Coherent generation and detection of acoustic phonons in topological nanocavities Cite as: APL Photonics 4, 030805 (2019); https://doi.org/10.1063/1.5082728 Submitted: 23 November 2018 . Accepted: 15 February 2019 . Published Online: 14 March 2019 G. Arregui , O. Ortíz, M. Esmann, C. M. Sotomayor-Torres, C. Gomez-Carbonell, O. Mauguin, B. Perrin, A. Lemaître, P. D. García, and N. D. 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Gomez-Carbonell,3 O. Mauguin,3 B. Perrin, A. Lemaître,3 P. D. García,1,b) and N. D. Lanzillotti-Kimura3,c) 5 AFFILIATIONS 1 Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and BIST, Campus UAB, Bellaterra 08193, Barcelona, Spain 2 Dept. de Física, Universitat Autonoma de Barcelona, 08193 Bellaterra, Spain 3 Centre de Nanosciences et de Nanotechnologies (C2N), CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91120 Palaiseau, France 4 ICREA—Institució Catalana de Recerca i Estudis Avançats, 08010 Barcelona, Spain 5 Institut de Nanosciences de Paris, Sorbonne Université, CNRS UMR 7588, 75005 Paris, France Note: This article is part of the Special Topic “Optoacoustics—Advances in high-frequency optomechanics and Brillouin scattering” in APL Photonics. a) Contributions: G. Arregui and O. Ortíz contributed equally to this work. b) Electronic mail: david.garcia@icn2.cat c) Electronic mail: daniel.kimura@c2n.upsaclay.fr ABSTRACT Inspired by concepts developed for fermionic systems in the framework of condensed matter physics, topology and topological states are recently being explored also in bosonic systems. Recently, some of these concepts have been successfully applied to acoustic phonons in nanoscale multilayered systems. The reported demonstration of confined topological phononic modes was based on Raman scattering spec- troscopy [M. Esmann et al., Phys. Rev. B 97, 155422 (2018)], yet the resolution did not suffice to determine lifetimes and to identify other acoustic modes in the system. Here, we use time-resolved pump-probe measurements using an asynchronous optical sampling (ASOPS) tech- nique to overcome these resolution limitations. By means of one-dimensional GaAs/AlAs distributed Bragg reflectors (DBRs) used as building blocks, we engineer high frequency (∼200 GHz) topological acoustic interface states. We are able to clearly distinguish confined topological states from stationary band edge modes. The generation/detection scheme reflects the symmetry of the modes directly through the selection rules, evidencing the topological nature of the measured confined state. These experiments enable a new tool in the study of the more complex topology-driven phonon dynamics such as phonon nonlinearities and optomechanical systems with simultaneous confinement of light and sound. © 2019 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5082728 I. INTRODUCTION Distributed Bragg reflectors (DBRs), the finite version of the infi- nite crystal, act as mirrors for frequencies inside the band gap of The atomic monolayer accuracy of molecular beam epitaxy the underlying crystal and constitute the building block for the stan- (MBE) allows nanostructures based on acoustic impedance mod- dard Fabry-Perot type acoustic cavity.1 This confinement approach ulation in the growth direction, resulting in a very precise con- is based on a spacer layer, bounded by two equal DBRs, leading to trol of sub-THz mechanical motion.1–5 Spectral and spatial tailor- a confined mode which exponentially decays along the mirrors, in ing of the acoustic excitations supported in these nanostructures an analogous manner to the electronic wave function in a quantum mostly relies on exploiting the energy band structure of periodic well. Recently, another type of cavity based on an adiabatic confine- one-dimensional phononic crystals. The artificial periodicity leads to ment potential rather than a spacer has been proposed.7 These two frequency band gaps within which elastic waves cannot propagate.6 strategies solely use the frequency band structure of their underlying APL Photon. 