Optoelectronic devices, plasmonics and photonics with topological insulators

Topological insulators are innovative materials with semiconducting bulk together with surface states forming a Dirac cone, which ensure metallic conduction in the surface plane. Therefore, topological insulators represent an ideal platform for optoelectronics and photonics. The recent progress of science and technology based on topological insulators enables the exploitation of their huge application capabilities. Here, we review the recent achievements of optoelectronics, photonics and plasmonics with topological insulators. Plasmonic devices and photodetectors based on topological insulators in a wide energy range, from Terahertz to the ultraviolet, promise outstanding impact. Furthermore, the peculiarities, the range of applications and the challenges of the emerging fields of topological photonics and thermoplasmonics are discussed.

Topological surface states (TSS) form a Dirac cone 6,7 , in analogy with graphene . In addition, TSS are chiral 8 and protected from back-scattering by time-reversal symmetry 9 . Therefore, TSS are robust against scattering processes from nonmagnetic impurities 10 . As a consequence of the timereversal symmetry, charge carriers from topologically protected surface states carry current with minimal dissipation with subsequent reduction of the low-frequency electronic noise 11, 12 . Moreover, unlike graphene, TSS naturally offer a two-dimensional (2D) Dirac fermion system 13 , without the complication of the physical implementation of atomically thin layers.
TIs can be divided into two classes: three-dimensional (3D) 14 and 2D 15 TIs. TSS are present in the bulk energy gap of 3D TIs, while 2D TIs consist of 1D gapless conductive edge channels with a 2D area exhibiting an energy gap.
The crystalline quality of samples is crucial in the road map for a technological exploitation of the novel topological phases of matter. As a matter of fact, it is now well established that TSS exist in crystals and films possessing high structural quality (both stoichiometric/ composition and The coexistence of several privileged conditions for applications-fast charge carriers 24,25 , sensitivity to applied fields 26 , reduced fluctuation of mobility 12 , and robustness to disorder 27 -makes TIs promising for technological use in quantum computing 28 , high-speed electronic devices 29 , as well as spintronic devices with novel functionalities 30 . Moreover, the amazing optical properties of TIs 31-37 allow their versatile and multifunctional use in signal emission, transmission, modulation, and detection.
Herein, we review applications based on plasmons supported by TIs in the fields of optoelectronics, photonics and thermoplasmonics.

Terahertz (THz) plasmonics and photodetectors
The two-dimensional electron gas (2DEG) formed by surface-state electrons support a Dirac plasmon [45][46][47] with energy ω , whose dispersion in the local approximation follows 48 : where q is the momentum. The vacuum permittivity and the relative dielectric constant are represented in (1) by ε0 and ε, respectively. D is the Drude weight 49 , which in a 2DEG with particle density n and electron mass me is defined as: D = e 2 n/me (2) with e the electron charge.
The dispersion relation in Equation (1) yields the square-root dispersion with the momentum q predicted by Stern for a 2DEG 50 .
Di Pietro et al. 47 measured the Dirac plasmon in Bi2Se3 TI with infrared spectroscopy. Plasmons cannot be directly excited by light in smooth samples 51 , since photon momentum is always minor than that of plasmons, but the momentum mismatch can be compensated by grating 52 .
Therefore, in order to let the light to couple with plasmons, Di Pietro et al. 47 fabricated thin micro-ribbon arrays of Bi2Se3 with different width to change the plasmon wavevector q up to 0.00015 Å -1 . An extension of the momentum range up to ~0.3 Å -1 has been achieved by exciting the Dirac plasmon with probing electrons 45,53 .
Successively, the plasmonic response of ring structures patterned in Bi2Se3 films has been studied by THz spectroscopy 54 . The rings exhibit a bonding and an antibonding plasmon modes, whose frequency can be tuned by changing their diameter. The bonding plasmon strongly couples with an optical phonon at about ~2 THz, leading to Fano profiles in the measured extinction spectra.
The most promising application of THz plasmonics with TIs is related to the rectification of THz radiation via the excitation of plasma waves in the active channel of antenna-coupled field-effect transistors (FETs). This photodetection mechanism, proposed by Dyakonov and Shur [55][56][57] , is based on the fact that a FET hosting a 2DEG can act as a cavity for plasma waves, which are launched at the source by means of a modulation of the potential difference between gate and source. Plasma waves propagating in the active channel of FETs cannot be merely recognized with the plasmonic resonance of a 2DEG (2D plasmon) because of the presence of a metal gate 58 . When q<1/d (with d the distance between the 2DEG and the gate), the dispersion relation is modified into ωp = sq (at T=0 K and neglecting friction and viscosity) with s the plasma-wave group velocity. Such a linear dispersion relation resembles that of acoustic surface plasmons 59 and sound waves 58 .
If a plasma wave reaches the drain contact in a time inferior with respect to the momentum relaxation time, constructive interference in the cavity allows frequency-resolved detection of the incoming radiation ("resonant regime"). In such regime, the dc photoresponse is characterized by peaks at the odd multiples of the lowest plasma-wave frequency. For typical device lengths and carrier densities, the fundamental frequency of plasma waves is in the THz range, so that photodetectors based on the Dyakonov-Shur mechanism are used for THz photodetection.
FET THz detectors conventionally operate at room temperature in the overdamped plasma waves regime, since the length of the FET channel is bigger than the propagation length of plasma waves at room temperature. The electromagnetic ac field, coupled to the source (S) and gate (G) electrodes, simultaneously modulates the carrier density and drift velocity. The resulting current exhibits a dc component, whose magnitude is proportional to the intensity of the incoming radiation and can be measured at the drain (D) contact either in a short circuit (photocurrent mode) or in an open circuit (photovoltage mode) configuration.
where η represents the antenna-dependent coupling efficiency of the incoming radiation 62 , σ is the channel conductivity, Rch is the total S-to-D resistance and ZL is the complex impedance of the readout circuitry. Equation (3)

