REVIEW OF SCIENTIFIC INSTRUMENTS 86, 044901 (2015)
A four-probe thermal transport measurement method for nanostructures Jaehyun Kim, Eric Ou, Daniel P. Sellan, and Li Shia) Department of Mechanical Engineering, The University of Texas at Austin, Austin, Texas 78712, USA
(Received 6 February 2015; accepted 18 March 2015; published online 6 April 2015) Several experimental techniques reported in recent years have enabled the measurement of thermal transport properties of nanostructures. However, eliminating the contact thermal resistance error from the measurement results has remained a critical challenge. Here, we report a different four-probe measurement method that can separately obtain both the intrinsic thermal conductance and the contact thermal resistance of individual nanostructures. The measurement device consists of four microfabricated, suspended metal lines that act as resistive heaters and thermometers, across which the nanostructure sample is assembled. The method takes advantage of the variation in the heat flow along the suspended nanostructure and across its contacts to the four suspended heater and thermometer lines, and uses sixteen sets of temperature and heat flow measurements to obtain nine of the thermal resistances in the measurement device and the nanostructure sample, including the intrinsic thermal resistance and the two contact thermal resistances to the middle suspended segment of the nanostructure. Two single crystalline Si nanowires with different cross sections are measured in this work to demonstrate the effectiveness of the method. This four-probe thermal transport measurement method can lead to future discoveries of unique size-dependent thermal transport phenomena in nanostructures and lowdimensional materials, in addition to providing reliable experimental data for calibrating theoretical models. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4916547]
I. INTRODUCTION
Advances are continuously made in both top-down nanofabrication and bottom-up syntheses of nanostructures, including carbon nanotubes, inorganic and organic nanowires and nanofibers, graphene, and, more recently, a variety of twodimensional (2D) materials. Because the characteristic dimension of these nanostructures is reduced to the scale of the scattering mean free path or even the wavelength of the charge or heat carriers, their electrical, thermal, and thermoelectric properties can be very different from their bulk counterparts. For example, elimination of inter-layer interaction is expected to result in enhanced basal plane thermal conductivity of clean suspended single-layered graphene (SLG) and singlewalled carbon nanotubes (SWCNTs) compared to the already high values found for graphite.1 Scattering is especially suppressed for low-frequency phonons in clean suspended SLG and SWCNTs, so that the thermal conductivity contribution from these long-wavelength modes can be limited mainly by scattering at the lateral edges instead of phonon-phonon processes.2 This and other size-dependent thermal properties have received increasing interest because they can lead to both new insights into thermal transport physics and potential applications such as thermal management and thermoelectric energy conversion. The interest in the unique thermal properties of nanostructures has motivated the development of several methods for thermal transport measurements of carbon nanotubes, nanowires, nanofilms, graphene, and other 2D layered materials.3–21 These methods have explored different thermometry a)Author to whom correspondence should be addressed. Electronic mail:
[email protected]. 0034-6748/2015/86(4)/044901/7/$30.00
techniques including suspended micro-devices with resistance thermometry, Raman spectroscopy, thermal reflectance, and bimaterial cantilevers, as discussed in a recent review.22 While the invention of scanning tunneling microscopy,23 atomic force microscopy,24 and other new experimental techniques nucleated the broader field of nanoscience, the development of these thermal transport measurement techniques has significantly enhanced the capability for the study of nanoscale thermal transport. A number of these measurements have revealed that scattering by either the surface roughness or a disordered layer in contact with the nanostructure can result in considerable reduction of the axial or in-plane thermal conductivity.6,12–14,19,20 Meanwhile, some measurement results have been used to suggest quasi-ballistic phonon transport in carbon nanotubes and graphene.7,25 In addition, several other experimental works have suggested that the thermal conductivity of carbon nanotube, graphene, or SiGe nanowire samples increases with the sample length as a result of the breakdown of Fourier’s law in these nanostructure samples.26–29 While these experimental reports of ballistic phonon transport and length-dependent thermal conductivity have suggested a new regime of thermal transport physics, outstanding questions have remained on uncertainties in the challenging thermal transport measurements of nanostructures.30–32 Among these challenges, the inability to separate the contact thermal resistance from the intrinsic thermal conductance of nanostructure samples has presented a major source of error, which has given rise to critical questions on the exact cause of the lengthdependent thermal conductivities or abnormally low thermal conductivity values found in a number of measurements on nanostructures. Several efforts have been devoted to the development of measurement methods for independently probing both the
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contact thermal resistance and intrinsic thermal conductance of nanostructures. In one method based on a suspended microdevice that consists of two membranes with built-in serpentine Pt resistance thermometers, two Pt electrodes were patterned on each membrane to measure the thermovoltage drop along the nanostructure segment in contact with the membrane.11 The measured thermovoltage was then used to determine the temperature drop at the contact and the contact thermal resistance. This method is limited to a conducting nanostructure sample with a large and uniform Seebeck coefficient compared to that of the Pt electrodes, and requires Ohmic electrical contact between the Pt electrode and the sample. A different approach is based on focused laser or electron beam heating along the nanostructure sample suspended between the two membranes.9,33 While the localized heating approach can potentially provide spatial distribution of the thermal resistance along a nanostructure sample, further efforts are required to probe thermal transport at different temperatures, especially in the low temperature regime where intriguing nanoscale thermal transport phenomena are pronounced, and to address experimental complications such as alteration of the structure or thermal properties of the nanostructure by the focused laser or electron beam. In addition, there have been efforts for developing the thermal analogue of the electrical van der Pauw method,34 which is limited to isotropic thin film samples and is not applicable to nanowire and nanoribbon structures. Therefore, there is still a lack of an effective and general experimental method for probing both the contact thermal resistance and intrinsic thermal conductance of nanostructures at different temperatures. Establishment of such experimental methods is necessary not only for advancing the field of nanoscale thermal transport but also for establishing the intrinsic thermal properties of a wide variety of nanostructures, which are continuously being synthesized or discovered by the nanomaterials research community. Here, we report a different four probe thermal resistance measurement method for determining both the contact thermal resistances and the intrinsic thermal conductivity of a nanostructured sample at different temperatures. The effectiveness of this method is demonstrated by employing this method for measuring patterned Si nanowire samples. The obtained measurement results are compared to theoretical calculation results obtained from two different thermal conductivity models.
II. MEASUREMENT METHOD
The four-probe thermal measurement device consists of four suspended Pt/SiNx resistance thermometer lines (RT1, RT2, RT3, and RT4), as shown in Figs. 1(a) and 1(b). While the i th Pt/SiNx line is electrically heated by a direct current (I) during the measurement, the average temperature rise (θ j,i ) in the j th Pt/SiNx line is obtained from the thermally induced increase of the electrical resistance measured using a fourprobe configuration.35,36 With i and j both ranging from 1 to 4, a total of 4 × 4 measurement data of the θ j,i /(IV )i ratio can be obtained, where V and (IV )i are the measured voltage drop and Joule heating rate in the i th heating line, respectively. In the following, we show that these 16 measurement data
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FIG. 1. (a) Optical micrograph (left panel) and scanning electron micrographs (SEMs, right panel) of a 240 nm wide, 220 nm thick patterned Si nanowire sample assembled across four suspended Pt/SiNx lines that serve as resistive heaters and thermometers. As shown in the top SEM, a small V-shape protrusion is patterned at the center of the first thermometer line from the left to assist in the measurement of the deviation (di and dj ) between the center of each thermometer line and the contact point to the nanostructure. (b) Optical micrograph of a 740 nm wide, 220 nm thick patterned Si nanowire sample assembled across four suspended Pt/SiNx lines together with a schematic of the temperature profiles along the Pt/SiNx heater line (i th line) and one Pt/SiNx resistance thermometer line ( j th line, j , i). (c) Thermal resistance circuit of the measurement device when the first Pt/SiNx line is electrically heated at a rate of (IV )1. R 1, R 2, and R 3 are the thermal resistances of the left, middle, and right suspended segments of the suspended Si nanowire sample, respectively. R c, j is the contact thermal resistance between the j th Pt/SiNx line and the sample. R b, j is the thermal resistance of the j th Pt/SiNx resistance thermometer line. θ c, j, i is the j th Pt/SiNx line temperature at the contact to the sample when the i th line is used as the heater line. The temperature rise θ 0 at the two ends of each of the suspended Pt/SiNx lines is assumed to be negligible.
