Molecular orbitals of an elastic artificial benzene

A. M. Martínez-Argüello, M. P. Toledano-Marino, A. E. Terán-Juárez, E. Flores-Olmedo, G. Báez, E. Sadurní, and R. A. Méndez-Sánchez

Phys. Rev. A 105, 022826 – Published 28 February 2022

Abstract

Benzene, a hexagonal molecule with formula $C_{6} H_{6}$ , is one of the most important aromatic hydrocarbons. Its structure arises from the $s p^{2}$ hybridization from which three in-plane $σ$ bonds are formed. A fourth $π$ orbital perpendicular to the molecular plane combines with those arising from other carbon atoms to form $π$ bonds, very important to describe the electronic properties of benzene. Here this $π$ system is emulated with elastic waves. The design and characterization of an artificial mechanical benzene molecule, composed of six resonators connected through finite phononic crystals, are reported. The latter structures trap the vibrations in the resonators and couple them through evanescent waves to neighboring resonators, establishing a tight-binding regime for elastic waves. Our results show the appearance of a spectrum and wave amplitudes reminiscent of that of benzene. Finite element simulations and experimental data show excellent agreement with the Hückel model for benzene.

Received 7 July 2021
Revised 10 December 2021
Accepted 28 January 2022

DOI:https://doi.org/10.1103/PhysRevA.105.022826

Physics Subject Headings (PhySH)

Electrical properties Electronic structure

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

A. M. Martínez-Argüello ^1,*, M. P. Toledano-Marino¹, A. E. Terán-Juárez¹, E. Flores-Olmedo², G. Báez², E. Sadurní ³, and R. A. Méndez-Sánchez ¹

¹Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Apartado Postal 48-3, 62210, Cuernavaca Morelos, Mexico
²Departamento de Ciencias Básicas, Universidad Autónoma Metropolitana-Azcapotzalco, Avenida San Pablo 180, Colonia Reynosa Tamaulipas, 02200 Ciudad de México, Mexico
³Benemérita Universidad Autónoma de Puebla, Instituto de Física, Apartado Postal J-48, 72570 Puebla, México

^*alael@icf.unam.mx

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Vol. 105, Iss. 2 — February 2022

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Images

Figure 1
(a) Elastic artificial atomic site composed of a hexagonal-shaped resonator of length $L_{h}$ and connected to three bonds with an angle of $120^{\circ}$ between them. (b) Finite phononic crystal (FPC), formed by the repetition of a unit cell, that plays the role of bonds. (c) The unit cell is composed of a square-shaped aluminum plate of thickness $e$ and side $L$ , and two smaller plates, one on each side of the larger plate, of size $a / 2$ and length $b$ . The FPCs give rise to bands and gaps in its level dynamics (frequency spectrum as a function of $L$ ), as shown in panel (d) for a structure composed of five unit cells. The band gap is indicated by a red oval. The geometrical parameters are $e = 6.05$ mm, $L = 26.0$ mm, $a / 2 = 2.5$ mm, $b = 8.2$ mm, and $L_{h} = 30.4$ mm.
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Figure 2
By choosing a frequency of the resonator in the gap of the FPCs [see Fig. 1], the vibration is trapped in the resonator as shown in panels (a) and (b), and the maximum (minimum) amplitude in panel (a) is shown in red (blue) color. In the lower right part of (a), this vibration is shown in three dimensions (3D). An exponential decay of the amplitude of vibration along the bonds, in the positions indicated in black-filled circles, is also evident. This exponentially decaying wave amplitude, panel (b) in red-filled circles, emulates a quantum atomic orbital and is used to build the artificial benzene molecule. The amplitude is measured at the points indicated in black-filled circles in panel (a). An exponential decay of the frequency level splitting as a function of the number of FPC cells [see panel (c)] is obtained, as shown in panel (b) in blue-filled squares. The dashed lines are fits to the corresponding data. The geometrical parameters are the same as those in Fig. 1.
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Figure 3
(a) Experimental setup: 1, work station; 2, vector network analyzer Anritsu MS4630B; 3, Cerwin Vega CV-2800 high-fidelity audio amplifier; 4, electromagnetic acoustic transducer (EMAT) exciter; 5, elastic artificial benzene; and 6, EMAT detector. (b) Numerical frequency spectrum of artificial benzene as a function of $L_{h}$ ; the FPCs give rise to forbidden bands that trap the vibrations in the resonators. For $L_{h} = 30.4$ mm, in the interval from 20500 to 21500 Hz, some natural frequencies associated with the resonators (red horizontal lines) are observed within the forbidden band. (c) Experimental frequency spectrum, six resonances are observed, two singlets and two doublets; the vertical lines correspond to the numerical predictions. An excellent agreement is observed between the experimental and numerical results.
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Figure 4
Wave amplitudes from tight-binding model, artificial mechanical benzene, and experiments, in left, middle, and right columns of panels (a) and (b). The color scale is the same of Fig. 2. The blue-dashed circles in the center and right columns of panels (a) and (b) indicate equivalent points and help to identify the symmetry of the artificial orbital. The “E” in the experimental orbitals indicate the position of the EMAT exciter.
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Figure 5
Frequency spectra of the artificial benzene in the presence of impurities. When two impurities are added on one resonator, the degeneracies of the molecule are broken. This is observed by the appearance of another resonance in the spectrum around 21 160 Hz (a). When more impurities are added (six) the cyclic property of the artificial molecule is fully broken and five resonances in the spectrum show up (b). In both experiments the impurities (neodymium magnets) are located on the vertices of a resonator [see Fig. 1].
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