Spin-plasmons
Due to the spin-momentum locking 64,65 , plasmons in TIs are always accompanied by transverse spin oscillations 66 . The novel spin-plasmon mode has the peculiarity that density fluctuations induce transverse spin fluctuations and vice versa 67 . In a spin-polarized 2DEG, a random-phase approximation model 68 predicts the spin-plasmon lifetime to be sufficiently high in order to enable their technological exploitation in spin-wavegenerating devices, such as spin-torque oscillators. The influence of spin-plasmons has been invoked to be crucial in the coupling of Dirac-cone electrons of TIs with phonons 69 . However, attempts to directly probe spin-plasmons by means of electron energy loss spectroscopy have been unsuccessful 53 , since the predominating mode in the plasmonic spectrum is a surface plasmon arising from the bulk, free carriers 45,53 . This mode hybridizes with the spin-plasmon for momenta far from the optical limit 45 .
To selectively excite the spin-plasmon, a mechanism exploiting optical spin injection has been proposed by Raghu et al. 67 (Figure 2). These authors suggested that two cross-polarized optical beams of laser pulses can generate a transient spin grating on the surface, whose period is determined by the wavelength and the angle between the two incident beams. Spin-plasmon can be excited whenever the spin grating period matches the plasmon momentum 70,71 . Experiments reporting (i) the existence of selection rules ruling spin-dependent optical transitions in TIs 72 and (ii) helicitydependent photocurrents 73 have supported the viability of optical spin injection. However, calculations including both spin-orbit coupling and Zeeman coupling demonstrated 74 that helicity-independent photocurrent dominates over helicity-dependent contributions.

Magnetoplasmonics
Recently, magnetoplasmonics is attracting considerable interest for the intriguing prospect of its technological applications 76,77 . The 2D magnetoplasmons are collective excitations between Landau levels 78 due to electron-electron interactions, which can be observed through infrared optical absorption and inelastic light scattering [79][80][81][82][83][84] . K ( Figure 3) . Both the collective (plasmon) and the single particle (Drude) excitations have been tuned by an external magnetic field B ranging from 0 to 30 T.
At finite B, plasmon gives rise to a magnetoplasmon mode, whose frequency ω follows: where ω is the cyclotron frequency ±eB/mcc, defined as positive for electrons and negative for holes, with mc the cyclotron mass and c the speed of light.
The Drude term becomes a cyclotron resonance at finite energy even for low values of B. For magnetic fields higher than 10 T, the cyclotron resonance and the plasmonic modes merge into a unique Dirac magnetic excitation ( Figure 3).
Therefore, in TIs, it is possible to achieve a magnetic control of plasmonic modes, paving the way toward THz magnetoplasmonics.

