A triple-scan scanning thermal microscopy (SThM) method and a zero-heat flux laser-heated SThM technique are investigated for quantitative thermal imaging of flexible graphene devices. A similar local tip-sample thermal resistance is observed on both the graphene and metal areas of the sample, and is attributed to the presence of a polymer residue layer on the sample surface and a liquid meniscus at the tip-sample junction. In addition, it is found that the tip-sample thermal resistance is insensitive to the temperature until it begins to increase as the temperature increases to 80 °C and exhibits an abrupt increase at 110 °C because of evaporation of the liquid meniscus at the tip-sample junction. Moreover, the variation in the tip-sample thermal resistance due to surface roughness is within the experimental tolerance except at areas with roughness height exceeding tens of nanometers. Because of the low thermal conductivity of the flexible polyimide substrate, the SThM measurements have found that the temperature rise in flexible graphene devices is more than one order of magnitude higher than those reported for graphene devices fabricated on a silicon substrate with comparable dimensions and power density. Unlike a graphene device on a silicon substrate where the majority of the electrical heating in the graphene device is conducted vertically through the thin silicon dioxide dielectric layer to the high-thermal conductivity silicon substrate, lateral heat spreading is important in the flexible graphene devices, as shown by the observed decrease in the average temperature rise normalized by the power density with decreasing graphene channel length from about 30 μm to 10 μm. However, it is shown by numerical heat transfer analysis that this trend is mainly caused by the size scaling of the thermal spreading resistance of the polymer substrate instead of lateral heat spreading by the graphene. In addition, thermoelectric effects are found to be negligible compared to Joule heating in the flexible graphene devices measured in this work.
I. INTRODUCTION
Besides high electron mobility, the transparency, stretchability, and elasticity of graphene have attracted significant interests in the application of this two-dimensional material for flexible electronics.1–5 Major advances have been made recently in chemical vapor deposition (CVD) of large-area graphene on a metal catalyst layer, and in transfer of the CVD graphene film from the metal layer to a flexible substrate.1,2,6,7 These advances have enabled the fabrication of flexible radio-frequency (RF) graphene devices with charge carrier mobility values comparable to those of exfoliated graphene on a Si substrate.8–11 In addition, graphene transparent conductors have achieved optical transparency and conductivity comparable to those of indium tin oxide (ITO) films, which are commonly used in optoelectronic devices, and offer significant improvement in the mechanical flexibility, stretchability,1,2 and cost.3,5,12
As rapid progresses are made in flexible graphene electronics, thermomechanical reliability issues have emerged and are becoming as critical as in silicon nanoelectronic devices, where the high power density results in local hot spots and consequently reduced performance and thermomechanical failures.13–16 In particular, in graphene electronic devices fabricated on a flexible and transparent polyimide substrate, device failures have been observed when the power density is increased to a level that was found to be safe for graphene devices fabricated on a silicon substrate.4 It is suspected that the very low thermal conductivity of the polyimide substrate results in local hot spots exceeding the glass transition temperature of the polyimide substrate even at a low or moderate power density. However, there has been a lack of knowledge on the temperature distribution in these flexible graphene electronic devices. Direct measurements of the temperature distribution of flexible graphene devices can be useful for rational thermal design of flexible graphene electronic devices and developing thermal management solutions of such devices, where theoretical prediction of the thermal response becomes challenging because of a complicated structure and unknown properties such as interface thermal resistances, anisotropic thermal properties of the substrate, and the effects of the substrate on electronic and thermal transport in graphene.17,18
Scanning thermal microscopy (SThM) techniques19 have been developed for direct probing of the temperature distribution in nanoelectronic devices with superior lateral spatial resolutions compared to optical thermometry methods such as micro-Raman thermometry,20–22 optical emission thermometry,21 and infrared thermal microscopy,22,23 where a diffraction-limited spatial resolution of ∼1 μm or larger prevents the observation of thermal phenomena occurring in smaller length scales. In addition, SThM techniques can provide much better temperature resolution and allow for investigation of temperature distribution in operating devices at lower power densities than those for Raman thermometry.24 In one of the SThM methods, a miniature thermocouple located at the apex of an atomic force microscopy (AFM) tip is scanned over the surface of the device. The measured thermovoltage map can be influenced by a number of factors that affect the tip-sample heat transfer, such as the sample topography and surface chemistry. Moreover, parasitic heat transfer through the air gap between the probe and the device may considerably distort the measured thermovoltage for measurements in ambient condition.25 In recent years, a double-scan or triple-scan technique26,27 and a null-point measurement approach28,29 have been developed to address these challenges in order to obtain quantitative thermal imaging results. The double-scan method has been used to obtain the temperature distribution in graphene devices fabricated on a silicon substrate with a spatial resolution close to 100 nm.24
In addition to these advances made in thermocouple SThM technique, progresses have also been made in Scanning Joule Expansion Microscopy (SJEM)30 and resistive thermometer and heater tip31 for quantitative SThM. SJEM operates by scanning an AFM tip in contact with the surface of a device under periodical heating to obtain the temperature distribution from the measured thermomechanical expansion distribution of the device, which is often coated with a thin polymer sensing layer with a much larger thermal expansion coefficient than the substrate.30 SJEM measurements have been used to study Joule heating and thermoelectric phenomena at the graphene-metal contacts and resistive heating at the graphene wrinkles and grain boundaries.32,33 In comparison, heated silicon tips have been used to measure the tip-sample thermal resistance distribution and surface temperature distribution of nanowire and graphene samples.31,34
Despite these recent progresses in different methods for quantitative SThM, further research is needed to establish a better understanding of several outstanding questions associated with each of these SThM methods. In terms of the triple-scan SThM method, an important assumption is that the tip-sample thermal resistance is constant and can be calibrated. This assumption is not needed for the proposed zero-heat flux SThM measurement in vacuum with active feedback control of the tip heating rate.35 However, the reported null-point measurements have been based on the linear extrapolation of the SThM measurement results under a non-zero heat flux to find the zero heat flux condition.28,29 The linear extrapolation is rigorous only if the tip-sample thermal resistance is constant when the tip temperature is varied. Therefore, it is necessary to develop a better understanding of the influence of the sample properties, temperature, and surface roughness on the tip-sample thermal resistance.
