We develop a generalized framework based on a Green's function formalism to calculate the efficiency of multiexciton gen-eration 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.
We study the role of the effective mass, band mixing and phonon emission on multiexciton generation in IV-VI nanocrystals. A 4-band k dot 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 …
The use of ultrafast gating techniques allows us to resolve both spectrally and temporally the emission from short-lived neutral and negatively charged biexcitons in ultrasmall (sub-10 nm) CdSe nanocrystals (nanocrystal quantum dots). Because of forced overlap of electronic wave functions and reduced dielectric screening, these states are characterized by giant interaction energies of tens (neutral biexcitons) to hundreds (charged biexcitons) of meV. Both types of biexcitons show extremely short lifetimes (from sub-100 picoseconds to sub-picosecond time scales) that rapidly shorten with decrea…
Electron-phonon and exciton-phonon interactions in nanoclusters are formulated and computed under the framework of GW-BSE (Bethe-Salpeter equation) approach. The phonon effect is modeled with the two-particle representation for the first time. The nonradiative relaxation rates of electrons and excitons are calculated. It is uncovered that both single-phonon relaxation and multiple-phonon relaxation are significant in nanocrystals, and correspond to two types of physical processes that have totally different spectral lineshapes. Furthermore, the multiple-phonon relaxation always occurs and its …
We study how shape affects multiexciton generation (MEG) rates in a semiconducting nanocrystal by considering CdSe nanorods with varying diameters and aspect ratios. The calculations employ an atomistic semiempirical pseudopotential model combined with an efficacious stochastic approach applied to systems containing up to 20,000 atoms. The effect of nanorod diameter and aspect ratio on multiexciton generation rates is analyzed in terms of the scaling of the density of trion states and the scaling of the Coulomb couplings. Both show distinct scaling from spherical nanocrystals leading to a surp…
The rates of elementary photophysical processes in nanocrystals, such as carrier cooling, multiexciton generation, Auger recombination etc., are determined by monitoring the transient occupation of the lowest exciton band. The underlying assumption is that hot carriers relax rapidly to their lowest quantum level. Using femtosecond transient absorption spectroscopy in CdSe/CdS nanodots we challenge this assumption. Results show, that in nanodots containing a preexisting cold exciton "spectator", \emph{only half of the photoexcited electrons}relax directly to the band-edge and the complementary …
We experimentally investigate the non-resonant feeding of photons into the optical mode of a zero dimensional nanocavity by quantum dot multiexciton transitions. Power dependent photoluminescence measurements reveal a super-linear power dependence of the mode emission, indicating that the emission stems from multiexcitons. By monitoring the temporal evolution of the photoluminescence spectrum, we observe a clear anticorrelation of the mode and single exciton emission; the mode emission quenches as the population in the system reduces towards the single exciton level whilst the intensity of the…
We propose a novel mechanism for photogeneration of multiexcitons by single photons (carrier multiplication) in semiconductor nanocrystals. In this mechanism, the Coulomb interaction between two valence-band electrons involving their transfer to the conduction band creates a virtual biexciton from vacuum that is then converted into a real biexciton by photon absorption on an intraband optical transition. This mechanism is inactive in bulk semiconductors as momentum conservation suppresses intraband absorption. However, it becomes highly efficient in zero-dimensional nanocrystals and can provid…
Semiconductor nanocrystals are being used as hosts to trap and manipulate single spins. Spins in nanocrystals can have different properties than their bulk counterparts, owing both to quantum confinement and surface effects. We will show that spins can be generated in nanocrystals by the insertion of impurities, either magnetic or not, or by magnetic polarons present at dangling bonds at their surfaces. This chapter reports theoretical contributions to this field, where the Density Functional Theory is used to simulate these functionalized nanocrystals.
