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ACS Appl Mater Interfaces
ACS Appl Mater Interfaces
am
aamick
ACS Applied Materials & Interfaces
1944-8244
1944-8252
American Chemical Society

37695862
10.1021/acsami.3c05070
Research Article
High-Accuracy Contact Resistance Measurement Method for Liquid Metal by Considering Current-Density Distribution in Transfer Length Method Measurement
https://orcid.org/0000-0002-8860-0531
Sato Takashi †
https://orcid.org/0000-0003-1770-4883
Iwase Eiji *†‡§
† Department of Materials Science, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan
‡ Department of Applied Mechanics and Aerospace Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan
§ Kagami Memorial Research Institute for Materials Science and Technology, Waseda University, 2-8-26 Nishiwaseda, Shinjuku-ku, Tokyo 169-0051, Japan
* Email: iwase@waseda.jp. Phone: +81-03-5286-2741.
11 09 2023
20 09 2023
11 09 2024
15 37 4440444412
11 04 2023
28 08 2023
© 2023 The Authors. Published by American Chemical Society
2023
The Authors
https://creativecommons.org/licenses/by-nc-nd/4.0/ Permits non-commercial access and re-use, provided that author attribution and integrity are maintained; but does not permit creation of adaptations or other derivative works (https://creativecommons.org/licenses/by-nc-nd/4.0/).

Liquid metals (LMs) are used as stretchable conductors in various stretchable electronic devices. Moreover, such devices using Ga-based LMs have attracted considerable attention. Herein, we propose a method for accurately determining the contact resistance (Rc) between galinstan and Cu electrodes by considering the current-density distribution in transfer length method (TLM) measurement. Conventional TLM measurements assume that the sheet resistance of the metal electrode (Rshe) is negligible compared with that of the object (Rsho), such as Si. However, this assumption may be problematic because the Rsho of Ga-based liquid metals (LMs) is close to the Rshe. Therefore, we developed a method of applying current to each measuring electrode and compared it with the conventional method of applying current to the outer electrodes. Simulation results indicated that Rshe cannot be ignored for galinstan, and the measured resistance in the contact area (RcTotal) included <10% of the Rc component when current was applied to the outer electrodes. In contrast, RcTotal included the entire Rc component when current was applied to each electrode. Furthermore, we found that the volume resistances of the object and electrode included in RcTotal cannot be ignored. Therefore, for accurate measurement, current must be applied to each electrode, and Rc must be determined from the intersections of the measured and simulated RcTotal. The obtained contact resistivity (ρc), i.e., the contact resistance per unit contact area, was 0.115 mΩ·mm2. The maximum error was 0.085 mΩ·mm2, which was lower than the ρc of the solders (≥10–1 mΩ·mm2) with the lowest ρc among the electrical interface materials between the electronic components and wiring. This study provides valuable insight into the Rc measurement of LMs, along with new opportunities for the development of stretchable electronics using LMs.

galinstan
contact resistance
transfer length method
stretchable electronics
finite-element method
KIOXIA Holdings Corporation 10.13039/100019355 NA ACT-X 10.13039/501100020962 JPMJAX21K6 Waseda University 10.13039/501100004423 NA Japan Society for the Promotion of Science 10.13039/501100001691 JP22J13665 document-id-old-9am3c05070
document-id-new-14am3c05070
ccc-price
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pmcIntroduction