In this article, we report a study of the variation of the tip-sample thermal resistance in triple-scan SThM measurements of the temperature distribution in electrically biased graphene devices supported on flexible polyimide substrates. The triple-scan SThM measurement is cross validated with a zero-heat flux SThM measurement based on variable laser heating of the thermal probe. Large parasitic non-local heat conduction through the air gap between the sample and the SThM probe is found in SThM measurements of the flexible devices, which represent an interesting challenge for testing the assumptions and applicability of these quantitative SThM measurements. The measured temperature rise in the flexible graphene devices is more than one order of magnitude higher than that reported for graphene devices fabricated on a SiO2/Si substrate with similar lateral dimensions and power density. In addition, lateral heat spreading is found to be more important in these flexible graphene devices than those on the silicon substrate, mainly because of the size effect on the thermal spreading resistance of the flexible substrate instead of heat spreading by the graphene layer. Compared to Joule heating, thermoelectric effects are negligible in the devices measured in this work according to numerical analysis of the experiments. These findings can be useful for improving the thermal design of flexible graphene devices. Meanwhile, the observed variation of the tip-sample thermal resistance at different temperatures and on different sample materials and surface roughness can be valuable for the further development of quantitative SThM methods.
II. EXPERIMENTAL METHODS
A. Flexible graphene devices
The flexible graphene devices consist of graphene channels contacted by metal electrodes on a flexible polyimide substrate, as shown in Figure 1. During the fabrication process, liquid polyimide is spun over a 125 μm thick polyimide sheet and cured at 300 °C to achieve the root-mean-square (RMS) roughness of about 1 nm or better. The process is repeated for the other side of the polyimide sheet to prevent bending of the substrate via thermal stress in the curing process. Atomic layer deposition (ALD) is employed to deposit 20 nm of Al2O3 on the substrate. Single layer graphene, grown on copper foil by CVD, is transferred onto the Al2O3/polyimide substrate via poly(methyl methacrylate) (PMMA) assisted wet transfer process.6 Electron-beam lithography (EBL) followed by oxygen plasma is employed to define the active graphene channels. Another EBL step followed by e-beam evaporation of 2 nm Ti and 40 nm Au is used to define the source and drain metal contacts.
Optical micrograph of a flexible graphene device on a polyimide substrate. The graphene channels are highlighted by the false blue color. The channel width for the graphene channels and metal fingers is 10 μm.
Optical micrograph of a flexible graphene device on a polyimide substrate. The graphene channels are highlighted by the false blue color. The channel width for the graphene channels and metal fingers is 10 μm.
B. Triple-scan SThM
The SThM probe is a custom-made AFM probe with a V-shaped SiNx cantilever and a SiO2 tip.36 A sub-micron Pt-Cr thermocouple is fabricated at the apex of the SiO2 tip. The tip radius is typically less than 50 nm. The thermopower of the Pt-Cr junction has been measured to be 13.4 μV/K.25 A thin native oxide layer of Cr electrically insulates the thermocouple junction from the surface of the device. In the current experiments, the electrical insulation is further improved by ALD of 20 nm Al2O3 on the probe at 250 °C using trimethylaluminum (TMA) and water as the precursors. Before SThM measurements, the effectiveness of the insulating layer is verified by measuring the electrical resistance between the tip and a conducting sample. Furthermore, raising the electrostatic potential of the sample without any electrical current flow in the graphene sample does not change the measured thermovoltage of the thermocouple.
