Thermal conductivity measurements of sub-surface buried substrates by steady-state thermoreflectance
稳态热反射法测量地下埋藏基底的热导率

Thin films with thicknesses ranging from nanometer to micrometer length scales have become an integral part of transistors,^1,2 thermoelectric devices,³ optical coatings,⁴ solar cells,⁵ and memory devices.^6–8 As the device efficiency and reliability are often dictated by the thermal performance, it is of crucial importance to properly characterize the thermal properties of the thin films and substrates.⁹ Traditional optical pump–probe and electrothermal methods, such as time-domain thermoreflectance (TDTR),^10–13 frequency-domain thermoreflectance (FDTR),¹⁴ and the 3ω method,^15,16 are widely used to measure the thermal conductivity of thin films.¹⁷ However, both TDTR and FDTR have shallow thermal penetration depths (∼0.2 to 3 µm) under standard operating conditions.^18,19 As a result, they often cannot measure thermal conductivity of buried substrates located beyond these length scales. The disadvantages of the 3ω method, on the other hand, include its requirement of large and flat sample surface,²⁰ complex microfabrication,²¹ and challenging sample preparation, particularly for electrically conducting and semiconducting materials.¹⁸ These requirements can often limit the applicability of the 3ω method for buried substrate measurements. The recently developed optical pump–probe technique steady-state thermoreflectance (SSTR) offers a solution to these issues as its thermal penetration depth can be much larger than those produced during TDTR and FDTR measurements.^19,22 In addition, SSTR is well suited for electrically conducting or semiconducting materials and can operate on an optically smooth surface of 100 × 100 µm² area or less.^19,23 Therefore, a detailed study into the thermal penetration depth of the SSTR technique and its ability to measure the thermal conductivity of sub-surface buried substrates is highly warranted.
厚度从纳米到微米尺度的薄膜已成为晶体管、热电器件、光学涂层、太阳能电池和存储器件的重要组成部分。由于器件的效率和可靠性通常由热性能决定，因此正确表征薄膜和基底的热性能至关重要。传统的光泵-探测和电热方法，如时域热反射（TDTR）、频域热反射（FDTR）和 3ω方法，被广泛用于测量薄膜的热导率。然而，TDTR 和 FDTR 在标准工作条件下具有较浅的热穿透深度（∼0.2 至 3 微米）。因此，它们通常无法测量超出这些长度尺度的埋藏基底的热导率。另一方面，3ω方法的缺点包括对大而平坦的样品表面、复杂的微加工和具有挑战性的样品制备的要求，特别是对于电导和半导体材料。 ¹⁸ 这些要求通常会限制 3ω方法用于埋藏基底测量的适用性。最近开发的光泵-探测技术稳态热反射（SSTR）提供了解决这些问题的方案，因为其热穿透深度可以比 TDTR 和 FDTR 测量期间产生的要大得多。 ^19,22 此外，SSTR 非常适用于电导或半导体材料，并且可以在 100×100 µm ² 面积或更小的光滑表面上运行。 ^19,23 因此，对 SSTR 技术的热穿透深度以及其测量地下埋藏基底的热导率能力进行详细研究是非常必要的。

Using the principle of thermoreflectance,²⁴ SSTR employs co-axially focused pump and probe beams from continuous wave (CW) lasers to directly measure the thermal conductivity of a material by applying Fourier’s law. Schematics of the SSTR measurement configuration and principle are shown in Figs. 1(a) and 1(b), respectively. Using a low modulation frequency (f → 0), the pump laser generates a periodic heat flux at the sample surface for an extended period. The low modulation frequency provides enough time for the system to reach steady-state. The probe beam then measures the resultant steady-state temperature rise by monitoring the change in surface reflectivity with a balanced photodetector and a lock-in amplifier. A linear relation between the heat flux and temperature rise is established by varying the pump power and monitoring the resulting change in reflectivity at each pump power. From this relation, the thermal conductivity of any material can be determined by using Fourier’s law. The thermal conductivity tensor measured by SSTR is different from that of TDTR or FDTR. For bulk materials, whereas TDTR and FDTR usually measure the cross-plane thermal conductivity, SSTR measures $\sqrt{k_r{k}_z}$⁠, where, k_z and k_r are cross-plane and in-plane thermal conductivities, respectively.¹⁹
利用热反射原理， ²⁴ SSTR 采用同轴聚焦泵浦和探测光束来直接测量材料的热导率，通过应用傅里叶定律。SSTR 测量配置和原理的示意图分别显示在图 1(a)和 1(b)中。使用低调制频率（f → 0），泵浦激光在样品表面产生周期性热流，持续一段时间。低调制频率提供足够的时间使系统达到稳态。然后，探测光束通过平衡光电探测器和锁相放大器监测表面反射率的变化，测量结果的稳态温升。通过改变泵浦功率并监测每个泵浦功率下反射率的变化，建立了热流和温升之间的线性关系。通过这种关系，可以使用傅里叶定律确定任何材料的热导率。SSTR 测得的热导率张量与 TDTR 或 FDTR 的不同。 对于大块材料，而 TDTR 和 FDTR 通常测量横向热导率，SSTR 测量 $\sqrt{k_r{k}_z}$ ，其中，k _z 和 k 分别是横向和纵向热导率。 ¹⁹  

In alignment with TDTR and FDTR, in SSTR, the thermal penetration depth (TPD) is defined as the distance normal to the surface at which the temperature drops to the 1/e value of the maximum surface temperature (T_max).^25–28 According to this definition, the 1/e² heater (pump) radius represents the upper limit of the TPD when the modulation frequency is low, such as in SSTR.²⁸ However, such a description of the SSTR TPD fails for multilayer material systems (e.g., a thin film on a substrate). In such systems, the TPD can change widely based on the ratio of thin film to substrate thermal conductivity and the thermal boundary conductance (G) between the thin film and substrate. This is further complicated by the presence of thin metal film transducers at the sample surface, which are often a requirement in optical pump–probe techniques for optothermal transduction.^29–33
根据 TDTR 和 FDTR 的规定，在 SSTR 中，热穿透深度（TPD）被定义为垂直于表面的距离，温度下降到最大表面温度（T _max ）的 1/e 值的距离。根据这个定义，1/e ² 加热器（泵）半径代表了 TPD 的上限，当调制频率较低时，比如在 SSTR 中。然而，对于多层材料系统（例如，衬底上的薄膜），对 SSTR TPD 的描述失败了。在这种系统中，TPD 可以根据薄膜与衬底的热导率比和薄膜与衬底之间的热边界导热（G）而有很大变化。薄金属膜换能器存在于样品表面，这进一步复杂化了情况，这些薄金属膜换能器通常是光泵-探测技术中光热换能的要求。 

