Enhancement of light extraction from light emitting diodes

Physics Reports 498 (2011) 189–241

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Enhancement of light extraction from light emitting diodes A.I. Zhmakin ∗ A.F. Ioffe Physical Technical Institute, Russian Academy of Sciences, Polytechnicheskaya 26, 194021, Saint-Petersburg, Russia Soft-Impact Ltd., Engelsa 27, 194156, Saint-Petersburg, Russia

article

info

Article history: Accepted 18 April 2010 Available online 17 November 2010 editor: G.I. Stegeman Keywords: Light emitting diode Light extraction Modelling

abstract The large amount of light emitted from a light emitting diode (LED) being trapped inside the semiconductor structure is the consequence of the large value of the refractive index. The total internal reflection (light incident on a planar semiconductor/air interface is totally internally reflected if the angle of incidence exceeds the critical value determined by Snell’s law) is the major factor responsible for the small light extraction efficiency (other important contributions to the losses are the internal absorption and blocking of the light by contacts). The typical LED structure comprising a number of layers most of which have high refractive index could be considered as a multilayer waveguide that could support a large number of trapped guided modes. The paper reviews approaches to enhanced light extraction grouped into two sets depending on whether their application results in the change in the spontaneous emission (either the spontaneous emission rate or the angular distribution, or both): (1) molding of the flow of light emitted from the active region by the modification of the chip shape or the surface morphology to increase the light intensity; and (2) modification of spontaneous emission, for example, by placing of the light emitting region inside the optical cavity. Special attention is given to LEDs made from nitrides of elements of group III (InAlGaN) that cover a large part of visible and ultraviolet (UV) spectra and are considered as a major candidate for sources for the solid-state general illumination. An Appendix contains review of numerical models used to study the light extraction. © 2010 Elsevier B.V. All rights reserved.

Contents 1. 2.

3.

4.

∗

Introduction............................................................................................................................................................................................. 190 Molding of the flow of emitted light...................................................................................................................................................... 197 2.1. Chip shaping................................................................................................................................................................................ 197 2.2. Surface modification................................................................................................................................................................... 199 2.3. Reflective submount ................................................................................................................................................................... 201 2.3.1. Distributed Bragg reflectors (DBRs)............................................................................................................................ 202 2.3.2. Omnidirectional reflectors (ODRs) ............................................................................................................................. 203 2.4. Photon recycling ......................................................................................................................................................................... 204 2.5. Photonic crystals ......................................................................................................................................................................... 205 Modification of spontaneous emission .................................................................................................................................................. 209 3.1. Microcavity effect ....................................................................................................................................................................... 209 3.2. Photonic crystals ......................................................................................................................................................................... 211 3.3. Surface plasmons ........................................................................................................................................................................ 212 Conclusions.............................................................................................................................................................................................. 216 Acknowledgements................................................................................................................................................................................. 216

Corresponding address: A.F. Ioffe Physical Technical Institute, Russian Academy of Sciences, Polytechnicheskaya 26, 194021, Saint-Petersburg, Russia. E-mail address: [email protected].

0370-1573/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physrep.2010.11.001

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Appendix A. Numerical simulation of light extraction .................................................................................................................... 216 A.1. High frequency asymptotic methods ........................................................................................................................................ 216 A.1.1. Ray and beam tracing .................................................................................................................................................. 216 A.1.2. Geometric optics and its extensions........................................................................................................................... 217 A.2. Computational electromagnetics............................................................................................................................................... 217 A.3. FDTD method & extensions ........................................................................................................................................................ 218 A.4. Boundary conditions................................................................................................................................................................... 220 A.5. Other CEM methods.................................................................................................................................................................... 220 A.5.1. Transmission line modelling....................................................................................................................................... 220 A.5.2. Plane wave expansion method (PWEM) .................................................................................................................... 220 A.5.3. Plane wave admittance method (PWAM) .................................................................................................................. 221 A.5.4. Time domain integral equation method .................................................................................................................... 221 A.5.5. Transfer matrix method .............................................................................................................................................. 221 A.5.6. The finite difference beam propagation method ....................................................................................................... 222 A.5.7. Effective index method ............................................................................................................................................... 223 Appendix B. Acronyms....................................................................................................................................................................... 223 Appendix C. Nomenclature................................................................................................................................................................ 224 References................................................................................................................................................................................................ 225

1. Introduction

Build a better mousetrap and the world will beat a path to your door Ralph Waldo Emerson The progress of light emitting diodes [1–3] is impressive. A few years ago, similar to well-known Moore’s Law of IC packaging, Haitz’s law was formulated [4] stating that in the past thirty years the lumen flux per package has increased twenty fold while the cost has decreased ten fold each decade; now these trends are even surpassed [5]. Still, especially for LEDs for general lighting [6,7,3,8–12], considerable efforts are being devoted now to further increase of the efficiency of light emitting diodes. Another concern is the reduction of the operating voltage while keeping the same current densities and, hence, the same luminance that is mainly achieved by reducing the charge injection barriers and minimizing the LED thickness. Characterisation of light sources. The intensity of the light sources of any kind is assessed using one of the following techniques [13]:

• radiometry — the measurement of electromagnetic radiation in the very wide wavelength range (usually the range from 10 to 106 nm is mentioned); the radiant flux Φe is defined as the radiant energy per unit time in Watt; radiometric units are also referred to as ‘‘energetic’’ ones

• photometry — the measurement of visible light as perceived by human in the wavelength range of 360–830 nm; the luminous flux Φv is defined as the total light emission per unit time detected by the human eye in lumen units that is equivalent to the radiant flux of 1/683 W at the most sensitive wavelength for human vision of 555 nm (this value corresponds to the photopic (day) vision; the maximum sensitivity for scotopic (dark-adapted) vision occurs at 505 nm). The luminous flux Φv is related to the radiant flux Φe by the spectral luminous efficacy Kλ as Kλ =

Φv (λ) . Φe (λ)

Evidently, Kλ reaches maximum Kmax = 683 lm/W at a wavelength λ of 555 nm. At other wavelength, Kλ = Kmax Vλ is expressed via the relative sensitivity of the human eye called the luminous function or spectral luminous efficiency Vλ [14]. The power efficiency (or radiant efficiency) ηe defined as (P is the total electrical input power)

ηe =

Φe P

is a dimensionless quantity. The luminous efficacy ηv

ηv =

Φv P

is measured in lm/W. It is clear that ηv < Kλ . LED efficiencies. Light emitting diodes (LEDs) considered below are frequently referred to as ‘‘LED devices’’, i.e. the packaged light emitting chips or dice including the mounting substrate, encapsulant, phosphor if applicable, and electrical connections.

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There also other names used to refer to light emitting devices depending on the presence of additional elements such as LED Module, LED Lamp, LED Light Engine and LED Luminaire, defined in Ref. [15]. Packaging main purpose is to protect LED chips from electrostatic discharge, high temperature that is considered, in particularity, as the cause of generation of defects in the active region [16] (especially in the devices subjected to high injection currents [17]), moisture that can induce delamination [18], chemical oxidation [19,9]. In addition, packaging could enhance light extraction [5]: in particularity, encapsulant shape and its material are optimized for this goal. In addition to the intensity, light sources are characterized by the angular distribution of radiation. The radiation pattern of the LED is frequently approximated as the sum of two or three Gaussian or cosine-power functions [20]. The external quantum efficiency ηext of light emission from a light emitting diode is defined as [21] free Nemit

ηext =

Ninj

free Nemit

where is the number of photons emitted into free space per second, Ninj is the number of electrons injected into LED per second. The wall plug efficiency is obtained by multiplying ηext by the electrical efficiency ηel which is the ratio of the energy of the emitted quantum Ephoton to the applied forward voltage V (q is the electron charge)

ηel =

Ephoton qV

.

The external quantum efficiency could be written as a product of the injection efficiency ηinj (the ratio of the number of electron injected into the active region to the number of electrons injected into the device), the internal quantum efficiency (IQE) ηint (that is known to strongly depend on the injected carrier density, especially in InGaN quantum well system [22])

ηint =

Nemit Ninj

where Nemit is the total number of photons emitted from active layer per second, and the light extraction efficiency (LEE)

ηextr

ηextr =

free Nemit

Nemit

.

Since the internal quantum efficiency ηint is the ratio of the radiative recombination rate krad to the total recombination rate krad + knon , the equation for ηext could be re-written as

ηext = ηextr × ηinj × ηint = ηextr × ηinj ×

krad krad + knon

.

The measurements of the total light emission from the LED are relatively easy since they are essentially the same as for other light sources; the recently developed method of confocal microscopy based on the ability to analyse the object properties at a specified depth provide a unique possibility to study light emission from the LED in detail, including different elements of the emitter and the near-field light intensity distribution [23] as well as angular beam divergences [24]. On the problem of the experimental separation of different contribution to the total efficiency see, for example, Ref. [25]. An important issue for white LEDs is the colour control [26] that is characterized by the correlated colour temperature (CCT) and the colour rendering index (CRI) [5]. To assess the efficiency of white LED, one has to consider both the primary light intensity from the LED chip and the intensity of the light converted (emitted) by the phosphor [5]. Some energy is inevitably lost due to the conversion of the UV or blue photons to the photons of longer wavelength — the so-called Stokes shift

ηstokes =

λchip . λphosphor

For example, for typical InGaN emission of 455–465 nm and YAG:Ce phosphor [27,28] with the wavelength of the converted light 550–560 nm, ηstokes is lower than 85%. Note that an alternative to phosphor conversion is the use of polychromatic LEDs [29] designed as stacked QW structures emitting basic colours [30,31] with electrical pumping of active layers with the widest bandgap (usually blue) and exploiting re-emission of light absorbed by other active regions (photon recycling) [32]. LED efficiency enhancement. There are different ways to increase the light output such as to decrease knon , to increase krad , to increase ηextr . Evidently, the main approach aiming at maximizing ηext is the LED design involving the choice of active layer structure (the sequence of layers, their thickness, composition, doping, the number of quantum wells, the nature of the electron blocking layer (EBL) etc.) [33–37], advancements in design of contacts with better optical (higher transparency) and electrical properties (smaller specific contact resistance) [38–45], using nitride layers grown along non-polar wurtzite directions [46–49], exploiting additional elements of the chip structure as, for example, a SiO2 current blocking layer beneath the

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p-pad electrode [50] or an n-doped InGaN electron reservoir layer inserted to influence the electron capture process [51] (an increase of the efficiency of electron capture into QWs could also be achieved via so-called ‘‘phonon engineering’’ that exploits high polar optical energy in AlInGaN materials for confining electrons in the QWs [52]). Decreasing knon is achieved by growing higher quality layers using advanced buffer layers [53] (such as double Mgx Ny /AlN [54] and GaN/SiN [55] buffer layers on the low-defect foreign substrates or using native GaN ones [56–60] (manufactured, for example, by epitaxial lateral overgrowth [61–64], pendeo epitaxy [65] or cantilever epitaxy [66])1 and bulk AlN substrates. There is still no agreement on the role of the In composition fluctuations observed in the active layers made from the ternary solid solution InGaN and in the corresponding quantum wells [69–71] (these fluctuations also affect emission of laser diodes [72]). Sometimes these fluctuations resulting from the poor In incorporation in the epitaxial layer during the growth or from the phase separation due to the large miscibility gap [73–75] are considered to be responsible for the unusual properties of the nitride-based light emitters – their ability to emit light in spite of the extremely high (compared to the devices based on the conventional III–V materials) level of the dislocation density2 – via the localized states related to the composition fluctuations [82,83] or the quantum-dot-like structures [84,85]. This carrier localization suppress the in-plane carrier diffusion reducing the probability of the nonradiative recombination [86]. The presence of such features is registered as spatial inhomogeneities of both photoluminescence intensity and spectra with scale of the order of 200 nm or less by confocal scanning laser microscopy [87], by confocal microphotoluminescence [88] and by near-field optical microscopy [89] as well as by transmission electron microscopy [90]. The presence of In-rich separated phases in cubic InGaN epitaxial layers is evident from resonant Raman scattering and X-ray diffractoscopy experiments [91]. The existence of the distribution of InGaN localized states is also supported by the analysis of the temperature and current dependence of the optical intensity and energy shift in InGaN-based LEDs via comparison between electroluminescence and cathodoluminescence [92]. Quantum dots could be introduced into InGaN active layer intentionally exploiting the strain and the affinity difference between the InGaN and SiCN layer [93]; self-assembled InGaN quantum dots as strongly localized recombination sites are exploited in UV LEDs [94]. Compositional fluctuation of In could also be created by anti-surfactant effect using SiN nanoislands [95]. There is, however, an opinion that there is no need to invoke the concept of the localized states to explain, using the spontaneous and piezo polarization effects [96–99], all the effects observed in INGaN quantum well (QW) such as the emission energy shifts towards the lower energy even below the bulk bandgap and increase of the lifetime of emission lines with increasing the well width [100,101]. This point of view is supported by the experimental evidence of similar behaviour of the emission from the GaN/AlGaN QW, where neither composition fluctuations nor quantum dot structures occur. Still, Hirayama et al. [102] observed carrier confinement in the In segregation region from cathodoluminescence measurements and suggested an indirect evidence supporting the existence of localized states — a drastic increase of room temperature ultraviolet emission by introducing a small concentration of In into AlGaN active region. However, note that the localized states could also be related to the variation of the thickness of the QW. ‘‘Light escape’’ problem. The large amount of emitted light being trapped inside the semiconductor structure is the consequence of the large value of the refractive index. This value for nitrides of elements of the III group (about 2.5 at blue wavelengths [103]) is smaller than for the traditional III–V materials (between 3.0 and 3.5 for AlGaInP system [104]) but it is still large enough for the total internal reflection (TIR) to be the major factor responsible for the small light extraction efficiency (other important contribution to the losses are the internal absorption and blocking of the light by contacts). In organic LEDs (OLEDs) there are two TIRs at the active region/substrate and at substrate/air interfaces [105], thus sometimes one distinguish waveguide and glass (substrate) modes [106]. Light incident on a planar semiconductor/air interface is totally internally reflected if the angle of incidence exceeds the critical value θc = arcsin(n1 /n2 ) determined by Snell’s law (Fig. 1) that describes the relationship between the angles of incidence θi and refraction θt when light is passing the boundary between media with refractive indices n1 and n2 [107] sin θi sin θt

=

n2 n1

.

The fraction of the incident power that is reflected from the boundary is given by the reflectance R and the fraction that is refracted is given by the transmittance T that depend on the light polarization. Evidently, R + T = 1. If the media are

1 While GaN substrates provide significant improvement over sapphire ones [67], they are inferior in comparison to AlN substrates for the UV LEDs due to light absorption into the GaN substrate [68]. 2 Threading dislocations with vacancies at Ga sites [76] acting as nonradiative recombination centers [77] present more serious problem for GaN/AlGaN system than for GaN/InGaN one [78]. Some experiments, however, indicate that threading dislocation density does not directly affect the photoluminescence decay time and the observations could be explained by the reduction of the net volume of the light emitting region [79]. V-defects and the associated dislocations are responsible for the leakage current observed in LEDs [80]. Linear defects are also considered to be the source of the failure of nitride-based LEDs, that was demonstrated for GaN RCLEDs by experiments of Roycroft et al. [81].

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193

Fig. 1. Snell’s law.

Fig. 2. Total internal reflection.

non-magnetic, the reflectance is determined by the Fresnel equations







 n1 cos θi − n2 1 −  

Rs =  

n1 cos θi + n2







 n1 1 −  

Rp =  

n1

1−

n1 n2 n1 n2

1−

n1 n2 n1 n2

2 2 sin θi   2  , sin θi

2

2

− n2 cos θi    2 sin θi + n2 cos θi sin θi

where subscripts ‘‘s’’ and ‘‘p’’ refer to s- and p-polarized light. Strictly speaking, these relations are valid as the short wavelength limit for a plane wave incident on an infinite flat interface; the wavelength-dependent corrections should be introduced for the incident beam having curved wavefronts or/and boundaries with finite curvature or sharp corners [108]). For angles of incidence greater than θc the value of θt becomes complex. The total internal reflection generate an evanescent wave in the low index material. This wave propagates along the interface and decreases exponentially with distance perpendicular to the interface; this wave does not transfer energy along the normal to the interface and thus all the power is reflected (Fig. 2). The critical angle corresponding to TIR defines the so-called escape cone. The solid angle of the escape cone is Ωc = 2π(1 − cos θc ). Photons emitted outside the escape cone get trapped in the structure (Fig. 3). The typical LED structure comprising a number of layers most of which have high refractive index could be considered as a multilayer waveguide [109] that could support a large number (up to over fifty [110]) of the trapped guided modes. The extraction efficiency is determined by the fraction of the photons emitted into the escape cone and for the semiconductor source with the refractive index n emitting into air (nair = 1) is

 arcsin(1/n) ηextr =

0

π 0

p(θ )2π sin(θ )dθ

2π sin(θ )dθ

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Fig. 3. Trapped light in the LED.

where p(θ ) is the light power distribution; for an isotropic emission (which is characteristic for the conventional double heterostructure structures) extraction efficiency is about 1 4n2

.