4, 030805 (2019); doi: 10.1063/1.5082728 4, 030805-1 © Author(s) 2019 APL Photonics ARTICLE scitation.org/journal/app periodic counterpart as a confinement strategy, the spatial distribu- a widely used approach to access complex wave dynamics and the tion of the Bloch modes having no particular role in the control of modal structure of acoustic nanoresonators.37–45 In such exper- the density of states. iments, tailoring of the generated and detected spectrum can be A recent work8 has evidenced a completely novel approach achieved by design of both the sample structure and experimental to confine acoustic vibrations in multilayered structures, based on conditions.46,47 As compared to the resolution-limited (13 GHz) topological band inversion in GaAs/AlAs superlattices. The idea is broad peak of Fig. 3(c) in Ref. 8, here we unveil the detailed structure based on the fact that two concatenated semi-infinite superlattices of nanoacoustic modes around the band gap region, allowing a clear exhibiting a common band gap region and having inverted sym- assignment of the peaks to topological and other stationary modes. metries of the Bloch modes at the minigap edges will give rise to an interface state in the band gap region.8,9 The finite-size ver- sion of such structures (two concatenated DBRs) will inherit the II. TOPOLOGICAL NANOPHONONIC RESONATORS presence of this topological interface state. This kind of resonator We studied two different nanoacoustic multilayer structures exploits symmetry protected topological (SPT) phases for confine- using three different types of DBRs (labelled A, B, and C) as build- ment purposes.10–17 States protected by one or more symmetries ing blocks. Each DBR contained 20 GaAs/AlAs layer pairs with have been studied in multiple systems like those of the quantum 10.76 nm/15.57 nm, 13.15 nm/12.74 nm, and 15.54 nm/9.91 nm spin Hall effect,18–20 the spin 1 antiferromagnetic chain (Haldane nominal thickness, respectively. Both samples were grown on a chain),21 the spin Chern insulator,22 and the Su-Schreiffer-Heger 200 µm thick GaAs substrate starting with a DBR of type B fol- (SSH) model.23,24 For the latter one, in particular, a feasible imple- lowed by DBR of type A (we label this the topological sample) or mentation within nanophononics has been theoretically proposed type C (control sample), respectively. While the topological sample already.25 Topological acoustic effects have been experimentally is designed to confine a topological acoustic mode at the interface observed in a variety of systems,26–33 most of them based on kHz– between the two DBRs based on band inversion, the control sam- MHz sound waves, with typical wavelength of the order of the cen- ple merely supports extended propagating and stationary band edge timeter. The approach proposed in Ref. 8 has successfully transferred modes. This difference in the topological properties of the two sam- these effects to GHz acoustic phonons in nanoscale multilayered ples can be understood from an analysis of the DBR bandstructures systems. The confined topological state was experimentally studied and the associated mode symmetries8,28 as shown in Figs. 1(a)–1(d). using Raman scattering measurements, with limited spectral reso- The band structures of the DBRs A and C are displayed in Figs. 1(a) lution. In this paper, we present time-resolved pump-probe differ- and 1(d), respectively, with common acoustic minigaps centered ential reflectivity experiments.34–36 Optical generation and detec- around 100 GHz and 200 GHz. For each acoustic band, the spa- tion of acoustic vibrations with ultrafast pulsed lasers constitutes tial Bloch mode symmetries at the edge and the center of the first FIG. 1. Topological interface state through band inversion in GaAs/AlAs superlattices. (a) Band structure corresponding to DBR A. Zak phase values of different bands depicted on top of them. Minigap edges at the center and border of the Brillouin zone are marked with green and black dots. Black (green) dots represent edge modes with a symmetric (anti-symmetric) displacement profile as shown in the insets. (b) and (c) present plots of the edge positions as a function of parameters x A−B and x C−B , respectively. Edges corresponding to DBR B are represented as dots at x A−B = x C−B = 1 using the same symmetric/antisymmetric color code for both lines and dots. (d) Band structure corresponding to DBR C using the same representation used in (a). (e) Reflectivity spectra for DBR A, B, and C. (f) Reflectivity spectra for topological and control samples. (g) Displacement profile of the topological interface state. APL Photon. 