First-principles calculations, within the framework of density functional theory, have been performed on the well-studied 2H and the less explored 1T$^{\prime}$ phase of single-layer MoS$_{2}$. We have addressed the strain-induced tunability of the electronic and phononic properties of both phases, and compared their stability against the applied strain. By considering a large number of strain profiles for both tensile and compressive stress, we have found that the electronic properties of both 2H and 1T$^{\prime}$ phases are sensitive to the direction of the applied strain and can be tuned in a controlled way. For the 2H phase, in most cases, a direct to indirect band gap transition at lower strain and a semiconductor to metal transition at higher strain is observed. The applied strain destroys the semimetallic nature of the 1T$^{\prime}$ phase via the overlapping of the bulk states with the topologically protected edge states. Significant strain-induced changes in the phononic properties, in the frequency of the phonon branches, as well as in the nature of the dispersion curves, are observed. A systematic change in the frequency of the optical phonon modes at the zone centre is seen for both phases. With increasing strain, the out-of-plane acoustic mode (ZA) turns imaginary, indicating a possibility of phase transition or instability of the crystal structure. The 2H phase appears to withstand a larger amount of strain and therefore possesses better stability compared to the 1T$^{\prime}$ phase since the imaginary branch starts to appear at much lower values of strain in the latter case. We highlight the significance of strain engineering in tuning the electronic and phononic properties and the safe limit of the strain application in different polymorphs of single-layer MoS$_{2}$.
Recently a stable monolayer of antimony in buckled honeycomb structure called antimonene was successfully grown on 3D topological insulator Bi$_2$Te$_3$ and Sb$_2$Te$_3$, which displays semiconducting properties. By first principle calculations, we systematically investigate the phononic, electronic and optical properties of $α-$ and $β-$ allotropes of monolayer arsenene/antimonene. We investigate the dynamical stabilities of these four materials by considering the phonon dispersions. The obtained electronic structures reveal the direct band gap of monolayer $α-$As/Sb and indirect band gap of …
Molecular dynamics simulations of a Xe monolayer sliding on Ag(001) and Ag(111) are carried out in order to ascertain the microscopic origin of friction. For several values of the electronic contribution to the friction of individual Xe atoms, the intra-overlayer phonon dissipation is calculated as a function of the corrugation amplitude of the substrate potential, which is a pertinent parameter to consider. Within the accuracy of the numerical results and the uncertainty with which the values of the relevant parameters are known at present, we conclude that electronic and phononic dissipation…
We investigate the electric response of chiral phonons on the low-buckled group-IVA monolayers by performing first-principles calculations. The vertical electric field breaks the degeneracy of phonon modes at high-symmetry $\pm K$ points of the phonon Brillouin zone, and the size of the phononic gap is proportional to the strength of the electric field. The gapped phonon modes at $\pm K$ possess chiralities with considerable phonon circular polarizations and discrete phonon pseudoangular momenta. The chiralities of phonons are robust against the variation of the field strength, but reversed by…
The field of topological mechanics has recently emerged due to the interest in robustly transporting various types of energy in a flaw and defect-insensitive fashion. While there have been a significant number of studies based on discovering and proposing topological materials and structures, very few have focused on tuning the resulting topological bandgaps, which is critical because the bandgap frequency is fixed once the structure has been fabricated. Here, we perform both lattice dynamical calculations and molecular dynamical simulations to investigate strain effects on the phononic topolo…
