The strong coupling effect of thermal and plasma waves disturbs the accurate characterization of thermal properties of semiconductors under super bandgap energy photon excitation. We propose a time-resolved laser scanning photothermal microscopy to decouple the thermo-electronic effect for accurate determination of the thermal diffusivity of a semi-insulating GaAs sample. The distinct advantage of the decoupling principle relies on that the scanning excitation of a laser beam on the sample surface introduces different transient and steady-state characteristics into the thermal and electronic parts of the photothermal response, where the transients between the thermal and electronic parts have a large time-scale separation, and the steady states show an enhanced thermal effect over the electronic effect. Therefore, the plasma wave is simply responsible for a negligible constant background in the thermo-electronic coupling. The theoretical and experimental results confirmed that the time-resolved photothermal signal is insensitive to the electronic transport parameters varying even by several orders of magnitude and can be used to determine the thermal diffusivity from its best fit. Moreover, the scanning excitation nature of this method allows for high-efficiency photothermal imaging of the sample to identify the thermal defects.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Semi-insulating (SI) GaAs is an important III-V compound semiconductor that has been widely used in photodetectors, LEDs and lasers due to its high carrier mobility and light-emission capability. The power handling performance and reliability of those devices rely on the thermal and electronic transport properties of SI-GaAs materials, which can be characterized by photoexcited techniques with super bandgap excitation photons. Following absorption of super bandgap photons, electrons will be excited from the valance band to an energy above the conduction band edge. These photogenerated carriers (PGC) are thermalized by giving the excess energy to the lattice through rapid (∼10−12 s) nonradiative transitions to the conduction band edge. The short time scale of this process results in a prompt thermal source having a distribution equivalent to the optical absorption profile. After a much longer time (∼10−3 s) , the PGC produce a delayed thermal source by giving up their remaining energy to the lattice through electron-hole recombination. The thermal source gives rise to a temperature profile having the properties of a critically damped wave, i.e. a thermal wave. Prior to the electron-hole recombination, there exists a plasma of electrons and holes whose density diffuses in a manner analogous to the thermal wave, i.e., a plasma wave . Therefore, the thermal and plasma waves are strongly coupled, known as thermo-electronic wave , in semiconductors.
Among photoexcited characterization techniques, photothermal radiometry (PTR) [4,5] has been widely used to determinate the electronic transport parameters by collecting infrared emissions within the blackbody spectral range at the excited site because of the dominant contribution of the plasma wave over the superposed thermal wave to the PTR signal. Photo-carrier radiometry (PCR) [6–8] following the PTR was proposed as a purely plasma wave detection technique by filtering out the thermal-infrared contributions to the emission spectral range, and has been recognized as a promising tool for electronic transport characterization [9–11]. On the other hand, the thermal transport properties are able to be characterized by another representative collection of photothermal techniques, such as photothermal deflection (PTD) [12–17] and surface thermal mirror/lens (STM/STL)  employing frequency modulation of the excitation beam as well as transient thermal mirror (TTM) [19–23]. These approaches detect either the deflection or phase shift introduced to the probe beam by the surface deformation, which is contributed by the thermoelastic and electronic-strain responses of the sample [2,24], instead of detecting the infrared emissions. The thermo-electronic decoupling with PTD and STM/STL requires the best fit to the frequency response of the amplitude and phase of the photothermal signal, in which the thermal properties dominate for low frequencies, 2πfτ << 1, while the electronic properties dominate for high frequencies, 2πfτ >> 1 , here f is the modulation frequency and τ is the carrier recombination lifetime. While, the spectra acquisition over several orders of magnitude of frequency is time-consuming. Decoupling using TTM with a continuous wave laser depends on the temporal evolution of the transient photothermal signal within the first few milliseconds and its best fit with respect to the thermal properties. However, the transient signal results from the superposition of the fast time-varying thermal and PGC density fields and, in addition, is susceptible to various errors compared with the frequency-scanned spectra, so that it is not conductive to determine the thermal properties accurately .
