Engineering valley quantum interference in anisotropic van der Waals heterostructures

In this paper, we present a novel route to manipulate the spontaneous valley coherence in two-dimensional valleytronic materials interfaced with other layered materials hosting anisotropic polaritonic modes. We propose two implementations—one, using anisotropic plasmons in phosphorene and another with hyperbolic phonon polaritons in $\alpha \text{−}{\mathrm{MoO}}_3$. In particular, we show the electrostatic tunability of the spontaneous valley coherence achieving robust valley coherence values in the near-infrared wavelengths at room temperature. The tunability of this valley coherence shown in these heterostructures would enable the realization of active valleytronic quantum circuitry.

Figure 1

Schematic of the heterostructure on a substrate. The heterostructure is designed such that an anisotropic vacuum of the electromagnetic field is created due to the 2D material, and radiation is focused back. The excitons in the valleys are modeled as in-plane circular dipoles.

Figure 2

Temporal evolution of the DoLP for different values of ${\gamma }_y$. The inset figure shows the dependence of the steady-state DoLP with the intervalley decoherence rate which is equal to the sum of ${\gamma }_s$ and ${\gamma }_{\mathrm{dep}}$. For the entire figure, ${\gamma }_x=10{\gamma }_0$, and the pumping rate is taken as $(R={\gamma }_0)$. Initial conditions are taken as ${\varrho }_{K{K}^{\prime }}\left(0\right)=0,{\varrho }_{KK}\left(0\right)=0,{\varrho }_{K^{\prime }{K}^{\prime }}\left(0\right)=0,{\varrho }_{00}\left(0\right)=1$.

Figure 3

(a)–(d) $Q$ maps. Dependence of quantum interference $\left(Q\right)$ on the diagonal conductivities for two dipole distances and exciton frequencies in low loss limit when the dipole is standing above the 2D anisotropic material. ${\sigma }_0=\frac{e^2}{\hslash }$ and $\mathrm{Re}\left[\sigma \right]=\frac{1}{1000}\mathrm{Im}\left[\sigma \right]$ for all the plots.

Figure 4

(a)–(c) are for the TMDC/hexagonal boron nitride (hBN)/phosphorene heterostructure: (a) Schematic of the heterostructure. (b) Imaginary parts of conductivities of phosphorene (solid and dotted lines are for armchair and zigzag directions, respectively) vs chemical potential ($\mu$) at ${\mathrm{MoTe}}_2$ exciton frequency 1.1 eV. The blue and gray shaded regions represent the elliptical insulator regime and elliptical metallic regime, respectively. (c) Color plot of quantum interference (Q) with hBN thickness and chemical potential ($\mu$). The Fermi level is defined, and $E_f=E_c\left(\mathrm{\Gamma }\right)+\mu$. The bright region represents the high valley coherence region. (d)–(f) are for the BLG/mica/$\alpha \text{−}{\mathrm{MoO}}_3$ heterostructure: (d) Schematic of the heterostructure. (e) Dielectric function of $\alpha \text{−}{\mathrm{MoO}}_3$. The white region represents the elliptical insulating regime. The orange and blue regions represent two hyperbolic bands. (f) Quantum interference (Q) as a function of the BLG exciton frequency in the heterostructure. The thickness of mica is equal to twice the thickness of ${\mathrm{MoO}}_3$.