Phonons and thermal transport in Si/SiO$_2$ multishell nanotubes: Atomistic study

Thermal transport in the Si/SiO$_2$ multishell nanotubes is investigated theoretically. The phonon energy spectra are obtained using the atomistic Lattice Dynamics approach. Thermal conductivity is calculated using Boltzmann transport equation within the relaxation time approximation. Redistribution of the vibrational spectra in multishell nanotubes leads to a decrease of the phonon group velocity and the thermal conductivity as compared to homogeneous Si nanowires. Phonon scattering on the Si/SiO$_2$ interfaces is another key factor of strong reduction of the thermal conductivity in these structures (down to 0.1 W/mK at room temperature). We demonstrate that phonon thermal transport in Si/SiO$_2$ nanotubes can be efficiently suppressed by a proper choice of nanotube's geometrical parameters: lateral cross-section, thickness and number of shells.


INTRODUCTION
Rapid miniaturization of electronic devices and increasing power consumption require an efficient heat management at nanoscale [1,2]. Thermal properties of nanostructures and different ways of their optimization have been widely investigated both experimentally and theoretically [3][4][5][6][7][8]. Phonon engineering, i.e. targeted modification of phonon modes in nanostructures to enhance their thermal and/or electrical properties, manifests itself as a powerful tool for the optimization of nanoscale thermal transport [5,9,10]. Nanomaterials with high thermal conductivity, such as graphene, are promising candidates as heat spreaders and interconnects [11][12][13][14], while nanomaterials with low thermal conductivity and high electrical conductivity can be used for thermoelectric applications. The efficiency of the thermoelectric energy conversion, figure of merit ZT, is directly proportional to the electrical conductivity and inversely proportional to the total thermal conductivity: the Seebeck coefficient, is the electrical conductivity, T is the absolute temperature, ℎ and are the phonon and electron thermal conductivities, respectively. Acoustic phonons are the main heat carriers in bulk semiconductors at room temperature (RT) and above. Strong spatial confinement of acoustic phonons in nanostructures significantly changes phonon energy spectra in comparison with the bulk case, resulting in a decrease of phonon group velocities [15][16][17][18][19][20][21].
The latter, in combination with an enhancement of phonon boundary scattering, stipulates a reduction of the lattice thermal conductivity in nanostructures as compared with bulk materials [9,15,16,18,20]. A significant reduction of the thermal conductivity leads to an increase of ZT, e.g. in Bi2Te3 quantum-well structures ZT increases up to 13 times in comparison with the bulk value [22]. The silicon-based nanostructures are also prospective for thermoelectric applications despite the fact that bulk silicon is a poor thermoelectric with ZT ~ 0.01 at RT [23].
Hochbaum et al. [24] and Boukai et al. [25] demonstrated that significant increase of ZT occurs in Si nanowires due to suppression of phonon transport. Strong reduction of lattice thermal conductivity in cross-section modulated Si nanowires [26][27][28] may also lead to improvement of their thermoelectric efficiency as compared with bulk Si. A reduction of thermal conductivity was also reported for SiGe nanocomposite materials [3]. It was shown that rise of the Seebeck coefficient is stronger than a possible increase of the electrical resistivity.
Efficient engineering of the acoustic phonon energy spectrum is carried out in multishell tubular structures. The strain-driven roll-up procedure is a powerful high-tech instrument for fabrication of multi-layer micro-and nano-superlattices and their arrays [29][30][31]. The acoustic phonon dispersion in multishell tubular nanostructures was analyzed within the framework of elastodynamics [32]. It was 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 shells. An increase of the number of shells was shown to lead to an appreciable decrease of the average and root-mean-square phonon group velocities.
In the present paper, we employ an atomistic calculation to tackle the phonons and thermal transport in multishell Si/SiO 2 nanotubes (NTs). We demonstrate that phonon thermal transport in Si/SiO2 nanotubes can be efficiently suppressed by a proper choice of nanotube's geometrical parameters: lateral cross-section, thickness and number of shells.
The rest of the paper is organized as follows. In Section 2 we describe our theoretical model employed for calculations of phonon modes and the lattice thermal conductivity in multishell Si/SiO2 NTs. Discussions of phonon modes and thermal transport in multishell Si/SiO2 NTs are provided in Section III. Conclusions are given in Section IV.

