Function phononic crystals

We propose a novel type of phononic crystal for which the materials parameters are continuous functions of space coordinates without discontinuities corresponding to a seamless fusion of the constituent materials within the crystal lattice. With the help of an adaptation of this fundamental approach, we extend the well-established concept of phononic crystals, allowing an investigation of the transition from conventional phononic crystals with a regulated step-like parameter function to the realm of so-called function phononic crystals. Our study is based on a first-principle theory assisted by high-performance computer simulations and focuses on an understanding of the effects of a deviation from the typical parameter step function on the phononic density of states (DOS). Our exploration of the DOS reveals a characteristic rapid convergence: even a slight deviation from an ideal step function has the potential to induce radical changes in the band structure leading to the emergence of desirable features, especially multiple complete phononic band gaps.

Introduction.-Photonic crystals are structures with a periodic variation in their dielectric constant, giving rise to a photonic band gap that can prohibit the propagation of light within certain frequency ranges [1,2].This is analogous to the crystal lattice in solid-state physics and its effect on electrons.However, unlike the crystal lattice which does not exhibit a step-like potential, photonic crystals consist of two or more materials, each of which with fixed dielectric constants, usually lacking a smooth transition between them.Photonic crystals have been extensively studied due to their ability to control light propagation, which has led to the development of a plethora of optical devices and applications [3][4][5].A critical research issue in the field of photonics occuring recently was the particular introduction of morphological hierarchy [6], (a) E-mail: david.roehlig@physik.tu-chemnitz.de(corresponding author) (b) E-mail: thomas.blaudeck@main.tu-chemnitz.departial order or randomness [7][8][9][10][11] as well as controlled orientational and positional disorder [12,13] as parameter dimensions, describing the morphology as a kind of "degree of deviation from order".In this respect, the photonic band gap structure and the optical properties of the components containing photonic crystals showed interesting optical effects such as multiple spectral band gaps [14][15][16][17][18] and angle-dependent iridescence [19].
The special concept of a photonic crystal consisting of materials the refractive indexes of which are functions of spatial position instead of constants, was proposed by Wu et al. [20,21].The resulting structures were called function photonic crystals.With these, the concept of conventional photonic crystals is extended a step further: so-called "function dielectrics" enable rapid control of material properties, e.g., the electro-optical Kerr effect [22] or the magneto-optical Faraday effect [23], in a highly tailored and customizable manner [24][25][26][27][28][29][30][31].
The generation of a smooth transition at a material interface is a concept that is also employed in smoothing techniques for the dielectric function [32,33].Utilizing such methods can substantially enhance the computational accuracy for systems that exhibit discontinuities in the dielectric function.However, these approaches fundamentally differ from the one of function photonic crystals, as their objective is to approximate the real system (possibly with abrupt transitions).In this context, the emphasis is on minimizing errors and achieving rapid convergence, not on changing the physical system.
Throughout this paper, we focus on an investigation of the immediate transition from the conventional case, i.e., the case of a step-like to the case of a continuous parameter function.In other words, we aim to describe the hitherto unknown impact of a small deviation from the step-like function on the band structure.It is emphasized that there are no comparable studies of this approach in photonics, and even less so in the analogous context of phononic crystals.The latter are defined as artificial structures with periodic variation of the elastic parameters that can manipulate the propagation of elastic waves, which are elastic deformations propagating in solids [34][35][36].Building a function crystal of this type requires materials that satisfy the condition of a nonlinear parameter function, not having a well-defined boundary layer, but merging continuously into each other.The structures we investigate can be realized as functionally graded materials, i.e., multifunctional matter abundant in nature or created artificially [37][38][39][40][41][42] with a spatial variation in composition or structure which relates to a variation of the functional properties.
In this paper, we also consider the simplified special case sometimes referred to as sonic crystals applying to acoustic waves (propagating in fluids) instead of elastic waves (propagating in solids).Elastic wave propagation can be neglected in these structures if there is a significant contrast in density and velocity between scatterers and matrix material [43].This ensures that waves in fluids (such as air in our case) will have a minimal effect on inducing motion in a solid (such as steel in our case).
In analogy to function photonic crystals, function phononic crystals can be seen as a promising class of phononic materials with tailored functionalities that can be designed to suit innovative applications.Similar to the Kerr effect in function photonic crystals, external manipulation of material properties could also be realized in function phononic crystals by means of smart materials [44,45] or in general by nonlinear mechanical phenomena [46].As research in this field starts, we anticipate the development of new tunable devices harnessing the unique properties of function phononic crystals.