In an outstanding experimental advance in the field of two-dimensional nanomaterials, cuprous iodide (CuI) and silver iodide (AgI) monolayers have been grown via a novel graphene encapsulation synthesis approach [Adv.Mater.2022, 34, 2106922]. Inspired by this accomplishment, we conduct first-principles calculations to investigate the elastic, phononic thermal transport, electronic, and optical properties of the native CuI and AgI and Janus Cu2BrI and Ag2BrI monolayers. Electronic and excitonic optical properties are elaborately studied using the many-body perturbation theory on the basis of GW…
Phosphorene, an emerging elemental two-dimensional (2D) direct band gap semiconductor with fascinating structural and electronic properties distinctively different from other 2D materials such as graphene and MoS2, is promising for novel nanoelectronic and optoelectronic applications. Phonons, as one of the most important collective excitations, are at the heart for the device performance, as their interactions with electrons and photons govern the carrier mobility and light-emitting efficiency of the material. Here, through a detailed first-principles study, it is demonstrated that monolayer …
Anisotropy is a general feature in materials. Strong anisotropy could lead to interesting physical properties and useful applications. Here, based on first-principles calculations and theoretical analysis, we predict a stable two-dimensional (2D) material---the monolayer MoOCl$_2$, and show that it possesses intriguing properties related to its high anisotropy. Monolayer MoOCl$_2$ can be readily exfoliated from the van der Waals layered bulk, which has already been synthesized. We show that a high in-plane anisotropy manifests in the structural, phononic, mechanical, electronic, and optical pr…
Germanium selenide (GeSe) is a unique two-dimensional (2D) material showing various polymorphs stable at ambient condition. Recently, a new phase with a layered hexagonal lattice (γ-GeSe) was synthesized with ambient stability and extraordinary electronic conductivity even higher than graphite while its monolayer is semiconducting. In this work, via using first-principles derived force constants and Boltzmann transport theory we explore the lattice thermal conductivity (κ_l) of the monolayer γ-GeSe, together with a comparison with monolayer α-GeSe and β-GeSe. The κ_l of γ-phase is relatively l…
Transition Metal Dichalcogenide (TMD) monolayers have most widely studied due to their unique physical properties. Recently, Janus TMD Monolayer MoSSe with sandwiched S-Mo-Se structure has been synthesized by replacing the top S atomic layer in $\mathrm{MoS_2}$ with Se atoms. In this work, we systematically investigate the phonon transport and lattice thermal conductivity ($κ_L$) of MoSSe monolayer by first-principles calculations and linearized phonon Boltzmann equation within the single-mode relaxation time approximation (RTA). Calculated results show that the $κ_L$ of MoSSe monolayer is ver…
Intrinsic electrical and excitonic properties of monolayer transition metal dichalcogenides are studied on CYTOP fluoropolymer substrates with greatly suppressed unintentional doping and dielectric screening. Ambipolar transport behavior is observed in monolayer WS2 by applying solid states backdates. The excitonic properties of monolayer MoS2 and WS2 are determined by intricate interplays between the bandage renormalization, Pauli blocking and carrier screening against carrier doping.
Self-assembled topological structures of post-processed two-dimensional materials exhibit novel physical properties distinct from those of their parent materials. Herein, the critical role of desulphurization on self-assembled topological morphologies of molybdenum disulfide ($MoS_2$) monolayer sheets is explored using molecular dynamics (MD) simulations. MD results show that there are differences in atomic energetics of $MoS_2$ monolayer sheets with different desulphurization contents. Both free-standing and substrate-hosted $MoS_2$ monolayer sheets show diversity in topological structures su…
We demonstrate the tunability of the photoluminescence (PL) properties of monolayer (1L)-MoS2 via chemical doping. The PL intensity of 1L-MoS2 was drastically enhanced by the adsorption of p-type dopants with high electron affinity, but reduced by the adsorption of n-type dopants. This PL modulation results from switching between exciton PL and trion PL depending on carrier density in 1L-MoS2. Achievement of the extraction and injection of carriers in 1L-MoS2 by this solution-based chemical doping method enables convenient control of optical and electrical properties of atomically thin MoS2.
