Exciton-photon interactions in semiconductor nanocrystals: {radiative transitions, non-radiative processes,} and environment effects
EExciton-photon interactions in semiconductor nanocrystals: radiative transitions,non-radiative processes, and environment effects
Vladimir A. Burdov
N. I. Lobachevsky State University of Nizhny Novgorod,23 Gagarin avenue, 603950 Nizhny Novgorod, Russian Federation
Mikhail I. Vasilevskiy
Centro de F´ısica, Universidade do Minho, Campus de Gualtar, Braga 4710-057, Portugal andInternational Iberian Nanotechnology Laboratory, Braga 4715-330, Portugal (Dated: January 12, 2021)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 otherelectronic systems such as neighbouring nanocrystals and polarisable 3D (semi-infinite dielectric ormetal) 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 mechanismssuch as the Auger process, electron capture on dangling bonds and interaction with phonons. Theemphasis is on explaining the underlying physics and illustrating it with calculated and experimentalresults in a comprehensive, tutorial manner.
I. INTRODUCTION
Semiconductor nanostructures form a basis for modernelectronic technologies. Nowadays, they are employed ina wide range of applications in the fields of optoelectron-ics, photonics, photovoltaics, biosensing, photocatalysis,etc. By virtue of the quantum confinement effect, theirelectronic spectra and, consequently, the related opticalproperties depend on the nanostructure size. This featureis most pronounced for zero-dimensional (0D) objects,nanocrystals (NCs), where the conduction band electronmotion is fully localized in all directions. Consequently,at least a part of the NC energy spectrum is completelydiscrete. In the limiting case of strong quantum con-finement effect, when the NC size is much less than theeffective exciton Bohr radius, the former strongly influ-ences the electron and hole energies. It leads to the size-dependent energies of photons emitted or absorbed by thenanocrystals and allows one to control their optical spec-tra. For this reason, they sometimes are called “artificialatoms” . A more scientific term to distinguish semicon-ductor NCs with strongly size-dependent electronic andoptical properties is Quantum Dot (QD), which will beused in this article. After the pioneering works by Efros and Efros andBrus explaining the origin of size-dependent opticalspectra of nanocrystals, the interest to these objects grewexponentially through the 1980s and 1990s. A varietyof methods has been used for preparation of crystalliteswith sizes not greater than several nanometers, such asion implantation , chemical vapour deposition from agas phase, magnetron sputtering , colloidal synthesis ,electron beam epitaxy , etc. One of these methods,chemical growth in colloidal solutions, allows for obtain-ing good quality NCs of a broad range of semiconduc-tors with an almost spherical shape and a rather narrowsize distribution, characterized by rather narrow emis- sion bands, which is, as a rule, extremely desirable forapplications. This method was first suggested for II–VI NCs by Murray et al. and became widely employedlater (see, e.g., chapters by Kudera et al., Reiss, Gaponikand Rogach in the book in for review). The colloidalchemistry methods work fairly well for II–VI and III–Vmaterials such as CdSe, CdS, CdTe, InAs, InP, etc. Theauthors of and several other groups have been ableto produce good QDs of some IV–VI materials andsilicon QDs (see, e.g., reviews in ).Colloidal QDs of II–VI materials (especially CdSe andCdTe) are probably the most studied. During the lastdecade, a considerable research activity has been focusedon the synthesis and investigation of the optical proper-ties of ZnSe QDs. These nanocrystals exhibit large blueshift of the photoluminescence and high enough (up to50 percent) quantum yield . Similarly to CdSe NCs, theZnSe QDs can be synthesized with a sufficiently narrowsize distribution, which is possible to control by temper-ature . Due to ZnSe’s lower toxicity, compared to manyother II–VI semiconductors, these QDs turn out to beattractive objects also for biosensing .Further improvement of the light emission properties ofsuch QDs was achieved by fabrication of core–shell struc-tures, successful for several pairs of II–VI and IV–VI materials, where the shell made of a wider band gap ma-terial provides a better protection of the quantized elec-tronic states in the QD core . For ZnSe/ZnS core–shellQDs it was revealed, in particular, that thermodynamical(slow) growth of the ZnS-shell on the colloidal ZnSe-coreleads to the quantum yield increase because of decreasingamount of the traps at the core–shell interface .The maturity of this technology of synthesis of semi-conductor nanocrystals is witnessed by the incorpo-ration of colloidal QDs into real-world products such ascolour displays and light-emitting diodes for lighting ,both already commercialized by large companies such a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n as Samsung and OSRAM , respectively (see Ref. for a recent review covering both colloidal and epitax-ial QDs). More on the scientific research side, a broadvariety of nanostructures can be prepared using colloidalNCs as building blocks, in particular, multilayer struc-tures of QDs of different average size, deposited on differ-ent substrates . Combining these structures with othermaterials, such as organic dielectrics , epitaxial quan-tum well (QW) heterostructures , metallic nanoparti-cles , patterned metallic surface or graphene can result in new interesting effects and applications,which physics is related to the coupling of QD excitonswith elementary excitations (such as surface plasmons orQW excitons) in the surrounding materials. One interest-ing possibility is the excitation of colloidal QDs by pump-ing energy through a nearby epitaxial QW, as demon-strated in . Another research topic with QDs, popu-lar in the past two decades is the modification of theiremission properties caused by the strong light–matter in-teraction, which has been achieved for a variety of com-posite structures that include point emitters embeddedin Fabry–Perot microcavities, micropillars and photoniccrystals as recently reviewed in .In this article, we overview several aspects of theexciton–photon interactions in nanocrystal QDs with fo-cus on the role of non-radiative processes and environ-ment, which affect the light emission. The QD photolu-minescnce is a result of competition between the radia-tive and various non-radiative processes, such as captureon dangling bonds, multi-phonon intra-band relaxation,Auger recombination, etc. In the past decade, a grow-ing interest has been paid to the multi-exciton dynamicsin nanocrystals because of their high potential for photo-voltaic applications. Such processes as carrier multiplica-tion (or multi-exciton generation) in nanocrystals, as wellas the Auger recombination (which is just a fast reverseprocess), are widely discussed in the literature andwe will consider these effects below. Special attentionis traditionally paid to nanostructured silicon be-cause of its widest use in microelectronics, high purity,natural abundance, low cost, and non-toxicity; we shallalso dedicate some space to the exciton–photon interac-tions and main non-radiative processes mentioned above,in Si NCs and related nanostructures.As a rule, one deals with nanocrystal ensembles (ratherthan with isolated QDs). Therefore, non-radiative en-ergy exchange between the nanocrystals is possible viathe F¨orster-type and Dexter-type exciton migra-tion. Both mechanisms were originally proposed for flu-orophores. The second one, based on the exchange inter-action between electrons located on different sites can beneglected in the presence of allowed dipole transitions .Recently, a third, so-called exciton tandem tunnelingmechanism was proposed , which is specific of connectednanocrystals. The most universal process named F¨orsterResonant Energy Transfer (FRET) was first observed forQDs by Kagan et al. in specially designed films contain-ing two different sizes of nanocrystals acting as donors and acceptors, respectively . Later it was shown in anumber of works performed on systems composedof two different QD species that the luminescence of thesmaller dots (donors) is quenched by the large dots (ac-ceptors), whose emission in turn is enhanced. Thesestudies demonstrated a dependence of the FRET effecton the NC density and spatial arrangement. Results ofother experiments performed on multilayer SiO x /SiO structures , porous Si , three-dimensional (3D) en-sembles of Si crystallites were interpreted as a man-ifestation of exciton migration via FRET-type mecha-nisms. The effect has a potential interest for photon-ics (e.g., photoluminescence upconversion in QD ensem-bles ), sensing , lighting and energy harvesting (e.g.,by unidirectional energy transfer in size-gradient layeredQD assemblies or fractal aggregates ). We shalldiscuss it below, in particular, addressing the questionwhether the FRET rates and length scales can be tunedby the photonic environment .The influence of the environment on the exciton-related optical properties of NCs is the third topic thatwill be discussed in this overview. The most commoneffect is the energy transfer between a photoexcited QDand surrounding materials, which can be reversible ornot. It has the same physical nature as FRET. In mostcases, it is responsible for the photoluminescence (PL)quenching . However, the environment can be usedfor engineering the photonic density of states (DOS) inthe vicinity of the QDs, often referred to as the Pur-cell effect . Observed for the first time in QDs abouttwenty years ago , it has recently been discussed withrespect to radiative decay rates of embedded point emit-ters . Moreover, strong near-field effects associated withlocalized surface plasmon resonance (LSPR) in metallic(nano-)structures can enhance PL emission . More-over, one can think of energy transfer from a recombiningQD exciton to propagating surface plasmons that wouldcarry the energy over a large distance and then eventuallytransfer it to another QD (by creating an exciton), thusallowing for a long-range exciton transport between twodots, much more efficient than if it occurred directly. Be-low we consider the influence of a flat interface betweentwo media on the PL emission and FRET rates for a QDemitter located in the vicinity of the interface .The article is organized as follows. In Section II, weintroduce basic notions of radiative transitions and dis-cuss their rates (i.e., probabilities per unit time) for NCsof materials possessing direct or indirect band structure.In particular, the rates are calculated for intrinsic anddoped silicon NCs, which are either hydrogen-coated orhalogenated. Section III is devoted to non-radiative tran-sitions and considers the Auger recombination, dangling-bond traps and phonon-assisted relaxation of hot carri-ers. In Section IV, multiple exciton generation initiatedby a highly excited electron or hole is considered. Sec-tion V is dedicated to the exciton transfer processes be-tween two QDs via the FRET mechanism. In SectionVI, emission decay and FRET rates near a plane inter-face between two dielectrics or a dielectric and a metalare discussed. The last section is left for summary andconclusions. II. LIGHT EMISSION IN NANOCRYSTALSA. Spontaneous Emission Rate
The emission rate of a point dipole emitter located inan infinite dielectric medium with the permittivity ε isgiven by γ = 4 k | d | ε ¯ h , (1)where k = √ ε ω/c is the modulus of the wavevector, ω is the oscillation frequency, c is the velocity of light invacuum, ¯ h denotes the Planck constant and d is the tran-sition dipole moment matrix element. This expression,with some adaptation, describes the radiative decay rateof an exciton in a QD made of a direct band gap semicon-ductor. We shall write it as the inverse of the radiativelifetime: τ − R ≡ γ ( QD )0 = 4 e E if | p if | κ √ ε m ¯ h c . (2)Here, m and − e are the free electron mass and charge,respectively, and E if and p if denote the transition en-ergy between the initial and final states and the momen-tum matrix element, respectively. The latter is morenatural to use in the electronic band theory of crys-tals; it can be related to d appearing in Equation (1)as p = − im ω d /e . The parameter κ in Equation (2)arises due to the difference in the dielectric constants ofthe crystallite ( ε ) and its surrounding ( ε ) and takes ac-count of the so called depolarization effects . Hereafter,we consider the crystallite as a sphere with the radius R and dielectric constant ε embedded in a homogeneousmedium with the dielectric constant ε . In this case, κ isgiven as κ = 9 ε (2 ε + ε ) . (3)This parameter varies within a wide range of valuesdepending on the dielectric constants of the two materi-als. B. Nanocrystals of Direct-Band-Gap Materials
The majority of III–V and II–VI semiconductors aredirect band gap materials. In this case, in the frameworkof the envelope function approximation, p if = p cv (cid:104) F c,i | F v,f (cid:105) , (4) where p cv is the standard bulk-like momentum matrixelement of inter-band transitions, and F c,i and F v,f , incase of the photon emission, stand for the envelope func-tions of the initial state in the conduction band and thefinal state in the valence band, respectively. The scalarproduct of these envelope functions determines selectionrules for such transitions.In the simplest case, the QD is treated as an infinitelydeep spherical potential well with radius R and theconduction and valence bands are assumed simple andparabolic. Consequently, the electron and hole groundstates are described by the same envelope function andthe scalar product in (4) equals to unity. It means thatthe radiative recombination rate turns out to be almostindependent of the QD radius. This dependence man-ifests only in the transition energy, E if . In the limit-ing case of a strong quantum confinement regime, where R (cid:28) a ex ( a ex denotes the effective exciton Bohr radius),the exciton binding energy can be neglected. Then, themodel of infinitely deep spherical potential well yields E if ( R ) = E ∞ g + ¯ h π / µR , where E ∞ g is the band gapenergy of the bulk semiconductor and µ is the reducedeffective mass of the electron–hole pair.The above assumption concerning the extreme strongconfinement regime is not quite realistic; usually the QDradius can be just slightly smaller than a ex (for instance, a ex = 5.6 nm for CdSe ) and the exciton binding en-ergy can be estimated as E b ≈ . e / ( εR ). Further-more, the most common semiconductors have a degen-erate valence band with light and heavy hole sub-bands,so that the confined holes are described by two envelopefunctions, which are completely different from the elec-tron’s one . Thus, the matrix element (4) does dependon R (see, e.g., chapter by Vasilevskiy in the book ).Still, the dependence of τ − R on the QD size is ratherweak for direct-gap materials as compared to, e.g., sili-con NCs considered below. Using typical parameters forCdSe QDs, ε ≈ ε ≈ p cv /m ≈
10 eV , τ − R [s − ] ≈ × E if ( R ) . (5)Thus, τ R is of the order of a nanosecond, in accordancewith experimental observations. C. Silicon Nanocrystals
Since the 1990s, the optical properties of Si NCs havebeen widely discussed . Many methods havebeen proposed to increase their emissivity, such as dop-ing with shallow impurities , growth in different ma-trices , plasma , and colloidal solutions . As aresult, numerous theoretical predictions and experimen-tal observations of enhanced PL intensity , radia-tive recombination rates and quantum efficiencyof photon generation have been reported.It is well known that introduction of shallow impuritiesis capable of modifying electronic properties of bulk sili-con. Similarly, doping with shallow impurities influencesthe electronic structure of silicon NCs , which, inturn, affects the electron–hole radiative recombination.It was revealed that doping of Si nanocrystals with Por Li is (under certain conditions) capable of improvingtheir emittance . In particular, the performedcalculations show that doping with P or Lican essentially increase the radiative transition rates.For NCs with
R > ∼ ortight binding model . The performed calculations ofthe phonon-assisted radiative recombination rates in Sicrystallites in the SiO matrix yielded the values vary-ing from ∼ to ∼ s − as the crystallite radiusdecreases from 3 to 1 nm. Within a simple model of aninfinitely deep spherical potential well, the dependenceof the radiative decay rate on the crystallite radius is τ − R ∝ R − [122]. For the no-phonon radiative transi-tions, the rates turn out to be much lower, by one tothree orders of magnitude with increasing the size withinthe same range. In this case, τ − R ∝ R − [122], i.e., therate sharply drops as R increases. At the same time,doping with phosphorus provides a several orders of mag-nitude increase of the radiative recombination rates forthe no-phonon transitions. The rates become even 1–3orders greater than those of the phonon-assisted transi-tions. They slowly decrease with increasing the NC size,especially for high concentration of phosphorus .As calculations show, the radiative transitions includ-ing both no-phonon and phonon-assisted ones becomefaster in small Si crystallites. However, for crystalliteswith R < ∼ T within the Casida’s ver-sion of the time-dependent density functional the-ory (TDDFT) are depicted for H-passivated P-doped(Si H P, Si H P, Si H P, Si H P, Si H P,Si H P), Li-doped (Si H Li, Si H Li, Si H Li,Si H Li, Si H Li, Si H Li), and undoped(Si H , Si H , Si H , Si H , Si H ,Si H ) Si nanocrystals. Here, NCs are considered invacuum, therefore, ε = 1, while for Si ε = 12. In thiscase, the rate values become ∼
16 times lower comparedto the case of Si NCs embedded in silicon dioxide due tothe factor κ √ ε .As seen in Figure 1, generally, doped crystallitesdemonstrate 1–2 orders of magnitude faster transitionsthan intrinsic ones. For Si crystallites doped with phos-phorus, increasing rates of the radiative transitions havebeen explained by the efficient mixing of the electronicstates in the Γ and X points of the Brillouin zone (Γ-Xmixing) caused by the short-range field of the phospho-rus ion . Meanwhile, the radiative decay in the Li-doped crystallites becomes faster due to the high densityof states above the inter-band energy gap (in the range¯ hω − E g < ∼ k B T , where k B is the Boltzmann constant) . RoomTemperature6 7 8 9 10100010 Radius H Þ L R a t e H (cid:144) s L Figure 1: Calculated rates of radiative transitions for undoped(small violet dots), P-doped (big blue dots) and Li-doped (bigred dots) Si nanocrystals.
