Arrangement and apparatus for dispersionless optical modes

Abstract

According to an example aspect, there is provided an arrangement for producing at least one flat band effect for at least one wavelength range of electromagnetic radiation, the arrangement comprising: a first set of first linear sequences of nanostructures, said first linear sequences being separated from one another such that an inter-sequence distance of adjacent first linear sequences is greater than a distance of adjacent nanostructures within the first linear sequences.

Description

ARRANGEMENT AND APPARATUS FOR DISPERSIONLESS OPTICAL MODES FIELD

[0001] At least some embodiments relate to the field of optics, such as organic lightemitting devices or diodes (OLEDs).BACKGROUND

[0002] Optoelectronic devices may need dispersionless optical modes for optimal operation. An example of an optoelectronic device is an organic light emitting diode, OLED. As an example, in the display sector, the success of OLEDs as illuminants is due to their simplified architecture (e.g., no need for a backlight), superior power efficiency, and exceptional visual quality, including enhanced color accuracy, wider viewing angles, and faster response times compared to liquid crystal displays (LCD). The OLED excels particularly in low-brightness use cases, such as smartphone displays - today, most smartphones are equipped with an OLED display. Notably, the display is usually the most expensive and power-consuming component in a smartphone; as such, any improvements in the cost and power efficiency of the display have the potential to drive down manufacturing costs and increase the power efficiency of smartphones.SUMMARY

[0003] The invention is defined by the features of the independent claims. Some specific embodiments are defined in the dependent claims.

[0004] According to a first aspect, there is provided an arrangement for producing at least one flat-band effect for at least one wavelength range of electromagnetic radiation, the arrangement comprising: a first set of first linear sequences of nanostructures, said first linear sequences being separated from one another such that an inter-sequence distance of adjacent first linear sequences is greater than a distance of adjacent nanostructures within the first linear sequences.

[0005] According to a second aspect there is provided an arrangement for producing at least one flat-band effect for at least one wavelength range of electromagnetic radiation, the arrangement comprising: parallel linear sequences of nanostructures, said parallel linear sequences separated from one another such that an inter-sequence distance between adjacentparallel linear sequences is greater than a distance of adjacent nanostructures within the parallel linear sequences.

[0006] According to a third aspect there is provided an arrangement for producing at least one flat-band effect for at least one wavelength range, the arrangement comprising nanostructures in a pattern defined by a periodic arrangement of unit cells, such that at least one lattice constant of the unit cells is more than two and a half times, such as three times, for example five times, larger than a distance of adjacent nanostructures.

[0007] According to a fourth aspect there is provided an arrangement for producing at least one flat-band effect for at least one wavelength range of electromagnetic radiation, the arrangement comprising: nanostructures in an ordered structure having a first dimension and a second dimension perpendicular to the first dimension, the ordered structure formed by periodically repeated patterns of nanostructures; each pattern covering the first dimension of the ordered structure; the patterns being repeated along the second dimension of the ordered structure at intervals having a length defined by the smallest period between the periodically repeated patterns; each pattern comprising a first linear sequence of nanostructures, wherein a length of the first linear sequence has a longer component along the first dimension than along the second dimension, and wherein the nanostructures in the first linear sequences has a first distance between consecutive nanostructures in the sequence; and the length of the interval being at least two and a half times, such as five times, the first distance.

[0008] According to a fifth aspect, there is provided an apparatus comprising an arrangement according to any one of the embodiments of the first, second, third or fourth aspect; and an optoelectronic device.

[0009] According to a sixth aspect, there is provided a light emitting diode, an organic light emitting diode, OLED, comprising a first electrode and a second electrode; an emissive layer between the first and second electrodes; and an arrangement of nanostructures, said arrangement being configured to, based on the geometric arrangement of the nanostructures, provide at least one flat-band effect for at least one wavelength of electromagnetic radiation.BRIEF DESCRIPTION OF THE DRAWINGS

[0010] FIG. 1 illustrates an example of a dispersion relation with a flat band;

[0011] FIG. 2A, 3A, 3B, 4, 5 and 6 illustrate example arrangements according to at least some embodiments;

[0012] FIG 2B illustrates dispersion relation and reciprocal lattice of an arrangement according to at least some embodiments;

[0013] FIG. 7 illustrate an example apparatus according to at least some embodiments;

[0014] FIG. 8A, 9A illustrate example measurements and obtained dispersion relation graphs with one or more flat bands in accordance with at least some embodiments;

[0015] FIG. 8B, 9B, 10 A, 10B illustrate examples of producing one or more flat bands in accordance with at least some embodiments.EMBODIMENTS

[0016] The present disclosure discusses arrangements capable of generating flat band(s) so as to improve or modify the behavior of electromagnetic radiation. In the present disclosure, an arrangement, or plurality thereof, suitable for generating at least one flat band via a repetitive arrangement of nanostructures is provided. With such repetitive and ordered array of nanostructures, said array having suitable dimensional characteristics, a flat band, or plurality thereof, may be produced for at least one wavelength, or at least one wavelength range of electromagnetic radiation. Electromagnetic radiation may be visible light, ultraviolet, and / or infrared light, for example. At least some of the embodiments presented herein include arrangements of particles, in specified arrangements, which are configured to produce at least one flat band at at least the first wavelength, or at least the first range of wavelengths. Thus, angle-independent emission at the first wavelength or the first range of wavelengths, or higher intensities may be output from media, with at least some of the arrangements of the present disclosure. Thus, issues, such as total internal reflections, in a medium may be overcome. Wavelengths in a medium surrounding the nanostructures may be between 200 nm and 2000 nm, such as from 250 nm to 1750 nm, for example, from 300 nm to 1550 nm.

[0017] In the present disclosure, a flat band, or flat-band effect, refers to a feature of electromagnetic dispersion, such as optical dispersion, of a structure comprising an arrangement of nanostructures. Such a feature is substantially invariant with respect to the component of the wavevector that is parallel to a plane of an arrangement, or, equivalently, propagation angle with respect to a normal of a plane of an arrangement over a range of inplane wavevector or angle. In other words, a flat-band feature is substantially invariant over a plurality of values (such as all values) or a range of values of in-plane wavevector or angle. The component of the wavevector which is parallel to the plane may be known as an inplane wavevector. The propagation angle, may be denoted here as an angle for simplicity.

[0018] Flat bands are especially beneficial in that, as the energy is not dependent on the wavevector, electromagnetic radiation may be manipulated in a controlled manner. Flat bands are beneficial in that dispersionless optical emission from a device may be produced, for example. On the other hand, flat bands provide a benefit in absorption as well, where dispersionless optical absorption to a device may be obtained. Flat bands are beneficial at least in that a dispersionless electromagnetic modes may be achieved at at least one energy range (i.e., at least one wavelength range). For example, dispersion and / or total internal reflections of electromagnetic radiation in a medium may be avoided or minimized with at least some embodiments. Therefore, an arrangement capable of producing such flat bands of energy, is especially beneficial.

[0019] FIG. 1 is provided to assist in understanding the concept of a flat-band effect. In FIG. 1, a dispersion relation E(k) as a function of wavevector k is presented. The flat-band effect may be formed by dispersive bands. For simplicity, three dispersive bands 110A, HOB, and HOC are shown in FIG. 1. As may be appreciated, even though here flat bands are characterized in terms of energy, similar definitions apply, equivalently, in view of wavelength or frequency of the respective electromagnetic radiation. Moreso, definition in view of a wavevector k, also equivalently applies to (propagation) angle of electromagnetic radiation. The flat-band effect may be characterized at least by a central wavelength (of a flat band). The central wavelength of a flat band refers to a wavelength at which the superposition of the dispersive bands appears as substantially invariant of wavevector k. The central wavelength of the flat band, together with a linewidth of the dispersive band(s) as well as flatness, may be used to characterize a flat-band effect.

[0020] Linewidth (of a dispersive band) refers to full width at half maximum (FWHM) of the dispersive band. In the present disclosure, linewidths are given with respect to the wavelength (energy) of the maximum. As may be appreciated, this is the inverse of Q factor. An exaggerated example of linewidth 115C of the band 110C is shown in FIG. 1.

[0021] Flatness refers to a maximal variation in wavelength of a dispersive band within a Brillouin zone divided by the flat-band wavelength. An example of flatness 125 of a flat band and first Brillouin zone, 1BZ, are illustrated in FIG. 1.

[0022] The flat-band effect may be observed in angle-resolved or wavevector-resolved optical measurements, comprising transmission, reflection, extinction, scattering, photoluminescence, electroluminescence, or lasing emission spectra, for example. The flatband effect may be characterized in absolute terms or relative to a reference structure. Such a reference structure may be without an arrangement capable of producing a flat band.

