Photonic Crystal LEDs- Designing Light Extraction

262 Laser & Photon. Rev. 3, No. 3, 262–286 (2009) / DOI 10.1002/lpor.200810053

Abstract Photonic crystals (PhCs) have attracted much atten-

tion during the last decade as a solution to overcome the low

extraction efficiency of as-grown light-emitting diodes (LEDs).

In this review we describe the underlying physics and summa-

rize recent results obtained with PhC LEDs. Here, the main

focus is on diffracting PhC. In order to quantify the benefit from

the incorporation of PhCs for diffracting light a comparison by

simulations between a PhC LED and a standard state-of-the-

art LED is carried out. Finally, the impact of the PhC on the

LEDs emission characteristics will be discussed with respect to

etendue-limited applications.

Azimuthal emission pattern of an InGaN/GaN LED with hexag-

onal PhC at a single wavelength. Background: SEM (scanning

electron microscope) image of the PhC structure.

© 2009 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim

Photonic crystal LEDs – designing light extraction

Christopher Wiesmann1,2,*, Krister Bergenek1, Norbert Linder1, and Ulrich T. Schwarz2

1 Osram Opto Semiconductors GmbH, Leibnizstr. 4, 93055 Regensburg, Germany2 Department of Physics, University of Regensburg, 93040 Regensburg, Germany

Received: 15 September 2008, Revised: 7 November 2008, Accepted: 13 November 2008

Published online: 29 December 2008

Key words: Photonics crystal, light emitting diode, semiconductor optoelectronics, diffraction.

PACS: 42.70.Qs, 78.67.-n, 85.60.Bt, 85.60.Jb, 85.60.Pg

1. Introduction

From the very beginnings of light-emitting diodes (LEDs)in the 1960s a continuous increase in their efficiency wasachieved. Due to this enhancement, LEDs were able to findtheir way into a wide field of applications ranging from theirroots as indicators and display signs to high-performanceapplications like automotive head lamps or projection sys-tems. Furthermore, owing to their efficiency, reliability, lowpower consumption and long lifetime LEDs are supposedto be the upcoming light source for general lighting [1].

One precondition for the introduction of LEDs into thesenew application areas was overcoming the low extractionefficiency of light from as-grown LEDs due to total internalreflection. In recent years numerous publications [2–24] in-vestigated the application of photonic crystals (PhCs) for ad-dressing this issue. Enhancements ranging from 1.5 [8, 21]to 2.5 [9] compared to unstructured references reveal thatPhCs have the potential to increase the extraction efficiencyof LEDs. However, a commercial application of PhCs willadditionally depend on the properties of state-of-the-art de-vices. First, the performance of PhC LEDs has to competewith standard LEDs already addressing the problem of total

* Corresponding author: e-mail: [email protected]

© 2009 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim

Laser & Photon. Rev. 3, No. 3 (2009) 263

internal reflection, like volume emitters or thin-film LEDs.The latter almost reaches 80% extraction efficiency [25,26].But also their manufacturing has to be as marketable as thatof standard techniques. However, PhCs offer advantagesbeyond the pure extraction enhancement. Due to their pe-riodicity it is possible to shape the emission profile of aLED [8]. This could be of great advantage for applicationsbased on imaging, like headlamps, or nonimaging optics,like projectors, as their overall efficiency is etendue lim-ited [27,28]. These applications accept light only within aspecific solid angle, thus it is preferable to emit as muchlight as possible into these directions. Here, apart fromthe total emitted radiant flux, measured in [W], the radiantemittance, the radiant intensity and the radiance play an im-portant role [29]. The radiant emittance describes the powerper area [W/m2] emitted from the LED’s surface. The ra-diant intensity gives the power per solid angle [W/sr] andthe radiance gives the radiant intensity per effective sourcearea [W/sr/m2]. Furthermore, PhCs have been proven topreserve the polarization of the extracted light [20]. If anet overall polarization could be achieved in this way, theperformance of LED back-lit LCD displays could be sig-nificantly enhanced.In general, the benefits of PhC for LEDs stem from

their influence on the dispersion relation of light, i.e. bandfolding due to Bloch’s theorem. However, due to technolog-ical limitations altering the dispersion of light within a PhCLED is restricted. In order to achieve reasonable electri-cal performance and homogeneous current distribution theepitaxial structure has to contain sufficiently thick unperfo-rated current spreading layers. Thus, most PhC LEDs onlyincorporate shallow etched holes. In this case of weak PhCsthe problem, in general three-dimensional, can be separatedinto a 2D+1D problem. The two-dimensional part coversthe lateral properties of the PhC and the one-dimensionalpart consists of the vertical layer composition of the LEDincluding the etch depth of the PhC. This separation enablesa clear insight into the mechanisms determining the extrac-tion efficiency and gives a route for optimizing PhC LEDs.Hence, we focus in this review on weak PhCs. The case ofstrongly coupled PhCs, i.e. PhCs etched vertically throughthe whole LED, has to be addressed separately. Here, themost popular property of PhCs, the photonic bandgap, hasto be taken into account. But apart from the electrical issuesa loss of active area equal to the area of holes has to beovercome for achieving high absolute brightness.

Apart from enhancing the extraction efficiency of LEDs,PhCs could also be incorporated into the LED as omnidi-rectional mirrors for decreasing the absorption losses. Ad-ditionally, the use of metallic gratings in the vicinity of theLED’s active region enables the use of surface plasmonpolaritons for enhancing the efficiency of light generation.In the upcoming section we will review the basics on

LEDs and discuss the problem of light extraction fromunstructured LEDs along with traditional approaches to im-prove light extraction. These approaches are the benchmarkfor PhC LEDs. In Sect. 3 first the influence of PhCs on thelight propagation inside a LED will be investigated and

from this the main mechanisms for enhancing the perfor-mance of LEDs will be derived.

Here, two regimes can be distinguished: the weakly andthe strongly coupled PhC. The former will be scrutinizedin Sect. 4 with a detailed investigation of the propertiesof the PhC, the vertical layer structure of the LED andsetups of PhC LEDs. Sect. 5 deals with the latter. Apartfrom the direct use of PhCs for light extraction its applica-tion as an omnidirectional mirror will be summarized inSect. 6.1. In Sect. 6.2 we will focus on the specific use ofmetallic PhCs in combination with surface plasmons forenhancing the internal quantum efficiency. In Sect. 7 themost widely used patterning techniques will be briefly sum-marized for manufacturing of PhCs for the visible range ofthe spectrum. Thereafter, in Sect. 8 a comparison betweena state-of-the-art LED and a PhC LED will be carried outin order to investigate the potential of PhC LEDs by usingthree-dimensional finite-difference time-domain (FDTD)simulations [30]. Furthermore, a detailed investigation onthe use of PhC LEDs in etendue-limited applications willbe given in Sect. 9.

2. LEDs – basics

In general, LEDs cover the whole range of visible wave-lengths. For the near-UV to green, gallium-nitride (GaN)-based LEDs are used, where the emission wavelength is de-fined by the amount of indium (In) content within the activeregion – the more In the longer the emission wavelength.For yellow up to the red emission the quaternary system(AlxGa1−x)yIn1−yP is common. The infrared part of thespectrum is opened up by the AlxGa1−xAs system. Butregardless of the emission wavelength, the overall powerconversion efficiency or wall-plug efficiency of a LED isone of the most important properties and is given by emit-ted radiant flux with respect to electrical input power, bothquantities measured in Watts [W]. Four main loss mech-anisms that cause heating of the LED determine its wall-plug efficiency

ηwall = ηel · ηinj · ηIQE · ηextr︸︷︷︸ηEQE

. (1)

Here, ηel describes the electrical efficiency and the externalquantum efficiency ηEQE is given by the product of theinjection efficiency ηinj, the internal quantum efficiencyηIQE and the extraction efficiency ηextr.The electrical efficiency ηel is determined by ohmic

losses that come with applying voltage across the p-n junc-tion due to the resistance of the contacts and the epitaxiallayers. For the nitrides, losses due to piezoelectric barri-ers are included, too. The injection efficiency ηinj takesinto account the efficiency of capturing the electrons andholes within the p-n junction (the active region). The in-ternal quantum efficiency ηIQE of photon generation fromelectron-hole pairs depends on the ratio between the ra-diative recombination rate Λrad (a photon is generated in

www.lpr-journal.org © 2009 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim

264 C. Wiesmann, K. Bergenek, et al.: Photonic crystal LEDs – designing light extraction

the active region) and the nonradiative recombination ratesΛnrad (no photon is generated):

ηIQE =Λrad

Λrad + Λnrad. (2)

For instance, nonradiative recombination can be caused bydefects within the semiconductor, the so-called Shockley-Read-Hall recombination [31, 32], or by Auger recombi-nation [33]. Finally, not every photon is emitted into theambient medium but gets absorbed inside the LED (ηextr).Experimentally, the external quantum efficiency is definedby the number of emitted photons divided by the num-ber of injected electrons and serves as an upper bound-ary for the extraction efficiency with the assumption ofηIQE = ηinj = 1.The internal quantum efficiency is mostly determined

by the epitaxial quality and the electronic band engineeringof the active region. Nevertheless, it is not an intrinsicproperty of the semiconductor but is related to the opticalenvironment. Both the photon density of states and thelocal electromagnetic field amplitudes alter the radiativerecombination rate – this is called the Purcell effect [34] –according to Fermi’s Golden Rule

Λrad (ω, r) =2π�〈d ·E (r)〉2 ρ (ω) , (3)

with d the electric dipole of the electron-hole transition,E(r) the local electric field strength, and ρ(ω) the localdensity of photon states. Due to the dispersion relation oflight ω(k) the radiative recombination rate is consequentlydependent on the propagation direction of light. As we willsee during this review, both the extraction efficiency andthe internal quantum efficiency can be affected by pho-tonic crystals.

The main limitation for an as-grown LED comes fromthe low extraction efficiency due to the high refractiveindex of the semiconductor (nSC = 2.5–3.5) relative tothe ambient medium (typically air with namb = 1). WithSnell’s law (nSC sin θSC = namb sin θamb) in mind, onlylight with incident angle smaller than the critical angleθSC < θc = asin(namb/nSC) can radiate into the ambi-ent medium, whereas the remaining light is total internallyreflected and never escapes from the structure as depictedin Fig. 1. Snell’s law is equivalent to the conservation of theinplane k-vector length β at the interface as can be seen bymultiplying with the vacuum k-vector length k0 = 2π/λ0.Therefore, only light with an inplane k-vector smaller thanβ < k0nSC sin θc = nambk0 can radiate into the ambientmedium. Light with β > nambk0 is called evanescent inthe ambient medium as the corresponding angle θamb isimaginary. The inplane k-vector is related to the total k-vector k by k2 = (nk0)2 = β2 + γ2, with γ the k-vectorcomponent perpendicular to β.

