**1. Introduction**

With reductions in size to the micro-/nanometer level, temperature probing and thermal measurement have become difficult to conduct using traditional contact-based methods and equipment (thermal couples and thermistor, etc.). Non-contact thermal measurement methods have thus become prevalent for the thermal characterization of micro-/nanomaterials [1,2]. Non-contact methods mainly benefit from the laser heating source and thermally induced phenomena, which can be detected from a distance [3–11]. Based on the features of the phenomena, the widely adopted non-contact thermal methods are typically divided into three types: time-domain techniques, frequency-domain techniques, and spectroscopy [1].

Among these techniques, time-domain thermoreflectance (TDTR) [7,12,13] and frequencydomain thermoreflectance (FDTR) [14,15] detect the temperature rise by sensing changes in the surface optical properties in the time and frequency domains. They utilize an ultrafast heating pulse to generate a nanometer-level thermal penetration depth and thus have a good ability to measure the in-plane and out-of-plane thermal conductivity for thin films and bulks. The obvious drawbacks of these two methods are that they require smooth surfaces and post-processing. The photoacoustic (PA) method is a frequency-domain method that measures the surface temperature by detecting the sound waves produced by the work done by the periodical thermal expansion of the heated surface [3]. Avoiding the mechanical piston effect induced by the thermal expansion of the heated surface, the modulation frequency of the heating laser is limited so that it is lower than the order of

**Citation:** Zhou, J.; Xu, S.; Liu, J. Review of Photothermal Technique for Thermal Measurement of Micro-/Nanomaterials. *Nanomaterials* **2022**, *12*, 1884. https://doi.org/ 10.3390/nano12111884

Academic Editor: Enyi Ye

Received: 8 May 2022 Accepted: 30 May 2022 Published: 31 May 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

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10 kHz [9,11,16]. The PA method can work well with micrometer-thick films and bulks due to the long thermal penetration depth at a lower frequency [9]. Furthermore, as it is limited by the microphone, the PA method is usually deployed at room temperature [3,9,11]. The laser flash method [17,18] involves heating a suspended material on the front side and analyzing the transient temperature rise in the time domain from the back based on the thermal radiation. Compared with the PA method, the laser flash method can be used with a wide temperature range from −125 to 2800 ◦C [1]. However, this method has thickness limitations with regard to samples [19,20]. More recently, Raman-based thermal methods have become popular due to its feature of being material-specific. The steady-state Raman method has a simple physical mechanism for thermal characterization [21,22], while the transient Raman method offers high accuracy in measurement results [8,23–31]. It is notable that the temperature probing depth with the Raman method is usually tens of nanometers. It is less often used to measure the thermal properties of thick films or bulks. More comprehensive reviews of general photothermal technologies can be found in [1,2].

In this paper, a short review is provided for the photothermal (PT) technique based on thermal radiation established in Wang's lab [32,33]. This PT technique stems from PA technology. However, in contrast to PA technology, PT technique acquires the frequencydomain thermal radiation instead of sound waves, which can reduce the complexity of the measurement setup and widen the temperature range for thermal measurement, as the microphone is no longer necessary. Furthermore, it has no frequency limitation and, thus, can theoretically measure samples with thicknesses ranging from nanoscale to bulk. The PT method also has a low requirement for smooth surfaces when compared to thermoreflectance methods because it detects thermal radiation rather than reflections. In Sections 2 and 3, the theory and a typical experimental setup for PT technique are introduced. Section 4 discusses the application of PT technique in the thermal characterization of micro-/nanomaterials, especially the measurement of thermophysical properties and structure probing. Furthermore, considerations for the thermal measurement of micro- /nanomaterials using PT technique are also discussed.

#### **2. PT Theory for Thermal Property Measurements**

The PT technique developed in Wang's lab employs a periodically modulated laser source to heat a solid surface. In each period, the surface temperature immediately rises after heating is applied. The speed and intensity of the thermal response of the surface are strongly dependent on the thermal properties of the materials under the surface. Thermal radiation due to surface temperature rise carries important information regarding the thermal properties of the materials and structures beneath, both for homogeneous and multilayered structures.

