Tensor network methods for bound electron-hole complexes beyond strong and weak confinement in nanoplatelets

Abstract

In semiconductor nanostructures, optical excitation typically creates bound electron-hole states, such as excitons, trions, and larger complexes. Their relative motion is described by the Wannier equation, which is valid only for spatially extended motion in the Coulomb-dominated, weak-confinement limit. Other small nanostructures, such as quantum dots, are in the confinement-dominated strong confinement regime, where the wavefunction factorizes into independent electron and hole parts. Nanoplatelets are in between the two regimes and require solving an unfactorized higher-dimensional Schrödinger equation, which is computationally expensive. This work demonstrates how tensor networks can partially overcome this problem, using CdSe nanoplatelets as an example. The method is also applicable to related two-dimensional systems. As a demonstration, we calculate the excitonic and trionic ground states, as well as several excited states, for nanoplatelets of varying sizes, including their energies and oscillator strengths. More importantly, overall strategies for using tensor networks in real space for systems under intermediate confinement have been developed.

Figures (8)

• Figure 1: Shift operator applied to the QTT representation of $f$ consisting of the addition network (full adder tensors FA), the one-hot tensors representing the shift $s$, and the termination tensors on both ends of the addition network.
• Figure 2: QTT representation of a two-particle potential built from the effective single particle potential $U(x, y)$ (top) and a subtraction network (bottom). The tensors that are contracted into a single tensor in the QTT are indicated by the dotted contour boxes.
• Figure 3: Contraction scheme for computing the oscillator strengths of: a) the transition from the crystal ground state into the exciton state $\psi_E$ and b) the transition from the single electron state $\psi_{Electron}$ into the trion state $\psi_{Trion}$.
• Figure 4: Contraction scheme for computing the electron density $|\tilde{\psi}_e(\mathbf{r}_e)|^2=\int d^2\mathbf{r}_h\left|\psi_E(\mathbf{r}_e, \mathbf{r}_h)\right|^2$ from the QTT representation of $\psi_E$. $T$ denotes a tensor filled with ones that is used to perform a summation (integration) over an index, here they are applied to all indicies belonging to the hole coordinates. Converseley, for the hole density $|\tilde{\psi}_h(\mathbf{r}_h)|^2$ they would be applied to the electron coordinates. If desired $T$ tensors are also applied to the least significant bits to reduce the resolution.
• Figure 5: Construction of the a) relative $|\tilde{\psi}_{\text{r}}(\mathbf{r})|^2$ and b) center-of-mass densities $|\tilde{\psi}_{\text{COM}}|^2$ from a QTT representing the original density $\left|\psi_E(\mathbf{r}_e,\mathbf{r}_h)\right|^2$ (top). The layers below the density QTT are applied one after another by interpreting them as MPOs.