can be used to obtain and separate nine of the eleven thermal resistances in the system (Fig. 1(c)), including the two contact thermal resistances and the intrinsic thermal resistance of the middle suspended segment of the nanostructure sample. First, the temperature rise (θ c, j,i ) of the j th thermometer line at the contact point to the nanostructure can be obtained from the measured θ j,i when the i th line is electrically heated. Because the spreading thermal resistance in the Si substrate is two orders of magnitude lower than that of the suspended thermometer line, the temperature rise (θ 0) at the two ends of each suspended thermometer line is negligible. In addition, the measurement is conducted with the sample in high vacuum to eliminate surface heat loss to gas molecules. Based on a fin heat transfer analysis similar to that reported in two prior works,37,38 the ratio between radiation heat loss from the surface of a beam and the heat conduction inside the beam can be obtained as α = cosh βL − 1, where β = 4εσT 3 P/κ A, κ, L, P, and A are the thermal conductivity, length, perimeter of the cross section, and cross section area of the beam, respectively, T is the average temperature of the system, and ε is the surface emissivity. For the same aspect ratio PL/A, this ratio decreases with decreasing L and becomes less than 1.5 × 10−3 for the thermometer line and less than 1 × 10−6 for the silicon nanowire measured in this work at temperatures lower than
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400 K. Hence, radiation loss is negligible. Therefore, if the j th thermometer line is not the i th heating line, the temperature distribution in the j th line is linear, as illustrated in Fig. 1(b), so that θ c, j,i = 2θ j,i , for j , i.
(1)
In addition, the heat flow from each thermometer line into the nanostructure can be obtained from the thermal resistance circuit of Fig. 1(c) as Q j,i = −θ c, j,i /Rb, j ,
for
j ,i
(2a)
Equations (1), (2), and (7) yield 16 pairs of θ c, j,i /(IV )i and Q j,i /(IV )i data, with both i and j ranging from 1 to 4. These 16 pairs of data can be used to obtain five of the thermal resistances in the circuit in Fig. 1(c), where R1, R2, and R3 are the intrinsic thermal resistances of the left, middle, and right suspended segments of the nanostructure, respectively, and Rc,1, Rc,2, Rc,3, and Rc,4 are the contact thermal resistances at the four contacts between the nanostructure and the thermometer lines. Based on the thermal resistance circuit in Fig. 1(c), θ c,2,i − θ c,3,i = Q2,i Rc,2 + (Q1,i + Q2,i )R2
and Q i,i
=−
j, j,i
Q j,i .
Here, the thermal resistance of each of the four thermometer lines is given as ( ) 2 d j L j 1− , (3) Rb, j = 2κ j A j L j where κ j , A j , and 2L j are the effective thermal conductivity, cross-sectional area, and length of the j th suspended thermometer line, respectively, and d j is the deviation of the center of the j th line from its point of contact to the nanostructure sample, as shown in Fig. 1(b). From the Joule heating in the i th line and the heat flow boundary condition at the contact point given by Eq. (2b), the solution to the heat conduction equation yields the following parabolic temperature distribution in the line as a function of distance (x) from its center: d i + (−1) H (d i −x) L i (IV )i 2 2 Li − x + θ i,i (x) = 4κ i Ai L i 2κ i Ai L i H (d i −x) × x − (−1) L i Q i,i , (4) where the Heaviside step function H( χ) takes the value of 0 and 1 for χ < 0 and χ ≥ 0, respectively. The obtained temperature distribution can be used to calculate the average temperature rise in the i th heater line as Li 1 θ i,i = θ i,i (x)dx. (5) 2L i −L i The integration can be carried out to obtain the following relationship: 4 θ j,i /(IV )i 1 , for i = 1, 2, 3, and 4. = j=1 Rb, j 3 1 − (d i /L i )2 (6) This set of four equations can be written in a matrix form and used to obtain the thermal resistances of the four thermometer lines, Rb,1, Rb,2, Rb,3, and Rb,4 based on the 16 sets of θ j,i /(IV )i measurement data, which are the elements of an invertible 4 × 4 matrix. With the four obtained Rb, j values, the contact point temperature rise for the heater line can now be calculated from Eq. (4) as θ c,i,i ≡ θ i,i (x = d i ) 4 θ j,i θ i,i + (IV )i = Rb,i − 2* − . j=1 Rb, j Rb,i - 2 ,
− Q3,i Rc,3, for i = 1, 2, 3, 4.
(2b)
(7)
(8)
Among this set of four equations, three are independent from each other and can be used to obtain the three unknown thermal resistances R2, Rc,2, and Rc,3 based on the measured θ c, j,i /(IV )i and Q j,i /(IV )i data. In addition, the thermal resistance circuit can be used to obtain the following: θ c,1,i − θ c,2,i = Q1,i (R1 + Rc,1) − Q2,i Rc,2, for i = 1, 2, 3, 4. (9) One of these four equations can be used to obtain the one unknown resistance (R1 + Rc,1) based on the above-obtained Rc,2 and the measured θ c, j,i /(IV )i and Q j,i /(IV )i data, whereas the other three equations are redundant. Similarly, (R3 + Rc,4) can be obtained from one of the following four equations derived from the thermal resistance circuit: θ c,4,i − θ c,3,i = Q4,i (R3 + Rc,4) − Q3,i Rc,3, for i = 1, 2, 3, 4. (10) Therefore, with the use of a linear set of equations, the 16 θ j,i /(IV )i measurement data can be used to obtain the nine thermal resistances shown in the thermal circuit in Fig. 2(b), including Rb,1, Rb,2, Rb,3, Rb,4, Rc,2, R2, Rc,3, R1 + Rc,1, and R3 + Rc,4. Because the heat flow across each end contact (Rc,1 or Rc,4) is the same as that through the adjacent suspended sample segment (R1 or R3), the contact thermal resistance component cannot be separated from R1 + Rc,1 and R3 + Rc,4. In comparison, the heat flow rates across the two middle contacts (Rc,2 and Rc,3) and the middle sample segment (R2) are different and depend on which Pt/SiNx line is used as the heater line. This feature allows separate determination of Rc,2, Rc,3, and the intrinsic thermal resistance (R2) of the middle suspended segment.
III. RESULTS
To demonstrate the effectiveness of this four-probe thermal transport measurement method, we have employed the method to measure two patterned Si nanowire samples with cross sections of 240 nm × 220 nm and 740 nm × 220 nm. The cross sections were intentionally chosen to be relatively large, so that the results obtained from the new method can be compared with established theoretical models without the need to consider quantum confinement or anomalous surface scattering effects that are not yet well understood.
