# Properties of Spatially Indirect Excitons in Nanowire Arrays
## Appendix A
### Electron Tunnelling between the NWs  
The recombination lifetime of the SIE was assumed to be determined by the tunneling of the electron between the NWs, thus neglecting the much slower tunneling rate of the heavy and much more localized hole. For estimating the tunneling time \(\tau_{t}\), we used an idea about oscillating probability to find the electron in one of two potential wells [30]. For simplicity, consider an electron tunneling between the two 1D potential wells separated by a potential barrier. The oscillations of the probability amplitude for the electron to be located in the one or the another potential well would take place with the period \(T=2\pi/\Delta E\), where \(\Delta E\) is the splitting between the symmetric and anti-symmetric electronic states. We associate this period \(T\) with the decay time \(\tau_{t}=T/4\) of the initial electronic state.  
In our case, the problem is more complicated: besides non-symmetric potential wells associated with the NWs, there is also an attraction Coulomb potential between the electron and hole. Therefore, we first calculated the hole probability density \(\rho_{h}(r_{h})(r_{h}=\{x_{h},y_{h}\})\) using the eigenfunctions of Equation (1). Then, by integrating the overall probability density over the hole variables \(x_{h}\) and \(y_{h}\), we obtained the stationary Schrodinger equation for the electron  
\[\begin{split}&-\frac{1}{2m_{e}}\Delta\psi_{t}+\left(U-\frac{e^{2}} {\varepsilon}\int\frac{\rho_{h}(r_{h})}{r_{eh}}dr_{h}\right)\cdot\psi_{t}=E_{t }\cdot\psi_{t},\\ & U=B_{e}\cdot\left(1-\Theta\left(\sqrt{\left(x-X_{e}\right)^{2}+ y^{2}}-R\right)-\Theta\left(\sqrt{\left(x-X_{h}\right)^{2}+y^{2}}-R\right) \right)\end{split} \tag{2}\]  
The eigen values \(E_{i}\) and eigen functions \(\psi(x,y,z)\) were determined using the virtual lattice method [29]. The four lowest eigen energy states of Equation (2) were found to be localized in the hole-hosting (below referred to as 'right') NW because of the strong electron-hole attraction. The fifth eigen state is mainly localized in the initial electron-hosting NW referred to below as 'left'. We want to estimate the tunnelling time from the left NW to the right one. To do this, consider the wave function \(\psi_{t}(x,y,z,t=0)\), which is non-zero only inside and around the left NW. We are interested in the transformation of this initial wave function into those localized in the right NW, namely  
\[\begin{split}&\psi_{t}(x,y,z,t)=\sum\limits_{k=1}^{5}C_{k}\cdot \varphi_{k}(x,y,z)\cdot\exp[-i\cdot E_{k}\cdot t],\\ & C_{k}=\underset{x,y,z=-\infty}{\iint\limits_{-\infty}}\psi_{t}( x,y,z,0)\cdot\varphi_{k}(x,y,z)dxdydz,\\ & C_{5}^{2}\approx 1,\ \ C_{k}^{2}<<1\ \ \ \text{if}\ \ k\
eq 5\end{split}\]  
where \(\psi_{t}(x,y,z,t=0)\) is actually close to \(\varphi_{5}\).  
Omitting the higher-degree terms, we can write for the probability for the electron to remain in the left NW \(P_{l}(t)\)  
\[P_{l}(t)=1-4\sum_{k=1..4}C_{k}^{2}\cdot Sin^{2}\left(\frac{(E_{k}-E_{5})\cdot t}{ 2}\right)\]  
The electron in the initial state \(\psi_{l}\) can tunnel to any of the four states \(\varphi_{i}\ \left(i=1,\ldots,4\right)\), which we will refer to as the tunnelling channels. Each such tunnelling channel reaches its maximum amplitude in the time period \(\Delta t_{i}=\pi/\Delta E_{i}\), where \(\Delta E_{i}=|E_{i}-E_{5}|\). The tunnelling time for each tunnelling channel can then be estimated by analogy with the 1D case of symmetric quantum wells [30], which has an additional factor \(C_{i}^{2}\). Depending on the considered parameters, some of the tunnelling channels can dominate. For all parameters considered in this work, the dominant channels are those to the states \(\varphi_{3}\) and \(\varphi_{4}\). The estimates of the resultant tunnelling time \(\tau_{i}=\left(\tau_{3}^{-1}+\tau_{4}^{-1}\right)^{-1}\) are presented in Table 1.  
\[\tau_{ti}=4\frac{C_{i}^{2}}{C_{5}^{2}}\frac{\pi}{\Delta E_{i}}\approx 4C_{i}^{ 2}\frac{\pi}{\Delta E_{i}}\]