[0023] Referring to FIG. 1, the flat-band effect may be formed by one or more dispersive bands, illustrated there with dispersive bands 110A, HOB, HOC. A flat-band effect may be defined based on a central wavelength 120. It may be appreciated that such a flat band may have a finite width (along a wavevector axis and range thereof 130), such that the flat band may be defined by a central wavelength value and by the linewidth of the flat band around the central wavelength 120. A superposition of the dispersive bands (e.g., dispersive bands 110A, 110B and 110C illustrated in FIG. 1) produces the flat band. At the flat band, the wavelength is invariant, or substantially invariant, with respect to the angle of incident or emitted electromagnetic radiation, such as visible light. Thus, with a respective flat band wavelength (or range), electromagnetic radiation of such wavelength (or wavelength range due to finite width of the flat band) behaves identically irrespective of the angle.

[0024] Referring to FIG. 1, the flat band may be characterized by its flatness. As a flat band is produced as a superposition of dispersive bands (see bands 110A, 110B, 110C as an example), their superposition may be less than ideal. That is, the flat band may deviate from the ideal flat band (a flat line with a finite width).

[0025] In an embodiment, the linewidth of one or more dispersive bands (with respect to the central wavelength) is at most 0.1, such as at most 0.075, for example at most 0.05 orat most 0.025. In an embodiment, Q factor for linewidth of the one or more dispersive bands may be more than 10, such as more than 13, for example more than 20, or more than 40.

[0026] In an embodiment, the flatness of the flat band is at most 0.1, such as at most 0.075, such as at most 0.05, for example at most 0.025.

[0027] A flat band is invariant of the wavevector at a wavelength range, such that the energy is independent of the wavevector, E(k) = constant. Flat bands would therefore be beneficial for various applications where efficient or increased performance in terms of transmission of electromagnetic radiation is sought. The flat band ensures that the emitted light is of the same color independent of the viewing angle - beneficial in many applications, such as displays and lighting. Furthermore, the diffractive origin of the flat band confers to it a narrow linewidth, typically on the order of a few or few tens of nanometers, depending on the material, size and shape of the nanostructures and the optical properties of the surrounding environment.

[0028] In accordance with the present disclosure, there is provided an arrangement of nanostructures, suitable for producing one or more flat-band effects at one or more wavelength ranges. An arrangement here refers to a physical arrangement comprising nanostructures. The arrangements include linear sequences of nanostructures, which are sufficiently apart (that is, separated) from one another. Within a linear sequence, individual nanostructures lie on a line, and the distance to the nearest-neighboring nanostructures is constant. The nanostructures in a form of linear sequences may also be known as chains. Thus-arranged nanostructures may be used to provide at least one flat band at a corresponding angle. The angle is perpendicular to the line of the linear sequence, when the sequence has a shorter interparticle distance than an inter-sequences distance (distance between the adj acent linear sequences). In such cases, a flat band is observable perpendicular to the linear sequence. Similar linear sequences, parallel to one another, and adjacent linear sequences separated by an intersequence distance, may be understood as a set of linear sequences. There may be more than one set of linear sequences. There may be two or more, such as three or more linear sequences that form a set of linear sequences. Such may be obtained by repetition of the unit cell, for example. The linear sequences in a set may be parallel to one another and separated by an inter-sequence spacing. The arrangement may be described via formation of chain-like structures of nanostructures, or via repetition of unit cells of specified dimensions, said unit cells comprising one or more nanostructures. Thenanostructures may form a lattice, a repetitive structure which may be defined by a unit cell and lattice vectors (defining the unit cell). The nanostructures may be in a pattern, such as a pattern defined by repetitive and periodic unit cells and the lattice geometry.

[0029] Nanostructures may be nanoparticles or nanoholes, for example. In an embodiment, the nanostructures are nanoparticles. Examples of nanoparticles include (but are not limited to) nanocylinders or nanospheres. The nanostructures may be asymmetrical, for example. An example of an asymmetrical nanostructure is a nanostructure which is L-shaped in the plane of the arrangement. Without a loss of generality, in the embodiments disclosed with reference to figures and embodiments therewith, nanoparticles are typically depicted although such nanostructures may be nanostructures other than nanoparticles, such as nanoholes, in at least some embodiments.

[0030] In an arrangement, the nanostructures may be same or similar in terms of shape with respect to one another. For example, nanostructures may be nanoparticles, such as nanocylinders. Nanoparticle here is used as a generic term, referring to, for example, spherical particles and / or cylindrical particles but includes other material shapes as well. In an embodiment, nanoparticle refers to a cylindrical nanoparticle, such as a nanocylinder. The size and shape of the nanostructures, such as nanoparticles, should be such that they support at least one localized surface plasmon or Mie resonance. Nanoparticles here refer to size and dimensions of particles in the nanometric range of such a particle. The particles in an arrangement may be same or similar in terms of size. There may be more than 5, such as more than 10, for example more than 30 nanostructures in a linear sequence, such as a first linear sequence and / or a second linear sequence.

[0031] The nanostructures, such as nanoparticles, may be characterized by their height. Height here refers to a measure of nanostructure length perpendicular to the plane defined by an arrangement, such as maximum height perpendicular to the plane of the arrangement. Heights of the nanostructures, such as nanoparticles, may be from 0.03 times the minimum wavelength to 1 times the maximum wavelength. For example, the height may be from 0.04 times the minimum wavelength to 0.8 times the maximum wavelength, such as 0.05 times the minimum wavelength to 0.6 times the maximum wavelength, such as from 0.075 times the minimum wavelength to 0.5 times the maximum wavelength. Wavelengths here refer to wavelengths of electromagnetic radiation at or near the central wavelength of a flat-band effect in a medium surrounding the nanostructures. Heights of the nanostructuresmay be from 2 nm to 2000 nm, such as from 10 nm to 1250 nm, for example 20 nm to 1000 nm, or 30 nm to 800 nm. The dimensions of a nanostructure, such as a nanoparticle, should be such that it supports at least one localized surface plasmon or Mie resonance, for example.

[0032] The nanostructures may be characterized by their width. Such a width may be a maximum diameter, or side length of the particles, for example. Width may be the largest average length of a plurality nanostructures along the lateral dimension of the arrangement. In an embodiment, the arrangement is a planar arrangement. The widths are measured appropriately according to the needs of the embodiment, taking into account the shape and number of the particles, for instance. In some embodiments, such a width is a maximum diameter of the particles. In another embodiments the width is a maximum side length of the particles. In some embodiments, the width is the largest average length of a plurality nanostructures along the lateral dimension of the arrangement. The width may be from 0.05 times the minimum wavelength to 2 times the maximum wavelength of at least one wavelength range, such as 0.075 times the minimum wavelength to 1.8 times the maximum wavelength, for example, from 0.1 times the minimum wavelength to 1.75 times the maximum wavelength. The wavelengths refer to wavelengths in a medium surrounding the nanostructures. Width may refer to a dimension along a plane defined by the arrangement. Widths of the particles may be from 5 nm to 2500 nm, such as from 10 nm to 2250 nm, for example from 20 nm to 2000 nm, or from 30 nm to 1750 nm. The dimensions of a nanostructure, such as a nanoparticle, should be such that it supports at least one localized surface plasmon or Mie resonance, for example. A set of linear sequences of nanostructures may be configured to, together with medium surrounding said nanostructures, to provide at least one flat band based on the distance of adjacent nanostructures within the respective linear sequence being smaller than the intersequence distance of the linear sequences of the set.

[0033] The nanoparticles in an arrangement may have same or similar composition with respect to one another. The nanoparticles may be metal, such as metallic nanoparticles, for example gold. The particles may be metal alloys. The particles may be composite materials, comprising for example two or more materials. The nanoparticles may be plasmonic nanoparticles and / or metallic nanoparticles, arranged as an array may provide dispersive plasmonic-photonic modes called surface lattice resonances (SLRs) that can efficiently couple to light (visible, or otherwise) from emitters, for example. The arrangement and particles therein should support some form of collective lattice resonance.In an embodiment, the (nano)particles comprise, are, or made of, a metal, such as including but not limited to one or more of: gold, silver, or aluminium. Here, the collective resonances are known as surface lattice resonances to the art. In an embodiment, the (nano)particles comprise, are, or made of, a high-index (refractive index greater than or equal to 2.5 in the visible wavelengths) dielectric or semiconducting material. The material may be germanium, silicon, SiCh, TiCh or ITO, for example. The material may be selected such that the nanostructure, together with its surroundings, produce a suitable refractive index contrast. The collective resonances may couple to the guided modes of the structure where the arrangement is embedded in. The nanostructures may be surrounded by media, such as medium comprising one or more of glass, quartz, silicon, and / or plastic. The media may be inorganic or organic materials suitable for one or more optoelectronic devices, such as LEDs, for example, OLEDs. The nanostructures may be embedded in a medium. The composition of the nanostructures should be such that, in relation to such medium, the refractive index contrast is suitable for providing resonance. The (absolute) refractive index contrast should thus be a suitable value, such as greater than 0.3. The refractive index contrast may be the absolute difference between the refractive index of the nanostructure material of the nanostructures and the refractive index of the medium, i.e., |m-n2| = refractive index contrast, where m is the refractive index of the nanostructure and the (effective) refractive index of the surrounding media, or vice versa. The refractive indices are taken at a relevant wavelength range. The refractive index contrast may be more than 0.35, such as more than 0.4, for example more than 0.45. The refractive index contrast may be more than 0.5. In an embodiment, the nanostructures, such as dielectric or semiconducting nanoparticles, having a refractive index contrast with respect to a surrounding medium greater than or equal to 0.3, such as defined by an absolute difference of refractive indices (i.e., |m-n2|)., of the nanostructures and the medium surrounding the nanostructures.