For an isotropic light source integrating over all possibledirections, while neglecting losses due to Fresnel reflection

Figure 1 (online color at: www.lpr-journal.org) Only light

within a narrow escape cone of θSC < θc can radiate from the

semiconductor to the ambient medium. Along with the different

k-vectors also their projection on the plane parallel to the interface(β = nSCk0 sin θSC) is illustrated.

at the boundary (i.e. perfect antireflection coating)

ηextr =12

∫ θc

0

sin θdθ (4)

reveals an extraction efficiency of typically 2% for Al-GaAs/AlGaInP material systems with nSC = 3.5 and 4%for GaN-based ones with nSC = 2.5 to air (in the followingthe ambient medium will be assumed as air, namb = 1). Soeven with 100% internal quantum efficiency the externalquantum efficiency of the LED is limited to 4% in the bestcase. An early idea for enhancing the extraction efficiencyis to encapsulate the LED with a high-index hemisphericaldome [35]. In order to obtain a large critical angle the en-capsulation should be index matched to the semiconductor.In commercial devices epoxy is widely used with a refrac-tive index of n ≈ 1.5 as it is easy to process and remainstransparent over the lifetime of the LED. But still 90% forthe nitrides and 95% for the arsenides/phosphides of thegenerated light are trapped within the semiconductor.Another way for enhancing the extraction efficiency

uses five of the six facets of the LED chip for extractinglight, as shown in Fig. 2a. For achieving reasonable efficien-cies, highly transparent substrates or window layers haveto be used in order to avoid absorption losses. Thus, it ispossible to obtain 10% and 20% extraction efficiency toair, respectively. The limitation of the extraction efficiencyin this case stems from the rectangular cross section ofthe LED. Hence, the total internally reflected light cannotchange its incident angle upon any of the facets. There-fore, additionally tilting of the angle between these facetsenables high extraction efficiencies, e.g. of up to 50% forGaN-based LEDs [36] and 55% for AlGaInP LEDs [37],as rays totally internally reflected at one facet have thechance to escape from the LED at another facet as the in-cident angle is changed; see Fig. 2b. However, the overallsize of these LEDs is limited as only the facets redistribute

© 2009 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim www.lpr-journal.org


Figure 2 (online color at: www.lpr-journal.org) Summary of

standard LED setups for overcoming the low extraction efficiency

along with images of real devices under operation. (a) Five facets

of the chip contribute to the extraction of light. The image shows

a GaN-based LED grown on SiC substrate. (b) A further im-

provement of a) with tilted angles. The image is taken from a

GaN-based LED grown on sapphire substrate. (c) A typical setup

of a thin-film LED with a rough surface and back-mirror (gray

area). The image shows a red-emitting AlGaInP thin-film LED.

The bars for current injection are visible as horizontal black lines.

(d) Schematic illustration of the altered internal emission due to

the mirrors of a RCLED. The extraction cone is depicted by the

straight black lines.

the light and the extraction competes with the absorptionlosses within the chip. Furthermore, in this setup the lightoutput is spread over the whole chip surface, implyinga volume-emitting LED along with strong side emission.Therefore, special packaging has to redirect the light intothe forward direction.

Another way to break the paths of the light is by incor-porating a scattering mechanism on top of the LED like therough surface shown in Fig. 2c [38, 39]. This has two ad-vantages: light can escape partially from the structure moreor less independent of its incident angle due to scattering atthe rough surface and the reflected light gets redistributed(similar effect to the tilted facets). With the incorporation ofa mirror at the opposite side of the LED the reflected lighthas several chances to escape from the LED through onesingle facet after hitting the mirror. These so-called thin-film LEDs are processed by bonding a second substrate,for instance Ge, Si or GaAs, on top of the last epitaxiallayer. Between the two a metallization like Au, Ag or Alforms the mirror. The removal of the primary substrate is ac-complished by either wet-etching in the AlGaInP/AlGaAsmaterial system [40] or laser lift-off in the GaNmaterial sys-

Figure 3 (online color at: www.lpr-journal.org) Required ex-

traction length 1/κ as a function of the absorption length 1/α inorder to achieve 50% (black), 70% (blue), or 90% (red) extrac-

tion efficiency.

tem [41,42]. Thin-film LEDs have shown high extractionefficiencies of 75–80% for blue GaN-based LEDs [25, 26]and 50% external efficiency for AlGaInP [43]. Both valuesare obtained from encapsulated chips. Secondly, this effi-ciently generates a surface-emitting LED with an increasedradiant emittance compared to a volume-emitting LED [29]as the same amount of radiant flux or even more is emit-ted from a drastically reduced area. A further advantageespecially of interest for projection applications – wherelarge chip areas are favorable – is the scalability of surfaceemitters [44]. Besides surface roughening there are someadditional solutions for scattering the light internally, likepatterned substrates [45] or buried microreflectors [44].In general, the extraction mechanisms have to compete

with absorption losses within the epitaxial layers, at the mir-ror, or at the contacts. Thus, the extraction efficiency reads

ηextr =κ

κ+ α, (5)

with κ the extraction coefficient due to the extraction mech-anism and α the absorption coefficient. Any redistributionof internally propagating light, either surface texturing ortilted facets, results in 100% extraction efficiency in thelimit of vanishing absorption. Therefore, the route to high-performance LEDs is to reduce the absorption within theLED to a negligible amount while simultaneously pullingthe light out of the LED as fast as possible (see Fig. 3).

Apart from these redistribution techniques, in resonant-cavity LEDs (RCLEDs) [46–48] as much light as possibleis generated internally within the extraction cone by meansof interference in order to obtain high extraction efficien-cies, as sketched in Fig. 2d. The vertical cavity of the LEDis tailored in a way that the radiative recombination rateaccording to Eq. (3) is increased within the extraction coneand decreased for the trapped light. The formation of theinterferences is typically achieved by embedding the ac-tive layer between two mirrors (metallic mirrors, Braggreflectors, or just the semiconductor/air interface). Withthis setup extraction efficiencies of 22% (GaN materialsystem) [49], 23% (AlGaInP material system and encap-sulated) [50], and 30% (AlGaAs material system) [51]



Figure 4 (online color at: www.lpr-journal.org) (a) Dispersion of the

two fundamental TE modes for a slab of thickness 0.5a, with a anarbitrary length within the system and the slab’s refractive index of

n = 2.5. The steep black line illustrates the light line and the secondless steep line the semiconductor light line. The dispersion of the funda-

mental TE-polarized guided mode and the following higher-order mode

are shown in red and blue, respectively. The horizontal dashed line at

ωa/2πc = 0.55 corresponds to the kx-ky-diagram shown in (c). The

straight vertical line depicts the edge of the 1. BZ in ΓM direction of

the hexagonal lattice used in (b). (b) Dispersion relation of the slab in (a)

with a hexagonal lattice with lattice constant a in the limit of infinitelyweak perturbation. (c) Illustration of the folding according to Bloch’s the-

orem for the frequency corresponding to the dashed horizontal line in (a)

and (b). The red and blue circles have radius equal to the inplane k-vectorobtained from a) and the centers are located at the reciprocal lattice points (black dots). The thin gray lines indicate the BZs and the

thick gray lines enclose the irreducible BZ with the symmetry points Γ ,M , andK. The evolution of the intersection of the red and bluecircles with the edges of the irreducible BZ gives the dispersion relation as shown in b). The black circle represents the light line and

encloses the extraction cone.

have been demonstrated. But as the layer composition ofRCLEDs is optimized for a specific emission spectrumof the active layer, their performance is heavily tempera-ture dependent [52]. Typically, with increasing temperaturethe emission of semiconductor-based active layers tends toshift to longer wavelengths (due to a decreased electronicbandgap) causing a temperature-dependent mismatch be-tween the emission and the surrounding cavity. For thenitrides the wavelength shifts to shorter wavelength withincreasing current density due to screening of the piezo-electric fields.

3. Photonic crystals

A PhC1 in general is a one-, two-, or three-dimensionalperiodic arrangement of materials with different refractiveindex, where the characteristic length scale is of the order of

1 For a more detailed description of PhCs the reader is referred

to [53–56].

the wavelength. Therefore, similar to the electronic bandsin quantum mechanics that are generated by the periodicpotential offered by the atoms in a crystal, a periodic re-fractive index alters the propagation behavior of photonsin a photonic crystal. The propagation properties of pho-tons within any system are summarized by its dispersionrelation ω(k) relating the allowed photon energies to thecorresponding propagation directions. In free space the dis-persion relation simply reads ω = k0c. In Fig. 4a this socalled light line for free-space propagation (steep blackline) is plotted as a function of the inplane k-vector lengthβ. This is convenient for systems with either continuous ordiscrete translational invariance in the lateral directions2

2 Please note the axes labelling. As Maxwell’s equations are

invariant under scaling – there is no fundamental length scale

like Bohr’s radius in quantum-mechanics – it is convenient to

use the reduced frequency ωL/2πc = L/λ0 and the reduced

k-vector kL/2π with L a characteristic length of the system.Therefore, photons with wavelength λ1 propagating in a system 1

with a characteristic length L1 behave in exactly the same way as



x and y. According to Fig. 1 light with a given frequencyand inplane k-vector β < k0 is able to propagate in air. Forinstance, the part of the dashed horizontal line in Fig. 4a leftto the intersection with the light line contains all propagat-ing states at the reduced frequency ωa/2πc = 0.55, wherea is some characteristic length defined later. In contrast tothis, light with β > k0 is evanescent in air but propagat-ing inside the semiconductor as long as β < nSCk0. Thissecond boundary is given by the semiconductor light lineω = k0c/nSC. Light with β > nSCk0 is evanescent in thesemiconductor, therefore nonpropagating, and is not takeninto account. This is valid as long as there is no materialwith higher refractive index present or material supportingsurface modes like surface plasmons.

In the case of a semi-infinite air slab on top of a semi-infinite semiconductor slab an infinite number of states withk0 < β < nSCk0 exist. By truncating the semiconductorslab to a finite thickness and introducing a material withlower refractive index, e.g. air, at the bottom, only a discretenumber of states are allowed, depending on the thicknessand the refractive index of the slab. This is similar to thefinite number of allowed electronic states in a potential wellof finite depth. The dispersion of these so-called guidedmodes is shown in Fig. 4a for TE-polarized light (electricfield parallel to the xy-plane also often referred to as s-polarization). The solution for this symmetric slab is givenfor example in [57], and a general solution for an arbitrarystructure can be found in [58]. At this point the readershould keep in mind that in such a vertical layer stack aguided mode with inplane k-vector β subsums an infinitenumber of degenerate modes with the same frequency butpropagating with different inplane directions β/β due thehomogeneity of the slab in the x and y directions. Wheneverit can simplify the treatment, this degenerate set of modesis referred to as a single mode.

After these general remarks on dispersion relations, letus introduce a periodicity offered by a PhC. For the sake ofsimplicity, we first deal with an artificial hexagonal latticewith lattice constant a; no refractive-index contrast definesthe periodicity. The resulting dispersion relation is oftenreferred to as free-photon dispersion. According to Bloch’stheorem the modes of the slab are folded at the edges of thefirst Brillouin zone (BZ). Typically, in a dispersion relationonly k-vectors along the edges of the irreducible BZ areconsidered representing the main directions of the lattice.The formation of the different bands is simply derived fromthe dispersion of the uncorrugated slab and the geometricalconsideration as shown in Fig. 4c. Starting from ω = 0 thedispersion for the corrugated slab is equal to that of theuncorrugated slab as the first BZ encloses the mode’s circlein k-space. As soon as the inplane k-vector of the mode islarger than the first BZ, e.g. the dashed blue circle around

photons with λ2 in a system 2 withL2 as long asL1/λ1 = L2/λ2.

But as the characteristic length is an arbitrary choice all spatial

dimensions in both systems have to satisfy the above equation.

Typically, in the case of PhCs the pitch a of the lattice is used asthe characteristic length L, i.e. L = a.

(0, 0) in Fig. 4c, circles from the neighboring BZ enter thefirst one. Due to Bloch’s theorem these are equivalent tothe mode starting from (0, 0). In the dispersion relationfor the corrugated slab the evolution of the intersections ofthe different mode circles with the irreducible BZ edgesare drawn. Now, the formerly guided mode lies above thelight line for reduced frequencies ωa/2πc > 0.38. There-fore, the periodicity enables this mode to be radiative in theambient medium and extracts guided light. An equivalentdescription of the process is given by Bragg’s law of diffrac-tion βd = βi + G, where the sum of the incident k-vectorβi and a reciprocal lattice vector G yields the diffractedk-vector βd. The reciprocal lattice vectorsG can be derivedanalytically or by Fourier transformation of the real lattice.As soon as βd < k0 some harmonics originating from afundamental mode with inplane k-vector βi are diffractedinto air.

