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Thermal conductivity of highly porous Si in the temperature range 4.2 to 20 K.

Valalaki K, Nassiopoulou AG - Nanoscale Res Lett (2014)

Bottom Line: The reported results are the first in the literature for this temperature range.This behavior is attributed to the presence of a majority of non-propagating vibrational modes, resulting from the nanoscale fractal structure of the material.The above results complement previous results by the authors in the temperature range 20 to 350 K.

View Article: PubMed Central - HTML - PubMed

Affiliation: NCSR Demokritos/INN, Terma Patriarchou Grigoriou, Aghia Paraskevi, Athens 15310, Greece.

ABSTRACT

Unlabelled: We report on experimental results of the thermal conductivity k of highly porous Si in the temperature range 4.2 to 20 K, obtained using the direct current (dc) method combined with thermal finite element simulations. The reported results are the first in the literature for this temperature range. It was found that porous Si thermal conductivity at these temperatures shows a plateau-like temperature dependence similar to that obtained in glasses, with a constant k value as low as 0.04 W/m.K. This behavior is attributed to the presence of a majority of non-propagating vibrational modes, resulting from the nanoscale fractal structure of the material. By examining the fractal geometry of porous Si and its fractal dimensionality, which was smaller than two for the specific porous Si material used, we propose that a band of fractons (the localized vibrational excitations of a fractal lattice) is responsible for the observed plateau. The above results complement previous results by the authors in the temperature range 20 to 350 K. In this temperature range, a monotonic increase of k with temperature is observed, fitted with simplified classical models. The extremely low thermal conductivity of porous Si, especially at cryogenic temperatures, makes this material an excellent substrate for Si-integrated microcooling devices (micro-coldplate).

Pacs: 61.43.-j; 63.22.-m; 65.8.-g.

No MeSH data available.


Related in: MedlinePlus

Porous Si SEM images used for the calculation of Hausdorff dimension.Examples of cross-sectional SEM images (a1)and top view images (b1) of the studiedporous Si layer with their corresponding binary images(a2) and(b2), used for the calculation of thebox counting dimension.
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Figure 3: Porous Si SEM images used for the calculation of Hausdorff dimension.Examples of cross-sectional SEM images (a1)and top view images (b1) of the studiedporous Si layer with their corresponding binary images(a2) and(b2), used for the calculation of thebox counting dimension.

Mentions: We derived the Hausdorff dimension of our porous Si material using scanning electron microscopy (SEM)images and the box counting algorithm [31]. The SEM images reflect the fractal microstructure of the material. Thebox counting dimension is then defined, which is a type of fractal dimension and isbased on the calculation of a scaling rule (using the negative limit of the ratio ofthe log of the number of boxes at a certain scale over the log of that scale). Theopen-access software ‘ImageJ’ [32] was used for the SEM image processing, while the open-access software‘FracLac’ [33] was used to calculate the Hausdorff dimension of our SEM images using thestandard non-overlapping box counting method. We used the maximum possible differentgrid positions for every image in order to ensure the accuracy of the calculation,while we calculated the box counting dimension for both cross-sectional and top viewSEM images of different magnifications. The results were similar from both top-viewand cross-sectional images. We also used SEM images from different samples that wereprepared with the same electrochemical conditions. In all cases, the calculatedHausdorff dimension was found to be less than two, including the standard error.Some examples of the images used and their corresponding binary ones are shown inFigure  3. The average of values was approximately 1.822 ± 0.084. Since is less than two, it is evident from expression (1) that is also lower than two, since θ is a positivequantity. The condition for the existence of fractons in our system is thusfulfilled.


Thermal conductivity of highly porous Si in the temperature range 4.2 to 20 K.

Valalaki K, Nassiopoulou AG - Nanoscale Res Lett (2014)

Porous Si SEM images used for the calculation of Hausdorff dimension.Examples of cross-sectional SEM images (a1)and top view images (b1) of the studiedporous Si layer with their corresponding binary images(a2) and(b2), used for the calculation of thebox counting dimension.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4109793&req=5

Figure 3: Porous Si SEM images used for the calculation of Hausdorff dimension.Examples of cross-sectional SEM images (a1)and top view images (b1) of the studiedporous Si layer with their corresponding binary images(a2) and(b2), used for the calculation of thebox counting dimension.
Mentions: We derived the Hausdorff dimension of our porous Si material using scanning electron microscopy (SEM)images and the box counting algorithm [31]. The SEM images reflect the fractal microstructure of the material. Thebox counting dimension is then defined, which is a type of fractal dimension and isbased on the calculation of a scaling rule (using the negative limit of the ratio ofthe log of the number of boxes at a certain scale over the log of that scale). Theopen-access software ‘ImageJ’ [32] was used for the SEM image processing, while the open-access software‘FracLac’ [33] was used to calculate the Hausdorff dimension of our SEM images using thestandard non-overlapping box counting method. We used the maximum possible differentgrid positions for every image in order to ensure the accuracy of the calculation,while we calculated the box counting dimension for both cross-sectional and top viewSEM images of different magnifications. The results were similar from both top-viewand cross-sectional images. We also used SEM images from different samples that wereprepared with the same electrochemical conditions. In all cases, the calculatedHausdorff dimension was found to be less than two, including the standard error.Some examples of the images used and their corresponding binary ones are shown inFigure  3. The average of values was approximately 1.822 ± 0.084. Since is less than two, it is evident from expression (1) that is also lower than two, since θ is a positivequantity. The condition for the existence of fractons in our system is thusfulfilled.

Bottom Line: The reported results are the first in the literature for this temperature range.This behavior is attributed to the presence of a majority of non-propagating vibrational modes, resulting from the nanoscale fractal structure of the material.The above results complement previous results by the authors in the temperature range 20 to 350 K.

View Article: PubMed Central - HTML - PubMed

Affiliation: NCSR Demokritos/INN, Terma Patriarchou Grigoriou, Aghia Paraskevi, Athens 15310, Greece.

ABSTRACT

Unlabelled: We report on experimental results of the thermal conductivity k of highly porous Si in the temperature range 4.2 to 20 K, obtained using the direct current (dc) method combined with thermal finite element simulations. The reported results are the first in the literature for this temperature range. It was found that porous Si thermal conductivity at these temperatures shows a plateau-like temperature dependence similar to that obtained in glasses, with a constant k value as low as 0.04 W/m.K. This behavior is attributed to the presence of a majority of non-propagating vibrational modes, resulting from the nanoscale fractal structure of the material. By examining the fractal geometry of porous Si and its fractal dimensionality, which was smaller than two for the specific porous Si material used, we propose that a band of fractons (the localized vibrational excitations of a fractal lattice) is responsible for the observed plateau. The above results complement previous results by the authors in the temperature range 20 to 350 K. In this temperature range, a monotonic increase of k with temperature is observed, fitted with simplified classical models. The extremely low thermal conductivity of porous Si, especially at cryogenic temperatures, makes this material an excellent substrate for Si-integrated microcooling devices (micro-coldplate).

Pacs: 61.43.-j; 63.22.-m; 65.8.-g.

No MeSH data available.


Related in: MedlinePlus