[0034] An arrangement according to an embodiment may also be defined using a unit cell comprising one or more nanostructures. A unit cell here refers to a repeatable arrangement of one or more nanostructures, such that a repetitive pattern may be formed. The unit cell is defined by two lattice vectors, (and their length, i.e. lattice constants), which when repeated form a repetitive periodic pattern, i.e., an arrangement. In an arrangement according to an embodiment, the unit cell has at least one lattice constant (lattice vector length) greater than the distance of adjacent nanostructures, such as nanostructures defining a linear sequence. The at least one lattice constant (which is greater than distance of adjacentnanostructures may be along a lattice vector perpendicular to a linear sequence of nanostructures having said distance of adjacent nanostructures. A unit cell refers to dimensions (size) of a smallest repetitive structure of an arrangement of nanostructures. In the present disclosure, a two-dimensional unit cell may be used at least for a two-dimensional arrangement (i.e. an arrangement defined on a plane). In at least some of the figures, the unit cell lattice constants are illustrated as ai and a2 while axand aydepict distances of adjacent nanostructure distances (i.e., for nanoparticles, interparticle distances). A set of linear sequences may be formed with repetitive unit cells such that one lattice constant is substantially smaller than a lattice constant in a perpendicular direction. Then, a set of linear sequences having an interparticle distance smaller than an intersequence distance may be formed.

[0035] An arrangement according to an embodiment comprises nanostructures in a pattern defined by a periodic arrangement of unit cells, such that at least one lattice constant of the unit cells is more than two and a half times, such as three times, for example, five times, larger than a distance of adjacent nanostructures. The nanostructures may be comprised in linear sequences formed via repetition of the unit cells. Such parallel linear sequences may be understood as a set of linear sequences. Via such periodic arrangement, linear sequences of nanostructures, such as nanoparticles, may be formed such that the intersequence distance between adjacent linear sequences is greater than distances of nanostructures within the linear sequences. Thus, a flat band is formed at an angle perpendicular to a line defining the linear sequence(s). The wavelength of the flat band may be directly proportional to the distance of adjacent nanostructures (in a linear sequence) and the refractive index of a medium surrounding the nanostructures. If more sets of linear sequences at different angle with respect to one another, and / or with different spacing of nanostructures are provided in an arrangement, different flat band effects may be generated, such as flat bands at different angles and / or more than one flat band at different (central) wavelengths.

[0036] An arrangement according to an embodiment comprises nanostructures, such as nanoparticles, in an ordered structure having a first dimension and a second dimension perpendicular to the first dimension, the ordered structure formed by periodically repeated patterns of nanoparticles. Dimensions here refer to physical dimensions, for example, x and y dimensions of the arrangement. Each pattern covers the first dimension of the ordered structure. Then, a pattern, comprising at least one linear sequence, may extend along the firstdimensions. The arrangement comprises patterns such that said patterns are repeated along the second dimension of the ordered structure. The repetition is done at intervals having a length defined by the smallest period between the periodically repeated patterns. Therefore, for example, a linear sequence may be repeated at said interval. Each pattern comprising a first linear sequence of nanoparticles, wherein a length of the first linear sequence has a longer component along the first dimension than along the second dimension. In other words, the line formed by the first linear sequences is substantially along the first dimension. The nanostructures in the first linear sequences have a first distance between consecutive nanoparticles in the sequence. This may be referred to as aj, for example. The nanostructures in the linear sequences may be equispaced (i.e., spaced apart at equal distances). The length of the interval being at least two and a half times, such as three times, for example five times, the first distance. Thus, a flat band is formed along a direction perpendicular to the directions along which the first linear sequences extend. Second linear sequences may be provided along the second dimensions, such that they are repeated at the respective interval and have a second distance between consecutive nanostructures, such as nanoparticles, in said second linear sequence.

[0037] In a linear sequence, the nanostructures are arranged so as to form a substantially straight line. However, as may be appreciated, the linear sequence of the nanostructures need not necessarily be an ideal straight line but deviations in position, within a linear sequence may occur, for example, within manufacturing tolerances. For example, there may be a tolerance of, at maximum, 0.5 times the maximum wavelength, such as at most 0.45, for example at most 0.4. There may be first set(s) of second linear sequences at an angle to that of the (first) set of (first) linear sequences, for example. Linear sequences, such as first linear sequences, in a set of linear sequences may be parallel to one another.

[0038] Spacing of adjacent nanostructures in a linear sequence may be understood as an interparticle distance. Such a spacing between adjacent nanostructures may be measured from a center of a nanostructure to a center of an adjacent nanostructure in the arrangement. For example, center-of-volume of a nanostructure to a center-of-volume of an adjacent nanostructure, or for example, center-of-mass of a nanostructure to a center-of-mass of an adjacent nanostructure. Inter-sequence distance between two adjacent sequences (of a set of linear sequences) may be the minimum distance between two adjacent linear sequences.

[0039] A flat band is formed based on at least the distance of adjacent nanostructuresin a linear sequence as well as repetition of the nanostructure linear sequence. The flat band is formed at an angle perpendicular to the linear sequence direction when the distance between adjacent nanostructures in the sequence is sufficiently small compared to a component of the lattice vector perpendicular to the sequence. By selecting the direction of the linear sequences, different flat bands at different angles may be generated.

[0040] Linear sequences may be used to provide flat bands at desired angles and spectral positions. The spectral positions of flat bands may be characterized as follows. A unit cell comprising a set of linear sequence, with Ncnumber of linear sequences. The linear sequences each have Nj number of nanostructures in a sequential arrangement (i.e., a chain) having a constant, or near constant, spacing ciy between adjacent nanostructures. The chain subtends an angle 0j E [0, TT) with the x axis, where j labels the chain. The chain may be translated by A) = (A%y, Ayy). The positions of the lattice sites, i.e., locations of nanostructures in an arrangement, of the / th linear sequence may be represented as:

[0041] For a set of linear sequences there are therefore Nu=Nj lattice sites (i.e., nanostructure locations) within a unit cell. In order to avoid double-counting, if there are any overlapping lattice sites between the linear sequences, they are to be subtracted from the number of lattice sites in the unit cell Nu.An angle 0j at which a flat band is generated may be provided by:arctan(— cot (In other words, the flat band(s) may be observed at right angles (perpendicular) to the linear sequences (said sequence at an angle 0y) that generate said flat band. The generation of flat band is governed by the interparticle distance Oy, and energy of said flat band corresponds to the spacing ay, (i.e., interparticle distance, Oy) of the corresponding linear sequence. The energy of the flat band is therefore:c0Em= - m, m E Nnajwhere h is the Planck constant, c0the speed of light in vacuum and n the refractive index of the background medium.

[0042] Therefore, one or more flat bands with defined spectral positions may be generated with suitable arrangement of linear sequences of nanostructures. As may be appreciated from the equation above, the spectral position of the flat band can be selected by tuning the interparticle distance. These features enable at least some embodiments to be used for selectively guiding emission (or absorption) to a narrow, desired wavelength range.

[0043] Underlying mechanism of the flat band creation is an arrangement, where the unit cell is much larger than the interparticle distance of neighboring nanostructures, such as NPs. An arrangement according to an embodiment comprises a rectangular array of linear sequences of nanostructures. The unit cell may be much larger than the interparticle distance in view of at least one lattice constant of the unit cell, for example. Rectangular array of linear sequences of nanostructures refers to an arrangement of nanostructures as linear sequences such that the distance of adjacent nanostructures is different to a distance of adjacent nanostructures in an orthogonal direction. In other words, a set of linear sequences is formed, where interparticle distance in the linear sequences is less than the intersequence distance of adjacent linear sequences of nanostructures.

[0044] FIG. 2A illustrates an arrangement according to an embodiment comprising nanoparticles. It is to be understood that the embodiment is generalizable to other nanostructures as well. There, a rectangular array of nanoparticles is illustrated forming thus, an arrangement 2. Referring to FIG. 2A, an interparticle distance in one direction is a multitude (for example multiple greater than 2.5-fold, such as 3-fold, for example 5-fold) of the interparticle distance in the other direction, thereby forming linear sequences, which may also be construed as lines, of NPs in one direction. As may be appreciated, one lattice constant a2of the unit cell 26 is larger than the interparticle distance axof adjacent particles. In at least such a configuration, a longer distance may be therefore understood as an intersequence distance, such as an inter-sequence distance of adjacent linear sequences. In at least such a configuration a shorter distance, thereby forming said linear sequence, may be understood as an inter-particle distance.

