Topological insulator-based devices can transport electrons/photons at the surfaces of materials without any back reflections, even in the presence of obstacles. Topological properties have recently been investigated using non-reciprocal materials, such as gyromagnetics, or using bianisotropy. However, these effects usually saturate at the optical frequencies and limit our ability to scale down the devices. In order to implement topological devices that we introduce in this paper for the terahertz range, we show that the semiconductors can be utilized via their cyclotron resonance in combination with small magnetic fields. We propose two terahertz operating devices such as the topological tunable power splitter and the topological circulator. This work opens up the perspectives in designing the terahertz integrated devices and circuits with high functionality.

The recent discovery of topological insulators intertwines the symmetry of crystals and the electronic topology to create an intriguing interaction between wave and matter and topological order. Topological insulators can transport electrons without any back-scattering at the surface of the materials, even in the presence of obstacles. This phenomenon was shown in the quantum Hall effect, where the electrons in a magnetic field are transported without any back-scattering.^{1} These systems have been extensively studied, and properties such as immunity to disorder have been predicted and demonstrated. The recent demonstration by Raghu and Haldane^{2} on the possibility to transfer some topological properties from the fermionic system to the bosonic system gives new degrees of freedom over the control of photons.

A number of robust waveguiding experiments have been proposed and demonstrated subsequently.^{3–10} By applying an external magnetic field (EMF) to a 2D gyromagnetic photonic crystal (PhC), the chiral boundary modes at microwave frequencies were induced as a result of time-reversal symmetry breaking in analogy with the integer quantum Hall effect.^{6} At higher frequencies where gyromagnetism is weak, the implementation of these ideas is challenging and 2D lattices with bianisotropy, temporal modulation, and ring resonators,^{4,9} in both strongly and weakly coupled periodic arrays, have all been proposed.

Besides weak gyromagnetism, which results in devices with large footprint, material losses represent an issue to break the time-reversal symmetry at the optical frequencies. Graphene was suggested to overcome this limitation, and an inhomogeneous strain can induce pseudo-magnetic fields.^{11}

In the terahertz spectral range, materials such as $Bi2Se3$ generating strong spin-orbit interaction, also protected by time-reversal symmetry, can be used.^{12,13} Integrated devices have not yet been used for topological insulator applications in the terahertz spectral range. This frequency band ($0.1\u201330\u2009\u2009THz$) is of interest for a number of applications, such as information and communication technologies, biology and medical sciences, non-destructive evaluation, and even global environmental monitoring.^{14}

A possible way to break the time reversal symmetry in this spectral range is to use the cyclotron resonance effect^{15–18} in the semiconductor materials.^{19} In these CMOS compatible materials, the cyclotron resonance can be triggered in the terahertz band with a small EMF.^{20}

In this paper, we report and numerically characterize the topological circuits operating in the terahertz frequency range. Using the propagation of one-way modes at the boundaries of semiconductor PhCs due to the topological effect under the small magnetic fields, we propose a tunable topological power splitter and a topological circulator.

Materials containing free electrons can exhibit the cyclotron resonance effect, i.e., free electrons circle in the plane perpendicular to the direction of the applied magnetic field. The cyclotron frequency of the electrons is proportional to the strength of the EMF. This phenomenon results in a non-reciprocal interaction between the material and the electromagnetic wave propagating in the forward and backward directions. Because of this effect, the relative permittivity tensor of the material presents the off-diagonal terms with opposite signs.^{21}

The cyclotron resonance frequency, $\omega c$, depends on the effective mass of the electrons, $m*$, and the strength of the EMF, $B$, as $\omega c=eB/m*$, where *e* is the charge of electron.^{21} The dielectric permittivity tensor for a dispersive material exposed to an EMF in the $z$ direction, defined as Transverse Electric mode,^{22,23} can be described by^{21}

with

where $\epsilon \u221e$ is the high frequency permittivity, $\omega p$ the bulk plasma frequency, and $\tau $ the decay time characterizing the material loss.

In order to have a cyclotron frequency, $\omega c$, comparable with the plasma frequency, $\omega p$ in metals, it is necessary to apply very large EMF (in the order of several Teslas). For small EMF, $\omega c$ is negligible (Eqs. (1) and (2)); hence, $\epsilon xy\u22480$ and non-reciprocity are inexistent. This represents a limitation for on-chip devices.

