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Abstract
Heat-sensitive materials have great applications in sensor, detector, and tunable photoelectric devices. However, the wave-thermal effect of the heat-sensitive material is rarely been investigated in the THz range. Here, we propose the incorporation of heat-sensitive material (strontium titanate (STO)) within a THz absorber. The simulated results show that the absorptance and frequency can be dynamically controlled by the temperature of STO. Because the absorbed THz waves are finally converted into heat, then we research the theoretical mechanism of heat generation. Theoretical analysis shows that there are two reasons for the temperature rise: surface plasmon polariton (SPP) and ohmic loss of gold patch; Electromagnetic energy consumption inside the loss materials. To verify the theory, finally, we use COMSOL Multiphysics to research the nanosecond wave-thermal effect. The transient temperature of the wave-thermal effect is calculated quantitatively. The quantitative prediction of temperature variation can provide good guidance for thermal regulation and wave-thermally tunable THz devices.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The terahertz spectrum ranges from 0.1 THz to 10 THz, which has great potential applications in communication, radiation detection, medical treatment, and other fields [1–4]. As one of the hottest research branches, THz metamaterial absorbers have attracted increasing attention because of their flexible design and wide applications. Typical THz metamaterial absorber structures are composed of a periodic frequency selective surface on the upper layer, the dielectric isolation layer, and a metal ground plane. Recently, all-dielectric metamaterial absorbers have been investigated widely and significant progress has been made [5,6]. However, one drawback of these conventional absorbers is that most of the reported absorbers have the fixed absorption spectra once they were fabricated [7], which hampers their practical applications and functionalities. To realize the tunable characteristics, increasing attention has been focused on novel materials with electromagnetic parameters that can be adjusted and controlled.
More recently, many dynamically tunable materials have been developed by researchers and introduced into the design of absorbers. These materials include graphene [8,9], black phosphorus [10,11], Dirac semimetal [12,13], vanadium dioxide (VO2) [14,15] and strontium titanate (STO) [16,17]. For graphene and Dirac semimetal, designed absorbers can be dynamically tuned by applying a bias voltage. The permittivity of VO2 and STO can be controlled by temperature. When an absorber is exposed to electromagnetic radiation, according to the law of conservation of energy, the energy of the incident wave will be converted into heat through the Joule heat effect. However, to our knowledge, the thermal characterization of the microwave or THz absorbers has rarely been discussed, which is very important in the conditions where the temperature increases sharply [18]. The impact of rising temperature is significant because it likely leads to mismatch impedance. Besides, some dielectric materials for the substrate cannot operate in high-temperature conditions because of their low melting point. For example, when the temperature is above 500 K, FR-4, Rogers, and polyimide can’t remain solid. Therefore, it is necessary to quantitatively evaluate the wave-thermal effect for designing and optimizing terahertz absorbers.
In this paper, a temperature-tunable THz absorber based on STO dielectric substrate and gold patch is proposed to research the wave-heat effect. When T=300 K and under normal incidence, the simulated results show that the absorption frequency is located at 4.6THz, with an absorptivity of 99.98%. By tuning the temperature of STO from 300 K to 900 K, the simulated results show that the center frequency can be changed from 4.6 THz to 6.5 THz. The electric field intensity and current density distributions are plotted to reveal the mechanism of electromagnetic energy loss. Finally, a heat transfer model is used to study the mechanism of the temperature rise of the terahertz absorber. The numerical calculation of the wave-thermal effect can quickly reveal the temperature information in complex micro-nano structures. This study paves the way for understanding the internal mechanism of THz wave-heat transfer and the relationship between temperature rise and incident wave intensity.
2. Design and simulation results
The schematic diagrams of the design sample are sketched in Fig. 1. The proposed structure is composed of periodic gold nanoscale patches, which is supported by a STO layer. A SiO2 layer is overlaid below the STO medium. The bottom is a gold layer, which acts as a reflective mirror. 3D schematic and top view are shown in Figs. 1(a) and 1(b), respectively. We used COMSOL Multiphysics software to optimize the geometric structure, and the values are P = 25 µm, R = 6.25 µm, h = 6 µm. The relative dielectric constant of SiO2 is 4.2. The thickness of the STO film is 0.1 µm, and the temperature is set at 300 K. The thicknesses of the top and bottom layers are 0.08 µm and 0.2 µm, respectively. The permittivity of gold can be calculated by Drude model [19]:
As a temperature-controlled material, the relative dielectric constant of STO can be calculated by [20]:
In the simulation setup, periodic boundary conditions are employed along the x and y directions, and wave propagation is along the z-direction. Angle φ is defined as the angle between E-field and x-axis. The total absorptance is calculated by A(ω) = 1 – R(ω) = 1 – |S11(ω)|2. Figure 2(a) shows the absorption and reflection spectra for both transverse electric (TE) polarization (electric field is parallel to the x-axis) and transverse magnetic (TM) polarization (electric field is parallel to the y-axis) under normal incidence. From the black curve, it can find that the absorption frequency is located at 4.6 THz with absorptivity of 99.98%. In order to investigate the sensitivity of the polarization angle to absorptance, we plotted the color map of absorption spectra corresponding to different polarization angles in Fig. 2(b). It can be seen that the absorption spectrum is polarization insensitivity. This is because the structure is axisymmetric.
