University of Groningen. Photonic bandgap optimization in inverted fcc photonic crystals Doosje, M; Hoenders, B.J; Knoester, Jasper

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1 University of Groningen Photonic bandgap optimization in inverted fcc photonic crystals Doosje, M; Hoenders, B.J; Knoester, Jasper Published in: Journal of the Optical Society of America B-Optical Physics DOI: /JOSAB IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 2000 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Doosje, M., Hoenders, B. J., & Knoester, J. (2000). Photonic bandgap optimization in inverted fcc photonic crystals. Journal of the Optical Society of America B-Optical Physics, 17(4), DOI: /JOSAB Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from the University of Groningen/UMCG research database (Pure): For technical reasons the number of authors shown on this cover page is limited to 10 maximum. Download date:

2 600 J. Opt. Soc. Am. BIVol. 17, No. 41April 2000 Doosjeet al. Photonic bandgap optimization in inverted fcc photonic crystals Marcel Doosje, Bernhard J. Hoenders, and Jasper Knoester Institute for Theoretical Physics and Materials Science Centre, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands Received June 9, 1999; revised manuscript received November 1, 1999 We present results of photonic band-structure calculations for inverted photonic crystal structures. We consider a structure of air spheres in a dielectric background, arranged in an fcc lattice, with a cylindrical tunnel connecting each pair of neighboring spheres. We derive (semi)analytical expressions for the Fourier coefficients of the dielectric susceptibility, which are used as input in a standard plane-wave expansion method. We optimize the width of the photonic bandgap by applying a gradient search method and varying two geometrical parameters in the system: the ratios Ria and R c IR, where a is the lattice constant, R is the sphere radius, and R c is the cylinder radius. It followsfrom our calculations that the maximal gap width in this type of photonic-crystal structure with air spheres and cylinders in silicon is Δωi ω0 = 9.59%. 2000Optical Society of America [S (00) ] OelS codes: , , , , INTRODUCTION The effect that photonic crystals, i.e., materials with a spatially periodic dielectric function ε(r), have on the propagation of electromagnetic radiation has been studied intensively over the past few decades.1,2 The dispersion relation ω(k) of these crystals is described by the photonic band structure, which consists of different photonic bands with gaps between them at the boundary of the Brillouin zone. The photonic bandgap is characterized by its central frequency ωo and its spectral width Δω and generally depends on the direction of wave propagation. It is well known that, for sufficient dielectric contrast, fcc photonic crystals may develop a full photonic bandgap, which overlaps the entire surface of the Brillouin zone, between the eighth and ninth bands of the band structure. Besides, a pseudogap develops between the fifth and sixth bands, which overlaps completely only a part of the Brillouin-zone surface. One of the main motivations to produce dielectric structures that possess large photonic bandgaps is the desire to be able to control (i.e., to suppress or, in some cases, to enhance) the spontaneous emission of radiation from atoms.3-5 Other possibilities for practical applications of these photonic crystals include the realization of lowthreshold lasers, high-quality resonant cavities, perfectly reflecting mirrors, and substrates for antennas. In artificial photonic crystals, such as those created by Yablonovitch et al.6 and by Özbay et al.,7 a complete photonic bandgap has been found for frequencies in the microwave regime. Until now, the smallest length scale at which artificial photonic crystals have been realized was of the order of the wavelength of infrared light.8,9 Much effort is taken nowadays to develop photonic crystals that have a complete photonic bandgap for frequencies of visible light. A new type of photonic crystal is the inverted fcc crystal, which consists of a regular structure of spherical air holes in a dielectric medium.1o-16 Such structures, with lattice constants of the order of several hundreds ofnanometers, may be created by infiltration of dielectric material into the empty voids in fcc crystals that consistof colloidal spheres.17 The photonic properties of structures thus created have been studied by several groups of researchers One obtains an inverted crystal if the original colloidal material is etched away after the infiltration process. 10 Such crystals hold great promise for having complete bandgaps in the visible range. 