Yun
Zhang
,
Wenkai
Zhu
and
Theodorian
Borca-Tasciuc
*
Mechanical, Aerospace, and Nuclear Engineering Department, Rensselaer Polytechnic Institute, Troy, NY 12180, USA. E-mail: borcat@rpi.edu
First published on 14th December 2020
Thermal conductivity measurements using Scanning Thermal Microscopy (SThM) usually involve heat transfer across the mechanical contact and liquid meniscus between the thermal probe and the sample. However, variations in contact conditions due to capillary effects at probe–sample contact and probe and sample wear due to mechanical contact interfere with accurate determination of the thermal conductivity. This paper presents measurements of thin film thermal conductivity using a SThM method employing a Wollaston probe in non-contact mode in synergy with detailed heat transfer analysis. In this technique, the thermal probe is scanned above the sample at a distance comparable with the mean free path of the ambient gas molecules. A Three-Dimensional Finite Element Model (3DFEM) that includes the specifics of the heat transfer between the sample and the probe in transition heat conduction regime was developed to predict the SThM probe thermal resistance and fit the thermal conductivity of the measured thin films. Proof-of-concept experimental in-plane thermal conductivity results for 240 nm and 46.6 nm Au films deposited on glass and silicon substrates were validated by experimental measurements of their electrical conductivity coupled with the Wiedemann–Franz law, with a discrepancy < 6.4%. Moreover, predictions based on a kinetic theory model for thin-film thermal conductivity agreed with the experimental results for the Au films with <6.6% discrepancy. To reduce the time and complexity of data analysis and facilitate experimental planning, an analytical model was also developed for the thermal transport between the Wollaston probe, ambient, and film-on-substrate samples. The accuracy of thin film thermal conductivity measurements using the analytical model was investigated using 3DFEM simulations. Fitted functions were developed for fast data analysis of thermal conductivity of thin films in the range of ∼100–600 W m^{−1} K^{−1} and thickness between ∼50–300 nm deposited on the two types of substrates investigated in this work, which yielded results with a discrepancy of 6–16.7% when compared to the Au films' thermal conductivity values.
3ω-SThM determines the thermal conductivity of the thin film quantitatively by deducing the probe temperature from the voltage signal at the 3rd harmonic and solving an analytical expression of sample spreading thermal resistance.^{12,17–19} The analytical expression assumes dominant cross-plane heat transfer since samples involved in these works are mostly low k_{f} thin films on a substrate with high thermal conductivity (k_{s}).^{6,12–14,20,21} However, this method has a limited measuring range of k_{f} < 100 W m^{−1} K^{−1}, where the deviation and uncertainty compared to literature increases dramatically when the in-plane heat transfer within the thin film starts to dominate the overall thin film thermal transport.
Alternatively, a calibration curve can be constructed by fitting the SThM signal, such as probe voltage, probe power ratio, or average probe temperature drop on reference bulk samples with known thermal conductivities to help interpolate the thermal conductivity of unknown bulk samples.^{2,4} For a film-on-substrate sample, a finite element simulation was used to decouple the thin film thermal conductivity from the interpolated total effective thermal conductivity of the entire sample. In this case, the total effective thermal conductivity should be within the sensitive region of the reference curve (1–100 W m^{−1} K^{−1}).^{15,16} Ref. ^{22} combined classical heat transfer finite element modeling and the calibration curve for a Wollaston wire probe to determine the effective thermal conductivity of suspended silicon film samples in contact mode, with the method being sensitive for film thermal conductivities up to 80 W m^{−1} K^{−1}. However, suspending the films may not be feasible in many situations. In addition, the fitted calibration curve for bulk samples contains uncertainties that may lead to enhanced uncertainty of the measured k_{f}.