The SThM measurements are conducted under ambient conditions due to the concern that the polymeric substrate may outgas and contaminate an ultrahigh vacuum (UHV) SThM setup. In ambient SThM measurements, a liquid meniscus can form at the tip-sample junction by condensation of water molecules and other adsorbates on the surface of the tip and the sample. The liquid meniscus can increase the local tip-sample thermal conductance and improve the temperature sensitivity compared to UHV SThM measurements, where such liquid meniscus is expected to be absent. However, the heat conduction through the tip-sample air gap generates a non-local parasitic signal which prevents direct interpretation of the thermocouple temperature.25
Figure 2 shows the cantilever deflection and thermovoltage of the SThM probe when a Joule-heated graphene device or an Au line on the polyimide substrate approaches or retracts from the probe. The thermovoltage signal can be converted to the thermocouple temperature rise using the measured thermopower of the Pt-Cr junction. Before the tip makes contact to the sample, the thermocouple reading is already large, indicating significant non-local parasitic heat transfer between the probe and the sample. As the graphene sample approaches the tip, the thermocouple reading increases nearly linearly. Once the sample is in close proximity of about a few nanometers to the tip, the tip jumps to contact the surface due to the capillary force. This phenomenon causes a jump in the thermocouple temperature from point C to point D in Figure 2(a). Given the observed rate of change of air conduction signal with the distance and the mean free path of air molecules, which is ∼68 nm under ambient pressure and room temperature,37 this jump can only be attributed to the local heat transfer through the solid-solid contact as well as the liquid meniscus formed at the tip-sample junction. The length scale of this local heat transfer phenomenon is comparable to the size of the liquid meniscus, which is on the order of the tip-sample contact size.19
Cantilever deflection and thermovoltage signals of the SThM probe as a function of the sample position when the sample approaches (blue curve) or retracts (red curve) from the probe. The samples are a graphene channel in (a) and an Au line in (b). Points A and B denote the measured thermovoltage signal when the tip is lifted by 400 nm (h2) and 100 nm (h1) from point D′, respectively. Points C and D denote the measured thermovoltage signals immediately before and after the jump-to-contact occurs. Point D′ is the position set for the contact mode SThM measurement.
Cantilever deflection and thermovoltage signals of the SThM probe as a function of the sample position when the sample approaches (blue curve) or retracts (red curve) from the probe. The samples are a graphene channel in (a) and an Au line in (b). Points A and B denote the measured thermovoltage signal when the tip is lifted by 400 nm (h2) and 100 nm (h1) from point D′, respectively. Points C and D denote the measured thermovoltage signals immediately before and after the jump-to-contact occurs. Point D′ is the position set for the contact mode SThM measurement.
The observed non-local tip-sample heat transfer is mainly from heat conduction through the tip-sample air gap. In comparison, far-field and near-field radiative heat transfers have been calculated to be negligible compared to the air heat conduction when the tip and the sample are close to room temperature.19 The observed non-local tip-sample heat transfer is more pronounced than those found in a prior measurement of graphene devices supported on a silicon substrate.24 As it has been found that the non-local parasitic heat transfer increases with increasing heated area of the sample,25 this comparison indicates that the heated area is larger for the flexible graphene device than the graphene-on-silicon device, although the lateral dimensions are comparable for the two different types of graphene devices.
Double-scan26 and triple-scan27 techniques have been reported to eliminate the effect of parasitic air conduction in recent works. In these methods, the local sample temperature rise () is obtained by measuring the temperature rise of the thermocouple in the contact mode () and a non-thermal contact mode (). In the contact mode where the tip is in contact with the surface of the device, there is heat transfer across the tip-surface contact. In comparison, the non-thermal contact mode represents the situation where the local heat transfer through the solid-solid and liquid film conduction is absent, namely, , even though the cantilever-sample spacing remains the same as that in the contact mode. It has been derived in a recent report that the local surface temperature rise of the device can be obtained via the following relationship27
where is the thermal resistance at the local tip-sample contact and is a constant with units of K W−1. The constant depends on the probe dimensions and material properties, and effective local heat transfer coefficient between the probe and its surroundings including the sample. If can be determined from a calibration and remains constant, can be obtained by measuring and .
When the tip scans the sample during the triple-scan SThM measurement, can be measured at point D′ near point D where the tip jumps to contact the sample, because the thermocouple reading remains nearly the same during the tip-sample contact at the small contact force. Measurement of requires additional efforts since the cantilever-sample distance should remain the same as that at point D, yet the local tip-sample heat transfer needs to be absent. Although this condition is nearly met at point C, the tip cannot be held in this state during scanning the sample. Some prior studies have used the tip temperature when the probe is lifted at a fixed height above the sample to obtain .24,26 However, Figure 2 shows that this assumption can lead to a considerable error when the parasitic heat transfer increases rather rapidly with decreasing tip-sample distance. This error can be reduced with a triple-scan technique reported in a recent work.27 In this method, the thermovoltage signals measured at two different lift heights, points A and B in Figure 2, are used for a linear extrapolation to find the thermovoltage signal at point C, which is used to obtain .
