Implicit calibration

Calibrating by the implicit method, the trend in measured probe voltage signal at a given joule heating value is determined by measuring several samples with well-characterized values of thermal conductivity, then curve fitting to find the form of the equation that describes the trend.^{[Reference Lefèvre, Volz, Saulnier, Fuentes and Trannoy11,Reference Lefèvre, Saulnier, Fuentes and Volz20]} Here, a Wheatstone bridge configuration is used. With a known value of electrical resistance in the bridge cable (R _ref), the current across the probe (I _P) may be readily calculated, thus probe resistance (R _P) and Joule heating power $(P_{\rm P} = I_{\rm P}^{\rm 2} R_{\rm P})$ at a given output voltage (V _out) are obtained. The bridge was balanced such that the voltage across the reference side of the bridge, V _r was equal in magnitude to the voltage across the measurement side of the bridge, V _S. The measured signal is (V _S − V _r), so this corresponds to the case where (V _S − V _r) = 0. Then the probe was brought near to contact (roughly 1 mm above the surface, where the tip reflection comes into focus). After finding a location on the surface which was free from observable debris and topographical artifacts, the probe was brought into contact. The change in probe voltage (and electrical resistance) was observed, then the probe was removed from the sample and brought back to >25 mm above the surface. This was repeated for a range of V _out, from 0.5 to 3.5 V so that the trends in resistance with Joule heating power may be obtained (discussed in section “Explicit thermal exchange calibration”). This output voltage range corresponded to 0.5–25 mA current across the probe leg of the Wheatstone bridge circuit. This was done for three reference samples with known values of thermal conductivity. The resulting graph (taken at V _out = 1.5 V) is shown in Fig. 1. The curve fits the following three-parameter trend line: $\lpar {V_{\rm s}-V_{\rm r}} \rpar = Ak_{\rm S}^B -C$, where (V _s − V _r) is the measured probe voltage signal, and A, B, and C are the parameters to be fit. In this instance, A = 0.3489 ± 0.004, B = −0.2865 ± 0.004, and C = −0.7763 ± 0.003, with units such that the input units of W/m/K get converted to V. Note that confidence when employing empirical curve fitting tends to improve if we include more data points, as described in separate works,^{[Reference Juszczyk, Kazmierczak-Balata, Firek and Bodzenta5,Reference Lefèvre, Saulnier, Fuentes and Volz20]} but three data points (owing to three unknowns in the fit) are the minimum number required to calibrate this way. Effective thermal conductivity (k _eff) determination of unknown samples is demonstrated by finding thermal conductivity that gives the probe voltage values measured on the thin-film Al₂O₃-on-silicon samples (with V _out = 1.5 V). Clearly, thicker Al₂O₃ leads to lower k _eff (because the intrinsic thermal conductivity of Al₂O₃^{[Reference DeCoster, Meyer, Piercy, Gaskins, Donovan, Giri, Strnad, Potrepka, Wilson, Losego and Hopkins9]} is much lower than silicon,^{[Reference Glassbrenner and Slack21]} and with thicker Al₂O₃, less of the thermal interaction volume encompasses the substrate). Note that this strategy yields an effective (bulk-like) thermal conductivity, and to decouple the film from the substrate, the area of thermal exchange must be determined. In this work, we describe the area of thermal exchange as a circular isothermal disc, so the parameter to calibrate is the radius of that disc, b.

Figure 1.

Implicit method for determining measured probe signal versus thermal conductivity. Curve fits take the following form: $\lpar {V_{\rm s}-V_{\rm r}} \rpar = Ak_{\rm S}^B -C$.