In this study, we numerically and experimentally analyze the TPD definition of SSTR measurements. We also discuss the implications of metal film transducers, thermal boundary conductances, and role of multilayer material systems on the heuristic approximations for the SSTR TPD. Notably, our experimental results indicate that although the traditional TPD definition provides a convenient estimate of how far beneath the surface SSTR can probe, it does not represent the absolute upper limit of the SSTR probing depth. In specific cases, SSTR can probe beyond both the 1/e temperature drop distance and the heater radius. Furthermore, the thermal conductivity measurements of buried layers or substrates by SSTR are not solely dictated by the TPD. The uncertainties associated with such measurements are often governed by the transducer thermal conductivity, the thermal boundary conductances, and the thermal resistances offered by different layers of the multilayer material system. Thus, when determining whether the material of interest at some depth under the surface is measurable within acceptable limits of uncertainty, sensitivity calculations provide the best means. Moreover, we show that due to the application of continuous wave pump laser and low modulation frequency, SSTR can measure the thermal conductivities of buried substrates that are traditionally challenging by TDTR and FDTR. This is illustrated by measuring the thermal conductivities of buried substrates in three different samples: (i) ∼130 nm amorphous silicon dioxide (a-SiO₂) thin film on a silicon (Si) substrate, (ii) ∼2.05 µm gallium nitride (GaN) thin film on a n-type GaN substrate, and (iii) ∼2 µm aluminum nitride (AlN) thin film on a sapphire substrate. In addition, it is also established that using large 1/e² pump and probe radii, SSTR can measure the thermal conductivity of highly resistive buried films.
在这项研究中，我们对 SSTR 测量的 TPD 定义进行了数值和实验分析。我们还讨论了金属薄膜换能器、热边界导热和多层材料系统对 SSTR TPD 的启发式近似的影响。值得注意的是，我们的实验结果表明，尽管传统的 TPD 定义提供了一个方便的估计，可以了解 SSTR 可以探测到表面以下多远，但它并不代表 SSTR 探测深度的绝对上限。在特定情况下，SSTR 可以探测超出 1/e 温度降距离和加热器半径的范围。此外，通过 SSTR 测量埋藏层或基底的热导率并不完全由 TPD 决定。与这些测量相关的不确定性通常由换能器热导率、热边界导热和多层材料系统中不同层的热阻决定。因此，在确定感兴趣的材料在表面以下某个深度是否可在可接受的不确定性限度内测量时，灵敏度计算提供了最佳手段。 此外，我们展示了由于连续泵浦激光和低调制频率的应用，SSTR 可以测量传统上由 TDTR 和 FDTR 难以测量的埋藏基底的热导率。这是通过测量三种不同样品中埋藏基底的热导率来说明的：(i)约 130 纳米的二氧化硅非晶薄膜（a-SiO ₂ ）在硅（Si）基底上，(ii)约 2.05 微米的氮化镓（GaN）薄膜在 n 型氮化镓基底上，以及(iii)约 2 微米的氮化铝（AlN）薄膜在蓝宝石基底上。此外，还建立了使用大的 1/e ² 泵浦和探测半径，SSTR 可以测量高电阻埋藏薄膜的热导率。

We have divided the findings of this study into several sections. The TPD of the SSTR technique is first numerically investigated and then validated by experimental measurements. Section II A presents numerical predictions of SSTR TPD for a two-layer system (metal transducer/substrate), which are then extended to a three-layer system (metal transducer/thin film/substrate) in Sec. II B. This section shows how different ratios of thin film to substrate thermal conductivity can influence the TPD of the SSTR technique. The traditional definition of the thermal penetration depth is experimentally verified in Secs. II C and II D for a three-layer system where both the thin film and substrate are considered to possess the same thermal conductivity. In Sec. II E, experimental measurements of buried substrates for three control samples are presented. The control samples possess different combinations of thin film and substrate thermal conductivities. This is followed by a brief discussion on the required criteria for SSTR measurements of buried films in Sec. II F. To guide future measurements of buried substrates, in Sec. II G, we have presented detailed sensitivity calculations for multiple heater radii and different ratios of thin film to substrate thermal conductivity. A few limitations of the SSTR technique for the measurements of buried films and substrates are discussed in the subsequent Sec. II H.
我们将这项研究的发现分成几个部分。首先对 SSTR 技术的 TPD 进行了数值研究，然后通过实验测量进行了验证。第二部分 A 节展示了金属换能器/基底两层系统的 SSTR TPD 的数值预测，然后在第二部分 B 节将其扩展到金属换能器/薄膜/基底三层系统。本节展示了不同薄膜与基底热导率比例如何影响 SSTR 技术的 TPD。在第二部分 C 和第二部分 D 中，对一个三层系统的传统热穿透深度定义进行了实验验证，其中薄膜和基底被认为具有相同的热导率。在第二部分 E 中，呈现了三个控制样本的埋藏基底的实验测量结果。这些控制样本具有不同的薄膜和基底热导率组合。在第二部分 F 中，对埋藏薄膜的 SSTR 测量所需标准进行了简要讨论。为了指导未来对埋藏基底的测量，第二部分。 II G，我们已经针对多个加热器半径和薄膜与基底热导率不同比例进行了详细的灵敏度计算。关于 SSTR 技术用于测量埋藏薄膜和基底的一些限制在随后的第 II H 节中进行了讨论。

We first review how the TPD of SSTR changes as a function of substrate thermal conductivity in a two-layer system: metal transducer/substrate. The substrate here represents a bulk isotropic material. The TPD is calculated by solving the cylindrical heat diffusion equation, detailed descriptions of which are provided elsewhere.²⁶ For these calculations, 1/e² pump and probe radii (r_o and r₁, respectively) of 10 µm are used. The modulation frequency is chosen to be 100 Hz as it represents a realistic value usable in an experiment. We further assume that all the energy is absorbed in an infinitesimal thin layer on the surface (i.e., surface boundary condition).
我们首先回顾了在双层系统中，即金属换能器/基底中，SSTR 的 TPD 随基底热导率的变化而变化的情况。这里的基底代表着一个块状各向同性材料。通过求解圆柱热扩散方程来计算 TPD，其详细描述已在其他地方提供。对于这些计算，使用了 1/e 泵浦和探测器半径（分别为 r2 和 r3，均为 10 微米）。调制频率选择为 100 赫兹，因为它代表了在实验中可用的一个现实值。我们进一步假设所有能量都被吸收在表面的一个无穷小薄层中（即表面边界条件）。

In Fig. 2(a), the TPD corresponding to the 1/e temperature drop distance from the surface is presented for two scenarios, with and without the inclusion of a transducer. When no transducer is present, the change in the TPD is very small with respect to the substrate thermal conductivity. The small decrease in the TPD with substrate thermal conductivity reduction can be attributed to the choice of modulation frequency. For a given pump and probe radii, the lower the substrate thermal conductivity, the longer it takes for the system to reach steady-state.¹⁹ Thus, as the substrate thermal conductivity decreases, the system slightly deviates from the ideal steady-state condition (f = 0). To keep the TPD constant, the modulation frequency needs to be lowered in accordance with the substrate thermal conductivity reduction. However, for the chosen modulation frequency of 100 Hz, the deviation from the ideal steady-state condition is quite small for the substrate thermal conductivities considered here, and therefore, the system can still be reasonably approximated to be in steady-state.¹⁹
在图 2(a)中，给出了与和不包括换能器的两种情况对应的距离表面的 1/e 温度降的 TPD。当没有换能器时，与基底热导率相比，TPD 的变化非常小。基底热导率降低导致的 TPD 略微减小可以归因于调制频率的选择。对于给定的泵浦和探测半径，基底热导率越低，系统达到稳态所需的时间就越长。因此，随着基底热导率的降低，系统略微偏离理想的稳态条件（f = 0）。为了保持 TPD 恒定，调制频率需要根据基底热导率的降低而降低。然而，对于所选的 100 Hz 调制频率，对于这里考虑的基底热导率，系统偏离理想稳态条件的程度非常小，因此，系统仍然可以合理地近似为处于稳态。 