21 Usually, the original colloidal crystal is closely packed (volume filling of 74%), so one would expect a closely packed inverted air-sphere crystal as final result. This does not happen, however, because during thermal annealing (sintering) of the colloidal crystal22 the colloidal particles become firmly attached, so the filling fraction increases above 74%. This effect may be modeled as the formation of cylindrical tunnels that connect neighboring spheres. In this spirit, the dependence of the photonic bandgap on the radius of the cylindrical air tunnels has been studied theoretically by Busch and John.23 Those authors also studied the effect of incomplete infiltration of the opal. The fact that in these inverted opal structures both the dielectric region and the air region form a connected network (as a result of the air tunnels) is favorable for increasing the width of the photonic gap.24,25 However, a complete, omnidirectional bandgap has not been measured yet, because the materials used, such as titania 10,11and carbon,12 have indices of refraction that are not sufficiently high. Here we present results of the optimization of the calculated photonic bandgap in the fcc structure of air spheres with connecting air tunnels, surrounded by silicon. The photonic band structures of these crystal structures were calculated by means of the plane-wave expansion method (Subsection 2.A below).1,26-29 A part of these calculations is the derivation of expressions for the $ Optical Society of America

3 Doosje et al. Vol. 17, No. 4/ApriI 2000/J. Opt. Soc. Am. B 601 Fouriercoefficients of the dielectric susceptibility of this structure (Subsection 2.B). In the band-structure calculations, the results of which are presented in Section 3, wevaried two parameters, namely, the ratios R/a and Rc/R. Here R is the sphere radius, Rc is the cylinder radius,and a is the lattice constant of the fcc lattice. We findthat the width of the bandgap depends strongly on thetotal volume-filling fraction of the air regions. To be morespecific, the gap width tends to increase substantiallywhen the ratio R c /R increases from 0 to 0.4, that is, whenin this way the amount of highly refractive material decreases. By applying a gradient search method in combination with the variation of the two parameters mentioned above, we obtain the geometrical parameters for whichthe relative gap width Δω/ωo reaches its maximum. We discuss our results and conclude in Section METHOD OF CALCULATION A. Plane-Wave Expansion Method Thecalculations were performed by means of the classic plane-wave expansion method. The periodic dielectric structure is modeled by a Fourier series for the inverse dielectricfunction η( r) : 1 η(r) = _( ) = Σ ηm exp( -igm. r), (1) ε r m where m labels the three-dimensional set of reciprocallattice vectors gm' In principle, the summation Σm is carriedout over all reciprocal-lattice vectors that exist in reciprocal space. The Fourier coefficients ηm are given by ηm = 1/Ω In η(r)exp(igm. r)dr, (2) where the integration extends over the volume Ω of one unit cell in the direct lattice. Because of the inversion symmetry possessed by the unit cell and the fact that the dielectric function ε (r) is assumed real, the Fourier coefficients ηm are also real. In photonic crystals, where ε (r) is periodic and µ(r) is constant, Maxwell's equations are solved preferentially in terms of the magnetic field H. The reason for this lies in the transversality of the magnetic field and in the Hermiticity of the operator, applied to the magnetic field in the left-hand side of the wave equation 1 a2h x [η(r) x H] = -_2-2 ' (3) c at as explained by Joannopoulos et al.l The magnetic field is written as a Fourier series as well: 3 H(r,t) = exp(iωt) Σ Σ Σ hmλ(k)ûmλexp( -ik m. r), m k λ=l (4) which is known as the Bloch-wave expansion. Here, for any label m, {ûm1, ûm2,ûm3}is a right-handed orthonormal basis for Euclidian space, chosen in such a way that ûm3 km. Furthermore, km = k + gm, and the summation over k denotes the summation over all wave vectors within the first Brillouin zone. The length of the vector k m will be denoted k m. Substituting the Fourier expansions for η(r) and H(r,t) into Maxwell's equations, one arrives at the matrix equation um ^2. Ul ^2 _^2. um U^l][hl(k)] l l ) Σ klk m ηm-l [ _ûl. û2 ûl. ûll hl2(k) l m l m = (ω/)2[h1m2(k)]. (5) c hm(k) Note that the A = 3 components of the magnetic field vanish because. H = O. In practical cases, a finite number of N reciprocallattice vectors is selected, and the set of fundamental equations (5) is commonly written as a 2N X 2N matrix equation. Then the eigenvalues of the 2N x 2N matrix must be identified with (ω/c)2, which immediately gives us the photonic band structure. This 2N X 2N matrix M is constructed in the following way: We regard the matrix M as an N x N matrix with elements M ml, where each of the labels m and l corresponds to one of the reciprocal-lattice vectors gm and gl selected. In turn, each element M ml is a 2 X 2 matrix, assigned according to the following prescription: U^2 m. Ul ^2 -U^2 m. Ul ^l] Mml = klkm ηm-l [ ^l ^2 ^l ^l' (6) -U m. Ul U m Ul Because of the inversion symmetry mentioned above and the assumption that the index of refraction of the material is a real quantity, matrix M is real and symmetric. Its eigenvalues are real-valued quantities, to be identified with (ω/c)2. Our calculations show that the eigenvalues are indeed nonnegative. B. Including the Material Properties In the plane-wave expansion method we need to substitute into Eq. (5) the Fourier coefficients ηm of the inverse dielectric function η(r). These coefficients may, of course, be calculated directly from Eq. (2). It turns out, however, that an indirect calculation by means of the Fourier coefficients ε m of the dielectric function ε (r) leads to a more efficient band-structure calculation (see below). To this end, one first calculates the Fourier coefficients of the dielectric susceptibility χ(r) according to χm = 1/Ω In χ(r)exp(igm. r)dr, (7) [compare Eq. (2)] and obtains the Fourier coefficients ε m as ε m = δm,o + χm' Next, one defines the Fourier coefficients matrix with matrix elements εm,m' εm-m,. Inverting this matrix, one obtains the Fourier coefficient matrix that contains the desired coefficients ηm :

4 602 J. Opt. Soc. Am. BIVol. 17, No. 41April2000 Doosjeet al. ηm,m' ηm-m' = εm,m' -1. Both εm,m' and ηm,m' are symmetric matrices. We note that, strictly speaking, this method to obtain the coefficients ηm is incorrect, as it holds only in the limit of infinite matrix sizes (N -7 (0), whereas one is restricted to a finite matrix size in performing numerical computations. Yet it has been shown numerically that, with the above procedure (referred to in Ref. 30 as Ho's method), the eigenvalues of M converge to the correct values much faster for increasing N than if one used the direct approach (referred to in Ref. 30 as the slow method). This motivates us to use the matrix-inversion method. However, to our knowledge a convincing mathematical explanation for this artifact still does not exist. Next we turn to the actual calculation of the coefficients χm' Motivated by their potential to exhibit large photonic bandgaps, we consider the inverted opals described in Section 1, consisting of air spheres and air tunnels surrounded by a dielectric. To be specific, we consider an fcc structure, with lattice constant a, of nonoverlapping air spheres of radius R, each of them connected to its 12 nearest neighbors by a cylindrical air tunnel of radius R c. A three-dimensional visualization of this structure is shown in Fig. l(a). The dimensions of lengths a, R, and R c are indicated in Fig. l(b). For these crystals, evaluating the coefficients χm according to Eq. (7) involves an integration over the dielectric regions that occur between the air spheres and the air tunnels. It is much more convenient, however, to perform integrations over the spheres and the cylinders themselves. To achieve this, we note that the band structure of a photonic crystal that is a composite of two dielectric materials (ε = εs for the spheres and ε = εb for the background) is determined not by the values of εs and εb independently but only by the dielectric contrast εs / εb' The only way in which εb enters the band structure as a separate parameter is by a trivial rescaling of the frequency scale by a factor of εb. Using the above values, we may consider the equivalent crystal with the effective dielectric constants lattice of overlapping spheres, where R/a > 1/4 2. First we give the volume-filling fraction φ of air in our (original) crystal: 80 φ = -- π(r/a)3 + 12π 2(Rc /a)2 3 R2 _ R2) π ( a c (10) The condition for this expression to be valid is that Rc R/2. For Rc = 0, Eq. (10) reduces to φ = 16π(R/a)3/3. εs 1 + χs εe = - = XE (8) εb 1 + χb for the spheres and ε = 1 (i.e., χ = 0) for the background. From Eq. (8) we obtain the effective dielectric susceptibility of the spheres and cylinders in the equivalent crystal: χs - χb (9) X E = 1 + χb. Note that, for our case of air spheres and air cylinders in the original crystal, X E < 0 (χs = 0). As the effective susceptibility for the regions between the spheres and the cylinders vanishes, the calculation of the Fourier coefficients for the equivalent crystal, according to Eq. (7), indeed involves only integrations over the spheres and the cylinders. These integrations can be performed semianalytically, in close analogy to the calculations performed by Leung and Liu 31 for a different geometry, namely, that of an fcc Fig. 1. (a) Three-dimensional visualization of an fcc air-sphere crystal with nonoverlapping air spheres, each of them connected to all of its 12 nearest neighbors by cylindrical air tunnels. The particular structure shown here has geometrical parameters, which for a silicon background (gray) optimize the bandgap (see Section 3). (b) Schematic cross section of the structure in (a) with definitions of the lengths a, R, and R c indicated.

