Emulating tightly bound electrons in crystalline solids using mechanical waves

Solid state physics deals with systems composed of atoms with strongly bound electrons. The tunneling probability of each electron is determined by interactions that typically extend to neighboring sites, as their corresponding wave amplitudes decay rapidly away from an isolated atomic core. This kind of description is essential in condensed-matter physics, and it rules the electronic transport properties of metals, insulators and many other solid-state systems. The corresponding phenomenology is well captured by tight-binding models, where the electronic band structure emerges from atomic orbitals of isolated atoms plus their coupling to neighboring sites in a crystal. In this work, a mechanical system that emulates dynamically a quantum tightly bound electron is built. This is done by connecting mechanical resonators via locally periodic aluminum bars acting as couplers. When the frequency of a particular resonator lies within the frequency gap of a coupler, the vibrational wave amplitude imitates a bound electron orbital. The localization of the wave at the resonator site and its exponential decay along the coupler are experimentally verified. The quantum dynamical tight-binding model and frequency measurements in mechanical structures show an excellent agreement. Some applications in atomic and condensed matter physics are suggested.

www.nature.com/scientificreports www.nature.com/scientificreports/ experiments with classical waves, however, have only a limited use in the context of quantum mechanical systems since they have their own physical constants, such as hopping amplitudes, a sui generis dispersion relation and, as a consequence, their own dynamics.
In the structured elastic systems presented here, opposite to the cases mentioned above, the site energies and couplings can be engineered to get a "quantum tight-binding result" with first and second neighbours at least. The results shown here represent a breakthrough in the area of structured elastic systems since the application of the tight-binding model in this area has remained open for exploration due to several difficulties. One of them is that the typical coupling between connected vibrating solids is very strong and thus has a long range; an elastic evanescent coupling was introduced only very recently 25 . Also the different kinds of elastic waves, as for instance transverse and longitudinal, are strongly coupled; the selective excitation or detection of these elastic waves has been solved also recently 26,27 .
In this paper, the emulation of evanescent (or weak) couplings via locally periodic structures is presented. In a similar fashion as the coupled-resonator optical waveguides are engineered with defects inside photonic crystals [11][12][13][14][15] , here coupled mechanical resonators are engineered in an otherwise periodic structure. The idea is to use the gaps of crystalline mechanical structures that will be taken as couplers. As it is well known, a periodic system, even an elastic one, shows bands and gaps. Therefore connecting neighbouring resonators through a locally periodic structure, allows them to communicate weakly with each other through the coupler when their resonant frequency lies within the coupler's badgap. The maximum wave amplitude will be located at the position of the resonator and from there, the amplitude will decay exponentially through the coupler 25,28 . This result is used in order to build a mechanical metamaterial with transport through such trapped states. Here five mechanical vibrating systems, that obey the quantum tight-binding model and emulate a finite quantum 1D crystal, are reported. The tight-binding model here developed is a generalization to mechanical waves of the TB model that emerges for coupled-resonator optical waveguides 13,14 . In that model a set of periodic defects inside a photonic crystal are treated as high-Q resonators weakly coupled each other. The elastic systems were constructed on aluminum beams and are composed of n resonators joined by couplers (see Fig. 1). Each coupler is formed by m unit cells of length . In turn, each one of these unit cells is composed of a large cuboid of cross-sectional area W × W and length ℓ − ϵ and two small cuboids of cross-sectional area w × w and length /2 where W, w,  (see Fig. 1a, left inset). The resonator also has length , and is composed of one cuboid of cross-sectional area W × W and length  − − l d and two small cuboids of cross-sectional area × w w and length + d ( )/2  (see right inset of Fig. 1a). Two couplers of M(>m) unit cells are used as terminators to avoid finite size coupler's border effects. The vibrating system can also be understood as a crystal composed by n coupled supercells (see shadow zone in the beam of Fig. 1). Each system was constructed by machining a solid aluminum piece.
The experimental spectra of the structured rods, for torsional waves, corresponding to two up to six artificial elastic atoms or supercells are shown in Fig. 1b. These spectra were measured using acoustic resonant spectroscopy by exciting at the position of one resonator and detecting at the position of another resonator. In this figure, the spectra show the emergence of a band, located within the second bandgap of the coupler (See ref. 25 ), as the number of supercells increases. As it can be seen in this figure, the band emerges symmetrically approximately www.nature.com/scientificreports www.nature.com/scientificreports/ at 26450 Hz, i.e. at a resonant frequency of the spectrum of a single artificial elastic atom. Each band is formed by well resolved Breit-Wigner line shapes with a number of resonances given by the number of artificial elastic atoms. The width of the band is approximately 150 Hz and it is possible to notice that the level spacings at the borders of the band are smaller than the level spacings at the center of the band, in agreement with the energy spectrum of a 1D atomic crystal.