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The Si nanowire sample was patterned from a silicon-oninsulator wafer with a 220 nm thick top Si layer specified with 5 × 1016 cm−3 boron doping on top of a 3 µm thick buried SiO2 layer. After patterning the Si nanowire with electron beam lithography and reactive ion etching, the SiO2 underlying the Si nanowire is etched in diluted hydrogen fluoride acid. The suspended Si nanowire sample was transferred to another wafer coated with a Polyvinyl alcohol (PVA) layer, followed by the spin coating of a Poly(methyl methacrylate) (PMMA) layer on top of the Si nanowire and the PVA layer. After the PVA layer is dissolved in deionized water, the Si nanowire was detached from the wafer together with the PMMA carrier layer, which was transferred and aligned to the four suspended Pt/SiNx thermometer lines fabricated on another Si wafer. The carrier PMMA layer was removed by heating to a temperature of 350 ◦C in a low-pressure tube furnace with argon and hydrogen flows. This process resulted in the Si nanowire sample suspended across the four Pt/SiNx lines. Each of the four Pt/SiNx lines was made of 60 nm thick Pt deposited on 300 nm thick SiNx with a 10 nm Cr adhesion layer in between. For the device shown in Fig. 1, each suspended Pt/SiNx line is 200 µm long and 2 µm wide, with the width limited by the photolithographic process. The line width can be further reduced with the use of electron beam lithography. When the Pt/SiNx line width is much smaller than the length of the middle suspended segment of the nanostructure sample and the length of the Pt/SiNx line is much longer than the width of the nanostructure, the contact between the thermometer line and the nanostructure can be accurately approximated as a point as assumed in the analysis. For measurements of a wider sample than the Si nanowires, such as a graphene flake, the length of the four thermometer lines can be increased to be much larger than the sample width, so that the contact area can be treated as a point contact. In addition, this measurement obtains the thermal resistance (R2) in the sample region between the two middle point contacts. The sample thickness and the thermometer line width need to be small compared to the length of the middle suspended segment, in order to assume one-dimensional temperature profile in the middle sample segment, which is necessary for obtaining the sample thermal conductivity as κ = L/R2 A, where L and A are the length and cross-sectional area of the middle segment, respectively, when the transport is diffusive in the middle segment. The sample was placed on the sample stage of an evacuated cryostat for the thermal measurements. During the measurement with the heating current I ramped between −350 µA and +350 µA in the i th heating line, the four-probe electrical resistance (Re,i ) of the i th heating line was obtained directly as the ratio of the measured voltage drop V and the heating current I, whereas a lock-in amplifier was used to measure the four-probe resistance (Re, j ) of each of the three other thermometer lines with a 1 µA sinusoidal excitation current. The sinusoidal sensing current is about a factor of 350 smaller than the maximum direct heating current applied to the heating line, so that the corresponding Joule heating is five orders of magnitude smaller. Moreover, the constant sinusoidal sensing current influences only the absolute temperature, but not the measured temperature rise (θ j,i ) of a thermometer line caused
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by the direct heating current in the heater line. As such, the theoretical results of θ j,i distributions are not affected by the presence of the sinusoidal current. In addition, it has been shown in a previous work that the theoretical θ j,i distribution is much more immune to errors caused by ignoring the surface radiation loss than the absolute temperature distribution.39 The silicon nanowire samples reported in this study did not make electrical contact to the Pt lines because of the presence of an amorphous native oxide on the nanowire. A thin dielectric can also be deposited on the Pt line to prevent electrical contact to the sample. However, this four-probe method can also be used to measure a nanostructure sample that makes electrical contacts to the Pt lines. In such a measurement, a floating current source is used to provide the heating current to the heating Pt line. At the same heating current, the electrical resistance of each of the other three Pt lines is measured separately with a small sinusoidal excitation current and a lock-in amplifier while the other two Pt lines are electrically floating. Such a measurement procedure prevents electrical current flow in the nanostructure sample. Figure 2(a) shows a set of measurement data of the resistance rise, ∆Re, j = Re, j (I) − Re, j (I = 0), as a function of the heating current I applied to the first Pt/SiNx line. The temperature coefficient of resistance (TCR) was obtained from the Re, j (I = 0) measured at different sample stage temperature (T0) values, as shown in Fig. 2(b). The obtained TCR values were used to convert the measured resistance rises ∆Re, j into the average temperature rise (θ j,i ) in each thermometer line. The θ j,i /(IV )i ratio was obtained as the slope of the linear θ j,i versus (IV )i data, as shown in Fig. 2(c). The θ j,i /(IV )i data were used to obtain the Rb,1, Rb,2, Rb,3, and Rb,4 from Eq. (6), and 16 pairs of θ c, j,i /(IV )i and Q j,i /(IV )i data according to Eqs. (1), (2), and (7). The obtained θ c, j,i /(IV )i and Q j,i /(IV )i data are then used in conjunction with Eqs. (8)–(10) to obtain Rc,2, R2, Rc,3, R1 + Rc,1, and R3 + Rc,4. It is worth noting that knowledge of the aforementioned nine thermal resistance values is sufficient to completely determine the 16 θ j,i /(IV )i measurement data based on the thermal resistance circuit. Hence, some of the 16 θ j,i /(IV )i measurement data are interdependent. While all 16 measurement data are used to obtain Rb,1, Rb,2, Rb,3, and Rb,4 from Eq. (6), it is sufficient to use a subset of these 16 measurement data to obtain Rc,2, R2, Rc,3, R1 + Rc,1, and R3 + Rc,4 from some of the interdependent equations. Without measurement errors, the use of different subsets of the measurement data in the corresponding equations should yield the same answer. The presence of errors in the actual measurements can cause variation in these resistance values obtained from different subsets of the measurement data. Such variation can be used to evaluate the measurement uncertainty. Specifically, four different combinations of three equations out of the set of four equations given by Eq. (8) can yield four values each for Rc,2, R2, and Rc,3. Similarly, four values can be obtained for R1 + Rc,1 and R3 + Rc,4 from the set of four equations expressed in Eqs. (9) and (10), respectively. For each of these five thermal resistances, we have calculated the average and range of variation of the four obtained resistance values, which are shown in Figs. 3(c) and 3(d) as symbols and the associated error bars, respectively.
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FIG. 2. Measurement results for the 740 nm wide, 220 nm thick Si nanowire sample. (a) Measured electrical resistance increases of RT1, RT2, RT3, and RT4 as a function of the heating current in RT1 at sample stage temperature T0 = 350 K. (b) Measured R e, j (I = 0) at different sample stage temperatures (T0). The TCR is obtained from the linear fit of each set of the measurement data. (c) Measured average temperature rise of each RT as a function of Joule heating in the first line when T0 = 350 K. (d) Net heat flow rate across the contact point from each RT into the nanowire sample as a function of the heating current (I ) in the first line.
For both samples, the two contact thermal resistances Rc,2 and Rc,3 are much smaller than the intrinsic thermal resistance R2 of the middle suspended segment of the sample.
Normalization of the measured room-temperature Rc,2 and Rc,3 values with the apparent contact areas of the two samples would yield interface thermal conductance in the range
FIG. 3. Measured thermal resistances of four suspended RTs ((a) and (b)) and five sample thermal resistances ((c) and (d)) for the 740 nm ((a) and (c)) and 240 nm ((b) and (d)) wide Si nanowire samples. The error bar of some data points is smaller than the symbol size.
