Breaking the dilemma between small size and room temperature stability is a necessary prerequisite for skyrmion-based information technology. Here we demonstrate by means of rate theory and an atomistic spin Hamiltonian that the stability of isolated skyrmions in ultrathin ferromagnetic films can be enhanced by the concerted variation of magnetic interactions while keeping the skyrmion size unchanged. We predict film systems where the lifetime of sub-10 nm skyrmions can reach years at ambient conditions. The long lifetime of such small skyrmions is due to exceptionally large Arrhenius pre-exponential and the stabilizing effect of the energy barrier is insignificant at room temperature. A dramatic increase in the pre-exponential is achieved thanks to softening of magnon modes of the skyrmion, thereby increasing the entropy of the skyrmion with respect to the transition state for collapse. Increasing the number of skyrmion deformation modes should be a guiding principle for the realization of nanoscale, room-temperature stable skyrmions.
We study the modification of micromagnetic configurations in nanostructures, due to the Dzyaloshinskii-Moriya interaction (DMI) that appear at the interface of an ultrathin film. We show that this interaction leads to new micromagnetic boundary conditions that bend the magnetization at the edges. We explore several cases of ultrathin film nanostructures that allow analytical calculations (1D systems, domain walls, cycloids and skyrmions), compare with fully numerical calculations, and show that a good physical understanding of this new type of micromagnetics can be reached. We particularly foc…
We report the results of simulations of magnetic skyrmion stability in ultrathin magnetic multilayer nanodots with interfacial Dzyaloshinskii-Moriya exchange interaction (DMI). We found that in presence of the lateral confinement the magnetostatic energy significantly influences the skyrmion stability and leads to stabilization of large radius skyrmion even at low values of the DMI strength, in addition to small radius skyrmion stabilized by DMI. In particular, stabilization of the skyrmion state with two different radii (bi-stability) is found in dipolarly-coupled (Pt/Co/Ir)$_{n}$ circular na…
We investigate the dependence of the Néel skyrmion size and stability on perpendicular magnetic field in ultrathin circular magnetic dots with out-of-plane anisotropy and interfacial Dzyaloshinskii-Moriya exchange interaction. Our results show the existence of two distinct dependencies of the skyrmion radius on the applied field and dot size. In the case of skyrmions stable at zero field, their radius strongly increases with the field applied parallel to the skyrmion core until skyrmion reaches the metastability region and this dependence slows down. More common metastable skyrmions demonstrat…
Understanding the physical properties of magnetic skyrmions is important for fundamental research with the aim to develop new spintronic device paradigms where both logic and memory can be integrated at the same level. Here, we show a universal model based on the micromagnetic formalism that can be used to study skyrmion stability as a function of magnetic field and temperature. We consider ultrathin, circular ferromagnetic magnetic dots. Our results show that magnetic skyrmions with a small radius compared to the dot radius are always metastable, while large radius skyrmions form a stable gro…
The static and dynamic states of Néel skyrmions in ultrathin ferromagnetic films with interfacial Dzyaloshinskii-Moriya interaction (DMI) have been micromagnetically simulated as functions of the interfacial DMI strength and applied static magnetic field. Findings reveal that while the breathing, counterclockwise (CCW) and clockwise (CW) rotational eigenmodes exist in the skyrmion lattice (SkL) phase, only the first two modes are present in the isolated skyrmion (ISk) phase. Additionally, the eigenfrequency of the CCW mode is insensitive to the magnetic-field driven SkL-ISk phase transition, a…
We show that thermal stability of magnetic skyrmions can be strongly affected by entropic effects. The lifetimes of isolated skyrmions in atomic Pd/Fe bilayers on Ir(111) and on Rh(111) are calculated in the framework of harmonic transition state theory based on an atomistic spin model parametrized from density functional theory. Depending on the system the attempt frequency for skyrmion collapse can change by up to nine orders of magnitude with the strength of the applied magnetic field. We demonstrate that this effect is due to a drastic change of entropy with skyrmion radius which opens a n…
Skyrmions have attracted significant interest due to their topological spin structures and fascinating physical features. The skyrmion phase arises in materials with Dzyaloshinskii-Moriya (DM) interaction at interfaces or in volume of non-centrosymmetric materials. However, although skyrmions were generated experimentally, one critical intrinsic relationship between fabrication, microstructures, magnetization and the existence of skyrmions remains to be established. Here, two series of CoFeB ultrathin films with controlled atomic scale structures are employed to reveal this relationship. By in…
