Phonon Spectrum Engineering in Rolled-up Nano- and Micro-Architectures

We report on a possibility of efficient engineering of the acoustic phonon energy spectrum in multishell tubular structures produced by a novel high-tech method of self-organization of nano- and micro-architectures. The strain-driven roll-up procedure paved the way for novel classes of metamaterials such as single semiconductor radial micro- and nano-crystals and multi-layer spiral micro- and nano-superlattices. The acoustic phonon dispersion is determined by solving the equations of elastodynamics for InAs and GaAs material systems. It is shown that the number of shells is an important control parameter of the phonon dispersion together with the structure dimensions and acoustic impedance mismatch between the superlattice layers. The obtained results suggest that rolled up nano-architectures have potential for thermoelectric applications owing to a possibility of significant reduction of the thermal conductivity without degradation of the electronic transport.

The initial work on the phonon confinement effects in thin films and nanowires was performed using the elastic continuum approximation [10,11,14,[29][30][31]38]. The results obtained with the elastic continuum approach have been confirmed using other techniques such as lattice dynamics [34,39] and molecular dynamics (MD) [40][41][42]. All computational approaches proved that the acoustic impedance η= ρ×v of the barrier shells of nanostructures presents an important tuning parameter for phonon transport, which can be used together with the lateral dimensions and shape for phonon engineering of material properties (ρ is the mass density and v is the sound velocity of the material). In the core-shell nanostructures with the acoustically mismatched barrier shells new types of phonon modes appear. Some of them are mainly concentrated in the nanostructure core while others are localized in the shell layers [11,14,34,38,39]. Controlling the acoustic impedance mismatch and thickness of the shell layers one can tune the electronphonon interaction and the electron mobility [32,33]. Similarly, one can engineer the phonon group velocity and thermal conductivity in such nanostructures [11,14,34,38,39].
Further progress in phonon engineering of material properties depends on availability of nanostructures with layered structures and substantial acoustic impedance mismatch. Multilayer tubes have attracted a special attention in newly developed acoustic metamaterials and phononic crystals at the micro-and nanoscale [43,44]. In particular, a cylindrical structure from 40 alternating layers of 0.36 mm thick natural latex rubber film and 0.38 mm thick silicone elastomers containing boron nitride particles serves as a thermal shield [45]. In a two-layered tube with the weak interfaces between the layers, the dispersion characteristics of longitudinal guided acoustic wave provide a tool for detecting and exploring defects [46]. The microstructures are shown to play a decisive role in the dispersion of both flexural and longitudinal waves in single-and multiwall carbon nanotubes [47].
Acoustic metamaterials and phononic crystals are formed by periodic variation of the acoustic properties of the materials (elasticity and/or density), what leads to the occurrence of the phononic band gaps and provides powerful tools to control the phonon velocity spectra. The phonon crystals were proposed theoretically for elastic waves propagating in a composite material consisting of identical spheres [48] and infinite cylinders [49,50] with parallel axes embedded in a periodic way within a host. First sonic materials with effective negative elastic constants were fabricated as lead coated spheres arranged in a simple cubic crystal [51]. They acted as total wave reflectors within certain adjustable acoustic frequency ranges. Split-ring resonator periodic arrays [52] and double negative (with the negative effective bulk modulus and the negative effective density) acoustic metamaterials [53] were suggested in a close analogy with electromagnetic metamaterials. For diverse experimental realizations of phononic crystals and their applications, see [44].
A novel method of self-assembly of micro-and nanoarchitectures was designed on the base of the strain-driven roll-up procedure [54,55]. It paved the way for novel classes of metamaterials: single semiconductor micro-and nanotubes (or radial crystals) [56] and multilayer spiral micro-and nanotubes (or radial superlattices) [57]. A comprehensive structural study was provided for semiconductor/oxide, semiconductor/organic as well as semiconductor/metal hybrid radial superlattices [58]. A combined "roll-up press-back" technology has been recently presented to fabricate novel acoustic metamaterialsmechanically joined nanomembrane superlattices [59], which reveal a significant reduction of the measured cross-sectional phonon transport compared to a single nanomembrane layer.
The optical phonon spectra in multilayer cylindrical quantum wires manifest a geometric structural effect [4], which is of immanent importance for understanding of the pairing of charge carries in quantum wires [60] as well as the electron-phonon phenomena in multilayer coaxial cylindrical AlxG1−xAs/GaAs quantum cables [61] and double coupled nanoshell systems [62]. The aim of the present work is to investigate feasibility of controlling the acoustic phonon energy spectra and corresponding phonon velocity dispersion in rolled-up micro-and nanoarchitectures. Of fundamental importance in this context is the experimental evidence [63], that due to oxide formation during fabrication, a single period of a radial superlattice is represented by a semiconductor/amorphous oxide/polycrystalline metal/amorphous oxide layer rather than a semiconductor/metal layer. It implies a necessity to investigate multilayer tubes. The elastodynamic boundary conditions on spiral interfaces of a rolled-up microtube with multiple windings or on cylindrical interfaces of a multilayer tube, which consists of coaxial cylindrical shells, (multishell) immediately affect acoustic phonon energy spectrum and, hence, phonon group velocities for propagation along the tube. Since the effect of these boundary conditions depends on the number of shells along with the geometric parameters, multishells are qualified into acoustic metamaterials. This introduces, in particular, extra capability for tuning the phonon spectrum, engineering the phonon transport and advancement of thermoelectric materials [9].