Explicit thermal exchange calibration

In SThM in DC mode, R _P directly correlates to the change in temperature ΔT _P, as expressed by:

(1)$${\rm \Delta} T_{\rm P} = \displaystyle{{R_{\rm P}-R_0} \over {{\rm TCR} \times R_0}}$$

where TCR is the temperature coefficient of resistance, and R ₀ is the nominal electrical resistance across the probe at room temperature. In this case, the temperature coefficient of resistance was calibrated to be 0.003/K, consistent with other studies using the same type of probe.^{[Reference Kaźmierczak-Bałata, Bodzenta, Krzywiecki, Juszczyk, Szmidt and Firek22]} The average probe thermal resistance is determined by taking the slope of ΔT _P with P _P:

(2)$$R_{{\rm P,meas}}^{{\rm th}} = \displaystyle{{d\lpar {{\rm \Delta} T_{\rm P}} \rpar } \over {dP_{\rm P}}}$$

with calculated $R_{\rm P}^{{\rm th}} $ from measured probe resistance, the probe-to-sample thermal exchange parameters may be explicitly calibrated by matching $R_{{\rm P,meas}}^{{\rm th}} $ to that project by an analytical probe–sample heat transfer model (PSHTM). A brief summary of PSHTM is described in the next section, and is depicted schematically in Fig. 2.

Figure 2.

Analytical model probe–sample heat transfer schematic. Reprinted from Ref. [Reference Wilson, Munoz Rojo, Abad, Perez, Maiz, Schomacker, Martin-Gonzalez, Borca-Tasciuc and Borca-Tasciuc8] with permission from the Royal Society of Chemistry, © 2015.

Analytical model derivation: salient points

The PSHTM described here has been solved several times in previous works.^{[Reference Puyoo, Grauby, Rampnoux, Rouviere and Dilhaire23,Reference Zhang, Hapenciuc, Castillo, Borca-Tasciuc, Mehta, Karthik and Ramanath24]} Thus, the most important points are highlighted here, foregoing the full derivation. For a probe with perimeter, p, and cross-sectional area, A _C, being Joule-heated by DC power, or by AC power at frequencies significantly above the probe's cut-off frequency, the probe's temperature distribution may be described by:

(3)$$T^{^\ast}\lpar x \rpar = C_1e^{\lambda x}-C_2e^{-\lambda x} + \displaystyle{{\rm \Gamma} \over {\lambda ^2}}$$

where ${\rm \Gamma} = ((I_{\rm P}^2 \rho _0)/(k_{\rm p}A_{\rm C}^2 ))$, ${\rm \lambda} ^2 = \displaystyle{{ph_{{\rm eff}}} \over {k_{\rm p}A_{\rm C}}}-\displaystyle{{I_{\rm P}^2 \rho _0{\rm TCR}} \over {k_{\rm p}A_{\rm C}^2}} $, and ρ ₀, k _p, and TCR are, respectively, the probe's electrical resistivity, thermal conductivity, and temperature coefficient of resistance, and h _eff is the effective heat transfer coefficient from the probe to the surroundings, including convective and radiative effects, and C ₁ and C ₂ are the constants arising from integration. Note that Eq. (3) is the temperature rise above ambient, i.e., T* = T _P − T ₀ where T _P is the probe absolute temperature and T ₀ is the temperature of the surroundings. The constants C ₁ and C ₂ are easily obtained by applying the two boundary conditions, defined as follows. The first assumes that where the probe meets the cantilever base, the probe is anchored to room temperature [i.e., T(0) = T ₀]. The second assumes that all of the heat transferred to the sample is contained within a disc of radius b, of uniform temperature. Performing energy balance in the probe yields:

(4)$$\eqalign{&-k_{\rm p}A_{\rm C}\displaystyle{{dT^{^\ast}} \over {dx}} \vert _{x = L/2-b} + I_{\rm P}^2 \rho _0\lpar {1 + {\rm TCR} \times T^{^\ast} \vert_{x = L/2-b}} \rpar \displaystyle{b \over {A_{\rm C}}} \cr & \quad = \displaystyle{{Q_{\rm s}} \over 2}}$$

where b is the thermal exchange radius (i.e., the radius of the isothermal disc) and Q _S is the heat transfer rate to the sample. It is convenient to define the tip thermal resistance $(R_{{\rm Tip}}^{{\rm th}} )$, which is the intrinsic thermal resistance of the sample $(R_{\rm S}^{{\rm th}} )$ plus the contact thermal resistance $(R_{\rm C}^{{\rm th}})$ as the temperature rise in the tip region (i.e., L/2−b < x < L/2) divided by the heat transfer rate to the sample [i.e., $R_{{\rm Tip}}^{{\rm th}} = \displaystyle{{T^*\vert _{x = (L/2)-b}} \over {Q_{\rm S}}} = R_{\rm C}^{{\rm th}} + R_{\rm S}^{{\rm th}} $]. Note that since the solution is for one probe leg, assuming symmetry, the temperature profile solution may be projected to both sides of the probe. Taking the integral average of the temperature profile solution,

