Valley polarization fluctuations, bistability, and switching in two-dimensional semiconductors

We theoretically study the nonlinear valley polarization dynamics of excitons in atom-thin semiconductors. The presence of significant polarization slows down valley relaxation due to an effective magnetic field resulting from exciton-exciton interactions. We address temporal dynamics of valley-polarized excitons and study the steady states of the polarized exciton gas. We demonstrate bistability of the valley polarization where two steady states with low- and high-valley polarization are formed. We study the effects of fluctuations and noise in such a system. We evaluate valley-polarization autocorrelation functions and demonstrate that for a high-polarization regime the fluctuations are characterized by high amplitude and long relaxation time. We study the switching between the low- and high-valley polarized states caused by the noise in the system and demonstrate that the state with high-valley polarization is preferential in a wide range of pumping rates.

Figure 1

Temporal dependence of the $z$-component of the pseudospin calculated at different initial valley polarizations: $\alpha S_0=1$ (blue), 5 (red), and 10 (orange). Dotted, dashed, and dash-dotted lines show analytical asymptotes Eqs. (9), (10), and (11), respectively.

Figure 2

(a) Valley pseudospin as function of the generation rate calculated after solution of Eq. (12). The area with bistable behavior is shown by the light yellow shading. (b) Graphical illustration of Eq. (13). Solid curves show the left-hand side $f\left(S_z\right)$ for different values of the generation rate $\alpha {\tau }_{v,0}{G}_z=0.1$ (blue), 0.3 (dark red), 1 (orange). Dashed magenta curve shows the right-hand side $S_z$. ${\tau }_0/{\tau }_{v,0}=100$.

Figure 3

Range of pumping rates where the bistable behavior is possible. Solid red and blue curves show upper and lower bounds for $G_z$, see Eqs. (22) and (23). Dashed green curve shows critical transition rate $G_{cr}$, Eq. (34), corresponding to the predominant realization of the $S_z^{\left(1\right)}$ or $S_z^{\left(3\right)}$ in the case of the noisy pumping.

Figure 4

Relaxation rates of small fluctuations in the vicinity of the steady-state solutions, Eq. (21). Blue curve shows ${\lambda }_1$ corresponding to the low valley imbalance $S_z^{\left(1\right)}$, red curve shows ${\lambda }_3$ corresponding to the hight alley imbalance $S_z^{\left(3\right)}$. Inset shows ${\lambda }_3$ in a linear scale. ${\tau }_0/{\tau }_{v,0}=100$.

Figure 5

Pseudospin autocorrelation functions, Eq. (27) calculated for the steady-state solutions with low valley imbalance (a) $S_z^{\left(1\right)}$ and (c) high valley imbalance $S_z^{\left(3\right)}$. Corresponding fluctuating time-dependent traces $S_z\left(t\right)$ found by direct numerical solution of Eq. (4) with fluctuating $G_z\left(t\right)$ according to Eq. (25). $\alpha \langle G_z\left(t\right)\rangle {\tau }_{v,0}=0.3, \beta =3\times {10}^{-4}/\left({\alpha }^2{\tau }_{v,0}\right), {\tau }_0/{\tau }_{v,0}=100$.

Figure 6

Fraction of time when the high $S_z$ solution ($\langle S_z\rangle =S_z^{\left(3\right)}$) is realized. Blue color shows the stability area of the low-$S_z$ solution ($\langle S_z\rangle =S_z^{\left(1\right)}$), red color shows the high $S_z$ solution ($\langle S_z\rangle =S_z^{\left(3\right)}$) is realized. Vertical dashed line shows $G_{cr}$ found from Eq. (34) condition. Inset shows typical time dependence of $S_z\left(t\right)$ at $\alpha {\tau }_{v,0}\langle G\rangle =0.21$ and sufficiently high noise amplitudes where switching between the high-$S_z$ and low-$S_z$ regimes is observed. ${\tau }_0/{\tau }_{v,0}=100$.