We investigate the effectiveness of the triple-scan method by calibrating it on a 550 μm long, 10 μm wide Au line heater and thermometer. For this calibration sample, is obtained from the change in the electrical resistance of the Au line measured as a function of the electrical current in addition to the measured temperature coefficient of resistance (TCR) of the Au line. Figure 3(a) shows the analysis of the results of measurements determined from double-scan, triple-scan, and quadruple-scan measurements. For quadruple-scan measurements, the measured thermovoltage at three lift heights is used for a linear extrapolation to point C. According to Equation (1), the calibration factor can be determined from the slope of a plot of versus . According to Figure 3(a), the value obtained from double-scan measurements depends on the lift height, which is indicative of the presence of large parasitic air conduction signal in the double-scan measurements of flexible devices with a large heated area. In comparison, the triple-scan and quadruple-scan measurements find similar values of 110 ± 5, within the uncertainty range of the measurements. This finding suggests that the triple-scan approach is effective in eliminating the parasitic signal induced by the air conduction.
(a) Calibration measurement results obtained on an Au heater and thermometer line based on double-scan measurements with lift heights of 400 nm (filled circles), 200 nm (filled triangles), and 100 nm (filled diamonds); triple-scan measurements with lift heights of 100 nm and 200 nm (unfilled diamonds) and 100 nm and 400 nm (unfilled triangles); and quadruple-scan measurements with lift heights 100 nm, 200 nm, and 400 nm (unfilled circles). (b) The measured tip-sample thermal resistance normalized by its value near room temperature as a function of the average temperature rise in the thermocouple and the metal line for two Au line samples. (c) The measured variation of the calibration factor along a line scan over the Au line with the temperature rise of 35 K (blue curve), 46 K (green curve), and 72 K (red curve). The corresponding height profile (black curve) and the average calibration factor (gray dashed line) are also shown. Also shown is the measured temperature rise from triple-scan SThM using the average calibration factor (dark red curve) and resistance thermometry (dotted black line) for the case of 72 K temperature rise.
(a) Calibration measurement results obtained on an Au heater and thermometer line based on double-scan measurements with lift heights of 400 nm (filled circles), 200 nm (filled triangles), and 100 nm (filled diamonds); triple-scan measurements with lift heights of 100 nm and 200 nm (unfilled diamonds) and 100 nm and 400 nm (unfilled triangles); and quadruple-scan measurements with lift heights 100 nm, 200 nm, and 400 nm (unfilled circles). (b) The measured tip-sample thermal resistance normalized by its value near room temperature as a function of the average temperature rise in the thermocouple and the metal line for two Au line samples. (c) The measured variation of the calibration factor along a line scan over the Au line with the temperature rise of 35 K (blue curve), 46 K (green curve), and 72 K (red curve). The corresponding height profile (black curve) and the average calibration factor (gray dashed line) are also shown. Also shown is the measured temperature rise from triple-scan SThM using the average calibration factor (dark red curve) and resistance thermometry (dotted black line) for the case of 72 K temperature rise.
C. Variation of tip-sample thermal resistance
The parameter needs to be invariant in order to allow quantitative SThM measurements with the triple-scan technique. The parameter is defined as , where can be assumed to be constant for a specific probe.27 In this work, we have investigated the dependence of and thus on the contact force, temperature, sample properties, and surface roughness. The contact area of a sphere and a plane under a normal force is expected to increase by .38 However, Figure 2 shows that increasing the tip-sample contact force does not change appreciably the temperature of the thermocouple within the small range of the contact force tested. Similar measurements at higher temperatures show similar insensitivity of the measured temperature to the contact force. The observed insensitivity on the contact force can be attributed to a dominant local heat transfer pathway through the liquid meniscus around the tip compared to the solid-solid conduction. In addition, the calibration parameter measured before and after SThM mapping is nearly the same and has not changed due to tip wear. This finding can be attributed to the insensitivity of the liquid meniscus to minor changes of the tip shape.
The important contribution from the liquid meniscus is shown clearly in our additional investigation of the effect of temperature on tip-sample heat transfer. With the use of the Au line sample, we have measured , which is proportional to , as a function of the average temperature of the liquid meniscus. Figure 3(b) shows that the obtained tip-sample thermal resistance between the SThM tip and the Au line () does not change noticeably until the average temperature reaches about 60 K, upon which there appears to be a small increase of with increasing average temperature rise up to about 85 K. An apparent jump in occurs at the interval of 85−95 K temperature rise, beyond which increases appreciably with increasing temperature. The observed jump can be attributed to the evaporation of the liquid meniscus. The result indicates that the temperature rise needs to be lower than about 60 K for assuming a constant in triple-scan SThM measurements.