The potential of semiconductors assembled from nanocrystals (NC semiconductors) has been demonstrated for a broad array of electronic and optoelectronic devices, including transistors, light emitting diodes, solar cells, photodetectors, thermoelectrics, and phase charge memory cells. Despite the commercial success of nanocrystals as optical absorbers and emitters, applications involving charge transport through NC semiconductors have eluded exploitation due to the inability for predictive control of their electronic properties. Here, we perform large-scale, ab-initio simulations to understand …
To fully deploy the potential of semiconductor nanocrystal films as low-cost electronic materials, a better understanding of the amount of dopants required to make their conductivity metallic is needed. In bulk semiconductors, the critical concentration of electrons at the metal-insulator transition is described by the Mott criterion. Here, we theoretically derive the critical concentration $n_c$ for films of heavily doped nanocrystals devoid of ligands at their surface and in direct contact with each other. In the accompanying experiments, we investigate the conduction mechanism in films of p…
Heavily-doped semiconductor nanocrystals characterized by a tunable plasmonic band have been gaining increasing attention recently. Herein, we introduce this type of materials focusing on their structural and photo physical properties. Beside their continuous-wave plasmonic response, depicted both theoretically and experimentally, we also review recent results on their transient, ultrafast response. This was successfully interpreted by adapting models of the ultrafast response of gold nanoparticles.
We have explored the influence of different matrices on the emission line shape of individual homogeneously coated CdSe/CdS/ZnS nanocrystals. The results obtained corroborate previous observations of a correlation between blinking events and spectral diffusion but in addition we have found that the extent of spectral diffusion is almost independent of the dielectric environment of the NC. Additionally, we report the observation of a correlation between the line width and emission energy which is not expected to occur in the spherical - symmetric NCs employed in this work. The implications of t…
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 a discrete spectrum at low excitation energies, similar to atoms and molecules, that converges 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…
In a recent paper [E. P. Pokatilov et al, PRB 65, 075316 (2002) the Raman selection rules in spherical nanocrystals with a degenerate valence band are analyzed. Some precisions are given here.
Some basic radiative and non-radiative processes taking place in semiconductor nanocrystals are discussed, and rates of these processes are calculated. In particular, in the present review we explore both intra-crystallite processes, such as the photon emission, Auger recombination, phonon-assisted exciton relaxation, capture on surface defects, multi-exciton generation, and inter-crystallite processes realized through the exciton migration in ensembles of nanocrystals. Exciton transitions, creation, and annihilation accompanied by the photon and (or) phonon emission or absorption are consider…
The dependences of the fundamental transition on the semiconductor quantum dot size obtained experimentally at various temperatures using different measuring methods are analyzed and compared. The possibility to extrapolate the results for the case of arbitrary temperature is discussed.
We present a theory of highly excited interacting carriers confined in a semiconductor nanostructure, incorporating Auger coupling between excited states with different number of excitations. The Coulomb matrix elements connecting exciton, bi-exciton and tri-exciton complexes are derived and an intuitive picture of breaking neutral multi-exction complexes into positively and negatively charged multi-exciton complexes is given. The general approach is illustrated by analyzing the coupling of biexciton and exciton in CdSe spherical nanocrystals. The electron and hole states are computed using at…
In this review we discuss several fundamental processes taking place in semiconductor nanocrystals (quantum dots, QDs) when their electron subsystem interacts with electromagnetic (EM) radiation. The physical phenomena of light emission and EM energy transfer from a QD exciton to other electronic systems such as neighbouring nanocrystals and polarisable 3D (semi-infinite dielectric or metal) and 2D (graphene) materials are considered. The cases of direct (II-VI) and indirect (silicon) band gap semiconductors are compared. We also cover the relevant non-radiative mechanisms such as the Auger pr…
We calculate the rate of non-radiative, Forster-type energy transfer (ET) from an excited epitaxial quantum well (QW) to a proximal monolayer of semiconductor nanocrystal quantum dots (QDs). Different electron-hole configurations in the QW are considered as a function of temperature and excited electron-hole density. A comparison of the theoretically determined ET rate and QW radiative recombination rate shows that, depending on the specific conditions, the ET rate is comparable to or even greater than the radiative recombination rate. Such efficient Forster ET is promising for the implementat…
Engineering semiconductors to enhance carrier multiplication (CM) could lead to increased photovoltaic cell performance and a significant widening of the materials range suitable for future solar technologies. Semiconductor nanocrystals (NCs) have been proposed as a favourable structure for CM enhancement, and recent measurements by transient absorption have shown evidence for highly efficient CM in lead chalcogenide and CdSe NCs. We report here an assessment of CM yields in CdSe and CdTe NCs by a quantitative analysis of biexciton and exciton signatures in transient photoluminescence decays. …
