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-equidistant Zeeman splitting of the four-fold degenerate hole state may take place in cube and spheroidal nanocrystals. We present a comparison of the calculated hole $g$-factors in nanostructures based on II-VI and III-V semiconductors for different sets of the Luttinger parameters and analyze the main effects contributing to the $g$-factor renormalization with the respect to the bulk value. We discuss different approaches to the definition of the hole and exciton $g$-factors which should be taken into account during the analysis of the experimental data and compare our results of $g$-factor calculations with the experimental data for semiconductor spherical nanocrystals and thin nanoplatelets available in the literature.
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 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…
The interaction between exciton and acoustic phonons via deformation potential in PbS nanocrystals is calculated in the ${\bf k} \cdot {\bf p}$ model. The size-dependence of the Huang-Rhys factors for the fundamental breathing mode and its overtones as well as for spheroidal vibrations with the total angular momentum $j=2$ are evaluated.
Recently, epitaxially connected at facets semiconductor nanocrystals (NCs) have been introduced to fascilitate the electron transport between nanocrystals. To fully deploy their potential a better understanding of the exciton transfer between connected NCs is needed. We go beyond the two well-known transfer mechanisms suggested by Förster and Dexter and propose a third mechanism of exciton tandem tunneling. The tandem tunnelling occurs through the intermediate state in which electron and hole are in different NCs. The corresponding rate for exciton hops is larger than the Dexter rate and for S…
We demonstrate the optical orientation and alignment of excitons in a two-dimensional layer of CsPbI$_3$ perovskite nanocrystals prepared by colloidal synthesis and measure the anisotropic exchange splitting of exciton levels in the nanocrystals. From the experimental data at low temperature (2K), we obtain the average value of anisotropic splitting of bright exciton states of the order of 120μeV. Our calculations demonstrate that there is a significant contribution to the splitting due to the nanocrystal shape anisotropy for all inorganic perovskite nanocrystrals.
Silicon nanocrystals are studied by time-resolved fluorescence spectroscopy. After laser excitation the bright and dark exciton ground state levels are populated at random, but subsequently the decay curves reveal a thermalization between these levels. The characteristic thermalization time is found to be approximately 100 ns for temperatures below 100 K and surprisingly increases for higher temperatures. The decay curves are analyzed using a simple two-state model for the bright and dark exciton ground states.
We study the exciton dynamics in an optically excited nanocrystal quantum dot. Multiple exciton formation is more efficient in nanocrystal quantum dots compared to bulk semiconductors due to enhanced Coulomb interactions and the absence of conservation of momentum. The formation of multiple excitons is dependent on different excitation parameters and the dissipation. We study this process within a Lindblad quantum rate equation using the full many-particle states. We optically excite the system by creating a single high energy exciton $E_{SX}$ in resonance to a double exciton $E_{DX}$. With Co…
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…
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…
We study the coherent mixing between two-particle (single exciton) and four-particle (biexciton) states of a semiconductor nanocrystal resulting from the coulomb coupling between states with different numbers of electron-hole pairs. Using a simple model of the nanocrystal wave functions and an envelope function approach, we estimate the efficiency of the multiple exciton generation (MEG) process resulting from such coherent admixture mechanism, including all the relevant states in a very broad energy interval. We show that in a typical ensemble of nanocrystals with 3~nm average radius, the ons…
We present a systematic experimental study along with theoretical modeling of the energy transfer in an ensemble of closely-packed CdTe colloidal nanocrystals identified as the Förster resonant energy transfer (FRET). We prove that at low temperature of 4.2 K, mainly the ground dark exciton states in the initially excited small-size (donor) nanocrystals participate in the dipole-dipole FRET leading to additional excitation of the large-size (acceptor) nanocrystals. The FRET becomes possible due to the weak admixture of the bright exciton states to the dark states. The admixture takes place eve…
The recombination dynamics and spin polarization of excitons in CdSe nanocrystals synthesized in a glass matrix are investigated using polarized photoluminescence in high magnetic fields up to 30 Tesla. The dynamics are accelerated by increasing temperature and magnetic field, confirming the dark exciton nature of the low-temperature photoluminescence (PL). The circularly polarized PL in magnetic fields reveals several unusual appearances: (i) a spectral dependence of the polarization degree, (ii) its low saturation value, and (iii) a stronger intensity of the Zeeman component which is higher …
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 …
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 …
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 …
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.
