Threefold Symmetry Research Articles

(Invited) Unidirectional Epitaxy of TMD Monolayers on Sapphire

Wafer-scale synthesis of semiconducting transition metal dichalcogenide (TMDs) monolayers is of significant interest for device applications to circumvent size limitations associated with the use of exfoliated flakes. Promising results have been demonstrated for epitaxial films deposited by vapor phase techniques such as CVD and MOCVD. However, the three-fold symmetry of TMDs such as MoS2 and WSe2, results in two energetically equivalent domain alignments, often referred to as 0o and 60o domains, when grown on substrates such as c-plane sapphire and graphene. The oppositely oriented domains give rise to inversion domain boundaries (IDBs) upon coalescence which exhibit a metallic character and are generally undesirable. In this study, we demonstrate the epitaxial growth of unidirectional TMD monolayers on 2” diameter c-plane sapphire substrates with a significantly reduced density of inversion domains. Steps on the sapphire surface are shown to break the surface symmetry giving rise to a preferred domain orientation.Metalorganic chemical vapor deposition (MOCVD) was used for the epitaxial growth of WSe2 and WS2 monolayers on c-plane sapphire in a cold-wall horizontal quartz-tube reactor. The as-received sapphire substrates, which are miscut ~0.2o toward <112 ̅0>, consist of steps with sub-1 nm step height separated by 50-70 nm wide terraces. A three-step nucleation-ripening-lateral growth process, carried out at temperatures ranging from 850oC to 1000oC, was used to achieve epitaxial films using W(CO)6, H2Se and H2S as precursors in a H2 carrier gas. Nucleation was observed to occur at the terrace edge and the growing domains align epitaxially with the underlying (0001) sapphire lattice. As a result of the nucleation process, the domains grow with a zig-zag edge facing the terrace edge which imparts a preferential direction to the domains. The percentage of domains with a preferred direction ranges from 75%-86% depending on MOCVD growth conditions. Continued lateral growth for times ranging from 10-30 minutes results in fully coalesced TMD monolayers that are epitaxially oriented on the sapphire, as assessed by in-plane x-ray diffraction, with a reduced density of inversion domain boundaries. The results demonstrate the important role of surface structure in nucleation and epitaxial growth of TMD monolayers.

Single tryptophan Y160W mutant of homooligomeric E. coli purine nucleoside phosphorylase implies that dimers forming the hexamer are functionally not equivalent

E. coli purine nucleoside phosphorylase is a homohexamer, which structure, in the apo form, can be described as a trimer of dimers. Earlier studies suggested that ligand binding and kinetic properties are well described by two binding constants and two sets of kinetic constants. However, most of the crystal structures of this enzyme complexes with ligands do not hold the three-fold symmetry, but only two-fold symmetry, as one of the three dimers is different (both active sites in the open conformation) from the other two (one active site in the open and one in the closed conformation). Our recent detailed studies conducted over broad ligand concentration range suggest that protein–ligand complex formation in solution actually deviates from the two-binding-site model. To reveal the details of interactions present in the hexameric molecule we have engineered a single tryptophan Y160W mutant, responding with substantial intrinsic fluorescence change upon ligand binding. By observing various physical properties of the protein and its various complexes with substrate and substrate analogues we have shown that indeed three-binding-site model is necessary to properly describe binding of ligands by both the wild type enzyme and the Y160W mutant. Thus we have pointed out that a symmetrical dimer with both active sites in the open conformation is not forced to adopt this conformation by interactions in the crystal, but most probably the dimers forming the hexamer in solution are not equivalent as well. This, in turn, implies that an allosteric cooperation occurs not only within a dimer, but also among all three dimers forming a hexameric molecule.

Electric-field-tunable electronic nematic order in twisted double-bilayer graphene

