Photonic crystals are optical media containing a periodic nanostructure – typically, a periodic variation of the refractive index, where the occurring periods are of the order of the optical wavelength. While in some cases there is a quite large refractive index contrast, for example between a glass or a semiconductor and air, much lower refractive index contrasts occur in other cases. Some of the particularly remarkable photonic crystal properties – for example complete photonic band gaps – arise primarily in cases with large index contrast, and this is why some authors suggest to demand a large index contrast for photonic crystals.
This area of research, which is highly important both in terms of fundamental science and practical applications, has been pioneered by Eli Yablonovitch  (who also introduced the term photonic crystals) and Sajeev John , and followed by a large number of other researchers. Early research on one-dimensional periodic structures (multilayer dielectric stacks → dielectric coatings as Bragg mirrors) has already been performed by Lord Rayleigh in 1887. While photonic crystals are mostly dealt with in areas of modern technology, photonic crystal structures have also been observed in nature, for example and certain minerals (e.g. opal) and in living organisms, e.g. in butterfly wings.
There is an obvious similarity between photonic crystals and photonic metamaterials: the latter also usually involve a periodic arrangement of structures with optical effects. However, photonic metamaterials have a structuring on a sub-wavelength scale, such that those structures appear like being homogeneous for optical fields, and the resulting remarkable properties are not explained with photonic band structures, but essentially with unusual values of the refractive index.
Analogy With Electrons in Crystalline Media
Electron States and Bloch Functions
In the early 20th century, many of the fundamental properties of solids such as their widely varying electrical and thermal conductivity, optical transmissivity or absorption were still not understood at all. Substantial progress was then achieved by considering the wave functions of electrons in periodic electrostatic potentials, as are created by the periodic arrangements of atomic nuclei in a crystalline material.
While initially it had to be expected that the very strongly varying electric potentials should lead to strong scattering of electrons in such structures, it was discovered that such periodic structures exhibit eigenstates for electrons which can be described with Bloch functions. A Bloch function has the form
where k is a variable wave vector, determining the phase evolution over long distances, and the u function has the periodicity of the electric potential. A Bloch wave is a standing wave – note that the intensity profile, determined by the u function, remains fixed in space. Different solutions are distinguished with a discrete index j. The wave vector can in principle be chosen arbitrarily, but it is sufficient to consider only wave vectors inside the first Brillouin zone; vectors outside that zone only lead to the same solutions. With suitable mathematical methods, for a given potential one can calculate the resulting Bloch functions, and for each one the corresponding electron energy (as the eigenvalue). Considering an infinitely large crystal, one obtains a continuum of Bloch functions and energy values.
For each value of the index j, the variation of the wave vector leads to a certain range of electron energies, which is called an energy band. These bands often overlap with each other, leading to wide continuous ranges of possible electron energies, but in certain situations band gaps occur, i.e., ranges of electron energies which do not occur for any of the available eigenstates.
Each Bloch state is completely delocalized, meaning that an electron in such a state is not corresponding to a certain atomic nucleus, but can in principle be found anywhere in the whole crystal. Localized electronic states only arise for certain lattice defects.
An electron would in principle stay in its eigenstate forever, unless additional effects cause transitions between the states – either inside a band or across energy bands. Examples are spontaneous emission, absorption of light and stimulated emission, also interactions with phonons, with other electrons or with lattice imperfections.
Without such additional coupling effects, the calculated states are eigenstates despite the strong scattering at the atomic nuclei. One consider the eigenstates as superpositions of a set of plane waves, moving in very different directions, such that despite the strong coupling between those their overall pattern remains stationary.
The resulting structure of the electron eigenstates in conjunction with Pauli's principle has various important consequences:
- Absorption of light (in the linear regime, i.e., without excessive optical intensities) is possible only if the photon energy is suitable for connecting a populated initial state with a non-populated higher-lying energy state. In addition, there are restrictions concerning the involved wave vectors, in other words based on the principle of momentum conservation: the initial and final energy level must have quite similar wave vectors, since the wave vectors of photons in the relevant energy range (e.g. infrared, visible or ultraviolet light) are relatively small.
- Similar restrictions apply for spontaneous and stimulated emission, which brings electrons to lower-lying states.