If losses due to the Fresnel reflections are considerable, this value should be multiplied by the Fresnel transmission factor [111]. If emission dipoles are not isotropic (as, for example, in the case of QWs — the preferable emission is in the plane of the QW [112]), the dependence on the refractive index is conserved while the numerical coefficient is different [113]. For the case of GaN bounded by air θc ≈ 23.5° [114,115] and the fraction of light that is emitted into the escape cone is about ≈0.04. The situation with emission from Al-rich AlGaN alloys (UV emission) differs considerably from the case of blue and green LEDs. The reason is in the unique optical properties of AlN: the recombination between conduction band electrons and the holes in the top valence band produced photons that are polarized along the direction of E⃗ ‖ ⃗ c in contrast to GaN where polarization is along E⃗ ⊥⃗ c ; thus UV photon with polarization parallel to the c axis can no be easily extracted from the escape cone [116]. The trapped optical modes – whispering gallery modes (the authorship of this term usually is attributed to Lord Rayleigh who studied the sound propagation close to the curved walls without being audible in its centre in the circular hall in St. Paul cathedral in London [117]) – that are confined by multiple total internal reflections could be observed in the LED chip layers as well as in the encapsulant. Various means introducing stochasticity into the light propagation (a diffuse reflector cap, a textured encapsulant dome, diffuser added to the encapsulant) could be used for suppression of these modes [118]. Xi et al. [119] estimated the probability of extraction of a guided mode from a waveguide with a flat top surface and a bottom diffuse omnidirectional reflector after a reflection event as p=R

 θc

Pdiff

0

Pdiff + Pspec

Idiff cos(θ ) sin(θ )2π dθ

 π/2 0

Idiff cos(θ ) sin(θ )2π dθ

,

where R is the mirror reflectivity, Pdiff and Pspec are the powers of diffusive and specular reflections, Idiff is the intensity of the diffusive reflection along the normal to the interface. For R = 1 and value of θc given by Snell’s law this expression is simplified to [119] p=

Pdiff

n2e

Pdiff + Pspec n2s

,

where subscripts ‘‘e’’ and ‘‘s’’ refer to the environment and semiconductor, respectively. Each reflection reduces the power of light trapped in the waveguide (1 − p) times, thus the number of reflections needed to reduce it e times is given as

[  N = − ln 1 −

Pdiff

n2e

Pdiff + Pspec n2s

]

.

Evidently, a diffusive component of the reflection should be maximized to increase the extraction. The angle of reflectance is equal to the angle of incidence for a specular reflector surface while for the diffuse reflector the intensity of the reflected light could be written as [119]

[  ] [  ] 1 θ − θi 1 φ − φi I (θ , φ) = Idiff cos(θ ) + Ispec 2 exp − exp − σ 2π 2 σ 2 σ 1

where θ and φ are the polar and azimuthal angles of reflection and θi and φi are the polar and azimuthal angles of incidence, Idiff is the maximum intensity for diffuse reflection, Ispec cos(θi )/(σ 2 2π ) is the maximum intensity for specular reflection. The diffuse reflection intensity follows the Lambertian distribution; the specular reflection intensity is assumed to be broadened as a Gaussian function. The study of the phosphorescence efficiency for different proximate and remote phosphor configurations for the cases of flat, convex or semispherical shapes of the encapsulant top surface showed that the remote phosphor [120–123] and

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Fig. 4. Effect of surface roughening on light extraction.

the semispherical encapsulant dome provide the highest phosphorescence extraction efficiency [118]. The ray tracing simulation had been used neglecting the scattering of light by the phosphor particles that is justified if either the size of the particle is small compared to the light wavelength or the refractive indices of phosphor and the encapsulant are approximately equal. If in the computations of the radiation pattern the computational domain is bounded by the exterior phosphor surface, the latter is usually assumed to be the homogeneous [124] or inhomogeneous [125] Lambertian light source. The blue passing/yellow reflecting filter made from quarter-wave films of alternate high and low refractive index dielectric layers (TiO2 /SiO2 ) placed between the phosphor and a blue LED provides 1. 64 and 1.95 fold increase in efficiency and luminous efficacy of the forward white emission by recycling of yellow light emitted by phosphor downwards toward the InGaN chip [126]. The issues related to the optimization of the encapsulant such as its loading with nanoparticles to increase the refractive index [127], an additional scattering layer around the chip [128] and light-converting phosphor [129–132] or alternative conversion aids such as fluorescent nanospheres [133,134] as well as secondary optics are beyond the present work. Approaches to LEE enhancement. There are a number of approaches aimed to the increasing of light extraction efficiency from the LED chip:

• chip-shaped LED design (chip shaping) [104,135–144]; • placing a back-surface mirror – either a distributed Bragg reflector (DBR) [145–149] or omnidirectional reflector (ODR) [150–152] – between the lower cladding layer and the substrate [153,154]; such LEDs are sometimes are referred to as to Reflective Submount LEDs (RSLEDs) [154]; • surface modification: – one-dimensional [155–159] or two-dimensional [160–163] periodic corrugation acting via Bragg scattering; these surface corrugations could either cover all the chip area or be etched around the periphery of the light emitting structure (for example, circular Bragg gratings [164]), avoiding thus a possible drop of the internal quantum efficiency due to the introduction of new recombination centers during fabrication [165], for example via plasma etching [166,109]; – random surface texturing or roughening [167–183] that reduces internal light reflection and scatters the light outward (Fig. 4); roughening could have a characteristic scale in the nano range [184–188] and be applied to the different surfaces involved in the light propagation, for example: * to one [135,189–197] or both [198] sides of the sapphire substrate; * to the bottom of 6H-SiC substrate [199]; * to the tin oxide electrode [200–202]; * to the undoped-GaN surface [203]; * to the p-type GaN surface [204–209]; * to the n-type GaN surface [210]; * to both the p-type GaN surface and the undoped-GaN surface [211–213]; * to encapsulant [214,215]; * to sidewalls [138,216]; * to both substrate sides and p-GaN layer (‘‘triple light scattering layer’’) [217]. In the OLEDs the indium-doped tin oxide (ITO) [218] region and the metal cathode could be structured [219–221] as well as backside substrate [222–224,106] or self-organized porous film could be used to get the ‘‘meshed’’ surface [225]; – placing arrays of the hexagonally packed silica microspheres [226] (Fig. 5) or other optical diffractive elements (ODE), or simply diffractives such as plastic micropyramids [227], ellipsoidal Ag nanoparticles [228], ZnO nanowire array [229], concave microstructures [115] or microstructured films [230] on the top emitting surface [231]; – integrating LEDs with microlenses [103,232–240,115] made from either sapphire [241], diamond [242], GaN [243], Si2 /polystyrene [244] or glass (plastic) lens-shaped elements [222]; – placing a close-packed monolayer of highly monodispersive CdSe/ZnS core/shell nanocrystals on the top emitting surface [245] (these nanocrystals are pumped by Förster-like nonradiative energy transfer based on Coulomb interactions rather than a wavefunctions overlap and activate an additional relaxation channel for the QW excitations, also acting as the colour-conversion mechanism);

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Fig. 5. Effect of microspheres on the top LED surface on the light extraction. Table 1 Enhancement of light extraction.

• •

•

• • • •

Spontaneous emission is unaltered

Spontaneous emission is modified

Chip shaping Photonic crystals Surface texturing/roughening Reflective submount Buried micro-reflectors Microlenses & diffusive optical elements

Microcavity effect Photonic crystals Surface plasmons

– placing either a patterned high-index layer [109] that provides the deeper penetration of guided mode profile into the upper layer of the structure or a layer with graded refractive index (GRIN layer) that eliminates Fresnel reflection [246,247] onto the top emitting surface; exploiting photonic crystals [248–254,166,255–271] (sometimes photonic light emitting diodes are referred to by an acronym PXLED [272]); placing the light emitting region inside an optical cavity where enhancement of spontaneous emission occurs [273] if mirror separation is of the order of the optical wavelength [274–280,111,281–293] (the microcavity enclosing the light emitting region could be formed ‘‘unintentionally’’ by the multilayer structure designed from other considerations [294]3 ); the so-called ‘thin film’ (TF) LED design, including thin GaN [296–298] LEDs based on the reducing internal absorption and the use of buried micro-reflectors (BMRs); the disadvantage of the TF-LED structure is the small ratio of the current injection area to the chip area that could be less than 10% [153]; moreover, if the micro-reflectors penetrate the active region, the nonradiative recombination could be increased. Nevertheless, this approach proved to be highly efficient for both AlGaInP [299] and InGaN [300,301] LEDs; placing a thin Si-doped AlGaN layer beneath the MQW in UV LED as an electron tunneling barrier allowing low-energy electron injection [302]; exploiting the surface plasmon resonance [303–325]; (in OLEDs) high refractive index substrate [326,327,105] or additional scattering layer comprising high refractive index dielectric spheres embedded into a low refractive index material [328] or nanoporous alumina film [329]; (in GaN LEDs) placing self-assembly of poly(styrene-b methacrylate) block copolymers as nanoporous pattern on the GaN surface [330]; inserting non-periodic dielectric stacks between the substrate and transparent anode in OLED [331].

These approaches could be grouped into two sets (Table 1) depending on whether their application results in the change in the spontaneous emission (either the spontaneous emission rate or the angular distribution, or both):

• modification of the emitted light; • modification of the spontaneous emission itself. This classification is, certainly, not perfect since, for example, photonic crystals could be used to either just mold the light emitted by the unaffected source acting as a diffraction grating to extract the guided modes or to modify both the emission by preventing emission into certain modes and light propagation and significantly modify the far-field pattern [257]; the microcavity effects both increase the spontaneous recombination rate and considerably change the directionality of the emitted light; the regular surface corrugations could act as photonic crystals if their characteristic period length is appropriately related to the light wavelength. 3 For example, Fabry–Perot effects in InGaN/GaN heterostructures on Si substrate results from the high refractive index contrasts at the GaN/Si and air/InGaN interfaces [295], no bottom or top mirrors are included. A simple three-layer Fabry–Perot resonator model consisting of Si/(GaN + InGaN)/air stack satisfactorily describes measurements of the modulation of light intensity.

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2. Molding of the flow of emitted light Molding of the flow of light emitted from the active region of the device by the modification of the chip shape or the surface morphology to increase the light intensity also frequently modifies the nature of the source in such a way that the latter does not work anymore as Lambertian source, but rather as a highly non-Lambertian, directional source [332,333]. 2.1. Chip shaping The ‘‘light escape’’ problem – a problem of an extraction of light through a planar face from semiconductor with high refractive index – is known for over a half of a century [3,334,118]. The optimum LED shape would be spherical with a point-like emitting region in the centre but this design is unacceptable technologically. Usually the LED structure has a shape of a rectangular parallelepiped and thus there are six escape cones, two of them perpendicular to the active layer plane, and four of them parallel to this plane [118,335] that in case of the cylindrical LED transform into the side escape ring [2,143]. The light emitted into the bottom escape cone completely and into the inplane cones partially will be absorbed by the substrate if the bandgap of the latter is lower than that of the active region. Employing a very thick conductive window layer maximize the side extraction efficiency; this approach is frequently used in the design of AlGaInP LEDs [336,104], providing additionally the current spreading beneath the top contact [153]. The amount of photons escaping through the side facets of GaN layer could be very small for large-area LEDs due to the large value of the extinction coefficient [166]. Lee et al. [337] used a Monte Carlo ray tracing simulation [338] to study the dependence of the light extraction efficiency of the GaN-based LED on the position of the light source over the active layer. It was assumed that the current spreading is uniform4 and the photon recycling effect is negligible. The authors have computed the fraction of emitted light through the top, bottom and side facets for the centre and corner emitting point sources as a function of the absorption coefficient of the active layer and the chip size. For the thin GaN LED the authors found that the greater part of the light is emitted through the top surface so the LED looks brighter while the total LEE turned out to be similar to that of the conventional sapphire-based LED, being, however, more sensitive to the absorption coefficient of the active layer. This study also showed the effect of the pyramid pattern introduced on the surface to be greater than that inside the LED in contrast to some other investigations. The authors do not discussed whether the surface texturing in the latter case (inside the LED) could results in appearance of new nonradiative recombination centers in the course of the manufacturing process that could cause the degradation of the IQE [342]. Schad and Neubert [343] have studied absorption in InGaN-on-sapphire LEDs using the photocurrent measurement methods described by the authors earlier [344]. They have found that a smaller bottom part of the GaN layer has the absorption coefficient an order of magnitude larger than the rest of the layer and this sublayer which thickness was estimated to be about 75 nm is mainly responsible for the power absorption in the GaN layer. The authors assume that this layer corresponds to the low quality buffer layer grown on sapphire and thus, probably, the observed behaviour will be absent in the LED structures grown on the native substrates. A very strong nonlinear optical absorption in the green GaInN/GaN multiple quantum well structures has been recently studied by Zhao et al. [345]. Boroditsky [346,347] used two models

• a photon gas model based on the statistical properties of the completely randomized photon in the semiconductor device • a ray tracing computations to study effects of the internal quantum efficiency, the absorption coefficient and the chip thickness on the LEE of the LED. The author found that for the high IQE thin LED design is preferable, while for the low IQE thick layers are better due to the large fraction of light escaping through the side walls. Comparison of these two models showed that the photon gas model overestimates the LEE. A considerable fraction of light in the top emitting LEDs is absorbed by the bondpads, the semitransparent p-contact, and wire bonds. These losses are essentially prevented by the use of a reflective p-type metal contact and the flip-chip bonding scheme (FCLED) [296,348–352] with light being mainly extracted through the transparent substrate (Fig. 6). The advantage of this configuration is the absence of absorption in the metallic current-spreading layer and/or the wirebond pads. FCLED design is particularly important for the UV LEDs since the p-GaN contact layer and p-metal are strongly absorbing in this wavelength range [103]. A problem specific for InGaAlP LEDs on GaAs substrate is the light absorption by the latter (such LEDs are sometimes are referred to as absorbing-substrate LEDs — AS LEDs) is solved by the transfer of the LED layers to the transparent GaP substrate—the so-called transparent-substrate (TS) LEDs. GaN LEDs that are rarely grown on GaAs substrates encounter the same problem. Sun et al. [353] cope with it using wafer-bonding technique and transfer the LED structure to Si substrate through intermediate Ni/Ag/Au layers, getting 2.66 fold enhancement of LEE. 4 This approximation is in most cases a very crude one: numerical simulations show very nonuniform distribution of the current and, hence, light intensity [339,340]; this conclusion is confirmed by the light intensity measurements by the charged camera device (CCD) [341] and by the recent measurements of the emission characteristics of LED using confocal microscopy. The latter showed that light intensity pattern on the external surface of the sapphire substrate for the flip-chip LED (FCLED) reflects the electrode structure of the LED [23]; moreover, this pattern is conserved in the near-field light intensity distribution at distances up to a few hundreds micrometers from the LED surface.

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Fig. 6. Flip-chip LED (FCLED).

Fig. 7. Truncated Inverted Pyramid (TIP) LED.

Among the approaches to increase the light extraction by the die shaping the following ones should be mentioned: the pedestal shaped chips [118], the tapered LEDs (having radial outcoupling tapers that has the shape of a shallow truncated cone with the angle equal or smaller than the critical angle for TIR) and the four-facet Truncated Inverted Pyramid (TIP) LEDs (Fig. 7), the latter being more efficient, with best results found for a taper angle of 22° and a sidewall angle of TIP of 44° [212,354]; LEDs with deep-angled mesa sidewalls [355] (sometimes called the sidewall-deflector-integrated (SDI) LEDs [139, 140]); sometimes acronym GSS-LED (geometric sapphire shaping LED) is used [356]; Al-deposited V-shaped sapphire facet mirror [357]. TIP-LEDs are common for devices based on traditional III-V materials (see, for example, [358–360]) and ZnO [361]; realization of such structures using nitrides is more difficult [354]. The comparative analysis of several chip shapes including a hemispherical chip, a Weierstrass sphere (a truncated sphere having a flat surface), a paraboloid, a truncated ellipsoid was performed by Carr [362] under assumptions that absorption could be neglected and there is a single internal reflection at the semiconductor/air interface. A truncated cone also provides rather high extraction efficiency, with optimal apex half-angle being equal to (π − θc )/2. In SDI-LED photons that are guided laterally are deflected by the angled mesa sidewalls; evidently this will result in re-distribution of the emission from the LED but will not alter the total amount of light emitted (with the accuracy to the possible difference in the internal absorption). For the InGaN-based LEDs the fraction of the photons escaping through four side facets of the GAN layer is small due to the absorption while it could be considerable for the side facets of the transparent sapphire substrate [363,166]. Kim et al. [364] compared the emission of a triangular and quadrangular LEDs and found the increase of the radiant flux from the former by 48% and 24% at input currents of 20 and 100 mA, respectively, and considerably higher extraction in the far-field distribution in the horizontal direction due to the enhancement of light emission from the side walls of the triangular LED. Wang [365] performed computations for the standard square die and the rhombic one and found the latter to be more efficient for the same reasons. Yang et al. [366] fabricated InGaN-based light emitting diodes through a photoelectrochemical process, selective wet ¯ planes and n-type (1 0 1¯ 0) planes oxidation and wet etching, producing cone-shaped-sidewall LED with p-type (1 0 1¯ 1) included at an angle of 27°; the InGaN/GaN multiple quantum well (MQW) active region was located at the cone-shaped tips of this structure. The light output power was increased 2.3 times at a 20 mA operating current. Shaping also could be applied to the sapphire [367] or SiC [368] substrate, leaving the active region of the LED intact. Kim reported manufacturing of a sapphire-etched vertical-electrode nitride semiconductor (SEVENS) LED fabricated by selective wet etching that produces a hole in the sapphire substrate and the buffer layer up to a thick p-contact layer [369]. Micro LEDs. The extreme example of the chip shaping approach is the use of the arrays of micro-light-emitting diodes (microdisks or microrings) [370–377,24,378,379] that provide increase in the extraction efficiency through the enhanced sidewall area and reduced optical absorption — it is well known that LED efficiency decreases with increasing chip size because of internal absorption and increased heating [380,381]. These device structures are also characterized by far more uniform current distribution compared to the conventional broad-area LEDs. The optical properties of such structures are strongly affected by strain of the material: for example, experimental study of the freestanding InGaN nanopillar fabricated by focused ion beam milling showed that spatial variation of strain results in the energy shift and spectrum broadening [382].