[0045] FIG. 2A illustrates an arrangement 2 according to an embodiment. The illustrated arrangement comprises a set of linear sequences 20 of particles 21 having an interparticle distance ax24 within said sequences 20, and an intersequence distance ay22 of adjacent linear sequences 20. The linear sequences 20 have equidistant spacing, that is, the interparticle distant is constant. The set shown in FIG. 2A has three linear sequences 20.There may be a different number of linear sequences in a set in an embodiment. One of the periods in x- and y-direction, ax24 and ay22 respectively, is a multitude, that is a multiple, of the other, as depicted in FIG. 2A. Also depicted in FIG. 2A, is a unit cell 26, which defines periodicity of the arrangement 2. The unit cell 26 has lattice constants a and a2. As may be appreciated, lattice constant a2is larger than the interparticle distance axperpendicular to it. In the current example, the lattice constant a2therefore corresponds to the intersequence distance aybetween adjacent linear sequences. When ay= Max, where M describes the number of times the unit cell 26 (or at least one lattice constant, i.e., here a2) is larger compared to the shorter period, linear sequences of nanostructures are created wherein the interparticle distance is shorter than the intersequence distance. Flat band may thus be created perpendicular to the line of the linear sequence(s). In this case, one or more flat bands occur at wavelengths P fb= axn, where P is a natural number and n is the refractive index of the surrounding medium. The refractive index n may be the effective index of a waveguided mode, which can be calculated using methods known in the art. Moreover, refractive index of surrounding medium may affect the flat band generation.

[0046] The one or more flat bands may occur starting from, approximately, M = 2.5, such as from M=5, but may become more pronounced for larger value of M. However, there may be a trade-off as the flat band will become weaker due to a reduced coupling between the particles, such as NPs, for larger distances. It may be that M = 60 to be the largest feasible size of the unit cell, for example. In an embodiment, multiple M of a first lattice constant (e.g., ax) is a second lattice constant (e.g., ay). M may be more than 2.5, such as more than 3, for example more than 5, such as more than 10, for example, more than 15. Additionally or alternatively, M may be less than 60, such as less than 50, for example, less than 40. In an embodiment M is between 2.5 and 60, such as 5 and 50, such as between 10 and 50, for example, between 15 and 40.

[0047] If nanostructures, such as nanoparticles, for example plasmonic nanoparticles, radiate light as electric dipoles, the resulting flat-band effect is predominantly linearly TM-polarized.

[0048] An arrangement may include three, such as five, for example, eight unit cell repetitions along at least one of the lattice vectors (directions), so as to provide an arrangement capable of generating a flat band. In an embodiment, the arrangement comprises more than 3 unit cells along one of the lattice vectors, such as more than 5 or morethan 8, for example more than 10 unit cells, so as to form at least one flat band. Such repetition of unit cells along one of the lattice vectors may be suitable, for example, for rectangular lattices, such as the arrangement illustrated in FIG. 2A. The repetition of unit cells may be for both lattice vectors directions along the plane of the arrangement, fore example.

[0049] To illustrate how an arrangement produces flat band(s) such as the one shown in FIG. 2A, an explanation in view of FIG. 2B, is provided. The band structure of the upper graph in FIG. 2B illustrates a dispersion relation graph, of TM and TE-polarized modes. The inset of the upper graph of FIG. 2B illustrates a rectangular lattice, similar to that of FIG.2A, for example. In the inset of FIG. 2B, three chains (linear sequences) of nanoparticles 21 are shown. The linear sequences of FIG. 2B have an inter-sequence spacing of 5-times more than the inter-particle distance in said sequences. One or more flat bands may form from the narrowing of the first Brillouin zone (1BZ) in the kydirection and the repetition of the mainly TM-polarized modes. At larger angles, more TE polarization mixes in.

[0050] The inset shows a respective arrangement (a lattice structure of particles 21), and, as a dashed line rectangle, a unit cell 26. The arrangement, shown in the inset of FIG.2B, is a rectangular lattice having lattice constants a and a2, and interparticle and intersequence spacing axand ay, respectively. There, one of the lattice constants a2(along y direction) is larger than the interparticle distance ax(along x direction) in each sequence, that is a2= ay= 5axin the example provided. The arrangement of the inset of FIG. 2B shows three linear sequences parallel to one another and separated by an inter-sequence distance. The inter-sequence distance of adjacent linear sequences is greater than the interparticle distance in the linear sequences, such that a flat band may occur at an angle perpendicular to the linear sequences.

[0051] Referring to FIG. 2B, a flat band is formed by the modes kycorresponding to the diffraction orders (mx= ±1, my) near the energy corresponding to the interparticle distance ax, where myis an integer. These modes are relatively flat near their vertices and are separated in kyby the length of the reciprocal lattice vector corresponding to y-direction lattice constant a2. Thus, increasing the aspect ratio a2 / axnarrows the width of the Brillouin zone, BZ, yielding a flatter band. The lattice vector defined by lattice constant a2is (substantially) perpendicular to the interparticle distance ax. Because these modes are basedon the diffractive orders mx= ±1, they lie along the lines kx= +2TVIQ.Xdepicted in the lower graph of FIG. 2B.

[0052] The area encompassed by a dashed-line rectangle indicates the 1 BZ in the lower graph of FIG. 2B. Also, the reciprocal lattice vectors of the arrangement, bi and b2, are indicated in FIG. 2B. The horizontal, dashed lines in the lower graph highlight the linelike features. The distance between the lines is \b1\. This distance determines the energy (i.e., central wavelength) of the flat band. Note that the axes in the lower graph of FIG. 2B are not to scale; in the example, the 1BZ is five times wider in the kxdirection than in the kydirection. The minor ticks on the kxaxis in the lower graph of FIG. 2B correspond to the width of the 1BZ in the kydirection. As shown, flat bands in the band structure correspond to line-like features in the reciprocal lattice. The lattice sites in the reciprocal lattice of the arrangement therefore correspond to the maxima in the structure factor that may produce a suitable flat band.

[0053] FIG. 3 A illustrates an arrangement having a cross-lattice structure. In FIG. 3 A, two intersecting linear sequences of nanostructures are shown, and the intersecting linear sequences are at a perpendicular angle. A so-called cross geometry (in a unit cell 30) is depicted in FIG. 3A, while a repetitive periodic arrangement of such a cross geometry is highlighted in FIG. 3B. Referring to FIG. 3 A, NPs 31, shown as dark circles, are arranged in two lines crossing in the center of a unit cell 30 shown as a dashed line rectangle. This has at least the advantage of obtaining additional scattering due to an increased number of NPs 31 compared to a single set of linear sequences, such as the one shown in FIG. 2 A and 2B. As such, at least in order to account for reduced coupling, more scatterers (that is, particles) may be introduced to an arrangement according to an embodiment. In such an embodiment, two lattice constants have lengths greater than the interparticle distance of nanoparticles in a linear sequence. A cross-lattice may comprise at least two linear sequences perpendicular to one another. The spectral position of the one or more flat bands may be tuned by adding more sets of linear sequences.

[0054] Referring to FIG. 3 A, the NPs 31 are placed on two perpendicular lines (i.e., linear sequences). The lines may be at an angle, for example, at an angle from 80 to 100 degrees, such as 90 degrees. Therefore, particles 31, form a cross-lattice arrangement. The particles 31 may have constant nearest-neighbor distances axin the x-direction 32 and ayin the y-direction 34, an example of which is shown in FIG. 3A. The unit cell 30 of thearrangement contains Nxnumber of NPs along a horizontal line 36 and Nynumber of NPs 38 along a vertical line, intersecting at the centre of the unit cell 30. The unit cell has two dimensions (depicted with lattice constants a238 and a 36) delimiting the NPs. As with the embodiments discussed in relation to FIG. 2A and 2B, also here, (at least one) side of the unit cell 30 must be sufficiently greater than the perpendicular nearest-neighbor distance, comprising a first set of second linear sequences of nanostructures.

[0055] The geometry of the cross lattice, such as the ones shown in FIG. 3 A and 3B, may support a relation ayNy= MaxNx. The design constraints as discussed in relation to FIG. 2 A and 2B apply to M for a cross lattice, such as the one shown in FIG. 3 A as well. For example, AT may be more than 2.5, such as more than 5, for example more than 10. AT may be greater than 5, for example. For a lattice vector defined by a lattice constant a2, being substantially perpendicular to an interparticle distance axin a linear sequence, said a2should be greater than ax(i.e., a2= Nyay> ax). For example, the lattice constant a2may be 2.5 times greater than, such as 5 times greater than, the interparticle distance ax.

