Introduction
Time-domain thermoreflectance (TDTR) is a non-contact, laser-based optical method for measuring thermal properties of thin films and bulk materials. The technique was developed in 1970s and became wider used in 1980s. It can be used to characterize multilayer systems, individual layers or bulk materials. Many types of materials can be measured, such as polymers, metals and ceramics.[1]
TDTR has also been used to study for example optical properties, picosecond acoustics, electron-phonon interaction, coherent photon transport and thermal transport properties of interfaces and surfaces [1]. Nowadays, the method is used more to study thermal properties of materials. One of them is thermal conductivity which indicates how fast heat flows in a material[2, p.9]. TDTR is useful in thermal conductivity measurements because it can measure a wide range of values, from ~0.05 W/(m*K) (WSe2 crystals) to ~2000 W/(m*K) (diamond, graphite)[1]. TDTR is based on the principle of thermoreflectance which means that the reflectivity of a material changes due to a change in temperature. It is especially useful for developing and characterizing new materials.[3]
Thermoreflectance
Reflectance and reflectivity are often used interchangeably as both refer to the fraction of incident light reflected by a surface. Reflectivity refers to the specular reflection (mirror-like reflection where the light is reflected only into a single direction) and reflectance refers to the diffuse scattering of light into many directions. In simple case they are often considered the same.[4, p.180-183]
Reflectivity (and reflectance) is measured by comparing the intensity of incident light with the intensity of reflected light and it describes how much of the incident light is reflected from the surface[3]. Thermoreflectance refers to the change of reflectivity due to change of temperature. Temperature dependence of reflectivity can be represented by thermoreflectance coefficient CTR
\[ C_{TR} = \frac{dR}{dT} \]where R reflectance and T is temperature. CTR describes reflection as a function of temperature. It is wavelength dependent parameter and strongly influenced by the measurement apparatus. Thermoreflectance coefficient is useful in determining many thermal properties but must be calibrated for each measurement.[5]
In TDTR there is a metal film on the surface (discussed more in section Sample preparation) so dependence of reflectivity on temperature can be explained with the Drude model. According to the model, in metals the phonon population determines the temperature. Number of phonons increases with temperature, which increases the electron-phonon collision frequency. Increased collision frequency reduces the absorption of light which causes increased reflection of light.[6]However, the dependence of reflectivity on temperature for a sample is difficult to predict because it depends on multiple factors such as phonon-carrier interactions, light polarization and the effect of electronic bands and transitions[5].
Working principle
The main component in the system is the laser source, usually a pulsed Ti:Sapphire laser. Another important component is a photodetector which is used to detect the reflected probe beam and to convert the laser power into electrical current.[1] The TDTR system is quite complicated and includes also multiple mirrors to guide the laser beam and other optical components. A simplified version of the device is presented in figure 1. It contains only the components that are important for understanding the principle of operation of the device.
Figure 1. Simplified schematics of TDTR device. The system contains two different laser beams: a pump beam for heating the sample and a probe beam for measuring the effect of the heating. (Figure: Ella Kultalahti)
The initial beam from a laser source is first split into a pump beam (red color in the figure) and a probe beam (orange color in the figure). The pump beam is modulated with electro-optic modulator which turns off some of the pulses in selected pattern (e.g. four pulses on, four pulses off). The pattern is presented in figure 2. with number 2. After that the beam is directed to the surface of the sample where it causes local heating.[1][3][7]
The probe beam is delayed by increasing its path length so that it hits the surface after the pump beam. Time delay is made with a delay stage shown in figure 1. A typical time delay is 100 fs and above. Speed of light is a constant so an accurate time delay can be calculated form laser beam path length. For example, a 30 µm change in path length corresponds to time delay of 100 fs. The probe beam is reflected from the sample’s surface and the reflection is detected by a photodetector.[8]
Figure 2 presents the intensity of the laser pulse at different stages numbered 1-3 in both figures. Number 1 is the intensity of the laser output. Number 2 is the intensity of modulated laser pulses after the electro-optic modulator. Number 3 is the intensity of the reflected probe beam. Lower values correspond to the off-phase of modulation which means that there are no pump pulses.