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The arrays of micro-light-emitting devices could be monolithically integrated with optical elements — diamond [242] or sapphire [241] microlenses on the surface at locations aligned to emitters. The arrays of microlenses could be either ordered [232,383] or irregular [384]. The microlenses studied by Choi et al. [241] have the diameter and height about 20 and 1.5 µm, respectively, giving a focal length from the apex of the convex surface of the lens array of 43 µm (calculated) or 44 µm (measured from AFM image). Experiments showed that the integration of microlenses into device does not significantly affect the extraction efficiency, but considerably increases the directionality of the light emitted, thus reducing the need for the secondary optics. The authors note that effect of the microlenses could be tailored by the reducing of the substrate thickness or changing of the focal length [241]. Kim et al. [243] reported increase of the measured light emission intensities at 460 nm in the direction normal to the surface of the device by 40% and 100% for the vertical GaN-based LED with microlenses of 10 µm and 5 µm diameter, respectively. The microlenses were formed by plasma etching and have coverage on the surface about 32% (for 5 µm microlenses). Microlenses also could be formed by casting SiO2 nanospheres onto a polymer (benzocyclobutene resin) layer forming a hexagonal pattern on the GaN surface that provide enhancement of the light extraction efficiency of UV LEDs by 23% [385]. This method is relatively simple and in contrast to various etching processes (except, probably, photo-enhanced chemical etching [386]) does not damage the surface. On the other hand, application of microlens arrays to increase brightness of LEDs incorporated into displays is limited due to resulting image blurring [240]. 2.2. Surface modification AR coating. The anti-reflection (AR) coating action is based on the so-called destructive interference (the mutual annihilation of the waves reflected from the external surface of AR and of the underlying semiconductor surface) thus it is efficient if (1) the thickness of AR layer is a multiple of the quarter of the wavelength and (2) its refractive index is the geometrical average of the refractive indices of the semiconductor and external medium (air); evidently, AR coating operates in a narrow range of wavelengths [387]. If AR coating is made as a phase diffraction grating, it in addition to the reducing the reflection of the light from the surface increases the efficiency of the optoelectronic device (both one that emits light and one that absorbs light) due to the conversion of the normally incident light to the one at the angle with the surface enlarging thus the light path’s length [388]. AR is applied to the semiconductor surface provide the broadening of the escape cone, but the second escape cone from the AR material into the free space arises and some of the rays that escape from semiconductor will be trapped in the AR coating. The net effect of these system of two escape cones is the same fraction of the extracted light that is determined by the overall refractive index ratio between medium where the light is generated and the free space [346]. AR could greatly reduce (and in some cases eliminate) losses caused by the Fresnel reflection. The optimal value of the √ refractive index of the one-layer AR coating is nar = ns ne and thickness of λ/4 [275]. Surface texturing. The aim of the surface texturing is to provide numerous scattering events for photons within the structure allowing them finally to escape the chip, coming to the semiconductor/air interface with the incidence angle inside the escape cone. The idea of the surface texturing could be considered as the downscaling of the chip shaping approach [104]. It should be mention that the disadvantage of the surface texturing is the suppression of the current spreading in the layer underneath the etched structures [153]. Rough surface could be considered as ‘‘multicorrugated’’ [389] and represented by a sum of periodic surface corregulations with appropriate weightings. A quantitative description of the surface texture effect could be based on the total integrated scatter (TIS) [365] defined as the total power scattered into a hemisphere divided by the incident power as TIS (θ , φ) =

2π

∫

dφ

0

′

π/2

∫

BSDF (θ , φ, θ ′ , φ ′ )dθ ′ ,

0

where bi-directional scatter distribution function defined for the light power incident on the rough surface with an incident angle of θ as via the irradiance E (θ , φ) =

dPI (θ , φ) dS cos θ

and the radiance B for the scattered power Ps at certain direction θ ′ B(θs , φs ) =

dPs (θs , φs ) dSs cos θs dΩ

as BSDF =

B(θs , φs ) E (θ , φ)

.

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There is a phenomenological relation of TIS to the surface roughness based on the assumption that the scattering mainly occurs at directions close to the specular direction, i.e. the regular reflection direction:

 TIS =

2π δ ∆n

λ

2

,

where δ is the RMS of the surface roughness and ∆n is the refractive index difference between the surface material and the ambient medium [365]. The multirough-interface optical model [390] is based on the assumptions that

• the rough surface reflects and transmit the same amount of light as a flat surface, thus Fresnel amplitude coefficients could be used;

• total reflected (transmitted) light at a rough interface is the sum the specular and diffuse fractions relation between which is determined by the so-called wavelength-dependent haze parameters that should be find by fitting the experimental data to some phenomenological models such as dependence of the specular reflected fraction due to Bennet and Porteus [390]; If the scattering at the individual interfaces is assumed to be incoherent, this approach could be extended to the multilayer structures. More rigorous (however, more cumbersome) approach to the treatment of d-periodic rough surfaces has been developed recently by Kurkcu and Reitch [391] which is based on the use of quasi-periodic Green functions and implemented some exact integral arising from judicious transformations of the integral representations of the Green functions allowing authors to make this otherwise quite prohibitively expensive algorithm useful in practice. Shapes of surface features. Surface roughness could be, for example, cone-shaped [169] or has a form of the truncated tetrahedron [153]; the density of the self-assembled hillocks that could be approximated as a truncated hexagonal pyramid could be rather high — up to 109 cm−2 [392]. The light rays emitted from the active region below the base of the cone will undergo a number of reflections until they eventually have a near-normal incidence at the semiconductor–air interface and thus escape from the chip. For example, Fujii et al. [169] reported a four-fold increase of the extraction efficiency for the InGaN MQW LED with 900 × 900 µm emitting area due to the cone-like surface roughness with feature width and depth about 500–1000 nm; Huang et al. [393] registered increase of the light output power by 160% with self-assembled GaN nano-cone structures with variable density of 1.5 · 107 to 1.4 · 109 cm−2 and depth of 0.56–1.34 µm. A more complex surface processing was used for GaInN LEDs by Xi et al. [394], who combined SiO2 pyramid array and silver layer; the pyramids have a side slope of 45° and base length about 3.5 µm. InGaN LED grown on a sapphire substrate with SiO2 hexagonal patterned mask using MOCVD contains artificial inverted polygonal pyramids (AIPPs) having six R plane and six N plane deflectors with inclined angles 57° and 61°, respectively, that provide multiple chances for photon to escape from the LED [395]; the authors named such LED ‘‘periodic deflector embedded structure LED (PDE-LED)’’. Lin et al. [396] used a micron polystyrene ball array as a template to produce an Al oxide honeycomb structure on the n-GaN surface of a GaN LED that consists of the networking hexagonal Al oxide nanowall. With the Al oxide honeycomb nanostructure, the total light output of thin GaN LED was enhanced by 35%. The authors suppose that the net of the Al oxide nanowalls acts as a waveguide to extract the light emitted to the outer medium effectively. Other pattern such as, for example, donut and waffle, are also used in practice. Kim et al. [162] reported an increase of the output power up to 54% for GaN-based LED by texturing n-type layer (the hexagonally arranged pyramids without a roof with the base length equal to 5 µm and the spacing equal to 8 µm). The authors note that LEE strongly depends on the reflectivity of p electrode. Degradation of the I–V characteristic of the LED was observed when texturing was performed by etching at high temperature. Nano-roughened surface prepared by plasma etching using self-assembled nickel Ni nano-sized islands as masks resulted in 2- to 10-fold enhancement of PL intensity, depending on the etching time [397]. Two kinds of nano-patterns were observed. The average size and density of the larger circular islands were about 250 nm and 5 · 108 cm−2 , respectively. Roughening of GaN surface could also be obtained by the laser lift-off technique [398]. Similar technique (Ni nano-mask and laser etching) [184] and the nanoimprint lithography (etching depth from 130 to 150 nm) [185] was used by Huang et al. to nano-roughen the p-GaN surface of InGaN/GaN LED. The light output was increased 1.55 times and the wall plug efficiency was 68% higher for the LED with roughened surface [185]. Due to the increase of the contact area of the nano-roughened surface, 32% reduction in the serial resistance was measured. Chiu et al. [399] achieved an enhancement of the output power over 50% via nanoscaled epitaxial lateral overgrowth of nitride layers on a SiO2 nanorod-array patterned sapphire substrate. Chen et al. used a hexagonal closely packed monolayer of polystyrene nanospheres and dry etching to produce pattern on the sapphire substrate. Roughening of sapphire surface also serves another purpose – to decrease the density of dislocation in the grown nitride layer [189,400,401] – higher crystalline quality is confirmed by high-resolution X-ray diffraction and grazing incidence Xray diffraction [402]; curved corrugation structure of the sapphire surface turned out to be preferable as providing greater LEE [192–195]. In case of parallel grooves of different orientation it was found that aligning grooves along (1 0 1¯ 0) direction results in slightly better light extraction than aligning along (1 1 2¯ 0) [194]. Growth of GaN film on the patterned sapphire substrate could yield a two-dimensional photonic crystal structure [403].

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Ee et al. [404] used polydimethylsiloxane (PDMS) to fabricate an array of concave microstructures on the surface of IIInitride LEDs to increase LEE by 1.5–2.0 times via imprinting method with SiO2 microsphere array as a template. Simulation of the surface texture effect. Lee [338] used ray tracing to model the effect of textured surface on the light extraction. Assuming that the effect of the texture is to randomize the photon trajectory, the author emulated the presence of textures surface by introduction of the random process to define the orientation of the trajectory of the photon that crosses this surface. Evidently, however, that such approach, allowing the estimate of the gross effect of texturing on light extraction, does not permit to study the effect of the texture parameters such as shape, dimensions, surface density. Lee et al. [212] used Monte Carlo ray tracing to assess the effect of surface texture and/or substrate patterning on the light extraction. This approach was used for similar purposes earlier by Ting and McGill [405,406], by Pufal and Nakwaski [407] and by Badano and Kanicki [408,409], among others. Lee et al. have considered three LED types: the conventional LED referred to as ‘‘the wire bonding LED’’, the flip-chip LED and the thin GaN LED [410,296,297] for the cases of both the bare LED chips and the ones encapsulated into the epoxy lens. It was assumed that the emitted light is monochromatic and unpolarized. The light extraction efficiency of the thin GaN LED was found to be smaller than that of other LED types due to the smaller area of side facets. Studying the 4-facet inverted pyramid of the microstructure introduced in the top surface, the authors found that the slanted angle of 30° provides the greatest enhancement of LEE and note that 6-facet pyramids that are frequent in the practical surface texturing produce the similar effect on the LEE. The effect of the patterned substrate was found to exceed that of the textured surface for all LED types. It should be noted that the classical ray tracing is known to substantially over-predict the light in waveguided modes [223]. Buss et al. [179] used in-house 3D FDTD code to study the effect of surface roughening on the vertical emission of a point dipole embedded into the centre of GaAs block (5 µm × 5 µm × 5 µm) with a mirror behind the light source. The considered roughness pattern was formed by a pseudo-random grid of 100 nm square air holes with depth varied randomly from 0 to 200 nm. The computational grid containing 2 million FDTD cells was nonuniform. All six field components at each time step was saved to used later to produce the emission as a function of the wavelength via discrete Fourier transform. Only very slight improvement over the flat surface was observed in the absence of the mirror; the authors stress an importance of the roughness and mirror recycling working together. A numerical analysis of a similar problem – enhancement of light absorption of photodiode using spherical pits on active surface – exploiting the multiple internal reflections was performed in Ref. [411]. The pits arranged in an equidistant rectangular grid are characterized by the radius of curvature and the depth. The incident radiation was approximated by a number of beams called ‘‘parent’’. After each interaction with the surface, each beam splits into two that are explicitly monitored and so on until the energy in the descendant beam is greater that 0.1% of the initial parent energy. The authors found the very weak effect of the pit depth on the efficiently enhancement if the ratio of the radius of the curvature and the distance between pits is fixed. Ee et al. [404] used Monte Carlo ray tracing to study the optimal parameters of PDMS concave microstructures on the surface of GaN LED that is justifiable since their features are in the micron-size range. The computational domain was 100 µm × 100 µm with periodic boundary conditions, the active region contained 4 quantum wells grown on 2.5 µm thick n-GaN virtual template on the sapphire substrate, total number of rays being about 1.5 millions. Lee et al. [412] also used ray tracing to study effect of the micropyramid array with slanted surfaces on the LEE for the case with and without epoxy lenses. The computation showed that such microstructure could provide up to threefold increase of the LEE. Bienstman et al. [413] used both rigorous coupled wave analysis and FDTD method to analyse in three-dimensional formulation effect of gratings on either side of substrate on light extraction accounting for multiple coherent reflections inside the active region and incoherent reflections inside the optically thick substrate. The authors observed no resonant phenomena indicating that grating functions as a scatterer. Lu and Sturm [414] simulated light extraction from OLEDs on shaped substrates using combined classical and quantum mechanical microcavity model developed by Bulović et al. [415] to estimate the light emitted into different modes (externally emitted, substrate waveguide, ITO/organic waveguide). The author obtained an increase by a factor 2.2 for the average light intensity and a factor 3.2 for the intensity in the normal direction for the substrate with attached lens of the same refractive index as substrate. The authors note that classical ray optics greatly underestimate the fraction of the extracted light. 2.3. Reflective submount Placing an ideally reflective mirror at the back should approximately (neglecting the role of the side facets) double the intensity of the light extracted through the top surface of the LED. In addition to high reflectivity, the mirror should provide low contact resistance and reliable adhesion. Metal reflectors are insensitive to the incident angle and reflect light over wide range of wavelengths, however, their reflectivity is not very high, for example, for Al and Ag about 95% for visible spectral range [416]. Metallic reflectors are absorbing, that presents a problem unless their thickness is extremely small. The phase change upon reflection at metal surface is negative due to the complex refractive index, thus sometimes an additional phase-matching layer is used to compensate this effect. Thus metal alone is not a good mirror [278].

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Besides metal mirrors such as silver [353] or multilayer reflectors comprising several metals, for example Ni/Ag/Au [353], there are two kind of such highly reflecting mirrors known:

• Distributed Bragg reflectors and • Omnidirectional reflectors. Hybrid reflectors consist of a metal mirror and a DBR, combining high reflectivity and small losses of DBR with large angular and spectral reflectivity of metal surface. 2.3.1. Distributed Bragg reflectors (DBRs) A distributed Bragg Reflector is a multilayer structure comprising N pairs of dielectric layers with high (h) and low (l) refractive index (that can be frequently grown monolithically [417,418]) with the layer thickness satisfying ‘‘quarter-wave’’ condition at a certain wavelength λ: 1

nl dl = nh dh =

4

λ.

DBRs exploit successive reflections at dielectric interfaces. Constructive interference of multiple reflective waves increases the reflectivity with the number of low/high refractive index pairs. The key parameters of the DBR are the refractive index difference ∆n/neff (∆n = nh − nl ) and the effective refractive index of the system

 neff = 2

1 nl

+

1

−1

nh

.

The major characteristics of the DBR is the maximum reflection at the design wavelength depending on N which for the case of DBR between the substrate with the refractive index ns and environment with the refractive index n0 is written as

 Rmax

 =

1−

ns n0

1+

ns n0

n2h

 2N 2 nl nh

  2N  nl nh

n2h

and a bandwidth of reflection (stopband width) [419]. The latter does not depend on the number of layers and approximately given as [420]

∆λ = λ0

4

π

 arcsin

∆n nl + nh



.