Continuous parameter function.
Considering a twodimensional system consisting of scatterers embedded in a matrix medium M with lattice spacing a, the parameter function ‫(ג‬r, η) can be written as where ‫ג‬ M denotes the parameter of the matrix material and ‫ג‬ S represents the parameter of the scatterer.For a phononic crystal, the material parameter ‫ג‬ might be the density ρ(r), the bulk modulus K(r) or the first or second Lamé parameter λ(r) or μ(r) (shear modulus).
For continuous function phononic crystals we define the broadening parameter η.For the case of conventional phononic crystals, we set η to zero, meaning that the parameter function corresponds to a piecewise-defined step function: For the function phononic crystal (η > 0) we use the following expression: where R is the radius of the scatterers.In the range r ∈ R ∈ [0, a/2] the function takes values between f (a/2, η) = 0 and f (0, η) = 1, as seen in fig. 1.For η → 0 the structural function corresponds to eq. ( 2), while for η → ∞ it matches a line corresponding to the case of a linear function crystal.In the literature, as mentioned, this case has already been considered for photonic crystals.For the case of cylinders with axes parallel to the z direction, we set r = x 2 + y 2 .
Phononic density of states.The propagation of transverse and longitudinal elastic waves in an isotropic and homogeneous medium can be described by the elastic wave equation The two Lamé constants, λ(r) and μ(r), describe the elastic properties of the respective medium.In this study, we calculated these parameters based on the bulk modulus K and Poisson's ratio ν.The density of the medium is represented by ρ(r).In the case of a two-dimensional crystal with an infinite number of rods parallel to the z-axis and a wave vector in the xy-plane, the wave equation can be separated into equations for xy and the z modes [47].
The sonic special case arises in the propagation of purely longitudinal modes in fluids, which can be described by the acoustic wave equation here solely described in terms of the pressure p.Besides the density, the bulk modulus K is a required parameter for this expression.To obtain the phononic and sonic frequency eigenvalues each from eqs. ( 4) and ( 5), we employed the plane-wave expansion method (PWE), widely used for calculating band structures.
The phononic density of states (DOS) is defined as the sum of delta distributions, with peaks at the N calculated frequency eigenvalues f i .For better control over the smoothness of the curve, we introduced Gaussian functions with a standard deviation σ, as follows: A normalization factor can be chosen as F = √ 2πσ 2 .However, we later refer to it in relation to a chosen maximum value.
Results.-For our investigations, we used a structure of steel cylinders (with density ρ S = 7860 kg/m 3 , bulk modulus K S = 160 GPa and Poisson's ratio ν S = 0.28) arranged in a square lattice with lattice constant a = 0.24 m and radius R = a/4.The scatterers are embedded in high-density polyethylene (HDPE) with ρ H = 950 kg/m 3 , K H = 0.2 GPa and ν H = 0.4.For the sonic crystal we embedded steel cylinders in air with parameters ρ A = 1.2 kg/m 3 and K A = 142000 Pa, allowing only longitudinal modes.This is a good approximation because transverse modes in the metal scatterers are not significantly excited by the low acoustic pressure in air.
The color plot of the DOS for 100 broadenings η between zero and 0.2 R/a is shown in fig. 2 for the sonic crystal and in fig. 3 for the phononic xy and z modes.Each DOS was calculated according to eq. ( 6) for frequencies up to 3000 Hz for the common path between the symmetry points Γ, X and M in reciprocal space with a cutoff radius of 76 π/a.Each curve was normalized to the maximum value of all DOS curves of the different broadenings.A frequency range of zero DOS corresponds to a band gap.
For the conventional sonic crystal no band gap can be found.But as the increasing broadening η softens the parameter step functions ρ(r, η) and K(r, η) according to eqs. ( 1) and ( 3), two complete band gaps occur, including a broad one around 1000 Hz, which opens from about η = 0.05 R/a.For the phononic crystal, we consider waves propagating in the plane of periodicity (xy modes) and in the perpendicular direction (z modes), which corresponds to the direction parallel to the scatterers.In total, the evolution of the DOS with increasing broadening parameter shows a similar trend for both types of modes (shown in fig.3): an opening of band gaps can be observed in the given frequency range.
Starting from step-like parameter functions ρ(r, η → 0), λ(r, η → 0) and μ(r, η → 0), one frequency gap can be observed for xy modes and two for z modes.Above a broadening of 0.05 R/a, there are two gaps for xy and four for z states.
In general, it can be stated that, starting from zero, even a small increase of the broadening parameter η leads to a radical change of the sonic and phononic band structures.For small deviations from the step-like parameter function (η > 0), the system can be viewed as an array of in-plane acoustic (or phononic) lenses, as opposed to sharp-edged scatterers (η = 0).The focusing of sound by means of spatially variable material parameters is a recognized phenomenon in the realm of gradient acoustics [48,49].In this respect, Jin et al. engineered acoustic gradient-index metasurfaces from soft graded-porous silicone rubber, effectively shaping wavefronts in water.When employing a hyperbolic index profile, both experimental observations and simulations distinctly demonstrate pronounced focusing effects [50].