We review the existing and present the new results of $\bf kp$ calculations of the electron, hole, and exciton effective $g$-factors in semiconductor nanocrystals of different shape and symmetry. We propose a simple yet accurate method for calculation of electron $g$-factor size dependence in bare nanocrystals within the eight-band Kane model. Using the spherical approximation for Luttinger Hamiltonian we find the dependence of hole $g$-factor on light to heavy hole effective mass ratio in semiconductor nanostructures with spherical, axial, and cubically symmetric shape. We show that the non-e…
The adiabatic theory of resonance one-phonon Raman scattering in semiconductor nanocrystals is revised and extended with perturbative non-adiabatic corrections, given by the Albrecht's B term. This theory is confronted with the time-dependent perturbation approach, pointing at their differences and similarities. It is shown that both theories are equivalent in the limit of weak electron-phonon coupling and non-degenerate or uncoupled resonant states. Evaluations of the A and B terms for the confined LO phonon in CdSe and CdS nanocrystals are reported. These evaluations show that the B term can…
In our recent paper [Phys. Rev. B 65, 075316 (2002)], the selection rules for the Raman scattering in spherical nanocrystals are analyzed. Here we show, that a physically expedient choice of the coordinate system, related to the polarization vector of the incoming or scattered light, simplifies and clarifies the analysis.
Laser cooling of matter through anti-Stokes photoluminescence, where the emitted frequency of light exceeds that of the impinging laser by virtue of absorption of thermal vibrational energy, has been successfully realized in condensed media, and in particular with rare earth doped systems achieving sub-100K solid state optical refrigeration. Studies suggest that laser cooling in semiconductors has the potential of achieving temperatures down to ~10K and that its direct integration can usher unique high-performance nanostructured semiconductor devices. While laser cooling of nanostructured II-V…
We describe the process of Foerster transfer between semiconductor nanoparticles in the presence of a metal subsystem (metal nanocrystals). In the presence of metal nanocrystals, the Foerster process can become faster and more long-range. The enhancement of Foerster transfer occurs due to the effect of plasmon-assisted amplification of electric fields inside the nanoscale assembly. Simultaneously, metal nanocrystals lead to an increase of energy losses during the Foerster transfer process. We derive convenient equations for the energy transfer rates, photoluminescence intensities, and energy d…
The paraphrasis of Feynman's statement is used to highlight the inherent relevance of surfaces -- and interfaces -- at the nanoscale. Here, the marked impact of the surface chemistry on the light absorption by colloidal inorganic semiconductor nanocrystals is discussed and demonstrated as general. Chemical species at the surface (ligands) of colloidal metal chalcogenide quantum dots (QDs) are shown to induce broadband absorption enhancement and band gap reduction. A comprehensive library of chalcogenol(ate) ligands is exploited to infer the role of surface chemistry on the QD optical absorptio…
Zero-dimensional nanocrystals, as obtained by chemical synthesis, offer a broad range of applications, as their spectrum and thus their excitation gap can be tailored by variation of their size. Additionally, nanocrystals of the type ABC can be realized by alloying of two pure compound semiconductor materials AC and BC, which allows for a continuous tuning of their absorption and emission spectrum with the concentration x. We use the single-particle energies and wave functions calculated from a multiband sp^3 empirical tight-binding model in combination with the configuration interaction schem…
Using previously developed exciton scattering model accounting for the interband, i.e., exciton-biexciton, Coulomb interactions in semiconductor nanocrystals (NCs), we derive a closed set of equations for 2D double-quantum coherence signal. The signal depends on the Liouville space pathways which include both the interband scattering processes and the inter- and intraband optical transitions. These processes correspond to the formation of different cross-peaks in the 2D spectra. We further report on our numerical calculations of the 2D signal using reduced level scheme parameterized for PbSe N…
Multi-phonon Raman scattering in semiconductor nanocrystals is treated taking into account both adiabatic and non-adiabatic phonon-assisted optical transitions. Because phonons of various symmetries are involved in scattering processes, there is a considerable enhancement of intensities of multi-phonon peaks in nanocrystal Raman spectra. Cases of strong and weak band mixing are considered in detail. In the first case, fundamental scattering takes place via internal electron-hole states and is participated by s- and d-phonons, while in the second case, when the intensity of the one-phonon Raman…
In this letter, we propose a new model to determine the recombination oscillator strength of trions and biexcitons for bound and unbound cases in the effective mass approximation. The validity of our model has been confirmed by the radiative lifetime of the trion and biexciton in a spherical quantum dot. The results show that the model works sufficient accuracy in comparison with results of more complex methods such as quantum monte carlo techniques and atomistic calculations.