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 propose a method for calculating Coulomb matrix elements between exciton and biexciton states in semiconductor nanocrystals based on the envelope function formalism. We show that such a calculation requires proper treatment of the Bloch parts of the carrier wave functions which, in the leading order, leads to spin selection rules identical to those holding for optical interband transitions. Compared to the usual (intraband) Coulomb couplings, the resulting matrix elements are additionally scaled by the ratio of the lattice constant to the nanocrystal radius. As a result, the Coulomb couplin…
The photoluminescence spectra of spherical CdTe nanocrystals with zincblende structure are studied by size-selective spectroscopic techniques. We observe a resonant Stokes shift of 15 meV when the excitation laser energy is tuned to the red side of the absorption band at 2.236 eV. The experimental data are analyzed within a symmetry-based tight-binding theory of the exciton spectrum, which is first shown to account for the size dependence of the fundamental gap reported previously in the literature. The theoretical Stokes shift presented as a function of the gap shows a good agreement with the…
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 study the band-edge exciton fine structure and in particular its bright-dark splitting in colloidal semiconductor nanocrystals by four different optical methods based on fluorescence line narrowing and time-resolved measurements at various temperatures down to 2 K. We demonstrate that all these methods provide consistent splitting values and discuss their advances and limitations. Colloidal CdSe nanoplatelets with thicknesses of 3, 4 and 5 monolayers are chosen for experimental demonstrations. The bright-dark splitting of excitons varies from 3.2 to 6.0 meV and is inversely proportional to …
Large surface to volume ratios of semiconductor nanocrystals cause susceptibility to charge trapping, which can modify luminescence yields and induce single-particle blinking. Optical spectroscopies cannot differentiate between bulk and surface traps in contrast to spin-resonance techniques, which in principle avail chemical information on such trap sites. Magnetic resonance detection via spin-controlled photoluminescence enables the direct observation of interactions between emissive excitons and trapped charges. This approach allows the discrimination of two functionally different trap state…
In semiconductors, exciton or charge carrier diffusivity is typically described as an inherent material property. Here, we show that the transport of excitons (i.e., bound electron-hole pairs) in CsPbBr3 perovskite nanocrystals (NCs) depends markedly on how recently those NCs were occupied by a previous exciton. Using fluence- and repetition-rate-dependent transient photoluminescence microscopy, we visualize the effect of excitation frequency on exciton transport in CsPbBr3 NC solids. Surprisingly, we observe a striking dependence of the apparent exciton diffusivity on excitation laser power t…
The exciton spin dynamics and polarization properties of the related emission are investigated in colloidal CdSe/CdS dot-in-rod (DiR) and spherical core/shell nanocrystal (NC) ensembles by magneto-optical photoluminescence (PL) spectroscopy in magnetic fields up to 15 T. It is shown that the degree of circular polarization (DCP) of the exciton emission induced by the magnetic field is affected by the NC geometry as well as the exciton fine structure and can provide information on nanorod orientation. A theory to describe the circular and linear polarization properties of the NC emission in mag…
We have shown theoretically that efficient multiple exciton generation (MEG) by a single photon can be observed in small nanocrystals (NCs). Our quantum simulations that include hundreds of thousands of exciton and multi-exciton states demonstrate that the complex time-dependent dynamics of these states in a closed electronic system yields a saturated MEG effect on a picosecond timescale. Including phonon relaxation confirms that efficient MEG requires the exciton--biexciton coupling time to be faster than exciton relaxation time.
The recombination and spin dynamics of excitons in colloidal CdTe nanocrystals (NCs) are studied by time-resolved photoluminescence in high magnetic fields up to 15 T and at cryogenic temperatures. The recombination decay shows a nonexponential temporal behavior, with the longest component corresponding to the dark excitons having 260 ns decay time at zero magnetic field and 4.2 K temperature. This long component shortens to 150 ns at 15 T due to the magnetic-field-induced mixing of the bright and dark exciton states. The spin dynamics, assessed through the evolution of the magnetic-field-indu…
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…
Theory of electronic and optical properties of exciton and bi-exciton complexes confined in CdSe spherical nanocrystals is presented. The electron and hole states are computed using atomistic $sp^3d^5s^*$ tight binding Hamiltonian including an effective crystal field splitting, spin-orbit interactions, and model surface passivation. The optically excited states are expanded in electron-hole configurations and the many-body spectrum is computed in the configuration-interaction approach. Results demonstrate that the low-energy electron spectrum is organized in shells ($s$, $p$, ...), whilst the …
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.
