The coupled optical interface modes in a Fibonacci dielectric superlattice are studied. In the dielectric continuum approximation, the dispersion relation is found to have two bands of dual triadic Cantor structures, each being nonuniform scaling. For most of the eigenfrequencies, the amplitude profiles of electrostatic potential in this quasiperiodic structure are critical. Moreover, an invariant is analytically derived and is used to describe the general features of the frequency spectra and potential profiles.published_or_final_versio

Hu, A

Jiang, SS

Jin, GJ

Kang, SS

Wang, ZD

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Title Coupled optical interface modes in a Fibonacci dielectricsuperlatticeAuthor(s) Jin, GJ; Kang, SS; Wang, ZD; Hu, A; Jiang, SSCitation Physical Review B - Condensed Matter And Materials Physics,1996, v. 54 n. 17, p. 11883-11886Issued Date 1996URL http://hdl.handle.net/10722/43176Rights Creative Commons: Attribution 3.0 Hong Kong LicenseCoupled optical interface modes in a Fibonacci dielectric superlatticeG. J. Jin and S. S. KangDepartment of Physics and National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093,People’s Republic of ChinaZ. D. WangDepartment of Physics, The University of Hong Kong, Pokfulam Road, Hong KongA. Hu and S. S. JiangDepartment of Physics and National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093,People’s Republic of China~Received 25 March 1996; revised manuscript received 22 May 1996!The coupled optical interface modes in a Fibonacci dielectric superlattice are studied. In the dielectriccontinuum approximation, the dispersion relation is found to have two bands of dual triadic Cantor structures,each being nonuniform scaling. For most of the eigenfrequencies, the amplitude profiles of electrostatic po-tential in this quasiperiodic structure are critical. Moreover, an invariant is analytically derived and is used todescribe the general features of the frequency spectra and potential profiles. @S0163-1829~96!02742-7#In recent years, there has been considerable interest inelementary excitations in artificial multilayers orsuperlattices.1–3 In particular, the optical phonon problem inalkali halide or polar semiconductor superlattices is ratherattractive.2 Usually, the optical modes can be divided intotwo types: one is a type of bulklike excitations, the other is atype of interface mode. Interestingly, the interface modeswill be coupled to give the collective excitation of the wholesuperlattice when the layer thickness of the system is rela-tively small. On the other hand, since the discovery of aquasicrystalline phase in Al-Mn alloys, great experimentaland theoretical efforts have been devoted to physical proper-ties in one-dimensional quasiperiodic structures.4–7 As iswell known, the quasiperiodicity of Fibonacci structure hassubstantial impact on the properties of elementary excita-tions. Therefore, it is worthwile to explore the properties ofcoupled optical interface modes in Fibonacci dielectric su-perlattices. In this paper, we first derive the basic formulasfor transfer matrices, an invariant, and the dispersion rela-tion. Then the numerical results as well as relevant discus-sions are presented.A Fibonacci superlattice is a simple one-dimensional qua-siperiodic structure with two building blocks denoted by Land S . For the structures considered here, each of them isconstructed by two layers with materials A and B . The Blayers in L and S blocks have the same thickness d , but theA layers have thickness dL in L blocks and dS in S blocks,respectively. Using these two blocks, a Fibonacci dielectricsuperlattice is formed according to the rule: S j115$S j ,S j21%,S15L ,S25LS . A and B are two kinds of di-electric materials with different dielectric functions «A and«B , which are the same as those in the corresponding infinitemedia