4, 030805 (2019); doi: 10.1063/1.5082728 4, 030805-2 © Author(s) 2019 APL Photonics ARTICLE scitation.org/journal/app Brillouin zone are indicated with black (symmetric) and green (anti- Both samples were grown by MBE on a [001]-oriented GaAs symmetric) dots. It has been shown that a Zak phase48 (equivalent of substrate and pre-characterized by means of high resolution x-ray a Berry’s phase in one-dimensional periodic systems) can be associ- diffraction (HRXRD). As an important tool for structural character- ated with each band. It acquires a value of π if the mode symmetries ization, HRXRD provides valuable information on the periodicity, at the edge and the center of the Brillouin zone are opposite (second layer sizes, and overall quality of both samples. A θ − 2θ HRXRD band of DBR A); otherwise, the topological phase is 0 (second band scan using Cu K-α 1 radiation was performed, diffractograms were of DBR C). As a consequence, for frequencies within this inverted measured, and further analysis of their peaks provided information minigap, the reflection phase has opposite sign for the two structures about the different parts of the structure. For the topological sample, (either from −π to 0 or from 0 to π across the minigap). Therefore, the results showed that DBR A is formed by GaAs/AlAs layers of upon concatenating the two, there will always exist a frequency in the 11.1/15.5 nm, whereas DBR B of 13.5/12.7 nm. For the control sam- minigap for which the reflection phases add up to 0 (i.e., fulfill the ple, DBR C presents layers of 16/10 nm of GaAs/AlAs, while DBR B resonance condition). That is, the Zak phases of the acoustic bands of 13.5/12.8 nm. For the topological sample, these thickness values directly determine the existence of a confined interface state. In turn, represent a reduction of the AlAs layers by 0.6% of their nominal the resonant interface mode can only be destroyed by fluctuations values, while for the GaAs layers, the change corresponds to an big enough to alter the energetic order of the band edge modes. That increase in 2.8%. For the control sample, both AlAs and GaAs lay- is, the mode is topologically protected by symmetry. In Figs. 1(b) ers present an increase in 0.5% and 2.8%, respectively. Despite this and 1(c), we illustrate that upon a topological phase transition, the deviation from the design values, the band structure analysis for energetic order of the band edge modes changes and we compare both samples and, as consequence, their predicted phonon dynamics the topological phase of DBR B to that of A and C. To that end, we remain valid as we show in Sec. III. define two continuous parameters xA−B (xC−B ) that deform DBR A (DBR C) into DBR B. That is, we define the thickness of a DBR’s GaAs layers as d = dY ⋅ (1 − xY −B ) + dB ⋅ xY −B and of the AlAs layers III. PHONON DYNAMICS as e = eY ⋅ (1 − xY −B ) + eB ⋅ xY −B with xY −B ∈ [0, 1], dY and eY being In order to access experimentally the phonon dynamics of the nominal sizes of the layers of DBR Y, and Y being A or C. In the two samples presented in this work, we need to resolve phys- Figs. 1(b) and 1(c), we plot the evolution of the band edges bounding ical processes at a picosecond time scale. For this purpose, we use the second acoustic minigap as a function of xA−B (b) and xC−B (c). a reflection-type pump-probe experiment at room temperature.34 As depicted in Fig. 1(c), continuously transforming DBR C into B The pump-probe technique relies on the use of ultrafast laser pulses merely implies a change in the size of the common minigap, but the for both coherent phonon generation and detection. The experiment energetic order of mode symmetries persists. In contrast, Fig. 1(b) can be described as composed of two stages. First, a pump pulse is shows that a continuous transition between DBR A and B necessar- focused on the sample with enough power to impulsively photoin- ily implies a band crossing, i.e., the minigap closes and re-opens, and duce a stress σpump (z, t) around the interaction region, basically con- an associated exchange of the band edge symmetries. DBRs B and C verting the optical energy into mechanical energy through various are hence in the same topological phase, whereas DBRs B and A can- processes, generating a coherent acoustic phonon wave packet.36 not be continuously transformed into one another, that is, they are Second, another, time-delayed pulse with significantly less power in different topological phases. is used to probe the instantaneous reflectivity of the sample. The Using a transfer matrix simulation, we furthermore compute presence of acoustic excitations will modify the local optical proper- acoustic transmission spectra of the individual DBRs [Fig. 1(e)] and ties, therefore modifying its reflectivity and allowing us to withdraw the two concatenated structures [Fig. 1(f)]. For each individual DBR, information of the phonons present in the nanostructure. By system- we find a broad dip in transmission centered around 200 GHz, which atically sampling measurements at increasing delay times between directly reflects the position and size of the common acoustic mini- the pump and probe, we are able to monitor in time the tran- gap. For the control sample [dashed line, panel (f)], only a broad stop sient