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Abstract
Terahertz logic gates play a vital role in optical signal processing and terahertz digitization. Herein, a strategy to design an all-optical terahertz logic gate device composed of metamaterials with a semiconductor-metal hybrid is proposed; accordingly, a concrete logic gate composed of Ge embedded-in Au stripe supported by a Si board is presented theoretically. Simulation results reveal the dependence of the terahertz transmission spectra on the different illuminations in the device. Based on the illumination-transmission response, the designed device can realize the NOR or OR Boolean operation. The effects of the width of the Ge-Au stripe as well as the Si board on the transmission spectra and logic performance were also investigated.
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1. Introduction
Terahertz (THz) waves generally refer to electromagnetic waves in the frequency range of 0.1–10 THz and are characterized by a high transmission rate, broad bandwidth, and low energy. They are expected to play a key role in medical testing, broadband communication, and security applications and have recently attracted considerable attention [1–4]. Manipulation of THz waves is crucial for realizing various THz applications. In recent years, studies have shown the ability of micro-scale metamaterials, which consist of a type of artificial metal array structures with a micro-level size, to modulate the parameters (amplitude, frequency, phase, polarization, etc.) of THz waves [5–8]. To date, numerous THz devices have been designed and fabricated based on metamaterials, and diverse functions such as wave absorption [9,10], biosensing [11,12], and polarization [13,14] have been realized.
An optical logic gate is an optical device that implements Boolean operations with light as an information carrier. Compared to conventional electrical logic gates in digital circuits, optical logic gates have the advantages of low power consumption attributed to the lack of current-induced Joule heating as well as fast signal transmission owing to the extremely high speed of light [15–17]. Many efforts have been devoted to the study of THz logic gates thus far, and several significant works have been reported recently. Y-structure waveguides are one way to realize THz logic gates, of which logic functions are performed based on interferences of coupled plasmon modes from input arms [18–20]. In 2020, Ortiz-Martinez et al. fabricated three Y-structure optical logic gates by three-dimensional printing, which can implement OR, AND, and XOR Boolean operations at the THz waveband. In 2022, Blessan et al. proposed various all-optical THz logic gates, which consist of Y-structure waveguides made by the bulk Dirac semimetals. However, the size of such THz logic gates usually reaches up to thousands of microns. It is not conductive to device miniaturization. On the other hand, the ability of metamaterials to manipulate THz waves can offer another way to realize THz logic gates. Additionally, micro-scale patterns in metamaterials are also beneficial for the current trend of device miniaturization. Therefore, the use of metamaterials to design and fabricate THz logic gates can help realize the digitization of THz waves [21–24]. In 2016, Kim et al. fabricated optical logic gates with the hybridization of graphene, ferroelectric materials, and metamaterials; the response of the transmission amplitude (regarded as the output signal) to the gating voltages (regarded as the input signal) enabled AND/OR Boolean operations in THz band [21]. In 2020, with counter THz waves as input signals and transmitted waves as output signals, Granpayeh et al. designed a logic gate system using the coherent perfect adsorption phenomenon that occurs on a metasurface. The system can perform AND, OR, and XOR Boolean operations by adjusting the phase difference [22]. Recently, Lin et al. have proposed an optical logic gate consisting of micro-electro-mechanical system (MEMS)-based metamaterials. With the polarization angle of incident light as well as the driving bias voltage as the input and the absorption efficiency as the output, the designed device can realize OR (AND) Boolean operations at 0.33 (0.88) THz [24]. Despite these advancements, the structures of these proposed logic gates are usually complex. Furthermore, Boolean operations in these devices cannot be manipulated purely by optical signals, which may lead to incompatibility between logic gates and optical systems. Therefore, the development of THz logic gates remains an ongoing challenge.
In this study, we propose a strategy to utilize semiconductor-metal hybrid metamaterials to build THz logic gates; accordingly, a specific THz device comprising of germanium (Ge) embedded-in gold (Au) stripe arrays and silicon (Si) back board is theoretically proposed. Using an incident optical pump, the conductivities of Ge and Si in the device can be modulated, and the consequential transmission characteristics of the device in THz band can be numerically simulated. The results reveal that the designed device can function as a NOR logic gate or an OR logic gate according to specific frequencies. Further investigations demonstrate that changes in the width of the metal stripe or Si board can affect the reliability and functions of the logic gate.