[0056] Referring to FIG. 3B, the unit cell 30 is repeated so as to form an arrangement 3B. In FIG. 3B, the arrangement 3B comprises a first set of first linear sequences 31A comprising two first linear sequences 310A. Moreover, the arrangement 3B comprises a first set of second linear sequences 3 IB comprising two second linear sequences 310B. The linear sequences 310A, 310B comprise NPs 31. To further highlight the linear sequences of the arrangement 3B, that is chains of NPs, respective dashed lines are provided in FIG. 3B. The linear sequences 310A, 310B therefore form chains of nanoparticles 31, and together with the two perpendicular (orthogonal) sets of linear sequences, and arrangement 3B is obtained. Length of a first lattice constant a is longer than a second lattice constant a2defining therefore, a rectangular unit cell 30. It is noted that, in a cross-lattice of FIG. 3 A and 3B, both lattice constants c^and a2are greater than the interparticle distance 32, 34 within the linear sequences. Thus, in at least cross lattice arrangements, such as the one shown in FIG.3B, a flat band may be generated at both viewing angles kx= 0 and ky= 0. As with embodiments depicted for example in FIG. 2A and 2B, the spectral positions of the flat bands along kx= 0 are fb= axn!P, where P is a natural number and n is the refractive index. Due to the symmetric nature of the cross arrays, flat bands occur also along ky= 0 at wavelengths Xjb= ayn!P as defined by the interparticle distance in y-direction.

[0057] In an embodiment, an arrangement comprises a first set of first linear sequences and first set of second linear sequences are at an angle with respect to the first set of first linear sequences. The angle may be perpendicular, such as between 80 to 110 degrees. Such may be understood as the cross lattice arrangement, shown for example in FIG. 3 A In an embodiment, the angle is selected such that two flat bands occur at specified angles dependent on the angle of the linear sequences in the sets of linear sequences. There may be a plurality of sets of linear sequences, each set comprising at least three parallel linear sequences having each therein an intersequence spacing greater than the interparticle spacing.

[0058] The same or similar design principles as discussed in view of FIG. 2A, 2B, 3A and 3B, for generation of at least one flat band effect, may be applied to create lattice designs with flat bands at multiple target energies (or energy ranges). In other words, a plurality of flat band effects, at respective different wavelength ranges may be generated with a suitable arrangement. Such arrangement may comprise linear sequences offset from one another.

[0059] Choice of a (chain) offset parameters can affect the strength of particular modes. A duplicate set of linear sequences may be formed such that it is displaced by an offset (Ax, Ay) with respect to the other sublattice.

[0060] An arrangement, for example those depicted in FIG. 3 A and 3B, may comprise more than one set of linear sequences offset from one another. For example, two crosslattices may be superposed so as to form lattices with two or more flat bands at two or more target wavelengths. An example of offset linear sequences is illustrated in FIG. 4. In FIG. 4, arrangement comprises multiple “cross arrays”. In FIG. 4, NPs 41 (shown as dark circles in FIG. 4) for a first set of first linear sequences and a first set of second linear sequences are shown (two linear sequences perpendicular to one another). NPs 42 (shown as white circles in FIG. 4) for a second set of first linear sequences and a second set of second linear sequences (two linear sequences perpendicular to one another) are also shown. The first and second sets are offset from one another. In other words, two sets comprise parallel linear sequences, but they are offset by a distance, as shown in FIG. 4 as distances 48, 47 to two perpendicular (x, y) directions. An arrangement may therefore comprise a first pattern of nanoparticles and a second pattern of nanoparticles, such that the patterns are offset from one another along two dimensions perpendicular to one another.

[0061] FIG. 4 also illustrates a unit cell 40 of a combined cross array comprises two or more cross lattices of NPs 41 and 42, each with their own interparticle distances in the x-direction ax43 and 45 and in the y-direction ay44 and 46, and own numbers of particles along the x-axis Nxand y-axis Ny. A first (sub)lattice 410 is shown as dark circles and comprises NPs 41. A second (sub)lattice 420 is shown as white circles and comprises NPs 42. Without loss of generality, the centrepoint of one of the sublattices can be taken to coincide with the origin of the unit cell (sublattice 410). The centrepoint(s) of the other sublattice(s) 420 are displaced from the origin of the unit cell by Ax in the x-direction 47 and Ay in the y-direction 48. To ensure that the sublattices share a common unit cell 40, the product Nxax(i.e., be equal to first lattice constant a ) should be the same for all sublattices and the product Nyay(i.e., be equal to second lattice constant a2) should be the same for all sublattices. For example, in view of FIG. 2A and 2B, the geometry of the cross lattice may follow a relation ayNy= MaxNx. Regarding selection of value M, Mmay be selected similarly than in embodiments discussed in relation to FIG. 2A-2B and / or FIG. 3 A-3B, for example. The same design constraints as in FIG.2A-2B and FIG. 3A-3B may apply to M. The spectral positions of the flat bands along kx= 0 are fb= axfifP , where P is a natural number, ax iis the interparticle distance in the x-direction of the ith sublattice and n is the refractive index.

[0062] FIG. 2A, 2B, 3A, 3B and 4 depict embodiments having therein, orthogonal lattice vectors a and a2of a respective unit cell. In other words, there, an angle between sets of linear sequences of particles are perpendicular (orthogonal). Whereas the (sub)lattices presented, for example, in relation to FIG. 2A, 2B, 3A, 3B and 4, have orthogonal lattice vectors, lattice arrangements whose lattice vectors are non-orthogonal can be created following a similar procedure.

[0063] In an embodiment, the sets of first and second linear sequences of particles are at an angle to one another. Such an angle may be, for example, 80 to 100 degrees, making thus, the angle of the sets of linear sequences substantially perpendicular to one another.

[0064] In an embodiment, the sets of first and second linear sequences of particles are at an angle to one another. Such an angle may be, for example, 50 to 70 degrees. The angle may be from 45 to 80 degrees, such as 60 to 70 degrees. Thus, non-orthogonal lattice vectors are obtained. There may be more than two sets of linear sequences such that they are at anangle to one another. For example, an embodiment may comprise sets of first, second and third linear sequences which are at an angle to one another.

[0065] An arrangement according to an embodiment, such as nanoparticle lattices, may also be constructed with non-orthogonal unit vectors and more than one linear sequence at an angle with respect to one another. Example of such an arrangement is depicted in FIG.5A. In FIG. 5A, two intersecting (at a non-orthogonal angle) linear sequences of nanostructures are shown. Referring to FIG. 5 A, there is provided a cross-lattice arrangement of NPs having non-orthogonal lattice vectors. The lattice vectors 56, 58 therefore define a non-orthogonal unit cell 54. FIG. 5 A depicts an illustrative, non-exhaustive example of such a lattice. Nanoparticles 50 are equispaced (interparticle distance “a” 52) along three lines that lie parallel to the lattice vectors 56 (a- and 58 (a2) and their sum (a + a2) and intersect at the origin of the unit cell 54 (origin is in the bottom left comer). Note that for reasons of clarity, in FIG. 5A, both lattice vectors 56, 58 are slightly displaced from the origin. The ratio of the length of the unit vector to the lattice constant, M =may be needed for the emergence of one or more flat bands, and may be critical for the appearance of the flat bands; similar design constraints as for, for example, FIG. 2A, apply also here. The flat bands occur at wavelengths fb= an / P, where P is a natural number and n is the refractive index. The angles at which the flat bands are observed depend on the choice of the unit vectors.

[0066] The angle of the first and second linear sequences may be used to adjust the angle of the flat bands. There may be more than two sets of linear sequences. For example, three sets linear sequences, may form an arrangement. An example is provided as a triangular chain lattice. A third chain may be therefore included in an arrangement. Such a third chain may align with at least some of the sites of the reciprocal lattice. An embodiment of a triangular chain lattice comprises three chains with N lattice sites spaced distance a apart, at an angle, such as at angles 0°, 60°, and 120° to the x axis, for example. The angles at which the flat bands appear coincide with the reciprocal lattice vectors bi, b2 and bi - b2 of the arrangement. Furthermore, in a triangular lattice, |bi| = |b2| = |bi— b2|, and therefore, three flat bands have equally many contributing modes.

[0067] FIG. 5B illustrates an example of a triangle chain lattice arrangement 5B-1, corresponding reciprocal lattice 5B-2 and obtained extinction spectrum flat bands with such an arrangement 5B-3. A part of the lattice is depicted as 5B-1 and includes a unit cell 54shown as a dashed line parallelogram. The reciprocal lattice vectors bi and b2 and the difference bi-bi are shown in 5B-2. Referring to 5B-3 of FIG. 5B, which depicts an experimental result, the measured extinction spectrum is viewed through a bandpass filter with central wavelength of 880±5 nm of a triangular chain lattice, with N = 25 number of particles and a = 570 nm interparticle distance (in a linear sequence) in a background with refractive index n = 1.52. The extinction spectrum shows three flat bands at the predicted angles. The example experimental arrangement, from which the extinction spectrum of FIG.5B is measured, comprises, as the nanostructures, gold (nano)cylinders with approximate diameter of 150 nm and a height of 50 nm.

[0068] As with orthogonal sets of linear sequences, cross lattices with non-orthogonal unit vectors may also be combined to create (super)lattices with flat bands at multiple target energies (i.e, a plurality of flat bands each having a different central wavelength). Example of such a design is illustrated in FIG. 6. Similarly to an embodiment as illustrated, for example, in FIG. 4, two or more cross lattices with non-orthogonal unit vectors of an embodiment may be superposed (i.e., offset) as long as they share the same lattice vectors. In other words, two or more sets of linear sequences may be separated by an offset. For example, first and second sets of first linear sequences may be separated by an offset.