#### *2.1. Physical Model Derivation*

PT technique stems from the physical model of PA technology proposed by Rosencwaig et al. [3], which is a one-dimensional cross-plane heat conduction model in a multilayered structure, as shown in Figure 1a. The model requires that the size of the heating source be much larger than heat diffusion length in each layer, so that the in-plane heat conduction can be safely neglected and the generated heat conducts one-dimensionally along the cross-plane direction. Furthermore, the surface temperature rise should be moderate, and the heat loss through thermal convection and radiation is reasonably negligible. Hence, the governing equation of 1D cross-plane heat conduction under periodical heating is

$$\frac{\partial^2 \theta\_i}{\partial \mathbf{x}^2} = \frac{1}{\alpha\_i} \frac{\partial \theta\_i}{\partial t} \quad - \frac{\beta\_i I\_{l0}}{2\kappa\_i} \exp\left(\sum\_{m=i+1}^N -\beta\_m L\_m\right) \times e^{\mathcal{G}\_i(\mathbf{x} - l\_i)} (1 + e^{j\omega t}),\tag{1}$$

The subscript *i* means that the physical properties are for a certain layer *i*; therefore, *θi = Ti* − *Tamb* is the temperature rise of layer *i* and *Tamb* is the ambient temperature. *I*0 is the incident laser power and *ω* is the angular frequency (2 *πf*) corresponding to the modulation frequency *f*. *αi*, *κi,* and *βi* are the thermal diffusivity, thermal conductivity, and

optical absorption coefficient for layer *i*. *Li = li* − *li*−<sup>1</sup> is the thickness of layer *i*, where *li* is the surface location of layer *i* on the *x* axis in Figure 1. *j* is √−1. A detailed derivation of Equation (1) is provided in [9].

**Figure 1.** Physical schematics of PT technique: (**a**) mechanism of modulated heating and 1D heat conduction across the multilayered structure; (**b**) a typical experimental setup for PT technique.

The resultant surface temperature rise *θi* for layer *i* gradually increases from zero to a new steady state with fluctuations. Thus, *θi* can be divided into three components: the transient component, *θi,t*; the steady DC component, *θi*,*<sup>s</sup>*; and the steady AC component, *θ i*,*s*. *θi,t* represents the initial temperature rise immediately after the laser heating is applied. When the sample reaches the steady state, *θi*,*<sup>s</sup>* indicates the steady state temperature, while *θ i*,*<sup>s</sup>* is the fluctuation in temperature due to the modulated heating source. *θ i*,*<sup>s</sup>* is easily determined by a lock-in amplifier at a set modulation frequency. It has an explicit expression as follows:

$$\widetilde{\boldsymbol{\theta}}\_{i,\mathbf{s}} = \left[ A\_i \mathbf{e}^{\sigma\_i(\mathbf{x} - l\_i)} + B\_i \mathbf{e}^{-\sigma\_i(\mathbf{x} - l\_i)} - E\_i \mathbf{e}^{\boldsymbol{\beta}\_i(\mathbf{x} - l\_i)} \right] \mathbf{e}^{\mathbf{i}\omega t},\tag{2}$$

where *Ei* = *Gi*/(*β*<sup>2</sup>*i* − *σ*2*i* ) with *Gi* = *βiI*0/(2*ki*) exp− *N*∑*<sup>m</sup>*=*i*+1 *<sup>β</sup>mLm* , and for *I* < *N*, *GN* = *βN I*0/2*kN*, and *GN*+1 = 0. *σI* is (1 + *j*)·*ai*, where *ai* = 1/*μi* is the thermal diffusion coefficient and *μi* = *<sup>α</sup>i*/*<sup>π</sup> f* is the thermal diffusion length.

*Ai* and *Bi* are important coefficients derived from the interfacial transmission matrix of heating *U* and the absorption matrix of light *V*:

$$
\begin{bmatrix} A\_i \\ B\_i \end{bmatrix} = \mathcal{U}\_i \begin{bmatrix} A\_{i+1} \\ B\_{i+1} \end{bmatrix} + \mathcal{V}\_i \begin{bmatrix} E\_i \\ E\_{i+1} \end{bmatrix} \tag{3}
$$