In Fig. 2, top left (top right, bottom left, bottom right) the measurements of the absolute value of the torsional wave amplitudes, as a function of the position, for the elastic structure with three (four, five, six) coupled resonators are shown. Each wave amplitude has a one-to-one correspondence with each frequency level in the emergent band of Fig. 1. The obtained data were recorded by moving the detector along the beam; only two thirds of the total length of the structure were measured to avoid the saturation of the detector by the exciter's magnetic field. Note that the experimental wave amplitudes are localized at the position of the resonators and that they show an exponential decay in the couplers. This is characteristic of states associated to frequencies in the bandgap and are compatible with the case of electrons strongly bounded to their atoms (tight-binding electrons). This fact is in agreement with the wave amplitudes obtained from the finite element numerical calculation in which the deformations of the structured beam can be seen mainly localized at the resonator positions.
The experimental results given in Figs. 1 and 2 allow us to use the tight-binding approximation, à la quantum mechanics, to calculate the torsional spectrum of the elastic crystal. In other words, a mechanical system which can be described by the Anderson model is built. The basis of the mechanical TB model will be formed by the torsional wave amplitudes φ x { ( )} n , localized at the position of the defect, with frequency f n lying inside a gap of the coupler, associated to each isolated supercell on site n (see Fig. 3a). The torsional wave amplitude θ x ( ) of the elastic crystal, as function of position, can be expanded in terms of the basis as Then the tight-binding model for the 1D elastic crystal can be written as where, f n , is the resonant frequency of one isolated supercell on site n (Fig. 3a); C is the coupling coefficient between nearest neighbour resonators, which depends also on the properties of the coupler. The resonant frequency of the complete crystal will be f. This is the simplest TB model, since it assumes only nearest-neighbour www.nature.com/scientificreports www.nature.com/scientificreports/ couplings that are identical for all resonators. The generalization of this mechanism to more complex situations including second neighbours, as well as 2D and 3D systems, is straightforward. We should note that this model is mathematically equivalent to the quantum-mechanical and to the resonant-cavity coupled waveguide cases 13 , and it differs from the mass-spring tight-binding model (previously considered [29][30][31] ), since the latter has ω 2 , instead of f, being ω the angular frequency. Thus f, f n , φ n and C, in the TB model given in Eqs. (1) and (2) for mechanical waves, take the role of the energy, the site energy, the orbital function, and the hopping amplitude in the quantum TB model, respectively. The model represented in Eq. (2) has two free parameters, the site-frequency f n and the mechanical hopping C. Physically f n roughly corresponds to the resonant frequency of the isolated resonator ( Fig. 3a) whereas C is related to the localization length of the wave amplitude of one resonator within the periodic structure. Both values should be determined to generate the TB torsional frequency spectrum for an elastic crystal with an arbitrary number of cells, as those given in Fig. 1. The values of f n and C were obtained from the numerical calculations, made with the transfer matrix method, for a structure with two coupled resonators (Fig. 3b). In this case the TB model gives two solutions = ± ± f f C 0 , thus giving the site-frequency f 0 and the mechanical hopping C just as in the tight-binding model for the coupled-resonator optical waveguides 15 . Figure 3c gives the ratio r between the total maximum and the central local maximum of the symmetric wave amplitude of Fig. 3b, a measure related to the localization length for the two-resonators case, as a function the number of cells of the coupler. The level splitting Δf = 2C is also given in Fig. 3c. As it can be seen in the same figure, the localization (level splitting) increases (decreases) as the number of cells in the coupler increases and that both quantities roughly show an exponential behaviour. One can also notice that the slope of the level splitting is approximately minus twice the slope of the ratio r, since the level repulsion can be understood as the overlap of two wave amplitudes φ i of the basis located at different sites. In Fig. 3d the emergent band, appearing around 25.89 kHz, is given as function of number of supercells. As can be seen, the results obtained with the tight-binding model agree with those obtained with the transfer matrix method. It is possible to observe that the frequency levels split symmetrically around the level of frequency f 0 . This is in contrast with other elastic crystals studied in the literature 32 , whose levels are distributed asymmetrically towards one side of the first frequency level. This figure shows that the spectrum of the elastic crystal constructed here is completely analog to a spectrum of an 1D atomic crystal. In the mechanical case the resonators interact through the bandgap of the couplers. One might also consider an ideal elastic crystal, constructed from an infinite number of supercells. In this case it is possible to obtain the dispersion relation of the crystal. The wave amplitude on site n is written as a plane wave, www.nature.com/scientificreports www.nature.com/scientificreports/ = − A e n inka , where a is the distance between consecutive sites on the