Stretchable electronic devices using Ga-based liquid metals (LMs) have attracted considerable attention.1−9 Ga-based LMs are composed of Ga, In, and Sn. LMs have low volume resistivity (gallinstan: 0.297 mΩ·mm)10 almost identical to that of metallic materials (Cu: 0.0168 mΩ·mm). They are highly stretchable (≥500% strain)11,12 because they exist in liquid form at room temperature owing to their low melting points (−19 to 25 °C). Additionally, LMs have low viscosity (2 mPa·s), low toxicity,13 and negligible vapor pressure.3,14 However, they have a high surface tension (534.6 mN/m),9 and an oxide layer tends to form on their surfaces,4,15−20 making it difficult to control the shape of the LMs. LMs are used as conductors in stretchable electronic devices, such as stretchable wirings,11,12 flexible sensors,6,21 flexible antennas,22−24 flexible actuators,25−27 flexible gas barrier films,28 and integrated devices with chip light-emitting diodes,29 temperature sensors,30,31 acceleration sensors,30 and thermoelectric generators.32 In such devices, the contact resistance (Rc) between the LM and metal electrodes, such as electronic components or wiring, is critical for achieving a low energy-consumption rate and high signal-to-noise ratio. The Rc measurement of LMs is mainly used to compare the Rc values of LMs for different contact conditions, e.g., materials and contact procedures, with those of conventional electrical interface materials (EIMs) between electronic components and wiring, such as solders and conductive adhesives. Although the Rc values of LMs have been measured,30,33−37 accurate measurement is difficult, and accurate methods for measuring the Rc of LMs have not been developed. The transfer length method (TLM)38−42 and cross-bridge Kelvin resistor (CBKR)43 method are well-established methods for measuring Rc. To measure the Rc of LMs, the current-density distribution must be considered because LMs have the same sheet resistance (Rshe) as metal electrodes. The TLM is suitable for considering the current-density distribution because the distribution is two-dimensional. Thus, we studied the current distribution in the TLM to achieve high accuracy. There are two types of TLM measurements: point contact and surface contact. Surface contact is used to measure Rc. The contact area between the object and the electrode has a finite value to include Rc in the measured R. Surface contact requires the fabrication of electrodes on the object before measurement. The measured R includes object sheet resistance (Rsho) and Rc; thus, Rsho and Rc can be obtained. In contrast, point contact is used to measure the sheet resistance of the object Rsho. The contact area is infinitely small to ignore Rc. The measured R includes only Rsho; thus, Rsho can be obtained but Rc cannot. Although only Rsho can be measured, point contact is beneficial for simplicity because electrode fabrication is not required. In this study, surface contacts were used to obtain Rc.

In the TLM, metal electrodes are brought into contact with an object, a current (I) is applied to the outer electrodes, and the voltage (V) between adjacent electrodes is measured. The measured resistance between adjacent electrodes R (=V/I) is the sum of Ro and RcTotal. Generally, RcTotal is treated as Rc. R is measured for different object length (Lo), and the relationship between Lo and R is approximated linearly. The resistance in the contact area (RcTotal) is obtained as the Y-intercept of the linear approximation without knowing Rsho. Current should be applied to each measuring electrode to achieve high accuracy regardless of the sheet and contact resistances. In the TLM measurement in the ISO standard (ISO 16525-2:2014), current is applied to the outer electrodes for simplicity within a limited range of sheet and contact resistances. However, the conventional TLM may not be applicable because LMs have low sheet resistance. Furthermore, the conventional TLM ignores the object and electrode resistances included in RcTotal. However, Rc between LMs and metal electrodes may be low; in this case, the electrode and object resistances cannot be ignored.