(5)$$\bar{T}^{^\ast} = \displaystyle{2 \over L}\mathop \int \limits_0^{L/2} T^{^\ast}\lpar x \rpar dx$$

and rewriting as,

(6)$$R_{{\rm P,model}}^{{\rm th}} = \displaystyle{{{\bar{T}}^{^\ast}A_{\rm C}} \over {I^2\rho _0\lpar {1 + {\rm TCR} \times {\bar{T}}^{^\ast}} \rpar L}}$$

the model may be more readily compared with experimentally obtained values. This is a general equation for self-heated probes which develop a temperature gradient along the length of the active region, and has been applied to both nanopatterned probes with an active palladium element,^{[Reference Puyoo, Grauby, Rampnoux, Rouvière and Dilhaire25]} as well as to Wollaston wire probes.^{[Reference Zhang, Hapenciuc, Castillo, Borca-Tasciuc, Mehta, Karthik and Ramanath24]}

Parameters to calibrate: h _eff, b, and $\curr R_{\rm \curr C}^{{\rm\curr th}} $

The parameters in PSHTM which must be calibrated are h _eff, b, $R_{\rm C}^{{\rm th}} $, and probe geometry (p and A _C). To calibrate for h _eff, the probe temperature versus Joule heating power is measured in the air when the probe is far from the sample, or no sample is present. The value of h _eff is then adjusted in Eqs. (3)–(6) until $R_{{\rm P,model}}^{{\rm th}} = R_{{\rm P,meas}}^{{\rm th}} $. Then, with a calibrated value of h _eff, when inputting $R_{{\rm P,meas}}^{{\rm th}} $ for a sample with known thermal conductivity, the only unknowns in Eqs. (3)–(6) are b and $R_{\rm C}^{{\rm th}} $. The following discussions are centered on how to determine these parameters.

Step method

A simple solution to this problem can be obtained by measuring the thermal response of the probe as it goes over a patterned edge feature, as proposed initially by Puyoo et al.^{[Reference Puyoo, Grauby, Rampnoux, Rouviere and Dilhaire23]} This is a strategy depicted in Figs. 3(a)–3(c). The spatial extent over which the thermal signal deviates from its behavior far from the edge is taken to be the diameter of the isothermal disc discussed in the previous section (i.e., it has the dimension 2b). This method has been used in several recent studies, both for A-SThM, as well as P-SThM.^{[Reference Juszczyk, Kazmierczak-Balata, Firek and Bodzenta5,Reference Kim, Chung, Hwang, Kwon and Lee18,Reference Puyoo, Grauby, Rampnoux, Rouviere and Dilhaire23]} With a calibrated value of b, a reference sample with a known thermal conductivity may be measured to determine $R_{\rm C}^{{\rm th}} $. This assumes that both b and $R_{\rm C}^{{\rm th}} $ are independent of the sample being measured. Recent studies have suggested that the thermal exchange radius may change as a function of thermal conductivity,^{[Reference Wilson6–Reference Wilson, Munoz Rojo, Abad, Perez, Maiz, Schomacker, Martin-Gonzalez, Borca-Tasciuc and Borca-Tasciuc8]} so in this study, b is determined this way for all of the samples found in Table I, covering a wide range of thermal conductivity (0.2–146 W/m/K). Each of these samples had 1 mm × 1 mm patterned metal squares of ~80 nm thickness, which were used to calibrate the probe via the step method. The step calibration was performed at several values of V _out, but there was no discernable difference in b with Joule heating power on any sample. However, the samples with lower thermal conductivity values demonstrated a noticeably smaller value of b compared with the higher thermal conductivity reference samples. This is depicted in Fig. 3(d). The effect of b increasing with thermal conductivity was explored in previous works by the authors,^{[Reference Wilson6,Reference Wilson and Borca-Tasciuc7]} and was observed (without additional comment) by Kim et al.^{[Reference Kim, Chung, Hwang, Kwon and Lee18]} However, the opposite trend (decreasing b with increasing k _ref) has also been observed.^{[Reference Wilson, Munoz Rojo, Abad, Perez, Maiz, Schomacker, Martin-Gonzalez, Borca-Tasciuc and Borca-Tasciuc8]} The apparent discrepancy may be explained by the relative contributions of the probe-to-sample heat transfer: it has been demonstrated that the nanopatterned probes transfer heat to the sample primarily across the air gap, and that high thermal conductivity samples facilitate more heat spreading across the air gap, leading to a larger thermal exchange area.^{[Reference Wilson6]} On the other hand, the larger Wollaston probes (with which the opposite trend was observed^{[Reference Wilson, Munoz Rojo, Abad, Perez, Maiz, Schomacker, Martin-Gonzalez, Borca-Tasciuc and Borca-Tasciuc8]}) have a larger solid contact area, which, for high thermal conductivity samples may lead to a much larger relative contribution from solid–solid conduction, which occurs only across the point of physical contact, much smaller than the area across which the heat is transferred to the sample across the air gap.^{[Reference Wilson6]}