- Completely filled bands cannot contribute to electrical conduction, because no redistribution of the level population is possible. Partially filled bands, as occur particularly in metals, can contribute to electrical and thermal conduction. The thermal and electrical conductivity can in principle be very high, but can be strongly reduced by lattice defects causing scattering.
Many essential properties of insulators (dielectrics), semiconductors and metals can be explained based on their electronic band structures.
Photon States in Periodic Media
In photonics, one deals with the propagation of light, i.e., electromagnetic waves. In the case of a periodic photonic nanostructure, described by a periodically varying refractive index n or dielectric constant ε, one can set up the wave equation under such conditions, where plane waves are of course not eigensolutions (as in a homogeneous medium).
Just as in solid-state physics for electrons, one can show that there are again Bloch waves, but this time with functions describing the complex electric field amplitude instead of electron wavefunctions. However, in various respects the circumstances are different:
- While the electron wave functions are scalar functions, the electric field is a vector field, leading to the possibility of different polarization states.
- On the other hand, while the description of electron states with band structures is somewhat inaccurate due to neglected electron–electron interactions, that complication does not occur in photonic crystals.
- Because electrons have a mass while photons do not, their dispersion relations in free space are different.
- Photons are bosons, while electrons are fermions, and that difference has profound implications on the possible occupation of states.
Still, substantial parts of the insight from solid-state physics can be used in quite similar form for the analysis of photonic crystals:
- There are Bloch functions describing the possible eigensolutions for wave vectors within the first Brillouin zone, and the corresponding eigenvalues are optical frequencies. Of course, the dimensions of the first Brillouin zone are about three orders of magnitude smaller than those for the electron states, since the modulation periods are much longer.
- There can again be band gaps (here called photonic band gaps or photonic bandgaps, PBG, or sometimes less accurately stop bands), where the optical frequency bands do not overlap. For frequencies within such a band gap, light cannot propagate in the material – in the sense that there are no suitable Bloch states, but only solutions with an exponential decay of amplitude, which means that appreciable light amplitudes can occur only over a limited distance. Incident light with such frequencies must therefore be reflected by the photonic crystal. For a complete photonic band gap, one even obtains omnidirectional reflection.
- Within a band, the propagating light has peculiar properties. It always has components propagating in different directions, which are coupled to each other. That also has profound implications on the transport of optical energy and on the group velocity. In those parts of such a photonic band structure where the contours become very flat, group velocities far below the ordinary velocity of light can occur. Due to the substantial frequency dependence of possible k values, highly anomalous chromatic dispersion can occur as well.
- Further peculiar effects can occur at boundaries between photonic crystals and ordinary optical materials. For example, there can be negative refraction and superprism effects.
In cases where the existence of a photonic band gap is essential, such materials are also often called photonic bandgap materials. They are also sometimes called photonic insulators or photonic semiconductors, because electric insulators and semiconductors are those where the electronic band structure exhibits band gaps.
One may wonder whether the periodicity on the atomic scale (for crystalline materials) could also play a role. However, that length scale is much too short to be relevant for light, having wavelengths on a much longer scale.
Photonic Crystals in One, Two and Three Dimensions
Photonic crystals can be realized in different dimensions, which are discussed in the following sections.
One-dimensional Photonic Crystals
It is debatable whether one-dimensional periodic structures should be qualified as photonic crystals; the pioneer Yablonovitch rejected that. However, in any case they are a good starting point for analyzing fundamental properties of photonic crystals without additional complications which come into play for higher dimensions.
As mentioned above, one-dimensional periodic structures in the form of dielectric multilayer stacks (Figure 1) have already been considered in the late 19th century by Lord Rayleigh. Nowadays, they are routinely used as dielectric mirrors, in the simplest case with a design of a Bragg mirror. One frequently just ignores the additional spatial dimensions by considering only light propagating into exactly opposite directions, coupled to each other by the periodic refractive index modulation.
The mathematically simplest case actually occurs for a sinusoidal (rather than rectangular) modulation of the dielectric constant, although for technical applications the step-wise modulation of a laminated material is much more common – realized by using layers of two different materials, rather than a continuous modulation as used in the much less common rugate filters. It can be shown quite easily that for a vanishing amplitude of the modulation of dielectric constant one obtains a simple band structure without any band gaps, but band gaps of increasing size then arise for an increasing strength of the modulation. In such frequency regions, one obtains an exponential decay of electric field amplitude in one direction (ignoring the modulation on short length scales), meaning that light cannot propagate over substantial distances into such a structure.