The dispersion within the extraction cone is directlyaccessible from the spectrally and angle-resolved measure-ment of emitted light of a PhC LED [11]. The folding of theguided modes due to the hexagonal PhC results for a singlefrequency in a Star-of-David-shaped pattern, as shown inthe title figure. This pattern directly reflects the dispersionaround the Γ point, as seen in Fig. 4c. The measurementwas performed on a GaN-based LED with ∼ 850 nm thick-ness including a more than 400 nm deep PhC with 274 nmpitch and an ITO cover layer [23]. The diffracted modecorresponds to an inplane k-vector length of βi ≈ 2.4k0.

Reconsidering the basics of bandgaps for electrons insolid state physics, the two degenerate photonic states at,e.g., the M-point do have the same periodicity but theirfield profiles are shifted by half of the lattice constant. Arefractive-index contrast as depicted in Fig. 5 for a one-dimensional grating lifts the degeneracy as the modes “see”different refractive indices. The frequency of a mode readsω = kc/〈n〉 with 〈n〉 the average refractive index seen bythat mode. Thus, a mode mainly located within the high-index material has lower frequency. The correspondingband is often called the dielectric band (as the high-indexmaterial is typically a dielectric), whereas the band relatedto the mode mainly located in the low-index material (hav-ing higher frequency) is referred to as the air band.

Figure 5 (online color at: www.lpr-journal.org) Intensity pro-

files of two modes at the BZ edge in the case of a one-dimensional

grating. When the refractive indices of the green and orange re-

gions are different the degeneracy at the BZ edge is lifted.



An intuitive interpretation of this degeneracy-lift andthe associated shift in frequency is given in [59]. Considerfor instance a material with refractive index n that is per-turbed in some regions with a refractive index shift Δn.The related shift in frequency can be estimated in first orderto be

Δω

ω= −Δn

n·(fraction of

∫n2|E|2 in perturbed regions

),

(6)with E the electric field amplitude of the modes and ω thefrequency. This shift causes band bending at the symmetrypoints, resulting in a bandgap. Therefore, propagation ofphotons at this frequency and the corresponding k-vector isinhibited. If the propagation is inhibited for k-vectors in alldirections, a complete bandgap arises. Additionally to thebandgap, the band bending causes a change of the photonicdensity of states as the number of available k-vectors withina given frequency range is altered. As a consequence ofEq. (3) the spontaneous emission is enhanced for certainfrequencies and directions and suppressed for others.

As long as either the index shift is small or the depth ofthe PhC’s holes is small compared to the slab’s thickness,the band-bending effects can be neglected as the frequencyshift is tiny and the formation of the dispersion relationas shown in Fig. 4b is still valid, see for instance [60].Hence, from Eq. (6) naturally two extremes of PhC LEDscan be distinguished:

1. The PhC is a perturbation of the LED and acts as adiffraction grating but does not alter the photonic den-sity of states. This is valid in most PhC LEDs and wemainly remain with our treatment of PhCs in this limit.

2. The PhC introduces a large modulation into the LEDand band-bending effects cannot be neglected. There-fore, the density of photonic states can be altered dras-tically and bandgaps play an important role. This willbe discussed in Sect. 5.

In general, a PhC LED is a three-dimensional problem re-quiring the solution of Maxwell’s equations fully vectorialin three dimensions due to the feature size of the PhC. Here,the FDTD method [30] is widely used. To achieve reliableresults, the dimension of the simulated volume has to beof the order of the third power of extraction or absorptionlength, resulting in a tremendous demand on computationalresources. For the huge parameter space of PhC LEDs –covering different material systems, the vertical layer com-position, the PhC lattice itself, etc. – a general optimizationprocedure is hardly obtainable. On the other hand, methodslike scattering matrix algorithms [18, 61, 62] taking advan-tage of the periodicity of the PhC are more suitable forthe solution of PhC LEDs but are restricted to periodicalsystems. Nevertheless, these methods enable a clear insightinto the mechanisms of arbitrary PhC LED geometries. Asimplification to the whole three-dimensional problem ispresented in [63], where transmission and reflection at thePhC are treated rigorously but the coupling of the sourcewith the cavity is neglected.

Figure 6 (online color at: www.lpr-journal.org) Schematic for

a general PhC LED in the case of diffraction. (a) Thin-film LED

with back-mirror; (b) LED with transparent substrate. The inten-

sity profiles (black curves) of the different modes will be discussed

in Sect. 4.2. The unetched core thickness is t.

However, in the case of weak PhCs, a lot can alreadybe learned from considerations of the vertical LED compo-sition itself and the diffraction of the guided modes by theBragg condition.

4. Weak photonic crystals –diffraction of light

A general setup for this regime is shown in Fig. 6 for twocases, a thin-film LED with a back-mirror and a LED withtransparent substrate. The PhC is typically etched into thetop surface of the LED. Certainly, the PhC can be posi-tioned arbitrarily within the vertical LED stack as long asit is technologically feasible [16, 19]. In any case the gen-erated light can either escape from the structure directlywithout the help of the PhC (roughly 2–4% depending onthe refractive index of the semiconductor) or gets diffractedby the PhC according to Fig. 4c. As soon as the bands ofguided modes are folded above the light line they are ac-cessible by measuring the far field of the PhC LED [11].Hence, it is possible to prove the operating regime of thePhC LED, whether the PhC is weak or not.Fig. 7 shows a spectrally resolved far field of a PhC

LED [23] with the inplane k-vector length β = k0 sin θairas the x-coordinate instead of the far-field angle θair. Thefour lines intersecting the y-axis at ωa/2πc = 0.485 agree



Figure 7 (online color at: www.lpr-journal.org) Dispersion re-

lation measured in ΓM direction of a hexagonal lattice PhC

LED [11]. Data taken from the same LED as used in the title

figure [23]. The four lines crossing at ωa/2πc = 0.485 stemfrom diffraction of the same guided mode as sketched in Fig. 4b.

with the dispersion relation close to the dashed line inFig. 4b. In both cases the bands stem from folding of a sin-gle guided mode. The next higher-order branches intersectwith the y-axis at ωa/2πc = 0.515. As band bending orsplitting at the Γ -point is hardly observed, the PhC canbe considered weak and the dispersion is related to theband folding of the free-photon dispersion as shown inFig. 4 (see also [12]). Also, in the azimuthal far field asshown in the title figure no local bandgaps appear, but aStar-of-David-shaped diffraction pattern similar to Fig. 4c.

After the operation principle of this regime – diffraction– has been proven, the question arises, what LED structureand what PhC lattice should be chosen in order to obtainas much extraction as possible. As the weak PhC can behandled as a perturbation, we can separate the whole 3Dproblem into a 2D+1D problem that facilitates the inter-pretation. The two-dimensional part is represented by thetwo-dimensional PhC and describes the diffraction process.Here, the parameters of the PhC, i.e. the lattice type, thelattice constant and the filling fraction, have to be takeninto account. In contrast, the one-dimensional part arisesfrom the vertical layer stack of the LED and contains theetch depth of the PhC. The two-dimensional and the one-dimensional part are coupled to each other by the fillingfraction F of the PhC. For the one-dimensional calculationthe effective refractive index of the PhC is approximatedby n2

PhC = Fn2amb + (1− F )n2

SC.

4.1. 2D part: lattice type, pitch andfilling fraction

As band-bending effects can be neglected, only Bragg’s lawof diffraction βd = βi + G determines whether a guided

Figure 8 (online color at: www.lpr-journal.org) Geometrical

derivation of all diffraction processes according to Bragg’s law of

diffraction βd = βi +G. The dashed blue line is a reminder of theguided mode with inplane k-vector length βi that gets diffracted.

The dark blue (light blue) line depicts all diffraction processes

from βd = βi + G1 (βd = βi + G2). The initial k-vector lengthβi was chosen to be larger than k0 (outer gray circle). The inner

gray circle encloses inplane k-vectors within some apex angle inair smaller than 90◦ (see Sect. 9).

mode with inplane k-vector length βi > k0 gets foldedabove the light line by a reciprocal lattice vectorG. In thefollowing paragraphs, we will investigate properties of thePhC that have an impact on the diffraction process, likethe pitch itself, the lattice type, the photonic strength ofthe lattice and the filling fraction. For the illustration of thediffraction process we will use a basic geometrical consid-eration as depicted in Fig. 8. Here, all resulting k-vectors βdfrom diffraction of all guided modes of the degenerate setwith inplane k-vector length βi by two different reciprocallattice vector lengths G1 and G2 are depicted.

Apparently, the length of the reciprocal-lattice vectordetermines whether a guided mode gets diffracted abovethe light line or not. As soon as βi −G < k0 an arc withhalf-angle ϕ of the blue circle lies within the extractioncone with radius k0 and some part of the formerly guidedmode gets extracted. The length 2ϕβi of this arc corre-sponds to the fraction of different inplane direction βi/βi

that gets diffracted to air. In order to extract as much inplanedirections as possible the reciprocal lattice vector shouldbe chosen around G ≈ βi. In Fig. 9 the fraction of the in-plane directions diffracted to air with respect to all inplanedirections is shown as a function of the inplane k-vectorlength βi and assumingG = βi. For small inplane k-vectorlengths βi ≈ k0 the arc within the extraction disk is almostone third of the circumference. In contrast, for the largestinplane k-vector possible within the AlInGaP-material sys-tem, i.e. βi ≈ 3.5k0, the fraction drops to only 9%.

Depending on the lattice type additional reciprocal lat-tice vectors are present with the same length |G| = G butdifferent directions. For instance, a hexagonal lattice asshown in Fig. 10a offers six reciprocal lattice vectors withlength equal to the main reciprocal lattice vector length G0.



Figure 9 (online color at: www.lpr-journal.org) Fraction of

the inplane directions diffracted to air (black) as a function of

the inplane k-vector. From a basic geometrical consideration the

fraction of the guided mode diffracted to air by a reciprocal lattice

vector G is given by 2ϕ/360◦ with cosϕ = 1 − 1/(2β2i /k

20)

and G = βi. In red the minimum number of reciprocal lattice

vectors with lengthG is shown for extracting all inplane directionsto air. Here, it has been assumed that all G are homogenously

spaced on a circle with radius G. The dashed vertical line atβi = 2.5k0 (3.5k0) indicates the maximum inplane k-vector forGaN (AlGaInP or AlGaAs).

Figure 10 Hexagonal lattices with (a) 1 atom per unit cell (com-mon hexagonal lattice) (b) 7 atoms per unit cell (Archimedean

tiling A7) and (c) 13 atoms per unit cell (Archimedean tiling A13)

with the corresponding Fourier transform. The diameter of the

discs in the Fourier transform represents the amplitudes’ squared

modulus of the corresponding reciprocal lattice vectorG. G0 is

referred to as the main reciprocal lattice vector length. In the case

of the hexagonal lattice, G0 is equal to the length of the primitive

reciprocal lattice vectors.