2. Theoretical model and computational details
It is well known that the photoelectric effect can alter the carrier concentration of semiconductors, thereby changing their conductance. Based on this characteristic, illumination will have a great influence on the transmission properties of THz waves on semiconductor embedded-in metamaterials [25,26]. From this perspective, it is a feasible strategy to realize all-optical logic gate devices by utilizing metamaterials containing semiconductors. To verify this strategy, a specific metamaterial-based device was designed. The proposed device is composed of periodically arranged unit cells. A schematic of the unit cell is depicted in Fig. 1. As can be seen from Fig.1a and Fig.1b, the unit cell consists of Ge embedded in a Au stripe on a dielectric substrate, and the other side of the substrate is partially covered by a Si board. The lossy Au stripe is 0.5 µm thick, and its conductivity is 4.56${\times} $104 S/m. The embedded Ge and Si board are also 0.5 µm thick, and their relative permittivity values are 16.3 and 11.7, respectively. The sandwiched dielectric substrate is composed of lossy polyimide; its thickness is 5 µm, relative permittivity is 3.5, and loss tangent is 0.03. The geometric parameters of a unit pattern are as follows: Px = Py = 100 µm, L1 = 40 µm, L2 = 20 µm, L3 = 100 µm, W1 = 8 µm, W2 = 35 µm. Here, Px and Py are sides of the unit cell; L1 is the length of Au stripe, L2 the length of the Ge component, W1 the width of the Ge-Au stripe; L3 and W2 are the length and width of the Si board. Previous studies have reported that the conductance of Si (Ge) can be modulated by a pump beam with a wavelength less than 1100 nm (1600 nm) [27,28]. When the energy flux of the pump beam is ca. 180 µJ/cm2, the conductivity of Si (σSi) can reach up to 5 ${\times} $104 S/m [29,30]. The conductivity of Ge (σGe) is assumed to has similar variations to that of σSi[31]. Therefore, it is feasible to use 1550-nm and 800-nm pump beams to illuminate the Ge-contained Au stripe and Si board, respectively, to alter the conductance of the microstructure and thus modulate the transmission of THz waves. According to different illuminations on the Ge-Au stripe and Si board, four cases of the device are considered in this study, Dss, Dsm, Dms, and Dmm, respectively. The first and second subscripts in these terms respectively denote the conductance of Ge and Si. Subscript s denotes that Ge or Si behaves as a semiconductor under no illumination, and subscript m denotes that Ge or Si exhibits metallicity under appropriate illumination.
Fig. 1. (a) Schematic of the designed device; (b) back view of the designed device. Yellow color represents Au, blue represents Ge, pink represents PI, and green represents Si. Size parameters are as follows: Px = Py = 100 µm, L1 = 40 µm, L2 = 20 µm, L3 = 100 µm, W1 = 8 µm, W2 = 35 µm. The incident THz waves propagate along the z direction with an x-polarized electric field and a y-polarized magnetic field.
The simulation was implemented using the frequency domain solver in the commercial software CST Microwave Studio. The x and y directions were adopted as periodic boundary conditions for the unit cell, and open (add space) is applied along the z direction. The incident THz waves propagate along the z direction with an x-polarized electric field and a y-polarized magnetic field. Here, the transmission coefficient (reflection coefficient) extracted from S-parameters is introduced to characterize the transmission (reflection) capability of THz waves, which is calculated by two ports respectively applied at both sides of the unit cell along the z direction. One port is applied at -50 µm from the Au-Ge stripe and the other one is applied at 50 µm from the Si board. The conductivities of Si and Ge in the semiconductor states (the metal states) are set to 0 S/m (5 ${\times} $104 S/m).
3. Results and discussion
The transmission spectra for Dss, Dsm, Dms, Dmm as a function of frequency are shown in Fig. 2. When no optical pump illuminates the device, both Ge and Si act as semiconductors. Figure 2(a) clearly shows that the transmission coefficient for Dss exceeds 0.90 in the frequency range of 0.01–0.71 THz and 1.65–2.00 THz. The device exhibits high transmission capability in the broad band. In addition, a noticeable resonant dip (marked as dip A) appears at 1.26 THz with the transmission coefficient of 0.02, which indicates that incident waves with frequencies close to 1.26 THz barely penetrate the device. When Si board is irradiated with 800nm pump beam, the conductivity of Si sharply increases. The consequent transmission spectra for Dsm presented in Fig. 2(b) indicate that the transmission of low-frequency THz waves is drastically suppressed, and that the transmission coefficient is less than 0.40 at frequencies between 0.01 and 1.10 THz. Furthermore, a small resonant dip (marked as dip B) can be observed at 0.95 THz, corresponding to a transmission coefficient of 0.27. When the 1550-nm pump beam is applied on the side of the Ge-Au stripe, the conductivity of the Ge part is enhanced. The corresponding transmission spectra in Fig. 2(c) show Dms still retains a low transmission coefficient at a low frequency. A transmission coefficient plateau (over 0.90) is observed at frequencies between 1.59 and 2.00 THz. When the 800-nm and 1550-nm pump beams simultaneously illuminate on Dmm, a weak transmission of the incident THz waves is observed at a low frequency range.