Specifically, when current is applied to the outer electrodes, the Rc component included in the measured RcTotal is Rc × Ii/I. Here, Ii represents the current passing through the interface between the electrode and object. For accuracy, all applied currents I must pass through the interface (Ii = I), and all Rc components must be included in the measured RcTotal. The current should be applied to each measuring electrode to achieve Ii = I regardless of the sheet resistance ratio (Rsho/Rshe) and the contact resistivity ratio (ρc/Rshe); however, this doubles the number of connections. Here, the specific contact resistivity (ρc) is the contact resistance per unit contact area. Conventional Rc measurements are primarily used for semiconductors and conductive pastes. Thus, the conventional TLM assumes that Rshe (Cu:10–3 Ω□) is negligible compared with Rsho (Si, graphene, conductive adhesive: ≥102 Ω□). In this case, current can be applied to the outer electrodes for simplicity because all of the applied current passes through the interface (Ii = I) for both current application methods. In previous studies, Rc was obtained via finite-element method (FEM) simulation alone when the real contact area between the object and electrode and the formation of a surface layer on the object and electrode were known.42 However, the Rc values of LMs are difficult to obtain via FEM simulations alone because the formation of an oxide layer on LMs and the real contact area are unknown. Furthermore, the Re and Ro included in RcTotal cannot be ignored when Rc is lower than Re or Ro. The measurement and simulation results should be compared to obtain Rc, ignoring Re and Ro. However, the simulation of each measurement is computationally demanding. Therefore, the conventional TLM assumes that RcTotal ≑ Rc for semiconductors and conductive pastes. Under this assumption, current is applied to the outer electrodes and Rc is obtained only via measurement, based on the ISO standards (ISO 16525-2:2014) and a book38 cited by many papers on conventional TLMs. In contrast, as the Rsho of the LMs (10–2 to 10–4 Ω□) is largely the same as the Rshe, only a part of the applied current may pass through the interface (Ii ≪ I) when current is applied to the outer electrodes. In this case, the Rc component included in RcTotal is Rc × Ii/I, which is undesirable for accuracy. Furthermore, the Rc between the LM and metal electrode may be lower than Re and Ro, in which case the Ro and Re included in RcTotal cannot be ignored. Thus, for accuracy, the measured RcTotal must not be regarded as Rc.

In this study, we examined the current application methods for highly accurate TLM measurement of LMs (galinstan) and metal (Cu) electrodes. The current-density distribution was analyzed using FEM simulations for different current application methods. Furthermore, the RcTotal between galinstan and Cu electrodes was measured for the current application methods. According to the simulation results, Ii/I was <10–1 for galinstan when current was applied to the outer electrodes. In contrast, Ii was equal to I, regardless of the sheet resistance and contact resistance, when current was applied to each electrode. Furthermore, ρc was obtained by comparing the measured RcTotal and simulated RcTotal for ρc. When current was applied to the outer electrodes, the measured RcTotal of 0.125 ± 0.056 mΩ and the simulated RcTotal of 0.111 to 0.122 mΩ did not intersect, and ρc could not be obtained. In contrast, when the current was applied to each electrode, the measured RcTotal of 0.120 ± 0.043 mΩ and the simulated RcTotal of 0.063 to 9.823 mΩ had an intersection. The obtained ρc was 0.115 mΩ·mm2 (max: 0.185 mΩ·mm2, min: 0.030 mΩ·mm2). The accuracy when current was applied to each electrode was sufficient for practical use compared with the ρc of the solders (10–1 mΩ·mm2).

Simulations

Figure 1a shows a schematic of the TLM measurements. The conventional TLM assumes that Rshe is negligible compared with Rsho. In this case, current applied to the outer electrode completely passes through the interface between the electrode and object (Ii = I), and the entire Rc component is included in the measured RcTotal. Furthermore, assuming that Ro and Re are negligible compared with Rc, the measured RcTotal is regarded as Rc. Figure 1b–i shows a schematic of the TLM measurement for the LMs in which current is applied to the outer electrodes. Rsho may not be negligible compared with Rshe. In this case, Ii < I, and part of the Rc component (Rc × Ii/I) is included in the measured RcTotal. Figure 1b-ii shows a schematic of the TLM measurement for the LMs in which current is applied to each measuring electrode. Ii is equal to I regardless of the sheet resistance and contact resistivity, and the entire Rc component is included in the measured RcTotal. For both current application methods, the Ro and Re included in RcTotal are not negligible compared with the Rc of the LMs. Figure S1 shows the equivalent circuit diagrams of the TLM measurement for the different current application methods.

