phonons at the boundary planes of the Brillouin zone in the vicinity of the M high-symmetric point. In addition to this, straight-line TPs are also found along the Γ-X and Γ-R high symmetric directions. Moreover, for TiS, six Weyl node points (WP1, WP2, WP3, WP4, WP5 and WP6) are found along H-K high-symmetric direction. In CdTe, it is found that Weyl points are located along the X-W high-symmetry direction. In the case of CsTe, a TP and a quadratic contact TP are found along the Γ-X direction and at the R high-symmetry point, respectively. The results obtained from the PH-NODE code are in good agreement with the experimentally and theoretically reported data for each material. Program summaryProgram title: PH-NODECPC Library link to program files:https://doi.org/10.17632/sjydzn49nw.1Licensing provisions: GNU General Public License 3.0Programming language: Python 3External routines/libraries: Math, Time, NumPy, SciPyNature of problem: Searching for the phonon-node points corresponding to the given number of phonon-branch using Nelder-Mead's simplex approach.Solution method: We present a density-functional perturbation theory and finite displacement supercell based Python 3 software package called PH-NODE for efficiently computing phonon nodes present in real material through a first-principles approach.

Optical manipulation of the charge-density-wave state in RbV3Sb5.

Broken time-reversal symmetry in the absence of spin order indicates the presence of unusual phases such as orbital magnetism and loop currents1-4. The recently discovered kagome superconductors AV3Sb5 (where A is K, Rb or Cs)5,6 display an exotic charge-density-wave (CDW) state andhave emerged as astrong candidate for materialshosting a loop currentphase. The idea that the CDW breaks time-reversal symmetry7-14 is, however, being intensely debated due to conflicting experimental data15-17. Here we use laser-coupled scanning tunnelling microscopy to study RbV3Sb5. By applying linearly polarized light along high-symmetry directions, we show that the relative intensities of the CDW peaks can be reversibly switched, implying a substantial electro-striction response, indicative of strong nonlinear electron-phonon coupling. A similar CDW intensity switching is observed with perpendicular magnetic fields, which implies an unusual piezo-magnetic response that, in turn, requires time-reversal symmetry breaking. We show that the simplest CDW that satisfies these constraints is an out-of-phase combination of bond charge order and loop currents that we dub a congruent CDW flux phase. Our laser scanning tunnelling microscopy data open the door to the possibility of dynamic optical control of complex quantum phenomenon in correlated materials.

Strain-induced shape transition of VSi2 clusters on Si(111)

We investigated the growth and shape of vanadium silicide clusters on Si(111) using scanning tunneling microscopy, atomic force microscopy and scanning electron microscopy. The deposition of 8 monolayers of vanadium on Si(111) and subsequent annealing at 1200 K leads to the formation of large elongated VSi2 clusters. This is in marked contrast to the deposition of submonolayer amounts of V where the VSi2 islands have a compact shape. The shape transition can be explained by a competition between edge formation and strain relaxation energy terms. The VSi2 clusters, which are aligned along the three high symmetry directions of the Si(111) surface and surrounded by a denuded zone, form a two-dimensional network.

Effect of next-nearest neighbor hopping on the single-particle excitations at finite temperature

In the half-filled one-orbital Hubbard model on a square lattice, we study the effect of next-nearest neighbor hopping on the single-particle spectral function at finite temperature using an exact-diagonalization + Monte-Carlo based approach to the simulation process. We find that the pseudogap-like dip, existing in the density of states in between the Néel temperature T_NTN and a relatively higher temperature scale T^*T*, is accompanied with a significant asymmetry in the hole- and particle-excitation energy along the high-symmetry directions as well as along the normal-state Fermi surface. On moving from (\pi/2, \pi/2π/2,π/2) toward (\pi, 0)(π,0) along the normal state Fermi surface, the hole-excitation energy increases, a behavior remarkably similar to what is observed in the dd-wave state and pseudogap phase of high-T_cTc cuprates, whereas the particle-excitation energy decreases. The quasiparticle peak height is the largest near (\pi/2, \pi/2π/2,π/2) whereas it is the smallest near (\pi, 0)(π,0). These spectral features survive beyond T_NTN. The temperature window T_N ≲ T ≲ T^*TN≲T≲T* shrinks with an increase in the next-nearest neighbor hopping, which indicates that the next-nearest neighbor hopping may not be supportive to the pseudogap-like features.