Surface chemistry is another method capable of mod-ifying electronic structure of nanocrystals . Thismethod is especially efficient for small crystallites, wherethe role of surface rises. It has been demonstrated ex-perimentally that chemical synthesis allows tofabricate Si nanocrystals with various ligand coatings,which emit light in visible range with typical nanosec-ond radiative lifetimes . Calculations based onthe tight-binding model performed for CH -capped Sinanocrystals confirmed the conclusion on the increase ofthe radiative recombination rate due to the formation ofdirect-like electronic structure of the crystallite.Let us now consider some other kind of the surface re-construction of the crystallites, the halogenation. Table1 presents calculated rates of radiative transitions be-tween the highest occupied molecular orbital (HOMO)and the lowest unoccupied molecular orbital (LUMO) incompletely halogenated with Cl-, Br- and H-coated sili-con crystallites. Table 1.
Calculated rates (s − ) of the radiative HOMO-LUMO transitions in H-, Cl-, and Br-passivated siliconnanocrystals. X = H X = Cl X = BrSi X . × X . × . × . × Si X . × . × X . × . × . × Si X . × . × X . × . × . × Si X . × . × . × Si X . × . × . The separation turns out to be morepronounced in the Br-passivated crystallites, in which Figure 2: Auger recombination in nanocrystals via eeh and ehh processes. Arrows indicate Auger transitions between theinitial and final states. the HOMO states are strongly concentrated near the Bratoms. Presumably, the bromine atoms produce stronglattice distortions, and the stress fields have an addi-tional localizing effect on the electron density. An ef-fect of the rate reduction was also found for completelyfluorine-passivated Si nanocrystals . III. NONRADIATIVE PROCESSES
As is well known, the emissivity of any light-emittingentity depends not only on the radiative transitions’ rate,but also on the speed of nonradiative decay processes. Inthe case of NCs, these include Auger recombination, cap-ture of photo-electrons on dangling bonds and cooling ofhot carriers. Below, we briefly discuss all these processes.
A. Auger Recombination
In order for the Auger process to occur, at least onenegative or positive trion (exciton + one electron or exci-ton + one hole) has to be initially excited in a nanocrys-tal, as shown in Figure 2. The trion turns into a high-energy single-particle excitation as indicated by the ar-rows in the figure.The rate of an Auger-type transition can be written as τ − A = 2Γ¯ h (cid:88) f | U if | Γ + ( E f − E g ) , (6)where U if is the matrix element of the two-particleCoulomb interaction operator, the two-electron wavefunctions are approximated by products of the single-electron ones and the δ -function expressing the energyconservation in the Fermi’s Golden Rule has been broad-ened with the half-width Γ = 10 meV. All the possiblefinal states of the Auger electron with energy E f (shownin Figure 2) are summed up. The initial states were cho-sen so that electrons and holes are in the LUMO andHOMO states, respectively.Usually, the Auger recombination is an efficient pro-cess that can “shunt” other photon-assisted processessuch as light emission or multiple exciton generation. For instance, in Si crystallites, the Auger rates can be suf-ficiently high, as the measurements and calcu-lations show. Therefore, suppression of theAuger process is an important practical problem.It turns out that coating with halogens can slow downthe Auger processes in Si NCs. Below we present the cal-culated Auger rates and demonstrate their reduction dueto the surface halogenation of Si crystallites. The calcu-lations take into account both the long-range ( U ( r , r ))and the short-range ( U ( r , r )) parts of the carrier–carrier interaction. The first one is a macroscopic elec-trostatic field modified by the nanocrystal boundary, U ( r , r ) = e ε | r − r | + e ( ε − εR × ∞ (cid:88) l =0 ( l + 1) P l (cos θ ) lε + l + 1 r l r l R l , (7)while the second one describes at a microscopic level thepoint charges interaction at short distances: U ( r ) = e r [ Ae − αr + (1 − A − /ε ) e − βr ] . (8)Here, r = | r − r | , A = 1 . α = 0 . /a B and β = 5 /a B , where a B stands for the Bohr radius, andwe set (cid:15) = 1 as before. The calculated values of τ − A arelisted in Table 2 for both eeh and ehh trion annihilationin the Si n X m crystallites considered here. Table 2.
Calculated Auger rates ( × s − ) for eeh and ehh processes in halogenated silicon nanocrystals.X=H X=Cl X=Br X=H X=Cl X=Br( eeh ) ( eeh ) ( eeh ) ( ehh ) ( ehh ) ( ehh )Si X
390 3 4 40 163 10Si X
78 0.3 7 5 27 19Si X
120 4 2 1780 59 11Si X X
59 84 2 474 20 65Si X
120 21 100 47 9 19Si X
270 2 2 161 90 0.8Si X τ − A for the Br-,Cl- and H-passivated nanocrystals. In the case of the eeh Auger recombination one has obtained: (cid:104) lg( τ − A (H) ) (cid:105) =11 . (cid:104) lg( τ − A (Cl) ) (cid:105) = 10 .
55 and (cid:104) lg( τ − A (Br) ) (cid:105) = 10 . ehh Auger recombination the ratesalso decrease but not so strongly: (cid:104) lg( τ − A (H) ) (cid:105) = 11 . (cid:104) lg( τ − A (Cl) ) (cid:105) = 11 .
44 and (cid:104) lg( τ − A (Br) ) (cid:105) = 11 .
20. It shouldbe noted also that strong reduction of the Auger recom-bination rates was theoretically shown by Califano forGaSb NCs whose surfaces were passivated with atoms ofelectronegative elements.
AugerRadiativeCapture1.0 1.5 2.0 2.5 3.0100010 R H nm L R a t e H (cid:144) s L Figure 3: Calculated rates of carrier trapping on danglingbonds compared to the rates of radiative phonon-assistedtransitions and Auger recombination in Si nanocrystals.
B. Capture on Dangling Bonds
There is one more relatively fast process inhibiting ef-ficient light emission from NCs, which is photo-carriertrapping on surface defects called P b -centres or danglingbonds. The P b -centres produce rather deep energy lev-els within the nanocrystal band gap, which act as elec-tron traps. The capture on neutral dangling bonds is amulti-phonon process, and its rate strongly depends ontemperature. The earlier performed calculations for Sicrystallites yielded τ − C shown in Figure 3 as function ofthe nanocrystal radius in comparison with the rates ofthe Auger recombination and radiative transitions.It can be seen from Figure 3 that for the NC radiusgreater than ∼ R close to 1.5 nm, the trapping on danglingbonds turns out to be even faster than the Auger re-combination. Obviously, for nanocrystals whose radiiare greater than 1.5 nm, the nonradiative trapping onthe surface defects suppresses all the other competing re-laxation processes inside the NC if the surface danglingbonds are not passivated. C. Phonon-Assisted Relaxation of Hot Carriers
It is well known that highly excited (“hot”) electrons,holes or excitons may relax to the lower states in order tominimize the system energy. In bulk semiconductors andQWS, where the electronic energy spectra are continu-ous, such a relaxation is accompanied by the emission ofphonons caused by the interaction between electrons andlattice vibrations. This interaction also plays a majorrole in QDs and it is generally accepted that the mecha-nisms are essentially the same in nanocrystals and bulkmaterials.The most universal mechanism of coupling betweenthe electrons (or holes) and long-wavelength acousticphonons is through the volume deformation potential.The bottom of a non-degenerate band (e.g., conduction band in materials with zinc-blend structure) is shiftedproportionally to the (local) relative variation of the vol-ume , ˆ H e − AP = a c (cid:88) ν ( ∇ · ˆ u ν ( r )) , where a c is the bulk deformation potential constant, andˆ u ν denotes the atomic displacement operator correspond-ing to an acoustic phonon mode ν (it is expressed in termsof phonon creation and annihilation operators in usualway). For optical phonons, there are two mechanisms:(i) a universal optical deformation potential (ODP) cou-pling and (ii) the Fr¨ohlich one characteristic of polar ma-terials. In bulk semiconductors with cubic structure, forlong-wavelength optical phonons, the ODP coupling van-ishes by symmetry for any non-degenerate band but it isnon-zero for holes near the top of the valence band .Its expression can be found in the book (Chapter 8) orin . The Fr¨ohlich interaction Hamiltonian for electronsis simply given byˆ H e − OP = − e (cid:88) ν ˆ ϕ ν ( r ) , where ˆ ϕ ν is the electrostatic potential operator corre-sponding to an optical phonon mode ν . With the knownelectron and hole wavefunctions and phonon displace-ments obtained from lattice dynamics equations, the cal-culation of the coupling matrix elements is straightfor-ward. Even though the Fr¨ohlich interaction usually dom-inates in QDs made of polar semiconductors, the ODPmechanism is important for some of them (e.g., InP) asrevealed by modelling of QD resonant Raman spectra .The so-called rigid-ion model is more suitable when de-scribing the electron–phonon interaction in multi-valleysemiconductors, such as Si and Ge , where the pro-cesses of inter-valley scattering are often important.Within this model, the electron–phonon interaction is de-scribed by the following operator (which describes bothacoustic and optical phonons),ˆ H e − P = − (cid:88) ν ; n ,s ˆ u ν ( R n s ) · ∇ V at ( r − R n s ) . Here, V at ( r − R n s ) stands for the electron potentialenergy in the field of the s -th atom situated in the n -th primitive cell of the lattice and R n s is the position-vector of this atom in equilibrium. In practice, atomicpseudopotentials are used.The discreteness of the energy spectrum in quantumdots implies that the rate of the single-phonon emissionshould strongly depend on the level spacing ; in partic-ular, electronic transitions causing the phonon emissionshould be impossible if the spacing is larger than thephonon energy. This simple idea gave rise to the “phononbottleneck” concept, which predicts the inefficiency ofhot carrier relaxation by emission of phonons in QDs .However, this prediction relies on the assumption thatthe phonon emission is irreversible, with a probabilitydescribed by the perturbation theory (Fermi’s GoldenRule), which may not be reliable in QDs. The electron–phonon interaction can be enhanced in nanocrystals be-cause of the spatial confinement of both electrons andphonons and, therefore, multiple scattering processes areimportant. It means that the electron–phonon interac-tion in QDs, in principle, must be treated in a non-perturbative and non-adiabatic way, leading to the en-ergy spectra described by polaronic quasi-particle excita-tions . In other words, virtual transitions betweendifferent electronic levels, assisted by phonons and not re-quiring energy conservation may be important enough toguarantee a significant modification of the exciton energyspectrum and dynamics . Thus, does the phonon bot-tleneck in QDs really exist? Experimental results stillare not completely conclusive. On the one hand, anefficient relaxation of optically created excitons was re-ported in a number of works studying self-assembled QDs(SAQDs) , with both PL emission rise time and photo-induced intraband absorption decay time below or of the order of 10 ps. Studies performed onnanocrystal QDs , where exciton energy level spac-ings are larger than in SAQDs, also revealed ultrafastintraband relaxation with a characteristic time in the pi-coseconds’ domain, which is characteristic of CdSe andCdTe QDs in general . On the other hand, there arepublished experimental results indicating that the relax-ation of optically created excitons can be slow (in thenanoseconds’ range), both in self-assembled andnanocrystal QDs.It is important to realize that the polaron concept byitself cannot explain intraband relaxation of carriers inQDs as polaron is a stationary state of an electron (orexciton) coupled to optical phonons. Some additional in-teractions should therefore be responsible for the polaronrelaxation . Several possible mechanisms of hot carrierrelaxation in QDs have been proposed:(i) The polaron has a rather short lifetime becauseof the anharmonic effects that lead to a fast decay ofnanocrystal’s optical phonons forming the polaron.(ii) Acoustic phonons can provide the possibility oftransitions between different (exciton-) polaronstates formed mostly by the interaction with opticalphonons; the polaron spectrum is discrete but rel-atively dense owing to the non-adiabaticity of thisinteraction . If the acoustic phonon spectrum iscontinuous, this additional interaction would drivethe polaron dynamics towards equilibrium.(iii) In the strong confinement regime where the elec-tron’s kinetic energy is larger than the electron–hole interaction, the electron (eventually dressedby phonons and forming the polaron) can relaxby an Auger-type mechanism. The excess energyis first transferred from the electron to the holethrough their Coulomb interaction and the subse-quent hole’s cooling occurs via emission of acoustic phonons . It can be feasible because the holelevel spacings are relatively small in QDs and matchthe continuum of acoustic phonon energies.There is no final conclusion concerning the relevanceof each of the three mechanisms to the breaking of thephonon bottleneck in QDs; however, the first one seemsto be the most popular in the literature, even thoughcalculations yielded too low relaxation rates for thismechanism, of the order of 1 ns − , for a typical quan-tum dot considering anharmonic (Gr¨uneisen) parame-ters characteristic of bulk materials and the Fr¨ohlich in-teraction with spherically confined optical phonons andelectrons . Moreover, the electron–phonon interactionin nanocrystals of non-polar materials (see below) maybe too weak to make the polaron effect important. Ifthis is the case, the relaxation proceeds via multi-phononprocesses (like temperature-induced interband excitationof charge carriers in bulk semiconductors). Usually,multi-phonon processes are much slower compared to thesingle-phonon transitions. Therefore, the bottleneck ef-fect does not disappear completely.Estimations of the single-phonon relaxation rates inSi nanocrystals yield typical values of the or-der of 10 –10 s − . At the same time, the multi-phonon relaxation rates calculated within the Huang–Rhys model for various transitions vary from10 s − to 10 s − for nanocrystals whose diametersdo not exceed 5 nm. Note that the rates sharply reduceas the nanocrystal diameter decreases. Similar values ofthe multi-phonon relaxation rates were obtained exper-imentally for InGaAs/GaAs SAQDs . Therefore, theprecise reason for very fast exciton dynamics character-istic of CdSe and CdTe colloidal QDs remains an openquestion, in our view. IV. MULTIPLE EXCITON GENERATION