Figure 1 Schematic of contact resistance (Rc) measurements of Ga-based LMs using the TLM. (a) Conventional TLM measurement, in which current is applied to the outer electrodes. The sheet resistance of the metal electrode (Rshe) is assumed to be negligible compared with that of the object (Rsho), and all of the applied current (I) passes through the interface between the electrode and the object. The object and electrode resistances (Ro and Re) included in the resistance in the contact area (RcTotal) are assumed to be negligible compared with Rc. (b-i) TLM measurement of LMs in which current is applied to the outer electrodes. Because the Rsho of the LM is equal to Rshe, part of the applied current (Ii) may pass through the interface. (b-ii) TLM measurement of LMs in which current is applied to each measuring electrode. The applied current completely passes through the interface, regardless of the sheet and contact resistances. For both current application methods, the Ro and Re included in RcTotal may not be negligible compared with the Rc of LMs.

In this study, the current-density distribution for the current application methods was analyzed through TLM measurements based on FEM simulations. Additionally, Ii/I was analyzed with respect to the sheet resistance ratio (Rsho/Rshe) and contact resistivity ratio (ρc/Rshe). The Ii/I for the Rsho of the LMs was compared with those of conventional objects, such as Si, graphene, and conductive adhesives. Furthermore, the RcTotal between galinstan and Cu electrodes for the current application methods was measured using the TLM. ρc, Ii/I, and Rc were obtained by comparing the FEM simulations with the measurement results. The measurement accuracies of the current application methods were compared. Although RcTotal can be measured using many conventional methods, the results of different methods cannot be compared. The Re and Ro included in RcTotal depend on the measurement configuration and dimensions of the devices.

Figure 2a shows the electrical contact model of the TLM measurement in which current is applied to the outer electrodes (COMSOL Multiphysics ver. 6.0, COMSOL Inc., MA). Two metal electrodes are placed in contact with the object. Current is applied uniformly to the cross section of one end of the object (hatched red). The cross section of the other end of the object (hatched green) is grounded. These current application conditions represent the situation where current is applied to one of the outer electrodes and flows through the object toward the measuring electrodes and then toward the other outer electrode. The interface between the object and the electrodes is characterized by contact impedance in COMSOL. This contact condition provides a uniform contact resistivity ρc [mΩ·mm2]. In the transmission-line model, which is the basis of the TLM, the upper surface of the electrode is grounded such that the current completely passes through the interface. However, in conventional TLM measurement, the current applied to the outer electrodes flows from the object to the electrode and then back to the object. Therefore, the cross section of the end of the object hatched green is grounded in our model. Here, the potential distribution on the cross sections at the end of the measuring electrodes hatched yellow is considered uniform because the electrode length (Le) exceeds the object width (Wo) and electrode width (We). Thus, the difference between the average potentials on the cross sections of the ends of the measuring electrodes is obtained as V. We analyzed the current-density distributions on the central cross section of the object, as indicated by the dotted line in Figure 2a. Ii was obtained by integrating the normal current density at the interface between the object and the electrodes. Figure 2b shows the electrical contact model of the TLM measurement, in which current is applied to each measuring electrode. Two metal electrodes are brought into contact with the object. Current is applied uniformly to the cross section of one end of the electrode (hatched red), while the cross section of the other end (hatched green) is grounded. The difference between the average potentials on the cross sections of the opposite ends of the measuring electrodes hatched yellow is measured. We also analyzed the current-density distribution on the same side of the object where the current was applied, as indicated by the dotted line. Ii was obtained by integrating the normal current density at the interface.

Figure 2 Electrical contact models for the simulation of the current-density distribution in TLM measurement using the FEM. Two metal electrodes are brought into contact with the object. The voltage difference between the two electrodes (V) is measured. Ii is obtained by integrating the normal current density at the interface. (a) Model of current application to the outer electrodes. The current is applied uniformly to the cross sections of both ends of the object. The current-density distribution is analyzed at the central cross section of the object. (b) Model of the current application to each electrode. A current is applied to the ends of the electrodes, opposite to the voltage measurement. The current-density distribution is analyzed on the same side of the object where the current was applied.

