# Silicon Nanocrystals: Fundamental Theory and Implications for Stimulated Emission

Authors: V. A. Belyakov; V. A. Burdov; R. Lockwood; A. Meldrum
Journal: Advances in Optical Technologies
 (2008)

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## Abstract

Silicon nanocrystals (NCs) represent one of the most promising material systems for light emission applications in microphotonics. In recent years, several groups have reported on the observation of optical gain or stimulated emission in silicon NCs or in porous silicon (PSi). These results suggest that silicon-NC-based waveguide amplifiers or silicon lasers are achievable. However, in order to obtain clear and reproducible evidence of stimulated emission, it is necessary to understand the physical mechanisms at work in the light emission process. In this paper, we report on the detailed theoretical aspects of the energy levels and recombination rates in doped and undoped Si NCs, and we discuss the effects of energy transfer mechanisms. The theoretical calculations are extended toward computational simulations of ensembles of interacting nanocrystals. We will show that inhomogeneous broadening and energy transfer remain significant problems that must be overcome in order to improve the gain profile and to minimize nonradiative effects. Finally, we suggest means by which these objectives may be achieved.


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## Body

## 1. Introduction

Silicon is the most widespread
semiconductor in modern microelectronics technologies. Its natural abundance,
low cost, and high purity, as well as the high electronic quality of the Si/SiO2 interface, have led to its overwhelming dominance in microelectronic devices.
Nevertheless, the use of silicon in optoelectronics remains highly limited. This
state of affairs has remained, in fact, almost unchanged because of a
fundamental property of the silicon band structure—the indirect band gap.The indirect radiative interband transitions in bulk Si are strongly
suppressed because an emitted photon cannot satisfy the momentum conservation
law for transitions from the conduction-band minimum (Δ-point) to the top of the valence band (Γ-point). The photon wave vector is about three orders of magnitude less
than that required for the transition between the Δ- and Γ-points. This difference in k-space is kΔ=0.86×2π/a0,
with a0 being the lattice constant of silicon, equal
to 5.43 Å. The electron-hole radiative recombination in the bulk material is
exactly forbidden unless additional mechanisms allowing the momentum to be
conserved are involved in the recombination process. The most probable means to
have a radiative indirect transition without breakdown of the momentum
conservation law is via phonon absorption or emission. However the
electron-phonon interaction is weak; consequently, phonon assistance is a low-probability
(and hence slow) process. This leads to a substantial increase of the total
recombination time, and a decrease of the recombination probability compared to
the direct no-phonon Γ−Γ radiative transitions in direct-gap
semiconductors. In this sense, we call such transitions in Si “strongly
suppressed.”The discovery of visible-range emission from nanocrystalline [1]
and porous [2–4] silicon in the early 1990s suggested that some of the problems
associated with the silicon band structure might be overcome in nanoscale
crystallites hosted in a widegap dielectric matrix like SiO2, in
order to create a strong confining potential for carriers inside the
nanocrystal. Electronic states become localized within the NC and the momentum
distribution spreads due to the Heisenberg uncertainty relations. In other
words, the wave functions consist of plane waves with all possible wave vectors
including both k~kΔ for holes and k~0 for electrons, respectively. Thus, the
momentum conservation law is not violated in this case, which yields a nonzero
probability of the Δ−Γ radiative transition even in the absence of
phonons.Indeed, some time later, efficient visible photoluminescence (PL) from
silicon nanocrystals was experimentally demonstrated, (e.g., [5]) and attributed to the transitions
between confined electron and hole states inside the nanocrystal [6, 7]
(the so-called quantum confinement effect) or between interface states [8, 9].
However, the emission quantum efficiency in Si nanocrystals remains low
compared to the direct gap III-V or II-VI materials. This is naturally
explained by the small “weights” of the plane waves with k~kΔ for the valence states and k~0 for the conduction states. Thus, improvement of
the light emission efficiency of Si quantum dots remains a challenge for
optoelectronic technologies.As a means to modify the optical properties of silicon
crystallites, doping with shallow impurities has been suggested [10–18]. In some cases (depending on the
conditions and methods of preparation), the emitting properties of the dots
were improved significantly. In particular, the PL intensity was several times
greater when the nanocrystals were doped with phosphorus [10–12, 14] or codoped with
phosphorus and boron [15, 17]. The origin of this
phenomenon is not fully understood at the present time, and we will touch on
the problem of impurity states in silicon nanocrystals in this review.PL experiments are not usually carried out with a
single quantum dot but rather with large ensembles. As a result, interpretation
of the experimental data is attendant with difficulties because of the
associated inhomogeneous broadening and various collective effects that can
occur as a result of the mutual influence of the nanocrystals in the ensemble.
Some recent studies have reported on the PL spectra of individual silicon
quantum dots [19, 20] and this is shedding new light on
the physics involved in the light emission process. A principal distinction in the emission of a single NC and a
nanocrystal ensemble lies in the various mechanisms of interaction amongst the
nanocrystals in an ensemble. In particular, in solid nanocrystal arrays, some
collective effects caused by electron, photon, and phonon transfer between the
dots can strongly influence the luminescence dynamics of the nanocrystals in
comparison with the case of isolated NCs. The size distribution of the
nanocrystals can play an essential role in the excitation exchange between the clusters.One possible mechanism of the NC-NC interaction is the direct tunneling
of excited carriers from one quantum dot to another [21–23]. As an example, we can imagine two
(or more) closely separated quantum dots with different sizes. Presumably,
excited carriers in the smaller nanocrystal, having higher quantized energy
levels, may relax either to the valence band of this nanocrystal or to the
conduction band of the adjacent nanocrystal with a larger size. The transition
to the valence band is indirect, and therefore suppressed. Meanwhile, the
transfer to the neighboring quantum dot occurs as if within the same
(conduction) band, and may be more probable in the case where the nanocrystals
are sufficiently close. If so, then the smaller nanocrystals will inject their
own excited carriers into the larger nanocrystals, in which radiative interband
transition will subsequently take place. This idea implies the possibility of
an optical nanofountain [24]—a device emitting photons from the area where quantum
dots with larger sizes are concentrated.An additional nonradiative mechanism of energy transfer between adjacent
nanocrystals becomes possible in dense arrays. This is the so-called Forster
mechanism originating from the dipole-dipole (or higher-order multipole)
interaction between excitons in different quantum dots [25–27]. Because of the dipole-dipole interaction, the electron-hole excitations can
“travel” throughout the nanocrystals without charge transfer. Such transitions
have been observed in solid arrays of CdSe nanocrystals [28, 29]. Some theoretical aspects of the Forster transfer in silicon quantum dots were
recently discussed by Allan and Delerue
[30].Various collective effects in quantum-dot arrays can result in a
complicated time dependence of the experimental PL decay. As a rule, the decay
is described by the stretched exponential functionexp{−(t/τ)β} with β less than unity [31–35]. Evidently, the value of β can be due to a size distribution
of the nanocrystals (and hence a distribution of the recombination rates).
Theoretical explanations of the stretched exponential from the point of view of
the multiexponential decay and associated Laplace transforms have been given in
[36].This is, in short, the framework in which we will address the present
review. At first we will briefly describe the experimental situation in this
field in Section2. Then the single-particle electronic structure and many-body
corrections will be considered in Section 3 for a single silicon quantum dot as
a basis for treatment of more complicated problem of solid quantum-dot
ensembles. The latter will be reviewed in Section 4. Finally, Section 5
presents a general conclusion and a commentary on the potential for stimulated
emission in silicon nanocrystal ensembles.


## 2. Experimental Results: Overview

Different methods can be used for preparation of
silicon nanocrystals, for instance, Si ion implantation [37–41],
chemical vapor deposition [42],
magnetron sputtering [43, 44], colloidal synthesis [45],
electron beam evaporation [46, 47], and some others. A high-temperature
thermal treatment is generally required in order to precipitate the
crystallites. All these techniques allow one to form silicon NCs with sizes
predominantly ranging from 2–6 nm. Their
electronic structure and optical properties depend, of course, on the
preparation conditions and method of fabrication. However, there are some
common properties typical for silicon NCs, independent of the fabrication
technique employed. In particular, the nanocrystals' surroundings, either
vacuum or some host material like silicon dioxide, represent a high potential
barrier for carriers of both kinds. Such a barrier is often referred to as a
confining potential that mainly defines the energy spectrum of the nanocrystal.
In what follows we will discuss some manifestations of the quantum-confinement
effect in experiments and theoretical models. First, let us briefly discuss
experimental data on the PL spectra in silicon quantum dots.


### 2.1. Size Dependence

As has been mentioned above, silicon NCs are capable
of emitting electromagnetic energy in the visible spectrum. This is in contrast
with bulk silicon, in which energy of the interband transition corresponds to the
silicon bandgap energy of 1.12 eV. The increase (decrease) of the photon
frequency (wavelength) in nanocrystals compared to the bulk material is a
universal phenomenon taking place in various semiconductor materials and
quantum dots. In the general case one may say that the energy of the emitted
photon increases as the nanocrystal size decreases. Such an increase is usually
called a “blueshift” because the photon energy shifts toward the shorter-wavelength
side of the visible spectrum. This blueshift is illustrated for Si NCs in Figure1. Here, the mean NC size is controlled via the excess Si concentration, with
the smallest NCs occurring in the most silicon deficient samples. The PL
intensity has not been normalized here; the drop in intensity on the
silicon-poor side of the compositional map is due to the lower number density
of NCs, and on the silicon-rich side it is due to the opening of nonradiative
pathways in large and highly interconnected nanoclusters.Figure 1
PL spectra of silicon NCs in SiO2. The 200-nm-thick samples were approximately identical except for the amount of excess silicon.In the simplest model of an infinitely strong confining potential (i.e.,
infinitely high potential barriers at the dot boundary) it is possible to
estimate the energies of electrons and holes localized inside the nanocrystal
as proportional toR−2,
where R is the nanocrystal radius. Thus, the optical
gap may be calculated as εg(R)=εg+A/R2,
where εg is the bandgap of bulk silicon, A is some positive constant, and A/R2 represents the total energy of the non-interacting
electron-hole pairs inside the dot. It is obvious that: (i) the nanocrystal gap
must be always greater than the bulk one due to the additional positive term A/R2, and (ii) the photon energy, equal to the energy of the lowest electron-hole
transition, increases as the dot size decreases.In experimental work carried out over the past fifteen
years with silicon nanocrystals, the optical-gap dependence on the dot size was
measured and discussed extensively. Although it is impractical to cite all the
papers dealing with this topic, a sampling is given in [8, 23, 48–56].
The data in these papers is summarized in Figure 2. There is a fairly large
spread in the calculated values of the optical gaps as a function of NC
diameter. Presumably, several factors influencing the accuracy of the
optical-gap measurements are as follows. First, it is difficult to determine
exactly the dot sizes and the size distribution in a luminescent ensemble of NCs.
Second, using the mean size in an ensemble of NCs in a diagram like Figure 2
can be misleading, since it is possible that the observed PL peak does not
correspond exactly to the mean size but instead to the largest PL rate. Third
and more practically, the nanocrystals studied by different research groups
have been prepared by different methods. As a result, the NCs have different
surroundings, surface bonds, and shapes, all of which could lead to scatter in the
experimental data. Finally, as shown in later sections, NC-NC interactions can
play a dominant role in the emission spectrum.Figure 2
Experimental data on the size dependence of the optical gap of silicon crystallites.
Hydrogen-passivated surface—Wolkin et al. [8] and Garrido et al. [53] (blue squares); SiO-bonds at the surface—Wolkin et al. [8], Kanemitsu et al. [51] (diamonds);
deposition on quartz substrate—Ledoux et al. [48] (filled triangles); nanocrystals embedded in Si3N4 matrix—Kim et al. [49] (empty triangles).Furthermore, the blue-shift energyΔεg(R),
determined from the experiments, does not obey the law A/R2 following from the simplest quantum-mechanical
model. This dependence is rather Δεg(R)~R−b with 1<b<1.5 or an even weaker dependence on R in some cases. Such behavior has been a reason
for supposing a key role for interface states in the radiative recombination
process [51]. The question about the origin of
the radiative electron-hole transitions in the nanocrystals is remains under
extensive debate even today.Nevertheless, employing the principle of Occam's
razor, the deviations from the simplest model predictions for high-energy
luminescence (when the peak-position energy exceeds the bulk gapεg) may be explained within the quantum confinement framework without
necessarily the need to invoke sub-gap radiative centers. Let us, first, take
into account the finiteness of the potential barriers, leading to some
weakening of the quantum confinement and, as a consequence, to a more
complicated and less steep size-dependence of Δεg(R) other than R−2.
In addition, electron-hole Coulomb interaction contributes a further −R−1 dependence in the blue-shift energy. Both
these mechanisms result in a more gradual dependence of the optical gap on the
dot radius, which is in fact observed in experiments.At the same time, low-energy PL with photon energies
less than the gap of bulk silicon are also observed in experiments, see, for
example, [57]. In this work an extra luminescence
peak arose at about 0.9 eV at low temperatures, and its position remained
almost unchanged for nanocrystals of various sizes. This may be indeed treated
as an attribute of the surface states which can appear inside the band gap of
bulk silicon. This possibility and the associated pair-wise trapping rates have
been discussed and derived theoretically by Lannoo et al. [58].