Hyperfine versus exchange interaction in the spin dynamics of spatially indirect excitons in CsPbI$_{3}$ perovskite nanocrystals
Rzhanov Institute of Semiconductor Physics, Siberian Branch of the Russian Academy of Sciences（俄罗斯科学院西伯利亚分院热瓦诺夫半导体物理研究所）；Technische Universität Dortmund（多特蒙德工业大学）；Ioffe Institute, Russian Academy of Sciences（俄罗斯科学院伊俄费研究所）；Spin Optics Laboratory, St. Petersburg State University（圣彼得堡大学旋光实验室）；ITMO University（ITMO大学）；St. Petersburg State Institute of Technology（圣彼得堡国立技术学院）
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…
Unique properties of lead chalcogenides have enabled multiple exciton generation (MEG) in their nanocrystals that can be beneficial in enhancing the efficiency of the third generation solar cells. Although the intrinsic electric field plays an imperative role in a solar cell, its effect on the multiple exciton generation (MEG) has been overlooked, so far. Using EOM-CCSD as a many-body approach, we show that any electric field can affect the absorptivity spectra of the lead chalcogenide nanocrystals (Pb4Te4, Pb4Se4, and Pb4S4). The same electric field, however, has insignificant effects on the …
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…
Lead-halide perovskite nanocrystals (PNCs) exhibit unique optoelectronic properties, many of which originate from a purported bright-triplet exciton fine-structure. A major impediment to measuring this fine-structure is inhomogeneous spectral broadening, which has limited most experimental studies to single-nanocrystal spectroscopies. It is shown here that the linearly-polarized single-particle selection rules in PNCs are preserved in nonlinear spectroscopies of randomly-oriented ensembles. Simulations incorporating rotational-averaging demonstrate that techniques such as transient absorption …
The optical spectra of CdSe nanocrystals up to 55 A in diameter are analyzed in a wide range of energies from the fine structure of the low-energy excitations to the so-called high-energy transitions. We apply a symmetry-based method in two steps. First we take the tight-binding (TB) parameters from the bulk sp^{3}s^{*} TB model, extended to include the spin-orbit interaction. The full single-particle spectra are obtained from an exact diagonalization by using a group-theoretical treatment. The electron-hole interaction is next introduced: Both the Coulomb (direct) and exchange terms are consi…
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…
A photo-excited II-VI semiconductor nanocrystal doped with a few Mn spins is considered. The effects of spin-exciton interactions and the resulting multi-spin correlations on the photoluminescence are calculated by numerical diagonalization of the Hamiltonian, including exchange interaction between electrons, holes and Mn spins, as well as spin-orbit interaction. The results provide a unified description of recent experiments of photoluminesnce of dots with one and many Mn atoms as well as optically induced ferromagnetism in semimagnetic nanocrystals.
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…
We measure the photoluminescence (PL) lifetime, $τ$, of excitons in colloidal PbSe nanocrystals (NCs) at low temperatures to 270~mK and in high magnetic fields to 15~T. For all NCs (1.3-2.3~nm radii), $τ$ increases sharply below 10~K but saturates by 500~mK. In contrast to the usual picture of well-separated ``bright" and ``dark" exciton states (found, e.g., in CdSe NCs), these dynamics fit remarkably well to a system having two exciton states with comparable - but small - oscillator strengths that are separated by only 300-900 $μ$eV. Importantly, magnetic fields reduce $τ$ below 10~K, consist…
Coherent manipulation of the exciton wave function in a single semiconductor colloidal nanocrystal (NC) has been actively pursued in the past decades without any success, mainly due to the bothersome existences of the spectral diffusion and the photoluminescence (PL) blinking effects. Such optical deficiencies can be naturally avoided in the newly-developed colloidal NCs of perovskite CsPbI3, leading to the PL spectrum with a stable intensity at the single-particle level. Meanwhile, from the first-order photon correlation measurement, a PL linewidth smaller than 20 μeV is estimated for the emi…
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…
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.
We discuss a variety of many-body approaches, within effective-mass and $\mathbf{k}\cdot\mathbf{p}$ envelope-function formalisms, for calculating correlated single excitons in semiconductor nanocrystals (NCs) to all orders in the electron-hole Coulomb interaction. These approaches are applied to NCs of the lead-halide perovskite CsPbBr$_{3}$, which typically present excitons in intermediate confinement with physical observables often strongly renormalized by correlation (e.g., radiative decay rate enhanced by a factor of about 7 relative to a mean-field approach, for a NC of edge length 11 nm)…
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…
Although both epitaxial quantum dots (QDs) and colloidal nanocrystals (NCs) are quantum-confined semiconductor nanostructures, so far they have demonstrated dramatically-different exciton fine structure splittings (FSSs) at the cryogenic temperature. The single-QD photoluminescence (PL) is dominated by the bright-exciton FSS, while it is the energy separation between bright and dark excitons that is often referred to as the FSS in a single NC. Here we show that, in single perovskite CsPbI3 NCs synthesized from a colloidal approach, a bright-exciton FSS as large as hundreds of μeV can be resolv…
We present a theoretical description of excitons and positively and negatively charged trions in "giant" CdSe/CdS core-shell nanocrystals (NCs). The developed theory provides the parameters describing the fine structure of excitons in CdSe/CdS core/thick shell NCs as a function of the CdSe/CdS conduction band offset and the CdSe core radius. We have also developed a general theory describing the fine structure of positively charged trions created in semiconductor NCs with a degenerate valence band. The calculations take into account the complex structure of the CdSe valence band and inter-part…