optical reflectivity, therefore gaining access to the coherent band as for the individual DBRs is found. For the topological sample dynamics of the generated phonons with temporal resolution given (solid line), however, a clear peak appears at 199.24 GHz, indicating by the pulse length. Here, we used an asynchronous optical sam- the presence of a confined mode. The corresponding spatial acoustic pling (ASOPS) method, where 2 fs Ti:sapphire lasers of repetition displacement profile |u(z)| is shown in Fig. 1(g) superimposed with rate f R ∼ 1 GHz are actively stabilized—via a single electronic feed- the layer structure of the topological sample. We observe that the back loop—to have a small repetition rate difference of ∆f R = 2 kHz. mode is indeed centered at the interface between the two DBRs and This rate difference realizes the time delay between the pump and decays exponentially to both extremes of the structure. The reflectiv- probe pulse pairs without the need of any mechanical stage. The ity of the two individual DBRs at the frequency of the mode deter- delay between the pump pulses remains 1 ns, while the pump and mines the strength of such decay, i.e., its localization properties. In probe pulses coincide every 500 ms. The acquired signal is processed this way, there are three parameters that can be tuned: (i) the mode’s to unequivocally assign actual time delays between pump and probe frequency to make it lie as deep as possible inside the correspond- pulses (in the fs-ps time scale) to measured time (on the microsec- ing band gap region; (ii) the contrast in the acoustic impedances of ond scale). Even though time traces with resolution of nearly 2 the two materials forming the DBR; and (iii) the relative thickness of fs can be achieved, the time resolution is set by the pulse dura- the two materials in the unit cell. By maximizing the reflectivity, the tion (∼50 fs), the accumulated timing jitter, and the photo-detector decay length inside the two mirrors shrinks and the mode becomes bandwidth.49 The measurements were done at a fixed central wave- more strongly localized. For the control sample, no occurrence of a length of λ = 800 nm for the pump beam (40 mW) and a varying topological interface mode is expected. central wavelength λ = 760-840 nm for the probe (4 mW) beam, APL Photon. 4, 030805 (2019); doi: 10.1063/1.5082728 4, 030805-3 © Author(s) 2019 APL Photonics ARTICLE scitation.org/journal/app collinearly focused to a 2 µm spot on the sample surface. Measure- As shown above, a purely electrostrictive model for the phonon ments at different magnifications and powers were done to rule out generation process and photoelasticity-based detection reproduces the presence of additional power density dependent temperature the main features of the measured spectra for the samples and exper- variations of the sample reflectivity. The Fourier transform of the imental conditions considered in this article. For GaAs/AlAs infinite differential reflectivity signals in the spectral region below 300 GHz superlattices in the transparency region, selection rules associated is shown in Fig. 2 for both topological (top) and control (bottom) with such generation/detection mechanisms have been largely stud- samples and compared to the expected differential reflectivity spec- ied.46,52 Generated coherent phonons correspond to phonons usu- tra extracted from a simple model based on electrostrictive forces ally detected in forward scattering (FS) q = 0 Raman experiments, for the generation and photoelasticity for the detection. The spe- i.e., zone-center acoustic excitations. Due to the well-defined sym- cific measurements shown Fig. 2 were performed at pump/probe metries of the Bloch modes at the band edges, an additional selection central wavelengths of 800/760 nm for the topological sample and rule can be added: only modes with odd symmetry with respect to 800/800 nm for the control sample. In our simulations, we used the bisecting plane (A1 group symmetry) of the composing layers the formalism presented in Refs. 50 and 51 assuming an impulsive is accessible. The spectral response for detection is itself peaked at generation mechanism. In order to reproduce the experimental con- q ∼ 2keff , with keff being the effective wave-vector of the electro- ditions, the limited time window (1 ns) has been taken into account magnetic field. The spectral mismatch between the two processes by convoluting the theoretical spectrum with a sinc function. The is relaxed in finite realistic samples (DBRs) both by finite-size and frequency cutoff induced by the finite size of the pulses (∼50 fs) has absorption