5 Doosjeet al. Vol. 17, No. 4/ April 2000/J. Opt. Soc. Am. B 603 Obviously, the zeroth Fourier coefficient of the equivalent crystal is given by χ0 = φxe. For the derivation of the other coefficients χm of the equivalent crystal we need to define six vectors Qm, two of which have the following representations in cylindrical coordinates (Qmρ, Qmz): Qmρ = [g2mz + l/2(gmx ± gmy)2]1/2, Qmz = 1/2 2(gmx gmy); (11) the remaining four are given by cyclic permutations of x, y, and z. Then the analytical expression for χm(m 0) isgiven by where 16πR3 ] χm = XE [_a3s(gmr) + ΣQm I(Qm) (12) sin x - x cos x S( x) 3 x (13) Thesummation in Eq. (12) is over all six vectors Qm definedabove, and the integral I(Qm) is given by I(Qm) = 8 2π R e la 1/4 2 φ=0 ρ=0 z=[(rla)2-ρ2l1/2 X ρ exp(iρqmρ cos φ) whichwe can evaluate to Re/a I(Qm) = 16π 0 ρj0(ρqmρ) f f0rqmz = 0 and to X cos(qmzz)dρdzdφ, (14) X {1/4 2 - [(Rla)2 - ρ2]1/2}dρ (15) In this section we discuss some characteristics of the phot0nicband structure of air-sphere crystals,10-16 which we calculated by the plane-wave expansion method as described in Section 2. The crystals under consideration are the inverted opals described in detail in subsection 2.B(see Fig. 1). We shall take silicon [Si;n = (Ref. 32)]as our choice for the dielectric that surrounds the air spheres and cylinders in this crystal. For explicitness, Fig. 2 shows the first Brillouin zone of the fcc lattice, in which the important symmetry points are indicated by the conventional symbols. A typical band structure for a closely packed fcc crystal 0fair spheres in silicon (without the connecting cylinders) is shown in Fig. 3. Clearly seen are the pseudogap between the fifth and sixth bands and the full gap between the eighth and ninth bands. In what follows, we shall concentrate mainly on the full gap. One usually characterizes this gap by the quantity Δωi ω0, which is the relative width of the frequency range that is common to this bandgap in all directions in reciprocal space. The lower bound of this frequency range (ω_) is obtained at the point where the eighth band of the photonic band structure reaches its maximum, usually at point W of the Brillouin zone. The upper bound of this frequency range (ω+) is obtained at the point where the ninth band reaches its minimum, usually at point X of the Brillouin zone. However, for some values of the geometrical parameters of the system, for instance in air-sphere crystals without the air cylinders, when the volume-filling fraction of the air spheres decreases below 67% it turns out that the lower and upper bounds of the full gap occur at different high-symmetry points. The relative gap width is given by Δω ω+ - ω_ (17) - = ω0 ω+ + ω_ Our main interest is in the dependence of Δωl ω0 on the geometrical parameters Ria and Rcla. To this end, we have calculated band structures by varying these two parameters. In Fig. 4 we plot the relative bandgap thus ob- I(Qm) 16π = - fr e /a ρj0(ρqmρ) ( ( Q ) sin mz_ 2 Qmz sin{qmz[(rla)2 - ρ2]1/2})dρ (16) f0r Qmz O. The remaining integrals that appear in Eqs.(15) and (16) contain the Bessel function J 0 and have t0be evaluated numerically. 3. RESULTS Fig. 2. First Brillouin zone for the fcc lattice (body-centered cubic in reciprocal space). The important symmetry points are denoted by the conventional symbols. Fig. 3. Photonic band structure of a closely packed fcc crystal of air spheres in silicon, calculated with N = 339 plane waves. This particular example has no cylinders connecting neighboring spheres.