In addition, we have investigated the effect of surface roughness on the calibration factor by measuring the local over the Au calibration line. The results, shown in Figure 3(c), show that the surface roughness of less than few nanometers introduces a fluctuation to the local calibration factor, and that the fluctuation is much smaller than the average value of the calibration factor, . When two rough surfaces come into contact, the actual solid-solid contact area is smaller than the apparent contact area and voids are formed at the interface. However, in ambient conditions, these voids are filled with water or other liquids due to capillary condensation.39 Consequently, the tip-sample heat transfer is insensitive to the surface roughness compared to the effect on tip-sample solid-solid conduction when the surface roughness is less than a few nanometers. In comparison, at a hill feature with a relatively large roughness height of 25 nm, the measured is considerably larger on the peak of the hill and smaller on the valley. The result indicates larger and smaller at the peak and the valley, respectively, based on Equation (1). This finding is consistent with the literature.19 Therefore, the obtained on areas with more than tens of nanometers roughness can contain apparent topologic artifacts. In comparison, the artifacts introduced to the obtained surface temperature using are reduced compared to that in the measured , even at the rough hill feature as shown in Figure 3(c), to produce a value that is relatively insensitive to the roughness.
Furthermore, we have investigated whether the calibration factor can change as the tip moves from the Au surface to an adjacent graphene surface. This change can be due to the difference in the material and surface properties, which may modify the tip-sample thermal resistance, , and the thermal spreading resistance of the sample, . We performed calibration measurements at the edge of the Au heater-thermometer line where the line crosses over a graphene sample. Because of the approximately cylindrical heat dissipation pathway from the electrically heated Au line with 10 μm width through the relatively thick polyimide substrate with a low thermal conductivity, the temperature of the graphene at a location few hundreds of nanometers away from the edge of the metal line can be calculated to be close to the temperature of the metal line. If the graphene temperature is assumed to be the same as that of the Au line, the obtained calibration factor for graphene () is found to be similar to that for Au () despite their different surface properties. It is worth noting that graphene is supported on a layer of ALD-deposited Al2O3 with a thickness of 20 nm, which can affect the contact angle of the liquid meniscus with graphene.40,41 In addition, a layer of 1–2 nm thick residual PMMA can be present on the graphene surface.42,43 PMMA residue is also likely present on the Au surface. The similar surface condition can result in a similar size of the liquid meniscus. This possibility is confirmed by additional measurements of the capillary force between the SThM probe and the samples. As the sample is moving out of contact with the cantilever, the cantilever deflects to a negative value due to the capillary force until a point where the cantilever snaps out of the contact, as shown in Figure 2. Near room temperature, the maximum negative deflection signal of the cantilever by the capillary force is 633 mV and 607 mV for the Au surface and the graphene surface, respectively. From this measurement, we conclude that the liquid meniscus contact angles on the graphene/Al2O3 and Au surfaces are similar.
Although the liquid meniscus size and thermal conductance can be similar on both the Au and graphene surfaces, is a function of thermal properties of the sample and is not expected to be uniform on the different areas of the sample. Using three-dimensional numerical analysis of heat transfer in the measurement device, we found that the effect of variation of on the measured temperature is negligible and the tip-sample heat transfer imposes only a small perturbation on the local temperature of the sample, which can be explained by the following calculations. The thermal spreading resistance in a layered orthotropic compound can be calculated using Dryden formula44 with modifications for anisotropy of the thermal properties,45 as explained elsewhere.34 The value of for single-layer graphene can be assumed to be similar to that of bilayer graphene.34 Using the data collected by Menges et al.34 for bilayer graphene, a tip-sample contact radius of 50 nm, cross-plane and in-plane thermal conductivity of 300 and 150 W m−1 K−1 for Au with 40 nm thickness, cross-plane and in-plane thermal conductivity of 0.2 and 0.6 W m−1 K−1 for polyimide, and the interface resistance of 2 10−8 m2 K W−1 for Au-polyimide and graphene-polyimide interfaces,46 the thermal spreading resistance for Au () and graphene () surfaces is obtained to be 2.5 104 and 1.1 106 K W−1, respectively. While and are different, both of them are negligible compared to . For a SThM probe similar to the SThM probe employed in this work, is found to be 1.5 108 K W−1.25 Therefore, the calibration parameter is dominated by the much larger and is insensitive to the variation in the much smaller .
D. Zero-heat flux laser-heated SThM
In order to provide a cross validation of the triple-scan SThM method, we have measured the same sample with a variation of the zero-heat flux method reported in the literature.28,35 In our implementation, we vary the power of the laser beam that is used to measure the AFM cantilever deflection. Here, we have used laser heating instead of electrical heating of the thermocouple to avoid contamination of the thermocouple voltage by the electrical heating current, and to confine the heating to the end of the cantilever. The confined heating can help to reduce indirect heating of the sample via parasitic heat transfer between the cantilever and the sample, which can still occur for the measurements conducted in air even when the tip-sample heat transfer is nullified in the zero-heat flux measurement. At each laser heating rate of the cantilever, the thermovoltage change due to jump to contact is obtained from the measured approach curve as shown in Figure 2. Increasing the temperature of the tip by laser heating decreases the temperature jump, as shown in Figure 4 for measurements made at the center of an electrically heated 10 × 10 μm2 graphene channel on polyimide. A linear extrapolation of the measurement data yields that the thermovoltage jump is zero when the thermovoltage in contact mode is about 396 μV, corresponding to of about 30 K. The zero jump indicates the same and values, which further implies the same values for and , regardless of the value for , according to Equation (1). However, Equation (1) also indicates that a constant or is required for assuming a linear relationship between the thermovoltage jump and . Hence, the linear extrapolation for finding is rigorous only when the temperature rise is lower than 60 K, where evaporation of the liquid meniscus starts. The obtained of about 30 K satisfies this condition and is close to the 29 ± 1 K value obtained from the triple-scan measurement, which finds a similar calibration parameter for the Au and graphene surface.
