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MICRO/NANOSCALE HEAT TRANSFER—PART II

# Atomic-Scale Three-Dimensional Phononic Crystals With a Very Low Thermal Conductivity to Design Crystalline Thermoelectric Devices

[+] Author and Article Information
Jean-Numa Gillet1

Laboratoire d’Energétique Moléculaire et Macroscopique, Combustion and Centre National de la Recherche Scientifique (EM2C, CNRS UPR 288), Ecole Centrale Paris, Grande Voie des Vignes, 92295 Châtenay-Malabry Cedex, Francejngillet@gmail.com

Yann Chalopin

Laboratoire d’Energétique Moléculaire et Macroscopique, Combustion and Centre National de la Recherche Scientifique (EM2C, CNRS UPR 288), Ecole Centrale Paris, Grande Voie des Vignes, 92295 Châtenay-Malabry Cedex, France

Sebastian Volz2

Laboratoire d’Energétique Moléculaire et Macroscopique, Combustion and Centre National de la Recherche Scientifique (EM2C, CNRS UPR 288), Ecole Centrale Paris, Grande Voie des Vignes, 92295 Châtenay-Malabry Cedex, Francevolz@em2c.ecp.fr

1

Corresponding author. Present address: Department of Physics, Institut d'Electronique, de Microélectronique et de Nanotechnologie (IEMN, CNRS UMR 8520), Université des Sciences et Technologies de Lille 1, Av. Poincaré, BP 60069, 59652 Villeneuve d'Ascq Cedex, France.

2

Present address: Institute of Industrial Science (IIS), University of Tokyo, Center for International Research on MicroMechatronics (CIRMM), and Centre National de la Recherche Scientifique, LIMMS, UMI CNRS 2820-IIS, Dw 304, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan.

J. Heat Transfer 131(4), 043206 (Feb 20, 2009) (10 pages) doi:10.1115/1.3072927 History: Received May 23, 2008; Revised October 27, 2008; Published February 20, 2009

## Abstract

Superlattices with thermal-insulating behaviors have been studied to design thermoelectric materials but affect heat transfer in only one main direction and often show many cracks and dislocations near their layer interfaces. Quantum-dot (QD) self-assembly is an emerging epitaxial technology to design ultradense arrays of germanium QDs in silicon for many promising electronic and photonic applications such as quantum computing, where accurate QD positioning is required. We theoretically demonstrate that high-density three-dimensional (3D) arrays of molecular-size self-assembled Ge QDs in Si can also show very low thermal conductivity in the three spatial directions. This physical property can be considered in designing new silicon-based crystalline thermoelectric devices, which are compatible with the complementary metal-oxide-semiconductor (CMOS) technologies. To obtain a computationally manageable model of these nanomaterials, we investigate their thermal-insulating behavior with atomic-scale 3D phononic crystals: A phononic-crystal period or supercell consists of diamond-cubic (DC) Si cells. At each supercell center, we substitute Si atoms by Ge atoms in a given number of DC unit cells to form a boxlike nanoparticle (i.e., QD). The nanomaterial thermal conductivity can be reduced by several orders of magnitude compared with bulk Si. A part of this reduction is due to the significant decrease in the phonon group velocities derived from the flat dispersion curves, which are computed with classical lattice dynamics. Moreover, according to the wave-particle duality at small scales, another reduction is obtained from multiple scattering of the particlelike phonons in nanoparticle clusters, which breaks their mean free paths (MFPs) in the 3D nanoparticle array. However, we use an incoherent analytical model of this particlelike scattering. This model leads to overestimations of the MFPs and thermal conductivity, which is nevertheless lower than the minimal Einstein limit of bulk Si and is reduced by a factor of at least 165 compared with bulk Si in an example nanomaterial. We expect an even larger decrease in the thermal conductivity than that predicted in this paper owing to multiple scattering, which can lead to a $ZT$ much larger than unity.

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## Figures

Figure 1

Schematics at two different scales of an atomic-scale 3D phononic crystal with N=5 and M=3. In the continuous mediumlike drawing in (a), the nanoparticles with an edge length of 3a=1.6293 nm and spacing of d=5a=2.7155 nm are shown by periodic black cubes with highlighted edges. In (a), the central transparent cube with bold thick edges displays one of the nanomaterial supercells. The discrete mediumlike drawing of a supercell is presented in (b), where the 344 Ge atoms forming each boxlike nanoparticle in (a) are colored in black while the 656 peripheral Si atoms are colored in gray in the remainder of the supercell.