To this end, we demonstrate a time-resolved laser scanning photothermal microscopy (TLS-PTM) to decouple the thermo-electronic effect for accurate determination of the thermal diffusivity of a semi-insulating GaAs sample. A time-resolved theoretical model is presented for describing the dynamic processes of the thermal and plasma waves in the SI-GaAs sample as well as the heat coupling between the sample and the surrounding fluid under the excitation of a laser beam scanning at a constant velocity relative to the sample surface. The phase shifts introduced to the probe beam through the deformations of the sample surface and the refractive index gradient of the fluid are given by the thermoelasticity and electroelasticity theories as well as the sample-fluid heat coupling effect. We will reveal theoretically and experimentally that the photothermal signal in TLS-PTM is insensitive to the electronic transport parameters and the best fit of the signal can be used to accurately determine the thermal diffusivity of the SI-GaAs sample. We also show high-efficiency photothermal imaging to identify thermal defects due to the scanning excitation nature of the TLS-PTM. The discussions on thermal imaging resolution and different materials are finally given before the conclusion is made.
2.1 Temperature and PGC density distributions
The principle of the TLS-PTM is shown in Fig. 1(a). The excitation beam (532 nm) and the probe beam (635 nm) propagating in the common path are focused by an objective lens on the sample-fluid interface (z = 0) with spot radii of we and wp, respectively. The waist radius of the probe beam is wp0 and has a distance of zp from the interface. The probe beam reflected from the interface is detected with a pinhole-avalanche photodetector (APD) assembly placed at the far-field detection plane with a distance of zd from the interface. The excitation and probe beams are scanned simultaneously in the x direction at a constant velocity of v relative to the sample.
The interaction of the excitation beam with the sample is illustrated in Fig. 1(b). Because the energy per excitation photon (Eep = 2.33 eV) is greater than the band gap energy of SI-GaAs (Ebg = 1.424 eV), the plasma and thermal waves are coupled in the sample, and thermal wave is involved in the surrounding fluid as well due to the heat coupling effect. Then, the governing equations describing the time-resolved plasma and thermal diffusion waves in a thermally and electronically isotropic sample as well as a non-absorbing fluid can be given by [26–28]
The generation of the heat source H in Eq. (2) involves complex mechanisms [8,30], mainly including (i) lattice heating due to direct absorption of photons, (ii) energy relaxation of PGC due to nonradiative intraband transition, (iii) nonradiative recombination of PGC, (iv) Auger recombination, and (v) Joule heating due to electron-phonon interaction. Nonlinear Auger effect normally occurs under high carrier concentrations, typically > 1×1025 m-3 [31,32], so it can be safely neglected in SI-GaAs. Joule heating is expected to be negligible as well due to the high carrier recombination lifetime in SI-GaAs (within nanoseconds) [8,33]. Therefore, the heat source H can be reduced to the sum of the first three terms [19,30]
The time-resolved PGC density and temperature fields can be calculated numerically using above solutions. The parameters used for simulations are listed in Table 1. Figures 2(a) and 2(b) show the time-resolved x-profiles of the PGC density, Ns, and the temperature rise, Ts, at the sample-fluid interface, z = 0, respectively, when the excitation beam moves at a velocity of v = 0.1 m/s in the x direction. The PGC profile follows the Gaussian excitation source profile because of the ultrafast carrier recombination lifetime, whereas the temperature profile extends a lot in the lateral direction due to thermal diffusion. The ridges in Figs. 2(a) and 2(b) are the time-resolved peaks of Ns and Ts, which are plotted in Fig. 2(c). Both ridges show an ultrashort transient (∼22 ns for Ns and ∼28 ms for Ts) followed by a steady state due to establishment of locally dynamic thermal equilibrium. Based on the observation that the PGC density has entered the steady state when the temperature starts to rise, a useful finding is that the electronic transport parameters, s0, τ and αc, has little effect on the thermodynamic process because s0, τ and αc are only closely related to the transient characteristics of the PGC density, indicating a possible way for thermo-electronic decoupling.
Figure 2(d) shows the temperature rise amplitude of the proposed TLS-PTM at steady state, t = 0.1 s, compared with that of the STM aforementioned, in which the thermal properties dominate for low modulation frequencies. Here, we show a low frequency range from f = 10 Hz to 10 kHz in the STM, which corresponds to a scanning velocity from v = 10−3 m/s to 1 m/s in the TLS-PTM by the relation of 1/2f = 2we/v. It can be observed a greatly enhanced thermal effect of the TLS-PTM over the conventional STM for 0.05 m/s < v < 1 m/s. This is so because the TLS-PTM provides continuous and stable heating at steady state, while the STM provides periodic heating with a duty cycle of 50%. The enhancement of the thermal effect will allow for a stronger thermoelastic response of the sample and, in turn, a higher signal-to-noise ratio of the photothermal signal.
