FRONTIERS ARTICLE
Theory of multiexciton generation in semiconductor nanocrystals

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Abstract

We develop a generalized framework based on a Green’s function formalism to calculate the efficiency of multiexciton generation in nanocrystal quantum dots. The direct/indirect absorption and coherent/incoherent impact ionization mechanisms, often used to describe multiexciton generation in nanocrystals, are reviewed and rederived from the unified theory as certain approximations. In addition, two new limits are described systematically – the weak Coulomb coupling limit and the semi-wide band limit. We show that the description of multiexciton generation in nanocrystals can be described as incoherent process and we discuss the scaling of multiexciton generation with respect to the photon energy and nanocrystal size. Illustrations are given for three prototype systems: CdSe, InAs and silicon quantum dots.

Research highlights

► Unified theory of multiexciton generation in nanocrystals. ► Energy- and size-dependence of the efficiency of charge multiplication in nanocrystals. ► Multiexciton generation in CdSe, InAs, and silicon.

Introduction

The development of efficient and cheap devices that utilize solar energy is one of the grand challenges in modern science [1]. In recent years, much attention has been given to the development of light-harvesting devices based on nanostructured thin-film materials [2], [3], [4], [5]. These materials offer the promise of low cost, small dimensions, light weight, and efficiencies up to the Shockley-Queisser (SQ) limit of 31% for single junction devices [6].

While reaching the SQ limit still remains a challenge for thin film nanostructured technology, there exist several concepts that hold the potential to move efficiencies beyond the SQ limit, to as much as 66% [7], [8]. One approach, which will be covered in the present work, is based on the generation of multiple pairs of charge carriers from a single absorption event. This process has been referred to as ‘Multiexciton Generation’ (MEG) which can lead to ‘Carriers Multiplication’ (CM) [7].

The key idea behind the generation of multiexciton upon the absorption of one photon is sketched in Fig. 1, for the special case of generating a biexciton. The absorbed photon creates an exciton composed of two charge carriers: a negative electron and a positive hole, each having an effective mass depending on the band structure of the nanomaterial. The exciton can either decay, typically by phonon emission, to the band edge with a timescale of γ-1 (γ is also the imaginary part of the phonon self energy, see below) [9]. The competing process, which is the one of interest in the present work, is the transformation of the excitonic state into a resonant biexcitonic state with a timescale ΓS-1 (ΓS is also the imaginary part of the biexciton self-energy, see below). This biexcitonic state can further decay to the biexcitonic band edge with a timescale γ-1 assumed to be independent of the number of charge carriers. The decay of the exciton/biexciton from the corresponding band edge occurs on much longer timescales and is not described here [10].

In this picture, MEG will become efficient and may lead to CM when the timescale ΓS-1 is significantly shorter than the timescale γ-1 associated with the relaxation of the initial exciton by other means. Furthermore, MEG can only occur at energies for which the initial excitation is at least twice above the material’s band gap, Eg to meet energy conservation. If excitation at energies twice above the band gap will result in 100% conversion to the biexcitonic state, then in principle, the energy efficiency of solar cell utilizing this process can exceed the SQ limit and reach values of 45% [11]. Thus, materials which exhibit large CM efficiencies require that ΓS-1γ-1 for all exciton energies above 2Eg.

The MEG phenomenon is known to occur in bulk semiconductors and has been studied for nearly 50 years [12]. Strict selection rules and other competing processes in the bulk allow generation of multiexcitons at energies of n × Eg where Eg is the band gap and n > 3, however, truly efficient MEG is observed only for n > 5 [13], [14]. In semiconducting nanocrystals (NCs) it was suggested that quantum confinement effects are important [7], enlarging Coulomb coupling and enabling a ‘phonon bottleneck’ phenomenon that reduces the rates of electronic excitation decay. This engendered the concept that MEG in NCs may be efficient at lower values of n (typically 2–3) [7]. Indeed, MEG in semiconducting NCs has been reported recently for several systems [8], [15], [16], [17], [18], [19], [20], [21], [22], [23], showing that the threshold is size and band-gap independent [16], [17], [22], [23]. However, more recent studies have questioned the efficiency of MEG in semiconducting NCs, in particular for CdSe [24] and InAs [25].