Figure 2

80 branches chosen among the 3000 dispersion curves of an atomic-scale 3D phononic crystal with N=5, M=3, and d=5a=2.715 nm

Figure 3

2D view in the reciprocal plane formed by the directions [1 0 0] and [0 1 0] of a cube with the edge length 2Kx. The smallest sphere circumscribing this cube has a radius K=21/2Kx, which explains the use of the constant geometric factor of fk=2 in the DOS gk in Eq. 2.

Figure 4

Diagrams in (a) of the incoherent-scattering efficiency Qk(inc) versus wave number k for three atomic-scale 3D phononic crystals: The curves displayed by the dashed-dotted, dashed, and solid black lines are for the nanomaterials with the smallest (M=1), middle (M=2), and largest (M=3) nanoparticle sizes. The thin top dashed horizontal line in (a) references to the geometrical limit with the value of 2. The thin dashed vertical lines in (a) give the right boundaries of the first, second, third, fourth, and fifth folded BZs from the left-hand side to the right-hand side. In (a), the allowed modes are only in the first folded BZ at the left of the first thin dashed vertical line (from the left-hand side). In (b), the maximal allowed Qk(inc) at the right boundary of the first folded BZ, where k=π/d, is given as a function of the Ge relative concentration x in the supercells, where x varies from 0 for bulk Si to 0.344 for the nanomaterial with the largest nanoparticle size (M=3).

Figure 5

Diagrams in (a) of the upper limits λmax of the thermal conductivity versus temperature T for three atomic-scale 3D phononic crystals: The curves displayed by the dashed-dotted, dashed, and solid black lines are for the nanomaterials with the smallest (M=1), middle (M=2), and largest (M=3) nanoparticle sizes, respectively. The thin bottom dashed horizontal line in (a) is a reference to the Einstein limit of 0.99 W/mK (for disordered bulk Si). The top circles interpolated by the thin top dashed line are experimental measurements of the thermal conductivity of bulk Si for comparison. In (b), the upper limits ⟨l⟩max of the effective MFP versus T are displayed by the dashed-dotted, dashed, and solid black lines for nanomaterials with M=1, M=2, and M=3, respectively. The top circles interpolated by the thin top dashed line give the curve of the effective MFP of bulk Si for comparison.

Figure 6

Diagram of the upper limits λmax of the thermal conductivity versus temperature T for the atomic-scale 3D phononic crystal with the largest nanoparticle size (M=3). This curve shows a peak at T=65 K with the value of 1.23 W/mK, which is referenced by the thin top horizontal dashed line. The middle and bottom thin dashed horizontal lines are references to the Einstein limit of 0.99 W/mK (for disordered bulk Si) and upper limit λmax=0.50 W/mK (for the nanomaterial) close to the melting point, respectively.

Figure 7

Diagrams in (a) of the ratios λmax/λbulk (black circles), ⟨l⟩max/⟨l⟩bulk (black upward triangles) and ⟨Cv⟩/⟨Cv⟩bulk (black downward triangles) versus temperature T for the atomic-scale 3D phononic crystals with the largest nanoparticle size (M=3). Data of the same group are interpolated by a black dashed line. The thin top dashed horizontal line is a reference to the intersection point of the curves ⟨l⟩max/⟨l⟩bulk and ⟨Cv⟩/⟨Cv⟩bulk at T=300 K, where these ratios have the value of 1/13=0.077. In (b), the ratios λmax/λbulk (black circles), ⟨l⟩max/⟨l⟩bulk (black upward triangles) and ⟨Cv⟩/⟨Cv⟩bulk (black downward triangles) are plotted at T=300 K versus the Ge relative concentration x, where x varies from 0 for bulk Si to 0.344 for the atomic-scale 3D phononic crystal with the largest nanoparticle size (M=3). In (b), the curves ⟨l⟩max/⟨l⟩bulk and ⟨Cv⟩/⟨Cv⟩bulk intersect when x=0.344, as also seen in (a).

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