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between 38 and 230 × 106 W m−2 K−1. These values are comparable to those obtained from a reported time domain thermal reflectance (TDTR) measurement between sputtered Au and Au(Pd) alloys on SiO2, which are higher than that measured by TDTR between transferred Au and SiO2.40 The relatively high interface thermal conductance values for the Si nanowire sample indicate that the high-temperature annealing process for removing the PMMA layer could have improved the thermal contact. In addition, the obtained Rc,2 and Rc,3 also increase slightly as the temperature decreases from 400 K to 100 K. Such temperature variation is consistent with the dependence of the thermal interface conductance on the specific heat, which decreases with decreasing temperature.41 The noise and resolution of the reported measurement are currently limited by the 10–50 mK temperature fluctuation of the cryostat sample stage, similar to an earlier measurement that used serpentine Pt resistance thermometer devices.5 If the maximum applied temperature rise is about 5 K for measurements near room temperature, the largest sample or contact resistance that can be measured would be about 100 times the beam thermal resistance (Rb ), or about 300 K/µW for the current design, while the smallest sample or contact resistance that can be measured would be about 100th of Rb , or about 0.03 K/µW for the current design, according to an analysis similar to that reported in an earlier work.5 In comparison, the thermal resistance is 6 × 104 K/µW and 2 K/µW, respectively, for a 10-nm-diameter, 5-µm-long nanowire with a low thermal conductivity of 1 W m−1 K−1 and a 3-µm-wide and 5-µmlong single-layer graphene with a high thermal conductivity of about 2500 W m−1 K−1. Therefore, the current method is already suitable for measuring the graphene sample. In addition, a differential method can be used to considerably reduce the impact of the cryostat temperature fluctuation, as reported in recent studies.37,42 The dimension of the measurement device can also be modified to match Rb, j with the sample resistance. With these additional efforts, this method should be able to measure a sample thermal resistance as large as the order of 1 × 104 K/µW, comparable to those achievable with the serpentine Pt resistance thermometer devices,37,42 as well as a contact thermal resistance as small as approaching 1 × 10−3 K/µW. Figure 4 shows the thermal conductivity obtained from the measured intrinsic thermal resistance R2 and measured dimensions of the two Si nanowires. For comparison, we have used two Si thermal conductivity models, one by Morelli et al.43 and the other by Wang and Mingo,44 to calculate the thermal conductivity of bulk Si crystals and nanowires. In the calculation, the equivalent boundary scattering mean free path is taken as l b = 2(wt/π)1/2, which is incorporated with phononphonon and phonon-isotope scatterings via Matthiessen’s rule. Although the specified impurity doping for the parent Si film is low, it is unclear whether additional point defects could have been generated by the reactive ion etching process in the patterned Si nanowire sample similar to that found in an earlier study on Si etching.45 Hence, the calculation has not accounted for phonon scattering by impurity dopants and other point defects. In addition, we found a slight discrepancy between our calculated thermal conductivity based on the parameters given by Morelli et al.43 and their reported calculation result.
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FIG. 4. Measured thermal conductivity of the 740 nm wide (circles) Si nanowire sample and the 240 nm wide (squares) Si nanowire sample, both with 220 nm thickness. Also included for comparison are the measured thermal conductivity data of bulk Si crystals (diamonds) reported by Glassbrenner and Slack,46 and two different theoretical calculation results for bulk Si and Si nanowires with the same equivalent diameter as the two samples measured here. The theoretical calculations are based on the models reported by Morelli, Heremans, and Slack43 (dashed lines) and Wang and Mingo44 (solid lines). Thin green, red, and thick blue lines and symbols are used for the results for bulk, 740 × 220 nm2, and 240 × 220 nm2 cross section. The uncertainty of some data points is smaller than the symbol size.
For both Si nanowires, our measurement data lie between the results of the two models and are close to those calculated with the model of Wang and Mingo,44 where the upper phonon frequency limit was adjusted according to a previous measurement result of a Si nanowire. With a better knowledge of the impurity and defect concentrations, the intrinsic thermal conductivity values obtained in the four-probe measurement method can be useful for further refinement of these theoretical models. IV. SUMMARY
These results demonstrate that the new four-probe thermal measurement method reported here is highly effective in probing the intrinsic thermal transport property of nanostructures. The measurement device consists of only four suspended resistance thermometer lines, which can be readily fabricated with the use of photolithography and simple etching processes. The nanostructure can be assembled on the measurement device with a high yield via the use of a carrier layer. Most importantly,
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the contact thermal resistance, which has given rise to many outstanding questions on existing thermal measurement results of nanostructures, can be independently determined together with the intrinsic thermal conductance of the nanostructure. Hence, we expect that this four-probe thermal transport measurement method for nanostructures will enable future discoveries of unique size-dependent thermal transport phenomena that have or have not been predicted by existing theoretical studies, in addition to providing reliable experimental data for calibrating theoretical models. ACKNOWLEDGMENTS
Work by J.K. and L.S. was supported by Office of Naval Research Award No. N00014-14-1-0258. Work by E.O. was supported by Department of Energy Office of Basic Energy Office Award No. DE-FG02-07ER46377. D.P.S. acknowledges the support from a NSERC Postdoctoral Fellowship. The authors thank Arden Moore for helpful discussion on the thermal conductivity models. 1L.
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W. Cai, A. L. Moore, Y. W. Zhu, X. S. Li, S. S. Chen, L. Shi, and R. S. Ruoff, Nano Lett. 10(5), 1645-1651 (2010). 17S. Shen, A. Henry, J. Tong, R. Zheng, and G. Chen, Nat. Nanotechnol. 5(4), 251-255 (2010). 18Z. Wang, R. Xie, C. T. Bui, D. Liu, X. Ni, B. Li, and J. T. L. Thong, Nano Lett. 11(1), 113-118 (2011). 19M. T. Pettes, I. S. Jo, Z. Yao, and L. Shi, Nano Lett. 11(3), 1195-1200 (2011). 20I. Jo, M. T. Pettes, J. Kim, K. Watanabe, T. Taniguchi, Z. Yao, and L. Shi, Nano Lett. 13(2), 550-554 (2013). 21C. Canetta, S. Guo, and A. Narayanaswamy, Rev. Sci. Instrum. 85(10), 104901 (2014). 22A. Weathers and L. Shi, in Annual Review of Heat Transfer, edited by G. Chen (2013), Vol. 16, pp. 101-134. 23G. Binnig, H. Rohrer, C. Gerber, and E. Weibel, Phys. Rev. Lett. 49(1), 57-61 (1982). 24G. Binnig, C. F. Quate, and C. Gerber, Phys. Rev. Lett. 56(9), 930-933 (1986). 25M.-H. Bae, Z. Li, Z. Aksamija, P. N. Martin, F. Xiong, Z.-Y. Ong, I. Knezevic, and E. Pop, Nat. Commun. 4, 1734 (2013). 26Z. L. Wang, D. W. Tang, X. B. Li, X. H. Zheng, W. G. Zhang, L. X. Zheng, Y. T. Zhu, A. Z. Jin, H. F. Yang, and C. Z. Gu, Appl. Phys. Lett. 91(12), 123119 (2007). 27C. W. Chang, D. Okawa, H. Garcia, A. Majumdar, and A. Zettl, Phys. Rev. Lett. 101(7), 075903 (2008). 28T. K. Hsiao, H. K. Chang, S. C. Liou, M. W. Chu, S. C. Lee, and C. W. Chang, Nat. Nanotechnol. 8(7), 534-538 (2013). 29X. Xu, L. F. C. Pereira, Y. Wang, J. Wu, K. Zhang, X. Zhao, S. Bae, C. T. Bui, R. Xie, J. T. L. Thong, B. H. Hong, K. P. Loh, D. Donadio, B. Li, and B. Özyilmaz, Nat. Commun. 5, 3689 (2014). 30L. Shi, Appl. Phys. Lett. 92(20), 206103 (2008). 31M. T. Pettes and L. Shi, Adv. Funct. Mater. 19(24), 3918-3925 (2009). 32L. Shi, Nanoscale Microscale Thermophys. Eng. 16(2), 79-116 (2012). 33D. Liu, R. Xie, N. Yang, B. Li, and J. T. L. Thong, Nano Lett. 14(2), 806-812 (2014). 34J. de Boor and V. Schmidt, Adv. Mater. 22(38), 4303-4307 (2010). 35J. H. Seol, A. L. Moore, Z. Yao, and L. Shi, J. Heat Transfer 133, 022403 (2011). 36M. M. Sadeghi, I. Jo, and L. Shi, Proc. Natl. Acad. Sci. U. S. A. 110(41), 16321-16326 (2013). 37A. Weathers, K. D. Bi, M. T. Pettes, and L. Shi, Rev. Sci. Instrum. 84(8), 084903 (2013). 38K. D. Bi, A. Weathers, S. Matsushita, M. T. Pettes, M. Goh, K. Akagi, and L. Shi, J. Appl. Phys. 114(19), 194302 (2013). 39A. L. Moore and L. Shi, Meas. Sci. Technol. 22(1), 015103 (2011). 40D.-W. Oh, S. Kim, J. A. Rogers, D. G. Cahill, and S. Sinha, Adv. Mater. 23(43), 5028-5033 (2011). 41E. Swartz and R. Pohl, Rev. Mod. Phys. 61(3), 605-668 (1989). 42M. C. Wingert, Z. C. Y. Chen, S. Kwon, J. Xiang, and R. Chen, Rev. Sci. Instrum. 83(2), 024901 (2012). 43D. T. Morelli, J. P. Heremans, and G. A. Slack, Phys. Rev. B 66(19), 195304 (2002). 44Z. Wang and N. Mingo, Appl. Phys. Lett. 97(10), 101903 (2010). 45G. S. Oehrlein, Mater. Sci. Eng.: B 4(1–4), 441-450 (1989). 46C. J. Glassbrenner and G. A. Slack, Phys. Rev. 134(4A), A1058-A1069 (1964).