MoS2 atomic layers have recently attracted much interest because of their two-dimensional structure as well as tunable optical, electrical, and mechanical properties for next generation electronic and electro-optical devices. Here we have achieved facile fabrication of MoS2 thin films on CdS nanowires by cation exchange in solution at room temperature and importantly observed their extraordinary magnetic properties. We establish the atomic structure of the MoS2/CdS heterostructure by taking atomic images of the MoS2/CdS interface as well as performing first-principles density functional geomet…
Elastic properties of materials are an important factor in their integration in applications. Chemical vapor deposited (CVD) monolayer semiconductors are proposed as key components in industrial-scale flexible devices and building blocks of 2D van der Waals heterostructures. However, their mechanical and elastic properties have not been fully characterized. Here we report high 2D elastic moduli of CVD monolayer MoS2 and WS2 (~ 170 N/m), which is very close to the value of exfoliated MoS2 monolayers and almost half the value of the strongest material, graphene. The 2D moduli of their bilayer he…
In this work, we investigated the electronic structure and the quantum capacitance of the functionalized MoS$_2$ monolayer. The functionalizations have been done by using different ad-atom adsorption on Mo$S_2$ monolayer. Density functional theory calculations are performed to obtain an accurate electronic structure of ad-atom doped MoS$_2$ monolayer with a varying degree of doping concentration. The quantum capacitance of the systems was subsequently estimated. A marked quantum capacitance above 200 $μ$F/cm$^2$ has been observed. Our calculations show that the quantum capacitance of MoS$_2$ m…
Various types of defects in MoS2 monolayers and their influence on the electronic structure and transport properties have been studied using the Density-Functional based Tight-Binding method in conjunction with the Green's Function approach. Intrinsic defects in MoS2 monolayers significantly affect their electronic properties. Even at low concentration they considerably alter the quantum conductance. While the electron transport is practically isotropic in pristine MoS2, strong anisotropy is observed in the presence of defects. Localized mid-gap states are observed in semiconducting MoS2 that …
In a recent work, new two-dimensional materials, the monolayer MoSi$_{2}$N$_{4}$ and WSi$_{2}$N$_{4}$, have been successfully synthesized in experiment, and several other monolayer materials with the similar structure, such as MoSi$_{2}$As$_{4}$, have been predicted [{\color{blue}Science 369, 670-674 (2020)}]. Here, based on first-principles calculations and theoretical analysis, we investigate the electronic and optical properties of monolayer MoSi$_{2}$N$_{4}$, WSi$_{2}$N$_{4}$ and MoSi$_{2}$As$_{4}$. We show that these materials are semiconductors, with a pair of Dirac-type valleys located …
By a combined study with first-principles calculations and symmetry analysis, we theoretically investigate the electronic properties of monolayer MoSi$_2$N$_4$. While the spin-orbital coupling results in bands splitting, the horizontal mirror symmetry locks the spin polarization along z-direction. In addition, a three-band tight-binding model is constructed to describe the low-energy quasi-particle states of monolayer MoSi$_2$N$_4$, which can be generalized to strained MoSi$_2$N$_4$ and its derivatives. The calculations using the tight-binding model show an undamped $\sqrt{q}$-dependent plasmo…
Density functional theory is used to systematically study the electronic and magnetic properties of doped MoS$_2$ monolayers, where the dopants are incorporated both via S/Mo substitution or as adsorbates. Among the possible substitutional dopants at the Mo site, Nb is identified as suitable p-type dopant, while Re is the donor with the lowest activation energy. When dopants are simply adsorbed on a monolayer we find that alkali metals shift the Fermi energy into the MoS$_2$ conduction band, making the system n-type. Finally, the adsorption of charged molecules is considered, mimicking an ioni…
Based on the Bethe-Salpeter equation eigenstates, we present a first-principles many-body formalism for calculating the two-photon absorption (TPA) coefficient of semiconductors. We apply this formalism to calculate the TPA spectra of MoS2 and WS2 monolayers. The all-electron full-potential linearised augmented-plane wave based functions are used for solving the Bethe-Salpeter equation. The calculated spectra are in good agreement with the available experimental ones for WS2 monolayer. The calculated TPA spectra exhibit significant excitonic effects when compared to those based on the independ…
In this letter, the effect of four types of defects (ONout, ONin, VNout and VNin) on the structural, electronic and magnetic properties of MoSi2N4 monolayer were investigated using first-principles calculations. The calculated results reveal that all the four types of defects lead to structural distortions around the O-dopant or N-vacancy, and thereby change the lattice parameter and monolayer height h. Specifically, ONout or ONin increases the lattice parameter, but VNout or VNin is on the contrary. ONout or VNout increases the monolayer height, whereas the height decreases for ONin or VNin. …