THEORETICAL MODEL
We study multishell NTs formed from alternating layers of silicon and silicon dioxide. The number of Si/SiO2 bilayer shells is varied. A scheme of a multishell NT is shown in Figure 1.
The external surface of the nanotube is assumed to be free [8,17,19]. The X and Y axes of the Cartesian coordinate system are located in the cross-sectional plane of the NT and are parallel to its sides, while the Z axis is directed along the NT axis. We assume that the NT is infinite along the Z axis. The thicknesses of the shells are denoted , and , for silicon ( , 2 and , 2 for silicon dioxide), while the cavity dimensions are , and , . The number of Si/SiO 2 bilayer shells is denoted by L. The phonon energy spectra in the Si/SiO2 multishell NTs and Si NWs are calculated using an atomistic Face-Centered Cubic Cell (FCC) model within the Lattice Dynamics approach [8]. In the FCC model, the diamond-type crystal lattice, consisting of two shifted face-centered cubic Bravais sublattices, is replaced with one face-centered cubic lattice with all atoms possessing a doubled mass. The equations of motion for atoms in the harmonic approximation is [26]: where i enumerates atoms in a NT translation period, is the mass of the i-th atom, ⃗ and ⃗ are the radius vectors of the i-th and j-th atoms, respectively, ⃗ is the phonon wavevector and where is the mass of the j-th atom and � ⃗ , ⃗ � is the matrix of force constants. In the FCC model, the summation in Eq. (1) is performed over all the nearest and second-nearest atoms of the i-th atom: 4 nearest atoms at ⃗ = ⃗ + ℎ �⃗ (j=1,…,4) and twelve second-nearest atoms at ⃗ = ⃗ + ℎ �⃗ (j=1,…,12) [8,26]. The components of the vectors ℎ �⃗ and ℎ �⃗ are presented in Table I of Ref. [26]. The force constant matrix � ⃗ , ⃗ � used in our calculations is taken from Ref. [8]: This matrix depends on 3 independent force constants 1 , 2 and 3 , which can be expressed through the elastic moduli 11 , 12 and 44 of a bulk cubic crystal [8] , where a is the lattice constant.
For calculation of the thermal conductivity of Si/SiO2 multishell NTs and Si NWs, we use the following expression, which was derived from the Boltzmann transport equation within the relaxation time approximation [6,20,33,34] taking into account quasi one-dimensional density of phonon states: Here, or, alternatively, with the relaxation rate for the boundary scattering −1 : where p is the specularity parameter, . . = 0.235 nm is the SiO2 bond length, ⟨ ⟩ is the mean vibrational frequency (the mean vibrational energy ℏ⟨ ⟩ = 34 is taken from Ref. [31]), = 0.33 according to Ref. [35].

DISCUSSION OF RESULTS
We calculated the phonon energy spectra in Si/SiO2 multishell NTs and Si NWs by solving Eq. (1) numerically. Our calculations were performed for all qz in the interval (0, ). In 1D case, the phonon density of states (DOS) per unit length in real space can be found from the relation: . Hence, The phonon energy spectra and DOS for Si NW and Si/SiO2 multishell NTs are shown in   The number of confined phonon branches in the NTs (Nb = 6600 for a NT with single shell and N b =15600 for a NT with double shell) is substantially larger than that in a NW (Nb =1323).
The slope of the lowest phonon branches in NTs is smaller than that in a NW due to acoustic mismatch between silicon and silicon dioxide. A great number of phonon modes in the NTs with energy ℏ > 10 meV are nearly dispersionless and possess group velocities close to zero.
As compared to a NW, the DOS maximum in NTs is shifted toward the lower energy interval, where the drop of the average phonon group velocity is more significant (see Figure 3). The DOS maximum in a NT with double shell is more prominent due to a larger number of phonon modes concentrated in SiO2, which possess a smaller maximal phonon energy than Si.

CONCLUSIONS
Phonon and thermal properties of Si/SiO 2 multishell NTs are studied within the Lattice Dynamics approach. The thermal conductivity in the Si/SiO 2 NT is lower than that in the Si NW with the same lateral dimensions due to the acoustic mismatch of the materials and a stronger phonon confinement. A significant number of phonon modes are scattered on Si/SiO2 interfaces, what enhances the influence of boundary scattering on the thermal conductivity. As a result, the thermal conductivity decreases with increase of the number of shells. Low values of the thermal conductivity in Si/SiO2 NTs make them prospective candidates for thermoelectric applications.