تجزیه و تحلیل عملکرد از روش حرارتی بازتاب گذرا برای اندازه گیری هدایت حرارتی مواد تک جداره

This work shows that the Fourier number (Fo) defines the shape and amplitude of the thermal response of a semi-infinite layer sample. The introduction of the responsivity, Rs, of the TTR method provides the ability to assess the performance of the thermal conductivity measurements. A simplified heat transfer analysis of a finite layer sample revealed that the properties ratio, (ρCpK)S/(ρCpK)L, and the layer thickness, h/δP, uniquely define both temperature response and measurement responsivity. If the material under test is the substrate, this work can help improve the measurement accuracy by selecting the appropriate thickness of the top layer. If the material is a layer on top of a known substrate, this work suggests that the accuracy of the TTR measurements can be fully maximized.

The high rate of innovation in the electronics and telecommunications fields has raised expectations for increased performance and functionality. Most advances have evolved from smart engineering and efficient manufacturing practices. Equally substantial gains, however, can be made from the introduction of innovative materials. Indeed, miniaturization and performance requirements have forced the use of existing materials beyond initially envisioned ranges and have spurred the development of special materials [1]. Knowledge of material properties is fundamental to the design process, especially for electronic and telecommunication devices, where performance depends heavily on electro-thermal interactions. Higher performance is only possible by significant reductions in the size of active features, which in turn can increase heat generation densities to critical levels. With the use of submicron devices came the realization that bulk and thin-film thermal properties differ markedly [2]. However, since no universal behavior is expected for these differences and since they cannot be predicted from theory [3], the properties of each material must be measured individually. Also, as films are typically layered and deposition techniques differ by manufacturer, it is important to measure the interface resistance of stacked layers [4]. The transient thermo-reflectance method (TTR) [5] is preferred among the various experimental techniques [6] used to determine the thermal conductivity of thin-film and multi-layered materials. The main advantage of the TTR method is that it is a non-contact and non-destructive optical approach, both for heating a sample under test and for probing the variations of its surface temperature [7]. Because the method is non-invasive, it is attractive for the measurement of the thermal properties of thin-layer materials whose investigation by invasive methods would present the difficulties of having to fabricate a measuring device into a sample, and then having to isolate and exclude the influence of the measuring device itself. The basic principle of the transient thermal reflectance method is to heat a sample by laser irradiation and probe the changes in the surface reflectivity of the heated material. The schematic in Fig. 1(a) depicts the square heating and round probing spots produced by the TTR system built by the authors at SMU (http://engr.smu.edu/netsl). The source of energy in the TTR method is normally provided by a pulsed laser with short pulse duration. During each pulse, a given volume on the sample surface heats up to a temperature level above ambient due to the laser light energy absorbed into the sample. The heating area is specified by adjusting the pulsing laser aperture and the optics of the system. The depth of the volumetric heating, on the other hand, is determined by the optical penetration depth, which is a function of laser wavelength and surface material properties. The heating level through the light penetration depth (δλ) obeys an exponential decay law, as described later. After each laser pulse is completed, the sample begins to cool down to the initial ambient temperature. During this process, the probing CW laser light reflected from the sample surface at the heating spot center (probing spot in Fig. 1(a)) is collected on a photodetector that reads the instantaneous surface reflectivity. The changes in surface reflectivity are linearly proportional to the changes in surface temperature, within a wide but finite temperature range [8]. Full-size image (30 K) Fig. 1. Problem geometry and important parameters: (a) heating and probing spots on a sample; and (b) different heat penetration depths imply either semi-infinite or finite layer behavior. Figure options The influence of a pulsed laser irradiation on a given material depends both on the optical properties of that material as well as on the wavelength and pulse duration of the laser itself. Thus, the wavelength and pulse width are important parameters in the determination of the effectiveness of the TTR method for different materials. Although several articles in the literature address the application of the TTR method for different stacked materials [9], a systematic investigation of the influence of the laser wavelength and pulse width on the performance of the method has not yet been presented even for the simplest bulk material measurements. Before proceeding, it is important to define what is meant by bulk material. Traditionally, a bulk sample, as opposed to a thin-film sample, has been used to denote a piece of material that is large enough for its size not to affect its thermal conductivity. In contrast, a thin-film sample has at least one of its dimensions in the sub-micron range and its thermal conductivity (K) is dependent on the size of that smallest dimension (normally K decreases as the sample gets thinner) [2]. The bulk and thin-film terms qualitatively characterize sample material properties in terms of the atomic structure of the material and the dimensions of the sample. As shown by Cahill [2] the thickness of the thin-film should be less than 10 times that of the mean free path of the energy carriers of the thin-film material. The differentiation between bulk and thin-film is of practical importance because the heat transfer mechanism in thin-films is influenced by boundary effects on the energy carriers, and, as a result, is governed by a more complex heat transfer equation and requires more effort for analysis than that in bulk. With an ultimate target of investigating thin-film samples, this work is concerned with the necessary first step of assessing the performance and determining the range of applicability of the TTR method only for bulk materials. In addition to the bulk and thin-film definitions, two other terms, semi-infinite and finite layers are introduced here to characterize the behavior of layers of a sample in the context of the TTR method (See Table 1 and Fig. 1(b)). Samples of interest are often formed by one or more layers of different materials deposited on the substrates. Since the TTR method depends on heating a sample under test with a pulsed laser of a specific pulse width, the method has a characteristic time associated with the duration of the temperature response within the sample. This transient nature of the TTR method in turn justifies considering a layer of sample as semi-infinite or finite layer. Specifically, if the overall heat penetration process experienced during a measurement cycle only partially involves a particular layer of the sample, this layer can be considered as a semi-infinite layer, irrespective of the structure of the underlying part (left side of Fig. 1(b)). If, on the other hand, the heat energy penetrates entirely through a layer, it becomes imperative to take into account the thermal properties of the material making up the affected layer as well as the interface resistance between this layer and its neighbors. As such, the layer of the sample should be classified as a finite layer (right side of Fig. 1(b)). Additional details of the approach to classifying samples will be given later. Table 1. Classification of material samples based on the ratio between the heat penetration depth and the thickness of the top layer Thermal conductivity behavior Transient temperature response behavior Semi-infinite layer Finite layer Bulk (I) K=const; h>δH (II) K=const; h<δH Thin-film (III) K=f(h); h>δH (IV) K=f(h); h<δH Table options The ultimate impact of this work would be to extend the TTR approach for the non-destructive measurement of the thermal properties of new and existing multi-layered materials, including metals, semiconductors, and dielectrics. However, in this first step, the investigation will focus on applying the TTR method to the measurement of the thermal conductivity of bulk samples having single semi-infinite or finite layers on the substrate, assuming that the effects of the interface resistance between them are negligible; future work will incorporate this additional complexity. This first step will make it possible to determine the method's range of applicability, to establish the criterion for distinguishing between semi-infinite and finite layers on the basis of an analytical solution of the governing heat transfer system, and to assess the performance of the TTR method for the type of layers mentioned above.

