Valley quantum interference modulated by hyperbolic shear polaritons

Recently, symmetry-broken polaritons within low-symmetry crystals have triggered extensive research interest since they present enhanced directionality of polariton propagation for nanoscale manipulation and steering of photons. The latest discovery of hyperbolic shear polaritons (HShPs) in low-symmetry Bravais crystals provides great promise for innovating valleytronics. Herein, we theoretically demonstrate the coherent manipulation of the valley degree of freedom in a two-dimensional valleytronic material interfaced with monoclinic $\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3$ and $\mathrm{CdW}{\mathrm{O}}_4$ crystals. Robust and wideband tunable valley interference values are achieved in the mid- to far-infrared wavelengths. By virtue of stronger shear effect in monoclinic $\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3$, the valley quantum interference fringes modulated by $\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3$ crystal are more than those tuned via $\mathrm{CdW}{\mathrm{O}}_4$ crystal. After the monoclinic $\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3$ crystal is doped by free charge carriers, the number of HShP modes gradually decreases accompanied by the blueshifts and broadening of some hyperbolic dispersion bands as the doping concentration increases. In consequence, main fringes of valley quantum interference are broadened and shift toward the short wavelengths. Additionally, the doping increases the optical losses which limit the effective propagation of shear polaritons in monoclinic crystals. Therefore, the valley quantum interference is gradually reduced to a smaller and smaller negative value range as the doping concentration increases in the monoclinic crystal. Finally, the azimuthal dispersion of the HShP propagation direction gives rise to symmetry-broken valley quantum interference patterns when tuning the azimuth and twist angles of monoclinic hybrid structures. The azimuth angles of quantum interference fringes are susceptive to the variation of lattice displacement direction induced via the doping concentration. Thus, the valley quantum interference has great potential in estimating the doping concentration and the propagation directions of HShPs in monoclinic crystals.

Figure 1

(a) A schematic of hyperbolic shear polariton (HShP)-modulated valley quantum interference in a two-dimensional (2D) valley material. Crystal1 (or crystal2) stands for monoclinic $\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3$ or $\mathrm{CdW}{\mathrm{O}}_4$ crystal. Monoclinic crystal1 and crystal2 create a local in-plane anisotropic environment which allows a finite nonzero coupling between mutually orthogonal valley excitons (see the inset) in the 2D valley material. The emission from the $K$ valley with ${\sigma }_{+}$ polarization can radiatively excite the orthogonal $K^{\prime }$ valley with ${\sigma }_{-}$ polarization. Such interaction leads to the spontaneous generation of valley coherence and yields quantum interference among their emissions. Unit cells of (b) $\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3$ and (c) $\mathrm{CdW}{\mathrm{O}}_4$ with monoclinic angle β and the Cartesian coordinate system ($x, y, z$) fixed to the unit cells. (d) Schematic view from the top of the twist-induced monoclinic crystals. The surface of $\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3$ or $\mathrm{CdW}{\mathrm{O}}_4$ is the monoclinic (010) plane ($x\text{−}y$ plane). The azimuth angle Φ (or ${\mathrm{\Phi }}^{\prime }$) of crystal1 (or crystal2) is the angle between the $x_0$ and a axes, where $\left(x_0,y_0,z_0\right)$ is the laboratory coordinate system. The twist angle between crystal1 and crystal2 is $\theta ={\mathrm{\Phi }}^{\prime }-\mathrm{\Phi }$.

Figure 2

Real parts of permittivity elements for pure monoclinic (a) $\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3$ and (f) $\mathrm{CdW}{\mathrm{O}}_4$ crystals, and $n$-type doped $\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3$ crystals with (b) $\mathrm{lo}{\mathrm{g}}_{10}N=18.0$, (c) $\mathrm{lo}{\mathrm{g}}_{10}N=18.5$, (d) $\mathrm{lo}{\mathrm{g}}_{10}N=19.0$, and (e) $\mathrm{lo}{\mathrm{g}}_{10}N=19.5$. The wavelength ranges in which $\mathrm{Re}\left({\varepsilon }_{xx}\right), \mathrm{Re}\left({\varepsilon }_{xy}\right), \mathrm{Re}\left({\varepsilon }_{yy}\right)$, and $\mathrm{Re}\left({\varepsilon }_{zz}\right)$ have different signs are shaded by different colors, and the corresponding signs of each color are shown in the left table.