effects.46 When DBRs are used as building blocks for also been taken into account, as is the full spectral width of the pump more complex multilayered structures, these simple selection rules and probe pulses by doing a weighted average of the generation and are still extremely useful in understanding the observed spectra53,54 detection functions. It is clear from the time traces and the spec- and allow for a qualitative understanding of the spectral components tra in Fig. 2(b) that specific coherent acoustic vibrations have been present in the transient reflectivities of Fig. 2. Several acoustic modes excited and that those can be read in the reconstructed differential are observed in Fig. 2(b). The Brillouin peak55 of both superlattices reflectivity. and the substrate at ∼40 GHz, as well as two groups of peaks at 150 and 245 GHz, are precisely linked to the detection q ∼ 2keff selec- tion rule for the two different respective superlattices composing the samples. This is evidenced in Fig. 2(a), where the band structure of one of the superlattices (SL B) is depicted, with the horizontal line representing the q = 2keff condition at λ = 800 nm. The analysis of the region around the first zone-center minigaps of the underlying superlattices A, B, and C (185-210 GHz) is depicted separately in Fig. 3. The expected differential reflectivity spectrum ∆R/Ro (ω) for the topological sample [Fig. 3(d)] exhibits a modal structure involv- ing three modes; owing to the inverted symmetry of the band edge modes of the underlying superlattices B and A and to the mentioned selection rules, the lower band edge mode of DBR B (LB ) and the upper band edge mode of DBR A (U A ) are present in the spectra. As shown in Fig. 3(b), these two peaks correspond indeed to the measured modes when the two DBRs are assessed individually. The additional central peak in Fig. 3(d) is associated with the topolog- ical mode (T) that arises precisely from this inversion of the sym- metry when the two DBRs are concatenated. The derivative of the reflection phase from the substrate side depicted in Fig. 3(c) and the calculated mode profiles u(z) of Fig. 3(f) clearly evidence the band edge and confined nature of the observed modes, respectively. The spectrum obtained after Fourier-transforming the differential reflec- tivity time trace obtained [Fig. 3(d)] shows qualitative agreement with the theoretical spectra. The right-hand side of Fig. 3 confirms FIG. 2. Reflection-type pump-probe coherent acoustic phonon experiments. (a) that the simple selection rules for photoelasticity-based infinite non- Band diagram of one of the building superlattices (SL B) with the frequencies absorbing superlattices are also conclusive for the control sample; related to q = 0 and q = 2k eff highlighted with circles. (b) Differential reflectivity the two peaks LC and LB corresponding to the lower zone-center time-traces (insets) obtained using an asynchronous optical sampling (ASOPS) band edge modes of DBRs C and B, respectively. When looking at technique on the topological (top) and control (bottom) samples. The used mea- surement conditions for pump and probe lasers are 800 nm (40 mW) and 765 nm the displacement profile of the q = 0 (FS) mode of DBR B (LB ) in (4 mW) for the topological sample and 800 nm (40 mW) and 800 nm (4 mW) for both the topological and control samples, we observe a different the control sample. The as-obtained data were treated by cutting the initial elec- spatial profile. At this point, it must be noted that the FS peak is tronic peak and using appropriate Savitzy-Golay filtering to extract low-frequency mainly generated in the second DBR (DBR B) and thus its intensity backgrounds. The Fourier transform of the experimental traces after treatment is almost independent of the spatial profiles in the first DBR (DBRs are given (blue-shaded) and compared to the theoretical spectra obtained from A or C). In the first DBR, the LB mode exponentially grows from a simple electrostriction-photoelastic model. the surface to the interface for the control sample while it is fully APL Photon. 