Expeditious stochastic calculation of multiexciton generation rates in semiconductor nanocrystals

Simulations of nonradiative processes in semiconductor nanocrystals

The description of carrier dynamics in spatially confined semiconductor nanocrystals (NCs), which have enhanced electron-hole and exciton-phonon interactions, is a great challenge for modern computational science. These NCs typically contain thousands of atoms and tens of thousands of valence electrons with discrete spectra at low excitation energies, similar to atoms and molecules, that converge to the continuum bulk limit at higher energies. Computational methods developed for molecules are limited to very small nanoclusters, and methods for bulk systems with periodic boundary conditions are not suitable due to the lack of translational symmetry in NCs. This perspective focuses on our recent efforts in developing a unified atomistic model based on the semiempirical pseudopotential approach, which is parameterized by first-principle calculations and validated against experimental measurements, to describe two of the main nonradiative relaxation processes of quantum confined excitons: exciton cooling and Auger recombination. We focus on the description of both electron-hole and exciton-phonon interactions in our approach and discuss the role of size, shape, and interfacing on the electronic properties and dynamics for II-VI and III-V semiconductor NCs.

Stochastic Vector Techniques in Ground-State Electronic Structure

We review a suite of stochastic vector computational approaches for studying the electronic structure of extended condensed matter systems. These techniques help reduce algorithmic complexity, facilitate efficient parallelization, simplify computational tasks, accelerate calculations, and diminish memory requirements. While their scope is vast, we limit our study to ground-state and finite temperature density functional theory (DFT) and second-order perturbation theory. More advanced topics, such as quasiparticle (charge) and optical (neutral) excitations and higher-order processes, are covered elsewhere. We start by explaining how to use stochastic vectors in computations, characterizing the associated statistical errors. Next, we show how to estimate the electron density in DFT and discuss highly effective techniques to reduce statistical errors. Finally, we review the use of stochastic vector techniques for calculating correlation energies within the second-order Møller-Plesset perturbation theory and its finite temperature variational form. Example calculation results are presented and used to demonstrate the efficacy of the methods. Expected final online publication date for the Annual Review of Physical Chemistry, Volume 73 is April 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

Modeling Non-adiabatic Dynamics in Nanoscale and Condensed Matter Systems

Rapid, far-from-equilibrium processes involving excitation of electronic, vibrational, spin, photon, topological, and other degrees of freedom form the basis of modern technologies, including electronics and optoelectronics, solar energy harvesting and conversion to electrical and chemical energy, quantum information processing, spin- and valleytronics, chemical detection, and medical therapies. Such processes are studied experimentally with various time-resolved spectroscopies that allow scientists to track system's evolution on ultrafast time scales and at close to atomistic level of detail. The availability of various forms of lasing has made such measurements easily accessible to many experimental groups worldwide, to study atoms and small molecules, nanoscale and condensed matter systems, proteins, cells, and mesoscopic materials. The experimental work necessitates parallel theoretical efforts needed to interpret the experiments and to provide insights that cannot be gained through measurements due to experimental limitations.Non-adiabatic (NA) molecular dynamics (MD) allows one to study processes at the atomistic level and in the time domain most directly mimicking the time-resolved experiments. Atomistic modeling takes full advantage of chemical intuition and principles that guide design and fabrication of molecules and materials. It provides atomistic origins of quasi-particles, such as holes, excitons, trions, plasmons, phonons, polarons, polaritons, spin-waves, momentum-resolved and topological states, electrically and magnetically polarized structures, and other abstract concepts. An atomistic description enables one to study realistic aspects of materials, which necessarily contain defects, dopants, surfaces, interfaces, passivating ligands, and solvent layers. Often, such realistic features govern material properties and are hard to account for phenomenologically. NA-MD requires few approximations and assumptions. It does not need to assume that atomic motions are harmonic, that electrons are Drude oscillators, that coupling between different degrees of freedom is weak, that dynamics is Markovian or has short memory, or that evolution occurs by exponential kinetics of transitions between few states. The classical or semiclassical treatment of atomic motions constitutes the main approximation of NA-MD and is used because atoms are 3-5 orders of magnitude heavier than electrons. NA-MD is limited by system size, typically hundreds or thousands of atoms, and time scale, picoseconds to nanoseconds. The quality of NA-MD simulations depends on the electronic structure method used to obtain excited state energies and NA couplings.NA-MD has been largely popularized and advanced in the chemistry community that focuses on molecules. Modeling far-from-equilibrium dynamics in nanoscale and condensed matter systems often has to account for other types of physics. At the same time, condensed phase NA-MD allows for approximations that may not work in molecules. Focusing on the recent NA-MD developments aimed at studying excited state processes in nanoscale and condensed phases, this Account considers how the phenomena important on the nanoscale can be incorporated into NA-MD and what approximations can be made to increase its efficiency with complex systems and processes.

