# Exciton-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 Física, Universidade do Minho, Campus de Gualtar, Braga
4710-057, Portugal  International Iberian Nanotechnology Laboratory, Braga
4715-330, Portugal

(August 27, 2024)

###### Abstract

In this review we discuss several fundamental processes taking place in
semiconductor nanocrystals (quantum dots, QDs) when their electron subsystem
interacts with electromagnetic (EM) radiation. The physical phenomena of light
emission and EM energy transfer from a QD exciton to other electronic systems
such as neighbouring nanocrystals and polarisable 3D (semi-infinite dielectric
or metal) and 2D (graphene) materials are considered. The cases of direct (II-
VI) and indirect (silicon) band gap semiconductors are compared. We also cover
the relevant non-radiative mechanisms such as the Auger process, electron
capture on dangling bonds and interaction with phonons. The emphasis is on
explaining the underlying physics and illustrating it with calculated and
experimental results in a comprehensive, tutorial manner.

nanocrystal, quantum dot, light-matter interaction, emission, recombination,
energy transfer, interface, 2D material

##  I Introduction

Semiconductor nanostructures form a basis for modern electronic technologies.
Nowadays, they are employed in a wide range of applications in the fields of
optoelectronics, photonics, photovoltaics, biosensing, photocatalysis, etc. By
virtue of the quantum confinement effect, their electronic spectra and,
consequently, the related optical properties depend on the nanostructure size.
This feature is most pronounced for zero-dimensional (0D) objects,
nanocrystals (NCs), where the conduction band electron motion is fully
localized in all directions. Consequently, at least a part of the NC energy
spectrum is completely discrete. In the limiting case of strong quantum
confinement effect, when the NC size is much less than the effective exciton
Bohr radius, the former strongly influences the electron and hole energies. It
leads to the size-dependent energies of photons emitted or absorbed by the
nanocrystals and allows one to control their optical spectra. For this reason,
they sometimes are called “artificial atoms” Ashoori (1996). A more scientific
term to distinguish semiconductor NCs with strongly size-dependent electronic
and optical properties is Quantum Dot (QD), which will be used in this
article.111Let us note that the term Quantum Dot was originally proposed for a
lithographic lateral nanostructure based on an AlGaAs/GaAs heterostructure
with 2D electron gas, probably by M. A. Reed in 1986. Beyond such structures
and isolated nanocrystals, it covers also self-assembled epitaxial QDs and
small transistor-like structures where discrete energy levels are created by
electrostatic confinement. Here we shall employ this term just for
semiconductor nanocrystals.

After the pioneering works by Efros and Efros Efros and Efros (1982) and Brus
Brus (1983) explaining the origin of size-dependent optical spectra of
nanocrystals, the interest to these objects grew exponentially through the
1980s and 1990s. A variety of methods has been used for preparation of
crystallites with sizes not greater than several nanometers, such as ion
implantation Kanemitsu (1996); Zhuravlev et al. (1998); Tetelbaum et al.
(2001), chemical vapour deposition from a gas phase, Negro et al. (2003)
magnetron sputtering Tsybeskov et al. (1998), colloidal synthesis Nayfeh et
al. (2002), electron beam epitaxy Meldrum et al. (2006), etc. One of these
methods, chemical growth in colloidal solutions, allows for obtaining good
quality NCs of a broad range of semiconductors with an almost spherical shape
and a rather narrow size distribution, characterized by rather narrow emission
bands, which is, as a rule, extremely desirable for applications. This method
was first suggested for II–VI NCs by Murray et al. Murray et al. (1993) and
became widely employed later (see, e.g., chapters by Kudera et al., Reiss,
Gaponik and Rogach in the book in Rogach (2008) for review). The colloidal
chemistry methods work fairly well for II–VI and III–V materials such as CdSe,
CdS, CdTe, InAs, InP, etc. The authors of Rogach (2008); Bruchez et al. (1998)
and several other groups have been able to produce good QDs of some IV–VI
materials Voznyy et al. (2017); Ahmad et al. (2019) and silicon QDs (see,
e.g., reviews in Dohnalova et al. (2014); McVey et al. (2017)).

Colloidal QDs of II–VI materials (especially CdSe and CdTe) are probably the
most studied. During the last decade, a considerable research activity has
been focused on the synthesis and investigation of the optical properties of
ZnSe QDs. These nanocrystals exhibit large blue shift of the photoluminescence
Memon et al. (2014) and high enough (up to 50 percent) quantum yield
Senthilkumar et al. (2012). Similarly to CdSe NCs, the ZnSe QDs can be
synthesized with a sufficiently narrow size distribution, which is possible to
control by temperature Baum et al. (2019). Due to ZnSe’s lower toxicity,
compared to many other II–VI semiconductors, these QDs turn out to be
attractive objects also for biosensing Moura et al. (2018).

Further improvement of the light emission properties of such QDs was achieved
by fabrication of core–shell structures, successful for several pairs of II–VI
Talapin et al. (2001) and IV–VI Maiti et al. (2020) materials, where the shell
made of a wider band gap material provides a better protection of the
quantized electronic states in the QD core Rogach (2008). For ZnSe/ZnS
core–shell QDs it was revealed, in particular, that thermodynamical (slow)
growth of the ZnS-shell on the colloidal ZnSe-core leads to the quantum yield
increase because of decreasing amount of the traps at the core–shell interface
Ji et al. (2020).

The maturity of this technology of synthesis of semiconductor
nanocrystals222We are not aware of successful colloidal chemistry synthesis of
matrix-free silicon nanocrystals. However, they can be fabricated by other
methods Pavesi and Turan (2010); Pereira et al. (2013); Otsuka et al. (2020)
and processed in a way similar to colloidal NCs of other semiconductors.
Holman et al. (2010) They will be discussed below. is witnessed by the
incorporation of colloidal QDs into real-world products such as colour
displays Bourzac (2013) and light-emitting diodes for lighting Qasim et al.
(2013), both already commercialized by large companies such as Samsung Sam
and OSRAM OSR , respectively (see Ref. Liu et al. (2020) for a recent review
covering both colloidal and epitaxial QDs). More on the scientific research
side, a broad variety of nanostructures can be prepared using colloidal NCs as
building blocks, in particular, multilayer structures of QDs of different
average size, deposited on different substrates Rogach (2008). Combining these
structures with other materials, such as organic dielectrics Basko et al.
(2000), epitaxial quantum well (QW) heterostructures Achermann et al. (2004),
metallic nanoparticles Theuerholtz et al. (2013); Lunz et al. (2012),
patterned metallic surface Pompa et al. (2006) or graphene Chen et al. (2010);
Biehs and Agarwal (2013) can result in new interesting effects and
applications, which physics is related to the coupling of QD excitons with
elementary excitations (such as surface plasmons or QW excitons) in the
surrounding materials. One interesting possibility is the excitation of
colloidal QDs by pumping energy through a nearby epitaxial QW, as demonstrated
in Achermann et al. (2004). Another research topic with QDs, popular in the
past two decades is the modification of their emission properties caused by
the strong light–matter interaction, which has been achieved for a variety of
composite structures that include point emitters embedded in Fabry–Perot
microcavities, micropillars and photonic crystals as recently reviewed in
Dovzhenko et al. (2018).

In this article, we overview several aspects of the exciton–photon
interactions in nanocrystal QDs with focus on the role of non-radiative
processes and environment, which affect the light emission. The QD
photoluminescnce is a result of competition between the radiative and various
non-radiative processes, such as capture on dangling bonds, multi-phonon
intra-band relaxation, Auger recombination, etc. In the past decade, a growing
interest has been paid to the multi-exciton dynamics in nanocrystals because
of their high potential for photovoltaic applications. Such processes as
carrier multiplication (or multi-exciton generation) in nanocrystals, as well
as the Auger recombination (which is just a fast reverse process), are widely
discussed in the literature Maiti et al. (2020); Klimov (2014); Bruhn et al.
(2016); Haverkort et al. (2018) and we will consider these effects below.
Special attention is traditionally paid to nanostructured silicon Pavesi and
Turan (2010); Khriachtchev et al. (2012); Ray et al. (2013); Barbagiovanni et
al. (2014); Priolo et al. (2014) because of its widest use in
microelectronics, high purity, natural abundance, low cost, and non-toxicity;
we shall also dedicate some space to the exciton–photon interactions and main
non-radiative processes mentioned above, in Si NCs and related nanostructures.

As a rule, one deals with nanocrystal ensembles (rather than with isolated
QDs). Therefore, non-radiative energy exchange between the nanocrystals is
possible via the Förster-type Förster (1948) and Dexter-type Dexter (1953)
exciton migration. Both mechanisms were originally proposed for fluorophores.
The second one, based on the exchange interaction between electrons located on
different sites can be neglected in the presence of allowed dipole transitions
Dexter (1953). Recently, a third, so-called exciton tandem tunneling mechanism
was proposed Reich and Shklovskii (2016), which is specific of connected
nanocrystals. The most universal process named Förster Resonant Energy
Transfer (FRET) was first observed for QDs by Kagan et al. in specially
designed films containing two different sizes of nanocrystals acting as donors
and acceptors, respectively Kagan et al. (1996). Later it was shown in a
number of works Crooker et al. (2002); Franzl et al. (2004); Lunz et al.
(2010, 2011); Yu (2003); Kawazoe et al. (2005) performed on systems composed
of two different QD species that the luminescence of the smaller dots (donors)
is quenched by the large dots (acceptors), whose emission in turn is enhanced.
These studies demonstrated a dependence of the FRET effect on the NC density
and spatial arrangement. Results of other experiments performed on multilayer
SiOx/SiO2 structures Linnros et al. (1999); Heitmann et al. (2004); Glover and
Meldrum (2005), porous Si Ben-Chorin et al. (1995), three-dimensional (3D)
ensembles of Si Priolo et al. (2001); Balberg et al. (2004) crystallites were
interpreted as a manifestation of exciton migration via FRET-type mechanisms.
The effect has a potential interest for photonics (e.g., photoluminescence
upconversion in QD ensembles Santos et al. (2008)), sensing Wargnier et al.
(2004), lighting and energy harvesting (e.g., by unidirectional energy
transfer in size-gradient layered QD assemblies Crooker et al. (2002); Franzl
et al. (2004) or fractal aggregates Sukhanova et al. (2006); Bernardo et al.
(2014)). We shall discuss it below, in particular, addressing the question
whether the FRET rates and length scales can be tuned by the photonic
environment Rabouw et al. (2014).

