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Commensuration Effects in Layered Nanoparticle Solids.
We have developed a hierarchical nanoparticle transport simulator (HINTS) and adapted it to study commensuration effects in two classes of nanoparticle (NP) solids: (1) a bilayer NP solid (BNS) with an energy offset and (2) a BNS as part of a field effect transistor (FET). HINTS integrates the ab initio characterization of single NPs with the phonon-assisted tunneling transition model of the NP-NP transitions into a kinetic Monte Carlo based simulation of the charge transport in NP solids. First, we studied a BNS with an interlayer energy offset Delta, possibly caused by a fixed electric field. Our results include the following. First, in the independent energy-offset model, we observed the emergence of commensuration effects when scanning the electron filling factor (FF) across integer values. These commensuration effects were profound as they reduced the mobility by several orders of magnitude. We analyzed these commensuration effects in a five-dimensional parameter space, as a function of the on-site charging energy E-C, energy offset Delta, disorder D, the electron FF, and temperature k(B)T. We demonstrated the complexity of our model by showing that at integer FFs commensuration effects are present in some regions of the parameter space, while they vanish in other regions, thus defining distinct dynamical phases of the model. We determined the phase boundaries between these dynamical phases. Second, using these results as a foundation, we shifted our focus to the experimentally much-studied NP-FETs. NP-FETs are also characterized by an interlayer energy offset Delta, which, in contrast to our first model, is set by the gate voltage V-G and thereby related to the electron FF. We repeated many of our simulations and again demonstrated the emergence of commensuration effects and distinct dynamical phases in these NP-FETs. Notably, the commensuration effects in the NP-FETs showed many similarities to those in the independent energy-offset BNS.
Commensuration effects on skyrmion Hall angle and drag for manipulation of skyrmions on two-dimensional periodic substrates
Here, we examine the dynamics of an individually driven skyrmion moving through a background lattice of skyrmions coupled to a 2D periodic substrate as we vary the ratio of the number of skyrmions to the number of pinning sites across commensurate and incommensurate conditions. As the skyrmion density increases, the skyrmion Hall angle is nonmonotonic, dropping to low or zero values in commensurate states and rising to an enhanced value in incommensurate states. Under commensuration, the driven skyrmion is channeled by a symmetry direction of the pinning array and exhibits an increased velocity. At fillings for which the skyrmion Hall angle is zero, the velocity has a narrow band noise signature, while for incommensurate fillings, the skyrmion motion is disordered and the velocity noise is broad band. Under commensurate conditions, multi-step depinning transitions appear and the skyrmion Hall angle is zero at low drives but becomes finite at higher drives, while at incommensurate fillings there is only a single depinning transition. As the gyrotropic component of the skyrmion dynamics, called the Magnus force, increases, peaks in the velocity that appear in commensurate regimes cross over to dips, and new types of directional locking effects can arise in which the skyrmion travels along other symmetry directions of the background lattice. At large Magnus forces, and particularly at commensurate fillings, the driven skyrmion can experience a velocity boost in which the skyrmion moves faster than the applied drive due to the alignment of the Magnus-induced velocity with the driving direction. In some cases, an increase of the Magnus force can produce regimes of enhanced pinning when the skyrmion is forced to move along a nonsymmetry direction of the periodic pinning array. This is in contrast to systems with random pinning, where increasing the Magnus force generally reduces the pinning effect. We demonstrate these dynamics for both square and triangular substrates and map out the different regimes as a function of filling fraction, pinning force, and the strength of the Magnus force in a series of dynamic phase diagrams.
Active matter commensuration and frustration effects on periodic substrates
In this report we show that self-driven particles coupled to a periodic obstacle array exhibit active matter commensuration effects that are absent in the Brownian limit. As the obstacle size is varied for sufficiently large activity, a series of commensuration effects appear in which the motility induced phase separation produces commensurate crystalline states, while for other obstacle sizes we find frustrated or amorphous states. The commensuration effects are associated with peaks in the amount of sixfold ordering and the maximum cluster size. When a drift force is added to the system, the mobility contains peaks and dips similar to those found in transport studies for commensuration effects in superconducting vortices and colloidal particles.