The extant SThM techniques discussed above were used to measure k_{f} from 0.1 W m^{−1} K^{−1} to 1000 W m^{−1} K^{−1} for films with thicknesses from tens of nanometres to several μm.^{4,23} However, these techniques involve mechanical contact between the probe and samples and possible variations of tip-sample thermal interactions during scanning due to varying mechanical contact conditions and sample-dependent liquid meniscus^{2,24–28} which induces additional uncertainties.^{29,30}
A non-contact SThM method that avoids probe–sample mechanical contact and liquid meniscus was demonstrated in ref. ^{31} to measure low thermal conductivity samples. This paper extends the method to high thermal conductivity nanoscale thin films on substrate. In this technique, the thermal probe thermal resistance is measured above thin films at a distance comparable with the Mean Free Path (MFP) of air molecules, which leads to heat transfer in the non-classical transition regime between the probe and the sample. We adapted for thin-film thermal conductivity measurements a Three-Dimensional Finite Element Model (3DFEM) initially developed in ref. ^{32} for sample temperature profile sensing experiments using non-contact SThM. The newly developed 3DFEM captured the heat transport phenomena in the SThM probe with Joule heating, in the multilayer film-on-substrate sample, and through the gap between the probe and sample in the transition heat conduction regime. This numerical model was used to determine the film thermal conductivity for film on substrate samples. The technique is demonstrated by experimental thermal conductivity measurements of 240 nm and 46.6 nm Au thin films deposited on glass and silicon substrates respectively, and also for a low thermal conductivity silicon dioxide film deposited on the silicon substrate. The experimental thermal conductivity results for the Au films are validated by four-probe electrical conductivity measurements^{33} coupled with Wiedemann–Franz (WF) law^{34,35} and are also confirmed by a theoretical model of k_{f}.^{36} To reduce the time and complexity of data analysis and facilitate experimental planning, an analytical model was developed for the thermal transport between the Wollaston probe and film-on-substrate samples under ambient conditions and was validated by 3DFEM. Thin film thermal conductivity measurement results using the analytical model and the 3DFEM were simulated and compared. Fitted functions between the SThM probe thermal resistance and film thermal conductivity were developed in order to enable faster data analysis of thermal conductivity for films deposited on the two types of substrates investigated in this work.
Middle-section length (μm) | Line width (μm) | Film thickness (nm) | |
---|---|---|---|
Au on glass | 1000 | 40 | 240 ± 4.1 |
Au on silicon | 1000 | 40 | 46.6 ± 5.8 |
Fig. 2 shows there are separate channels used to monitor the voltages across each probe and each shunt resistor (2.2 Ω) in the two branches. This allows to monitor the powers dissipated and electrical resistances of the measurement and reference probes independently.
(1) |
Fig. 3 The micrograph of a Wollaston wire probe in the proximity of a sample surface. The plane of the probe tip is normal to the sample surface. |
In general, the heat conduction between the probe, the sample, and the surrounding air, and the Joule heating of the probe are all presented in the form of the heat diffusion equation,
−∇⋅(k∇T) = Q | (2) |
Q = J^{2}⋅ρ_{0}(1 + TCR_{p}(T − T_{ref})) | (3) |
The initial temperature was set to T_{ref} = 299.05 K for all the computational domains. Mesh independence studies were conducted with 12956804, 14636046, 16348113, 17039584 and 20286444 triangular meshes, respectively. Since the difference of computation results (probe thermal resistance) from 14636046 meshes to 17039584 meshes was only in the order of 1 × 10^{−5}, 16348113 meshes were chosen.
Since the tip-sample clearance at the apex (50–150 nm) is on the same order as the MFP of ambient air, particular attention is given to modeling the heat transfer in the air gap between the Wollaston probe tip and sample surface to accurately capture the heat transfer in the transition regime. Similar to the method used in ref. ^{32}, the air gap at the probe apex is discretized in concentric cylindrical sections as shown in Fig. 4a. Each section is then assigned an effective thermal conductivity according to its local tip-sample clearance D_{i} and a thermal conductivity model for heat conduction in transition regime,^{42–44}
(4) |
Fig. 4 (a) Schematic showing decomposition of probe–sample air gap used for 3DFEM. In the transition regime, eqn (4) is used to assign the effective thermal conductivity of each discretized cylindrical layer based on its local probe–sample clearance; (b) 3DFEM simulations of the probe thermal resistance above a Au film sample (same geometry shown in Table 1) on a glass substrate as a function of thin film thermal conductivity, when the apex tip-sample clearance is 50–150 nm. The curves overlap due to the transition heat conduction regime across apex tip-sample clearances which are similar to the mean free path of the air. |
In 3DFEM the probe thermal resistance is obtained by taking the ratio of volumetric averages of probe temperature rise and its thermal power where the probe power is the same as the thermal power in the SThM experiment. Fig. 4b shows a comparison between 3DFEM predictions of R^{th}_{p} for apex tip-sample clearances of 50 nm, 100 nm, and 150 nm and as a function of thin film thermal conductivity for a 240 nm film on the glass substrate.