Acoustic and optical phonons in quasi-two-dimensional MPS3 antiferromagnetic semiconductors

We report the results of the investigation of the acoustic and optical phonons in quasi-two-dimensional antiferromagnetic semiconductors of the transition metal phosphorus trisulfide family with Mn, Fe, Co, Ni, and Cd as metal atoms. The Brillouin–Mandelstam and Raman light scattering spectroscopies were conducted at room temperature to measure the acoustic and optical phonon frequencies close to the Brillouin zone center and the Γ−A high symmetry direction. The absorption and index of refraction were measured in the visible and infrared ranges using the reflectometry technique. We found an intriguing large variation, over ∼28%, in the acoustic phonon group velocities in this group of materials with similar crystal structures. Our data indicate that the full-width-at-half-maximum of the acoustic phonon peaks is strongly affected by the optical properties and the electronic bandgap. The acoustic phonon lifetime extracted for some of the materials was correlated with their thermal properties. The results are important for understanding the layered van der Waals semiconductors and assessing their potential for optoelectronic and spintronic device applications.

Electronic structure and effective mass of pristine and Cl-doped CsPbBr3

Organic–inorganic lead halide perovskites (LHPs) have attracted great interest owing to their outstanding optoelectronic properties. Typically, the underlying electronic structure would determinate the physical properties of materials. But as for now, limited studies have been done to reveal the underlying electronic structure of this material system, comparing to the huge amount of investigations on the material synthesis. The effective mass of the valance band is one of the most important physical parameters which plays a dominant role in charge transport and photovoltaic phenomena. In pristine CsPbBr3, the Fröhlich polarons associated with the Pb–Br stretching modes are proposed to be responsible for the effective mass renormalization. In this regard, it would be very interesting to explore the electronic structure in doped LHPs. Here, we report high-resolution angle-resolved photoemission spectroscopy (ARPES) studies on both pristine and Cl-doped CsPbBr3. The experimental band dispersions are extracted from ARPES spectra along both Γ¯ – M¯ – Γ¯ and X¯ – M¯ – X¯ high symmetry directions. DFT calculations are performed and directly compared with the ARPES data. Our results have revealed the band structure of Cl-doped CsPbBr3 for the first time, which have also unveiled the effective mass renormalization in the Cl-doped CsPbBr3 compound. Doping dependent measurements indicate that the chlorine doping could moderately tune the renormalization strength. These results will help understand the physical properties of LHPs as a function of doping.

Nonsymmorphic symmetry protected nodal lines in layered topological semimetal Ta3GeTe6

Topological semimetals have garnered significant attention due to their distinctive physical properties. However, the ideal material platforms for studying these phenomena remain limited. Here, we report the synthesis and nontrivial topological properties of high-quality van der Waals material Ta3GeTe6. Angle-resolved photoemission spectroscopy studies performed systematically on the as-grown Ta3GeTe6 single crystal along different high-symmetry directions reveal the formation of two nodal lines located near 0.07 and 0.55 eV below EF along the loop X-U-R-S, which arises from band crossings protected by nonsymmorphic symmetry. Furthermore, the nodal lines near the Fermi level along the S-R and S-X directions exhibit a flat feature. The discovery of this material enriches the family of topological nodal line semimetals and provides a promising platform for future investigations into exotic electronic correlation phenomena and potential device applications.

Vertical and lateral manipulation of single Cs atoms on the semiconductor InAs(111)A

The tip of the scanning tunneling microscope can be used to position atoms and molecules on surfaces with atomic scale precision. Here, we report the controlled vertical and lateral manipulation of single Cs atoms on the InAs(111)A surface. The Cs adatom adsorbs on the In-vacancy site of the InAs(111)A—(2x2) surface reconstruction. Lateral manipulation is possible in all directions over the surface, not just along high-symmetry directions. Both pushing and pulling modes were observed in the height profile of the tip. We assembled two artificial structures, demonstrating the reliability of the manipulation procedures. Structures remained intact to a temperature of at least 44 K.