By calculating the optical properties of a Bragg mirror with 100 layer pairs, one can already explore to some extent the approximate properties of a photonic bandgap of an infinite periodic structure, extending in z direction from zero to infinity. Figure 2 shows the field penetration into such a structure. Within the wavelength range from approximately 875 nm to 1165 nm, one obtains strong reflection and penetration of the optical field into the structure only to a quite limited extent (a few microns). Light outside that wavelength range can propagate deep into the structure. It is only that the intensity pattern in that “propagating region” would still get finer and finer with an increasing number of layer pairs; of course, one does not get the true band structure of the infinite photonic crystal with that type of calculation. (Anyway, existing structures are always finite.) However, a more and more increasing number of layer pairs does not have a substantial influence on the width of the reflecting wavelength range; that is essentially determined by the refractive index contrast (in the example case between titanium dioxide and silica). One can show that in terms of optical frequency relative to the center frequency it is
Two-dimensional Photonic Crystals
Two-dimensional photonic crystal structures are understood to be periodic in two spatial dimensions, while there is no modulation of dielectric constant in the third dimension. For example, one can consider a hexagonal structure, where Figure 3 shows a cross section (which would look exactly the same for all z values, assuming that the z direction is perpendicular to the drawing plane). The air holes (shown in white) run through the structure infinitely in z direction.
Figure 3 shows as an example a hexagonal structure, which belongs to the particularly common ones. One may also invert such a structure, i.e., use a pattern of cylinders surrounded by voids, and held in position either by very thin strands (with negligible optical effects) or with two solid planes at the edges of a device.
Again, one may ignore the possibility that the k vector could have a component in the z direction. That would be appropriate for a situation where one tries to inject only such kind of light into the structure. However, such a situation is rarely found in practical applications. It is more common that one realizes a relatively flat 2D photonic crystal (a photonic crystal slab) which is enclosed on two sides e.g. with a homogeneous material of substantially lower refractive index, confining the light in that dimension with the classical waveguide approach. At least light hitting such an interface with a large enough angle of incidence, one obtains total internal reflection. If the photonic crystal slab is thin enough, it can have single-mode properties in the transverse dimension.
Another common situation is that the considered light even more or less propagates in the z direction. That is the case for photonic crystal fibers – which, however, also require an additional lattice defect for obtaining a waveguide – to be discussed further below.
An important point to emphasize is that it is not only more difficult to calculate band structures of 2D photonic crystals, but also more difficult to find designs which exhibit complete band gaps in the sense that within a certain range of optical frequencies there is no direction at all in which light can propagate without an exponential decay of amplitude. That is particularly the case when different polarization directions are also considered. It can easily happen, for example, that one obtains a band gap only for TE polarization, but not for TM polarization, or vice versa. Only with carefully chosen structures and a relatively large refractive index contrast (which precludes calculations with purely scalar field models), complete band gaps are possible. Such difficulties help one to understand that it took about a century from first thoughts on one-dimensional structures to a good understanding of two-dimensional photonic crystals.
Various technologies have been developed for the fabrication of 2D photonic crystal structures. Compared with 3D structures (see below), this is substantially simpler, basically because one has good access to the whole structure from one side and may later on deposit additional homogeneous material, if not simply using an interface to air. For an example for such technology, see the article on photonic crystal surface-emitting lasers. Often, one utilizes technologies which have been well developed in the context of semiconductor processing for optoelectronics.
Three-dimensional Photonic Crystals
It is easy to imagine various kinds of 3D photonic crystal structures, for example
- with a pattern of spherical pieces of glass or semiconductor, arranged with lattice types which are well known from solid-state physics, e.g. hexagonal, face-centered cubic or diamond structure (for the moment ignoring the necessity of holding the solid pieces in place),
- with an inverted pattern of voids surrounded by a solid medium (removing the problem of mechanical stability), or
- with other geometrical shapes, obtained e.g. with a regular pattern of cylindrical voids of different orientations.