Thus, a larger fraction of the different inplane directions ofa guided mode can be diffracted to air. Assuming that thesereciprocal lattice vectors are evenly distributed on a circlewith radius G, Fig. 9 also shows the minimum number ofreciprocal lattice vectors required for extracting all inplane

directions of the guided mode. Hence, all inplane directionsof guided modes with k0 < βi < 1.9k0 can be extracted bythe six main reciprocal lattice vectors of a hexagonal lattice.In contrast, for diffraction of guided modes with βi > 1.9k0

a higher number of reciprocal lattice vectors with almostthe same length is necessary. In the case of a hexagonallattice higher-order diffraction processes, i.e. diffraction by

G =√

3G0 along with G = 2G0, can be used but as wewill see later, diffraction by higher-order processes is lessefficient compared to diffraction by G = G0.To address the issue for large inplane k-vectors, lat-

tices with higher local symmetry have been suggested, likeArchimedean tilings [7], which are especially of interestfor AlGaInP- or AlGaAs-based LEDs as their refractiveindex is rather high (nSC ≈ 3.5). The omnidirectionaldiffraction properties of Archimedean lattices have beenproven experimentally in [13] by measuring the dispersionabove the light line. In order to investigate the propertiesof these lattices, hexagonal lattices with 1 atom per unitcell (the traditional hexagonal lattice), 7 atoms per unit cell(Archimedean tiling referred to as A7) and 13 atoms perunit cell (Archimedean tiling referred to as A13) along withtheir Fourier transform are shown in Fig. 10. For the hexag-onal lattice the length of the main reciprocal lattice vectoris G0 = 4π/31/2a with the pitch a; for the Archimedeantilings G0 ≈ 2π/a. The lattices A7 and A13 offer twelvedominant reciprocal lattice vectors with large Fourier co-efficients and therefore, extract all inplane directions ofguided modes with inplane k-vector length of up to 3.8k0

according to Fig. 9. But apart from a more ring-like Fouriertransform a significant reduction of the main Fourier peaksis observed for the A7 and A13 lattices. Therefore, onemight expect that the extraction length of a specific guidedmode βi is increased as the index contrast is reduced. Forconvenience, we only compare the hexagonal lattice withthe A13. The A7 was proven to yield almost the sameresults as the A13 tiling.

In order to take this effect into account we introduce anextraction coefficient κ that is proportional to the squaredmodulus of the amplitude of the Fourier transform of the lat-tice, |ΔεG|2, [17,18] as a measure for the photonic strengthof the lattice

κ (βi,G) ∝{|ΔεG|2 if |βi −G| < k0

0 else. (7)

If the diffracted k-vector length |βd| = |βi −G| > k0 theextraction coefficient is zero. We further assume that diffrac-tion of light from the extraction cone back to guided modescan be neglected. As discussed in [6], this assumption isvalid if the confinement of radiation modes within the ver-tical cavity is weak compared to the confinement of theguided modes. Therefore, light within the extraction conegets extracted more rapidly than backdiffracted, resulting ina net power flow from the guided mode to air. In Fig. 11 thetotal extraction coefficient κtot of the whole guided modeis shown. To calculate κtot, all inplane directions are takeninto account by weighting each |ΔεG|2 with the arc length



Figure 11 (online color at: www.lpr-journal.org) Total extrac-

tion coefficient κtot as a function of the reciprocal lattice vector

length of the main Fourier peak. The air filling fraction of the PhC

is F = 0.4. Black/red/blue lines correspond to guided modes withinplane k-vector length βi = 1.3k0, βi = 2.4k0 (e.g. a highly

guided mode in the GaN system), and βi = 3.4k0 (AlGaInP or

AlGaAs). The solid/dashed line corresponds to a hexagonal/A13

lattice. The A7 lattice shows nearly the same dependency as the

A13 and is left out for clarity.

2ϕβi within the extraction cone. This length depends onthe inplane k-vector and the reciprocal lattice vector length,see Fig. 8. As a guided mode might be diffracted by morethan one reciprocal lattice vector into the extraction disk,the whole Fourier transform of the lattices is considered bysummation over all reciprocal lattice vectors,

κtot(βi) ∝ 2βi

∑G

ϕ (βi,G) |ΔεG|2 . (8)

Thus, we take into account both, the photonic strength ofthe lattices and the inplane diffraction behavior.Before focusing on the comparison between the dif-

ferent lattices, let us briefly investigate the main char-acteristics of the extraction coefficient in Fig. 11 for thehexagonal lattice. In the case of the guided mode withinplane k-vector length βi = 2.4k0 diffraction to air byG0 < 1.4k0 only takes place by high-order diffraction pro-

cesses, i.e. diffraction by G =√

3G0, 2G0, 3G0, etc., as|βd| = |βi−G| < k0 only if |G| > G0. As the correspond-ing Fourier intensities are low at F = 0.4, the extractioncoefficient is small in this regime. The guided mode is sig-nificantly diffracted to air as soon as |βd| = |βiG| < k0 for|G| = G0. Here, light gets diffracted by the main reciprocallattice vectors having the highest Fourier intensity. The opti-mum main reciprocal-lattice vector is slightly smaller thanβi (G0 ≈ 2.1k0) as the fraction of inplane directions ex-tracted is maximized in this case. As soon asG0 > βi +k0,diffraction into the extraction disk is not possible becausethe hexagonal lattice supports no reciprocal lattice vec-tors with |G| < G0. The calculations for βi = 1.3k0 ingeneral show the same dependency except that also high-order diffraction processes result in diffracted k-vectorswith βd < k0 as the optimum main reciprocal lattice lengthvector is G0 ≈ 0.7k0. Thus, small inplane k-vectors getdiffracted more efficient compared to longer ones.

From the reduced Fourier intensities of the A13 latticecompared to the hexagonal lattice a decreased extractioncoefficient is observed despite the more omnidirectionaldiffraction behavior of the Archimedean tilings. But nev-ertheless almost the same amount of flux gets coupled outof the guided mode after some propagation distance. How-ever, according to Fig. 9 the hexagonal lattice can onlydiffract 80% (56%) of the inplane directions of the guidedmode with βi = 2.4k0 (βi = 3.4k0) to air. Therefore, itempties these parts, i.e azimuthal directions, of the modesvery rapidly while leaving the other parts unchanged. Thislight can only be recovered by diffraction processes be-tween guided modes. But during this internal redistributionof light it is subject to absorption losses. In contrast, thehigher symmetry of the Archimedean lattices enables di-rect extraction of the light as every inplane direction getsdiffracted to air.Hence, three different regimes can be distinguished.

(i) If the absorption length resulting from the absorptionpresent in the structure is comparable to the extractionlength due to diffraction, the hexagonal lattice diffractsmore light to air than the Archimedean lattices owing to itshigher Fourier coefficient. In particular, guided modes areextracted faster by a hexagonal lattice, that are diffractedonly by |G| = G0 into the extraction disk without thecontribution of higher-order diffraction processes. If, addi-tionally, higher-order diffraction processes contribute to theextraction of a guided mode, both lattices perform equally.(ii) For systems with decreased absorption, Archimedeanlattices can extract more light due to their omnidirection-ality, regardless of the Fourier intensities. (iii) In the limitof vanishing absorption the gap in extraction between thehexagonal lattice and the Archimedean lattices decreasesas the redistribution process is less prone to absorption.The last parameter of this section – the filling fraction

F of the lattice – determines the photonic strength and iseasily optimized by calculating first the Fourier transformof the lattice and then using Eq. (8) to get the total extrac-tion coefficient. Fig. 12a shows the extraction coefficientκtot as a function of the filling fraction for the hexago-nal lattice and the A13 tiling. In both limits F → 0 andF → 1 the PhC vanishes and the modes are not diffracted.Here, the main reciprocal lattice vector was chosen to yieldmaximum diffraction per length according to Fig. 11. Theoptimum filling fraction depends on the guided mode beingextracted. If higher-order diffraction processes do not re-sult in diffraction within the extraction disk only the mainreciprocal lattice vectors G = G0 contribute to the diffrac-tion process. According to Fig. 12b these have maximumintensity for F = 0.4. Please keep in mind that the longerthe inplane k-vector of the guided mode being diffractedthe longer the optimum reciprocal lattice vector and con-sequently fewer high-order diffraction processes result indiffracted k-vectors βd < k0. Thus, in the case of diffrac-tion within the extraction disk by high-order diffractionprocesses their Fourier intensity also has to be taken intoaccount. The resulting summed Fourier intensity is maxi-mum for F = 0.5.



Figure 12 (online color at: www.lpr-journal.org) (a) Total extrac-

tion coefficient as a function of the filling fraction for the hexago-

nal (solid line) lattice and the A13 (dashed line). Black/red/blue

lines correspond to guided modes with inplane k-vector length

βi = 1.3k0, βi = 2.4k0 and βi = 3.4k0. The reciprocal lattice

has been adjusted for maximum diffraction in each case. (b) Sum

over the Fourier intensities of all reciprocal lattice vectors with

length G = G0 (black) and G � 2G0 (red) as a function of the

filling fraction of the PhC.

In summary, the optimum PhC lattice type dependson the inplane k-vector length βi of the guided mode thatshould get diffracted and on the losses present in the LEDunder study. In general, the larger the inplane k-vectorlength, the higher the required number of reciprocal latticevectors with the same length in order to diffract every in-plane direction of the guided mode. These reciprocal latticevectors G should have the highest Fourier peaks |ΔεG|2of the PhC and length G ≈ βi. However, the need forhigher symmetry lattices depends on the absorption losses.Maximized diffraction efficiency is achieved with a fillingfraction of the lattices around F ≈ 0.45.

4.2. 1D part: vertical layer stack and etch depth

As we have seen in the last section, the choice of the recip-rocal lattice vector length G relies on the inplane k-vectorlength of the guided mode that should get diffracted. Typ-ically, in real LEDs one is not in the lucky situation tohave a layer stack that supports only a single guided mode,but rather one has to deal with a distribution of modes.In Fig. 13 the basic problem of several guided modes isdepicted. The guided mode with inplane k-vector length β1

certainly gets diffracted very efficiently, whereas diffractionof guided mode β2 mainly results in diffracted k-vectors in

Figure 13 (online color at: www.lpr-journal.org) Geometrical

derivation of all diffraction processes for two guided modes with

inplane k-vector lengths β1 and β2 by one reciprocal lattice vector

G. The dashed blue (red) line is a reminder of the guided modethat gets diffracted. The solid blue (red) line depicts all diffraction

processes from βd = β1 +G(βd = β2 +G).

the vicinity of the edge of the extraction disk βd ≈ k0. Onehas to mention here, that all guided modes with inplanek-vector βi ∈ [G− k0;G+ k0] get partially diffracted intothe extraction disk. However, the question arises, whichlattice pitch should be chosen in order to extract as muchlight as possible from the LED. Hence, what are the prop-erties of the guided modes that determine how much lightis diffracted to air before it is absorbed and how can weoptimize these?

In Fig. 6 different types of modes are summarized. Fromthese we will derive the main properties determining theextraction of a guided mode, namely the spontaneous emis-sion into the guided mode, its interaction with the PhC, theabsorption and the Fabry-Perot resonances of the radiationmodes. Additionally, some limits for the optimization dueto technological feasibility and performance will be givenalong with results found in the literature.

The first property that comes to mind that should in-fluence the choice of the lattice constant is the amount oflight emitted into different modes. A guided mode carryinga large fraction of the active layer’s emission should bediffracted instead of a guided mode that no light is emittedinto. In general, the larger the inplane k-vector length ofthe guided mode the more light is emitted into that mode3.From that point of view, guided modes with large inplane

3 From Eq. (3) we notice that the density of photonic states

plays an important role for the spontaneous emission. With Fig. 4c

in mind we conclude that the “red” mode covers a larger area in

k-space than the “blue” mode and thus offers a higher number ofphoton states. This number scales with β/kSC and is taken into

account in Eq.(4) by the factor sinθ = β/kSC. Therefore, more

light is emitted into guided modes with large in-plane k-vectorlength compared to modes with smaller k-vector length.



k-vector are more preferable for diffraction than guidedmodes with smaller inplane k-vector.