and may be frequency dependent.In the electrostatic limit, the electrostatic potential F sat-isfies the Laplace equation ¹2F(r,t)50. If the z axis ischosen to be perpendicular to the superlattice planes, withoutloss of generality, we assume that only a plane-waveexp(ikx) propagates along the x direction with k as the in-plane wave vector. It is reasonable to write F(r,t)5f(z)exp$i(kx2vt)%, and thus,S d2dz2 2k2Df~z !50. ~1!Denoting n as a layer index, the electrostatic continuum con-ditions at the interface takes the formfn~z !5fn11~z !, «ndfn~z !dz 5«n11df~z !dz . ~2!The solutions of Eq. ~1! can be written as f l(z)5glekz1hle2kz in the A layers, and f l(z)5plekz1qle2kzin the B layers, where l denotes the block index. If we writeS gl11hl11D 5Tl11,lS glhlD ~3!for A layers, it is straightforward to obtainTl11,l5S aek~dl111dl!/2, bek~dl112dl!/22be2k~dl112dl!/2, ge2k~dl111dl!/2D , ~4!wherea5coshkd112 S «B«A 1 «A«BD sinhkd ,b512 S «B«A 2 «A«BD sinhkd , ~5!g5coshkd212 S «B«A 1 «A«BD sinhkd .PHYSICAL REVIEW B 1 NOVEMBER 1996-IVOLUME 54, NUMBER 17540163-1829/96/54~17!/11883~4!/$10.00 11 883 © 1996 The American Physical SocietyOne can find that there are three types of transfer matricesTL ,L , TS ,L , TL ,S , which are all unimodular. As usual, we setM 15TL ,L and M 25TL ,STS ,L , and have recursion relationsM j115M j21M j , from which all M j8s can be obtained,where j is the Fibonacci generation number.Defining x j5 12TrM j , one can find that the quantityI5x j112 1x j21x j212 22x j11x jx j2121 is invariant. For ourFibonacci dielectric superlattice,I514 S «B«A 2 «A«BD2sinh2kdsinh2k~dL2dS!. ~6!This analytic formula is different from those for electronsand acoustic phonons in Fibonacci chains.8 Comparing withthat for magnetostatic modes in Fibonacci multilayers,6 bothinvariants appear to have the same wave-vector dependence,but different frequency dependence. The invariant can beused to characterize the structure of energy spectra as well asthe properties of the states of Fibonacci structures.8In the calculation of the frequency spectra of Fibonaccidielectric superlattice, we here use the free-boundary condi-tion: the electrostatic potentials at the left and theright boundaries, which contact with the enviroment ofdielectric function «C , are FL and FR with FL ,R5fL ,Rexp$i(kx2vt)%. In detail, the constraint equations arewritten as~«A2«C!e2kdL/2g12~«A1«C!ekdL/2h150,~«A2«C!ekdL/2gN112~«A1«C!e2kdL/2hN1150. ~7!On the other hand, the global equation for the quasiperiodicstructure can be written asS gN11hN11D 5M jS g1h1D 5Sm11 m12m21 m22D S g1h1D , ~8!where m11 , m12 , m21 , and m22 are all complicated func-tions of the wave-vector, thicknesses, and frequency. Thelinear equations of g1 , h1 , gN11 and hN11 in Eqs. ~7! and~8! have nontrivial solutions only if the coefficient determi-nant vanishes. Thus the dispersion equation becomes~«A22«C2 !e2kdLm111~«A2«C!2ekdLm122~«A1«C!2ekdLm212~«A22«C2 !m2250. ~9!Equation ~9! is the central result of this paper, from which allrelevant information regarding the optical interface modes inthe present quasiperiodic structure can be extracted. We willsee later on, in a specified case, this equation is of 2F j-thorder in v , which gives 2F j eigenfrequencies for each valueof k , where F j is a Fibonacci number satisfying the relationF j5F j211F j22 with F15F051.Each of the eigenfrequencies can create a special distribu-tion of potential. First, we consider the amplitudes in A lay-ers. By using Eqs. ~3! and ~4!, the potential amplitudesgl11 ,hl11 of l11 block can be recursively obtained if g1and h1 are known.9 After gl and hl are determined, the po-tential distributions in the B layers, characterized by pl andql , can also be obtained.To get the concrete dispersion relation from Eq. ~9!, wechoose «A as frequency independent, but «B(v)5«B ,`(v22vB,LO2 )/(v22vB ,TO2 ), as for alkali