The above result suggests that the optical pump has a significant impact on the transmission as it modulates the device conductivity. It is essential to elucidate the mechanism behind it. At a low frequency, only Dss has a high transmission coefficient, while Dsm, Dms, and Dmm have a low transmission coefficient. To explain this phenomenon, the transmittivity (denoted T, here $T = S_\textrm{t}^2$, where ${S_\textrm{t}}$ is the transmission coefficient), reflectivity (denoted R, here $R = S_\textrm{r}^2$, where ${S_\textrm{r}}$ is the reflection coefficient) and absorptivity (denoted A, here $A = 1 - T - R$) for the investigated devices were calculated. The results are presented in Figs. 3(a)-(d). One can clearly see that the values of R and A are nearly negligible at a low frequency for Dss (see Fig. 3(a)), whereas they are larger for the other three devices (Figs. 3(b)-(d)). The high R and A for Dsm, Dms, and Dmm might origin from the strengthened conductivity due to the pump beams; thus, the Ge or/and Si parts behave like a metal. A large portion of the incident THz waves is reflected or absorbed by the metallic micro unit cell rather than transmitted; this results in low transmission for Dsm, Dms, and Dmm. In addition, dip A at 1.26 THz for Dss and dip B at 0.95 THz for Dsm can be illustrated through the distribution of the electric field as shown in Figs. 3(e)-(h). Figure 3(e) displays the intense electric field locates on the upper and lower splits of the Au stripe, which demonstrates that dip A in Dss stems from the coupling between incident THz waves and dipole modes. For Dsm, only the conductance of the Si part improves, while that of the Ge-Au stripe remains unchanged. Therefore, the electric field distribution on the Ge-Au stripe for Dsm (see Fig. 3(f)) is similar to that for Dss, contributing to dip B. For Dms and Dmm, since the increasing conductivity of the Ge part alters the electromagnetic response of the device, the resonant modes almost vanish (see Fig. 3(g) and Fig. 3(h)). Thus, there are no resonant dips in Dms as well as Dmm.
Fig. 3. Top: transmittivity, reflectivity, and absorptivity spectra for (a) Dss, (b) Dsm, (c) Dms, and (d) Dmm. Bottom: electric field distributions for (e) Dss at 1.26 THz, (f) Dsm at 0.96 THz, (g) Dms at 1.26 THz, and (h) Dmm at 1.26 THz.
From the aforesaid discussion, it is clear that the transmission performance of THz waves for the designed device can be switched by changing the illumination. Such a response in the device enables logic operations. Based on this point, a schematic of the designed device acting as a logic gate is shown in Fig. 4. Here, we assume two optical pump beams as input signals illuminating the devices: a 1550-nm pump beam acts as input I, and an 800-nm pump beam acts as input II. When the device is irradiated, the input signal is defined as 1. When no pump illuminates the device, the input signal is 0. Meanwhile, the transmission coefficient at 0.1 THz is assumed as the output signal, where a large value (larger than 0.5) and a small value (less than 0.5) of the output are defined as 1 and 0, respectively. Based on the above definition, the truth table for the device is obtained (Table 1). When no pump beam illuminating the device (i.e., the case of Dss), the input signals are [0, 0]. The transmission coefficient at 0.1 THz is 1 (see Fig. 2(a)), which indicates that the output signal is 1. When only the 800-nm pump beam illuminating on the Si board (i.e., the case of Dsm), input signal is [0, 1], and the transmission coefficient is 0.38 (Fig. 2(b)); the corresponding output signal is 0. When the input signal for Dms is [1, 0], i.e., when only the 1550-nm pump beam illuminates the Ge-Au stripe, the transmission coefficient is 0.36 (see Fig. 2(c)), which indicates that the output signal is 0. Finally, when both 1550-nm and 800-nm pump beams simultaneously irradiate the device (i.e., the case of Dmm), the input signal is [1,1], and the transmission coefficient is 0.23 (Fig. 2(d)); the device output is 0. Apparently, a high-level output signal 1 for Dss can only be generated when the input signal is [0, 0]; thus, the designed device can realize the NOR Boolean operation.