Stopband edges, however, steepen with increasing N. Another important characteristic of the DBR is a penetration length defined as a distance between the first DBR interface and the location of the imaginary perfect mirror inside the DBR. DBR for GaN-based devices. Ideally, DBR should possess the following properties [420]:

• a large ∆n; • a small lattice and thermal expansion coefficient between the materials used for the quarter-wave layers; • high electric conductivity. For GaN-based applications, the most commonly used DBRs are based on either

• HfO2 /SiO2 or • AlN/GaN. There are two difficulties in fabricating AlN/GaN DBRs: (1) AlN is usually an insulator due to the low formation energy of Al vacancies in n-type AlN (low mobility of electrons in DBRs in AlGaAs system in layers with high Al content was marked also in computations by Winston et al. [421]); (2) there is a large lattice mismatch (2.4%) between GaN and AlN that can cause cracks. The dielectric DBR made from HfO2 and SiO2 provides a large bandwidth of reflection about 80 nm due to the relatively large difference of the refractive indices HfO2 and SiO2 , AlN/GaN DBR – a bandwidth about 45 nm [419] while in situ grown AlGaN/GaN DBR – usually about 20 nm [365]; the reflectance of the MOCVD grown AlN/GaN DBRs so far considerably lower then that of the MBE grown structures [422,419]. AlN/GaN DBRs grown by Ive demonstrate high reflectivity (0.99 for 454 and 457 nm, 0.98 for 561 nm) with stopband about 46 nm while 20 period AlInGa/GaN DBRs show reflectivity far below the expected values (0.47 at 457 nm) that is probably attributed to residual absorption or scattering [420]. DBRs have high reflectivity which, however, strongly depends on both the incidence angle [118] and the polarization of the incident light; thus DBRs become transparent for oblique angles of incidence because semiconductors used for DBR have Brewster angles much less than 90° [154]. Several approaches are known to improve DBR properties [154], including

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203

Fig. 8. Omnidirectional reflector (ODR).

introduction of disorder into the structure [423–425]. In addition, applications of DBR in the LED structures is limited by the electrically resistant and thermally insulating properties of the dielectric stack. Tuning of DBR. DBRs could be tuned to the wavelength lightly longer than the emission peak of the active region in order to maximize the radiated power integrated over the top surface, for example, by adding a half period to the conventional DBR [426]. Compositional grading within the DBR allows one to enjoy additional degrees of freedom in engineering the optical properties of Bragg reflectors [427]. 2.3.2. Omnidirectional reflectors (ODRs) Omnidirectional reflectors have lower losses than metallic reflectors and DBRs and thus provide greater enhancement of the LEE [118,416]. They possess the following properties:

• • • •

high reflectivity; omnidirectionality; broad spectral range of the high reflectivity range; electrical conductivity for the current-injected structures. A typical ODR comprises of three layers:

1. a semiconductor layer with a refractive index ns ; 2. a low refractive index (nl ) layer and 3. a metal layer with a complex refractive index Nm = nm + ikm . The normal-incidence reflectance of the triple-layer ODR is given as [154,153] R=

[(ns − nl )(nl + nm )(ns + nl )km ]2 + [(ns − nl )km + (ns + nl )(nl − nm )]2 . [(ns + nl )(nl + nm )(ns − nl )km ]2 + [(ns + nl )km + (ns − nl )(nl − nm )]2

Since frequently LED active region emits light isotropically, the figure-of-merit of the reflector should be reflectivity averaged over the solid angle [416] R′ (λ) =

1 2π

π/2

∫

R(λ, θ )2π sin θ dθ ,

0

where θ is the angle of incidence in the semiconductor. This expression is obtained for the thickness of the low index layer corresponding to λ/(4nl ) (‘‘quarter-wave layer’’). The triple-layer ODR could be considered as a degenerate hybrid DBR/metal reflector [428]. The low index layer in practice is penetrated by the small-area alloyed ohmic contacts, for example, NiZn/Ag microcontacts in the ODR made from GaN, SiO2 and Al [151] (Fig. 8). A metal layer covered by a semiconductor one could serve as ODR since the Brewster angle (note that the Brewster effect does not occur for transverse electric waves [429]) tan θB =

nm − ikm ns

has an imaginary part which is much larger than the real part [154]. In spite of absorption in the metal layer, high reflectivity is achieved if nm < ns . The effect of ODR for the enhancement of the LEE from the GaN-based LEDs was demonstrated in a number of works (see, for example, Refs. [430–432,428]). Lee et al. [433] fabricated stripe patterned ODR and used ray tracing Monte Carlo simulation to optimize the dimensions of the stripe pattern finding that best configuration for their AlGaInP LED is 3 µm wide, 2 µm deep and 3 µm space in an agreement with experimental results. The higher the refractive index contrast of ODR, the higher it reflectivity and the wider the spectral width of the stopband [416]. The progress in this direction depends on the ability to produce materials with low value of the refractive index. One is approaches is to produce nanoporous SiO2 by sol–gel process with the average pore size about 2 nm [416,434]. A low index layer for the ODR for GaN-based LEDs could be made from indium–tin oxide (ITO) via the so-called obliqueangle deposition [435] that results in the nanorod layer (the column angle is less than the flux incident flux; empirical

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relations between these two angle for small and large incident angles are well established [435]) with the refractive index as low as 1.34 at 461 nm which is significantly lower that of the dense ITO layer (n = 2.06) [428] and of SiO2 . This method also called glancing-angle deposition used to fabricate ODR from other materials, for example, silicon [436] and nanorod SiO2 [416]. The refractive index of porous ITO is well described by the Bruggemann effective medium approximation. ODR for the UV GaInN LEDs have been considered in Ref. [118]. An ingenious application of ODR – as a ‘‘superstrate’’ in a white LED that increases the probability of interaction of UV photons with phosphor and prevent them from leaving the device – was described by Chang et al. [437] who also used ray tracing to simulate all the processes of light interaction with randomly distributed phosphor particles. 2.4. Photon recycling Two similar terms – light recycling and photon recycling – are used in the literature, the first one being usually referred to the superposition of some part of the light returned via the reflection to the active region and the original direct radiation [438] while the latter is related to the photon re-emission after absorption that results in a contribution to the extraction efficiency (an absorption of the photons propagating outside an escape cone and re-emission into photons directed into the escape cone). Photons emitted into guided modes could be reabsorbed in the active region and either cause an instant re-emission or generate electron–hole pair; in the latter case carriers could drift/diffuse elsewhere. This effect is usually does not observed in devices with QW active region due to the extremely small thickness and low probability of the photon absorption unless these modes have large overlap with the QWs [439], as, for example, occur in the microcavity. Photon recycling can be accounted for as an increase of the external quantum efficiency by a factor Fpr [281] Fpr =

1 1 − g ηint

,

where g is the reabsorption factor. The efficiency of the photon recycling effect depends on the internal quantum efficiency ηint as well as on the thickness of the active layer l and its absorption coefficient α . These parameters enter into the expression for the efficiency as the optical thickness of the active layer α × l. A thin active layer is favorable in materials with low internal efficiency [104]; however, the poor carrier confinement in thin active layer forces one to search a compromise. The term ‘‘photon recycling’’ is sometimes used to describe the processes in thin film LEDs where photons that does not belong to the escape cone are reflected, reabsorbed and partially re-emitted in the active region and to describe ‘‘recycling of radiative modes’’ in case of coupling of the fundamental waveguide mode with radiative modes in the Fabry–Perot microcavities [440]. Photon recycling is significant in solar cells based on wide-gap semiconductors [441] and integrated quantum well infrared photodetector-LED devices since at the operating temperature (70–80 K) the internal quantum efficiency of LED is close to unity [442]. Photon recycling white LEDs. Photon recycling with lowering the energy of the re-emitted photon is the basis of the photon recycling semiconductor light emitting diode (PRS-LED) [443] (dual wavelength LED [365]). Light emitted by the blue GaInN/GaN LED is absorbed by the AlGaInP active region and re-emitted at the so-called complimentary colour [444]. The latter notion is based on the possibility to generate white light by the generation of the light with two or three distinct colours, accounting for the existence of only three types of colour-sensitive receptors (cones) in the human eye. This approach for the white light generation could be more efficient [443] than the tradition light conversion by conventional [129], nanocrystal-based phosphor [130] or CdSe nanocrystals [445] and allows extension in two directions [443]:

• fabrication of the secondary light source from the same InGaN solid solution, but with different indium molar fraction – an approach used successfully in the multijunction solar cells – allowing to produce white LED as the monolytic chip obtained in a single growth process; • introduction of an additional – ternary light source – emitting in the third spectral band. The advantage of such ‘‘all-semiconductor’’ white LED is also an easy adjustment of the required ratio between the different colour intensities by the tuning of the thickness of the secondary active layer thickness [443]. Note, however, that LEE enhancement methods depending on the wavelength will be useless for the dual LEDs. Photon recycling in the white emitting diode based on InGaN multiple quantum well heterostructure was studied by Nikolaev et al. [32] with the use of the transfer matrix method in the plane waves basis. The authors considered the case of several active layers within the single structure and calculated the rate of absorption by the quantum well of the spontaneous emission from the other QW and took into account that one of the channels for the photon to escape from the recycling process is to be absorbed in a metallic mirror. The authors considered GaN microcavity that contained three QWs covered by a metallic mirror. The structure was designed in such a way that it could serve as 5λ/2, 6λ/2, 7λ/2 resonators for the normally propagating red-light waves, green-light waves and blue-light waves, respectively. The QWs were placed at the antinode of the corresponding resonant mode. The computations showed that both the emission spectra and the external efficiency strongly depend on the active layer position.

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Fig. 9. Photonic crystal formed by an array of air holes in semiconductor.

2.5. Photonic crystals Wave behaviour in photonic crystals (PCs) – structures with a periodic modulation of the refractive index in one, two or three dimensions [446] – is similar in many respects to the electron wave behaviour in semiconductor materials.5 Photonic crystal is a low-loss periodic structure that has a photonic bandgap preventing light in a certain wave range from propagation in certain directions [446]; if this gap covers all possible directions which is possible if the refractive index contrast exceeds some threshold value, photonic crystal is said to have a complete photonic bandgap. This case is also referred to as to photonic bandgap (PBG) materials (the term electromagnetic crystal is also used, however rarely [450,451]). Photonic crystals could be considered as an extension to other frequency range, including visible, of devices well-known in microwave technique such as metallic waveguides, cavities, interference filters [452]; multilayer dielectric mirrors such as a quarter-wave stack, dielectric Fabry–Perot filters and other similar structures common in optoelectronics are onedimensional photonic crystals [446]. Photonic quasicrystals. Along with regular photonic crystals the so-called photonic quasicrystals (PQC) are considered [110] that differ from the photonic crystals by the absence of the short range order; PQC, however, being precisely define by the so-called tiling rules possess long range order; PQC usually have a more isotropic far-field pattern and are free from artifacts that are frequently related to the regular PC [453]. An example of PQC is the so-called sunflower lattice constructed on the basis of Fibonacci series [110]. PCs have different degree of symmetry, with diminishing the directional properties with increasing the symmetry level. The highest degree of symmetry found in the natural PCs is given by the triangular lattice (six-fold symmetry). Evidently, a crystal with a complete photonic bandgap could serve as an omnidirectional reflector — Young stated as early as in 1801 that a multilayer stack of two alternating dielectric plates exhibits a near total reflection in a range of wavelengths (see Ref. [454]). The length scale in which the dielectric function variation takes place (lattice parameter) determines the spectral range of functioning of the PC. PC structure. The classical photonic crystal comprises the periodic structure made from two dielectrics (one of them could be air) with different refractive indices (Fig. 9). If one of the dielectric constituents is replaced by semiconductor or metal, new phenomena are observed. For example, in the metals the frequency dependence of the dielectric constant should be accounted for [455]; in the semiconductor/dielectric photonic crystal an exciton and a photon couple strongly to form a polariton (such crystals are sometimes called polaritonic crystals [456]); these crystals are shown to provide ultrafast optical switching [457]. Semiconductor photonic crystals could be used for compact optical amplifiers [458]; metallic photonic crystals (MPC) could provide simultaneously photonic and electronic resonances in the same spectral range and strong coupling between resonances via formation of a polaritonic-type quasiparticle [459]. Strong resonances in the optical transmission spectra at normal incidence are associated with so-called Fano coupling between free space and guided modes [460]. Photonic crystals occur in nature, for example in the peacock feathers [461,462], wings of some insects (for example, jewel beetles such as Chrysochroa Fulgidissima and butterflies such as Morpho Rhetenor) and natural opals [463–465]. Vukusic and Sambles note, however, that often ‘natural systems offer technologically unrealized photonic structures’ [464]. Photonic crystals with bandgaps in near infrared, optical and ultraviolet regions could be manufactured from nitrides of the III group [466].

5 It is worth to mention that another example of analogous wave behaviour could be found for elastic waves in a high-contrast, two-component elastic media [447,448]; recently, analysis of phononic band structures in such ‘‘phononic’’ crystals for the case of large contrast of the shear modulus and that of the density was performed by Ammari et al. [449].

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Experimentally the usefulness of PCs for the light manipulation was first demonstrated in the late eighties [467,468]. The simplest example of two-dimensional photonic crystal is made by the lithographically etching an array of holes in the semiconductor or dielectric layer [469]. Strictly speaking, not photonic crystals but the so-called photonic crystal slabs (PCS) [469,254,470] are used in practice truncating the infinite thickness of the true two-dimensional PC down to a fraction of the lattice constant. As a rule, the modes of PCS-structure shifted towards higher frequencies due to its lower effective index [471]. The major difference between PC and PCS is an existence of a light cone in the latter: all modes that have frequencies lying below the light cone are confined to the slab by TIR [465]. Gao and Zhou [472] used the effective medium approximation to define via the volume filling fraction f the effective index of the planar two-dimensional PC as n22D n2h

=

n2i (1 + f ) + n2h (1 − f ) n2i (1 − f ) + n2h (1 + f )

,

where nh and ni are the refractive indices of the host medium and of inclusions, respectively. Assuming that the dispersion relation in both PC and PCS could be expressed via the effective index as ω/k = c /n, the authors derived the nonlinear equation that relates the eigenfrequencies of these photonic structures. Fabrication of PCs. A number of different approaches — both top-down fabrication methods and up–down approaches [473] such as self-assembling [463,474] are used to fabricate three-dimensional photonic structures: advanced semiconductor processing [475], for example a combination of photoelectrochemical etching (such as, for example,, reactive ion etching [476]) and subsequent focused ion beam drilling [477], self-assembly of colloidal spheres from suspension [478] or via ink-jet printing [479] (also for two-dimensional photonic crystals [480]), holography in a polymer photoresist [481]. Fabrication methods of such structures often referred to as metamaterials (and as planar metamaterials if the propagation direction is normal to the plane of the metamaterial layers [482]) including standard electron-beam lithography, focused ion beam milling, interference lithography, nanoimprint lithography have been recently reviewed by Boltasseva and Shalaev [483]. Analysis of PCs. The analogy between electronic and photonic crystals is exploited in creation of the photonic heterostructures – concatenations of photonic crystals differing in the refractive index or the lattice geometry (the lattice period or even the lattice type) – that could be used in devices based on either perpendicular (across the interface) or parallel light transport [484,485]. Graded (or chirped) photonic heterostructures are also considered [486]. Photonic crystal heterostructures could be considered as an alternative to the common approach of introduction of defects breaking the translational symmetry in the perfectly periodic crystals to create various devices [487,488], for example, photonic crystal point defect nanolasers [489] – the defect in the periodic array behaves as a microcavity resonator [490]. The introduction of defects (for example, microcavities [491]) – which is in some respects is similar to the doping of the semiconductor crystals [492] (for example, adding some extra material to the photonic crystal corresponds to donor doping [493]) — into two-dimensional crystals [494] is relatively easy, while inserting defects into three-dimensional photonic crystals is more involved [495,496]. The analogy between the wave behaviour in electronic and photonic crystals extends into the analysis approaches such as the envelope function description [486,487,497]. In photonic crystals, light travels as Bloch waves; the strongly modulated photonic crystals demonstrate such peculiar properties as the negative refractive index near the bandgap edges [498–503] that could be mimicked by a homogeneous material with the negative permittivity and permeability [504] or ultraslow propagation and even stopping of light [505,506] as well as superluminal group velocities and pulse separation into distinct parts [507]. Sometimes, the bandgaps of photonic crystals are referred to as ‘‘dielectric band’’ and ‘‘air band’’ [508], depending on the location of high power in the heterogeneous media whether it is in the high-ε region (dielectric) or the low-ε region (frequently air) [509]. Some well-known models of electronic crystals, for example the Kronig–Penney model, are adapted to description of the corresponding photonic crystal structures [510]. Analysis of the optical properties of photonic crystals is frequently based on the Floquet–Bloch theorem that forms the basis for the representation of eigenvectors as a product of a plane wave exp(i⃗ k·⃗ r ) and a Fourier series. This theorem also exploited in numerical studies justifying use of the unit cell of the material and imposing periodic boundary conditions [511]. Photonic crystals are unique in their ability to both increase the light extraction using different means of the light manipulation they provide and, if they are placed near enough to the active layer, to increase the rate of spontaneous light emission [500]. Effective medium approximation. Whenever the photonic crystal made from two dielectrics with the dielectric constants εd1 √ and εd2 could be considered as a continuum effective medium, its refractive index neff ≡ εeff could be determined via the approximation due to either Bruggemann or Maxwell–Garnett from one of the following relations [479]: f or

εd1 − εeff εd − εeff + (1 − f ) 2 =0 εd1 + 2εeff εd2 + 2εeff

εeff − εd1 εd − εd 1 =f 2 , εeff + 2εd1 εd2 + 2εd1

where f is the filling factor.

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The first of these models is thought to better describe media in which both constituents are equal while the latter is better suits for the description of medium that could be represented as a connected matrix with alien inclusions [512]. Effect of 2D PCs on the LEE. There are a number of ways to exploit PC called ‘‘light extractors for LEDs’’ [513] for the LEE enhancement such as

• inhibition of guided modes emission by PBG; • enhancement of spontaneous emission (Purcell effect); • light extraction on the whole surface by leaky mode coupling. The effect of two-dimensional PCs on the LEE is determined, in addition to the feature shape, by four parameters: 1. 2. 3. 4.

the type of the PC lattice (triangular, square, honeycomb, Archimedean [514]); the lattice constant; the filling factor; and the etch depth.