Due to their exceptional topological and dynamical properties magnetic skyrmions - localized stable spin structures on the nanometre scale - show great promise for future spintronic applications. To become technologically competitive, isolated skyrmions with diameters below 10 nm that are stable at zero magnetic field and room temperature are desired. Despite finding skyrmions in a wide spectrum of materials, the quest for a material with these envisioned properties is still ongoing. Here we report zero field isolated skyrmions with diameters smaller than 5 nm coexisting with 1 nm thin domain …
We calculate low-frequency gyrotropic spin excitation modes of the skyrmion ground state ultrathin cylindrical magnetic dots. The skyrmion is assumed to be stabilized at room temperature and zero external magnetic field due to an interplay of the isotropic and Dzyaloshinskii-Moriya exchange interactions, perpendicular magnetic anisotropy and magnetostatic interaction. We consider Bloch- and Neel-type magnetic skyrmions and assume that the dot magnetization does not depend on the thickness coordinate. The skyrmion gyrotropic frequencies are calculated in GHz range as a function of the skyrmion …
We use the continuum micromagnetic framework to derive the formulas for compact skyrmion lifetime due to thermal noise in ultrathin ferromagnetic films with relatively weak interfacial Dzyaloshinskii-Moriya interaction. In the absence of a saddle point connecting the skyrmion solution to the ferromagnetic state, we interpret the skyrmion collapse event as ``capture by an absorber'' at microscale. This yields an explicit Arrhenius collapse rate with both the barrier height and the prefactor as functions of all the material parameters, as well as the dynamical paths to collapse.
Ultrathin ferromagnets with frustrated exchange and the Dzyaloshinskii-Moriya interaction can support topological solitons such as skyrmions and antiskyrmions, which are metastable and can be considered particle-antiparticle counterparts. When spin-orbit torques are applied, the motion of an isolated antiskyrmion driven beyond its Walker limit can generate skyrmion-antiskyrmion pairs. Here, we use atomistic spin dynamics simulations to shed light on the scattering processes involved in this pair generation. Under certain conditions a proliferation of these particles and antiparticles can appea…
Magnetic skyrmions are topologically nontrivial spin textures which hold great promise as stable information carriers in spintronic devices at the nanoscale. One of the major challenges for developing novel skyrmion-based memory and logic devices is fast and controlled creation of magnetic skyrmions at ambient conditions. Here we demonstrate the single ultrafast (35-fs) laser pulse-induced generation of skyrmion bubbles and skyrmion bubble lattices from a ferromagnetic state in sputtered ultrathin magnetic films at room temperature. The skyrmion bubble density increases with the laser fluence …
Magnetic skyrmions are chiral spin textures that hold great promise as nanoscale information carriers. Since their first observation at room temperature, progress has been made in their current-induced manipulation, with fast motion reported in stray-field-coupled multilayers. However, the complex spin textures with hybrid chiralities and large power dissipation in these multilayers limit their practical implementation and the fundamental understanding of their dynamics. Here, we report on the current-driven motion of Néel skyrmions with diameters in the 100-nm range in an ultrathin Pt/Co/MgO …
An understanding of the pinning of magnetic skyrmions to defects is crucial for the development of future spintronic applications. While pinning is desirable for a precise positioning of magnetic skyrmions it is detrimental when they are to be moved through a material. We use scanning tunneling microscopy to study the interaction between atomic scale defects and magnetic skyrmions that are only a few nanometers in diameter. The studied pinning centers range from single atom inlayer defects and adatoms to clusters adsorbed on the surface of our model system. We find very different pinning stren…
We determined the parameters of a classical spin Hamiltonian describing an Fe monolayer on Pd(111) surface with a Pt_{1-x}Ir_{x} alloy overlayer from ab initio calculations. While the ground state of the system is ferromagnetic for x=0.00, it becomes a spin spiral state as Ir is intermixed into the overlayer. Although the Dzyaloshinsky-Moriya interaction is present in the system, we will demonstrate that the frustrated isotropic exchange interactions play a prominent role in creating the spin spiral state, and these frustrated couplings lead to an attractive interaction between skyrmions at sh…