The steady-state depth-dependent distributions of the PGC density and the temperature rise as well as the corresponding profiles are shown in Figs. 2(e)–2(g), respectively. The PGC do not diffuse away and concentrate only a few microns deep at the photo-injection spot as a result of the large surface recombination velocity. The temperature rise originating from the surface absorption decays exponentially as the thermal wave diffuses. The thermal diffusion length (i.e., the distance at which the initial magnitude of the thermal wave reduces by a 1/e factor) of tens of microns is observable in the sample, which is slightly larger than that in the fluid (i.e. air in this study), because SI-GaAs has a larger thermal diffusivity than air.
A notable feature is observed when we recall Fig. 2(b) that the temperature profile follows Gaussian at the interface, z = 0, whereas it loses the symmetry as the depth, |z|, increases, as indicated by the dashed contour lines in Fig. 2(f). This is so because, on the one hand, the absorption coefficient of SI-GaAs at the excitation wavelength is so large that the photon energy is absorbed almost entirely by the interface, making the interface act as a surface heat source. On the other hand, the scanning excitation nature of the TLS-PTM makes the thermal diffusion of the surface heat source asymmetrical in the scanning direction. Even so, the asymmetrical temperature distributions in the sample and air remains stable at steady state.
2.2 Thermal and electronic elastic deformations
The depth-dependent thermal wave in the sample will cause elastic vibrations, i.e., the thermo-elastic (TE) mechanism of elastic deformation . Moreover, semiconductor materials show a mechanical strain when the plasma wave is generated. The depth-dependent plasma wave produces elastic deformation in the sample, i.e., electronic deformation (ED) . Both deformations can be obtained by solving the elastic equation [18,19]
Figure 3(a) shows the time-resolved amplitudes of the thermal and electronic elastic deformations, uzTE and uzED, respectively. The time evolutions of the peak temperature, Ts, and the peak PGC density, Ns, shown in Fig. 2(c) are also plotted in the dashed lines. The amplitudes of Ts and Ns are adjusted to be equal to those of uzTE and uzED, respectively, while the time axis of Ts and Ns remain unchanged for comparison. The parameters used can be found in Table 1.
It is shown that the characteristics of the time evolutions of uzTE and uzED are similar to those of Ts and Ns, except for the delay of the steady state arrival time, which is around 29 ms for uzTE and 36 ns for uzED. This can be explained by the mechanisms of TE and ED. The TE mechanism shows that it takes a certain amount of time for the thermal wave originating from the surface heat source to diffuse to adjacent regions inside the sample. Following the reestablishment of the local thermal equilibrium in the temperature field, the surface deformation then approaches the steady state due to the thermoelastic response. The ED mechanism is based on the fact that the photogenerated plasma leads to stress in the semiconductor. The consequence of the stress is a deformation of the crystal lattice (i.e., electronic-strain response). Under low power optical excitation, the electronic-strain changes linearly with PGC density [37,38]. Thus, electronic transport parameters, s0, τ and αc, governing the time evolution of the PGC density in the bulk and at the interface are directly responsible for the time delay of uzED. Because of the higher s0, αc and ultrashort τ in SI-GaAs, the time delay of uzED is much smaller than that of uzTE.
Figure 3(b) shows the x-profiles of uzTE, uzED and uzTE + uzED at steady state, t = 0.1 s. Both amplitude and spatial distribution of uzED are negligible compared with those of uzTE, where uzTE accounts for 99.9575% of the total deformation and uzED only for 0.0425%. The asymmetry is shown in the profile of uzTE as expected because of the asymmetrical temperature distribution in the sample. In addition, uzTE and uzED are opposite in sign since the thermal and electronic properties of SI-GaAs, at and ae, are opposite in sign, which means that the interaction of the thermal wave with the crystal lattice causes an expansion of the sample in the negative z direction, while the plasma wave interaction results in a contraction in the positive z direction. The definition of the z direction can be found in Fig. 1(a).