Some types of PC lattices lack the omnidirectional extraction properties except the Archimedean tilings that form rotation-invariant PCs [513]. Ichikawa and Baba [515] studied numerically using two-dimensional FDTD method and experimentally the light extraction efficiency enhancement in LED with one-dimensional rectangular surface grating photonic crystal and found for the constant grating depth equal to 0.6 µm the enhancement factor increases with the lattice size a of the photonic crystal and saturates at approximately a = 0.8 µm that correspond to a/λ = 0.516. The authors note that such 2D photonic crystals could be manufactured relatively easily without process-induced nonradiative recombination centers. The enhancement factor could be further increased by the combination of PC and a round shape plastic mold with a refractive index about 1.5 and by using a high reflectivity backside mirror. Effect of the photonic structures fabricated on the p-type GaN with the different arrangement of the circular holes (the diameter was around 6 µm and the depth 1.1 µm or 100 nm) was studied experimentally by Wu et al. [516]. The following cases were considered (the period of the lattice array was about 15 µm for all arrangements of the holes):

• square lattice array; • triangular lattice array; • honeycomb lattice array. The etching depth was sufficient for the penetration of MQW layers. The measured enhancement of the photoluminescent intensity using optical pumping was 208%, 188% and 90% for the square, triangular and honeycomb lattices, respectively. Ryu et al. [254] have studied the light extraction enhancement from the two-dimensional photonic crystal slab structures using six pairs of the strain-compensated InGaAsP QWs experimentally and using FDTD computations for the different photonic crystal lattice constant and different arrangement and found PL enhancement in the square lattice to be much larger than in the triangular one at a = 800 nm. Hexagonal array of circular holes was used by Tian et al. to increase the light extraction efficiency of GaN [114]. The hole diameters range from 120 to 700 nm with the depths up to 1 µm and the spacing of 1 µm were considered. The measurements of the cathodoluminescence showed increase of the light intensity with the pore diameter that was attributed to the change of the extraction efficiency. The authors state that the light extraction from the unetched part of the GaN surface depends on the area fraction of pores not directly only due to the emitting surface reduction, but also through the change of the effective refractive index of the porous layer that could be accurately described by the Maxwell–Garnett effective medium model. There are three sources of the specific optical properties of the nanocomposites made from dielectrics or semiconductors [512]:

• quantum effects; • surface effects since the surface area could increased by orders of magnitude; • local fields in the nanocomposite medium that are determined by the size, the shape and arrangement of nanoobjects. An effect of the square lattice two-dimensional photonic crystals on the surface of the GaN LED was studied using FDTD simulations by Choi et al. [517]. Photonic crystals with the lattice constant Λ 300, 500 and 700 nm were considered. To reduce the computation costs, the authors considered the 5 × 5 air-hole array and reduced the actual GaN thickness of 5 µm to 1 µm; to account for the thick sapphire substrate, the perfectly matched layer (PML) was used as an absorbing boundary condition. The active layer emission was simulated by the 250 × 250 array of the TE-polarized point dipoles (the light emitted from the MQW is mainly TE-polarized [112]) uniformly distributed in the active layer plane. Each dipole was driven by a sinusoidal function with a fixed frequency. Reabsorption of light in the active layer was computed using an effective absorption coefficient for the GaN MQW layer. The computed enhancement of the light extraction was 1.8 and 2.3 for the photonic crystal lattice Λ = 700 nm and Λ = 500 nm, respectively. The Si-gel-encapsulated GaN-based blue LEDs with the square lattice photonic crystal pattern introduced into the top layer of the LED were studied numerically (FDTD method and the Monte Carlo ray tracing procedure were used) and experimentally by Kim et al. [518]. The depth of the hole in the p-GaN was 140 nm, the lattice constant Λ of the photonic crystal was varied from 200 to 3000 nm. The enhancement of light extraction was found to depend weakly on the lattice

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constant, achieving the maximum of 1.8 at Λ = 700 nm. The authors note that the holes in GaN may result in increasing the serial resistance. Ray tracing computation results can cause doubts whether the resolution was sufficient since the authors state: ‘‘a large number of random directional rays (104 ) were generated, occupying the entire volume of the MQWs’’. The effect of the etch depth in fabricating PC was studied by Charlton et al. [110]. Shallow etched PC structures will provide coupling with only a small number of high order modes trapped inside the waveguide, thus increasing of the hole depth increase the LEE. Choi et al. [270] studied green Inx Ga1−x N/Iny Ga1−y N light emitting diodes with the square lattice arrays of photonic crystals with diameter/periodicity of 200/500 nm and etch depths 65, 95 and 175 nm and reported up to about 28-fold enhancement of the luminescence intensity. Cho et al. [166] investigated the effect of the wide-area (375 × 330 µm) PC pattern incorporated onto the top surface of the GaN-based LED using nanoimprint lithography and FDTD computations. The authors found, as was expected, an increase of the LEE with the etch depth that saturates when the depth of the holes becomes comparable to the wavelength of the guides modes (≈ λ/n). The authors also noted that the relative enhancement effect of PC in comparison to the LED with the unpatterned surface decreases after encapsulation due to the reduction of the effective index contrast of the PC. The dependence of the position of the guided resonances – leaky in-plane guided modes, coupled to the free space modes – of photonic crystals were studied by Prasad et al. [519] who found the strong dependence on the ratio of the hole radius to the slab thickness (which determines the effective refractive index of the slab) and relatively weak dependence on the slab thickness; an increase of the latter shifts resonances to lower frequencies. Kim et al. [520] note that plasma etching process of GaN layer to produce holes could degrade the electrical characteristics and proposed to deposit a high refractive index TiO2 -patterned layer onto the top emitting surface of GaN slab. The guided mode profile penetrates deeper into the higher-index layer increasing light extraction. The authors performed 3D FDTD computations using randomly polarized dipole sources as earlier was proposed by So et al. [521] and Riel et al. [522]. Peripheral PCs. Usually 2D photonic crystals are formed over the whole dye area including the metal transparent layer, except the contact pads. In such a way Oder et al. achieved 20-fold enhancement of the light extraction efficiency using optical pumping at 475 nm [249] and 63% and 95%, respectively, enhancement of optical power under current injection in the blue (460 nm) and UV (340 nm) nitride-based LEDs [250]. Shakya et al. [252] reported increase of the optical power by a factor of 2.5 for the UV (333 nm) by the use of photonic crystals configuration that separates the light generation region and the extraction region, photonic crystals being present in the latter only in order to scatter the guided light into space [346]. The triangular arrays of holes with the diameter from 100 to 200 nm and the lattice constant from 300 to 600 nm were considered. The authors note that the extraction enhancement is accompanied by the forward current increase that is attributed to the increase of the surface recombination due to the increased surface around the etched holes. In this case the photonic crystal do not influence the spontaneous recombination, but just improves the extraction of the emitted light. There are a few requirements for the optimal design of such LED [523]:

• the size of the unpatterned region should not exceed the reabsorption length of the guided light; • the spontaneous emission band of the active region should overlap with the frequency of the leaky modes of photonic band;

• the leakage length of the guided modes inside photonic crystal should be smaller than the reabsorption length. PCs cannot out-perform the best LEDs based on surface roughening in term of the total LEE, but they can provide significant benefits in term of the light directionality (‘‘beam shaping’’) that could be crucial in many applications [110]. Kaneko et al. [524] increased LEE of InGaN/GaN LED by 85% using the reverse-tapered microholes of diameter 4 µm and the exterior angle of the sidewalls 41° fabricated within the GAN-based layer. Favorable effect of two-dimensional array of microcolumns on the LEE was first demonstrated for GAInAsP LEDs [525]. The array was designed as a two-dimensional photonic crystal with a honeycomb arrangement of columns. The authors state that the favorable effect of such array is threefold. First, the escape cone of the internal light is expanded since array of microcolumns acts as an effective medium (‘‘diluted’’ semiconductor—a media having a small filling factor of semiconductor material) with a low refractive index. The second effect is the conversion of the lateral light to the vertical direction due to the diffraction in the air space between microcolumns and scattering at air/semiconductor interfaces. Finally, the lateral emission is essentially inhibited by the photonic bandgaps of microcolumns. The authors stress that the array of microcolumns has advantage over the semiconductor slab with holes since, for the same filling factor, the sidewall area exposed to air, and, hence, surface recombination is smaller. Favorable effect on LEE was reported for indium–tin oxide [428] and zinc oxide [526] nanorod arrays deposited on top of GaN-based LEDs. Recently this approach was successfully applied to the GaN–InGaN vertical-injection LEDS by fabricating high-aspect-ratio nanorod arrays with heights over 1 µm [527]. These arrays, in addition to providing an omnidirectional escaping zone for the emitted light, serve also as waveguiding channels improving collimated beam profile. GaN-nanorod array behaves as low-effective-refractive-index (low-n) media in the entire visible spectrum [528].

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3. Modification of spontaneous emission

3.1. Microcavity effect Placing of the light emitting region inside the optical cavity having the thickness of one-half or one times the wavelength of light results in the increase of spontaneous emission due to the change of the optical mode density at or near the emission wavelength [529–531]. Spontaneous emission is described as a transfer of a quantum of energy from an oscillating dipole to modes of the electromagnetic field. A spatial distribution of the optical field in the cavity arises from the standing wave formed by the two counter propagating waves. The presence of the cavity changes the boundary conditions for the electromagnetic field. Planar microcavity favors emission into some privileged modes. If the emitter is located at a node of mode, there is no coupling, if at antinode of mode, the coupling is optimal. Microcavity can be formed by the mirrors of different nature, including metal mirrors [532], for example, H. Peng et al. [533] fabricated microcavity top emitting OLED (TOLED), using semitransparent LiF/Al/Ag as cathode and highly reflective Ag as anode. An ideal cavity with perfectly reflecting mirrors would be useless since no light could be extracted. As a rule, a highly reflecting back mirror is used while an outcoupling mirror should be moderately reflecting. When DBR is used as one or both mirrors, the effective cavity length should be corrected by the penetration depth of the mirror(s). The spontaneous emission decay constant τ for a radiating dipole in the Markovian approximation [534] is given by Fermi’s golden rule as [304,150] 1

τ

=

2π h¯

|⟨f |d⃗ · E⃗ |i⟩|2 ϱ(h¯ ω),

⃗ · E⃗ |i⟩ is the dipole emission matrix element and ϱ(h¯ ω) is the photon density of states (DOS). where ⟨f |d Placing QW at the antinodes in semiconductor microcavity results in appearance of polaritons due to the interaction of excitons with the optical field; polariton could be considered as a linear superposition state of a cavity photon and a QW exciton [535]. Purcell effect. Increasing the spontaneous emission rate via the photon DOS is called the Purcell effect (and the measure of this enhancement is called the Purcell factor) after Purcell who was the first to report this phenomenon in 1946 for the case of nuclear magnetic moment transitions at radio frequencies [536,335]. This effect is observed also in the case of a single surface close to the emitter [389]. The reviews of the mechanisms involved as well as of description approaches – either classical or quantum mechanical which are equivalent if only linear interactions are considered [537,530] – for this so-called ‘‘fluorescence near interfaces’’ or ‘‘surface enhanced fluorescence’’ with a special attention to the photonic density of states and its modification were given by Barnes in the cited paper and by Fort and Grésillion [538]. Placing emitter inside a planar cavity concentrate its emission into circular fringes (lobes); achieving of the maximum light extraction is formulated as a minimum spacing between the peaks of the Airy function, i.e. a sufficiently small cavity order mc = L/(λ/2n) [335]. However, it should be noted that planar microcavities could not remove the photon leakage through the guided modes propagating in the active layer plane. A possible solution to this problem is to place the twodimensional photonic crystals at the periphery of the active region [335]. The optical field penetration into a DBR [539] Lp =

λ

nh nl

2n 2nav e ∆n

≈

λ 4∆n

results in an increase of the effective cavity length Leff = L0 + 2Lp for the symmetric cavity restricted by two DBRs. Resonant cavity light emitting diode (RCLED) or resonant cavity enhanced LED (RCE LED), also called microcavity LED (MCLED), was the first device using the spontaneous emission enhancement occurring in the resonant cavity [540,541,118]. In some cases RCLEDs are considered as an alternative for laser diodes for plastic optical fiber applications [542]. RCLEDs, in contrast to conventional LEDs [543], possess a unique property—a variable coherence [544] that is indispensable for many applications such as optical coherence tomography, optical time domain reflectometry, surface mapping in integrated circuits where the coherence of a light source should be simultaneously large enough to resolve details on the relevant length scale and small enough to exclude coherent reflections from distant objects [291]. While in conventional LEDs the spectral linewidth is determined by thermal effects and usually is limited to ≈ 1.8 kT [278], linewidth of RCLED is depend on the cavity parameters and could be sufficiently small.

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Three types of mirrors could be used to form the cavity 1. metallic, including semitransparent TIO [218], NiO [545], Ag [546] 2. oxide-based (e.g., SiO2 –TiO2 ) 3. distributed Bragg reflector. The first two options require difficult lift-off process, thus epitaxial growth of DBR along with other layers of the LED structure is preferable [542]. In case of GaN-based LEDs, however, growth of DBR layers with high content of Al is difficult [145,547]. In practice, Al content is less than 0.4 resulting in a relatively small refractive index variation [542]. The QW should be located near the antinode of a Fabry–Perot mode for good coupling with this node, thus only a few QW could be sufficiently strongly coupled [539]. Optical cavity effects are observed in InGaN quantum well heterostructure flip-chip light emitting diodes (FCLEDs) [548,110]. In a FCLED a thick, highly reflective metallic contact is deposited on the p-type GaN surface acting as a mirror. If the quantum well is close to the mirror (within a few wavelengths), light emitted directly from QW interferes with light reflected from the mirror, forming regions of low and high intensity. The interference pattern is described by the expression [548]

 2 ⃗ E  = w02 + wr2 + 2w0 wr cos(π + Φ + Φ ′ ), where w0 and wr are the amplitudes of the emitted and reflected light, Φ is the phase shift upon reflection off the mirror, Φ ′ is the phase shift due the difference in the optical path length, the latter being a function of the distance d from QW to the mirror, of the wavelength in GaN λn = λ/n (n is the refractive index of GaN) and of the angle θ from the normal which is given (under assumption that neither wr nor Φ depends on the angle) as

Φ ′ = 2π

2d cos θ

λn

.

In organic light emitting diodes (OLEDs) the microcavity is also used to deal with the impurity of the primary colour—to reduce the full-width at half maximum (FWHM) [549]; excessive microcavity effects result in undesirable angular-emission characteristics, thus the so-called weak microcavity OLED (WMOLED) has been recently proposed by Cho et al. [292]. An organic RCLED was reported that emit simultaneous at three wavelengths using a multi-mode Fabry–Perot cavity [278]. Analysis of multilayer structures. There are two major approaches to analysis of spontaneous emission [281]:

• a quantum mechanical description accounting for quantization of the electromagnetic field and mode density, being rather complex, is applicable to idealized problems;

• Classical approach of modelling of spontaneous emission via an electric dipole that is valid in the so-called weak coupling regime which is the operating regime of practical RCLEDs. The traditional approach to analysis of the multilayer structure is based on the representation of the electromagnetic field of the radiation mode as a superposition of two plane waves incident on the structure from opposite directions. The relation between the amplitudes of these waves is found from the condition that standing waves are formed outside the structure that no power flow occurs. This approach is found to fail in the limiting case when the refractive indices of all the layers are equal to 1; an approach free from this problem called ‘‘the effective resonator model’’ was suggested recently by Rudziński [550]. Processes in microcavity could be simulated using the transmission line model with every semiconductor layer considered as a transmission line segment, the transfer matrix method or the recursive Green function technique, especially for the linear regime of the normal mode coupling [531]. Results of analysis, in particularity, explain a three- or morepeaked normal mode coupling spectra by coupling of the fundamental cavity mode to the guided or leaky modes of the microcavity [531]. FDTD computations for the semiconductor cavity using full-wave vectorial Maxwell–Bloch equations for a two-level quantum system were performed by Slavcheva et al. [551]. FDTD method also was used to calculate the spontaneous emission lifetime in microcavity by Xu et al. [552], perfectly matched layer proposed by Berenger [553] was used to restrict the computational domain. The electric field of an oscillating dipole in a microcavity was separated into a longitudinal and transverse parts. The former is the unretarded Coulomb field due to the dipole source and the microcavity surrounding it; this part does not contribute to the dipole radiation power (a strict proof of this fact is contained in the Appendix A to the sited paper). The authors used FDTD in 3D Cartesian coordinates and in cylindrical coordinates to analyse three cases: (1) a dipole between two parallel perfect-metal plates; (2) a dipole in a dielectric waveguide and (3) a dipole in a dielectric microdisk. In the latter case the authors observed a peak in the spontaneous emission rate for the radial dipole that is probably attributed to the whispering gallery modes in the microdisk. GaN hexagonal microsized pyramids could serve as highly efficient microcavities that could support several different types of the optical resonance modes. Such pyramids with a side length of 8 µm ware fabricated and studied by Jiang et al. [554]. Bienstman [276] used eigenmode expansion to analyse processes in RCLED. To represent light emission inside the cavity, the author introduced dipole point current sources formulated via Dirac δ -function. He showed that the excitation strength of a particular mode is proportional to its field amplitude at the point where the dipole source is located. The author

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considered the modification of the spontaneous emission rate caused by the optical environment and found a factor 3 enhancement over the vacuum value for a source placed between two perfectly conducting metallic mirrors. P. Bienstman notes that cavity should be designed in such a way that radiation perpendicular to the interface will be enhanced due to constructive interference while oblique radiation will be suppressed via destructive interference. The condition for the constructive interference is formulated using a round-trip phase change that should be an integer multiple of 2π . The also author consider in detail cases of changing the cavity length, i.e. so-called undertuned and overtuned cavities. Grating assisted RCLED was analysed by Delbeke et al. [555] who used plane wave matrix formalism to compute the dipole radiation in a two-dimensional periodically corrugated multilayer structure. Detailed computations were performed for RCLED with a DBR bottom mirror and a partially metallic covered two-dimensional air–semiconductor grating. Dipole radiation into grating structures was also considered by Rigneault et al. [556] using the scattering-matrix algorithm who showed that strong corrugation could results in the energy concentration in directions close to the normal to the structure. Design of RCLEDs could be performed with different goal functions, for example, maximum efficiency, minimum beam divergence or some optimal spectral properties. Doubly resonant cavity LED. There also more complex cavity structure referred to as a doubly resonant cavity LED (RC 2 LED) which comprises two resonant cavity. It could be constructed from the conventional RCLED by replacing the bottom outcoupling mirror with two parts, a usual DBR and a symmetric cavity that has the same resonance wavelength as the cavity containing the active region [276]. For example, in OLEDs in addition to the primary cavity formed by metal, organic layers and ITO, the second cavity is formed by the introduction of three extra layers, for example, two high refractive index layers made from Nb2 O5 enclosing low refractive index layer (SiO2 ) [557,558]. A doubly resonant cavity LED (RC 2 LED) has a very narrow radiation profile; the extraction efficiency of RC 2 LED is slightly lower than of the RCLED. However, RC LEDs could not provide enhancement over the entire emission spectrum, just in the vicinity of the resonance, in contrast to photonic crystals which allow a significant enhancement over a broad range of frequencies via modification of the spontaneous emission [559]. Note also that microcavity effect could be used in devices based on light absorption such as resonant cavity enhanced Schottky photodiode [560] and resonant cavity enhanced MSM photodetector [561]. An alternative to the standard λ/2 cavity is two λ/8 low refractive index phase-shift layers around a λ/2 high index containing the active region. Such virtual λ/2 cavity – ‘‘phase-shift cavity’’ – reduces coupling to guided modes that leads to increasing LEE [275]. It should be noted that spectral broadening of the source is detrimental to extraction and results in a directionality reduction [111,562]. Microcavity OLEDs [563–565], as a rule, have a dielectric DBR as a bottom mirror and a top metal mirror. However, both mirrors could be metallic as well [566]. 3.2. Photonic crystals Effective volume. The spontaneous emission rate of an initial state is proportional to the square of the matrix element and the density of final states. In free space, the free-photon density of states per unit volume Df scales as Df = const ×

1 1

ω λ3

where ω is the frequency of transition and λ is the wavelength of light. If the photonic band gap is around ω, there are no allowed modes to couple to and the spontaneous emission is inhibited. If, however, the photonic crystal has a point defect with a localized or resonant state at ω, the spontaneous emission rate will be greatly increased due to the increase of the density of states per unit volume Dr which is [509] Df = const ×

1

1

∆ω Veff

where ∆ω is the frequency width of the resonance and Veff is its effective volume. The effective mode volume is defined as [346]

ε(⃗r )E⃗ 2 (⃗r )d3 ⃗r . (ε(⃗r )E⃗ 2 (⃗r ))max

 Veff =

Thus the enhancement factor could estimated as Dr Df

≈

ω λ3 Q · = ∆ω V V /λ3

where Q ≡ ω/∆ω is the quality factor of the cavity [509]. Since the smallest volume is of the order λ3 (see, for example, [567,568] and Table 1.1 in Ref. [569]), the largest enhancement is of the order Q .

