# Properties of Spatially Indirect Excitons in Nanowire Arrays
## 2 Results and Discussion
### SIE Life Time  
The lifetime of the SIE in the NWs is affected by (i) tunnelling of charges (predominantly electron) followed by electron-hole recombination, and (ii) exciton dissociation. The calculated (see Appendix A) tunnelling times are presented in Table 1 below.  
The dissociation times \(\tau_{d}(D,h)\) can be estimated using the expression \(\tau_{d}^{-1}(D,h)\approx\
u(D,h)\cdot\exp(-E_{x}(D,h)/k_{B}T)\), where \(\
u(D,h)\) is the frequency of the exciton internal oscillations along the NWs, \(k_{B}\) is the Boltzman constant, and \(T\) is absolute temperature. Assuming for simplicity that these oscillations are harmonic, we can estimate the frequency as \(\
u(D,h)=\sqrt{K/\mu}/2\pi\), where \(K\) is an effective harmonic force constant, and \(\mu=m_{e}m_{h}/\left(m_{e}+m_{h}\right)\) is an effective exciton mass. The value of \(K\) can be estimated using Equation (1) with effective electron-hole spacing \(h^{\prime}=\sqrt{h^{2}+x^{2}}\approx h+0.5\cdot x^{2}/h\), where \(x\) is the electron-hole instantaneous distance along the NW (in equilibrium \(x=0\)). Then, expanding the binding energy in a series of \(x\)-powers and retaining the first two terms, we obtain  
\[E_{x}(D,h^{\prime})\approx-\frac{A}{\hbar+0.5\cdot x^{2}/h+ \cdot(1-\exp(-\beta\cdot D))}\approx E_{x}(D,h)+\frac{1}{2}K\cdot x^{2},\] \[K=\frac{1}{h}\cdot\frac{(E_{x}(D,h))^{2}}{A}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \(D\), nm & \(h\), nm & \(\tau_{d}(D,h)\), s & \(\tau_{d}(D,h)\), s \\ \hline
1.5 & 1.0 & \(2\times 10^{-11}\) & \(5\times 10^{-6}\) \\ \hline
3.0 & 1.0 & \(2\times 10^{-9}\) & \(5\times 10^{-8}\) \\ \hline
1.5 & 2.0 & \(10^{-6}\) & \(4\times 10^{-8}\) \\ \hline
3.0 & 2.0 & \(7\times 10^{-6}\) & \(4\times 10^{-9}\) \\ \hline \end{tabular}
\end{table}
Table 1: Tunnelling times \(\tau_{t}\) for electrons from the home NW into the hole NW (see Appendix A) and exciton dissociation times \(\tau_{d}\) estimated using Equation (2) for T = 77 K.  
Figure 2: Charge densities of electron and hole across corresponding neighbouring parallel NWs of diameter \(D\) obtained by direct quantum mechanical calculations (see Appendix A) for different values of the NW diameter \(D\) and \(h\). (**A**) \(h\) = 1 nm, \(D\) = 1.5 nm; (**B**) \(h\) = 3 nm, \(D\) = 1.5 nm; (**C**) \(h\) = 1 nm \(D\) = 5 nm; (**D**) \(h\) = 3 nm, \(D\) = 5 nm.  
With the help of this expression for \(K\), we estimated the exciton dissociation time as a function of \(D\) and \(h\) for \(m_{h}=5m_{e}\) and \(T=77K\) (\(\approx 0.0067\) eV)  
\[\tau_{d}(D,h)=1/\
u(D,h)\cdot\exp\biggl{(}\frac{E_{x}(D,h)}{k_{B}T}\biggr{)} \approx\frac{10^{-14}\sqrt{h[\text{nm}]}}{E_{x}(D,h)[\text{eV}]}\cdot\exp \biggl{(}\frac{E_{x}(D,h)}{k_{B}T}\biggr{)} \tag{2}\]  
Corresponding values of \(\tau_{d}(D,h)\) are presented in Table 1. If the tunnelling times determine the exciton recombination times, then, according to Table 1, the SIE's lifetime in the case of \(D=3\) nm and \(h=1\) nm is determined by recombination and is restricted to a few nanoseconds. Most likely, this value can be significantly increased by applying some bias voltage to the NWs [22,23]. In contrast, for \(h=2\), the exciton lifetime is determined by dissociation, which can be increased by decreasing the system's temperature. The spatially indirect excitons can be generated by a number of ways, e.g., by optical pumping [25,26], applying a strain gradient [27], and possibly, in analogy with superlattices [28], from ordinary excitons by applying a sufficiently high electric field to the NWs. Keeping in mind that the NW parameters can be predetermined during their fabrication and that the NWs can subsequently be arranged in various structures, there is a hope that SIEs can potentially be useful for information processing.