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Bi-tunable terahertz absorber based on strontium titanate and Dirac semimetal

Han Xiong, YueHong Peng, Fan Yang, Zhijing Yang, and ZhenNi Wang

Open Access Open Access

Abstract

We proposed a polarization-insensitive absorber based on strontium titanate (STO) and bulk Dirac semimetal (BDS) in the terahertz (THz) region. The center frequency of the absorption peak can be independently regulated by temperature or Fermi energy level of STO or BDS, respectively. The numerical simulation result reveals that the peak absorptivity reaches to 99.98% at 2.16 THz when the temperature and Fermi energy were set at 300 K and 20 meV, respectively. Interestingly, by adjusting the temperature of STO from 250 to 400 K, the simulation results indicate that the center frequency can be tuned from 1.94 to 2.53 THz, and peak absorptivity can be maintained above 99% at normal incident. As the Fermi energy EF of Dirac semimetal increases from 10 to 60 meV, the center frequency can be changed from 2.14 to 2.44 THz and the amplitude of absorption peaks can be tuned from 99.9% to 82.8%. Impedance matching theory was used to understand the tunable performance. Furthermore, interference theory was employed to further explain the absorption mechanism of the proposed absorber. The absorber achieves bi-controlled absorptance via two independently controllable methods, which may provide guidance to research tunable, smart and multifunctional terahertz devices.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

During the last decades, metamaterial absorbers (MMAs) have drawn numerous attentions for their potential applications in solar energy harvesting [1,2], plasmonic sensors [3,4], thermal emitting [5,6], stealth technology [7–9], and so on. As an important branch of MMAs, the terahertz (THz) absorber has drawn special concern due to its versatile utilization [10–12]. However, once these structures are fabricated, the absorption bandwidths and absorption peak positions will be fixed, which is the drawback in many practical applications. Therefore, designing the arbitrarily tunable absorbers is urgently needed. For the development of dynamically tunable metamaterial absorber, active media, such as, vanadium dioxide (VO2) [13–15], graphene [16–18], germanium telluride (GeTe) [19] or graphene/liquid crystal [20] have been introduced into the absorber structures to realize the tunable property. Because their conductivities can be tuned by thermal, chemical potential or optical. However, the tunable modes in all these absorbers are limited, that cannot meet the application requirement.

With the development of materials science, many new materials have been developed by researchers. In recent years, STO has received great interest from researchers owing to its permittivity is temperature-dependent, as demonstrated by Y. J. Zhang and X. Huang [21,22]. More recently, as the 3D analogs of graphene, BDS metasurfaces have been introduced to the metamaterial absorber [23–25]. This is because its permittivity can also be dynamically tuned by adjusting its Fermi energy EF via gate voltage modulation [26,27]. Compared with graphene, BDS not only has a much higher mobility under the same conditions but also are easier to fabricate and stable [28,29]. These features imply that Dirac semimetal may be more suitable for design THz devices that can be effectively tuned than graphene, and many researchers have been attracted by it [30,31]. As the electromagnetic energy of the THz waves can be converted into heat by the absorber, the temperature of substrate may increase to some extent. Therefore, when we design an absorber with STO and BDS, the absorption performance of this absorber can be controlled by temperature and Fermi energy. Although tunable devices with Dirac semimetal or strontium titanate have been extensively studied [32–34], the combination of these two materials has never been reported.

In this paper, we proposed a temperature and Fermi energy tunable absorber for the first time. The competitive configuration, which is composed of a BDS patch array, a STO dielectric layer, and a gold ground plane. The simulated results show that by turning the Fermi energy of BDS pattern from 10 to 60 meV, the absorption frequency can be tuned from 2.13 to 2.43 THz while the absorption peak is greater than 80%. In addition, the center frequency can be changed from 1.94 to 2.53 THz when the temperature of STO varies from 250 to 400 K. Furthermore, we have investigated the absorber under normal and oblique incidence for both TE and TM polarizations with different Fermi energies and temperatures. To quantitatively understand the physical mechanism of the proposed absorber, interference theory was applied to analyze numerical simulation results. The proposed absorber has potential applications for selective heat emitter and solar photovoltaic field in the future.