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FDTD simulation of PC effect. Koenderink et al. studied the modification of the spontaneous emission rates of the dipoles in the photonic crystal membranes using 3D FDTD computations and found over 7 times inhibition and 15 times enhancement compared with the vacuum values [570]. The spontaneous emission rate of an emitting dipole with the position ⃗ r and

ˆ

ˆ

⃗ depends on the local radiative density of states (LRDOS) ϱ(⃗r , d⃗, ω) that could be determined for a wide the orientation d frequency range from a single FDTD computation via the broadband temporal excitation of a dipole point source in terms of its vacuum value using the Fourier transform as [570] ⃗j(⃗r0 , ω) · E⃗ (⃗r0 , ω) ϱ(⃗r , d⃗ˆ , ω) = , ⃗j(⃗r0 , ω) · E⃗vac (⃗r0 , ω) where ⃗j(⃗ r0 , ω) is the frequency spectrum of the dipole source and E⃗ (⃗ r0 , ω) is the Fourier transform of the time trace of the electric field E⃗0 (⃗ r0 , t ) . By comparison with the analytical solutions for the dipole in a dielectric sphere and for the dipole near a planar interface between two dielectrics the authors has found that over 10 grid points per wavelength at the frequency of interest in the highest refractive index medium are needed to provide the accuracy of computation of the spontaneous emission rate not worse than 5%. Lee et al. [571] used 3D FDTD method to analyse the light extraction from the OLED via the photonic crystal sandwiched between the glass substrate and the transparent ITO electrode. The computational domain was restricted to 7 × 7 × 3 µm with the perfect mirrors placed at four sides. The emission of the diode was simulated as a large number of the Gaussian dipole pulses. Tangential components of both electric and magnetic fields computed at each time step were stored to be used later for the estimate of the far-field pattern via the Fourier transform. The computed increase of the extraction efficiency in excess of 60% was confirmed by the experimental study. Photonic crystals also could be used to steer emission into the specific direction and influence the light polarization. The active medium with low surface-related losses is preferable for the photonic crystal light emitters because of large surface-to-volume ratio of PC structures; quantum dots present a good candidate for this role [572]. 3.3. Surface plasmons Metal support collective oscillation of the conduction electrons (bulk plasmons) at the plasma frequency ωP and surface plasmons of a lower frequency ωSP [573] that could be excited by high energy electron beams or by light. Surface plasma wave could exist only for TM polarization [429]. By definition surface plasmons are the quanta of surface-charge-density oscillations but the same term is also used to refer to collective oscillations in the density of electrons at the metal surface [574] (less attention so far has get a close relative of SPs —surface phonon polaritons formed by the strong coupling of light and optical phonons in polar semiconductors that are of interest for the devices operating at mid-infrared frequencies [575]). Resonance condition. In silver, the bulk plasmon energy is 3.76 eV and the surface plasmon energy is below 3 eV, depending on the refractive index of the adjacent dielectric material (h¯ ωSP = 2.92 eV for the Ag/GaN interface) [576]; the plasmon energy of gold and aluminum on GaN is lower than 2 eV and higher than 5 eV, respectively [577]. The surface plasmon resonance condition is written as ′ εm + εs′ = 0, ′ where εm and εs′ are the real parts of the dielectric constants of metal and semiconductor, respectively. The surface plasmons described above are frequently referred to as surface plasmon polaritons (SPP). Another type of surface plasmons – localized surface plasmons (LSP) – are associated with bound electron plasmas in nanovoids or particles with dimensions much smaller than the light wavelength [578–581]. In contrast to SPP, LSP can be directly coupled to the free-propagating light [581,582] and thus to serve as optical antennas. The specific plasmon modes are observed for the metal membrane sandwiched between two layers of the same dielectric media [583]. In this case, if the thickness of the metal film is small enough, the field associated with each metal–dielectric interface will overlap [584] and propagating surface-bound optical excitations will be coupled to produce either symmetric or odd symmetric modes, differing in the level of ohmic losses that are smaller for odd symmetry modes [583].

SPP length scales. There are four length scales connected with the SPPs modes [585]: 1. 2. 3. 4.

the SSP wavelength the SPP propagation length the penetration depth of the electromagnetic field associated with SPP mode into the dielectric medium the penetration depth of the electromagnetic field into the metal.

These length scales span seven orders of magnitude, from nanometers to centimeters [585]. It is known that SPs could enhance absorption of light in and fluorescence of macromolecules [586], increase intensity of the Raman scattering [587] and light transparency thought the metal gratings [588], generate photonic band gaps [589]. Surface plasmons act as mediators in the visible light transmission through the homogeneous thin metallic films [590–595]. In this case an optimal film thickness exists that maximizes the film transparency. To explain it, some researches

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invoke the process of the regeneration of evanescent waves [596,597]; however, this is hardly justified (Occam’s razor) since such behaviour of the transmittance as a function of the metal film thickness could be explained by the competition between increasing optical field enhancement due to SPP excitation with the film thickness and increasing absorption in the metal layer [593]. Plasmonics is considered as a link between electronics and photonics relieving the size mismatch between electronic and photonic components [598,599]. Plasmonic waveguides can simultaneously carry optical and electric signals [600]; propagation distance of SPP could be several tens of micrometers and could be further increased through the use of subwavelength grooves cut into the metal surface (the so-called channeling surface plasmons) [601,602]. Surface plasmons [603,599,604] could increase the density of states and the spontaneous emission rate in the semiconductor active layer and thus lead to the enhancement of the light emission due to SP-QW coupling [605–607,576]. The excitation of surface plasmon polaritons at metallic electrodes of organic LEDs was studied by Nowy et al. [316]. The authors used a transfer matrix method (TMM) for simulation, assuming that all interfaces are sharp and without roughness. Two OLEDs were considered: (1) with transparent indium–tin oxide anode and a metallic cathode, (2) with two metallic electrodes (in the latter case a microcavity is formed). The dipoles embedded in the multilayer stack of the diode were represented as forced damped harmonic oscillators d2 p dt 2

+ ω02 =

e2

E r − b0

dp

,

m dt where p is the electric dipole moment, ω0 is the resonant angular frequency, m is the effective mass, Er is the reflected field at the dipole position and b0 is the damping constant related to the finite radiative time of the emitter. The authors were able to identify different regions in the power dissipation spectra and estimate the fractions of light that is emitted to substrate, to waveguide modes, to surface plasmon polaritons as well as to identify transverse electric (TE) and transverse magnetic (TM) modes. SPs in InGaN LEDs. Significant enhancement of the internal quantum efficiency ηint was measured when a silver or aluminum layer was placed 10 nm above an InGaN light emitting layer while no enhancement was registered for gold layer: the peak PL intensity was increased by 14 times for Ag and 8 times for Al, for the PL intensity integrated over the emission spectrum 17-fold and six-fold increase was found, respectively [305,577]. Later these authors [306] showed the increase of the photoluminescence for the green and blue wavelength in the case of the nano-grating structures on the gold surface and the dependence of the emission enhancement on the grating period which has a staircase appearance (Fig. 3 of Ref. [306]). The nano-grating structures modify the SP frequencies and thus provide conditions for coupling of the SPs and the blue/green InGaN emission. The electron–hole pair excited within a quantum well is coupled to the electron vibrations at the metal/semiconductor interface if the energies of the electron–hole pair in InGaN h¯ ωInGaN and of the metal SP h¯ ωSP are similar. Thus, the electron–hole recombination produces SPs instead of photons, and this new recombination path increases the spontaneous recombination rate. The wavelength-dependent efficiency ηint (ω) enhanced due to the coupling between QW and SP is written as [305] ⋆ ηint (ω) =

′ krad (ω) + ηext (ω)kSP (ω)

krad (ω) + knon (ω) + kSP (ω)

,

′ where ηext (ω) is the probability of the photon extraction from the SP’s energy that is determined by the ratio of the light ′ scattering and dumping of the electron vibration (ηext (ω) depends on the roughness and structure of the metal surface), kSP (ω) is the coupling rate between QW and SP, krad (ω) and knon (ω) are the radiative and nonradiative recombination rates, respectively. The increase of the spontaneous emission rate is characterized by the Purcell enhancement factor that is determined for ′ the ideal extraction ηext (ω) = 1 as

FP (ω) =

krad (ω) + knon (ω) + kSP (ω) krad (ω) + knon (ω)

≈

1 − ηint (ω) ⋆ 1 − ηint (ω)

.

For the high quality samples similar to ones used by Gontijo et al. [576] the contribution of the nonradiative recombination is negligible and the recombination rate enhancement could be estimated as FP (ω) ≈ 1 +

kSP (ω) krad (ω)

.

Later Okamoto et al. have established an ‘‘exponential’’ increase in the intensity of photoluminescence with the spacer thickness decrease [608,306]. Probably, such strong dependence resulted from the fact that the range of thickness considered in the experiments contained the penetration depth Z (ω) of the SP fringing field into GaN from metal — SP is an evanescent wave that exponentially decays with distance from the metal surface. The penetration depth was estimated as [609,272,577] Z (ω) =

c

ω



εGaN (ω) − εmetal (ω) . εmetal (ω)2

This parameter is about 40 nm for the Ag/GaN interface at the wavelength corresponding the energy h¯ ω = 2.92 eV.

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Even higher enhancement (92-fold) was registered in the experiments by Neogi et al. [304] with InGaN/GaN QW covered by a 8 nm silver film, the thickness of GaN cap layer was 12 nm. The authors considered the SP dispersion relation for the system in question and noted that it contains two branches — tangential and normal modes, depending on the dominant direction of the current in the silver film. The estimates made indicate that resonant enhancement could be as high as 104 for the separation of the metal surface from the QW smaller than 4 nm [304]. Dispersion engineering. The evident limitation of the SP enhanced approach is the need to find metal with the SP energy in the required spectral range. To some extent this problem is alleviated (at the price of greater fabrication complexity) by the socalled dispersion engineering that is just using the metal/dielectric stacks instead of a single metal film [311]: for example, no single metal film based on materials commonly used with nitride semiconductors could provide SPP with resonance frequency in the green spectral range about 2.4 eV, while it is possible with multilayer structure Ag/TiO2 /Au/Ti2 . Henson et al. [311] have also considered other metal (Al, Ag, Au)/dielectric (TiO2 and HfO2 ) combinations: Al is needed as a bottom metal film due its large plasma frequency (greater than 5 eV for Al on AlGaN) while Hf is preferable over Ti because its higher bandgap prevent absorption of the near-UV light. Use of multiple metallo-dielectric layers allows one to introduce singularities in the density of states of the SP at the energies of interest via the anticrossing of SP modes of different metal films [610]. Enhancement of light emission of LED by surface plasmons (SPP-mediated LEDs [611]) could be considered as a two-stage process: 1. Spontaneous recombination of the electron–hole pair in the active layer with the energy transfer to the SPPs; 2. Extraction of light from the excited SPPs by scattering them on a corrugated or structured surface. Since the spontaneous emission into the SP could be extremely high, the efficiency of the emitter as a light source depends on the ability effectively couple the SP to free radiation. Extraction efficiency of confined SPP to far-field radiation has been recently studied by Yoon et al. [611] who proposed a method to determine the upper limit of radiative extraction efficiency from SPPs, based on measurement of the spectral bandwidth of grating-induced absorption spectrum as a function of the dielectric constant of the metal. Gontijo et al. [573] who studied emission of InGaN/GaN QW positioned 12 nm below the silver surface report also another feature of SPP-mediated LED — considerably faster (over 50 times) LED modulation. The effect of the QW-SP coupling is more significant than the enhancement of the InGaN emission observed in the semipolar GaInN/GaN LED with reduced piezoelectric field [612] or in the piezoelectric-free QW grown on M-plane of GaN substrate [613] and depends on the emission wavelength, being stronger for the shorter wavelengths [608]. This coupling was found to be more effective at the higher temperature [608]. SPs in other light emitters. Coupling with SPs could be used for other types of emitters such as bulk ZnO [614], small organic molecules [615,616], dye-doped polymers [617,618], conjugated dendrimers [616] and conjugate polymers [615,619], Si quantum dots [620]; it is also frequently used in design of the biological sensors of the autofluorescence emission [621]. Neal et al. [617] studied the surface plasmon enhanced emission from dye-doped polymer layers; Okamoto et al. reported a 30-fold enhancement of the photoluminescence intensity for the ‘‘naked’’ CdSe-based quantum dot nanocrystals [622,623] while smaller enhancement was registered for the crystals that contain CdSe core and ZnS shell [623], and 70-fold enhancement of emission from silicon nanocrystals (≈3 nm in diameter) in silicon dioxide, getting the value of the internal quantum efficiency of 38% which is comparable to that of compound semiconductors with direct transition. The authors note that effect of the SP coupling on the spontaneous emission is now an only feasible technique to developed bright light emitting devices based on silicon (on the development of the low-power silicon light sources needed for the optoelectronic integration see, for example [624–626]). Brolo et al. [627] measured PL enhancement by 2 orders of magnitude due to the coupling of semiconductor quantum dots to the surface plasmon modes of nanohole arrays in a metal (gold) film. The authors studied polystyrene-b-poly(acrylic acid)stabilized CdS quantum dots; arrays of nanoholes with diameter from 86 to 131 nm have lattice constant from 350 to 660 nm. These block copolymer-coated QDs as emitters were fixed while the structure parameters were varied to tune to the SP resonances that are closely connected to the geometrical parameters of the metallic nanostructure, more exactly, to provide the best overlap between the SP resonance and PL emission. The enhancement (Purcell factor) up to 300 was measured. The surface plasmon resonance of metal film nanohole arrays has been studied recently by Zhou et al. [628] who showed that the enhancement transmission of light through a gold film with a periodic array of subwavelength rectangular holes is related to two different SP resonance features: 1. localized waveguide resonance where each hole acts as a truncated rectangular waveguide with both ends open to free space, and 2. well-known photonic crystal resonance effect due to the periodicity. The authors performed the full-vectorial 3D FDTD computations for a number of hole array configurations. Enhancement via the SPs and photonic crystals could be combined in a single device as was demonstrated by Rebound et al. [629] who studied a two-dimensional nanoimprinted photonic crystal containing dye in the vicinity of a silver surface, obtaining 27-fold increase of PL intensity compared to the reference sample. Coupling of SPs to free radiation. If the metal surface is perfectly flat, surface plasmons are nonradiative [630]—the momentum (wave vector) of surface plasmon on planar or cylindrical metal interface is larger than the momentum associated with free-