The influence of the environment on the exciton-related optical properties of
NCs is the third topic that will be discussed in this overview. The most
common effect is the energy transfer between a photoexcited QD and surrounding
materials, which can be reversible or not. It has the same physical nature as
FRET. In most cases, it is responsible for the photoluminescence (PL)
quenching Chance et al. (1978); Koppens et al. (2011). However, the
environment can be used for engineering the photonic density of states (DOS)
in the vicinity of the QDs, often referred to as the Purcell effect Purcell
(1946). Observed for the first time in QDs about twenty years ago Gérard et
al. (1998), it has recently been discussed with respect to radiative decay
rates of embedded point emitters Senden et al. (2015). Moreover, strong near-
field effects associated with localized surface plasmon resonance (LSPR) in
metallic (nano-)structures can enhance PL emission Pompa et al. (2006);
Shimizu et al. (2002). Moreover, one can think of energy transfer from a
recombining QD exciton to propagating surface plasmons that would carry the
energy over a large distance and then eventually transfer it to another QD (by
creating an exciton), thus allowing for a long-range exciton transport between
two dots, much more efficient than if it occurred directly. Below we consider
the influence of a flat interface between two media on the PL emission and
FRET rates for a QD emitter located in the vicinity of the interface 333We
will not cover here the broad topic of interactions between QDs and metallic
nanostructures supporting localized surface plasmons because it deserves a
separate article. Refs. Lunz et al. (2012); Theuerholtz et al. (2013);
Schreiber2014 ; Rakovich2015  can provide an introduction to this topic..

The article is organized as follows. In Section II, we introduce basic notions
of radiative transitions and discuss their rates (i.e., probabilities per unit
time) for NCs of materials possessing direct or indirect band structure. In
particular, the rates are calculated for intrinsic and doped silicon NCs,
which are either hydrogen-coated or halogenated. Section III is devoted to
non-radiative transitions and considers the Auger recombination, dangling-bond
traps and phonon-assisted relaxation of hot carriers. In Section IV, multiple
exciton generation initiated by a highly excited electron or hole is
considered. Section V is dedicated to the exciton transfer processes between
two QDs via the FRET mechanism. In Section VI, emission decay and FRET rates
near a plane interface between two dielectrics or a dielectric and a metal are
discussed. The last section is left for summary and conclusions.

##  II Light Emission in Nanocrystals

###  II.1 Spontaneous Emission Rate

The emission rate of a point dipole emitter located in an infinite dielectric
medium with the permittivity $\varepsilon_{1}$ is given by Novotny and Hehct
(2012)

$\gamma_{0}=\frac{4k_{1}^{3}|\mathbf{d}|^{2}}{3\varepsilon_{1}\hbar}\,,$ (1)

where $k_{1}=\sqrt{\varepsilon_{1}}\omega/c$ is the modulus of the wavevector,
$\omega$ is the oscillation frequency, $c$ is the velocity of light in vacuum,
$\hbar$ denotes the Planck constant and $\mathbf{d}$ is the transition dipole
moment matrix element. This expression, with some adaptation, describes the
radiative decay rate of an exciton in a QD made of a direct band gap
semiconductor. We shall write it as the inverse of the radiative lifetime:

$\tau_{R}^{-1}\equiv\gamma_{0}^{(QD)}=\frac{4e^{2}E_{if}|{\bf
p}_{if}|^{2}\kappa\sqrt{\varepsilon_{1}}}{3m_{0}^{2}\hbar^{2}c^{3}}\,.$ (2)

Here, $m_{0}$ and $-e$ are the free electron mass and charge, respectively,
and $E_{if}$ and $\mathbf{p}_{if}$ denote the transition energy between the
initial and final states and the momentum matrix element, respectively. The
latter is more natural to use in the electronic band theory of crystals; it
can be related to $\mathbf{d}$ appearing in Equation (1) as
$\mathbf{p}=-im_{0}\omega\mathbf{d}/e$. The parameter $\kappa$ in Equation (2)
arises due to the difference in the dielectric constants of the crystallite
($\varepsilon$) and its surrounding ($\varepsilon_{1}$) and takes account of
the so called depolarization effects Jackson (1998). Hereafter, we consider
the crystallite as a sphere with the radius $R$ and dielectric constant
$\varepsilon$ embedded in a homogeneous medium with the dielectric constant
$\varepsilon_{1}$. In this case, $\kappa$ is given as Thränhardt et al. (2002)

$\kappa=\frac{9\varepsilon_{1}^{2}}{(2\varepsilon_{1}+\varepsilon)^{2}}.$ (3)

This parameter varies within a wide range of values depending on the
dielectric constants of the two materials.

###  II.2 Nanocrystals of Direct-Band-Gap Materials

The majority of III–V and II–VI semiconductors are direct band gap materials.
In this case, in the framework of the envelope function approximation,

${\bf p}_{if}={\bf p}_{cv}\langle F_{c,i}|F_{v,f}\rangle,$ (4)

where ${\bf p}_{cv}$ is the standard bulk-like momentum matrix element of
inter-band transitions, and $F_{c,i}$ and $F_{v,f}$, in case of the photon
emission, stand for the envelope functions of the initial state in the
conduction band and the final state in the valence band, respectively. The
scalar product of these envelope functions determines selection rules for such
transitions.

In the simplest case, the QD is treated as an infinitely deep spherical
potential well with radius $R$ and the conduction and valence bands are
assumed simple and parabolic. Consequently, the electron and hole ground
states are described by the same envelope function and the scalar product in
(4) equals to unity. It means that the radiative recombination rate turns out
to be almost independent of the QD radius. This dependence manifests only in
the transition energy, $E_{if}$. In the limiting case of a strong quantum
confinement regime, where $R\ll a_{ex}$ ($a_{ex}$ denotes the effective
exciton Bohr radius), the exciton binding energy can be neglected. Then, the
model of infinitely deep spherical potential well yields
$E_{if}(R)=E_{g}^{\infty}+\hbar^{2}\pi^{2}/2\mu R^{2}$, where $E_{g}^{\infty}$
is the band gap energy of the bulk semiconductor and $\mu$ is the reduced
effective mass of the electron–hole pair.

The above assumption concerning the extreme strong confinement regime is not
quite realistic; usually the QD radius can be just slightly smaller than
$a_{ex}$ (for instance, $a_{ex}$ = 5.6 nm for CdSe Rogach (2008)) and the
exciton binding energy can be estimated as Brus (1983) $E_{b}\approx
1.79e^{2}/(\varepsilon R)$. Furthermore, the most common semiconductors have a
degenerate valence band with light and heavy hole sub-bands, so that the
confined holes are described by two envelope functions, which are completely
different from the electron’s one Efros et al. (1996). Thus, the matrix
element (4) does depend on $R$ (see, e.g., chapter by Vasilevskiy in the book
Rogach (2008)). Still, the dependence of $\tau_{R}^{-1}$ on the QD size is
rather weak for direct-gap materials as compared to, e.g., silicon NCs
considered below. Using typical parameters for CdSe QDs, $\varepsilon\approx
10$, $\varepsilon_{1}\approx 2$ and ${\bf p}_{cv}^{2}/m_{0}\approx 10$ eV
Efros et al. (1996),

$\tau_{R}^{-1}[\text{s}^{-1}]\approx 3\times 10^{8}E_{if}(R)\,.$ (5)

Thus, $\tau_{R}$ is of the order of a nanosecond, in accordance with
experimental observations.

###  II.3 Silicon Nanocrystals

Since the 1990s, the optical properties of Si NCs have been widely discussed
Pavesi and Turan (2010); Khriachtchev et al. (2012); Ray et al. (2013);
Barbagiovanni et al. (2014); Priolo et al. (2014); Holman et al. (2010);
Belyakov et al. (2004); Pereira et al. (2013); Otsuka et al. (2020). Many
methods have been proposed to increase their emissivity, such as doping with
shallow impurities Tetelbaum et al. (1998); Fujii et al. (1999); Tetelbaum et
al. (2004); Belov et al. (2009); Belyakov et al. (2009); Nomoto et al. (2017);
Klimešová et al. (2012), growth in different matrices Kim et al. (2004);
Klangsin et al. (2008), plasma Nozaki et al. (2007); Zhou et al. (2014), and
colloidal solutions Baldwin et al. (2002); Wolf et al. (2013); Bell et al.
(2014); Zhou et al. (2015); Carroll et al. (2018). As a result, numerous
theoretical predictions and experimental observations of enhanced PL intensity
Tetelbaum et al. (1998); Fujii et al. (1999); Belyakov et al. (2009); Nomoto
et al. (2017), radiative recombination rates Klimešová et al. (2012); Belyakov
and Burdov (2009); Kůsová et al. (2010); Ma et al. (2011, 2012); Wang et al.
(2013); Dohnalová et al. (2013); Poddubny and Dohnalová (2014); Derbenyova and
Burdov (2017, 2018) and quantum efficiency of photon generation Sangghaleh et
al. (2015) have been reported.

It is well known that introduction of shallow impurities is capable of
modifying electronic properties of bulk silicon. Similarly, doping with
shallow impurities influences the electronic structure of silicon NCs Belyakov
and Burdov (2007a, b, c); Chelikowsky et al. (2011); Oliva-Chatelain et al.
(2016); Arduca and Perego (2017); Nomoto et al. (2016); Marri et al. (2017);
Hiller et al. (2017), which, in turn, affects the electron–hole radiative
recombination. It was revealed that doping of Si nanocrystals with P or Li is
(under certain conditions) capable of improving their emittance Fujii et al.
(1999); Belyakov et al. (2009); Klimešová et al. (2012); Yang et al. (2017).
In particular, the performed calculations Belyakov et al. (2009); Belyakov and
Burdov (2009); Derbenyova and Burdov (2017, 2018) show that doping with P or
Li can essentially increase the radiative transition rates.