6 604 J. Opt. Soc. Am. B/Vol. 17, No. 4/April 2000 Doosjeet al. Fig. 4. Relative width of the full photonic bandgap between bands eight and nine in the fcc inverted opals as a function of Rc/a. The background material is silicon (Si;n = 3.415). The number of plane waves used is N = 339. The four curves have been calculated for (a) R/a = , (b) R/a = , (c) R/a = , and (d) R/a = For R c = 0 these values correspond to φ = 68%, φ = 71%, φ = 7 2%,and φ = 74%, respectively. tained as a function of the cylinder radius (Rc/a) for four different values of the sphere radius (R/a). These results were obtained for N = 339 plane waves. It is known that, for this type of crystal structure, one needs N ~ 103 plane waves to reach convergence of the band structure well below the 1% level.23 We used such high values for N only when we were optimizing the bandgap and calculating the density of states (see below). Comparison has taught us that the relative errors in the bandgap at N = 339 are of the order of 6%. Whereas this is by no means full convergence, it does suffice to provide considerable insight into the trends of the band structure as the parameters are varied, as we shall see below. Returning to Fig. 4, one clearly observes that at first, when there are only very narrow cylindrical air holes present, as is the case in the crystal structure created by Wijnhoven and Vos (Rc/R 0.2, as we estimate from the scanning-electron microscope pictures in Ref. 10), their effect on the gap width is rather small. If we go toward larger values of the radius R c, the positive influence on the gap width becomes increasingly prominent. From our calculations we conclude that, for the type of geometry under consideration here, the largest relative gap width is achieved near R c /R = 0.4 and that it tends to increase when the value of R/a decreases, i.e., when also the volume-filling fraction of the air spheres decreases. Curve (d) in Fig. 4, which gives the dependence of the gap width on the cylinder radius for closely packed spheres (R/a = ), agrees well with the results of Busch and John.23 We finally note that a remarkable feature of Fig. 4 is that the four curves nearly cross at one point for Rc/a 0.052, where the gap width is Δω/ωo 4.2%. We were unable to establish whether this feature is sheer coincidence or whether it has a deeper significance. Next, we consider the optimization of the relative width of the bandgap. To find the values of R/a and R c /R for which Δω/ωo reaches its maximum, we performed a twoparameter variation calculation based on a gradient search method. This method consists of successively determining the gap width f Δω/ ωo and the gradient f [df/d(r/a),df/d(rc/r)] for certain values ofr/a and R c /R and adjusting R/a and Rc/R such that a step is taken in the direction of 'If. Thus we reach the maximal value of f by following the path of steepest ascent. In a first, crude approach toward the maximal gap, we used band-structure calculations with N = 339 plane waves, as before. We started at R/a = and Rc/R = 0.37, which yielded the maximal gap width as shown in Fig. 4, curve (a), and initially the adjustment of the parameters was a decrease of R/a of the order of 0.02 per step. Smaller step sizes were used as the value of IV'f decreased. Thus, following the path prescribed by the gradient search method, using of the order of 15 steps, we finally reached R/a = and R c /R = 0.398, for which f = 9.3% and f O. The central frequency of the gap obtained in this way is ωo/2π = 0.744c/a. At all points along this path of steepest ascent, the width of the full photonic bandgap turned out to be determined by point W in band eight and by point X band nine. To obtain a more reliable value for the bandgap in this geometry, i.e., closer to convergence, we also calculated the gap width of the structure with R/a = and Rc/R = using N = 1037 plane waves. The result Fig. 5. Photonic band structure of an fcc inverted opal with silicon (n = 3.415) as the dielectric medium and with geometrical parameters R/a = andR c /R = Theseparameters optimize the relative width of the full gap between the eighth and ninth bands to Δω/ ω0 = 9.59% at ω0/2π = 0.746c/a. The volume-filling fraction of air spheres and cylinders in this structure is φ = 66.3%. The number of plane waves used is N = Fig. 6. Density of states (DOS) corresponding to the photonic band structure depicted in Fig. 5.