2.3 Phase shift and photothermal signal
As shown in Fig. 1(b), the surface deformations, uzTE and uzED, will introduce phase shifts, φsTE = 2uzTE(2π/λp) and φsED = 2uzED(2π/λp), to the complex electric field of another weak Gaussian probe beam reflected from the excited site, respectively, where λp is the wavelength of the probe beam. The depth-dependent temperature field in the fluid will generate a refractive index gradient of air, resulting in an additional phase shift to the probe beam
The probe beam after interaction with the sample and fluid propagates to the detection plane (x′, y′) placed at a distance of zd from the sample-fluid interface (x, y, z = 0), as shown in Fig. 1, and its complex electric field can be treated using Fresnel diffraction theory
The time-resolved phase shifts due to TE, ED and refractive index gradient of the fluid as well as the time-resolved total phase shift are shown in Fig. 4(a). The characteristics of the time evolutions of φsTE and φsED are similar to those of uzTE and uzED because the phase shift is linearly related to the surface deformation. The transient and steady state of φf closely follow those of φsTE due to the heat transfer from the sample interface to the fluid. Therefore, the large time-scale separation of the transients between (φsTE + φf) and φsED indicates the unique capability of thermo-electronic decoupling.
Figure 4(b) shows the x-profiles of φsTE, φsED and φf at steady state, t = 0.1 s. The phase shift amplitude is 9.8×10−2 rad for φsTE, 0.01 rad for φf, while only 4.2×10−5 rad for φsED, which makes a negligible contribution to the total phase shift. Besides, the profiles of φsTE and φf show asymmetry because of the asymmetrical temperature distributions in the sample and the fluid due to the scanning excitation nature. Whereas, the profile of φsED keeps following that of the excitation beam owing to the ultrafast free carrier recombination rate of the sample. As a result, the electronic effect acting as a weak constant background at steady state is negligible both in terms of phase shift amplitude and spatial distribution.
Since the photothermal signal results immediately from the variation of the phase shift and the phase shift amplitude is so small that the photothermal signal varies approximately linearly with the phase shift, the characteristics of the time evolution of the photothermal signal will be very similar to those of the phase shift, as shown in Fig. 4(a), and are not shown here.
3. Experimental results and discussions
3.1 Experimental arrangement
The experimental arrangement is shown in Fig. 5(a). Briefly, both continuous wave excitation beam (Coherent, Verdi V10, 532 nm, TEM00) and probe beam (Coherent, Lablaser, 635 nm, TEM00) were expanded and then directed to a common path by a dichroic beam splitter (DBS) and a 50/50 beam splitter (BS), respectively. Both beams can be scanned by a galvo scanner (Thorlabs, GVS212) and then focused by a telecentric scan lens (Thorlabs, CLS-SL). The sample was placed at the focal plane of the telecentric scan lens. Both beams were reflected off along their original paths and the probe beam was only detected by a longpass filter-pinhole (diameter of 200 μm)-APD (Thorlabs, APD430A2/M) assembly.
The spot radii of both beams on the sample surface were adjusted by the expanders to be equivalent to those used in simulations, we = 20 μm and wp = 60 μm. The reflectances of SI-GaAs are 37.5% at 532 nm and 34.7% at 635 nm, which were used to correct the incident power of both beams. The telecentric scan lens has poor focusing spots at the edge of the field of view, so we use only diffraction limited field of view of 18 × 18 mm2. The detection plane was placed around optical path length of -500 mm away from the sample interface. The sample used to verify the proposed method was a vertical gradient freeze grown SI-GaAs wafer with a thickness of 625 μm, diameter of 50 mm, crystal orientation of (1 0 0), etch pit density < 8000 cm-2 and resistivity > 1×107 Ω·cm as provided by the vendor (Powerway Advanced Material). The front surface was polished and the back surface was left to be matte.
3.2 Results and discussions
We first tested the thermal diffusivity of the SI-GaAs sample from the best fit to the time-resolved normalized photothermal signal, which was acquired when the galvo scanner scanned at v = 0.1 m/s in the x direction from t = 0 s to t = 0.05 s, as shown in Fig. 5(b). The sampling rate was 10 kHz. The thermal, electronic and mechanical properties of the sample given by the vendor were used to fit the experimental data, which are listed in Fig. 5(b). The best fits for ten repeated experiments employing the Levenberg-Marquardt nonlinear fitting algorithm gave the thermal diffusivity to be αs = 2.49×10−5 ± 0.14×10−5 m2·s-1, which is in good agreement with the value of 2.58×10−5 given by the vendor. We blame the measurement error to the estimation errors of the laser beam parameters and the fluctuations of the sample parameters.