Optimal skyrmion stability in antisymmetric ultrathin ferromagnetic bilayers
Université de Toulouse, CNRS, CEMES（图卢兹大学，法国国家科学研究中心，电子与分子材料中心）；School of Mathematics, University of Bristol（布里斯托尔大学数学学院）；Dipartimento di Matematica, Università di Pisa（比萨大学数学系）
Spin-polarized scanning tunneling microscopy has been applied to study non-collinear spin textures of a Mn monolayer on a four-fold symmetric W(001) substrate revealing a zero-field spin spiral ground state and two different types of rotational domain walls. With an applied magnetic field of 9 T, we observe a coexistence of the spin spiral and the skyrmion phase, even though a previous theoretical study reported that the phase transition occurs at 18 T. The skyrmions show a roughly hexagonal arrangement despite the square lattice symmetry of the W(001) substrate. Based on a reduced set of ener…
We demonstrate that magnetic skyrmions with a mean diameter around 60 nm can be stabilized at room temperature and zero external magnetic field in an exchange-biased Pt/Co/NiFe/IrMn multilayer stack. This is achieved through an advanced optimization of the multilayer stack composition in order to balance the different magnetic energies controlling the skyrmion size and stability. Magnetic imaging is performed both with magnetic force microscopy and scanning Nitrogen-Vacancy magnetometry, the latter providing unambiguous measurements at zero external magnetic field. In such samples, we show tha…
Magnetic skyrmions are prime candidates as information carriers for spintronic devices due to their topological nature and nanometric size. However, unavoidable inhomogeneities inherent to any material leads to pinning or repulsion of skyrmions that, in analogy to biology concepts, define the phenotype of the skyrmion-defect interaction, generating complexity in their motion and challenging their application as future bits of information. Here, we demonstrate that atom-by-atom manufacturing of multi-atomic defects, being antiferromagnetic or ferromagnetic, permits the breeding of their energy …
We characterize skyrmions in ultrathin ferromagnetic films as local minimizers of a reduced micromagnetic energy appropriate for quasi two-dimensional materials with perpendicular magnetic anisotropy and interfacial Dzyaloshinskii-Moriya interaction. The minimization is carried out in a suitable class of two-dimensional magnetization configurations that prevents the energy from going to negative infinity, while not imposing any restrictions on the spatial scale of the configuration. We first demonstrate existence of minimizers for an explicit range of the model parameters when the energy is do…
Skyrmions in frustrated magnets have the helicity degree of freedom, where two different configurations of Bloch-type skyrmions are energetically favored by the magnetic dipole-dipole interaction and characterized by opposite helicities. A skyrmion must become a quantum-mechanical object when its radius is of the order of nanometer. We construct a qubit based on the two-fold degeneracy of the Bloch-type nanoscale skyrmions in frustrated magnets. It is shown that the universal quantum computation is possible based on nanoscale skyrmions in a multilayered system. We explicitly show how to constr…
The atomic-scale spin structure of individual isolated skyrmions in an ultrathin film is investigated in real space by spin-polarized scanning tunneling microscopy. Their axial symmetry as well as their unique rotational sense is revealed by using both out-of-plane and in-plane sensitive tips. The size and shape of skyrmions change as a function of magnetic field. An analytical expression for the description of skyrmions is proposed and applied to connect the experimental data to the original theoretical model describing chiral skyrmions. Thereby, the relevant material parameters responsible f…
An enhancement of the spin-orbit effects arising on an interface between a ferromagnet (FM) and a heavy metal (HM) is possible through the strong breaking of the structural inversion symmetry in the layered films. Here we show that the introduction of an ultrathin W interlayer between Co and Ru in Ru/Co/Ru films enables to preserve perpendicular magnetic anisotropy (PMA) and simultaneously induce a large interfacial Dzyaloshinskii-Moriya interaction (iDMI). We find that the Ru/Co/W/Ru films have PMA up to 0.35 nm of the nominal thickness of W (tW). The study of the spin-wave propagation in the…
Employing Brillouin spectroscopy, strong interfacial Dzyaloshinskii-Moriya interactions have been observed in an ultrathin Pt/CoFeB film. Our micromagnetic simulations show that spin-wave nonreciprocity due to asymmetric surface pinning is insignificant for the 0.8nmthick CoFeB film studied. The observed high asymmetry of the monotonic spin wave dispersion relation is thus ascribed to strong Dzyaloshinskii-Moriya interactions present at the Pt/CoFeB interface. Our findings should further enhance the significance of CoFeB as an important material for magnonic, spintronic and skyrmionic applicat…