Nanocrystal Quantum Dots: From Discovery to Modern Development

This review traces nanocrystal quantum dot (QD) research from the early discoveries to the present day and into the future. We describe the extensive body of theoretical and experimental knowledge that comprises the modern science of QDs. Indeed, the spatial confinement of electrons, holes, and excitons in nanocrystals, coupled with the ability of modern chemical synthesis to make complex designed structures, is today enabling multiple applications of QD size-tunable electronic and optical properties.

Stochastically Realized Observables for Excitonic Molecular Aggregates

We show that a stochastic approach enables calculations of the optical properties of large 2-dimensional and nanotubular excitonic molecular aggregates. Previous studies of such systems relied on numerically diagonalizing the dense and disordered Frenkel Hamiltonian, which scales approximately as O(N3) for N dye molecules. Our approach scales much more efficiently as O(Nlog(N)), enabling quick study of systems with a million of coupled molecules on the micrometer size scale. We calculate several important experimental observables, including the optical absorption spectrum and density of states, and develop a stochastic formalism for the participation ratio. Quantitative agreement with traditional matrix diagonalization methods is demonstrated for both small- and intermediate-size systems. The stochastic methodology enables the study of the effects of spatial-correlation in site energies on the optical signatures of large 2D aggregates. Our results demonstrate that stochastic methods present a path forward for screening structural parameters and validating experiments and theoretical predictions in large excitonic aggregates.

Dielectric Confinement and Excitonic Effects in Two-Dimensional Nanoplatelets

Quasi-two-dimensional (2D) semiconductor nanoplatelets manifest strong quantum confinement with exceptional optical characteristics of narrow photoluminescence peaks with energies tunable by thickness with monolayer precision. We employed scanning tunneling spectroscopy (STS) in conjunction with optical measurements to probe the thickness-dependent band gap and density of excited states in a series of CdSe nanoplatelets. The tunneling spectra, measured in the double-barrier tunnel junction configuration, reveal the effect of quantum confinement on the band gap taking place mainly through a blue-shift of the conduction band edge, along with a signature of 2D electronic structure intermixed with finite lateral-size and/or defects effects. The STS fundamental band gaps are larger than the optical gaps as expected from the contributions of exciton binding in the absorption, as confirmed by theoretical calculations. The calculations also point to strong valence band mixing between the light- and split-off hole levels. Strikingly, the energy difference between the heavy-hole and light-hole levels in the tunneling spectra are significantly larger than the corresponding values extracted from the absorption spectra. Possible explanations for this, including an interplay of nanoplatelet charging, dielectric confinement, and difference in exciton binding energy for light and heavy holes, are analyzed and discussed.