The presence of a high thermal conductivity (100 W m⁻¹ K⁻¹) metallic transducer drastically changes the TPD. For instance, when the substrate thermal conductivity is low (<10 W m⁻¹ K⁻¹), the TPD with the transducer is higher than the TPD without the transducer. This stems from the radial heat spreading in the transducer and a corresponding increase in the overall heater radius.²⁸ On the other hand, when the substrate thermal conductivity is high (>100 W m⁻¹ K⁻¹), the TPD with the transducer sharply deceases. With the increase in substrate thermal conductivity, the thermal resistance offered by the probed region gradually decreases. This leads to an increasingly important role of the interfacial thermal resistance. As a result, for high thermal conductivity substrates, the temperature drop at the transducer/substrate interface can become comparable to or even greater than that in the substrate.²⁸ This is exemplified in Fig. 2(b), where we present the normalized temperature drop as a function of depth for a substrate thermal conductivity of 1000 W m⁻¹ K⁻¹. In this example, the temperature decreases by nearly 41% at the transducer/substrate interface, leading to a TPD of 2.48 µm.
具有高热导率（100 W m ⁻¹ K ⁻¹ ）的金属换能器的存在显着改变了 TPD。例如，当基底热导率较低（<10 W m ⁻¹ K ⁻¹ ）时，带有换能器的 TPD 高于没有换能器的 TPD。这源于换能器中的径向热传播以及整体加热器半径的相应增加。另一方面，当基底热导率较高（>100 W m ⁻¹ K ⁻¹ ）时，带有换能器的 TPD 急剧减少。随着基底热导率的增加，被探测区域提供的热阻逐渐减小。这导致了界面热阻的作用日益重要。因此，对于高热导率基底，换能器/基底界面的温度降可以变得与基底中甚至更大。这在图 2(b)中得到了体现，我们展示了对于热导率为 1000 W m ⁻¹ K ⁻¹ 的基底，温度降的归一化函数深度。 在这个例子中，温度在传感器/基底界面降低了近 41%，导致 TPD 为 2.48 µm。

We also calculate the distance normal to the surface at which the temperature drops to 1/e² value of maximum surface temperature and present it in Fig. 2(a). It is evident that the 1/e² distance (calculated with a transducer) is much higher than the 1/e distance for all substrate thermal conductivities. This is to be expected as the temperature decay increases with depth.
我们还计算了温度下降到最大表面温度的 1/e ² 值的垂直于表面的距离，并在图 2(a)中展示。显然，1/e ² 距离（用传感器计算）远高于所有基底热导率的 1/e 距离。这是可以预料的，因为随着深度增加，温度衰减也会增加。

We now extend the TPD discussion to a three-layer system with the following geometry: metal transducer/thin film/substrate. When the thin film and substrate thermal conductivities are nearly equal, the TPD will closely follow those shown in Fig. 2(a) with a minor influence from the thermal boundary conductance between the thin film and substrate. Thus, we consider two extreme cases of this hypothetical geometry: an insulating film on a conductive substrate (k₂ = 10 W m⁻¹ K⁻¹ and k₃ = 100 W m⁻¹ K⁻¹) and a conductive film on an insulating substrate (k₂ = 100 W m⁻¹ K⁻¹ and k₃ = 10 W m⁻¹ K⁻¹).
我们现在将 TPD 讨论扩展到一个具有以下几何形状的三层系统：金属换能器/薄膜/基底。当薄膜和基底的热导率几乎相等时，TPD 将紧密地遵循图 2(a)中显示的情况，受到薄膜和基底之间热边界导热的轻微影响。因此，我们考虑这种假设几何形状的两种极端情况：绝缘薄膜在导电基底上（k ₂ = 10 W m ⁻¹ K ⁻¹ 和 k ₃ = 100 W m ⁻¹ K ⁻¹ ）以及导电薄膜在绝缘基底上（k ₂ = 100 W m ⁻¹ K ⁻¹ 和 k ₃ = 10 W m ⁻¹ K ⁻¹ ）。

In Figs. 3(a) and 3(b), we present the TPD corresponding to the 1/e temperature drop distance as a function of thin film thickness for the first and second cases, respectively. It is evident that the TPD with and without presence of a transducer are nearly identical. This is due to the fact that the thin film thermal conductivities are 10 and 100 W m⁻¹ K⁻¹ for the two cases considered here. As shown in Fig. 2(a), for this range of thermal conductivities, the transducer does not have a significant impact on the TPD. Similar to the two-layer system, the 1/e² distance is much higher than the 1/e distance in the three-layer system. From Figs. 3(a) and 3(b), it is also clear that the TPD changes greatly with the film thickness when there is a significant difference between thin film and substrate thermal conductivities. Interestingly, the influence of the thin film on the TPD does not subside until the film thickness is approximately four times the heater radius. To understand the rationale behind this, it is necessary to review how the ratio of thin film to substrate thermal conductivity influences the heat flow direction.
在图 3(a)和 3(b)中，我们分别展示了与薄膜厚度相关的 1/e 温度降距离作为第一和第二情况的函数。显而易见，存在换能器和不存在换能器时的 1/e 温度降距离几乎相同。这是因为这里考虑的两种情况下，薄膜的热导率分别为 10 和 100 W m ⁻¹ K ⁻¹ 。如图 2(a)所示，在这个热导率范围内，换能器对 1/e 温度降距离没有显著影响。与双层系统类似，1/e ² 距离远高于三层系统中的 1/e 距离。从图 3(a)和 3(b)可以清楚地看出，当薄膜和衬底的热导率之间存在显著差异时，TPD 随着薄膜厚度的变化而发生很大变化。有趣的是，薄膜对 TPD 的影响直到薄膜厚度约为加热器半径的四倍时才消失。要理解这背后的原因，有必要回顾薄膜与衬底热导率比对热流方向的影响。

In Figs. 3(c) and 3(d), we study the normalized temperature drop as a function of depth for the first and second cases, respectively. When the thin film is insulating and the substrate is conductive, the bulk of the heat flows along the cross-plane direction of the thin film. Due to this, a large temperature gradient exists in the thin film along the cross-plane direction, as shown in Fig. 3(c). Therefore, in this case, the TPD is much lower than the heater radius unless the thin film thickness is too high or too low. On the other hand, when the thin film is conductive and the substrate is insulating, the majority of the heat flows along the in-plane direction of the thin film. Thus, the temperature gradient along the cross-plane direction of the thin film is quite small. As a result, the TPD can be much higher than the heater radius as evident in Fig. 3(d).
在图 3(c)和 3(d)中，我们分别研究了第一和第二情况下深度作为函数的归一化温度降。当薄膜绝缘且衬底导电时，大部分热量沿着薄膜的横向方向流动。由于这个原因，如图 3(c)所示，薄膜沿着横向方向存在很大的温度梯度。因此，在这种情况下，除非薄膜厚度过高或过低，否则 TPD 要比加热器半径低得多。另一方面，当薄膜导电而衬底绝缘时，大部分热量沿着薄膜的平面方向流动。因此，沿着薄膜横向方向的温度梯度非常小。因此，如图 3(d)所示，TPD 可以比加热器半径高得多。