Graphene-based moiré systems have attracted considerable interest in recent years as they display a remarkable variety of correlated phenomena. Besides insulating and superconducting phases in the vicinity of integer fillings of the moiré unit cell, there is growing evidence for electronic nematic order both in twisted bilayer graphene and twisted double-bilayer graphene (tDBG), as signaled by the spontaneous breaking of the threefold rotational symmetry of the moiré superlattices. Here, we combine symmetry-based analysis with a microscopic continuum model to investigate the structure of the nematic phase of tDBG and its experimental manifestations. First, we perform a detailed comparison between the theoretically calculated local density of states and recent scanning tunneling microscopy data (arXiv:2009.11645) to resolve the internal structure of the nematic order parameter in terms of the layer, sublattice, spin, and valley degrees of freedom. We find strong evidence that the dominant contribution to the nematic order parameter comes from states at the moiré scale rather than at the microscopic scale of the individual graphene layers, which demonstrates the key role played by the moiré degrees of freedom and confirms the correlated nature of the nematic phase in tDBG. Secondly, our analysis reveals an unprecedented tunability of the orientation of the nematic director in tDBG by an externally applied electric field, allowing the director to rotate away from high-symmetry crystalline directions. We compute the expected fingerprints of this rotation in both STM and transport experiments, providing feasible ways to probe it. Rooted in the strong sensitivity of the flat bands of tDBG to the displacement field, this effect opens an interesting route to the electrostatic control of electronic nematicity in moiré systems.

Ruga mechanics of soft-orifice closure under external pressure

Here, we report the closure resistance of a soft-material bilayer orifice increases against external pressure, along with ruga-phase evolution, in contrast to the conventional predictions of the matrix-free cylindrical-shell buckling pressure. Experiments demonstrate that the generic soft-material orifice creases in a threefold symmetry at a limit-load pressure of p / μ ≈ 1.20, where μ is the shear modulus. Once the creasing initiates, the triple crease wings gradually grow as the pressure increases until the orifice completely closes at p / μ ≈ 3.0. By contrast, a stiff-surface bilayer orifice initially wrinkles with a multifold symmetry mode and subsequently develops ruga-phase evolution, progressively reducing the orifice cross-sectional area as pressure increases. The buckling-initiation mode is determined by the layer's thickness and stiffness, and the pressure by two types of the layer's instability modes—the surface-layer-wrinkling mode for a compliant and the ring-buckling mode for a stiff layer. The ring-buckling mode tends to set the twofold symmetry for the entire post-buckling closure process, while the high-frequency surface-layer-wrinkling mode evolves with successive symmetry breaking to a final closure configuration of two- or threefold symmetry. Finally, we found that the threefold symmetry mode for the entire closure process provides the orifice's strongest closure resistance, and human saphenous veins remarkably follow this threefold symmetry ruga evolution pathway.

Imaging the Néel vector switching in the monolayer antiferromagnet MnPSe3 with strain-controlled Ising order.

Antiferromagnets are interesting materials for spintronics because of their faster dynamics and robustness against perturbations from magnetic fields. Control of the antiferromagnetic order constitutes an important step towards applications, but has been limited to bulk materials so far. Here, using spatially resolved second-harmonic generation, we show direct evidence of long-range antiferromagnetic order and Ising-type Néel vector switching in monolayer MnPSe3 with large XY anisotropy. In additional to thermally induced switching, uniaxial strain can rotate the Néel vector, aligning it to a general in-plane direction irrespective of the crystal axes. A change of the universality class of the phase transition in the XY model under uniaxial strain causes this emergence of strain-controlled Ising order in the XY magnet MnPSe3. Our discovery is a further ingredient for compact antiferromagnetic spintronic devices in the two-dimensional limit.

Two-fold symmetric superconductivity in few-layer NbSe2

Two-dimensional transition metal dichalcogenides (TMDs) have been attracting significant interest due to a range of properties, such as layer-dependent inversion symmetry, valley-contrasted Berry curvatures, and strong spin-orbit coupling (SOC). Of particular interest is niobium diselenide (NbSe2), whose superconducting state in few-layer samples is profoundly affected by an unusual type of SOC called Ising SOC. Combined with the reduced dimensionality, the latter stabilizes the superconducting state against magnetic fields up to ~35 T and could lead to other exotic properties such as nodal and crystalline topological superconductivity. Here, we report transport measurements of few-layer NbSe$_2$ under in-plane external magnetic fields, revealing an unexpected two-fold rotational symmetry of the superconducting state. In contrast to the three-fold symmetry of the lattice, we observe that the magnetoresistance and critical field exhibit a two-fold oscillation with respect to an applied in-plane magnetic field. We find similar two-fold oscillations deep inside the superconducting state in differential conductance measurements on NbSe$_2$/CrBr$_3$ superconductor-magnet junctions. In both cases, the anisotropy vanishes in the normal state, demonstrating that it is an intrinsic property of the superconducting phase. We attribute the behavior to the mixing between two closely competing pairing instabilities, namely, the conventional s-wave instability typical of bulk NbSe$_2$ and an unconventional d- or p-wave channel that emerges in few-layer NbSe2. Our results thus demonstrate the unconventional character of the pairing interaction in a few-layer TMD, opening a new avenue to search for exotic superconductivity in this family of 2D materials.

