Interaction of Periodically Arranged Point Defects in a Two Dimensional Photonic Crystal -The Photonic Analogue to a Doped Semiconductor

S. Richter1, S. L. Schweizer2, R. Hillebrand1, C. Jamois1, R. B. Wehrspohn2, M. Zacharias1 andU. Goesele11Max-Planck-Institut fur Mikrostrukturphysik, Weinberg 2, D-06120 Halle, Germany2Universtat Paderborn, Warburgerstr. 100, D-33098 Paderborn

ABSTRACT

We present and characterize hexagonal point defects in a two dimensional photonic crystal basedon macroporous silicon. These point defects are prepatterned periodically, forming asuperstructure within the photonic crystal after electrochemical etching. Spatially resolved,optical investigations related to morphological properties, like defect concentration and poreradius, are compared to bandstructure calculations. The confined defect states are identified andtheir interaction is evaluated quantitatively.

INTRODUCTION

The concepts of photonic bandgap materials and photonic crystals have been extensively studiedover the recent years. By applying theoretical methods of solid state physics to periodic dielectricstructures, the prediction of their properties and behavior is possible. So the dispersion relation ofa photonic crystal is a bandstructure similar to semiconductors.[1]Structures disturbing the periodicity of the photonic crystal are of special interest, because of theiracting as guiding or confining elements for light within the crystal. If, e.g., a point defect isintroduced, defect levels within the photonic bandgap will arise. Their behavior can be donor likeor acceptor like, regarding on the type of the defect.[2]A first theoretical approach of interacting resonators was proposed by Yariv et.al.[3] Thecoupled-resonator optical waveguide (CROW) relies on evanescent field coupling of resonatormodes of adjacent resonators. Similar structures were realized by a linear arrangement of pointdefects in photonic crystals. [4, 5] A formation of bands within the photonic bandgap wasobserved. The linear arrangement of the point defects causes a breaking of the translationalsymmetry within the crystal. Therefore, only a projection of the bandstructure along the directionof the waveguides is possible.A periodic arrangement of point defects in low concentration, where interaction with each othercan be neglected, would correspond to doping of a semiconductor. A successive increase of thedefect concentration allows the investigation of the point defect interaction. An approach forexperimental and theoretical realisation of periodic point defects is presented in this work.

EXPERIMENTAL

Macroporous silicon is a very suitable model system for photonic crystals. The high refractiveindex contrast allows the fabrication of structures with complete photonic bandgaps. Twodimensional (2D) photonic crystals in the near infrared have been already established. [6, 7, 8]The 2D photonic crystals consist of ordered macropores prepared by an electrochemical etching

Mat. Res. Soc. Symp. Proc. Vol. 797 © 2004 Materials Research Society W3.2.1

Figure 1: Variation of the defect concentration by setting different distances inΓ-K direction; a)1.86 · 104cm−2, distance 13 lattice constants; b)4.91 · 104cm−2, distance 8 lattice constants; c)8.73 · 104cm−2, distance 6 lattice constants.

process. [9, 10] The pores are etched in hydrofluoric acid applying a photoelectrochemicaldissolution process. Using lithographic prestructuring the nucleation spots of the pores can bedefined at the surface of the n-type silicon wafer. This allows to control pore pattern and latticeconstant. During the etching process the backside of the wafer must be illuminated to generateelectronic holes in the silicon which are consumed by the dissolution of silicon. Due toelectrochemical passivation of the pore walls very high aspect rations (length/diameter) of100-500 are obtained. Therefore, macroporous silicon represents an excellent system to study 2Dphotonic crystal properties.The lithographic prestructuring before the etching process allows setting and also leaving outsingle pores in the hexagonal lattice of the photonic crystal as well as the creation of periodicstructures. The investigated structure is a hexagonal cavity similar to H2 cavities, reported byOlivier et.al.[5] A H2 cavity is a point defect in a hexagonal 2D photonic crystal and consists ofseven missing pores. In contrast to the H2 structure, a central pore is introduced to create ahexagonal ring, shown in Fig. 1a. The lattice constant of the host crystal is 700 nm.This point defect geometry causes two types of disturbances. The six missing holes in thehexagonal ring introduce more dielectric material into the crystal. Furthermore, the pores aroundthe defect as well as the center pore show slightly larger diameters than all the other pores of thecrystal. This is caused by the so called proximity effect. During the etching process, a missingpore offers additional electronic holes which are consumed by surrounding pores, leading tolarger pore diameter.The defects studied here, are arranged in a hexagonal lattice according to the symmetry of thephotonic crystal. A variation of the defect concentration is achievable by setting differentdistances between the hexagonal rings. As shown in Fig. 2a-c, the distance between the defectsvaries from 13 to 6 lattice constants inΓ-K direction. In principle, any integer value larger 3 ofthe lattice constant can be chosen. The corresponding defect concentrations are1.86 · 104cm−2, 4.91 · 104cm−2 and8.73 · 104cm−2.To investigate the optical properties of the photonic crystal structures, FTIR (Fourier TransformInfrared) reflection measurements were made by coupling light perpendicular into the cleavedfacet of the pores of the 2D photonic crystal and detecting the reflected light using an IRmicroscope coupled to the FTIR spectrometer.

