In-Plane Thermal Conductivity of Radial and Planar

Transcription

1 Supporting Information In-Plane Thermal Conductivity of Radial and Planar Si/SiO x Hybrid Nanomembrane Superlattices Guodong Li, *,,, Milad Yarali,, Alexandr Cocemasov, ǁ, Stefan Baunack, Denis L. Nika, ǁ Vladimir M. Fomin, ǁ, Shivkant Singh, Thomas Gemming, Feng Zhu, *,, Anastassios Mavrokefalos, *, and Oliver G. Schmidt, Material Systems for Nanoelectronics, Technische Universität Chemnitz, Chemnitz, Germany Department of Mechanical Engineering, University of Houston, Houston, United States of America ǁ E. Pokatilov Laboratory of Physics and Engineering of Nanomaterials, Department of Physics and Engineering, Moldova State University, Chisinau, MD-2009, Republic of Moldova Institute for Integrative Nanosciences, IFW Dresden, Dresden, Germany Institute for Complex Materials, IFW Dresden, Dresden, Germany * g.li@ifw-dresden.de; f.zhu@ifw-dresden.de; amavrokefalos@uh.edu G. Li, M. Yarali and A. Cocemasov contributed equally to this work. 1

2 I. Dimensions and thermal conductivity at 300K of the measured radial and planar Si/SiO x HNMSLs samples Table S1. Dimensions and thermal conductivity at 300K of the measured radial and planar Si/SiO x HNMSLs samples Sample Suspended Length (µm) Width (µm) Thickness (µm) Outer Diameter (µm) Inner Diameter (µm) Thermal conductivity at 300K (Wm -1 K -1 ) Radial 1w L=8µm 7.7 ± ± ± ±0.62 Radial 1w L=16µm 15.8 ± ± ± ±0.6 Radial 2w L=8µm 8.1 ± ± ± ±1.08 Radial 2w L=12µm 12 ± ± ± ±1.07 Radial 5w L=11µm 11.2 ± ± ± ±0.18 Radial 5w L=16µm 16.2 ± ± ± ±0.13 Planar L=8µm 8 ± 0.3 4± ± ±0.77 Planar L=11µm 11.2 ± 0.5 4± ± ±0.6 Planar L=16µm 16.2 ± 0.2 4± ± ±0.6 Additionally, Figure S1 shows the AFM topography image of the planar Si/SiO x HNMSL and its corresponding thickness profiles. 2

3 Figure S1. (a) Surface topological scan using AFM. (b) Average thickness profile II. Calculating contact thermal resistance We follow two approaches to quantify the contact thermal resistance between the Si/SiO x HNMSLs and the suspended membranes of the microdevice. Both approaches require multiple measurements with different suspended lengths of the same HNMSLs. In the first approach (subtraction method) developed by Yang et al, 1 the relation between measured total thermal resistance, R t, the intrinsic thermal resistance of the suspended Si/SiO x HNMSLs segment between the two membranes per unit length, R s/l, and the contact thermal resistance between the Si/SiO x HNMSLs and the two membranes, R c, is defined as: / 1 where L s is the suspended length shown in the Table S1. We can assume that R c is the same in different measurements if the Si/SiO x HNMSLs segment on each membrane (contact length) is long enough so that it is fully thermalized with the supporting membrane. 1 Therefore, R s/l, R c, and the minimum contact length, L c,min that is used to verify the aforementioned assumption can be extracted from the measured total thermal resistance of two different measurements as below: 1 / / 2 / 3, / / 4 As an example of our calculation, Figure S2a depicts temperature dependent total, intrinsic and contact thermal resistances calculated by this procedure for the Si/SiO x HNMSL sample of one winding and suspended lengths of 16 µm. It can be seen that the contribution of the contact resistance to the total resistance is negligible. Also, the maximum value for L c,min is less than 1 3

