Properties of Spatially Indirect Excitons in Nanowire Arrays

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Determining parameters of spatially indirect excitons in semiconductor nanowires for nanoelectronics.

Abstract

This paper deals with the excitons formed by electrons and holes located in different, closely placed semiconducting nanowires (spatially indirect excitons). We calculated the charge densities and the binding energies of the excitons for different nanowire diameters D and distances h between the nanowires. Together with the estimated exciton lifetimes, these results suggest that at certain h and D, the spatially indirect excitons in the nanowire arrays may have the potential to serve as information-processing units. Possible ways of exciton generation in the nanowire arrays are discussed.

1. Introduction

Unprecedented progress in the fabrication of semiconductor nanowires (NWs) has stimulated their application in various areas, such as nano- and optoelectronics [1,2,3,4], photovoltaics (PV) [5,6], random lasers [7,8], etc. NWs have been used as building blocks for fabricating various nanodevices including FETs [1,9], p-n diodes, and inverters [1,2]. These NW devices can be assembled in a predictable way and methods exist for their parallel assembly [1,10]. The control of NW sizes achieved by a number of growth mechanisms including the widely used vapour–liquid–solid (VLS) process [11,12] also allows for the study of exciton properties confined in two dimensions, with various degree of confinement. The Coulomb interaction between an electron and a hole located in neighbouring NWs may at certain conditions result in the bound state called ‘spatially indirect exciton’ (SIE) [13,14,15]. The controlled arrangement of NWs into three dimensional ordered structures might potentially allow generating arrays of SIEs, the properties of which would be very interesting to investigate.

Spatially indirect excitons are thought to be more long-lived than the usual (direct) excitons because of the spatial separation of electrons and holes [16,17,18]. They have been discussed in conjunction with quantum Bose gas in quasi-2D systems [13,19,20], capacitance in layered transition metal dichalcogenide structures [18,21], and as the information units in quantum posts [22,23]. So far, SIEs have been considered either for flat interfaces (the 2D case), or in pairs of quantum dots, the so-called quantum posts (the 0D case). In this article, we consider SIEs formed in parallel NWs (the 1D case) and analyse their binding energy and lifetime as functions of the NW diameter.

2. Results and Discussion

2.1. 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 charges mediates 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 $D<10 \mathrm{nm}$ 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 $E_x\left(D,h=3\right)$ 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 $E_x\left(D\to \infty ,h\right)$ 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 $E_x\left(D,h\right)$ with the following ad hoc analytical expression

$$E_x\left(D,h\right)=\frac{A}{h+\alpha ·\left(1-\exp \left(-\beta ·D\right)\right)}, A=\frac{e^2}{4\pi {\epsilon }_0\epsilon }=0.359 \mathrm{eV}·\mathrm{nm}$$

(1)

where e stands for the electronic charge, ${\epsilon }_0\epsilon$ is an effective dielectric constant ($\epsilon =4$) of the NW-vacuum spacing-NW system, and $\alpha \hspace{0.33em}\mathrm{and}\hspace{0.33em}\beta$ 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 β, 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 α. 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.

2.2. 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. Tunnelling times ${\tau }_t$ for electrons from the home NW into the hole NW (see Appendix A) and exciton dissociation times τ_d estimated using Equation (2) for T = 77 K.

D, nm	h, nm	${\mathit{\tau }}_{\mathit{t}}\left(\mathit{D},\mathit{h}\right), \mathbf{s}$	${\mathit{\tau }}_{\mathit{d}}\left(\mathit{D},\mathit{h}\right), \mathbf{s}$
1.5	1.0	2 × 10−11	5 × 10−6
3.0	1.0	2 × 10−9	5 × 10−8
1.5	2.0	10−6	4 × 10−8
3.0	2.0	7 × 10−6	4 × 10−9

below.

The dissociation times ${\tau }_d\left(D,h\right)$ can be estimated using the expression ${\tau }_d^{-1}\left(D,h\right)\approx \nu \left(D,h\right)·\exp \left(-E_x\left(D,h\right)/k_{B}T\right)$, where $\nu \left(D,h\right)$ 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 $\nu \left(D,h\right)=\sqrt{K/\mu }/2\pi$, 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 $h\prime =\sqrt{h^2+x^2}\approx h+0.5·x^2/h$, 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

$$\begin{array}{l}{E}_x\left(D,h\prime \right)\approx -\frac{A}{h+0.5·x^2/h+\alpha ·\left(1-\exp \left(-\beta ·D\right)\right)}\approx E_x\left(D,h\right)+\frac{1}{2}K·x^2, \hspace{0.17em}\\ K=\frac{1}{h}·\frac{{\left(E_x\left(D,h\right)\right)}^2}{A}\end{array}$$

With the help of this expression for K, we estimated the exciton dissociation time as a function of D and h for $m_h=5m_e \mathrm{and} T=77K \left(\approx 0.0067 \mathrm{eV}\right)$

$${\tau }_d\left(D,h\right)=1/\nu \left(D,h\right)·\exp \left(\frac{E_x\left(D,h\right)}{k_{B}T}\right)\approx \frac{{10}^{-14}\sqrt{h\left[\mathrm{nm}\right]}}{E_x\left(D,h\right)\left[\mathrm{eV}\right]}·\exp \left(\frac{E_x\left(D,h\right)}{k_{B}T}\right)$$

(2)

Corresponding values of ${\tau }_d\left(D,h\right)$ 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 $D=3 \mathrm{nm} \mathrm{and} h=1 \mathrm{nm}$ 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.

3. Conclusions

We calculated the binding energies of spatially indirect excitons in semiconductor NWs and estimated the exciton’s lifetime. The binding energies are found to be in the range $0.02÷0.1$ eV and strongly dependent on the NW diameter and inter-NW spacing for small (below 10 nm) diameter values. The SIE lifetimes are found to be in the range ${10}^{-11}÷{10}^{-6} s$, depending on the NW diameter and inter-NW spacing. Considering recent progress in controlling the NW parameters during the fabrication and assembling of NW structures, the obtained SIE parameters illustrate their significant potential for applications in nanoelectronics and information processing.