To provide a more visual representation of this, the temperature profiles²⁶ of SSTR measurements are shown for a 3 µm thin film corresponding to the first and second cases in Figs. 3(e) and 3(f), respectively. For the insulating thin film case, temperature decreases greatly along the cross-plane direction of the film, whereas for the conductive film case, such a temperature decrease is much smaller. However, for the conductive thin film case, the temperature decrease is significant along the in-plane direction. This is in alignment with our previous discussion.
为了更直观地表示这一点，图 3(e)和 3(f)分别显示了对应于第一和第二情况的 3 µm 薄膜的 SSTR 测量温度剖面 ²⁶ 。对于绝缘薄膜情况，温度沿薄膜的横向方向大幅下降，而对于导电薄膜情况，这种温度下降要小得多。然而，对于导电薄膜情况，温度沿平面方向显著下降。这与我们先前的讨论一致。

Thus far, we have numerically predicted the TPD of two-layer and three-layer systems according to the conventional definition. We now conduct a series of experiments to check the validity of this conventional TPD definition for SSTR measurements. Specifically, we address the following questions: (i) can SSTR probe up to the 1/e temperature drop distance defined by the traditional TPD description and (ii) whether this 1/e distance or the heater radius represents the absolute upper limit of how deep beneath the surface SSTR can probe?
到目前为止，我们根据传统定义对双层和三层系统的 TPD 进行了数值预测。我们现在进行一系列实验，以检验这种传统 TPD 定义在 SSTR 测量中的有效性。具体而言，我们探讨以下问题：(i) SSTR 能否探测到传统 TPD 描述中定义的 1/e 温度下降距离，以及(ii)这个 1/e 距离或者加热器半径是否代表 SSTR 可以探测到表面以下多深的绝对上限？

For this purpose, the thermal conductivities (⁠$\sqrt{k_r{k}_z}$⁠) of three bulk samples are measured by SSTR: SiO₂ glass, z-cut quartz, and Si. The 1/e temperature drop distance of these samples are ∼10, 9, and 7 µm, respectively, according to the conventional TPD definition presented in Fig. 2(a). Prior to the measurements, the samples are coated with an ∼80 nm aluminum (Al) film to serve as an optical transducer. The SSTR experimental proportionality constant, γ,¹⁹ is determined from a reference sapphire sample (35 ± 2 W m⁻¹ K⁻¹).^34,35 Co-axially focused pump and probe radii of ∼10 µm along with a modulation frequency of 100 Hz are used for these SSTR measurements. To minimize measurement noise at this low modulation frequency, we use a combination of factors including a balanced photodetector with a path-matched reference beam, equivalent powers of the reference and sample beams, high-quality laser diodes, low probe power, high pump power, and longer averaging and lock-in times. With the optimum settings of each of these parameters, a signal-to-noise ratio up to 100 can be obtained in our setup.¹⁹ Details of our SSTR setup and measurement procedure have been thoroughly discussed in previous publications.^19,36
为此目的，通过 SSTR 测量了三个大样品的热导率：SiO ₂ 玻璃，z 切割石英和 Si。根据图 2(a)中提出的传统 TPD 定义，这些样品的 1/e 温度降距离分别为∼10、9 和 7 微米。在测量之前，样品涂覆了约 80 纳米的铝（Al）薄膜，用作光学传感器。SSTR 实验比例常数γ， ¹⁹ 是从参考蓝宝石样品（35±2 W m ⁻¹ K ⁻¹ ）中确定的。 ^34,35 用于这些 SSTR 测量的同轴聚焦泵浦和探测半径约为 10 微米，调制频率为 100 赫兹。为了在这种低调制频率下最小化测量噪声，我们使用了一系列因素的组合，包括平衡的光电探测器与路径匹配的参考光束，参考光束和样品光束的等效功率，高质量的激光二极管，低探测功率，高泵浦功率以及更长的平均和锁定时间。通过调整这些参数的最佳设置，我们的设置中可以获得高达 100 的信噪比。 ¹⁹ 我们的 SSTR 设置和测量程序的详细内容已在先前的出版物中进行了彻底讨论。 ^19,36

The SSTR best-fit curves for the thermal conductivities of the three samples are shown in Fig. 4(a). The SSTR-measured thermal conductivities of the SiO₂ glass, z-cut quartz, and Si are ∼1.14 ± 0.16, 8.87 ± 0.64, and 141 ± 10 W m⁻¹ K⁻¹, respectively. The uncertainties of the measured values stem from the uncertainty associated with the γ value (sapphire reference), Al transducer thermal conductivity, and the thermal boundary conductance. Details of these parameters are listed in Table I. The measured thermal conductivities of the three specimen are in agreement with the literature.^{12,19,44–46}
三个样品的热导率的 SSTR 最佳拟合曲线如图 4(a)所示。SiO ₂ 玻璃、z-cut 石英和 Si 的 SSTR 测得的热导率分别为∼1.14 ± 0.16、8.87 ± 0.64 和 141 ± 10 W m ⁻¹ K ⁻¹ 。测量值的不确定性来自于与γ值（蓝宝石参考）、Al 换能器热导率和热边界导热的不确定性相关的不确定性。这些参数的详细信息列在表 I 中。三个样品的测得热导率与文献中的结果一致。 ^{12,19,44–46}

In Fig. 4(b), the SiO₂ glass, z-cut quartz, and Si samples are approximated as a three-layer material system: Al transducer/second layer/third layer. Here, the second and third layers represent the thin film and buried substrate of the same material, respectively. We fit for the thermal conductivity of the third layer, assuming that the second layer possesses the value presented in Fig. 4(a). The thermal boundary conductance between the second and third layers is kept fixed at 1000 MW m⁻² K⁻¹. Figure 4(b) shows that with the increase in the second layer thickness, the uncertainty of the third layer thermal conductivity increases. When the second layer thickness is equal to the 1/e distance (∼10, 9, and 7 µm for SiO₂ glass, z-cut quartz, and Si, respectively), the third layer thermal conductivities are ∼1.15 ± 0.71, 8.88 ± 2.74, and 142.5 ± 33.7 W m⁻¹ K⁻¹, respectively. Furthermore, when the second layer thickness is 14 µm, the third layer thermal conductivities are ∼1.16 ± 1.11, 8.89 ± 4.35, and 142.7 ± 61 W m⁻¹ K⁻¹, respectively.
在图 4(b)中，SiO ₂ 玻璃，z-cut 石英和 Si 样品被近似为一个三层材料系统：Al 传感器/第二层/第三层。这里，第二层和第三层分别代表相同材料的薄膜和埋藏基底。我们拟合第三层的热导率，假设第二层具有图 4(a)中呈现的数值。第二层和第三层之间的热边界导热度保持在 1000 MW m ⁻² K ⁻¹ 。图 4(b)显示，随着第二层厚度的增加，第三层热导率的不确定性增加。当第二层厚度等于 1/e 距离（分别为 SiO ₂ 玻璃，z-cut 石英和 Si 的∼10、9 和 7 µm）时，第三层的热导率分别为∼1.15 ± 0.71、8.88 ± 2.74 和 142.5 ± 33.7 W m ⁻¹ K ⁻¹ 。此外，当第二层厚度为 14 µm 时，第三层的热导率分别为∼1.16 ± 1.11、8.89 ± 4.35 和 142.7 ± 61 W m ⁻¹ K ⁻¹ 。