### 2.2. Decay Rate

Time-resolved studies of the PL of silicon NCs demonstrate
near-exponential decays of the photoluminescence intensity. Typical decay
timesτPL vary within a wide range of about 1 to1000 μs
depending on the particle size, temperature, detected wavelength, method of
preparation, and so forth [31, 32, 51, 59–62]. Such
lifetimes are indeed large, and this is an explicit indication of an indirect-band-gap
material. Contrary to direct-gap III-V or II-VI compounds, the drop of
luminescence intensity for silicon crystallites is sufficiently slow to result
in a low PL irradiance compared to direct-gap NCs, even for comparable quantum
efficiencies.The characteristic time of the PL decay is determined
by two different recombination mechanisms: radiative, with typical timeτR; and nonradiative, with typical time τNR. The rate of the photoluminescence decay is the sum of the radiative and
nonradiative recombination rates: 1/τPL=1/τR+1/τNR. In the case where one of the two times is much less than the other, the
photoluminescence lifetime coincides with the smallest time. Usually in
silicon, the nonradiative channel turns out to be faster compared to the
radiative one (see, e.g., [31]), so that the PL
lifetime equals τNR. The quantum yield η, which is proportional to the PL intensity, may be defined as the “weight” of
the radiative channel in the recombination process: η=τR−1/τPL−1. If τR≫τNR,
then η=τNR/τR≪1.
Obviously, one can use two physically different ways to increase η. The first one is an increase of the radiative transitions, while the second is
a decrease of the nonradiative processes. The second way appears easier in
practice because the nonradiative processes can be influenced by preparation
conditions, as has been recently demonstrated by Miura et al. [62]. On the contrary,
according to [62], radiative
recombination cannot be controlled by the preparation conditions. Therefore, control
of the radiative channel efficiency seems to be fairly difficult to achieve in
practice.Sometimes, both radiative and nonradiative channels
contribute comparably in the interband recombination in silicon nanocrystals.
In these cases the quantum efficiency becomes very high—possibly on the order of tens of percent [62]. It should be noted,
however, that these situations are possible, presumably, at pumping levels
corresponding to no more than one excited carrier in every quantum dot. Then, traps
such as surface defects may be the main sources of nonradiative recombination.
Alternatively, when the excitation power is very high and several electrons are
excited in the dots, the Auger process becomes possible. Since Auger
interactions are fast, the radiative channel will be “shunted” in this case.
Accordingly, η tends to some small value, on the order of a
fraction of a percent.In cases when the nonradiative channel is mostly “closed,”
that is,τR≪τNR,
silicon nanocrystals demonstrate strong temperature dependence of the PL lifetime [23, 33, 57, 63, 64]. In
particular, the lifetime becomes smaller as the temperature decreases from ~300
to 4 K. This behavior is
explained within the framework of the singlet-triplet two-level model suggested
by Calcott et al. [3, 4]
for porous silicon.
According to this model the exciton state splits due to the electron-hole
exchange interaction into the upper singlet with relatively short lifetime τ↑, and lower triplet with about 2-3 orders-of-magnitude longer lifetime τ↓. The decay rate is then defined by
(1)1τPL≈1τR=(3/τ↓)+(1/τ↑)exp[−(ΔS-T/kT)]3+exp[−(ΔS-T/kT)], where ΔS-T is the energy of the singlet-triplet
splitting, k is the Boltzmann constant, and T is the temperature. The energy of the
singlet-triplet splitting depends on the nanocrystal size and increases from a
few meV to 10–20 meV for photon energies increasing from 1.4 eV to
2.2 eV [57, 64]. At low temperature, when ΔS-T≫kT the decay of the singlet state is strongly
suppressed, and the total lifetime τPL coincides with the longer time τ↓. In the opposite case (when ΔS-T≪kT), τPL is close to 4τ↑. As a result, the PL decay occurs substantially faster at low temperature. More
rigorous theoretical analysis [65–68] reveals a rich fine structure in the
excitonic spectrum of silicon NCs. Exciton states with different symmetry may
be ascribed to so-called bright and dark excitons which have essentially
different recombination lifetimes.


### 2.3. Enhancement and Quenching of Photoluminescence due to Impurities

During the past decade, doping Si NCs with shallow
impurities has been explored as a potential route for modifying the luminescent
properties of silicon nanocrystals. The main questions are the following: are
we able to improve the emission of silicon quantum dots by incorporating donors
or acceptors (or both)? If so, then the next question is what is the mechanism for
the improvement? The results obtained by different research groups indicate
that the PL spectra of silicon nanocrystals are, indeed, sensitive to doping.
However, both an increase [10–13] and a decrease [18] of the PL intensity have been observed. This, again,
depends on the sample fabrication method and the type of doping.Silicon NCs doped with phosphorus [10–13, 16, 69], boron [18], or co-doped with P and
B [14, 15, 17] have recently been
investigated. At the same time, the authors of [70–72]
reported on
experiments with silicon NCs doped not only with phosphorus or boron but also
with hydrogen and nitrogen. Summarizing the results, one can conclude that
doping with phosphorus or codoping with phosphorus and boron can enhance the PL
intensity of the nanocrystals by several times, as initially reported in [10]. Meanwhile, no
enhancement is observed for NCs doped with boron, hydrogen, or nitrogen.
Moreover, the degree of quenching of the luminescence depends on the annealing
procedure [18, 70–72].In accordance with the results of Miura et al. [12],
Fujii et al. [13], and Mikhaylov et al. [72]
reported that an increase of the phosphorus concentration leads to an initial rise of the
photoluminescence peak, whereupon the PL intensity subsequently decreases with
a further increase in doping. Moreover, the P concentration corresponding to
the maximum intensity becomes greater with decreasing NC size [11–13]. Therefore, in some
cases, where the NCs had small sizes of about 3-4 nm [11, 70], only a monotonic rise
of the PL peak was observed as P concentration increased. One possible reason
for this phenomenon lies in the passivation of dangling bonds (i.e., neutral or
charged Pb
centers) at the NC surfaces. The absence of the enhancement effect for heavily P-doped
crystallites at a given size has been explained by more extensive Auger
recombination [14, 15, 17] or a coalescence
mechanism [72]. The authors of [72] also pointed out the
essential distinction in the photolumine scence of the P-doped nanocrystals
formed by ion implantation at 1000°C and 1100°C. The former demonstrate up to
5 to 6 times intensity enhancement, while for the latter ony PL quenching was
found (Figure 3).Influence of doping with P, B, N on the PL intensity at 750 nm in Si NC layers synthesized by ion implantation and annealed at either 1000 or 1100°C. Samples were subsequently implanted with impurity ions and then re-annealed. The dashed line represents the intensity of the undoped reference sample. (The figures have been kindly provided by D. I. Tetelbaum, A. N. Mikhaylov, O. N. Gorshkov, D. I. Kambarov, V. K. Vasiliev, and A. I. Belov, who did the experiments).
(a)(b)As a method to reduce the Auger recombination,
simultaneous codoping with phosphorus and boron, thereby compensating the
numbers of donors and acceptors inside the crystallite, has been proposed [14, 15, 17]. PL enhancement was successfully obtained for larger nanocrystals
of ~6–8 nm in diameter. In this case, a large shift of the PL peak, even below the optical
gap of bulk silicon, takes place when increasing P concentration with fixed
concentration of boron [14]. This may be evidence of
the formation of bulk-like impurity states. At the same time, codoping with
high concentration of both donor- and acceptor-type impurities result in a
strong chemical shift of the ground-state energy levels in the conduction and
valence bands [73], which may substantially reduce the NC
optical gap. In the latter case no assumptions on the bulk-like form of
electron states in silicon quantum dots are required, because the effect of the
strong chemical shift is exclusively due to the confining potential of the dot.


## 2.1. Size Dependence

As has been mentioned above, silicon NCs are capable
of emitting electromagnetic energy in the visible spectrum. This is in contrast
with bulk silicon, in which energy of the interband transition corresponds to the
silicon bandgap energy of 1.12 eV. The increase (decrease) of the photon
frequency (wavelength) in nanocrystals compared to the bulk material is a
universal phenomenon taking place in various semiconductor materials and
quantum dots. In the general case one may say that the energy of the emitted
photon increases as the nanocrystal size decreases. Such an increase is usually
called a “blueshift” because the photon energy shifts toward the shorter-wavelength
side of the visible spectrum. This blueshift is illustrated for Si NCs in Figure1. Here, the mean NC size is controlled via the excess Si concentration, with
the smallest NCs occurring in the most silicon deficient samples. The PL
intensity has not been normalized here; the drop in intensity on the
silicon-poor side of the compositional map is due to the lower number density
of NCs, and on the silicon-rich side it is due to the opening of nonradiative
pathways in large and highly interconnected nanoclusters.Figure 1
PL spectra of silicon NCs in SiO2. The 200-nm-thick samples were approximately identical except for the amount of excess silicon.In the simplest model of an infinitely strong confining potential (i.e.,
infinitely high potential barriers at the dot boundary) it is possible to
estimate the energies of electrons and holes localized inside the nanocrystal
as proportional toR−2,
where R is the nanocrystal radius. Thus, the optical
gap may be calculated as εg(R)=εg+A/R2,
where εg is the bandgap of bulk silicon, A is some positive constant, and A/R2 represents the total energy of the non-interacting
electron-hole pairs inside the dot. It is obvious that: (i) the nanocrystal gap
must be always greater than the bulk one due to the additional positive term A/R2, and (ii) the photon energy, equal to the energy of the lowest electron-hole
transition, increases as the dot size decreases.In experimental work carried out over the past fifteen
years with silicon nanocrystals, the optical-gap dependence on the dot size was
measured and discussed extensively. Although it is impractical to cite all the
papers dealing with this topic, a sampling is given in [8, 23, 48–56].
The data in these papers is summarized in Figure 2. There is a fairly large
spread in the calculated values of the optical gaps as a function of NC
diameter. Presumably, several factors influencing the accuracy of the
optical-gap measurements are as follows. First, it is difficult to determine
exactly the dot sizes and the size distribution in a luminescent ensemble of NCs.
Second, using the mean size in an ensemble of NCs in a diagram like Figure 2
can be misleading, since it is possible that the observed PL peak does not
correspond exactly to the mean size but instead to the largest PL rate. Third
and more practically, the nanocrystals studied by different research groups
have been prepared by different methods. As a result, the NCs have different
surroundings, surface bonds, and shapes, all of which could lead to scatter in the
experimental data. Finally, as shown in later sections, NC-NC interactions can
play a dominant role in the emission spectrum.Figure 2
Experimental data on the size dependence of the optical gap of silicon crystallites.
Hydrogen-passivated surface—Wolkin et al. [8] and Garrido et al. [53] (blue squares); SiO-bonds at the surface—Wolkin et al. [8], Kanemitsu et al. [51] (diamonds);
deposition on quartz substrate—Ledoux et al. [48] (filled triangles); nanocrystals embedded in Si3N4 matrix—Kim et al. [49] (empty triangles).Furthermore, the blue-shift energyΔεg(R),
determined from the experiments, does not obey the law A/R2 following from the simplest quantum-mechanical
model. This dependence is rather Δεg(R)~R−b with 1<b<1.5 or an even weaker dependence on R in some cases. Such behavior has been a reason
for supposing a key role for interface states in the radiative recombination
process [51]. The question about the origin of
the radiative electron-hole transitions in the nanocrystals is remains under
extensive debate even today.Nevertheless, employing the principle of Occam's
razor, the deviations from the simplest model predictions for high-energy
luminescence (when the peak-position energy exceeds the bulk gapεg) may be explained within the quantum confinement framework without
necessarily the need to invoke sub-gap radiative centers. Let us, first, take
into account the finiteness of the potential barriers, leading to some
weakening of the quantum confinement and, as a consequence, to a more
complicated and less steep size-dependence of Δεg(R) other than R−2.
In addition, electron-hole Coulomb interaction contributes a further −R−1 dependence in the blue-shift energy. Both
these mechanisms result in a more gradual dependence of the optical gap on the
dot radius, which is in fact observed in experiments.At the same time, low-energy PL with photon energies
less than the gap of bulk silicon are also observed in experiments, see, for
example, [57]. In this work an extra luminescence
peak arose at about 0.9 eV at low temperatures, and its position remained
almost unchanged for nanocrystals of various sizes. This may be indeed treated
as an attribute of the surface states which can appear inside the band gap of
bulk silicon. This possibility and the associated pair-wise trapping rates have
been discussed and derived theoretically by Lannoo et al. [58].


## 2.2. Decay Rate

Time-resolved studies of the PL of silicon NCs demonstrate
near-exponential decays of the photoluminescence intensity. Typical decay
timesτPL vary within a wide range of about 1 to1000 μs
depending on the particle size, temperature, detected wavelength, method of
preparation, and so forth [31, 32, 51, 59–62]. Such
lifetimes are indeed large, and this is an explicit indication of an indirect-band-gap
material. Contrary to direct-gap III-V or II-VI compounds, the drop of
luminescence intensity for silicon crystallites is sufficiently slow to result
in a low PL irradiance compared to direct-gap NCs, even for comparable quantum
efficiencies.The characteristic time of the PL decay is determined
by two different recombination mechanisms: radiative, with typical timeτR; and nonradiative, with typical time τNR. The rate of the photoluminescence decay is the sum of the radiative and
nonradiative recombination rates: 1/τPL=1/τR+1/τNR. In the case where one of the two times is much less than the other, the
photoluminescence lifetime coincides with the smallest time. Usually in
silicon, the nonradiative channel turns out to be faster compared to the
radiative one (see, e.g., [31]), so that the PL
lifetime equals τNR. The quantum yield η, which is proportional to the PL intensity, may be defined as the “weight” of
the radiative channel in the recombination process: η=τR−1/τPL−1. If τR≫τNR,
then η=τNR/τR≪1.
Obviously, one can use two physically different ways to increase η. The first one is an increase of the radiative transitions, while the second is
a decrease of the nonradiative processes. The second way appears easier in
practice because the nonradiative processes can be influenced by preparation
conditions, as has been recently demonstrated by Miura et al. [62]. On the contrary,
according to [62], radiative
recombination cannot be controlled by the preparation conditions. Therefore, control
of the radiative channel efficiency seems to be fairly difficult to achieve in
practice.Sometimes, both radiative and nonradiative channels
contribute comparably in the interband recombination in silicon nanocrystals.
In these cases the quantum efficiency becomes very high—possibly on the order of tens of percent [62]. It should be noted,
however, that these situations are possible, presumably, at pumping levels
corresponding to no more than one excited carrier in every quantum dot. Then, traps
such as surface defects may be the main sources of nonradiative recombination.
Alternatively, when the excitation power is very high and several electrons are
excited in the dots, the Auger process becomes possible. Since Auger
interactions are fast, the radiative channel will be “shunted” in this case.
Accordingly, η tends to some small value, on the order of a
fraction of a percent.In cases when the nonradiative channel is mostly “closed,”
that is,τR≪τNR,
silicon nanocrystals demonstrate strong temperature dependence of the PL lifetime [23, 33, 57, 63, 64]. In
particular, the lifetime becomes smaller as the temperature decreases from ~300
to 4 K. This behavior is
explained within the framework of the singlet-triplet two-level model suggested
by Calcott et al. [3, 4]
for porous silicon.
According to this model the exciton state splits due to the electron-hole
exchange interaction into the upper singlet with relatively short lifetime τ↑, and lower triplet with about 2-3 orders-of-magnitude longer lifetime τ↓. The decay rate is then defined by
(1)1τPL≈1τR=(3/τ↓)+(1/τ↑)exp[−(ΔS-T/kT)]3+exp[−(ΔS-T/kT)], where ΔS-T is the energy of the singlet-triplet
splitting, k is the Boltzmann constant, and T is the temperature. The energy of the
singlet-triplet splitting depends on the nanocrystal size and increases from a
few meV to 10–20 meV for photon energies increasing from 1.4 eV to
2.2 eV [57, 64]. At low temperature, when ΔS-T≫kT the decay of the singlet state is strongly
suppressed, and the total lifetime τPL coincides with the longer time τ↓. In the opposite case (when ΔS-T≪kT), τPL is close to 4τ↑. As a result, the PL decay occurs substantially faster at low temperature. More
rigorous theoretical analysis [65–68] reveals a rich fine structure in the
excitonic spectrum of silicon NCs. Exciton states with different symmetry may
be ascribed to so-called bright and dark excitons which have essentially
different recombination lifetimes.