halide or polar semicon-ductor materials, where vB ,LO and vB ,TO are thelongitudinal- and transverse-optical frequencies. We take«A52.1, as the value of SiO2; «B ,`52.34,«B ,055.9,vB ,TO532.01 THz and vB ,LO550.74 THz, which correspond tothe values of NaCl; «C51 ~the value of vacuum!.Figure 1 shows the dispersion relation of coupled opticalinterface modes for 12th order Fibonacci dielectric superlat-tice, where dL54d ,dS52d , and d is fixed. The spectra aredevided into two branches, namely v1 and v2 , which areseparated by a gap as in the periodic superlattices.1 For lowerkd , the spectra form two bands, while for higher kd , themodes are highly degenerate. Between these two limits, be-ing also different from periodic superlattices, there are manygaps to appear. It can be seen more clearly from Fig. 2 thatthe allowed frequencies form two branches of Cantor sets,which are singular continuous. For the j th order Fibonaccidielectric structure, the subbands of v1 or v2 haveF j22 ,F j21 ,F j22 eigenfrequencies, respectively. One canFIG. 1. Dispersion relation of the coupled optical interfacemodes for 12th order Fibonacci superlattice. Here dL52dS54d .FIG. 2. Eigenfrequency versus number of modes for a 12th or-der Fibonacci superlattice with kdL52.0, kdS51.0, and kd50.5.Two enlarged local regions are shown in the insets.11 884 54BRIEF REPORTSalso see that for the v1 band, the low-frequency subband iswider than high-frequency subband, but for the v2 band, thesituation is reversed. This feature reflects the strength of qua-siperiodicity, as can be illustrated by Eq. ~6!: the lower re-gion of the v2 band and the higher region of the v1 bandhave larger values of I , while the higher region of v2 bandand the lower region of the v1 band have smaller values ofI .The relative thicknesses of d , dL , and dS have importanteffects on the frequency spectra. One example is shown inFig. 3. When k , dS and d are fixed (kdS51.0,kd50.5), thefrequency spectra are all threefold branches as dL /dSÞ1.Notice that, for dL /dS!1, two continuous bands are promi-nant, which stems from the fact that the structure becomesperiodic ~at this time I!0 as expected!. However, the qua-siperiodicity is more prominant for small or large values ofdL /dS . Particularly for very large dL /dS , the spectra be-come six highly degenerate branches, which seem to be dif-ferent from the existing results in the literatures. For ex-ample, when dL /dS@10.0, numerical calculations show thatthere are only six limiting frequencies v5 35.388 308 194 9,36.862 657 159 6, 38.023 154 200 9, 47.175 283 229 3,47.985 416 306 7, 48.936 159 968 8. These six limiting fre-quencies are the isolated modes when the thickness dL ap-proaches infinity; they are actually the solutions of the fol-lowing three equations:«B212«A«Bcothkd1«A250,~12e2kdS!«B212«A«Bcothkd1~11e2kdS!«A250,~10!~11e2kdS!«B212«A«Bcothkd1~12e2kdS!«A250.Figure 4 shows another example, where k , dL and dS arefixed (kdL52.0 and kdS51.0). For d/dS!1 , the eigenfre-quencies approach vB ,LO and vB ,TO , while for d/dS@1,there also exist six limiting frequencies v5FIG. 3. The variation of eigenfrequency distribution with thethickness ratio dL /dS for kdS51.0 and kd50.5.FIG. 4. The variation of eigenfrequency distribution with thethickness ratios d/dS for kdL52.0 and kdS51.0.FIG. 5. The amplitude profiles of electrostatic potential for twocritical states: ~a! v535.218 148 127 7; ~b! v549.040 631 708 8.FIG. 6. The amplitude profiles of electrostatic potential for twoquasilocalized states corresponding to the band-edge frequencies:~a! v538.085 539 634 3; ~b! v538.119 253 211 6.54 11 885BRIEF REPORTS39.427 153 712 9, 41.716 274 108 1, 43.026 124 350 3,43.430 412 732 0, 44.165 714 517 2, 46.114 221 825 4 cor-responding to another kind of isolated modes, which satisfyanother set of equations«B52«AtanhkdS , «B52«AcothkdS ,«B52«AtanhkdL , «B52«AcothkdL , ~11!