Photoinduced interfacial chiral modes in threefold topological semimetal

TL;DR

This paper explores how circularly polarized light induces chiral electronic modes at the interface of a threefold topological semimetal, revealing phase-dependent chirality and anomalous Hall effects through Floquet theory analysis.

Contribution

It introduces a novel analysis of photoinduced chiral interface modes in topological semimetals, highlighting phase-dependent chirality and associated Hall responses.

Findings

Two distinct chiral interface modes identified

Chirality depends on the phase of incident light

Anomalous Hall response linked to band transitions

Abstract

We investigate the chiral electronic modes at the interface between two regions of a threefold topological semimetal, which is illuminated by left and right handed elliptically polarized waves. The radiation effects on the band structure of semimetal is analyzed by using Floquet theory. Two distinct solutions of the interface modes are found with the chirality depending on the phase of the irradiation. We also consider the anomalous Hall response which is attributed to the transition between dispersionless flat band and conic bands.

Figures

Equations

$$H_{3f}=v\left[\begin{array}[]{ccc}0&e^{i\phi}k_{z}&e^{-i\phi}k_{y}\\ e^{-i\phi}k_{z}&0&e^{i\phi}k_{x}\\ e^{i\phi}k_{y}&e^{-i\phi}k_{x}&0\end{array}\right],$$

$$\mathbf{S}=i\left[\begin{array}[]{ccc}0&\hat{e}_{z}&-\hat{e}_{y}\\ -\hat{e}_{z}&0&\hat{e}_{x}\\ \hat{e}_{y}&-\hat{e}_{x}&0\end{array}\right].$$

$$H_{3f}(t)=H_{3f}+V(t),$$

$$V(t)=-ev[S_x{A}_x\sin (\Omega t)+S_y{A}_y\sin (\Omega t-\delta )].$$

$${\tilde{H}}_{3f}\simeq H_{3f}+\frac{[V_{+},V_{-}]}{\Omega },$$

$$V_m=\frac{1}{T}{\int }_0^{T}V(t)e^{-im\Omega t}.$$

$${\tilde{H}}_{3f}=H_{3f}+S_{z}v\gamma ,$$

$$\gamma =-{\gamma }_0[2\Theta (x)-1],$$

$$iv\left[\begin{array}[]{ccc}0&k_{z}+\gamma&-k_{y}\\ -k_{z}-\gamma&0&-i\partial_{x}\\ k_{y}&i\partial_{x}&0\end{array}\right]\Psi=E\Psi.$$

$$\frac{{\partial }^2{\psi }_2}{\partial x^2}+U(x){\psi }_2=\left\{k_y^2+[k_z+\gamma (x){]}^2-\frac{E^2}{v^2}\right\}{\psi }_2,$$

$$\sum _{s=\pm }{\left[k_y^2+(k_z+s{\gamma }_0{)}^2-\frac{E^2}{v^2}\right]}^{1/2}=2{\gamma }_0\frac{v{k}_y}{E}.$$

$$E_1=v{k}_y\text{sgn}{\gamma }_0,$$

$$E_2=\frac{v{k}_y{\gamma }_0}{\sqrt{k_y^2+k_z^2}}.$$

$$\gamma (x)=-{\gamma }_0\tanh (x/L),$$

$$\frac{E^2}{v^2}-k_y^2-{\gamma }_0^2=\frac{L^2}{4}{\left[\sqrt{1+4L^2{U}_0}-(1+2n)\right]}^2,$$

$${\Pi }_{xy}({\omega }_m)=\text{tr}\sum _{n}T\int \frac{d^{3}k}{(2\pi {)}^3}G(i{\omega }_n+i{\omega }_m,\mathbf{k})j_{x}G(i{\omega }_n,\mathbf{k})j_y,$$

$$G(i{\omega }_n,\mathbf{k})$$

$$M_{0}(\mathbf{k})=\frac{1}{k^{2}}\left[\begin{array}[]{ccc}k_{x}^{2}&k_{x}k_{y}&k_{x}k_{z}\\ k_{x}k_{y}&k_{y}^{2}&k_{y}k_{z}\\ k_{x}k_{z}&k_{y}k_{z}&k_{z}^{2}\end{array}\right],$$

$$M_{1}(\mathbf{k})=\frac{1}{k}\left[\begin{array}[]{ccc}0&k_{z}&k_{y}\\ k_{z}&0&k_{x}\\ k_{y}&k_{x}&0\end{array}\right].$$