Figure 2. Intensity of the laser pulse at different stages. Numbers 1-3 correspond to the numbers in figure 1. (Figure: Ella Kultalahti)
Due to very low intensity changes it is important that the pump beam is not scattered so that the photodetector collects it. The pump beam has very high intensity that would hide the small intensity variations in the probe beam. Usually, the beams are polarized differently and optical filter is used to filter off the pump beam. It is also possible to change the frequency of one of the beams so that they can be distinguished from another if the pump beam is scattered so that it reaches the detector.[9]
Sample preparation
Compared to other methods used to measure thermal properties, such as 3ω-method, TDTR requires minimal sample preparation. A typical structure of the sample is presented in figure 3.
Figure 3. Typical structure of the sample. (Figure: Ella Kultalahti)
Silicon is a typical substrate material. A metallic transducer with thickness of ~100 nm is deposited on the surface of the sample, usually by sputtering or evaporating. It ensures even heating of the sample layer. Aluminum is often used as a transducer because it has a large change in reflectivity at given temperature change compared to many other materials. The most accurate measurement is achieved when the change in reflectivity is as large as possible even when the temperature change is small. If temperature change is over 10 K the change in reflectivity is not linear since other temperature dependent properties may change.[1][7]
The sample's surface must be fairly smooth. A general rule-of-thumb is that the RMS of the surface roughness should be <15 nm. If the surface is rough, the pump beam is diffusely scattered. If the scattered light reaches the photodetector it renders the data invalid.[1]
Signal processing
Basically, there exist two levels of signals: on and off modes of the pump beam. A lock-in amplifier is used to detect the difference between them. Lock-in amplifier is an amplifier which can extract signal with known frequency even from strong background noise. The laser beam frequency was modulated earlier in the process and now it is easier to detect the right signal since the modulation frequency is known. The cleaned data can be used to determine different thermal properties.[1][7]
The experimental data is compared to a theoretical model which predicts what is expected to happen during the measurement. Several thermal properties must be known to form the model, depending on what is measured. These values are taken from literature or the sample is measured using for example differential scanning calorimetry. If the model doesn’t fit the data, the assumed values aren’t correct. The values used in the model are optimized so that the model fits the measurement data. At that point the values in the model are correct. With this method different thermal properties can be derived from the data.[1][10]
The theory behind TDTR software is quite complicated and understanding it requires a solid background in math. If you want to learn more about it, a good foundational article is Analysis of heat flow in layered structures for time-domain thermoreflectance by D. G. Cahill.[10]It contains the basic theory while other articles are usually extensions for it.
Ultralow thermal conductivity measured by TDTR
In 2006 Chiritescu et al.[11] measured ultralow thermal conductivity of 0.05 W/(m*K) in layered WSe2 crystals using time-domain thermoreflectance. It was at the time one of the lowest thermal conductivities observed in fully dense solids. The WSe2 thin film consisted of disordered crystal layers with ~3% of tungsten intercalated between the layers. The spacing between the interfaces in layered structure was only a few nanometers. Thermal resistivity of the interfaces reduced the thermal conductivity considerably compared to amorphous structure. After the measurements ion irradiation was used to disrupt the crystalline order in films which made the structure amorphous. It caused increase in thermal conductivity because amorphous film has no grain boundaries in it.
Thermal conductivity measurements
The only important unknown parameter in this experiment was the thermal conductivity of the bulk crystal (and amorphous film in the second measurement). Other parameters were either known, measured or assumed. Thickness of the transducer Al thin film was measured by Rutherford backscattering spectrometry. Thermal conductivity was calculated based on the electrical resistivity at 300 K measured by 4-point probe system. Thermal conductivity at T<300 K was estimated. Heat capacities of Al and Si substrate were taken from the literature and heat capacity of WSe2 was measured.
TDTR measurement data was compared with thermal model from previously mentioned article[10] by D. G. Cahill. TDTR was the most suitable measuring technique due to the unknown parameters and the large number of measurements for each sample.
References
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T. Favaloro, J. H. Bahk, A. Shakouri, Characterization of the temperature dependence of the thermoreflectance coefficient for conductive thin films, Rev. Sci. Instrum., 2015, 86, 024903 (https://doi.org/10.1063/1.4907354). |
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C. Chiritescu, D. G. Cahill, N. Nguyen, D. Johnson, A. Bodapati, P. Keblinski, P. Zschack, Ultralow Thermal Conductivity in Disordered, Layered WSe2 Crystals, Science, 2006, 315, 351-353 (https://doi.org/10.1126/science.1136494).
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