The overlapping profiles indicate the gap-independence of R^{th}_{p} when apex tip-sample clearance is in the transition regime, similar to the experimental results shown in ref. ^{31}. Thus, small fluctuations of apex tip-sample clearance should cause negligible uncertainties (∼0.2%) onto the value of R^{th}_{p}.
Fig. 5 Probe thermal resistance vs. apex tip-sample clearance for Au thin films on silicon and glass substrates from the experiments and 3DFEM simulations. The non-contact thermal resistance is measured at an apex tip-sample clearance of 100 ± 50 nm. The error bars are (max − min)/2 that are calculated based on four separate scanning experiments. The 3DFEM simulations are performed using the fitted values of the film thermal conductivities as measured at 100 nm tip-sample clearance (Table 2). The simulations indicate good agreement with measurements over the entire range of tip-sample distance. |
The comparisons shown in Fig. 5 between the experimental probe thermal resistance and 3DFEM predictions indicate that the discrepancy between the experiment and modeling is within the uncertainty of the experimental results. The 3DFEM simulations are performed using the fitted values of the film thermal conductivities based on the probe thermal resistances measured at 100 nm tip-sample distance (the values reported in Table 2). An abrupt change occurs in the trend of the probe thermal resistance vs. position at distances <300 nm because the transition heat conduction in the tip-sample gap starts to become important. The numerical simulations predict an invariant probe thermal resistance in the 50–200 nm height interval which practically eliminates effects due to height uncertainty (see ESI†).
Bulk Au sample | 240 nm Au film on the glass substrate | 46.6 nm Au film on the silicon substrate | |
---|---|---|---|
ρ _{b}, ρ (Ω m) | 2.2 × 10^{−8} (ref. ^{48}) | (3.02 ± 0.01)·10^{−8} | 4.98 × 10^{−8} ± 1.20 × 10^{−11} |
k _{b}, k^{WF}_{f} (W m^{−1} K^{−1}) | 317 (ref. ^{33}) | 229.6 ± 1.0 | 140 ± 0.03 |
k _{f} (W m^{−1} K^{−1}) | — | 240.2 ± 5.9 | 131.0 ± 9.9 |
If the heat flow into the sample is low because the sample has a high thermal resistance, the probe remains at a high temperature and results in a higher R^{th}_{p}. Although the Si substrate with SiO_{2} film has a lower thermal resistance than the glass substrate, the overall higher probe thermal resistance in Fig. 5 shown for Au on Si sample indicates that the probe–sample heat transfer mainly depends, as discussed later, on the in-plane film thermal resistance and the Au film on Si substrate sample has a higher thermal resistance than the Au film on glass substrate due to the much smaller film thickness.
The jump of the R^{th}_{p} from non-contact point to contact point is caused by the additional solid–solid conduction and conduction through the water meniscus. According to ref. ^{31} and ^{45}, the jumps of the probe thermal resistance were observed to be smaller for samples with smaller thermal conductivities, which implied a dependence of solid–solid heat conduction on sample thermal conductivities and surface artifacts. Thus, the smaller jump for the Au on Si substrate sample shown in Fig. 5 may be a result of less heat transfer through the solid–solid heat conduction. More importantly, the non-contact experimental R^{th}_{p} in Fig. 5 remains almost constant in the transition regime (for the range between 100–200 nm tip-sample clearance at apex), which means it is solely correlated to the sample thermal properties rather than contact artifacts. The probe thermal resistance value at the tip-sample clearance of 100 nm is fitted with 3DFEM simulations in order to obtain the thermal conductivity of the thin film sample. The example in Fig. 6 shows 3DFEM simulations of R^{th}_{p} performed as a function of k_{f}. By locating the value of the experimental R^{th}_{p} on the 3DFEM curve, the experimental thin film thermal conductivity was obtained. The red solid line indicates the average thin film thermal conductivity and the red dotted line shows the upper and lower boundaries of the experimental uncertainties of the measured film thermal conductivity due to the uncertainties in the probe thermal resistance. The results for Au thin film thermal conductivity are summarized in Table 2. 3DFEM of the Au film on glass substrate shows that the dominant thermal transport during our experiments is in the radial direction. This is due to the spreading effect of a large thermal conductivity film deposited on a low thermal conductivity substrate (see ESI†). A similar configuration occurs for the Au film deposited on a low thermal conductivity silicon dioxide layer (see ESI†). Therefore the thermal conductivity measurements reported here for the Au films are for in-plane thermal conductivity. The measured thermal conductivity of silicon dioxide film is 1.19 W m^{−1} K^{−1} and has a maximum discrepancy of 3.48% vs. the literature values.^{46,47} For the silicon dioxide film, the dominant thermal transport is in cross-plane direction (see ESI†).