Measured thermovoltage jump as a function of measured thermovoltage reading in the contact mode for different laser heating powers incident on the SThM probe. The measurement is performed at the center of a 10 × 10 μm2 graphene channel with a power dissipation density of 895 W cm−2. The red dashed line shows the linear extrapolation of the data for finding the thermovoltage of the thermocouple that corresponds to zero thermovoltage jump. At this point, the local temperature of the sample is equal to temperature of the thermocouple.
Measured thermovoltage jump as a function of measured thermovoltage reading in the contact mode for different laser heating powers incident on the SThM probe. The measurement is performed at the center of a 10 × 10 μm2 graphene channel with a power dissipation density of 895 W cm−2. The red dashed line shows the linear extrapolation of the data for finding the thermovoltage of the thermocouple that corresponds to zero thermovoltage jump. At this point, the local temperature of the sample is equal to temperature of the thermocouple.
III. THERMAL IMAGING RESULTS OF FLEXIBLE GRAPHENE DEVICES
The SThM results for four flexible graphene devices are discussed in this section. The channel width for all of the devices is 10 μm, and the channel lengths are 10 μm, 10 μm, 25 μm, and 30 μm. Before SThM measurements, the graphene channels and metal lines are annealed by passing a DC current. The applied electrical bias is increased and decreased repeatedly until there is no more change in the obtained IV curves.
Figure 5 shows the SThM measurement results for a 25 × 10 μm2 graphene channel with a power dissipation density of 1276 W cm−2. The thermovoltage maps in the contact and lift modes and their difference are shown. The lateral extent of the heated area in the flexible device is larger than that found in graphene devices on a Si substrate.22–24 In particular, large temperature rises occur on the exposed polyimide surface surrounding the graphene channel. For graphene devices fabricated on a SiO2/Si substrate, in comparison, the large thermal conductivity of the Si substrate below the 300 nm top SiO2 layer effectively sets the SiO2/Si interface temperature to be close to the ambient temperature. Because of this effect, the heated area is confined in the electrically heated graphene and does not spread much laterally along the exposed SiO2 surface.
SThM measurement results for a 25 × 10 μm2 graphene channel with a power dissipation density of 1276 W cm−2. Topography (a), thermovoltage map in contact mode (b), and thermovoltage map when the tip is lifted 1 μm above the device surface (c) are shown. The color bars are identical in panels (b) and (c). Panel (d) shows the difference between (b) and (c). In panel (a), the graphene channel is highlighted by the false blue color. A positive electrical bias is applied to the left metal line, and the right metal line is grounded.
SThM measurement results for a 25 × 10 μm2 graphene channel with a power dissipation density of 1276 W cm−2. Topography (a), thermovoltage map in contact mode (b), and thermovoltage map when the tip is lifted 1 μm above the device surface (c) are shown. The color bars are identical in panels (b) and (c). Panel (d) shows the difference between (b) and (c). In panel (a), the graphene channel is highlighted by the false blue color. A positive electrical bias is applied to the left metal line, and the right metal line is grounded.
Figure 6 shows the temperature profile along the centerline of two graphene channels for a range of power dissipation densities. The temperature rise for these flexible devices is much higher than the temperature rise in the graphene devices on SiO2/Si substrates with similar lateral dimensions and under a similar power density.20–24 The higher temperature in the flexible device is caused by the lower thermal conductivity of polyimide compared to SiO2 and Si. Because of the high thermal conductance of the metal electrodes, the temperature rise measured on top of the metal contacts is lower than that measured on the graphene channel. As the power density increases, the peak temperature moves from the middle of the channel toward the positively biased source contact.
Measured temperature profile along the centerline of the 10 × 10 μm2 (a) and 30 × 10 μm2 (b) graphene channels. The positively biased and grounded electrodes are shown by the shaded area on the left and right ends, respectively. The measured temperature rise increases with increasing power dissipation density of 0, 305, 497, 760, 1425, 1787, and 2044 W cm−2 for (a) and 0, 23, 141, 311, 398, and 650 W cm−2 for (b).
Measured temperature profile along the centerline of the 10 × 10 μm2 (a) and 30 × 10 μm2 (b) graphene channels. The positively biased and grounded electrodes are shown by the shaded area on the left and right ends, respectively. The measured temperature rise increases with increasing power dissipation density of 0, 305, 497, 760, 1425, 1787, and 2044 W cm−2 for (a) and 0, 23, 141, 311, 398, and 650 W cm−2 for (b).