Fig. 4. Schematic of the designed device working as a logic gate. Here, a 1550-nm pump beam illuminating the Ge-Au stripe is assumed as input I, and an 800-nm pump beam illuminating the Si board is assumed as input II. The THz wave propagating through the device is assumed as the output signal. (Only illuminated areas can work as logic gates).
Table 1. Truth table for the meta device, where input signals are manipulated by illuminations on the device surfaces and output signals are defined as transmission coefficients at 0.1 THz.a
In fact, different logic functions can be realized by sampling input and output signals at different frequencies. Table 2 shows that different transmission coefficients can be realized at different resonant frequencies, and specific logic functions can be achieved. Here, the definition of input signals is kept unchanged while transmission coefficient at 1.26 THz is defined as output signal, where the large value more (less) than 0.5 represents high-level value 1 (low-level value 0). As exhibited in Table 2, for Dss, i.e., the case of input signals [0, 0], the resonant dip with 0.02 indicates output signal is 0. For Dsm, Dms, and Dmm, i.e., the case of input signals [0, 1], [1, 0], and [1,1], transmission coefficients at 1.26 THz are 0.53, 0.82, and 0.56, respectively, which all correspond to output signal 1. Obviously, the device now can work as OR logic gate.
Table 2. Truth table for the meta device, where input signals are manipulated by illumination on the device surfaces and output signals are defined as transmission coefficients at 1.26 THz.a
As is known to all, pattern parameters play a key role in the electromagnetic response of meta devices. This implies that the performance of the designed optical logic gate can be modulated by the pattern parameters. Therefore, we studied the effect of pattern width on the transmission properties as well as logic functions of the designed device.
3.1 Effect of the width of the Ge-Au stripe on the designed logic gate
The relationship between the transmission coefficient of the devices and W1 (ranging from 5 to 11 µm with a step of 1 µm) for W2 = 35 µm was investigated, and the corresponding transmission spectra are shown in Fig. 5. As shown in Fig. 5(a), for Dss, the transmission spectra with different W1 are nearly identical, which indicates that the transmission of THz waves is barely affected when no pump beam illuminates the device. Figure 5(b) shows that the increase in W1 causes a slight red shift of the resonant dip for Dsm, where the resonant frequency shifts from 0.98 THz for W1 = 5 µm to 0.93 THz for W1 = 11 µm. For Dms, the transmission coefficient monotonously decreases with increasing W1 at a low frequency, while it almost overlaps with each other for different W1 at a high frequency (Fig. 5(c)). Thus, an increase in W1 can suppress the low-frequency transmission capability. From Fig. 5(d), the trend of transmission spectra vs. W1 for Dmm is similar to that for Dms, and only the transmission coefficient decreases with increasing W1 at a low frequency.
Fig. 5. Transmission spectra for (a) Dss, (b) Dsm, (c) Dms, and (d) Dmm with different values of W1 (ranging from 5 µm to 11 µm with a step of 1 µm).
We further analyzed the influence of the width of the Ge-Au stripe on the logic performance of the device. Table 3 lists the transmission coefficients at 0.1 THz and 1.26 THz, extracted from Fig. 5, for W1 = 5 µm, 8 µm, and 11 µm as well as the corresponding input/output signals. Apparently, NOR and OR Boolean operations always hold when W1 changes from 5 µm to 11 µm. Nevertheless, for W1 = 5 µm, the transmission coefficient of for Dms at 0.1 THz is closer to the threshold (i.e., 0.5). This is a drawback as the output signal can be easily disturbed by external factors. Contrarily, when W1 = 11 µm, the transmission coefficient for Dms at 0.1 THz is farther from the threshold; this indicates that the output signal is distinguishable, and the logic reliability of the device improves.
Table 3. Truth table for the designed device with W1 = 5 µm, 8 µm, and 11 µm, where input signals are manipulated by illumination on the device surfaces and the output signals are defined as the transmission coefficients at 0.1 THz and 1.26 THz.a
3.2 Effect of the width of the Si board on the designed logic gate
The relationship between the transmission coefficient of the device and W2 (ranging from 20 µm to 50 µm with the step of 5 µm) for W1 = 8 was investigated. The corresponding transmission spectra are exhibited in Fig. 6. For the cases of Dss and Dms, the overlap in the transmission spectra with an increase in W2 (see Fig. 6(a) and Fig. 6(c)) shows that the change in W2 has no effect on the transmission properties of THz waves in the studied frequency range. From Fig. 6(b) and Fig. 6(d), it can be seen that the overall transmission spectra of Dsm and Dmm decrease with the increasing W2, but the position of dip B for Dsm is unchanged.