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propagating electromagnetic radiation [631]; SPPs have electromagnetic field that decay exponentially into both metal and dielectric that bound the interface and the SP energy will be eventually thermally dissipated. However, recovering of the energy transferred to the SP modes as light is possible by providing the surface roughness or nanostructuring [576,303], for example, metal nanoparticles coupled to a flat metal layer [578,632], a system of unidirectional nanoslits [633,634], annular apertures in the metal film [635] or nanoporous metallic films—a buried nanovoid lattice [580,581]. The use of metallic gratings instead of planar film breaks up the SP dispersion relation into series of bands [610]. The transmission efficiency of a periodic array of subwavelength holes does strongly depend on the hole shape and increases by an order of magnitude with the shape change from circular to rectangular [636]. Plasmon-assisted enhancement of LEE of the InGaN green LED using deposited ellipsoidal silver nanoparticles was reported by Butun et al. [228]. Optical antennas. Enhancement of light emission through the surface plasmons is similar in some respect to the well-known approach to enhance radiation by coupling to the secondary body that is frequently referred to as antenna (in this case also as to ‘optical antenna’ [637–641]) — a device that efficiently converts the localized energy to the energy of free-propagating radiation. Probably the simplest configuration of this concept is the enhancement of the emission of the single molecule by the gold particle placed in the near vicinity [642–645] or of the dye–polymer mixture [646]; the optical properties of metal nanoparticles strongly depend on their size, their shape and their environment (proximity to other particles, for example, in dimmer nanoantennas [647]) [648]. The dependence of the spontaneous emission rate enhancement via the QW-SP coupling on the emission wavelength and the antenna geometry was studied by Rogobete et al. [649]. The authors used FDTD computations, assuming that both the emitter and the antenna are embedded into the homogeneous medium of refractive index equal to 1.7. A number of possible antenna’s configurations were considered, including a nanosphere, a system of two spheres, a system of two ellipsoids and the bow-tie antenna [639], accounting for the known fact that the electric field between two nanostructures can be much stronger than the near field of a single structure [649]. The authors have found that while the bow-tie geometry provides the best effect (the computed enhancement of the radiative recombination rate is over 1700), this optimum design could not be realized within the current technology due to the difficulties in fabricating sharp corners and avoiding tip snipping [650] and the double-ellipse structure seems to be the best practical choice. Experimental study of the plasmonic nanoantenna comprising two gold nanoparticles showed increase on fluorescence of a dye around the antenna up to 100 times [651]. By using electromagnetic resonance related to surface plasmons any nanostructure can be considered as an optical antenna [630]. A simple metal strip also can function as an optical antenna, with the resonance frequency being tuned by the strip dimensions [591]. The linear antenna structures such as nanostrips and nanorods similar to Fabry–Perot resonators in addition to the fundamental dipole mode could support higher-order resonances [641]. The aim of the optical antenna is to convert optical radiation into localized energy, and vice versa and thus to boost the efficiency of the optoelectronic device that emits (for example, LED) or absorb (for example, solar cell) radiation [637]. Optical antennas involve controlling light on the subwavelength scale, in contrast to the usual in optical science and engineering means such as lenses, mirrors, and diffractive elements. In the classical antenna theory, antenna parameters are directly related to the wavelength λ of the incident radiation: the characteristic lengths of the antenna elements could be written as L = const × λ. This scaling fails at optical frequencies since the incident radiation is no perfectly reflected from the surface, but penetrates into the metal causing oscillations of the free electron gas [630]. Recently Novotny [652] proposed to retain the classical design approaches using an effective (shorter) wavelength determined by the simple linear scaling law

λeff = n1 + n2

λ , λp

where λp is the plasma wavelength and ni are the coefficients with dimensions of length that depend on the geometry of antenna and the static dielectric properties. This model is applicable for antennas made from straight linear segments with the radius much smaller than the wavelength under assumption that the free electron gas in metal obeys the Drude model. Results of three-dimensional simulations of a number of different configurations of optical antennas including two- and multiple-sphere antennas, silver spheres protected by a gold coating and nanorods have been recently reported by Kappler et al. [653] while Taminiau et al. [654] have performed 3D computations of the so-called Yagi-Uda antenna that comprises five cylindrical elements. Liu et al. [655] have recently investigated gold arrays composed of paired elliptical cylinders as plasmonic nanoantenna for the visible spectral range. The analysis was based on the Drude–Lorentz model (this model contains the Drude term frequently used to characterize the optical properties of metal at low frequencies that describes the free electron response [538] and the sum of Lorentzian terms containing the contribution from m Lorentzian oscillators) for the metal–dielectric constant

ε =1−

ωp2 ω + iΓp ωp 2

+

−

2 fm ωm

m

ω − ω 2 − i Γm ω 2 m

with parameters for gold being determined by fitting the experimental data; here ωp is the plasmon frequency, Γp is the damping constant, and ωm and Γm are the resonant frequency and the damping constant of the mth Lorentzian oscillator, respectively. The authors found the field enhancement to be inversely proportional to the size of the gap between cylinders.

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4. Conclusions There are numerous ways to increase the light extraction efficiency and it is hardly possible to indicate the single best approach, except for applications where the high directionality of the light is of primary importance (as, for example, for short-haul polymer optical fiber (POF) [656] data transmission systems [539,111] and low-power high-speed photonic circuits [657] — in this case RCLED is an evident winner). From the practical point of view, two additional issues could be considered:

• possible damage to the active layer or/and degradation of the electrical properties of the LED6 ; • the fabrication cost unless the additional cost is small compared to the cost of the conventional LED dye, as, for example, is the case of 2D photonic crystal patterning by imprint lithography which is estimated to cost less than $0.01 per square millimeter [658]. Fortunately, most of the LEE enhancement approaches are not mutually exclusive and thus can be combined, for example

• geometric sapphire shaping is easily combined with surface texturing [356]; • Thin film design with buried micro-reflectors is combined with surface texturing; • simultaneous use of omnidirectional reflector on the backside and nanoimprinted 2D photonic crystal pattern was

•

• • •

reported for the InGaN-based green LED by Cheng et al. [257], of ODR and textured micropillar arrays on the bottom side of sapphire substrate by Lee et al. [659], of An InGaN–GaN thin film vertical-type light emitting diode with a twodimensional photonic crystal (PC) on the emitting surface and a TiO2 –SiO2 omnidirectional reflector on the bottom by Lin et al. [300]; planar microcavity could have walls with wavelength periodic texturing; an additional effect of such texture could be the appearance of the photonic bandgaps in the dispersion of the microcavity modes depending on whether the surface profiles of two walls are symmetric [660]; such device is also called ‘‘grating assisted resonant cavity light emitting diode’’ [555]; periodic defects in thin film 2D photonic crystal slabs prohibits the propagation of the lateral modes and thus both increase the light extraction from the cavity and directionality of the emitted light [661]; photon recycling (reabsorption–emission) is non-exclusive of microcavity effects; the use of Archimedean photonic lattice for the enhancement of the LEE from RC LED was reported by Rattier et al. [514] and of the irregular microlens array by Kim et al. [384]; introduction of periodical effects into the two-dimensional photonic crystal slab to form a microcavity with inhibition of the lateral modes and thus high extraction efficiency [661]; Lee [213] studied the light extraction enhancement from vertical GaN LED using both double surface roughening and ODR.

Acknowledgements The author is grateful to M.N. Nemtseva for the help with manuscript preparation and to M.V. Bogdanov, K.A. Bulashevich, I.Yu. Evstratov, V.F. Mymrin, S.Yu. Karpov and M.S. Ramm for useful discussions. Appendix A. Numerical simulation of light extraction There are two approaches to simulation of light propagation in optoelectronic devices. The first one is the high frequency asymptotic approximation such as geometric optics and its extensions. It is valid if the geometric features are large compared to the wavelength (more detailed analysis of the corresponding conditions, including necessary and sufficient ones formulated via the so-called ‘‘Fresnel volume’’ could be found in [662]). Nevertheless, it is widely used beyond its region of applicability because of the simplicity, producing sometimes reasonable results. The strict approach to the problem is based on the solution of Maxwell equations. A.1. High frequency asymptotic methods A.1.1. Ray and beam tracing The best known method of this class is the geometric optics used in applications most frequently in the framework of ray tracing method [663]. Its main weak point is the danger of the ‘‘undersampling’’ errors [664]. Ray tracing method is also known to be unstable in case of geometry displaying numerous small features [662]. Beam ray tracing, or simply beam tracing, method [665,666] is free from undersampling artifacts of raytracing at the price of loosing simplicity: geometric operations needed to trace polyhedral beam (e.g., intersection, clipping) are rather complex. 6 Dry etched-induced damage could be sometimes recovered by thermal annealing [159]; electrical properties could also be improved due to the pattern fabrication, for example, the corrugated photonic crystal structure fabricated by the etching of an indium–tin oxide anode layer in OLED is reported to lower the operating voltage (by 30% at constant current compared to the conventional structure) due to the partial reduction of the thickness of the organic layer [267] as well similar results for InGaN-based LED due to the increased the effective area of contact between the n electrode and the n-GAN layer [520]; however, Cheng et al. reported a slight increase of the forward voltage of the InGaN-based grebe LED due to the nanoimprinted photonic crystal pattern [257].

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A.1.2. Geometric optics and its extensions The geometric theory of diffraction [667] as well as its modification – uniform theory of diffraction [668] – supplements the standard geometric optics with the ability to model diffraction by introducing additional (diffracted) rays when the original ray strikes an edge or a vertex. These methods incorporate closed-form analytical solutions for a number of diffraction cases [669]. Diffraction is a local phenomenon at high frequencies. Thus, the behaviour of the diffracted wave at edges, corners etc. could be approximately determined from the exact solutions for simple canonical problems. For example, the diffraction around an edge is based on the asymptotic form of the solution for an infinite wedge. An efficient algorithm for the determination of the propagation path of multiple diffracted rays was proposed recently by Bagnerini et al. [670]. Budaev and Bogy reformulate the problem as the mathematical expectation of some functional in the space of Brownian trajectories with the Wiener probabilistic measure [671]. One more extension of the geometric optics should be mentioned — ‘‘complex geometric optics’’ [672] that is based on consideration of complex values of the wave vector as means to describe nonuniform waves. A.2. Computational electromagnetics Computational Electromagnetics (CEM) problems include both low frequency and high frequency ones that mathematically differ by the nature of the kernel in the integral equation (a fundamental solution or Green function) that could be either smooth or oscillatory [673]. In the former case the higher-order derivatives become increasingly smaller as the distance from the source increases while in the case of an oscillatory kernel all derivatives decay as the inverse of the distance. Thus, the spatial bandwidth of an oscillatory kernel does not decrease away from the source enhancing the complexity of the numerical solution. The field of Computational Electromagnetics could be subdivided into three subfields [674,675,446]: Frequency domain eigensolvers Find the bandstructure ω(⃗ k). ⃗ 0 (⃗r ) of the fields E⃗ (⃗r , t ) = E⃗0 exp−iωt and H⃗ (⃗r , t ) = Frequency domain simulations Find the amplitudes E⃗0 (⃗ r ) and H − i ω t − i ω t ⃗ 0 exp H for a given current distribution ⃗J exp at a fixed frequency ω. ⃗ (⃗r , t ), usually for a given ⃗J (⃗r , t ). Time domain simulations Simulate the time evolution of the fields E⃗ (⃗ r , t ) and H A so-called complex frequency hopping method based on the low-order multi-point Padé approximation is in some sense intermediate since it is able to produce simultaneously both the frequency and time domain results [676]. Additionally, there are some approximate approaches that are usually restricted to low index contrasts or to structures that are slowly varying in at least one direction such as the Beam Propagation Method (BPM) [446] or the Slowly Varying Envelope Approximation (SVEA) [534]. ⃗ (see, Maxwell equations for the frequency domain methods are, as a rule, re-written in terms of one field (E⃗ or H) however, [677]), for example,

[

(∇ × ∇×) −

ω2 c

] ε(⃗ r ) E⃗0 (⃗ r ) = iωµ0⃗J (⃗ r ). 2

In contrast, in time domain methods initial (first-order) Maxwell equations are usually discretized. A single simulation by the time domain method can be used to study the material response over a wide bandwidth. To simulate harmonic (sinusoidal) steady state, one can either directly prescribe a single frequency incident plane wave or perform a Fourier transform step on the pulse waveform response [678]. In the latter case, a too short pulse suffers from accumulating errors due to overshoot and ringing during propagation requiring filtration of the numerical noise before Fourier transformation. Anyway, integration time should be not lesser than several wave periods until the sinusoidal steady state is achieved in every element. To reduce the continuum problem to the discrete one, a number of approaches are used such as finite differences, finite elements, spectral methods, boundary elements giving several closely related methods:

• Finite difference time domain method (FDTD) [679,680,490]7 • Finite element time domain method (both acronyms FEMTD and TDFEM are used) [682–687] and partition of unity FEM [688]

• Transmission line modelling method (TLM) • Time domain integral equation method (TDIE) [689–691] (on the integral equation formulation in the frequency domain see, for example, [692])

• Rigorous coupled wave analysis (RCWA), also called Fourier model method [471,693] • Plane wave expansion method (PWEM) [471,694] • Plane wave admittance method (PWAM) [465,695]

7 This method is also applicable to the analysis of the wave phenomena of different nature, for example, waves in elastic media [681].

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• Mode matching (MM) method (also called eigenmode expansion method [471]) based on the subdivision of the structure into a number of sections and finding the transmissive and reflective properties as a product of the corresponding matrices for sections [696] • Transfer matrix method (TMM) [697,698,32,365] • Finite difference beam propagation method (FD-BPM) [699–703] • Special methods valid under additional simplifying assumptions – Spectral index (SI) method [699,704] – Free space radiation mode method (FSRM) [705] – Effective index approximation [706] – Coupled modes approximation [707,517,708]. A.3. FDTD method & extensions Yee scheme. The original FDTD method was suggested by Yee for the isotropic nondispersive media in 1966 [679] and implemented for the two-dimensional Cartesian computational grid. The explicit formulation of FDTD method (the extension to the higher dimensions in the case of Cartesian coordinates is trivial) is based on the central difference approximation of the Maxwell equations in free space that in the one-dimensional case are written as ∂ Ex 1 ∂ Hy =− , (A.1) ∂t ε0 ∂ z

∂ Hy 1 ∂ Ex =− , ∂t µ0 ∂ z

(A.2)

FDTD scheme is formulated as follows n+ 12

Ex

n− 21

(k) − Ex ∆t

(k)

=−

1 1 n n 1 Hy (k + 2 ) − Hy (k − 2 )

ε0

Hyn+1 (k + 12 ) − Hyn (k + 12 )

∆t

∆x n+ 21

=−

1 Ex

µ0

, n+ 12

(k + 1) − Ex ∆x

(k)

.

It can be seen that the electric and magnetic fields are separated in both space and time by half a step. In FDTD algorithm only two of Maxwell equations are used to update the electromagnetic field. It can be shown that if the remaining two equations are satisfied at the initial time moment, this property is conserved in the course of computations [552]. On the benchmarks used for the verification of the FDTD computations see, for example, Ref. [709]. Extensions. Extension of the Yee scheme to multidimensional problems is trivial [710] as well as to orthogonal curvilinear grids [711]. Its advantage is simplicity (allowing, for example, implementation using graphics processors that at present are an order of magnitude faster than CPUs of common computers [712]) as well as, due to the nearest grid-neighbour dependencies, a straightforward parallelization [713–715] (an example of large scale FDTD computations of the photonic crystal LED using AMD cluster comprising 128 dual node could be found in Ref. [110]). Certainly, some kind of postprocessing of the FDTD data is required to get some modal or integral information such as, for example, intrinsic waveguide losses, dispersion curves etc.; for example, J. Witzens et al. applied an inner product to the previously computed mode profiles in the simulated cross section to get intrinsic waveguide losses in photonic crystal waveguides since it is otherwise difficult to distinguish insertion losses arising due to the mode mismatch from the propagation losses [716]. Another difficulty with applying FDTD method to light extraction problem was reported by A. Chutinan et al. [717] who used FDTD and mode-expansion methods to study light extraction from OLEDs: in the FDTD computation it is difficult to separate radiation modes with radiation angle close to π /2 from the waveguide modes since both propagate along the same direction. The Yee scheme could be interpreted as a mass-lumped version of mixed finite element formulation [718–720] that is a special case of Whitney forms [721]. The latter has a significant advantages over simple pure scalar nodal formulations that are known for significant problems with spurious eigenmodes that could be corrected, among other approaches, by using edge-elements [722,723] that were pioneered by Nedelec [724]. Staircasing. The main weakness of the classical explicit FDTD approach is the need to have a grid fine enough to resolve both the smallest electromagnetic wavelength and the smallest geometrical feature in the problem, with the size of the time step being determined by the smallest space scale and the need to update the fields in the whole computational domain each time step. The time step is determined by the classical Courant condition [725] which for the uniform Cartesian grid is written as 1 ∆t ≤  1 c (∆x)2 + (∆1y)2 + (∆1z )2 where c is the speed of wave propagation in free space.