We introduce a new class of primitive building blocks for realizing quantum logic elements based on nanoscale magnetization textures called skyrmions. In a skyrmion qubit, information is stored in the quantum degree of helicity, and the logical states can be adjusted by electric and magnetic fields, offering a rich operation regime with high anharmonicity. By exploring a large parameter space, we propose two skyrmion qubit variants depending on their quantized state. We discuss appropriate microwave pulses required to generate single-qubit gates for quantum computing, and skyrmion multiqubit s…
Topological spin textures can serve as non-volatile information carriers. Here we study the current-induced dynamics of an isolated magnetic skyrmion on a nanoscale square-grid pinning pattern formed by orthogonal defect lines with reduced magnetic anisotropy. The skyrmion on the square grid can be pixelated with a quantized size of the grid. We demonstrate that the position, size, and shape of skyrmion on the square grid are electrically configurable. The skyrmion center is quantized to be on the grid and the skyrmion may show a hopping motion instead of a continuous motion. We find that the …
We experimentally demonstrate the formation of room-temperature skyrmions with radii of about 25\,nm in easy-plane anisotropy multilayers with interfacial Dzyaloshinskii-Moriya interaction (DMI). We detect the formation of individual magnetic skyrmions by magnetic force microscopy and find that the skyrmions are stable in out-of-plane fields up to about 200 mT. We determine the interlayer exchange coupling as well as the strength of the interfacial DMI. Additionally, we investigate the dynamic microwave spin excitations by broadband magnetic resonance spectroscopy. From the uniform Kittel mode…
Spin-polarized scanning tunneling microscopy (SPSTM) was used to directly image nanoscale Néel skyrmions in a SrIrO3 / SrRuO3 bilayer system that are among the smallest reported to date in any system. Off-axis magnetron sputtering was used to cap epitaxial films of the oxide ferromagnet SRO with 2 unit cells of SrIrO3, intended to provide interfacial spin orbit coupling. Atomic resolution STM imaging and tunneling spectroscopy were used to identify island-like SrIrO3 grains and small regions of bare SrRuO3. Isolated skyrmions were only observed in SrIrO3-covered regions of the film, and exhibi…
There is an ever increasing interest in the development of plasmonic 2D nanomaterials, with widespread applications in optoelectronics, high resolution microscopy, imaging and sensing, among others. With the current ability of ultrathin noble metal film deposition down to a few monolayers in thickness, there is a need for an analytical expression of the thickness dependent complex dielectric function for predicting optical properties for arbitrary thicknesses. The free and bound electron contributions to the dielectric function are dealt with independently, since their influences affect separa…
Magnetic skyrmions have attracted considerable interest, especially after their recent experimental demonstration at room temperature in multilayers. The robustness, nanoscale size and non-volatility of skyrmions have triggered a substantial amount of research on skyrmion-based low-power, ultra-dense nanocomputing and neuromorphic systems such as artificial synapses. Room-temperature operation is required to integrate skyrmionic synapses in practical future devices. Here, we numerically propose a nanoscale skyrmionic synapse composed of magnetic multilayers that enables room-temperature device…
Transition-metal interfaces and multilayers are a very promising class of systems to realize nanometer-sized, stable magnetic skyrmions for future spintronic devices. For room temperature applications it is crucial to understand the interactions which control the stability of isolated skyrmions. Typically, skyrmion properties are explained by the interplay of pair-wise exchange interactions, the Dzyaloshinskii-Moriya interaction and the magnetocrystalline anisotropy energy. Here, we demonstrate that higher-order exchange interactions -- which have so far been neglected -- can play a key role f…
The discovery of magnetic skyrmion bubbles in centrosymmetric magnets has been receiving increasing interest from the research community, due to the fascinating physics of topological spin textures and its possible applications to spintronics. However, key challenges remain, such as how to manipulate the nucleation of skyrmion bubbles to exclude the trivial bubbles or metastable skyrmion bubbles that usually coexist with skyrmion bubbles in the centrosymmetric magnets. Here, we report having successfully performed this task by applying spatially geometric confinement to a centrosymmetric frust…