The volume resistivity, thickness, length, and width of the object were ρo = 10–10 to 10–4 Ωm, To = 10 μm, Lo = 10 mm, and Wo = 5 mm, respectively. The simulated RcTotal was constant regardless of Lo because the current density was uniformly distributed in the object between the measuring electrodes. Thus, any Lo value could be used in the simulation. The volume resistivity, thickness, length, and width of the metal electrodes were ρe = 10–8 Ωm, Te = 5 μm, Le = 10 mm, and We = 5 mm, respectively. ρc ranged from 10–12 to 10–6 Ωm2. Ii was simulated for different values of the sheet resistance ratio Rsho/Rshe and contact resistivity ratio ρc/Rshe. Rsho varied because the metal electrodes were in contact with objects having different Rsho values. Additionally, ρc varied under the assumption that Ii decreased as ρc increased because the resistance along the current path increased.

Figure 3a,b shows the current-density distribution and current ratio Ii/I with respect to the sheet resistance ratio Rsho/Rshe and contact resistivity ratio ρc/Rshe when current was applied to the outer electrodes. As indicated by Figure 3a-iii,a-iv, most of the applied current passed through the interface between the electrode and object (Ii ≑ I) for high Rsho/Rshe values. The degree of current crowding around the interface edge decreased as ρc/Rshe increased. In contrast, a small amount of applied current passed through the interface (Ii ≪ I) for low Rsho/Rshe values, and Ii decreased slightly as ρc/Rshe increased (Figure 3a-i,ii). Figure 3b shows that Ii/I decreased from 100 to <10–2 as Rsho/Rshe decreased from 105 to 10–2 for all of the ρc/Rshe values. The reduction in Ii/I accelerated as ρc/Rshe increased from 10–9 to 10–3. In addition, Ii was equal to I for conventional objects such as Si (Rsho ≥ 104 Ω□), graphene (Rsho ≥ 103 Ω□), and conductive pastes (Rsho ≥ 102 Ω□) for all of the ρc/Rshe values. This result confirms that the assumptions in the TLM measurement that Rshe is negligible relative to Rsho and that Ii is equal to I are valid for conventional objects. In contrast, Ii/I was <10–1 for LMs (Rsho = 10–2 to 100 Ω□) for all of the ρc/Rshe values. This result indicates that the assumptions that Rshe is negligible compared with Rsho and that Ii is equal to I are invalid for LMs. Overall, current application to the outer electrodes yields accurate measurement when Rsho/Rshe ≥ 103 because Ii is equal to I and the entire Rc component is included in RcTotal. However, to ensure high accuracy, the current must not be applied to the outer electrodes when Rsho/Rshe is <103 because Ii < I and part of the Rc component is included in RcTotal.

Figure 3 FEM simulation results for (a) the current-density distribution and (b) the ratio of the current passing through the interface (Ii/I) with respect to the sheet resistance ratio (Rsho/Rshe) and contact resistivity ratio (ρc/Rshe). Current was applied to the outer electrodes. The lines in the current-density distribution indicate the current flow. Ii was equal to I for conventional objects (Si, graphene, conductive adhesive: Rsho ≥ 102 Ω□). In contrast, Ii/I was <10–1 for LMs (Rsho = 10–2 to 100 Ω□).

Figure 4a,b presents the current-density distribution and Ii/I with respect to Rsho/Rshe and ρc/Rshe when current was applied to each electrode. As shown in Figure 4a, all of the applied currents passed through the interface between the electrode and the object, regardless of Rsho/Rshe and ρc/Rshe. The degree of current crowding around the interface edge decreased as Rsho/Rshe decreased and ρc/Rshe increased. Figure 4b shows that Ii was equal to the applied I for all of the Rsho/Rshe and ρc/Rshe values when current was applied to each electrode. Therefore, to ensure high accuracy, the current should be applied to each electrode because Ii = I and the entire Rc component is included in RcTotal, regardless of Rsho/Rshe and ρc/Rshe. However, Ii = I and the entire Rc component is included in RcTotal for both current application methods when Rsho/Rshe is ≥103. In this case, for simplicity, current can be applied to the outer electrodes.