(5) |
The electrical resistivity was measured using the four-probe setup shown in Fig. 7. The thin-film line was connected electrically in series with an 11.2 Ω resistor. A small DC current (1.5 mA) was applied to the Au thin film to minimize Joule-heating induced resistance change. The voltage drop on the middle section of the line was collected along with the voltage on the series resistor to determine the sample's electrical resistance (R_{f}). Since the full geometry of the sample is known (Table 1), the electrical resistivity (ρ) of the thin film can be calculated from,
(6) |
A summary of the Wiedemann–Franz law validation results is shown in Table 2. SThM thermal conductivity measurements for both Au films agree very well with the thermal conductivity results from Wiedemann–Franz law, with a maximum discrepancy of 6.4%.
A theoretical model for in-plane thin-film thermal conductivity based on kinetic theory was proposed by Qiu and Tien,^{36}
(7) |
Fig. 8 Comparisons between the SThM experimental results (dots) and theoretical model predictions (lines) for thin film thermal conductivity based on ref. ^{36}. Theoretical predictions employ two grain size models, De Vries^{49} and Power law.^{50} |
Qiu and Tien's theoretical model helps to explain the physical mechanisms responsible for the observed thickness dependence of k_{f} for the Au metallic thin films investigated in this work. For the 240 nm film l/t_{f} and l/D_{g} are 0.171 and 0.854 respectively, which makes the contribution of grain size the dominant term in eqn (7). For the 46.6 nm film l/t_{f} and l/D_{g} are 0.88 and 4.40 respectively, which also makes the contribution of grain size the dominant term in eqn (7). When comparing the two thin films, the reduction of thickness likely created smaller size grains and the additional scattering of the heat carriers resulted in lower in-plane thermal conductivity.^{51} Similar size effects have been observed for other types of 2D materials using different measurement techniques.^{41,52}
The thermal model shown in Fig. 9 is used to connect into a thermal resistance network all three heat transfer domains (probe, sample, ambient), and includes models for the thermal resistances of the probe, the air gap, and the film-on-substrate thermal resistances. An analytical model for a self-heated Wollaston probe in air with the probe–sample heat transfer confined to the apex region was developed in ref. ^{12} and ^{31}, but for different sample configurations than used in this work (for substrate only samples in ref. ^{30}, and for film-on-substrate samples where heat transfer is assumed unidirectional along the film thickness direction in ref. ^{12}). For the Wollaston probe, the model solves a 1D fin heat conduction equation along the length of the Joule heated probe wire to obtain the temperature rise at probe apex () as a function of probe geometry, probe thermal properties, effective heat transfer coefficient h and tip-sample heat transfer rate (Q_{C}). In this analytical model a correlation is employed for the tip-sample thermal exchange resistance (R^{th}_{C}) as a function of thermal exchange radius b, probe–sample clearance and probe wire diameter, which was developed in ref. ^{32} for a Wollaston probe in air. This correlation considers both the classical and non-classical heat conduction effects in the small tip-sample air gap. Then, using the heat transfer rate Q_{C} through the tip-sample thermal resistance network and the tip to back-of-substrate temperature difference, the sum of the tip-sample thermal exchange resistance with the thermal resistance of the film-on-substrate sample (R^{th}_{S}) can be written as,
(8) |
Fig. 9 Schematic of the probe–sample system used for the analytical modeling of the probe–sample-ambient heat transfer. |
For large thermal conductivity films on low thermal conductivity substrates, in-plane heat conduction is expected to be significant and the model developed in ref. ^{12} and ^{19–21} will not apply for the sample configurations measured here. The key challenge remaining for the analytical model is then finding an analytical thermal resistance model for the film-on-substrate sample (R^{th}_{S}), when in-plane heat conduction in the film is significant.