It is possible to answer the previously posed questions from Fig. 4(b). As shown in this figure, SSTR can measure the thermal conductivities of layers located at 1/e temperature drop distance although such measurements have relatively large uncertainty. However, it is also evident that SSTR can probe beyond this conventional 1/e distance and the heater radius. This indicates that immediately beyond the 1/e distance or the heater radius, SSTR measurement sensitivity does not drop to zero. This phenomenon can be explained by reviewing Fig. 2(a), which shows that the temperature does not drop to the 1/e² value of the maximum surface temperature until the distance is much higher than the 1/e distance or the heater radius. Thus, even though the traditional TPD definition can be used as a convenient estimate of SSTR probing depth, neither the 1/e distance nor the heater radius should be taken as an absolute upper limit of how far beneath the surface SSTR can probe. Figure 4(b) also shows that at the 1/e temperature drop distance, the uncertainty associated with the third layer thermal conductivity varies widely among the measured samples. This can be attributed to the fact that SSTR measurements of buried substrates are not solely dictated by the TPD. Therefore, the uncertainty associated with such measurements cannot be estimated from the TPD alone. This is discussed in more detail in Sec. II D.
可以回答之前提出的图 4(b)中的问题。如图所示，SSTR 可以测量位于 1/e 温降距离的层的热导率，尽管这些测量具有相对较大的不确定性。然而，显然 SSTR 可以探测超出传统的 1/e 距离和加热器半径。这表明在 1/e 距离或加热器半径之外，SSTR 测量灵敏度并不降至零。这种现象可以通过查看图 2(a)来解释，该图显示温度直到距离远高于 1/e 距离或加热器半径时才降至最大表面温度的 1/e ² 值。因此，即使传统的 TPD 定义可以用作 SSTR 探测深度的便捷估计，1/e 距离或加热器半径都不应被视为 SSTR 可以探测到表面以下多远的绝对上限。图 4(b)还显示，在 1/e 温降距离处，第三层热导率的不确定性在测量样本中有很大差异。 这可以归因于埋藏基板的 SSTR 测量不仅仅由 TPD 决定。因此，与这些测量相关的不确定性无法仅从 TPD 中估计。这在第 II D 节中更详细地讨论。

To empirically study how different parameters of multilayer material systems impact the thermal conductivity measurements of buried layers or substrates, we use the same examples used in Sec. II C. Figure 5(a) shows the % uncertainty of the third layer thermal conductivity as a function of the second layer thickness corresponding to Fig. 4(b). The uncertainty of the third layer thermal conductivity is highest for SiO₂ glass, followed by z-cut quartz and Si. This may seem counterintuitive as SiO₂ glass has the highest 1/e temperature drop distance among the three materials. Thus, one might expect the third layer of the SiO₂ glass to have the lowest uncertainty among the samples. To understand this apparent anomaly, it is necessary to review how sensitivities to different parameters influence the SSTR measurements of third layer thermal conductivity.
为了从实证角度研究多层材料系统的不同参数如何影响埋藏层或基底的热导率测量，我们使用了第 II-C 节中使用的相同示例。图 5(a)显示了第三层热导率的%不确定性与第二层厚度的函数对应于图 4(b)。SiO ₂ 玻璃的第三层热导率的不确定性最高，其次是 z-cut 石英和硅。这可能看起来有些反直觉，因为 SiO ₂ 玻璃在这三种材料中具有最高的 1/e 温度降距离。因此，人们可能会期望 SiO ₂ 玻璃的第三层在样本中具有最低的不确定性。为了理解这一明显的异常，有必要回顾不同参数的敏感性如何影响第三层热导率的 SSTR 测量。

The sensitivities of SSTR measurements to different parameters for SiO₂ glass, z-cut quartz, and Si are presented in Figs. 5(b)–5(d), respectively. It is evident from these sensitivity calculations that when the second layer thickness is high, measurements of the third layer are greatly impacted by the second layer thermal conductivity. However, for SiO₂ glass, there is also significant sensitivity to the transducer thermal conductivity, whereas for z-cut quartz and Si, sensitivity to nearly all other parameters is very small. Due to the influence of second layer and transducer thermal conductivity, the uncertainty of SiO₂ glass is highest. For similar reasons, the uncertainty of the third layer thermal conductivity is higher for z-cut quartz compared to Si when the second layer thickness is high. At such thicknesses, SSTR measurements of z-cut quartz become sensitive to the transducer thermal conductivity. Although Si measurements also become sensitive to the Al/Si interface conductance, the sensitivity of Si to this thermal boundary conductance is lower compared to the sensitivity of z-cut quartz to the transducer thermal conductivity. As a result, the uncertainties of Si measurements are relatively lower than z-cut quartz at high second layer thicknesses.
SSTR 测量对 SiO ₂ 玻璃、z 切割石英和硅的不同参数的敏感性分别在图 5(b)–5(d)中呈现。从这些敏感性计算中可以明显看出，当第二层厚度较大时，第三层的测量受第二层的热导率影响很大。然而，对于 SiO ₂ 玻璃，对换能器热导率也有显著的敏感性，而对于 z 切割石英和硅，对几乎所有其他参数的敏感性都非常小。由于第二层和换能器热导率的影响，SiO ₂ 玻璃的不确定性最高。出于类似的原因，当第二层厚度较大时，与硅相比，z 切割石英的第三层热导率的不确定性更高。在这种厚度下，z 切割石英的 SSTR 测量变得对换能器热导率敏感。尽管硅的测量也对 Al/Si 界面导热有敏感性，但与 z 切割石英对换能器热导率的敏感性相比，硅对这种热边界导热的敏感性较低。 因此，Si 测量的不确定性相对于高第二层厚度的 z 切石英较低。

From the above discussion, it can be concluded that the TPD cannot provide an estimation of the uncertainty associated with SSTR measurements of a buried substrate. Instead, such uncertainty depends on how sensitive SSTR measurements are to different parameters, such as the transducer thermal conductivity, the thermal boundary conductances, and the thermal resistances of different layers of the multilayer material system. Therefore, sensitivity calculations can provide the best means for estimating the uncertainty of a buried layer or substrate thermal conductivity.
从上面的讨论可以得出结论，TPD 无法提供与埋藏基板的 SSTR 测量相关的不确定性的估计。相反，这种不确定性取决于 SSTR 测量对不同参数的敏感性，例如换能器的热导率、热边界导热和多层材料系统中不同层的热阻。因此，灵敏度计算可以提供估算埋藏层或基板热导率不确定性的最佳方法。