Figure 4 Simulation results of (a) the current-density distribution and (b) Ii/I for Rsho/Rshe and ρc/Rshe. Current is applied to each electrode. Ii is equal to I regardless of Rsho/Rshe and ρc/Rshe.

Measurements and Results

Figure 5a,b shows schematics and photographs, respectively, of the TLM measurement devices. Galinstan (Changsha Rich Nonferrous Metals, China) was employed as the LM material because it has been widely used owing to its low melting point (−19 °C) among Ga-based LMs. Electroplated Cu with submicron surface roughness (Ra) (Toray, Japan) was used as the metal electrode material, in accordance with the ISO standard (ISO 16525-2:2014) for TLM measurement of conductive adhesives. Galinstan was brought into contact with a Cu substrate. The Cu substrate was patterned into electrode shapes using an ultraviolet laser cutter. Then, cutouts on a poly(ethylene terephthalate) (PET) substrate were filled with galinstan, and the resulting PET substrate and a Cu substrate were sandwiched between acrylic substrates. These measurement configurations were designed to ignore the effects of the wetting behavior of galinstan. Galinstan has high surface tension and forms an oxide layer on its surface. Thus, the cross-sectional shape of galinstan is unknown if it is deposited on a substrate in the same manner as conventional TLM measurements. Therefore, we filled a rectangular channel with galinstan to determine the cross-sectional shape of the galinstan. Although galinstan largely filled the channel, it may not have filled the corners of the channel because of its wettability and the oxide layer. Thus, the channel length exceeded the distance between the measuring electrodes, to ignore the edge shape of the galinstan. In addition, the structure could seal galinstan without changes in the surface conditions of the metal electrodes. The thickness, length, and width of the galinstan were To = 1 mm, Lo = 7.5–17.5 mm, and Wo = 1 mm, respectively. The thickness and width of the Cu electrodes were Te = 8 μm and We = 1 mm, respectively.

Figure 5 (a) Schematics and (b) photographs of the TLM measurement devices. LM (galinstan) was brought into contact with metal electrodes (Cu substrate). Current was applied to the outer electrodes or each electrode of the same device. (c) Fabrication procedures for the devices. (c-i, c-ii) A simple channel was fabricated by sandwiching the PET substrate with a cutout and Cu substrate between acrylic substrates. (c-iii, c-iv) Galinstan was injected into the channel and sealed.

Figure 5c presents the fabrication process for the measurement devices. As shown in Figure 5c-i,c-ii, two acrylic substrates, a PET substrate with cutouts, and a Cu substrate were assembled and fixed. Prior to the assembly, the Cu substrate was deoxidized using solder flux (NS-30, Nihon Superior, Japan). The acrylic substrate was drilled to create the inlet and outlet for injecting galinstan into the cutouts. As shown in Figure 5c-iii,iv, galinstan was injected into the cutouts using a syringe. The resistance of the device (R) was measured using an ohm meter (RM3545, Hioki, Japan). The measurement conditions were determined to ignore the time-dependent change in R during TLM measurements. For LMs, the TLM is primarily used to measure the time-dependent change in Rc for different contact conditions, such as materials and contact procedures. The TLM measurement of LMs can be affected by alloying, Joule heating, electromagnetic interference, and ground loops. To ignore the effect of Joule heating, pulse currents were applied to the measuring electrodes for 200 ms by the SLOW2 mode. To ignore the effect of alloying, the duration of the TLM measurements was 150 s using an automatic switching machine (SW1002, Hioki, Japan). The applied current I was as high as 1.5 A for accurately measuring the low resistance. In one measurement cycle, we measured R for five different Lo values and waited until 15 s had passed from the start of the measurement. This measurement cycle was performed 10 times consecutively (150 s). Figure S2 shows the time-dependent changes in R. R was reduced by 0.45% after 150 s of contact between the galinstan and Cu electrodes and by 0.68% after 500 min of contact. These results indicated that the effect of noise during the measurement was <0.45% immediately after contact. Assuming measurement under various contact conditions, the effect of noise is <0.68%, regardless of the time from contact to measurement. An approximate straight line was drawn for the measurement results (200 plots) of the four devices, and the Y-intercept was calculated. The simulation and measurement results were compared, and ρc was obtained from the intersections of the measured RcTotal and the simulated values for different ρc. Ii/I and Rc were calculated from the obtained ρc. The accuracy was evaluated from the error range of the obtained ρc. The measured RcTotal = (R – Ro)/2 was obtained from the Y-intercept (=2RcTotal) of the TLM results. The simulated RcTotal = Rc × Ii/I was obtained using the Ii/I and Rc from the simulations. In the simulations, we used the models shown in Figure 2. The channel width is used as the galinstan width, and the sheet resistance obtained experimentally from the slope of the TLM measurement results is used as that of galinstan.