2. Permittivity of STO and BDS

STO is considered as a thermal active material, whose relative permittivity can be calculated by [21,22,35]:

$${\varepsilon _w} = {\varepsilon _\infty } + \frac{f}{{\omega _0^2 + {\omega ^2} - i\omega \gamma }}$$
where ε∞ is high-frequency bulk permittivity and the value is 9.6, f = 2.3×106 cm2 is a temperature impendent oscillator strength, ω is the angular frequency, ω0 is soft mode frequency fitting by the Cochran law, and γ is soft mode damping parameter. These two factors ω0 and γ can be calculated by:
$${\omega _0}(T)[\textrm{c}{\textrm{m}^{ - 1}}] = \sqrt {31.2(T - 42.5)}$$
$$\gamma (T)[\textrm{c}{\textrm{m}^{ - 1}}] ={-} 3.3 + 0.094T$$
where T is temperature (unit is K). Because ω0(T) and γ(T) are dependent on temperature, the relative permittivity of STO can be controlled by temperature.

Furthermore, the relative permittivity of BDS can be expressed as [24]:

$$\varepsilon = {\varepsilon _\textrm{b}} + i\frac{{\sigma (\Omega )}}{{{\varepsilon _0}\omega }}$$
where εb = 1 (for degeneracy factor is equal to 40) is the effective background dielectric, ε0 is the permittivity of vacuum. σ(Ω) = Re(σ(Ω)) + Im(σ(Ω)) is the conductivity of BDS, Re(σ(Ω)) and Im(σ(Ω)) are the real and imaginary parts of dynamic conductivity, respectively, which can be written as [27,29]:
$${\mathop{\rm Re}\nolimits} \sigma (\Omega ) = \frac{{{e^2}}}{\hbar }\frac{{g{k_F}}}{{24\pi }}\Omega G({\Omega \mathord{\left/ {\vphantom {\Omega 2}} \right.} 2})$$
$${\mathop{\rm Im}\nolimits} \sigma (\Omega ) = \frac{{{e^2}}}{\hbar }\frac{{g{k_F}}}{{24{\pi ^2}}}\left\{ {\frac{4}{\Omega }\left[ {1\textrm{ + }\frac{{{\pi^2}}}{3}{{\left( {\frac{T}{{{E_F}}}} \right)}^2}} \right] + 8\Omega \int_0^{{\varepsilon_c}} {\left( {\frac{{G(\varepsilon ) - G(\varOmega /2)}}{{{\Omega ^2} - 4{\varepsilon^2}}}} \right)\varepsilon d\varepsilon } } \right\}$$
where $G(E )= n({ - E} )- n(E )\textrm{ = }\frac{{\sinh ({E \mathord{\left/ {\vphantom {E T}} \right.} T})}}{{\cosh ({{{E_F}} \mathord{\left/ {\vphantom {{{E_F}} T}} \right.} T}) + \cosh ({E \mathord{\left/ {\vphantom {E T}} \right.} T})}}$ with n(E) being the Fermi distribution function, the degeneracy factor g = 40, ${k_F} = {{{E_F}} \mathord{\left/ {\vphantom {{{E_F}} {\hbar {\upsilon_F}}}} \right.} {\hbar {\upsilon _F}}}$ is the Fermi momentum, ћ is the reduced Planck constant, EF is the Fermi energy, and ${\upsilon _F} \approx {10^6}$m/s, $\Omega = {{\hbar \omega } \mathord{\left/ {\vphantom {{\hbar \omega } {{E_F}}}} \right.} {{E_F}}}$ and ${\varepsilon _c} = {{{E_c}} \mathord{\left/ {\vphantom {{{E_c}} {{E_F}}}} \right.} {{E_F}}}$ (Ec = 3 is the cutoff energy). According Eqs. (4)–(6), it can find that the relative permittivity of BDS is relative with Fermi energy and completely independent of temperature.

