Temperature coefficient of resistance and thermal boundary conductance determination of ruthenium thin films by micro four-point probe

Accurate characterization of the temperature coefficient of resistance (α TCR) of electrically conductive materials is pertinent for reducing self-heating in electronic devices. In-situ non-destructive measurements of α TCR using the micro four-point probe (M4PP) technique have previously been demonstrated on platinum (Pt) thin films deposited on fused silica, assuming the thermal conductivity of the substrate as known. In this study, we expand the M4PP method to obtain the α TCR on industrially relevant stacks, comprising ruthenium (Ru) thin films (3.3 nm and 5.2 nm thick) deposited on bulk silicon (Si), separated by a 90 nm SiO2 spacer. The new M4PP methodology allows simultaneous determination of both α TCR and the total thermal boundary conductance (G TBC) between the metallic film and its substrate. We measured the α TCR and the G TBC to be 542 ± 18 ppm K−1 and 15.6 ± 1.3 MW m−2K−1 for 3.3 nm Ru, and 982 ± 46 ppm K−1 and 19.3 ± 2.3 MW m−2K−1 for 5.2 nm Ru. This is in good agreement with independent measurements of α TCR. Our methodology demonstrates the potential of M4PP to characterize thermal properties of metallic thin films used in semiconductor technology.


Introduction
The temperature coefficient of resistance (α TCR ) defines the fractional change in the electrical resistivity of a material due to an increase in temperature.For most metals near room temperature, α TCR is positive, which entails an unwanted heating of electronic devices and sensors.Thus, to ensure device performance, α TCR must be considered during their design and monitored during production to avoid losses in efficiency, performance and reliability [1][2][3].In addition, α TCR can be used as an indirect measure of critical dimension [4], and as an early warning indicator of thermal failure [5,6].
The prevalent approach for measuring α TCR involves recording the electrical resistivity at various steady-state temperatures, and determining the slope ∂R/R∂T through linear fitting of the experimental data [7].Although this is a simple and effective method, it is time-consuming since it requires sample preparation and significant measurement time to allow the temperature to stabilize at several set points.Furthermore, it usually involves additional steps of sample preparation (e.g.use of conductive paint) to improve the electrical contacts.Thus, development of a method for fast and accurate measurement of α TCR is highly relevant.
Micro four-point probe (M4PP) is commonly utilized for spatial characterization of material properties, to enhance process control and optimization.The M4PP technique has been extensively used for measuring sheet resistance [8,9], electron mobility [10,11], carrier density [10,11], and tunneling magnetoresistance [12,13] mainly at wafer level in semiconductor manufacturing and research.
Recently, we have demonstrated the potential of the M4PP technique to determine several thermal and thermoelectric properties.These developments include the determination of α TCR of metallic thin films deposited on fused silica [14], extraction of the ratio of the Seebeck coefficient to the thermal conductivity [15], spatial probing of microscopic thermal fields [16], and measurement of thermal diffusivity [17].Previously, we conducted α TCR measurements on platinum (Pt) thin films deposited on fused silica, as documented in [14].However, these samples do not align with standard silicon (Si) substrates widely used in semiconductor research and manufacturing.Commonly, thin-film metals are deposited on a thin SiO 2 layer (less than 500 nm) grown on a silicon wafer.M4PP measurements on such wafers present a considerable challenge due to the high thermal conductivity of the substrate, which diminishes all the temperature-dependent signals.Furthermore, the presence and influence of the SiO 2 layer are non-negligible factors that add complexity to the analytical model required to accurately derive α TCR .
In this study, we present an improved method that allows the characterization of α TCR even for samples with standard Si substrates.In addition, instead of using a semianalytical model, we use the finite element method (FEM), which decreases the computational time.The new approach is demonstrated on two ruthenium (Ru) thin films deposited on 90 nm of SiO 2 grown on a Si substrate.Measurements of α TCR using M4PP and a physical property measurement system (PPMS) are compared, and found to be in excellent agreement.The M4PP measurements also allow the determination of the total thermal boundary conductance between the metallic thin films and their Si substrate (G TBC ).