A model for a spin-torque nano-oscillator based on the self-sustained oscillation of a magnetic skyrmion is presented. The system involves a circular nanopillar geometry comprising an ultrathin film free magnetic layer with a strong Dzyaloshinkii-Moriya interaction and a polariser layer with a vortex-like spin configuration. It is shown that spin-transfer torques due to current flow perpendicular to the film plane leads to skyrmion gyration that arises from a competition between geometric confinement due to boundary edges and the vortex-like polarisation of the spin torques. A phenomenology fo…
Magnetic skyrmions are topologically protected spin textures that exhibit many fascinating features. As compared to the well-studied cryogenic Bloch skyrmions in bulk materials, we focus on the room-temperature Néel skyrmions in thin-film systems with an interfacial broken inversion symmetry in this article. Specifically, we show the stabilization, the creation, and the implementation of Néel skyrmions that are enabled by the electrical current-induced spin-orbit torques. Towards the nanoscale Néel skyrmions, we further discuss the challenges from both material optimization and imaging charact…
Nanoscale chiral skyrmions in noncentrosymmetric helimagnets are promising binary state variables in high-density, low-energy nonvolatile memory. Skyrmions are ubiquitous as an ordered, single-domain lattice phase, which makes it difficult to write information unless they are spatially broken up into smaller units, each representing a bit. Thus, the formation and manipulation of skyrmion lattice domains is a prerequisite for memory applications. Here, using an imaging technique based on resonant magnetic x-ray diffraction, we demonstrate the mapping and manipulation of skyrmion lattice domains…
Magnetic skyrmions are promising candidates as information carriers in a new generation of memories. How to generate and stabilize skyrmions is essential for their successful application to technology. Here we theoretically demonstrate that arrays of skyrmions can be imprinted in ultrathin ferromagnetic films in large numbers by bringing a magnetic nanostructured template close to the film. Two kind of templates, allowed by present-day nanotechnologies, are studied: arrays of ferromagnetic nanorods and superconducting vortices. Skyrmions are generated when exposing magnetic films to the templa…
In quantum matters hosting electron-electron correlation and spin-orbit coupling, spatial inhomogeneities, arising from competing ground states, can be essential for determining and understanding topological properties. A prominent example is Hall anomalies observed in SrRuO3 films, which were interpreted in terms of either magnetic skyrmion-induced topological Hall effect (THE) or inhomogeneous anomalous Hall effect (AHE). To clarify this ambiguity, we systematically investigated the AHE of SrRuO3 ultrathin films with controllable inhomogeneities in film thickness (tSRO). By harnessing the st…
A theoretical study of the current-driven dynamics of magnetic skyrmions in disordered perpendicularly-magnetized ultrathin films is presented. The disorder is simulated as a granular structure in which the local anisotropy varies randomly from grain to grain. The skyrmion velocity is computed for different disorder parameters and ensembles. Similar behavior is seen for spin-torques due to in-plane currents and the spin Hall effect, where a pinning regime can be identified at low currents with a transition towards the disorder-free case at higher currents, similar to domain wall motion in diso…
Magnetic skyrmions are topologically-distinct swirls of magnetic moments which display particle-like behaviour, including the ability to undergo thermally-driven diffusion. In this paper we study the thermally activated motion of arrays of skyrmions using temperature dependent micromagnetic simulations where the skyrmions form spontaneously. In particular, we study the interaction of skyrmions with grain boundaries, which are a typical feature of sputtered ultrathin films used in experimental devices. We find the interactions lead to two distinct regimes. For longer lag times the grains lead t…
We address, by means of ab-initio calculations, the origin of the correlation that has been observed experimentally between the chemisorption energy of CO on nanoscale Cu(001) supported films and quantum-size effects. The calculated chemisorption energy shows systematic oscillations, as a function of film thickness, with a periodicity corresponding to that of quantum-well states at the surface-Brillouin-zone center crossing the Fermi energy. We explain this trend based on the oscillations, with film thickness, of the decay length on the vacuum side of the quantum-well states at the Fermi energ…
The skyrmion racetrack is a promising concept for future information technology. There, binary bits are carried by nanoscale spin swirls -- skyrmions -- driven along magnetic strips. Stability of the skyrmions is a critical issue for the realization of this technology. Here we demonstrate that the racetrack skyrmion lifetime can be calculated from first principles as a function of temperature, magnetic field and track width. Our method combines harmonic transition state theory extended to include Goldstone modes, with an atomistic spin Hamiltonian parametrized from density functional theory ca…