## 2.3. Enhancement and Quenching of Photoluminescence due to Impurities

During the past decade, doping Si NCs with shallow
impurities has been explored as a potential route for modifying the luminescent
properties of silicon nanocrystals. The main questions are the following: are
we able to improve the emission of silicon quantum dots by incorporating donors
or acceptors (or both)? If so, then the next question is what is the mechanism for
the improvement? The results obtained by different research groups indicate
that the PL spectra of silicon nanocrystals are, indeed, sensitive to doping.
However, both an increase [10–13] and a decrease [18] of the PL intensity have been observed. This, again,
depends on the sample fabrication method and the type of doping.Silicon NCs doped with phosphorus [10–13, 16, 69], boron [18], or co-doped with P and
B [14, 15, 17] have recently been
investigated. At the same time, the authors of [70–72]
reported on
experiments with silicon NCs doped not only with phosphorus or boron but also
with hydrogen and nitrogen. Summarizing the results, one can conclude that
doping with phosphorus or codoping with phosphorus and boron can enhance the PL
intensity of the nanocrystals by several times, as initially reported in [10]. Meanwhile, no
enhancement is observed for NCs doped with boron, hydrogen, or nitrogen.
Moreover, the degree of quenching of the luminescence depends on the annealing
procedure [18, 70–72].In accordance with the results of Miura et al. [12],
Fujii et al. [13], and Mikhaylov et al. [72]
reported that an increase of the phosphorus concentration leads to an initial rise of the
photoluminescence peak, whereupon the PL intensity subsequently decreases with
a further increase in doping. Moreover, the P concentration corresponding to
the maximum intensity becomes greater with decreasing NC size [11–13]. Therefore, in some
cases, where the NCs had small sizes of about 3-4 nm [11, 70], only a monotonic rise
of the PL peak was observed as P concentration increased. One possible reason
for this phenomenon lies in the passivation of dangling bonds (i.e., neutral or
charged Pb
centers) at the NC surfaces. The absence of the enhancement effect for heavily P-doped
crystallites at a given size has been explained by more extensive Auger
recombination [14, 15, 17] or a coalescence
mechanism [72]. The authors of [72] also pointed out the
essential distinction in the photolumine scence of the P-doped nanocrystals
formed by ion implantation at 1000°C and 1100°C. The former demonstrate up to
5 to 6 times intensity enhancement, while for the latter ony PL quenching was
found (Figure 3).Influence of doping with P, B, N on the PL intensity at 750 nm in Si NC layers synthesized by ion implantation and annealed at either 1000 or 1100°C. Samples were subsequently implanted with impurity ions and then re-annealed. The dashed line represents the intensity of the undoped reference sample. (The figures have been kindly provided by D. I. Tetelbaum, A. N. Mikhaylov, O. N. Gorshkov, D. I. Kambarov, V. K. Vasiliev, and A. I. Belov, who did the experiments).
(a)(b)As a method to reduce the Auger recombination,
simultaneous codoping with phosphorus and boron, thereby compensating the
numbers of donors and acceptors inside the crystallite, has been proposed [14, 15, 17]. PL enhancement was successfully obtained for larger nanocrystals
of ~6–8 nm in diameter. In this case, a large shift of the PL peak, even below the optical
gap of bulk silicon, takes place when increasing P concentration with fixed
concentration of boron [14]. This may be evidence of
the formation of bulk-like impurity states. At the same time, codoping with
high concentration of both donor- and acceptor-type impurities result in a
strong chemical shift of the ground-state energy levels in the conduction and
valence bands [73], which may substantially reduce the NC
optical gap. In the latter case no assumptions on the bulk-like form of
electron states in silicon quantum dots are required, because the effect of the
strong chemical shift is exclusively due to the confining potential of the dot.


## 3. Single-Dot Problem

We next turn to the quantum-mechanical analysis of
various aspects of light emission in silicon nanocrystals. It is quite clear,
of course, that a consideration of electronic structure, electron-hole
recombination, and the role of impurities in a single quantum dot provides a
basis for understanding optical properties of the quantum-dot ensembles.
Therefore the present chapter first discusses some theoretical models
concerning the interband transitions in an individual quantum dot. First, we
will discuss the optical gap and energy spectra of undoped nanocrystals. Then
the “band” structure and charge distribution in doped dots will be analyzed.
Finally, the recombination lifetimes will be calculated for undoped and doped
crystallites.


### 3.1. Electronic Structure of Perfect Silicon Nanocrystals

The first step is a calculation of the ground-state
energies both for the conduction and valence bands of a nanocrystal. Note that
here and throughout the paper we will use terminology that is typical for the
bulk material. Of course, no genuine energy bands exist in the nanocrystal
because of its finite size. Nevertheless, the states above and below the NC
optical gap originate from the size-quantized states of conduction and valence
bands of the bulk crystal, respectively. Moreover, in the following we mainly
intend to employ the envelope-function approximation (k⋅p method) to
describe electronic states of the nanocrystals. Consequently, such the
terminology is convenient and should not lead to any misunderstandings. The
states above and below the optical gap will be further referred to as
conduction and valence bands, as in the bulk. Different approaches are used
when calculating the gap of silicon nanocrystals. Among them, the simplest is
the single-particle approximation. Usually, the single-particle gap exceeds
real optical gap because the electron-hole Coulomb interaction reduces the
total energy of the system. For this reason, the values of the single-particle
gap may be treated as an upper bound for the optical gap. In this section we
discuss electron and hole states of a “pure” crystallite, which has no defects
and a perfect diamond lattice all over its whole volume. The results are
compared with the simpler approximations discussed in the introduction.


#### 3.1.1. Optical Gap. Numerical Results

It is possible to separate all calculation methods
applied for computation of the electronic spectra in pure NCs into two groups. In
the first group we have the numerical first-principles, empirical, and semi-empirical
methods, such as density functional theory (DFT) [68, 74–78], pseudopotential (PP) method (or various combinations
of PP and DFT) [79–82],
tight-binding (TB) models [83–86], combined Green's function
(GW-approximation), and the
Bethe-Salpeter equation (BSE) technique [87–89]. All these methods require considerable
computational time, a problem that worsens as the nanocrystal size is
increased. Analytical methods such as the effective mass approximation or k⋅p method [67, 90–95] may be
ascribed to the second group. In following we combine these two methods into a
single envelope function approximation (EFA). The latter allows one to perform
calculations without any restrictions on cluster sizes. Additionally, this
approach can be used to find the higher excited states of the nanocrystal.In the present subsection, we summarize the results of
some of the methods discussed above, which illustrate dependence of the optical
gap on the nanocrystal size (Figure4). In all calculations, any dangling bonds
at the nanocrystal surface were considered to be H-passivated. As is seen in
the figure, there is some scatter of data due to the different computational
methods and different approximations utilized in the computational procedure.Figure 4
Optical gap calculated by PP [65],  TB [85, 86], and DFT [68, 74] methods. Solid line: EFA calculations, see (22) below [93].In particular, taking into account the self-energy
correction or electron-hole interaction can considerably change the magnitude
of the nanocrystal optical gap, in opposite directions, as has been pointed out
by many authors (see, e.g., [81, 88, 89]). In particular, it is known that the local density
approximation of the DFT tends to underestimate the optical-gap (similar
results happen for silicon nanowires) [96, 97] compared to those obtained in experiments. For a more accurate account of the
self-energy corrections and excitonic effects, the GW+BSE approach is often
applied. The combination of the GW and BSE methods yields almost complete
compensation of the exciton and the self-energy corrections to the
single-particle optical gap [88]. The same effect was
also found by Franceschetti and Zunger [81] within the framework of
the PP method. Unfortunately, the GW+BSE approach is employed for nanocrystals
whose diameters do not exceed 1 nm [87–89]. Therefore, we did not
depict the results of this approach in Figure 4.


#### 3.1.2. Electron and Hole Spectra in a Dot

Let us now discuss electronic spectra in both the valence
and conduction bands of the quantum dot. As has been pointed out, for this
purpose it is convenient to use the EFA. Here we describe in more detail not
only the lowest but also the higher excited electronic states as a function of
size.We assume the total potential
energy of the electron inside the quantum dot to be of the following form(2)U(r)=U0(r)+Vsp(r). Here U0(r) is the confining potential equal
to zero inside and infinity outside the dot. The second part Vsp(r) describes an interaction between
the electron and its image, arising due to the charge polarization on the
boundary between the silicon nanocrystal and its dielectric surrounding. Vsp(r) is often referred to as a
self-polarization term. It can be represented as(3)Vsp(r)=e2(εs−εd)2εsR∑l=0∞l+1lεs+(l+1)εdr2lR2l, where εs and εd are the static dielectric constants of silicon
and the dielectric matrix, respectively. In order to find the electronic states
in the dot, we have to solve the single-particle Schrödinger-like equation for
the envelope function vector |Φ〉:(4)(H+U(r))|Φ〉=E|Φ〉. Here, H is the bulk k⋅p Hamiltonian operator acting on the
six-dimensional (6D) envelope-function vectors |Φ〉,
and E is the electron energy. The components of the
6D-vector |Φ〉 are slowly varied expansion coefficients Φj(r) of the total wave function in the Bloch-state
basis.The electronic states in the
valence and conduction bands can be determined separately. Starting with the
valence band, the Bloch-state for theΓ-point is |YZ〉|↑〉, |XZ〉|↑〉, |XY〉|↑〉, |YZ〉|↓〉, |XZ〉|↓〉, |XY〉|↓〉,
where |↑〉 and |↓〉 are “up” and “down” spinors, respectively, and
the Bloch functions |YZ〉, |XZ〉, |XY〉 belong to the irreducible representation Γ25′.
The k⋅p Hamiltonian
matrix H in 4 is the sum of three
parts: H=H(0)+H(1)+H(so).
Here H(0)=(p2/2mh)×I is the isotropic part obtained as the average
of the total k⋅p matrix
over all angles in p-space. H(1) is the anisotropic part that can be
represented by two equivalent 3×3 blocks situated on the main diagonal of the
total 6×6  k⋅p matrix:(5)H(1)=(H300H3),H3=−12m0(Q1NpxpyNpxpzNpxpyQ2NpypzNpxpzNpypzQ3), where Q1 denotes ((L−M)/3)(3px2−p2), Q2 denotes ((L−M)/3)(3py2−p2), and Q3 denotes ((L−M)/3)(3pz2−p2).Finally, the term(6)H(so)=13(0−iΔ000ΔiΔ0000−iΔ000−ΔiΔ000−Δ0iΔ000−iΔ−iΔ00ΔiΔ0000) describes the spin-orbit
interaction. We have introduced above the spin-orbit energy Δ equal to 44 meV for silicon, the 6×6 identity matrix I,
and the hole effective mass mh=3m0/(L+2M),
where the numbers L, M, N are dimensionless empirical parameters in the k⋅p Hamiltonian operator in the valence band. For
silicon they equal 5.8, 3.43, and 8.61, respectively [98].Because of the isotropic and
diagonal form of the operatorH(0)+U0(r),
it is possible to classify its eigenstates similarly to atomic systems using
common terminology such as s-, p-, and d-type states, and so forth [92]. Accordingly, one may expand the components of the envelope-function
vectors over these eigenstates as(7)Φj(r)=∑αCjα|α〉, where |α〉 stands for the s-, p-, d-, … states and Cjα are the expansion coefficients. For instance,
the 1s-, 1p-, 1d-, (in the following simply s-, p-, and d-) and 2s-type states
are described by the following functions:(8)|s〉=π2R3j0(πr/R),|pa〉=32πR3j1(μ1r/R)j0(μ1)xar,|dab〉=152πR3j2(μ2r/R)j1(μ2)xaxbr2,a≠b,|dx2−y2〉=158πR3j2(μ2r/R)j1(μ2)x2−y2r2,|d3z2−r2〉=158πR3j2(μ2r/R)j1(μ2)3z2−r2r2,|2s〉=π2R3j0(2πr/R), where jn(x) are the spherical Bessel functions of argument x,
and μn are the first roots of jn(x).
Below, we restrict the basis of envelope functions |α〉 with these states only, and determine the
electron and hole spectra. Substitution of Φj(r) into (4) yields algebraic equations for Cjα:(9)(E−Eα)Ciα=∑β〈α|Hij(1)+Hij(so)+Vsp(r)δij|β〉Cjβ, where Eα denotes the energy of the state |α〉, δij stands for the Kronecker delta, and the Einstein
convention has been used when summing over j.
Considering all operators in the right side of (9)) as perturbations [92], one can solve the equation and find
the energies in the valence band.Restricting the basis of the
envelope states|α〉 by 1s, 2s, 1p, and 1d functions, and
neglecting the spin-orbit interaction, we obtain the following “hierarchy” of
energies (for convenience, we write down the hole energies differing in the
sign from the electron energies, which are determined with (9))). The
ground-state energy(10)Eh(s)=ℏ2π22mhR2 is triply degenerate (or sixfold,
if spin is taken into account). It is the energy of the 1s state. Hybridization
of the p-type states yields four different levels with the energies(11)Eh1(p)=ℏ2μ122mhR2[1−3N+2L−2M5L+10M],Eh2(p)=ℏ2μ122mhR2[1−3N−4L+4M5L+10M],Eh3(p)=ℏ2μ122mhR2[1+3N−2L+2M5L+10M],Eh4(p)=ℏ2μ122mhR2[1+6N+4L−4M5L+10M], where μ1=4.4934.
The energies of the d-d hybridized states are as follows:(12)Eh1(d)=ℏ2μ222mhR2[1−6N7L+14M],Eh2(d)=ℏ2μ222mhR2[1−3N+4L−4M7L+14M],Eh3(d)=ℏ2μ222mhR2[1+3N−2L+2M7L+14M],Eh4(d)=ℏ2μ222mhR2[1+6N−4L+4M7L+14M], where μ2=5.7634.
Finally, the energies of the 1d-2s hybrids are written as(13)Eh1(2sd)=ℏ22mhR2[4π2+μ222+3μ2214N+2L−2ML+2M−1265πμ24π2−μ22NL+2M],Eh2(2sd)=ℏ2μ222mhR2,Eh3(2sd)=ℏ22mhR2[4π2+μ222+3μ2214N+2L−2ML+2M+1265πμ24π2−μ22NL+2M]. The contribution of the
self-polarization term from (10)–(13) has been
omitted since it represents, in fact, a common shift of all the levels by e2(1/εd−1/εs)/2R.In Figure5 the energy levels from (10)–(13) have been
plotted as functions of the NC radius. The two lowest hole levels have very
similar energies. The lowest level, being sixfold degenerate, corresponds to
six electron states with an s-type envelope function. These states have an
isotropic macroscopic (averaged over the unit-cell volume) charge distribution
inside the dot. The second level corresponds to six electron states with p-type
envelope functions. Correspondingly, the macroscopic charge has an anisotropic
distribution for these states (see [92] for details).Figure 5
Hole energy spectrum according to (10)–(13) in case of no spin-orbit interaction. Solid line: s-type level. Short dashed lines: p-p hybridized levels. Dots: d-d hybridized levels. Long dashed lines: 2s-d hybridized levels. All energies are counted deep into the valence band with respect to the valence band maximum. The degeneracy degree including spin (from bottom to top) is 6; 6; 6; 4; 6; 4; 6; 6; 2; 6; 2; 6.In the conduction
band the Bloch-state basis consists of six Bloch functions
of three differentX-points in the Brillouin zone
[99]. We denote three pairs of these
functions as |X〉, |X′〉; |Y〉, |Y′〉; |Z〉, |Z′〉.
Each pair belongs to the twofold degenerate irreducible representation X1 of the corresponding X-point. The unprimed
Bloch functions have a nonzero value at the lattice site, while the primed
functions are zero at those points. The choice of the X1-functions as the Bloch basis allows us to describe correctly a
dispersion-law nonparabolicity that originates mainly from the energy-branch
crossing in the X-points.Thek⋅p Hamiltonian operator in the conduction band
represents a 6×6 matrix having the form(14)H=(Hx000Hy000Hz). Each element of the matrix H is a block 2×2 matrix defined by the following expressions [99]:(15)Ha=(pa22ml+p2−pa22mt(1mt−1m0)pbpc+ip0mlpa(1mt−1m0)pbpc−ip0mlpapa22ml+p2−pa22mt). Here p0≈0.14(2πℏ/a0) is the distance in p-space from any of the energy minima to the nearest X-point, a0 is the lattice constant of silicon. ml=0.92m0 and mt=0.19m0 are the “longitudinal” and “transverse”
effective masses. The origin of the E-axis coincides with the X-point energy. Each of the
small indices (a, b, c) runs over the values x, y,
or, z and always differs from the others.Solving (4) for the conduction
band is done in the same way as for the valence band, as in (9)–(13) above. One
can then obtain the following groups of energies. The ground states, as well as
the second excited ones, are the 1s-1p hybrids. Their energy levels are given
by(16)Ee1(sp)=Es+Ep−2Hpp2−(Ep−Es−2Hpp2)2+Hsp2,Ee2(sp)=Es+Ep−2Hpp2+(Ep−Es−2Hpp2)2+Hsp2, where(17)Hsp=ip0ml〈s|pz|pz〉=2πℏμ1p0[3mlR(μ12−π2)],Hpp=112(1mt−1ml)〈pz|p2−3pz2|pz〉=ℏ2μ12(ml−mt)15mtmlR2 are s-p and p-p type matrix
elements of anisotropic part Ha, Es=ℏ2π2/2meR2,
and Ep=ℏ2μ12/2meR2 are the energies of the s- and p-type states
of the isotropic unperturbed Hamiltonian operator. The isotropic average
electron effective mass is defined by me−1=(2mt−1+ml−1)/3≈(0.26m0)−1. The p-p hybridization forms states with energies:(18)Ee1,2(p)=Ep+Hpp∓Hxy. Here Hxy=ℏ2μ12(m0−mt)/5mtm0R2.
The next three levels are the result of the d-d hybridization:(19)Ee1(d)=ℏ2μ222meR2(1−27ml−mt2ml+mt−67mlm0m0−mt2ml+mt),Ee2(d)=ℏ2μ222meR2(1+47ml−mt2ml+mt),Ee3(d)=ℏ2μ222meR2(1−27ml−mt2ml+mt+67mlm0m0−mt2ml+mt). Finally, the 2s-d hybridized
energies are obtained numerically:(20)Ee1(2sd)=ℏ22meR2(4π2+2μ223−0.477μ22),Ee2(2sd)=ℏ22meR2(4π2+2μ223−0.455μ22),Ee3(2sd)=ℏ22meR2(4π2+2μ223+0.932μ22). The results for the
single-particle electron spectrum in the conduction band are shown in Figure 6.Figure 6
The lowest energy levels in the conduction band as functions of the NC radius, from (16)–(20). The energies are counted from the X-point energy. Solid line: s-p type levels. Short dashed lines: p-p hybridized levels. Dots: d-d hybridized levels. Long dashed lines: 2s-d hybridized levels. All the levels are 12-fold degenerate including spin degeneracy.The wave function of the ground state turns out to be
a superposition of the terms withs- and p-type envelope functions (see [92] for details). The “weight” of the
s-type envelope function is greater than that of the p-type function. Therefore
the macroscopic charge for the ground state has a distribution close to
isotropic.From the above calculations, the optical gap of the NC
can be obtained as the sum of the lowest energies in both bands and the energy
difference betweenX- and Γ-points in bulk ΔXΓ=1.24 eV:(21)εg(R)=ΔXΓ+Eel(sp)+Eh(s). This energy somewhat overestimates value compared to
that obtained with the numerical methods, as indicated by the solid line in Figure 7.
This is a consequence of two main factors.Figure 7
NC gap versus reciprocal dot radius. Solid line: infinite barriers (21). Circles: finite barriers (3.2 eV for electrons and 4.3 eV for holes, which corresponds to Si/SiO2 interface) and constant effective mass. Disks—finite barriers and free electron mass outside the dot. Both of the latter are numerical results by EFA [92]. The dashed line is a guide to the eye.The first factor is that we have assumed infinitely
high potential barriers at the interface. The authors of  [91, 93, 95] calculated the carrier energies supposing the
potential barriers to be finite and equal to approximately 3 eV (or a
little more) for electrons and 4-5 eV for holes. Moreover, the discontinuity
of the effective mass has been taken into account. In Figure 7 we have plotted
the optical gap of silicon nanocrystal versus the confining parameter 1/R for three models differeing according the
nanocrystal’s surroundings. The solid line corresponds to infinitely high
potential barriers. Circles represent the finite barriers and constant
effective mass throughout the sample. Finally, in case of the mass discontinuity
with effective mass outside the dot coinciding with free electron mass, the
optical gap has been represented by disks. For the last two cases, the
dependence Δεg(R) becomes weaker than R−2,
especially for the case of the discontinuous effective mass where the
dependence is closer to 1/R.
In the last case the gap dependence is well fitted by the function:(22)εg(1)(R)=εg2+D1R2 with D1=4.8 eV2·nm2. In general, one may conclude that the calculated values of
the optical gap are reduced due to the barrier finiteness and mass
discontinuity, and agree better with those shown in Figure 4.The second factor, strongly influencing the dot gap,
is electron-hole Coulomb interaction. Its contribution to the optical gap will
be discussed in the next subsection.