Since the DOS curves quickly converge, frequency gaps are quite stable from a broadening parameter of around 0.15 R/a for sonic and phononic modes, which is still sufficiently far from the line function case f (r, η → ∞) of eq. ( 3).Thus, a linear transition between two materials (as seen in fig. 1 for η → ∞) is not required for practical applications.With increasing η for all cases (see fig. 2 and fig.3), the states concentrate within a narrow range, leading to pronounced DOS peaks.This phenomenon indicates that the bands are flattening out, resulting in an increased probability for a band gap in the given frequency range.To observe the evolution of band gaps more clearly with increasing η, we sorted all frequency eigenvalues 0 ≤ f i < f i+1 < . . .< f N of the band structure and calculated a density of the differences between adjacent eigenvalues defined analogously to eq. ( 6).The results, exemplary for xy modes, are shown in fig. 4 (excluding frequency distances less than 100 Hz).We disregarded small frequency distances as they are attributed to numerics.For zero broadening there is only a single frequency distance peak located just above 200 Hz, which corresponds to the first band gap (see fig. 3 for xy modes).As η increases, this peak gradually moves to higher distances, reaching approximately 800 Hz.The shift basically indicates a widening of the frequency gap.Furthermore, Fig. 4: Density of eigenvalue differences (fi+1 − fi) of phononic crystals composed of steel cylinders embedded in HDPE (depicted here for σ = 10).The calculations were performed for xy modes, which are oriented perpendicularly to the scatterers.The peaks correspond to complete band gaps that widen significantly or even appear due to the deviation from the parameter step function (defined by the broadening parameter η).
we observe the emergence of new gaps occurring above 0.05 R/a.Overall, the transition from a conventional to a function phononic crystal is characterized by a widening of band gaps, which is a result of the flattening of the phononic bands.These results occur also for the sonic and phononic z modes.
Conclusions.-In summary, introducing the concept of function phononic crystals and studying theoretically the transition from the conventional case of phononic (and sonic) crystals we discovered that small deviations of the materials parameters can have a profound impact on the band structure.The bands tend to flatten as the broadening increases, leading to the emergence of larger (in all cases up to complete) band gaps.Notably, we discovered that the DOS exhibits a rapid convergence with the broadening parameter.Even a slight deviation from the ideal step function has the potential to induce significant and transformative changes in the band structure.This finding holds promise for a potential fabrication in the future (e.g., facilitated by functionally graded materials), as it can lead to a viable forecast of desirable properties of components through design loops and a guide to a simplification of manufacturing strategies on account of the said rapid convergence of the DOS with broadening.Our results are also relevant with respect to the fact that it is never possible to produce a strict boundary layer, since surface roughness or the like prevent this.With that, among others, the inclusion of external nonlinear effects can make the materials concept of function phononic crystals highly interesting for various fields in applied physics, materials science and engineering.* * * This work was partially funded by a PhD scholarship awarded by the Free State of Saxony (DR).The authors thank Sibylle Gemming, Jörg Schuster and Erik E. Lorenz for granting computation time and resources on the MAINSIM High-Performance Computer Cluster.

Fig. 1 :
Fig.1: One-dimensional plot depicting the structural function f (x, η) as defined in eq.(3) of a crystal with lattice constant a and scatterer radius R. The displayed curves represent states between two extreme cases: the usual parameter step function (η → 0) and the linear function (η → ∞).

Fig. 2 :
Fig. 2: Density of states (DOS) curves (depicted here for σ = 50) of sonic crystals composed of steel cylinders embedded in air.By applying broadening η, the parameter step function is softened, leading to the emergence of two new complete band gaps (vanishing DOS), including even a surprisingly large one around 1000 Hz.

Fig. 3 :
Fig. 3: Density of states of phononic crystals consisting of steel cylinders embedded in high-density polyethylene (HDPE), depicted here for σ = 50.Separate calculations were performed for two distinct cases: xy modes perpendicular (left) and z modes parallel to the scatterers (right).Notably, when broadening η softens the parameter step function, numerous new complete band gaps (vanishing DOS) appear for both modes.