However, due to the smaller effective mass of electrons in the semiconductors, it is possible to obtain high $\omega c$ comparable to $\omega p$ at THz frequencies by applying a reasonable EMF.^{19} For instance, indium antimonide ($InSb$) is a semiconductor with a small energy gap of $0.17\u2009\u2009eV$ and a large electronic mobility^{24} ($\u223c7.7\xd7104\u2009cm2\u2009V\u22121\u2009s\u22121$) that has been used in the THz frequency range.^{19}

In this work, we used the dispersive relative permittivity of $InSb$ with a very small loss and the material parameters at room temperature given by $m*=0.014m0$ ($m0$ is the free electron mass in vacuum), $\omega p=1.26\xd71012\u2009\u2009Hz$, and $\epsilon \u221e=15.68$.^{25,26}

We consider the periodic array of $InSb$ rods with square lattice shape, as depicted in Fig. 1(a). The rods are invariant along the $z$ direction, with a periodicity of $a1=140\u2009\mu m\u2009$ and a radius of $r1=0.25a1$. It is worth noting that the geometrical parameters of the non-trivial PhC mirror (lattice and unit cell) should be modified if a different semiconductor material is used.

Breaking the time-reversal symmetry can result in one-way edge modes that depend on the topological properties of the Bloch bands. These topological properties are characterized by the Chern number, which, for the n-th band, is given by^{2}

where $A$ is called the Berry connection with a definition as

where the brackets denote spatial integration over a unit cell.

In the calculated band diagram of Fig. 1(b), we do not observe a complete bandgap. Ref. 2 demonstrated that a bandgap with a non-zero Chern number can be opened by breaking the time-reversal symmetry of a gapless band diagram formed by two modes linearly touching each other (Dirac-shaped modes). Ref. 3 showed that it is also possible to open a bandgap with a non-zero Chern number using the quadratically crossing modes.

Here, we demonstrate that none of these conditions are necessary. In general, it is only required to have a closed gap in the entire Brillouin zone, regardless of the shape of the modes in the band diagram. For example, in Fig. 1(b), the second and third modes (red curves) close the bandgap without touching each other. Furthermore, we will show that it is possible to open a bandgap with a non-zero Chern number using these two modes.

The Chern number is zero for all the modes in Fig. 1(b), since time-reversal symmetry has not been broken.^{27}

By applying an EMF of $B=0.5\u2009T$ to the structure in the $z$ direction, the free electrons of $InSb$ rotate in the $xy$ plane with a frequency of $\omega c$, breaking the time reversal symmetry. According to Eqs. (1) and (2), the relative permittivity tensor becomes asymmetric. The resulting band diagram presents a bandgap between the second and third modes (red curves in Fig. 1(c)).

Once the bandgap is opened, the Chern number below the gap becomes a non-zero integer number ($\Delta C=+1$). Since the sum of the Chern numbers on the entire band diagram is invariantly zero,^{27} the Chern number above the gap is $\u2212\Delta C=\u22121$. Thus, the resulting periodic structure is a non-trivial mirror that allows the propagation of edge modes in only one direction. The integration of this structure with a trivial mirror such as perfect electric conductor (PEC) will lead to the guided electromagnetic waves propagating in one direction and immune to any back reflection. To corroborate this one-way propagation, we placed a PEC close to the designed PhC to form a waveguide (inset in Fig. 2(a)). The waveguide is excited with a point source ($S$) located at the boundary of the PhC and PEC mirrors.

Fig. 2(a) shows the spectrum of the normalized power transmitted from the point source to the ports. In Fig. 1(c), the shaded area indicates the bandgap region of the PhC. We can observe an isolation between the ports in this region.

This isolation is corroborated in the intensity maps at the frequencies $f1=1.11\u2009\u2009THz$ (Fig. 2(b)) and $f2=1.125\u2009\u2009THz$ (Fig. 2(c)) within and without the bandgap, respectively. The transmission to the ports is not symmetric, as the presence of the EMF breaks the time-reversal symmetry. The transmission at lower frequencies is not unity because of the finite numerical aperture of the port (the ports only collect part of the light from the source).