Now, how much light is actually emitted into the differ-ent guided modes heavily depends on the vertical positionof the active layer with respect to the vertical mode profile(RCLED effect). As depicted in Fig. 6a the guided modelabelled I has an antinode at the position of the active layerand thus will carry away a lot of emission. The local electricfield strength is maximum resulting in a high radiative rate,see Eq. (3). In contrast, no photons will be emitted into theguided mode II with a node at the position of the activelayer. From this we conclude that the vertical layer structurehas to be optimized in order to reveal guided modes bestsuitable for diffraction by adjusting the cavity itself and theposition of the active region. The guided mode carryingmost emitted light should be chosen for diffraction. For amore detailed discussion regarding the layer optimizationthe reader is referred to [64].

Secondly, the photons emitted into a guided mode haveto interact with the PhC in order to couple with radiationmodes above the light line. In [15] a thorough investigationreveals that the extraction coefficient κ of guided modesscales with t−3 – with t the thickness of the remainingunetched core (see Fig. 6) – as long as the guided modeis confined within this core and evanescent in the PhC. Itcan be further shown that the overlap of the guided modewith the PhC mainly entails this dependence. Therefore,we assume that the interaction depends to first order onthe overlap of the guided mode with the PhC layer. Hence,along with the amount of emitted light into the guidedmodes a second property of the guided mode determinesthe choice of the optimum reciprocal lattice vector. Forinstance, mode I in Fig. 6a has less overlap with the PhCcompared to mode III. In general, guided modes with 1 <β/k0 < nPhC have good overlap with the PhC layer asthey are not evanescent in the PhC layer. In contrast, guidedmodes with β/k0 > nPhC are evanescent in the PhC layerand therefore interact poorly with the PhC. But typicallythese carry the highest fraction of emitted light and thus,there is a need for extracting them.From this consideration and according to the t−3-

dependence it seems preferable to etch through the wholeLED in order to maximize the overlap of the guided modeswith the PhC. But this implies perforation of the active re-gion resulting in a decrease of the active area according tothe filling fraction of the lattice. Furthermore, the nonradia-tive recombination of electron-hole pairs is increased as theactive region is prone to surface-recombination processes.Thus, the active region should be embedded within thecore having thickness t > λ/2nSC. This lower boundaryresults from the cutoff thickness of the lowest order mode,i.e. mode I in Fig. 6a. Cores thinner than λ/2nSC do notconfine any mode and thus, all the guided modes are mainlyguided within the PhC. As opposed to a maximized overlapwith the PhC, in this case a reduced recombination ratedecreases the internal quantum efficiency, as none of theguided modes has significant field amplitude at the positionof the active region, see e.g. mode III in Fig. 6a. Therefore,

the optimum PhC LED consists of a core slightly thickerthan λ/2nSC and a PhC with an etch depth deeper than thevertical attenuation length 1/γ of the guided mode, withγ =

√n2

PhCk20 − β2

i . As this results in an overall thick-ness of the LED on the order of 100–300 nm technologicalissues like current spreading and current injection becomea severe problem. To accomplish reasonable current spread-ing while remaining with thin LEDs in [23] a transparentconductive oxide (TCO) was used. This has also been uti-lized in [24]. However, the absorption introduced by theseTCOs has to be carefully considered.

Similar to the amount of spontaneous emission into amode and its interaction with the PhC, the absorption ofthe mode relies on its overlap with the absorbing layers,like a TCO, the mirror, the contacts or the epitaxial layersthemselves. In general, the vertical layer design should notsupport guided modes with both high spontaneous emissionand high absorption, as some of the light will probably belost. If such modes cannot be avoided, the PhC should beadjusted to these in order to extract them as fast as possible.

An implementation of PhC LEDs explicitly addressingthe issue of a thin core is given in [12]. The vertical layerstack of the presented GaN-on-sapphire LED is similar tothe one shown in Fig. 6b except that a low-refractive indexlayer of AlGaN is inserted between the active layer and thesapphire substrate. This layer along with the∼ 250 nm deepPhC squeezes a single guided mode within a 350 nm thickcore. The remaining guided modes are mainly confinedunderneath the AlGaN layer and thus, only a little light isemitted into them. With a back-mirror behind the sapphiresubstrate (mimicking a thin-film device) an enhancementof at least 70% compared to a flat LED is reported. Theresulting extraction efficiency is estimated to be 20–24%.In [24] an adjustment of these layers results in a 2.3-timesenhancement compared to a flat reference.

Also, the PhC itself can be used to define the desiredsmall cavity, as has been shown by embedding the PhC intothe vertical layer structure of the LED [16,19] – in this casethe core is sandwiched between the PhC layer and the semi-conductor/air interface. This has been obtained by lateralepitaxial overgrowth, even though this is a challenging task.In [16] a 500 nm thick core showed twice as much lightoutput compared to a ∼ 1 μm thick one. A 1.5-fold of theextraction efficiency compared to an as-grown reference isreported in [19].

On the other hand, microcavities only a few wave-lengths thick could also be a solution as they only support afew guided modes and some of these modes can be chosento have optimum properties for diffraction by adjusting theoverall thickness and the position of the active layer insidethe LED. In the case of AlGaInP-based PhC RCLEDs en-hancement factors compared to a flat reference of 2.6 arepossible [22]. For the GaAs material system the extractionefficiency of a RCLED with a metallic grating has beencalculated as 43% [6]. PhC LEDs only ∼ 850 nm thickwith a metallic back-mirror have shown high extractionenhancements of 1.8 [23] in the nitride material system.



Figure 14 (online color at: www.lpr-journal.org) Impact of the

pitch of a∼ 300 nm deeply etched, hexagonal PhC on the emissionpattern of a ∼ 6 μm thick InGaN/GaN thin-film LED with a peakwavelength of 520 nm. Azimuthal angle and wavelength averaged

data taken from [17]; different colors represent different pitches.

The overall flux is normalized to unity. A Lambertian emission

profile is shown in black.

But even for thick LEDs supporting numerous guidedmodes enhancement factors compared to flat LEDs of 1.5in the blue emission range [8], 2.5 in the UV [9] and 1.5 inthe AlGaInP system [21] can be found.

Along with these extraction enhancements the influenceof the lattice pitch on the emission pattern of the LED hasbeen proven [8, 17, 24]. In Fig. 4 farfields obtained froma ∼ 6 μm thick InGaN/GaN thin-film LED with a peakemission of 520 nm are shown for different pitches of the∼ 300 nm deeply etched hexagonal PhC. The basic impactof the pitch on the far field can easily be seen by consid-ering Fig. 4b or Fig. 8. Imagine an emission wavelengthof 520 nm and pitch of 208 nm for the slab presented inFig. 4. The leaky mode at a/λ = 0.4 is diffracted intothe vicinity of the light line. The associated external angleθair = asin(β/k0) is close to 90◦ and thus, the resultingemission pattern is rather flat. In contrast, a pitch of 281 nm(a/λ = 0.54) causes diffraction into the vicinity of the Γ -point and therefore a strong forward emission as θair ≈ 0.In Fig. 8 the azimuthal diffraction can also be seen, wherethe longer (shorter) pitch corresponds to the shorter (longer)reciprocal lattice vectorG1 (G2). In the case of the thickLED of Fig. 14 many guided modes get diffracted, someinto the vicinity of the light line and some close to thesurface normal according to Fig. 13. The net far field is asuperposition of all diffraction processes from all guidedmodes, where the main contribution comes from guidedmodes with high overlap with the active region and rea-sonable interaction with the PhC. Thus, a PhC alters theradiance from the LED. Applications favoring a forwardemission will benefit from this, if the radiant emittance iscomparable to standard LEDs. This will be discussed indetail in Sect. 9.

Up to now we investigated properties of the guidedmodes but neglected the impact of the radiation modes(β < k0). Apart from their contribution to the extractionefficiency that is optimized in RCLEDs (see Sect. 2) theirresonances also have an impact on the diffraction efficiency

of guided modes [15]. Consider for instance, the radiationmode IV in Fig. 6a and the radiation mode V. The differencebetween the two relies on their overlap with the PhC due tothe Fabry-Perot resonances. Thus, diffraction from guidedmodes into mode IV will be more efficient compared todiffraction into mode V.

In the case of substrate LEDs a detailed investigation isgiven in [15] and [18] revealing that the Fabry-Perot reso-nances can suppress diffraction into the substrate. Diffrac-tion into so-called substrate modes, e.g. modes labelledII and III in Fig. 6b with 1 < β/k0 < nsubs, has to beavoided as this light can hardly be recovered due to the lowinteraction of these modes with the PhC. The amount ofpower radiated into the substrate by the radiation modes, forinstance mode IV and mode V in Fig. 6b, can be redirectedby placing a mirror at the bottom side of the substrate.In summary, the vertical layer stack of the LED should

support guided modes with both a high fraction of thetotal emitted light and good interaction with the PhC. Thereciprocal lattice vector length should be adjusted to thosemodes. Thus, a thin core containing the active region ismost preferable, as the interaction of the guided modeswith the PhC can be increased, and the total number ofguided modes is reduced. In the case of substrate LEDs therange of guided modes for diffraction is limited to nsubs <G/k0 < nSC compared to the range of 1 < G/k0 < nSC

in the case of thin-film LEDs, as substrate modes haveonly small interaction with the PhC and hence, cannot berecovered. Furthermore, the diffraction efficiency can befurther improved by adjusting the Fabry-Perot resonances.The latter requires a precise control over the thickness ofthe layers, especially that of the PhC layer.

4.3. Setups for PhC LEDs

In general, two setups for PhC LEDs in the diffractionregime are possible, as shown in Fig. 15 [64]. In the scheme(a) the whole LED area is used for light generation and ex-traction. This can be incorporated into thin-film LEDs andtherefore has the advantages of scalability and a large light-emitting area with only a few dark regions due to contact

Figure 15 (online color

at: www.lpr-journal.org)

Two possible setups for

PhC LEDs (see text).



shadowing. In addition, as the whole chip area is used forlight generation the current density can be minimized. Thisis of great importance since the internal quantum efficiencytends to decrease with increasing current density becauseof carrier leakage and/or nonradiative recombination pro-cesses of electron-hole pairs, like Auger recombination [65].Decreased internal quantum efficiency results in strongerheating of the chip forcing a further increase of nonradia-tive processes. Of course, in this case etching through thewhole LED is not favorable (see Sects. 4.2 and 5).

In contrast, by separating the area of light generationand light extraction as depicted in Fig. 15b, etching throughthe active area incurs no penalty. As the light is generatedwithin the unperforated part the radiative recombinationrate is not reduced and no active region is lost due to etching.However, the active area of the LED is limited as the light-extraction area cannot be used for light generation. Thus,the efficiency decreases rapidly with increasing current dueto current crowding and the resulting high nonradiativerecombination rates. Nevertheless, this setup provides agood insight into the diffraction behavior of the PhC, asonly the guided light interacts with the PhC and the directlyemitted light is either already coupled out or absorbed atthe contact.

In both cases the LED area should be large enough toensure complete light out-coupling. The extraction lengthof the fundamental guided mode in the case of GaN-on-sapphire LEDs has been calculated to be 80–200 μm [11,12]and for GaN-based laser lift-off LEDs it has been esti-mated to be 5–10 μm [14]. Therefore, thin-film LEDs offerstronger interaction of the guided modes with the PhC.It should be mentioned here that these values do not in-clude the extraction of all guided modes, resulting in longeroverall extraction lengths that have to be considered. Ingeneral, short extraction lengths are preferable as the lightextraction has to compete with absorption processes at themirror or inside the epitaxial structure [63] as they takeplace within the same length scale (see Fig. 3). Here, espe-cially the reflectivity of the mirror plays an important role([14] and Sect. 8).