4 µm, that are much smaller than actual contact length between the Si/SiO x HNMSLs segment and each membrane. In the second approach (linear fitting method) introduced in Ref. [2], the measured total thermal resistance (R t ) multiplied by cross sectional area (A) is plotted against the different suspended lengths of the samples. If the contact resistance is negligible, the linear fitting and extrapolation of the data will intercept the y-axis at the origin. As an example, Figure S2b depicts this for the planar Si/SiO x HNMSL samples at room temperature. The linear intercept with the origin validate the negligible contact thermal resistance between the samples and the membranes. Figure S2. (a) Temperature dependent total, intrinsic and contact thermal resistances of Si/SiO x HNMSL sample with one winding and suspended length of 16 µm. For the whole temperature range the contact thermal resistance is negligible compare to measured total thermal resistance. (b) Total thermal resistance multiplied by cross-sectional area as a function of the Si/SiO x HNMSL length for planar samples at room temperature. Approximately zero y-axis interception of the linear relationship extrapolating indicates negligible thermal contact resistance. In summary, Figure S3 lists total, intrinsic and contact thermal resistances of different samples measured in this work. The maximum value for L c,min is 1 µm, 7.64 µm, 1.9 µm and 2.72 µm for Radial 1w, 2w, 5w and planar samples, respectively that are much smaller than actual contact length between the Si/SiO 2 HNMSLs segment and each membrane for all samples. 4

5 Figure S3. Total, intrinsic and contact resistances of all samples versus temperature. 5

6 III. Parametric analysis of thermal conductance of the interfaces for the planar and radial Si/SiO x HNMSLs. Finite element software (COMSOL) was used to carry out parametric analysis of thermal conductance at the interfaces for the planar and radial Si/SiO x HNMSLs. 3-dimensional analysis of the planar and radial Si/SiO x HNMSLs was considered as illustrated in the Figure S4a and S4b which also depicts the representative boundary conditions and dimensions of the model taken from experimental conditions. For the planar and the radial samples interfacial thermal conductance of two adjacent silicon layers was calculated as G t = 2(G 1 + G 2 ) + G int where G 1 is the interfacial thermal conductance of Si/SiO x and G 2 is thermal conductance of SiO x layer. G 1 was taken from literature 3 to be 250 MWm -2 K -1 and G 2 for 2 nm thickness SiO x was taken as 500 MWm -2 K -1. G int is the interfacial thermal conductance of mechanically joined oxide-oxide interfaces and was reported to be 30 MWm -2 K -1 for the planar sample. 4 For the radial sample we expect a weak van-der-waals interaction at the interfaces and the typical range of the interfacial thermal conductance for such interaction is reported to be 4 < G int < 30 MWm -2 K -1. 5,6 Cross plane temperature profile was calculated by sweeping the G int magnitude from 0.03 MW m -2 K -1 to MWm -2 K -1, where 0.1 MWm -2 K -1 indicates a very poor contact for macroscopically joined interfaces 7,8 while interfacial thermal conductance in range of 30 to 500 MW m -2 K -1 is typically indicative of strong chemically bonded interfaces. 4,9 Figure S4c and S4d shows the temperature profile along the thickness of the planar and radial sample near the heating side of the suspended segment, respectively. No temperature variation was observed for the interfacial thermal conductance of 30 MWm -2 K -1 for the planar sample as well as for the range of 0.1 < G int < MWm -2 K -1 for the radial sample. 6

7 Figure S4. 3D model showing temperature distribution, geometric dimensions and applied boundary conditions taken from experimental measurement for (a) Planar and (b) radial samples. Heat flux equal to 3458 Wm -2 was applied to outer surfaces of heating side and the sensing side outer surfaces were kept at T S. Temperature profile along the thickness of the Si/SiO x HNMSLs near the heating side of the suspended segment for different interfacial thermal conductance for (c) Planar and (d) radial samples. IV. Thickness determination of Si and SiO x layer from TEM images 7