where *Ui* and *Vi* from layer *i* + 1 to *I* are

$$\mathcal{U}\_{i} = \frac{1}{2} \begin{bmatrix} \boldsymbol{u}\_{11,i} & \boldsymbol{u}\_{12,i} \\ \boldsymbol{u}\_{21,i} & \boldsymbol{u}\_{22,i} \end{bmatrix}; \; V\_{i} = \frac{1}{2} \begin{bmatrix} \boldsymbol{v}\_{11,i} & \boldsymbol{v}\_{12,i} \\ \boldsymbol{v}\_{21,i} & \boldsymbol{v}\_{22,i} \end{bmatrix} \tag{4}$$

where

$$u\_{1n,i} = (1 \pm k\_{i+1}\sigma\_{i+1} / k\_i \sigma\_i \mp k\_{i+1}\sigma\_{i+1} R\_{i,i+1}) \times \exp(\mp \sigma\_{i+1} L\_{i+1}), \ n = 1,2,\tag{5}$$

$$\mu\_{2n,i} = (1 \pm k\_{i+1}\sigma\_{i+1} / k\_i \sigma\_i \mp k\_{i+1}\sigma\_{i+1} \mathcal{R}\_{i,i+1}) \times \exp(\mp \sigma\_{i+1} L\_{i+1}), \ n = 1,2,\tag{6}$$

$$
v\_{n1,j} = 1 \mp \beta\_i / \sigma\_{i\prime} \ n = 1 \text{, } 2 \text{,} \tag{7}$$

and

$$w\_{n2,i} = (-1 \mp k\_{i+1} \beta\_{i+1} / k\_i v\_i \mp k\_{i+1} \beta\_{i+1} R\_{i,i+1}) \times \exp(-\beta\_{i+1} L\_{i+1}), \ n = 1, 2. \tag{8}$$

*Ri*,*i*+1 is the thermal contact resistance between layer *i* and (*i* + 1). It is noticeable that the thermal and optical properties of the materials in the multilayered structure are all included in Equations (5)–(8). Thus, the thermal properties are closely related to the temperature rise *θ i*,*<sup>s</sup>* of the layer *i*.

Under the assumption that the front air layer and back substrate are thermally thick—that is, |*<sup>σ</sup>*0*L*0| 1 and |*<sup>σ</sup>N*+<sup>1</sup>*LN*+<sup>1</sup>| 1—*AN*+1 and *B*0 are equal to zero. Then, applying the interfacial condition between layer *i* and (*i* + 1),

$$k\_i \frac{\partial \widetilde{\theta}\_{i,s}}{\partial \mathbf{x}} - k\_{i+1} \frac{\partial \widetilde{\theta}\_{i+1,s}}{\partial \mathbf{x}} = 0 \tag{9}$$

$$\text{and } k\_i \frac{\partial \bar{\theta}\_{i,s}}{\partial x} + \frac{1}{R\_{i,i+1}} \left( \check{\theta}\_{i,s} - \check{\theta}\_{i+1,s} \right) = 0,\tag{10}$$

the solved *Ai* and *Bi* are

$$
\begin{bmatrix} A\_i \\ B\_i \end{bmatrix} = (\prod\_{m=i}^N \mathcal{U}\_m) \begin{bmatrix} 0 \\ B\_{N+1} \end{bmatrix} + \sum\_{m=i}^N (\prod\_{k=i}^{m-1} \mathcal{U}\_k) V\_m \begin{bmatrix} E\_m \\ E\_{m+1} \end{bmatrix} \tag{11}
$$

$$B\_{N+1} = -\frac{\begin{bmatrix} 0 & 1 \end{bmatrix} \sum\_{m=0}^{N} \left( \prod\_{i=0}^{m-1} \, \mathcal{U}\_i \right) V\_m \begin{bmatrix} E\_m \\ E\_{m+1} \end{bmatrix}}{\begin{bmatrix} 0 & 1 \end{bmatrix} \left( \prod\_{i=0}^{m-1} \, \mathcal{U}\_i \right) \begin{bmatrix} 0 \\ 1 \end{bmatrix}} \tag{12}$$

By substituting *Ai*, *Bi*, and *Ei* into Equation (2), we can obtain the temperature distribution in any layer of interest. This greatly increases the flexibility of the PT method. For the purpose of non-contact measurement, an infrared detector is usually employed to gather the surface radiation from either the front or back.

