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

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{x}G

_{1−x}As/GaAs quantum cables [61] and double-coupled nanoshell systems [62]. The aim of the present work is to investigate the 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. This 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 the 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].

## 2. Theoretical Model

**Figure 1.**Microtube with multiple windings fabricated using roll-up technology, a scheme (

**a**) and a STEM image of a radial superlattice from a rolled-up 20 nm InGaAs/10 nm Ti/46 nm Au layer (

**b**) (From Ref. [58] © IOP Publishing. Reproduced with permission. All rights reserved). Cross-section in the plane orthogonal to the X-axis of a multilayer tube consisting of coaxial shells (multishell) (

**c**). The tube core and the outer medium are vacuum. The axis of the structure is selected as the X-axis. The polar coordinates in the YZ-plane are (r,φ). The picture corresponds to a multishell with a periodic alternation of two materials.

**u**

_{m}in each layer (m = 1,…, N), treated as an elastic continuum, obeys the equations of elastodynamics [64]:

_{m}, (m = 1,…, N−1) the following six boundary conditions represent continuity of the stress tensor components:

_{0,}the following three boundary conditions represent vanishing of the stress tensor components:

_{N}

_{,}the following three boundary conditions represent vanishing of the stress tensor components:

_{1m}and ν

_{2m}correspond to dilatational and shear waves in the material of the m-th layer. The solutions to the wave equations are sought as plane waves traveling along the axis of the structure.

_{0}= 100 nm. All odd shells (m = 2k + 1) consist of one and the same material with elastic properties λ

_{1}, μ

_{1}, density ρ

_{1}and have the same thickness: Δr

_{1}. All even shells (m = 2k) consist of the same material with elastic properties λ

_{2}, μ

_{2}, density ρ

_{2}and have the same thickness: Δr

_{2}. They are represented in Table 1. In what follows, the number of layers is denoted by N

_{L}.

Parity i of the layer number m | 1 | 2 |
---|---|---|

Material | InAs | GaAs |

λ_{i}, dyn/cm^{2} | 4.54 × 10^{11} | 5.34 × 10^{11} |

μ_{i}, dyn/cm^{2} | 1.90 × 10^{11} | 3.285 × 10^{11} |

ρ_{i}, g/cm^{3} | 5.68 | 5.317 |

Δr_{i}, nm | 5 | 5 |

Characteristic of the layer with parity i of the number m | Denotation |
---|---|

Velocity of a dilatational wave | v_{1i} = [(λ_{i} + 2μ_{i})/ ρ_{i}]^{1/2} |

Velocity of a shear wave | v_{2i} = [μ_{i}/ ρ_{i}]^{1/2} |

Squared radial wave number for a dilatational wave | α_{i}^{2} = ω^{2}/v_{1i}^{2} − ζ^{2} |

Squared radial wave number for a shear wave | β_{i}^{2} = ω^{2}/v_{2i}^{2} − ζ ^{2} |

Dispersion of the Rayleigh waves α_{i}^{2} = 0 | ω = v_{1i}ζ |

Dispersion of the Rayleigh waves β_{i}^{2} = 0 | ω = v_{2i}ζ |

_{m}(r) and h

_{xm}(r) at m = 1, 2,…, N

_{L}:

_{rm}(r) and h

_{φm}(r). Due to gauge invariance of the vector potential, one of the functions h

_{rm}(r), h

_{φm}(r), h

_{xm}(r) can be selected arbitrarily; we use, like in [65], the calibration h

_{rm}(r) = −h

_{φm}(r) ≡ h

_{1m}(r). The differential Equations (9) are Bessel equations (see Chapter 9 in [67]). Their solutions can be represented in the general form:

_{n}(W

_{n}) are the Bessel functions of the first kind J

_{n}(second kind Y

_{n}) (p. 358 of [67]) for the real radial wave numbers α

_{m}and β

_{m}and the modified Bessel functions I

_{n}(K

_{n}) (p. 374 of [67]) for the imaginary wave numbers α

_{m}and β

_{m}. In the set of solutions (10), there are in total 6N

_{L}unknown coefficients, which can be represented as a 6N

_{L}-dimensional vector $\Xi \equiv \left\{{A}_{1m},{B}_{1m},{A}_{2m},{B}_{2m},{A}_{3m},{B}_{3m}|m=1,2,\dots ,{N}_{\text{L}}\right\}$. After substituting the solutions (10) in the set of 6N

_{L}boundary conditions (2) to (5), we arrive at a homogeneous system of 6N

_{L}linear algebraic equations with respect to 6N

_{L}components Ξ

_{j}(j = 1,…, N

_{L}) of the vector Ξ:

Physical quantity | Unit |
---|---|

Longitudinal wave vector ζ | 1/Δr_{2} |

Frequency ω | πv_{22}/Δr_{2} |

Group velocity dω/dζ | πv_{22} |

## 3. Results for N_{L} = 2

_{L}= 2 in Figure 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 (u

_{x}= u

_{r}= 0, u

_{θ}≠ 0) and non-torsional (associated with the displacement components u

_{x}and u

_{r}, u

_{θ}= 0) modes. Dispersion of the phonon group velocity for the dispersion curves in Figure 2 is represented in Figure 3 (see Figure A1 for a detailed graph). A group velocity dispersion curves stops when the eigenfrequency ω goes beyond the window [0, 1.5].

