Title Resonant interaction of phonons with surface vibrational modes in a finite-size superlattice Author(s) Mizuno, Seiji; Tamura, Shin-ichiro Physical Review B, 53(8), 4549-4552 Citation https://doi.org/10.1103/PhysRevB.53.4549 Issue Date 1996-02-15 Doc URL http://hdl.handle.net/2115/5708 Rights Copyright © 1996 American Physical Society Type article File Information PRB53-8.pdf Instructions for use Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP PHYSICAL REVIEW B VOLUME 53, NUMBER 8 15 FEBRUARY 1996-II Resonant interaction of phonons with surface vibrational modes in a finite-size superlattice Seiji Mizuno and Shin-ichiro Tamura Department of Engineering Science, Hokkaido University, Sapporo 060, Japan ~Received 13 September 1995! We study theoretically the interaction of phonons with surface vibrational modes in a finite-size superlattice with a free surface. A phonon incident normally on the superlattice from a substrate is perfectly reflected, i.e., the reflection rate is unity irrespective of frequency. However, it comes back to the substrate with a large time delay when the frequency coincides with an eigenfrequency of the surface mode. This result is attributable to the resonant interaction of incident phonons with a vibrational mode localized near the surface. Acoustic vibrations in superlattices ~SL’s! with various We can formulate the reflection of phonons in the present stacking order, such as periodic, quasiperiodic, and random system as a stationary scattering problem in one superlattices have been investigated extensively during the dimension.10,11 We adopt a continuum model and consider last decade.1–11 The SL’s actually grown are not ideal and the phonon propagation normal to the layer interfaces. Thus, they usually possess both natural and artificial defects. In the displacement field of incident phonons is represented by ikx particular, an inhomogeneity embedded in a SL with perfect the plane wave e (k is a wave number and x is normal to periodicity ~e.g., a defect layer or a free surface! is shown to the layer interfaces!, and that of reflected phonons is ex- 2ikx cause localized vibrations within the frequency gaps induced pressed as r(k)e in the substrate region, where r(k)is by the periodicity of a SL. The localized vibrational modes the reflection coefficient. For a finite-size SL grown on a due to defect layers have also been studied theoretically for substrate and capped with a ‘‘detector layer,’’ the analytical both infinite- and finite-size SL systems. For an infinite, pe- expression of the reflection coefficient is derived in a previ- 10 riodic SL with a defect layer, it is possible to derive analyti- ous paper. Setting the acoustic impedance of the detector cally the frequencies of these localized modes.6 For a finite- layer to zero in Eq. ~45! of Ref. 10, we can obtain the ana- size system, on the other hand, the transmission and lytical form of r for a finite-size SL with a free surface. The reflection rates have been calculated, and their local enhance- result is ments are indicative of the presence of the localized vibra- tional states.7 However, the study of the surface vibrations or r5eiQ, ~1! surface acoustic waves in SL’s was, hitherto, carried out only for a semi-infinite system.8,9 where In the present paper, we study the interaction of the sur- face vibrational modes with acoustic phonons injected to a ZX F Q52 tan21 , ~2! finite-size superlattice grown on a substrate ~see Fig. 1!. Spe- S Z z D A cifically, we consider a possibility of observing surface vi- brations by a phonon reflection experiment. The phonons m2l C~N! incident on a SL with a frequency inside a gap are Bragg F5 1 , ~3! reflected, i.e., the associated lattice displacement decays ex- 2 S~N! ponentially in the SL with the distance from the substrate. and The phonons within a frequency band can propagate up to the surface of the SL but they are also perfectly reflected from the surface. In other words, the reflection rate in this system is unity for the whole frequency range. Thus, one may naively think that it is impossible to observe evidence of surface vibrations by a phonon reflection experiment. How- ever, this is not the case. Although the magnitude of the reflection amplitude does not give us any information on the surface mode, its phase contains important information on the interaction with the surface vibrations, which is basically accessible in a phonon reflection experiment. This is because the time needed for a phonon to complete the reflection pro- cess is given by the derivative of this phase with respect to FIG. 1. Schematic of the system consisting of a substrate (X) 12–14 the frequency, or ‘‘phase time.’’ In particular, we show and a finite-size, periodic superlattice with alternate stacking of A that for an incident phonon with a frequency near an eigen- and B layers. N denotes the number of bilayers and n indicates the frequency of surface vibrations the phase time is enhanced boundaries of unit period. Phonons with a unit amplitude incident to strongly due to the resonant interaction with the surface the superlattice are reflected back to the substrate with reflected mode. amplitude ~reflection coefficient! r. 0163-1829/96/53~8!/4549~4!/$06.0053 4549 © 1996 The American Physical Society 4550 SEIJI MIZUNO AND SHIN-ICHIRO TAMURA 53 ZB z52sinkAdAcoskBdB2 coskAdAsinkBdB . ~4! ZA ZB m5coskAdAcoskBdB2 sinkAdAsinkBdB , ~5! ZA ZA l5coskAdAcoskBdB2 sinkAdAsinkBdB . ~6! ZB In these equations, N is the number of the periodicity of a SL; dA and dB are the thicknesses of A ~the layer adjacent to the substrate X! and B layers, respectively; Zi5rivi (i5A,B, and X! is the acoustic impedance given by the product of the mass density ri and the sound velocity vi ; ki5v/vi and v is the angular frequency. For phonons within a frequency gap ~the frequency gap is determined by the condition um1lu/2.1), S(N) and C(N) have the following forms: m1l N11 sinhNu S~N!5 , ~7! S um1luD sinhu FIG. 2. Frequency dependence of the phase time of the longitu- m1l N C~N!5 coshNu, ~8! dinal phonons reflected from a ~100!GaAs/AlAs superlattice. The S um1luD solid line and dots are the phase times calculated from the exact 21 expression Eq. ~2! and the approximate expression Eq. ~15!, respec- and u5cosh (um1lu/2). For a phonon within a frequency tively. Narrow frequency range labeled 1 to 3 indicates the lowest band (um1lu/2<1), Eqs. ~7! and ~8! take different forms: three frequency gaps of phonons. The unit period assumed in these 21 S(N)5sinNu/sinu, C(N)5cosNu, and u5cos @(m1l)/2#. calculations is ~GaAs! ~AlAs! : the surface layer is assumed to 2 15 15 In both cases, the reflection rate R defined by uru is unity. be ~a! AlAs, i.e., A5GaAs and B5AlAs, and ~b! GaAs, i.e., Thus, the reflection rate gives us no information on the in- A5AlAs and B5GaAs. The parameters used are as follows: the teraction of phonons with the superlattice system. The rel- number of periods is eight; the thickness of one monolayer is 2.83 evant information is included in the phase Q of the reflection Åinthe~100! direction for both GaAs and AlAs; the mass densities coefficient. This phase is related to the dynamical properties and longitudinal sound velocities are 5.36 g/cm 3 and 4.71 km/s for of phonons propagating through the system. Explicitly, the GaAs, and 3.76 g/cm 3 and 5.65 km/s for AlAs. In ~a!, the phase phase time t defined by dQ/dv describes the temporal delay time within the lowest frequency gap is enlarged in the inset. of a reflected phonon packet. The phase time was originally introduced12–14 for electrons tunneling through a potential SL’s!. Thus, there is a possibility of z vanishing at a fre- barrier and later applied to the photon15 and phonon quency in the vicinity of the mth-order Bragg frequency 16,17 propagations in multilayered dielectric and elastic media. vm5mv1 . Putting z5zm1(v2vm)z18 for v around vm , For recent reviews, see Refs. 18–21. the frequency v˜ m satisfying z(v˜ m)50 is evaluated as As an example, we show in Fig. 2~a! the frequency de- v˜ m5vm2zm /zm8 , where pendence of the phase time t calculated numerically for the ~AlAs! 15~GaAs! 15 superlattice with N58. The plotted phase « z [z~v !5~21!m sin~2v d /v !, ~10! time oscillates around t052N(dA /vA1dB /vB)50.0264 ns m m 2 m A A ~the time needed for a round trip! and exhibits sharp peaks in the lowest and third frequency gaps. These peaks, or large dz p d time delays associated with reflected phonons, are due to the m11 A 2 zm8 [ 5~21! 1« sin ~vmdA /vA! resonant interaction with the localized surface vibrations dv U Fv1 H vA vm which exist in these frequency gaps. d To understand this striking feature of the phase time, we B 2 1 cos ~vmdA /vA!%]. ~11! derive an approximate expression of t. The function z(v) vB defined in Eq. ~4! can be rewritten as Therefore, up to the order of «, we have z52sin~pv/v1!2«cos~vdA /vA!sin~vdB /vB!, ~9! « v˜ 5v 1 v sin~2v d /v !. ~12! where v1 is the frequency at the center of the lowest fre- m m 2p 1 m A A quency gap satisfying cosu(v )5(m1l)/2 51, or 1 uv1 21 v15p(dA /vA1dB /vB) and «5ZB /ZA21 measures the On the other hand, the function F(v) defined by Eq.