For NCs with $R\gtrsim 1$ nm, semi-empirical methods are usually employed for
computing their electronic structure, such as envelope function approximation
Hybertsen (1994); Moskalenko et al. (2007); Belyakov et al. (2008) or tight
binding model Delerue et al. (2001). The performed calculations of the phonon-
assisted radiative recombination rates in Si crystallites in the SiO2 matrix
yielded the values varying from $\sim 10^{2}$ to $\sim 10^{5}$ $\text{s}^{-1}$
as the crystallite radius decreases from 3 to 1 nm. Within a simple model of
an infinitely deep spherical potential well, the dependence of the radiative
decay rate on the crystallite radius is $\tau_{R}^{-1}\propto R^{-3}$
[Belyakov et al., 2008]. For the no-phonon radiative transitions, the rates
turn out to be much lower, by one to three orders of magnitude with increasing
the size within the same range. In this case, $\tau_{R}^{-1}\propto R^{-8}$
[Belyakov et al., 2008], i.e., the rate sharply drops as $R$ increases. At the
same time, doping with phosphorus provides a several orders of magnitude
increase of the radiative recombination rates for the no-phonon transitions.
The rates become even 1–3 orders greater than those of the phonon-assisted
transitions. They slowly decrease with increasing the NC size, especially for
high concentration of phosphorus Belyakov and Burdov (2009).

As calculations show, the radiative transitions including both no-phonon and
phonon-assisted ones become faster in small Si crystallites. However, for
crystallites with $R\lesssim 1$ nm, all semi-empirical methods of calculation
turn out to be essentially less accurate. In this case, first-principle
calculations have to be carried out. In Figure 1, the radiative recombination
rates computed at room temperature $T$ within the Casida’s version Casida
(1996) of the time-dependent density functional theory (TDDFT) are depicted
for H-passivated P-doped (Si46H60P, Si70H84P, Si86H76P, Si122H100P,
Si166H124P, Si238H196P), Li-doped (Si42H64Li, Si66H64Li, Si82H72Li,
Si106H120Li, Si172H120Li, Si220H144Li), and undoped (Si47H60, Si71H84,
Si87H76, Si123H100, Si167H124, Si239H196) Si nanocrystals. Here, NCs are
considered in vacuum, therefore, $\varepsilon_{1}=1$, while for Si
$\varepsilon$ = 12. In this case, the rate values become $\sim$16 times lower
compared to the case of Si NCs embedded in silicon dioxide due to the factor
$\kappa\sqrt{\varepsilon_{1}}$.

As seen in Figure 1, generally, doped crystallites demonstrate 1–2 orders of
magnitude faster transitions than intrinsic ones. For Si crystallites doped
with phosphorus, increasing rates of the radiative transitions have been
explained by the efficient mixing of the electronic states in the $\Gamma$ and
X points of the Brillouin zone ($\Gamma$-X mixing) caused by the short-range
field of the phosphorus ion Belyakov et al. (2009); Belyakov and Burdov
(2009). Meanwhile, the radiative decay in the Li-doped crystallites becomes
faster due to the high density of states above the inter-band energy gap (in
the range $\hbar\omega-E_{g}\lesssim k_{B}T$, where $k_{B}$ is the Boltzmann
constant) Derbenyova and Burdov (2018).

Figure 1: Calculated rates of radiative transitions for undoped (small violet
dots), P-doped (big blue dots) and Li-doped (big red dots) Si nanocrystals.

Surface chemistry is another method capable of modifying electronic structure
of nanocrystals Li and Jin (2017); Heuer-Jungemann et al. (2019). This method
is especially efficient for small crystallites, where the role of surface
rises. It has been demonstrated experimentally Dohnalová et al. (2013);
Wheeler et al. (2015); Purkait et al. (2016) that chemical synthesis allows to
fabricate Si nanocrystals with various ligand coatings, which emit light in
visible range with typical nanosecond radiative lifetimes Dohnalová et al.
(2013); De los Reyes et al. (2015); Sinelnikov et al. (2017). Calculations
based on the tight-binding model Poddubny and Dohnalová (2014) performed for
CH3-capped Si nanocrystals confirmed the conclusion on the increase of the
radiative recombination rate due to the formation of direct-like electronic
structure of the crystallite.

Let us now consider some other kind of the surface reconstruction of the
crystallites, the halogenation. Table 1 presents calculated rates of radiative
transitions between the highest occupied molecular orbital (HOMO) and the
lowest unoccupied molecular orbital (LUMO) in completely halogenated with Cl-,
Br- and H-coated silicon crystallites.

Table 1. Calculated rates (s-1) of the radiative HOMO-LUMO transitions in H-,
Cl-, and Br-passivated silicon nanocrystals.

| X = H | X = Cl | X = Br
---|---|---|---
Si35X36 | $3.7\times 10^{5}$ | 3.3 | 2.7
Si59X60 | $2.9\times 10^{4}$ | $4.2\times 10^{4}$ | $3.0\times 10^{2}$
Si87X76 | $1.6\times 10^{3}$ | $2.4\times 10^{2}$ | 36
Si123X100 | $3.6\times 10^{3}$ | $4.2\times 10^{3}$ | $1.7\times 10^{5}$
Si147X100 | $2.2\times 10^{4}$ | $4.3\times 10^{2}$ | 15
Si163X116 | $6.1\times 10^{3}$ | $3.0\times 10^{4}$ | $3.1\times 10^{3}$
Si175X116 | $3.5\times 10^{4}$ | $1.3\times 10^{2}$ | $3.5\times 10^{2}$
Si317X172 | $1.4\times 10^{4}$ | $2.4\times 10^{3}$ | 63

It is possible to see from Table 1 that the surface halogenation significantly
slows down the radiative recombination in silicon nanocrystals, especially in
the case of the bromine coating. Such a slowing down is due to the spatial
separation of the electron densities of the HOMO and LUMO states, which takes
place in halogenated Si crystallites Derbenyova et al. (2017, 2019); Derbeneva
et al. (2019). The separation turns out to be more pronounced in the Br-
passivated crystallites, in which the HOMO states are strongly concentrated
near the Br atoms. Presumably, the bromine atoms produce strong lattice
distortions, and the stress fields have an additional localizing effect on the
electron density. An effect of the rate reduction was also found for
completely fluorine-passivated Si nanocrystals Ma et al. (2012).

##  III Nonradiative Processes

As is well known, the emissivity of any light-emitting entity depends not only
on the radiative transitions’ rate, but also on the speed of nonradiative
decay processes. In the case of NCs, these include Auger recombination,
capture of photo-electrons on dangling bonds and cooling of hot carriers.
Below, we briefly discuss all these processes.

###  III.1 Auger Recombination

In order for the Auger process to occur, at least one negative or positive
trion (exciton + one electron or exciton + one hole) has to be initially
excited in a nanocrystal, as shown in Figure 2. The trion turns into a high-
energy single-particle excitation as indicated by the arrows in the figure.

The rate of an Auger-type transition can be written as

$\tau_{A}^{-1}=\frac{2\Gamma}{\hbar}\sum_{f}\frac{|U_{if}|^{2}}{\Gamma^{2}+(E_{f}-E_{g})^{2}},$
(6)

where $U_{if}$ is the matrix element of the two-particle Coulomb interaction
operator, the two-electron wave functions are approximated by products of the
single-electron ones and the $\delta$-function expressing the energy
conservation in the Fermi’s Golden Rule has been broadened with the half-width
$\Gamma=10$ meV. All the possible final states of the Auger electron with
energy $E_{f}$ (shown in Figure 2) are summed up. The initial states were
chosen so that electrons and holes are in the LUMO and HOMO states,
respectively.

Figure 2: Auger recombination in nanocrystals via eeh and ehh processes.
Arrows indicate Auger transitions between the initial and final states.

Usually, the Auger recombination is an efficient process that can “shunt”
other photon-assisted processes such as light emission or multiple exciton
generation. For instance, in Si crystallites, the Auger rates can be
sufficiently high, as the measurements Stolle et al. (2017); Pevere et al.
(2015) and calculations Derbenyova and Burdov (2017); Sevik and Bulutay
(2008); Delerue et al. (1995); Mahdouani et al. (2009); Kurova and Burdov
(2010) show. Therefore, suppression of the Auger process is an important
practical problem.

It turns out that coating with halogens can slow down the Auger processes in
Si NCs. Below we present the calculated Auger rates and demonstrate their
reduction due to the surface halogenation of Si crystallites. The calculations
take into account both the long-range ($U_{0}({\bf r}_{1},{\bf r}_{2}$)) and
the short-range ($U_{1}({\bf r}_{1},{\bf r}_{2}$)) parts of the
carrier–carrier interaction. The first one is a macroscopic electrostatic
field modified by the nanocrystal boundary,

$\displaystyle U_{0}({\bf r}_{1},{\bf r}_{2})$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{\varepsilon|{\bf r}_{1}-{\bf
r}_{2}|}+\frac{e^{2}(\varepsilon-1)}{\varepsilon R}$ (7)
$\displaystyle\times\sum_{l=0}^{\infty}\frac{(l+1)P_{l}(\cos\theta)}{l\varepsilon+l+1}\frac{r_{1}^{l}r_{2}^{l}}{R^{2l}}\,,$

while the second one describes at a microscopic level the point charges
interaction at short distances:

$U_{1}(r)=\frac{e^{2}}{r}[Ae^{-\alpha r}+(1-A-1/\varepsilon)e^{-\beta r}]\,.$
(8)

Here, $r=|{\bf r}_{1}-{\bf r}_{2}|$, $A=1.142$, $\alpha=0.82/a_{B}$ and
$\beta=5/a_{B}$ Belyakov and Burdov (2009), where $a_{B}$ stands for the Bohr
radius, and we set $\epsilon_{1}=1$ as before. The calculated values of
$\tau_{A}^{-1}$ are listed in Table 2 for both eeh and ehh trion annihilation
in the SinXm crystallites considered here.

Table 2. Calculated Auger rates ($\times 10^{10}$ s-1) 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)
Si35X36 | 390 | 3 | 4 | 40 | 163 | 10
Si59X60 | 78 | 0.3 | 7 | 5 | 27 | 19
Si87X76 | 120 | 4 | 2 | 1780 | 59 | 11
Si123X100 | 8 | 3 | 280 | 108 | 25 | 24
Si147X100 | 59 | 84 | 2 | 474 | 20 | 65
Si163X116 | 120 | 21 | 100 | 47 | 9 | 19
Si175X116 | 270 | 2 | 2 | 161 | 90 | 0.8
Si317X172 | 0.3 | 0.4 | 0.02 | 5 | 3 | 78

The surface halogenation of Si nanocrystals also slows down the Auger
recombination according to the data presented in Table 2. In order to
quantitatively analyse the effect of halogenation on the Auger rates, we have
calculated the average decimal logarithms of $\tau_{A}^{-1}$ for the Br-, Cl-
and H-passivated nanocrystals. In the case of the eeh Auger recombination one
has obtained: $\langle\lg(\tau_{A(\text{H})}^{-1})\rangle=11.66$,
$\langle\lg(\tau_{A(\text{Cl})}^{-1})\rangle=10.55$ and
$\langle\lg(\tau_{A(\text{Br})}^{-1})\rangle=10.60$. In the case of the ehh
Auger recombination the rates also decrease but not so strongly:
$\langle\lg(\tau_{A(\text{H})}^{-1})\rangle=11.86$,
$\langle\lg(\tau_{A(\text{Cl})}^{-1})\rangle=11.44$ and
$\langle\lg(\tau_{A(\text{Br})}^{-1})\rangle=11.20$. It should be noted also
that strong reduction of the Auger recombination rates was theoretically shown
by Califano Califano (2018) for GaSb NCs whose surfaces were passivated with
atoms of electronegative elements.