Figure 3

(a) Real and (b) imaginary parts of Fresnel reflection coefficient $r_{pp}$ for pure monoclinic $\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3$ and $\mathrm{CdW}{\mathrm{O}}_4$ crystals, and doped $\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3$ crystals at different free charge-carrier densities. Quantum interferences ($Q$) for two orthogonal dipoles positioned at a distant height from the monoclinic $\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3$ crystal with (c) $N =0.0$, (d) $\mathrm{lo}{\mathrm{g}}_{10}N=18.0$, (e) $\mathrm{lo}{\mathrm{g}}_{10}N=18.5$, (f) $\mathrm{lo}{\mathrm{g}}_{10}N=19.0$, (g) $\mathrm{lo}{\mathrm{g}}_{10}N=19.5$, and (h) pure $\mathrm{CdW}{\mathrm{O}}_4$ crystal. For comparison, their respective spectral curves of $\mathrm{Im}\left(r_{pp}\right)$ are replotted by white solid lines in (c)−(h).

Figure 4

Isofrequency surfaces for pure monoclinic (a) $\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3$ and (f) $\mathrm{CdW}{\mathrm{O}}_4$ crystals at the work wavelengths $\lambda =30.5$ and 24.5 µm, respectively. (b)−(e) are the isofrequency surfaces for doped $\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3$ crystals at electron-doping concentrations $\mathrm{lo}{\mathrm{g}}_{10}N=18.0$, 18.5, 19.0, and 19.5, respectively. Their corresponding work wavelengths are $\lambda =30.0$, 28.8, 28.6, and 28.3 µm, respectively. The red and blue surfaces denote the positive and negative signs of $k_z/k_0$. The contour lines at the bottom of each subgraph are the projections of isofrequency surfaces.

Figure 5

Variations of quantum interference $Q$ with respect to the wavelength λ and the azimuth angle Φ. The radial axis and angle stand for the factors of λ in µm and Φ in degrees, respectively. (a) and (f) are for the cases of pure monoclinic $\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3$ and $\mathrm{CdW}{\mathrm{O}}_4$ crystals, respectively. (b)−(e) are for the conditions of doped $\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3$ crystals with $\mathrm{lo}{\mathrm{g}}_{10}N=18.0$, 18.5, 19.0, and 19.5, respectively.

Figure 6

(a) The azimuth and the major polarizability angles at which the quantum interferences $Q$ have local maximum (see the circular symbols) and minimum (see the square symbols) values for pure ($N =0$) monoclinic $\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3$ and $\mathrm{CdW}{\mathrm{O}}_4$ crystals, and doped $\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3$ crystals. The solid lines are just drawn as a guide to the eye. (b) Dispersion curves of the major polarizability angle α(λ) for pure monoclinic $\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3$ and $\mathrm{CdW}{\mathrm{O}}_4$ crystals, and doped $\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3$ crystals at different $N$ values. Solid (or open) circles indicate the α (or Φ) values at $\lambda =34.6\mu \mathrm{m}$ for $\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3$ and $\lambda =26.8\mu \mathrm{m}$ for $\mathrm{CdW}{\mathrm{O}}_4$. Solid (or open) squares indicate the α (or Φ) values at $\lambda =30.5$, 30.0, 28.8, 28.2, 28.1, and 24.4 µm for $\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3$ crystal with $N=0.0, \mathrm{lo}{\mathrm{g}}_{10}N=18.0$, 18.5, 19.0, 19.5, and pure $\mathrm{CdW}{\mathrm{O}}_4$ crystal, respectively.

Figure 7

Variations of quantum interference $Q$ with respect to the wavelength λ and the twist angle θ for the hybrid architectures of (a) $\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3\text{−}\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3$, (b) $\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3\text{−}\mathrm{CdW}{\mathrm{O}}_4$, (c) $\mathrm{CdW}{\mathrm{O}}_4\text{−}\mathrm{CdW}{\mathrm{O}}_4$, and (d) $\mathrm{CdW}{\mathrm{O}}_4\text{−}\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3$. The radial axis and angle stand for the factors of λ in µm and θ in degrees, respectively. The azimuth angle Φ of crystal1 is tuned from $0^{\circ }$ to ${180}^{\circ }$, while the azimuth angle ${\mathrm{\Phi }}^{\prime }$ of crystal2 is fixed at ${90}^{\circ }$. Both $\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3$ and $\mathrm{CdW}{\mathrm{O}}_4$ crystals are pure monoclinic crystals.