1. INTRODUCTION

Due to the constantly growing number of transistors per unit area and the emergence of additional tasks that require constantly increasing computing power, experts predict the integration of tens and even hundreds of computing cores into processors over the next ten years. Systems with such a large number of cores present significant problems in designing the network architecture, since their performance is increasingly limited by the exchange of information between computational units rather than by the computations themselves [1]. The communication requirements of multicore processors cannot be met by the standard electronic interconnects because of the disadvantages of high power dissipation and limited bandwidth. Therefore, active work is underway to develop communication systems based on new principles, including using photonic nanodevices that perform the function of optical interconnects, capable of providing higher bandwidth and lower power consumption [2–4]. The basic element of such systems is a plasmon antenna.

Incident light or an IR wave induces a localized surface plasmon in the nanoantenna at a certain wavelength, and oscillating charges concentrated at the antenna leads lead to an increase in the field. Due to the low response of nanoantennas in the optical frequency range, they were previously studied in order to improve the characteristics of optical detectors and devices for collecting IR energy [5, 6]. Now, due to the reasons listed above, nanoantennas are considered as devices for receiving and transmitting data in the terahertz frequency range.

It should be noted that at this stage, the creation of a communication system on a chip is being considered, but only on the surface of the crystal. The possibilities of implementing a wireless communication system in a three-dimensional integrated circuit between different layers have not yet been considered. In relation to this, a proposal is being put forward to integrate optical antennas with Through Silicon Via (TSV) connections. TSV connections are usually filled with copper to carry out the signal transfer process between the 3D IC layers. However, this method has certain disadvantages. During operation, copper oxidizes [7] and heats up, which leads to a significant deterioration in the transfer characteristics. It is proposed to replace copper columns with optical nanoantennas, thus eliminating the key disadvantages of this technology.

The main problem of nanoantennas lies in the range of data transmission. The width of the through channels of TSV connections varies from 0.5 to 5 µm, and their height varies from 1 to 50 µm [7]. Therefore, optical antennas should achieve a signal transmission range of at least several micrometers for their potential use as far-field data transceivers.

2. DESCRIPTION OF FAR-FIELD ANTENNAS

The performance of a long-range antenna can be obtained by considering the power flow under conditions of limited power consumption. However, the distance from the antenna to its far field depends on the geometrical dimensions of the antenna, and it is usually assumed that the far-field region begins beyond the distance

$$R = r = \frac{{2{{D}^{2}}}}{{{\lambda }}},$$
(1)

where D is the size of the largest inclusion of the antenna. This is due to the different propagation distances of the field contributions from different parts of the antenna to the observation point. In the far field, each antenna is considered to be a point source, and the far-field criterion in Eq. (1) is derived from the assumption that the phase errors due to changing signal propagation distances are less than π/8. We assume that the antenna is in the center of the sphere. We also assume that the antenna is transmitting a signal. It has the following parameters:

• Rt is the power received by the antenna, W;

• Rrad is the power radiated by the antenna, W;

• η is the radiation efficiency.

The presented parameters are related by Eq. (2):

$${{\eta }} = \frac{{{{P}_{{{\text{rad}}}}}}}{{{{P}_{{\text{t}}}}}}.$$
(2)

Antennas also have the parameter St(θ, φ), the power density, which does not depend on the distance from the beginning of the far field to antenna r. The total radiated power can be obtained by calculating the integral of the power density within the surface containing the antenna. Such an area can take any shape. For simplicity, this area is represented as a sphere.

$${{P}_{{{\text{rad}}}}} = \mathop \smallint \limits_0^{2\pi } \mathop \smallint \limits_0^\pi {{S}_{{\text{t}}}}\left( {\theta ,~\varphi } \right){{r}^{2}}{\kern 1pt} \sin {\kern 1pt} \theta ~d\theta ~d\varphi .$$
(3)

The average power density takes on the value

$${{P}_{{{\text{avg}}}}} = \frac{{{{P}_{{{\text{rad}}}}}}}{{4\pi {{r}^{2}}}}.$$
(4)

We assume that the following parameter Dt is directivity; i.e., the ability of the antenna to concentrate the radiated power in a certain direction. It is related to power density as follows:

$${{D}_{t}}\left( {{{\theta }},{{\;\varphi }}} \right) = \frac{{{{S}_{{\text{t}}}}\left( {\theta ,~\varphi } \right)}}{{{{P}_{{{\text{avg}}}}}}} = \frac{{{{S}_{{\text{t}}}}\left( {\theta ,~\varphi } \right)}}{{{{P}_{{{\text{rad}}}}}{\text{/}}4\pi {{r}^{2}}}}.$$
(5)

An antenna’s directivity is the ratio of the achieved power density in a certain direction to the power density of an isotropic antenna.

Gt is the antenna gain. This indicator is related to the directivity and radiated power density by Eq. (6):

$${{G}_{t}}\left( {\theta ,~\varphi } \right) = {{\eta }}{{D}_{t}}\left( {{{\theta }},{{\;\varphi }}} \right) = \frac{{{{\eta }}{{S}_{t}}\left( {\theta ,~\varphi } \right)}}{{{{P}_{{{\text{rad}}}}}{\text{/}}4\pi {{r}^{2}}}}.$$
(6)

From this it follows that

$${{G}_{{\text{t}}}}\left( {\theta ,~\varphi } \right) = \frac{{{{S}_{{\text{t}}}}\left( {\theta ,~\varphi } \right)}}{{{{P}_{{\text{t}}}}{\text{/}}4\pi {{r}^{2}}}}.$$
(7)

If a lossless antenna is modeled, the directivity and gain will be equal.