Figure 6a shows the relationship between Lo and the measured and simulated R = αLo + 2RcTotal values when current was applied to the outer electrodes. The slope α and Y-intercept (=2RcTotal) of the approximate line between Lo and the measured R were 0.243 mΩ/mm and 0.250 mΩ, respectively. Here, the Rsho of galinstan obtained from the measured α was 2.43 × 10–4 Ω□. Figure 6b shows the relationship between Lo and the measured and simulated resistance in the contact area RcTotal = (R – Ro)/2. The measured RcTotal was 0.125 ± 0.056 mΩ. In contrast, the simulated RcTotal increased from 0.111 to 0.122 mΩ as ρc increased from 10–4 to 104 mΩ mm2. The measured and simulated RcTotal did not intersect, and ρc could not be obtained. Figure 6c shows the simulated relationship between Ii/I and ρc. Ii/I decreased from 0.099 to 0.000 as ρc increased from 10–4 to 104 mΩ·mm2. Ii was far lower than I, which reduced the slope of the simulated curve of RcTotal, as shown in Figure 6b. This simulation result indicates that Rshe is not negligible compared with Rsho for LMs. Thus, even if they intersect, the accuracy of the Rc obtained from the interface will be low because the range of the simulated RcTotal is lower than that of the measured RcTotal.

Figure 6 (a) Relationship between the object length (Lo) and the measured resistance (R) (N = 200). Current was applied to the outer electrodes. (b) Relationship between ρc and the resistance in the contact area (RcTotal = (R – Ro)/2). RcTotal is given as Y-intercept/2 in the TLM measurement results. The RcTotal for ρc was simulated via the FEM. The measured and simulated RcTotal did not intersect, and ρc could not be obtained. (c) Simulation results for the current-density distribution and Ii/I for ρc. Ii/I was <0.1.

Figure 7a shows the relationship between Lo and the measured and simulated R = αLo + 2RcTotal when current was applied to each electrode. The slope α and Y-intercept (=2RcTotal) of the approximate line between Lo and the measured R were 0.241 mΩ/mm and 0.239 mΩ, respectively. Here, the Rsho of galinstan obtained from the measured α was 2.41 × 10–4 Ω□. Figure 7b shows the relationship between Lo and the measured and simulated resistances in the contact area (RcTotal = (R – Ro)/2). The measured RcTotal was 0.120 ± 0.043 mΩ. In contrast, the simulated RcTotal increased from 0.063 to 9.823 mΩ as ρc increased from 10–4 to 101 mΩ·mm2. The ρc at the intersection of the measured and simulated RcTotal was 0.115 mΩ·mm2 (min: 0.030 mΩ·mm2, max: 0.185 mΩ·mm2). The Rc obtained using ρc was 0.230 mΩ (min: 0.060 mΩ, max: 0.370 mΩ). These results indicate that RcTotal is not equal to Rc even when a current is applied to each electrode. Therefore, Rc must be obtained using the intersection of the measured and simulated RcTotal to achieve high accuracy. Figure 7c shows the simulated relationship between Ii/I and ρc. I was equal to I regardless of ρc. This result indicated that Ii being equal to I increased the slope of the simulated curve of RcTotal in Figure 7b compared with that in Figure 6b and increased the accuracy. The Rc measurement of LMs is mainly used to compare the ρc values of LMs with those of conventional EIMs. Solders have the lowest contact resistivity (ρc ≥ 10–1 mΩ·mm2) among conventional EIMs. The ρc of galinstan was 0.115 mΩ·mm2 and the maximum ρc error was 0.085 mΩ·mm2. The obtained ρc value was of the same magnitude as the solders. ρc close to that of the solders can be measured under the magnitude of the errors in the proposed method. Therefore, the proposed method is practical.