[0069] FIG. 6 illustrates an embodiment where two or more cross lattices have non-orthogonal unit vectors, and are also offset from one another. A unit cell 60 of a so-called combined cross array comprises two or more cross lattices of NPs 61 and 62. A first lattice 610 is shown as a collection of dark circles and comprises NPs 61. A second (sub)lattice 620 is shown as a collection of white circles and comprises NPs 62. As may be appreciated, two sublattices 610 and 620 that share the same unit cell 60 and lattice vectors 65 and 66 are superposed so that one of the sublattices is offset by Ax in the x-direction 67 and Ay in the y-direction 68. Each such sublattice has its own interparticle distance aL, where i is the sublattice index; the interparticle distances are indicated by 63 and 64 in FIG. 6. The ratio ofthe length of the unit vectors to the lattice constants, M = may be critical for theaiappearance of the flat bands; similar design constraints as for, for example, embodiments described in FIG. 2A, 2B, apply also here. The flat bands occur at wavelengths fb=where cq is the lattice constant of the ith lattice, P is a natural number and n is the refractive index. The angles at which the flat bands are observed depend on the choice of the latticevectors. When applying an offset, part of the particles having said offset are folded from adjacent unit cells, so as to form a single unit cell for both sublattices.

[0070] At least some embodiments relate to the field of optics, for example, optoelectronic devices, such as organic light-emitting devices or diodes (OLEDs). For example, at least some embodiments related to OLEDs, a technology which has, over the past two decades, assumed a transformative role in display technology and solid-state lighting, for example. The production of at least one flat-band effect may be beneficial for optoelectronic devices. In the following, although an OLED is shown as an example, other optoelectronic devices may be suitable as well. Examples of optoelectronic devices include light-emitters and photodetectors, for example. Light-emitters include, for example, lightemitting diodes. Photodetectors include, for example, photodiodes.

[0071] At least some embodiments provide an arrangement as part of an apparatus. For example, the arrangement may be part of an optoelectronic device or coupled to an optoelectronic device. An arrangement according to an embodiment may be located inside the optoelectronic device. For example, an arrangement may be located inside an LED, such as an OLED, for example adjacent to, or in contact with, an emissive layer of an OLED. However, in an embodiment, an arrangement may be located outside an optoelectronic device. Example of such is a photodiode, with which an arrangement may be electromagnetically coupled so as to diffract electromagnetic radiation to said photodiode. For example, an arrangement may be located between two electrodes (anode and cathode) of an OLED. For example, the arrangement of nanostructures, such as NPs, may be embedded in the organic layer(s) of an OLED, and / or at the surface of an electrode, such as an anode. The arrangement may be embedded into the electrode. For example, electrode, such as an anode or a cathode, may be etched so as to form nanostructures in an arrangement according to an embodiment.

[0072] A side view of an organic light emitting diode, OLED, structure is shown in FIG. 7. FIG. 7 illustrates an example sketch of an OLED (stack) with an embedded arrangement 7, such as a nanoparticle array. The arrangement 7 may be of any of the embodiments of the present disclosure, for example. In FIG. 7, the particles 78 are illustrated as cylinders. The OLED depicted in FIG. 7 is a bottom-emitting OLED, comprising a transparent, or non-opaque, substrate 70 and a transparent or non-opaque electrode, here in FIG. 7, an anode 72. Adjacent to the anode 72 are disposed one or more organic layers 74,which comprise at least the emissive layer (EML) and possibly other layers (as non-limiting examples, hole / electron injection layers, hole / electron transport layers or hole / electron blocking layers). Disposed adjacent to the topmost organic layer is another electrode, depicted here as a cathode 76. The (nano)particle arrangement 7 of nanoparticles 78 is embedded in the organic layer(s) 74. The relative thicknesses of the layers 70, 72, 74, and 76, and the size and shape of the nanoparticle arrangement 7, and nanoparticles 78 therein, are merely illustrative examples. It is noted that other OLED designs may be applicable. The arrangement 7 may be, for example, one shown in FIG. 2A, 3 A, 3B, 4, 5A, 5B, or 6. It may be appreciated that the presented bottom-emitting OLED is an example of a device capable of supporting at least some embodiments. In an embodiment, the apparatus is a light emitting diode, LED, such as an OLED, for example a bottom emitting OLED diode, or a top-emitting OLED diode. It is also noted, that an arrangement according to at least some embodiments is not necessarily dependent on a particular OLED design (number of layers, thicknesses and materials of the layers, top and / or bottom emitting device design) as long as the NPs 78 are situated close enough to the emissive layer (such as inside, or embedded in, the organic layer(s) 74). Therefore, any common OLED design known to the art can be used in conjunction with an arrangement according to an embodiment.

[0073] The emitting material should be sufficiently close to the arrangements (NP arrays) in order to enable efficient coupling between the emitted light and the SLRs of the array. In accordance with the present disclosure, one or more flat bands may stem from the dispersive modes of a (nano)particle array caused by the specified arrangement(s) of the nanoparticles in the array.

[0074] The dimensions of an arrangement may be selected such that they correspond to the (emitted) wavelength(s) of an optoelectronic device, such as an OLED. The arrangement may be situated, for example, at most 3500 nm, such as at most 2500 nm, for example at most 2250 nm or at most 2000 nm, from the emissive layer. The arrangement may be situated, for example, a distance of at most 3 times the maximum wavelength, such as at most 2 times, for example at most 1.75 times, such as at most 1.5 times, from the emissive layer where the maximum wavelength is the maximum wavelength of the OLED emission. The distance of the arrangement (and nanostructures therein) to the emissive layer may be at most 50 nm in an embodiment.

[0075] An OLED comprises at least a first electrode and a second electrode, and an emissive layer therebetween. The arrangement may be arranged on a, surface of the first electrode (of an OLED stack) such as on the inner surface of the electrode (i.e., inside the OLED. In other words, nanostructures, such as nanoparticles may be provided on a surface of a cathode or an anode (depending on the OLED type, for example). The arrangement may be in contact with the first electrode. The electrode may comprise additional layers on which the arrangement may be. The arrangement may be embedded in the first electrode. Then, the material surrounding the first electrode (such as an anode or cathode), may encompass the nanostructures of the arrangement. The arrangement may be separated from the second electrode by a distance of more than 20 nm, such as more than 50 nm, for example. Such may be beneficial at least in that a short circuit may be avoided, for example.

[0076] Regarding manufacturing of arrangements and nanostructures, common methods may be used. The nanostructures, for example nanoparticles, may be produced with various methods known in the art. For example, lithography-based methods may be used to obtain an arrangement according to an embodiment. The arrangements may be made with lithography-based methods, for example. The arrangements and / or NP arrays can be fabricated using, e.g., electron beam lithography or an alternative method. Embedding arrangements of nanostructures, such as nanoparticles, in optoelectronic devices, such as OLEDs, is compatible with industrial-scale fabrication processes, such as via lithography, e.g. via substrate conformal imprint lithography (SCIL) or nanoimprint lithography (NIL).

[0077] The arrangements may be experimentally shown to provide the proposed effect of one or more flat bands. One or more such flat bands, or a flat band effect may be produced, based on an arrangement according to an embodiment. FIG. 8A and 9A illustrate dispersion relations obtained with arrangements of gold nanoparticles. Dimensions of the particles were in the 100 nm range and of the arrays in the 200x200 square microns range. The samples are fabricated on glass substrates by electron-beam lithography. A symmetric refractive index environment (n = 1.52) was realized by covering the sample with index-matching oil corresponding to the glass substrate. Dispersions were measured via transmission of white light through the sample, analyzed by Fourier imaging, in some cases polarization filtered. TE polarization may be filtered out, for example.

[0078] FIG. 8A and 8B illustrate examples of dispersion relation as obtained with an arrangement according to an embodiment. The dispersion relation is obtained with a so-called cross geometry, such as similarly to cross geometry depicted in FIG. 2A and 2B, for example. FIG. 8A illustrates measured dispersion relation E(k), for a cross-lattice geometry. FIG. 8B illustrates a simulated dispersion relation E(k) for such a cross-lattice geometry.

[0079] FIG. 8A shows an experimentally measured extinction spectrum of the arrangement illustrated in the inset. As may be appreciated, in FIG. 8A, a flat band 820 produced by a rectangular chain cross-lattice geometry (also known as a cross geometry), such as the one illustrated in FIG. 3A and 3B, is observable. The arrangement (cross geometry) comprised particles 811. Inset in FIG. 8 A illustrates such a rectangular chain lattice. In-plane momentum kycorresponds to the angle of light extinction by the arrangement (i.e., a lattice). The extinction spectrum is filtered to show only TM polarization. The linewidth of the flat band 820 indicated by bar is 6 meV (equivalent of 3.7 nm in wavelength), corresponding to a Q-factor of 235.