In this work, we have systematically studied the role of point defects in the recombination time of monolayer MoS$_2$ using time-dependent ab initio non-adiabatic molecular dynamics simulations. Various types of point defects, such as S vacancy, S interstitial, Mo vacancy and Mo interstitial have been considered. We show that defects strongly accelerate the electron-hole recombination, especially interstitial S atoms do that by 3 orders of magnitude higher compared to pristine MoS$_2$. Mo defects (both vacancy and interstitial) introduce a multitude of de-excitation pathways via various defect…
Studying the reason, why single-layer molybdenum disulfide (MoS$_2$) appears to fall short of its promising potential in flexible nanoelectronics, we found that the nature of contacts plays a more important role than the semiconductor itself. In order to understand the nature of MoS$_2$/metal contacts, we performed ab initio density functional theory calculations for the geometry, bonding and electronic structure of the contact region. We found that the most common contact metal (Au) is rather inefficient for electron injection into single-layer MoS$_2$ and propose Ti as a representative examp…
Multi Moire Networks in Engineered Lateral Hetero-Bilayers: Programmable Phononic Reconfiguration and Second Harmonic Generation
Indian Institute of Technology Kharagpur（印度理工学院卡拉格普尔分校）；Universidade Federal de Minas Gerais（米纳斯吉拉斯联邦大学）；Aarhus University（奥胡斯大学）；Indian Institute of Technology Goa（印度理工学院果阿分校）；Indian Institute of Science Education and Research Tirupati（蒂鲁帕蒂科学教育研究所）；KIIT Deemed to be University（KIIT认定大学）
We utilize density functional theory to calculate the edge energy and edge stress for monolayer MoS$_{2}$ nanoribbons. In contrast to previous reports for graphene, for both armchair and zigzag chiralities, the edge stresses for MoS$_{2}$ nanoribbons are found to be tensile, indicating that their lowest energy configuration is one of compression in which Mo-S bond lengths are shorter than those in a bulk, periodic MoS$_{2}$ monolayer. The edge energy and edge stress is found to converge for both chiralities for nanoribbon widths larger than about 1 nm.
We propose a theoretical model for describing the operation of a transistor with a MoS2 monolayer channel, which allows to obtain an analytical approximation of the potential in the channel. This potential depends on the drain and gate voltages. On this basis we make estimates for the minimum channel lengths due to the fundamental restriction of quantum tunneling through the barrier. It is shown that the relatively large effective mass of electrons in the MoS2 monolayer allows to predict the creation of devices with channels of a significantly shorter (2.5 - 3 nm) length than in traditional si…
We report on the integration of monolayer molybdenum disulphide with silicon nitride microresonators assembled by visco-elastic layer transfer techniques. Evanescent coupling from the resonator mode to the monolayer is confirmed through measurements of cavity transmission. The absorption of the monolayer semiconductor flakes in this geometry is determined to be 850 dB/cm, which is larger than that of graphene and black phosphorus with the same thickness. This technique can be applied to diverse monolayer semiconductors for assembling hybrid optoelectronic devices such as photodetectors and mod…
Photo-excited electrons and holes in insulators, above a critical density and below a critical temperature, can condense to form an electron-hole liquid (EHL) phase. However, observing the EHL phase at room temperature is extremely challenging. Here, we introduce the monolayer MoSi$_2$Z$_4$ (Z= N, As, P) series of compounds as a promising platform for observing the EHL phase at room temperature. The higher impact of the Coulomb interactions in two dimensions helps these monolayers support the EHL phase with an increased EHL binding energy and transition temperature, along with strongly bound e…
Understanding the magnetic properties of the various Mn doping configurations that can be encountered in $2H$-MoS$_2$ monolayer could be beneficial for its use in spintronics. Using density functional theory plus Hubbard U (DFT$+$U) approach, we study how a single isolated, double- and triple-substitution configurations of Mn atoms within a MoS$_2$ monolayer could contribute to its total magnetization. We find that the doping-configuration plays a critical role in stabilizing a ferromagnetic state in a Mn-doped MoS$_2$ monolayer. Indeed, the Mn-Mn magnetic interaction is found to be ferromagnt…
We investigate electrical gating of photoluminescence and optical absorption in monolayer molybdenum disulfide (MoS$_2$) configured in field effect transistor geometry. We observe an hundredfold increase in photoluminescence intensity and an increase in absorption at ~660 nm in these devices when an external gate voltage is decreased from +50 V to -50 V, while the photoluminescence wavelength remains nearly constant. In contrast, in bilayer MoS$_2$ devices we observe almost no changes in photoluminescence with gate voltage. We propose that the differing responses of the monolayer and bilayer d…
