Appendix A
The ceramic substrates are lapped with cubic B
4C slurry for 20 min and polished for 90 min using diamond slurry with 1 µm grain size. The final substrate thickness was 0.9 mm. The roughness of the polished areas is measured using laser scanning microscopy (LSM) and atomic force microscopy (AFM).
Figure A1a shows a LSM image and
Figure A1b an AFM scan.
Figure A1.
Surface topography of a polished LTCC ceramic substrate, captured by (a) laser scanning microscopy (LSM); (b) atomic force microscopy (AFM) beside pores.
Figure A1.
Surface topography of a polished LTCC ceramic substrate, captured by (a) laser scanning microscopy (LSM); (b) atomic force microscopy (AFM) beside pores.
The LSM (OLS 4100, Olympus, Hamburg, Germany) has a spot size of 200 nm. Surface scans were carried out over 130 µm × 130 µm (field of view) on 9 substrates. The roughness of the whole substrate area including pores was obtained from the primary data after applying a filter with a cut-off length of 8 µm. Root mean square height of the surface
Sq is evaluated using the device software OLS4100 (Version). The results
Sq (LSM) are depicted in
Figure A2. Three of these LSM surface scans were chosen and 6-line scans along closed polished areas were evaluated to obtain root mean square roughness
Rrms of non-porous regions as illustrated in
Figure A1a.
Figure A2 presents the obtained values
Rq (LSM). The cut-off length for this evaluation amounts to 2.5 µm. As a third parameter, the portion of pores, was calculated on the base of the LSM surface scans. The threshold was defined at a level of 300 nm underneath the maximum values of the topography. All regions lower than the threshold were interpreted as pore. The obtained porosity is 23% and the standard deviation is 5%.
Additionally, AFM scans (VECCO AFM measurements, Veeco Instruments Inc., Plainview, NY, USA) were carried out on regions with an area of 1.4 µm × 1.4 µm. As indicated in
Figure A1a, regions without pores were selected. The obtained
Sq values base on seven measurements,
Figure A2 depicts the values in under category
Sq (AFM).
Figure A2.
Surface parameter obtained from LSM and AFM.
Figure A2.
Surface parameter obtained from LSM and AFM.
Appendix C
The rough calculation of thermal conditions bases on the consideration of an area element on the respective substrate.
Figure A3 illustrates the conditions. With the beginning sputtering process, heat is generated on the substrate surface by ion bombardment. The generated heat is dissipated through the substrate towards the substrate holder. Thermal resistivity
Rth and heat capacitance
Cth of the substrate characterize the heat flow.
Figure A3.
Thermal conditions and geometric parameters determining the heat dissipation: Tsurface—temperature at the substrate surface; Tholder—temperature at the substrate holder; th—substrate thickness.
Figure A3.
Thermal conditions and geometric parameters determining the heat dissipation: Tsurface—temperature at the substrate surface; Tholder—temperature at the substrate holder; th—substrate thickness.
Rth is calculated using Equation (A1):
The presumed thermal conductivity
kth of the substrates is 150 W m
−1 K
−1 for silicon [
44] and 3.3 W m
−1 K
−1 for ceramic [
45]. Considering the different substrate thickness
th of 0.5 mm (silicon) and 0.9 mm (ceramic),
Rth of a representative area element
A of 1 mm
2 amounts to 3.3 KW
−1 for silicon and 273 KW
−1 for ceramics.
Cth of the substrates can be calculated using Equation (A2):
csp is the specific thermal capacity. It can be experimentally determined or calculated using Equation (A3):
where
kth is the specific thermal conductivity in W m
−1 K
−1, α represents the thermal diffusivity in m
2 s
−1 and
is the density in kg m
−3.
Reliable data
csp for the LTCC material are not available. Since LTCC ceramics are a composite of different glasses and ceramic fillers, the specific heat capacity is roughly estimated as a 1:1 mixture of alumina as frequently used ceramic filler and glass. Reference [
46] specifies the
csp of glasses to be 850 J kg
−1 K
−1 with a tolerance of 10%. Other internet sources specify values between 503 J kg
−1 K
−1 (lead glass, [
47]) and 720–800 J kg
−1 K
−1 (soda lime glass, [
48]). Values for silicon vary from 703 J kg
−1 K
−1 [
49] to 741 J kg
−1 K
−1 [
50]. For alumina, sources specify 990 J kg
−1 K
−1 [
51] and 850–1050 J kg
−1 K
−1 [
52]. The calculation of
csp based on thermal diffusivity (Equation (A3)) is an alternative way to obtain data. Necessary material constants are gathered in
Table A2.
Table A2.
Thermal material data for silicon, alumina and lead oxide.
Table A2.
Thermal material data for silicon, alumina and lead oxide.
Material | kth (W/mK) | α (10−6 m2/s) | (103 kg/m3) |
---|
Silicon | 156 [53] ~150 [44] | 88.0 [54,55] 76.9 [56] | 2.329002 [57] |
Alumina | 36.96 [58] 30.5 * [59] | 12 [58] 13.7 [56] | 3.48 * [59] |
The calculated heat capacity based on these data for silicon and alumina results in the data range presented in
Table A3.
Table A3.
Data calculated using Equation (A3) for silicon and alumina.
Table A3.
Data calculated using Equation (A3) for silicon and alumina.
Material | min | csp [J kg−1] average | max |
---|
Silicon | 761 | 816 | 871 |
Alumina | 640 | 710 | 781 |
Considering calculated values for
csp and those found in the literature, a minimum-maximum assessment for
Cth was carried out. These valued and resulting time constant τ and period 5τ considering the above calculated thermal resistance are presented in
Table A4, whereby the time constant is calculated with Equation (A4):
Table A4.
Calculated values for Cth using Equation (A2), based on different csp values.
Table A4.
Calculated values for Cth using Equation (A2), based on different csp values.
Specific Heat Capacity Csp [J kg−1 K−1] | Cth | τ [s] | 5τ [s] | 5τ [min] |
---|
Si | – | – | – | – |
871 | 1.017 | 3.4 | 17 | 0.3 |
703 | 0.821 | 2.7 | 13.7 | 0.2 |
Alumina | glass | ceramic * | | | | |
990 | 800 | 895 | 2.497 | 681 | 3405 | 57 |
640 | 503 | 571 | 1.594 | 435 | 2174 | 36 |
Despite of huge deviations due to the uncertainty of material data, it is evident that the time constant τ of the ceramic substrate is significantly higher than that of silicon. The thermal resistance, which amounts to 3.3 KW−1 for silicon and 273 KW−1 for ceramic, has the strongest influence. As a consequence, the silicon substrate reaches the steady state in a few seconds and the heat rapidly dissipated. The ceramic substrate accumulates the heat; it can take up to an hour until the steady state is reached.