7 Doosjeet al. Vol. 17, No. 4/Apri1 2000/J. Opt. Soc. Am. B 605 ofthis calculation is Δω/ωo = 9.59%. Further fine tuning of the optimum by applying the gradient search methodonce more, but now with N = 1037 plane waves are starting from the optimum obtained above, yield negligible changes: The maximally achievable gap width was found to be Δω/ωo = 0.95% at R/a = and Re/R = The central frequency of the gap is ω0/2π = 0.746c/a. The photonic band structure for this geometry, calculated with N = 1037 plane waves, is shownin Fig. 5. The corresponding density of states, calculated by use of the Monkhorst-Park discretization schemefor the Brillouin zone,33is shown in Fig. 6. This figureclearly demonstrates the existence of a full gap. 4. DISCUSSION AND CONCLUSIONS Wehave studied the photonic band structure of inverted fcc photonic crystals, using the plane-wave expansion method. The crystals under consideration consist of an fcc structure of spherical air regions surrounded by a highlyrefractive material. The air spheres may be connectedby cylindrical air tunnels. Recently such inverted fcc photonic crystals, with lattice dimensions at the length scale of the wavelengths of visible and nearinfrared light, were created experimentally by Wijnhoven andvos.10 Although the main purpose of the presence of the tunnels between neighboring air spheres in these inverted opal structures is to serve as passageways through which the original colloidal material is removed,10,12it turns out from our calculations that these holes also have a positive influence on the width of the full photonic bandgap, provided that their radius is large enough. Applying a gradient search method for these inverted opals, we optimized the relative width of the photonic bandgap by varying two parameters, R/a and Re/R. The optimal value that we obtained for the gap width is Δω/ωo = 9.59% for a structure of air spheres in silicon, with sphere radius R = a and cylinder radius R e = 0.398R. The central frequency of this bandgap is ω0/2π = 0.746c/a. In general, the width of the photonic bandgap depends strongly on the dielectric contrast and on the volumefilling fraction. We point out that, for the optimized structure described above, the volume-filling fraction of siliconis only 33.7%; i.e., the optimal bandgap occurs in a rather empty structure [see Fig. 1(a)]. The controlled preparation and manipulation of such empty crystal structure will be a real challenge for material scientists and experimentalists. It will be worthwhile to investigate whether also for other classes of geometries such empty structures optimize the bandgap. As is well known, the calculated band structures also depend significantly on the number N of plane waves considered. It is accepted quite generally that, for N ~ 10 3,the error in the results for the lowest-lying photonic bands is well below 1% (for the type of crystal that we consider). Our calculations reveal that for N = 339 (which allows for much faster computing), the relative error in the bandgap is still of the order of6%. At the same time, however, we have demonstrated that, even with this rather small number of plane waves, the dependence of the bandgap on the geometrical parameters can be studied with appreciable accuracy. In particular, we have shown that, using the gradient search method with N = 339 plane waves, one may determine the geometrical parameters that optimize the band gap quite accurately. By fine tuning these results, using N = 1037 plane waves, we found that the values of the optimal parameters change by only a small fraction, namely, less than 0.1% for the present case. We note that the crystal structure considered by us formally belongs to the class of A 7 structures discussed by Chan et al. 34 The specific examples of this class considered by those authors consist of cylinders connecting neighboring lattice sites, for which they reported relative band gaps of up to 30% between the second and third bands. Our sphere-cylinder crystals would reduce to the specific class considered in Ref. 34 if R < R c This parameter range, however, lies outside the range considered by us [see below Eq. (10)] and also seems to be less relevant to the experimentally realized inverted opals. We finally note that an alternative method for calculating photonic band structures is a photonic analog of the on-shell multiple scattering theory and the Korringa- Kohn-Rostoker method,35 which was used by Moroz36 and by Moroz and Sommers.37 This method is known to yield convergence of the results much more rapidly than the plane-wave expansion method. 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