Auger Recombination Lifetime Scaling for Type I and Quasi-Type II Core/Shell Quantum Dots

Having already achieved near-unity quantum yields, with promising properties for light-emitting diode, lasing, and charge separation applications, colloidal core/shell quantum dots have great technological potential. The shell thickness and band alignment of the shell and core materials are known to influence the efficiency of these devices. In many such applications, improving the efficiency requires a deep understanding of multiexcitonic states. Herein, we elucidate the shell thickness and band alignment dependencies of the biexciton Auger recombination lifetime for quasi-type II CdSe/CdS and type I CdSe/ZnS core/shell quantum dots. We find that the biexciton Auger recombination lifetime increases with the total nanocrystal volume for quasi-type II CdSe/CdS core/shell quantum dots and is independent of the shell thickness for type I CdSe/ZnS core/shell quantum dots. To perform these calculations and compute Auger recombination lifetimes, we developed a low-scaling approach based on the stochastic resolution of identity. The numerical approach provided a framework for studying the scaling of the biexciton Auger recombination lifetimes in terms of the shell thickness dependencies of the exciton radii, Coulomb couplings, and density of final states in quasi-type II CdSe/CdS and type I CdSe/ZnS core/shell quantum dots.

Electron-Hole Correlations Govern Auger Recombination in Nanostructures

The fast nonradiative decay of multiexcitonic states via Auger recombination is a fundamental process affecting a variety of applications based on semiconductor nanostructures. From a theoretical perspective, the description of Auger recombination in confined semiconductor nanostructures is a challenging task due to the large number of valence electrons and exponentially growing number of excited excitonic and biexcitonic states that are coupled by the Coulomb interaction. These challenges have restricted the treatment of Auger recombination to simple, noninteracting electron-hole models. Herein we present a novel approach for calculating Auger recombination lifetimes in confined nanostructures having thousands to tens of thousands of electrons, explicitly including electron-hole interactions. We demonstrate that the inclusion of electron-hole correlations are imperative to capture the correct scaling of the Auger recombination lifetime with the size and shape of the nanostructure. In addition, correlation effects are required to obtain quantitatively accurate lifetimes even for systems smaller than the exciton Bohr radius. Neglecting such correlations can result in lifetimes that are two orders of magnitude too long. We establish the utility of the new approach for CdSe quantum dots of varying sizes and for CdSe nanorods of varying diameters and lengths. Our new approach is the first theoretical method to postdict the experimentally known "universal volume scaling law" for quantum dots and makes novel predictions for the scaling of the Auger recombination lifetimes in nanorods.

Effect of Aspect Ratio on Multiparticle Auger Recombination in Single-Walled Carbon Nanotubes: Time Domain Atomistic Simulation

Many-particle Auger-type processes are common in nanoscale materials due to a combination of high densities of states that can support multiple excitations and substantial Coulomb coupling between charges enhanced by quantum confinement. Auger decay dynamics in (10,5) semiconductor carbon nanotubes (CNT) with different aspect ratios and particle densities are simulated in time domain using global flux surface hopping, recently developed and implemented within Kohn-Sham tight-binding density functional theory. Despite an increasing density of states, the multiparticle Auger recombination rate decreases in longer CNTs. The atomistic simulation shows that the effect is directly related to the coupling between electronic states, which decreases as the aspect ratio becomes larger. The dependence on tube length is stronger for three-exciton than two-exciton recombination and the calculated time scale ratio approaches the experimental value measured for long CNTs. Phonon-assisted transitions play a particularly important role during Auger recombination. Electron-phonon relaxation is faster than the recombination, and Auger transitions are assisted by phonons over a range of frequencies up to the G-mode. The involvement of phonons strongly enhances the probability of transitions involving asymmetric electron-hole pairs. The time-domain atomistic simulation mimics directly time-resolved optical experiments and provides a detailed, systematic analysis of the phonon-assisted Auger dynamics.

Theory of highly efficient multiexciton generation in type-II nanorods

Multiexciton generation, by which more than a single electron–hole pair is generated on optical excitation, is a promising paradigm for pushing the efficiency of solar cells beyond the Shockley–Queisser limit of 31%. Utilizing this paradigm, however, requires the onset energy of multiexciton generation to be close to twice the band gap energy and the efficiency to increase rapidly above this onset. This challenge remains unattainable even using confined nanocrystals, nanorods or nanowires. Here, we show how both goals can be achieved in a nanorod heterostructure with type-II band offsets. Using pseudopotential atomistic calculation on a model type-II semiconductor heterostructure we predict the optimal conditions for controlling multiexciton generation efficiencies at twice the band gap energy. For a finite band offset, this requires a sharp interface along with a reduction of the exciton cooling and may enable a route for breaking the Shockley–Queisser limit.