3D photonic crystals are generally more challenging to fabricate, as is fairly obvious particularly because of the tiny dimensions dictated by the short optical wavelengths, and the resulting extreme demands in terms of resolution and accuracy:
- One possible approach is to build up such structures layer by layer, e.g. with techniques of electron-beam lithography or laser 3D printing. It is then also possible to flexibly introduce lattice defects as required. However, such techniques are limited in terms of the materials which can be processed that way, and by the number of layers which can be made with sufficiently high quality in various respects, such as precise regularity, absence of irregular forms, etc. In some cases, only a few layers can be obtained with an overall thickness of only a few micrometers, and not more extended photonic crystals.
- One may also start with a piece of solid material and then create a pattern of voids in some way, e.g. by drilling or by reactive ion etching. (A particular structure of that type has been named Yablonovite.) Naturally, such techniques are practical only for a quite limited size of a photonic crystal, since the number of required operations and the difficulty e.g. of drilling over long distances rapidly increase with increasing size.
- One may first fabricate a template e.g. with closely packed silica spheres, which are then sintered together to obtain mechanical stability. The voids are then filled with silicon (using a precursor gas), and finally one removes the silica template by acid etching .
- There are methods of holographic lithography, using interference patterns to obtain the wanted periodicity over large volumes.
- Another possibility is to use self-assembled colloidal crystals , which may in principle be made with large sizes, but it is difficult to reliably obtain the wanted crystal structure.
After the pioneering work on 2D photonic crystals, it took only a couple of years to find the first 3D photonic crystal structures with complete band gaps, and to fabricate them. The first one contained dielectric spheres arranged in the diamond structure . A previous report concerning a face-centered cubic (fcc) structure  had to be revised. It turned out that fcc structures usually exhibit a semimetallic band structure (i.e., not with real band gaps), although a complete band gap is possible for a structure with very large refractive index contrast of greater than 3:1 . Another possibility is the “woodpile” structure, which was first demonstrated for microwave and mid-infrared frequencies, later also for the near infrared including the telecom wavelength region around 1.5 μm [20, 24]. With a 3D tungsten photonic crystal structure, it became possible to substantially control the emission of thermal radiation such that it is strong for wavelength around 1.5 μm while it is suppressed for long wavelength beyond 3 μm.
Omnidirectional Reflection; Lattice Defects
We have seen that a photonic band gap essentially prohibits the propagation of light in the medium within a certain optical frequency range. This allows a photonic crystal to act as a kind of omnidirectional reflector, assuming that the band gap exists for all relevant propagation directions – ideally, for any directions in three dimensions. A photonic crystal material is thus suitable for enclosing structures in order to prevent light from escaping. Indeed, the pioneering paper of Yablonovitch in 1987  already described the use of a three-dimensional photonic crystal for suppressing spontaneous emission of excited atoms or ions inside the medium. It had already been known from quantum optics that the possibility of emission of radiation is subject to the availability of suitable modes of the light field, and a photonic crystal can thus suppress spontaneous emission by removing such propagation modes.
Similarly, a photonic crystal can be used to confine light to different kinds of structures, which can be considered as some kind of intentionally introduced lattice defects. Those provide localized defect states within a photonic bandgap :
- If a photonic crystal is made as a pattern of voids (air holes) in a solid medium, for example, one can create an isolated region where one or a couple of voids are missing. The result can be a tiny optical resonator, called micro-resonator or microcavity . Particularly high Q factors of such resonators are possible if one only uses air holes of reduced diameter rather than totally missing holes, or other features with only modified dimensions or positioning . Optimization concerning fabrication tolerances can also be important , as the imperfections of fabrication techniques cannot be reduced arbitrarily. High Q factors in conjunction with a light-amplifying element (typically a quantum dot) also allow the formation of micro-lasers with low threshold pump power . While other types of micro-resonators, e.g. toroid microcavities, may reach even higher Q factors, photonic crystal micro-cavities have the crucial advantage of easier well-controlled coupling to waveguides made with the same technology.
- A long sequence of such removed voids can act as a channel waveguide. Similarly, a kind of planar waveguide can be obtained by extending the lattice defect in two dimensions. Channel waveguides can usually run only in certain directions, and cannot be arbitrarily shifted in the transverse direction, because there is the requirement of conformity with the photonic crystal structure. However, it is also possible to realize even sudden turns into other directions (e.g. right angle turns) , which for conventional waveguides would lead to massive power losses. Also, one can realize various types of waveguide junctions (e.g. T junctions) and beam splitters , for example for use in interferometers. Nearby waveguides can exhibit very strong coupling within a short length.