As demonstrated above, the surface halogenationsuppresses both radiative and Auger recombination innanocrystals. These processes are reverse ones with re-spect to the carrier multiplication (or multi-exciton gen-eration): they tend to decrease the number of excitons ina system, while the process of multi-exciton generation,shown schematically in Figure 4, has an opposite trend.Initially, a high-energy photon creates a highly excitedelectron–hole pair which then reduces its energy creatingone more electron-hole pair with lower energy. As a re-sult, two (or even more) excitons can arise in the systemafter absorption of a single photon.The multi-exciton generation is a fundamental processfor photovoltaics, where light energy transforms into elec-tric current. Its occurrence in NCs has been experimen-tally confirmed . In order to be more efficient,this process should be faster than other competitive pro-cesses taking place along with exciton generation, suchas inter-band radiative recombination or Auger recombi-nation. From this point of view, slowing the latter down
Figure 4: Schematic representation of the electron- ( left )and hole-initiated ( right ) process of multi-exciton generationin a nanocrystal. Initial electron configurations are shown.Dashed arrow indicates highly excited exciton with the ex-citation energy E created by an absorbed photon. Verticalsolid arrows indicate electron transitions to the final states.After the transition, the number of excitons increases by one. caused by surface halogenation is an extremely positivefactor. It is important to understand how halogen passi-vation influences the multi-exciton generation itself.The rates of the multi-exciton generation in theSi X crystallite (X = H, Cl, and Br) can be cal-culated using the following relation, τ − G = 2Γ¯ h (cid:88) f (cid:88) i (cid:88) f |(cid:104) Ψ i i | ˆ U | Ψ f f (cid:105)| Γ + ( E i + E i − E f − E f ) , (9)where ˆ U = ˆ U + ˆ U , as before, while Ψ i i and Ψ f f are the products of the single-particle Kohn–Sham wavefunctions of the initial or final electron states participat-ing in the transition, as shown in Figure 4. Here, wepresent calculated the rates for the Si X crystallitewithin the range of excess energy 0 < ∆ E < . E = E i − E LUMO − E g for the process initiatedby highly excited electron, and ∆ E = E HOMO − E f − E g for the hole-initiated process. The calculated rates areshown in Figure 5. It is evident that the rates rise glob-ally as the excess energy increases, because of the con-siderable increase in the number of possible states par-ticipating in the transitions with increasing ∆ E , whichopens up many new channels for the realization of excitongeneration.It is important to emphasize that the bromination of aSi crystallite increases the exciton generation rates com-pared to an H-passivated crystallite, especially if the pro-cess is initiated by a highly excited hole. The rates ofexciton generation in chlorinated Si crystallite turn outto be lower than those in the hydrogenated one at smallexcess energies. Meanwhile, upon approaching ∆ E ∼ . τ − G in the Si Cl crystallite increases and tendsto the typical values observed in Si H crystallite.Accordingly, it is possible to conclude that halogen coat-ing of Si crystallites, at least, does not reduce their abilityto generate excitons, particularly when the excess ener-gies are not too small. This is in contrast with the radia-tive and Auger recombination processes, where the ratesbecame substantially lower due to the halogenation. H a L e - initiated0.0 0.1 0.2 0.3 0.4 0.5 0.610 D E H eV L R a t e H s - L H b L h - initiated0.0 0.1 0.2 0.3 0.4 0.5 0.610 D E H eV L Figure 5: Calculated rates of the exciton generation initiatedby highly excited ( a ) electron and ( b ) hole (as shown in Figure4) in Si X crystallite for X = H (small blue dots), X =Cl (medium green dots) and X = Br (big red dots). It means that the halogenation of Si NCs can increasethe efficiency of the photon-to-exciton conversion, whichis defined by an excess of the number of created exci-tons ( n ) over the number of absorbed photons ( N ) : η = n/N > , as wellas in crystallites of IV–VI, II–VI or III–V semiconduc-tors . The authors reported on the observationof multi-exciton generation in the investigated systems.On the theoretical side, the consideration of the exci-ton kinetics in the halogen-coated Si crystallites hasrevealed strong dependence of η on the quantitative rela-tionship between the rates τ − G and τ − A . According to theobtained results the decrease in the Auger rate (causedby the halogenation), and its absence in the multi-excitongeneration rate, is accompanied by a gradual increase inthe quantum efficiency of the order of a few tens of per-centage points. V. FRET IN ENSEMBLES OF NANOCRYSTALS
All the processes considered above may occur in iso-lated nanocrystals. Meanwhile, usually in experiments,as was already pointed out in the Introduction, one dealswith ensembles of NCs, where a non-radiative energy ex-change between them takes place and strongly influencesthe ensemble photoluminescence . Such an en-ergy transfer occurs by means of the electron tunnellingif the NCs are connected or through the F¨orster-typemigration of excitons if the NCs are separated in space.Below we consider the latter mechanism in some detail.The F¨orster resonant exciton transfer (FRET) takesplace mainly through the dipole–dipole interaction oftwo QDs (a donor and an acceptor). The Quantum Elec-trodynamics theory of FRET developed in reproducesthe results obtained in a simpler way by F¨orster andDexter who considered the electrostatic interaction oftwo dipoles in two adjacent crystallites , V ( r , r ) = κe (cid:15) b (cid:20) r · r − r · b )( r · b ) b (cid:21) , (10)where b is the inter-crystallite centre-to-centre vector.In one crystallite, the electron–hole pair annihilates andtransfers its energy into a neighbouring crystallite wherea new electron–hole pair is excited. Thus, virtual trans-fer of excitons between two crystallites can be realizedwithout charge transfer.In order to calculate the rate of the F¨orster excitontransition from a QD with a radius R into a neighbour-ing one with a radius R , we use, as before, the Fermi’sGolden Rule: k F = 2Γ¯ h |(cid:104) Ψ i | ˆ V | Ψ f (cid:105)| Γ + ( E g ( R ) − E g ( R )) . (11)Here, ˆ V is the dipole interaction operator identical to(10), Ψ i = ψ c ( r ) ψ v ( r ) is the wave function of the ini-tial two-particle state with the energy coinciding withthe energy gap of the first crystallite, E g ( R ), and Γ is aphenomenological damping parameter. Initially, there isone electron in the conduction band of the first crystal-lite, with the wave function ψ c ( r ), while a hole exists inthe valence band. In the second QD, the electron occu-pies a valence band state described by the wave function ψ v ( r ). In the final state, the system has the wave func-tion Ψ f = ψ v ( r ) ψ c ( r ), corresponding to the electron-hole pair transferred to the second QD.The result is essentially the same as if one consideredthe interaction between two transient point dipoles, d D and d A (located at points r and r ) assuming that theyare stationary. The square of the matrix element in (11)(the donor–acceptor coupling parameter) can be writtenas J = ν κ d D d A (cid:15) b , (12)where ν is a factor of the order of unity that takes intoaccount the relative orientation of the dipoles , ν = if these orientations are completely random.The broadening of the exciton transfer resonance, Γcan be related to the spectral overlap of the emissionand absorption spectra of the donor and acceptor, re-spectively , and the transfer rate can be expressed as k F γ ( D )0 = 3 κ c π(cid:15) b Q A (cid:90) + ∞−∞ dωω I A ( ω ) L D ( ω ) , (13)where γ ( D )0 is the spontaneous emission rate of the donor, I A ( ω ) is the absorption lineshape function of the accep-tor, L D ( ω ) is the emission lineshape function of the donor(both are normalized to unity) and Q A is the frequency-integrated absorption cross section of the acceptor QD, Q A = (cid:18) π ¯ hcE g ( R ) (cid:19) γ ( A )0 (14) with γ ( A )0 being the spontaneous emission rate of the ac-ceptor. Notice that the depolarization factor κ , Equa-tion (3), which distinguishes QD donors and acceptorsfrom, e.g., molecules, has been included explicitly inEquation (13).The FRET rate decreases rather quickly with thedonor-acceptor distance, k F = γ ( D )0 ( b b ) , where b is aparameter of the dimension of length named “F¨orster ra-dius” whose definition is clear from Equation (13).In ensembles formed of direct-band-gap II–VI or III–V semiconductor crystallites (QDs), the exciton transferhas the rates ∼ –10 · s − , as measurements and calculations show. Notice that k F de-pends on b and the above values probably correspondto somewhat different donor–acceptor distances. Exper-imental observations of the exciton transfer were carriedout with ensembles of closely packed monodisperse CdSenanocrystals as well as of two-size three-dimensional mix-tures and bilayered CdSe nanocrystal systems . Itwas shown that the inter-layer transfer in bilayered en-sembles with controlled donor-acceptor separation turnsout to be more efficient than in the 3D ensembles of themonodisperse and the two-size-mixed crystallites . Notethat the measured and computed FRET rates are of thesame order of magnitude as the radiative recombinationrates in high-density ensembles of colloidal II–VI crys-tallites. Moreover, a few measurements of the F¨orsterradius have been performed by controlling the distancebetween two different groups of QDs of different size, act-ing as donors and acceptors , or by preparing a homo-geneously blended solid-state films composed of twogroups of dots. In both works, CdSe/ZnS core–shell QDswere used and the reported results for the F¨orster ra-dius are 14–22 nm and 8–9 nm, respectively; notice thatthe former work used larger nanocrystals. Despite theuncertainty in experimental conditions and difficulty toevaluate the spectral overlap in Equation (13) for indi-vidual QDs, there is a consensus that R F typically isof the order of 10 nm for highly luminescent colloidalnanocrystals.In ensembles of silicon crystallites the exciton trans-fer turns out to be much slower than the radiative re-combination, its rates are two to three orders of magni-tude lower (smaller than ∼ · s − ). Doping ofsilicon nanocrystals with phosphorus allows to increasethe rates up to the values comparable with those of ra-diative recombination ( < ∼ s − for nanocrystal ra-dius R > ∼ . Such a process was termed electronic-to-0vibrational energy transfer (EVET) . The importanceof EVET was demonstrated by comparing the lumines-cence properties of HgTe QDs dissolved in two chemicallyidentical solvents: H O and D O . VI. QD EMITTERS NEAR A FLAT INTERFACE
Now, we shall discuss the influence of a flat interfacebetween two media on the emission and FRET rates fora QD emitter located at a certain distance from it. Ifthe characteristic distances (such as that between theemitter and the interface) are small compared to the EMwavelength, one can treat the problem in the electrostaticapproximation where one neglects both the retardationeffects and the magnetic field associated with the electricfield present in the media; then, the image dipole methodcan be used to take into account the effect of surfacepolarization induced by the dipole . A more generalapproach consists in using the dyadic Green’s functionformalism . If both dielectrics are dispersionless, thespontaneous decay and FRET are affected through thephotonic density of states (DOS) renormalization due tothe reflection of electromagnetic (EM) waves at the inter-face. However, even in this geometrically simplest situa-tion, there is a less trivial effect produced by EM wavescreated by the polarization in the second medium, totallyreflected at the interface so that only an exponentiallydecreasing field amplitude of theirs reaches the dipole.Yet, this effect can be dominating at small distances, notonly for radiative decay but also for FRET. When thesecond medium is a metal, several contributions to bothdecay and transfer processes arise that can be associ-ated with (i) propagating EM waves (so-called radiativelosses), affected by the presence of the second medium;(ii) coupling to propagating surface plasmons (SPs); and(iii) Ohmic losses (when the exciton energy is irreversiblytransferred to heat via electron scattering in the metal).Even though the subject has been studied duringseveral decades and is described in topical re-views and books , we find some significant effectsthat were overlooked in previously published discussionsand, in our opinion, are important for potential applica-tions of FRET in photonics and energy harvesting. Inparticular, we will show that (1) if a QD embedded in adielectric matrix is located near an interface with anotherdielectric (a substrate) having a lower dielectric constant,its emission is polarized parallel to the interface; (2) inthe opposite case of substrate’s dielectric constant higherthan that of the matrix, the emission and transfer ratesare strongly enhanced (without dissipation) near the in-terface; and (3) the resonant coupling between SPs prop-agating along a metal/dielectric interface and excitonsconfined in QDs located at a distance of the order ofthe light wavelength from the interface, can be used forlong-range FRET. We present these rates calculated fortypical colloidal nanocrystal QDs. A. Radiative Lifetime Near Interface
We shall consider the QD emitter as a point dipole.