Plasmons in topological crystalline insulators (TCIs)
In a TCI, the space group symmetries of a crystal replace the role of time-reversal symmetry in an ordinary TI (see Refs. 91-95 for more information on TCI). Plasmons of TCI SnTe with nanostructured patterns have been investigated by Wang et al. 96 . Four plasmon resonances excited on the TCI SnTe nanogratings are found in the visible-near-infrared (vis-NIR) spectral region. By variating the grating heights, periodic shifts of resonance wavelengths are observed.
Localized surface plasmons (LSP), i.e. plasmon modes in nanostructures, notably enhance the local electric field near the surface of the nanostructures, so as to achieve an enhanced optical response. Photothermal effects activate nanoscale thermal hotspots by light irradiation 104,105 , where plasmon energy is transformed into heat, which increases the temperature of the surrounding medium 106 .
of a several Bi2Se3 nanocrystals. In addition, the heat obtained by irradiating Bi2Se3 nanoplates with a laser in the NIR has been exploited by Li et al. 108 in the fields of cancer imaging and therapy.

Photonic TIs
The emerging field of topological photonics 109 promises to introduce into optical physics the various topological phenomena in condensed matter physics, such as the quantum Hall effect. In addition, topological photonics aims to protect photons from undesirable random scattering in their transport from one place to another, similarly to the topologically protected transport of electrons in condensed-matter TIs.
Using appropriately devised electromagnetic media (metamaterials) it is possible to realize topologically non-trivial photonic states, similar to those that have been identified for solid-state TIs. Photonic TIs could have a major impact on optical devices (couplers, waveguides, and so on), making them (i) more robust against scattering from defects or disorder and (ii) more energy efficient.
Recently, photonic topological transport has been reported in many innovative photonic systems 110,111 . Firstly, quantum-Hall-type phenomena in optics have been predicted by the groups of Raghu & Haldane 112,113 and Soljačić 114 . The first experimental realization was obtained in the microwave frequency range by means of magneto-optic materials 115 . Nevertheless, magnetic effects are too weak at optical frequencies to achieve photonic TIs with scatter-free edge states. However, it is now demonstrated that it is possible to obtain spin-polarized one-way photon transport without applying external magnetic fields, as shown by Khanikaev et al. 111 by using metacrystals, which consist in superlattices of metamaterials with appropriately designed properties.
Floquet TIs represent a suitable alternative to obtain helical edge states. In Floquet TIs, the temporal modulation of a photonic crystal breaks the time-reversal symmetry, so as to induce topological edge states. Floquet TI are based on an array of evanescently coupled helical waveguides arranged in a honeycomb lattice 110 , as sketched in Figure 5. The lattice is needed in order to support Dirac points (a prerequisite for topological phenomena), whereas the helicity of the waveguides disrupts the degeneracy between clockwise and counterclockwise diffraction near the array. the inter-ring coupling strength, it is possible to induce a "topological phase transition" from an ordinary insulator to a TI.

Saturable absorbers
Bismuth chalcogenides represent promising candidate to generate L-band ultrafast pulses for applications as saturable absorbers 117,118 for modelocking. They can be also used as a high-nonlinear medium for mitigating the mode competition of erbium-doped fiber and stabilizing the dualwavelength oscillation 119 . In many cases, bismuth chalcogenides are used in the form of powders 120

Conclusions and Outlook
The development of disruptive technologies based on the novel topological phases of matter, especially in the field of optoelectronics and photonics, is a great challenge. However, current prototypes of THz photodetectors, photonic devices and thermoplasmonic applications based on TIs have solid possibilities to overcome the up scaling issues.
Even in their first implementation, THz photodetectors using TI-based FETs showed very promising performances. The use of heterostructures of TIs and, moreover, topological Weyl and Dirac semimetals could afford new possibilities for next-generation THz photodetectors. The main target is represented by the design, fabrication and testing of detectors that can be used as pixels in a high-efficiency, high-resolution array for THz imaging.
Furthermore, the physics of photonic topological states is exceptionally promising, with potential implications in various other areas ranging from cold atoms to solid-state electronic systems. New topological phenomena, such as quantum effects with entangled photons and non-equilibrium phenomena, can be explored by optics. Thus, topological photonics is expected to continue engaging researchers in next years, for its intriguing capability to highly improve the performance of optical devices and to limit power consumption.
Future challenges of topological photonics are related to the observation of the nonlinear effects of topological systems, of topological pumping and Bloch oscillations in photonic TIs with novel implementations of one-way waveguiding.
Moreover, the facile control of the plasmons of TCI nanopatterns in the vis-NIR spectral region may have several potential applications. Finally, novel applications could come from thermoplasmonics, which is now ready to concretize the pioneer attempts in cancer therapy.