According to Eqs. (1)–(6), we have calculated permittivity of STO and BDS at different temperatures and Fermi energies and shown in Fig. 1. From Fig. 1(a), it can be found that the real part of permittivity increases slowly with increasing of frequency, and the imaginary part increases sharply across the whole research frequencies. However, the imaginary parts under different temperatures are much lower than the real parts. The resonance frequency is mainly affected by the real part of the permittivity, and the loss is mainly affected by the imaginary. Therefore, the central frequency of the absorption peak will change significantly, while the intensity of the absorptivity will keep almost unchanged. Figure 1(b) displays the permittivity of BDS as a function of different Fermi energies. It is obvious that the permittivity of the BDS is sensitive to the Fermi energies. the real part of BDS is gradually increase from negative values to zero, which indicates that BDS exhibits metallic characteristics in this frequency range. While the imaginary part of the permittivity remarkably decreases until it approaches zero with increasing frequency, which means that the loss is very low in high-frequency range.

 figure: Fig. 1.

Fig. 1. Real and imaginary parts of the permittivity for (a) STO under different temperatures and (b) BDS under different Fermi energies as a function of frequency.

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3. Design and simulation

The unit cell of the proposed dual-tunable absorber is presented in Fig. 2, which consists of BDS circle with a discontinuous ring and a gold ground plane, spaced by the dielectric layer of STO material. The BDS pattern with Fermi energy EF = 20 meV has a thickness 0.2 μm. The conductivity of gold film is 4.56×107 S/m with thickness 0.2 μm, which is greater than maximum skin depth to suppress any transmission through the absorber. The temperature of STO substrate is set as 300 K and the thickness (h) is 2 μm. Outer and inner radiuses of the circle are 1.4 μm and 1.2 μm, respectively. The widths of connector and interspace are set as 0.2 μm and 0.1 μm, respectively. Moreover, the unit cell has a periodic constant (L) of 3 μm, and the angle α is taken to be 45°.

 figure: Fig. 2.

Fig. 2. Schematic diagram of STO-and BDS-based tunable absorber (Here, the parameters of absorber are set as L = 3 μm, h = 2 μm, R = 1.4 μm, r = 1.2 μm, c = 0.2 μm, w = 0.1 μm, and α=45°).

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The numerical computation of the proposed tunable absorber is performed by commercial simulation software CST microwave package with periodic boundary conditions in the x-axis and y-axis and an open boundary condition in the z-direction. The incident waves with electric field polarized along the x-axis were illuminated normally to the surface. Due to the electromagnetic shielding by the gold ground plane, absorptivity of the absorber can be calculated as $A(\omega )= 1 - {|{{S_{11}}(\omega )} |^2}$, where ${S_{11}}(\omega )$ is the reflection coefficient.

4. Results and discussion

Figure 3(a) presented the reflection and absorption spectrums and for both transverse-electric (TE)-mode (the electric field is parallel to x-axis) as well as transverse-magnetic (TM)-mode (the electric field is parallel to the y-axis) polarized plane electromagnetic wave under the normal incidence. According to this figure, we can find that the absorption spectrum for TE polarization is the same as TM polarization. The perfect absorption frequency located at 2.16 THz with 99.98% peak absorption and the bandwidth with absorption over 90% is 0.1 THz from 2.11 to 2.21 THz. Figure 3(b) is the color map of the absorption spectra with different polarization angles. It can be found that the absorption spectra are independent of the variation of φ for both TE and TM polarizations due to the four-fold rotational symmetry of the designed structure.

 figure: Fig. 3.

Fig. 3. (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.