Multiple exciton generation could help limit thermalization losses in solar cells, but the efficiency of the process is still limited. Here, the authors show by atomistic calculations that type-II interfaces in nanostructures along with a change in exciton cooling rate favour multiple exciton generation.

Size, dimensionality, and strong electron correlation in nanoscience

In electronic structure theory, electron-electron repulsion is normally considered only in an average (or mean field) sense, for example, in a single Hartree-Fock determinant. This is the simple molecular orbital model, which is often a good approximation for molecules. In infinite systems, this averaging treatment leads to delocalized electronic bands, an excellent description of bulk 3D sp(3) semiconductors. However, in reality electrons try to instantaneously avoid each other; their relative motion is correlated. Strong electron-electron repulsion and correlation create new collective states and cause new femtosecond kinetic processes. This is especially true in 1D and 2D systems. The quantum size effect, a single electron property, is widely known: the band gap increases with decreasing size. This Account focuses on the experimental consequences of strong correlation. We first describe π-π* excited states in carbon nanotubes (CNTs). To obtain the spectra of individual CNTs, we developed a white-light, right-angle resonant Rayleigh scattering method. Discrete exciton transitions dominate the optical absorption spectra of both semiconducting and metallic tubes. Excitons are bound neutral excited states in which the electron and hole tightly orbit each other due to their mutual Coulomb attraction. We then describe more generally the independent roles of size and dimensionality in nanoelectronic structure, using additional examples from graphene, trans-polyacetylene chains, transition metal dichalcogenides, organic/inorganic Pb iodide perovskites, quantum dots, and pentacene van der Waals crystals. In 1D and 2D chemical systems, the electronic band structure diagram can be a poor predictor of properties if explicit correlation is not considered. One- and two-dimensional systems show quantum confinement and especially strong correlation as compared with their 3D parent systems. The Coulomb interaction is enhanced because the electrons are on the surface. One- and two-dimensional systems can exhibit essentially molecular properties even though they are infinite in size. Zero-dimensional Qdots show quantum confinement and modest electron correlation. Correlation is weak in 3D bulk semiconductors. Strongly correlated electronic states can behave as if they have fractional charge and effectively separate the spin and charge of the electron. This is apparent in the "soliton" state of polyacetylene, the fractional charge quantum Hall state of graphene, and the Luttinger electrical conductivity of metallic CNTs.

Breaking the theoretical scaling limit for predicting quasiparticle energies: the stochastic GW approach

We develop a formalism to calculate the quasiparticle energy within the GW many-body perturbation correction to the density functional theory. The occupied and virtual orbitals of the Kohn-Sham Hamiltonian are replaced by stochastic orbitals used to evaluate the Green function G, the polarization potential W, and, thereby, the GW self-energy. The stochastic GW (sGW) formalism relies on novel theoretical concepts such as stochastic time-dependent Hartree propagation, stochastic matrix compression, and spatial or temporal stochastic decoupling techniques. Beyond the theoretical interest, the formalism enables linear scaling GW calculations breaking the theoretical scaling limit for GW as well as circumventing the need for energy cutoff approximations. We illustrate the method for silicon nanocrystals of varying sizes with N_{e}>3000 electrons.

Multiexciton Generation in Seeded Nanorods

The stochastic formulation of multiexciton generation (MEG) rates is extended to provide access to MEG efficiencies in nanostructures containing thousands of atoms. The formalism is applied to a series of CdSe/CdS seeded nanorod heterostructures with different core and shell dimensions. At energies above 3Eg (where Eg is the band gap), the MEG yield increases with decreasing core size, as expected for spherical nanocrystals. Surprisingly, this behavior is reversed for energies below this value, and is explained by the dependence of the density of states near the valence band edge, which increases with the core diameter. Our predictions indicate that the onset of MEG can be shifted to lower energies by manipulating the density of states in complex nanostructure geometries.