We then investigated the sensitivity of the electronic transport parameters, s0, τ and αc, to the photothermal signal. Different values of s0, τ and αc were substituted into the theoretical model and other parameters remain unchanged. The root mean squared errors (RMSE) between the theoretical time-resolved signals and the experimental signal, as shown in Fig. 5(b), were calculated to evaluate the sensitivity. The results are shown in Figs. 5(c)–5(e). The fluctuations of the RMSE are very small when s0, τ and αc vary by several orders of magnitude, confirming that the photothermal signal is insensitive to the variation of the electronic transport parameters. However, Fig. 5(f) shows that the RMSE fluctuates strongly although the thermal diffusivity, αs, only changes a little bit, verifying that the thermal imperfections in the sample will prevent normal heat diffusion and, in turn, lead to an obvious fluctuation in the photothermal signal, as shown in the signal spikes in Fig. 5(b).
As a proof of high-efficiency photothermal imaging to identify thermal defects, we carried out a two-dimensional scan on the SI-GaAs sample surface line by line with the galvo scanner. The scanning velocity and the sampling rate were set to 0.1 m/s and 2.5 kHz, respectively. The signal acquisition started after 0.05 s of scan for each line to exclude the transient data. The effective imaging area was obtained to be 10 × 10 mm2, as shown in Fig. 6(a), and the imaging time was less than 80 s. The hot spots and lines show the thermal defects in the sample such as digs, contaminants and scratches, which cause local heat accumulation and, in turn, temperature rise in local sites.
As a discussion on the thermal imaging resolution, we performed the same scan, as shown in Fig. 6(b), except at the different velocity of 0.05 m/s and sampling rate of 1.25 kHz (to keep the sampling interval unchanged). The imaging took around 160 s. We can see from the low-velocity scan image that it has a lower thermal imaging resolution and a higher signal-to-noise ratio (SNR) than the high-velocity scan image. This can be understood by having a look at the thermal diffusion lengths for different scanning velocities. The thermal diffusion length, μt, is calculated to be around 80 μm for v = 0.1 m/s and 112 μm for v = 0.05 m/s by the relation of μt = [4weαs/(πv)]1/2. Thus, the large thermal diffusion length in the sample degrades the thermal resolution in the low-velocity scan image, but causes a stronger thermal effect (favorable for thermo-electronic decoupling) and consequently a higher SNR. In this sense, we conclude that if high-accuracy determination of the thermal properties is required, the top scanning velocity should be limited.
In addition, other semiconductor materials with different electronic transport parameters may have different performances on thermo-electronic decoupling. A much longer free carrier lifetime (> 1 ms) combined with a much lower surface recombination velocity (< 10 m·s-1) may lead to inseparable transients between thermal and electronic parts as well as a large electronic contribution, and, in turn, the enhancement of the thermo-electronic coupling effect, which decreases the accuracy in the determination of the thermal properties.
We demonstrated theoretically and experimentally an effective TLS-PTM as an alternative means of thermo-electronic decoupling for accurate determination of the thermal properties of semiconductors. The advantages of this method rely on three aspects. One is that the scanning excitation provides the time-resolved photothermal signal with transient and steady-state periods for both thermal and electronic contributions, respectively, so that we can recognize the large time-scale separation of the transients between thermal and electronic parts, indicating a unique advantage in the thermo-electronic decoupling. The other is that the steady-state period in the scanning excitation enhances the thermal effect over the electronic effect, which is realized by providing a high peak temperature as well as a stable temperature field deep inside the sample and the fluid, while the PGC only concentrate near the photo-injection spot due to the high carrier recombination rate. Thus, the electronic part is only responsible for a negligible constant background in the thermo-electronic coupling. Another is that the scanning excitation nature combined with the feature of insensitivity to the electronic effect allows for high-efficiency photothermal imaging of the sample to identify the thermal defects.
National Natural Science Foundation of China (51805138), (41875158), (61675005); Fundamental Research Funds for the Central Universities (JZ2018HGBZ0126, JZ2019HGTB0085); Open Fund of Key Laboratory of Opto-Electronic Information Acquisition and Manipulation of Ministry of Education (2016YFC0302202).
The authors declare that there are no conflicts of interest related to this article.