This controversy calls for theoretical assessment of the processes of MEG in nanostructures. In recent years several different theoretical treatments have been proposed [8], [26], [27], [28], [29], [30], [31], [32], [33], [34] to address the efficiency of MEG in NCs. These can be classified to two groups: (a) direct/indirect absorption into the biexcitonic manifold [8], [31] and (b) coherent/incoherent impact excitation [26], [27], [28], [29], [30], [34], [35]. The purpose of the present review is to present a unified theory to calculate the efficiency of MEG and to derive the former approaches as approximations to the unified framework. We will argue that one approach based on the concept of incoherent impact excitation is the most suitable for MEG in semiconducting NCs. Our calculations support recent experiments on various systems reporting low efficiencies of <20% at exciton energies near 3Eg[24], [25], [34], [36], [37].

Section snippets

Theory

Several different theoretical approaches have been suggested to describe MEG in semiconducting NCs. They can be classified to direct/indirect absorption [8], [31] and coherent/incoherent impact ionization [26], [27], [28], [29], [30], [34], [35]. In this section we will derive a unified theoretical approach to MEG based on the Green’s function formalism and show how the different treatments emerge as approximations to the proposed framework. Within the unified approach, we will compute the

Density of single and biexciton states

The calculation of the MEG efficiencies as defined in section 0, by solving directly Eqs. (9), (10), (11) or by referring to one of the approximations, e.g., weak coupling limit, indirect absorption, and semi-wide band limit, requires as input the screened Coulomb matrix elements between single and biexcitonic states WSB. In some cases these Coulomb matrix elements between single excitonic states WS is also required. Therefore, one has to specify a framework which provides an accurate account

Summary

We have developed a unified approach to the treatment of MEG in nanocrystals. Our approach is based on the Green’s function formalism, which in principle, leads to an exact description of MEG. It accounts for the screened Coulomb couplings between single and biexcitons, and between the exciton manifolds themselves. In addition, the formalism allows for the description electron–phonon couplings that are crucial for a complete description of MEG. Within this formalism, the efficiency of MEG is

Acknowledgments

We would like to thank Louis Brus and David Reichman for stimulating discussion and for helpful suggestions regarding this review. This research was supported by the Converging Technologies Program of The Israel Science Foundation (grant number 1704/07) and The Israel Science Foundation (grant number 962/06). We would like to thank the Center for Re-Defining Photovoltaic Efficiency through Molecule Scale Control, an Energy Frontier Research Center funded by the US Department of Energy, Office

Eran Rabani is a Professor of Chemistry at the School of Chemistry , Tel Aviv University. Rabani started his academic career at the Hebrew University of Jerusalem. He received his B.Sc. special program “Amirim” in 1991 (summa cum laude). He continued his graduate studied at the Hebrew University under the guidance of Professor Raphael D. Levine and received his Ph.D. in 1996. After completing 3 years at Columbia University as a post-doctoral research scientist with Professors Bruce J. Berne and

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    Eran Rabani is a Professor of Chemistry at the School of Chemistry , Tel Aviv University. Rabani started his academic career at the Hebrew University of Jerusalem. He received his B.Sc. special program “Amirim” in 1991 (summa cum laude). He continued his graduate studied at the Hebrew University under the guidance of Professor Raphael D. Levine and received his Ph.D. in 1996. After completing 3 years at Columbia University as a post-doctoral research scientist with Professors Bruce J. Berne and Louis E. Brus, he joined the faculty at Tel Aviv University in 1999. Rabani’s research focuses on the fundamental physical properties of nano-systems with emphasis on the electronic and conductance properties of nano-materials and the theory of self-assembly at the nano-scale.

    Roi Baer is a Professor of theoretical chemistry and the director of the Fritz Haber Center for Molecular Dynamics at the Institute of Chemistry, The Hebrew University of Jerusalem. Baer received his M.Sc and Ph.D. (1996) at the Hebrew University of Jerusalem under guidance of Professor Ronnie Kosloff. His post-doc at University of California Berkeley, with Professor Martin Head-Gordon, dealt with linear response density functional methods. He joined the faculty at the Hebrew University in 1998. Baer’s research focuses on electronic structure, and especially on the development of new density functional theories that widen the scope of chemical and physical problems accessible to them.

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