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A four-probe thermal transport measurement method for nanostructures Jaehyun Kim, Eric Ou, Daniel P. Sellan, and Li Shia) Department of Mechanical Engineering, The University of Texas at Austin, Austin, Texas 78712, USA
(Received 6 February 2015; accepted 18 March 2015; published online 6 April 2015) Several experimental techniques reported in recent years have enabled the measurement of thermal transport properties of nanostructures. However, eliminating the contact thermal resistance error from the measurement results has remained a critical challenge. Here, we report a different four-probe measurement method that can separately obtain both the intrinsic thermal conductance and the contact thermal resistance of individual nanostructures. The measurement device consists of four microfabricated, suspended metal lines that act as resistive heaters and thermometers, across which the nanostructure sample is assembled. The method takes advantage of the variation in the heat flow along the suspended nanostructure and across its contacts to the four suspended heater and thermometer lines, and uses sixteen sets of temperature and heat flow measurements to obtain nine of the thermal resistances in the measurement device and the nanostructure sample, including the intrinsic thermal resistance and the two contact thermal resistances to the middle suspended segment of the nanostructure. Two single crystalline Si nanowires with different cross sections are measured in this work to demonstrate the effectiveness of the method. This four-probe thermal transport measurement method can lead to future discoveries of unique size-dependent thermal transport phenomena in nanostructures and lowdimensional materials, in addition to providing reliable experimental data for calibrating theoretical models. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4916547]
I. INTRODUCTION
Advances are continuously made in both top-down nanofabrication and bottom-up syntheses of nanostructures, including carbon nanotubes, inorganic and organic nanowires and nanofibers, graphene, and, more recently, a variety of twodimensional (2D) materials. Because the characteristic dimension of these nanostructures is reduced to the scale of the scattering mean free path or even the wavelength of the charge or heat carriers, their electrical, thermal, and thermoelectric properties can be very different from their bulk counterparts. For example, elimination of inter-layer interaction is expected to result in enhanced basal plane thermal conductivity of clean suspended single-layered graphene (SLG) and singlewalled carbon nanotubes (SWCNTs) compared to the already high values found for graphite.1 Scattering is especially suppressed for low-frequency phonons in clean suspended SLG and SWCNTs, so that the thermal conductivity contribution from these long-wavelength modes can be limited mainly by scattering at the lateral edges instead of phonon-phonon processes.2 This and other size-dependent thermal properties have received increasing interest because they can lead to both new insights into thermal transport physics and potential applications such as thermal management and thermoelectric energy conversion. The interest in the unique thermal properties of nanostructures has motivated the development of several methods for thermal transport measurements of carbon nanotubes, nanowires, nanofilms, graphene, and other 2D layered materials.3–21 These methods have explored different thermometry a)Author to whom correspondence should be addressed. Electronic mail:
[email protected]. 0034-6748/2015/86(4)/044901/7/$30.00
techniques including suspended micro-devices with resistance thermometry, Raman spectroscopy, thermal reflectance, and bimaterial cantilevers, as discussed in a recent review.22 While the invention of scanning tunneling microscopy,23 atomic force microscopy,24 and other new experimental techniques nucleated the broader field of nanoscience, the development of these thermal transport measurement techniques has significantly enhanced the capability for the study of nanoscale thermal transport. A number of these measurements have revealed that scattering by either the surface roughness or a disordered layer in contact with the nanostructure can result in considerable reduction of the axial or in-plane thermal conductivity.6,12–14,19,20 Meanwhile, some measurement results have been used to suggest quasi-ballistic phonon transport in carbon nanotubes and graphene.7,25 In addition, several other experimental works have suggested that the thermal conductivity of carbon nanotube, graphene, or SiGe nanowire samples increases with the sample length as a result of the breakdown of Fourier’s law in these nanostructure samples.26–29 While these experimental reports of ballistic phonon transport and length-dependent thermal conductivity have suggested a new regime of thermal transport physics, outstanding questions have remained on uncertainties in the challenging thermal transport measurements of nanostructures.30–32 Among these challenges, the inability to separate the contact thermal resistance from the intrinsic thermal conductance of nanostructure samples has presented a major source of error, which has given rise to critical questions on the exact cause of the lengthdependent thermal conductivities or abnormally low thermal conductivity values found in a number of measurements on nanostructures. Several efforts have been devoted to the development of measurement methods for independently probing both the
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contact thermal resistance and intrinsic thermal conductance of nanostructures. In one method based on a suspended microdevice that consists of two membranes with built-in serpentine Pt resistance thermometers, two Pt electrodes were patterned on each membrane to measure the thermovoltage drop along the nanostructure segment in contact with the membrane.11 The measured thermovoltage was then used to determine the temperature drop at the contact and the contact thermal resistance. This method is limited to a conducting nanostructure sample with a large and uniform Seebeck coefficient compared to that of the Pt electrodes, and requires Ohmic electrical contact between the Pt electrode and the sample. A different approach is based on focused laser or electron beam heating along the nanostructure sample suspended between the two membranes.9,33 While the localized heating approach can potentially provide spatial distribution of the thermal resistance along a nanostructure sample, further efforts are required to probe thermal transport at different temperatures, especially in the low temperature regime where intriguing nanoscale thermal transport phenomena are pronounced, and to address experimental complications such as alteration of the structure or thermal properties of the nanostructure by the focused laser or electron beam. In addition, there have been efforts for developing the thermal analogue of the electrical van der Pauw method,34 which is limited to isotropic thin film samples and is not applicable to nanowire and nanoribbon structures. Therefore, there is still a lack of an effective and general experimental method for probing both the contact thermal resistance and intrinsic thermal conductance of nanostructures at different temperatures. Establishment of such experimental methods is necessary not only for advancing the field of nanoscale thermal transport but also for establishing the intrinsic thermal properties of a wide variety of nanostructures, which are continuously being synthesized or discovered by the nanomaterials research community. Here, we report a different four probe thermal resistance measurement method for determining both the contact thermal resistances and the intrinsic thermal conductivity of a nanostructured sample at different temperatures. The effectiveness of this method is demonstrated by employing this method for measuring patterned Si nanowire samples. The obtained measurement results are compared to theoretical calculation results obtained from two different thermal conductivity models.