Symmetry analysis of tensors in the honeycomb lattice of edge-sharing octahedra

We obtain the most general forms of rank-2 and rank-3 tensors allowed by the crystal symmetries of the honeycomb lattice of edge-sharing octahedra for crystals belonging to different crystallographic point groups, including the monoclinic point group $2/m$ and the trigonal (or rhombohedral) point group $\bar{3}$. Our results are relevant for two-dimensional materials, such as $\alpha$-RuCl$_3$, CrI$_3$, and the honeycomb iridates. We focus on the magnetic-field-dependent thermal conductivity tensor $\kappa_{ij}(\mathbf{H})$, which describes a system's longitudinal and thermal Hall responses, for the cases when the magnetic field is applied along high-symmetry directions, perpendicular to the plane and in the plane. We highlight some unexpected results, such as the equality of fully-longitudinal components to partially-transverse components in rank-3 tensors for systems with three-fold rotational symmetry, and make testable predictions for the thermal conductivity tensor.

Intramolecular O-H⋯S hydrogen bonding in threefold symmetry: Line broadening dynamics from ultrafast 2DIR-spectroscopy and ab initio calculations.

The dynamics of intramolecular hydrogen-bonding involving sulfur atoms as acceptors is studied using two-dimensional infrared (2DIR) spectroscopy. The molecular system is a tertiary alcohol whose donating hydroxy group is embedded in a hydrogen-bond potential with torsional C3-symmetry about the carbon-oxygen bond. The linear and 2DIR-spectra recorded in the OH-stretching region of the alcohol can be simulated very well using Kubo's line shape theory based on the cumulant expansion for evaluating the linear and nonlinear optical response functions. The correlation function for OH-stretching frequency fluctuations reveals an ultrafast component decaying with a time constant of 700 fs, which is in line with the apparent decay of the center line slopes averaged over absorption and bleach/emission signals. In addition, a quasi-static inhomogeneity is detected, which prevents the 2DIR line shape to fully homogenize within the observation window of 4 ps. The experimental data were then analyzed in more detail using a full ab initio approach that merges time-dependent structural information from classical molecular dynamics (MD) simulations with an OH-stretching frequency map derived from density functional theory (DFT). The latter method was also used to obtain a complementary transition dipole map to account for non-Condon effects. The 2DIR-spectra obtained from the MD/DFT method are in good agreement with the experimental data at early waiting delays, thereby corroborating an assignment of the fast decay of the correlation function to the dynamics of hydrogen-bond breakage and formation.

Engineering interface polarization

Interface Physics Many properties can emerge at the interface of van der Waals materials created by rotating the layers of a single material or by creating heterointerfaces between different materials. Akamatsu et al. formed an interface that intentionally broke in-plane inversion symmetry by combining crystals of tungsten diselenide with threefold rotational symmetry and black phosphorus with twofold rotational symmetry. This interface creates in-plane electronic polarization that results in a spontaneous photovoltaic effect only along the polarization direction. This effect was explained in terms of a shift current mechanism. Science , this issue p. [68][1] [1]: /lookup/doi/10.1126/science.aaz9146

A van der Waals interface that creates in-plane polarization and a spontaneous photovoltaic effect.

Van der Waals interfaces can be formed by layer stacking without regard to lattice constants or symmetries of individual building blocks. We engineered the symmetry of a van der Waals interface of tungsten selenide and black phosphorus and realized in-plane electronic polarization that led to the emergence of a spontaneous photovoltaic effect. Spontaneous photocurrent was observed along the polar direction and was absent in the direction perpendicular to it. The observed spontaneous photocurrent was explained by a quantum-mechanical shift current that reflects the geometrical and topological electronic nature of this emergent interface. The present results offer a simple guideline for symmetry engineering that is applicable to a variety of van der Waals interfaces.