Two-dimensional materials such as transitional metal dichalcogenides exhibit unique optical and electrical properties. Here we report on the varying optical properties of CVD grown MoS2 monolayer flakes with different shapes. In particular, it is observed that the perimeter and the central region of the flakes have non-uniform photoluminescence (PL) energy and intensity. We quantified these effects systematically and propose that thermally induced strain during growth is the origin. The strain relaxation after transfer of the MoS2 flakes supports this explanation. Detailed investigations of th…
Ab-initio density functional theory calculations are performed to study the electronic properties of a MoS2 monolayer deposited over a SiO2 substrate in the presence of interface impurities and defects. When MoS2 is placed on a defect-free substrate the oxide plays an insignificant role, since the conduction band top and the valence band minimum of MoS2 are located approximately in the middle of the SiO2 band-gap. However, if Na impurities and O dangling bonds are introduced at the SiO2 surface, these lead to localized states, which modulate the conductivity of the MoS2 monolayer from n- to p-…
Monolayers (ML) of Group-6 transition-metal dichalcogenides (TMDs) are semiconducting two-dimensional materials with direct bandgap, showing promising applications in various fields of science and technology, such as nanoelectronics and optoelectronics. These monolayers can undergo strong elastic deformations, up to about 10\%, without any bond breaking. Moreover, the electronic structure and transport properties, which define the performance of these TMDs monolayers in nanoelectronic devices, can be strongly affected by the presence of point defects, which are often present in the synthetic s…
In this letter we show by means of first principle numerical simulations that electron-hole interaction significantly contributes to the second-harmonic generation spectrum of h-BN or MoS$_2$ monolayers. Specifically, it doubles the signal intensity at the excitonic resonances with respect to the contribution from independent electronic transitions. This result hints that the intensity of second-harmonic signal of those materials can be tuned by changing the dielectric screening, that controls the strength of the electron-hole interaction.
Real-time monitoring is essential for understanding and eventually precise controlling of the growth of two dimensional transition-metal dichalcogenides (2D TMDCs). However, it is very challenging to carry out such kind of studies on chemical vapor deposition (CVD). Here, we report the first real time $in-situ$ study on the CVD growth of the 2D TMDCs. More specifically, CVD growth of molybdenum disulfide (MoS$_2$) monolayer on sapphire substrates has been monitored $in-situ$ using differential transmittance spectroscopy (DTS). The growth of the MoS$_2$ monolayer can be precisely followed by lo…
Monolayer Molybdenum Disulfide (MoS2) with a direct band gap of 1.8 eV is a promising two-dimensional material with a potential to surpass graphene in next generation nanoelectronic applications. In this letter, we synthesize monolayer MoS2 on Si/SiO2 substrate via chemical vapor deposition (CVD) method and comprehensively study the device performance based on dual-gated MoS2 field-effect transistors. Over 100 devices are studied to obtain a statistical description of device performance in CVD MoS2. We examine and scale down the channel length of the transistors to 100 nm and achieve record hi…
We use micro-Raman and photoluminescence (PL) spectroscopy at 300K to investigate the influence of uniaxial tensile strain on the vibrational and optoelectronic properties of monolayer and bilayer MoS2 on a flexible substrate. The initially degenerate E^1_{2g} Raman mode is split into a doublet as a direct consequence of the strain applied to MoS2 through Van der Waals coupling at the sample-substrate interface. We observe a strong shift of the direct band gap of 48meV/(% of strain) for the monolayer and 46meV/% for the bilayer, whose indirect gap shifts by 86meV/%. We find a strong decrease o…
In this work we present a study on the impact of various intrinsic deformations like ripples, twist, wrap on the electronic properties of ultra-short monolayer MoS2 channels. The effect of deformation (3-7o twist or wrap and 0.3-0.7 buckling amplitude) on a 3.5 nm planar monolayer MoS2 MOSFET is evaluated by the density functional theory and the non-equilibrium Green`s function (DFT-NEGF) approach. We study the channel density of states, transmission spectra and the ID-VD characteristics under the varying conditions, with focus on the negative differential resistance (NDR) behavior. Our result…