Dynamical commensuration effect in a two-dimensional Yukawa solid modulated by periodic substrates
Transverse depinning dynamics of a periodic-square-substrate modulated two-dimensional dusty plasma solid driven by a constant force in the longitudinal direction are investigated using Langevin dynamical simulations. When the commensuration ratio (the number ratio of particles to substrate potential wells) is increased, there is a nonmonotonic variation of the critical transverse depinning force, and the local maxima and minima of the critical transverse depinning force precisely correspond to the dynamical commensurate and incommensurate conditions, respectively. Here, the dynamical commensuration effect is also clearly visible in the stable one-dimensional channel particle trajectories and the highly ordered structure, while both the particle trajectories and the structure are more disordered under the incommensurate conditions. The nonmonotonic variation of the critical transverse depinning force is attributed to the stability of the lattice structure at specific commensuration ratios.
Moiré commensurability and the quantum anomalous Hall effect in twisted bilayer graphene on hexagonal boron nitride
The quantum anomalous Hall (QAH) effect is sometimes observed in twisted bilayer graphene (tBG) when it is nearly aligned with an encapsulating hexagonal boron nitride (hBN) layer. We propose that the appearance or absence of the QAH effect in individual devices could be related to commensurability between the graphene/graphene and graphene/hBN moiré patterns. We identify a series of points in the (θ GG , θ GBN ) twist-angle space at which the two moiré patterns are commensurate, allowing moiré band theory to be applied, and we show that the band Chern numbers are in this case sensitive to a rigid in-plane hBN displacement. Given this property, we argue that the QAH effect is likely only when (i) the (θ GG , θ GBN ) twist-angle-pair is close enough to a commensurate point that the two moiré patterns yield a supermoiré pattern with a sufficiently long length scale, and (ii) the supermoiré has a percolating topologically nontrivial QAH phase. For twist angles far from commensurability, the hBN layer acts as a source of disorder that can destroy the QAH effect. Our proposal can explain a number of current experimental observations. Finally, further experimental studies that can test this proposal more directly are suggested.
Electronic coherence evolution at the nearly commensurate-incommensurate CDW boundary of 1T−TaS 2
Transition-metal dichalcogenides host a variety of charge-density-wave phases that couple lattice, charge, and correlation effects. In 1𝑇−TaS 2 , the commensurate and nearly commensurate states are well characterized, yet the transition near 350 K into the incommensurate phase has lacked direct momentum-resolved insight. In this work, we use temperature-dependent angle-resolved photoemission spectroscopy to track the electronic structure across this transition. We observe a suppression of quasiparticle spectral weight at the Brillouin-zone center, coincident with the transport anomaly, but without clear evidence of a full band-gap opening. The transition appears to involve momentum-dependent redistribution of spectral weight, consistent with a loss of coherence that reshapes the Fermi surface while leaving conduction dispersions largely intact. These results suggest that the nearly commensurate–incommensurate transition may not align with a conventional metal-insulator transition picture, but rather as an electronic reconstruction driven by loss of coherence. Our work provides new microscopic insight into the resistivity anomaly near room temperature and may guide design principles for collective electronic switching in transition-metal dichalcogenides.
Heterobilayer moiré magnets: Moiré skyrmions and commensurate-incommensurate transitions
In this work, we study untwisted heterobilayers of ferromagnetic and antiferromagnetic van der Waals materials, with in particular a Dzyaloshinskii-Moriya interaction in the ferromagnetic layer. A continuum low-energy field theory is utilized to study such systems. We develop a phase diagram as a function of the strength of interlayer exchange and Dzyaloshinskii-Moriya interactions, combining perturbative and strong coupling analyses with numerical simulations using Landau-Lifshitz-Gilbert equations. Various moiré-periodic commensurate phases are found, and the commensurate-incommensurate transition is discussed. Among the commensurate phases, we observe an interesting skyrmion lattice phase wherein each moiré unit-cell hosts one skyrmion.