#### *2.2. Phase Shift and Amplitude*

The AC temperature rise component *θ i*,*<sup>s</sup>* has two critical properties, amplitude and phase. Compared with the original periodical heating source, the occurrence of temperature rises and thermal radiation is delayed by the heat conduction inside the multilayered structure. Correspondingly, the phase of the radiation is slower than the phase of the heating source, and the difference (phase shift) between these two can be deducted to be *Arg*(*BN*+1) − *π*/4. According to Equation (12), the thermal properties are included in *BN*+1. Measurement based on the phase shift—the phase shift method—can be used to accurately evaluate the thermal properties of a specific layer and the interfacial thermal conductance in the multilayered structure due to the high sensitivity of ~0.1◦ [32]. However, for bulk materials with a smooth surface, it becomes a constant of −45◦ [16]. In contrast, the

amplitude of thermal radiation is proportional to the temperature rise. Since the thermal diffusion depth is different with different modulation frequencies, the amplitude of the thermal radiation changes against the frequency. Alternatively to the phase shift method, measurement based on the amplitude is able to determine the thermal properties of the bulk materials.

#### **3. Experimental Implementation of PT Method for Thermal Property Measurements** *3.1. Experimental Setup*

A typical measurement setup using the PT technique developed in Wang's lab [11] is shown in Figure 1b. The function generator-modulated diode laser (at a visible wavelength) is focused on a sample surface by using a focal lens to heat the sample. Then, a pair of offaxis parabolic mirrors collect the raised thermal radiation resulting from the temperature rise and send it to an infrared (IR) detector. Along with the radiation collection, the diffuse reflection of the incident laser from the surface is also gathered. Though the IR detector is much less sensitive to the visible wavelength, the reflection is still much stronger than the radiation. Thus, an IR window (germanium (Ge) window in Figure 1b) is placed in front of the detector to eliminate the visible diffuse reflection and let only the thermal radiation enter into the detector. The radiation-converted voltage signal is then intensified in a preamplifier and finally analyzed in a lock-in amplifier to extract the phase shift and the amplitude when compared with the reference signal from the function generator.

When using PT technique, given that the unexpected, complex photon–electron– phonon process may occur under laser irradiation in the sample, especially for semiconductors, a metal coating (usually gold, aluminum, etc.) is applied to the sample surface to act as a well-defined energy absorber and heater. The coating is optically thick and can totally absorb the incidence. At the same time, it is thermally thin and has a negligible effect on heat conduction (phase shift). It is physically understandable that the heat diffusion length/depth in the multilayered structure should be controllable by changing the modulating frequency. The selection of the modulating frequency of the heating laser needs to be evaluated in advance because the sample layer in the multilayered structure should be involved in the heat conduction.

#### *3.2. System Calibration*

The raw data recorded by the lock-in amplifier—the phase shift *φraw* and amplitude *Araw*—are not available for direct analysis because the measurement system induces additional errors in the raw data (the phase shift and amplitude). For example, the optical path and electric devices involved raise an additional time delay in the phase shift, and the fluctuations in the laser power, as well as the optical path, cause unexpected variations in amplitude. Thus, calibration of the measurement system is necessary to exclude these effects from the raw data. The diffuse reflection of the incident laser is measured to calibrate the measurement system, since it passes through the same path as the thermal radiation does. The calibrated phase shift *φcal* and amplitude *Acal* are shown in Figure 2. To calibrate *φraw*, the absolute phase shift due to heat conduction is quickly determined as *φnor* = *φraw* − *φcal*. For the amplitude, it is more complicated. The amplitude is affected by not only the laser power and attenuation in the optical path but also the modulation frequency *f*. Xu et al. proposed the equation *Anor* = *Araw* · *f* /*Acal* to normalize *Araw* and exclude all the possible errors induced by the system, detailed in [33]. The calibrated *Anor* is approximated to *ζ*/*et*, where *ζ* is a system-related constant and the effusivity *et* = √*κρcp*. It directly correlates *Anor* with the thermal properties. After calibration, *φnor* and *Anor* can be used to determine the thermal properties of the materials of interest.

**Figure 2.** The calibration of a typical experimental setup for PT technique: (**a**) phase shift and (**b**) amplitude. The black square is the measured phase shift and amplitude, and the red dot denotes the measurement uncertainty of the phase shift and amplitude.