Theoretical model
M4PP measurements are performed on a metallic thin film by forcing an alternating current I (t) = √ 2I RMS sin (ωt) at an angular frequency ω between two electrodes on a sample.Here, t is the time and I RMS is the root mean square of the current.The forced current results in a potential difference V(t) that is probed with two other electrodes, as represented in figure 1(a).The measurement current causes a local temperature increase at an angular frequency 2 ω ∆T (r) ∝ I 2 due to Joule heating at the location r, as can be seen in figure 1(b).This temperature increase results in a change in the local sheet resistance R S (r), which for small temperature changes are well described by the linear approximation R S (r) = R S,0 [1 + α TCR ∆T (r)], where R S,0 is the sheet resistance at reference (room) temperature.
The resulting four-point voltage, which is represented in figure 1(c), can be described as where R 0 is the zero-current four-point resistance (without Joule heating) and Ψ is a function of material properties and geometry [18], including the thermal boundary conductance G TBC between metal thin film and Si substrate.Typically, the four-point voltage is measured via lock-in amplification, and we define the first and third harmonic resistances R 1ω and R 3ω via V (t) = √ 2I RMS [R 1ω sin (ωt) + R 3ω sin (3ωt)] [18].It should be clarified that the origin of R 3ω is temperature fluctuating at an angular frequency 2ω, and not temperature at an angular frequency 3ω.To estimate α TCR , it is convenient to measure R 3ω = − 1 2 R 0 α TCR Ψ I 2 RMS [18] as an alternative to using the resistance difference obtained from measurements at two different currents [14].Using R 3ω is particularly advantageous when measuring weak signals, as is the case of samples with Si as substrate.
Figure 2 presents FEM simulations of R 3ω as function of pitch s in configurations A ′ and A for different combinations of α TCR and G TBC .The use of A and A ′ configurations is justified since it minimizes unwanted thermoelectric contributions [14,15].The insets in figure 2 show a schematic of the configurations used and the definition of pitch.It can be seen that weaker R 3ω signals are produced in both configurations when (i) measuring with larger electrode pitch and (ii) in A configuration compared with A ′ configuration (notice the difference in scale between figures 2(a) and (b)).Furthermore, the influence of α TCR and G TBC on R 3ω depends on the pitch and the measurement configuration (see figure 2).To summarize, the simulations in figure 2 suggest that it should be possible to determine simultaneously α TCR and G TBC by performing two or more measurements with different configuration and/or pitch.Although a semi-analytical expression for α TCR of a thin film on a bulk substrate has been derived [14], the approach is inapplicable to samples with an additional electricallyinsulating yet thermally-conductive layer between the metallic thin film and the substrate, as the case with our current samples.To overcome this drawback, in the present study, we use FEM to simulate the electrical and thermal responses of the sample (see appendix A for simulation details).For data fitting, the FEM simulations were further performed with direct current I DC ≡ I RMS to reduce execution time.Each FEM simulation provided the four-point resistance R DC and the zerocurrent resistance R DC,0 (by setting α TCR = 0).The misfit to experimental data for an individual measurement i is then given by, where the factor 1/2 arises from the definition I DC ≡ I RMS [14].
Notice that in the limit of a small current, the first harmonic resistance R 1ω = R 0 and the third harmonic term vanishes, R 3ω ≈ 0. At larger currents in the low-frequency limit, where heat transport is considered instantaneous within the measured volume, the zero-current four-point resistance can be obtained from R 0 = R 1ω + 3R 3ω [18].The main factors that affect R 3ω are: (1) R S,0 and (2) thermal conductivity of the electrically conductive thin film, (3) electrode pitch, (4) electrode contact radii, (5) α TCR and (6) G TBC .The R S,0 and the accurate position of the electrodes were calculated by regression, using R 0 from several configurations [13].Since the top layer is metallic, its thermal conductivity was estimated using the Wiedemann-Franz law once R S,0 was determined.Then, the contact radii of the current electrodes were calculated from the two-point load resistance of multiple configurations [15].Finally, the remaining two parameters (α TCR and G TBC ) were fitted simultaneously, provided measurements in A ′ and A configurations for different electrode pitch.A summary of the workflow is shown in table 1.