###  III.2 Capture on Dangling Bonds

There is one more relatively fast process inhibiting efficient light emission
from NCs, which is photo-carrier trapping on surface defects called Pb-centres
or dangling bonds. The Pb-centres produce rather deep energy levels within the
nanocrystal band gap, which act as electron traps. The capture on neutral
dangling bonds is a multi-phonon process, and its rate strongly depends on
temperature. The earlier performed calculations Delerue and Lannoo (2013) for
Si crystallites yielded $\tau_{C}^{-1}$ shown in Figure 3 as function of the
nanocrystal radius in comparison with the rates of the Auger recombination and
radiative transitions.

It can be seen from Figure 3 that for the NC radius greater than $\sim$1 nm,
the capture rate becomes larger than the radiative recombination rate.
Moreover, starting from $R$ close to 1.5 nm, the trapping on dangling bonds
turns out to be even faster than the Auger recombination. Obviously, for
nanocrystals whose radii are greater than 1.5 nm, the nonradiative trapping on
the surface defects suppresses all the other competing relaxation processes
inside the NC if the surface dangling bonds are not passivated.

Figure 3: Calculated rates of carrier trapping on dangling bonds compared to
the rates of radiative phonon-assisted transitions and Auger recombination in
Si nanocrystals.

###  III.3 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 to minimize the system energy. In bulk
semiconductors and QWS, where the electronic energy spectra are continuous,
such a relaxation is accompanied by the emission of phonons caused by the
interaction between electrons and lattice vibrations. This interaction also
plays a major role in QDs and it is generally accepted that the mechanisms are
essentially the same in nanocrystals and bulk materials.

The most universal mechanism of coupling between the electrons (or holes) and
long-wavelength acoustic phonons is through the volume deformation potential.
The bottom of a non-degenerate band (e.g., conduction band in materials with
zinc-blend structure) is shifted proportionally to the (local) relative
variation of the volume Blacha et al. (1984),

$\hat{H}_{e-AP}=a_{c}\sum_{\nu}{\left(\nabla\cdot\hat{\bf u}_{\nu}(\bf
r)\right)}\;,$

where $a_{c}$ is the bulk deformation potential constant, and $\hat{\bf
u}_{\nu}$ denotes the atomic displacement operator corresponding to an
acoustic phonon mode $\nu$ (it is expressed in terms of phonon creation and
annihilation operators in usual way). For optical phonons, there are two
mechanisms: (i) a universal optical deformation potential (ODP) coupling and
(ii) the Fröhlich one characteristic of polar materials. In bulk
semiconductors with cubic structure, for long-wavelength optical phonons, the
ODP coupling vanishes by symmetry for any non-degenerate band but it is non-
zero for holes near the top of the valence band Blacha et al. (1984). Its
expression can be found in the book Rogach (2008) (Chapter 8) or in Rolo et
al. (2008). The Fröhlich interaction Hamiltonian for electrons is simply given
by

$\hat{H}_{e-OP}=-e\sum_{\nu}{\hat{\varphi}_{\nu}(\bf r)}\;,$

where $\hat{\varphi}_{\nu}$ is the electrostatic potential operator
corresponding to an optical phonon mode $\nu$. With the known electron and
hole wavefunctions and phonon displacements obtained from lattice dynamics
equations, the calculation of the coupling matrix elements is straightforward.
Even though the Fröhlich interaction usually dominates in QDs made of polar
semiconductors, the ODP mechanism is important for some of them (e.g., InP) as
revealed by modelling of QD resonant Raman spectra Rolo et al. (2008).

The so-called rigid-ion model is more suitable when describing the
electron–phonon interaction in multi-valley semiconductors, such as Si and Ge
Glembocki and Pollak (1982), where the processes of inter-valley scattering
are often important. Within this model, the electron–phonon interaction is
described by the following operator (which describes both acoustic and optical
phonons),

$\hat{H}_{e-P}=-\sum_{\nu;\,{\bf n},s}{\hat{\bf u}}_{\nu}({\bf R}_{{\bf
n}s})\cdot\nabla V_{at}({\bf r}-{\bf R}_{{\bf n}s}).$

Here, $V_{at}({\bf r}-{\bf R}_{{\bf n}s})$ stands for the electron potential
energy in the field of the $s$-th atom situated in the ${\bf n}$-th primitive
cell of the lattice and ${\bf R}_{{\bf n}s}$ is the position-vector of this
atom in equilibrium. In practice, atomic pseudopotentials are used.

The discreteness of the energy spectrum in quantum dots implies that the rate
of the single-phonon emission should strongly depend on the level spacing
Inoshita and Sakaki (1992); in particular, electronic transitions causing the
phonon emission should be impossible if the spacing is larger than the phonon
energy. This simple idea gave rise to the “phonon bottleneck” concept, which
predicts the inefficiency of hot carrier relaxation by emission of phonons in
QDs Bockelmann and Bastard (1990). However, this prediction relies on the
assumption that the phonon emission is irreversible, with a probability
described by the perturbation theory (Fermi’s Golden Rule), which may not be
reliable in QDs. The electron–phonon interaction can be enhanced in
nanocrystals because of the spatial confinement of both electrons and phonons
and, therefore, multiple scattering processes are important. It means that the
electron–phonon interaction in QDs, in principle, must be treated in a non-
perturbative and non-adiabatic way, leading to the energy spectra described by
polaronic quasi-particle excitations Stauber et al. (2000); Jacak et al.
(2003); Vasilevskiy et al. (2004); Sauvage et al. (2002); Miranda et al.
(2006); Sarkar et al. (2005); Stauber and Vasilevskiy (2009). In other words,
virtual transitions between different electronic levels, assisted by phonons
and not requiring energy conservation may be important enough to guarantee a
significant modification of the exciton energy spectrum and dynamics Rogach
(2008). Thus, does the phonon bottleneck in QDs really exist? Experimental
results still are not completely conclusive. On the one hand, an efficient
relaxation of optically created excitons was reported in a number of works
studying self-assembled QDs (SAQDs) Marcinkevičius et al. (2001); Sun et al.
(2006); Müller et al. (2003), with both PL emission rise time Marcinkevičius
et al. (2001); Sun et al. (2006) and photo-induced intraband absorption decay
time Müller et al. (2003) below or of the order of 10 ps. Studies performed on
nanocrystal QDs Schaller et al. (2005); Hendry et al. (2006), where exciton
energy level spacings are larger than in SAQDs, also revealed ultrafast
intraband relaxation with a characteristic time in the picoseconds’ domain,
which is characteristic of CdSe and CdTe QDs in general Rogach (2008). On the
other hand, there are published experimental results indicating that the
relaxation of optically created excitons can be slow (in the nanoseconds’
range), both in self-assembled Heitz et al. (2001); Urayama et al. (2001) and
nanocrystal Guyot-Sionnest et al. (2005); Nozik (2001) QDs.

It is important to realize that the polaron concept by itself cannot explain
intraband relaxation of carriers in QDs as polaron is a stationary state of an
electron (or exciton) coupled to optical phonons. Some additional interactions
should therefore be responsible for the polaron relaxation Stauber and
Vasilevskiy (2009). Several possible mechanisms of hot carrier relaxation in
QDs have been proposed:

  1. (i)

The polaron has a rather short lifetime Li et al. (1999) because of the
anharmonic effects that lead to a fast decay of nanocrystal’s optical phonons
forming the polaron.

  2. (ii)

Acoustic phonons can provide the possibility of transitions between different
(exciton-) polaron states formed mostly by the interaction with optical
phonons; the polaron spectrum is discrete but relatively dense owing to the
non-adiabaticity of this interaction Vasilevskiy et al. (2004). If the
acoustic phonon spectrum is continuous, this additional interaction would
drive the polaron dynamics towards equilibrium.

  3. (iii)

In the strong confinement regime where the electron’s kinetic energy is larger
than the electron–hole interaction, the electron (eventually dressed by
phonons and forming the polaron) can relax by an Auger-type mechanism. The
excess energy is first transferred from the electron to the hole through their
Coulomb interaction and the subsequent hole’s cooling occurs via emission of
acoustic phonons Efros et al. (1995); Narvaez et al. (2006). It can be
feasible because the hole level spacings are relatively small in QDs and match
the continuum of acoustic phonon energies.

There is no final conclusion concerning the relevance of each of the three
mechanisms to the breaking of the phonon bottleneck in QDs; however, the first
one seems to be the most popular in the literature, even though calculations
yielded too low relaxation rates for this mechanism, of the order of 1 ns-1,
for a typical quantum dot considering anharmonic (Grüneisen) parameters
characteristic of bulk materials and the Fröhlich interaction with spherically
confined optical phonons and electrons Stauber and Vasilevskiy (2009).
Moreover, the electron–phonon interaction in nanocrystals of non-polar
materials (see below) may be too weak to make the polaron effect important. If
this is the case, the relaxation proceeds via multi-phonon processes (like
temperature-induced interband excitation of charge carriers in bulk
semiconductors). Usually, multi-phonon processes are much slower compared to
the single-phonon transitions. Therefore, the bottleneck effect does not
disappear completely.

Estimations of the single-phonon relaxation rates in Si nanocrystals Prokofiev
et al. (2014); Poddubny et al. (2010) yield typical values of the order of
$10^{10}$–$10^{12}$ s-1. At the same time, the multi-phonon relaxation rates
calculated within the Huang–Rhys model Delerue and Lannoo (2013); Prokofiev et
al. (2009); Moskalenko et al. (2012) for various transitions vary from
$10^{7}$ s-1 to $10^{11}$ s-1 for nanocrystals whose diameters do not exceed 5
nm. Note that the rates sharply reduce as the nanocrystal diameter decreases.
Similar values of the multi-phonon relaxation rates were obtained
experimentally for InGaAs/GaAs SAQDs Ohnesorge et al. (1996). Therefore, the
precise reason for very fast exciton dynamics characteristic of CdSe and CdTe
colloidal QDs remains an open question, in our view.