#### 3.1.3. EFA Calculations: Exciton Corrections

The electron-hole interaction reduces the energy of
the electron-hole pair. As a result, the exciton optical gap is also reduced.
In order to estimate the exciton correction to the total electron-hole
single-particle energy one must solve the two-particle problem. This has been
done for silicon NCs embedded in an infinitely wide-band-gap material [67, 90, 94]. The exciton corrections
were found to be about 0.4 eV [67], 0.25 eV [90], and 0.3 eV [94] for 2-nm-diameter NCs, with the
correction decreasing approximately as R−1 with increasing size.Meanwhile, as has been pointed out in the preceding section, the
finiteness of the potential barriers and effective mass discontinuity can
appreciably influence the electron and hole energies. Therefore, it is possible
to expect the same for the excitonic energies. Ferreyra and Proetto [91] have studied excitonic states for a
quantum dot in vacuum. Accordingly, they accepted the barrier heights equal to
the electron affinity of the corresponding bulk material for electrons and
infinity for holes, and the effective mass being m0.
They found an exciton correction of the order of 0.2 eV for a 2-nm-diameter
dot. As the nanocrystal size increases, the magnitude of this correction
decreased as R−0.7.The authors of some of the previous work (e.g., [67, 90, 91]) treated the Coulomb potential energy as VC=−e2/εsreh similar to bulk silicon. However, in a quantum
dot the electron and hole interact not only with each other but also with their
“images” due to the difference in the permittivity of the materials inside and
outside the dot. These are the so-called polarization fields arising because of
the charge polarization at the dot boundary. The interaction with the
polarization fields has been taken into account in [94]. Nevertheless, the
polarization fields did not lead to any significant corrections to the optical
gap. As mentioned above, the correction does not exceed 0.3 eV, which is
close to the values obtained by other authors.Evidently, the calculations carried out with the EFA [67, 90, 91, 94] provide sufficiently good coincidence with both experimental and
theoretical data. For comparison, we have depicted in Figure 4 with the
solid line the EFA single-particle gap according to (22). The gap values εg(1)(R) agree well with those computed by PP, TB, and
DFT methods. Obviously, a small exciton correction of about 0.3–0.1 eV
for 2 to 5 nm diameter crystallites, respectively, does not significantly
change the single-particle gap values.It should be noted, in conclusion, that all the authors employed the
perturbation theory considering the electron-hole interaction to be weak
compared to the typical quantum confinement energies in the dot. Such an
approach is justified if the exciton Bohr radius is significantly larger than
the dot radius. This requirement remains valid for small quantum dots with the
sizes of about 5-6 nm or less.


#### 3.1.4. Interface States

We have already emphasized in the introduction that radiative
transitions between interface states are also often considered as the origin of
NIR, or even visible, emission from silicon nanocrystals. In a certain sense,
this point of view is alternative to the idea of quantum confinement.
Description of the interface states with EFA is difficult because the real
potential existing in the vicinity of the nanocrystal surface has in principle
a microscopic nature that manifests itself at a distance on the order of a bond
length. Meanwhile, the EFA is, in fact, a “macroscopic” method which is not
able to distinguish the spatial structure on scales smaller than the size of
the primitive cell. Therefore TB and DFT methods are essentially more suitable
for this goal.As a rule, interface states originate from various-type defects existing
at the surface of the crystallite. These include, mainly, dangling bonds (Pb centers) which can be neutral, positively, or negatively charged, and different
SiO bonds (backbonded and double bonded oxygen) arising at Si/SiO2 interface. Charged dangling bonds corresponding to zero or two electrons in the
bond level, respectively, produce deep levels inside the band gap of bulk
silicon. Their energies are approximately 0.3 eV below the conduction band
minimum for the negatively charged bond, and 0.3 eV above the valence band
maximum if the bond is positively charged [100]. Consequently, the effective gap
between these levels is about 0.5-0.6 eV [101]. For nanocrystals, the effective gap gradually rises as the nanocrystal size
decreases. In particular, for a 5-nm-diameter NC this shift equals
approximately 0.6 eV, while for a 2 nm crystallite it is 1.3-1.4 eV [100]. Thus, we see that the dangling bond levels are sensitive to the NC size.Nanocrystal oxidation has been studied theoretically by TB [8, 102], DFT [103, 104],
and GW+BSE [89]
methods. The role of the so-called
backbonded (Si–O–Si) and double bonded (Si=O) oxygen atoms has been examined
for the case where the spherical NC surface is passivated with hydrogen. In the
first case (Si–O–Si), oxygen forms two bonds with two different silicon atoms
which already have four bonds with other silicon or hydrogen atoms. In the
second case (Si=O), oxygen replaces two hydrogen atoms bonded with one silicon
atom. It is natural to suppose that the system perturbation in the second case
should be more significant because of the stronger distortions of the structure.
In fact, it was found that the presence of one Si=O bond at the NC surface
provides an optical gap almost independent of the dot size [8].
At the same time, the nanocrystal
with a Si–O–Si bond has a gap that gradually decreases with increasing size.
Nevertheless, this gap remains smaller, and varies more slowly than the gap of
perfect nanocrystal with hydrogen passivated surfaces. The
calculations carried out by different methods [89, 102, 103]
for oxidized crystallites show an essential difference
in the energy of the ground electron-hole transition in crystallites with
Si–O–Si and Si=O bonds. All the methods used exhibit the smallest gap for the
case of double-bond oxidation, while the crystallites with backbonded oxygen
have the greater gap. In turn, the latter is less than the gap of a perfect NC.
The authors of [89] have found that only the bridge-type bond (Si–O–Si) can
provide a satisfactory agreement with experiments on the photoluminescence in
nc-Si/SiO2 system. This fact may be explained by the considerably
different oscillator strengths of the ground transitions for
NCs with Si=O and Si–O–Si defects. As has been shown by Nishida [102], the oscillator strength for Si–O–Si
case is several orders of magnitude greater than that for Si=O case, especially
for NCs smaller than 1 nm in diameter. This is precisely the case studied
by Luppi et al. [89], where NCs of 0.5 nm and 0.9 nm in
diameter were considered. However, according to [102]
the difference between the
size-quantized energy levels and those of the Si=O and Si–O–Si defects, as well
as the difference in the oscillator strengths of the ground electron-hole
transitions, are essential only for NC diameters less than 2 to 2.5 nm [8, 102].
For larger sizes, the energies and
the transition intensities for perfect and oxidized crystallites almost
coincide.Finally, nonradiative trapping
rates are also critical in the light emission properties of silicon NCs. The
group of neutral or chargedPb centers at silicon dangling bonds is the most important nonradiative trap in
bulk silicon [105],
and acts also in the case of Si NCs [58]. In a detailed theoretical investigation, Lannoo et al.
[106] found that the carrier trapping at neutral Pb centers is nonradiative, whereas capture at charged
centers can lead to photon emission at energies smaller than the bandgap of
bulk silicon. The rate for carrier trapping at a neutral dangling bond defect
at the Si–SiO2 interface in a NC with a single Pb center was established theoretically as a function of
the NC size [106]:(23)wnr=σ0vV12π((E0ℏω)2+z2)−0.25×exp[−Scoth(ℏω2kT)+E02kT+((E0ℏω)2+z2)0.5−E0ℏωasinh(E0ℏωz)]. In (23), σ0 is the capture cross section given by c0/v where c0 is the capture coefficient and v is the thermal velocity equal to 8kt/πm*, E0 is the ionization energy of the defect
(approximately equal to the Franck-Condon energy plus the carrier confinement
energy), S is the Huang-Rhys factor (S=~15) [107], ℏω is the average phonon energy, V is the nanocrystal volume, and z=S/[sinh(ℏω/2kT)]. The capture rate increases strongly as a function of NC size due to the
decreasing defect ionization energy, until eventually the volume term begins to
dominate the trapping rate, which then begins to decrease again. Since, the
calculated rate is for a single isolated defect, in practice the rate should be
modified for the likelihood of multiple defects on larger clusters, as
discussed in Section 4.


### 3.2. Shallow Impurities in Silicon Nanocrystals

There is one more type of defect which can play an
important role in the optical and transport properties of bulk semiconductors
and their low-dimensional counterparts (in particular, silicon). These are
impurity centers, of which some experimental results were discussed previously. Let
us now consider NCs doped with shallow impurities and discuss impurity states
in silicon quantum dots.Various aspects of the problem
have been explored. One of them—the central-cell effect [108–111]—causes splitting of the sixfold degenerate
ground energy level in bulk silicon into a singlet, doublet, and triplet with a
typical energy splitting of about 10–20 meV
for various donors [112]. According to a number of theoretical
studies [73, 113–119], in quantum dots doped with donors or acceptors the central-cell
potential strengthens the level splitting compared to (i) bulk silicon, and
(ii) undoped dots [66, 85, 120, 121]. Investigations of
the spatial charge distribution [122, 123], the formation of impurity centers
inside silicon nanocrystals from the energy point of view [114–117, 124, 125], intervalley scattering [126], hyperfine splitting and optical gap effects [127], and the screening of the point-charge field [123, 128–136] have been
performed. Below, we present EFA calculations of the electronic structure of
both donor- and acceptor-doped silicon quantum dots and examine the dependence
on the impurity position inside the dot within the framework of the hydrogenic-impurity
model. Also, we discuss the effect of the central-cell potential in screening
the point charge in NCs.