Due to its topological dependence, this edge mode features unidirectional robust propagation, ensuring the absence of back reflections due to the defects or disorders. To illustrate this robustness, we studied the one-way propagation of a structure with sharp bends (Fig. 3(a)). For this structure, we used a PhC mirror instead of a PEC. The PhC mirror consists of a square lattice shape of $InSb$ rods. The periodicity and radius of the rods are $a2=75.5\u2009\mu m$ and $r2=0.35a2$, respectively, with a plasma frequency of $\omega p=1.26\xd71011\u2009\u2009Hz$. With these parameters, the sum of the Chern number of the modes below the bandgap is zero (see supplementary material). Thus, this PhC can be used as a trivial mirror. This periodic structure was designed such that its bandgap overlaps with the bandgap of the PhC non-trivial mirror, as previously studied. The spectra of the normalized power transmitted from the point source to each port are depicted in Fig. 3(b), showing the isolation between the ports. The robustness of the edge mode is clearly observed in the intensity map in Fig. 3(c) at a frequency within the bandgap ($f1=1.11\u2009\u2009THz$).

Based on these results, we propose a unidirectional power splitter formed by the combination of trivial (purple rods) and non-trivial (greed rods) PhC mirrors, as depicted in Fig. 4(a). The point source is placed between the two mirrors to excite the one-way edge mode propagating in the indicated direction (black arrow). The power splits into two edge-modes at the crossing point between the four edge waveguides. The first one propagates clockwise toward port 1 (around the top-right PhC), while the second mode propagates counterclockwise toward port 2 (around the bottom-right PhC). Therefore, due to this asymmetric propagation, the amount of power flowing to each port is different ($\kappa 1$ and $\kappa 2$).

Both $\kappa 1$ and $\kappa 2$ can be controlled by varying the radius of a defect ($InSb$ rod) placed at the center of the power splitter (see supplementary material). This first approach is just a static control, since for a fixed radius of the defect the power flowing to the ports will remain unchanged.

We can provide a dynamic control of the power splitter by changing the phase of a control signal ($C$) injected from the fourth remaining channel (left port), as illustrated in Fig. 4(a). The amplitude of the control and source signals is the same. This control signal constructively or destructively interferes with the fixed source signal ($S$), modifying the propagation of the edge mode toward the ports. In Fig. 4(b), we show the normalized power at each port as a function of the phase of the control signal, $\Phi $. For example, for $\Phi =0\u2009\u2009rad$, the power is equally transmitted to each port (Fig. 4(c)). For $\Phi =1.57\u2009rad$, the mode propagates toward port 1 (Fig. 4(d)), while for $\Phi =4.71\u2009rad$, the mode is mainly propagating toward port 2 (Fig. 4(e)).

Following the previous results, we present a topological circulator operating at the terahertz frequencies (Fig. 5(a)). The device consists of a symmetric array of trivial (purple rods) and non-trivial (green rods) PhC mirrors of $InSb$ with four ports. At the corners of the central PhC mirror–core of the circulator–four defects of radius $r=80\u2009\mu m$ (orange rods) are placed to control the power flowing toward each port. The circulator is excited from port 1 with a point source at a frequency $f=1.11\u2009THz$. As can be observed from the intensity map of Fig. 5(b), the unidirectional edge mode is mainly transmitted to port 2. The isolation between port 2 and 3 is $15.8\u2009dB$ and between port 2 and 4 is $18.2\u2009dB$.

We have noted that the number of channels can be modified by changing the geometrical shape of the core, such as triangular or pentagonal, to operate with three or five ports. Also, by adding a second port at each section of the circulator, one may inject a control signal to dynamically control the power transmitted to the other ports. Another possibility is to dynamically control the defect that can be made of a phase-change material.

In conclusion, we demonstrated the topological terahertz devices using the one-way propagation of edge modes in periodic structures based on the concept of topological electromagnetic insulator. By applying a small EMF and breaking the time-reversal symmetry of $InSb$ semiconductor through its cyclotron resonance in the terahertz frequency range, a variety of topological devices can be realized. We also demonstrated that, in order to open a non-trivial gap between two modes, it is not necessary to have the modes touch each other either linearly or quadratically in the band diagram. We proposed a power splitter capable of dynamically tuning the amount of power flowing to the ports by varying the phase of a control signal. Based on the robustness of the power splitter, we also proposed a topological circulator operating at terahertz frequencies.

Due to their material properties and compatibility with the CMOS manufacturing technology, the semiconductors open up perspectives in designing a new generation of integrated circuits with high functionality based on the topological effects.

See supplementary material for more detail about the designing of trivial mirror and power splitter with and without defect.

This work was partially supported by the NSF Career Award (1554021) and the Office of Naval Research Multi-University Research Initiative (N00014-13-1-0678).