Materials and instrumentation
Two Ru thin films (3.3 nm and 5.2 nm thick) were deposited by atomic layer deposition with an adhesion layer of TiN (0.3 nm) on 90 nm of SiO 2 grown on 300 mm Si (100) wafers (see inset figure in figure 1(a)).Further information about the sample preparation can be found in [20,21].Both samples were cut in a square shape of approximately 1 cm in edge length to facilitate reference measurements with a Quantum Design PPMS.For these measurements, the surface of each Ru thin film was contacted at four locations by thin copper wires using silver (Ag) paint.The samples were then positioned in a temperature-regulated chamber, with their electrical resistance being continuously monitored as the temperature incrementally increased from 290 K to 310 K in 5 K steps.The α TCR for each sample was obtained from a linear fitting of the resistance-temperature data (see figure A1 in appendix B for further details).
M4PP measurements were performed using a modified CAPRES A301 microRSP ® tool equipped with a digital lockin amplifier [18], allowing the extraction of R 3ω .A sinusoidal current at a low frequency (3.01 Hz) was used, and the current amplitude was stepped from I RMS = 2 mA to I RMS = 5 mA in all measurements.The use of current steps enables to monitor the linearity of R 3ω with current squared and eventually can be used to reduce measurement error.The M4PP used had eight equidistant collinear electrodes with a separation of 4 µm so that it was possible to perform equidistant four-point measurements with a pitch of s = 4 µm and s = 8 µm at each probe engage with the thin-film surface.All measurements were performed at atmospheric pressure and at room temperature.