Fig. 2. (a) Reflection and absorption spectrums of the absorber under normal incidence for TE and TM polarizations. (b) color map of the absorption spectra with different polarization angles.
Since the permittivity of STO can be dynamically tuned by temperature, which is influenced by the environment and wave-heat conversion. The influence of the temperature of STO on the absorptivity under different situations is studied and shown in Fig. 3(a). With increasing of the temperature of STO from 300 K - 900 K, the central frequency of the absorption peak shows a blue shift, and the operating band with absorption over 80% gradually becomes wider. To display the relationship more clearly, Fig. 3(b) shows the dependence of the relationship among temperature T, central frequency fc, and absorptivity Ac. It is obvious that the central frequency increases non-linearly, and the absorptivity displays a nonlinear descent, respectively. Specifically, when the temperature is 300 K, the absorption peak locates at 4.6 THz with 99.98% peak absorption. However, when the temperature increases to 900 K, the central frequency is 6.52 THz with 93% absorptivity. To show the relationship between discrete data more clearly, we used the method of nonlinear curve fitting to study their changing trend. By the discrete data points, Eqs. (5) and (6) were deduced, which represent the relationship between temperature and central frequency point, and the relationship between temperature and absorptivity, respectively.
where fc and Ac represent the central frequency and the corresponding absorptance, respectively. T is the operating temperature.Fig. 3. (a) Absorption map of the proposed absorber as a function of temperature and frequency, (b) the center frequency and absorptance with different temperatures.
To reveal the role of each part in the absorption of the incident THz wave, the electric field distributions of xy, xz and yz planes at the resonance frequency of 4.6 THz are plotted in Fig. 4. For TE polarization (the electric field is fixed along the x-direction), the electric field is mainly concentrated in STO medium, which is not covered by gold patch, as shown in Fig. 4(a). It is well known that the surface plasmon polariton (SPP) can be generated via a resonant interaction between free electrons in the gold patch and THz wave. Besides, the lossy nature of gold can also dissipate the incident wave. According to the law of conservation of energy, the temperature of the gold patch will rise. The volume density of the converted thermal energy by gold patch can be calculated by [21]
where ω represents the angular frequency, ε1 is the relative dielectric constant of gold, and E1 is the electric fields in the gold patch. Eq. (7) indicates that the upper layer has an important role for dissipating electromagnetic wave.Fig. 4. Electric field distributions of this tunable absorber for TE polarization at 4.6 THz on (a) xy-plane, (b) xz-plane and (c) yz-plane, respectively.
To further study the effect of the substrate on the absorption for TE polarization, the electric field distributions on the xz and yz planes were plotted and shown in Figs. 4(b) and 4(c), respectively. From these two figures, it can find that the electric field distributes in the STO and SiO2 layers. However, the distributions of the electric field are non-uniform. The color of the STO layer is darker than the SiO2 layer at the selected frequency points, which indicated that the power dissipation in the STO layer is more than the lower layer. This is because the electromagnetic energy consumption inside the loss materials is calculated by [22]
where ε2 is the permittivity of the substrate, E2 is the electric field inside the loss materials, and V is the volume of STO and SiO2. In the range of 3 to 8 THz, the imaginary part of STO is larger than SiO2. Therefore, the electromagnetic energy of THz wave dissipated in STO layer is larger than SiO2 substrate. Due to the symmetry of this structure, the situation of TM polarization is similar to the TE polarization.Another important factor for efficient absorption may be the ohmic loss of gold used here. So, it is necessary to explore the surface current distributions at the absorption peak frequency of 4.6 THz. The surface current distributions at this frequency point were plotted in Fig. 5. From the perspective view, Fig. 5(a) shows that the excited current evenly distributed on the upper surface of gold patch. To display the surface currents on two metallic layers more clearly, we plotted them on the top and back layers, as shown in Figs. 5(b) and 5(c). The red arrows and density represent the direction and strength of the surface current, respectively. According to Figs. 5(b) and 5(c), it can find that the excited surface current on the top layer is anti-parallel to the metallic backplane. This is because the upper gold patch is closed to the metallic backplane and the near field coupling was induced. In general, the surface current on the upper layer combining the ohmic resistance can also contribute to the effective absorption performance. The conversion of electromagnetic energy into heat can be calculated by ${Q_{loss}} = {I^2}Rt$. Where I is the induced current, R is the effective resistance, and t is the effective time.
