## Abstract

We show that the second-harmonic generation (SHG) is enhanced in the chiral one-dimensional electron currents in a broad frequency range. The origin of the enhancement is twofold: first, the linear dispersion of the quasiparticles and the associated plasmonic mode as well as the quasi-linear dispersion of plasmon-polariton result in the lift of the phase-matching condition. Moreover, the strong field localization leads to the further increase of the SHG in the structure. The results suggest that the chiral currents localized at the domain walls of magnetic topological insulators can be an efficient source of the second-harmonic signal in the terahertz frequency range.

## 1 Introduction

In recent years, the field of plasmonics has been enjoying the exploration of plasmonic excitations in novel topological materials [1], [2], [3]. The appeal of the topological materials for plasmonics is largely dictated by the fact that the topologically protected surface currents in these materials support plasmonic excitations inheriting the immunity to backscattering, resulting in the suppression of the net plasmonic loss rate, which is of paramount importance for enabling applications of plasmonics in various fields [4]. The field of topological plasmonics is now rapidly evolving, and a plethora of novel low-loss plasmonic excitations have been predicted and observed in various topological insulators [5], [6], [7], [8], [9] and other topological materials, such as, e.g., Weyl semimetals [10], [11], [12], [13].

One of the most promising applications of plasmonics is the enhancement of the nonlinear optical processes, specifically second-harmonic generation (SHG) and higher harmonic generation [14], [15], [16], [17]. The amplification of the nonlinear signal is achieved owing to the plasmon-assisted field enhancement. One of the limiting factors for the harmonic generation efficiency is the ubiquitous ohmic losses. In this perspective, exploitation of topological plasmons with suppressed loss rates for the nonlinear frequency conversion could significantly enhance the conversion efficiencies.

Noteworthy, the topologically nontrivial photonic structures have been recently proposed for the enhancement of the higher harmonic generation (see the review by Smirnova et al. [18] and the references within). While in most of these studies the nonlinear current is produced by the conventional optically nonlinear media (such as lithium niobate or GaAs), the topologically nontrivial edge and surface states emerging in these structures facilitate the strong field enhancement, extended lifetime, and unidirectional mode propagation which cumulatively increase the conversion efficiency [19].

At the same time, it has been shown in a number of papers that the linear electronic dispersion arising in topologically protected surface states as well as in low-dimensional Dirac materials such as graphene may result in drastic enhancement of the nonlinear current [20], [21], [22].

Topological plasmon-polariton, composite quasiparticle, a superposition of the topologically protected surface or edge current and an electromagnetic field could thus provide a twofold source for the enhancement of the nonlinear signal since they emerge owing to the interaction of the electrons with linear dispersion, hence, with immunity to backscattering, and the subwavelengthly localized electromagnetic field.

In this letter, we exploit this simple idea by studying the SHG by the edge plasmon-polariton (EPP) localized at the domain wall in the magnetic topological insulator (MTI).

The MTI can be realized, e.g., in the form of a ferromagnet thin film in close proximity to a surface of the 3D topological insulator or topological semimetal [23], [24], [25]. A perpendicular-to-plane magnetization component in the ferromagnet induces a finite effective mass of the otherwise massless surface electrons. This results in a bandgap in the spectrum of the surface states. A one-dimensional domain wall in the ferromagnet is, however, imaged in the Dirac electron system as a zero mass line that supports a helical electronic state. These quasi-one-dimensional edge states are characterized by the linear electron dispersion and are anomalous Hall counterparts of the quantum Hall edge states.

It has been recently demonstrated that the MTI can be realized in a single material without the need for the realization of the proximity effect, namely, in the study by Yin et al. [26], it has been shown that

In the study by Iorsh et al. [27], we have shown that these currents support a strongly localized low-loss helical plasmon-polaritons with almost linear dispersion. Here, we consider a SHG supported by this EPP mode, namely, we consider the situation shown schematically in Figure 1. A linear EPP is excited by a point-like scatterer (it may be a tip of the scattering near-field optical microscope [28], [29], [30] or a deeply subwavelength resonant nanoantenna [31]). We then calculate the nonlinear conductivity, nonlinear current and the intensity of the second-harmonic signal in this setup.

## 2 Results and discussion

A helical electronic state arising at the single domain wall in the MTI structure is described by the Hamiltonian:

where *v* is the Fermi velocity,

where

In what follows, we assume that the Fermi energy lies in the center of the bulk gap and that the frequency of the electromagnetic field is smaller than the gap width

where **J** is the current operator, *Z* is the corresponding partition function. The time-dependent eigenstates in the interaction picture are simply

where

where *α* is the fine structure constant,

It can be seen that the linear dispersion has a resonance at the dispersion of the chiral plasmon

which in the limit

where

We can see that the plasmon is weakly hybridized by the electromagnetic field and the dispersion of the plasmon-polariton is close to the one of plasmon. Moreover, we can see that for the reasonable values of *ν*, the dispersion of EPP depends on *ν* only weakly. For the currently known experimental realizations of the MTI, the value of *ν* lies in the range *ν* approaches zero. Figure 2(b) shows the quantity describing the linearity of the dispersion *ν* approaches zero, the phase mismatch is small across all of the gap regions. As shown in the study by Iorsh et al. [27], the structure excited by a point-like scatterer such as a tip of scattering SNOM would support a long-living quasi-one-dimensional plasmon-polariton with the dispersion defined by (8).