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It is well known that the permittivity of STO is highly dependent on temperature, which can be seen as thermally tunable property. Because the incident electromagnetic waves are almost completely converted into heat in the absorption band and the change in ambient temperature, we investigated the temperature dependence of the proposed absorber. As shown in Fig. 4(a), with increasing of temperature from 250 K to 400 K, the central frequency of the absorption peak shows a blue shift, but the shape of the absorption curves has little change in the entire temperature. When the temperature is 250 K, the central frequency is 1.94 THz with 99.67% absorptivity. However, when the temperature is increased to 400 K, the absorption peak located at 2.53 THz with 99.8% peak absorption. The adjustment range of the central frequency reaches 0.59 THz for this absorber. To show more clearly and later discussion, we mark the abscissa of each center frequency with dotted lines and symbols in different colors, as shown in Fig. 4. This clearly confirms that the performance of this proposed absorber can control by temperature. The physical mechanism of continuous modulation is mainly caused by the variation of permittivity of STO with temperature.

 figure: Fig. 4.

Fig. 4. Absorption spectra of the absorber with different (a) temperatures and (b) Fermi energies under normal incidence.

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According to Fig. 1(b), it is known that the permittivity of BDS relates largely to the Fermi energy EF, which can be controlled by chemical doping. The real and imaginary parts of permittivity vary with Fermi energy. Therefore, the central frequency of the absorption peak will be controlled by EF. Figure 4(b) shows the simulated absorption spectra with different Fermi energies EF. It is obvious that the central frequency also displays a blue shift with increasing EF. When EF changes from 10 meV to 60 meV, the peak frequency shifts from 2.14 THz ($f_1^{\prime}$) to 2.44 THz ($f_6^{\prime}$), and the peak absorptivity is maintained over 99% when EF is in the range of 10 meV to 30 meV. However, the magnitude of absorption decreases clearly when EF is larger than 30 meV. The absorptivity is only 82.8% when EF = 60 meV. Therefore, by applying these independent control methods in STO and BDS, we can conclude that the proposed absorber is able to control by thermally and Fermi energy, independently.

Impedance matching theory is widely used to demonstrate the physical mechanism of the perfect absorber [36,37]. The relative impedance equation at normal incidence can be written as follows:

$${Z_r}(\omega ) = \sqrt {\frac{{{{(1 + {S_{11}}(\omega ))}^2} - {S_{21}}{{(\omega )}^2}}}{{{{(1 - {S_{11}}(\omega ))}^2} - {S_{21}}{{(\omega )}^2}}}}$$
and the corresponding absorption formula is
$$A(\omega ) = 1 - R(\omega ) = 1 - {\left|{\frac{{Z - {Z_0}}}{{Z + {Z_0}}}} \right|^2} = 1 - {\left|{\frac{{{Z_r} - 1}}{{{Z_r} + 1}}} \right|^2}$$
where Z0 = 120 Ω is the free space impedance, Z is the effective impedance of the absorber and ${Z_r} = Z/{Z_0}$ is the relative impedance. According to Eq. (8), we can find that the absorptivity reaches the peak value when Zr = 1. The real and imaginary parts of the relative impedances of the proposed absorber with different temperatures and Fermi levels under normal incidence are given in Fig. 5. As shown in Fig. 5(a), it can be found that the real parts are closer to 1. Meanwhile, the imaginary parts approach to 0 at the resonant frequency points f1 - f4 with high absorptivity. It means the effective impedances are nearly matched to free space, and the absorptivity is quite high, which can be seen from Fig. 4(a). While in Fig. 5(b), we can find that a similar situation is also observed at the absorption frequency points $f_1^{\prime}$ and $f_2^{\prime}$. However, as the increase of Fermi energy, the discrete distance between real and imaginary parts of the relative impedance from the dotted line increases at the absorption frequency points. It means that the impedance mismatch gets worse and worse, causing the absorption rate to gradually decrease, which is consistent with the results in Fig. 4(b).

 figure: Fig. 5.