5. Strong PhCs – molding the flow of light

In the regime of strong PhCs the impact on the dispersionrelation goes beyond the pure band folding and band bend-ing significantly alters the properties of light propagation.In principle, this setup solves the fundamental problem oflight extraction from LEDs – light emission into guidedmodes – as the emission into the guided modes is inhibitedby the complete photonic bandgap [2], as shown in Fig. 16.Thus, light can only radiate into the ambient medium result-ing in high extraction efficiencies. In the case of a photoniccrystal slab in air nearly 100% extraction efficiency [4]has been calculated, if the emission wavelength overlapswith the photonic bandgap. Therefore, an enhanced ver-tical emission is observed [10]. On the other hand, this

Figure 16 Dispersion relation for a photonic crystal slab [3].Emission around reduced frequencies of 0.35 is strongly inhib-

ited due to a complete bandgap for TE-like polarized light. For

frequencies of 0.5 to 0.6 emission occurs in leaky modes with a

high density of states.

causes a significant reduction of the radiative recombina-tion rate as the photon density of states is heavily reduced(see Eq. (3), [4], and [10]). This itself decreases the internalquantum efficiency of the devices as a consequence of thecompetition between radiative and nonradiative recombina-tion, see Eq. (2). Additionally, etching of holes through theactive layer enforces surface recombination of electron-holepairs, leading to an increase of the nonradiative recombi-nation rates and thus, to a further decrease of the internalquantum efficiency.

Therefore, instead of inhibiting spontaneous emissionby a photonic bandgap it is more preferable to use boththe enhancement of extraction due to Bragg scattering andan increased density of photonic states [3]. This can be ac-complished by adjusting the emission of the active layer tothe flat bands of leaky modes, for instance for frequenciesaround 0.5 in Fig. 16. These flat bands offer a large numberof possible states in k-space within a small frequency range.The resulting enhancement of radiative recombination ratehas been estimated to two experimentally [3]. In addition tothis, the Bragg scattering of photons adds up to an overallexternal efficiency of > 50%. In comparison, the Purcellfactor in typical LEDs is roughly 1, meaning that the radia-tive recombination of the source is basically unaltered bythe density of states provided by the optical environment.

From the folding of the light line an interesting factarises. In the case of a hexagonal lattice, for frequenciesa/λ > 2/3 all the modes lie above the light line. In thisregime the 1. BZ fits completely within the extraction disk.Therefore, high extraction efficiencies of up to 90% couldbe achieved [2].

Apart from these approaches for enhancing the effi-ciency, a combination of light generation and extraction,as described in [5], results in 80% of the overall emissionwithin a narrow escape cone of ±0.2◦. Here, firstly theemission is collimated inplane into the ΓK direction of ahexagonal PhC slab (for instance emission around frequen-



cies of 0.3 in Fig. 16). A second grating with twice the pitchof the first one is then used to redirect that light into thevicinity of the surface normal. However, this approach isextremely sensitive to the emission wavelength [18]. Forshorter wavelength the emission falls into the bandgap andfor longer wavelength the collimation of the emission is de-creased.Unfortunately, all of these encouraging results have

only been demonstrated on free-standing PhC slabs withoptical pumping. An electrically driven realization of suchPhC LEDs would probably suffer from weak current spread-ing and again from a loss of active area. Furthermore, asflat bands correspond to a low group-velocity of the Blochmodes, reinforced absorption will restrict the performanceof such PhC LEDs.

6. Special cases for PhCs

6.1. Omnidirectional mirror

According to Fig. 3 the absorption within an LED plays animportant role for the extraction efficiency. Especially inthin-film LEDs with a metallic back-mirror the losses intro-duced by the mirror limit the performance of these devices.Therefore, a lossless omnidirectional mirror based on a PhCcould be a significant improvement. However, this couldonly be accomplished by a three-dimensional PhC with aphotonic bandgap for every propagation direction and po-larization. To the best of our knowledge no implementationof a three-dimensional PhC on LEDs has been reported sofar that addresses this issue. A promising realization of athree-dimensional photonic crystal is reported in [66]. Ingeneral, besides its reflectivity the PhC should also offergood heat dissipation, at least.A first approximation is reported in [67]. Here, the

mirror is formed by a low-index layer followed by a one-dimensional PhC (or Bragg reflector). Due to total internalreflection at the interface between the semiconductor andthe low-index layer 100% reflection from the critical an-gle to glancing angles is achieved. The transmitted lightwithin the critical angle is reflected back at the Bragg mir-ror. An overall angle and polarization averaged reflectivityof R > 99.95% is reported. Compared to conventionalmetallic mirrors or Bragg reflectors the loss is two ordersof magnitude lower.

6.2. Metallic PhC

Depending on the dielectric and the metal, the correspond-ing interface guides light, so-called surface plasmons (SPs)are formed. Namely, the real part of the metal’s permittivityεM has to be smaller than minus the permittivity of thedielectric εD : Re{εM} < Re{−εD}. Microscopicallya SP is an oscillation of the electron plasma inside themetal. This oscillation can be expressed as a mode that is

Figure 17 (online color at: www.lpr-journal.org) Dispersion

(red) of a surface plasmon at GaN/silver interface (nGaN = 2.5,silver from [75]). Light line of GaN is shown in black. Right plot

shows the Purcell factor for an emitter placed 10 nm in front of

the GaN/silver interface.

evanescent in both the semiconductor and the metal. Thedispersion relation of a SP as shown in Fig. 17 reveals, thatdue to the flat band the SP offers a huge density of statesand therefore, significantly enhances the radiative rate of anemitter placed into the vicinity of the metal/semiconductorinterface [68]. The coupling of the emitter (quantum wells)with the SP forms the surface plasmon polariton (SPP [69]).But due to the confinement of the SP, the light emitted intoit never escapes and is absorbed in the metal. Here, a PhC ishelpful as it scatters the light out of the SP into the semicon-ductor [69, 70]. But the efficiency of a SPP coupled emitteris mainly limited by the efficiency of coupling light out ofthe SP [71,72]. This is determined by the absorption lengthwithin the metal and the extraction length due to scatteringby the PhC and limits the application of SPPs to emitterswith poor efficiency. In this case, the enhancement of inter-nal quantum efficiency due to the Purcell factor overcomesthe losses introduced by the metal.Instead of a metallic PhC localized metallic particles

have also been suggested to take advantage of the highPurcell factor [73]. But the main drawback of absorption inthe metal still limits the efficiency. In addition, in a realis-tic device many metallic particles should be incorporated,in order to use the whole chip area and therefore, get rea-sonable performance. Here, to maintain the properties oflocalized particle plasmons in spite of their mutual couplingstrong disorder has to be introduced [74].

7. Patterning techniques

As the periodicity of PhCs is on the order of the wave-length and LEDs emit from the near UV (360 nm) up tothe near IR (940 nm), the patterning techniques have tobe able to define holes in the same regime (100–1000 nm)depending on the filling fraction and the lattice. But conven-tional patterning techniques (typically contact lithographyfor semiconductors) are restricted to about a = 1 μm bythe diffraction limit or are extremely expensive, like deep-ultraviolet lithography. In this section, commonly used pat-



terning techniques capable of realizing such dimensionsare summarized.

7.1. Electron-beam lithography

Electron-beam lithography (e-beam) is probably the mostwidely used technique for patterning 1D and 2D PhCs as itamounts to an extended scanning electron microscope [76].In this setup an electron beam scans across the resist cov-ering the LEDs surface and exposes the resist accordingto the PhC pattern. After selectively removing either theexposed or the unexposed areas of the resist the pattern isetched into the semiconductor. With this method typicalfeature sizes down to 20 nm are achieved. The scanningtechnique offers a huge flexibility on the patterns but limitsthe overall throughput. Therefore, e-beam is mostly applica-ble for low-volume production and especially for researchand development.

7.2. Nanoimprint

Nanoimprint could be thought of as Gutenberg’s inventiontransferred to the nanoscale [77]. First, a stamp is generated,typically an e-beam-patterned material like quartz. As in themacroscopic world, the stamp is the inverse image of theone-dimensional or two-dimensional PhC and can be usedseveral times. This stamp prints its pattern into a resist castupon the LED’s surface. By pressure and capillary forcesthe PhC lattice is transferred into the resist and after UVcuring and/or heating (depending on the resist) the stampis removed. Now, the resist can act as an etching mask fortransferring the pattern into the semiconductor. As first thestamp or master has to be generated it is not as flexible ase-beam lithography. But nanoimprint is applicable to massproduction as it offers high throughput (full-wafer process)and reproducible results on the order of 10–20 nm. Also,every possible pattern can be printed, so it is not limited toperiodical ones. However, due to the stamping of the wholewafer, particles have to be avoided as these could damagethe master. Transferring the pattern into the resist suffersfrom rough wafer surfaces and from bow of the wafer, atleast if a rigid master is used [78].

7.3. Laser interference lithography

In this technique two laser beams are used for interfer-ence and the resulting one-dimensional intensity profiledefines the resist pattern [79]. By simply rotating this one-dimensional profile two-dimensional regular patterns arecreated, like rectangular or hexagonal lattices. Thus, thismethod is limited to regular patterns and is not capable ofcreating arbitrary lattices. The pitch of the lattice is variedby either the laser wavelength or the angle between the twolaser beams.

7.4. Self-organized lithography

Here, the natural affinity of spheres to pack to each otheras closely as possible is used [80]. Hence, with these tech-niques it is only possible to generate exactly these patterns,namely hexagonal in two dimensions and opal in threedimensions. Typically in two dimensions, spheres with di-ameter equal to the desired pitch are used to form the latticeupon a resist. These spheres are used as an etching maskin order to transfer the pattern into the resist. By a finaletching process the pattern is transferred from the resistinto the semiconductor. On the one hand, this is an easy andcheap way of fabricating a hexagonal lattice. On the otherhand, this method is limited by the availability of spherediameters, to hexagonal patterns and by the reproducibilityof the results. The latter is a result from multilayer stackingof the spheres and a lack of long-range ordering.

7.5. Nanorods by selective growth

Contrary to the techniques presented so far, here no mate-rial is removed but the LED is only grown on some definedareas [81]. In order to obtain a regular arrangement of thesemiconductor pillars first one of the above techniques hasto be used to define a pattern for the selective growth [82].This results in an inverted PhC structure compared to theones discussed so far: now pillars of semiconductor rise upinto air instead of air pillars etched into the semiconduc-tors. The main investigations in the area of nanorods arefocused on defect-free growth of semiconductor structures– especially important for the GaN system. But dependingon the pitch of the pattern clearly PhC effects should beobserved. The beauty of this approach lies in the possibilityof so-called core-shell structures [83]. Here, the active areaof the LED is meant to be grown on the vertical side wallsof the pillars resulting in a huge enhancement in active areacompared to a horizontal active area. This could overcomethe main disadvantage of fully etched through PhC LEDs:the reduced active area.

8. Comparison between PhC LEDs andrandomly textured LEDs

In this section we focus on the fundamental question forthe application of PhC in LEDs: Does a PhC LED extractas much light as a state-of-the-art commercial LED? So far,commercial available thin-film LEDs typically include arandom surface texturing as it is easy to fabricate and solvesthe problem of total internal reflection very efficiently. Thebenchmark presented in Sect. 2 for realized LEDs alreadyshows that this is a close competition. Here we shed lighton both approaches using FDTD modelling and – first – acomparison of both schemes in the weak scattering model.For the latter, the rough surface is modelled in analogy tothe PhC, yet with the holes distributed in a random pattern.



Figure 18 (online color at: www.lpr-journal.org) (left) Real

space image of a random distribution of holes. The image was

generated on a area of (40a)2 assuming the same filling fraction

F = 0.4 and the same number of holes per unit area as forthe hexagonal lattice. (right) The Fourier transform of the rough

surface along with a reminder of the main reciprocal lattice vector

length G0 = 4π/√

3a of the hexagonal lattice (indicated by thered dashed circle).