In order to better understand the distribution of heat flux on the sample surface, results from 3DFEM heat transfer simulations of the probe–sample-ambient system were first used to calculate the surface heat flux spatial profile due to SThM probe heating of a thin film-on-substrate sample. Although the geometry of a Wollaston SThM probe is not axisymmetric, the sample surface heat fluxes probed along two perpendicular directions in the plane of the sample (one parallel and one perpendicular to the plane of the SThM probe defined by its V-shape) were found to have similar Gaussian radii which are independent of the tip-sample distance when this is in 100–300 nm range (see ESI†). The characteristic radius of the heat flux Gaussian profile b_{GC} was then taken as the square root of the product of the two Gaussian radii (b_{G}). Thus, an analytical model developed for Gaussian laser heating of a multilayer film-on-substrate configuration^{6} was adapted here to determine the thermal resistance of film-on-substrate samples in general, and was exemplified here for thin films on glass substrate and on silicon substrate with an interfacial SiO_{2} film.
Based on ref. ^{6}, the governing equation for steady-state heat conduction in the thin film and substrate, assuming isotropic thermal properties and no temperature gradients in the azimuthal direction is given by eqn (9):
(9) |
Then, with the boundary conditions described in Section 12 of the ESI,† the steady-state solution for the temperature distribution on the thin film surface can be obtained by Hankel transformation in terms of the Hankel variable β as,
(10) |
(11) |
(12) |
(13) |
The value of the thermal resistance calculated from eqn (13) is independent of the peak flux. The temperature profile predicted by the analytical model was cross-validated by 3DFEM for the same sample geometry (see ESI†) using the Gaussian heat flux profile determined from 3DFEM. The thermal resistance discrepancy calculated from the analytical and 3DFEM models was within 0.4%.
The probe tip temperature rise can be determined from the analytical model of the Wollaston wire probe with circular cross-section and neglecting radiative heat loss as shown in eqn (14)–(16)^{12,29,32,53}
(14) |
(15) |
(16) |
The expressions for C_{3} and C_{4} and the values^{12,28,31,52} used for h, k_{p}, A_{p}, r_{p}, TCR_{p}, I, and ρ_{0} (effective heat transfer coefficient, thermal conductivity of the probe, cross-sectional area of the probe, radius of the probe, thermal coefficient of resistance, applied current and reference electrical resistivity of the probe respectively) are found in the ESI.† As the Wollaston probe comes closer to the surface, the effective heat transfer coefficient is still applied to the probe surface, except in the region undergoing heat transfer to the sample through the thermal exchange area. Eqn (17) connects the probe tip temperature rise to ambient temperature via the R^{th}_{C} and R^{th}_{S}, where R^{th}_{S} is based on eqn (13) and R^{th}_{C} is a function of b according to the correlation developed in ref. ^{32}.
(17) |
The analytical model assumes that the thermal exchange radius (b) of the probe is the same as b_{GC}. This assumption was validated (see ESI†) by 3DFEM of the SThM sample system with bulk samples for k_{b} = 1.1–50 W m^{−1} K^{−1} and finding b = b_{GC} = 5.4 μm ± 0.1 μm. Then (see ESI†) it was found that b_{G} for the thin film-on-substrate sample is also almost unchanged when k_{f} ranges from 10 to 1000 W m^{−1} K^{−1} for t_{f} = 240 nm and k_{s} = 1.1 W m^{−1} K^{−1}. It was concluded that b = b_{GC} = 4.6 μm ± 0.1 μm for thin film-on-glass samples.
Using the tip-sample thermal exchange resistance correlation developed in ref. ^{52}, the sample–probe thermal exchange resistance at the tip-sample clearance of 100 nm yields R^{th}_{C} of 126626 K W^{−1} for b = 4.6 μm. The unknown k_{f} can be then obtained based on eqn (17) by numerically fitting k_{f} as k_{s}, t_{f}, t_{s}, b and R^{th}_{C} are known.
The lateral spatial resolution in SThM measurements using heated probes is indicated by the diameter of the heat diffusion region in the sample, resulting from the heat transfer between the heated probe and the sample. This region is ultimately limited by 2b, the diameter of the probe–sample thermal exchange area,^{25} which is typically considered the limit of the spatial resolution. For contact mode SThM performed under ambient conditions, air conduction contributes significantly to the probe–sample heat transfer (∼10 μm thermal exchange diameter for a Wollaston probe^{22}) and therefore the spatial resolution of the contact mode may not be significantly smaller than in non-contact mode.^{54} The spatial resolution of SThM non-contact mode in air is controlled by the dimension and geometry of the probe tip. Smaller probe tips will generally have a smaller thermal exchange radius. For SThM experiments performed under vacuum conditions, the thermal exchange area for contact mode is controlled by the solid–solid contact area and thus results in an improved spatial resolution.^{4}
Based on the data from Fig. 10b, three numerical correlations were developed by fitting the 3DFEM and analytical results for t_{f}·k_{f} (m W m^{−1} K^{−1}) vs. R^{th}_{p} (K W^{−1}) with eqn (18).