Fig. 6. Transmission spectra for (a) Dss, (b) Dsm, (c) Dms, and (d) Dmm with different values of W2 (ranging from 20 µm to 50 µm with a step of 5 µm).
The transmission coefficients at 0.1 THz and 1.26 THz as well as the obtained input/output signals for W2 = 20 µm, 35 µm, 50 µm are listed in Table 4. At 0.1 THz, for W2 = 35 µm or 50 µm, the designed device can still realize NOR Boolean operations. Particularly, the values of Dsm and Dmm for W2 = 50 µm have lower transmission coefficients compared to those for W2 = 35 µm, implying high reliability. However, when W2 is 20 µm, the transmission coefficient for Dsm is 0.51, which exceeds the threshold of 0.5. This demonstrates that the input signals [0, 1] will generate an output signal 1 instead of 0; hence, the device cannot realize NOR Boolean operation anymore. For input signals based on transmission coefficients at 1.26 THz, the devices with W2 = 20 µm and 35 µm can implement OR Boolean operations. Nevertheless, when W2 increases to 50 µm, the relationship between the transmission coefficient and illumination cases (see Table 4) indicates that the OR Boolean operation cannot be performed anymore.
Table 4. Truth table for the meta device with W2 = 20 µm, 35 µm, and 50 µm, where input signals are manipulated by illumination on the device surfaces and output signals are defined as the transmission coefficients at 0.1 THz and 1.26 THz.a
In addition, signal processing speed is an important index for the logic gate. It is feasible to evaluate this index from the switch time of Si and Ge. Theoretically, the time with femtoseconds for electrons moving from valence band to conduction band in semiconductors due to the photoelectric effect means an extremely fast response time [32]. However, in the experiment, the switch time of these semiconductors is reported to be 101–104 ns with respect to different systems [33,34]. It indicates that the response time of the logic gate is at microsecond or below, and can be modified by different methods. It is expected to be explored in the future experiment.
4. Conclusion
In this work, a metamaterial-based THz device composed of Ge embedded in a Au stripe combined with a Si board was theoretically designed, and the corresponding transmission coefficients for different illuminations by pump beams were numerically simulated. The results reveal that the change in the conductivity of the Ge and Si parts induced by the pump beams plays a vital role in determining the transmission properties of the designed device. Specifically, when no pump beam illuminates the device, i.e., the case of Dss, THz waves at a low frequency can easily propagate through the device. Nevertheless, once the pump beam illuminates the device (i.e., the cases of Dsm, Dms, and Dmm), the transmission of THz waves at a low frequency is suppressed. This can be explained by the enhanced reflection and absorption, which are attributed to the high conductivity of the Ge and Si parts excited by the illumination. In addition, resonant dips for Dss and Dsm are evident, which stem from the coupling between THz waves and the excited dipolar modes. The designed device can implement NOR or OR operations with the illumination as the input signal and the transmission coefficients at different frequencies as the output. In addition, the effect of pattern width on the transmission properties and consequential logic performance was investigated. The results show that increasing the width of Ge-Au stripe can improve the reliability of logic functions at 0.1 THz. When the width of the Si board is 20 µm, the NOR Boolean operation performed at 0.1 THz is disabled; when the width is 50 µm, the OR Boolean operation implemented at 1.26 THz ceases. Thus, a new strategy to realize THz logic gates capable of implementing multiple Boolean operations is proposed, which will facilitate the development of THz devices.
Funding
National Natural Science Foundation of China (62201496, 61675147, 61701434, 61735010); Qingchuang Science and Technology Plan of Shandong Universities (2019KJN001); National Key Research and Development Program of China (2017YFA0700202, 2017YFB1401203); Natural Science Foundation of Shandong Province (ZR2020FK008, ZR202102180769, ZR2021MF014, ZR2022QF054); Taishan Scholar Project of Shandong Province (tsqn201909150).
Acknowledgments
This work was supported by the National Natural Science Foundation of China (NSFC), Special Funding of the Taishan Scholar Project, the Natural Science Foundation of Shandong Province, the National Key Research and Development Program of China, funding from the Qingchuang Science and Technology Plan of Shandong Universities. We thank languageediting.osa.org for editing the English text of a draft of this manuscript.
Disclosures
The authors declare no competing financial 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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