# Properties of Spatially Indirect Excitons in Nanowire Arrays
## 2 Results and Discussion  
### Binding Energy and Charge Distribution for SIEs  
Consider an electron and hole located in parallel NWs of a diameter \(D\) separated by the distance \(h\) between their surfaces. The Coulomb interaction between the chargesmediates the formation of spatially indirect excitons. The binding energy \(E_{x}\) of such SIEs as a function of \(D\) obtained by direct quantum mechanical calculations (see Appendix A) for different values of \(h\) is shown in Figure 1. The value of \(E_{x}\) at \(h=1\) nm and \(D=1.5\) nm slightly exceeds 100 mV, which is close to the binding energy of the Hybrid Charge Transfer Excitons at the organic-inorganic interface [24]. An increase in \(h\) understandably results in a noticeable decrease of \(E_{x}\) for all NW diameter values \(D\). It is worth pointing out the strong dependence of \(E_{x}\) upon \(D\) in the range of \(D<10\) nm and its levelling up at high \(D\)-values for all values of \(h\). The steep decrease in \(E_{x}\) at low \(D\)-values can be understood in terms of the charge density delocalization for the electron and hole across the corresponding NWs. Indeed, the strong confinement in the narrower NWs makes the charges look more point-like, and results in a stronger interaction between them and hence in a higher \(E_{x}\). The non-monotonic behaviour of \(E_{x}(D,h=3)\) at low \(D\)-values is most likely due to some minor transformation in the charge distributions. Delocalization of the charges in relatively thick NWs makes them look more like clouds and results in levelling up \(E_{x}(D\to\infty,h)\) down to the values characteristic for flat interfaces and determined by the spacing \(h\). It was found that the binding energies for SIEs in the crossed NWs are very close to those presented above.  
The outlined qualitative picture is illustrated by the charge density distributions shown in Figure 2. According to the figure, the charge densities in narrow NWs (panels A and B) are localized, while they are much more extended in the thicker NWs (panels C and D). This change in charge density localization for \(h=1\) (compare the panels A and C) significantly affects the binding energy-see the blue symbols in Figure 1-while for \(h=3\) (panels B and D), it creates a much smaller effect--see the black symbols in Figure 1. Based on the obtained results, it is useful to approximate the SIE binding energy \(E_{x}(D,h)\) with the following ad hoc analytical expression  
\[E_{x}(D,h)=\frac{A}{h+\alpha\cdot(1-\exp(-\beta\cdot D))},\ A=\frac{e^{2}}{4 \pi\epsilon_{0}\epsilon}=0.359\ \mathrm{eV}\cdot\mathrm{nm} \tag{1}\]  
Figure 1: Binding energy \(E_{x}\) of the SIE formed by an electron and hole located in neighbouring parallel NWs of diameter \(D\) separated by the distance \(h\). Symbols (triangles, circles, and squares) –results of calculations for \(m_{h}=5\cdot m_{e}\) and effective dielectric constant \(\epsilon=4\), as described in Appendix A. Solid lines show the corresponding approximations by Equation (1) (see text) with the fit parameters: \(\alpha=6.4\) nm, \(\beta=0.23\) nm\({}^{-1}\) for \(h=1\) nm; \(\alpha=8.1\) nm, \(\beta=0.19\) nm\({}^{-1}\) for \(h=2\) nm; and \(\alpha=8.9\) nm, \(\beta=0.18\) nm\({}^{-1}\) for \(h=3\) nm.  
where \(e\) stands for the electronic charge, \(\epsilon_{0}e\) is an effective dielectric constant (\(\epsilon=4\)) of the NW-vacuum spacing-NW system, and \(\alpha\) and \(\beta\) are the fit parameters. The fitting function in Equation (1) is constructed considering the fact that the system of two parallel NWs with constant inter-NW distance and increasing diameters \(D\) is converted, according to parameter \(\beta\), into a flat interface. Then, the binding energy between the electron and hole would be simply a Coulomb potential with some effective electron-hole distance, determined by the parameter \(\alpha\). According to Figure 1, the proposed analytical expression describes the calculation results quite well, suggesting that it can be used for estimating the SIE binding energies.