elastic lattice and k is the wave number. The dispersion relation 0 is the same as that of a 1D periodic array of tightly bound electrons. This dispersion relation is shown in Fig. 3e; one can notice that the emergent band of the locally periodic system of n supercells, Fig. 3d, is completely contained inside the limits of the dispersion relation, ± f C 2 0 , of the ideal elastic crystal. The group velocity can be obtained directly from the dispersion relation and it implies that, at the bottom of the band, there are free mechanical quasiparticles analog to the free electrons in an 1D atomic crystal. In the mechanical case the quasiparticles have effective mass = m Ca eff 1 2 2 . In summary: in this article it was shown that it is possible to emulate tightly bound electrons in atomic crystals using mechanical waves. This was feasible thanks to the design of an elastic crystal composed of resonators joined with couplers in such a way that the levels associated with the resonators lie within the coupler's bandgap. One should notice that the perturbative coupling used in refs. 25,28,[30][31][32][33] is not enough to emulate a quantum tight-binding model; a resonator-coupler engineering using the Bragg reflection is needed. A generalization of resonant acoustic spectroscopy was used to measure the spectra and wave amplitudes of the artificial elastic lattices. The experimental results are well reproduced by the quantum tight-binding model. A complete analogy between the quantities used in the quantum model appear in the mechanical system: the on-site frequency, the mechanical hopping, the mechanical orbitals, the "frequency" levels, quasi-particles and an effective mass. The envelope of the atomic orbitals and the dispersion relation can also be obtained. The structured elastic systems reported here can be easily generalized to more complex quantum systems in 2D or 3D, such as molecules graphene, graphene nanoribbons with an electric field 34 (changing the site size to emulate electric fields 33 ), among many others. Transport through molecular junctions can also be implemented using the techniques recently developed by our group 35 . For obvious reasons, with the advantages and challenges of the macroscopic systems used, this gives the possibility of analyzing problems that cannot be studied easily in atomic systems due to experimental limitations. The mechanism described here allows to create mechanical metamaterials that have the features of a quantum crystal. Although the results presented are novel and have their value by themselves, they also open the door to the study of other quantum properties using mechanical waves. In fact, the effects of second neighbours on spectra and wave amplitudes can studied with the structured systems designed here. The previous statements are supported by the fact that coupling engineering in structured mechanical systems is flexible enough to introduce length deformations that generalize simple periodic configurations. The coupling engineering also allows to tune in the coupling with the distance since the model in the site basis is tridiagonal. The realization of evanescent couplings extends well beyond the mathematical abstraction of Wannier functions 36,37 and the limitations of electronic transport in traditional solids.

Methods
Acoustic resonant spectroscopy. This technique is used to measure the mechanical vibration spectra for any elastic system and the setup is given in Fig. 1. The procedure starts with the generation of a harmonic signal, of frequency f 0 in the vector network analyzer (VNA). This signal from the VNA is sent to a high-fidelity audio amplifier (Cerwin-Vega 2800 was used) to increase its power. The output of the amplifier is sent to an acoustic electromagnetic transducer (EMAT) located at the vicinity of the artificial elastic crystals. The EMAT, by electromagnetic induction generates mechanical vibrations in the aluminum piece 26 . A second EMAT detects the mechanical response of the crystal, at other location the beam, and converts it into a voltage signal. This signal is captured by the VNA. A workstation is used for the automated storage and subsequent analysis of the data of the different measurements. Then f 0 is changed to f 0 + Δf and the procedure is repeated to obtain an spectrum. To excite and detect spectrum efficiently the transducers are located at the edges of the resonators. By moving the EMAT detector along the beam, it is possible measure the wave amplitudes as a function of the position. electromagnetic acoustic transducers (eMAtS). We will first discuss how these devices excite mechanical waves (See Fig. 1). A harmonic current is applied to the EMAT coil and the latter produces a magnetic field, also alternating, which induces eddy currents in the metallic rod. The interaction, via Lorentz force, between the magnetic field of the magnet and the eddy currents produce a force on the metal. The same device can be used as a detector: when the metal surface oscillates close to the EMAT's magnet, the magnetic flux through any loop of the paramagnetic metal will change. This, according to Faraday's law, originates an electromotive force in the loops which in turn generates a magnetic field measured by the detector's coil. A deeper explanation of the EMAT operation can be found in refs. 27,33 . coMSoL simulations. COMSOL Multiphysics was used to calculate the finite element method simulations of Fig. 2 with the parameters corresponding to aluminum: Young's module = .
E 68 6 GPa, Poisson's coefficient ν = .   More details about the transfer matrix method applied to the elastic beams can be found in refs. 32 and 33 .