Resonant Interaction of Phonons with Surface Vibrational Modes in a Finite-Size Superlattice

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

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 1995͒ 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’s͒ 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 surface͒ is shown to the layer interfaces͒, and that of reflected phonons is ex- Ϫikx 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. ͑45͒ 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 vibrational states.7 However, the study of the surface vibrations or rϭei⌰, ͑1͒ 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 surface vibrational modes with acoustic phonons injected to a ZX F ⌰ϭ2 tanϪ1 , ͑2͒ finite-size superlattice grown on a substrate ͑see Fig. 1͒. Spe- ͩ Z ␨ ͪ A cifically, we consider a possibility of observing surface vibrations by a phonon reflection experiment. The phonons ␮Ϫ␭ C͑N͒ incident on a SL with a frequency inside a gap are Bragg Fϭ ϩ , ͑3͒ reflected, i.e., the associated lattice displacement decays ex- 2 S͑N͒ 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 coefficient͒ r.

ZB ␨ϭϪsinkAdAcoskBdBϪ coskAdAsinkBdB . ͑4͒ ZA

ZB ␮ϭcoskAdAcoskBdBϪ sinkAdAsinkBdB , ͑5͒ ZA

ZA ␭ϭcoskAdAcoskBdBϪ sinkAdAsinkBdB . ͑6͒ 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 X͒ and B layers, respectively; Ziϭ␳ivi (iϭA,B, and X͒ is the acoustic impedance given by the product of the mass density ␳i and the sound velocity vi ; kiϭ␻/vi and ␻ is the angular frequency. For phonons within a frequency gap ͑the frequency gap is determined by the condition ͉␮ϩ␭͉/2Ͼ1), S(N) and C(N) have the following forms:

␮ϩ␭ Nϩ1 sinhN␪ S͑N͒ϭ , ͑7͒ ͩ ͉␮ϩ␭͉ͪ sinh␪ FIG. 2. Frequency dependence of the phase time of the longitu- ␮ϩ␭ N C͑N͒ϭ coshN␪, ͑8͒ dinal phonons reflected from a ͑100͒GaAs/AlAs superlattice. The ͩ ͉␮ϩ␭͉ͪ solid line and dots are the phase times calculated from the exact Ϫ1 expression Eq. ͑2͒ and the approximate expression Eq. ͑15͒, respec- and ␪ϭcosh (͉␮ϩ␭͉/2). For a phonon within a frequency tively. Narrow frequency range labeled 1 to 3 indicates the lowest band (͉␮ϩ␭͉/2р1), Eqs. ͑7͒ and ͑8͒ take different forms: three frequency gaps of phonons. The unit period assumed in these Ϫ1 S(N)ϭsinN␪/sin␪, C(N)ϭcosN␪, and ␪ϭcos ͓(␮ϩ␭)/2͔. calculations is ͑GaAs͒ ͑AlAs͒ : the surface layer is assumed to 2 15 15 In both cases, the reflection rate R defined by ͉r͉ is unity. be ͑a͒ AlAs, i.e., AϭGaAs and BϭAlAs, and ͑b͒ GaAs, i.e., Thus, the reflection rate gives us no information on the in- AϭAlAs and BϭGaAs. 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 ⌰ of the reflection Åinthe͑100͒ 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 ͑a͒, the phase phase time ␶ defined by d⌰/d␻ 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’s͒. Thus, there is a possibility of ␨ 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. ␻mϭm␻1 . Putting ␨ϭ␨mϩ(␻Ϫ␻m)␨1Ј for ␻ around ␻m , For recent reviews, see Refs. 18–21. the frequency ␻˜ m satisfying ␨(␻˜ m)ϭ0 is evaluated as As an example, we show in Fig. 2͑a͒ the frequency de- ␻˜ mϭ␻mϪ␨m /␨mЈ , where pendence of the phase time ␶ calculated numerically for the ͑AlAs͒ 15͑GaAs͒ 15 superlattice with Nϭ8. The plotted phase ␧ ␨ ϵ␨͑␻ ͒ϭ͑Ϫ1͒m sin͑2␻ d /v ͒, ͑10͒ time oscillates around ␶0ϭ2N(dA /vAϩdB /vB)ϭ0.0264 ns m m 2 m A A ͑the time needed for a round trip͒ and exhibits sharp peaks in the lowest and third frequency gaps. These peaks, or large d␨ ␲ d time delays associated with reflected phonons, are due to the mϩ1 A 2 ␨mЈ ϵ ϭ͑Ϫ1͒ ϩ␧ sin ͑␻mdA /vA͒ resonant interaction with the localized surface vibrations d␻ ͯ ͫ␻1 ͭ vA ␻m which exist in these frequency gaps. d To understand this striking feature of the phase time, we B 2 ϩ cos ͑␻mdA /vA͖͒]. ͑11͒ derive an approximate expression of ␶. The function ␨(␻) vB defined in Eq. ͑4͒ can be rewritten as Therefore, up to the order of ␧, we have