Four-dimensional structural description of phase transformations in titanium alloys

Titanium alloys display formation of $$\beta$$ , $$\alpha$$ , and $$\omega$$ phases under different processing conditions. Understanding structural transformations involving these phases on a unified basis and identifying a possible path of transformation seem to be interesting from the point of view of studies on phase transformations. An alternative and unified description pertaining to these is being presented here. This is accomplished by making an appeal to four dimensional structural description. It will be shown that such a model requires an angular distortion ( $$\theta$$ ) around the threefold axis. Such a distortion helps one define a scalar order parameter ( $$\eta$$ ) with values lying between 0 and 1. The two terminal values recover all the symmetrical characteristics of $$\beta$$ phase corresponding to $$\eta =0$$ and that of $$\omega$$ phase pertaining to $$\eta =1$$ . $$\alpha$$ phase formation in this model takes place for $$\theta = \cos ^{-1} (\frac{1}{4})$$ and $$\eta =\frac{13}{16}$$ . It is important to point out here that all structures related to choice of $$\eta$$ always preserve a minimal threefold symmetry. This led us to conclude that formation of trigonal phases may be taking place during the paths of transformation. In addition to all these advantages, the model also provides a general possibility of understanding commensurate and incommensurate modulations along threefold axis.

Cooperative hydrophobic core interactions in the β-trefoil architecture.

Symmetric protein architectures have a compelling aesthetic that suggests a plausible evolutionary process (i.e., gene duplication/fusion) yielding complex architecture from a simpler structural motif. Furthermore, symmetry inspires a practical approach to computational protein design that substantially reduces the combinatorial explosion problem, and may provide practical solutions for structure optimization. Despite such broad relevance, the role of structural symmetry in the key area of hydrophobic core-packing cooperativity has not been adequately studied. In the present report, the threefold rotational symmetry intrinsic to the β-trefoil architecture is shown to form a geometric basis for highly-cooperative core-packing interactions that both stabilize the local repeating motif and promote oligomerization/long-range contacts in the folding process. Symmetry in the β-trefoil structure also permits tolerance towards mutational drift that involves a structural quasi-equivalence at several key core positions.

Spin-Electric Coupling in a Cobalt(II)-Based Spin Triangle Revealed by Electric-Field-Modulated Electron Spin Resonance Spectroscopy.

A cobalt(II)‐based spin triangle shows a significant spin–electric coupling. [Co3(pytag)(py)6Cl3]ClO4⋅3 py crystallizes in the acentric monoclinic space group P21. The intra‐triangle antiferromagnetic interaction, of the order of ca. −15 cm−1 (H=−JSaSb), leads to spin frustration. The two expected energy‐degenerate ground doublets are, however, separated by a few wavenumbers, as a consequence of magnetic anisotropy and deviations from threefold symmetry. The Co3 planes of symmetry‐related molecules are almost parallel, allowing for the determination of the spin–electric properties of single crystals by EFM‐ESR spectroscopy. The spin–electric effect detected when the electric field is applied in the Co3 plane was revealed by a shift in the resonance field. It was quantified as ΔgE/E=0.11×10−9 m V−1, which in terms of frequency corresponds to approximately 0.3 Hz m V−1. This value is comparable to what was determined for a Cu3 triangle despite the antiferromagnetic interaction being 20 times larger for the latter.

Reconstruction of moiré lattices in twisted transition metal dichalcogenide bilayers

An important step in understanding the exotic electronic, vibrational, and optical properties of the moir\'{e} lattices is the inclusion of the effects of structural relaxation of the un-relaxed moir\'{e} lattices. Here, we propose novel structures for twisted bilayer of transition metal dichalcogenides (TMDs). For $\theta\gtrsim 58.4^{\circ}$, we show a dramatic reconstruction of the moir\'{e} lattices, leading to a trimerization of the unfavorable stackings. We show that the development of curved domain walls due to the three-fold symmetry of the stacking energy landscape is responsible for such lattice reconstruction. Furthermore, we show that the lattice reconstruction notably changes the electronic band-structure. This includes the occurrence of flat bands near the edges of the conduction as well as valence bands, with the valence band maximum, in particular, corresponding to localized states enclosed by the trimer. We also find possibilities for other complicated, entropy stabilized, lattice reconstructed structures.

Quantum frequency doubling in the topological insulator Bi2Se3

The nonlinear Hall effect due to Berry curvature dipole (BCD) induces frequency doubling, which was recently observed in time-reversal-invariant materials. Here we report novel electric frequency doubling in the absence of BCD on a surface of the topological insulator Bi2Se3 under zero magnetic field. We observe that the frequency-doubling voltage transverse to the applied ac current shows a threefold rotational symmetry, whereas it forbids BCD. One of the mechanisms compatible with the symmetry is skew scattering, arising from the inherent chirality of the topological surface state. We introduce the Berry curvature triple, a high-order moment of the Berry curvature, to explain skew scattering under the threefold rotational symmetry. Our work paves the way to obtain a giant second-order nonlinear electric effect in high mobility quantum materials, as the skew scattering surpasses other mechanisms in the clean limit.