Fig. 5. Real and imaginary parts of the relative impedance Zr with different (a) temperatures of STO and (b) Fermi levels of BDS.

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For the circle or ring structure, the relationship between absorption frequency and permittivity can be estimated by [38]:

$$2\pi {R_{eff}} = \frac{c}{{f\sqrt {{\varepsilon _{eff}}} }}$$
where ${R_{eff}} = ({R + r} )/2$ is effective radius of the ring, f is the absorption frequency, c and ${\varepsilon _{eff}}$ are the speed of light in free space and effective permittivity of the substrate. Therefore, as the decrease of εeff, absorption frequency shifts to the higher value when the radius is constant. This explanation is good agreement with the results in Fig. 5(b).

The above results were calculated under normal incidence. In the case of oblique incidence on the performances of absorptance for TE and TM polarizations are worthy to investigate. Figures 6(a) and (b) show the absorptance of this proposed absorber as a function of frequency and incidence angle when the temperature and Fermi level are set as 300 K and 20 meV. As shown in Fig. 6(a) under TE polarization, the absorption bandwidth decreases slightly with increasing of oblique angles. In the case of TM polarization, the absorption bandwidth is almost stable when θ varies between 0-60°. However, the central frequency remains unchanged for both modes. The stability of this proposed absorber versus varying angles φ and θ is an important advantage for sensing, detecting and optoelectronic applications [39].

 figure: Fig. 6.

Fig. 6. Color map of the absorption spectra with different incident angles for (a) TE and (b) TM polarizations, respectively.

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To deeply demonstrate the physical mechanism of the proposed absorber, we applied interference theory to explain the absorption mechanism, as shown in Fig. 7(a). When a THz wave incident from air with angle ϕ12, partial reflection and transmission occur at the air-spacer interface. The transmissive wave continues to propagate until it reaches the ground plane and then reflect. The overall reflection of the multiple reflections can be expressed as [40]:

$$\tilde{r} = {\tilde{r}_{12}} - \frac{{{{\tilde{t}}_{12}}{{\tilde{t}}_{21}}{e^{i2\tilde{\beta}}}}}{{1 + {{\tilde{r}}_{21}}{e^{i2\tilde{\beta}}}}}$$
where ${\tilde{r}_{12}} = {r_{12}}{e^{i{\phi _{12}}}}$ represents the reflection coefficient from the BDS array, ${\tilde{t}_{12}} = {t_{12}}{e^{i{\theta _{12}}}}$ is the transmission coefficient when the incident wave transmitted into the spacer. ${\tilde{r}_{21}} = {r_{21}}{e^{i{\phi _{21}}}}$ and ${\tilde{t}_{21}} = {t_{21}}{e^{i{\theta _{21}}}}$ are the reflection and transmission coefficients occur again at the air-spacer interface. $\tilde{\beta } = {\beta _r} + i{\beta _i} = \sqrt {{{\tilde{\varepsilon }}_{STO}}} {k_0}h$ is the complex propagation phase of the THz wave, where k0 is the free space wavenumber, h is the thickness of the STO layer, βr is the propagation phase, and βi relates to the absorption in the dielectric layer. Therefore, the absorption can be calculated through $A(\omega ) = 1 - R(\omega ) - T(\omega ) = 1 - {|{\tilde{r}(\omega )} |^2}$ since the transmission $T(\omega ) = 0$ due to the existence of the ground plane. As shown in Fig. 7(b), we compared the absorptance spectra calculated by interference theory and numerical simulations. It can be seen that the simulated results are in general agreement with theoretical calculation spectra.

 figure: Fig. 7.

Fig. 7. (a) Schematic of interference theory model. (b) Comparison of absorptance spectra produced by theoretical calculation and numerical simulation using the unit cell shown in Fig. 2. The Fermi energy of the BDS layer was set as20 meV, and the temperature of STO was set as 300 K.