␨ϭϪsin͑␲␻/␻1͒Ϫ␧cos͑␻dA /vA͒sin͑␻dB /vB͒, ͑9͒ ␧ ␻˜ ϭ␻ ϩ ␻ sin͑2␻ d /v ͒. ͑12͒ where ␻1 is the frequency at the center of the lowest fre- m m 2␲ 1 m A A quency gap satisfying cos␪(␻ )ϭ(␮ϩ␭)/2 ϭ1, or 1 ͉␻1 Ϫ1 ␻1ϭ␲(dA /vAϩdB /vB) and ␧ϭZB /ZAϪ1 measures the On the other hand, the function F(␻) deﬁned by Eq. ͑3͒ acoustic mismatch between the constituent layers of the SL. depends weakly on ␻ inside the frequency gap. Thus, we put For most of the SL’s, ZA is close to ZB , i.e., ͉␧͉Ӷ1 F(␻)ХF(␻m)ϵFm in the mth frequency gap. Conse- (͉␧͉ϭ0.188 for the longitudinal phonons in AlAs/GaAs quently, in the vicinity of ␻˜ m , Eq. ͑2͒ is approximated as 53 RESONANT INTERACTION OF PHONONS WITH SURFACE . . . 4551

n Fm Unϭ͓␮͑␻˜ m͔͒ U0 , Snϭ0, ͑nϭ0,1,2,...͒, ͑19͒ ⌰Х2tanϪ1 , ͑13͒ ͩ ͑␻Ϫ␻˜ m͒␨Ј ͪ m where and the phase time is expressed as a form characteristic of the resonance: cos͑␻˜ mdB /vB͒ ␮͑␻˜ m͒ϭ . ͑20͒ cos͑␻˜ mdA /vA͒ d⌰ 2⌫m ␶ϭ ϭ 2 , ͑14͒ Thus, if ␮ Ͻ1, the frequency ␻˜ corresponds to a vibration d␻ ͑␻Ϫ␻˜ ͒2ϩ⌫ ͉ ͉ m m m localized near the surface, whose displacement decays expo- where the width ⌫m (ϵϪFm /␨mЈ ) of the resonance peak is nentially with the distance away from the surface. Since the expressed as function ␮ depends only weakly on ␻ inside a frequency gap, we can approximate ␻1 coth͓N␧sin͑␻mdA /vA͔͒ ϭ sin2 d / 1ϩ , m 2 ⌫m ␧ ͑␻m A vA͒ ␮͑␻˜ m͒Х␮͑␻m͒ϭ͑Ϫ1͒ ͓1ϩ␧sin ͑␻mdA /vA͔͒. ͑21͒ ␲ ͭ sin͑␻mdA /vA͒ ͮ ͑15͒ This equation implies that the surface mode exists if ␧Ͻ0, i.e., Z ϾZ but not for Z ϽZ . Hence, the existence of the and we have chosen ZAϭZX for simplicity. A B A B The frequency dependence of the phase time ␶ calculated surface vibrations is a necessary condition for the resonance from the resonance formula Eq. ͑14͒ is compared in the inset enhancement of time delay of reflected phonons. We also of Fig. 2͑a͒ with the numerical result. Here we note that note that in the second gap ͉␮(␻m)͉ is substantially equal to unity because of sin2(␻ d /v ) 0, whereas ␮(␻ ) ␻˜ 2 satisfying ␨(␻˜ 2)ϭ0 is found inside the second, narrow m A A Х ͉ m ͉ gap close to the lower edge. For the parameters we have Х1ϩ␧ in the first and third gaps. Accordingly, at ␻ϭ␻˜ 2 the amplitude decays very slowly and the corresponding local- chosen (dAϭdB and vAХvB), sin(␻dA /vA)Х0 for a frequency ␻ within this gap. This leads to the fact that the ized vibration behaves rather like a bulk mode in a frequency band. This also explains why the enhancement of ␶ cannot be width of the frequency gap ͉⌬m͉ϭ(␻1 /␲)͉sin(␻mdA /vA)͉ becomes small for mϭ2. At the same time seen in the second frequency gap. For a semi-infinite SL system, Camley et al.8 derived the coth͓N␧sin(␻2dA /vA)͔ becomes very large though N is large (Nϭ8), leading to the width of the resonance much larger equation ␨(␻)ϭ0 giving the