1. J. Opsal and A. Rosencwaig, “Thermal and plasma wave depth profiling in silicon,” Appl. Phys. Lett. 47(5), 498–500 (1985). [CrossRef]
2. D. M. Todorović, “Plasma, thermal, and elastic waves in semiconductors,” Rev. Sci. Instrum. 74(1), 582–585 (2003). [CrossRef]
3. A. Mandelis, M. Nestoros, and C. Christofides, “Thermoelectronic Wave Coupling in Laser Photothermal Theory of Semiconductors at Elevated Temperatures,” Opt. Eng. 36(2), 459–468 (1997). [CrossRef]
4. D. Shaughnessy and A. Mandelis, “Carrier-density-wave transport property depth profilometry using spectroscopic photothermal radiometry of silicon wafers I: Theoretical aspects,” J. Appl. Phys. 93(9), 5236–5243 (2003). [CrossRef]
5. M. E. Rodríguez, J. A. Garcia, and A. Mandelis, “Infrared photothermal radiometry of deep subsurface defects in semiconductor materials,” Rev. Sci. Instrum. 74(1), 839–841 (2003). [CrossRef]
6. J. Batista, A. Mandelis, and D. Shaughnessy, “Temperature dependence of carrier mobility in Si wafers measured by infrared photocarrier radiometry,” Appl. Phys. Lett. 82(23), 4077–4079 (2003). [CrossRef]
7. A. Mandelis, J. Batista, and D. Shaughnessy, “Infrared photocarrier radiometry of semiconductors: Physical principles, quantitative depth profilometry, and scanning imaging of deep subsurface electronic defects,” Phys. Rev. B 67(20), 205208 (2003). [CrossRef]
8. J. Xia and A. Mandelis, “Radiative defect state identification in semi-insulating GaAs using photo-carrier Radiometry,” Semicond. Sci. Technol. 24(12), 125002 (2009). [CrossRef]
9. Q. Wang and B. Li, “Accurate determination of electronic transport properties of silicon wafers by nonlinear photocarrier radiometry with multiple pump beam sizes,” J. Appl. Phys. 118, 215707 (2015). [CrossRef]
10. M. Pawlak, “Photothermal, photocarrier, and photoluminescence phenomena in semiconductors studied using spectrally resolved modulated infrared radiometry: Physics and applications,” J. Appl. Phys. 126(15), 150902 (2019). [CrossRef]
11. X. Lei, B. Li, Q. Sun, J. Wang, and C. Gao, “Electronic transport characterization of B+ ion-implanted silicon wafers with nonlinear photocarrier radiometry,” J. Appl. Phys. 127(3), 035701 (2020). [CrossRef]
12. U. Zammit, F. Gasparrini, M. Marinelli, R. Pizzoferrato, F. Scudieri, and S. Martellucci, “Photothermal deflection spectroscopy study of defects in semi-insulating GaAs,” Appl. Phys. A 52(2), 112–114 (1991). [CrossRef]
13. M. Bertolotti, V. Dorogan, G. Liakhou, R. Li Voti, S. Paoloni, and C. Sibilia, “New photothermal deflection method for thermal diffusivity measurement of semiconductor wafers,” Rev. Sci. Instrum. 68(3), 1521–1526 (1997). [CrossRef]
14. S. D. George, P. Radhakrishnan, V. P. N. Nampoori, and C. P. G. Vallabhan, “Photothermal deflection measurement on heat transport in GaAs epitaxial layers,” Phys. Rev. B 68(16), 165319(2003). [CrossRef]
15. S. Ilahi, M. Baira, F. Saidi, N. Yacoubi, L. Auvray, and H. Maaref, “Non-radiative recombination process in BGaAs/GaAs alloys: Two layer photothermal deflection model,” J. Alloys Compd. 581, 358–362 (2013). [CrossRef]
16. S. Ilahi, F. Saidi, R. Hamila, N. Yacoubi, L. Auvray, and H. Maaref, “Photothermal deflection spectroscopy PDS investigation of optical and thermal properties of BGaAs/GaAs alloys,” Curr. Appl. Phys. 13(3), 610–613 (2013). [CrossRef]
17. W. Dickmann, J. Dickmann, F. F. Bruns, and S. Kroker, “Key signal contributions in photothermal deflection spectroscopy,” J. Appl. Phys. 126(2), 025106 (2019). [CrossRef]
18. J. W. Fang and S. Y. Zhang, “Modeling for laser-induced surface thermal lens in semiconductors,” Appl. Phys. B, Lasers Opt. (Germany) 67(5), 633–639 (1998). [CrossRef]
