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7-9 October 2009, Leuven, Belgium Directional Thermal Conductivity of a Thin Si Suspended Membrane with Stretched Ge Quantum Dots Jean-Numa Gillet, ∗ Bahram Djafari-Rouhani, Yan Pennec Université de Lille 1, Institut d'Electronique de Microélectronique et de Nanotechnologie (IEMN, CNRS UMR 8520), Département de Physique, Av. Poincaré, BP 60069, 59652 Villeneuve d'Ascq cedex, France (www.iemn.univ-lille1.fr); Members of the European Consortium NANOPACK Abstract-We model a nanomaterial showing a hybrid thermal behavior between dissipative and insulating regimes. The nanomaterial is made up of a thin Si suspended membrane covered by self-assembled Ge quantum dots (QDs) with facets. A membrane plane is constituted from the orthogonal [100] and [001] directions (x and z, respectively). The QDs are stretched in [001] forming nanoscale phonon waveguides. When hot and cold junctions are connected to the membrane following [001], the throughput thermal conductivity λ shows a significant exaltation with respect to the in-plane orthogonal direction [001] where QD constriction is defined. This property can be used for the design of nanoscale dissipaters to remove heat in only one main direction. Indeed, low leakage heat currents are obtained in other directions so that they cannot affect thermal budget in other parts of a device to cool as a silicon chip. In our theoretical model, a deflection angle β is taken in a membrane plane from the axis x. The anisotropic thermal conductivity is analyzed as a function of β. In an example molecular-scale device, λ can be exalted by 4 to 5 folds, or from 0.7 to 2.9 W/m/K, when β is increased from 0° (x) to 90° (z), respectively. Therefore, the QD-waveguide nanomaterial presents a different thermal insulating behavior in the direction [100] and can as well be used for the design of both dissipative and thermoelectric devices. The transition between both contra effects is obtained for the in-plane close-packed directions . Keywords: Heat dissipation, Thermoelectrics, Nanoscale devices, Quantum dots, Silicon, Germanium I. INTRODUCTION The design of nanostructured semiconducting devices with optimized thermal properties and indirect electronic band gap (as those using the Si/Ge IV-IV ∗ Corresponding author’s email: jean-numa.gillet@univ-lille1.fr couple) is currently one of the major challenges for onchip cooling in nanoscale silicon-based architectures [1]. These thermal nanodevices should enable continuation of the historical integration pace given by the Moore's law in CMOS microelectronics. With the fast and spectacular development of nanotechnology, self-assembly became a major technology for bottom-up fabrication of threedimensional (3D) nanostructured devices for various applications in drug design, biotechnologies, electronics and photonics, for instance [2-5]. Epitaxial self-assembly has been used to design germanium quantum-dot (QD) arrays in silicon [6,7]. The Ge QDs stand on or are sandwiched between diamond-cubic (dc) Si thin layers. In the classical Stranski-Krastanov growth mode, a thin wetting Ge layer is grown by epitaxy in the vertical direction with respect to a Si {010} substrate as investigated by experimentalists to fabricate twodimensional (2D) arrays of self-assembled (SA) Ge islands forming QDs with facets on Si. 3D ordering of a SA Ge-QD array in a Si matrix can be obtained, thereafter, by propagation of the stress field from a bottom layer to the superposed layers. To obtain sharper Ge QDs with lower size dispersion, more sophisticated technologies as e-beam and focus ion beam can be used. 3D QD Ge/Si nanocomposites were used for quantum applications (as single-electron or single-photon devices, e.g.) and for solar-energy conversion. Gillet et al. recently presented theoretical studies of 3D SA Ge-QD supercrystals in Si for the design of crystalline thermoelectric (TE) devices that can be CMOS-compatible [8,9]. These Si/Ge supercrystals can present an extreme reduction of the thermal conductivity λ that can be lower than only 0.04 W/m/K (i.e. less than twice the value of air) for different size parameters and Ge concentrations [9,10]. The aim of modeling this 3D Si/Ge supercrystal was to obtain λ as tiny as possible since the energy-conversion efficiency of a TE material is ©EDA Publishing/THERMINIC 2009 203 ISBN: 978-2-35500-010-2