##  IV Multiple Exciton Generation

As demonstrated above, the surface halogenation suppresses both radiative and
Auger recombination in nanocrystals. These processes are reverse ones with
respect to the carrier multiplication (or multi-exciton generation): they tend
to decrease the number of excitons in a 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 excited electron–hole pair
which then reduces its energy creating one more electron-hole pair with lower
energy. As a result, two (or even more) excitons can arise in the system after
absorption of a single photon.

Figure 4: Schematic representation of the electron- (left) and hole-initiated
(right) process of multi-exciton generation in a nanocrystal. Initial electron
configurations are shown. Dashed arrow indicates highly excited exciton with
the excitation energy $E_{0}$ created by an absorbed photon. Vertical solid
arrows indicate electron transitions to the final states. After the
transition, the number of excitons increases by one.

The multi-exciton generation is a fundamental process for photovoltaics, where
light energy transforms into electric current. Its occurrence in NCs has been
experimentally confirmed Klimov (2014); Stolle et al. (2017); Beard et al.
(2007); Nozik (2010); de Boer et al. (2011); Trinh et al. (2012). In order to
be more efficient, this process should be faster than other competitive
processes taking place along with exciton generation, such as inter-band
radiative recombination or Auger recombination. From this point of view,
slowing the latter down caused by surface halogenation is an extremely
positive factor. It is important to understand how halogen passivation
influences the multi-exciton generation itself.

The rates of the multi-exciton generation in the Si317X172 crystallite (X = H,
Cl, and Br) can be calculated using the following relation,

$\tau_{G}^{-1}=\frac{2\Gamma}{\hbar}\sum_{f_{1}}\sum_{i_{2}}\sum_{f_{2}}\frac{|\langle\Psi_{i_{1}i_{2}}|{\hat{U}}|\Psi_{f_{1}f_{2}}\rangle|^{2}}{\Gamma^{2}+(E_{i_{1}}+E_{i_{2}}-E_{f_{1}}-E_{f_{2}})^{2}},$
(9)

where ${\hat{U}}={\hat{U}}_{0}+{\hat{U}}_{1}$, as before, while
$\Psi_{i_{1}i_{2}}$ and $\Psi_{f_{1}f_{2}}$ are the products of the single-
particle Kohn–Sham wave functions of the initial or final electron states
participating in the transition, as shown in Figure 4. Here, we present
calculated the rates for the Si317X172 crystallite within the range of excess
energy $0<\Delta E<0.6$ eV, where $\Delta E=E_{i_{1}}-E_{LUMO}-E_{g}$ for the
process initiated by highly excited electron, and $\Delta
E=E_{HOMO}-E_{f_{1}}-E_{g}$ for the hole-initiated process. The calculated
rates are shown in Figure 5. It is evident that the rates rise globally as the
excess energy increases, because of the considerable increase in the number of
possible states participating in the transitions with increasing $\Delta E$,
which opens up many new channels for the realization of exciton generation.

It is important to emphasize that the bromination of a Si crystallite
increases the exciton generation rates compared to an H-passivated
crystallite, especially if the process is initiated by a highly excited hole.
The rates of exciton generation in chlorinated Si crystallite turn out to be
lower than those in the hydrogenated one at small excess energies. Meanwhile,
upon approaching $\Delta E\sim 0.5$ eV, $\tau_{G}^{-1}$ in the Si317Cl172
crystallite increases and tends to the typical values observed in Si317H172
crystallite. Accordingly, it is possible to conclude that halogen coating of
Si crystallites, at least, does not reduce their ability to generate excitons,
particularly when the excess energies are not too small. This is in contrast
with the radiative and Auger recombination processes, where the rates became
substantially lower due to the halogenation.

Figure 5: Calculated rates of the exciton generation initiated by highly
excited (a) electron and (b) hole (as shown in Figure 4) in Si317X172
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 increase the efficiency of the
photon-to-exciton conversion, which is defined by an excess of the number of
created excitons ($n$) over the number of absorbed photons ($N$) Klimov
(2014); Haverkort et al. (2018): $\eta=n/N>1$ (internal quantum efficiency).
There are many experimental works in which external quantum efficiency was
measured in Si crystallites Stolle et al. (2017); Beard et al. (2007); Nozik
(2010); de Boer et al. (2011); Trinh et al. (2012), as well as in crystallites
of IV–VI, II–VI or III–V semiconductors Klimov (2014); Schaller and Klimov
(2004); Nair and Bawendi (2007); Davis et al. (2015); Hu et al. (2016); Smith
et al. (2017). The authors reported on the observation of multi-exciton
generation in the investigated systems.

On the theoretical side, the consideration of the exciton kinetics in the
halogen-coated Si crystallites Derbenyova et al. (2019) has revealed strong
dependence of $\eta$ on the quantitative relationship between the rates
$\tau_{G}^{-1}$ and $\tau_{A}^{-1}$. According to the obtained results the
decrease in the Auger rate (caused by the halogenation), and its absence in
the multi-exciton generation rate, is accompanied by a gradual increase in the
quantum efficiency of the order of a few tens of percentage points.

##  V FRET in Ensembles of Nanocrystals

All the processes considered above may occur in isolated nanocrystals.
Meanwhile, usually in experiments, as was already pointed out in the
Introduction, one deals with ensembles of NCs, where a non-radiative energy
exchange between them takes place and strongly influences the ensemble
photoluminescence Crooker et al. (2002); Franzl et al. (2004); Lunz et al.
(2010, 2011); Yu (2003); Kawazoe et al. (2005); Meldrum et al. (2009);
Belyakov et al. (2014); Derbenyova et al. (2016). Such an energy transfer
occurs by means of the electron tunnelling if the NCs are connected Reich and
Shklovskii (2016) or through the Förster-type migration of excitons Förster
(1948) if the NCs are separated in space. Below we consider the latter
mechanism in some detail.

The Förster resonant exciton transfer (FRET) takes place mainly through the
dipole–dipole interaction444It has been shown Curutchet2008  that, in contrast
to the case of FRET between donor and acceptor molecules, where the dipole
approximation breaks down at lengthscales comparable to molecular dimensions,
it works fairly well when donor and/or acceptor is a spherical QD, even at
contact donor-acceptor separations. of two QDs (a donor and an acceptor). The
Quantum Electrodynamics theory of FRET developed in Andrews and Sherborne
(1987) reproduces the results obtained in a simpler way by Förster Förster
(1948) and Dexter Dexter (1953) who considered the electrostatic interaction
of two dipoles in two adjacent crystallites Allan and Delerue (2007),

$V({\bf r}_{1},{\bf r}_{2})=\frac{\kappa e^{2}}{\epsilon_{1}b^{3}}\left[{\bf
r}_{1}\cdot{\bf r}_{2}-3\frac{({\bf r}_{1}\cdot{\bf b})({\bf r}_{2}\cdot{\bf
b})}{b^{2}}\right]\,,$ (10)

where ${\bf b}$ is the inter-crystallite centre-to-centre vector. In one
crystallite, the electron–hole pair annihilates and transfers its energy into
a neighbouring crystallite where a new electron–hole pair is excited. Thus,
virtual transfer of excitons between two crystallites can be realized without
charge transfer.

In order to calculate the rate of the Förster exciton transition from a QD
with a radius $R_{1}$ into a neighbouring one with a radius $R_{2}$, we use,
as before, the Fermi’s Golden Rule:

$k_{F}=\frac{2\Gamma}{\hbar}\frac{|\langle\Psi_{i}|{\hat{V}}|\Psi_{f}\rangle|^{2}}{\Gamma^{2}+(E_{g}(R_{1})-E_{g}(R_{2}))^{2}}\,.$
(11)

Here, ${\hat{V}}$ is the dipole interaction operator identical to (10),
$\Psi_{i}=\psi_{c}({\bf r}_{1})\psi_{v}({\bf r}_{2})$ is the wave function of
the initial two-particle state with the energy coinciding with the energy gap
of the first crystallite, $E_{g}(R_{1})$, and $\Gamma$ is a phenomenological
damping parameter. Initially, there is one electron in the conduction band of
the first crystallite, with the wave function $\psi_{c}({\bf r}_{1})$, while a
hole exists in the valence band. In the second QD, the electron occupies a
valence band state described by the wave function $\psi_{v}({\bf r}_{2})$. In
the final state, the system has the wave function $\Psi_{f}=\psi_{v}({\bf
r}_{1})\psi_{c}({\bf r}_{2})$, corresponding to the electron-hole pair
transferred to the second QD.

The result is essentially the same as if one considered the interaction
between two transient point dipoles, $\bf d_{D}$ and $\bf d_{A}$ (located at
points $\bf r_{1}$ and $\bf r_{2}$) assuming that they are stationary. The
square of the matrix element in (11) (the donor–acceptor coupling parameter)
can be written as Stavola et al. (1985)

$J^{2}=\nu\frac{\kappa^{2}{d_{D}^{2}d_{A}^{2}}}{\epsilon_{1}^{2}b^{6}}\,,$
(12)

where $\nu$ is a factor of the order of unity that takes into account the
relative orientation of the dipoles Lyo (2009), $\nu=\frac{2}{3}$ if these
orientations are completely random.

The broadening of the exciton transfer resonance, $\Gamma$ can be related to
the spectral overlap of the emission and absorption spectra of the donor and
acceptor, respectively Dexter (1953), and the transfer rate can be expressed
as Jang et al. (2014)

$\frac{k_{F}}{\gamma_{0}^{(D)}}=\frac{3\kappa^{2}c^{4}}{4\pi\epsilon_{1}^{2}b^{6}}Q_{A}\int_{-\infty}^{+\infty}\frac{d\omega}{\omega^{4}}I_{A}(\omega)L_{D}(\omega)\,,$
(13)

where $\gamma_{0}^{(D)}$ is the spontaneous emission rate of the donor,
$I_{A}(\omega)$ is the absorption lineshape function of the acceptor,
$L_{D}(\omega)$ 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,

$Q_{A}=\left(\frac{\pi\hbar c}{E_{g}(R_{2})}\right)^{2}\gamma_{0}^{(A)}$ (14)

with $\gamma_{0}^{(A)}$ being the spontaneous emission rate of the acceptor.
Notice that the depolarization factor $\kappa^{2}$, Equation (3), which
distinguishes QD donors and acceptors from, e.g., molecules, has been included
explicitly in Equation (13).