Nonradiative processes limit optoelectronic functionality of nanocrystals and curb their device performance. Nevertheless, the dynamic structural origins of nonradiative relaxations in nanocrystals are not understood. Here, femtosecond electron diffraction measurements corroborated by atomistic simulations uncover transient lattice deformations accompanying radiationless electronic processes in semiconductor nanocrystals. Investigation of the excitation energy dependence shows that hot carriers created by a photon energy considerably larger than the bandgap induce structural distortions at nan…
The influence of the tip-substrate bias induced electric field in a scanning-tunneling-spectroscopy experiment on charged InAs nanocrystals is studied. Calculating the ground and first excited many-particle state for five electrons occupying the quantum dot reveals a Stark-induced reordering of states by increasing the electric field strength. It is shown that this reordering of states is accompanied by a symmetry change of the local density of states (LDOS), which in principal is observable in a wave-function mapping experiment. Since in the usually performed experiments the electric field ca…
The combined effect of finite potential barriers and dielectric mismatch between dot and matrix on excitonic properties of semiconductor quantum dots has been studied. To avoid the unphysical divergence in the self-polarization energy which arises for the simplest and profusely adopted step-like model of the dielectric interface, we proposed a realistic (finite size) smooth profile for the dielectric interface. We have found that the excitonic binding energy can be either higher than the corresponding one to complete confinement by infinite barriers or essentially zero for a wide range of dot …
Time-resolved Faraday rotation is used to study both transverse and longitudinal spin relaxation in chemically-synthesized CdSe nanocrystals (NCs) 22-80 Angstroms in diameter. The precession of optically-injected spins in a transverse magnetic field occurs at distinct frequencies whose assignment to electron and exciton spins is developed through systematic studies of the size-dependence and theoretical calculations. It is shown that the transverse spin lifetime is limited by inhomogeneous dephasing to a degree that cannot be accounted for by the NC size distribution alone. Longitudinal spin r…
In semiconductors, quantum confinement can greatly enhance the interaction between band carriers (electrons and holes) and dopant atoms. One manifestation of this enhancement is the increased stability of exciton magnetic polarons in magnetically-doped nanostructures. In the limit of very strong 0D confinement that is realized in colloidal semiconductor nanocrystals, a single exciton can exert an effective exchange field $B_{\rm{ex}}$ on the embedded magnetic dopants that exceeds several tesla. Here we use the very sensitive method of resonant photoluminescence (PL) to directly measure the pre…
The resistivity of a dense crystalline array of semiconductor nanocrystals (NCs) depends in a sensitive way on the level of doping as well as on the NC size and spacing. The choice of these parameters determines whether electron conduction through the array will be characterized by activated nearest-neighbor hopping or variable-range hopping (VRH). Thus far, no general theory exists to explain how these different behaviors arise at different doping levels and for different types of NCs. In this paper we examine a simple theoretical model of an array of doped semiconductor NCs that can explain …
Carrier spins in semiconductor nanocrystals are promising candidates for quantum information processing. Using a combination of time-resolved Faraday rotation and photoluminescence spectroscopies, we demonstrate optical spin polarization and coherent spin precession in colloidal CsPbBr3 nanocrystals that persists up to room temperature. By suppressing the influence of inhomogeneous hyperfine fields with a small applied magnetic field, we demonstrate inhomogeneous hole transverse spin-dephasing times (T2*) that approach the nanocrystal photoluminescence lifetime, such that nearly all emitted ph…
We study the dependence of the quantum yield of photoluminescence of a dense, periodic array of semiconductor nanocrystals (NCs) on the level of doping and NC size. Electrons introduced to NCs via doping quench photoluminescence by the Auger process, so that practically only NCs without electrons contribute to the photoluminescence. Computer simulation and analytical theory are used to find a fraction of such empty NCs as a function of the average number of donors per NC and NC size. For an array of small spherical NCs, the quantization gap between 1S and 1P levels leads to transfer of electro…
