Flexural deformations and collapse of bilayer two-dimensional crystals by interlayer excitons

We develop a consistent theory of the interlayer exciton-polaron formed in atomically thin bilayers. Coulomb attraction between an electron and a hole situated in the different layers results in their flexural deformation and provides an efficient mechanism of the exciton coupling with flexural phonons. We study the effect of layers tension on the polaron binding energy and effective mass leading to suppression of polaron formation by the tension both in the weak and strong coupling regimes. We also consider the role of the nonlinearity related to the interaction between the out- and in-plane lattice displacements and obtain the criterion of the layer sticking, where the exciton collapses due to the Coulomb attraction between the charge carriers.

Figure 1

Schematic illustration of the two-dimensional interlayer exciton.

Figure 2

(a) Interlayer exciton binding energy $E_B$ calculated after Eq. (5) and (b) deformation potential parameter $D$ calculated after Eq. (8) as a function of interlayer distance. Solid red curves correspond to the unscreened Coulomb interaction, dashed magenta curves correspond to the screened potential after Ref. [43] with the screening radii being $r_1=r_2=6a_B$, and dotted blue curves show large-$L$ asymptotics. An extended range of the $L$ axis is used to illustrate the asymptotics.

Figure 3

Absolute value of the polaron energy in the units of $\beta \hslash {\omega }_0$ as a function of the bilayer tension. Solid red curve has been calculated after Eq. (12), dashed green and dotted blue curves show, respectively, small and large tension asymptotics calculated after Eq. (14). Here the dimensionless bending rigidity $\varkappa /\mathcal{K}=0.001$. Inset demonstrates the renormalization of the polaron effective mass $m^{*}/m-1$ in the units of $\beta$ as a function of the bilayer tension. Solid red curve shows the full dependence, Eq. (15), dashed green and dotted blue curves demonstrate the asymptotics, Eq. (17). An extended range of the $\alpha$ axis is used to illustrate the asymptotics.

Figure 4

Absolute value of the polaron energy in the strong coupling regime in the units of $\beta \hslash {\omega }_0$ as a function of the bilayer tension. Solid red curve has been calculated after Eq. (21), dashed green and dotted blue curves show, respectively, small and large tension asymptotics calculated after Eq. (25). Here the dimensionless bending rigidity $\varkappa /\mathcal{K}=0.001$ and dimensionless coupling parameter $\beta =1000$. Inset demonstrates the renormalization of the polaron effective mass $m^{*}/m-1$ in the units of $\beta$ as a function of the bilayer tension. Solid red curve shows the full dependence, Eq. (26), dashed green and dotted blue curves demonstrate the asymptotics, Eq. (28).

Figure 5

Shape of the layer in anharmonic regime with point force at the center of circular flake for different Poisson's ratio. Solid lines show exact solution of Eqs. (34, 35, 36). Dashed lines demonstrate approximate form of the layers Eq. (40). Inset illustrates the bilayer exciton-polaron in nonlinear regime.

Figure 6

Absolute value of the exciton-polaron binding energy in the anharmonic regime as a function of the distance between layers in the units of critical distance $L_c$, Eq. (49). Solid blue curve has been calculated after minimization of Eq. (44), dashed red curve represents its asymptotic behavior at large distance Eq. (48). Inset demonstrates the total energy of the system as a function of the distance between the layers change for different interaction parameters $\gamma$ in Eq. (44). Solid curves demonstrate energy for the realizations, in which polaron exists. The polaron energies shown by dots are obtained by minimization of Eq. (44). A dashed black line shows the critical behavior with maximum interaction of polaron existence. Dotted curves represent large interaction with monotonic behavior of the energy, when the layers stick and the polaron collapses.

Figure 7

Polaron radius in the strong coupling regime in the units $\sqrt{\varkappa /{\omega }_0}$ as a function of the bilayer tension. Solid black curve has been calculated after minimization Eq. (21). Solid red curve is calculated using the first line of Eq. (B1). Dashed green and dotted blue curves show, respectively, small and large tension asymptotics Eq. (B2). Inset demonstrates the deflection of the polaron in the units of $\sqrt{m/\rho }$ as the function of bilayer tension. Solid black curve has been calculated after minimization Eq. (21). Solid red curve is found after substitution of $b$ from Eq. (B1) in Eq. (B4). Dashed green and dotted blue curves show, respectively, small and large tension asymptotics Eq. (B5).

Figure 8

Shape of the layers calculated after Eq. (B3) for different bilayer tension $\alpha$ (presented in the plot legend).