8 (a) (b) Figure S5. (a) TEM image of planar Si/SiO x HNMSL (i.e. Fig. 3c) and (b) image intensity profile perpendicular to the layers for the area marked in (a). From the profile the Si layer thickness was found to be (20.5±0.5) nm and the oxide layer thickness to be approx. 4 nm. V. Born von Karman lattice dynamics model for planar Si/SiO 2 HNMSLs Within the BvK theory of lattice dynamics the set of equations of motion for atoms from the s th monolayer of a HNMSL can be written in a harmonic approximation as 2 r r r r r mω U ( n, q) = Φ ( n, n )exp iq ( n ) ( n ) U ( n, q) i= x, y, z, (5) ( ( )) i s ij s s s s j s ns j= x, y, z where m is the mass of an atom, ω is the phonon frequency, q r is the phonon wave vector, U r r is the displacement vector, ( n ) is the radius-vector of the n th atom from the s th s monolayer and Φ is the force constants matrix. In our model we take into account the interaction between atoms from the nearest and second nearest neighbor atomic spheres, therefore the summation in Equation 5 is performed over all atoms n from two neighbor spheres of the atom. The force s' constant matrices contain three independent force constants: α, β and µ. Having solved n s Equation 5 in the long-wavelength limit q 0, we express the inter-atomic force constants 8

9 2 c11 c44 through independent elastic moduli of a bulk material and as α = aπ c /16 11, β = α( a( c 2 c ) + α) / 2 and µ = ( ac 11 α) / 8, where a is the lattice constant The translational symmetry of the SiO 2 material under the Virtual Crystal approximation results in the appearance of a pseudo-brillouin zone with defined wave numbers q x, q y, q z characterized by the microscopic reference state in the form of a crystal lattice with a lattice constant a. The pseudo-brillouin zone terminology for glasses was discussed in Ref. [14]. Using the pseudo-brillouin zone concept, the authors of Ref. [15] have associated the well-known boson peak, characteristic to all amorphous materials, with the singularities of transverse acoustic vibrations near the pseudo-brillouin zone boundary. Moreover, both longitudinal and transverse acoustic phonon dispersion in SiO 2, calculated within our BvK model (see Figure 4b in the main text), are in a good accordance with the inelastic X-ray (IXS) and neutron (INS) scattering experiments for amorphous silica for wave numbers up to ~6 nm For Si/SiO 2 HNMSL, the periodicity of the SiO 2 /Si/SiO 2 segments results in the occurrence of the pseudo-brillouin zone with the wave number q z between 0 and π/d, where 2 is the superlattice period. VI. Linearized Boltzman Transport Equation In the framework of a linearized BTE approach, the in-plane phonon thermal conductivity κ of a SL at a given temperature T can be expressed as: 1 κ T 2 ħω,,,,, exp ħω, exp ħω, 1, 6 where, is the dispersion relation for the th phonon branch with the in-plane and the cross-plane wave vectors,, and, are the in-plane phonon group velocity and the total relaxation time of the th phonon branch., and are the number of atomic layers, the thickness of a SL period and the lattice constant, respectively. ħ is the reduced Plank constant and is the Boltzmann constant. 9

10 The dominant phonon scattering mechanisms in the thermal conductivity of Equation 6 are determined by theoretically reproducing the experimental thermal conductivities of Si and SiO 2 materials individually (more details are in the VII and VIII section). To describe essentially different scattering mechanisms for vibrational modes within Si and SiO 2 layers, we introduce two scattering rates, and, correspondingly. The total phonon relaxation rate in the planar Si/SiO 2 HNMSLs is combined using Matthiessen rule 17 with weighting factors proportional to the thicknesses of the constituent layers: 18,,,, 2,,, 7 We consider the Si/SiO 2 and SiO 2 /SiO 2 interfaces in planar Si/SiO 2 HNMSLs as smooth enough, so that the influence of phonon scattering due to interfacial roughness on thermal conductivity is much weaker in comparison with phonon scattering in amorphous SiO 2 layers. Hence, we do not include in Eq. (7) an additional phonon scattering on interfaces 19 with a specularity factor as another fitting parameter of the model. VII. Scattering mechanisms in crystalline Si By comparison the experimental temperature dependence of thermal conductivity of bulk Si with our model, we find phonon scattering mechanisms characteristic of this material. A good agreement between theoretical and experimental thermal conductivities is obtained taking into account the Umklapp and point defect scattering (see Figure S6): The Umklapp scattering rate is calculated as 20,,,. 8,, 9 10