#### 3.2.1. Screening in Quantum Dots

In order to solve accurately
the Schrodinger-like equation for the envelope functions in the dot in the
presence of an impurity ion, we need the correct expression for the Coulomb
potential energyVC.
Its determination in the quantum dot is not a simple task because of the
complicated character of screening in NCs. Some authors described the screening
properties in an NC with a modified dielectric constant ε(R) that depends on the dot radius [137–139]. Such a simple model, indeed, gives a moderate increase of the dielectric
properties due to the finite size of the crystallite but it does not reflect
correctly the local structure of the electric field.The correct description of such
a structure requires first-principles calculations [123, 129–136]. Nevertheless, the microscopic
picture permits a clear qualitative macroscopic interpretation using the local
dielectric function ε(r) depending on electron position
vector, r.
Taking into account the short-range field and charge polarization at the dot
boundary due to different static dielectric constants εs and εd of an NC and its surroundings,
one can in principle obtain ε(r) (Figure 8) [73].Figure 8
Dielectric function of silicon NCs in silicon dioxide matrix (εd=3) for (1)  R=1 nm; (2)  R=1.75 nm; (3)  R=2.5 nm.An overall decrease ofε(r) in a nanocrystal takes place compared to the
bulk value of εs=12. As was pointed out earlier [123, 128, 129], this decrease is mainly due to the polarization charges at the dot
boundary. The monotonic decrease of ε(r) extends to
the nanocrystal boundary, where the value of the dielectric function becomes
equal to εd. At the same time, the sharp reduction of ε(r) toward unity occurring at small r is exclusively due to the short-range
central-cell potential.


#### 3.2.2. Donor States in Silicon Crystallite

First, we consider electronic
states in a silicon NC with a hydrogenlike donor placed in some arbitrary
position inside the dot. The total electron potential energy described by (2) transforms
into U(r,h)=U0(r)+VC where h stands for the impurity position vector, and VC is a Coulomb potential energy, including not
only the direct electron-ion interaction but also the interaction with
polarized charges arising at the dot boundary. Solving the eigenstate and
eigenvalue problem for the conduction band [140] yields the ground-state energy splitting shown in Figure 9 at nx=0.8, ny=0.5, and nz=0.33, where n=h/h,
and h stands for the absolute value of h.Figure 9
Fine structure of the energy spectrum with respect to the unperturbed sixfold degenerate energy level. Solid lines correspond to two states located in the Brillouin zone at theX-point (001). Dashed line: X-point (010). Dots: X-point (100). The direction of the donor's position vector is defined by  nx=0.8,  ny=0.5, and nz=0.33. The NC radius is indicated in the figure.In Figure10, the
envelope-function correction ΔΦ is shown for the ground state due to the
hydrogen-like donor, as a function of the angles θ and φ on the spherical surface r=h for the former values of na. Angles θ and φ are the spherical coordinates. The maximal
values of ΔΦ (light areas in the figure) are located around
the donor site marked with the cross. This is a natural result in that the
electron density shifts towards the donor site.Figure 10
Contour plot of the first-order correction to the envelope function for a 3-nm-diameter quantum dot. The value of the correction rises from dark to light. The cross indicates the donor position defined by:nx=0.8, ny=0.5, and nz=0.33. h/R=0.46.


#### 3.2.3. Acceptor States in Silicon Crystallite

It is interesting to compare the fine structure of the
spectrum for donor- and acceptor-doped dots. Obviously, the fine structure in
the valence band occurs because of the spin-orbit interaction and the asymmetry
of the Coulomb field inside the dot, when the acceptor occupies some noncentral
position. The energy splitting has been presented in Figure11. Similarly to
the donor case, the splitting turns out to be large compared to the magnitudes
of chemical shifts in bulk silicon [141–144].Splitting of the two lower hole levels shown in Figure5 as function of the dimensionless acceptor displacement from the dot center in (a), and with respect to the NC radius in (b). The direction of the acceptor position-vector is defined by  nx=0.8, ny=0.5, and nz=0.33. Dashed and solid lines represent the energies originating from doublet and quadruplet at  h=0, respectively. All the energies are counted from the mean quadruplet energy at  h=0.
(a)(b)The Coulomb and spin-orbit interactions remove the sixfold degeneracy of
both lowest levels shown in Figure5, keeping only the double spin degeneracy
of each the levels. Furthermore the splitting increases with decreasing the
nanocrystal size. This is evidence in favor of a quantum confinement effect, as
was the case in the donor-doped dots.The spatial distribution of the
hole density in B-doped crystallite for all the six doublets is shown in Figure12 for h/R=0.1 (upper panels) and h/R=0.46 (lower panels). We have plotted the average of
the squared absolute value of the total wave function over the unit cell. In
this case the Bloch-function oscillations do not appear in the electron density,
which reduces to the “density of envelope function” ρenv(r)=∑j=16|Φj(r)|2,
where Φj(r) is the jth element of the 6D envelope function vector |Φ〉 as before.Density plot of the probability distributionρenv(r)   of all the six doublets at    h/R=0.1 (upper images), and h/R=0.46 (lower images) for NCs containing a hydrogenic acceptor. The acceptor position is indicated with a red point. (a): ground state; (b) to (f): first to fifth excited states. For each state the hole density is normalized to its maximum in the state, and rises from dark to light. The nanocrystal boundary is marked by the yellow circle, with R=1.5 nm.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)As the calculations show, for
all the six doublets the envelope-function density has an axial symmetry with
respect to the line drawn through the NC center and the acceptor. Thereforeρenv(r) has the same distribution in any dot
cross-section to which the acceptor position-vector belongs. Figure 12 shows such
a central cross-section of the electron density averaged over the unit cell for
all the doublets. Brighter areas in the density plots correspond to higher
values of ρenv(r).
The circle represents the nanocrystal boundary, and the bold point situated at
the vertical axis indicates the acceptor location.It should be noted that the general trends in the location of the
electron density under the action of the acceptor electric field are, in fact,
similar for all the doublets in both casesh/R=0.1 and h/R=0.46.
In particular, the ground-state electron density shifts to the acceptor site,
while for the excited states, as the energy of the state increases and the
electron density gradually moves into the areas free of the acceptor.


### 3.3. Interband Recombination in Silicon Nanocrystals

The PL intensity and quantum efficiency is defined by the
relative contributionτR−1/τPL−1 of a radiative recombination channel in an
interband transition. In turn, the total decay rate is defined by both
radiative and nonradiative recombination rates: 1/τPL=1/τR+1/τNR.
Consequently, in order to discuss possible means of increasing the quantum
efficiency of silicon crystallites, we should understand what factors influence
the radiative and nonradiative rates. For this purpose in this section we
analyze theoretically both radiative and nonradiative recombination processes.


#### 3.3.1. Radiative Recombination Times in Undoped Si NCs

In bulk silicon, no-phonon radiative transitions between the
conduction-bandΔ-point and the valence-band Γ-point are forbidden because of indirect band-gap of silicon. In this
case, only phonon-assisted transitions may take place, as shown in Figure 13.
In a quantum dot no-phonon emission becomes possible but remains a low
probability process for reasons discussed below. Meanwhile, the phonon-assisted
radiative transitions have much greater rate. Here, we calculate both no-phonon
and phonon-assisted radiative lifetimes in silicon nanocrystal.Figure 13
Schematic representation of the silicon band structure with phonon-assisted radiative transitions from the conduction-band minimum to the valence-band maximum indicated by arrows.We first calculate the rateτ0−1 of the no-phonon radiative recombination.
Using Fermi's golden rule in the first-order perturbation theory, one can write
the decay rate for the transition between initial (I) and final (F) states in the conduction and valence bands, respectively, in the
following form:(24)τ0,IF−1=2πℏ∑Q,σ|WIF|2δ(εg(R)−ℏωσ(Q)). Here, ωσ(Q) and Q stand for the photon frequency and wave
vector. The Dirac delta-function reflects the energy conservation law for the
electron transition. The operator W describes the electron-photon interaction,
and has the form:(25)W=∑k,σ2πℏe2κ(εs;εd)m02ωσ(Q)V0εs(cQσ+cQσ+)eQσp, where p=−iℏ∇ is the
electron momentum operator, eQσ is the polarization vector, the operator cQσ annihilates and cQσ+ creates a photon with the wave vector Q and polarization σ,
and V0 stands for the volume of the electromagnetic
resonator. The function κ(εs;εd) is written as [145](26)κ(εs;εd)=εdεs(3εd2εd+εs)2. This function represents the
correction factor in the field magnitude due to the replacement of homogeneous
media with bulk permittivity εs by a spherical silicon nanocrystal surrounded
by silicon dioxide with a dielectric constant εd.The initial state in (24) corresponds
to an electron-hole pair being in its ground state, and an ensemble of photons
whose distribution over Q is described by the Bose-Einstein statistics.
In the final state, the valence band is completely occupied and the conduction
band is empty. The number of photons in the final state always increases by
one.As has already been mentioned
(see Section3.1.2), the ground spinless electron state in the conduction
band is sixfold degenerate. However, TB [85, 120], PP [66], and DFT [121] calculations revealed a weak
splitting of the ground state into the singlet, doublet, and triplet states due
to the tetrahedral symmetry of the spherical silicon nanocrystal. The authors
pointed out that as the NC size varies, A1-, E-, or T2-type states alternately become the ground level. Correspondingly, in
order to take into account tetrahedral symmetry of the quantum dot, we build
the six ground states in accordance with the symmetric transformations of the irreducible
representations A1 (singlet), E (doublet), and T2 (triplet) of the tetrahedral group. These functions
are defined as follows [73]:(27)ΨS=cos(λ)|A1〉|s〉+sin(λ)|X′〉|px〉+|Y′〉|py〉+|Z′〉|pz〉3,ΨD(1)=cos(λ)|E(1)〉|s〉+sin(λ)|X′〉|px〉−|Y′〉|py〉2,ΨD(2)=cos(λ)|E(2)〉|s〉+sin(λ)|X′〉|px〉+|Y′〉|py〉−2|Z′〉|pz〉6,ΨT(1)=cos(λ)|X′〉|s〉−sin(λ)2|A1〉+3|E(1)〉+|E(2)〉6|px〉,ΨT(2)=cos(λ)|Y′〉|s〉−sin(λ)2|A1〉−3|E(1)〉+|E(2)〉6|py〉,ΨT(3)=cos(λ)|Z′〉|s〉−sin(λ)|A1〉−2|E(2)〉3|pz〉. Here, the parameter λ is defined by the relationships [92](28)cos(2λ)=Ep−Es−2Hpp(Ep−Es−2Hpp)2+4Hsp2,sin(2λ)=2Hsp(Ep−Es−2Hpp)2+4Hsp2, where vectors |s〉 and |pa〉 stand for the s- and p-type envelope functions
as before. The Bloch states |A1〉=(|X〉+|Y〉+|Z〉)/3, |E(1)〉=(|X〉−|Y〉)/2, |E(2)〉=(|X〉+|Y〉−2|Z〉)/6,
and |T2〉=|X′〉, |Y′〉,
or |Z′〉 belong to the representations A1, E,
and T2 of the tetrahedral group. In what follows,
when computing the interband matrix elements of W we will choose one of the states given by (27)
as the initial one. For convenience, one can rewrite the wave functions of the
initial states in a general form:(29)ΨI=∑α,jBIαjFα(r)ψj(r), where the index α indicates the “values” of the s, px,
py, and pz, functions, and ψj(r) stands for the six Bloch functions |X〉, |Y〉, |Z〉, |X′〉, |Y′〉, |Z′〉 of the irreducible representation X1.
The s- and p-type envelope functions are denoted as F(r),
and Fx(r), Fy(r), Fz(r),
respectively. Selecting expansion coefficients BIαj,
we can construct any of the states defined by (27).The final state is one of the
three degenerate states (neglecting the spin-orbit coupling) in the valence
band [92] with the s-type envelope function:(30)ΨF=F(r)ψF(r), where ψF(r) stands for the Bloch state basis functions |YZ〉, |XZ〉,
or |XY〉 of the irreducible representation Γ25′.After some algebra, one can
obtain recombination rateτ0,IF−1 in the form:(31)τ0,IF−1=4e2εsκ(εs;εd)εg(R)3m02ℏ2c3|pIF|2, where pIF=〈ΨI|p|ΨF〉.
In order to compute the momentum matrix element, we expand the Bloch functions ψj(r) and ψF(r) over the reciprocal lattice vectors gn as follows:(32)ψj(r)=∑mCjmexp[2πi(gm+ej)r/a0],ψF(r)=∑nVFnexp[2πignr/a0], where a0 is the lattice constant of silicon, and unit
vectors ej define the X-point. We set e1,4=ex, e2,5=ey,
and e3,6=ez.
Expansion coefficients Cjm and VFn are determined with the local pseudopotential
method. They satisfy the normalizing condition ∑n|Cjn|2=∑n|VFn|2=1.
As a result, the momentum matrix element is given by(33)pIF=ℏ4πa0bIF(R)(a0R)4, where bIF(R)=∑α,jBIαjJα(R) with Jα(R) being oscillating vector functions [146]
of the dot radius. For s- and p-type subscripts, respectively, these functions
are(34)J(R)=∑n,mCjm∗VFndn+mdn−m4cos(2πdn−mRa0),Ja(R)=−i3μ1π∑n,mCjm∗VFndn+mdn−m(a)dn−m5sin(2πdn−mRa0). Here, a=x,y,z, dn±m=gn±gm±ej,
and dn−m(a) represents an ath component of dn−m.
The nonzero value of pIF is exclusively due to the Heisenberg
uncertainty relations which cause the wide distribution of the wave function in p-space.Averaging (31) over all
possible initial and final states yields(35)τ0−1=e2εsκ(εs;εd)εg(R)12π2m02c3a02(a0R)8〈|bIF(R)|2〉. In Figure 14 we show the
no-phonon recombination rate τ0−1 as function of the dot size. Although the
no-phonon decay rate has a nonzero value, its magnitude remains small (less
than 103 s−1) within the range of the NC sizes shown
in the figure.Figure 14
Recombination rates of no-phonon and phonon-assisted radiative transitions between the ground electron and hole states in silicon NCs, after averaging over all degenerate initial and final states.Next, we calculate the
phonon-assisted recombination rate [95, 147–149]. The total phonon-assisted
rate of the radiative electron-hole recombination is determined in the
second-order perturbation theory as(36)τR,IF−1=2πℏ∑Q,σ∑q,ℓ|∑aWFAUAI+UFAWAIεI−εA|2×[δ(εg(R)−ℏωσ(Q)−ℏνℓ(q))+δ(εg(R)−ℏωσ(Q)+ℏνℓ(q))]. Here the
matrix elements of the electron-photon (W) and electron-phonon (U) interaction operators are calculated between the
initial I,
the final F,
and an intermediate state A; εA and εI are the total energies of the
intermediate and initial states, respectively, including not only the energies
of the electrons (or holes) but also the energies of the photons and phonons.
The phonon frequency of ℓth polarization is denoted as νℓ(q) with q being the dimensionless phonon wave vector
taken in units of 2π/a0.The
electron-phonon interaction operator is treated within the framework of the
rigid-ion model and is given by(37)U=−∑q,ℓ∑n,sℏ2MNνℓ(q)∇Vns×(eqℓsexp{i2πqRna0}bqℓ+eqℓs*exp{−i2πqRna0}bqℓ+). Here, N is the number of primitive cells in the
crystal, M is the mass of a silicon atom, Vns=Vat(r−Rn−τs) is the atomic potential, where Rn stands for the position vector of the nth unit cell, and τs represents the position vectors of two atoms
within the unit cell: τ1=0,
and τ2=(1,1,1)×a0/4, bqℓ+ and bqℓ are the phonon creation and annihilation
operators, and the phonon polarization vectors are denoted as eqℓs.
Making the Fourier transformation of the atomic potential: Vat(r)=N−1∑pVpexp{ipr},
one can rewrite the operator U in the form (see also [150, 151]):(38)U=2πia0∑q,ℓ,s,mℏ2MNνℓ(q)(q+gm)×[eqℓs∗Vq+gm∗exp{−i2π(q+gm)(r−τs)a0}bqℓ+−eqℓsVq+gmexp{i2π(q+gm)(r−τs)a0}bqℓ]. Calculations
of the electron-photon and electron-phonon matrix elements yield(39)τR,IF−1=4π2ℏ2e2εsκ(εs;εd)εg(R)3Mm02c3a04(a0R)3×∑ℓPIF(ℓ)ℏνℓcoth[ℏνℓ2kT], where(40)PIF(ℓ)=BIαm(BIβn)∗xFm(ℓ)xFn(ℓ)2π2R3×∫drFα(r)Fβ(r)F2(r),xFm(ℓ)=∑A〈ψF|w|ψA〉〈ψA|u|ψm〉+〈ψF|u|ψA〉〈ψA|w|ψm〉εI−εA.ψA represents the Bloch functions of
the intermediate states and the operators w and u are directly proportional to W and U:
(41)W=w2πℏa02πℏe2κ(εs;εd)m02ω(Q)V0εs,U=u2πa0ℏ2MNνℓ(q). As has been
shown earlier, the electron subsystem in silicon nanocrystals interacts more
efficiently with TO and LO phonons [147, 148] similarly to bulk silicon [150, 151]. The contribution of TA phonons is
negligibly small. Therefore, in the sum over the phonon modes in (39), only TO
and LO modes may be taken into account.For
convenience, we have again averaged the calculated rates over all the possible
initial and final degenerate states and obtained the recombination rate in the
following form:(42)τR−1=π2ℏ2e2εg(R)εsκ(εs,εd)9Mm02c3a4(a0R)3×(7.527coth[ℏνLO/2kT]ℏνLO+32.768coth[ℏνTO/2kT]ℏνTO)×(4cos2λ+9.26). Here: ℏνLO=0.051eV and ℏνTO=0.0576eV represent the energies of LO and TO phonons
and the numbers 7.527, 32.768 come from numerical integration of
electron-phonon matrix elements. The size dependence of the recombination rate
has been presented in Figure 14. The dependence of τR−1 on the dot radius is close to R−3.
This is much slower than the rate obtained for no-phonon transitions (R−8).The results presented here agree well both qualitatively and
quantitatively with those obtained by other authors [95, 147, 148] for similar systems. In particular,
according to their results, the recombination rates for NCs from 2 to 6 nm in
diameter do not exceed 105 s−1 and are never
less than 5×102 s−1. It should be noted,
however, that the TB calculations [148] produce a considerably slower
decrease of the radiative decay rate than the EFA (as developed here and in
[95, 147, 149]) predicts.Finally, concluding this subsection, we would like to discuss briefly
the relation between our single-particle calculations and the singlet-triplet
model which gives a reduction of the radiative lifetime with increasing
temperature, as in (1). It is, of course, not applicable in the frame of a
single-particle treatment to obtain the state splitting caused by the exchange
electron-hole interaction. Therefore, the temperature dependence of τR−1 (42)) is defined exclusively by the phonons. As
a result, no sharp rise of the decay rate appears in our model as the
temperature increases up to the room values. At temperatures higher than about
150–200 K, both models nevertheless should give similar results because the
upper excitonic states become highly populated. Thus, the single-particle model
remains valid for T>200 K. As well, it should be noted that the separation of dark and bright
excitons is not always possible. The spin-orbit coupling transforms forbidden
exciton transitions into suppressed ones due to the entanglement of
single-particle states with different spin projections. The entanglement can be
crucially strengthened by imperfections of the crystallite structure such as
shape nonsphericity or, especially, by various point defects such as Pb centers, SiO bonds, and impurities. In the presence of the latter effects, all
the transitions can become allowed and consequently the single-particle model
is acceptable.


