Root-mean-squared-exciton-radius

Simple program to calculate rms radius of exciton from an a3dr file generated by plotxct.x in the BGW suite

Parent article to cite: K Ulman & SY Quek, Nano Lett. 2021, 21, 8888−8894

This code is written with a ZnPc/MoS2 organic-2D heterostructure system in mind:

1. The position vector of the hole is on the ZnPc molecule (C atom)
2. The electronic wavefunction is generated with this hole position using plotxct.x in the BGW suite.
3. a3dr file is used for the code
4. One can use a z-bound on the electronic wavefunction to consider only the part of the wavefunction that lies on the MoS2 layer (these lines are commented out)

The Mott critical density is identified as $n_{Mott} \sim \frac{1}{\pi r_x^2}$, where $r_x$ is the in-plane gyration radius of the exciton. (Ref. Fogler et al, Nat. Commun. 2014, 5, 4555). In our case for the CT exciton, the hole is localised on the molecule while the electron is quite delocalised but located on the MoS2. One way to estimate the size of the CT exciton (only valid in our case as the hole is localised), we can calculate the radius of the only the electronic part of the exciton wavefunction, keeping the hole fixed.

$$ r_x^2 \approx \frac{\int |\bar{r} - \bar{r_0} |^2 \rho( {\bar{r}}) d\bar{r} } {\int \rho( {\bar{r}}) d\bar{r} } = \frac{\sum_{ijk} [(x_{ijk} - x_0)^2 + (y_{ijk} - y_0)^2 + (z_{ijk} - z_0)^2 ]\rho_{ijk}}{\sum_{ijk} \rho_{ijk}} $$

Here, $\bar{r_0}$ is the centre of the electronic distribution, which needs to be defined after looking at the shape of electronic distribution. In our case, $\bar{r_0}$ is taken as the hole position. We have fixed the hole at C atom of ZnPc (chosen after looking at the maxima in wavefunction of HOMO). $\rho_{ijk}$ is read from the a3dr file (third column) written out by plotxct.f90 in BGW.