Fundamental collapse of the exciton-exciton effective scattering

The exciton-exciton effective scattering which rules the time evolution of two excitons is studied as a function of initial momentum difference, scattering angle, and electron-to-hole mass ratio. We show that this effective scattering can collapse for energy-conserving configurations provided that the difference between the two initial exciton momenta is larger than a threshold value. Sizeable scatterings then exist in the forward direction only. We even find that, for an electron-to-hole mass ratio close to 1/2, the exciton-exciton effective scattering stays close to zero in all directions when the difference between the initial exciton momenta has a very specific value. This unexpected but quite remarkable collapse comes from tricky compensation between direct and exchange Coulomb processes which originates from the fundamental undistinguishability of the exciton fermionic components.

Figure 1

Pauli scattering $\lambda \left(\genfrac{}{}{0ex}{}{nj}{mi}\right)$ for fermion exchange between two excitons starting in in states $\left(i,j\right)$ and ending in out states $\left(m,n\right)$. Electrons are represented by solid lines and holes by dashed lines.

Figure 2

(a) Direct interaction scatterings between two excitons starting in in states $\left(i,j\right)$ and ending in out states $\left(m,n\right)$ for carrier interaction in the absence of carrier exchange. (b) A $\left(m,n\right)$ permutation can be seen as the result of a double carrier exchange.

Figure 3

Coulomb exchange scatterings in ${\xi }^{\text{in}}$, the Coulomb processes take place between the in excitons while in ${\xi }^{\text{out}}$ they take place between the out excitons. These exchange Coulomb scatterings can be seen as a succession of a direct Coulomb scattering and a Pauli scattering for carrier exchange in the absence of carrier interaction.

Figure 4

Exciton momenta in 3D Bohr radius unit $\tilde{P}=a_{X}P$, for which the exciton kinetic energy is equal to its binding energy, as a function of mass ratio $\alpha =\frac{m_e}{m_h}$.

Figure 5

(a) In the laboratory frame, the in excitons with momenta ${\mathbf{K}}_i=\mathbf{K}+\mathbf{P}$ and ${\mathbf{K}}_j=\mathbf{K}-\mathbf{P}$ transform into out excitons with momenta ${\mathbf{K}}_m=\mathbf{K}+{\mathbf{P}}^{\prime }$ and ${\mathbf{K}}_n=\mathbf{K}-{\mathbf{P}}^{\prime }$. (b) In the center-of-mass frame, which corresponds to set $\mathbf{K}=0$, these excitons have momenta $\left(\mathbf{P},-\mathbf{P}\right)$ and $\left({\mathbf{P}}^{\prime },-{\mathbf{P}}^{\prime }\right)$. Energy-conserving processes, which are the relevant ones in the large time limit, lead to $P=P^{\prime }$: momenta follow a simple rotation in the center-of-mass frame, as shown by the dashed circle.

Figure 6

Scattering of two excitons with initial momenta $\left(\mathbf{P},-\mathbf{P}\right)$ and final momenta $\left({\mathbf{P}}^{\prime },-{\mathbf{P}}^{\prime }\right)$ in the center-of-mass frame.

Figure 7

Effective scatterings ${\xi }_{{\alpha }_e=0}^{\text{eff}}$ in the case of infinite hole mass, $m_e/m_h=0$, for three different values of half the initial momentum difference $P$, namely, $P_0^{\ast }/2$, $P_0^{\ast }$, $2P_0^{\ast }$, where $P_0^{\ast }$ is the threshold value of $P$ above which the effective exciton-exciton scattering can cancel.

Figure 8

Effective scatterings ${\xi }_{{\alpha }_e=0}^{\text{eff}}$ for energy-conserving processes as a function of half the initial momentum difference $P$ and the angle $\theta$ between initial and scattered momenta when the hole mass is infinite. The full lines correspond to constant ${\xi }_{{\alpha }_e=0}^{\text{eff}}$, the heavy one corresponding to ${\xi }_{{\alpha }_e=0}^{\text{eff}}=0$.

Figure 9

Initial half exciton momentum difference $P_{{\alpha }_e=0}^{\left(0\right)}\left(\theta \right)$ for effective scattering cancellation as a function of the angle between initial and scattered momenta when the hole mass is infinite. The dashed line corresponds to the treshold value $P_0^{\ast }\simeq 2.77$ above which cancellation can occur [see Eq. (35)].

Figure 10

Effective scatterings ${\xi }_{{\alpha }_e=1/2}^{\text{eff}}$ for energy-conserving processes as a function of half the initial momentum difference $P$ and the angle $\theta$ between initial and scattered momenta when the electron and hole masses are equal. The full lines correspond to constant ${\xi }_{{\alpha }_e=1/2}^{\text{eff}}$, the heavy one corresponding to ${\xi }_{{\alpha }_e=1/2}^{\text{eff}}=0$.

Figure 11

Initial half exciton momentum difference $P_{{\alpha }_e=1/2}^{\left(0\right)}\left(\theta \right)$ for effective scattering cancellation as a function of the angle between initial and scattered momenta for equal electron and hole masses.

Figure 12

Initial half exciton momentum difference $P_{{\alpha }_e}^{\left(0\right)}\left(\theta \right)$ for effective scattering cancellation as a function of the angle between initial and scattered momenta when $m_h=2.08m_e$, i.e., when ${\alpha }_e={\overline{\alpha }}_e=0.32$. The dotted line, $P=3.33$, is to guide the eyes for the weak oscillation of $P_{{\overline{\alpha }}_e}^{\left(0\right)}\left(\theta \right)$, the value of $P_{{\overline{\alpha }}_e}^{\left(0\right)}\left(\theta \right)$ being exactly equal to $\overline{P}\simeq 3.3$ for $\text{cos}\theta =\left(0,\pm 1\right)$

Figure 13

Half initial exciton momentum difference $P_{{\alpha }_e=1/3}^{\left(0\right)}\left(\theta \right)$ for cancellation of the effective scattering, as a function of the angle between initial and scattered momenta when $m_h=2m_e$.

Figure 14

Half initial exciton momentum difference $P_{{\alpha }_e}^{\left(0\right)}\left(\theta \right)$ for cancellation of the effective scattering, as a function of the angle between initial and scattered momenta when $m_h=2.16m_e$.