Figure 19 (online color at: www.lpr-journal.org) Comparison of

the extraction coefficient κtot for a hexagonal lattice (solid lines)

and a random surface texture (dashed lines) as a function of the

main reciprocal lattice vector. Black/red/blue lines correspond to

βi = 1.3k0, βi = 2.4k0, and βi = 3.4k0.

From the point of view sketched in Sect. 4.1 the Fourieramplitude of the random surface should give first informa-tion on the extraction efficiency. In Fig. 18 the real spaceimage used for the random surface texture is shown alongwith its Fourier transform. Clearly, no discrete pattern isobserved as for the lattices shown in Fig. 10. According toEq. (7), Fig. 19 shows the extraction coefficient for a hexag-onal lattice and the random surface texture. The randomtexture extracts less light from a single guided mode withina given propagation distance compared to the hexagonallattice due to its lower Fourier amplitude – this is similarto the Archimedean lattices. In contrast, the random sur-face texture is less sensitive on the exact inplane k-vectorof the guided mode as for large reciprocal lattice vectorsstill all the modes get diffracted to air. This is especiallyhelpful for thick LEDs with numerous guided modes. Fur-thermore, similar to the case of Archimedean lattices everyinplane propagation direction is diffracted to air as a huge

Figure 20 (online color at: www.lpr-journal.org) Cut through

the three-dimensional FDTD simulation setup. A single dipole is

excited and its overall emitted flux is detected (detection planes

illustrated by black lines). The detection plane in air yields the

extracted flux and thus, the extraction efficiency. The arrows in-

dicate the Poynting flux component used for integration. The

mirror is indicated by the gray layer, the uniform GaN layer by

the blue area.

number of reciprocal lattice vectors with |G| = G exists.In contrast to the Archimedean lattice, a rough surface of-fers omnidirectional inplane diffraction for every inplanek-vector length of the guided mode not just for guidedmodes with βi ≈ G0.

In order to compare a PhC LED with a standard LEDin terms of absolute efficiency we resort now to full three-dimensional calculations using the FDTD method. As avehicle we choose a GaN-based LED in a thin-film config-uration with a peak wavelength of 520 nm. The modellingresults shown represent a Gaussian average with a FWHMof 30 nm including wavelengths from 470 to 570 with 71supporting wavelengths. As the refractive index of GaNLEDs varies negligibly within the layer stack we assumeda constant index of n = 2.4 for the whole 3 μm thick LED,as depicted in Fig. 20. The active region is placed 160 nmabove an Ag-based mirror. The refractive index of this mir-ror was obtained by fitting a combined Plasma-Lorentzmodel [84] for the dielectric constant to the values of Agfrom [75]. In order to take into account thin absorptiveadhesion layers between the Ag and the GaN, we increasedthe damping constant within the plasma model by a factorof 10 and dropped the adhesion layer. Otherwise, the accu-rate description of such a thin absorptive layer with FDTDwould result in a huge demand on computer memory. Thereflectivity of this mirror under normal incidence is ∼ 90%compared to ∼ 98% of the pure Ag mirror at 520 nm. Theextracting surface roughness of standard GaN-based LEDstypically consists of pyramidal structures defined by thecrystallographic planes of GaN [85] with an average heightof around 1 μm.

For a first comparison we simply replaced the surfaceroughening by a hexagonal lattice with a typical etch depthof 300 nm and a lattice constant of a = 600 nm that hasbeen optimized for maximum extraction efficiency. As thislattice constant corresponds to a main reciprocal lattice



Figure 21 (online color at: www.lpr-journal.org) Extraction ef-

ficiency as a function of the integration radius for a PhC LED

(black) and a rough LED (red). The horizontal dashed line indi-

cates the extraction efficiency of the unstructured LED. Symbols

indicate the extraction efficiency calculated from the FDTD data

for increasing integration radius. Solid lines represent the ex-

trapolation with a function ∼ (1− exp(−αx)) starting from anintegration radius x = 8 μm. The inset shows a magnification forsmall propagation distances.

constant of G0 = k0 this PhC first diffracts guided modeswith high overlap with the PhC and secondly, guided modeswith higher inplane k-vectors by high-order diffraction pro-cesses. Thus, we have chosen the filling fraction to beF = 50%. In order to investigate the dependency of theextraction on the propagation distance of light we calculatethe Poynting flux through a disk parallel to the semiconduc-tor/air interface in air with respect to the overall emittedflux by the dipole. By successively increasing the radius ofthis disk it is possible to compare different surface texturesin terms of their extraction length.

In Fig. 21 the extraction efficiency for the PhC LEDand the rough LED is shown as a function of the integrationradius. Additionally, the extraction efficiency of an unstruc-tured LED is depicted for comparison. First, the extractionefficiency of the unstructured LED is significantly higherthan the 4% for an isotropic source. Here, the interferenceof the source with its mirror-image enhances the extractionefficiency (RCLED effect [86]). Secondly, due to the higherextraction coefficient, as shown in Fig. 19 the PhC LEDindeed extracts more light within an integration radius of5 μm, although the etch depth is only 300 nm compared to1 μm of the random surface texture. But due to its discretediffraction behavior and its restricted inplane omnidirec-tionality it cannot compete with the rough surface. After anextrapolation of the extraction efficiency by an exponentialfunction [17] we obtained an overall extraction efficiencyof 45% for the PhC LED and 60% for the rough LED. Theresulting enhancement factor of about 2.8 of the PhC LEDcompared to the unstructured reference is within the rangeof the reported enhancements summarized in Sect. 4.2.

Of course, the chosen structure is not optimized forthe implementation of the PhC as it is a rather thick LEDsupporting ∼ 50 guided modes and the interaction of thePhC with the guided modes carrying most of the spon-taneous emission is low. The interaction is increased by

Figure 22 (online color at: www.lpr-journal.org) Refractive-

index profile (black line) of the “optimal” diffracting PhC LEDs.

Figure 23 (online color at: www.lpr-journal.org) Extraction

efficiency as a function of the integration radius for the “optimal”

PhC LED (black) and a rough LED (red) with the same SiO2-

mirror combination. Symbols indicate the extraction efficiency

calculated from the FDTD data for increasing integration radius.

Solid lines represent the extrapolation with a function ∼ (1 −exp(−αx)) starting from an integration radius x = 8 μm.

decreasing the overall thickness of the LED (rememberthe t−3 dependence of the overlap with the PhC). In orderto demonstrate the potential of diffracting PhCs for lightextraction, we have chosen a LED with a reduced GaNthickness of 650 nm. The refractive-index profile of this“optimal” diffracting PhC LED is shown in Fig. 22. Thethick low-index layer (SiO2 with n = 1.5) pushes guidedmodes with 1.5k0 < β < 2.4k0 away from the mirror [87]and reduces their absorption to a negligible amount. There-fore, we have chosen a main reciprocal lattice vector lengthG0 = k0 and an hexagonal lattice in order to extract themainly absorbed modes at first. The position of the activeregion guarantees high extraction of the directly emittedlight into forward direction. Additionally, the shallow PhCensures staying within the weak regime. The extractionefficiency of this PhC LED is extrapolated to 70% (Fig. 23),whereas a roughened LED with this SiO2–mirror combina-tion but an overall GaN thickness of 3 μm has almost thesame efficiency with 71%, but further away along the ex-tracting ring. The PhC in combination with the thin verticalstructure extracts the light more rapidly, but still its effi-ciency is limited by the discrete diffraction behavior as canbe seen by the extrapolation of the data. A further reductionof the core thickness will probably enhance the extraction



Figure 24 (online color at: www.lpr-journal.org) Example of

an etendue-limited system. The light emitted from the source is

projected on the optical system with area AOS and acceptance

angle θOS. Depending on the source area ALS only light emitted

into a limited apex cone with angle θLS passes through the optical

system. In blue a Lambertian emission profile is shown and in red

a super-Lambertian one emitting more of its overall emission into

the limited apex cone with angle θLS.

efficiency further. But this would also mean leaving theregime of weak coupling.

9. Directional emission

One certain advantage of PhC over rough surface LEDs istheir capability to design the shape of the emission pattern.The variation of the emission pattern of a PhC LED withthe pitch of the lattice was shown in Fig. 14. An example ofan application taking advantage of directional emission – amore forward or super-Lambertian emission – is sketchedin Fig. 24. Here, only a fraction of the total emitted lightcan be coupled into the optical system and gets projectedonto the area AOS. Such applications, like projection-basedsystems or headlamps [27,28], have in common that theiroverall optical performance is limited due to etendue con-servation.

9.1. Etendue

The etendue or Huygen-Helmholtz invariant describes thephase space of light that can pass an optical component andis defined by

dE = n2 cos θdΩdA , (9)

with n the refractive index, θ the polar angle with respectto the optical axes, dΩ the solid angle and the surface areadA. Therefore, etendue is a limiting quantity for the func-tionality of a system. The performance of the whole setupis determined by that optical component having the small-est phase-space volume or etendue. In a projection-basedsystem the limiting component is the imager, e.g. a digitalmicromirror device (DMD), liquid crystal display (LCD) orliquid crystal on silicon (LCoS). In the case of a DMD theetendue is given by the area of all micromirrors and half ofthe angle between on- and off-state – typically this angle is

θOS = 12◦. For a 0.55′′ imager with an area of 93.7mm2the etendue is (with dΩ = sin θdθdϕ and n = 1).

EOS = πn2AOS sin2 θOS = 12.7mm2 sr . (10)

The limitation of the performance of the whole system bythe etendue of the imager can be seen by considering thefollowing example. Let us assume an ideal case, where allthe light emitted into the entire hemisphere is collectedby some optics and projected onto the DMD. From theetendue of the DMD of 12.7mm2 sr and a source angle ofθOS = 90◦, the source area is calculated to 4mm2, wherethe etendue of the source matches the etendue of the opticalsystem, ELS = EOS. Thus, by assuming some given radi-ant emittance of the source, the brightness on the imagercan be enhanced by increasing the source area up to 4mm2,as more source area simply generates more flux. For areaslarger than 4mm2 the etendue of the light source is largerthan that of the DMD, and light will be wasted as it is emit-ted beyond the phase space of the imager. Thus, an increaseof ELS over and above EOS by enlarging the source areastarting from 4mm2 does not enhance the brightness on theimager. In this case, the only way to increase the brightnesson the imager is to improve the radiant emittance from thesurface. Hence, the emitted radiant flux from a given sourcearea has to be increased.Consequently, a reduction of the imager area AOS de-

creases the maximum usable source area ALS and thus, theemitted radiant flux from the source. In terms of costs, thereduction of the imager area is favorable. However, this de-mands a further increase of the performance of high-powerLEDs in order to stay with the same amount of flux pro-jected onto the imager. This increase in efficiency wouldalso close the still existing gap in radiant emittance – thecentral property determining the performance of etendue-limited applications – between LEDs and common lightsources, like HID lamps, and would open up new applica-tion areas for LEDs.In general, depending on the projection system only

light emitted into a cone with apex angle θLS < 90◦ iscollected and projected onto the imager. In this case, thesource area has to be adjusted depending on θLS in order tomatch the etendue of the optical system

ALS =EOS

π sin2 θLS

. (11)

As we will see in the next section, the radiant flux projectedonto the imager for a limited source angle can be enhancedthanks to a more forward emission. On the one hand, thiscan be accomplished while remaining with constant radiantemittance or even with less radiant emittance. On the otherhand, the required source area increases.In Fig. 25 the required source area is illustrated as a

function of the used source angle θLS for different etenduevalues of the optical system. The smaller the apex anglethe larger the chip area causing higher costs of the wholesystem as more and/or larger chips along with bulky opticshave to be used. Here, the total area of several mm2 is not



Figure 25 (online color at: www.lpr-journal.org) Dependence

of the required source area ALS on the used apex angle θLS in

order to match different values of etendue of the optical system

EOS, ELS = EOS.

covered by a single chip but by several. However, largechips are favorable in order to reduce the area occupied bythe spacing between the individual chips.