#### 3.3.2. Radiative Recombination in Doped Dots

Experimental evidence of shallow impurity effects in
the PL of silicon NCs (see Section2.3) has no corresponding rigorous
theoretical grounds. Usually, when discussing various changes in optical
properties of the NCs due to doping, the nonradiative recombination channel is
considered more closely. For instance, doping may passivate dangling bonds, and,
as a consequence, essentially reduce the effectiveness of the nonradiative
channel. On the other hand, a large number of donors or acceptors in the
nanocrystal leads to the appearance of extra carriers that can recombine
through the Auger process. Thus, doping is an important factor influencing the
nonradiative recombination (a more detailed description of the nonradiative
recombination mechanisms can be found in the next subsection). The nonradiative
lifetime τNR can be varied by doping, which leads to the change
of the emission efficiency.The impact of doping on the radiative channel has not
yet reported. However, it is interesting to examine also the possibility of increasing
the quantum yield via enhancement of the radiative rateτR−1.
This is exclusively a quantum effect that can be explained by the
reconstruction of the wave functions in different symmetry (or asymmetry), and
changing selection rules for the interband transitions. We would like to touch
upon this question in the present subsection and compute the radiative
recombination rates in a silicon NC doped with a hydrogen-like donor and acceptor.In order to compute the decay rates for the doped dot
we follow (36). The main reason for the rate changes compared to the case of
the undoped dot lies in some corrections to the electronic wave
functions of both initial and final states (27), (30) due to the existence of a
shallow impurity in the dot. In the following, we discuss two different cases.The first case corresponds to an off-center position for the impurity
inside the dot. In the valence band such an impurity location creates a field
asymmetry in the system, which entangles the three spinlesss-type states (30))
with the three spinless p-type states [92](43)Ψp1=Fx(r)|XZ〉−Fy(r)|YZ〉2,Ψp2=Fz(r)|YZ〉−Fx(r)|XY〉2,Ψp3=Fy(r)|XY〉−Fz(r)|XZ〉2. These states (30) and (43) correspond to the closely spaced lowest triply
degenerate energy levels shown in Figure 5. As a result, the ground hole state becomes
a superposition of the states with s- and p-type envelope functions. In the
conduction band, the ground electron states are the s-p combinations even
without impurities, as seen from (27). Embedding the impurity into some
arbitrary position within the NC raises the number of the p-type envelope
functions in the ground states as well as in the valence band. However, PIF(ℓ) becomes smaller if the integrated envelope
functions are p-type. Consequently, the squared transition matrix element,
proportional to PIF(ℓ),
and the rate itself decrease relative to the case of an undoped NC, as shown in
Figure 15 for two different displacements of the impurity from the dot center.The rates of the phonon-assisted ground radiative transition as functions of the dot radius for a doped dot. Forh≠0, we have  nx=0.8, ny=0.5, and nz=0.33.
(a)(b)The ratioh/R=0.46 corresponds to the greatest possible mix of s-
and p-type states with maximal weight of the p-type envelope functions in the
ground states both in the conduction and valence bands. The strong mixing
results in maximal decrease of the decay rate independently of the impurity type.
When the impurity is situated near the nanocrystal boundary, h/R=0.9,
the weight of the p-type envelope functions reduces in the ground state of both
bands. As a consequence, the rate rises with respect to the case h/R=0.46,
but remains smaller than that for the undoped NC.The second case we consider is the case of the central-located impurity.
Ifh=0,
no s-p entanglement takes place in the valence band. The functions with the
s-type envelope (30)) describe the ground states of both donor- and
acceptor-doped NCs.The situation is different for the conduction band. Formally, the wave
functions of the six spinless ground states are described by (27) as before.
However, for doped dots, parameter λ has a form which differs from that given by (28).
In the presence of an impurity center, λ is defined as:(44)cos(2λ)=Ep−Es+δV−2Hpp(Ep−Es+δV−2Hpp)2+4Hsp2,sin(2λ)=2Hsp(Ep−Es+δV−2Hpp)2+4Hsp2, where δV=Vpp−Vss is the difference between the p-p and s-s type
matrix elements of the Coulomb potential energy VC.
The appearance of this term in the definition of λ is explained by the shift of the energy levels
of the s- and p-states Es and Ep,
respectively, due to the Coulomb interaction. The relative shift δV turns out to be positive for donors, and
negative for acceptors. Consequently, cos(λ) increases and sin(λ) decreases for donors, and vice versa for
acceptors.This implies that the weight of thes-type envelope function in the
electron ground state increases for the donor-doped dot and shrinks for the dot
doped with acceptor, with respect to its value in the undoped dot. Hence, the
coefficient PIF(ℓ) for the donor-doped dot is greater than that
for the undoped dot. In turn, the latter exceeds PIF(ℓ) for the acceptor-doped dot. Precisely the same
relationships take place for the recombination rates, as shown in Figure 15.Note in conclusion, that the central-symmetric caseh=0 qualitatively corresponds to some real
situation, where the NC has been highly doped with donors or acceptors. It is
natural to assume a homogeneous impurity distribution within the nanocrystal in
this case. Obviously, such a homogeneous density of charge induces a spherically
symmetric electric field that cannot mix the s- and p-type states. Moreover,
the relative Coulomb shift δV remains positive for donors and negative for
acceptors, as in the above case of a single impurity center in the dot [73]. Therefore, one can expect similar
behavior of τR−1 for highly doped NCs.


#### 3.3.3. Nonradiative Recombination

In parallel with radiative transitions, nonradiative processes occur in
silicon nanocrystals. As a rule, nonradiative processes play a problematic role
in silicon crystallites, due to their high recombination rates which
substantially exceed the rates of the radiative transitions. Among the possible
nonradiative processes, capture on dangling bonds and Auger recombination are the
most efficient. Both processes are phonon-assisted, possibly even multi-phonon.
The first one takes place both at low and high pumping levels, while Auger
recombination becomes possible exclusively at high excitation power when more
than one excited electron (hole) exist in the conduction (valence) band of the
nanocrystal.The capture on a neutral dangling bond has been studied earlier by TB
method [100, 106]. It was shown that excited
electron-hole pairs can recombine on a neutral dangling bond in two stages. First,
the electron (hole) is trapped on a neutral Pb center making it
negatively (positively) charged. Then, the hole (electron) is trapped on the
charged dangling bond. According to estimations [100], the capture rate on the neutral
dangling bond for holes is always much greater than that for electrons. In
turn, the electron capture rate at room temperature increases sharply as the
nanocrystal gap decreases from 2.7 eV to 1.5 eV (see (23)). Within
the same range of energy gaps, the hole capture rate also increases but less
sharply. We should emphasize the presence of two main features of the capture
process. First, there is a very stron dependence of the capture rates (even for
holes) on the energy gap, that is, the NC size.
The second feature is the overall high capture rates, which are much
greater than the radiative decay rates. Evidently, the presence of a dangling
bond in silicon crystallite drastically reduces the quantum efficiency of
photon generation. Therefore the passivation of dangling bonds is required for
achievement of relatively efficient luminescence.Let us now touch briefly on the role of SiO defects in the
photoluminescence. The calculations of Nishida [102] showed that the oscillator strength
is almost independent of the existence of Si–O–Si bonds. In contrast, the Si=O
bonds reduce the oscillator strength by several orders of magnitude. Recently,
however, Sa'ar et al. [64] demonstrated experimentally that
both back-bonded and double-bonded oxygen atoms can enhance the PL intensity
due to formation of coupled states of electrons (holes) and SiO vibrations.
Such vibrations induce electric polarization fields generating long-lived
coupled states named “vibrons”. Usually, the surface vibrations scatter the
carriers; however, vibrons, being coupled states, do not scatter the carriers
and thereby exclude one of possible nonradiative channels from the
recombination process.For highly excited
nanocrystals, Auger recombination may take place. It is sufficient to have only
three particles (two electrons and a hole, or vice versa) for this process to
occur. It is well known that Auger recombination has high efficiency in bulk
silicon. Therefore, it is natural to expect a similar behavior in silicon
crystallites. Calculations of the Auger-recombination rate in crystallites
carried out with TB [58, 152] and EFA [153] techniques confirm, indeed, the fast character of this mechanism. In
particular, the characteristic times have been found to be of the order
0.1 to 10 ns−1 and gradually rise with decreasing
nanocrystal size. Thus, Auger recombination, as well as dangling bonds, can
strongly quench the luminescence in silicon crystallites, as confirmed
experimentally [152].


## 3.1. Electronic Structure of Perfect Silicon Nanocrystals

The first step is a calculation of the ground-state
energies both for the conduction and valence bands of a nanocrystal. Note that
here and throughout the paper we will use terminology that is typical for the
bulk material. Of course, no genuine energy bands exist in the nanocrystal
because of its finite size. Nevertheless, the states above and below the NC
optical gap originate from the size-quantized states of conduction and valence
bands of the bulk crystal, respectively. Moreover, in the following we mainly
intend to employ the envelope-function approximation (k⋅p method) to
describe electronic states of the nanocrystals. Consequently, such the
terminology is convenient and should not lead to any misunderstandings. The
states above and below the optical gap will be further referred to as
conduction and valence bands, as in the bulk. Different approaches are used
when calculating the gap of silicon nanocrystals. Among them, the simplest is
the single-particle approximation. Usually, the single-particle gap exceeds
real optical gap because the electron-hole Coulomb interaction reduces the
total energy of the system. For this reason, the values of the single-particle
gap may be treated as an upper bound for the optical gap. In this section we
discuss electron and hole states of a “pure” crystallite, which has no defects
and a perfect diamond lattice all over its whole volume. The results are
compared with the simpler approximations discussed in the introduction.