[0080] FIG. 8B illustrates a calculated dispersion (in TM polarization) and the substantially non-zero values of the structure factor (in the inset of FIG. 8B) of a 200 pm x 200 pm rectangular chain lattice with Nx= 13, ax= 580 nm, 9X= 0° and Ny = 7, ay= 1160 nm, 9y= 90°, where Nxand Nyare the number of nanoparticles in the respective chains (linear sequences) and 0Xand 0ytheir angle with respect to x axis. The parameters axand ayrefer to interparticle distance of the respective linear sequences (chains along x and y axis). The corresponding rectangular chain lattice is shown in the inset of FIG. 8A. The arrangement comprises gold nanocylinders with diameter 130 nm and height 50 nm in an index-symmetric background with the refractive index n = 1.52. In the inset of FIG. 8B, the axes have been slightly displaced for clarity. As may be appreciated, a flat band is formed by the modes on the top and bottom horizontal rows of peaks (white circles). These modes correspond to the diffraction orders with qi = ±NXor equivalently, kx= ±27t / ax.

[0081] FIG. 9A and 9B illustrate flat band(s) as generated by at least some embodiments. FIG. 9A shows an experimentally measured TM polarized extinction spectrum. FIG. 9B shows a calculated dispersion of the lattice embedded in an index-symmetric environment. The simulated and experimental data are in good agreement, as the three flat bands appear at the predicted energies. The features may, however, be relatively weak for the modes constituting the flat bands. As can be appreciated, multiple flat bands 91A, 92A and 93A in FIG. 9A and 91B, 92B and 93B in FIG. 9B may be obtained with a suitable arrangement of linear sequences and nanostructures.

[0082] Similarly to FIG. 4, an offset of linear sequences was utilized to obtain such a flat-band effect with plurality of flat-bands at two or more frequencies. FIG. 9B illustrates the computational example of the dispersion relation, as well as the arrangement suitable for generating such a multi-flat band effect (in the inset). The inset of FIG. 9B shows a part of the unit cell of the lattice. The lattice of FIG. 9A comprises superimposed 41^21, 43x21, 46x21 rectangular chain lattices with lattice constants ax= 592 nm, 564 nm, 528nm and ay= 592 nm and offsets (Axi, Ayi) = (0 nm, 0 nm), (Ax2, Ay2) = (200 nm, 200 nm), (Ax3, Ay3) = (-210nm, -210 nm). The size of the unit cell was 24.3 pm x 12.4 pm. The particles are gold (nano)cylinders with a diameter of 120 nm and a height of 50 nm in an environment with a refractive index n = 1.52.

[0083] At least some embodiments utilize offsets of linear sequences, so as to provide two or more flat bands at respective wavevector ranges.

[0084] In an embodiment, the one or more flat bands may be restricted to a narrower wavevector range. FIG. 10A illustrates an example dispersion relation in relation, and the band structure of a lattice designed to have at least one flat band 1001 between ±15°, and in the inset an example arrangement with a constant offset along y direction. The inset also shows the unit cell 1016, as a rectangle with a dashed outline of the lattice. In the unit cell, there are four nanoparticles 1011, having a (interchain, offset) spacing of Ay ~ 0.389ax. Note that the inset is not to scale, ay= |a2| = 8ax(ax= |ai|); this value of ayas selected to best visualize the weakening of the flat band at greater angles. The axes in the inset have been slightly displaced for ease of viewing, the leftmost lattice site is actually at the origin. The spacing of the minor ticks of the kyaxis of the main figure corresponds to the width of the Brillouin zone, BZ. As may be appreciated, the arrangement of FIG. 10 therefore has at least one lattice constant (here ay) greater than, such as greater than 2.5 times, for example three times or five times, the interparticle distance in the orthogonal direction (here ax).

[0085] FIG. 10B illustrates a repeated arrangement of nanoparticles 1011 using a unit cell 1016 as also illustrated in the inset of FIG. 10A. The arrangement may therefore comprise an offset Ay along y direction as shown. This arrangement may be configured to reduce the extent of the wavevector range of a flat band. The lattice constant ayshould be sufficiently greater than ax. The arrangement comprises therefore a first set, a second set, third set, and a fourth set of first linear sequences, displaced by an offset of Ay, and repeatedfour times along a2 direction (y direction) of the unit cell 1016 and eight times along ai direction (x direction) of the unit cell 1016.

[0086] The offset may be selected based on interparticle spacing in the linear sequences and the desired angular range for a flat band. The angular range (for half-angle 0HA) for the wavevector, such as for example shown in FIG. 10A and 10B, may follow an equation (where N refers to number of lattice sites, i.e. number of nanoparticles) such as:

[0087] Therefore, the offset Ay may be based on at least the interparticle distance ax, and the desired half-angle (in which a flat band is to appear). Notably, the lattice constant aydoes not directly appear in the expression above. However, aymay control how many modes appear within the angular range determined by the half-angle 0HA. For example, for FIG.10A, have a flat band extending to, approximately, 0HA = 15 degrees.

[0088] The arrangements, as well as devices incorporating such arrangements, according to at least some embodiments provide several advantages and benefits. With a flat band, electromagnetic radiation, such as light, may be directed in a defined manner. The flat band(s) generated by the arrangement(s) is invariant to the in-plane wavevector having therefore an advantage in that light trapped in a medium by total internal reflection can couple to the SLRs, leading to increased efficiencies. A flat band provides viewing angle independent emission. A flat band enhances emission only at its specific wavelength, thereby narrowing the linewidth of the emitter and enhancing color purity. Such may be beneficial in display and lighting devices, for example. Moreover, in at least OLEDs, the rate of spontaneous emission depends on the density of states (Purcell effect) and at the flat band, the density of states approaches infinity. For emitter molecules that benefit from an increased rate of spontaneous emission, flat bands could improve the efficiency, such as the efficiency of an OLED. Similarly, a flat band can enhance absorption at the flat band wavelength, which could be beneficial to e.g., applications in sensing and photovoltaics. A flat band may be beneficial for lasing applications because of light localization, high density of states, and Q factor.

[0089] It is to be understood that the embodiments of the invention disclosed are not limited to the particular structures, process steps, or materials disclosed herein, but areextended to equivalents thereof as would be recognized by those ordinarily skilled in the relevant arts. It should also be understood that terminology employed herein is used for the purpose of describing particular embodiments only and is not intended to be limiting. Reference throughout this specification to “one embodiment” or “an embodiment” means that a particular feature, structure, or characteristic described in connection with the embodiment is included in at least one embodiment of the present invention. Thus, appearances of the phrases “in one embodiment” or “in an embodiment” in various places throughout this specification are not necessarily all referring to the same embodiment. As used herein, a plurality of items, structural elements, compositional elements, and / or materials may be presented in a common list for convenience. However, these lists should be construed as though each member of the list is individually identified as a separate and unique member. Thus, no individual member of such list should be construed as a de facto equivalent of any other member of the same list solely based on their presentation in a common group without indications to the contrary. In addition, various embodiments and example of the present invention may be referred to herein along with alternatives for the various components thereof. It is understood that such embodiments, examples, and alternatives are not to be construed as de facto equivalents of one another, but are to be considered as separate and autonomous representations of the present invention. Furthermore, the described features, structures, or characteristics may be combined in any suitable manner in one or more embodiments. In the following description, numerous specific details are provided, such as examples of lengths, widths, shapes, etc., to provide a thorough understanding of embodiments of the invention. One skilled in the relevant art will recognize, however, that the invention can be practiced without one or more of the specific details, or with other methods, components, materials, etc. In other instances, well-known structures, materials, or operations are not shown or described in detail to avoid obscuring aspects of the invention. While the forgoing examples are illustrative of the principles of the present invention in one or more particular applications, it will be apparent to those of ordinary skill in the art that numerous modifications in form, usage and details of implementation can be made without the exercise of inventive faculty, and without departing from the principles and concepts of the invention. Accordingly, it is not intended that the invention be limited, except as by the claims set forth below. The verbs “to comprise” and “to include” are used in this document as open limitations that neither exclude nor require the existence of also un-recited features. The features recited in depending claims are mutually freely combinable unless otherwise explicitly stated. Furthermore, it is to beunderstood that the use of "a" or "an", i.e. a singular form, throughout this document does not exclude a plurality.INDUSTRIAL APPLICABILITY

[0090] At least some embodiments find industrial application in relation to optics, such as optoelectronic devices, for example organic light-emitting diode, OLED, devices.ACRONYMS LISTREFERENCE SIGNS LIST

Claims

CLAIMS:

1. An arrangement for producing at least one flat-band effect for at least one wavelength range of electromagnetic radiation, the arrangement comprising:a first set of first linear sequences of nanostructures, said first linear sequences being separated from one another such that an inter-sequence distance of adjacent first linear sequences is greater than a distance of adjacent nanostructures within the first linear sequences.

2. The arrangement according to claim 1 further comprising a first set of second linear sequences of nanostructures.

3. The arrangement according to claim 2, wherein the first set of second linear sequences are at an angle with respect to the first set of first linear sequences.

4. The arrangement according to claim 3, wherein the angle is along a plane defined by the first set of first linear sequences.