II. MEASUREMENT METHOD
The four-probe thermal measurement device consists of four suspended Pt/SiNx resistance thermometer lines (RT1, RT2, RT3, and RT4), as shown in Figs. 1(a) and 1(b). While the i th Pt/SiNx line is electrically heated by a direct current (I) during the measurement, the average temperature rise (θ j,i ) in the j th Pt/SiNx line is obtained from the thermally induced increase of the electrical resistance measured using a fourprobe configuration.35,36 With i and j both ranging from 1 to 4, a total of 4 × 4 measurement data of the θ j,i /(IV )i ratio can be obtained, where V and (IV )i are the measured voltage drop and Joule heating rate in the i th heating line, respectively. In the following, we show that these 16 measurement data
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FIG. 1. (a) Optical micrograph (left panel) and scanning electron micrographs (SEMs, right panel) of a 240 nm wide, 220 nm thick patterned Si nanowire sample assembled across four suspended Pt/SiNx lines that serve as resistive heaters and thermometers. As shown in the top SEM, a small V-shape protrusion is patterned at the center of the first thermometer line from the left to assist in the measurement of the deviation (di and dj ) between the center of each thermometer line and the contact point to the nanostructure. (b) Optical micrograph of a 740 nm wide, 220 nm thick patterned Si nanowire sample assembled across four suspended Pt/SiNx lines together with a schematic of the temperature profiles along the Pt/SiNx heater line (i th line) and one Pt/SiNx resistance thermometer line ( j th line, j , i). (c) Thermal resistance circuit of the measurement device when the first Pt/SiNx line is electrically heated at a rate of (IV )1. R 1, R 2, and R 3 are the thermal resistances of the left, middle, and right suspended segments of the suspended Si nanowire sample, respectively. R c, j is the contact thermal resistance between the j th Pt/SiNx line and the sample. R b, j is the thermal resistance of the j th Pt/SiNx resistance thermometer line. θ c, j, i is the j th Pt/SiNx line temperature at the contact to the sample when the i th line is used as the heater line. The temperature rise θ 0 at the two ends of each of the suspended Pt/SiNx lines is assumed to be negligible.
can be used to obtain and separate nine of the eleven thermal resistances in the system (Fig. 1(c)), including the two contact thermal resistances and the intrinsic thermal resistance of the middle suspended segment of the nanostructure sample. First, the temperature rise (θ c, j,i ) of the j th thermometer line at the contact point to the nanostructure can be obtained from the measured θ j,i when the i th line is electrically heated. Because the spreading thermal resistance in the Si substrate is two orders of magnitude lower than that of the suspended thermometer line, the temperature rise (θ 0) at the two ends of each suspended thermometer line is negligible. In addition, the measurement is conducted with the sample in high vacuum to eliminate surface heat loss to gas molecules. Based on a fin heat transfer analysis similar to that reported in two prior works,37,38 the ratio between radiation heat loss from the surface of a beam and the heat conduction inside the beam can be obtained as α = cosh βL − 1, where β = 4εσT 3 P/κ A, κ, L, P, and A are the thermal conductivity, length, perimeter of the cross section, and cross section area of the beam, respectively, T is the average temperature of the system, and ε is the surface emissivity. For the same aspect ratio PL/A, this ratio decreases with decreasing L and becomes less than 1.5 × 10−3 for the thermometer line and less than 1 × 10−6 for the silicon nanowire measured in this work at temperatures lower than
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400 K. Hence, radiation loss is negligible. Therefore, if the j th thermometer line is not the i th heating line, the temperature distribution in the j th line is linear, as illustrated in Fig. 1(b), so that θ c, j,i = 2θ j,i , for j , i.
(1)
In addition, the heat flow from each thermometer line into the nanostructure can be obtained from the thermal resistance circuit of Fig. 1(c) as Q j,i = −θ c, j,i /Rb, j ,
for
j ,i
(2a)
Equations (1), (2), and (7) yield 16 pairs of θ c, j,i /(IV )i and Q j,i /(IV )i data, with both i and j ranging from 1 to 4. These 16 pairs of data can be used to obtain five of the thermal resistances in the circuit in Fig. 1(c), where R1, R2, and R3 are the intrinsic thermal resistances of the left, middle, and right suspended segments of the nanostructure, respectively, and Rc,1, Rc,2, Rc,3, and Rc,4 are the contact thermal resistances at the four contacts between the nanostructure and the thermometer lines. Based on the thermal resistance circuit in Fig. 1(c), θ c,2,i − θ c,3,i = Q2,i Rc,2 + (Q1,i + Q2,i )R2
and Q i,i
=−
j, j,i
Q j,i .
Here, the thermal resistance of each of the four thermometer lines is given as ( ) 2 d j L j 1− , (3) Rb, j = 2κ j A j L j where κ j , A j , and 2L j are the effective thermal conductivity, cross-sectional area, and length of the j th suspended thermometer line, respectively, and d j is the deviation of the center of the j th line from its point of contact to the nanostructure sample, as shown in Fig. 1(b). From the Joule heating in the i th line and the heat flow boundary condition at the contact point given by Eq. (2b), the solution to the heat conduction equation yields the following parabolic temperature distribution in the line as a function of distance (x) from its center: d i + (−1) H (d i −x) L i (IV )i 2 2 Li − x + θ i,i (x) = 4κ i Ai L i 2κ i Ai L i H (d i −x) × x − (−1) L i Q i,i , (4) where the Heaviside step function H( χ) takes the value of 0 and 1 for χ < 0 and χ ≥ 0, respectively. The obtained temperature distribution can be used to calculate the average temperature rise in the i th heater line as Li 1 θ i,i = θ i,i (x)dx. (5) 2L i −L i The integration can be carried out to obtain the following relationship: 4 θ j,i /(IV )i 1 , for i = 1, 2, 3, and 4. = j=1 Rb, j 3 1 − (d i /L i )2 (6) This set of four equations can be written in a matrix form and used to obtain the thermal resistances of the four thermometer lines, Rb,1, Rb,2, Rb,3, and Rb,4 based on the 16 sets of θ j,i /(IV )i measurement data, which are the elements of an invertible 4 × 4 matrix. With the four obtained Rb, j values, the contact point temperature rise for the heater line can now be calculated from Eq. (4) as θ c,i,i ≡ θ i,i (x = d i ) 4 θ j,i θ i,i + (IV )i = Rb,i − 2* − . j=1 Rb, j Rb,i - 2 ,
− Q3,i Rc,3, for i = 1, 2, 3, 4.