Optical second harmonic generation in anisotropic multilayers with complete multireflection of linear and nonlinear waves using ♯SHAARP.ml package

Optical second harmonic generation (SHG) is a nonlinear optical effect widely used for nonlinear optical microscopy and laser frequency conversion. Closed-form analytical solution of the nonlinear optical responses is essential for evaluating materials whose optical properties are unknown a priori. A recent open-source code, ♯SHAARP.si, can provide such closed form solutions for crystals with arbitrary symmetries, orientations, and anisotropic properties at a single interface. However, optical components are often in the form of slabs, thin films on substrates, and multilayer heterostructures with multiple reflections of both the fundamental and up to ten different SHG waves at each interface, adding significant complexity. Many approximations have therefore been employed in the existing analytical approaches, such as slowly varying approximation, weak reflection of the nonlinear polarization, transparent medium, high crystallographic symmetry, Kleinman symmetry, easy crystal orientation along a high-symmetry direction, phase matching conditions and negligible interference among nonlinear waves, which may lead to large errors in the reported material properties. To avoid these approximations, we have developed an open-source package named Second Harmonic Analysis of Anisotropic Rotational Polarimetry in Multilayers (♯SHAARP.ml). The reliability and accuracy are established by experimentally benchmarking with both the SHG polarimetry and Maker fringes using standard and commonly used nonlinear optical materials as well as twisted 2-dimensional heterostructures.

Spectrum of Gyrotropic Modes and Energy Transfer between Dipolar‐Coupled Magnetic Vortices in a Square Lattice via Triggered Vortex Gyration: A Promise for Spintronics Technology

AbstractThe atoms in a crystal lattice vibrate together and generate vibrational modes known as acoustic and optical phonon modes. Similarly, collective vortex gyration motion can be generated in artificial magnetic vortex lattices, created from periodically arranged nanodisks of specific dimensions and aspect ratios. This study uses micromagnetic simulations to characterize the spectra of gyrotropic modes in two‐dimensional magnetic vortex‐based, tailor‐made square lattice arrangements. The shifting of mode frequencies and mode splitting is witnessed depending on the vortex configurations in the lattice. The obtained gyrotropic modes are analogous to two‐dimensional square lattice structures exhibiting acoustic and optical phonon modes. The dynamics of magnetic vortices are also discussed in the transfer of magnetic energy at three points along the three high symmetry directions of the lattice when the central magnetic vortex is excited by a spin‐polarized current with an excitation frequency equal to the collective gyrotropic frequencies of the vortices. Similar to the higher contribution of acoustic phonon modes in the heat transmission process in a crystal lattice, acoustic‐like gyrotropic modes transfer more energy than optical‐like gyration modes in the vortex lattices. These magnetic metastructures hold promise for novel applications in magnetic waveguides and information processing devices, as well as for advancing spintronics.

Electronic Structure Evolution in the Temperature Range of Metal–Insulator Transitions on Sn/Ge(111)

One‐third of monolayer of Sn adatoms on a Ge(111) substrate forms a 2D triangular lattice with one unpaired electron per site. The system presents a metal–insulator transition when decreasing the temperature and it is known to exhibit strong electron–phonon coupling at 120–150 K. Herein, a study of the electronic band structure for α‐Sn/Ge(111) between 150 and 5 K is reported. Both the experimental Fermi surfaces and the energy dispersions along high symmetry directions as a function of the temperature are presented. At 5 K it is observed a weakly or low‐dispersing spectral feature, exhibiting an extended gap in the reciprocal space. This feature is derived from the topmost occupied band, which is metallic at high temperature and which develops a kink associated with the strong electron–phonon coupling. The spectral evolution is partially explained with an increase of the electron–phonon coupling when decreasing the temperature. The increase of the electron–phonon coupling at low temperatures gives light into the new physics of this 2D system. The bandwidth is progressively reduced when reducing the temperature, enhancing the electronic correlation effects, and triggering the Mott transition.

Hole subband dispersions in a cylindrical Ge nanowire: exact results based on the axial Luttinger–Kohn Hamiltonian

Based on the Luttinger–Kohn Hamiltonian in the axial approximation, the transcendental equations determining the hole subband dispersions in a cylindrical Ge nanowire are analytically derived. These equations are more general than that derived using the spherical approximation, and are suitable to study the growth direction dependence of the subband dispersions. The axial approximation almost gives rise to the accurate low-energy subband dispersions for high-symmetry nanowire growth directions [001] and [111]. The perturbation correction from the non-axial term is negligible for these two directions. The lowest two subband dispersions can be regarded as two shifted parabolic curves with an energy gap at kz=0 for both growth directions [001] and [111]. At the position of the energy gap, the eigenstates for growth direction [111] are inverted in comparison with the normal eigenstates for growth direction [001]. A nanowire growth direction where the energy gap closes at kz=0 is predicted to exist between directions [001] and [111].