Results and discussion
The measured R 3ω in a representative engage as function of current in configurations A ′ and A and pitches s = 4 µm and s = 8 µm are shown in figure 3. Current sweeps from 2 mA to 5 mA exhibited strong linearity of R 3ω with I RMS 2 , and were easily reproducible for all the four combinations of pitch and configuration by the FEM model using a single pair of (α TCR , G TBC ).In order to reduce measurement error, the individual data of each current sweep was fitted using R 3ω = aI RMS 2 to obtain a current-independent slope a; the FEM model sought to reproduce only this slope (and not the entire dataset), as realized at an arbitrary I RMS = 5 mA.Such a treatment resulted in largely indistinguishable estimates of α TCR and G TBC (as compared to the fitting of all data points), but further reduced their scatter, while also considerably decreasing the FEM runtime by about a tenfold.The parameters α TCR and G TBC were fitted simultaneously in COMSOL Multiphysics 6.1 with MATLAB to the four slopes (represented by R 3ω values regressed at I RMS = 5 mA), by minimizing the total misfit, ε T = ∑ ε i , where i represents each of the four independent measurements, and ε i is given by equation (1). Figure 4 shows the best-fit values of α TCR and G TBC with their respective uncertainties for 30 probe engages with a step size of 1 µm (color symbols), alongside the independent reference measurements of α TCR performed by PPMS (dashed lines in figure 4(a)), and the expected intrinsic specific thermal conductance of the SiO 2 layer alone (dashed line in figure 4(b)), considering a thickness of 90 nm and thermal conductivity of 1.33 W m K −1 [22].All engages were treated independently, and they did not consider thermoelectric effects (see appendix C for proof of this contribution being negligible).The means of the best-fitted α TCR and G TBC were 542 ± 18 ppm K −1 and 15.6 ± 1.3 MW m −2 K −1 for the 3.3 nm Ru thin film, and 982 ± 46 ppm K −1 and 19.3 ± 2.3 MW m −2 K −1 for the 5.2 nm Ru thin film.Although the microstructure of the thin films may affect the variation of α TCR with thickness in different ways [23], a lower α TCR for thinner thin films could be explained due to additional scattering processes at the surfaces of the film.In brief, when temperature increases, the mean free path of electrons decreases, reducing the relative contribution of the surface scattering to the phonon scattering [24].
The mean α TCR of the thinnest Ru sample is ≈9% lower than the PPMS reference value, although the reason could be an overestimation of the electrical resistance measured by PPMS due to the use of Ag paint [25].The mean α TCR of the 5.2 nm Ru thin film is ≈5% larger than the reference measurement, and most probe engages (≈2/3) overlap with the PPMS value when the fitting uncertainty is considered.Assuming the thermal boundary conductance is purely determined by the SiO 2 layer (neglecting the thermal interface resistances), the mean G TBC value obtained for the 3.3 nm Ru thin film corresponds to a SiO 2 thermal conductivity of 1.40 ± 0.12 W m K −1 .The thermal conductivity was calculated as κ SiO2 = G TBC h SiO2 , where κ SiO2 and h SiO2 are the thermal conductivity and thickness (90 nm) of the SiO 2 layer, respectively.This compares well with the intrinsic thermal conductivity of thin SiO 2 , 1.33 W m K −1 [22], indicating a negligible influence of the interface resistances.G TBC extracted from the 5.2 nm Ru thin film results in a thermal conductivity of 1.74 ± 0.21 W m K −1 .This value is unexpectedly larger than the 3.3 nm Ru thin film (since the thickness of the SiO 2 layer is the same in both samples), and the reason could be an overestimation of the contact radii, which mainly depends on the calibration measurement [15].The calibration measurement has a stronger influence on thicker thin films due to their lower sheet resistance (R S,0 ≈ 170 Ω and R S,0 ≈ 60 Ω for the 3.3 nm and 5.2 nm Ru thin films, respectively).The fitting uncertainty of both parameters (α TCR and G TBC ) is also systematically larger for the 5.2 nm thin film, which is likely due to weaker R 3ω signals also produced by the lower R S,0 of this thin film.
Finally, a sensitivity analysis in the form of Monte Carlo simulations was performed for both thin films as can be seen in figure 5. First, a FEM simulation for each sample was computed for the same pitches (s = 4 µm and s = 8 µm), configurations (A ′ and A) and current (I RMS = 5 mA) as the experimental measurements.For the 3.3 nm Ru thin film R S,0 = 170 Ω, α TCR = 550 ppm K −1 , and a thermal conductivity of 12.7 W m K −1 , while for the 5.2 nm Ru thin film R S,0 = 60 Ω, α TCR = 950 ppm K −1 , and a thermal conductivity  of 22.9 W m K −1 were used.The remaining parameters were identical for both samples and similar to the expected values (90 nm of κ SiO2 = 1.4 W m K −1 for the SiO 2 layer with negligible thermal interface resistances, a thermal conductivity of 130 W m K −1 for the Si substrate, and a probe contact radii of 130 nm).The simulations provided four values of R 3ω for each thin film (corresponding to all combinations of the 2 pitches and the 2 configurations).Then, a set of 1000 independent measurements were artificially generated by adding a normally distributed electrical noise with a standard deviation of 1% on the R 3ω values.Each of the 1000 artificial measurements (consisting of 4 values of R 3ω ) were numerically fitted to obtain simultaneously α TCR and κ SiO2 , as shown in figure 5.As expected, the mean values of α TCR were 550 ppm K −1 and 950 ppm K −1 for the 3.3 nm and 5.2 nm Ru thin films, respectively, while the mean value of κ SiO2 was 1.4 W m K −1 in both cases.The standard deviation of the fitted values for both samples was 1.1% for α TCR and 2.3% for κ SiO2 , indicating a significantly higher sensitivity of the method for determining α TCR compared to G TBC , which is in agreement with the experimental data.Figure 5 also shows a stronger covariation between α TCR and G TBC than experimentally observed, and a possible reason could be an experimental covariation between the electrical noise of different configurations during the same engage.It also seems that the experimental electrical noise is lower than the 1% simulated.