$${\sigma }_{xy}=\frac{e^2{\gamma }_0}{2{\pi }^2},$$

$$\displaystyle\Pi_{xx}=\frac{\omega_{m}^{2}}{6\pi^{2}v}\Big{[}\ln\frac{\Lambda^{2}}{\mu^{2}+\omega_{m}^{2}}+\frac{\mu^{2}}{\omega_{m}^{2}}\Big{]},$$

$$\mu >\omega \ge \frac{\alpha \mu }{(1+{\alpha }^2\ln |\Lambda /\mu \sqrt{1-{\alpha }^2}|{)}^{1/2}},$$

Full text

I Introduction

Recently, the possibility of manipulating the electronic band structure by applying a time dependent periodic perturbation in the form of irradiation/light has received much attention, especially after the proposals of light induced topological phase transition Oka and Aok (2009); Lindner et al. (2011); Inoue and Tanaka (2012); Kitagawa et al. (2011); López et al. (2012); Zhai and Jin (2014); Saha (2016); Ezawa (2013); Perez-Piskunow et al. (2014); Cayssol et al. (2013) (Floquet topological insulator) which has also been confirmed by experiments Peng et al. (2016); Zhang et al. (2014); Wang et al. (2013). Apart from this, optical pumping can also be used to control the spin and the valley degree of freedom in Dirac materials Kundu et al. (2016); Golub et al. (2011); Dey and Ghosh (2018); Tahir et al. (2014) which is the key requirement for the spin and valleytronics. Moreover, electromagnetic field can play the key role in $0-\pi$ phase transition in Josephson current in a NSN hybrid junction made of silicene Zhou and Jin (2016) and Weyl semimetal Khanna et al. (2017).

Topological semimetals with multiple bands have recently been predicted Bradlyn et al. (2016); Tang et al. (2017). These materials are also known as multifold fermions and can be described by the higher number (more than two) of band crossing degeneracy, which differs itself from the linear dispersion at the nodal band-touching points in Weyl semimetal. Several experiments Takane et al. (2019); Rao et al. (2019); Sanchez et al. (2019); Schröter et al. (2019) have also confirmed the existence of such multifold semimetals. It has recently been found that multifold fermions are very promising in exhibiting the quantized photogalvanic effect Chang et al. (2017); Flicker et al. (2018). The signature of multifold bands in such materials has also been explored in optical conductivity Sánchez-Martínez et al. (2019).

We particularly focus on three fold semimetal which can be considered as $3$D analog of $2$D dice lattice Dóra et al. (2011); Raoux et al. (2014). One of the distinct features of three fold semimetal is the existence of dispersionless flat band which has some unusual consequences in transport signature in its $2$D counterpart Vigh et al. (2013); Malcolm and Nicol (2015); Biswas and Ghosh (2016). The effects of light/irradiation in $2$D dice lattice Dey and Ghosh (2018, 2019) have been investigated very recently. On the other hand, much less attention has been paid for $3$D three fold semimetal in the context of its interaction with irradiation. It can be easily anticipated that shedding light can induce a mass term and subsequently opens a gap in the band dispersion, irrespective of the dimensionality. However, apart from the gap opening, irradiation can also induce a momentum shift along the extra dimension and that shifting can be manipulated in opposite direction in two different regions of the materials by controlling the phase of the irradiation. In this work, we follow this route to induce opposite momentum shifting in two different region by shedding irradiation with opposite phase. The effects of irradiation is included by means of Floquet theory Eckardt (2017) under the limit of high frequency expansion (Floquet - Magnus expansion). We find that there exist two chiral interfacial surface modes and calculate the anomalous Hall conductivity induced due to irradiation.

The occurrence of analogues optical interfacial surface modes is well studied topic Yariv and Yeh . In optics, the interface of two optically active isotropic medias with opposite sign of the gyrotropic coefficient, supports unidirectional surface electromagnetic waves Seshadri (1962); Zhukov and Raikh (2000); Zyuzin and Zyuzin (2015).

The remainder of the paper is presented as follows. The section II discusses the linearized Hamiltonian of the three band model and its band dispersion with the effect of irradiation. The dispersion relations of chiral modes localized at the interface where the irradiation changes phase are obtained in section III. The section IV is devoted to calculation of the anomalous Hall conductivity and the transparency region for irradiation. Finally, we summarize and conclude in section V.