- If two or more such structures are made with a sufficiently small distance between them, there can be some coupling between them. For example, a resonator can be coupled to a waveguide, such that one can inject light through the waveguide, which will then efficiently get into the resonator in a certain narrow frequency range – and possibly be coupled out to a second waveguide also coupled to the resonator. In that way, one can construct highly frequency-sensitive couplers.
The concept of intentionally introducing certain lattice defects, e.g. for resonators and waveguides, and also to integrated light-emitting features, is also applicable in three dimensions. Only, it is again quite difficult to implement, and also to optimally design such structures. Propagation losses of waveguides resulting from imperfections need to be considered . Nevertheless, there are some results:
- It has been demonstrated that light-emitting quantum wells can be integrated into 3D “woodpile” photonic crystal structures, and both the suppression of light emission by photonic crystal and the modification of emission properties by point defects acting as optical resonators has been demonstrated .
- Similar things have been achieved with semiconductor quantum dots in an inverse opal photonic crystal structure .
Various Special Effects
Beyond the suppression of propagation of light in certain frequency regions, photonic crystals can provide a wide range of quite special effects:
- It has already been mentioned that the group velocity of light in a photonic crystal (outside the band gap regions) can be very unusual, for example far lower than normal (slow light).
- There are cases where negative refraction occurs [41, 36] – as would be expected for a material with negative refractive index: after the interface to a photonic crystal material, the reflected beam occurs at the same side of the normal direction as the incident beam.
- Another possibility is ultra-refraction, where the medium behaves like one with an refractive index below unity.
- In some situations, the refraction of light at photonic crystals exhibits an unusually strong dependence of the propagation direction at the output on the input direction or the optical wavelength; such superprism effects [16, 18, 19] have been studied both theoretically and experimentally.
- Subwavelength imaging (superlensing) is feasible using a slab of photonic crystal material , which can also capture evanescent waves to allow for superresolution.
- Autocollimation means that an injected light beam (with a frequency inside a photonic band) propagates without substantial beam divergence – and this without using any waveguide structure. It thus works without any critical alignment to the input of the waveguide, and for beams with some range of propagation directions.
A detailed analysis of band structures of photonic crystal materials and related devices is of crucial importance not only for acquiring a good quantitative understanding of such structures and devices, but also for successful design optimization. It requires quite sophisticated mathematical and computational tools – particularly if a comprehensive 3D model is required. As was pointed out already, a full vectorial description of the electromagnetic field is usually needed due to the necessary large refractive index contrast. Rather different techniques have been developed, which can have a specific advantages (e.g. accuracy and computational efficiency) but also serious limitations such as applicability only to specific cases:
- Finite Difference Time Domain (FDTD) simulations  allow one to flexibly simulate time-dependent phenomena, such as the propagation of ultrashort pulses in photonic crystal waveguides, experiencing effects of chromatic dispersion and nonlinearities. Such methods are quite versatile, but tend to cause a substantial computational load and require large amounts of computer memory – particularly if devices with relatively large volumes are investigated.
- Beam propagation methods can be substantially more efficient, but are more limited in application. They usually work for special structures only, also with monochromatic fields, and can generally not be used to simulate time-dependent processes, particularly when involving nonlinear effects.
- Plane wave methods  work in the spatial frequency domain. Here, the optical field is decomposed into a limited (but substantial) number of plane waves with different propagation directions, which are coupled with each other by the photonic crystal structure. The complex coefficients of the different plane waves can be computed with a matrix method [7, 12].
- More efficient (but more difficult to implement) variants of such techniques replace the plane waves with a different set of functions which better reflect the properties of the propagation modes. For example, one may use Wannier functions or a multipole expansion. One then needs a smaller number of such functions and correspondingly smaller matrices.
- For 3D photonic crystals, there are transfer matrix methods, where the photonic crystal is considered to be a stack of different diffraction gratings. For each layer of such a stack (of which there are typically only a few types), one calculates a coupling matrix e.g. with a finite-element method. By multiplication of the obtained matrices, one can obtain a matrix representing the whole photonic crystal.
A particularly difficult area is the comprehensive study of the effects of 3D photonic crystal structures with lattice defects and random imperfections, as can result from deficiencies of fabrication methods. Even on modern computers, such computations can be fairly demanding in terms of computation time and required computer memory.
Because of the crucial importance of such computation methods and their difficulty, the progress of the scientific research and practical applications substantially depends on advances in that field.