Conclusions
By performing M4PP measurements with different electrode configurations and pitch, the α TCR and the G TBC were determined simultaneously.Here, we have characterized two Ru thin films (3.3 nm and 5.2 nm thick) on 90 nm of SiO 2 deposited on Si.The system was modeled using the FEM (COMSOL Multiphysics 6.1).The α TCR and the G TBC mean values obtained for 30 independent measurements on each sample were 542 ± 18 ppm K −1 and 15.6 ± 1.3 MW m −2 K −1 for the 3.3 nm Ru thin film as well as 982 ± 46 ppm K −1 and 19.3 ± 2.3 MW m −2 K −1 for the 5.2 nm Ru thin film.The mean G TBC of the thinnest sample, assumed to be the most accurate, corresponds to a SiO 2 thermal conductivity of 1.40 ± 0.12 W m K −1 .The values of the α TCR were found to be in good agreement with a PPMS, while the G TBC values also agreed with literature.This study shows the potential of M4PP as a powerful tool to perform local, fast, accurate α TCR and G TBC measurements of metallic thin films in multilayered stacks.By optimal choice of a metallic thin film with large sheet resistance and α TCR , this new M4PP method can be optimized for accurate evaluation of the effective thermal conductivity of individual layers in a multilayered stack.

Figure 1 .
Figure 1.Schematic view of the M4PP setup on a multilayered system consisting of an electrically conductive Ru thin film deposited on SiO 2 with Si as substrate.The (a) electric potential and (b) temperature when a constant current is applied/extracted at the outer electrodes were obtained using COMSOL Multiphysics 6.1.(c) Simulated waveforms for the extreme case of R 3ω /R 1ω = 0.1, the actual experimental ratios are several orders of magnitude lower.

Figure 2 .
Figure 2. FEM simulations of the third harmonic resistance for different pitches in (a) A ′ and (b) A configurations.Different combinations of temperature coefficient of resistance and thermal boundary conductance are plotted.All simulations were performed for a 3.3 nm Ru thin film (R S,0 = 171 Ω and thermal conductivity of 12.7 W m K −1 ) deposited on 90 nm of SiO 2 (1.5 W m K −1 ) with Si (130 W m K −1 ) as substrate, using I RMS = 5 mA, α TCR = 542 ppm K −1 , and considering a probe contact radii of 130 nm.The inset is a schematic of the configuration used.

Figure 3 .
Figure 3.The empty symbols represent the measured third harmonic resistance at different currents for a single engage of the 3.3 nm Ru thin film, while the lines show their respective best fits using R 3ω = aI RMS 2 .

Figure 4 .
Figure 4. (a) Temperature coefficient of resistance of the Ru thin films and (b) total thermal boundary conductance between the Ru and its Si substrate with their respective fitting errors for the two samples studied in this work.The dashed lines in (a) indicate the reference values given by PPMS, while the dashed line in (b) represents the intrinsic specific thermal conductance of 90 nm of a material with a thermal conductivity of 1.33 W m K −1 .

Figure 5 .
Figure 5. Output of the Monte Carlo simulations for the (a) 3.3 Ru and (b) 5.2 Ru thin films.For each thin film, 1000 independent fittings to artificially generated data with 1% normally distributed electrical noise in R 3ω were performed.All fittings were executed at the same pitches and configurations as measured experimentally (s = 4 µm and s = 8 µm in A ′ and A).The 3.3 nm Ru thin film (R S,0 = 170 Ω, α TCR = 550 ppm K −1 , and thermal conductivity of 12.7 W m K −1 ) and the 5.2 Ru thin film (R S,0 = 60 Ω, α TCR = 950 ppm K −1 , and thermal conductivity of 22.9 W m K −1 ) were considered in perfect contact with the 90 nm of SiO 2 (1.4 W m K −1 ) on the Si substrate (130 W m K −1 ).A current I RMS = 5 mA, and a probe contact radii of 130 nm were used.

Table 1 .
Summary of the steps followed in this work to determine the temperature coefficient of resistance of a metallic thin film and the total thermal boundary conductance between the thin film and its substrate.