The FRET rate decreases rather quickly with the donor-acceptor distance,
$k_{F}=\gamma_{0}^{(D)}(\frac{b_{0}}{b})^{6}$, where $b_{0}$ is a parameter of
the dimension of length named “Förster radius” whose definition is clear from
Equation (13).

In ensembles formed of direct-band-gap II–VI or III–V semiconductor
crystallites (QDs), the exciton transfer has the rates
$\sim$$10^{8}$–$10^{9}$$\cdot$s-1, as measurements Kagan et al. (1996);
Crooker et al. (2002); Lunz et al. (2010, 2011) and calculations Allan and
Delerue (2007); Scholes and Andrews (2005); Lyo (2009) show. Notice that
$k_{F}$ depends on $b$ and the above values probably correspond to somewhat
different donor–acceptor distances. Experimental observations of the exciton
transfer were carried out with ensembles of closely packed monodisperse CdSe
nanocrystals as well as of two-size three-dimensional mixtures Lunz et al.
(2010) and bilayered CdSe nanocrystal systems Lunz et al. (2011). It was shown
that the inter-layer transfer in bilayered ensembles with controlled donor-
acceptor separation turns out to be more efficient than in the 3D ensembles of
the monodisperse and the two-size-mixed crystallites Lunz et al. (2011). Note
that the measured and computed FRET rates are of the same order of magnitude
as the radiative recombination rates in high-density ensembles of colloidal
II–VI crystallites. Moreover, a few measurements of the Förster radius have
been performed by controlling the distance between two different groups of QDs
of different size, acting as donors and acceptors Shafiei et al. (2011), or by
preparing a homogeneously blended solid-state films Mork et al. (2014)
composed of two groups of dots. In both works, CdSe/ZnS core–shell QDs were
used and the reported results for the Förster radius are 14–22 nm and 8–9 nm,
respectively; notice that the former work Shafiei et al. (2011) used larger
nanocrystals. Despite the uncertainty in experimental conditions and
difficulty to evaluate the spectral overlap in Equation (13) for individual
QDs, there is a consensus that $R_{F}$ typically is of the order of 10 nm for
highly luminescent colloidal nanocrystals.

In ensembles of silicon crystallites the exciton transfer turns out to be much
slower than the radiative recombination, its rates are two to three orders of
magnitude lower Allan and Delerue (2007); Belyakov and Burdov (2011); Gusev et
al. (2011) (smaller than $\sim$$10^{3}$$\cdot$s-1). Doping of silicon
nanocrystals with phosphorus allows to increase the rates up to the values
comparable with those of radiative recombination Belyakov and Burdov (2013)
($\lesssim 10^{7}$ s-1 for nanocrystal radius $R\gtrsim 1$ nm). Nevertheless,
these values remain still smaller than the FRET rates for colloidal QDs made
of direct-gap semiconductors.

For the sake of completeness, Förster-type resonant energy transfer from a QD
exciton can occur also when the final elementary excitations have a different
nature, for instance, they are phonons or molecular vibrations. The latter can
involve ligand molecules present in the vicinity of a colloidal QD or solvent
molecules and is particularly important for quantum dots emitting in the
infrared Aharoni et al. (2008); Wen et al. (2016). Such a process was termed
electronic-to-vibrational energy transfer (EVET) Aharoni et al. (2008). The
importance of EVET was demonstrated by comparing the luminescence properties
of HgTe QDs dissolved in two chemically identical solvents: H2O and D2O Wen et
al. (2016).

##  VI QD Emitters Near a Flat Interface

Now, we shall discuss the influence of a flat interface between two media on
the emission and FRET rates for a QD emitter located at a certain distance
from it. If the characteristic distances (such as that between the emitter and
the interface) are small compared to the EM wavelength, one can treat the
problem in the electrostatic approximation where one neglects both the
retardation effects and the magnetic field associated with the electric field
present in the media; then, the image dipole method can be used to take into
account the effect of surface polarization induced by the dipole Santos et al.
(2014). A more general approach consists in using the dyadic Green’s function
formalism Amorim et al. (2017). If both dielectrics are dispersionless, the
spontaneous decay and FRET are affected through the photonic density of states
(DOS) renormalization due to the reflection of electromagnetic (EM) waves at
the interface. However, even in this geometrically simplest situation, there
is a less trivial effect produced by EM waves created by the polarization in
the second medium, totally reflected at the interface so that only an
exponentially decreasing field amplitude of theirs reaches the dipole. Yet,
this effect can be dominating at small distances, not only for radiative decay
Lukosz and Kunz (1977) but also for FRET. When the second medium is a metal,
several contributions to both decay and transfer processes arise that can be
associated with (i) propagating EM waves (so-called radiative losses),
affected by the presence of the second medium; (ii) coupling to propagating
surface plasmons (SPs); and (iii) Ohmic losses (when the exciton energy is
irreversibly transferred to heat via electron scattering in the metal).

Even though the subject has been studied during several decades Lukosz and
Kunz (1977); Drexhage (1970); Chance et al. (1978) and is described in topical
reviews Ford and Weber (1984) and books Novotny and Hehct (2012), we find some
significant effects that were overlooked in previously published discussions
and, in our opinion, are important for potential applications of FRET in
photonics and energy harvesting. In particular, we will show that (1) if a QD
embedded in a dielectric matrix is located near an interface with another
dielectric (a substrate) having a lower dielectric constant, its emission is
polarized parallel to the interface; (2) in the opposite case of substrate’s
dielectric constant higher than that of the matrix, the emission and transfer
rates are strongly enhanced (without dissipation) near the interface; and (3)
the resonant coupling between SPs propagating along a metal/dielectric
interface and excitons confined in QDs located at a distance of the order of
the light wavelength from the interface, can be used for long-range FRET. We
present these rates calculated for typical colloidal nanocrystal QDs.

###  VI.1 Radiative Lifetime Near Interface

We shall consider the QD emitter as a point dipole.555See Gordon2014  for
discussion of possible effects beyond this approximation. The radiative decay
rate in the presence of other bodies is determined by the total field that
acts on the emitter (created by itself and scattered by the bodies) and given
by Novotny and Hehct (2012)

$\gamma=\mbox{Im}\left[\bm{d}^{\star}\cdot\left(\mathbf{E}_{0}(\mathbf{r})+\mathbf{E}_{s}(\mathbf{r})\right)\right]/2\hbar\;,$
(15)

where $\mathbf{E}_{0}$ is the dipole field in infinite space. Here,
$\bm{d^{\star}}$ is the complex-conjugate of the classical dipole moment, the
necessary adaptation for the quantum mechanical transition dipole matrix
element consists in multiplying it by 2 in the final result,
$\bm{d}\rightarrow 2\mathbf{d}$ Novotny and Hehct (2012). The electric
component of the scattered field created by an emitting dipole located in the
origin of the reference frame can be written in terms of the Green’s dyadic
Novotny and Hehct (2012), which is specific of system’s geometry:

$\mathbf{E}_{s}(\mathbf{r}^{\prime})=4\pi
k_{1}^{2}\mathbf{G}(\mathbf{r}^{\prime},\mathbf{0})\cdot\bm{d}\,.$ (16)

The $3\times 3$ dyadic matrix $\mathbf{G}(\mathbf{r}^{\prime},\mathbf{r})$ is
determined by the Fresnel coefficients, $r^{(p)}$ and $r^{(s)}$, and its
explicit expression can be found in Amorim et al. (2017).

From Equation (15), one obtains for a dipole perpendicular to the interface
(i.e., “vertical”, see inset in Figure 6):

$\frac{\gamma^{(V)}}{\gamma_{0}}=1+\frac{3}{2}\mbox{Re}\int_{0}^{\infty}{\frac{s^{3}}{\sqrt{1-s^{2}}}r^{(p)}\exp{(2ik_{1}h\sqrt{1-s^{2}})}ds}\;,$
(17)

and for a dipole parallel to the interface (i.e., “horizontal”):

$\displaystyle\frac{\gamma^{(H)}}{\gamma_{0}}=1+\frac{3}{4}\mbox{Re}\int_{0}^{\infty}\frac{s}{\sqrt{1-s^{2}}}\left[r^{(s)}-(1-s^{2})r^{(p)}\right]$
$\displaystyle\times\exp{(2ik_{1}h\sqrt{1-s^{2}})}ds\,.$ (18)

Here, $\gamma_{0}$ is the decay rate in an infinite medium with dielectric
constant $\varepsilon_{1}$, given by Equation (1). The integrals in Equations
(17) and (18) with respect to the normalized in-plane wavevector, $s=q/k_{1}$,
can be divided into two parts, one from 0 to 1 corresponding to propagating
waves in the upper half-space and the other from 1 to $\infty$ representing
evanescent waves with imaginary wavevector component along $z$ axis
perpendicular to the interface, $k_{1z}$. The latter type of waves exist in
two cases: (i) if $\varepsilon_{1}<\varepsilon_{2}$ and both dielectric
constants are positive and (ii) if
$\varepsilon_{2}^{\prime}\equiv\mbox{Re}(\varepsilon_{2})<0$, i.e., when the
second medium is a metal.

In the case of two non-dispersive dielectrics, $\gamma$ given by Equations
(17) and (18) represents only radiative losses of dipole’s energy,
renormalized by the back action of the scattered waves. Yet, for
$\varepsilon_{1}<\varepsilon_{2}$ there are EM waves (excited by the dipole)
propagating in the lower half-space (with real $z$-component of the wavevector
inside medium 2, $k_{2z}$) and experiencing total internal reflection at the
interface with the upper medium, possessing imaginary $k_{1z}$. The Fresnel
coefficient $r^{(p)}$ has an imaginary part associated with these waves. If
the interaction of the dipole with these evanescent waves is neglected (dotted
lines in Figure 6), i.e., we set the integral from 1 to $\infty$ in Equations
(17) and (18) equal to zero, then $\gamma$ becomes independent of dipole’s
distance from the interface.

As can be seen from this figure, the emission decay rate is strongly enhanced,
$\gamma>>\gamma_{0}$, at small distances ($k_{1}h<1$) for both dipole
orientations because of the strong interaction with the evanescent waves. It
has been pointed out for plane interfaces by Lukosz and Kunz Lukosz and Kunz
(1977), and it is similar to the coupling to the whispering gallery modes (a
kind of surface EM waves) observed for emitters placed inside a micrometer-
size dielectric sphere Benner et al. (1980); Scheneipp and Sandoghbar (2002).
As we will show in the next section, this coupling can be used for FRET
enhancement.