Skyrmions are nanoscale magnetic structures with features promising for future low-power memory or logic devices. In this work, we demonstrate novel scanning techniques based on nitrogen vacancy center magnetometry that simultaneously probe both the magnetic dynamics and structure of room temperature skyrmion bubbles in a thin film system Ta/CoFeB/MgO. We confirm the handedness of the Dzyaloshinskii-Moriya interaction in this material and extract the helicity angle of the skyrmion bubbles. Our measurements also show that the skyrmion bubbles in this material change size in discrete steps, depe…
Room-temperature polar skyrmion bubbles that are recently found in oxide superlattice, have received enormous interests for their potential applications in nanoelectronics due to the nanometer size, emergent chirality, and negative capacitance. For practical applications, the ability to controllably manipulate them by using external stimuli is prerequisite. Here, we study the dynamics of individual polar skyrmion bubbles at the nanoscale by using in situ biasing in a scanning transmission electron microscope. The reversible electric field-driven phase transition between topological and trivial…
Magnetic skyrmions are nanoscale spin structures recently discovered at room temperature (RT) in multilayer films. Employing their novel topological properties towards exciting technological prospects requires a mechanistic understanding of the excitation and relaxation mechanisms governing their stability and dynamics. Here we report on the magnetization dynamics of RT Néel skyrmions in Ir/Fe/Co/Pt multilayer films. We observe a ubiquitous excitation mode in the microwave absorption spectrum, arising from the gyrotropic resonance of topological skyrmions, and robust over a wide range of tempe…
Bias-Engineered Synthetic Antiferromagnets Hosting sub-20 nm Zero-Field Skyrmions at Room Temperature
Empa, Swiss Federal Laboratories for Materials Science and Technology（瑞士联邦材料科技研究所）；Politecnico di Bari（巴里理工大学）；University of Basel（巴塞尔大学）；University of Messina（墨西拿大学）
Magnetic skyrmions are highly mobile nanoscale topological spin textures. We show, both analytically and numerically, that a magnetic skyrmion of an even azimuthal winding number placed in proximity to an s-wave superconductor hosts a zero-energy Majorana bound state in its core, when the exchange coupling between the itinerant electrons and the skyrmion is strong. This Majorana bound state is stabilized by the presence of a spin-orbit interaction. We propose the use of a superconducting tri-junction to realize non-Abelian statistics of such Majorana bound states.
Nanoscale magnetic skyrmions have interesting static and transport properties that make them candidates for future spintronic devices. Control and manipulation of the size and behavior of skyrmions is thus of crucial importance. Using a Ginzburg-Landau approach, we show theoretically that skyrmions and skyrmion lattices can be stabilized by a spatial modulation of the uniaxial magnetic anisotropy in a thin film of centro-symmetric ferromagnet. Remarkably, the skyrmion size is determined by the ratio of the exchange length and the period of the spatial modulation of the anisotropy, at variance …
External magnetic field, temperature, and spin-polarized current are usually employed to create and control nanoscale vortex-like spin configurations such as magnetic skyrmions. Although these methods have proven successful, they are not energy-efficient due to high power consumption and dissipation. Coupling between magnetic properties and mechanical deformation, the magnetoelastic (MEL) effect, offers a novel approach to energy-efficient control of magnetism at the nanoscale. It is of great interest in the context of ever-decreasing length scales of electronic and spintronic devices. Therefo…
Magnetic skyrmions are nanoscale whirls of magnetism that can be propagated with electrical currents. The repulsion between skyrmions inspires their use for reversible computing based on the elastic billiard ball collisions proposed for conservative logic in 1982. Here we evaluate the logical and physical reversibility of this skyrmion logic paradigm, as well as the limitations that must be addressed before dissipation-free computation can be realized.
We experimentally demonstrate efficient third harmonic generation from an indium tin oxide (ITO) nanofilm (lambda/42 thick) on a glass substrate for a pump wavelength of 1.4 um. A conversion efficiency of 3.3x10^-6 is achieved by exploiting the field enhancement properties of the epsilon-near-zero (ENZ) mode with an enhancement factor of 200. This nanoscale frequency conversion method is applicable to other plasmonic materials and reststrahlen materials in proximity of the longitudinal optical phonon frequencies.