(18) |
The fitted parameters in eqn (18) and the goodness of fit are summarized in Table 3 for each case. Plots comparing the 3DFEM and analytical results from Fig. 10bvs.eqn (18) fits are shown in ESI.†
AM b_{GC} = 4.6 μm | 3DFEM | AM b_{GC} = 4.78 μm | |
---|---|---|---|
A1 | 190848.80 | 210153.80 | 201554.71 |
A2 | 21855.57 | 21850.29 | 21847.14 |
A3 | 1424.99 | 1265.99 | 1406.76 |
A0 | −5318.05 | 822.06 | −3804.22 |
R | 0.99919 | 0.99951 | 0.99974 |
Using the experimentally obtained probe thermal resistance for the measured sample and eqn (18) fit of the analytical model with b = b_{GC} = 4.6 μm, the k_{f} of Au film on glass substrate was calculated as 205.2 W m^{−1} K^{−1}, which has a 10.6% discrepancy compared to k^{WF}_{f} shown in Table 2. Using eqn (18) fit of the 3DFEM yields k_{f} = 216.4 W m^{−1} K^{−1} which has a 5.7% discrepancy compared to k^{WF}_{f} shown in Table 2. The deviation from the 3DFEM simulation result from Fig. 6a is due to the different geometry of the film used in the modelling (patterned line vs. un-patterned film). If the 3DFEM results on glass substrate (Fig. 10b) are used to re-calibrate the value of b in the analytical model, then b = 4.78 μm. Then the discrepancy between the 3DFEM result and the result obtained using eqn (18) fitted function of the analytical model with the new b reduces to 0.8%.
The k_{f} for the Au film on the silicon substrate sample can be quickly evaluated using a fitted function developed by fitting the three-layer analytical model predictions (see ESI†). The fitted k_{f} = 116.6 W m^{−1} K^{−1} for the gold film on silicon substrate using this fitted function has a 16.7% discrepancy compared to the k^{WF}_{f} shown in Table 2.
While the data analysis based on 3DFEM provides the most accurate thermal conductivity measurement results, fitted functions avoid repeating the 3DFEM simulations for films deposited on the same substrates. The fitted functions reported in this work provide faster ways to perform data analysis or investigate feasibility of future experiments for films deposited on the two types of substrates investigated here. For films deposited on different substrates or for different probe geometries, the experimentalist has to find the new fit functions. Moreover, analytical modelling can prove useful (vs. 3DFEM and developed fitted functions) due to its speed and flexibility, for instance, new fitted functions can be developed faster based on analytical modeling for cases when the probe has different dimensions or thermal properties that are different from those used in this work, or for films on other types of substrates, or if the sample configuration requires additional layers to be added to the 3-layer model developed in this work. Then 3DFEM and fit-functions based on 3DFEM can be used for fine-tuning or validating the final thermal conductivity result.
Footnote |
† Electronic supplementary information (ESI) available: eqn (S1-S5) and Table S1: supplemental equations and coefficients for the analytical model of the active probe and values for Wollaston probe parameters. Fig. S1: Comparison of temperature profiles predicted by 3DFEM and analytical models. Fig. S2: Gaussian fitting of surface heat flux along different directions. Fig. S3: Gaussian fitting of surface heat flux for different bulk samples. Fig. S4: Gaussian fitting of surface heat flux for different films. Fig. S5: Gaussian fitting of surface heat flux for different probe-sample distances. Fig. S6: Probe thermal resistance and measured thermal conductivity for a silicon dioxide film on silicon substrate. Fig. S7 and S8: Comparisons between data and fitted functions of the product between film thickness and thermal conductivity vs. probe thermal resistance for Au on glass and Au on Si substrate samples, respectively. Fig. S9: photodetector signals for Au on glass and Si substrate samples. Fig. S10: Heat flux ratio (in-plane vs. cross-plane) for the thin film samples. Fig. S11: Optical microscopy image of the probe and 3DFEM temperature distributions for the Wollaston probe–air–sample system. Eqn (S7-S13): Boundary conditions for the analytical model of the film-on-substrate sample. See DOI: 10.1039/d0na00657b |
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