Figure 3.

Calibration of thermal exchange radius, b, by the step method. (a) Schematic depicting SThM tip as it approaches and moves over the edge of the patterned feature; (b) measured z-position and (c) measured SThM probe voltage as tip moves across the edge of the patterned feature; (d) determined value of b versus Al₂O₃ thickness, t _f, demonstrating that higher effective thermal conductivity samples show demonstrably larger thermal exchange radius by this method.

Intersection method

A calibration method that validates the probe for quantitative usage over a range of thermal conductivity values is the intersection method.^{[Reference Zhang, Hapenciuc, Castillo, Borca-Tasciuc, Mehta, Karthik and Ramanath24,Reference Zhang, Castillo, Mehta, Ramanath and Borca-Tasciuc26]} In this method, with $R_{{\rm P,meas}}^{{\rm th}} $ for a sample with known thermal conductivity, a value of b is assumed and plugged into Eqs. (3)–(6), and Q _S is adjusted until $R_{{\rm P,meas}}^{{\rm th}} = R_{{\rm P,model}}^{{\rm th}} $. That value of Q _S has a corresponding value of $R_{\rm C}^{{\rm th}} $, so for every b a value of $R_{\rm C}^{{\rm th}} $ is found. Repeating this for a total of two or more reference samples each with a known thermal conductivity and plotting $R_{\rm C}^{{\rm th}} $ versus b for each calibration sample, the points where the curves intersect may be found, corresponding to the calibrated pairs of (b _,$\; R_{\rm C}^{{\rm th}} $). This is depicted in Fig. 4. In previous studies, it was demonstrated that if thermal conductivity of the unknown sample is well outside of the range of thermal conductivity of the calibration samples, the probe calibration may no longer be valid,^{[Reference Wilson6,Reference Wilson, Munoz Rojo, Abad, Perez, Maiz, Schomacker, Martin-Gonzalez, Borca-Tasciuc and Borca-Tasciuc8]} and a follow-on study explored the effect of sample thermal conductivity on the calibrated parameters, showing that intersections using different calibration samples do not occur in the same location, meaning that b and $R_{\rm C}^{{\rm th}} $ may be a function of sample thermal conductivity.^{[Reference Wilson and Borca-Tasciuc7]}

Figure 4.

Calibration of contact thermal resistance ${\rm (}R_{\rm C}^{{\rm th}} {\rm )}$ versus thermal exchange radius, b. Intersections indicate a calibrated pair of $R_{\rm C}^{{\rm th}} $ and b.

To vet the intersection method, the calibration sample set was used in contact mode. The slope of temperature with respect to Joule heating power was used as the input for the thermal resistance of the probe in each case. The resulting plot is shown in Fig. 4.