Degenerately doped semiconductor nanocrystals (NCs) exhibit a localized surface plasmon resonance (LSPR) in the infrared range of the electromagnetic spectrum. Unlike metals, semiconductor NCs offer tunable LSPR characteristics enabled by doping, or via electrochemical or photochemical charging. Tuning plasmonic properties through carrier density modulation suggests potential applications in smart optoelectronics, catalysis, and sensing. Here, we elucidate fundamental aspects of LSPR modulation through dynamic carrier density tuning in Sn-doped Indium Oxide NCs. Monodisperse Sn-doped Indium Ox…
Spin-selective tunneling of holes in SiGe nanocrystals contacted by normal-metal leads is reported. The spin selectivity arises from an interplay of the orbital effect of the magnetic field with the strong spin-orbit interaction present in the valence band of the semiconductor. We demonstrate both experimentally and theoretically that spin-selective tunneling in semiconductor nanostructures can be achieved without the use of ferromagnetic contacts. The reported effect, which relies on mixing the light and heavy holes, should be observable in a broad class of quantum-dot systems formed in semic…
The epitaxial growth of germanium on silicon leads to the self-assembly of SiGe nanocrystals via a process that allows the size, composition and position of the nanocrystals to be controlled. This level of control, combined with an inherent compatibility with silicon technology, could prove useful in nanoelectronic applications. Here we report the confinement of holes in quantum-dot devices made by directly contacting individual SiGe nanocrystals with aluminium electrodes, and the production of hybrid superconductorsemiconductor devices, such as resonant supercurrent transistors, when the dot …
The effect of carrier multiplication (CM) in semiconductor nanocrystals is systematically treated by employing an exciton scattering approach. Using projection operators, we reduce the Coulomb coupled multi-exciton dynamics to scattering dynamics in the space spanning both single- and bi-exciton states. We derive a closed set of equations determining the scattering matrix elements. This allows us to interpret CM dynamics as a series of odd-order interband scattering events. Using the time-dependent density matrix formalism, we provide a rigorous description of the CM dynamics induced by a fini…
We study the surface conductivity of a field-effect transistor (FET) made of periodic array of spherical semiconductor nanocrystals (NCs). We show that electrons introduced to NCs by the gate voltage occupy one or two layers of the array. Computer simulations and analytical theory are used to study the array screening and corresponding evolution of electron concentrations of the first and second layers with growing gate voltage. When first layer NCs have two electrons per NC the quantization energy gap between its 1S and 1P levels induces occupation of 1S levels of second layer NCs. Only at a …
Theoretical description of electron spin orientation and detection by short optical pulses is proposed for the ensembles of the singly charged semiconductor nanocrystals. The complex structure of the valence band in spherical nanocrystals is taken into account. We demonstrate that the direction of electron spin injected by the pump pulse depends on both the pump pulse helicity and the pump pulse power. It is shown that the train of the optical pulses can lead to the complete orientation of the resident electron spin. The microscopic theory of the spin Faraday, Kerr and ellipticity effects is d…
We investigate theoretically the effects of interaction between an optical dipole (semiconductor quantum dot or molecule) and metal nanoparticles. The calculated absorption spectra of hybrid structures demonstrate strong effects of interference coming from the exciton-plasmon coupling. In particular, the absorption spectra acquire characteristic asymmetric lineshapes and strong anti-resonances. We present here an exact solution of the problem beyond the dipole approximation and find that the multipole treatment of the interaction is crucial for the understanding of strongly-interacting exciton…