The radiative decay rate in the presence of other bodiesis determined by the total field that acts on the emitter(created by itself and scattered by the bodies) and givenby γ = Im [ d (cid:63) · ( E ( r ) + E s ( r ))] / h , (15)where E is the dipole field in infinite space. Here, d (cid:63) isthe complex-conjugate of the classical dipole moment,the necessary adaptation for the quantum mechanicaltransition dipole matrix element consists in multiplyingit by 2 in the final result, d → d . The electric compo-nent of the scattered field created by an emitting dipolelocated in the origin of the reference frame can be writ-ten in terms of the Green’s dyadic , which is specific ofsystem’s geometry: E s ( r (cid:48) ) = 4 πk G ( r (cid:48) , ) · d . (16)The 3 × G ( r (cid:48) , r ) is determined by theFresnel coefficients, r ( p ) and r ( s ) , and its explicit expres-sion can be found in .From Equation (15), one obtains for a dipole perpen-dicular to the interface (i.e., “vertical”, see inset in Figure6): γ ( V ) γ = 1+ 32 Re (cid:90) ∞ s √ − s r ( p ) exp (2 ik h (cid:112) − s ) ds , (17)and for a dipole parallel to the interface (i.e., “horizon-tal”): γ ( H ) γ = 1 + 34 Re (cid:90) ∞ s √ − s (cid:104) r ( s ) − (1 − s ) r ( p ) (cid:105) × exp (2 ik h (cid:112) − s ) ds . (18)Here, γ is the decay rate in an infinite medium withdielectric constant ε , given by Equation (1). The in-tegrals in Equations (17) and (18) with respect to thenormalized in-plane wavevector, s = q/k , can be di-vided into two parts, one from 0 to 1 corresponding topropagating waves in the upper half-space and the otherfrom 1 to ∞ representing evanescent waves with imag-inary wavevector component along z axis perpendicularto the interface, k z . The latter type of waves exist intwo cases: (i) if ε < ε and both dielectric constants arepositive and (ii) if ε (cid:48) ≡ Re( ε ) <
0, i.e., when the secondmedium is a metal.In the case of two non-dispersive dielectrics, γ given byEquations (17) and (18) represents only radiative lossesof dipole’s energy, renormalized by the back action of thescattered waves. Yet, for ε < ε there are EM waves (ex-cited by the dipole) propagating in the lower half-space(with real z -component of the wavevector inside medium2, k z ) and experiencing total internal reflection at theinterface with the upper medium, possessing imaginary1 -1 0 1 2 3 4 5-10123-101234 r ad k h = 0.2 k h = 0.1no surface wave H V r ad Figure 6: Dependence of the radiative decay rate, γ , on thedielectric constant ratio ε /ε for two two different distancesto the interface between them, k h = 0 . < ε /ε <
1) is neglected; it is inde-pendent of h . The inset shows two orthogonal orientations ofthe dipole moment. The upper plot is for the vertical dipoleand the lower for the horizontal orientation. k z . The Fresnel coefficient r ( p ) has an imaginary partassociated with these waves. If the interaction of thedipole with these evanescent waves is neglected (dottedlines in Figure 6), i.e., we set the integral from 1 to ∞ inEquations (17) and (18) equal to zero, then γ becomesindependent of dipole’s distance from the interface.As can be seen from this figure, the emission decayrate is strongly enhanced, γ >> γ , at small distances( k h <
1) for both dipole orientations because of thestrong interaction with the evanescent waves. It has beenpointed out for plane interfaces by Lukosz and Kunz ,and it is similar to the coupling to the whispering gallerymodes (a kind of surface EM waves) observed for emittersplaced inside a micrometer-size dielectric sphere .As we will show in the next section, this coupling can beused for FRET enhancement.For ε > ε only propagating waves in the uppermedium (with k z real) are present and γ does not de-pend on h . As can be seen from Figure 6, the emissionrate of a dipole oriented perpendicular to the interface issuppressed ( γ → γ close to its value in an infinite dielectric, γ . As known, the emission of a spherical QD made of a semicon-ductor material with cubic crystal lattice does not haveany preferential polarization in empty space (if excitedby non-polarized light). If the dot is embedded in a ma-trix with sufficiently high ε and located near the surface,one can expect its luminescence to be strongly polarized.Note that this is not a quenching effect because there isno dissipation in the system (both ε and ε are real). B. Non-Radiative Losses to a Metal Substrate
If the substrate is lossy, e.g., a metal, several addi-tional decay channels arise. Mathematically, they arerelated to the imaginary part of the Fresnel coefficients.Direct evaluation of the integrals in Equations (17) and(18) yields the overall effect of all these channels. How-ever, their relative contributions depend on the distanceand the real and imaginary parts of ε . At very small dis-tances ( k h << , γ ( V, H ) Ol γ = 3(2 k h ) Im (cid:18) ε − ε ε + ε (cid:19) . (19)The coupling to propagating surface plasmons math-ematically is described by the pole of r ( p ) , which yieldsan additional imaginary part of the integral from 1 to ∞ in Equations (17) and (18). Indeed, the equation( r ( p ) ) − = 0 determines the dispersion relation of p -polarized SPs. The other Fresnel coefficient, r ( s ) , hasno poles, and accordingly there are no s -polarized SPsat a metal–dielectric interface. If we neglect the imagi-nary part of ε , the SP wavevector is real. The resonantcoupling occurs for q sp = ωc (cid:115) ε (cid:48) ε ε (cid:48) + ε (where ε (cid:48) ε < ε (cid:48) + ε < ∞ in Equations (17) and (18)is calculated explicitly: γ ( V ) sp γ = 3 πs sp √− u − u exp (cid:18) − k hs sp √− u (cid:19) , (20) γ ( H ) sp γ = 3 π s sp 1 √− u (1 − u ) exp (cid:18) − k hs sp √− u (cid:19) , (21)where u = ε (cid:48) /ε and s sp = q sp /k .Taking ε ( ω ) within the simple Drude model for bulkgold, we calculated γ Ol, γ sp, and the radiative decayrate (neglecting ε (cid:48)(cid:48) ). They are presented in Figure 7. Asexpected, at small distances the Ohmic losses dominate(see Figure 7, lower panel). However, as the distanceto the interface increases, for k h ∼
1, the coupling toSPs becomes the main mechanism that determines thelifetime (see Figure 7, upper panel). Although this effect2 -1 Total losses SP coupling ( =0) Radiative losses ( =0) Ohmic losses Photon energy, eV h = 11 nm h = 220 nm Figure 7: Spectral dependence of different contributions tothe emission decay rate for a vertically oriented emitter lo-cated in a dielectric with ε = 2 at two different distances(indicated on the plots) above a flat interface with gold. Notethat the contribution of the Ohmic losses (the main cause ofthe emission quenching near metal surfaces) is negligible inthe upper plot, while it is completely dominating in the lowerpanel. For the calculation of the radiative losses and couplingto surface plasmons, the damping parameter, Γ of the Drudemodel was set equal to zero. decreases with the distance exponentially, this decreasecan be rather slow, as | ε (cid:48) | /ε >> s sp ∼
1, so itis nearly constant over tens of nanometres. Of course, atstill large distances γ tot → γ .Thus, for intermediate distances of the order of a hun-dred nanometres the radiative decay is mostly due tothe coupling to SPs within a broad spectral range (ap-proximately from 2.5 to 4.5 eV). In this range, rele-vant for colloidal QDs, one can expect also plasmon-enhanced FRET. Even though the enhancement is mod-erate, the coupling to SPs is not dissipative unless we ap-proach the electrostatic SP resonance frequency, ω spr = ω p / (cid:112) ε ∞ + ε (here, ω p is the bulk plasma frquency and ε ∞ is the background dielectric constant of the metal).In other words, excitation can be transferred from theQD to SPs and back many times without dissipation. Itcan lead to strong coupling regime characteristic of plas-monic microcavities . C. Energy Transfer to a 2D Material
Graphene, an atomic-thick monolayer of carbon, isa semimetal (or gapless semiconductor) with unusualelectronic properties first demonstrated by Geim andNovoselov . Low-energy excitations in graphene aremassless, chiral, Dirac fermions. In neutral graphene, thechemical potential (hereafter called Fermi level) crossesexactly the Dirac point . Pristine graphene is trans-parent in a broad spectral range from the infrared (IR)to the ultraviolet (UV), with the residual absorptionof ≈ . TheFermi level, E F , can be shifted by up to ≈ . One can say that graphene is a transparentconductor with a tunable conductivity, both static andfrequency-dependent (for the frequencies up to the mid-IR) . It supports propagating surface plasmons whosedispersion relation can be controlled via gate voltage .Interestingly, a specific type of quantum dots can formin graphene, namely, mass profile QDs, a system withincurrent experimental reach , for which FRET can alsobe important .Numerous experiments have shown that the nanopar-ticle or dye molecule emission is quenched in a broadspectral range by a single graphene sheet . Itis understandable for the lower-energy part of the spec-trum, as graphene is a conductor, although with a rel-atively low free carrier concentrations (which can reach10 cm − ), with the plasma frequency in the THz-to-IR range; it also is supported by calculations. However,the situation is more complex than in the case of a pointemitter in the vicinity of a usual metal.In addition to the Ohmic losses and energy transferto surface plasmons, there is an additional mechanismthat may be named exciton transfer (ET) to graphene.It is similar to FRET and consists in energy transfer toelectron–hole pairs generated by inter-band transitions,symmetric with respect to the Dirac point (see Figure 8).Doped graphene absorbs EM radiation due to inter-bandtransitions only for energies above 2 E F , therefore, it alsoapplies to the ET. The main contribution of this mecha-nism scales with distance as h − . It can be demonstratedin a rather simple way that elucidates its connection toFRET.Let us assume that acceptor dipoles are distributedwithin a plane (representing the graphene sheet), thenthe ET rate can be obtained by integrating (13) overthis plane and written as follows, k ET ( h ) γ ( QD )0 = ρ A (cid:90) π dφ (cid:90) ∞ rdrb ( r + h ) , (22)where ρ A is the surface density of acceptor dipoles. Theemitter lineshape may be taken as a δ -function. As ρ A Q A I A = β ( ω ) is the absorbance of the graphene, we3 Figure 8: Schematics of the exciton energy transfer from aQD to graphene. The left part represents an excited QDwith an electron–hole pair. It can be transferred as a wholeto the graphene represented by its conduction and valencebands’ cones touching in the Dirac point. For the first-ordervertical interband transition in graphene, at low temperaturesthe transfer is possible only if the exciton energy exceeds 2 E F . obtain k ET ( h, ω ) γ ( QD )0 = 3 c (cid:15) ω h β ( ω ) . (23)This formula can be derived in a more rigorous way ,with just slightly different numerical coefficients.The h − scaling law was verified in several works andit seems to be obeyed for molecular emitters (down to h ≈ but not so much for QDs where deviationswere found for small distances . The above theory doesnot take into account the finite size of the QDs. As a firstapproximation, one can assume that the point dipole islocated at the centre of the QD. In this case, the dis-tance of the dipole from the graphene sheet can be ap-proximated as h ∗ = ( h + R QD + s ) where R QD is the ra-dius of the nanocrystal core of the QD and s is thicknessof the organic shell for a colloidal QD. The latter canbe estimated as the chain length of the capping agent, s ∼ . . Using this correction, h → h ∗ , indeed im-proves the agreement between the theory and experiment . However, one needs to bear in mind also the possi-bility of charge transfer from an excited QD to grapheneif the distance is sufficiently small for tunnelling. Indeed,the demonstration of hybrid graphene-QD phototransis-tors and solar cells where the light is absorbed inthe QDs and changes the electric current in grapheneimplies that such processes can take place in speciallydesigned structures.Calculations including both intra-band (i.e., Drudeplasmons) and inter-band transitions in graphene showthat there can be a no-quenching spectral window for anemitter over a strongly doped graphene sheet . Indeed,the graphene absorbance, β ( ω ), due to Drude plasmons decreases with the frequency, while the inter-band tran-sitions contribute to β (and, therefore, to the quenching)only for ω > E F / ¯ h . As the Fermi level in graphenecan be tuned electrically, it opens an interesting possi-bility of electrical switching of the QD emission by con-trolling the quenching rate; it has been demonstratedexperimentally in .Energy transfer (ET) from QDs to other 2D materials,namely, few-atomic-layer-thick semiconductors (transi-tion metal dichalcogenides (TMDs)) has also been in-vestigated in the recent years . These materialssupport robust excitons, which determine their opticalproperties in the visible range (along with the inter-bandtransitions) and they depend strongly on the number ofmonolayers in the material . The ET from a QD exci-ton to the continuum of states above the band gap in the2D material should work similar to the case of graphene(Equation (23)). However, it was found that the donorQD emission is quenched most strongly near monolayerMoS and then the effect decreases with the number ofmonolayers (presumably keeping the QD distance to thesemiconductor surface fixed), in contrast with the case offew-layer graphene . Meanwhile, the tendency sim-ilar to graphene (quenching increases with the number ofmonolayers) was found for the less studied TMD SnS .On the theoretical side, it has been suggested that the to-tal spontaneous emission rate of a quantum emitter canbe enhanced several orders of magnitude due to the exci-tation of surface exciton–polariton modes supported bythe 2D semiconductor . In the vicinity of an excitonictransition, the real part of the dielectric function of thesemiconductor material is negative, like in a metal, so onecan indeed expect an enhancement of the local photonicDOS and, consequently, the Purcell effect . However,it would require matching of the energies of the QD and2D excitons. D. FRET between QDs Near Interface
In the vicinity of a polarisable material, such asgraphene, the dipole–dipole interaction responsible forthe FRET between two QDs becomes renormalized be-cause of the additional interaction between the inducedcharges. In the electrostatic limit, these additional inter-actions can be modelled with image dipoles (see Figure9). This physical mechanism, of course, is not uniqueto graphene and can take place in the vicinity of plas-monic nanoparticles . The effect of plasmonic en-hancement of FRET has been detected experimentallyby observing a decrease of the emission decay rate ofdonor QDs and a corresponding increase of the lumines-cence intensity for acceptor dots when they are placedin the vicinity of gold nanoparticles . Of course, if theconcentration of the nanoparticles becomes too high, theemission is quenched for both donor and acceptor QDsbecause of the energy transfer to lossy modes in the metal(the Ohmic losses, Section VI B).4 Figure 9: Schematics of two QDs on top of a graphene sheet,interacting via their real transient dipoles and image dipolesrepresenting transient charge distributions in the graphenesheet.