4, 030805 (2019); doi: 10.1063/1.5082728 4, 030805-4 © Author(s) 2019 APL Photonics ARTICLE scitation.org/journal/app FIG. 3. Optical transient reflectivity spectra around 200 GHz. (Left) (a) Band diagram of the underlying superlattices A (black) and B (blue). (b) Pump-probe phonon spectra of the individual uncoupled DBRs A and B. The same color code as in (a). (c) Simulated derivative of the phase shift φ for a substrate-incident acoustic plane wave, allowing identification of supported resonances for a closed structure at the top layer. The three relevant modes are identified as LB , UA , and T and are also present in the theoretical ∆R/Ro (d) and obtained experimental (e) spectra. (f) Mode profiles of the topological nanophononic cavity (T) as well as the detected lower (upper) band edge mode LB (UA ). (Right) The same for the control sample. propagating for the topological sample. This difference arises from on very well established selection rules in Raman scattering that the different matching conditions with the second DBR. In Fig. 3(a), are mapped into photoelastic mediated processes in pump-probe we show that the lower band edge associated with DBR B lies inside coherent phonon generation/detection experiments. Even though a frequency region where propagating modes exist for DBR A, while the existence of the topological mode could also be probed by means Fig. 3(e) shows that the same band edge lies deep inside the band of transmission/reflection experiments, similar to the ones reported gap of DBR C. Note that the profiles in Fig. 3 directly give the dis- in Ref. 56, this kind of experiments does not provide any information placement field u(z), i.e., its real part, in contrast with Fig. 1, where on the mode symmetries, a critical feature to prove the topolog- we show the absolute value of the complex-valued displacement ical nature of the most intense observed central peak. Moreover, field |u(z)|. they require the growth of metallic transducers that would gener- ate additional acoustic modes. Therefore, the use of time-resolved pump-probe measurements directly on the semiconductor super- IV. CONCLUSIONS lattices was essential to spectrally resolve and distinguish the three We designed a topological nanoacoustic cavity with a reso- modes around 200 GHz. The possibility of identifying individual nance frequency of 200 GHz. The design is based on the inversion modes that are closely spaced in frequency is of central importance of the symmetries of the Brillouin zone edge modes of two con- for the study of dynamics in more complex topological acoustic catenated superlattices. This band inversion relies on a small vari- structures where the interaction with the optical field can be engi- ation of the thicknesses of the constituting layers, due to the short neered,37,57 for example, in topological resonators for light and wavelength of the considered phonons. We fabricated the sample acoustic phonons. and a control structure by MBE. We performed HRXRD charac- terization to confirm that the actual samples do not differ consid- erably from the nominal designs, i.e., the band structures associ- ACKNOWLEDGMENTS ated with the actual thicknesses preserve acoustic band inversion. This work was supported by the European Commission in the By means of pump-probe spectroscopy, we characterized the photo- form of the H2020 FET Proactive project TOCHA (No. 824140). The acoustic behavior of the two samples resulting in markedly dif- authors acknowledge funding by the European Research Council ferent phononic spectra. The main features were remarkably well Starting Grant No. 715939, Nanophennec; by the French RENAT- reproduced by transfer matrix simulations. ECH network, and through a public grant overseen by the ANR as In a pump-probe measurement in a semiconductor superlat- part of the “Investissements d’Avenir” program (Labex NanoSaclay tice, we observe peaks that are related to q = 0 and q = 2keff acoustic Grant No. ANR-10-LABX-0035). ICN2 was supported by the Severo phonons. For symmetry reasons, only one of the two FS modes is Ochoa program from Spanish MINECO (Grant No. SEV-2017- accessible by the experiment. In the case of the control sample, two 0706) and the project PHENTOM (Fis 2015-70862-P), as well FS peaks are expected to appear on the same side of the common as by the CERCA Programme/Generalitat de Catalunya, and by minigap. In the case of the topological cavity, the two FS peaks the European Commission in the form of the H2020 FET Open appear on opposite sides of the gap, validating the band inversion project PHENOMEN (No. 713450 to G.A.). G.A. was supported concept. The symmetry-dependence of the observed peaks is based by a BIST Ph.D. fellowship and P.D.G. by a Ramon y Cajal APL Photon. 4, 030805 (2019); doi: 10.1063/1.5082728 4, 030805-5 © Author(s) 2019 APL Photonics ARTICLE scitation.org/journal/app 30 Fellowship No. RyC-2015-18124. M.E. acknowledges funding from L. M. Nash, D. Kleckner, A. Read, V. Vitelli, A. M. Turner, and W. T. M. Irvine, the German Research Foundation DFG (Forschungsstipendium ES Proc. Natl. Acad. Sci. U. S. A. 112, 14495 (2015). 31 560/1-1). Z. Yang, F. 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