Deconvoluting the Magnetic Structure of the Commensurately Modulated Quinary Zintl Phase Eu 11– x Sr x Zn 4 Sn 2 As 12
The structure, magnetic properties, and 151 Eu and 119 Sn Mössbauer spectra of the solid-solution Eu 11– x Sr x Zn 4 Sn 2 As 12 are presented. A new commensurately modulated structure is described for Eu 11 Zn 4 Sn 2 As 12 ( R 3 m space group, average structure) that closely resembles the original structural description in the monoclinic C 2/ c space group with layers of Eu, puckered hexagonal Zn 2 As 3 sheets, and Zn 2 As 6 ethane-like isolated pillars. The solid-solution Eu 11– x Sr x Zn 4 Sn 2 As 12 (0 < x < 10) is found to crystallize in the commensurately modulated R 3 space group, related to the parent phase but lacking the mirror symmetry. Eu 11 Zn 4 Sn 2 As 12 orders with a saturation plateau at 1 T for 7 of the 11 Eu 2+ cations ferromagnetically coupled (5 K) and shows colossal magnetoresistance at 15 K. The magnetic properties of Eu 11 Zn 4 Sn 2 As 12 are investigated at higher fields, and the ferromagnetic saturation of all 11 Eu 2+ cations occurs at ~8 T. The temperature-dependent magnetic properties of the solid solution were investigated, and a nontrivial structure–magnetization correlation is revealed. The temperature-dependent 151 Eu and 119 Sn Mössbauer spectra confirm that the europium atoms in the structure are all Eu 2+ and that the tin is consistent with an oxidation state of less than four in the intermetallic region. The spectral areas of both Eu(II) and Sn increase at the magnetic transition, indicating a magnetoelastic effect upon magnetic ordering.
Locally commensurate charge-density wave with three-unit-cell periodicity in YBa 2 Cu 3 O y
In order to identify the mechanism responsible for the formation of charge-density waves (CDW) in cuprate superconductors, it is important to understand which aspects of the CDW’s microscopic structure are generic and which are material-dependent. Here, we show that, at the local scale probed by NMR, long-range CDW order in YBa 2 Cu 3 O y is unidirectional with a commensurate period of three unit cells ( λ = 3 b ), implying that the incommensurability found in X-ray scattering is ensured by phase slips (discommensurations). Furthermore, NMR spectra reveal a predominant oxygen character of the CDW with an out-of-phase relationship between certain lattice sites but no specific signature of a secondary CDW with λ = 6 b associated with a putative pair-density wave. These results shed light on universal aspects of the cuprate CDW. In particular, its spatial profile appears to generically result from the interplay between an incommensurate tendency at long length scales, possibly related to properties of the Fermi surface, and local commensuration effects, due to electron-electron interactions or lock-in to the lattice.
Possible quadrupole-order-driven commensurate-incommensurate phase transition in B20 CoGe
For this study, the B20-type cobalt germanide CoGe was investigated by measuring the specific heat, resistivity, and 59 Co nuclear magnetic resonance (NMR). We observed a phase transition at T Q = 13.7 K, evidenced by a very narrow peak of the specific heat and sharp changes of the nuclear spin-spin (T 2 –1 ) and spin-lattice ( T 1 – 1 ) relaxation rates. The fact that the entropy release is extremely small and the Knight shift is almost independent of temperature down to low temperatures as anticipated in a paramagnetic metal indicates that the T Q transition is of nonmagnetic origin. In addition, we detected a crossover scale T 0 ~ 30 K below which the resistivity and the NMR linewidth increase, and T 1 –1 is progressively distributed in space, that is, a static and dynamical spatial inhomogeneity develops. While the order parameter for the T Q transition remains an open question, a group-theoretical analysis suggests that the finite electric quadrupole density arising from the low local site symmetry at cobalt sites could drive the crystal symmetry lowering from the P2 1 3 symmetry that is commensurate to the R3 symmetry with an incommensurate wave vector, which fairly well accounts for the T Q transition. The quadrupole-order-driven commensurate-incommensurate phase transition may be another remarkable phenomenon arising from the structural chirality inherent in the noncentrosymmetric B20 family.