We consider a receiving antenna subjected to the power density radiated by some transmitting antenna. The ability of an antenna to receive energy is calculated using \({{A}_{{e,r}}}\), the effective area (m2), where the antenna is assumed to be located at the origin of the coordinate system.

The effective area of the antenna is related to the gain by formula (8):

$${{A}_{{e,r}}}\left( {\theta ,~\varphi } \right) = \frac{{{{\lambda }^{2}}}}{{4\pi }}{{G}_{r}}\left( {\theta ,~\varphi } \right),$$
(8)

where λ is the wavelength. Note that formula (7) depends on the wavelength and, consequently, on the frequency. Based on this, the antenna can be characterized either as a transmitter or as a receiver, with the behavior for the other case immediately known.

It should be noted that through the effective area of the antenna, it is possible to calculate the power of the antenna itself using formula (9):

$${{P}_{{{\text{ant}}}}} = ~\frac{{E_{0}^{2}{{A}_{{e,r}}}}}{{120\pi }},$$
(9)

where E0 is the field strength, V/m.

When eliminating the angular dependences of the transmitting and receiving antennas in their local coordinate systems, the received power is equal to the product of the power density of the incident wave and the effective aperture of the receiving antenna (formula (10)):

$${{P}_{r}} = {{S}_{t}}{{A}_{{e,r}}}.$$
(10)

From formula (10) it follows that

$${{P}_{r}} = \frac{{{{G}_{t}}{{P}_{t}}}}{{4\pi {{r}^{2}}}}\frac{{{{\lambda }^{2}}{{G}_{r}}}}{{4\pi }}{\text{ and}}$$
(11)
$${{P}_{r}} = {{\left( {\frac{\lambda }{{4\pi r}}} \right)}^{2}}{{G}_{t}}{{G}_{r}}{{D}_{t}},$$
(12)

where Gt is the gain of the transmitting antenna in the direction of the receiving and Gr is the gain of the receiving antenna in the direction of transmission [8].

The angular dependence of the transmitting and receiving properties of an antenna in the far field is often referred to as the antenna pattern. The template is thus a normalized plot of directivity, gain, or effective aperture versus angle, and is often given in the dB scale. Normally, the radiated normalized power density or radiated field is displayed in dB.

3. MODELING

Using the finite element method, the nanoantenna was simulated to identify the most optimal geometric parameters suitable for signal transmission in the far field. The resulting model is shown in Fig. 1. The antenna’s height is 20 nm. The simulation was carried out under the following conditions:

Fig. 1.
figure 1

Two-dimensional drawing of a nanoantenna model with specified geometric dimensions.

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• nanoantenna material, gold;

• substrate material, silicon;

• radiation propagation medium, air;

• the frequency of the incident radiation ranges from 1 to 600 THz (with a step of 0.05 THz).

The presented modification of the optical nanoantenna has good transmission characteristics, a resonant frequency of 392.3 THz, which corresponds to the infrared spectrum, a formed radiation propagation direction, and a signal gain of 7.3 (Fig. 2a), as well as low transmittance S11 of –19.95 dB (Fig. 2b). Judging by the radiation pattern shown in Fig. 2, the signal transmission range is ~5 µm. Since the height of the TSV channels varies from 1 to 50 µm, the presented modification of the nanoantenna can be applied in certain cases where the height of the through channel does not exceed 5 µm.

Fig. 2.
figure 2

(a) Directivity diagram of a nanoantenna with a signal amplification factor; (b) dependence of parameter S11 on the frequency of the incident radiation.

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Then, a system consisting of emitting and receiving plasmonic antennas located in the TSV communication channel was simulated. The geometric parameters of the antennas remained unchanged. The channel’s height was 5 µm and the diameter was 0.5 µm. The resulting model is shown in Fig. 3.

Fig. 3.
figure 3

Communication system of plasmonic antennas in the TSV communication channel.

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The transmitting antenna is at the bottom, while the receiving antenna is at the top of the channel. This step is necessary in order to understand that the signal successfully reached the receiving antenna, and also to check the efficiency of signal transmission. For this, the electric field strength inside each antenna was calculated. The obtained simulation results are presented in Fig. 4.

Fig. 4.
figure 4

Antenna’s electric field strength: (a) transmitting antenna; (b) receiving antenna.

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As can be seen from Fig. 4, the electric field strength inside the radiating antenna takes on a value of 1.68 × 109 V/m, while for the receiving nanoantenna (Fig. 4b) it is 1.66 × 109 V/m. Based on these results, it is possible to calculate the coefficient of performance (COP) of the transmitting antenna. Since the antenna efficiency is the ratio of the radiated power generated by the antenna to the power of the signal supplied to the antenna, it can be calculated through the ratio of the antenna field strengths (formula (9)). Thus, the signal transmission efficiency within the TSV channel at a distance of 5 µm is 97.63%.

CONCLUSIONS

As part of the study, a system of nanoantennas located in the TSV channel were simulated as devices for receiving and transmitting a signal. Gold nanoantennas were located on a silicon substrate and the radiation propagation medium was air. The frequency of the incident radiation was varied in the range from 1 to 600 THz. It is found that the presented optical antenna design was capable of transmitting a signal up to 5 µm with an efficiency of 97.63%, which makes it suitable for use with TSV technology in 3D integrated circuits for signal transmission between their layers. The resonant frequency of the nanoantenna takes the value 392.3 THz, which corresponds to the infrared range, the antenna gain takes the value 7.3, and the transmittance S11 is –19.95 dB.