an increasing function of its TE figure of merit ZT. Indeed, the non-dimensional ZT is given by ZT = S 2 σ T / λ, where S, σ and T denote the Seebeck coefficient, electrical conductivity and absolute temperature, respectively [11,12]. Therefore, ZT is inversely proportional to λ but directly proportional to the power factor S 2 σ. Discovery of a material with ZT higher than 3 would result in a TE yield that is higher than 42 % of the Carnot efficiency for hot and cold junctions at 800 K and 300 K, respectively [9,12]. Achievement of such a dream would have an enormous impact for energy conversion and renewable energies. In this theoretical study, we investigate a novel type of thermal membranous nanomaterials that is made up of a thin dc Si membrane covered by stretched SA Ge QDs with facets. The QDs are stretched in the [001] direction with respect to the dc symmetry to form phonon waveguides. The length L of the QD waveguides in the stretching direction [001] is defined as larger than the average phonon mean free path (MFP) in Si given by Λ 0 ∼ 100 nm. In contrast, the QDs are constricted in the orthogonal in-plane direction [100] with a bottom basis B 0 that is several folds smaller than Λ 0 . The anisotropic thermal conductivity λ of the QD device is obtained in a membrane plane as a function of the deflection angle β taken with respect to the axis x parallel to the QD constriction direction [100]. The throughput λ, when hot and cold junction are connected to both sides of the membrane, is computed with 3D lattice dynamics for β = 0° to β = 90°. The latter value is related to the axis z parallel to the QD stretching direction [001]. The discrete supercell to be encoded to obtain the phonon dispersion curves is taken as a molecular slab of the nanomaterial. Since L > Λ 0 , the slab has an out-ofequilibrium width given by the Si lattice parameter a = 0.5431 nm in the direction z. To respect the dc symmetry, four molecular planes with the same Miller indices (001) have to be used to define the supercell slab. The dispersion curves, computed by lattice dynamics [13-15], show flat behaviors in the direction x owing to QD constriction. Consequently, low phonon group velocities are obtained in this direction leading to a small throughput λ. In contrast, the slopes of the dispersion curves are usually higher in the direction z so that the stretched QDs form nanoscale phonon waveguides with axes parallel to z. Indeed, the QD average length L is large in z with L > Λ 0 . As a result, the throughput λ is much larger in the direction z with respect to that x. When hot and cold junctions are connected to the membrane following z, the QD-waveguide nanomaterial can be used for the design of efficient unidirectional thermal interface devices for heat sinking. Indeed, the leakage heat currents, which might affect thermal budget 7-9 October 2009, Leuven, Belgium of other parts of a device to cool, are small owing to the much smaller throughput λ in the in-plane orthogonal direction x. The proposed membranous material presents a hybrid behavior between thermal dissipation and insulation. The operation regime depends on the heatflux direction determined by the hot and cold junctions with a connection angle β with respect to x. Consequently, the same nanomaterial can be applied to the design of heat sinkers and dissipaters as well as TE generators and coolers. In the following sections, using an example molecular-scale QD-waveguide nanomaterial, we show exaltation of the throughput λ by a significant factor of 4 to 5 folds (i.e. from 0.7 to 2.9 W/m/K) when the connection angle β is increased from 0° to 90°, respectively. The thermal transition between the insulating and dissipative regimes is obtained for β = 45° that is related to the in-plane close-packed direction of the Si membrane. II. MODEL To keep a membrane-like geometry, which is responsible of directional effects on the thermal conductivity, we set the length L of the stretched Ge QDs in the in-plane direction z, or [001], as being of the order or larger than the average MFP Λ 0 ∼ 100 nm of the phonons in bulk dc Si, as depicted in Fig. 1(a). The QD bottom basis B 0 in the in-plane x direction, or [100], is defined as being several folds lower than Λ 0 . Therefore, the QDs are stretched in the z direction with respect to the orthogonal direction x so that they form phonon waveguides. In the continuous-medium representation of Fig. 1(b), the top basis of the QDs is denoted by B 1 while d x is the length of a nanomaterial supercell in the QDconstriction direction x. In the non-repetitive direction y, or [010], the height of the dc Si membrane is h while that of the Ge QDs is H. The throughput thermal conductivity λ of the suspended nanomaterial is computed from the dispersioncurve diagram in a range from 0 to ∼ 20 THz using 3D lattice dynamics and an incoherent approach of the phonon scattering relaxation times. A discrete model is derived from a supercell slab of the continuous-medium representation sketched in Fig. 1(b). As shown in Fig. 2 where the red and blue dots denote the locations of Si and Ge atoms, respectively, we encode a discrete supercellslab model at the molecular scale for lattice-dynamics calculations. Since L ≥ Λ 0 ≥ B 0 , the width in z of the supercell slab, which is parallel to the crystallographic plane with the Miller indices (001), can be taken as equal to the Si lattice parameter a = 0.5431 nm. Since the dc ©EDA Publishing/THERMINIC 2009 204 ISBN: 978-2-35500-010-2
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