A one-dimensional analytical solution of heat transfer in a bulk, semi-infinite layer sample was derived in non-dimensional form in order to analyze the temperature response of the sample's surface. Most importantly, it was shown that the shape of the temperature response can be uniquely described by a single parameter, namely the Fo number. The analytical solution shows that the “amplitude” approach for extracting the thermal conductivity from experimental data leads to high uncertainty because of the necessity to obtain a calibration curve of temperature versus reflectivity. In contrast, the “shape” approach does not require prior calibration, and is therefore more attractive for thermal conductivity measurements. On the basis of the analytical solution, it was also possible to determine both the responsivity of the TTR method and the measurement uncertainty of a particular TTR system. That, in turn, revealed the range of applicability of the “shape” method, namely, Fo<100 for any TTR method and Fo<10 for the particular TTR system used in the authors' laboratory. The TTR method provides the maximum accuracy for Fo<0.1, which corresponds to a maximum responsivity value of Rsmax=0.125. The introduction of the heat penetration depth (δH) of the laser pulse energy provides a basis for classifying a given sample that has a top layer whose thickness is larger than δH as a semi-infinite layer sample. Specifically, the thickness of a semi-infinite layer should be at least seven times larger than the light penetration depth (δλ) of the irradiation produced by the heating laser. Otherwise, the sample should be classified as a finite layer sample. While the main approach to the heat transfer problem of a finite layer sample is numerical, a simplified analytical analysis of the problem has revealed two essential parameters, namely, the ratio of the physical properties of the materials that make up the sample, Φ=(ρCpK)S/(ρCpK)L, and the dimensionless layer thickness, h/δP, which together entirely define the behavior of the normalized temperature response as well as the responsivity of the conductivity measurement. Since it was discovered that the responsivity, Rs, exhibits a maximum at a specific (optimal) layer thickness of a finite layer sample, it becomes possible to tie the maximum responsivity, Rsmax, and the optimal thickness, hmax/δP, to the properties ratio, Φ, for both cases when the layer material or the substrate material are under test. For each case, two universal relationships were obtained relating Rsmax and hmax/δP to Φ over wide ranges of specific heat values, ρCp∈[0.8×106,25×106] J/m3 K, and thermal conductivity values, K∈[0.1,1000] W/m K. It was determined that the applicability of those relationships is limited by the inequality K/ρCp>5×10−6 m2/s for the layer properties, which nevertheless covers the range of useful microelectronics materials. The main conclusion of the findings associated with finite layer samples is that the TTR measurement performance (maximum responsivity) is higher if the material under test is a layer rather than a substrate. For example, the use of diamond as a layer on a silicon dioxide substrate (with thin gold metallization for the light absorption) increases the measurement accuracy by a factor of two as compared with the case where the diamond is a substrate covered by an optimal layer of gold.