Theoretical Model
Spiral interfaces of a rolled-up nano-and microtube with multiple windings are modelled by cylindrical interfaces of a multilayer tube that consists of coaxial shells (multishell) as shown schematically in Fig. 1. In total, there are 6(N+1) boundary conditions. On that basis, the secular equation is derived for eigenmodes of phonons at an arbitrary number of the shell pairs. The boundary problem described by the Eq. (1) and the boundary conditions (2) to (5) satisfies the correspondence principle with respect to the case of a two-shell composite coaxial tube [65,66]. The Here the scalar potential of the dilatational motion and the vector potential of the shear motion along the axis of the structure (which has only two independent components) satisfy the equations: The velocities Every neighboring layers are assumed to be perfectly bonded. Therefore, the eigenwaves in all layers will have the same longitudinal wave number  and circular frequency : A multishell with a periodic alternation of two materials is further assumed (see Fig. 1, lower panel) with r0=100 nm. All odd shells (m=2k+1) consist of one and the same material with elastic properties 1, 1, density 1 and have the same thickness: r1. All even shells (m=2k) consist of the same material with elastic properties 2, 2, density 2 and have the same thickness: r2. They are represented in Table   1. In what follows, the number of layers is denoted by NL. The wave characteristics in the materials are defined in Table 2. Dispersion of the Rayleigh waves i 2 =0: The phonon dispersion curves are represented in the non-dimensional form, using the units, which are defined in Table 3.

Results for NL=2
The lowest phonon dispersion curves (in the window of eigenfrequencies [0, 1.5]) are shown for axially symmetric waves n=0, NL =2 in Fig. 2. (More time-consuming calculations for flexural waves with n=1,… are ongoing.) There are anticrossings of torsional or non-torsional modes, but there might occur crossings of torsional (ux=ur=0, u0) and non-torsional (associated with the displacement components ux and ur, u=0) modes. Dispersion of the phonon group velocity for the dispersion curves in Fig. 2 is represented in Fig. 3 (see Fig. A1 for a detailed graph). A group velocity dispersion curves stops when the eigenfrequency  goes beyond the window [0, 1.5].

Results for NL=4 and NL=6
The lowest phonon dispersion curves are shown for axially symmetric waves n=0, NL=4 in Fig. 4. For clarity, dispersion curves for non-torsional and torsional waves are represented also separately.  Fig. 2. Dispersion of the phonon group velocity for the lowest dispersion curves in Fig. 4 is represented in Fig. 5 (see Fig. A2 for a detailed graph).  A larger number of dispersion curves in multishells with 6 shells emerge within the same interval of energies and wave vectors as for multishells with 4 shells in Fig. 4 and even more so for multishells with 2 shells in Fig. 2. Dispersion of the phonon group velocity for the lowest dispersion curves in Fig. 6 is represented in Fig. 7 (see Fig. A3 for a detailed graph).