Applications of Photonic Crystal Structures
In the previous paragraphs, the focus was on the physical foundations, but some possible applications have already been mentioned. Indeed, a wide range of applications is at least conceivable, although only a small part has been demonstrated to be suitable already for widespread practical use. In the following, a brief summary of possible applications is given.
Quantum Optics Research
The substantial change of the electromagnetic mode structure caused by a photonic crystal has profound implications on spontaneous emission. That can be explained in quantum optics, and is the basis for the use of photonic crystals in various quantum optics experiments for fundamental research. However, there are also various practical uses, as explained in the following sections.
A light-emitting diode usually emits into a very large number of radiation modes, i.e., with basically no spatial coherence. That can be changed, however, by strongly modifying the mode structure of the radiation field with a photonic crystal. In the extreme case, one obtains single-mode LEDs, which emit most of the output power into a single mode. That allows efficient coupling into a single-mode waveguide. Such devices also allow very high modulation frequencies , which is important for applications in communications, for example. Such single-mode LEDs may thus compete with VCSELs, for example.
Photonic Integrated Circuits
For application in optical fiber communications, optical metrology and other areas, the prospect of obtaining powerful photonic integrated circuits has been a substantial driver of research for many years, although it has turned out to be quite difficult to fabricate well working devices at reasonable cost.
Typically, photonic band gaps are exploited for confining light to resonators or waveguides, which may be coupled to each other. A substantial amount of work has focused on 2D photonic crystals, mainly because those are substantially easier to fabricate than 3D structures, and sufficient for many purposes. 3D photonic chips, however, would offer a substantially increased potential for implementing complex functionalities in rather small volumes.
A range of physical effects beyond the formation of waveguides in resonators might be exploited. Some examples:
- Precise optical filters can be realized with structures containing high-Q resonators coupled to waveguides, with various kinds of interferometers, and possibly based on superprism effects . A typical application would be add-drop multiplexers for wavelength division multiplexing in optical fiber communications.
- High-Q resonators can exhibit substantial nonlinear effects due to the strong resonant enhancement of the circulating optical power [46, 48]. One might exploit that for signal processing or for generating additional wavelength components, for example.
- Lasers with very low pump threshold (essentially thresholdless lasers) can also be realized and conveniently used on a chip as coherent light sources.
Photonic Crystal Fibers
Photonic crystal fibers  are an example for two-dimensional photonic crystal structures, where light propagation is dominantly in directions which are approximately perpendicular to the photonic crystal plane. Typically, one has some regular pattern of air holes with sub-micrometer diameters running through the whole length of a fiber. A common design carries a missing hole in the center, which forms the fiber waveguide. However, there are also many more sophisticated designs – for example, of hollow-core fibers, which partly rely on photonic bandgaps.
Note that not all microstructure fibers are truly photonic crystal fibers, because some of them lack an actual crystal symmetry.
See the article on photonic crystal fibers for more details.
Photonic Crystal Surface-emitting Lasers
There is a type of semiconductor lasers, called photonic crystal surface-emitting lasers, where the gain region is a two-dimensional photonic crystal structure [22, 28]. This helps to ensure emission with high spatial coherence despite a large emitting area. As a consequence of that, one can achieve very high beam quality in combination with high output power levels.
See the article on photonic crystal surface-emitting lasers for more details.
Control of Thermal Emission
Thermal emission has been applied for a very long time for light generation in incandescent lamps. A fundamental problem has always been that the spectral shape of the emission can be controlled only to a quite limited extent through the operation temperature; the largest part of the emitted power is in unusable infrared spectral region. Although the efficiency of visible light generation can be improved by increasing the operation temperature, one then runs into problems of limited device lifetime. The situation could be much improved if one could suppress long-wavelength emission by using photonic band gap effects. Although only limited technical progress in that direction has been achieved so far, it is at least conceivable that in the future one could have light bulbs with massively improved energy efficiency, which may theoretically even compete with light-emitting diodes (LEDs).
Similar control of thermal emission could also be quite useful even if it worked only in the near infrared. One could then use such emission for generating electricity in photovoltaic cells. Theoretically, efficiencies above 30% would even be possible based on emitting structures which have been demonstrated already . It might thus become possible e.g. to convert waste heat into electricity with a reasonable efficiency.
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