The observed asymmetry can be caused by the spatial variation of charge carrier density22–24 as well as the Peltier effect32 at the graphene-Au contacts. To evaluate the impact of the Peltier effect, we performed three-dimensional finite element analyses of heat conduction in a device where a graphene channel is contacted by two 500 μm long Au lines atop 500 × 600 × 145 μm3 polyimide substrate, which is a simplified geometry compared to the measurement device that consists of multiple polyimide layers and additional electrodes. In the simulations, the in-plane and cross-plane thermal conductivities of the polyimide are assumed to be 0.6 and 0.2 W m−1 K−1, respectively. The thermal conductivity of graphene on polyimide is assumed to be 600 W m−1 K−1 based on a prior measurement of graphene on SiO2.47 Joule heating over the graphene channel is assumed to be uniform and equals the average value found in the measurement. The rate of Peltier heating or cooling at the graphene-Au junctions is calculated as where is the absolute temperature, and are the Seebeck coefficients of graphene47 and gold,48 respectively, and is the electrical current obtained from the measurements. is assumed to distribute uniformly at the graphene-Au contact areas. The simulation results for the 10 × 10 μm2 graphene channel are shown in Figure 7. The change in temperature due to Peltier effect is negligible at the lowest non-zero power density of the measurement results shown in Figure 6(a). Therefore, Peltier effect is not expected to be the main cause for the asymmetry in the measured temperature profiles shown in Figure 6. The impact of Peltier effect on the temperature distribution is expected to be even less important for longer channels and higher power densities, as shown in Figure 7. Therefore, the asymmetric hot spot observed in Figure 6 should be caused by spatial variation of the charge carrier density along the sample, instead of by thermoelectric effect at the contacts. Specifically, the measurement results suggest that the local carrier density and conductivity near the grounded drain electrode become higher than those near the source electrode when the positive bias applied to the source electrode is increased. This phenomenon can occur when the Fermi level is above the Dirac point so that the graphene channel is n-type.
Simulation of the temperature rise along centerline of a 10 × 10 μm2 graphene channel with uniform power dissipation density of 305 W cm−2 on a polyimide substrate with (red solid curve) and without (black dotted curve) including Peltier effect at graphene-Au contacts. The temperature rises are normalized by the maximum temperature in the graphene channel. The percentage changes in the temperature rise caused by Peltier effect are also shown for 305 W cm−2 (red dashed curve) and 2044 W cm−2 (blue dashed curve) power dissipation densities.
Simulation of the temperature rise along centerline of a 10 × 10 μm2 graphene channel with uniform power dissipation density of 305 W cm−2 on a polyimide substrate with (red solid curve) and without (black dotted curve) including Peltier effect at graphene-Au contacts. The temperature rises are normalized by the maximum temperature in the graphene channel. The percentage changes in the temperature rise caused by Peltier effect are also shown for 305 W cm−2 (red dashed curve) and 2044 W cm−2 (blue dashed curve) power dissipation densities.
Figure 8 shows the average resistance () for heat dissipation, defined as the ratio of the average temperature rise to the power dissipation density per unit lateral area of the graphene channel, in the flexible devices and graphene devices on Si substrates with the 300 nm thick SiO2. In the graphene/SiO2/Si devices, has been measured in prior works with different thermometry methods to be ∼3 K cm2 kW−1 irrespective of the channel length in the range of 1.5 and 7 μm,20–22,24 because the main pathway for heat dissipation is across the SiO2 layer toward the high thermal conductivity Si substrate. In comparison, for the flexible devices is found to increase by more than one order of magnitude, and increases with the channel length.
The average resistance for heat dissipation () as a function of the channel length for flexible graphene samples (filled squares). Also shown for comparison are the results for graphene samples on Si substrates covered with 300 nm SiO2 measured by SThM (unfilled square),24 infrared emission thermometry (unfilled diamond),22 and Raman thermometry (unfilled circles).20–22,24 The inset shows a comparison between the measured normalized by its value for the 30 μm long graphene channel (filled squares) and the simulation results for graphene channels on polyimide with (gray unfilled circles) and without (gray unfilled triangles) including the in-plane heat conduction in graphene.
The average resistance for heat dissipation () as a function of the channel length for flexible graphene samples (filled squares). Also shown for comparison are the results for graphene samples on Si substrates covered with 300 nm SiO2 measured by SThM (unfilled square),24 infrared emission thermometry (unfilled diamond),22 and Raman thermometry (unfilled circles).20–22,24 The inset shows a comparison between the measured normalized by its value for the 30 μm long graphene channel (filled squares) and the simulation results for graphene channels on polyimide with (gray unfilled circles) and without (gray unfilled triangles) including the in-plane heat conduction in graphene.