To further demonstrate the ability of SSTR to measure the thermal conductivity (⁠$\sqrt{k_r{k}_z}$⁠) of buried substrates, we choose three samples with the following three-layer geometry: Al transducer/thin film/substrate. In these samples, the thin film and substrate are separated by a physical interface contrary to those of Sec. II C. The schematics of the three samples are shown in Figs. 6(a)–6(c). The first sample is a ∼130 nm a-SiO₂ thin film on the Si substrate. This sample represents an insulating film on a conductive substrate. The second sample is a ∼2.05 µm unintentional doped (UID) GaN thin film on the hydride vapor phase epitaxy (HVPE) n-GaN substrate. This sample represents the case where the thin film and substrate thermal conductivities are nearly equal. The third sample is a ∼2 µm molecular beam epitaxy (MBE) grown AlN thin film on the sapphire substrate. This sample represents a conductive film on an insulating substrate. Under standard operating conditions, it is often challenging to measure the thermal conductivity of such buried substrates by TDTR and FDTR due to their limited thermal penetration depths.
进一步展示 SSTR 测量埋藏基底热导率（⁠ $\sqrt{k_r{k}_z}$ ⁠）的能力，我们选择了具有以下三层几何结构的三个样本：Al 换能器/薄膜/基底。在这些样本中，薄膜和基底通过物理界面分开，与第 II C 节的情况相反。这三个样本的示意图如图 6(a)– 6(c)所示。第一个样本是 Si 基底上的∼130 nm a-SiO ₂ 薄膜。该样本代表了导电基底上的绝缘薄膜。第二个样本是氮化镓（GaN）n-GaN 基底上的∼2.05 µm 无意掺杂（UID）GaN 薄膜。该样本代表了薄膜和基底热导率几乎相等的情况。第三个样本是蓝宝石基底上的∼2 µm 分子束外延（MBE）生长的 AlN 薄膜。该样本代表了绝缘基底上的导电薄膜。在标准操作条件下，由于其有限的热穿透深度，通常很难通过 TDTR 和 FDTR 测量这些埋藏基底的热导率。

In the ∼130 nm a-SiO₂ thin film on the Si sample, we use co-axially focused 1/e² pump and probe radii of ∼10 µm to measure the thermal conductivity of the buried Si substrate. The sensitivity calculation for this sample is shown in Fig. 6(d). As shown here, SSTR measurements of the Si substrate are sensitive to the cross-plane thermal conductivity of the SiO₂ thin film. TDTR is used to measure the cross-plane thermal conductivity of the SiO₂ thin film. Using this SiO₂ value as an input, the SSTR-measured thermal conductivity of the Si substrate is 141 ± 27 W m⁻¹ K⁻¹. Figure 6(d) indicates that the sensitivity to SiO₂ cross-plane thermal conductivity is very high when the pump and probe radii are 10 µm. As a result, the uncertainty associated with the Si thermal conductivity is also high, ∼19%. The sensitivity calculation also shows that using larger spot sizes, the sensitivity to SiO₂ and corresponding uncertainty of Si measurement can be reduced. To demonstrate this, we repeat the measurement with 1/e² pump and probe radii of ∼20 µm. The resultant Si thermal conductivity is 140 ± 18 W m⁻¹ K⁻¹. As predicted, the measurement with the 20 µm spot sizes has a reduced uncertainty of ∼13%.
在硅样品上的∼130 nm a-SiO ₂ 薄膜中，我们使用同轴聚焦的 1/e ² 泵浦和探测半径约为∼10 µm 来测量埋藏的硅衬底的热导率。该样品的灵敏度计算如图 6(d)所示。如所示，对 Si 衬底的 SSTR 测量对 SiO ₂ 薄膜的横向热导率敏感。TDTR 用于测量 SiO ₂ 薄膜的横向热导率。使用这个 SiO ₂ 值作为输入，测得的 Si 衬底热导率为 141 ± 27 W m ⁻¹ K ⁻¹ 。图 6(d)表明，当泵浦和探测半径为 10 µm 时，对 SiO ₂ 横向热导率的灵敏度非常高。因此，与 Si 热导率相关的不确定性也很高，∼19%。灵敏度计算还显示，使用更大的光斑尺寸，可以减小对 SiO ₂ 的灵敏度，从而减小 Si 测量的不确定性。为了证明这一点，我们重复了 1/e ² 泵浦和探测半径约为∼20 µm 的测量。得到的 Si 热导率为 140 ± 18 W m ⁻¹ K ⁻¹ 。 根据预测，具有 20 微米斑点尺寸的测量具有约 13%的减少不确定性。

In the ∼2.05 µm GaN thin film on the n-GaN substrate sample, we measure the thermal conductivity of the n-GaN substrate by SSTR using ∼10 and 20 µm spot sizes. GaN samples of similar geometries have received significant attention in recent years for thermal management applications of high-power and high-frequency electronic devices.^47–51 The sensitivity calculation for our sample is presented in Fig. 6(e). The sensitivities to the in-plane and cross-plane thermal conductivities of the GaN thin film are considerably lower when the spot sizes are 20 µm compared to the 10 µm spot sizes. The cross-plane thermal conductivities of the GaN thin film is measured by TDTR. At room temperature, the in-plane and cross-plane thermal conductivities of the GaN thin film can be considered to be the same.⁵² The SSTR-measured thermal conductivity of the GaN substrate is 194 ± 27 W m⁻¹ K⁻¹ when the spot sizes are 10 µm. Using spot sizes of 20 µm, the thermal conductivity of the GaN substrate is measured with a lower uncertainty to be 185 ± 16 W m⁻¹ K⁻¹.
在 n-GaN 衬底样品上的∼2.05 µm GaN 薄膜中，我们使用∼10 和 20 µm 的斑点尺寸通过 SSTR 测量 n-GaN 衬底的热导率。近年来，类似几何形状的 GaN 样品在高功率和高频电子器件的热管理应用中受到了重视。我们样品的灵敏度计算如图 6(e)所示。与 10 µm 斑点尺寸相比，当斑点尺寸为 20 µm 时，GaN 薄膜的平面和垂直热导率的灵敏度明显较低。GaN 薄膜的垂直热导率通过 TDTR 测量。在室温下，GaN 薄膜的平面和垂直热导率可以被认为是相同的。当斑点尺寸为 10 µm 时，通过 SSTR 测量的 GaN 衬底的热导率为 194 ± 27 W m ⁻¹ K ⁻¹ 。使用 20 µm 的斑点尺寸，GaN 衬底的热导率测量值的不确定性较低，为 185 ± 16 W m ⁻¹ K ⁻¹ 。

The thermal conductivity of the sapphire substrate is measured by SSTR in the ∼2 µm AlN thin film on the sapphire sample. The sensitivity calculation for this sample is shown in Fig. 6(f). SSTR measurement of the sapphire substrate thermal conductivity is most sensitive to the in-plane thermal conductivity of the AlN thin film. The cross-plane thermal conductivity of this AlN thin film is measured by TDTR. As the anisotropy in the AlN thermal conductivity of is very small at room temperature,⁵⁵ the in-plane and cross-plane thermal conductivities of the 2 µm AlN thin film can be assumed to be the same. Using SSTR, the thermal conductivity of the sapphire substrate is measured to be 35.1 ± 5.9 W m⁻¹ K⁻¹ with 1/e² pump and probe radii of 10 µm. Similar to the other two samples, with 20 µm spot sizes, the sapphire thermal conductivity can be determined with a lower uncertainty, 34.5 ± 4.2 W m⁻¹ K⁻¹.
蓝宝石衬底的热导率由 SSTR 测量，在蓝宝石样品上的∼2 µm 氮化铝薄膜。该样品的灵敏度计算如图 6(f)所示。蓝宝石衬底热导率的 SSTR 测量对氮化铝薄膜的平面热导率最为敏感。这种氮化铝薄膜的横向热导率由 TDTR 测量。由于在室温下氮化铝热导率的各向异性非常小，可以假定 2 µm 氮化铝薄膜的平面和横向热导率相同。使用 SSTR，测得蓝宝石衬底的热导率为 35.1 ± 5.9 W m ⁻¹ K ⁻¹ ，1/e ² 泵测半径和探测半径为 10 µm。与另外两个样品类似，使用 20 µm 的点尺寸，蓝宝石的热导率可以以更低的不确定性确定为 34.5 ± 4.2 W m ⁻¹ K ⁻¹ 。