### 3.1.1. Optical Gap. Numerical Results

It is possible to separate all calculation methods
applied for computation of the electronic spectra in pure NCs into two groups. In
the first group we have the numerical first-principles, empirical, and semi-empirical
methods, such as density functional theory (DFT) [68, 74–78], pseudopotential (PP) method (or various combinations
of PP and DFT) [79–82],
tight-binding (TB) models [83–86], combined Green's function
(GW-approximation), and the
Bethe-Salpeter equation (BSE) technique [87–89]. All these methods require considerable
computational time, a problem that worsens as the nanocrystal size is
increased. Analytical methods such as the effective mass approximation or k⋅p method [67, 90–95] may be
ascribed to the second group. In following we combine these two methods into a
single envelope function approximation (EFA). The latter allows one to perform
calculations without any restrictions on cluster sizes. Additionally, this
approach can be used to find the higher excited states of the nanocrystal.In the present subsection, we summarize the results of
some of the methods discussed above, which illustrate dependence of the optical
gap on the nanocrystal size (Figure4). In all calculations, any dangling bonds
at the nanocrystal surface were considered to be H-passivated. As is seen in
the figure, there is some scatter of data due to the different computational
methods and different approximations utilized in the computational procedure.Figure 4
Optical gap calculated by PP [65],  TB [85, 86], and DFT [68, 74] methods. Solid line: EFA calculations, see (22) below [93].In particular, taking into account the self-energy
correction or electron-hole interaction can considerably change the magnitude
of the nanocrystal optical gap, in opposite directions, as has been pointed out
by many authors (see, e.g., [81, 88, 89]). In particular, it is known that the local density
approximation of the DFT tends to underestimate the optical-gap (similar
results happen for silicon nanowires) [96, 97] compared to those obtained in experiments. For a more accurate account of the
self-energy corrections and excitonic effects, the GW+BSE approach is often
applied. The combination of the GW and BSE methods yields almost complete
compensation of the exciton and the self-energy corrections to the
single-particle optical gap [88]. The same effect was
also found by Franceschetti and Zunger [81] within the framework of
the PP method. Unfortunately, the GW+BSE approach is employed for nanocrystals
whose diameters do not exceed 1 nm [87–89]. Therefore, we did not
depict the results of this approach in Figure 4.


### 3.1.2. Electron and Hole Spectra in a Dot

Let us now discuss electronic spectra in both the valence
and conduction bands of the quantum dot. As has been pointed out, for this
purpose it is convenient to use the EFA. Here we describe in more detail not
only the lowest but also the higher excited electronic states as a function of
size.We assume the total potential
energy of the electron inside the quantum dot to be of the following form(2)U(r)=U0(r)+Vsp(r). Here U0(r) is the confining potential equal
to zero inside and infinity outside the dot. The second part Vsp(r) describes an interaction between
the electron and its image, arising due to the charge polarization on the
boundary between the silicon nanocrystal and its dielectric surrounding. Vsp(r) is often referred to as a
self-polarization term. It can be represented as(3)Vsp(r)=e2(εs−εd)2εsR∑l=0∞l+1lεs+(l+1)εdr2lR2l, where εs and εd are the static dielectric constants of silicon
and the dielectric matrix, respectively. In order to find the electronic states
in the dot, we have to solve the single-particle Schrödinger-like equation for
the envelope function vector |Φ〉:(4)(H+U(r))|Φ〉=E|Φ〉. Here, H is the bulk k⋅p Hamiltonian operator acting on the
six-dimensional (6D) envelope-function vectors |Φ〉,
and E is the electron energy. The components of the
6D-vector |Φ〉 are slowly varied expansion coefficients Φj(r) of the total wave function in the Bloch-state
basis.The electronic states in the
valence and conduction bands can be determined separately. Starting with the
valence band, the Bloch-state for theΓ-point is |YZ〉|↑〉, |XZ〉|↑〉, |XY〉|↑〉, |YZ〉|↓〉, |XZ〉|↓〉, |XY〉|↓〉,
where |↑〉 and |↓〉 are “up” and “down” spinors, respectively, and
the Bloch functions |YZ〉, |XZ〉, |XY〉 belong to the irreducible representation Γ25′.
The k⋅p Hamiltonian
matrix H in 4 is the sum of three
parts: H=H(0)+H(1)+H(so).
Here H(0)=(p2/2mh)×I is the isotropic part obtained as the average
of the total k⋅p matrix
over all angles in p-space. H(1) is the anisotropic part that can be
represented by two equivalent 3×3 blocks situated on the main diagonal of the
total 6×6  k⋅p matrix:(5)H(1)=(H300H3),H3=−12m0(Q1NpxpyNpxpzNpxpyQ2NpypzNpxpzNpypzQ3), where Q1 denotes ((L−M)/3)(3px2−p2), Q2 denotes ((L−M)/3)(3py2−p2), and Q3 denotes ((L−M)/3)(3pz2−p2).Finally, the term(6)H(so)=13(0−iΔ000ΔiΔ0000−iΔ000−ΔiΔ000−Δ0iΔ000−iΔ−iΔ00ΔiΔ0000) describes the spin-orbit
interaction. We have introduced above the spin-orbit energy Δ equal to 44 meV for silicon, the 6×6 identity matrix I,
and the hole effective mass mh=3m0/(L+2M),
where the numbers L, M, N are dimensionless empirical parameters in the k⋅p Hamiltonian operator in the valence band. For
silicon they equal 5.8, 3.43, and 8.61, respectively [98].Because of the isotropic and
diagonal form of the operatorH(0)+U0(r),
it is possible to classify its eigenstates similarly to atomic systems using
common terminology such as s-, p-, and d-type states, and so forth [92]. Accordingly, one may expand the components of the envelope-function
vectors over these eigenstates as(7)Φj(r)=∑αCjα|α〉, where |α〉 stands for the s-, p-, d-, … states and Cjα are the expansion coefficients. For instance,
the 1s-, 1p-, 1d-, (in the following simply s-, p-, and d-) and 2s-type states
are described by the following functions:(8)|s〉=π2R3j0(πr/R),|pa〉=32πR3j1(μ1r/R)j0(μ1)xar,|dab〉=152πR3j2(μ2r/R)j1(μ2)xaxbr2,a≠b,|dx2−y2〉=158πR3j2(μ2r/R)j1(μ2)x2−y2r2,|d3z2−r2〉=158πR3j2(μ2r/R)j1(μ2)3z2−r2r2,|2s〉=π2R3j0(2πr/R), where jn(x) are the spherical Bessel functions of argument x,
and μn are the first roots of jn(x).
Below, we restrict the basis of envelope functions |α〉 with these states only, and determine the
electron and hole spectra. Substitution of Φj(r) into (4) yields algebraic equations for Cjα:(9)(E−Eα)Ciα=∑β〈α|Hij(1)+Hij(so)+Vsp(r)δij|β〉Cjβ, where Eα denotes the energy of the state |α〉, δij stands for the Kronecker delta, and the Einstein
convention has been used when summing over j.
Considering all operators in the right side of (9)) as perturbations [92], one can solve the equation and find
the energies in the valence band.Restricting the basis of the
envelope states|α〉 by 1s, 2s, 1p, and 1d functions, and
neglecting the spin-orbit interaction, we obtain the following “hierarchy” of
energies (for convenience, we write down the hole energies differing in the
sign from the electron energies, which are determined with (9))). The
ground-state energy(10)Eh(s)=ℏ2π22mhR2 is triply degenerate (or sixfold,
if spin is taken into account). It is the energy of the 1s state. Hybridization
of the p-type states yields four different levels with the energies(11)Eh1(p)=ℏ2μ122mhR2[1−3N+2L−2M5L+10M],Eh2(p)=ℏ2μ122mhR2[1−3N−4L+4M5L+10M],Eh3(p)=ℏ2μ122mhR2[1+3N−2L+2M5L+10M],Eh4(p)=ℏ2μ122mhR2[1+6N+4L−4M5L+10M], where μ1=4.4934.
The energies of the d-d hybridized states are as follows:(12)Eh1(d)=ℏ2μ222mhR2[1−6N7L+14M],Eh2(d)=ℏ2μ222mhR2[1−3N+4L−4M7L+14M],Eh3(d)=ℏ2μ222mhR2[1+3N−2L+2M7L+14M],Eh4(d)=ℏ2μ222mhR2[1+6N−4L+4M7L+14M], where μ2=5.7634.
Finally, the energies of the 1d-2s hybrids are written as(13)Eh1(2sd)=ℏ22mhR2[4π2+μ222+3μ2214N+2L−2ML+2M−1265πμ24π2−μ22NL+2M],Eh2(2sd)=ℏ2μ222mhR2,Eh3(2sd)=ℏ22mhR2[4π2+μ222+3μ2214N+2L−2ML+2M+1265πμ24π2−μ22NL+2M]. The contribution of the
self-polarization term from (10)–(13) has been
omitted since it represents, in fact, a common shift of all the levels by e2(1/εd−1/εs)/2R.In Figure5 the energy levels from (10)–(13) have been
plotted as functions of the NC radius. The two lowest hole levels have very
similar energies. The lowest level, being sixfold degenerate, corresponds to
six electron states with an s-type envelope function. These states have an
isotropic macroscopic (averaged over the unit-cell volume) charge distribution
inside the dot. The second level corresponds to six electron states with p-type
envelope functions. Correspondingly, the macroscopic charge has an anisotropic
distribution for these states (see [92] for details).Figure 5
Hole energy spectrum according to (10)–(13) in case of no spin-orbit interaction. Solid line: s-type level. Short dashed lines: p-p hybridized levels. Dots: d-d hybridized levels. Long dashed lines: 2s-d hybridized levels. All energies are counted deep into the valence band with respect to the valence band maximum. The degeneracy degree including spin (from bottom to top) is 6; 6; 6; 4; 6; 4; 6; 6; 2; 6; 2; 6.In the conduction
band the Bloch-state basis consists of six Bloch functions
of three differentX-points in the Brillouin zone
[99]. We denote three pairs of these
functions as |X〉, |X′〉; |Y〉, |Y′〉; |Z〉, |Z′〉.
Each pair belongs to the twofold degenerate irreducible representation X1 of the corresponding X-point. The unprimed
Bloch functions have a nonzero value at the lattice site, while the primed
functions are zero at those points. The choice of the X1-functions as the Bloch basis allows us to describe correctly a
dispersion-law nonparabolicity that originates mainly from the energy-branch
crossing in the X-points.Thek⋅p Hamiltonian operator in the conduction band
represents a 6×6 matrix having the form(14)H=(Hx000Hy000Hz). Each element of the matrix H is a block 2×2 matrix defined by the following expressions [99]:(15)Ha=(pa22ml+p2−pa22mt(1mt−1m0)pbpc+ip0mlpa(1mt−1m0)pbpc−ip0mlpapa22ml+p2−pa22mt). Here p0≈0.14(2πℏ/a0) is the distance in p-space from any of the energy minima to the nearest X-point, a0 is the lattice constant of silicon. ml=0.92m0 and mt=0.19m0 are the “longitudinal” and “transverse”
effective masses. The origin of the E-axis coincides with the X-point energy. Each of the
small indices (a, b, c) runs over the values x, y,
or, z and always differs from the others.Solving (4) for the conduction
band is done in the same way as for the valence band, as in (9)–(13) above. One
can then obtain the following groups of energies. The ground states, as well as
the second excited ones, are the 1s-1p hybrids. Their energy levels are given
by(16)Ee1(sp)=Es+Ep−2Hpp2−(Ep−Es−2Hpp2)2+Hsp2,Ee2(sp)=Es+Ep−2Hpp2+(Ep−Es−2Hpp2)2+Hsp2, where(17)Hsp=ip0ml〈s|pz|pz〉=2πℏμ1p0[3mlR(μ12−π2)],Hpp=112(1mt−1ml)〈pz|p2−3pz2|pz〉=ℏ2μ12(ml−mt)15mtmlR2 are s-p and p-p type matrix
elements of anisotropic part Ha, Es=ℏ2π2/2meR2,
and Ep=ℏ2μ12/2meR2 are the energies of the s- and p-type states
of the isotropic unperturbed Hamiltonian operator. The isotropic average
electron effective mass is defined by me−1=(2mt−1+ml−1)/3≈(0.26m0)−1. The p-p hybridization forms states with energies:(18)Ee1,2(p)=Ep+Hpp∓Hxy. Here Hxy=ℏ2μ12(m0−mt)/5mtm0R2.
The next three levels are the result of the d-d hybridization:(19)Ee1(d)=ℏ2μ222meR2(1−27ml−mt2ml+mt−67mlm0m0−mt2ml+mt),Ee2(d)=ℏ2μ222meR2(1+47ml−mt2ml+mt),Ee3(d)=ℏ2μ222meR2(1−27ml−mt2ml+mt+67mlm0m0−mt2ml+mt). Finally, the 2s-d hybridized
energies are obtained numerically:(20)Ee1(2sd)=ℏ22meR2(4π2+2μ223−0.477μ22),Ee2(2sd)=ℏ22meR2(4π2+2μ223−0.455μ22),Ee3(2sd)=ℏ22meR2(4π2+2μ223+0.932μ22). The results for the
single-particle electron spectrum in the conduction band are shown in Figure 6.Figure 6
The lowest energy levels in the conduction band as functions of the NC radius, from (16)–(20). The energies are counted from the X-point energy. Solid line: s-p type levels. Short dashed lines: p-p hybridized levels. Dots: d-d hybridized levels. Long dashed lines: 2s-d hybridized levels. All the levels are 12-fold degenerate including spin degeneracy.The wave function of the ground state turns out to be
a superposition of the terms withs- and p-type envelope functions (see [92] for details). The “weight” of the
s-type envelope function is greater than that of the p-type function. Therefore
the macroscopic charge for the ground state has a distribution close to
isotropic.From the above calculations, the optical gap of the NC
can be obtained as the sum of the lowest energies in both bands and the energy
difference betweenX- and Γ-points in bulk ΔXΓ=1.24 eV:(21)εg(R)=ΔXΓ+Eel(sp)+Eh(s). This energy somewhat overestimates value compared to
that obtained with the numerical methods, as indicated by the solid line in Figure 7.
This is a consequence of two main factors.Figure 7
NC gap versus reciprocal dot radius. Solid line: infinite barriers (21). Circles: finite barriers (3.2 eV for electrons and 4.3 eV for holes, which corresponds to Si/SiO2 interface) and constant effective mass. Disks—finite barriers and free electron mass outside the dot. Both of the latter are numerical results by EFA [92]. The dashed line is a guide to the eye.The first factor is that we have assumed infinitely
high potential barriers at the interface. The authors of  [91, 93, 95] calculated the carrier energies supposing the
potential barriers to be finite and equal to approximately 3 eV (or a
little more) for electrons and 4-5 eV for holes. Moreover, the discontinuity
of the effective mass has been taken into account. In Figure 7 we have plotted
the optical gap of silicon nanocrystal versus the confining parameter 1/R for three models differeing according the
nanocrystal’s surroundings. The solid line corresponds to infinitely high
potential barriers. Circles represent the finite barriers and constant
effective mass throughout the sample. Finally, in case of the mass discontinuity
with effective mass outside the dot coinciding with free electron mass, the
optical gap has been represented by disks. For the last two cases, the
dependence Δεg(R) becomes weaker than R−2,
especially for the case of the discontinuous effective mass where the
dependence is closer to 1/R.
In the last case the gap dependence is well fitted by the function:(22)εg(1)(R)=εg2+D1R2 with D1=4.8 eV2·nm2. In general, one may conclude that the calculated values of
the optical gap are reduced due to the barrier finiteness and mass
discontinuity, and agree better with those shown in Figure 4.The second factor, strongly influencing the dot gap,
is electron-hole Coulomb interaction. Its contribution to the optical gap will
be discussed in the next subsection.