Graphene is a conductor with electrically controllableconductivity and, in principle, it promises a possibilityof controlling the inter-dot FRET rate. The exciton en-ergy transfer between two QDs located near a graphene-covered interface occurs not just through the directdipole–dipole interaction but also through polarizationcharges that they induce on the interface. One canthink that the image dipoles of Figure 9 have magnitudesthat depend on the optical conductivity of graphene and,therefore, on its Fermi energy . This interaction can bewritten in the form V F RET = 1 (cid:15) d D · ˆT · d A with ˆ T = ˆ T (0) + ˆ T (1) , whereˆ T (0) ( b ) = b − (3 n ⊗ n − ˆI )is the usual dipole–dipole interaction tensor ( n = b /b )andˆ T (1) ( b ) = (cid:90) d q (2 π ) ( − i e q + e z ) ⊗ ( i e q + e z ) qA ( q ) e − qh + i q · b (24)is the part due to the image dipoles. In Equation (24), e q = q /q , e z is the unit vector along z -axis, A ( q ) = ε − ε + f ( q ) ε + ε + f ( q ) , f ( q ) = i πσ g ( ω ) ω q , and σ g ( ω ) is the optical conductivity of graphene. Thecomponents of ˆ T (1) have an oscillatory dependence uponthe inter-dot distance , which arises from the Besselfunctions appearing as the result of the angular integra-tion in Equation (24). It means that instead of the usualmonotonic b − dependence of the FRET rate one mayhave an oscillatory behaviour, controllable by the gate voltage through the graphene Fermi energy (which de-termines σ g ( ω )). Beyond the non-monotonic dependenceupon the inter-dot distance, the emission/transfer fre-quency also should affect it in a rather complex way.If this interaction is strong enough, the emitters (QDsin our case) can be coupled to create a collective radia-tive mode, a phenomenon called superradiance , whichhas been observed for epitaxial quantum dots . Asthe coupling between the emitters can be strongly en-hanced or suppressed in the vicinity of graphene, depend-ing on the distance between them, the frequency and thegraphene Fermi energy, both superradiance and subra-diance regimes can be expected . So far, we are notaware of an experimental demonstration of such effects.Losses associated with the real part of graphene’s opticalconductivity can be one possible reason for this.The influence of a polarisable surface (such asgraphene) on the irreversible FRET should be an easier-to-observe effect than the superradiance. It can beseen as a surface plasmons’ effect. Huge FRET en-hancement factors of the order of 10 (compared to vac-uum) have been predicted for two dipole emitters ona graphene sheet, based on numerical electrodynamicscalculations , reaching a maximum when the distancebetween the dipoles equals twice the graphene plasmonpropagation length L p . This prediction is at variancewith the results presented in according to which theeffect must be considerably smaller and varying on scaleof the surface plasmon wavelength, λ p << L p . VII. CONCLUDING REMARKS
To summarize, we would like to emphasize, once again,that the radiative properties and efficiency of light emis-sion in semiconductor nanostructures are determined notonly by the radiative transitions, but also by various non-radiative processes which can be even more intense thanthe radiative transitions themselves. As a result, in thesecases, the radiative recombination can be significantlysuppressed. On the other hand, even the non-radiativeprocesses can be used in a constructive way. For instance,the exciton migration is accompanied by a non-radiativeenergy transfer (FRET) that can be channelled in a de-sired direction via creating a certain “architecture” of thenanocrystals in the ensemble, which allows to concen-trate and illuminate the energy inside a given area or direct it to a certain layer of a funnel-type heterostruc-ture of QD monolayers . Exciton energy transfer andemission with upconversion because of simultaneous ab-sorption of optical phonons has been suggested for QD-assisted cooling . The multi-exciton effects make itpossible to efficiently transform the absorbed photonsinto the rising number of electron–hole pairs capable ofparticipating in the electric current. Such processes un-derlie the operating principle of photosynthetic light har-vesting systems and can be mimicked in solar cells .The exciton transport can be strongly enhanced if me-5diated by polaritons arising under strong coupling of theexcitons to light in microcavities . Combining en-tities supporting localized excitons, such as QDs, dyemolecules or J-aggregates, with microcavities in order toachieve the strong coupling regime is an important areaof research envisaging various applications , in particu-lar, in the context of controllable quantum emitters. Aplasmonic surface considered in Section VI can be seen asa near-field microcavity if the lossy channels are not thedominating ones. A detailed discussion of the strong cou-pling between surface plasmon-polaritons and quantumemitters can be found elsewhere . Acknowledgments
Funding from the Ministry of Science and Higher Ed-ucation of the Russian Federation (State Assignment No 0729-2020-0058), the European Commission within theproject ”Graphene-Driven Revolutions in ICT and Be-yond” (Ref. No. 696656), from the Portuguese Foun-dation for Science and Technology (FCT) in the frame-work of the PTDC/NAN-OPT/29265/2017 ”Towardshigh speed optical devices by exploiting the unique elec-tronic properties of engineered 2D materials” project andthe Strategic Funding UID/FIS/04650/2019 is gratefullyacknowledged. Ashoori, R.C. Electrons in artificial atoms.
Nature , , 413–419. Efros, A.L.; Efros, A.L. Interband absorption of lightin a semiconductor sphere.
Sov. Phys. Semicond. , , 772–775. Brus, L.E. A simple model for the ionization poten-tial, electron affinity, and aqueous redox potentials ofsmall semiconductor crystallites.
J. Chem. Phys. , , 5566–5571. Kanemitsu, Y. Photoluminescence spectrum and dynam-ics in oxidized silicon nanocrystals: A nanoscopic disordersystem.
Phys. Rev. B , , 13515–13520. Zhuravlev, K.S.; Gilinsky, A.M.; Kobitsky, A.Y. Mecha-nism of photoluminescence of Si nanocrystals fabricatedin a SiO matrix. Appl. Phys. Lett. , , 2962–2964. Tetelbaum, D.; Trushin, S.; Burdov, V.; Golovanov, A.;Revin, D.; Gaponova, D. The influence of phosphorusand hydrogen ion implantation on the photoluminescenceof SiO with Si nanoinclusions. Nucl. Instrum. MethodsPhys. Res. Sect. B Beam Interact. Mater. Atoms , , 123–129, Negro, L.D.; Cazzanelli, M.; Pavesi, L.; Ossicini, S.; Paci-fici, D.; Franz`o, G.; Priolo, F.; Iacona, F. Dynamics ofstimulated emission in silicon nanocrystals.
Appl. Phys.Lett. , , 4636–4638, Tsybeskov, L.; Hirschman, K.D.; Duttagupta, S.P.;Zacharias, M.; Fauchet, P.M.; McCaffrey, J.P.; Lockwood,D.J. Nanocrystalline-silicon superlattice produced by con-trolled recrystallization.
Appl. Phys. Lett. , , 43–45, Nayfeh, M.H.; Rao, S.; Barry, N.; Therrien, J.; Belomoin,G.; Smith, A.; Chaieb, S. Observation of laser oscillationin aggregates of ultrasmall silicon nanoparticles.
Appl.Phys. Lett. , , 121–123, Meldrum, A.; Hryciw, A.; MacDonald, A.N.; Blois, C.;Marsh, K.; Wang, J.; Li, Q. Photoluminescence in thesilicon-oxygen system.
J. Vac. Sci. Technol. A Vac. Surf.Films , , 713–717, Murray, C.B.; Norris, D.J.; Bawendi, M.G. Synthesis andcharacterization of nearly monodisperse CdE (E = sulfur, selenium, tellurium) semiconductor nanocrystallites.
J.Am. Chem. Soc. , , 8706–8715, Rogach, A.L. (Ed.)
Semiconductor Nanocrystal QuantumDots ; Springer: Wien, Vienna, 2008. Bruchez, M.; Moronne, M.; Gin, P.; Weiss, S.; Alivisatos,A.P. Semiconductor Nanocrystals as Fluorescent Biolog-ical Labels.
Science , , 2013–2016, Voznyy, O.; Levina, L.; Fan, F.; Walters, G.; Fan, J.Z.;Kiani, A.; Ip, A.H.; Thon, S.M.; Proppe, A.H.; Liu, M.;et al. Origins of Stokes Shift in PbS Nanocrystals.
NanoLett. , , 7191–7195, Ahmad, W.; He, J.; Liu, Z.; Xu, K.; Chen, Z.; Yang, X.;Li, D.; Xia, Y.; Zhang, J.; Chen, C. Lead Selenide (PbSe)Colloidal Quantum Dot Solar Cells with ¿10% Efficiency.
Adv. Mater. , , 1900593, Dohnalova, K.; Gregorkiewicz, T.; Kusova, K. Siliconquantum dots: Surface matters.
J. Phys. Condens. Matter , , 173201. McVey, B.; Prabakar, S.; Gooding, J.; Tilley, R. Solu-tion Synthesis, Surface Passivation, Optical Properties,Biomedical Applications, and Cytotoxicity of Silicon andGermanium Nanocrystals.
ChemPlusChem , , 60–70. Memon, U.; Chatterjee, U.; Ganthi, M.; Tiwari, S.; Dut-tagupta, S. Synthesis of ZnSe Quantum Dots with Sto-ichiometric Ratio Difference and Study of its Optoelec-tronic Property.
Procedia Mater. Sci. , , 1027–1033. Senthilkumar, K.; Kalaivani, T.; Kanagesan, S.; Bala-subramanian, V. Synthesis and characterization studiesof ZnSe quantum dots.
J. Mater. Sci. Mater. Electron. , , 2048–2052. Baum, F.; da Silva, M.; Linden, G.; Feijo, D.; Rieder, E.;Santos, M. Growth dynamics of zinc selenide quantumdots: The role of oleic acid concentration and synthesistemperature on driving optical properties.
J. Nanopart.Res. , , 42. Moura, I.; Filho, P.; Seabra, M.; Pereira, G.; Pereira,G.; Fontes, A.; Santos, B. Highly fluorescent positivelycharged ZnSe quantum dots for bioimaging.
J. Lumin. , , 284–289. Talapin, D.V.; Rogach, A.L.; Kornowski, A.; Haase,M.; Weller, H. Highly Luminescent Monodis-perse CdSe and CdSe/ZnS Nanocrystals Synthe-sized in a Hexadecylamine-Trioctylphosphine Oxide-Trioctylphospine Mixture.
Nano Lett. , , 207–211, Maiti, S.; van der Laan, M.; Poonia, D.; Schall, P.; Kinge,S.; Siebbeles, L. Emergence of new materials for exploit-ing highly efficient carrier multiplication in photovoltaics.
Chem. Phys. Rev. , , 011302. Ji, B.; Koley, S.; Slobodkin, V.; Remennik, S.; Banin, U.ZnSe/ZnS Core/Shell Quantum Dots with Superior Op-tical Properties through Thermodynamic Shell Growth.
Nano Lett. , , 2387–2395. Pavesi, L.; Turan, R. (Eds.)
Silicon Nanocrystals ; WileyVCH Verlag GmbH: Weinheim, Germany, 2010. Pereira, R.N.; Niesar, S.; Wiggers, H.; Brandt, M.S.;Stutzmann, M.S. Depassivation kinetics in crystalline sil-icon nanoparticles.
Phys. Rev. B , , 155430, Otsuka, M.; Kurokawa, Y.; Ding, Y.; Juangsa, F.B.; Shi-bata, S.; Kato, T.; Nozaki, T. Silicon nanocrystal hybridphotovoltaic devices for indoor light energy harvesting.
RSC Adv. , , 12611–12618, Holman, Z.C.; Liu, C.Y.; Kortshagen, U.R. Germaniumand Silicon Nanocrystal Thin-Film Field-Effect Transis-tors from Solution.
Nano Lett. , , 2661–2666, Bourzac, K. Quantum dots go on display.
Nature , , 283, Qasim, K.; Lei, W.; Li, Q. Quantum dots for light emit-ting diodes.
J. Nanosci. Nanotechnol. , , 3173–3185, Available online: (accessed on20/12/2020). Available online: (accessedon 20/12/2020). Liu, Z.; Lin, C.H.; Hyun, B.R.; Sher, C.W.; Lv, Z.; Luo,B.; Jiang, F.; Wu, T.; Ho, C.H.; Kuo, H.C.; et al. Micro-light-emitting diodes with quantum dots in display tech-nology.
Light Sci. Appl. , , 83. Basko, D.M.; Agranovich, V.M.; Bassani, F.; Rocca,G.C.L. Colloidal metal films as a substrate for surface–enhanced spectroscopy.
Eur. Phys. J. B , , 653. Achermann, M.; Petruska, M.A.; Kos, S.; Smith, D.L.;Koleska, D.D.; Klimov, V.I. Energy-transfer pumping ofsemiconductor nanocrystals using an epitaxial quantumwell.
Nature , , 642–646. Theuerholtz, T.S.; Carmele, A.; Richter, M.; Knorr, A.Influence of F¨orster interaction on light emission statisticsin hybrid systems.
Phys. Rev. B , , 245313. Lunz, M.; Zhang, X.; Gerard, V.A.; Gunko, Y.K.;Lesnyak, V.; Gaponik, N.; Susha, A.S.; Rogach, A.L.;Bradley, A.L. Effect of Metal Nanoparticle Concentra-tion on Localized Surface Plasmon Mediated FoersterResonant Energy Transfer.
J. Phys. Chem. C , , 26529–26534. Pompa, P.P.; Martiradonna, L.; Torre, A.D.; Sala, F.D.;Manna, L.; Vittorio, M.D.; Calabi, F.; Cingolani, R.;Rinaldi, R. Metal-enhanced fluorescence of colloidalnanocrystals with nanoscale control.
Nat. Nanotechnol. , , 126–130. Chen, Z.; Berciaud, S.; Nuckolls, C.; Heinz, T.F.; Brus,L.E. Energy transfer from individual semiconductornanocrystals to graphene.
ACS Nano , , 2964–2968. Biehs, S.A.; Agarwal, G.S. Large enhancement of Foersterresonance energy transfer on graphene platforms.
Appl.Phys. Lett. , , 243112. Dovzhenko, D.S.; Ryabchuk, S.V.; Rakovich, Y.P.; Na-biev, I.R. Light-matter interaction in the strong cou-pling regime: Configurations, conditions and applications.
Nanoscale , , 3589–3605, Delerue, C.J.; Lannoo, M.
Nanostructures: Theoryand Modeling (NanoScience and Technology) ; Springer:Berlin/Heidelberg, Germany, 2013. Klimov, V.I. Multicarrier Interactions in SemiconductorNanocrystals in Relation to the Phenomena of Auger Re-combination and Carrier Multiplication.
Annu. Rev. Con-dens. Matter Phys. , , 285–316, Bruhn, B.; Limpens, R.; Chung, N.X.; Schall, P.; Gre-gorkiewicz, T. Spectroscopy of carrier multiplication innanocrystals.
Sci. Rep. , , 20538, Haverkort, J.E.M.; Garnett, E.C.; Bakkers, E.P.A.M.Fundamentals of the nanowire solar cell: Optimizationof the open circuit voltage.
Appl. Phys. Rev. , , 031106, Khriachtchev, L.; Ossicini, S.; Iacona, F.; Gourbilleau,F. Silicon Nanoscale Materials: From Theoretical Sim-ulations to Photonic Applications.
Int. J. Photoenergy , , 872576, Ray, S.K.; Maikap, S.; Banerjee, W.; Das, S. Nanocrystalsfor silicon-based light-emitting and memory devices.
J.Phys. D Appl. Phys. , , 153001, Barbagiovanni, E.G.; Lockwood, D.J.; Simpson, P.J.;Goncharova, L.V. Quantum confinement in Si and Genanostructures: Theory and experiment.
Appl. Phys. Rev. , , 011302, Priolo, F.; Gregorkiewicz, T.; Galli, M.; Krauss, T.F. Sil-icon nanostructures for photonics and photovoltaics.
Nat.Nanotechnol. , , 19–32, F¨orster, T. Zwischenmolekulare Energiewanderung undFluoreszenz.
Ann. Phys. , , 55–75, Dexter, D.L. A theory of sensitized luminescence in solids.
J. Chem. Phys. , , 836–850. Reich, K.V.; Shklovskii, B.I. Exciton Transfer in Arrayof Epitaxially Connected Nanocrystals.
ACS Nano , , 10267–10274, Kagan, C.R.; Murray, C.B.; Nirmal, M.; Bawendi, M.G.Electronic Energy Transfer in CdSe Quantum Dot Solids.
Phys. Rev. Lett. , , 1517–1520, Crooker, S.A.; Hollingsworth, J.A.; Tretiak, S.; Klimov,V.I. Spectrally Resolved Dynamics of Energy Transfer inQuantum-Dot Assemblies: Towards Engineered EnergyFlows in Artificial Materials.
Phys. Rev. Lett. , ,186802, Franzl, T.; Klar, T.A.; Scheitinger, S.; Rogach, A.L.; Feld-mann, J. Exciton Recycling in Graded Gap NanocrystalStructures
Nano Lett. , , 1599. Lunz, M.; Bradley, A.L.; Chen, W.Y.; Gerard, V.A.;Byrne, S.J.; Gun’ko, Y.K.; Lesnyak, V.; Gaponik, N. In-fluence of quantum dot concentration on F¨orster resonantenergy transfer in monodispersed nanocrystal quantumdot monolayers.