We propose an effective lattice Hamiltonian for monolayer MoS$_2$ in order to describe the low-energy band structure and investigate the effect of perpendicular electric and magnetic fields on its electronic structure. We derive a tight-binding model based on the hybridization of the $d$ orbitals of molybdenum and $p$ orbitals of sulfur atoms and then introduce a modified two-band continuum model of monolayer MoS$_2$ by exploiting the quasi-degenerate partitioning method. Our theory proves that the low-energy excitations of the system are no longer massive Dirac fermions. It reveals a differen…
Phonons are essential for understanding the thermal properties in monolayer transition metal dichalcogenides, which limit their thermal performance for potential applications. We investigate the lattice dynamics and thermodynamic properties of MoS2, MoSe2, and WS2 by first principles calculations. The obtained phonon frequencies and thermal conductivities agree well with the measurements. Our results show that the thermal conductivity of MoS2 is highest among the three materials due to its much lower average atomic mass. We also discuss the competition between mass effect, interatomic bonding …
With the advances in low dimensional transition metal dichalcolgenides (TMDCs) based metal oxide semiconductor field effect transistor (MOSFET), the interface between semiconductors and dielectrics has received considerable attention due to its dramatic effects on the morphology and charge transport of semiconductors. In this study, first principle calculations were utilized to investigate the strain effect induced by the interface between Al2O3 (0001) and monolayer MoS2. The results indicate that Al2O3 in 1.3nm thickness can apply the strain of 0.3% on MoS2 monolayer. The strain effect monoto…
Intrinsic electron spin relaxation due to the D'yakonov-Perel' mechanism is studied in monolayer Molybdenum Disulphide. An intervalley in-plane spin relaxation channel is revealed due to the opposite effective magnetic fields perpendicular to the monolayer Molybdenum Disulphide plane in the two valleys together with the intervalley electron-phonon scattering. The intervalley electron-phonon scattering is always in the weak scattering limit, which leads to a rapid decrease of the in-plane spin relaxation time with increasing temperature. A decrease of the in-plane spin relaxation time with the …
Two-dimensional MoS2 has emerged as promising material for nanoelectronics and spintronics due to its exotic properties. However, high contact resistance at metal semiconductor MoS2 interface still remains an open issue. Here, we report electronic properties of field effect transistor devices using monolayer MoS2 channels and permalloy (Py) as ferromagnetic (FM) metal contacts. Monolayer MoS2 channels were directly grown on SiO2/Si substrate via chemical vapor deposition technique. The increase in current with back gate voltage shows the tunability of FET characteristics. The Schottky barrier …
Transition metal dichalcogenide monolayers such as MoSe2,MoS2 and WSe2 are direct bandgap semiconductors with original optoelectronic and spin-valley properties. Here we report spectrally sharp, spatially localized emission in monolayer MoSe2. We find this quantum dot like emission in samples exfoliated onto gold substrates and also suspended flakes. Spatial mapping shows a correlation between the location of emitters and the existence of wrinkles (strained regions) in the flake. We tune the emission properties in magnetic and electric fields applied perpendicular to the monolayer plane. We ex…
We report circularly-polarized optical reflection spectroscopy of monolayer WS$_2$ and MoS$_2$ at low temperatures (4~K) and in high magnetic fields to 65~T. Both the A and the B exciton transitions exhibit a clear and very similar Zeeman splitting of approximately $-$230~$μ$eV/T ($g\simeq -4$), providing the first measurements of the valley Zeeman effect and associated $g$-factors in monolayer transition-metal disulphides. These results complement and are compared with recent low-field photoluminescence measurements of valley degeneracy breaking in the monolayer diselenides MoSe$_2$ and WSe$_…
When transistor gate insulators have nanometer-scale equivalent oxide thickness (EOT), the gate capacitance ($C_\textrm{G}$) becomes smaller than the oxide capacitance ($C_\textrm{ox}$) due to the quantum capacitance and charge centroid capacitance of the channel. Here, we study the capacitance of monolayer MoS$_\textrm{2}$ as a prototypical two-dimensional (2D) channel while considering spatial variations in the potential, charge density, and density of states. At 0.5 nm EOT, the monolayer MoS$_\textrm{2}$ capacitance is smaller than its quantum capacitance, limiting the single-gated $C_\text…
The optical susceptibility is a local, minimally-invasive and spin-selective probe of the ground state of a two-dimensional electron gas. We apply this probe to a gated monolayer of MoS$_2$. We demonstrate that the electrons are spin polarized. Of the four available bands, only two are occupied. These two bands have the same spin but different valley quantum numbers. We argue that strong Coulomb interactions are a key aspect of this spontaneous symmetry breaking. The Bohr radius is so small that even electrons located far apart in phase space interact, facilitating exchange couplings to align …