Fig. 5. Surface current distributions on the absorber at the resonant frequency 4.6 THz under normal incidence for TE polarization. (a) Perspective view, (b) on the top layer, and (c) back layer.
From the above analysis, we can conclude that incident electromagnetic energy is mainly converted into heat energy. According to Figs. 4 and 5, it is deduced that, under the irradiation of THz waves, the proposed absorber will have two heat sources, namely the gold nanoscale patch and the STO medium. In addition, from Figs. 4(b) and 4(c), we can infer that the heat energy converted by the dielectric loss will be stronger than that of SPP and ohmic loss. This will be directly reflected in the maximum local temperature of the gold nanoscale patch and the STO medium. Then we discuss the transient temperature variation in the absorber structure.
3. Nanosecond wave-thermal effect
The heat conduction of STO depends on the vibration of the crystal lattice, which is mainly accomplished by phonons. In this paper, Callaway model was used to fit the thermal conductivity data of STO, and it can be calculated from [23,24]
In this equation, kB = 1.380649 × 10−23 J/K is the Boltzmann’s constant. T is the temperature. x = $\hbar \omega/k_{\rm B}T $ is the reduced phonon frequency. θD is the Debye temperature with 513 K. vs is the speed of sound and τ is the relaxation time. For a single crystal material, the scattering is mainly dominated by phonon-phonon Umklapp and point defect. In this situation, the total relaxation time can be written as
where $\mathrm{\tau }_\textrm{U}^{\textrm{ - 1}}\; $= S0Tω2 is the relaxation time of the phonon-phonon Umklapp scattering, $\mathrm{\tau }_{\textrm{PD}}^{\textrm{ - 1}}$ = Aω4 is the relaxation time of the point defect scattering. S0 and A are the parameters of the sample. For phonon-phonon Umklapp scattering, the parameters S0 is given by where γ is the Grüneisen parameter. μ is the shear modulus. V0 is the volume per atom, and ωD is the Debye frequency. The scattering parameter of point defect is given by where V is the volume of the unit cell. vs is the velocity of phonon. Г is related to defect concentration and phonon scattering intensity. According to these equations, we obtain scattering parameters S0 = 0.95×10−19 s·K−1 and A = 1.96×10−41 s3. Thermal conductivity data of STO fitted by Callaway model and displayed in Fig. 6. It can be found that the thermal conductivity of STO decreases with the increase of temperature.THz waves absorbed by the proposed absorber will be converted into heat. To research the wave-thermal conversion, COMSOL Multiphysics was used to simulate the coupling physical phenomena. The Gaussian pulsed light with TE-polarized was used as the radiation source to analyze the nanosecond wave-thermal effect. We set the frequency range from 3 to 6 THz and periodic boundary conditions along the x- and y- directions. In addition, the convective heat flux was used in the solid heat transfer module and the heat transfer coefficient is 5 W/(m2 .K). The wave-thermal conversion process of the proposed terahertz absorber can be described by Eqs. (13) – (16) [21,25]. We set the absorber to be excited by a continuous wave source with pulse repetition frequency fr = 25 kHz. The power of the incident wave P0 = 2.5 mW, and Gaussian beam waist radius w is 35 µm. The calculation formula of the THz wave fluence irradiated on the surface of the THz absorber is
When the THz energy is absorbed by this absorber, it will be converted into heat in picosecond. According to the previous discussion, we know that a heat source is formed on the upper layer of gold patch and in the STO dielectric layer. For a one-unit cell, the converted thermal energy can be calculated by
After overlapping integral between the THz wave source power density and the absorption spectrum of this proposed absorber, the value of the absorption coefficient Rα = 0.325. From Eqs. (13) – (14) and the period length of the absorber unit, the converted thermal energy is Eth(0) = 1.06×104 pJ in the center of THz beam. The heat source power for different layers can be expressed as
Here, we set the time constant of the wave pulse τ = 1.5 ns, and the time delay of the pulse peak t0 = 3 ns. Due to the transient temperature in the unit cell and heat diffusion, there will be a temperature gradient between the different temperatures. Eq (16) can describe the transient heat diffusion process
Table 1. Thermal property of different materials used in the simulation