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5. Conclusion

In conclusion, we have designed and numerically investigated a polarization-independent and active tunable absorber utilizing patterned BDS sheet and STO material in the terahertz region. Through a detailed mathematic calculation, the permittivity of STO and BDS materials were obtained and used in the simulation. Due to the symmetry of the structure, the proposed absorber has an almost perfect absorption characteristics for both TE and TM polarizations at 2.16 THz when the temperature was set at 300 K and Fermi level was set at 20 meV. More importantly, the center frequency could be tuned from 1.94 to 2.53 THz by changing the temperature from 250 to 400 K, while the amplitude of the absorptance can be maintained above 99%. However, the peak absorptivity can be controlled from 99.9% to 82.8% by adjusting the Fermi energies from 10 to 60 meV, and the center frequency shifts from 2.14 to 2.44 THz. Angular tolerance of metamaterial absorber is also studied. The results show that the excellent absorption performance of the absorber remains stable for both TE and TM polarizations even the incident angle varies up to 60°. Furthermore, the physical mechanism was explored by interference theory and impedance matching theory. The simulated and theoretical calculation results are agreed well. This proposed absorber can provide guidance to research tunable, smart and multifunctional terahertz devices.

Funding

National Natural Science Foundation of China (51777023, 61501067); Fundamental Research Funds for the Central Universities (2019CDQYTX033).

Disclosures

The authors declare no conflicts of interest.

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Figures (7)
Fig. 1.
Fig. 1. Real and imaginary parts of the permittivity for (a) STO under different temperatures and (b) BDS under different Fermi energies as a function of frequency.

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Fig. 2.
Fig. 2. Schematic diagram of STO-and BDS-based tunable absorber (Here, the parameters of absorber are set as L = 3 μm, h = 2 μm, R = 1.4 μm, r = 1.2 μm, c = 0.2 μm, w = 0.1 μm, and α=45°).

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Fig. 3.
Fig. 3. (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.

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Fig. 4.
Fig. 4. Absorption spectra of the absorber with different (a) temperatures and (b) Fermi energies under normal incidence.

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Fig. 5.
Fig. 5. Real and imaginary parts of the relative impedance Zr with different (a) temperatures of STO and (b) Fermi levels of BDS.

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Fig. 6.
Fig. 6. Color map of the absorption spectra with different incident angles for (a) TE and (b) TM polarizations, respectively.

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Fig. 7.
Fig. 7. (a) Schematic of interference theory model. (b) Comparison of absorptance spectra produced by theoretical calculation and numerical simulation using the unit cell shown in Fig. 2. The Fermi energy of the BDS layer was set as20 meV, and the temperature of STO was set as 300 K.

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Equations (10)

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ε w = ε ∞ + f ω 0 2 + ω 2 − i ω γ
ω 0 ( T ) [ c m − 1 ] = 31.2 ( T − 42.5 )
γ ( T ) [ c m − 1 ] = − 3.3 + 0.094 T
ε = ε b + i σ ( Ω ) ε 0 ω
Re σ ( Ω ) = e 2 ℏ g k F 24 π Ω G ( Ω / Ω 2 2 )
Im σ ( Ω ) = e 2 ℏ g k F 24 π 2 { 4 Ω [ 1  +  π 2 3 ( T E F ) 2 ] + 8 Ω ∫ 0 ε c ( G ( ε ) − G ( Ω / 2 ) Ω 2 − 4 ε 2 ) ε d ε }
Z r ( ω ) = ( 1 + S 11 ( ω ) ) 2 − S 21 ( ω ) 2 ( 1 − S 11 ( ω ) ) 2 − S 21 ( ω ) 2
A ( ω ) = 1 − R ( ω ) = 1 − | Z − Z 0 Z + Z 0 | 2 = 1 − | Z r − 1 Z r + 1 | 2
2 π R e f f = c f ε e f f
r ~ = r ~ 12 − t ~ 12 t ~ 21 e i 2 β ~ 1 + r ~ 21 e i 2 β ~
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