Figure 6: Dependence of the radiative decay rate, $\gamma$, on the dielectric
constant ratio $\varepsilon_{1}/\varepsilon_{2}$ for two two different
distances to the interface between them, $k_{1}h=0.1$ and 0.2. The dotted
shows the dependence when the contribution of evanescent waves (that exists
for $0<\varepsilon_{1}/\varepsilon_{2}<1$) is neglected; it is independent of
$h$. The inset shows two orthogonal orientations of the dipole moment. The
upper plot is for the vertical dipole and the lower for the horizontal
orientation.

For $\varepsilon_{1}>\varepsilon_{2}$ only propagating waves in the upper
medium (with $k_{1z}$ real) are present and $\gamma$ does not depend on $h$.
As can be seen from Figure 6, the emission rate of a dipole oriented
perpendicular to the interface is suppressed ($\gamma\rightarrow 0$), while
that parallel to the interface has $\gamma$ close to its value in an infinite
dielectric, $\gamma_{0}$. As known, the emission of a spherical QD made of a
semiconductor material with cubic crystal lattice does not have any
preferential polarization in empty space (if excited by non-polarized light).
If the dot is embedded in a matrix with sufficiently high $\varepsilon_{1}$
and located near the surface, one can expect its luminescence to be strongly
polarized. Note that this is not a quenching effect because there is no
dissipation in the system (both $\varepsilon_{1}$ and $\varepsilon_{2}$ are
real).

###  VI.2 Non-Radiative Losses to a Metal Substrate

If the substrate is lossy, e.g., a metal, several additional decay channels
arise. Mathematically, they are related 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. However, their relative
contributions depend on the distance and the real and imaginary parts of
$\varepsilon_{2}$. At very small distances ($k_{1}h<<1$), the dominating
contribution is due to Ohmic losses Chance et al. (1978); Ford and Weber
(1984),

$\frac{\gamma^{(V,\>H)}_{\mbox{Ol}}}{\gamma_{0}}=\frac{3}{(2k_{1}h)^{3}}\mbox{Im}\left(\frac{\varepsilon_{2}-\varepsilon_{1}}{\varepsilon_{2}+\varepsilon_{1}}\right)\;.$
(19)

The coupling to propagating surface plasmons mathematically is described by
the pole of $r^{(p)}$, which yields an additional imaginary part of the
integral from 1 to $\infty$ in Equations (17) and (18). Indeed, the equation
$(r^{(p)})^{-1}=0$ determines the dispersion relation of $p$-polarized SPs.
The other Fresnel coefficient, $r^{(s)}$, has no poles, and accordingly there
are no $s$-polarized SPs at a metal–dielectric interface. If we neglect the
imaginary part of $\varepsilon_{2}$, the SP wavevector is real. The resonant
coupling occurs for

$q_{\mbox{sp}}=\frac{\omega}{c}\sqrt{\frac{\varepsilon_{2}^{\prime}\varepsilon_{1}}{\varepsilon_{2}^{\prime}+\varepsilon_{1}}}$

(where $\varepsilon_{2}^{\prime}\varepsilon_{1}<0$ and
$\varepsilon_{2}^{\prime}+\varepsilon_{1}<0$). The pole contribution to the
integral from 1 to $\infty$ in Equations (17) and (18) is calculated
explicitly:

$\displaystyle\frac{\gamma^{(V)}_{\mbox{sp}}}{\gamma_{0}}=3\pi
s_{\mbox{sp}}^{5}\frac{\sqrt{-u}}{1-u}\exp{\left(-\frac{2k_{1}hs_{\mbox{sp}}}{\sqrt{-u}}\right)}\,,$
(20)
$\displaystyle\frac{\gamma^{(H)}_{\mbox{sp}}}{\gamma_{0}}=\frac{3\pi}{2}s_{\mbox{sp}}^{5}\frac{1}{\sqrt{-u}(1-u)}\exp{\left(-\frac{2k_{1}hs_{\mbox{sp}}}{\sqrt{-u}}\right)}\,,$
(21)

where $u=\varepsilon_{2}^{\prime}/\varepsilon_{1}$ and
$s_{\mbox{sp}}=q_{\mbox{sp}}/k_{1}$.

Taking $\varepsilon_{2}(\omega)$ within the simple Drude model for bulk gold,
we calculated $\gamma_{\mbox{Ol}}$, $\gamma_{\mbox{sp}}$, and the radiative
decay rate (neglecting $\varepsilon_{2}^{\prime\prime}$). They are presented
in Figure 7. As expected, at small distances the Ohmic losses dominate (see
Figure 7, lower panel). However, as the distance to the interface increases,
for $k_{1}h\sim 1$, the coupling to SPs becomes the main mechanism that
determines the lifetime (see Figure 7, upper panel). Although this effect
decreases with the distance exponentially, this decrease can be rather slow,
as $|\varepsilon_{2}^{\prime}|/\varepsilon_{1}>>1$ and $s_{\mbox{sp}}\sim 1$,
so it is nearly constant over tens of nanometres. Of course, at still large
distances $\gamma_{\mbox{tot}}\rightarrow\gamma_{0}$.

Figure 7: Spectral dependence of different contributions to the emission decay
rate for a vertically oriented emitter located in a dielectric with
$\varepsilon_{1}=2$ at two different distances (indicated on the plots) above
a flat interface with gold. Note that the contribution of the Ohmic losses
(the main cause of the emission quenching near metal surfaces) is negligible
in the upper plot, while it is completely dominating in the lower panel. For
the calculation of the radiative losses and coupling to surface plasmons, the
damping parameter, $\Gamma$ of the Drude model was set equal to zero.

Thus, for intermediate distances of the order of a hundred nanometres the
radiative decay is mostly due to the coupling to SPs within a broad spectral
range (approximately from 2.5 to 4.5 eV). In this range, relevant for
colloidal QDs, one can expect also plasmon-enhanced FRET. Even though the
enhancement is moderate, the coupling to SPs is not dissipative unless we
approach the electrostatic SP resonance frequency,
$\omega_{spr}=\omega_{p}/\sqrt{\varepsilon_{2}^{\infty}+\varepsilon_{1}}$
(here, $\omega_{p}$ is the bulk plasma frquency and $\varepsilon_{2}^{\infty}$
is the background dielectric constant of the metal). In other words,
excitation can be transferred from the QD to SPs and back many times without
dissipation. It can lead to strong coupling regime characteristic of plasmonic
microcavities Giannini et al. (2011); Dovzhenko et al. (2018).

###  VI.3 Energy Transfer to a 2D Material

Graphene, an atomic-thick monolayer of carbon, is a semimetal (or gapless
semiconductor) with unusual electronic properties first demonstrated by Geim
and Novoselov Novoselov et al. (2004). Low-energy excitations in graphene are
massless, chiral, Dirac fermions. In neutral graphene, the chemical potential
(hereafter called Fermi level) crosses exactly the Dirac point Neto et al.
(2009). Pristine graphene is transparent in a broad spectral range from the
infrared (IR) to the ultraviolet (UV), with the residual absorption of
$\approx$2.3% originating from interband transitions Nair et al. (2008). The
Fermi level, $E_{F}$, can be shifted by up to $\approx$1 eV with respect to
the Dirac point by applying an appropriate gate voltage. The usual doping via
adsorbed impurities is also possible Neto et al. (2009). One can say that
graphene is a transparent conductor with a tunable conductivity, both static
and frequency-dependent (for the frequencies up to the mid-IR) Li et al.
(2008). It supports propagating surface plasmons whose dispersion relation can
be controlled via gate voltage Bludov et al. (2013). Interestingly, a specific
type of quantum dots can form in graphene, namely, mass profile QDs, a system
within current experimental reach Martínez-Galera et al. (2014), for which
FRET can also be important Gutiérrez-Rubio and Stauber (2015).

Numerous experiments have shown that the nanoparticle or dye molecule emission
is quenched in a broad spectral range by a single graphene sheet Chen et al.
(2010); Gaudreau et al. (2013); Federspiel et al. (2015); Gonçalves et al.
(2016); Raja et al. (2016). It is understandable for the lower-energy part of
the spectrum, as graphene is a conductor, although with a relatively low free
carrier concentrations (which can reach $10^{13}$ cm-2), 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 point emitter in
the vicinity of a usual metal.

In addition to the Ohmic losses and energy transfer to surface plasmons, there
is an additional mechanism that may be named exciton transfer (ET) to
graphene. It is similar to FRET and consists in energy transfer to
electron–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-band transitions only for energies above $2E_{F}$, therefore, it
also applies to the ET. The main contribution of this mechanism scales with
distance as $h^{-4}$. It can be demonstrated in a rather simple way that
elucidates its connection to FRET.

Let us assume that acceptor dipoles are distributed within a plane
(representing the graphene sheet), then the ET rate can be obtained by
integrating (13) over this plane and written as follows,

$\frac{k_{ET}(h)}{\gamma_{0}^{(QD)}}=\rho_{A}\int_{0}^{2\pi}d\phi\int_{0}^{\infty}\frac{rdrb_{0}^{6}}{(r^{2}+h^{2})^{3}}\,,$

where $\rho_{A}$ is the surface density of acceptor dipoles. The emitter
lineshape may be taken as a $\delta$-function. As
$\rho_{A}Q_{A}I_{A}=\beta(\omega)$ is the absorbance of the graphene, we
obtain

$\frac{k_{ET}(h,\omega)}{\gamma_{0}^{(QD)}}=\frac{3c^{4}}{8\epsilon_{1}^{2}\omega^{4}h^{4}}\beta(\omega)\,.$
(22)

This formula can be derived in a more rigorous way Gómez-Santos and Stauber
(2011), with just slightly different numerical coefficients.

Figure 8: Schematics of the exciton energy transfer from a QD to graphene. The
left part represents an excited QD with an electron–hole pair. It can be
transferred as a whole to the graphene represented by its conduction and
valence bands’ cones touching in the Dirac point. For the first-order vertical
interband transition in graphene, at low temperatures the transfer is possible
only if the exciton energy exceeds 2$E_{F}$.