The nonlinear conductivity responsible for the SHG can be calculated straightforwardly from the expression (4). The nonlocal nonlinear conductivity is found from the relation

The details of the calculation can be found in Appendix A, and the expression for

We see that according to the symmetry restrictions, since our system possesses the center of symmetry, the second-harmonic current should be proportional to the wavevector of light in the direction of propagation [32],

In calculation of the linear and nonlinear conductivity, we have neglected the processes of photoionization, i.e., the direct transitions between the edge states and the bulk states in the conduction and valence bands. This approximation is valid when (for the case of the Fermi energy in the center of the gap)

We now consider the situation similar to one considered in the study by Iorsh et al. [27]: a helical EPP is excited by a point-like scatterer and propagates along the domain wall. At the sufficient distance from the scatterer, the profile electric field is dominantly defined by the field of the EPP. Its *y* component, the only one responsible for the nonlinear current generation, reads for the plane

where

where the Green’s function *G* is given by

where

According to Eq. (9), the nonlinear current can be written as

where

However, the electric field at the second harmonic also gets renormalized owing to the hybridization with the linear EPPs at the second harmonic. Collecting all the terms together, we get the expression for the electric field at the second harmonic:

where

where *S* is defined by Eq. (8).

The profiles of the *x* direction owing to the finite localization length *l* of the topological current and that the field at second harmonic is more localized.

Different terms entering Eq. (17) are plotted in Figure 3(c), namely, the term *S* in the denominator can be regarded as the phase-matching factor. Naturally, owing to the almost linear dispersion of the EPP, *S* is quite small, and the resonant contribution to the SHG signal is significant. The terms *z* direction or introducing a low frequency cutoff which is done further in the manuscript. The value of *S*, is of the order of

Omitting the spatial profiles, the ratio of the field amplitudes at the second and fundamental harmonic can be presented as

We can see that the efficiency of the SHG is proportional to the ratio of the maximum kinetic energy gain by electron per the EM field period and the photon energy. First, let us recall that in the conventional conducting systems in the limit of low frequencies, the mean momentum is proportional to the relaxation time *τ* rather than to the field period. The relaxation time *τ* is defined by the impurity-assisted backscattering and by the thermal fluctuation–assisted ionization of the edge state electrons to the bulk conduction band. The latter processes are suppressed by a factor *L* is much larger than the effective wavelength of the plasmon-polariton, which can be approximated by

For the domain wall length of 10 μm, the dimensionless quantity can be as large as 20. For the adequate electric field amplitudes, a more accurate approximation may be made, namely, the characteristic timescale is defined not by momentum relaxation *τ* but rather by the time

Let us consider the specific case of *L* = 10 µm. In this case, *R*, we can estimate the effective nonlinear susceptibility as

We stress that the aforementioned response is broadband and does not require any additional photonic resonant structure, while it is evident that the latter would further increase the SHG signal. In the estimation of the effective nonlinear response, we did not account for the efficiency of coupling of the fundamental harmonic signal to the EPP mode, which is usually weak owing to the strong localization of the EPP. It is also noteworthy that the broadband response is achieved owing to the almost linear dispersion of EPPs in the structure providing the lift of the strict phase-matching conditions.

To conclude, we have considered the SHG in the chiral current localized at the domain wall of the MTI. Assisted by the excitation of the edge plasmon-polariton both at fundamental harmonic and second-harmonic frequency, the SHG process can be several orders more efficient than in graphene, in two-dimensional and bulk GaAs ϕT. The effect is broadband due to the linear dispersion of both the current and plasmon-polariton mode, and owing to the absence of the backscattering in the chiral current, its magnitude is virtually limited only by the domain wall length. Thus, we anticipate that the nanostructures comprising domain walls in the MTI can become a building block for the efficient sources of SHG in the terahertz range.

**Funding source: **RFBR

**Award Identifier / Grant number: **20-32-70185

**Funding source: **Russian Science Foundation

**Award Identifier / Grant number: **20-12-00224

## Acknowledgments

We acknowledge the support from the Russian Science Foundation, under Project 20-12-00224, and RFBR, project number 20-32-70185.

**Author contribution:**All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.**Research funding:**The reported study was funded by Russian Science Foundation (Project No. 20-12-00224) and RFBR (project number 20-32-70185).**Conflict of interest statement:**The authors declare no conflicts of interest.

## Appendix A Derivation of the expression for the nonlinear conductivity

The average current is given by (4), where the time evolution operator:

The interaction term is represented in a form

where

In the second order, there are two contributions to the nonlinear current:

Here, the first one comes from the averaging of the second-order perturbation of the current operator over the equilibrium distribution. From the Baker-Campbell-Hausdorff formula,

Besides, the SHG, the second-order response allows for the generation of the dc current. These terms would be proportional to

The first term in Eq. (A.4) results in the zero dc current (for the case of linear dispersion of the electron eigenstates), and the contribution to the second harmonic is given by

The analogous term comes for

The contribution from the second term in Eq. (A.4) is given by

We then sum the contributions in Eqs. (A.5) and (A.6) and assume the low temperature limit, when *n* is readily yielding the final answer presented in the main manuscript

where

The contribution to the dc current from the second term in Eq. (A.4) is given by

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**Received:**2020-05-15

**Accepted:**2020-09-09

**Published Online:**2020-09-25

© 2020 Gulnaz Rakhmanova and Ivan V. Iorsh, published by De Gruyter, Berlin/Boston

This work is licensed under the Creative Commons Attribution 4.0 International License.

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