19. P. P. González-Borrero, G. V. B. Lukasievicz, V. S. Zanuto, N. G. C. Astrath, and L. C. Malacarne, “Accessing thermo-mechanical properties of semiconductors using a pump-probe surface displacement method,” J. Appl. Phys. 121(19), 195101 (2017). [CrossRef]
20. A. Marcano, G. Gwanmesia, and B. Workie, “Photothermal Mirror Method for the Study of Thermal Diffusivity and Thermo-Elastic Properties of Opaque Solid Materials,” Int. J. Thermophys. 38(9), 136 (2017). [CrossRef]
21. C. Jian-chun and Z. Shu-yi, “Theoretical studies of pulsed photothermal phenomena in semiconductors,” J. Appl. Phys. 74(9), 5718–5725 (1993). [CrossRef]
22. I. A. Abbas, K. A. Aly, and A. Dahshan, “Analytical Solutions of Plasma and Thermoelastic Waves Photogenerated by a Focused Laser Beam in a Semiconductor Material,” Silicon 10(6), 2609–2616 (2018). [CrossRef]
23. A. Marcano, G. Gwanmesia, M. King, and D. Caballero, “Determination of thermal diffusivity of opaque materials using the photothermal mirror method,” Opt. Eng. 53(12), 127101 (2014). [CrossRef]
24. A. Hobiny and I. A. Abbas, “Analytical solutions of photo-thermo-elastic waves in a non-homogenous semiconducting material,” Results Phys. 10, 385–390 (2018). [CrossRef]
25. N. A. George, C. P. G. Vallabhan, V. P. N. Nampoori, and P. Radhakrishnan, “Photothermal deflection studies of GaAs epitaxial layers,” Appl. Opt. 41(24), 5179–5184 (2002). [CrossRef]
26. D. Fournier, C. Boccara, A. Skumanich, and N. M. Amer, “Photothermal investigation of transport in semiconductors: Theory and experiment,” J. Appl. Phys. 59(3), 787–795 (1986). [CrossRef]
27. V. N. Abakumov, V. l. Perel, and I. N. Yassievich, Nonradiative recombination in semiconductors (Elsevier, 1991).
28. J. Dong and R. Lu, “Heat coupling effect on photothermal detection with a moving Gaussian excitation beam,” Appl. Opt. 58(31), 8695–8701 (2019). [CrossRef]
29. S. Ozaki and S. Adachi, “Spectroscopic ellipsometry and thermoreflectance of GaAs,” J. Appl. Phys. 78(5), 3380–3386 (1995). [CrossRef]
30. W. Thielemann and B. Rheinlander, “Photoacoustic and currentacoustic effects in optoelectronic devices,” Solid-State Electron. 28(11), 1111–1116 (1985). [CrossRef]
31. U. Strauss, W. W. Ruhle, and K. Kohler, “Auger recombination in intrinsic GaAs,” Appl. Phys. Lett. 62(1), 55–57 (1993). [CrossRef]
32. R. K. Ahrenkiel, R. Ellingson, W. Metzger, D. I. Lubyshev, and W. K. Liu, “Auger recombination in heavily carbon-doped GaAs,” Appl. Phys. Lett. 78(13), 1879–1881 (2001). [CrossRef]
33. A. Mandelis and J. Xia, “Deep level photothermal spectroscopy: Physical principles and applications to semi-insulating GaAs band-gap multiple trap states,” J. Appl. Phys. 103(4), 043704 (2008). [CrossRef]
34. G. V. B. Lukasievicz, L. C. Malacarne, N. G. C. Astrath, V. S. Zanuto, L. S. Herculano, and S. E. Bialkowski, “A Theoretical and Experimental Study of Time-Resolved Thermal Mirror with Non-Absorbing Heat-Coupling Fluids,” Appl. Spectrosc. 66(12), 1461–1467 (2012). [CrossRef]
35. A. Mandelis, Diffusion-wave fields: mathematical methods and Green functions (Springer Science & Business Media, 2001).
36. W. Nowacki, Thermoelasticity (Pergamon, Oxford, 1982), Vol. 3.
37. G. J. Williams, S. Lee, D. A. Walko, M. A. Watson, W. Jo, D. R. Lee, and E. C. Landahl, “Direct measurements of multi-photon induced nonlinear lattice dynamics in semiconductors via time-resolved x-ray scattering,” Sci. Rep. 6(1), 39506 (2016). [CrossRef]
38. D. M. Todorović, P. M. Nikolić, and A. I. Bojičić, “Photoacoustic frequency transmission technique: Electronic deformation mechanism in semiconductors,” J. Appl. Phys. 85(11), 7716–7726 (1999). [CrossRef]