Macroscopic Degeneracy of Zero-Mode Rotating Surface States in 3D Dirac and Weyl Semimetals under Radiation

We investigate the development of novel surface states when 3D Dirac or Weyl semimetals are placed under circularly polarized electromagnetic radiation. We find that the hybridization between inverted Floquet bands opens, in general, a gap, which closes at so-called exceptional points found for complex values of the momentum. This corresponds to the appearance of midgap surface states in the form of evanescent waves decaying from the surface exposed to the radiation. We observe a phenomenon reminiscent of Landau quantization by which the midgap surface states get a large degeneracy proportional to the radiation flux traversing the surface of the semimetal. We show that all of these surface states carry angular current, leading to an angular modulation of their charge that rotates with the same frequency of the radiation, which should manifest in the observation of a macroscopic chiral current in the irradiated surface.

Figure 1

Left: Schematic representation of the first Floquet bands with quasienergy $\epsilon$ and $\epsilon \pm \hslash \mathrm{\Omega }$ ($\mathrm{\Omega }$ being the radiation frequency) for a model of 3D semimetal with two nodes along the momentum axis $k_z$ (red color denotes positive energy while light blue denotes negative energy, the direction transverse to $k_z$ represents additional components of the momentum). Right: Geometry of 3D semimetal with the surface exposed to the radiation where the evanescent states appear.

Figure 2

Band structure for model parameters $c_0=c_1=c_2=0.0$, $m_0=-0.5\mathrm{eV}$, $m_1=-0.605\mathrm{eV}{\AA }^2$, $m_2=-1.0\mathrm{eV}{\AA }^2$, $\hslash v=1.1\mathrm{eV}\AA$, $\hslash \mathrm{\Omega }=0.5\mathrm{eV}$, $\mathcal{A}=0.05{\AA }^{-1}$, in the bulk (left-hand panel) and for a wire of section $S=150\times 150{\AA }^2$ in the $x\text{−}y$ plane (right-hand panel). The position of the Dirac cone at zero field is $k_{c,z}=\sqrt{m_0/m_1}=0.909{\AA }^{-1}$. The color represents the predominant Floquet mode of the particular state in a RGB coding (RGB standing for red-green-blue) where the intensity of each mode is mapped to the intensity of each fundamental color; $n=-1$, 0, 1 correspond to red, green, and blue, respectively. The position of the evanescent waves obtained as strictly real solutions of the eigenvalue problem after including an imaginary part in $k_z$ of $\mathrm{Im}(k_z)=0.01{\AA }^{-1}$ is shown with black circles with a dot. A few high-energy bands with $n=-1$ index not affecting the avoided crossing structures have been erased for clarity of presentation.

Figure 3

(a) Plot of the gap of the Hamiltonian Eq. (8) (in eV) as a function of real $k_z$ (measured in units of the inverse of the typical microscopic length scale $a$ in the material) for $\hslash v/a=1.1\mathrm{eV}$, $\hslash \mathrm{\Omega }=0.5\mathrm{eV}$, $\mathcal{A}=0.005a^{-1}$, and radius $R=200a$ of the cylindrical geometry considered in the text. (b) Imaginary part of the lowest eigenvalue of the Hamiltonian Eq. (8) (in eV) as a function of $\mathrm{Re}(k_z)$, for complex momenta with $\mathrm{Im}(k_z)=0.08a^{-1}$ and the same parameters as in (a).

Figure 4

Left: Probability distribution $r|\chi {|}^2$ along the radial direction for evanescent states corresponding to three decreasing values of $\mathrm{Re}(k_z)$ with $\mathrm{Im}(ϵ)=0$ (starting with the outermost evanescent state) for the same parameters as in Fig. 3 and $R=400a$. Right: Time-dependent angular current $r{j}_{\mathrm{\Omega }}^{\theta }$ for the states in the upper part (top) and the middle part (bottom) of the left-hand panel, represented for fixed time $t=0$ at the irradiated surface of the cylindrical geometry.