In Table II, we present the SSTR-measured substrate thermal conductivities for the two spot sizes. The uncertainties of the measured values incorporate the uncertainty associated with the γ value (sapphire reference), Al transducer and thin film thermal conductivities, thin film thickness, and the thermal boundary conductances. The values of these parameters are tabulated in Table I. As shown in Table II, the measured substrate thermal conductivities are in excellent agreement with the literature.
在表 II 中，我们展示了两种斑点尺寸的 SSTR 测量基底热导率。测量值的不确定性包括与γ值（蓝宝石参考）、Al 换能器和薄膜热导率、薄膜厚度以及热边界导热的不确定性相关的不确定性。这些参数的值已在表 I 中列出。如表 II 所示，测得的基底热导率与文献中的值非常吻合。

For comparison, we also measure the substrate thermal conductivities with TDTR. These substrates are inaccessible with the typical modulation frequency (∼8 to 10 MHz) used in TDTR setups.^38,56–59 Therefore, to enable these measurements, we use a low modulation frequency of 1 MHz. Even with this low modulation frequency, the TDTR TPD^22,25,26 is much lower than that of SSTR. As a result, the uncertainties associated with the TDTR measurements are significantly higher compared to SSTR as evident in Table II. This proves the superiority of SSTR for accurately measuring the thermal conductivity of sub-surface buried substrates.
为了比较，我们还使用 TDTR 测量基底的热导率。这些基底在 TDTR 设置中使用的典型调制频率（∼8 至 10 兆赫）下无法访问。 ^38,56–59 因此，为了实现这些测量，我们使用了 1 兆赫的低调制频率。即使在这种低调制频率下，TDTR TPD ^22,25,26 远低于 SSTR。因此，与 SSTR 相比，TDTR 测量所涉及的不确定性显着更高，如表 II 所示。这证明了 SSTR 在准确测量地下埋藏基底的热导率方面的优越性。

We now discuss the required criteria for SSTR to measure the thermal conductivity of a buried film in a four-layer system: metal transducer/thin film/buried film/substrate. The measurement of such a buried film is possible when the thermal resistance of this layer is much greater than those of the top thin film and substrate. This stems from the fact that for SSTR to measure the thermal conductivity of any layer in a multilayered material system, a significant steady-state temperature gradient must exist in that layer, either in the cross-plane or in-plane direction. As the top thin film is in contact with the metal transducer, the temperature gradient of this layer is often large unless the film thickness is very low. On the other hand, since the substrate is a semi-infinite medium, a measurable temperature gradient exists in the substrate when large pump and probe radii are used. For a buried film, however, unless the thermal resistance is large, the resulting temperature gradient is relatively small compared to those of the thin film and substrate. Therefore, although SSTR probes through the buried film and is influenced by the thermal properties of this layer, the degree of such influence is relatively small. As a result, SSTR cannot isolate the thermal conductivity of a buried film with low thermal resistance.
我们现在讨论 SSTR 测量四层系统中埋藏薄膜的热导率所需的标准：金属换能器/薄膜/埋藏薄膜/基板。当这一层的热阻远大于顶部薄膜和基板的热阻时，就有可能测量这种埋藏薄膜。这源于 SSTR 测量多层材料系统中任何一层的热导率时，该层中必须存在显著的稳态温度梯度，无论是在横向还是纵向方向。由于顶部薄膜与金属换能器接触，除非薄膜厚度非常薄，否则该层的温度梯度通常很大。另一方面，由于基板是半无限介质，当使用大的泵浦和探测半径时，基板中存在可测量的温度梯度。然而，对于埋藏薄膜，除非热阻很大，否则与薄膜和基板相比，所得的温度梯度相对较小。 因此，尽管 SSTR 探针穿过埋藏的薄膜并受到该层热性能的影响，但这种影响程度相对较小。因此，SSTR 无法隔离具有低热阻的埋藏薄膜的热导率。

In addition, large pump and probe radii (>10 µm) are needed for buried film measurements. When the thermal resistance of the buried layer is much higher than those of thin film and substrate, bulk of the heat flows along the in-plane direction of the top thin film. For a sufficient thermal gradient to exist in the buried film, large spot sizes are required.
此外，需要较大的泵测半径（>10 µm）来进行埋藏薄膜的测量。当埋藏层的热阻远高于薄膜和衬底的热阻时，大部分热量沿着顶部薄膜的平面方向流动。为了在埋藏薄膜中存在足够的热梯度，需要较大的光斑尺寸。

To experimentally show this, we have selected a sample that fits this criteria: 85 nm Al transducer/2.5 µm Si film/1 µm SiO₂ layer/Si substrate. The sensitivity calculation for this sample is shown in Fig. 7. As exhibited here, SSTR can measure the thermal conductivity of the buried SiO₂ layer when large spot sizes are used. However, such measurements are also sensitive to the in-plane thermal conductivity of the top Si film. TDTR is used to measure the cross-plane thermal conductivity of the top Si film, as shown in Table I. The in-plane and cross-plane thermal conductivities of the 2.5 µm Si film can be considered to be the same.⁶⁰ Using 1/e² pump and probe radii of ∼20 µm, we measure the buried SiO₂ film thermal conductivity to be 1.34 ± 0.26 W m⁻¹ K⁻¹. This value is in agreement with the literature,^38,61 showing the capability of SSTR to measure the thermal conductivity of sub-surface buried layers.
为了实验性地展示这一点，我们选择了符合以下标准的样本：85 nm 的 Al 换能器/2.5 µm 的 Si 薄膜/1 µm 的 SiO ₂ 层/Si 衬底。该样本的灵敏度计算如图 7 所示。正如所展示的，当使用大的光斑尺寸时，SSTR 可以测量埋藏的 SiO ₂ 层的热导率。然而，这样的测量也对顶部 Si 薄膜的平面热导率敏感。TDTR 用于测量顶部 Si 薄膜的横向热导率，如表 I 所示。2.5 µm 的 Si 薄膜的平面和横向热导率可以被认为是相同的。使用约 20 µm 的 1/e ² 泵浦和探测半径，我们测量埋藏的 SiO ₂ 薄膜的热导率为 1.34 ± 0.26 W m ⁻¹ K ⁻¹ 。这个数值与文献中的数值一致， ^38,61 显示了 SSTR 测量亚表面埋藏层热导率的能力。