Phys. Rev. B , , 205316. Lunz, M.; Bradley, A.L.; Gerard, V.A.; Byrne, S.J.;Gun’ko, Y.K.; Lesnyak, V.; Gaponik, N. Concentrationdependence of F¨orster resonant energy transfer betweendonor and acceptor nanocrystal quantum dot layers: Ef-fect of donor-donor interactions.
Phys. Rev. B , ,115423, Yu, D. n-Type Conducting CdSe Nanocrystal Solids.
Sci-ence , , 1277–1280, Kawazoe, T.; Kobayashi, K.; Ohtsu, M. Opticalnanofountain: A biomimetic device that concentrates op-tical energy in a nanometric region.
Appl. Phys. Lett. , , 103102, Linnros, J.; Lalic, N.; Galeckas, A.; Grivickas, V. Analy-sis of the stretched exponential photoluminescence decayfrom nanometer-sized silicon crystals in SiO . J. Appl.Phys. , , 6128–6134, Heitmann, J.; M¨uller, F.; Yi, L.; Zacharias, M.; Kovalev,D.; Eichhorn, F. Excitons in Si nanocrystals: Confine-ment and migration effects.
Phys. Rev. B , ,195309, Glover, M.; Meldrum, A. Effect of “buffer layers” onthe optical properties of silicon nanocrystal superlattices.
Opt. Mater. , , 977–982, Ben-Chorin, M.; M¨oller, F.; Koch, F.; Schirmacher, W.;Eberhard, M. Hopping transport on a fractal: Ac conduc-tivity of porous silicon.
Phys. Rev. B , , 2199–2213, Priolo, F.; Franz`o, G.; Pacifici, D.; Vinciguerra, V.; Ia-cona, F.; Irrera, A. Role of the energy transfer in theoptical properties of undoped and Er-doped interactingSi nanocrystals.
J. Appl. Phys. , , 264–272, Balberg, I.; Savir, E.; Jedrzejewski, J. The mutual exclu-sion of luminescence and transport in nanocrystalline sil-icon networks.
J. Non-Cryst. Solids , , 102–105, Santos, J.R.; Vasilevskiy, M.I.; Filonovich, S.A. En-ergy transfer via exciton transport in quantum dot basedself-assembled fractal structures.
Phys. Rev. B , , 245422. Wargnier, R.; Baranov, A.V.; Maslov, V.G.; Stsiapura,V.; Artemyev, M.; Pluot, M.; Sukhanova, A.; Nabiev,I. Energy Transfer in Aqueous Solutions of OppositelyCharged CdSe/ZnS Core/Shell Quantum Dots and inQuantum Dot-Nanogold Assemblies.
Nano Lett. , , 451–457, Sukhanova, A.; Baranov, A.V.; Perova, T.S.; Cohen,J.H.; Nabiev, I. Controlled Self-Assembly of Nanocrystalsinto Polycrystalline Fluorescent Dendrites with Energy-Transfer Properties.
Angew. Chem. , , 2048. Bernardo, C.; Moura, I.; Fernandez, Y.; Pereira, E.;Coutinho, P.; Garcia, A.; Schellenberg, P.; Belsley, M.;Costa, M.; Stauber, T.; et al. Energy transfer via exci-ton transport in quantum dot based self-assembled fractalstructures.
J. Phys. Chem. C , , 4982–4990. Rabouw, F.T.; den Hartog, S.A.; Senden, T.; Meijerink,A. Photonic effects on the F¨orster resonance energy trans-fer efficiency.
Nat. Photonics , , 3610. Chance, R.P.; Prock, A.; Silbey, R. Molecular fluorescenceand energy transfer near interfaces.
Adv. Chem. Phys. , , 1–65. Koppens, F.H.L.; Chang, D.E.; Garcia de Abajo,F.J. Graphene Plasmonics: A Platform for StrongLight–Matter Interactions.
Nano Lett. , , 3370–3377. Purcell, E.M. Spontaneous emission probability at radiofrequencies.
Phys. Rev. , , 681. G´erard, J.M.; Sermage, B.; Gayral, B.; Legrand, B.;Costard, E.; Thierry-Mieg, V. Enhanced SpontaneousEmission by Quantum Boxes in a Monolithic Optical Mi-crocavity.
Phys. Rev. Lett. , , 1110–1113, Senden, T.; Rabouw, F.T.; Meijerink, A. Photonic Effects on the Radiative Decay Rate and Luminescence quantumYield of Doped Nanocrystals.
ACS Nano , , 1801–1808. Shimizu, K.T.; Woo, W.K.; Fisher, B.R.; Eisler, H.J.;Bawendi, M.G. Surface-Enhanced Emission from SingleSemiconductor Nanocrystals.
Phys. Rev. Lett. , ,117401, Schreiber, R.; Do, J.; Roller, E.-M.; Zhang, T.; Sch¨aler,V.J.; Nickels, P. C.; Feldmann, J.; Liedl, T. Hierarchical as-sembly of metal nanoparticles, quantum dots and organicdyes using DNA origami scaffolds.
Nature Nanotechnology , , 74. Rakovich, A.; Albella, P.; Maier, S. A. Plasmonic controlof radiative properties of semiconductor quantum dotscoupled to plasmonic ring cavities.
ACS Nano , ,2648. Novotny, L.; Hehct, B.
Principles of Nano-Optics ; Cam-bridge University Press: Cambridge, UK, 2012. Jackson, J.D.
Classical Electrodynamics ; John Wiley &Sons: New York, NY, USA, 1998. Thr¨anhardt, A.; Ell, C.; Khitrova, G.; Gibbs, H.M. Re-lation between dipole moment and radiative lifetime ininterface fluctuation quantum dots.
Phys. Rev. B , , 035327, Efros, A.L.; Rosen, M.; Kuno, M.; Nirmal, M.; Norris,D.J.; Bawendi, M. Band-edge exciton in quantum dotsof semiconductors with a degenerate valence band: Darkand bright exciton states.
Phys. Rev. B , , 4843. Belyakov, V.A.; Burdov, V.A.; Gaponova, D.M.;Mikhaylov, A.N.; Tetelbaum, D.I.; Trushin, S.A. Phonon-assisted radiative electron-hole recombination in siliconquantum dots.
Phys. Solid State , , 27–31, Tetelbaum, D.; Karpovich, I.; Stepikhova, M.; Shengurov,V.; Markov, K.; Gorshkov, O. Characteristics of photo-luminescence in SiO with Si nanoinclusions produced byion implantation. In Surface Investigation ; OPA (Over-seas Publishers Association) N.V.: New York, NY, USA,1998, Volume 14, pp. 601–604. Fujii, M.; Mimura, A.; Hayashi, S.; Yamamoto, K. Photo-luminescence from Si nanocrystals dispersed in phospho-silicate glass thin films: Improvement of photolumines-cence efficiency.
Appl. Phys. Lett. , , 184–186. Tetelbaum, D.I.; Gorshkov, O.N.; Burdov, V.A.; Trushin,S.A.; Mikhaylov, A.N.; Gaponova, D.M.; Morozov, S.V.;Kovalev, A.I. The influence of P + , B + , and N + ion im-plantation on the luminescence properties of the SiO :Nc-Si system. Phys. Solid State , , 17–21, Belov, A.I.; Belyakov, V.A.; Burdov, V.A.; Mikhailov,A.N.; Tetelbaum, D.I. Phosphorus doping as an efficientway to modify the radiative interband recombination insilicon nanocrystals.
J. Surf. Investig. X-ray SynchrotronNeutron Tech. , , 527–533, Belyakov, V.; Belov, A.; Mikhaylov, A.; Tetelbaum, D.;Burdov, V. Improvement of the photon generation ef-ficiency in phosphorus-doped silicon nanocrystals: Γ-Xmixing of the confined electron states.
J. Phys. Condens.Matter , , 045803, Nomoto, K.; Yang, T.C.J.; Ceguerra, A.V.; Zhang, T.;Lin, Z.; Breen, A.; Wu, L.; Puthen-Veettil, B.; Jia, X.;Conibeer, G.; et al. Microstructure analysis of siliconnanocrystals formed from silicon rich oxide with high ex-cess silicon: Annealing and doping effects.
J. Appl. Phys. , , 025102. Klimeˇsov´a, E.; K˚usov´a, K.; Vac´ık, J.; Hol´y, V.; Pelant, I. Tuning luminescence properties of silicon nanocrystals bylithium doping.
J. Appl. Phys. , , 064322, Kim, T.Y.; Park, N.M.; Kim, K.H.; Sung, G.Y.; Ok,Y.W.; Seong, T.Y.; Choi, C.J. Quantum confinement ef-fect of silicon nanocrystals in situ grown in silicon nitridefilms.
Appl. Phys. Lett. , , 5355–5357, Klangsin, J.; Marty, O.; Mungu´ıa, J.; Lysenko, V.;Vorobey, A.; Pitaval, M.; C´ereyon, A.; Pillonnet, A.;Champagnon, B. Structural and luminescent propertiesof silicon nanoparticles incorporated into zirconia matrix.
Phys. Lett. A , , 1508–1511, Nozaki, T.; Sasaki, K.; Ogino, T.; Asahi, D.; Okazaki,K. Microplasma synthesis of tunable photoluminescentsilicon nanocrystals.
Nanotechnology , , 235603, Zhou, S.; Ding, Y.; Pi, X.; Nozaki, T. Doped siliconnanocrystals from organic dopant precursor by a SiCl4-based high frequency nonthermal plasma.
Appl. Phys.Lett. , , 183110, Baldwin, R.K.; Pettigrew, K.A.; Garno, J.C.; Power, P.P.;yu Liu, G.; Kauzlarich, S.M. Room Temperature Solu-tion Synthesis of Alkyl-Capped Tetrahedral Shaped Sili-con Nanocrystals.
J. Am. Chem. Soc. , , 1150–1151, Wolf, O.; Dasog, M.; Yang, Z.; Balberg, I.; Veinot, J.G.C.;Millo, O. Doping and Quantum Confinement Effects inSingle Si Nanocrystals Observed by Scanning TunnelingSpectroscopy.
Nano Lett. , , 2516–2521, Bell, J.P.; Cloud, J.E.; Cheng, J.; Ngo, C.; Ko-dambaka, S.; Sellinger, A.; Williams, S.K.R.; Yang, Y.N-Bromosuccinimide-based bromination and subsequentfunctionalization of hydrogen-terminated silicon quantumdots.
RSC Adv. , , 51105–51110, Zhou, T.; Anderson, R.T.; Li, H.; Bell, J.; Yang, Y.;Gorman, B.P.; Pylypenko, S.; Lusk, M.T.; Sellinger, A.Bandgap Tuning of Silicon Quantum Dots by SurfaceFunctionalization with Conjugated Organic Groups.
NanoLett. , , 3657–3663, Carroll, G.M.; Limpens, R.; Neale, N.R. Tuning Con-finement in Colloidal Silicon Nanocrystals with SaturatedSurface Ligands.
Nano Lett. , , 3118–3124, Belyakov, V.; Burdov, V. Γ-X mixing in phosphorus-doped silicon nanocrystals: Improvement of photon gen-eration efficiency.
Phys. Rev. B , , 35302, K˚usov´a, K.; Cibulka, O.; Dohnalov´a, K.; Pelant, I.; Va-lenta, J.; Fuˇc´ıkov´a, A.; ˇZ´ıdek, K.; Lang, J.; Englich, J.;Matˇejka, P.; et al. Brightly Luminescent OrganicallyCapped Silicon Nanocrystals Fabricated at Room Tem-perature and Atmospheric Pressure.
ACS Nano , , 4495–4504, Ma, Y.; Chen, X.; Pi, X.; Yang, D. Theoretical Studyof Chlorine for Silicon Nanocrystals.
J. Phys. Chem. C , , 12822–12825, Ma, Y.; Pi, X.; Yang, D. Fluorine-Passivated SiliconNanocrystals: Surface Chemistry versus Quantum Con-finement.
J. Phys. Chem. C , , 5401–5406, Wang, R.; Pi, X.; Yang, D. Surface modification ofchlorine-passivated silicon nanocrystals.
Phys. Chem.Chem. Phys. , , 1815, Dohnalov´a, K.; Poddubny, A.N.; Prokofiev, A.A.;de Boer, W.D.; Umesh, C.P.; Paulusse, J.M.; Zuilhof,H.; Gregorkiewicz, T. Surface brightens up Si quantumdots: Direct bandgap-like size-tunable emission.
LightSci. Appl. , , e47, Poddubny, A.N.; Dohnalov´a, K. Direct band gap silicon quantum dots achieved via electronegative capping.
Phys.Rev. B , , 245439, Derbenyova, N.V.; Burdov, V.A. Effect of Doping withShallow Donors on Radiative and Nonradiative Relax-ation in Silicon Nanocrystals: Ab Initio Study.
J. Phys.Chem. C , , 850–858, Derbenyova, N.V.; Burdov, V.A. Radiative decay ratesin Si crystallites with a donor ion.
J. Appl. Phys. , , 161598, Sangghaleh, F.; Sychugov, I.; Yang, Z.; Veinot, J.G.C.;Linnros, J. Near-Unity Internal Quantum Efficiency ofLuminescent Silicon Nanocrystals with Ligand Passiva-tion.
ACS Nano , , 7097–7104, Belyakov, V.A.; Burdov, V.A. Chemical-shift enhance-ment for strongly confined electrons in silicon nanocrys-tals.
Phys. Lett. A , , 128–134, Belyakov, V.A.; Burdov, V.A. Fine Splitting of ElectronStates in Silicon Nanocrystal with a Hydrogen-like Shal-low Donor.
Nanoscale Res. Lett. , , 569–575, Belyakov, V.A.; Burdov, V.A. Anomalous splitting of thehole states in silicon quantum dots with shallow acceptors.
J. Phys. Condens. Matter , , 025213, Chelikowsky, J.R.; Alemany, M.M.G.; Chan, T.L.;Dalpian, G.M. Computational studies of doped nanos-tructures.
Rep. Prog. Phys. , , 046501, Oliva-Chatelain, B.L.; Ticich, T.M.; Barron, A.R. Dopingsilicon nanocrystals and quantum dots.
Nanoscale , , 1733–1745, Arduca, E.; Perego, M. Doping of silicon nanocrys-tals.
Mater. Sci. Semicond. Process. , , 156–170, Nomoto, K.; Sugimoto, H.; Breen, A.; Ceguerra, A.V.;Kanno, T.; Ringer, S.P.; Wurfl, I.P.; Conibeer, G.; Fujii,M. Atom Probe Tomography Analysis of Boron and/orPhosphorus Distribution in Doped Silicon Nanocrys-tals.
J. Phys. Chem. C , , 17845–17852, Marri, I.; Degoli, E.; Ossicini, S. First Principle Studiesof B and P Doped Si Nanocrystals.