frequencies of surface modes than the gap width, i.e., and the expression of the decay parameter ␮. Their result is essentially the same as our results. Equations ͑19͒ and ͑20͒ ͉⌫m /⌬m͉ϭ͉sin͑␻mdA /vA͒ϩcoth͓N␧sin͑␻mdA /vA͔͉͒ӷ1, are valid for both the finite and semi-infinite SL systems. ͑16͒ In order to see explicitly the time delay of the reflected phonons, we examine the time development of a phonon for mϭ2. Thus, the enhancement of ␶ is not found in the wave packet whose average wave number in the initial state second frequency gap. It is readily seen that ␧Ͻ0 ͑i.e., is k ϭ␻˜ /v, where vϵv . As an initial packet, we assume Z ϾZ ) is necessary for / Շ1. In addition, m m X A B ͉⌫m ⌬m͉ a Gaussian of the form of ͉coth͓N␧sin(␻mdA /vA)͔͉Շ2, or ͉N␧sin(␻mdA /vA)͉տ1/2 should be satisfied. This explains the fact that no peak of the phase 2 ͑xϪx0͒ time is seen inside the gaps when the stacking order of the ␺ ͑x,0͒ϭexp Ϫ ϩik x , ͑22͒ i 4͑⌬x͒2 m bilayer is interchanged, i.e., the surface layer B is replaced ͫ ͬ with the A layer ͓see Fig. 2͑b͔͒. where x0 is the coordinate at the center of the packet. The Now we show that the displacement field of the phonons Fourier component ␾(k) of this packet is given by with the resonance frequency ␻˜ m is localized near the sur- 2 face. In the continuum model, the lattice displacement and 2ͱ␲ ͑kϪkm͒ ␾͑k͒ϭ exp Ϫ ϩi͑k Ϫk͒x , ͑23͒ stress should be continuous at each interface of the adjacent ⌬k ͫ ͑⌬k͒2 m ͬ layers. This boundary condition can be expressed as where the magnitude of ⌬kϵ1/(⌬x) is chosen so that Unϩ1 ␮ Ϫ␴/͑␻ZA͒ Un ␾(k) is finite only inside the frequency window correspond- ϭ , ͑17͒ ͩ S ͪ ͩ Ϫ␻Z ␨ ␭ ͪͩS ͪ ing to the mth frequency gap. If the reflected phonon packet nϩ1 A n is well separated from the superlattice, it can be written as where, Un and Sn are the displacement and stress at the nth interface, respectively. The elements ␨,␮, and ␭ of the dk ␺ ͑x,t͒Х ␾͑k͒r͑k͒eϪi͑kxϩ␻t͒ ‘‘transfer matrix’’ are given by Eqs. ͑4͒ to ͑6͒, and ␴ is r ͵ 2␲ defined by dk ϭ ␾͑k͒eϪi͑kxϩ␻tϩ⌰͒. ͑24͒ ZA ͵ 2␲ ␴ϭsinkAdAcoskBdBϩ coskAdAsinkBdB . ͑18͒ ZB 2 The intensity Irϭ͉␺r͉ calculated numerically from Eq. ͑24͒ For a SL with a free surface, an additional boundary condi- supplemented by Eqs. ͑2͒ to ͑8͒ is plotted in Fig. 3 together tion is imposed, i.e., the stress at the surface should vanish, with the initial wave packet. In this calculation, we choose or S0ϭ0(nϭ0 denotes the surface͒, see Fig. 1. With the use x0ϭ0, mϭ1 ͑the lowest gap͒, and ⌬␻ϭv⌬k ϭ 10 GHz. of this condition together with the equation ␨ϭ0 satisfied by The reflected packets are illustrated as functions of xϩvt. the phonons of frequency ␻˜ m , Eq. ͑17͒ leads to From Fig. 3, we see the spatial delay of the packet which is 4552 SEIJI MIZUNO AND SHIN-ICHIRO TAMURA 53