«B52«A«AtanhkdL1«C«A1«CtanhkdL, «B52«A«AtanhkdL2«C«A2«CtanhkdL.The quasiperiodicity of the frequency spectra must be re-flected in the distribution of potential which is related to thelong-wavelength optical oscillations. If the averaged poten-tial over each layer is considered, thenf lA5~gl1hl!sinh~kdl!/~kdl/2! ~12!for the A layers, andf lB51kd H F sinhkd1 «A«B ~coshkd21 !Gekdl/2gl1F sinhkd2 «A«B ~coshkd21 !Ge2kdl/2hlJ , ~13!for the B layers. For the parameters chosen in Fig. 2, wehave examined all potential profiles of 2F125466 eigenfre-quencies and find that almost all states are critical. This factcan be illustrated by the values of the invariant which rangefrom 0.465 068 026 2 to 4.355 134 986 3. Figures 5~a! and5~b! show two critical states corresponding to N513, 454modes in Fig. 2. The potential profiles in these two figuresobey power laws , which is quite similar to the cases oftight-binding electrons, acoustic phonons or magnetostaticspin waves; but it is interesting to note that N5454 mode isthe 13th mode counted from the upper part of the v1 band.The similarity of these two states denotes the duality of thev2 and v1 bands, which is specific to the present system, incontrast to the well-studied systems.3–6 Actually, almost allthe states for these two bands are one to one correspondent inoverall characteristics, although there may be some differ-ences in detail.Among the 466 states, still a few states are quasilocaliz-ied. These states usually appear at the edges of the subbandsin Fig. 2. Figures 6~a! and 6~b! are two examples. Theirfrequencies are corresponding to the N5178, 179 modes,and both modes are at the two sides of a gap. These twostates are localized at the surfaces of the superlattices: one atthe left and the other at the right, and both are symmetric.Here the duality of the v1 band and v2 band also exists.This work was supported by the National Natural ScienceFoundation of China, the Provincial Natural Science Foun-dation of Jiangsu, and the RGC of Hong Kong under GrantNo. HKU262/95P.1R. E. Camley and D. Mills, Phys. Rev. B 29, 1695 ~1984!; B. L.Johnson and R. E. Camley, ibid. 38, 3311 ~1988!.2C. Colvard, T. A. Gant, M. V. Klein, R. Merlin, R. Fischer, H.Morkoc, and A. C. Gossard, Phys. Rev. B 31, 2080 ~1985!; A.K. Sood, J. Mene´ndez, M. Cardona, and K. Ploog, Phys. Rev.Lett. 54, 2115 ~1985!; E. P. Pokatilov and S. I. Beril, Phys.Status Solidi B 110, K75 ~1982!; Kun Huang and Bangfen Zhu,Phys. Rev. B 38, 13 377 ~1988!.3R. E. Camley, T. S. Rahman, and D. L. Mills, Phys. Rev. B 27,261 ~1983!; P. Gru¨nberg and K. Mika, ibid. 27, 2955 ~1983!.4M. Kohmoto and J. R. Banavar, Phys. Rev. B 34, 563 ~1986!; F.Nori and J. P. Rodriguez, ibid. 34, 2207 ~1986!.5P. Hawrylak and J. J. Quinn, Phys. Rev. Lett. 57, 380 ~1986!; B.L. Johnson and R. E. Camley, Phys. Rev. B 44, 1225 ~1991!; N.H. Liu, Y. He, W. G. Feng, and X. Wu, ibid. 52, 11 105 ~1995!.6 J. W. Feng, G. J. Jin, A. Hu, S. S. Kang, S. S. Jiang, and D. Feng,Phys. Rev. B 52, 15 312 ~1995!.7R. Merlin, K. Bajema, R. Clarke, F.-Y. Juang, and P. K. Bhatta-charya, Phys. Rev. Lett. 55, 1768 ~1985!; M. W. C. Dharma-wardana, A. H. MacDonald, D. J. Lockwood, J.-M. Barkeau,and D. C. Houghton, ibid. 58, 1761 ~1987!; K. Bajema and R.Merlin, Phys. Rev. B 36, 4555 ~1987!; A. Hu, C. Tien, X. Li, Y.Wang, and D. Feng, Phys. Lett. A 119, 313 ~1986!; X.-K.Zhang, H. Xia, G.-X. Cheng, A. Hu, and D. Feng, ibid. 136, 312~1989!.8M. Kohmoto, B. Sutherland, and C. Tang, Phys. Rev. B 35, 1020~1987!.9 J. E. S. Socolar and P. J. Steinhardt, Phys. Rev. B 34, 617 ~1986!.11 886 54BRIEF REPORTS

Coupled optical interface modes in a Fibonacci dielectric superlattice

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Coupled optical interface modes in a Fibonacci dielectric superlattice

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