Geometric Effects in the Phonon Dispersion and Group Velocities for Different Numbers of Layers
For the axially symmetric waves (n=0), as follows from Figs. 2, 4 and 6, the lowest group of the phonon dispersion curves, containing one torsional and two non-torsional modes, at the small wave vectors  is only slighltly changed by the number of layers NL. In the same region of wave vectors, the second, consisting of one torsional and one non-torsional modes (the third, consisting of one torsional and two non-torsional modes) group of the phonon frequencies  significantly decreases from 0.57 (1.17) for NL=2 to 0.29 (0.57) for NL=4 and 0.19 (0.38) for NL=6. Within the numerical accuracy, the decrease of the phonon frequencies in the long-wave limit is inversely proportional to NL. Away from the long-wave limit, a general trend of "compression" of the phonon energy spectrum towards lower values of phonon frequencies persists.
As seen from Figs. 3, 5 and 7, the phonon group velocity related to the fundamental (lowest) torsional mode is a weakly varying function of the wave vector , while for the higher torsional modes it monotonously increases with the wave vector , apparently towards a saturation. For a fixed value =0.05, the phonon group velocity related to the lowest (second lowest) torsional mode depends on the number of layers NL as follows: 1.17 (0.13) for NL=2; 1.17 (0.27) for NL=4; 1.17 (0.37) for NL=6. Within the numerical accuracy, the increase of the phonon group velocity for the second lowest torsional mode is directly propotional to NL.
The phonon group velocity related to the lowest two non-torsional modes is a weak function of the wave vector , while for higher torsional modes it always strongly depends on . For the same fixed value =0.05 as above, the phonon group velocity related to the lowest (second lowest and third lowest) non-torsional mode depends on the number of layers NL as follows: 0.75 (2.02 and 0.43) for NL=2; 0.75 (1.98 and 0.82) for NL=4; 0.78 (1.94 and 1.06) for NL=6. Within the numerical accuracy, the phonon group velocity for the third lowest torsional mode reveals a sublinear dependence on NL . Finally, in order to clearly vizualize the overall impact of the number of layers in a multishell on the group velocity dispersion, the average and rms phonon group velocities are calculated for the branches available from the results of the previous section. The results, shown in Fig. 8, demonstrate that an increase of NL from 2 to 4 leads to an appreciable decrease of the average and rms phonon group velocities. A further increase of NL from 4 to 6 has a smaller impact on the average and rms phonon group velocities. For the wave vector =0.05, the average phonon group velocity decreases from 0.82 for NL=2 to 0.54 for NL=4 and further to 0.53 for NL=6. At the same time, the rms phonon group velocity is reduced from 0.95 for NL=2 to 0.71 for NL=4 and further to to 0.65 for NL=6. At small wave vectors the trend persists: the average and rms phonon group velocities decrease with increasing NL.

Conclusions
We established a possibility of efficient engineering of the acoustic phonon energy dispersion in multishell tubular structures produced by a novel method of self-assembly of micro-and nanoarchitectures. A dependence on the number of layers in a multishell structure is a manifestation of geometric effects on phonon energy spectrum. Such geometric effects are features pertinent to acoustic metamaterials and phonon crystals. Based on the calculated energies, the phonon confinement effects should be directly observable using Brillouin spectrometry. The changes in the acoustic phonon spectrum affect phonon transport and can be experimentally detected in thermal conductivity measurements. The reduction of the phonon group velocity and phonon thermal conductivity can be achieved without significant roughness scattering and degradation of electron transport. Our results suggest that arrays of rolled-up multishell tubular structures are prospective candidates for advancement in thermoelectric materials and devices.

Appendix. Dispersion of the Phonon Group Velocity
Figures A1 to A3 represent the detailed phonon group velocity for the wavevector window [0, 0.5].
Step-like features provide a measure of precision for the exploited numerical procedure of solving the boundary problem of Eqs. (1) to (5).