II The model and Floquet theory

We start with the Hamiltonian, that describes the groups of SGs 199 and 214 materials as Bradlyn et al. (2016); Sánchez-Martínez et al. (2019)

[TABLE]

where $\phi$ is a real parameter, $v$ is the Fermi velocity, and we use $c=k_{B}=\hbar=1$ throughout the calculations. We consider only a single nodal point in the first Brillouin zone. The band is non-degenerate except at $k=0$, unless $\phi=n\pi/3$ with $n$ being integer. Another noticeable point is that for $\pi/3<\phi<2\pi/3$, the Hamiltonian is adiabatically connected to the one with $\phi=\pi/2$. The Hamiltonian at this point can be written as $H_{3f}=v{\bf k.S}$ with $S_{x,y,z}$ being the generators of the rotation group $SO(2)$ in the pseudospin-1 representation defined as

[TABLE]

The energy spectrum is given by dispersive bands $E_{\pm}=\pm vk$ and a disperionless flat band $E_{0}=0$, which is shown in Fig. (1)a. This is a 3D analog of a 2D dice lattice.

Let us now consider that the system is subjected to an external time-dependent periodic perturbation, which is described by a vector potential $\mathbf{A}(t)=[A_{x}\sin(\Omega t),A_{y}\sin(\Omega t-\delta),0]$. Here $\Omega$ is the frequency of the irradiation and $\delta$ is the phase. The Hamiltonian describing threefold semimetal in the presence of radiation can be written by utilizing the Peierls substitution $\mathbf{k}\rightarrow\mathbf{k}-e\mathbf{A}(t)$, where $e<0$ is the electron charge, as

[TABLE]

with

[TABLE]

The above Hamiltonian can be solved by using Floquet theory which states that the system under the time dependent periodic perturbation exhibits a complete set of solutions of the form $\psi(r,t)=\phi(r,t)e^{-i\epsilon t}$ where $\phi(r,t)=\phi(r,t+T)$ and $T$ is the periodicity of the field, the corresponding Floquet states Eckardt (2017). It resembles the Bloch theorem in the momentum space. The Floquet eigenstates can be inserted into the time dependent Schrodinger equation to obtain the Floquet eigenvalue equation as $H_{3f}^{F}\phi(r,t)=\epsilon\phi(r,t)$ with $H^{F}_{3f}=H_{3f}(t)-i\partial_{t}$.

The Floquet eigenstates can be further expressed in Fourier form as $\phi(r,t)=\sum_{n}\phi{{}_{n}}(r,t)e^{in\Omega t}$, where $n$ is the Fourier components or Floquet side bands. To obtain the Floquet energy spectrum, the Floquet Hamiltonian has to be diagonalized in the basis of Floquet side band ‘$n$’. However, in such case one must restrict the side band index by setting a cut-off. On the other hand, for high-frequency regime of the field, an effective Hamiltonian can be obtained by following Floquet - Magnus expansion up to the second order in field amplitude as Eckardt (2017)

[TABLE]

where the second term describes the irradiation induced correction with

[TABLE]

The effective Hamiltonian can be further simplified to

[TABLE]

where $\gamma=ve^{2}(2\Omega)^{-1}A_{x}A_{y}\sin\delta$. For linear polarizations, $\gamma=0$. Note that the extra term in the Eq. (7), $S_{z}\gamma$, causes a momentum shift along the $k_{z}$-direction and that can be further seen in the energy spectrum of the irradiated system as $E_{\pm}=\pm v\sqrt{(k_{z}+\gamma)^{2}+k_{x}^{2}+k_{y}^{2}}$.