We derive the matrix elements of Coulomb interaction between states with different number of electrons and holes in a semiconductor nanocrystal within the 8-band $\mathbf{k}\cdot\mathbf{p}$ theory. These matrix elements are responsible for multiple exciton generation which may contribute to the enhancement of the efficiency of solar cells. Our calculations are performed within the multi band envelope function formalism based on the states resulting from diagonalization of the 8-band $\mathbf{k}\cdot\mathbf{p}$ Hamiltonian. We study in detail and compare two contributions to the inter-band Coul…
We investigate theoretically the effect of injection of plasmonic carriers from an optically-excited metal nanocrystal to a semiconductor contact or to attached molecules. The distributions of optically-excited hot carriers are dramatically different in metal nanocrystals with large and small sizes. In large nanocrystals, most carriers have very small energies and the hot carrier distribution resembles the case of a plasmon wave in bulk. In nanocrystals smaller than 20nm, the carrier distribution extends to larger energies and occupies the whole region E_{F}<E<omega. The physical reason for th…
We report magneto-photoluminescence studies of strongly quantum-confined "0-D" diluted magnetic semiconductors (DMS), realized in Mn$^{2+}$-doped ZnSe/CdSe core/shell colloidal nanocrystals. In marked contrast to their 3-D (bulk), 2-D (quantum well), 1-D (quantum wire), and 0-D (self-assembled quantum dot) DMS counterparts, the ubiquitous yellow emission band from internal \emph{d-d} ($^4T_1 \rightarrow ^6A_1$) transitions of the Mn$^{2+}$ ions in these nanocrystals is \emph{not} suppressed in applied magnetic fields and \emph{does} become circularly polarized. This polarization tracks the Mn$…
The population dynamics of single- and bi-exciton states in semiconductor nanocrystals is modeled numerically in the presence of Coulomb coupling between single- and two-exciton states and a dissipation channel in order to study the transient bi-exciton population that occurs in an optically excited semiconductor nanocrystal. The results show that the system evolution strongly changes if the dissipation is included. In a certain range of parameters, the growth of the exciton number (MEG process) is fast (on picosecond time scale) and the following decay (Auger process) is much slower (hundreds…
We present theoretical and experimental investigations of the magnetism of paramagnetic semiconductor CdSe:Mn nanocrystals and propose an efficient approach to the exposure and analysis of the underlying anti-ferromagnetic interactions between magnetic ions therein. A key advance made here is the build-up of an analysis method with the exploitation of group theory technique that allows us to distinguish the anti-ferromagnetic interactions between aggregative Mn2+ ions from the overall pronounced paramagnetism of magnetic ion doped semiconductor nanocrystals. By using the method, we clearly rev…
The ability to produce nanowires through vapor- and solution-based processes has propelled nanowire material systems toward a wide range of technological applications. Conventional, vapor-based nanowire syntheses have enabled precise control over nanowire composition and phase. However, vapor-based nanowire growth employs batch processes with specialized pressure management systems designed to operate at high temperatures, limiting throughput. More recently developed solution-based nanowire growth processes have improved scalability but can require even more extensive pressure and temperature …
Optical techniques for the quantum control of the dynamics of multiexciton states in a semiconductor quantum dot are explored in theory. Composite bichromatic phase-locked pulses are shown to reduce the time of elementary quantum operations on excitons and biexcitons by an order of magnitude or more. Analytic and numerical methods of designing the pulse sequences are investigated. Fidelity of the operation is used to gauge its quality. A modified Quantum Fourier Transform algorithm is constructed with only Rabi rotations and is shown to reduce the number of operations. Application of the desig…
The optical extinction spectra arising from localized surface plasmon resonance in doped semiconductor nanocrystals (NCs) have intensities and lineshapes determined by free charge carrier concentrations and the various mechanisms for damping the oscillation of those free carriers. However, these intrinsic properties are convoluted by heterogeneous broadening when measuring spectra of ensembles. We reveal that the traditional Drude approximation is not equipped to fit spectra from a heterogeneous ensemble of doped semiconductor NCs and produces fit results that violate Mie scattering theory. Th…