In Fig. 8, we have provided detailed sensitivity calculations for a three-layer system: metal transducer/thin film/substrate corresponding to three different heater radii: 2, 20, and 50 µm. For these calculations, the thin film thermal conductivity is kept fixed at 10 W m⁻¹ K⁻¹, whereas the substrate thermal conductivity is varied from 1 to 100 W m⁻¹ K⁻¹. Thus, the first, second, and third columns of Fig. 8 represent thin film to substrate thermal conductivity ratios of 10, 1, and 0.1, respectively. As exhibited here, SSTR measurements are most sensitive to the substrate thermal conductivity when the heater radius is much larger than the thin film thickness, regardless of what the thin film to substrate thermal conductivity ratio is. When the heater radius is small (e.g., 2 µm), the sensitivities to the in-plane and cross-plane thermal conductivities of the substrate are nearly the same. However, as the heater radius increases, sensitivity to the in-plane thermal conductivity of the substrate keeps decreasing. This occurs because larger heater radius requires longer time to reach steady-state. Therefore, to increase the sensitivity to the in-plane thermal conductivity of the substrate, the modulation frequency needs to be lowered. From Fig. 8, it is evident that by changing the heater radius, it is possible to measure the thermal conductivity of buried substrates for different thin film thicknesses. This figure also shows that with the increase in heater radius, sensitivity to the substrate thermal conductivity continuously increases, while sensitivity to all other parameters decreases. Thus, it can be concluded that using a larger spot size (e.g., 20 or 50 µm), the uncertainty associated with any buried substrate measurement can be reduced. This conclusion is applicable to all the thermal conductivity measurements presented in Secs. II C and II E.
在图 8 中，我们提供了对三层系统进行的详细灵敏度计算：金属换能器/薄膜/衬底，对应于三种不同的加热器半径：2、20 和 50 微米。对于这些计算，薄膜热导率保持固定为 10 W m ⁻¹ K ⁻¹ ，而衬底热导率从 1 变化到 100 W m ⁻¹ K ⁻¹ 。因此，图 8 的第一、第二和第三列分别表示薄膜到衬底热导率比为 10、1 和 0.1。正如所展示的，当加热器半径远大于薄膜厚度时，SSTR 测量对衬底热导率最为敏感，而不管薄膜到衬底热导率是多少。当加热器半径较小时（例如，2 微米），对衬底的平面和垂直热导率的灵敏度几乎相同。然而，随着加热器半径的增加，对衬底的平面热导率的灵敏度不断降低。这是因为较大的加热器半径需要更长的时间达到稳态。 因此，为了增加对基板平面热导率的灵敏度，需要降低调制频率。从图 8 可以明显看出，通过改变加热器半径，可以测量不同薄膜厚度的埋藏基板的热导率。该图还显示，随着加热器半径的增加，对基板热导率的灵敏度不断增加，而对所有其他参数的灵敏度降低。因此，可以得出结论，使用更大的光斑尺寸（例如，20 或 50 微米），可以减少与任何埋藏基板测量相关的不确定性。这个结论适用于在第 II C 和 II E 节中提出的所有热导率测量。

The measurements presented in this study establishes SSTR as a suitable technique for measuring buried substrates and buried films in typical device geometries. However, there are a few scenarios where such measurements can become challenging. One such example is when the interfacial resistance between the thin film and substrate is very high (i.e., 200 m² K GW⁻¹ or greater). In this case, SSTR measurements have reduced sensitivity to the buried substrate, and therefore, the resulting uncertainty can be large. However, this also opens up new opportunities for SSTR. When the interface is highly resistive, the temperature drop at the interface becomes quite large. As SSTR fundamentally measures the steady-state temperature difference across an interface, for this case, SSTR becomes sensitive to the interfacial thermal resistance between the thin film and substrate. Buried interfaces are also known to be challenging. Measurements of such highly resistive buried interfaces might be of interest for a future study.
该研究中提出的测量结果将 SSTR 确立为在典型器件几何结构中测量埋藏基底和埋藏薄膜的合适技术。然而，在某些情况下，这些测量可能变得具有挑战性。一个这样的例子是当薄膜与基底之间的界面电阻非常高（即 200 mΩ·K GW 或更高）时。在这种情况下，SSTR 测量对埋藏基底的灵敏度降低，因此，由此产生的不确定性可能很大。然而，这也为 SSTR 带来了新的机遇。当界面具有很高的电阻时，界面处的温度降会变得非常大。由于 SSTR 基本上测量跨界面的稳态温度差异，对于这种情况，SSTR 对薄膜和基底之间的界面热阻具有灵敏度。埋藏界面也被认为是具有挑战性的。对这些高电阻埋藏界面的测量可能对未来的研究感兴趣。

Similarly, if the thermal boundary conductance between the Al transducer and thin film is very low, SSTR measurements become sensitive to this interfacial resistance. As a result, sensitivity to other parameters reduces.
同样，如果 Al 换能器和薄膜之间的热边界导热率非常低，SSTR 测量就会对这种界面电阻敏感。因此，对其他参数的敏感性会降低。

In addition, as discussed in Sec. II F, SSTR can isolate the thermal conductivity of a buried layer when the resistance offered by this layer is much higher than those of the top film and substrate. Therefore, if the buried film is conductive, then it is not possible to measure the thermal conductivity of the film with SSTR.
此外，如第 II F 节所讨论的，SSTR 可以在埋藏层的电阻远高于顶部薄膜和衬底的情况下隔离出热导率。因此，如果埋藏薄膜是导电的，那么就不可能用 SSTR 测量薄膜的热导率。

Despite these limitations, SSTR technique demonstrates significant advantages in buried film and substrate measurements compared to the traditional pump–probe techniques.
尽管存在这些限制，SSTR 技术与传统的泵测技术相比，在埋膜和基底测量方面表现出显著优势。

We experimentally and numerically investigate the influences of multilayer material systems, thin metal film transducers, and thermal boundary conductances on the TPD of the SSTR technique. The traditional TPD definition of the 1/e temperature drop distance from the maximum surface temperature does not represent the absolute upper limit of the SSTR probing depth. Thus, when estimating whether the thermal conductivity of a buried substrate is measurable within acceptable limits of uncertainty, sensitivity calculations provide the best means. The low modulation frequency of SSTR enables it to measure the thermal conductivity of buried substrates that are traditionally challenging by TDTR and FDTR, demonstrated by presenting experimental data on three control samples. In addition, SSTR has the capability to isolate the thermal properties of a buried film as long as the thermal resistance of this layer is much higher than those of the top thin film and substrate. This work marks an advancement in experimental metrology by establishing SSTR as a robust technique for thermal characterizations of sub-surface buried substrates.
我们在实验和数值上研究了多层材料系统、薄金属膜换能器以及热边界导热对 SSTR 技术的 TPD 的影响。传统的 TPD 定义中，从最大表面温度到 1/e 温度降的距离并不代表 SSTR 探测深度的绝对上限。因此，在估计埋藏基底的热导率是否在可接受的不确定性限度内可测量时，灵敏度计算提供了最佳手段。SSTR 的低调制频率使其能够测量传统上由 TDTR 和 FDTR 挑战性的埋藏基底的热导率，通过对三个对照样品的实验数据的展示来证明。此外，只要这一层的热阻远高于顶部薄膜和基底的热阻，SSTR 就有能力隔离埋藏薄膜的热性能。这项工作通过将 SSTR 确立为用于热特性表征埋藏基底的坚固技术，标志着实验计量学的进步。

The authors would like to acknowledge financial support from the U.S. Office of Naval Research under a MURI program (Grant No. N00014-18-1-2429). Z.C.L. acknowledges financial support from the Deanship of Scientific Research at the King Fahd University of Petroleum and Minerals under Project No. DF191001. J.K.H. acknowledges that the work at the Naval Research Laboratory (NRL) is supported by the Office of Naval Research.
作者们要感谢美国海军研究办公室在 MURI 计划（资助号码 N00014-18-1-2429）下的财政支持。Z.C.L.要感谢沙特阿拉伯国王法赫德石油与矿产大学科学研究院在项目编号 DF191001 下的财政支持。J.K.H.要感谢海军研究实验室（NRL）的工作得到海军研究办公室的支持。

The data that support the findings of this study are available from the corresponding author upon reasonable request.
本研究结果的支持数据可根据合理要求从相应作者处获得。