The size dependence of indicates different heat dissipation pathways in the flexible graphene devices. In a graphene/SiO2/Si device with a 10 μm × 10 μm lateral dimension of the graphene channel, the thermal resistance for vertical heat transfer through the 300 nm SiO2 dielectric can be calculated to be about 2100 K W−1, which is much larger than the 360 K W−1 thermal spreading resistance of the Si substrate, according to a calculation similar to that reported in a prior work.24 In comparison, the thermal resistance for lateral heat spreading in the graphene channel is on the order of 2.5 × 106 K W−1 based on a thermal conductivity of 600 W m−1 K−1.47 Therefore, lateral heat spreading in the graphene channel is negligible compared to vertical heat conduction through the SiO2 spacer, until the graphene channel length is reduced to well below 1 μm. The low thermal conductivity of polyimide in flexible devices can potentially make the contribution of graphene to heat dissipation more important. To investigate the effect of lateral heat spreading via graphene in flexible devices, we have used the aforementioned heat conduction simulations to calculate for different channel length and the thermal conductivity values. The results are shown in the inset of Figure 8 and agree well with the measurement results. By comparing the calculation results obtained with the basal-plane thermal conductivity of graphene assumed to be 600 and 0 W m−1 K−1, respectively, the contribution of graphene to lateral heat spreading is found to reduce by 5%, 7%, 14%, and 62% for channel lengths of 30, 20, 10, and 2 μm, respectively. According to the calculation results shown in inset of Figure 8, the size effect observed in Figure 8 is mainly caused by a different mechanism instead of heat spreading by the graphene. We note that the thermal spreading resistance from the electrically heated graphene channel into the polyimide substrate scales as 1/a, where a is the equivalent lateral dimension of the graphene channel. When this thermal spreading resistance is normalized by the graphene channel area, which is proportional to a2, the normalized resistance becomes proportional to a, similar to the approximately linear dependence of the measured normalized resistance on the channel length.
The observed lateral heat spreading in the flexible devices has significant implications for the design of such devices. For example, for the flexible device shown in Figure 5, the corresponding temperature rise on polyimide at a point 20 μm away from the center of the graphene channel is about half of the temperature rise at the center of graphene channel. As the heat diffusion equation is linear, this result suggests that if three parallel graphene channels are placed with an edge-to-edge distance of 10 μm, the temperature rise in the middle graphene channel when all three channels are electrically heated would become twice as high compared to the case where only one channel is electrically heated. For a similar situation in SiO2/Si devices, the effect of temperature rise in one channel has negligible effect on the temperature distribution of the other channels because of the dominantly vertical heat dissipation mechanism. Hence, the observed difference provides an important guideline in the design of the layout of flexible devices. In addition, the quantitative SThM measurements have been conducted in this work only in the relatively low power density corresponding to a temperature rise lower than the onset of the evaporation of the tip-sample meniscus. However, the approximately linear temperature rise versus power density trend can be extracted to predict the power density value corresponding to a temperature rise approaching the glass transition temperature of the flexible substrate, because the average temperature rise is largely dominated by the relatively temperature-insensitive and low thermal conductivity of the flexible substrate.
IV. SUMMARY
These experiments have provided useful insights into the influence of contact force, temperature, sample properties, and surface roughness in tip-sample heat transfer in SThM. It is found that the liquid meniscus dominates the local tip-sample heat transfer for the flexible devices measured in this work. The tip-sample thermal conductance remains nearly constant at temperatures up to 80 °C, above which it begins to decrease and exhibits an abrupt decrease at 110 °C because of evaporation of the liquid meniscus at the tip-sample junction. Due to the presence of polymer residue on the sample surface, the tip-sample thermal conductance is similar when the tip scans either the Au electrodes or the graphene channel. The presence of the liquid meniscus also decreases the effect of the sample surface roughness on the tip-sample thermal conductance, which is found to be nearly spatially invariant when the surface roughness is less than a few nanometers. These findings provide the conditions for assuming a constant tip-sample thermal conductance in triple-scan SThM measurements, which have yielded the same result as the zero-heat flux measurement when the conditions are met. In addition, the thermal spreading resistance in the measured sample is much lower than the tip-sample thermal resistance so that the tip-sample heat transfer is expected to result in negligible change in the sample temperature. According to the thermal imaging results obtained on the flexible graphene devices, the peak temperature at low power dissipation rates occurs at the center of graphene channels and moves toward the source contact by increasing the positive source bias. Instead of the Peltier effect, this asymmetry is mainly caused by non-uniform charge carrier concentration in the graphene channel, where the Fermi level is expected to be above the Dirac point. Compared to Si devices with 300 nm SiO2 where the temperature rise is found to be insensitive to the lateral sample size, the average temperature rise in flexible graphene devices is found to be higher by more than one order of magnitude and increases with increasing graphene channel length. The observed size dependence is mainly caused by that of the thermal spreading resistance of the flexible substrate, instead of heat spreading by the graphene channel that is effective only for channel lengths much smaller than 10 μm. These results are useful for designing the inter-channel spacing of flexible graphene devices to minimize the thermal cross talk due to lateral heat spreading.
ACKNOWLEDGMENTS
This work was supported in parts by the National Science Foundation Award No. EEC-1160494 (M.M.S. and L.S.) and the Office of Naval Research Award No. N00014-14-1-0200 (S.P. and D.A.).