9.2. Photonic crystals and etendue

In this section we will discuss the benefit from PhC LEDsfor etendue-limited applications due to a more forwardemission. In order to quantify the collimation of the farfield profile we define the directionality as the amount offlux emitted into the apex cone θLS with respect to theoverall emitted flux.But before presenting results for PhC LEDs let us

briefly investigate the amount of directionality achievableby applying diffracting PhCs. As every inplane directionβi/βi of a guided mode gets diffracted by a reciprocallattice vector G, consider for instance the blue circle inFig. 13, two groups of diffraction processes can be dis-tinguished. Firstly, diffraction processes with inplane k-vector βd within the cone of accepted light, 0 � βd �k0 sin θLS depicted by the inner gray circle in Fig. 13, en-hance the directionality. In contrast, diffraction processeswith k0 sin θLS < βd � k0 between the two gray circlesdecrease the directionality. The amount of diffracted inten-sity into the different groups relies on the coupling of theguided mode with the different radiation modes and thuson the Fabry-Perot resonances, see Sect. 4.2. However, ifthe LED supports numerous resonances within the extrac-tion cone and/or a wavelength average softens their impact,the diffracted intensity Id from every guided mode of thedegenerate set with |βi| = βi can be considered constant.Therefore, by assuming βi k0 with βi ≈ G the direc-tionality D resulting from diffraction of a single guidedmode reads [17]

D =2Idk0 sin θLS

2Idk0= sin θLS . (12)

Ignoring the impact of directly emitted light on the far-fieldpattern and assuming a monomode LED – additional guidedmodes will reduce the directionality due to the mismatch

Figure 26 (online color at: www.lpr-journal.org) (a) Emission

profiles of a Lambertian source (black), a thick PhC LED with

G0 = 1.3k0 (red dashed) and a thin PhC LED with G0 = 1.6k0

(red solid). Overall emission of the three LEDs normalized to

unity. (b) Flux within apex angle θLS for the thick PhC LED (red

dashed), the thin PhC LED (red solid), the rough LED (black

dashed) and the rough LED with SiO2 (black solid). (c) Flux

projected on the imager as a function of the apex angle θLS for the

same LEDs as in (b). The etendue of the imager is 12.7mm2 sr

and the required source area for matching this value is given on

the upper x-axis.

between their k-vector and the reciprocal lattice vector –Eq. (12) describes a fundamental limit for the directionality.A directionality within any θLS < 90◦ of 100% cannot beachieved because besides diffraction into the limited apexcone, diffraction outside this cone simultaneously takesplace. In comparison, for a Lambertian emitter sin2 θLS

of the total emitted light is located inside an apex angleθLS – here, the ratio between the area within k0 sin θLS

and the area within k0 determines the directionality insteadof a ratio of lengths as in Eq. (12). For an apex angle ofθLS = 30◦ diffraction of a guided mode results in 50%directionality in the optimum case, whereas only 25% ofthe Lambertian emission lies within the same cone.

In Fig. 26 four different LEDs are compared in termsof their emission profile, the flux within an apex angleθLS and the projected flux on the imager. As a reference



(black dashed line) we choose the rough LED from Fig. 21to have an efficacy of 100 lm per electrical input powerof 1W. For the other LEDs we assume the same internalquantum efficiency and the same electrical performance.Therefore, the overall efficiencies of these LEDs differonly owing to the individual extraction efficiency. Thus, therough LED with SiO2 in Fig. 23 (solid black line in Fig. 26)has an efficacy of 120 lm/W. The emission profile of therough LEDs is approximated by a Lambertian [25]. Thethird LED (red dashed line) is the same thick PhC LEDas in Fig. 21 except that the main reciprocal lattice vectorwas adjusted (G0 = 1.3k0) to yield maximum forwardemission. Thus, the extraction efficiency drops from 45%to 40% and the efficacy is about 68 lm/W. The last exampleis the thin PhC LED from Fig. 23, but again with correctedmain reciprocal lattice vector (G0 = 1.6k0) yielding anextraction efficiency of 64% and an efficacy of 106 lm/W.

Due to the lower number of different guided modes thethin PhC LED shows a more pronounced forward emissioncompared to the thick PhC LED, as depicted in Fig. 26a.However, both have a super-Lambertian emission profile fa-vorable for coupling their emission into an etendue-limitedoptical system. Apart from the emission profile the actualamount of flux within some apex angle θLS also depends onthe total emitted flux, as shown in Fig. 26b. Due to its highefficacy and forward emission the thin PhC LED is superiorto the other examples within an apex angle of θLS < 65◦.A more general remark should also be noted here: a limitedapex angle decreases the efficiency of the system as lightis wasted.After this summary of the performance of the bare

LEDs, in the following their impact on the performanceof the DMD-based projector will be investigated, that hasalready been used in the example of Eq. (10). Here, weassume that the area of a single chip is 1mm2 havingthe efficacy as shown in Fig. 26b. If the whole emissioncan be collected, the maximum applicable source area forthis example is 4mm2. Thus, 480 lm are generated on theimager by incorporation of four rough LEDs with SiO2driven at 1W electrical input power. For smaller apex an-gles θLS < 90◦ still the same amount of flux can be coupledinto the system, if the etendue of the imager is matchedby increasing the source area according to Eq. (11). Thisstems from the constant radiance of a Lambertian emitter.In contrast, for limited apex angles more light is projectedonto the imager by application of PhC LEDs due to thesuper-Lambertian emission profile. Even though the thinPhC LED has less radiant emittance compared to the roughLED with SiO2, more flux is coupled into the projectionsystem, resulting in a peak flux of almost 800 lm on theDMD. But even for rather large apex angles of about 40◦the system with the thin PhC LED still projects 1.3 timesmore flux on the imager compared to the system with thethick rough LED and SiO2. This enhancement is regardlessof the actual source area as only the total flux scales withthe chip area.

In summary, projection applications could benefit fromthe use of PhC LEDs due to their super-Lambertian emis-

sion profile and reasonable extraction efficiency. But theapplicability of a more forward emission is limited owing tothe required source areas and decreased efficiency. Further-more, diffraction outside the desired apex angle restrictsthe amount of directionality achievable with diffractingPhC LEDs.

10. Conclusions

We present an overview of the application of PhCs in LEDs:extraction of light, improving the internal quantum effi-ciency by either band effects or surface plasmon polari-tons, and reduction of mirror losses. In order to reach highwall-plug efficiency, both electrical efficiency and internalquantum efficiency have to be high in addition to extrac-tion efficiency. This puts limits on the vertical layer stackof the LED, as e.g. current spreading has to be taken intoaccount. In nearly all demonstrated designs the PhC is im-plemented as a scattering layer within the LED, acting as aweak perturbation to redistribute light from guided modesto the extraction cone. Thus, we focus on this regime anddiscuss the properties of PhCs determining the route forhigh-performance devices.In general, the vertical layer structure determines the

interaction of the guided modes with the PhC and the totalnumber of guided modes, and its optimization is crucial tothe PhC LED performance. Due to the discrete diffractionbehavior a low number of guided modes is most preferable.A reduction of absorption losses enhances the overall ef-ficiency further. In the end, the extraction efficiency boilsdown to the role of two numbers: the extraction length andthe absorption length. We demonstrate how a PhC LED canbe optimized, staying within the weak perturbation limit,which provides insight into the physical processes and isstill realistic enough to describe most cases of practical in-terest.The freedom to design the angular distribution of the

extracted light opens the road to enhance the performanceof etendue-limited applications owing to the collimation ofthe far-field pattern. From a basic geometrical considerationa fundamental limit for the directionality achievable bydiffracting PhCs is given.

Even though PhCs have demonstrated their potential forlight extraction the benchmark is given by state-of-the-artdevices, like thin-film LEDs with rough surfaces. UsingFDTD we compare thin-film and PhC LEDs, each withan optimized vertical structure. The differences are sub-tle and the overall extraction efficiency of the optimizedstructures is a close match. So, for the now standard thin-film roughened or PhC LED, where the scattering layeris implemented as a thin layer – usually at the semicon-ductor surface – other criteria, like directionality, electricalperformance and cost of production, are decisive.This may be different for PhC LEDs that leave the

regime of weak perturbation. For PhCs in the strong cou-pling regime, the most impressive electromagnetic mod-elling results have been shown. From the authors’ point of



view, the challenge of an electrically driven device alongwith a carving of the active region restricts their applica-tion in PhC LEDs. Here, radical new approaches, drivenby progresses in material science, like core-shell growth ofnanorods, offer new perspectives.

Acknowledgements The authors would like to thank Ross Stanleyfrom CSEM, Neuchatel, Switzerland, and Romuald Houdre from

EPFL, Lausanne, Switzerland, for the introduction to PhC LEDs

and fruitful discussions.

Christopher Wiesmann received thediploma in physics, in 2004 from theUniversity of Regensburg, Regens-burg, Germany. Currently, he is aPhD student working at Osram OptoSemiconductors GmbH, Regensburg,Germany. His research focuses on thesimulation and design of photoniccrystal light-emitting diodes.

Krister Bergenek received a degree inengineering physics from ChalmersUniversity of Technology, Goteborg,Sweden, in 2006. He is employed byOsram Opto Semiconductors GmbH,Regensburg, Germany and he isworking toward the Ph. D. degree atthe University of St Andrews, St An-drews, Scotland. His research inter-

ests include modelling, design and fabrication of pho-tonic crystal light-emitting diodes in the GaN and Al-GaInP material system.

Norbert Linder received his Ph.D.degree in Physics from the Universityof Erlangen, Germany, in 1996. From1996 to 1998 he worked as a post-graduate researcher at the Physics de-partment of the University of Cali-fornia, San Diego, investigating elec-tron spin dynamics in semiconduc-tor structures. In 1998 he joined the

R&D department at OSRAM Opto Semiconductors inRegensburg, Germany. Since then, he has held vari-ous positions as project manager for the developmentof high-brightness LEDs and semiconductor lasers. In2004, Dr. Linder became head of the device modellinggroup that supports all R&D activities in OSRAM OS.His interests are focused on the research and designof nitride and phosphide LEDs and laser diodes, withspecial emphasis on the application of nanooptical struc-tures. He has authored or coauthored more than 40 jour-nal publications and conference presentations and is amember of the German Physical Society (DPG).

Ulrich T. Schwarz received his Ph. D.degree in physical science in 1997from the University of Regensburg,Germany. In 2004, he concluded hishabilitation and since then joined thefaculty as assistant professor. His re-search is in the field of optical charac-terization of nitride semiconductorsand laser dynamics. Another focus is

on singular optics, in particular polarization singularities.Awarded by the Alexander von Humboldt foundationwith a Feodor Lynen scholarship he spent two postdoc-torial years (1997–1999) at Cornell University, Ithaca,NY, with research on intrinsically localized modes. In2001 he joined the group of Prof. R. Grober at YaleUniversity, New Haven, CT. In 2006/2007, he visitedKyoto University, Kyoto, Japan, with an invited fellow-ship (long-term) awarded by the Japanese Society forthe Promotion of Science. He is a member of the OpticalSociety of America (OSA), American Physical Society(APS), International Society for Optical Engineering(SPIE), and the German Physical Society (DPG).

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