A Guided Stochastic Energy-Domain Formulation of the Second Order Møller-Plesset Perturbation Theory

We develop an alternative formulation in the energy-domain to calculate the second order Møller-Plesset (MP2) perturbation energies. The approach is based on repeatedly choosing four random energies using a nonseparable guiding function, filtering four random orbitals at these energies, and averaging the resulting Coulomb matrix elements to obtain a statistical estimate of the MP2 correlation energy. In contrast to our time-domain formulation, the present approach is useful for both quantum chemistry and real-space/plane wave basis sets. The scaling of the MP2 calculation is roughly linear with system size, providing a useful tool to study dispersion energies in large systems. This is demonstrated on a structure of 64 fullerenes within the SZ basis as well as on silicon nanocrystals using real-space grids.

Self-averaging stochastic Kohn-Sham density-functional theory

We formulate the Kohn-Sham density functional theory (KS-DFT) as a statistical theory in which the electron density is determined from an average of correlated stochastic densities in a trace formula. The key idea is that it is sufficient to converge the total energy per electron to within a predefined statistical error in order to obtain reliable estimates of the electronic band structure, the forces on nuclei, the density and its moments, etc. The fluctuations in the total energy per electron are guaranteed to decay to zero as the system size increases. This facilitates "self-averaging" which leads to the first ever report of sublinear scaling KS-DFT electronic structure. The approach sidesteps calculation of the density matrix and thus, is insensitive to its evasive sparseness, as demonstrated here for silicon nanocrystals. The formalism is not only appealing in terms of its promise to far push the limits of application of KS-DFT, but also represents a cognitive change in the way we think of electronic structure calculations as this stochastic theory seamlessly converges to the thermodynamic limit.

Size and composition dependent multiple exciton generation efficiency in PbS, PbSe, and PbS(x)Se(1-x) alloyed quantum dots

Using ultrafast transient absorption and time-resolved photoluminescence spectroscopies, we studied multiple exciton generation (MEG) in quantum dots (QDs) consisting of either PbSe, PbS, or a PbSxSe1-x alloy for various QD diameters with corresponding bandgaps (Eg) ranging from 0.6 to 1 eV. For each QD sample, we determine the MEG efficiency, ηMEG, defined in terms of the electron-hole pair creation energy (εeh) such that ηMEG = Eg/εeh. In previous reports, we found that ηMEG is about two times greater in PbSe QDs compared to bulk PbSe, however, little could be said about the QD-size dependence of MEG. In this study, we find for both PbS and PbSxSe1-x alloyed QDs that ηMEG decreases lineally with increasing QD diameter within the strong confinement regime. When the QD radius is normalized by a material-dependent characteristic radius, defined as the radius at which the electron-hole Coulomb and confinement energies are equivalent, PbSe, PbS, and PbSxSe1-x exhibit similar MEG behaviors. Our results suggest that MEG increases with quantum confinement, and we discuss the interplay between a size-dependent MEG rate versus hot exciton cooling.

Carrier Multiplication in Quantum Dots within the Framework of Two Competing Energy Relaxation Mechanisms

The realization of high-yield, low-threshold carrier multiplication (CM) in semiconductor quantum dots (QDs) is a promising step toward third-generation photovoltaics (PV). Recent studies of QD solar cells have shown that CM can indeed produce greater-than-unity quantum efficiencies in photon-to-charge-carrier conversion, establishing the relevance of this process to practical PV technologies. While being appreciable, the reported CM yields are still not high enough for a significant increase in the power conversion efficiency over traditional bulk materials. At present, the design of nanomaterials with improved CM is hindered by a poor understanding of the mechanism underlying this process. Here, we present a possible solution to this problem by introducing a model that treats CM as a competition between impact-ionization-like scattering and non-CM energy losses. Importantly, it allows for evaluation of expected CM yields from fairly straightforward measurements of Auger recombination (inverse of CM) and near-band-edge carrier cooling. The validation of this model via a comparative CM study of PbTe, PbSe, and PbS QDs suggests that it indeed represents a predictive capability, which might help in the development of nanomaterials with improved CM performance.