5. The arrangement according to claim 3 or 4, wherein the angle is between 80 and 100 degrees.

6. The arrangement according to claim 3 or 4, wherein the angle is between 50 and 80 degrees, such as 50 to 70 degrees.

7. The arrangement according to any one of claims 2 to 6, wherein the distance of the adjacent nanostructures within one of the first linear sequences or the second linear sequences is different than, such as smaller than, a distance of the adjacent nanostructures within one other of the second linear sequences or first linear sequences; or wherein the distance of the adjacent nanostructures within the first linear sequences is different than a distance of the adjacent nanostructures within the second linear sequences.

8. The arrangement according to any one of the preceding claims, wherein the adjacent nanostructures within at least the first linear sequence have an equidistant spacing.

9. The arrangement according to any one of the preceding claims for producing at least oneflat-band effect for at least two separate wavelength ranges, the arrangement further comprising a second set of first linear sequences of nanostructures separated by an offset from at least the first set of first linear sequences.

10. The arrangement according to any one of the preceding claims further comprising a second set of first linear sequences of nanostructures separated by an offset from at least the first set of first linear sequences.

11. The arrangement according to any one of the preceding claims, wherein the nanostructures are nanoparticles.

12. The arrangement according to any one of the preceding claims, wherein the nanostructures are plasmonic nanoparticles, such as metallic plasmonic particles, for example, metallic plasmonic particles comprising at least one of: gold, aluminum, or silver.

13. The arrangement according to any one of claims 1 to 11, wherein the nanostructures are dielectric or semiconducting nanoparticles having a refractive index contrast with respect to a surrounding medium greater than or equal to 0.3.

14. The arrangement according to claim 13, wherein the dielectric or semiconducting nanoparticles comprise at least one of: germanium, silicon, or titania, TiCh.

15. The arrangement according to any one of claims 11 to 14, wherein the material of the nanoparticles is selected such that the nanoparticles possess a single particle resonance for forming collective modes in the linear sequence.

16. The arrangement according to any one of the preceding claims, wherein the dimensions of the nanostructures are such that said nanostructures support at least one of localized surface plasmon resonance or Mie resonance.

17. The arrangement according to any one of the preceding claims, wherein widths of the nanostructures are from 5 nm to 2500 nm, such as from 10 nm to 2250 nm, for example from 20 nm to 2000 nm or 30 nm to 1750 nm.

18. The arrangement according to any one of the preceding claims, wherein widths of the nanostructures are more than 0.05 times a minimum wavelength of the at least one wavelength range and less than twice the wavelength of a maximum wavelength of the at least one wavelength range, the wavelengths referring to wavelengths of electromagnetic radiation in a medium surrounding the nanostructures.

19. The arrangement according to any one of the preceding claims, wherein heights of the nanostructures are from 2 to 2000 nm, such as from 10 to 1250 nm, for example 20 nm to 1000 nm or from 30 nm to 800 nm.

20. The arrangement according to any one of the preceding claims, wherein heights of the particles are from 0.03 times a minimum wavelength to 1 time a maximum wavelength of the at least one wavelength range, the wavelengths referring to wavelengths of electromagnetic radiation in a medium surrounding the nanostructures.

21. The arrangement according to any one of the preceding claims, further comprising a surrounding medium and the nanostructures are embedded in the surrounding medium, such as a medium comprising at least one of: glass, quartz, silicon, organic material, plastic, organic or inorganic materials suitable for at least some optoelectronic devices.

22. The arrangement according to any one of the preceding claims, wherein the intersequence distance of at least the first linear sequences is at least 2.5 times, such as at least 3 times, for example at least 5 times, greater than the distance of adj acent nanostructures within the first linear sequences.

23. The arrangement according to any one of the preceding claims, wherein the nanostructures are nanoparticles, and wherein the inter-sequence distance of at least the first linear sequences is at least 2.5 times, such as at least 3 times, for example at least 5 times, greater than interparticle distances of adjacent nanoparticles within the first linear sequences.

24. An arrangement for producing at least one flat band effect for at least one wavelength range of electromagnetic radiation, the arrangement comprising:parallel linear sequences of nanostructures, said parallel linear sequences separated from one another such that an inter-sequence distance between adjacent parallellinear sequences is greater than a distance of adjacent nanostructures within the parallel linear sequence.

25. The arrangement according to claim 24 wherein the inter sequence distance is two and a half times, such as three times, for example five times, greater than the separation of adjacent nanostructures within the linear sequence.

26. An arrangement for producing at least one flat band effect for at least one wavelength range, the arrangement comprisingnanostructures in a pattern defined by a periodic arrangement of unit cells, such that at least one lattice constant of the unit cells is more than two and a half times, such as three times, for example, five times, larger than a distance of adjacent nanostructures.

27. The arrangement according to claim 26, wherein the nanostructures form at least two parallel linear sequences of nanostructures via repetition of the unit cells, such that the at least one lattice constant larger than two and a half, such as three times, for example five times, corresponds to the inter-sequence distance of adjacent linear sequences of the at least two parallel linear sequences.

28. The arrangement according to claim 26 or 27, wherein the nanostructures form at least two linear sequences at an angle to one another, such as at an angle of 50 to 140 degrees, for example from 80 to 100 degrees or from 50 to 80 degrees.

29. The arrangement according to any one of claims 26 to 28, wherein the arrangement comprises another pattern of nanostructures defined by the same unit cells, said another pattern separated by an offset from the pattern.

30. The arrangement according to any one of claims 26 to 29, wherein the nanostructures in each unit cell form at least two intersecting linear sequences of nanostructures.

31. The arrangement according to claim 30, wherein the two intersecting linear sequences intersect at an angle from 50 to 140 degrees, such as from 80 to 100 degrees or from 50 to 80 degrees.

32. The arrangement according to any one of claims 26 to 31, wherein the nanoparticles in each unit cell form two or more, such as three, intersecting linear sequences of nanostructures.

33. The arrangement according to any one of claims 26 to 32, wherein the arrangement comprises more than three, such as more than 5, for example more than 10 unit cells along at least one lattice vector, such as each lattice vector, for example both lattice vectors along a plane of the arrangement, in the periodic arrangement.

34. The arrangement according to any one of claims 26 to 33 wherein the nanostructures are nanoparticles.

35. An arrangement for producing at least one flat-band effect for at least one wavelength range of electromagnetic radiation, the arrangement comprising:nanostructures in an ordered structure having a first dimension and a second dimension perpendicular to the first dimension, the ordered structure formed by periodically repeated patterns of nanostructures;each pattern covering the first dimension of the ordered structure; the patterns being repeated along the second dimension of the ordered structure at intervals having a length defined by the smallest period between the periodically repeated patterns;each pattern comprising a first linear sequence of nanostructures, wherein a length of the first linear sequence has a longer component along the first dimension than along the second dimension, and wherein the nanostructures in the first linear sequences has a first distance between consecutive nanostructures in the sequence; andthe length of the interval being at least two and a half times, such as three times, for example five times, the first distance.

36. An apparatus comprising:an arrangement according to any one of claims 1 to 35; andan optoelectronic device.

37. The apparatus according to claim 36, wherein the optoelectronic device is electromagnetically coupled to the arrangement.

38. The apparatus according to any one of claims 36 or 37, wherein the optoelectronic device comprises the arrangement.

39. The apparatus according to any one of claims 36 to 38, wherein the optoelectronic device comprises at least one of:a photodetector, such as a photodiode, phototransistor, photomultiplier tube, or solar cell; ora light source, such as a light-emitting diode, LED.

40. The apparatus according to any one of claims 36 to 38, wherein the optoelectronic component comprises an organic light-emitting diode, OLED.

41. The apparatus according to claim 40, wherein the arrangement is optically coupled to an emissive layer of said OLED.

42. The apparatus according to claim 40 or 41, wherein a distance of the nanostructures to an emissive layer of the OLED is less than two times of a maximum wavelength of light configured to be emitted from the emissive layer, said maximum wavelength referring to the wavelength in a medium surrounding the nanostructures.

43. The apparatus according to any one of claims 40 to 42, wherein the arrangement is between an electrode of the OLED and the emissive layer of the OLED.

44. An organic light emitting diode, OLED, comprisinga first electrode and a second electrode;an emissive layer between the first and second electrodes; andan arrangement of nanostructures, said arrangement being configured to, based on the geometric arrangement of the nanostructures, provide at least one flat-band effect for at least one wavelength of electromagnetic radiation.

45. The OLED according to claim 44, wherein said arrangement is between the first electrode and the second electrode.

46. The OLED according to claim 44, wherein the arrangement is at least in part embeddedin the first electrode.

47. The OLED according to any one of claims 44 to 46, wherein the arrangement is the arrangement according to any one of claims 1 to 35.

48. The OLED according to any one of claims 44 to 47, wherein arrangement is arranged on the surface of the first electrode.

49. The OLED according to any one of claims 44 to 47, wherein the arrangement is in contact with the first electrode.

50. The OLED according to any one of claims 44 to 47, wherein the arrangement is embedded in the first electrode.

51. The OLED according to any one of claims 44 to 50, wherein said arrangement is separated from the second electrode by a distance of more than 20 nm, such as more than 50 nm.

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