(2b)
(7)
(8)
Among this set of four equations, three are independent from each other and can be used to obtain the three unknown thermal resistances R2, Rc,2, and Rc,3 based on the measured θ c, j,i /(IV )i and Q j,i /(IV )i data. In addition, the thermal resistance circuit can be used to obtain the following: θ c,1,i − θ c,2,i = Q1,i (R1 + Rc,1) − Q2,i Rc,2, for i = 1, 2, 3, 4. (9) One of these four equations can be used to obtain the one unknown resistance (R1 + Rc,1) based on the above-obtained Rc,2 and the measured θ c, j,i /(IV )i and Q j,i /(IV )i data, whereas the other three equations are redundant. Similarly, (R3 + Rc,4) can be obtained from one of the following four equations derived from the thermal resistance circuit: θ c,4,i − θ c,3,i = Q4,i (R3 + Rc,4) − Q3,i Rc,3, for i = 1, 2, 3, 4. (10) Therefore, with the use of a linear set of equations, the 16 θ j,i /(IV )i measurement data can be used to obtain the nine thermal resistances shown in the thermal circuit in Fig. 2(b), including Rb,1, Rb,2, Rb,3, Rb,4, Rc,2, R2, Rc,3, R1 + Rc,1, and R3 + Rc,4. Because the heat flow across each end contact (Rc,1 or Rc,4) is the same as that through the adjacent suspended sample segment (R1 or R3), the contact thermal resistance component cannot be separated from R1 + Rc,1 and R3 + Rc,4. In comparison, the heat flow rates across the two middle contacts (Rc,2 and Rc,3) and the middle sample segment (R2) are different and depend on which Pt/SiNx line is used as the heater line. This feature allows separate determination of Rc,2, Rc,3, and the intrinsic thermal resistance (R2) of the middle suspended segment.
III. RESULTS
To demonstrate the effectiveness of this four-probe thermal transport measurement method, we have employed the method to measure two patterned Si nanowire samples with cross sections of 240 nm × 220 nm and 740 nm × 220 nm. The cross sections were intentionally chosen to be relatively large, so that the results obtained from the new method can be compared with established theoretical models without the need to consider quantum confinement or anomalous surface scattering effects that are not yet well understood.
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The Si nanowire sample was patterned from a silicon-oninsulator wafer with a 220 nm thick top Si layer specified with 5 × 1016 cm−3 boron doping on top of a 3 µm thick buried SiO2 layer. After patterning the Si nanowire with electron beam lithography and reactive ion etching, the SiO2 underlying the Si nanowire is etched in diluted hydrogen fluoride acid. The suspended Si nanowire sample was transferred to another wafer coated with a Polyvinyl alcohol (PVA) layer, followed by the spin coating of a Poly(methyl methacrylate) (PMMA) layer on top of the Si nanowire and the PVA layer. After the PVA layer is dissolved in deionized water, the Si nanowire was detached from the wafer together with the PMMA carrier layer, which was transferred and aligned to the four suspended Pt/SiNx thermometer lines fabricated on another Si wafer. The carrier PMMA layer was removed by heating to a temperature of 350 ◦C in a low-pressure tube furnace with argon and hydrogen flows. This process resulted in the Si nanowire sample suspended across the four Pt/SiNx lines. Each of the four Pt/SiNx lines was made of 60 nm thick Pt deposited on 300 nm thick SiNx with a 10 nm Cr adhesion layer in between. For the device shown in Fig. 1, each suspended Pt/SiNx line is 200 µm long and 2 µm wide, with the width limited by the photolithographic process. The line width can be further reduced with the use of electron beam lithography. When the Pt/SiNx line width is much smaller than the length of the middle suspended segment of the nanostructure sample and the length of the Pt/SiNx line is much longer than the width of the nanostructure, the contact between the thermometer line and the nanostructure can be accurately approximated as a point as assumed in the analysis. For measurements of a wider sample than the Si nanowires, such as a graphene flake, the length of the four thermometer lines can be increased to be much larger than the sample width, so that the contact area can be treated as a point contact. In addition, this measurement obtains the thermal resistance (R2) in the sample region between the two middle point contacts. The sample thickness and the thermometer line width need to be small compared to the length of the middle suspended segment, in order to assume one-dimensional temperature profile in the middle sample segment, which is necessary for obtaining the sample thermal conductivity as κ = L/R2 A, where L and A are the length and cross-sectional area of the middle segment, respectively, when the transport is diffusive in the middle segment. The sample was placed on the sample stage of an evacuated cryostat for the thermal measurements. During the measurement with the heating current I ramped between −350 µA and +350 µA in the i th heating line, the four-probe electrical resistance (Re,i ) of the i th heating line was obtained directly as the ratio of the measured voltage drop V and the heating current I, whereas a lock-in amplifier was used to measure the four-probe resistance (Re, j ) of each of the three other thermometer lines with a 1 µA sinusoidal excitation current. The sinusoidal sensing current is about a factor of 350 smaller than the maximum direct heating current applied to the heating line, so that the corresponding Joule heating is five orders of magnitude smaller. Moreover, the constant sinusoidal sensing current influences only the absolute temperature, but not the measured temperature rise (θ j,i ) of a thermometer line caused
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by the direct heating current in the heater line. As such, the theoretical results of θ j,i distributions are not affected by the presence of the sinusoidal current. In addition, it has been shown in a previous work that the theoretical θ j,i distribution is much more immune to errors caused by ignoring the surface radiation loss than the absolute temperature distribution.39 The silicon nanowire samples reported in this study did not make electrical contact to the Pt lines because of the presence of an amorphous native oxide on the nanowire. A thin dielectric can also be deposited on the Pt line to prevent electrical contact to the sample. However, this four-probe method can also be used to measure a nanostructure sample that makes electrical contacts to the Pt lines. In such a measurement, a floating current source is used to provide the heating current to the heating Pt line. At the same heating current, the electrical resistance of each of the other three Pt lines is measured separately with a small sinusoidal excitation current and a lock-in amplifier while the other two Pt lines are electrically floating. Such a measurement procedure prevents electrical current flow in the nanostructure sample. Figure 2(a) shows a set of measurement data of the resistance rise, ∆Re, j = Re, j (I) − Re, j (I = 0), as a function of the heating current I applied to the first Pt/SiNx line. The temperature coefficient of resistance (TCR) was obtained from the Re, j (I = 0) measured at different sample stage temperature (T0) values, as shown in Fig. 2(b). The obtained TCR values were used to convert the measured resistance rises ∆Re, j into the average temperature rise (θ j,i ) in each thermometer line. The θ j,i /(IV )i ratio was obtained as the slope of the linear θ j,i versus (IV )i data, as shown in Fig. 2(c). The θ j,i /(IV )i data were used to obtain the Rb,1, Rb,2, Rb,3, and Rb,4 from Eq. (6), and 16 pairs of θ c, j,i /(IV )i and Q j,i /(IV )i data according to Eqs. (1), (2), and (7). The obtained θ c, j,i /(IV )i and Q j,i /(IV )i data are then used in conjunction with Eqs. (8)–(10) to obtain Rc,2, R2, Rc,3, R1 + Rc,1, and R3 + Rc,4. It is worth noting that knowledge of the aforementioned nine thermal resistance values is sufficient to completely determine the 16 θ j,i /(IV )i measurement data based on the thermal resistance circuit. Hence, some of the 16 θ j,i /(IV )i measurement data are interdependent. While all 16 measurement data are used to obtain Rb,1, Rb,2, Rb,3, and Rb,4 from Eq. (6), it is sufficient to use a subset of these 16 measurement data to obtain Rc,2, R2, Rc,3, R1 + Rc,1, and R3 + Rc,4 from some of the interdependent equations. Without measurement errors, the use of different subsets of the measurement data in the corresponding equations should yield the same answer. The presence of errors in the actual measurements can cause variation in these resistance values obtained from different subsets of the measurement data. Such variation can be used to evaluate the measurement uncertainty. Specifically, four different combinations of three equations out of the set of four equations given by Eq. (8) can yield four values each for Rc,2, R2, and Rc,3. Similarly, four values can be obtained for R1 + Rc,1 and R3 + Rc,4 from the set of four equations expressed in Eqs. (9) and (10), respectively. For each of these five thermal resistances, we have calculated the average and range of variation of the four obtained resistance values, which are shown in Figs. 3(c) and 3(d) as symbols and the associated error bars, respectively.