Figure 7 (a) Relationship between Lo and R (N = 200). Current was applied to each electrode. (b) Relationship between ρc and RcTotal = (R – Ro)/2. RcTotal is given as Y-intercept/2 in the TLM measurement results. The RcTotal for ρc was simulated via the FEM. The ρc at the intersection of the measured and simulated RcTotal was 0.115 mΩ·mm2 (min: 0.030 mΩ·mm2, max: 0.185 mΩ·mm2). (c) Simulation results for the current-density distribution and Ii/I for ρc. Ii was equal to I for all of the ρc values.

Conclusions

In this study, the current-density distribution in the TLM measurement of galinstan was analyzed using FEM simulations, and a method for applying current to each measuring electrode for accurate resistance measurements was developed. We simulated the ratio of the current passing through the interface between the electrode and the object Ii/I for different values of the sheet resistance ratio Rsho/Rshe and contact resistivity ratio ρc/Rshe. When current was applied to the outer electrodes, Ii was equal to I for conventional objects (Si, graphene, conductive adhesive: ≥102 Ω□) for all of the tested values of ρc/Rshe, and the assumption in the conventional TLM was valid. However, we found that Ii/I was <10–1 for LMs (Rsho = 10–2 to 100 Ω□) for all of the ρc/Rshe values, and the conventional assumption was invalid. In contrast, when current was applied to each electrode, Ii was equal to I, regardless of Rsho/Rshe and ρc/Rshe. Therefore, to ensure high accuracy, the current should be applied to each electrode because Ii is equal to I and the entire Rc component is included in RcTotal regardless of Rsho/Rshe and ρc/Rshe. Furthermore, ρc was obtained by comparing the measured and simulated RcTotal values. When current was applied to the outer electrodes, the measured RcTotal was 0.125 ± 0.056 mΩ, and the simulated RcTotal ranged from 0.111 to 0.122 mΩ for ρc values of 10–4–104 mΩ·mm. The measured and simulated RcTotal did not intersect, and ρc could not be obtained. In contrast, when the current was applied to each electrode, the measured RcTotal was 0.120 ± 0.043 mΩ, and the simulated RcTotal ranged from 0.063 to 9.823 mΩ for ρc values of 10–4–101 mΩ·mm2. The obtained ρc was 0.115 mΩ·mm2, and the maximum error was 0.085 mΩ·mm2, which is sufficient for practical use compared with the ρc of the solders (≥10–1 mΩ·mm2). This study can help to discover suitable electrode materials for LMs and provide new avenues for the development of stretchable electronics using LMs.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsami.3c05070.Details of the equivalent circuit diagrams of the TLM measurement for the different current application methods and the time-dependent changes in R (PDF)

Supplementary Material

am3c05070_si_001.pdf

Author Contributions

T.S. and E.I. conceived and designed the experiments. T.S. performed the experiments, analyzed the data, and wrote the paper. E.I. reviewed and edited the paper and supervised the research.

This work was partially supported by JST ACT-X (Grant Number JPMJAX21K6), JSPS KAKENHI (Grant Number JP22J13665), and cooperation between Waseda University and Kioxia Corporation.

The authors declare no competing financial interest.

Acknowledgments

The authors are grateful to T. Sugahara for advice on the TLM measurements.
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