III Interfacial modes

In this section, we obtain the chiral modes propagating along the interface between two regions with different $\gamma$. The two regions of the material exposed to the external irradiation as schematically shown in Fig. 1b. The phase factor of the wave is adjusted in such way that $\gamma$ changes sign in two region as

[TABLE]

where $\gamma_{0}=ve^{2}A_{x}A_{y}/2\Omega$, $\Theta(x)$ is the step function, and $\partial\gamma/\partial x=-2\gamma_{0}\delta(x)$. We seek for a solution of equation $\tilde{H}_{3f}\Psi=E\Psi$ in the form $\Psi=\left[\psi_{1},\psi_{2},\psi_{3}\right]^{T}e^{i(k_{y}y+k_{z}z)}$, which explicitly gives

[TABLE]

It is instructive to rewrite the above equation as

[TABLE]

where $U(x)=-2\gamma_{0}vk_{y}\delta(x)/E$. The above equation resembles $1$D Schrodinger equation in a delta potential, which supports a bound state provided $\gamma_{0}vk_{y}/E>0$. By utilizing the continuity conditions for the wave-function at the interface $x=0$, one obtains an equation to determine the dispersion relation

[TABLE]

Two solutions of this equation exist provided inequality $vk_{y}\gamma_{0}/E\geq 0$ holds:

[TABLE]

The first“Fermi-arc” like solution exists provided $|\gamma_{0}|\geq|k_{z}|$, while the second is determined in the region $k_{y}^{2}+k_{z}^{2}>|\gamma_{0}k_{z}|$. Note that the chirality of the modes is defined by the sign of parameter $\gamma_{0}$. The first dispersion is flat at $k_{y}=0$, while the second one becomes flat at $k_{z}=0$, as shown in Fig. (2)a. At finite values of momenta $k_{z}$ the second mode acquires a curvature with $k_{y}$ and this curvature increases further with $k_{z}$ as shown in Fig. (2)b. For completeness we present solution of Eq. (11) for finite $k_{z}$ in Fig. (2). This figure contains two subplots in which Fig. (2)b is for $k_{z}>1$ and Fig. (2)a is for $k_{z}\leq 1$.

The wave-function, which corresponds to the eigenvalue $E_{1,2}$, is proportional to $\Psi_{1,2}\propto e^{-x/\ell_{1,2;+}}$ at $x>0$ and $\Psi_{1,2}\propto e^{x/\ell_{1,2;-}}$ at $x<0$. The localization lengths are given by $\ell_{1,\pm}=|k_{z}\mp\gamma_{0}|^{-1}$ and $\ell_{2,\pm}=|k_{y}^{2}+k_{z}^{2}\mp\gamma_{0}k_{z}|^{-1}\sqrt{k_{y}^{2}+k_{z}^{2}}$.

So far we have addressed the scenario for an abrupt interface between two regions. However, in realistic case, interface may not be always abrupt rather a smooth, which may be modelled as

[TABLE]

where $L$ is the typical barrier width. This yields for $k_{z}=0$ the $1$D Schrodinger equation $\partial^{2}_{x}\psi_{2}-U_{0}\mathrm{sech}^{2}(x/L)\psi_{2}=(k_{y}^{2}+\gamma_{0}^{2}-E^{2}/v^{2})\psi_{2}$, with $U_{0}=\frac{vk_{y}}{E}\frac{\gamma_{0}}{L}-\gamma_{0}^{2}.$ By following the standard solution Landau and Lifshitz ,

[TABLE]

where $n=0,1,2,3...$ The dispersion of the interface modes can be found by solving the above equation, which hints that there exist many solutions depending on the values of $n<(-1+\sqrt{1+4L^{2}U_{0}})/2$ for $k_{z}=0$.

The basic recipe of engineering such interfacial modes is the momentum shift along the $k_{z}$ direction by applying irradiation.

It is worthwhile to mention at this stage that topological $1$D chiral mode was predicted on the $2$D surface of a $3$D topological insulator (TI) Calvo et al. (2015). The irradiation induced band gap opening on the surface of $3$D TI plays the key role there. The present study is based on $3$D materials, where the external irradiation results in the momentum shift along the extra dimension and formation of the Fermi arc interface modes. Additionally, a finite number of propagating $2$D interface modes evolved depending on the interface width, which can be found by simple quantum mechanical treatment.

IV Anomalous Hall response

Let us now briefly discuss signatures of the irradiation in the response function. We particularly focus on anomalous Hall conductivity, which is exclusively attributed to the irradiation unlike longitudinal conductivity.