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FIG. 2. Measurement results for the 740 nm wide, 220 nm thick Si nanowire sample. (a) Measured electrical resistance increases of RT1, RT2, RT3, and RT4 as a function of the heating current in RT1 at sample stage temperature T0 = 350 K. (b) Measured R e, j (I = 0) at different sample stage temperatures (T0). The TCR is obtained from the linear fit of each set of the measurement data. (c) Measured average temperature rise of each RT as a function of Joule heating in the first line when T0 = 350 K. (d) Net heat flow rate across the contact point from each RT into the nanowire sample as a function of the heating current (I ) in the first line.
For both samples, the two contact thermal resistances Rc,2 and Rc,3 are much smaller than the intrinsic thermal resistance R2 of the middle suspended segment of the sample.
Normalization of the measured room-temperature Rc,2 and Rc,3 values with the apparent contact areas of the two samples would yield interface thermal conductance in the range
FIG. 3. Measured thermal resistances of four suspended RTs ((a) and (b)) and five sample thermal resistances ((c) and (d)) for the 740 nm ((a) and (c)) and 240 nm ((b) and (d)) wide Si nanowire samples. The error bar of some data points is smaller than the symbol size.
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between 38 and 230 × 106 W m−2 K−1. These values are comparable to those obtained from a reported time domain thermal reflectance (TDTR) measurement between sputtered Au and Au(Pd) alloys on SiO2, which are higher than that measured by TDTR between transferred Au and SiO2.40 The relatively high interface thermal conductance values for the Si nanowire sample indicate that the high-temperature annealing process for removing the PMMA layer could have improved the thermal contact. In addition, the obtained Rc,2 and Rc,3 also increase slightly as the temperature decreases from 400 K to 100 K. Such temperature variation is consistent with the dependence of the thermal interface conductance on the specific heat, which decreases with decreasing temperature.41 The noise and resolution of the reported measurement are currently limited by the 10–50 mK temperature fluctuation of the cryostat sample stage, similar to an earlier measurement that used serpentine Pt resistance thermometer devices.5 If the maximum applied temperature rise is about 5 K for measurements near room temperature, the largest sample or contact resistance that can be measured would be about 100 times the beam thermal resistance (Rb ), or about 300 K/µW for the current design, while the smallest sample or contact resistance that can be measured would be about 100th of Rb , or about 0.03 K/µW for the current design, according to an analysis similar to that reported in an earlier work.5 In comparison, the thermal resistance is 6 × 104 K/µW and 2 K/µW, respectively, for a 10-nm-diameter, 5-µm-long nanowire with a low thermal conductivity of 1 W m−1 K−1 and a 3-µm-wide and 5-µmlong single-layer graphene with a high thermal conductivity of about 2500 W m−1 K−1. Therefore, the current method is already suitable for measuring the graphene sample. In addition, a differential method can be used to considerably reduce the impact of the cryostat temperature fluctuation, as reported in recent studies.37,42 The dimension of the measurement device can also be modified to match Rb, j with the sample resistance. With these additional efforts, this method should be able to measure a sample thermal resistance as large as the order of 1 × 104 K/µW, comparable to those achievable with the serpentine Pt resistance thermometer devices,37,42 as well as a contact thermal resistance as small as approaching 1 × 10−3 K/µW. Figure 4 shows the thermal conductivity obtained from the measured intrinsic thermal resistance R2 and measured dimensions of the two Si nanowires. For comparison, we have used two Si thermal conductivity models, one by Morelli et al.43 and the other by Wang and Mingo,44 to calculate the thermal conductivity of bulk Si crystals and nanowires. In the calculation, the equivalent boundary scattering mean free path is taken as l b = 2(wt/π)1/2, which is incorporated with phononphonon and phonon-isotope scatterings via Matthiessen’s rule. Although the specified impurity doping for the parent Si film is low, it is unclear whether additional point defects could have been generated by the reactive ion etching process in the patterned Si nanowire sample similar to that found in an earlier study on Si etching.45 Hence, the calculation has not accounted for phonon scattering by impurity dopants and other point defects. In addition, we found a slight discrepancy between our calculated thermal conductivity based on the parameters given by Morelli et al.43 and their reported calculation result.
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FIG. 4. Measured thermal conductivity of the 740 nm wide (circles) Si nanowire sample and the 240 nm wide (squares) Si nanowire sample, both with 220 nm thickness. Also included for comparison are the measured thermal conductivity data of bulk Si crystals (diamonds) reported by Glassbrenner and Slack,46 and two different theoretical calculation results for bulk Si and Si nanowires with the same equivalent diameter as the two samples measured here. The theoretical calculations are based on the models reported by Morelli, Heremans, and Slack43 (dashed lines) and Wang and Mingo44 (solid lines). Thin green, red, and thick blue lines and symbols are used for the results for bulk, 740 × 220 nm2, and 240 × 220 nm2 cross section. The uncertainty of some data points is smaller than the symbol size.
For both Si nanowires, our measurement data lie between the results of the two models and are close to those calculated with the model of Wang and Mingo,44 where the upper phonon frequency limit was adjusted according to a previous measurement result of a Si nanowire. With a better knowledge of the impurity and defect concentrations, the intrinsic thermal conductivity values obtained in the four-probe measurement method can be useful for further refinement of these theoretical models. IV. SUMMARY
These results demonstrate that the new four-probe thermal measurement method reported here is highly effective in probing the intrinsic thermal transport property of nanostructures. The measurement device consists of only four suspended resistance thermometer lines, which can be readily fabricated with the use of photolithography and simple etching processes. The nanostructure can be assembled on the measurement device with a high yield via the use of a carrier layer. Most importantly,
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the contact thermal resistance, which has given rise to many outstanding questions on existing thermal measurement results of nanostructures, can be independently determined together with the intrinsic thermal conductance of the nanostructure. Hence, we expect that this four-probe thermal transport measurement method for nanostructures will enable future discoveries of unique size-dependent thermal transport phenomena that have or have not been predicted by existing theoretical studies, in addition to providing reliable experimental data for calibrating theoretical models. ACKNOWLEDGMENTS
Work by J.K. and L.S. was supported by Office of Naval Research Award No. N00014-14-1-0258. Work by E.O. was supported by Department of Energy Office of Basic Energy Office Award No. DE-FG02-07ER46377. D.P.S. acknowledges the support from a NSERC Postdoctoral Fellowship. The authors thank Arden Moore for helpful discussion on the thermal conductivity models. 1L.
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