2 where xrϭxϩvt. The intensity Irϭ͉␺r͉ calculated from Eq. ͑25͒ is also plotted in Fig. 3 by dots. The small deviation from the numerical result ͑solid line͒ is due to the fact that we consider only the effect of the resonance and it comes from the difference between the phase times calculated from the exact and approximate expressions illustrated in Fig. 2͑a͒. In the present work, we have shown that the phase associated with the reflected amplitude describes the temporal delay for the phonon packet scattered off a finite-size SL system. This delay is caused as a result of the inter- FIG. 3. Asymptotic forms of the reflected intensities of the pho- action of incident phonons with a surface vibrational non wave packet calculated from the exact expression Eq. ͑24͒ mode. In the numerical example shown in Figs. 2͑a͒ and ͑solid line͒ and approximate expression Eq. ͑25͒͑dots͒ which are 3, the time delay at a resonant frequency becomes as large compared to the intensity of the initial Gaussian packet ͑dashed as 0.1 ns for the system of about 70 nm thickness. This line͒. The reflected packets are illustrated as functions of xϩvt. means that the magnitude of the time delay of reflected phonons associated with the resonant interaction with a equivalent to the corresponding time delay. An interesting surface vibrational mode is in the range detectable by a pho- feature is that the reflected packet has double peaks but the non experiment using picosecond laser technique.22 The con- former peak is much smaller than the latter one. The similar cept of the phase time is an old one but it proves to be also double-peak structure of the reflected packet has also been useful for studying the existence of surface vibrational obtained for the transmission of phonons through a double modes. We have shown that this phase time has large reso- barrier structure. However, in this case the former peak is nant peaks at eigenfrequencies of the surface modes in SL’s, much larger than the latter peak. Now, the distance between though the reflection rate is exactly unity for the whole range the peaks of the initial packet and the latter large peak of the of frequency. Our results suggest the observability of the reflected packet gives the time delay. surface vibrational modes by a time-resolved phonon reflec- For the approximated formula of ⌰ given by Eq. ͑10͒, the tion experiment. integral in Eq. ͑24͒ can be analytically performed, leading to S.M. was partly supported by a Special Grant-in-Aid for Promotion of Education and Science in Hokkaido Uni- 2 ͑xr⌬␻͒ 2ͱ␲⌫m versity provided by the Ministry of Education, Science, and ␺ ͑x,t͒ϭeϪikmxr exp Ϫ r ͫ ͩ v2 ͪ ⌬␻ Culture. S.T. was supported by the Murata Science Founda- tion and by a Grant-in-Aid for Scientific Research from the 2 ⌫m ⌫mxr ⌫m xr⌬␻ Ministry of Education, Science, and Culture ͑Grant No. ϫexp Ϫ erfc Ϫ , ͑25͒ ͩ ͑⌬␻͒2 v ͪ ͩ ⌬␻ 2v ͪͬ 05650002͒.