W3.2.2

Figure 2: a) Scanning electron micrograph of point defects within the photonic crystal, b) Fourier trans-formation of an SEM image of the photonic crystal with periodic point defects, c) Laser surface diffractionpattern with point defect spots between the host reflections.

Figure 3: a) Reflection measurement of a 2D photonic crystal (lattice constant 700 nm,Γ-M direction, TEpolarization), b) Comparison of a reflection measurement to the corresponding bandstructure (r/a= 0.36).

RESULTS and DISCUSSION

The periodic arrangement of the point defects can be verified by comparing a Fouriertransformation of scanning electron microscope (SEM) images, as shown in Fig. 2b, and a lasersurface diffraction pattern (Fig. 2c). A laser hits the surface of the crystal parallel to the pores, thedirectly reflected intensity is detected on a flat screen and imaged by a digital camera. This causesa diffraction pattern, because the wavelength of the laser (488 nm) is smaller than the latticeconstant of 700 nm. This pattern of the hexagonal surface lattice is similar to the Fouriertransformation of the SEM images (Fig. 2b). In between the six spots of the perfect photoniccrystal (distancea∗) a hexagonal pattern appears with distancesa∗/d, whered · a is the distancebetween two defects in real space. The distances of the additional pattern correspond to ahexagonal lattice with a lattice constant 8 times larger for this example than the lattice constant ofthe host crystal.For the unperturbed crystal the reflection spectrum features a broad frequency interval (from 0.22to 0.34 norm. frequency) of high reflectivity (Fig. 3a). Comparing this spectrum to thecorresponding bandstructure a matching of high reflectivity and fundamental bandgap is found,Fig. 3b. Light frequencies within the bandgap cannot enter the crystal and will be reflected byinterference total reflection.

W3.2.3

Figure 4: a) Reflection measurement of a PC including point defects (defect concentration1.86 · 104cm−2,distance 13 lattice constants) compared to a bandstructure calculation; b) same as a) but for a defect concen-tration of4.91 · 104cm−2, distance 8 lattice constants.

A reflection spectrum for a crystal including point defects (defect concentration1.86 · 104cm−2)is shown in Fig. 4a. The plateau of high reflectivity shows several notches of lower reflectivity. Acomparison of the reflection measurement to the corresponding bandstructure calculation clearlyidentifies these notches as defect states within the photonic bandgap. Light that enters the crystalcouples to resonators close to the surface of the cleaved pores and will not be reflected. Thebandstructure shows 19 states within the first photonic bandgap for TE polarization (Fig. 4a).These states are energetically separated and form discrete levels within the bandgap. This is aclear indication for separated point defects within the crystal. Due to their distance of 13 latticeconstants they are not interacting.The large number of defect levels can be explained by the complex defect geometry. Two types ofdefect states are expected, dielectric defects due to the six missing pores and air defects caused bylarger pores during the etching process [proximity effect]. Dielectric defects are similar to donorlevels in semiconductors. Introducing some more dielectric material to a crystal site, additionalstates are pulled from the air band into the bandgap. Vice versa for the so called air defects, theycorrespond to acceptor levels in semiconductors. Defect states from the dielectric band arecreated within the bandgap.Increasing the defect concentration to4.91 · 104cm−2 (corresponding to a defect distance of 8lattice constants) causes broader and deeper notches in the high reflectivity region (Fig. 4b).Light, entering the crystal, couples to more resonators close to the surface of the cleaved pores.After a further increase of the defect concentration (1.26 · 105cm−2, distance of 5 latticeconstants) only several peaks of high reflectivity remain in the spectrum (Fig. 5a). The reason forthese peaks is the overlapping of defect modes. This is shown via the correspondingbandstructure (Fig. 4b), discrete levels begin to form bands within the bandgap. A bending oflevels towards air and dielectric band, respectively, takes place. Only small gaps remain inbetween bands of defect levels (Fig. 5a).To extrapolate this behavior, a bandstructure for a defect density of1.96 · 105cm−2 (defectdistance of 4 lattice constants) is calculated (Fig. 5b). Within the host bandgap several groups ofbands are forming. Smaller gaps remain in between them. The host bandgap splits up in 2 smaller