**Figure 2.**Phonon dispersion curves for n = 0, N

_{L}= 2. The inner and outer radii of a multishell are r

_{0}= 100 nm and r

_{2}= 110 nm, respectively. Non-torsional and torsional modes are represented with filled and empty circles, correspondingly. Dashed lines indicate the dispersion curves for dilatational (α

_{I}= 0) ans shear (β

_{I}= 0) waves in the material with parity i of the number m.

**Figure 3.**Phonon group velocity dispersion curves for n = 0, N

_{L}= 2. Denotations of non-torsional and torsional modes are the same as in Figure 2.

## 4. Results for N_{L} = 4 and N_{L} =6

_{L}= 4 in Figure 4. For clarity, dispersion curves for non-torsional and torsional waves are represented also separately.

**Figure 4.**Phonon dispersion curves for n = 0, N

_{L}= 4 (

**a**). The inner and outer radii of a multishell are r

_{0}= 100 nm and r

_{4}= 120 nm, respectively. Phonon dispersion curves for non-torsional (

**b**) and torsional (

**c**) waves.

**Figure 5.**Phonon group velocity dispersion curves for n = 0, N

_{L}= 4 (

**a**). Phonon group velocity dispersion curves for non-torsional (

**b**) and torsional (

**c**) waves.

_{L}= 6 in Figure 6.

**Figure 6.**Phonon dispersion curves for n = 0, N

_{L}= 6 (

**a**). The inner and outer radii of a multishell are r

_{0}= 100 nm and r

_{6}= 130 nm, respectively. Phonon dispersion curves for non-torsional (

**b**) and torsional (

**c**) waves.

**Figure 7.**Phonon group velocity dispersion curves for n = 0, N

_{L}= 6 (

**a**). Phonon group velocity dispersion curves for non-torsional (

**b**) and torsional (

**c**) waves.

## 5. Geometric Effects in the Phonon Dispersion and Group Velocities for Different Numbers of Layers

_{L}. 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 ω significantly decreases from 0.57 (1.17) for N

_{L}= 2 to 0.29 (0.57) for N

_{L}= 4 and 0.19 (0.38) for N

_{L}= 6. Within the numerical accuracy, the decrease of the phonon frequencies in the long-wave limit is inversely proportional to N

_{L}. Away from the long-wave limit, a general trend of “compression” of the phonon energy spectrum towards lower values of phonon frequencies persists.

_{L}as follows: 1.17 (0.13) for N

_{L}= 2; 1.17 (0.27) for N

_{L}= 4; 1.17 (0.37) for N

_{L}= 6. Within the numerical accuracy, the increase of the phonon group velocity for the second lowest torsional mode is directly propotional to N

_{L}.

_{L}as follows: 0.75 (2.02 and 0.43) for N

_{L}= 2; 0.75 (1.98 and 0.82) for N

_{L}= 4; and 0.78 (1.94 and 1.06) for N

_{L}= 6. Within the numerical accuracy, the phonon group velocity for the third lowest torsional mode reveals a sublinear dependence on N

_{L}.

_{L}from two to four leads to an appreciable decrease of the average and rms phonon group velocities. A further increase of N

_{L}from four to six has a smaller impact on the average and RMS phonon group velocities. For the wave vector ζ = 0.05, the average phonon group velocity decreases from 0.82 for N

_{L}= 2 to 0.54 for N

_{L}= 4 and further to 0.53 for N

_{L}= 6. At the same time, the RMS phonon group velocity is reduced from 0.95 for N

_{L}= 2 to 0.71 for N

_{L}= 4 and further to to 0.65 for N

_{L}= 6. At small wave vectors the trend persists: the average and RMS phonon group velocities decrease with increasing N

_{L.}

**Figure 8.**Average (

**left**) and rms (

**right**panel) phonon group velocity dispersion curves for n = 0 at N

_{L}= 2, 4, and 6.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix

**Figure A1.**Detailed phonon group velocity dispersion curves for n = 0, N

_{L}= 2. Denotations of non-torsional and torsional modes are the same as in Figure 2.

**Figure A2.**Detailed phonon group velocity dispersion curves for n = 0, N

_{L}= 4 for non-torsional (

**a**) and torsional (

**b**) waves.

**Figure A3.**Detailed phonon group velocity dispersion curves for n = 0, N

_{L}= 6 for non-torsional (

**a**) and torsional (

**b**) waves.

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Fomin, V.M.; Balandin, A.A. Phonon Spectrum Engineering in Rolled-up Micro- and Nano-Architectures. *Appl. Sci.* **2015**, *5*, 728-746.
https://doi.org/10.3390/app5040728

**AMA Style**

Fomin VM, Balandin AA. Phonon Spectrum Engineering in Rolled-up Micro- and Nano-Architectures. *Applied Sciences*. 2015; 5(4):728-746.
https://doi.org/10.3390/app5040728

**Chicago/Turabian Style**

Fomin, Vladimir M., and Alexander A. Balandin. 2015. "Phonon Spectrum Engineering in Rolled-up Micro- and Nano-Architectures" *Applied Sciences* 5, no. 4: 728-746.
https://doi.org/10.3390/app5040728