Electric-field-tunable band gap in commensurate twisted bilayer graphene
Bernal bilayer graphene exhibits a band gap that is tunable through the infrared with an electric field. In this study, we show that sublattice odd commensurate twisted bilayer graphene (C-TBG) exhibits a band gap that is tunable through the terahertz with an electric field. We show that from the perspective of terahertz optics the sublattice odd and even forms of C-TBG are “inflated” versions of Bernal and AA-stacked bilayer graphene, respectively, with energy scales reduced by a factor of 110 for the 21.79° commensurate unit cell. This lower energy scale is accompanied by a correspondingly smaller gate voltage, which means that the strong-field regime is more easily accessible than in the Bernal case. Finally, we show that the interlayer coherence energy is a directly accessible experimental quantity through the position of a power-law divergence in the optical conductivity.
Dative Epitaxy of Commensurate Monocrystalline Covalent van der Waals Moiré Supercrystal
Realizing van der Waals (vdW) epitaxy in the 1980s represents a breakthrough that circumvents the stringent lattice matching and processing compatibility requirements in conventional covalent heteroepitaxy. However, due to the weak vdW interactions, there is little control over film qualities by the substrate. Typically, discrete domains with a spread of misorientation angles are formed, limiting the applicability of vdW epitaxy. In this study, the epitaxial growth of monocrystalline, covalent Cr 5 Te 8 2D crystals on monolayer vdW WSe 2 by chemical vapor deposition is reported, driven by interfacial dative bond formation. The lattice of Cr 5 Te 8 , with a lateral dimension of a few tens of micrometers, is fully commensurate with that of WSe 2 via 3 × 3 (Cr 5 Te 8 )/7 × 7 (WSe 2 ) supercell matching, forming a single-crystalline moiré superlattice. This work establishes a conceptually distinct paradigm of thin-film epitaxy, termed “dative epitaxy”, which takes full advantage of covalent epitaxy with chemical bonding for fixing the atomic registry and crystal orientation, while circumventing its stringent lattice matching and processing compatibility requirements; conversely, it ensures the full flexibility of vdW epitaxy, while avoiding its poor orientation control. Cr 5 Te 8 2D crystals grown by dative epitaxy exhibit square magnetic hysteresis, suggesting minimized interfacial defects that can serve as pinning sites.
Transition from an incommensurate spin density wave to a commensurate magnetic order in a triangular lattice compound Ho 2 PdAl 6 Ge 4
Rare-earth (RE) intermetallics on a triangular lattice are promising candidates for generating interesting magnetic phases due to the complex interplay between Ruderman-Kittel-Kasuya-Yoshida (RKKY) interaction and geometrical frustration. Here, in this work, we report the exotic magnetic structure of a layered compound Ho 2 PdAl 6 Ge 4 with triangular lanthanide nets. Magnetization and heat capacity measurements in zero magnetic field reveal two magnetic phase transitions at T N1 = 10.8 K and T N2 = 6.0 K. Neutron powder diffraction demonstrates a commensurate antiferromagnetic phase with k 1 = (0, 0, 1.5) below T N2 . With increasing temperature, another incommensurate vector appears and therefore, the magnetic structure of the intermediate state is identified as an unusual incommensurate spin density wave with two propagation vectors k 1 = (0, 0, 1.5) and k 2 = (0.0492, 0.0492, 1.5). The magnetic moments in the intermediate state rotate continuously and form an unusual S-shaped wave arrangement in the ab plane, sharing similarities with typical cycloid and helix magnetic orders. These results identify Ho 2 PdAl 6 Ge 4 as a candidate for exploring field-induced topological magnetic phases such as skyrmions, opening the way for further investigations on the family of RE 2 PdAl 6 Ge 4 materials.
