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In this paper we first give a necessary and sufficient condition for a group $G$ generated by $n$ elements to be a square commutative group and prove $G$ is a square commutative group if and only if $\widehat{G}$ is an abelian group, then we give conditions for a group generated by two elements, with additional conditions, to be a square commutative group., Comment: 10 pages, added Corollary 2.8, Corollary 3.5 and Theorem 4.8

Two-dimensional (2D) multiferroic materials with coexisting ferroelectricity and ferromagnetism have garnered substantial attention for their intriguing physical properties and diverse promising applications in spintronics. For example, multiferroic materials with electronically controlled broken central symmetry provide a versatile platform for designing and manipulating topological skyrmions and diverse spintronic applications. Here, we investigate the complex magnetic properties of room-temerature multiferroic material MoPtGe2S6 and its electrical control of topological skyrmions using first-principles calculations and atomistic micromagnetic simulations. A sizable Dzyaloshinskii-Moriya interaction (DMI) (2.1 meV) is found in the multiferroic material MoPtGe2S6 with an electrically polarized ground state. The magnetic skyrmions can be stabilized in monolayer MoPtGe2S6 under zero magnetic field, and the chirality of skyrmions can be reversed with electric field-induced flipping of electrical polarization due to the reversed chirality of the DMI. Furthermore, an external magnetic fielc can reverse the magnetization direction and topological charge of the skyrmions as well as tune the size of skyrmions. These results demonstrate that the monolayer MoPtGe2S6 can enrich the 2D skyrmion community and pave the way for electronically controlled spintronic devices., Comment: 24 pages, 7 figures

In the absence of time-reversal invariance, metals without inversion symmetry may exhibit nonreciprocal charge transport -- a magnetochiral anisotropy that manifests as unequal electrical resistance for opposite current flow directions. If superconductivity also sets in, the charge transmission may become dissipationless in one direction while remaining dissipative in the opposite, thereby realizing a superconducting diode. Through both direct-current and alternating-current measurements, we study the nonreciprocal effects in thin films of the noncentrosymmetric superconductor T$_d$-MoTe\textsubscript{2} with disorders. We observe nonreciprocal superconducting critical currents with a diode efficiency close to 20\%~, and a large magnetochiral anisotropy coefficient up to $\SI{5.9e8}{\per\tesla\per\ampere}$, under weak out-of-plane magnetic field in the millitesla range. The great enhancement of rectification efficiency under out-of-plane magnetic field is likely abscribed to the vortex ratchet effect, which naturally appears in the noncentrosymmetric superconductor with disorders. Intriguingly, unlike the finding in Rashba systems, the strongest in-plane nonreciprocal effect does not occur when the field is perpendicular to the current flow direction. We develop a phenomenological theory to demonstrate that this peculiar behavior can be attributed to the asymmetric structure of spin-orbit coupling in T$_d$-MoTe\textsubscript{2}. Our study highlights how the crystallographic symmetry critically impacts the nonreciprocal transport, and would further advance the research for designing the superconducting diode with the best performance., Comment: 7 pages, 5figures

Two-dimensional layered semiconductors have recently emerged as attractive building blocks for next-generation low-power non-volatile memories. However, challenges remain in the controllable sub-micron fabrication of bipolar resistively switching circuit components from these novel materials. Here we report on the scalable experimental realisation of lateral on-dielectric memtransistors from monolayer single-crystal molybdenum disulfide (MoS2) utilising a focused helium ion beam. Site-specific irradiation with the probe of a helium ion microscope (HIM) allows for the creation of charged defects in the MoS2 lattice. The reversible drift of these locally seeded defects in the applied electric field modulates the resistance of the semiconducting channel, enabling versatile memristive functionality on the nanoscale. We find the device can reliably retain its resistance ratios and set biases for hundreds of switching cycles at sweep frequencies of up to 2.9 V/s with relatively low drain-source biases. We also demonstrate long-term potentiation and depression with sharp habituation that promises application in future neuromorphic architectures. This work advances the down-scaling progress of memristive devices without sacrificing key performance parameters such as power consumption or its applicability for synaptic emulation., Comment: 37 pages, 5 main figures, 14 supplementary figures

Precise and scalable defect engineering of 2D nanomaterials is acutely sought-after in contemporary materials science. Here we present defect engineering in monolayer graphene and molybdenum disulfide (MoS$_2$) by irradiation with noble gas ions at 30 keV. Two ion species of different masses were used in a gas field ion source microscope: helium (He$^+$) and neon (Ne$^+$). A detailed study of the introduced defect sizes and resulting inter-defect distance with escalating ion dose was performed using Raman spectroscopy. Expanding on existing models, we found that the average defect size is considerably smaller for supported than freestanding graphene and that the rate of defect production is larger. We conclude that secondary atoms from the substrate play a significant role in defect production, creating smaller defects relative to those created by the primary ion beam. Furthermore, a similar model was also applied to supported MoS$_2$, another promising member of the 2D material family. Defect yields for both ions were obtained for MoS$_2$, demonstrating their different interaction with the material and facilitating comparison with other irradiation conditions in the literature., Comment: 11 pages, 6 figures

Precise tunability of electronic properties of 2D nanomaterials is a key goal of current research in this field of materials science. Chemical modification of layered transition metal dichalcogenides leads to the creation of heterostructures of low-dimensional variants of these materials. In particular, the effect of oxygen-containing plasma treatment on molybdenum disulfide (MoS$_2$) has long been thought to be detrimental to the electrical performance of the material. Here we show that the mobility and conductivity of MoS$_2$ can be precisely controlled and improved by systematic exposure to oxygen:argon plasma, and characterise the material utilising advanced spectroscopy and microscopy. Through complementary theoretical modelling which confirms conductivity enhancement, we uncover the role of a two-dimensional phase of molybdenum trioxide (2D-MoO$_3$) in improving the electronic behaviour of the material. Deduction of the beneficial role of MoO$_3$ will serve to open the field to new approaches with regard to the tunability of 2D semiconductors by their low-dimensional oxides in nano-modified heterostructures., Comment: 11 pages, 5 figures

Graphene is a promising candidate to succeed silicon based devices and doping holds the key to graphene electronics. Conventional doping methods through surface functionalization or lattice modification are effective in tuning carrier densities. These processes, however, lead to degradation of device performance because of structural defect creation. A challenge remains to controllably dope graphene while preserving its superlative properties. Here we show a novel method for tunable and erasable doping of on-chip graphene, realized by using a focused electron beam. Our results demonstrate site-specific control of carrier type and concentration achievable by modulating the charge distribution in the dielectric substrate. Thereby, the structural integrity and electrical performance of graphene are preserved, and the doping states are rewritable. Different logic operations were thus implemented in a single graphene sheet. By extending this method to other two-dimensional materials, this work lays out a blueprint for powerful yet simple means of incorporating two-dimensional materials into prospective electronic technologies., Comment: 19 pages, 5 figures