The $h^{-4}$ scaling law was verified in several works and it seems to be
obeyed for molecular emitters (down to $h\approx 5\,\text{nm}$) Gaudreau et
al. (2013) but not so much for QDs where deviations were found for small
distances Federspiel et al. (2015). The above theory does not take into
account the finite size of the QDs. As a first approximation, one can assume
that the point dipole is located at the centre of the QD. In this case, the
distance of the dipole from the graphene sheet can be approximated as
$h^{*}=(h+R_{QD}+s)$ where $R_{QD}$ is the radius of the nanocrystal core of
the QD and $s$ is thickness of the organic shell for a colloidal QD. The
latter can be estimated as the chain length of the capping agent, $s\sim
1.5\,\text{nm}$ Rogach (2008). Using this correction, $h\rightarrow h^{*}$,
indeed improves the agreement between the theory and experiment Federspiel et
al. (2015). However, one needs to bear in mind also the possibility of charge
transfer from an excited QD to graphene if the distance is sufficiently small
for tunnelling. Indeed, the demonstration of hybrid graphene-QD
phototransistors Konstantatos et al. (2012) and solar cells Sun et al. (2011)
where the light is absorbed in the QDs and changes the electric current in
graphene implies that such processes can take place in specially designed
structures.

Calculations including both intra-band (i.e., Drude plasmons) and inter-band
transitions in graphene show that there can be a no-quenching spectral window
for an emitter over a strongly doped graphene sheet Koppens et al. (2011).
Indeed, the graphene absorbance, $\beta(\omega)$, due to Drude plasmons
decreases with the frequency, while the inter-band transitions contribute to
$\beta$ (and, therefore, to the quenching) only for $\omega>2E_{F}/\hbar$
Bludov et al. (2013). As the Fermi level in graphene can be tuned
electrically, it opens an interesting possibility of electrical switching of
the QD emission by controlling the quenching rate; it has been demonstrated
experimentally in Lee et al. (2014); Salihoglu et al. (2016).

Energy transfer (ET) from QDs to other 2D materials, namely, few-atomic-layer-
thick semiconductors (transition metal dichalcogenides (TMDs)) has also been
investigated in the recent years Prasai et al. (2015); Raja et al. (2016);
Prins et al. (2014); Zang et al. (2016); Karanikolas et al. (2016). These
materials support robust excitons, which determine their optical properties in
the visible range (along with the inter-band transitions) and they depend
strongly on the number of monolayers in the material Wang et al. (2018). The
ET from a QD exciton to the continuum of states above the band gap in the 2D
material should work similar to the case of graphene (Equation (22)). However,
it was found that the donor QD emission is quenched most strongly near
monolayer MoS2 and then the effect decreases with the number of monolayers
(presumably keeping the QD distance to the semiconductor surface fixed), in
contrast with the case of few-layer graphene Raja et al. (2016); Prins et al.
(2014). Meanwhile, the tendency similar to graphene (quenching increases with
the number of monolayers) was found for the less studied TMD SnS2 Zang et al.
(2016). On the theoretical side, it has been suggested that the total
spontaneous emission rate of a quantum emitter can be enhanced several orders
of magnitude due to the excitation of surface exciton–polariton modes
supported by the 2D semiconductor Karanikolas et al. (2016). In the vicinity
of an excitonic transition, the real part of the dielectric function of the
semiconductor material is negative, like in a metal, so one can indeed expect
an enhancement of the local photonic DOS and, consequently, the Purcell effect
Gomes et al. (2020). However, it would require matching of the energies of the
QD and 2D excitons.

###  VI.4 FRET between QDs Near Interface

In the vicinity of a polarisable material, such as graphene, the dipole–dipole
interaction responsible for the FRET between two QDs becomes renormalized
because of the additional interaction between the induced charges. In the
electrostatic limit, these additional interactions can be modelled with image
dipoles (see Figure 9). This physical mechanism, of course, is not unique to
graphene and can take place in the vicinity of plasmonic nanoparticles Lunz et
al. (2012); Theuerholtz et al. (2013); Komarala et al. (2008). The effect of
plasmonic enhancement of FRET has been detected experimentally by observing a
decrease of the emission decay rate of donor QDs and a corresponding increase
of the luminescence intensity for acceptor dots when they are placed in the
vicinity of gold nanoparticles Lunz et al. (2012). Of course, if the
concentration of the nanoparticles becomes too high, the emission is quenched
for both donor and acceptor QDs because of the energy transfer to lossy modes
in the metal (the Ohmic losses, Section VI.2).

Figure 9: Schematics of two QDs on top of a graphene sheet, interacting via
their real transient dipoles and image dipoles representing transient charge
distributions in the graphene sheet.

Graphene is a conductor with electrically controllable conductivity and, in
principle, it promises a possibility of controlling the inter-dot FRET rate.
The exciton energy transfer between two QDs located near a graphene-covered
interface occurs not just through the direct dipole–dipole interaction but
also through polarization charges that they induce on the interface. One can
think that the image dipoles of Figure 9 have magnitudes that depend on the
optical conductivity of graphene and, therefore, on its Fermi energy Santos et
al. (2014). This interaction can be written in the form

$V_{FRET}=\frac{1}{\epsilon_{1}}\bf d_{D}\cdot\hat{\bf T}\cdot\bf d_{A}$

with $\hat{\bf T}=\hat{\bf T}^{(0)}+\hat{\bf T}^{(1)}$, where

$\hat{\bf T}^{(0)}(\bm{b})=b^{-3}(3\bf n\otimes\bf n-\hat{I})$

is the usual dipole–dipole interaction tensor (${\bf n}={\bf b}/b$) and

$\hat{\bf
T}^{(1)}(\bm{b})=\int\\!\frac{\mathrm{d}^{2}q}{(2\pi)^{2}}\,(-i\bm{e}_{q}+\bm{e}_{z})\otimes(i\bm{e}_{q}+\bm{e}_{z})qA(q)e^{-2qh+i\bm{q}\cdot\bm{b}}$
(23)

is the part due to the image dipoles. In Equation (23), $\bm{e}_{q}=\bm{q}/q$,
$\bm{e}_{z}$ is the unit vector along $z$-axis,

$A(q)=\frac{\varepsilon_{2}-\varepsilon_{1}+f(q)}{\varepsilon_{2}+\varepsilon_{1}+f(q)}\,,\qquad
f(q)=i\frac{4\pi\sigma_{g}(\omega)}{\omega}q\,,$

and $\sigma_{g}(\omega)$ is the optical conductivity of graphene. The
components of $\hat{\bf T}^{(1)}$ have an oscillatory dependence upon the
inter-dot distance Huidobro et al. (2012), which arises from the Bessel
functions appearing as the result of the angular integration in Equation (23).
It means that instead of the usual monotonic $b^{-6}$ dependence of the FRET
rate one may have an oscillatory behaviour, controllable by the gate voltage
through the graphene Fermi energy (which determines $\sigma_{g}(\omega)$).
Beyond the non-monotonic dependence upon the inter-dot distance, the
emission/transfer frequency also should affect it in a rather complex way. If
this interaction is strong enough, the emitters (QDs in our case) can be
coupled to create a collective radiative mode, a phenomenon called
superradiance Dicke (1954), which has been observed for epitaxial quantum dots
Scheibner et al. (2007). As the coupling between the emitters can be strongly
enhanced or suppressed in the vicinity of graphene, depending on the distance
between them, the frequency and the graphene Fermi energy, both superradiance
and subradiance regimes can be expected Huidobro et al. (2012). So far, we are
not aware of an experimental demonstration of such effects. Losses associated
with the real part of graphene’s optical conductivity can be one possible
reason for this.

The influence of a polarisable surface (such as graphene) on the irreversible
FRET should be an easier-to-observe effect than the superradiance. It can be
seen as a surface plasmons’ effect. Huge FRET enhancement factors of the order
of $10^{6}$ (compared to vacuum) have been predicted for two dipole emitters
on a graphene sheet, based on numerical electrodynamics calculations Biehs and
Agarwal (2013), reaching a maximum when the distance between the dipoles
equals twice the graphene plasmon propagation length $L_{p}$. This prediction
is at variance with the results presented in Huidobro et al. (2012) according
to which the effect must be considerably smaller and varying on scale of the
surface plasmon wavelength, $\lambda_{p}<<L_{p}$.

##  VII Concluding Remarks

To summarize, we would like to emphasize, once again, that the radiative
properties and efficiency of light emission in semiconductor nanostructures
are determined not only by the radiative transitions, but also by various non-
radiative processes which can be even more intense than the radiative
transitions themselves. As a result, in these cases, the radiative
recombination can be significantly suppressed. On the other hand, even the
non-radiative processes can be used in a constructive way. For instance, the
exciton migration is accompanied by a non-radiative energy transfer (FRET)
that can be channelled in a desired direction via creating a certain
“architecture” of the nanocrystals in the ensemble, which allows to
concentrate and illuminate the energy inside a given area Kawazoe et al.
(2005); Belyakov and Burdov (2011) or direct it to a certain layer of a
funnel-type heterostructure of QD monolayers Klar et al. (2005). Exciton
energy transfer and emission with upconversion because of simultaneous
absorption of optical phonons has been suggested for QD-assisted cooling
Rakovich et al. (2009). The multi-exciton effects make it possible to
efficiently transform the absorbed photons into the rising number of
electron–hole pairs capable of participating in the electric current. Such
processes underlie the operating principle of photosynthetic light harvesting
systems Mirkovic et al. (2016) and can be mimicked in solar cells Sambur et
al. (2010).

The exciton transport can be strongly enhanced if mediated by polaritons
arising under strong coupling of the excitons to light in microcavities Coles
et al. (2014); Schachenmayer et al. (2015). Combining entities supporting
localized excitons, such as QDs, dye molecules or J-aggregates, with
microcavities in order to achieve the strong coupling regime is an important
area of research envisaging various applications Dovzhenko et al. (2018), in
particular, in the context of controllable quantum emitters. A plasmonic
surface considered in Section VI can be seen as a near-field microcavity if
the lossy channels are not the dominating ones. A detailed discussion of the
strong coupling between surface plasmon-polaritons and quantum emitters can be
found elsewhere Törmä and Barnes (2014).

## Acknowledgments

Funding from the Ministry of Science and Higher Education of the Russian
Federation (State Assignment No 0729-2020-0058), the European Commission
within the project ”Graphene-Driven Revolutions in ICT and Beyond” (Ref. No.
696656), from the Portuguese Foundation for Science and Technology (FCT) in
the framework of the PTDC/NAN-OPT/29265/2017 ”Towards high speed optical
devices by exploiting the unique electronic properties of engineered 2D
materials” project and the Strategic Funding UID/FIS/04650/2019 is gratefully
acknowledged.

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