11 where B= s K -1 and C=177.0 K. The point defect scattering rate is calculated as 21, 4 Γ, 10 where V 0 is the volume per atom, Γ is the strength of the point-defect scattering (Γ for natural Si). Figure S6. Theoretical (solid line) vs. experimental (open circles) 22 crystalline Si. thermal conductivity of bulk VIII. Scattering mechanisms in amorphous SiO 2 In case of bulk amorphous SiO 2 or other amorphous materials, it is accepted 23 that the major part of thermal energy is transferred by diffusion between localized atomic vibrations, rather than by propagating waves (as in crystalline materials). Therefore, a rigorous description of the thermal transport should be carried out in terms of phonon diffusivity, rather than in terms of phonon lifetimes or group velocities. However, comparing the equation for thermal conductivity obtained in Ref. [23] for the phonon diffusion mechanism with that obtained from the Boltzmann transport equation (BTE), we have introduced an effective relaxation rate of phonons in amorphous SiO 2 in the form, 3.., 11 11

12 where a b.l. =0.235 nm is the bond length, is the mean vibrational frequency, is the group velocity of the vibrational mode,, and A=0.33 is taken from Ref. [23]. The authors of Ref. [23] have introduced ω to establish a correct frequency scale for the equation of phonon diffusivity and argued that the mean frequency of vibrational spectrum of an amorphous material can be taken as the value of ω. We have used it as a fitting parameter of our model. The validity of this approach is checked by a comparison between the theoretical and experimental 24 curves of the temperature-dependent thermal conductivity in bulk amorphous SiO 2 (see Figure S7). The best accordance between theory and experiment is achieved for the mean vibrational energy ħ 34. This value is in agreement with experimental dispersion relations of bulk amorphous SiO 2 from Ref. [16] as well as with our BvK lattice dynamics calculations, as is seen from Figure 4b in the main text (green curves). Figure S7. Theoretical (solid line) vs. experimental (open circles) 24 amorphous SiO 2. thermal conductivity of bulk The obtained good agreement between theoretical and experimental results confirms accuracy of the proposed model for the description of thermal processes in amorphous SiO 2. While we consider a model medium in the spirit of the Virtual Crystal approximation, average properties of vibrational excitations, which are crucial for the thermal transport, are described reasonably well in our model. 12

13 IX. Phonon scattering interplay in Si/SiO 2 HNMSLs In order to clarify the explanation behind the thermal conductivity reduction in our Si/SiO 2 HNMSLs we have carried out special theoretical investigations. We have calculated the in-plane thermal conductivity in planar Si/SiO 2 HNMSL using two different combinations of phonon scattering: (i) without point defect scattering in Si layers and (ii) without Umklapp scattering in Si layers. The phonon scattering in SiO 2 layers is included in both cases. The corresponding relaxation times of phonons are described with eqs.(9-11). Figure S8. Calculated in-plane thermal conductivity in the planar Si/SiO 2 HNMSL with a period of 2-nm SiO 2 / 20-nm Si / 2-nm SiO 2. Red, blue and black curves represent calculations without Umklapp scattering, without point defect scattering and with all scatterings included, respectively. It is seen from Figure S8 that exclusion of point defect scattering from calculation leads to ~45% rise of thermal conductivity in a wide temperature range. While it is a clear evidence of impact of point defects on thermal conductivity, still the obtained without point defect scattering values are approximately 3 times lower than those of 20-nm individual Si nanolayers. At the same time, the influence of Umklapp scattering is very weak: even at RT, exclusion of Umklapp scattering from calculation increases the thermal conductivity by less than 5%. Therefore, our calculations imply the dominant role of phonon scattering in SiO 2 layers in reducing the thermal conductivity of planar Si/SiO 2 HNMSLs. 13

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