We set the incident wave power P0 equal to 2.5 mW. Through multi-physics field coupling simulation, the distribution of Gaussian pulse heat source Qs and the maximum temperature of several material layers are shown in Fig. 7 (gold patch, STO, and SiO2). The red dash-dot line represents the power of a Gaussian pulse heat source irradiated on the upper surface of a unit cell. Moreover, the magenta, blue and olive lines represent the maximum temperature of STO layer, SiO2 layer and gold patch, respectively. It is clearly observed that the maximum temperature of the gold patch is 640 K at 3.7 ns. However, the situation of the layers of the STO, and SiO2 is quite different from that of the gold patch. At 4.9 ns and 5.5 ns, the maximum temperature of STO and SiO2 is 751.8 K and 685.2 K, respectively. The time sequence for each layer of material to reach the highest temperature is the gold layer first, then the STO layer, and finally the silicon dioxide layer. The temperature distribution can be explained by the power dissipation of the medium and the Joule heating effect of the gold patch. The main reason for the temperature gradient is that the power dissipation in the STO and SiO2 dielectric layer is stronger than the Joule heat loss in the gold patch. Moreover, the non-uniform distribution of the electric fields is another major factor too. Then, the heat diffusion process will take a period. About 40 μs later, the absorber will reach the thermal balance, which is just before the next Gaussian pulse.
Fig. 7. The heat power irradiating on this proposed absorber at the center of the Gaussian beam and the maximum temperature curves of the gold patch, STO, and SiO2.
To further display the transient temperature in this absorber, we plot the typical temperature field distribution at 4.9 ns and shown in Fig. 8. It can be found that the color in the STO layer is relatively light, indicating that it has a higher temperature. Compared with Fig. 5, the temperature distributions are very similar to the electric field distribution map. This is because the internal electric field is completely converted into heat.
Fig. 8. (a) xy-plane, (b) xz-plane and (c) yz-plane of the temperature distribution of the unit cell at 4.9 ns.
In addition, we calculate the influence of various incident wave power P0 on the rise of temperature and thermal diffusion process, as given in Fig. 9. Figure 9(a) is the maximum temperature of the gold patch, which indicates that the temperature rises sharply and then drops rapidly, until it approaches equilibrium. As P0 increases from 1.5 mW to 3 mW, the temperature difference interval at each time point is almost the same, and the time point of the highest temperature is around 3.68 ns. Figure 9(b) gives the P0 versus average temperature as a function of time. From Figs. 9(a) and 9(b), it can be noted that there is a big difference between the maximum and average temperatures, which is because of the high thermal conductivity of gold. High thermal conductivity is an important factor for high heat dissipation [26]. When a Gaussian pulse heat source irradiates the upper surface of the absorber, the gold patch can quickly conduct heat.
Fig. 9. Maximum and average temperatures under different incident THz power. (a)-(b) gold patch; (c)-(d) STO layer; (e)-(f) SiO2 layer.
Figures 9(c), and 9(d) are the maximum and average temperatures of the STO, respectively. Comparing all the temperatures in Fig. 9, it can be found that the maximum and average temperatures of the STO layer are the highest, which proves once again that STO has the greatest effect in converting THz energy into heat. The distributions of temperature in SiO2 are shown in Figs. 9(e) and 9(f), it can be seen that the difference between the maximum and average temperature is the largest at first. With the increase of time, the average temperature of SiO2 gets higher and higher, which is due to the temperature of the STO layer diffuses downward.
4. Conclusion
In summary, we have theoretically studied the wave-thermal effect of a temperature-tunable THz absorber based on STO. We first investigated the absorptance of the proposed absorber at room temperature. The results show that the absorptivity is 99.98% at the center frequency of 4.6 THz. When the temperature varies from 300 K to 900 K, the absorptance and center frequency can be dynamically controlled. To investigate the cause of the rising temperature qualitatively, the distributions of the electric field in the absorber and current on the gold patch are then analyzed. Finally, to study the wave-thermal effect, the transient temperature response in the wave-thermal process is quantitatively calculated. When the total power of the incident wave is 2.5 mW, the distributions of temperature in a unit structure are studied. Furthermore, we calculate the influence of various incident wave power P0 on the rise of temperature and the thermal diffusion process. This work can provide good guidance for thermal regulation and wave-thermally tunable THz devices.
Funding
State Key Laboratory of Millimeter Waves (K202204).
Disclosures
The authors declare no conflicts of interest
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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