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The failure to achieve the needed spatial resolution manifests itself as the so-called staircasing leading to large errors even for low frequencies [726]. Staircasing could be to some extent alleviated by the use of the effective properties for partially filled cells [727] or by artificial numerical aids such as introduction of the boundary penalty terms [726]. On the other hand, fine spatial grids force small time integration steps when explicit schemes are used, thus considerable efforts were directed to development of implicit FDTD schemes [728–730]. The time step of these schemes is limited by the level of the numerical dispersion error that is tolerated [731]; an alternative to the alternating-direction FDTD (ADI-FDTD) scheme is the so-called symplectic FDTD method [732] based on the use of fourth-order finite differencing for space and symplectic exponential differential operators for time. FDTD for metals. The simulation of materials having the real part of the dielectric constant lesser than unity such as metals requires the solution of the auxiliary differential equation [680] that is the time domain counterpart of the metal dispersion relation (which is the frequency domain entity) and for the Drude model has the form [261]

∂J J (t ) + = ε0 ωp2 E (t ), ∂t τ where τ is the relaxation time and ωp is the plasma frequency. FDTD-related errors frequently manifest themselves as numerical dispersion that depends on both space and time steps and the wave propagation direction relative to the grid axis [715] or anisotropy artifacts. Development of the FDTD method occurred in several directions, including lossy dielectric media [733], media having nonlinear material properties [734], local mesh refinement (sometimes called subgridding [735–739,685,740–742] or segmented FDTD (SFDTD) [743]) to treat small geometrical features, development of more accurate so-called dispersionrelation-preserving (DRP) finite difference schemes [744]. The difficulty of the implementation of local refinement is related to the spurious reflections and unconditional instabilities. An alternative approach is based on use of overlapping grids providing some advantages for the case of different propagation angles and broad range of frequencies. A variant of subgridding having a flavor of multigrid approach is the socalled embedded-grid FDTD method (EG-FDTD) with fine grids overlapping only selective regions of the computational domain [745]. The loss of accuracy of FDTD methods applied to the structures containing rectangular dielectric corners could be alleviated by use of special corner equations derived with account of the singular nature of the field [746]. An equivalent formulation of FDTD method as a lumped circuit also was suggested [747]. FDTD method could be also extended by inclusion of active devices within the passive structures simulated (lumped-FDTD) [748–751], with direct access to SPICE to model the lumped elements that could be an arbitrary large SPICE circuit [752]. It should be noted that hybridization of the continuum simulation methods with the lumped circuit models of components and device is also possible in the frame of TLM [753]. Novel three-dimensional FDTD schemes on dual staggered Cartesian grids have been recently proposed by Zygiridis and Tsiboukis [754], that can be optimized to get maximum accuracy for the narrow or wide frequency band computations. These schemes are based on symmetrical approximations of spatial derivatives (asymmetrical finite difference are known to introduce large artificial dissipation that is undesirable for Maxwell equations) of the form

 N /2   1 − ∂ f  = C + O(∆uM ), f 2m−1 2m−1 − f m i− 2 i+ 2 ∂ u i ∆u m=1 with maximal formal order of approximation achieved for N = M. Low-dispersion FDTD schemes based on the combination of the pseudospectral space approximation and either secondor fourth-order time stepping developed by Lee and Hagness [755] provide significant improvement in computational efficiency and accuracy, in particularity, much slower increase of the error in the phase velocity with frequency. Frequently such schemes are called nonstandard FDTD method [756–758]. Reduction of the dispersive errors also could be achieved by the introduction of the artificial anisotropy [759]. A general methodology to develop FDTD methods with controlled order of accuracy and dispersion (wave-equation-finite-difference-time domain (WE-FDTD)) that includes as special cases numerous known schemed was recently proposed by Finkelstein and Kastner [760]. In a discontinuous Galerkin time domain (DGTD) method numerical solutions are allowed to be discontinuous across element interfaces while numerical fluxes are defined in such a way as to provide local and global conservation of physical quantities [761]. A local spectral time domain method (LSTD) relies on the discrete singular convolution algorithm. An analysis of this method shows that it provides an extremely low requirements for the acceptable accuracy of computations down to approximately two grid points per wavelength [762]. FDTD method was used for the analysis of subwavelength diffractive optical elements by Mirotznik et al. [763]. Recent advances in FDTD/FETD methods have been reviewed by Govan et al. [699] and Teixera [744]. An equivalent circuit (EC) FDTD method has been recently derived by Rennings et al. [764] that is applicable to the analysis of surface plasmons coupling between two metal–dielectric interfaces, allowing dispersive materials, including ones described by either the extended-Drude or combined Drude/Lorentz permittivity function.

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An efficient approach to CEM in general and problems related to the light propagation in the semiconductor structures is based on the separation of the governing equations (using differential form formulations) into topological matrix-free equations and constitutional material equations pioneered by Tonti [765]. This approach is free from the numerous artifacts encountered in traditional approaches and is naturally extended to the higher-order methods [766–768]. A.4. Boundary conditions The essential element of FDTD approach to the solution of the outer problems is the artificial restriction of the computational domain by applying absorbing boundary conditions that prevent spurious reflection [769]. Such boundary conditions usually are based on one of the strategies [770]:

• mode annihilating (frequently using far-field expansions) or • one-way equation approximation that permit outward propagating waves to exit computational domain in such a way as if is of infinite extent. Mur was the first to propose such conditions in 1981 [771]; an important class of absorbing boundary conditions named perfectly matched layer (PML) was introduced by Bérenger [553] and later extended to the three-dimensional case [772]. Technically PML is rather an absorbing region rather than a boundary condition in the classical sense. The PML could be reformulated as a complex stretching (analytic continuation) on the spatial coordinates that forms a basis of its extension to the case of curvilinear coordinates and anisotropic media. In such extensions the normal coordinate to the grid boundary is considered. The field in the buffer region obeys modified Maxwell equations that map the original propagating field onto the exponentially decaying ones [744]. Since the fields inside the PML do not obey the classical Maxwell equations, one sometimes speak about non-Maxwellian PML. Maxwellian PML is used to refer to the dual formulation where the PML corresponds to an artificial anisotropic dispersive media that could be described by diagonal tensor constitutive relations. Other modifications of PML are known such as anisotropic-medium (uniaxial) (U)PML and complex frequency shifted (CFS)-PML [773]. Far-field radiation. Absorbing boundary conditions, however, do not allow one to determine the far-field characteristics of radiation. The known near-to-far-field transformation is based on the plane wave spectrum representation of fields (NFFT) involves numerical conversion between frequency domain and time domain representations [774], frequently being implemented using Fourier transform, for example, in calculation of the angular spectrum [715]. Lee et al. [571] described a procedure for computation of the far-field radiation power per solid angle (Nθ , Nφ , Lθ , Lφ are far fields represented in spherical coordinates, η is the characteristic impedance of air)

η P (θ , φ) = 8λ2

 2  2    Lθ  Lφ  Nθ +  + Nφ −    η η

on the basis of tangential field components (Ex , Ey , Hx , Hy ) on the specified surface near the LED structure (‘‘detection surface’’) obtained during FDTD computations and stored for each time step using two-dimensional Fourier transformation. A version of NFFT that can be directly incorporated into FDTD simulation has been proposed recently by Capoglu [775]. It is, unfortunately, rather mathematically involved, expressing the time domain electric field as a superposition of integrals of the time domain electric and magnetic surface currents and auxiliary functions (different for different polarizations) related to the so-called time domain transmission line Green functions. This approach is also very demanding numerically, for example, 6 or 10 arrays updated each time step should be stored for the cases of free space and lossless multilayered media, respectively, to be used after the FDTD computation to determine the far-field waveforms. Still more popular approach to compute the far-field pattern of the LED emission is the ray tracing (see, for example, [776]). A.5. Other CEM methods A.5.1. Transmission line modelling Transmission Line Modelling (TLM) method is based on the representation of the materials parameters of the media using lumped components and the equivalence between field variables and circuit parameters. The computational domain is discretized into sections of transmission lines, connected with each other at nodes. These TL sections are represented by the combination of equivalent circuit components such as series resistance R, series inductance L, shunt admittance G and shunt capacitance C , all per section of length ∆x, where R and G describe losses in the transmission line. Traditional approach to apply TLM method to the multidimensional problems uses Cartesian grids; however, a greater flexibility for two-dimensional problems could be attained with triangular TLM algorithm that exploit the local expansion in terms of cylindrical field harmonics for the mapping of the physical fields onto passive circuit components [777]. Absorbing boundary conditions devised for FDTD method (in particularity, PML) are applicable in TLM as lateral boundary conditions [778]. A.5.2. Plane wave expansion method (PWEM) This method is based on expanding the periodic functions in appropriate Fourier series and inserting the expansions into the wave equation. The dielectric function can be represented in Fourier space using either the direct method by

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computing the Fourier coefficients of the inverse dielectric function 1/ε(⃗ r ) or by the so-called HCS method (after Ho, Chan and Soukoulis) [471] based on Fourier transforming ε(⃗ r ). Since matrix inversion and Fourier transform commute, the results obtained by these methods should coincide for a complete (infinite) set of plane waves. For finite set of plane waves the convergence rate of HCS method was found greater [471]. A.5.3. Plane wave admittance method (PWAM) This method is an example of a frequency domain approach. It combines method of lines and PWEM aiming at development an efficient approach to the analysis of structures with large number of layers [465,695] for which such methods as FDTD are not efficient because of the nonlinear dependence of the computational work on the number of layers. Electromagnetic field is represented as a weighted sum of orthogonal basic functions as in PWEM. Two main elements of this method are semi-analytical relations for a single layer and a stable procedure of the admittance transfer for connecting separate layers into a single entity. PWAM is applicable to anisotropic materials if both permittivity and permeability can be represented as diagonal tensors. The weak point of PWAM that is most suitable for propagating waves is the large number of plane wave to accurately describe localized states. The are two ways to represent the dielectric function ϵ(⃗ r ) in Fourier space: (1) directly compute the Fourier coefficients of the inverse dielectric function 1/ϵ(⃗ r ) or (2) first perform Fourier transforming. The resulting complete (infinite) set of plane waves must coincide, since matrix inversion and Fourier transformation commute [471]. In practice, however, the latter approach is preferable due to the greater convergence rate. A.5.4. Time domain integral equation method Time domain integral equation (TDIE) method is based on the use of the Green function of the form G(ζ , ζ ) =

1 1

′

4π R

δ



R

 − (t − t ) , ′

ν √ = c / εb is the velocity of light in the background medium with the dielectric constant εb and R =  (x − x′ )2 + (y − y′ )2 + (z − z ′ )2 that could be used to define the vector potential to the electric field as the integral over where ν

the domain. TDIE has two main advantages:

• this method requires discretization of only those regions of the computational domain that contain material with properties different from the background material that frequently results in considerable memory savings;

• since radiating fields are treated analytically, there is no need in numerical absorbing boundary conditions. TDIE algorithms (frequently a method of moments [779] is used) are found to provide better than FDTD ones accuracy in the time-depend problems in the vicinity of the jump permittivity variation [689]. The problem in using this approach is related to the weak singularity that exists when the observation point coincide with the source location that is resolved by the introduction of a small spherical exclusion region and analytic calculation of the contribution of this region to the total integral [690]. Similar technique — Green tensor in the frequency domain was used by Pieruccini et al. [780] to study near-field light emission from arbitrary nano and microstructures. Green tensor was also used by Greiner and Martin to compute light emission and propagation in OLEDs [781]. A.5.5. Transfer matrix method This method requires separate calculations for TE and TM waves [365]; it is reported to be numerically unstable for multilayer media with large layer thickness [782]; Transfer matrix method is based on writing time-harmonic Maxwell equations in terms of (ω, ⃗ k) space as

⃗k × E⃗ = ωB⃗ ⃗k × H⃗ = −ωD ⃗ and using approximations of the form kx ≈ (−ia)−1 (exp(ikx a) − 1) and similar for other components, where a is the unit length of a discretization mesh in x direction. TMM solves Maxwell equations on cubic lattice (a, b, c ). Propagation in, for example, z direction is described, after Fourier transforming back into the real space, as F (⃗ r + c) =

−

Tˆ (⃗ r, ⃗ r ′ )F (⃗ r ′)

⃗r ′

where F (⃗ r ) = (Ex (⃗ r ), Ey (⃗ r ), Hx (⃗ r ), Hy (⃗ r ))T .

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Since ⃗ r and ⃗ r ′ belong to the same layer (z = z ′ ), a layer-by-layer computation of the field propagation in z direction is possible. A modification of the TMM based on the time-reversal symmetry was proposed by Guida et al. [783]. When an arbitrary inhomogeneous refractive index profile is considered, the differential-transfer-matrix method (DTMM) [784,785] is preferable. The field variation in this method is represented as A(x) = A+ (x) exp(−ikx (x)x) + A− (x) exp(ikx (x)x). The results obtained by this method are in agreement with exact solutions of some limiting cases [785]. The modified differential-transfer-matrix method (MDTMM) [786,787]) exploits a more general representation of field variables in the form



A(x) = A+ (x) exp −i

∫

x

kx (x′ )dx′



  ∫ x kx (x′ )dx′ . + A− (x) exp i

0

0

Note that modified form is in accordance to the Feynman’s path integral, where the phase variation of propagating waves follows the integral of the wavenumber kx (x) [787]. TMM is a general numerical method that can be used for a number of problems of different nature, for example, for to solve time independent Schrödinger equation [788]. A.5.6. The finite difference beam propagation method This method (FD-BPM) is used to compute the steady state evolution of the transverse field as it propagates along the structure in the axial direction via the solution of the Helmholtz equation

∇ 2 ϕ(x, y, z ) + k20 n2 ϕ(x, y, z ) = 0, and is based on the assumption that the electric and magnetic fields vary slowly along the chosen direction of propagation. The method relies on the slow continuous variation of the refractive index and is the most efficient for cases when it is not necessary to determine the propagation coefficient, only field distribution should be found. The refractive index is split into a constant and a cross-section dependent part n(x, y) = n0 + δ n(x, y). BPM involves propagation in both Fourier- and real space. The field is split into the envelope and a term responsible for the propagation and only the former is modeled

ϕ(x, y, z ) = ϕ(x, y) exp(−iβ z ), where β = k0 n0 , k0 = ω/c and n0 is the background refractive index. The difficulty inherent to BPM is the need to solve a large matrix problem at each propagation step. The so-called paraxial BPM [701] exploits the Taylor series expansion √ 1 1+L≈1+ L to simplify the governing equation. There is a variant of this method called a wide-angle propagation method [789] which is frequently used to model tapered laser cavities [699]. The intention of this extension of the BPM method is to override the constraints on the angular range of the principal propagation distance [790]. The wide-angle scheme is based on the equation for the slowly varying envelope that in two-dimensional case is written as

∂ 2ϕ ∂ 2ϕ ∂ϕ − j2k n + k20 (n(x, z )2 − n20 )ϕ = 0. + 0 0 ∂ z2 ∂z ∂ x2 This equation is factorized as



   √ √ ∂ ∂ + jβ 1 + X · − jβ 1 + X ϕ = 0, ∂z ∂z

(A.3)

where X =

1

β

[

] ∂ 2 2 2 + k ( n ( x , z ) − n ) . 0 0 ∂ x2

When only forward propagating waves are considered, Eq. (A.3) is reduced to

√  ∂ϕ = −jβ 1 + X − 1 ϕ. ∂z √ Different variants of the wide-angle BPM method could be obtained by the choice of the approximation for the operator 1 + X , the most popular being one based on the rational Padé approximation [791] that has the form √ Nm (X ) 1+X −1= , Dn (X ) where Nm (X ) and Dn (X ) are polynomials of degree m and n, respectively. An improved version of wide-angle BPM for nonlinear waveguides was developed by Alcantara et al. [792].

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A.5.7. Effective index method This method is applicable to devices that can be split into several regions that depend on z only. Similar to FD-BPM, effective index method is based on decomposition E (⃗ r ) = E (⃗ rt )φi (z ) and assumption that φi (z ) ≈ φj (z ) ≈ φ(z ), allowing to separate 3D equation into 1D and 2D equations. One of the modification of this method is known as ‘‘effective frequency method’’ differs in two aspects: (1) effective indices are replaced by effective resonance frequencies and (2) the method could be used for dispersive materials. There are also hybrid methods [793], for example, finite element finite difference time domain method [794], methods of moments combined with physical optics [795] or finite elements [796], FEM adjusted to integral equation approach [797]. Appendix B. Acronyms AFM Atomic force microscopy AIPP Artificial inverted polygonal pyramid AR Anti-reflection AS LED Absorbing-substrate light emitting diode BMR Buried micro-reflector CEM Computational Electromagnetics DBR Distributed Bragg reflector DGTD Discontinuous Galerkin time domain DOS Density of states DTMM Differential-transfer-matrix method EBL Electron blocking layer EQE External quantum efficiency EG-FDTD Embedded-grid finite difference time domain FCLED Flip-chip light emitting diode FDTD Finite difference time domain FEMTD Finite element time domain method FD-BPM Finite difference beam propagation method GRIN Graded index GSS-LED Geometric sapphire shaping LED IQE Internal quantum efficiency IR Infrared ITO Indium–tin oxide LED Light emitting diode LEE Light extraction efficiency LSTD Local spectral time domain MCLED Microcavity LED MDTMM Modified differential-transfer-matrix method MM Mode matching MQW Multiple quantum well NFFT Near-to-far-field transformation ODE Optical diffractive element ODR Omnidirectional reflector OLED Organic light emitting diode QW Quantum well PBG Photonic bandgap PC Photonic crystal PCS Photonic crystal slab PDE-LED Periodic deflector embedded structure LED PL Photoluminescence PML Perfectly matched layer POF Polymer optical fiber PQC Photonic quasicrystal PRS-LED Photon recycling semiconductor light emitting diode PWAM Plane wave admittance method PWEM Plane wave expansion method RC2LED doubly resonant cavity light emitting diode RCLED Resonant cavity light emitting diode RCWA Rigorous coupled wave analysis

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RMS Root-mean-square RSLED Reflective submount LED SDI-LED Sidewall-deflector-integrated LED SFDTD Segmented finite difference time domain method SP Surface plasmon SPP Surface plasmon polariton TDIE Time domain integral equation TF Thin film TIP Truncated inverted pyramid TIR Total internal reflection TIS Total integrated scatter TLM Transmission line modelling TMM Transfer matrix method TOLED Top emitting organic light emitting diode TS LED transparent-substrate light emitting diode UV Ultraviolet WMOLED Weak microcavity organic light emitting diode.

Appendix C. Nomenclature Df The free-photon density of states per unit volume E⃗ E⃗ (⃗ r0 , ω) f FP ⃗ H ⃗j(⃗r0 , ω) Kλ Vλ P krad knon Veff Z (ω)

α δ ∆n Γ ε ηv ηext ηel ηint ηinj ηextr ηe λ φi Φv Φe θi τ ⟨f |d⃗ · E⃗ |i⟩ ϱ(h¯ ω) mc = L/(λ/2n) Φ ω ∆ω ωP

Electric field strength The Fourier transform of the time trace of the electric field E⃗0 (⃗ r0 , t ) Filling factor Purcell enhancement factor Magnetic field strength The frequency spectrum of the dipole source Spectral luminous efficacy Spectral luminous efficiency Total electrical input power Radiative recombination rate Nonradiative recombination rate Effective volume Penetration depth Absorption coefficient The RMS of the surface roughness The refractive index difference between the surface material and the ambient medium Damping constant The dielectric constant Luminous efficacy External quantum efficiency Electric efficiency Internal quantum efficiency Injection efficiency Light extraction efficiency Power efficiency (or radiant efficiency) Wavelength Azimuthal angles of incidence Luminous flux Radiant flux Polar angle of incidence Spontaneous emission decay constant The dipole emission matrix element The photon density of states (DOS) Cavity order Phase shift upon reflection off the mirror The frequency of transition The frequency width of the resonance Plasma frequency.

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