W3.2.4

Figure 5: a) Reflection measurement of a PC including point defects (1.26 · 105cm−2, distance 5 lat-tice constants) compared to a bandstructure calculation; b) bandstructure for a defect concentration of1.96 · 105cm−2, distance 4 lattice constants.

gaps.To characterize the interaction of the resonators quantitatively, the bandwidth∆ of the defectstates within the original bandgap of the pure photonic crystal was calculated. To comparedifferent concentrations the bandwidths of all defect states were averaged. An increase of morethan one order of magnitude in the investigated range of defect distances was observed and isdisplayed in Fig. 6.An single exponential decay fit,∆avg = A + Be−

Dγ (decay constantγ = 0.81a), shows an

excellent agreement with the averaged bandwidth∆avg. This can be explained by an increasingcoupling of the resonator modes with decreasing distance of the resonators. Localized fields ofresonator modes can be treated similar to electron wavefunctions in a tight-binding model.[3, 11, 12] If the distance between two resonators decreases, the localized fields begin to overlapand to split. The tight-binding model predicts such an exponential decay of the interaction withincreasing distance.

CONCLUSION

A periodic arrangement of point defects in a photonic crystal is fabricated and analyzed. Thepoint defect concentration ranges from small perturbations of the crystal to strong modificationsof its original periodicity. The periodicity of the point defects is verified by laser surfacediffraction, an optical technique similar to diffraction experiments in solid state physics.The optical properties are investigated by a comparison of reflection measurements andbandstructure calculations. At low point defect concentrations, discrete levels inside the photonicbandgap appear. These levels are the resonator modes of the point defects. They can be clearlyidentified by reflection measurements.An increase of the defect concentration causes a broadening of the discrete levels to bands andresults in narrow gaps in between the bands. A quantitative evaluation of the defect bands showsan exponential decay of their bandwidth with increasing distance between them. This is ameasure for interaction of periodic point defects.

W3.2.5

Figure 6: Averaged bandwidth∆ of defect levels as function of the defect distance

ACKNOWLEDGEMENTWe gratefully acknowledge the DFG (Deutsche Forschungsgemeinschaft) for funding this project(No. Za 191/13-1).

[1] S. John.Phys. Rev. Lett., 58:2486–2489, 1987.[2] J.D. Joannopoulos, R.D. Meade, and J.N. Winn.Photonic crystals: Molding the flow of light.

Princeton University Press, 1995.[3] A. Yariv, Y. Xu, R. K. Lee, and A. Scherer.Opt. Lett., 24:711–713, 1999.[4] M. Bayindir, B. Temelkuran, and E. Ozbay.Phys. Rev. B, 61:11855, 2000.[5] S. Olivier, C. Smith, M. Rattier, H. Benisty, C. Weisbuch, T. Krauss, R. Houdre, and U. Oesterle.

Opt. Lett., 26:1019–1021, 2001.[6] U. Gruening, V. Lehmann, S. Ottow, and K. Busch.Appl. Phys. Lett., 68:747, 1996.[7] A. Birner, U. Gruening, S. Ottow, A. Schneider, F. Mueller, V. Lehmann, H. Foell, and U. Goesele.

Phys. Stat. Sol. (a), 165:111, 1998.[8] J. Schilling, A. Birner, F. Mueller, R. B. Wehrspohn, R. Hillebrand, U. Goesele, K. Busch, S. John,

S. W. Leonard, and H. M. van Driel.Opt. Mater., 17:7–10, 2001.[9] V. Lehmann and H. Foell.J. Electrochem. Soc., 137:653, 1990.

[10] V. Lehmann.J. Electrochem. Soc., 140:2836, 1993.[11] M. Bayindir, B. Temelkuran, and E. Ozbay.Phys. Rev. B, 84:2140, 2000.[12] E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis.Phys. Rev. Lett., 81:1405, 1998.

W3.2.6