Gapped commensurate antiferromagnetic response in a strongly underdoped model cuprate superconductor
It is a distinct possibility that spin fluctuations are the pairing interactions in numerous unconventional superconductors. In the high-transition-temperature (high-T c ) cuprates, superconductivity emerges upon doping antiferromagnetic Mott insulators, and spin fluctuations might furthermore drive unusual pseudogap phenomena. Here we use magnetic neutron scattering to study the highly underdoped cuprate HgBa 2 CuO 4+δ (hole concentration p ≈ 0.064). In contrast to prior results for other underdoped cuprates, we find no evidence of incommensurate magnetic order associated with spin-density-wave or stripe correlations. Instead, the antiferromagnetic response in both the superconducting and pseudogap states is gapped below Δ AF ≈ 6 meV, commensurate over a wide energy range, and disperses above about 55 meV. Given the pristine nature of HgBa 2 CuO 4+δ , which exhibits high structural symmetry and minimal point disorder effects, this behavior likely signifies the unmasked response of the underlying CuO 2 planes near the Mott-insulating state. These results serve as a benchmark for a refined theoretical understanding of the cuprates.
Tuning commensurability in twisted van der Waals bilayers
Moiré superlattices based on van der Waals bilayers created at small twist angles lead to a long wavelength pattern with approximate translational symmetry. At large twist angles ( θ t ), moiré patterns are, in general, incommensurate except for a few discrete angles. Here we show that large-angle twisted bilayers offer distinctly different platforms. More specifically, by using twisted tungsten diselenide bilayers, we create the incommensurate dodecagon quasicrystals at θ t = 30° and the commensurate moiré crystals at θ t = 21.8° and 38.2°. Further, valley-resolved scanning tunnelling spectroscopy shows disparate behaviours between moiré crystals (with translational symmetry) and quasicrystals (with broken translational symmetry). In particular, the K valley shows rich electronic structures exemplified by the formation of mini-gaps near the valence band maximum. These discoveries demonstrate that bilayers with large twist angles offer a design platform to explore moiré physics beyond those formed with small twist angles.
Commensurate states and pattern switching via liquid crystal skyrmions trapped in a square lattice
Using continuum based simulations we show that a rich variety of skyrmion liquid crystal states can be realized in the presence of a periodic obstacle array. As a function of the number of skyrmions per obstacle we find hexagonal, square, dimer, trimer and quadrimer ordering, where the n-mer structures are a realization of a molecular crystal state of skyrmions. As a function of external field and obstacle radius we show that there are transitions between the different crystalline states as well as mixed and disordered structures. Here, we discuss how these states are related to commensurate effects seen in other systems, such as vortices in type-II superconductors and colloids interacting with two dimensional substrates.
Correlated insulating phases of twisted bilayer graphene at commensurate filling fractions: A Hartree-Fock study
Motivated by the recently observed insulating states in twisted bilayer graphene, we study the nature of the correlated insulating phases of the twisted bilayer graphene at commensurate filling fractions. Iin this work, we use the continuum model and project the Coulomb interaction onto the flat bands to study the ground states by using a Hartree-Fock approximation. In the absence of the hexagonal boron nitride substrate, the ground states are the intervalley coherence states at charge neutrality (filling $\textit{ν}$ = 0 , or four electrons per moiré cell) and at $\textit{ν}$ = –1/4 and –1/2 (three and two electrons per cell, respectively) and the $C_2\mathcal{T}$ symmetry-broken state at $\textit{ν}$ = –3/4 (one electron per cell). The hexagonal boron nitride substrate drives the ground states at all $\textit{ν}$ into $C_2\mathcal{T}$ symmetry broken-states. Our results provide good reference points for further study of the rich correlated physics in the twisted bilayer graphene.