We should mention here that the expression for the current-current operator in the presence of the intense AC background has been formulated with the Floquet method in Oka and Aok (2009); Menon et al. (2018). It was shown that the time averaged operator can be written in terms of the Green function, where the energies are replaced with the Floquet quasi-energies

[TABLE]

where the Green function is given by

[TABLE]

with $\mu$ is the chemical potential, and $\omega_{n},\omega_{m}$ are the fermionic and bosonic Matsubara frequencies with $n,m\in\mathcal{Z}$. The different components of Green function are given by

[TABLE]

and

[TABLE]

The first term in Eq. (IV) corresponds to the dispersionless flat band whereas the second term corresponds to the conic bands. The components of the current density operator are given by $j_{\alpha}=e\partial\tilde{H}_{3f}/\partial k_{\alpha}.$ At zero temperature, using $k=\sqrt{k_{\bot}^{2}+k_{z}^{2}}$, $\int\frac{d^{3}k}{(2\pi)^{3}}...=\int_{-\Lambda_{\textrm{reg}}-\gamma_{0}}^{\Lambda_{\textrm{reg}}+\gamma_{0}}\frac{dk_{z}}{2\pi}\int\frac{d^{2}k_{\bot}}{4\pi^{2}}...$ with $\Lambda_{\textrm{reg}}$ introduced for the correct definition of the $k_{z}$ integral, and performing analytical continuation $i\omega_{m}\rightarrow\omega+i\delta$, it is straight forward to arrive at the anomalous Hall conductivity $\sigma_{xy}=\lim_{\omega\to 0}\Pi_{xy}(\omega)/i\omega$ as

[TABLE]

which is two times larger than anomalous Hall conductivity in the Weyl semimetal due to the doubling of the topological charge. The sign of $\sigma_{xy}$ depends on the phase of the incident radiation. Note that it is the transition between the dispersionless flat band and conic band which causes such result. The inter or intra conic band transition does not contribute to the anomalous Hall conductivity, as the corresponding matrix element vanishes in the polarization operator in Eq. (15). We also mention here that the above formalism is well justified as long as we treat the irradiated Hamiltonian via Floquet-Magnus high frequency approximation, where only the two nearest side bands are taken into account. It is instructive to adjust the chemical potential near to the band touching point in order to avoid any interference of higher Floquet side bands. To capture higher side bands, one must consider beyond high frequency approximation and Floquet Green function approach for transport study Oka and Aok (2009).

For completeness, we also comment on the diagonal components of the polarization matrix at zero temperature and wave-vector, which reduces to

[TABLE]

where $\Lambda$ is the energy cut-off and the diamagnetic contribution has been subtracted. The effect of flat band is to render the valence to conduction inter-band transitions, which leads to imaginary component already at $\omega=\mu$. This is in contrast to $\omega=2\mu$ condition in 3D Weyl semimetals. The transparency region to the external irradiation can be estimated by considering the conditions for the zero imaginary part and positive real components of the dielectric function. These require frequencies of the incident wave to be confined in the interval

[TABLE]

with $\alpha=N/(6\pi^{2}v\epsilon_{0})$ in which $\epsilon_{0}$ is the permittivity of free space and $N$ is the number of threefold band touching points.

V Discussion and conclusions

Let us estimate the parameters of the model. Typically, one can use the standard parameters of irradiation in the high frequency limit as: $evA=0.1-1$ eV and $0.1<evA/\Omega<1$. Under this regime, we can estimate $v\gamma_{0}=v^{2}e^{2}A_{x}A_{y}/2\Omega\approx 0.5(eAv/2)=0.12$ eV. The frequency $\Omega$ can be estimated to be $\approx 200$ THz for $eAv=0.5$ eV, which gives the wavelength of the irradiation as $1.2$ $\mu$m.

To summarize, we present a theoretical proposal of engineering a unidirectional mode propagating along the interface between two regions of threefold semimetal. The band structure of such material in the presence of the irradiation is obtained within the Floquet theory in the high frequency limit treating external field as a perturbation within the Floquet - Magnus expansion.

It is shown that time-dependent periodic perturbation, in the form of elliptically polarized field in $x-y$ plane, can cause a momentum shift along the $k_{z}$ direction in the electronic band structure. By utilizing several sources of irradiation, domain walls might be realized in the material at which phase factor of field changes. It is found that the interface between two regions, exposed to the irradiation with opposite phase, can host unidirectional Fermi-arc mode in addition to a dispersionless flat mode. The anomalous Hall conductivity and frequency interval at which the material becomes transparent to irradiation by evaluating the dielectric tensor are discussed.

Acknowledgements This work is supported by the Academy of Finland. A.A.Z. is grateful to the hospitality of the Pirinem School of Theoretical Physics.