1 V. Narayanamurti, H. L. Stormer, M. A. Chin, A. C. Gossard, and 11 S. Mizuno and S. Tamura, Phys. Rev. B 45, 13 423 ͑1992͒. W. Wiegmann, Phys. Rev. Lett. 43, 2012 ͑1979͒. 12 E. P. Wigner, Phys. Rev. 98, 145 ͑1955͒. 2 O. Koblinger, J. Mebert, E. Dittrich, S. Dottinger, W. Eisen- 13 E. H. Hauge, J. P. Falck, and T. A. Fjeldly, Phys. Rev. B 36, 4203 menger, P. V. Santos, and L. Ley, Phys. Rev. B 35, 9372 ͑1987͒. ͑1987͒. 3 14 D. C. Hurley, S. Tamura, J. P. Wolfe, and H. Morkoc, Phys. Rev. E. H. Hauge and J. A. Sto”vneng, Rev. Mod. Phys. 61, 917 ͑1989͒. Lett. 58, 2446 ͑1987͒. 15 A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, Phys. Rev. Lett. 4 D. C. Hurley, S. Tamura, J. P. Wolfe, K. Ploog, and J. Nagle, 71, 708 ͑1993͒. Phys. Rev. B 37, 8829 ͑1988͒. 16 S. Mizuno, M. Ito, and S. Tamura, Jpn. J. Appl. Phys. 33, 2880 5 S. Tamura, in Proceedings of the 3rd International Conference on ͑1994͒. Phonon Physics and the 6th International Conference on Pho- 17 S. Mizuno and S. Tamura, Phys. Rev. B 50, 7708 ͑1994͒. non Scattering in Condensed Matter, Heidelberg, 1989, edited 18 R. Landauer, Nature 365, 692 ͑1993͒. by S. Hunklinger, W. Ludwig, and G. Weiss ͑World Scientiﬁc, 19 R. Landauer, Nature 341, 567 ͑1989͒. Singapore, 1990͒, p. 703. 20 R. Landauer and Th. Martin, Rev. Mod. Phys. 66, 217 ͑1994͒. 6 S. Tamura, Phys. Rev. B 38, 1427 ͑1988͒. 21 M. Buttiker, in Electronic Properties of Multilayers and Low- 7 S. Tamura, Phys. Rev. B 39, 1261 ͑1989͒. Dimensional Semiconductors Structures, edited by J. M. Cham- 8 R. E. Camley, B. Djafari-Rouhani, L. Dobrzynski, and A. A. Ma- berlain, L. Eaves, and J. C. Portal ͑Plenum, New York, 1990͒,p. radudin, Phys. Rev. B 27, 1427 ͑1983͒. 297. 9 B. Djafari-Rouhani, L. Dobrzynski, O. Hardouin Duparc, R. E. 22 See, for example, H. J. Maris, C. Thomsen, and J. Tauc, in Pho- Camley, and A. A. Maradudin, Phys. Rev. B 28, 1711 ͑1983͒. non Scattering in Condensed Matter, edited by A. C. Anderson 10 S. Mizuno and S. Tamura, Phys. Rev. B 45, 734 ͑1992͒. and J. P. Wolfe ͑Springer, Berlin, 1986͒, p. 374.