Third generation photovoltaics based on multiple exciton generation in quantum confined semiconductors

Improving the primary photoconversion process in a photovoltaiccell by utilizing the excess energy that is otherwise lost as heat can lead to an increase in the overall power conversion efficiency (PCE). Semiconductor nanocrystals (NCs) with at least one dimension small enough to produce quantum confinement effects provide new ways of controlling energy flow not achievable in thin film or bulk semiconductors. Researchers have developed various strategies to incorporate these novel structures into suitable solar conversion systems. Some of these methods could increase the PCE past the Shockley-Queisser (SQ) limit of ∼33%, making them viable "third generation photovoltaic" (TGPV) cell architectures. Surpassing the SQ limit for single junction solar cells presents both a scientific and a technological challenge, and the use of semiconductor NCs to enhance the primary photoconversion process offers a promising potential solution. The NCs are synthesized via solution phase chemical reactions producing stable colloidal solutions, where the reaction conditions can be modified to produce a variety of shapes, compositions, and structures. The confinement of the semiconductor NC in one dimension produces quantum films, wells, or discs. Two-dimensional confinement leads to quantum wires or rods (QRs), and quantum dots (QDs) are three-dimensionally confined NCs. The process of multiple exciton generation (MEG) converts a high-energy photon into multiple electron-hole pairs. Although many studies have demonstrated that MEG is enhanced in QDs compared with bulk semiconductors, these studies have either used ultrafast spectroscopy to measure the photon-to-exciton quantum yields (QYs) or theoretical calculations. Implementing MEG in a working solar cell has been an ongoing challenge. In this Account, we discuss the status of MEG research and strategies towards implementing MEG in working solar cells. Recently we showed an external quantum efficiency for photocurrent of greater than 100% (reaching 114%) at ∼4Eg in a PbSe QD solar cell. The internal quantum efficiency reached 130%. These results compare favorably with ultrafast transient spectroscopic measurements. Thus, we have shown that one of the tenets of the SQ limit, that photons only produce one electron-hole pair at the electrodes of a solar cell, can be overcome. Further challenges include increasing the MEG efficiency and improving the QD device structure and operation.

Expeditious Stochastic Calculation of Random-Phase Approximation Energies for Thousands of Electrons in Three Dimensions

A fast method is developed for calculating the random phase approximation (RPA) correlation energy for density functional theory. The correlation energy is given by a trace over a projected RPA response matrix, and the trace is taken by a stochastic approach using random perturbation vectors. For a fixed statistical error in the total energy per electron, the method scales, at most, quadratically with the system size; however, in practice, due to self-averaging, it requires less statistical sampling as the system grows, and the performance is close to linear scaling. We demonstrate the method by calculating the RPA correlation energy for cadmium selenide and silicon nanocrystals with over 1500 electrons. We find that the RPA correlation energies per electron are largely independent of the nanocrystal size. In addition, we show that a correlated sampling technique enables calculation of the energy difference between two slightly distorted configurations with scaling and a statistical error similar to that of the total energy per electron.

Multiexciton Generation in IV-VI Nanocrystals: The Role of Carrier Effective Mass, Band Mixing, and Phonon Emission

We study the role of the effective mass, band mixing, and phonon emission on multiexciton generation in IV-VI nanocrystals. A four-band k · p effective mass model, which allows for an independent variation of these parameters, is adopted to describe the electronic structure of the nanocrystals. Multiexciton generation efficiencies are calculated using a Green's function formalism, providing results that are numerically similar to impact excitation. We find that multiexciton generation efficiencies are maximized when the effective mass of the electron and hole are small and similar. Contact with recent experimental results for multiexciton generation in PbS and PbSe is made.