Modulation of intertube band dispersion relation of carbon nanotube bundles by symmetry and intertube wave function coupling

Based on density functional theory with a local density approximation and the simple tight-binding approximation, we investigated an electronic structure of carbon nanotube bundles in terms of mutual nanotube arrangement. The dispersion relation near the Fermi level along the intertube direction was found to be sensitive to the nanotube species and their mutual orientation within the bundles. Nanotube bundles with three-fold symmetry exhibited a substantial orientation dependence in the band dispersion relation along the intertube direction. The tight-binding calculation and wave function analysis revealed that this orientation dependence arises from the intertube wave function coupling whether a node exists between nanotubes.

Evidence of Potts-Nematic Superfluidity in a Hexagonal sp^{2} Optical Lattice.

As in between liquid and crystal phases lies a nematic liquid crystal, which breaks rotation with preservation of translation symmetry, there is a nematic superfluid phase bridging a superfluid and a supersolid. The nematic order also emerges in interacting electrons and has been found to largely intertwine with multiorbital correlation in high-temperature superconductivity, where Ising nematicity arises from a four-fold rotation symmetry C_{4} broken down to C_{2}. Here, we report an observation of a three-state (Z_{3}) quantum nematic order, dubbed "Potts-nematicity", in a system of cold atoms loaded in an excited band of a hexagonal optical lattice described by an sp^{2}-orbital hybridized model. This Potts-nematic quantum state spontaneously breaks a three-fold rotation symmetry of the lattice, qualitatively distinct from the Ising nematicity. Our field theory analysis shows that the Potts-nematic order is stabilized by intricate renormalization effects enabled by strong interorbital mixing present in the hexagonal lattice. This discovery paves a way to investigate quantum vestigial orders in multiorbital atomic superfluids.

A chiral uranyl-Kemp's tricarboxylate cubic framework: structure-directing effect of counterions with three-fold rotational symmetry.

In the presence of PPh3Me+ cations, Kemp's tricarboxylate (kta3-) complexes the uranyl cation to give [PPh3Me][UO2(kta)] (1), a triperiodic framework with cubic symmetry and srs topology. The PPh3Me+ cation is held by weak interactions into cavities with matching three-fold rotational symmetry. Comparison with the diperiodic hemi-hydrate polymorph previously reported points to the disrupting role of OHO hydrogen bonds in the latter.

From starphenes to non-benzenoid linear conjugated polymers by substrate templating.

Combining on-surface synthetic methods with the power of scanning tunneling microscopy to characterize novel materials at the single molecule level, we show how to steer the reactivity of one anthracene-based precursor towards different product nanostructures. Whereas using a Au(111) surface with three-fold symmetry results in the dominant formation of a starphene derivative, the two-fold symmetry of a reconstructed Au(110) surface allows the selective growth of non-benzenoid linear conjugated polymers. We further assess the electronic properties of each of the observed product structures via tunneling spectroscopy and DFT calculations, altogether advancing the synthesis and characterization of molecular structures of notable scientific interest that have been only scarcely investigated to date, as applies both to starphenes and to non-benzenoid conjugated polymers.

Molecular dynamics simulation of nanoindentation on c-plane sapphire

Sapphire is a typical hard and brittle material, making it difficult to investigate its plastic deformation behavior using conventional mechanical experimental methods. Although nanoindentation and transmission electron microscopy (TEM) have been used to characterize its plastic deformation behavior, the dynamic deformation process remains difficult to observe, which limits our understanding of the mechanism of plastic deformation in sapphire. In this study, the plastic deformation mechanism of c-plane (0001) sapphire under nanoindentation was simulated using the molecular dynamics (MD) method. The relationship between the surface morphology of the nano-indented c-plane sapphire and the activation process of slip systems was studied. The simulation results are in good agreement with the nanoindentation experiment results. A hexagonal pattern was induced on the indented surface of c-plane sapphire by prismatic slip activation. A three-fold symmetry pile-up was observed because of rhombohedral slip activation, and the twin/slip systems of the r-plane and a-plane provided a nucleation condition for crack formation on c-plane sapphire in the experiments. In addition, the MD simulation results showed that the O atomic arrangement transforms from an HCP structure to an FCC structure, mostly between the two basal dislocation lines. This change in the arrangement of O atoms is attributed to the basal dislocations in sapphire. The study is expected to broaden our understanding of the deformation mechanism in c-plane sapphire and to help facilitate its analysis on a nanometer-scale.