Phys. Status Solidi A , , 1700414. Hiller, D.; Lopez-Vidrier, J.; Gutsch, S.; Zacharias, M.;Nomoto, K.; Konig, D. Defect-Induced luminescencequenching vs. Charge carrier generation of phosphorus in-corporated in silicon nanocrystals as function of size.
Sci.Rep. , , 863. Yang, T.C.J.; Nomoto, K.; Puthen-Veettil, B.; Lin, Z.;Wu, L.; Zhang, T.; Jia, X.; Conibeer, G.; Perez-Wurfl, I.Properties of silicon nanocrystals with boron and phos-phorus doping fabricated via silicon rich oxide and silicondioxide bilayers.
Mater. Res. Express , , 075004, Hybertsen, M.S. Absorption and emission of light innanoscale silicon structures.
Phys. Rev. Lett. , , 1514–1517, Moskalenko, A.S.; Berakdar, J.; Prokofiev, A.A.;Yassievich, I.N. Single-particle states in spherical Si/SiO quantum dots. Phys. Rev. B , , 085427, Belyakov, V.A.; Burdov, V.A.; Lockwood, R.; Meldrum,A. Silicon Nanocrystals: Fundamental Theory and Im-plications for Stimulated Emission.
Adv. Opt. Technol. , , 279502, Delerue, C.; Allan, G.; Lannoo, M. Electron-phonon cou-pling and optical transitions for indirect-gap semiconduc-tor nanocrystals.
Phys. Rev. B , , 193402, Casida, M.
Recent Developments and Applications ofModern Density Functional Theory (Volume 4) (The-oretical and Computational Chemistry (Volume 4)) ; Time–Dependent Density Functional Response Theoryof Molecular Systems: Theory, Computational Meth-ods, and Functionals; Elsevier Science: Amsterdam, TheNetherlands, 1996; Chapter 11.
Li, Q.; Jin, R. Photoluminescence from colloidal siliconnanoparticles: Significant effect of surface.
Nanotechnol.Rev. , , 601–602. Heuer-Jungemann, A.; Feliu, N.; Bakaimi, I.; Hamaly,M.; Alkilany, A.; Chakraborty, I.; Masood, A.; Casula,M.; Kostopoulou, A.; Oh, E.; et al. The Role of Ligandsin the Chemical Synthesis and Applications of InorganicNanoparticles.
Chem. Rev. , , 4819–4880. Wheeler, L.; Anderson, N.; Palomaki, P.; Blackburn, J.;Johnson, J.; Neale, N. Silyl radical abstraction in thefunctionalization of plasma-synthesized silicon nanocrys-tals.
Chem. Mater. , , 6869–6878. Purkait, T.; Iqbal, M.; Islam, M.; Mobarok, H.; Gonzalez,C.; Hadidi, L.; Veinot, J. Alkoxy-Terminated Si Surfaces:A New Reactive Platform for the Functionalization andDerivatization of Silicon Quantum Dots.
J. Am. Chem.Soc. , , 7114–7120. De los Reyes, G.; Dasog, M.; Na, M.; Titova, L.; Veinot,J.; Hegmann, F. Charge transfer state emission dynamicsin blue-emitting functionalized silicon nanocrystals.
Phys.Chem. Chem. Phys. , , 30125. Sinelnikov, R.; Dasog, M.; Beamish, J.; Meldrum, A.;Veinot, J. Revisiting an Ongoing Debate: What Role DoSurface Groups Play in Silicon Nanocrystal Photolumi-nescence?
ACS Photonics , , 1920. Derbenyova, N.V.; Konakov, A.A.; Shvetsov, A.E.; Bur-dov, V.A. Electronic structure and absorption spectraof silicon nanocrystals with a halogen (Br, Cl) coating.
JETP Lett. , , 247–251, Derbenyova, N.V.; Shvetsov, A.E.; Konakov, A.A.; Bur-dov, V.A. Effects of surface halogenation on exciton relax-ation in Si crystallites: Prospects for photovoltaics.
Phys.Chem. Chem. Phys. , , 20693–20705, Derbeneva, N.V.; Konakov, A.A.; Burdov, V.A. Effect ofHalogen Passivation of a Surface on Radiative and Nonra-diative Transitions in Silicon Nanocrystals.
J. Exp. Theor.Phys. , , 234–240, Stolle, C.J.; Lu, X.; Yu, Y.; Schaller, R.D.; Korgel,B.A. Efficient Carrier Multiplication in Colloidal SiliconNanorods.
Nano Lett. , , 5580–5586, Pevere, F.; Sychugov, I.; Sangghaleh, F.; Fucikova, A.;Linnros, J. Biexciton Emission as a Probe of Auger Re-combination in Individual Silicon Nanocrystals.
J. Phys.Chem. C , , 7499–7505, Sevik, C.; Bulutay, C. Auger recombination and car-rier multiplication in embedded silicon and germaniumnanocrystals.
Phys. Rev. B , , 125414, Delerue, C.; Lannoo, M.; Allan, G.; Martin, E.; Mihal-cescu, I.; Vial, J.C.; Romestain, R.; Muller, F.; Bsiesy, A.Auger and Coulomb Charging Effects in SemiconductorNanocrystallites.
Phys. Rev. Lett. , , 2228–2231, Mahdouani, M.; Bourguiga, R.; Jaziri, S.; Gardelis, S.;Nassiopoulou, A. Investigation of Auger recombination inGe and Si nanocrystals embedded in SiO matrix. Phys.E Low-Dimens. Syst. Nanostructures , , 57–62, Kurova, N.V.; Burdov, V.A. Resonance structure of therate of Auger recombination in silicon nanocrystals.
Semi-conductors , , 1414–1417, Califano, M. Suppression of Auger Recombination inNanocrystals via Ligand-Assisted Wave Function Engi- neering in Reciprocal Space.
J. Phys. Chem. Lett. , , 2098–2104, Blacha, A.; Presting, R.; Cardona, M. Deformation po-tentials of k = 0 states of tetrahedral semiconductors.
Phys. Stat. Sol. B , , 11–35. Rolo, A.G.; Vasilevskiy, M.I.; Hamma, M.; Trallero-Giner, C. Anomalous first-order Raman scattering in III-V quantum dots: Optical deformation potential interac-tion.
Phys. Rev. B , , 081304, Glembocki, O.J.; Pollak, F.H. Calculation of the Γ-∆electron-phonon and hole-phonon scattering matrix ele-ments in silicon.
Phys. Rev. Lett. , , 413–416. Inoshita, T.; Sakaki, H. Electron relaxation in a quantumdot: Significance of multiphonon processes.
Phys. Rev. B , , 7260–7263, Bockelmann, U.; Bastard, G. Phonon scattering and en-ergy relaxation in two-, one-, and zero-dimensional elec-tron gases.
Phys. Rev. B , , 8947–8951, Stauber, T.; Zimmermann, R.; Castella, H. Electron-phonon interaction in quantum dots: A solvable model.
Phys. Rev. B , , 7336–7343, Jacak, L.; Machnikowski, P.; Krasnyj, J.; Zoller, P. Coher-ent and incoherent phonon processes in artificial atoms.
Eur. Phys. J. D-Atomic Mol. Opt. Plasma Phys. , , 319–331, Vasilevskiy, M.I.; Anda, E.V.; Makler, S.S. Electron-phonon interaction effects in semiconductor quantumdots: A nonperturabative approach.
Phys. Rev. B , , 035318, Sauvage, S.; Boucaud, P.; Lobo, R.P.S.M.; Bras, F.; Fish-man, G.; Prazeres, R.; Glotin, F.; Ortega, J.M.; G´erard,J.M. Long Polaron Lifetime in InAs/GaAs Self-AssembledQuantum Dots.
Phys. Rev. Lett. , , 177402, Miranda, R.P.; Vasilevskiy, M.I.; Trallero-Giner, C. Non-perturbative approach to the calculation of multiphononRaman scattering in semiconductor quantum dots: Po-laron effect.
Phys. Rev. B , , 115317, Sarkar, D.; van der Meulen, H.P.; Calleja, J.M.; Becker,J.M.; Haug, R.J.; Pierz, K. quantum dots observed byoptical emission.
Phys. Rev. B , , 081302, Stauber, T.; Vasilevskiy, M.I. Polaron relaxation in aquantum dot due to anharmonic coupling within a mean-field approach.
Phys. Rev. B , , 113301, Marcinkeviˇcius, S.; Gaarder, A.; Leon, R. Rapid car-rier relaxation by phonon emission in In . Ga . As / GaAsquantum dots.
Phys. Rev. B , , 115307, Sun, K.W.; Kechiantz, A.; Lee, B.C.; Lee, C.P. Ultrafastcarrier capture and relaxation in modulation-doped InAsquantum dots.
Appl. Phys. Lett. , , 163117, M¨uller, T.; Schrey, F.F.; Strasser, G.; Unterrainer, K.Ultrafast intraband spectroscopy of electron capture andrelaxation in InAs/GaAs quantum dots.
Appl. Phys. Lett. , , 3572–3574, Schaller, R.D.; Pietryga, J.M.; Goupalov, S.V.; Petruska,M.A.; Ivanov, S.A.; Klimov, V.I. Breaking the PhononBottleneck in Semiconductor Nanocrystals via Multi-phonon Emission Induced by Intrinsic Nonadiabatic In-teractions.
Phys. Rev. Lett. , , 196401, Hendry, E.; Koeberg, M.; Wang, F.; Zhang, H.;de Mello Doneg´a, C.; Vanmaekelbergh, D.; Bonn, M. Di-rect Observation of Electron-to-Hole Energy Transfer inCdSe Quantum Dots.
Phys. Rev. Lett. , , 057408, Heitz, R.; Born, H.; Guffarth, F.; Stier, O.; Schliwa, A.;Hoffmann, A.; Bimberg, D. Existence of a phonon bottle- neck for excitons in quantum dots. Phys. Rev. B , , 241305, Urayama, J.; Norris, T.B.; Singh, J.; Bhattacharya, P.Observation of Phonon Bottleneck in Quantum Dot Elec-tronic Relaxation.
Phys. Rev. Lett. , , 4930–4933. Guyot-Sionnest, P.; Wehrenberg, B.; Yu, D. Intrabandrelaxation in CdSe nanocrystals and the strong influenceof the surface ligands.
J. Chem. Phys. , , 074709, Nozik, A.J. Spectroscopy and hot electron relaxationdynamics in semiconductor quantum wells and quantumdots.
Annu. Rev. Phys. Chem. , , 193–231. Li, X.Q.; Nakayama, H.; Arakawa, Y. Phonon bottleneckin quantum dots: Role of lifetime of the confined opticalphonons.
Phys. Rev. B , , 5069–5073. Efros, A.L.; Kharchenko, V.; Rosen, M. Breaking thephonon bottleneck in nanometer quantum dots: Role ofAuger-like processes.
Solid State Commun. , , 281–284. Narvaez, G.A.; Bester, G.; Zunger, A. Carrier relaxationmechanisms in self-assembled (In,Ga)As/GaAs quantumdots: Efficient P – S Auger relaxation of electrons
Phys.Rev. B , , 075403. Prokofiev, A.A.; Poddubny, A.N.; Yassievich, I.N.Phonon decay in silicon nanocrystals: Fast phonon re-cycling.
Phys. Rev. B , , 125409. Poddubny, A.N.; Prokofiev, A.A.; Yassievich, I.N. Opti-cal transitions and energy relaxation of hot carriers in Sinanocrystals.
Appl. Phys. Lett. , , 231116. Prokofiev, A.A.; Goupalov, S.V.; Moskalenko, A.S.; Pod-dubny, A.N.; Yassievich, I.N. Carrier relaxation in quan-tum dots.
Phys. E Low-Dimens. Syst. Nanostructures , , 969–971. Moskalenko, A.S.; Berakdar, J.; Poddubny, A.N.;Prokofiev, A.A.; Yassievich, I.N.; Goupalov, S.V. Mul-tiphonon relaxation of moderately excited carriers inSi/SiO nanocrystals. Phys. Rev. B , , 085432. Ohnesorge, B.; Albrecht, M.; Oshinowo, J.; Forchel,A.; Arakawa, Y. Rapid carrier relaxation in self-assembledInxGa1-xAs/GaAs quantum dots.
Phys. Rev.B , , 11532–11538. Beard, M.C.; Knutsen, K.P.; Yu, P.; Luther, J.M.; Song,Q.; Metzger, W.K.; Ellingson, R.J.; Nozik, A.J. Multi-ple Exciton Generation in Colloidal Silicon Nanocrystals.
Nano Lett. , , 2506–2512. Nozik, A.J. Nanoscience and Nanostructures for Photo-voltaics and Solar Fuels.
Nano Lett. , , 2735–2741. de Boer, W.D.A.M.; Trinh, M.T.; Timmerman, D.;Schins, J.M.; Siebbeles, L.D.A.; Gregorkiewicz, T. In-creased carrier generation rate in Si nanocrystals in SiO investigated by induced absorption. Appl. Phys. Lett. , , 053126. Trinh, M.T.; Limpens, R.; de Boer, W.D.A.M.; Schins,J.M.; Siebbeles, L.D.A.; Gregorkiewicz, T. Direct gener-ation of multiple excitons in adjacent silicon nanocrystalsrevealed by induced absorption.
Nat. Photonics , , 316–321. Schaller, R.D.; Klimov, V.I. High Efficiency Carrier Mul-tiplication in PbSe Nanocrystals: Implications for SolarEnergy Conversion.
Phys. Rev. Lett. , , 186601. Nair, G.; Bawendi, M.G. Carrier multiplication yieldsofCdSeandCdTenanocrystals by transient photolumines-cence spectroscopy.
Phys. Rev. B , , 081304. Davis, N.J.L.K.; B¨ohm, M.L.; Tabachnyk, M.;Wisnivesky-Rocca-Rivarola, F.; Jellicoe, T.C.; Ducati, C.; Ehrler, B.; Greenham, N.C. Multiple-exciton generationin lead selenide nanorod solar cells with external quan-tum efficiencies exceeding 120%.
Nat. Commun. , ,8259. Hu, F.; Lv, B.; Yin, C.; Zhang, C.; Wang, X.; Lounis, B.;Xiao, M. Carrier Multiplication in a Single SemiconductorNanocrystal.
Phys. Rev. Lett. , , 106404. Smith, C.T.; Leontiadou, M.A.; Clark, P.C.J.; Lydon, C.;Savjani, N.; Spencer, B.F.; Flavell, W.R.; O’Brien, P.;Binks, D.J. Multiple Exciton Generation and Dynamicsin InP/CdS Colloidal Quantum Dots.
J. Phys. Chem. C , , 2099–2107. Meldrum, A.; Lockwood, R.; Belyakov, V.; Burdov, V.Computational simulations for ensembles of luminescentsilicon nanocrystals: Implications for optical gain andstimulated emission.
Phys. E Low-Dimens. Syst. Nanos-tructures , , 955–958. Belyakov, V.A.; Sidorenko, K.V.; Konakov, A.A.; Er-shov, A.V.; Chugrov, I.A.; Grachev, D.A.; Pavlov, D.A.;Bobrov, A.I.; Burdov, V.A. Quenching the photolumi-nescence from Si nanocrystals of smaller sizes in denseensembles due to migration processes.
J. Lumin. , , 1–6. Derbenyova, N.V.; Konakov, A.A.; Burdov, V.A. Reso-nant tunneling of carriers in silicon nanocrystals.
J. Appl.Phys. , , 134302. Curutchet, C.; Franceschetti, A.; Zunger, A.; Scholes, G.D. Examining F¨orster energy transfer for semiconduc-tor nanocrystalline quantum dot donors and acceptorslstudy.
The Journal of Physical Chemistry C Letters , , 13336 – 13341. Andrews, D.L.; Sherborne, B.S. Resonant excitationtransfer: A quantum electrodynamical study.
J. Chem.Phys. , , 4011–4017. Allan, G.; Delerue, C. Energy transfer between semicon-ductor nanocrystals: Validity of F¨orster’s theory.
Phys.Rev. B , , 195311. Stavola, M.; Dexter, D.L.; Knox, R.S. Electron-hole pairexcitation in semiconductors via energy transfer from anexternal sensitizer.
Phys. Rev. B , , 2277–2289. Lyo, S.K. Spectral and spatial transfer and diffusion ofexcitons in multiple quantum dot structures.
Phys. Rev.B , , 125328. Jang, S.; Hossein-Nejad, H.; Scholes, G.D. GeneralizedF¨orster resonance energy transfer. In
Quantum Effectsin Biology ; Mohseni, M., Omar, Y., Engel, G.S., Plenio,M.B., Eds.; Cambridge University Press: Cambridge, UK,2014; Chapter 3, pp. 53–90.
Scholes, G.D.; Andrews, D.L. Resonance energy transferand quantum dots.
Phys. Rev. B , , 125331. Shafiei, F.; Ziama, S.P.; Curtis, E.D.; Decca, R.S. Mea-surement of the separation dependence of resonant en-ergy transfer between CdSe / ZnS core / shell nanocrystal-lite quantum dots. Phys. Rev. B , , 075301. Mork, A.; Weidman, M.; Prins, F.; Tisdale, W. Mag-nitude of the F¨orster Radius in Colloidal Quantum DotSolids.
J. Phys. Chem. C , , 13920–13928. Belyakov, V.A.; Burdov, V.A. Radiative Recombinationand Migration Effects in Ensembles of Si Nanocrystals:Towards Controllable Nonradiative Energy Transfer.
J.Comput. Theor. Nanosci. , , 365–374. Gusev, O.B.; Prokofiev, A.A.; Maslova, O.A.; Terukov,E.I.; Yassievich, I.N. Energy transfer between siliconnanocrystals.
JETP Lett. , , 147–150. Belyakov, V.A.; Burdov, V.A. Intensification of F¨orstertransitions between Si crystallites due to their doping withphosphorus.
Phys. Rev. B , , 045439. Aharoni, A.; Oron, D.; Banin, U.; Rabani, E.; Jortner,J. Long-Range Electronic-to-Vibrational Energy Transferfrom Nanocrystals to Their Surrounding Matrix Environ-ment.
Phys. Rev. Lett. , , 057404. Wen, Q.; Kershaw, S.V.; Kalytchuk, S.; Zhovtiuk, O.;Reckmeier, C.; Vasilevskiy, M.I.; Rogach, A.L. Impactof D O/H O Solvent Exchange on the Emission of HgTeand CdTe Quantum Dots: Polaron and Energy TransferEffects.
ACS Nano , , 4301–4311, Santos, J.E.; Vasilevskiy, M.I.; Peres, N.M.R.; Smirnov,G.; Bludov, Y.V. Renormalization of nanoparticle polar-izability in the vicinity of a graphene-covered interface.
Phys. Rev. B , , 235420. Amorim, B.; Gon¸calves, P.A.D.; Vasilevskiy, M.I.; Peres,N.M.R. Impact of Graphene on the Polarizability ofa Neighbour Nanoparticle: A Dyadic Green’s FunctionStudy.
Appl. Sci. , , 1158–1188. Lukosz, W.; Kunz, R.E. Light emission by magnetic andelectric dipoles close to a plane interface.
J. Opt. Soc.Am. , , 1607–1615. Drexhage, K. Influence of a Dielectric Interface on Fluo-rescence Decay Time.
J. Luminescence , , 693–701. Ford, G.W.; Weber, W.H. Electromagnetic Interactionsof Molecules with Metal Surfaces.
Phys. Rep. , , 195–287. Gordon, J. M.; Cartstein, Y. N. Local field effects forspherical quantum dot emitters in the proximity of a pla-nar dielectric interface.
J. Opt. Soc. Am. B , , 2029– 2035. Benner, R.E.; Barber, P.W.; Owen, J.F.; Chang, R.K.Observation of Structure Resonances in the FluorescenceSpectra from Microspheres.
Phys. Rev. Lett. , , 475–478. Scheneipp, H.; Sandoghbar, V. Spontaneous Emissionof Europium Ions Embedded in Dielectric Nanospheres.
Phys. Rev. Lett. , , 257403. Giannini, V.; Fern´andez-Dom´ıngueez, A.I.; Heck, S.C.;Maier, S.A. Plasmonic Nanoantennas: Fundamentals andTheir Use in Controlling the Radiative Properties of Na-noemitters.
Chem. Rev. , , 3888–3912. Novoselov, K.S.; Geim, A.K.; Morozov, S.V.; Jiang, D.;Zhang, Y.; Dubonos, S.V.; Grigorieva, I.V.; Firsov, A.A.Electric field effect in atomically thin carbon films.
Sci-ence , , 666–669. Neto, A.H.C.; Guinea, F.; Peres, N.M.R.; Novoselov,K.S.; Geim, A.K. The electronic properties of graphene.
Rev. Mod. Phys. , , 109. Nair, R.R.; Blake, P.; Grigorenko, A.N.; Novoselov, K.S.;Booth, T.J.; Stauber, T.; Peres, N.M.R.; Geim, A.K.Fine Structure Constant Defines Visual Transparency ofGraphene.
Science , , 1308–1308. Li, Z.Q.; Henriksen, E.A.; Jiang, Z.; Hao, Z.; Martin,M.C.; Kim, P.; Stormer, H.L.; Basov, D.N. Dirac chargedynamics in graphene by infrared spectroscopy.
Nat.Phys. , , 532–535. Bludov, Y.V.; Ferreira, A.; Peres, N.M.R.; Vasilevskiy,M. A Primer on Surface Plasmon-Polaritons in Graphene.
Int. J. Mod. Phys. B , , 1341001. Mart´ınez-Galera, A.J.; Brihuega, I.; Guti´errez-Rubio, A.;Stauber, T.; G´omez-Rodr´ıguez, J.M. Towards scalable nano-engineering of graphene.
Sci. Rep. , , 7314. Guti´errez-Rubio, A.; Stauber, T. Mass-profile quantumdots in graphene and artificial periodic structures.
Phys.Rev. B , , 165415. Gaudreau, L.; Tielrooij, K.J.; Prawiroatmodjo, C.E.D.K.;Osmond, J.; de Abajo, F.J.G.; Koppens, F.H.L. Uni-versal distance-scaling of non-radiative energy transfer tographene.
Nano Lett. , , 2030. Federspiel, F.; Froehlicher, G.; Nasilowski, M.; Pedetti,S.; Mahmood, A.; Doudin, B.; Park, S.; Lee, J.O.; Halley,D.; Dubertret, B.; et al Distance Dependence of the En-ergy Transfer Rate from a Single Semiconductor Nanos-tructure to Graphene.
Nano Lett. , , 1252–1258. Gon¸calves, H.; Bernardo, C.; Moura, C.; Ferreira, R.A.S.;Andr´e, P.S.; Stauber, T.; Belsley, M.; Schellenberg, P.Long range energy transfer in graphene hybrid structures.
J. Phys. D Appl. Phys. , , 315102. Raja, A.; Montoya-Castillo, A.; Zultak, J.; Zhang, X.X.;Ye, Z.; Roquelet, C.; Chenet, D.A.; van der Zande, A.M.;Huang, P.; Jockusch, S.; et al. Energy Transfer fromQuantum Dots to Graphene and MoS2: The Role ofAbsorption and Screening in Two-Dimensional Materials.
Nano Lett. , , 2328–2333. G´omez-Santos, G.; Stauber, T. Fluorescence quenchingin graphene: A fundamental ruler and evidence for trans-verse plasmons.
Phys. Rev. B , , 165438. Konstantatos, G.; Badioli, M.; Osmond, J.; Gaudreau,L.; de Arquer, F.P.G.; Gatti, F.; Koppens, F.H.L. Hy-brid graphene-quantum dot phototransistors with ultra-high gain.
Nat. Nanotechnol. , , 363–368. Sun, S.; Gao, L.; Liu, Y.; Sun, J. Assembly of CdSenanoparticles on graphene for low-temperature fabrica-tion of quantum dot sensitized solar cell.
Appl. Phys.Lett. , , 093112. Lee, J.; Bao, W.; Ju, L.; Schuck, P.J.; Wang, F.; Weber-Bargioni, A. Switching Individual Quantum Dot Emissionthrough Electrically Controlling Resonant Energy Trans-fer to Graphene.
Nano Lett. , , 7115–7119. Salihoglu, O.; Kakenov, N.; Balci, O.; Balci, S.; Kocabas,C. Graphene as a Reversible and Spectrally Selective Flu-orescence Quencher.
Sci. Rep. , , 33911. Prasai, D.; Klots, A.R.; Newaz, A.; Niezgoda, J.S.; Or-field, N.J.; Escobar, C.A.; Wynn, A.; Efimov, A.; Jen-nings, G.K.; Rosenthal, S.J.; et al. Electrical Controlof near-Field Energy Transfer between Quantum Dotsand Two-Dimensional Semiconductors.
Nano Lett. , , 4374–4380. Prins, F.; Goodman, A.J.; Tisdale, W.A. Reduced Dielec-tric Screening and Enhanced Energy Transfer in Single-and Few-Layer MoS2.
Nano Lett. , , 6087–6091. Zang, H.; Routh, P.K.; Huang, Y.; Chen, J.S.; Sutter, E.;Sutter, P.; Cotlet, M. Nonradiative Energy Transfer fromIndividual CdSe/ZnS Quantum Dots to Single-Layer andFew-Layer Tin Disulfide.
ACS Nano , , 1936–0851. Karanikolas, V.D.; Marocico, C.A.; Eastham, P.R.;Bradley, A.L. Near-field relaxation of a quantum emitterto two-dimensional semiconductors: Surface dissipationand exciton polaritons.
Phys. Rev. B , , 195418. Wang, G.; Chernikov, A.; Glazov, M.M.; Heinz, T.F.;Marie, X.; Amand, T.; Urbaszek, B. Colloquium: Exci-tons in atomically thin transition metal dichalcogenides.
Rev. Mod. Phys. , , 021001. Gomes, J.N.S.; Trallero-Giner, C.; Peres, N.M.R.;Vasilevskiy, M.I. Exciton–polaritons of a 2D semicon- ductor layer in a cylindrical microcavity. J. Appl. Phys. , , 133101. Komarala, V.K.; Bradley, A.L.; Rakovich, Y.P.; Byrne,S.J.; Gun’ko, Y.K.; Rogach, A.L. Surface plasmon en-hanced F¨orster resonance energy transfer between theCdTe quantum dots.
Appl. Phys. Lett. , , 123102. Huidobro, P.A.; Nikitin, A.Y.; Gonz´alez-Ballestero, C.;Martin-Moreno, L.; Garc´ıa-Vidal, F.J. Superradiance me-diated by graphene surface plasmons.
Phys. Rev. B , , 155438. Dicke, R.H. Coherence in Spontaneous Radiation Pro-cesses.
Phys. Rev. , , 99–110. Scheibner, M.; Schmidt, T.; Worschech, L.; Forchel, A.;Bacher, G.; Passow, T.; Hommel, D. Superradiance ofquantum dots.
Nat. Phys. , , 106–110. Klar, T.A.; Franzl, T.; Rogach, A.L.; Feldmann, J. Super-Efficient Exciton Funneling in Layer-by-Layer Semicon-ductor Nanocrystal Structures.
Adv. Mater. , , 769–773. Rakovich, Y.P.; Donegan, J.F.; Vasilevskiy, M.I.; Rogach,A.L. Anti-Stokes cooling in semiconductor nanocrystalquantum dots: A feasibility study.
Phys. Status Solidi A , , 2497–2509. Mirkovic, T.; Ostroumov, E.E.; Anna, J.M.; van Gron-delle, R.G.; Scholes, G.D. Light Absorption and EnergyTransfer in the Antenna Complexes of Photosynthetic Or-ganisms.
Chem. Rev. , , 249–293. Sambur, J.B.; Novet, T.; Parkinson, B.A. Multiple Exci-ton Collection in a Sensitized Photovoltaic System.
Sci-ence , , 63–66. Coles, D.M.; Somaschi, N.; Michetti, P.; Clark, C.;Lagoudakis, P.G.; Savvidis, P.G.; Lidzey, D.G. Polariton-mediated energy transfer between organic dyes in astrongly coupled optical microcavity.
Nat. Mater. , , 712–719. Schachenmayer, J.; Genes, C.; Tignone, E.; Pupillo, G.Cavity-Enhanced Transport of Excitons.
Phys. Rev. Lett. , , 196403. T¨orm¨a, P.; Barnes, W.L. Strong coupling between surfaceplasmon polaritons and emitters: A review.
Rep. Prog.Phys. , , 013901. Let us note that the term Quantum Dot was originallyproposed for a lithographic lateral nanostructure based onan AlGaAs/GaAs heterostructure with 2D electron gas,probably by M. A. Reed in 1986. Beyond such structuresand isolated nanocrystals, it covers also self-assembledepitaxial QDs and small transistor-like structures wherediscrete energy levels are created by electrostatic confine-ment. Here we shall employ this term just for semicon-ductor nanocrystals.
We are not aware of successful colloidal chemistry synthe-sis of matrix-free silicon nanocrystals. However, they canbe fabricated by other methods and processed in away similar to colloidal NCs of other semiconductors. They will be discussed below.
We will not cover here the broad topic of interactionsbetween QDs and metallic nanostructures supporting lo-calized surface plasmons because it deserves a separatearticle. Refs. can provide an introduction to thistopic.
It has been shown that, in contrast to the case of FRETbetween donor and acceptor molecules, where the dipoleapproximation breaks down at lengthscales comparableto molecular dimensions, it works fairly well when donorand/or acceptor is a spherical QD, even at contact donor-acceptor separations.
See201