### 3.1.3. EFA Calculations: Exciton Corrections

The electron-hole interaction reduces the energy of
the electron-hole pair. As a result, the exciton optical gap is also reduced.
In order to estimate the exciton correction to the total electron-hole
single-particle energy one must solve the two-particle problem. This has been
done for silicon NCs embedded in an infinitely wide-band-gap material [67, 90, 94]. The exciton corrections
were found to be about 0.4 eV [67], 0.25 eV [90], and 0.3 eV [94] for 2-nm-diameter NCs, with the
correction decreasing approximately as R−1 with increasing size.Meanwhile, as has been pointed out in the preceding section, the
finiteness of the potential barriers and effective mass discontinuity can
appreciably influence the electron and hole energies. Therefore, it is possible
to expect the same for the excitonic energies. Ferreyra and Proetto [91] have studied excitonic states for a
quantum dot in vacuum. Accordingly, they accepted the barrier heights equal to
the electron affinity of the corresponding bulk material for electrons and
infinity for holes, and the effective mass being m0.
They found an exciton correction of the order of 0.2 eV for a 2-nm-diameter
dot. As the nanocrystal size increases, the magnitude of this correction
decreased as R−0.7.The authors of some of the previous work (e.g., [67, 90, 91]) treated the Coulomb potential energy as VC=−e2/εsreh similar to bulk silicon. However, in a quantum
dot the electron and hole interact not only with each other but also with their
“images” due to the difference in the permittivity of the materials inside and
outside the dot. These are the so-called polarization fields arising because of
the charge polarization at the dot boundary. The interaction with the
polarization fields has been taken into account in [94]. Nevertheless, the
polarization fields did not lead to any significant corrections to the optical
gap. As mentioned above, the correction does not exceed 0.3 eV, which is
close to the values obtained by other authors.Evidently, the calculations carried out with the EFA [67, 90, 91, 94] provide sufficiently good coincidence with both experimental and
theoretical data. For comparison, we have depicted in Figure 4 with the
solid line the EFA single-particle gap according to (22). The gap values εg(1)(R) agree well with those computed by PP, TB, and
DFT methods. Obviously, a small exciton correction of about 0.3–0.1 eV
for 2 to 5 nm diameter crystallites, respectively, does not significantly
change the single-particle gap values.It should be noted, in conclusion, that all the authors employed the
perturbation theory considering the electron-hole interaction to be weak
compared to the typical quantum confinement energies in the dot. Such an
approach is justified if the exciton Bohr radius is significantly larger than
the dot radius. This requirement remains valid for small quantum dots with the
sizes of about 5-6 nm or less.


### 3.1.4. Interface States

We have already emphasized in the introduction that radiative
transitions between interface states are also often considered as the origin of
NIR, or even visible, emission from silicon nanocrystals. In a certain sense,
this point of view is alternative to the idea of quantum confinement.
Description of the interface states with EFA is difficult because the real
potential existing in the vicinity of the nanocrystal surface has in principle
a microscopic nature that manifests itself at a distance on the order of a bond
length. Meanwhile, the EFA is, in fact, a “macroscopic” method which is not
able to distinguish the spatial structure on scales smaller than the size of
the primitive cell. Therefore TB and DFT methods are essentially more suitable
for this goal.As a rule, interface states originate from various-type defects existing
at the surface of the crystallite. These include, mainly, dangling bonds (Pb centers) which can be neutral, positively, or negatively charged, and different
SiO bonds (backbonded and double bonded oxygen) arising at Si/SiO2 interface. Charged dangling bonds corresponding to zero or two electrons in the
bond level, respectively, produce deep levels inside the band gap of bulk
silicon. Their energies are approximately 0.3 eV below the conduction band
minimum for the negatively charged bond, and 0.3 eV above the valence band
maximum if the bond is positively charged [100]. Consequently, the effective gap
between these levels is about 0.5-0.6 eV [101]. For nanocrystals, the effective gap gradually rises as the nanocrystal size
decreases. In particular, for a 5-nm-diameter NC this shift equals
approximately 0.6 eV, while for a 2 nm crystallite it is 1.3-1.4 eV [100]. Thus, we see that the dangling bond levels are sensitive to the NC size.Nanocrystal oxidation has been studied theoretically by TB [8, 102], DFT [103, 104],
and GW+BSE [89]
methods. The role of the so-called
backbonded (Si–O–Si) and double bonded (Si=O) oxygen atoms has been examined
for the case where the spherical NC surface is passivated with hydrogen. In the
first case (Si–O–Si), oxygen forms two bonds with two different silicon atoms
which already have four bonds with other silicon or hydrogen atoms. In the
second case (Si=O), oxygen replaces two hydrogen atoms bonded with one silicon
atom. It is natural to suppose that the system perturbation in the second case
should be more significant because of the stronger distortions of the structure.
In fact, it was found that the presence of one Si=O bond at the NC surface
provides an optical gap almost independent of the dot size [8].
At the same time, the nanocrystal
with a Si–O–Si bond has a gap that gradually decreases with increasing size.
Nevertheless, this gap remains smaller, and varies more slowly than the gap of
perfect nanocrystal with hydrogen passivated surfaces. The
calculations carried out by different methods [89, 102, 103]
for oxidized crystallites show an essential difference
in the energy of the ground electron-hole transition in crystallites with
Si–O–Si and Si=O bonds. All the methods used exhibit the smallest gap for the
case of double-bond oxidation, while the crystallites with backbonded oxygen
have the greater gap. In turn, the latter is less than the gap of a perfect NC.
The authors of [89] have found that only the bridge-type bond (Si–O–Si) can
provide a satisfactory agreement with experiments on the photoluminescence in
nc-Si/SiO2 system. This fact may be explained by the considerably
different oscillator strengths of the ground transitions for
NCs with Si=O and Si–O–Si defects. As has been shown by Nishida [102], the oscillator strength for Si–O–Si
case is several orders of magnitude greater than that for Si=O case, especially
for NCs smaller than 1 nm in diameter. This is precisely the case studied
by Luppi et al. [89], where NCs of 0.5 nm and 0.9 nm in
diameter were considered. However, according to [102]
the difference between the
size-quantized energy levels and those of the Si=O and Si–O–Si defects, as well
as the difference in the oscillator strengths of the ground electron-hole
transitions, are essential only for NC diameters less than 2 to 2.5 nm [8, 102].
For larger sizes, the energies and
the transition intensities for perfect and oxidized crystallites almost
coincide.Finally, nonradiative trapping
rates are also critical in the light emission properties of silicon NCs. The
group of neutral or chargedPb centers at silicon dangling bonds is the most important nonradiative trap in
bulk silicon [105],
and acts also in the case of Si NCs [58]. In a detailed theoretical investigation, Lannoo et al.
[106] found that the carrier trapping at neutral Pb centers is nonradiative, whereas capture at charged
centers can lead to photon emission at energies smaller than the bandgap of
bulk silicon. The rate for carrier trapping at a neutral dangling bond defect
at the Si–SiO2 interface in a NC with a single Pb center was established theoretically as a function of
the NC size [106]:(23)wnr=σ0vV12π((E0ℏω)2+z2)−0.25×exp[−Scoth(ℏω2kT)+E02kT+((E0ℏω)2+z2)0.5−E0ℏωasinh(E0ℏωz)]. In (23), σ0 is the capture cross section given by c0/v where c0 is the capture coefficient and v is the thermal velocity equal to 8kt/πm*, E0 is the ionization energy of the defect
(approximately equal to the Franck-Condon energy plus the carrier confinement
energy), S is the Huang-Rhys factor (S=~15) [107], ℏω is the average phonon energy, V is the nanocrystal volume, and z=S/[sinh(ℏω/2kT)]. The capture rate increases strongly as a function of NC size due to the
decreasing defect ionization energy, until eventually the volume term begins to
dominate the trapping rate, which then begins to decrease again. Since, the
calculated rate is for a single isolated defect, in practice the rate should be
modified for the likelihood of multiple defects on larger clusters, as
discussed in Section 4.


## 3.1.1. Optical Gap. Numerical Results

It is possible to separate all calculation methods
applied for computation of the electronic spectra in pure NCs into two groups. In
the first group we have the numerical first-principles, empirical, and semi-empirical
methods, such as density functional theory (DFT) [68, 74–78], pseudopotential (PP) method (or various combinations
of PP and DFT) [79–82],
tight-binding (TB) models [83–86], combined Green's function
(GW-approximation), and the
Bethe-Salpeter equation (BSE) technique [87–89]. All these methods require considerable
computational time, a problem that worsens as the nanocrystal size is
increased. Analytical methods such as the effective mass approximation or k⋅p method [67, 90–95] may be
ascribed to the second group. In following we combine these two methods into a
single envelope function approximation (EFA). The latter allows one to perform
calculations without any restrictions on cluster sizes. Additionally, this
approach can be used to find the higher excited states of the nanocrystal.In the present subsection, we summarize the results of
some of the methods discussed above, which illustrate dependence of the optical
gap on the nanocrystal size (Figure4). In all calculations, any dangling bonds
at the nanocrystal surface were considered to be H-passivated. As is seen in
the figure, there is some scatter of data due to the different computational
methods and different approximations utilized in the computational procedure.Figure 4
Optical gap calculated by PP [65],  TB [85, 86], and DFT [68, 74] methods. Solid line: EFA calculations, see (22) below [93].In particular, taking into account the self-energy
correction or electron-hole interaction can considerably change the magnitude
of the nanocrystal optical gap, in opposite directions, as has been pointed out
by many authors (see, e.g., [81, 88, 89]). In particular, it is known that the local density
approximation of the DFT tends to underestimate the optical-gap (similar
results happen for silicon nanowires) [96, 97] compared to those obtained in experiments. For a more accurate account of the
self-energy corrections and excitonic effects, the GW+BSE approach is often
applied. The combination of the GW and BSE methods yields almost complete
compensation of the exciton and the self-energy corrections to the
single-particle optical gap [88]. The same effect was
also found by Franceschetti and Zunger [81] within the framework of
the PP method. Unfortunately, the GW+BSE approach is employed for nanocrystals
whose diameters do not exceed 1 nm [87–89]. Therefore, we did not
depict the results of this approach in Figure 4.


## 3.1.2. Electron and Hole Spectra in a Dot

Let us now discuss electronic spectra in both the valence
and conduction bands of the quantum dot. As has been pointed out, for this
purpose it is convenient to use the EFA. Here we describe in more detail not
only the lowest but also the higher excited electronic states as a function of
size.We assume the total potential
energy of the electron inside the quantum dot to be of the following form(2)U(r)=U0(r)+Vsp(r). Here U0(r) is the confining potential equal
to zero inside and infinity outside the dot. The second part Vsp(r) describes an interaction between
the electron and its image, arising due to the charge polarization on the
boundary between the silicon nanocrystal and its dielectric surrounding. Vsp(r) is often referred to as a
self-polarization term. It can be represented as(3)Vsp(r)=e2(εs−εd)2εsR∑l=0∞l+1lεs+(l+1)εdr2lR2l, where εs and εd are the static dielectric constants of silicon
and the dielectric matrix, respectively. In order to find the electronic states
in the dot, we have to solve the single-particle Schrödinger-like equation for
the envelope function vector |Φ〉:(4)(H+U(r))|Φ〉=E|Φ〉. Here, H is the bulk k⋅p Hamiltonian operator acting on the
six-dimensional (6D) envelope-function vectors |Φ〉,
and E is the electron energy. The components of the
6D-vector |Φ〉 are slowly varied expansion coefficients Φj(r) of the total wave function in the Bloch-state
basis.The electronic states in the
valence and conduction bands can be determined separately. Starting with the
valence band, the Bloch-state for theΓ-point is |YZ〉|↑〉, |XZ〉|↑〉, |XY〉|↑〉, |YZ〉|↓〉, |XZ〉|↓〉, |XY〉|↓〉,
where |↑〉 and |↓〉 are “up” and “down” spinors, respectively, and
the Bloch functions |YZ〉, |XZ〉, |XY〉 belong to the irreducible representation Γ25′.
The k⋅p Hamiltonian
matrix H in 4 is the sum of three
parts: H=H(0)+H(1)+H(so).
Here H(0)=(p2/2mh)×I is the isotropic part obtained as the average
of the total k⋅p matrix
over all angles in p-space. H(1) is the anisotropic part that can be
represented by two equivalent 3×3 blocks situated on the main diagonal of the
total 6×6  k⋅p matrix:(5)H(1)=(H300H3),H3=−12m0(Q1NpxpyNpxpzNpxpyQ2NpypzNpxpzNpypzQ3), where Q1 denotes ((L−M)/3)(3px2−p2), Q2 denotes ((L−M)/3)(3py2−p2), and Q3 denotes ((L−M)/3)(3pz2−p2).Finally, the term(6)H(so)=13(0−iΔ000ΔiΔ0000−iΔ000−ΔiΔ000−Δ0iΔ000−iΔ−iΔ00ΔiΔ0000) describes the spin-orbit
interaction. We have introduced above the spin-orbit energy Δ equal to 44 meV for silicon, the 6×6 identity matrix I,
and the hole effective mass mh=3m0/(L+2M),
where the numbers L, M, N are dimensionless empirical parameters in the k⋅p Hamiltonian operator in the valence band. For
silicon they equal 5.8, 3.43, and 8.61, respectively [98].Because of the isotropic and
diagonal form of the operatorH(0)+U0(r),
it is possible to classify its eigenstates similarly to atomic systems using
common terminology such as s-, p-, and d-type states, and so forth [92]. Accordingly, one may expand the components of the envelope-function
vectors over these eigenstates as(7)Φj(r)=∑αCjα|α〉, where |α〉 stands for the s-, p-, d-, … states and Cjα are the expansion coefficients. For instance,
the 1s-, 1p-, 1d-, (in the following simply s-, p-, and d-) and 2s-type states
are described by the following functions:(8)|s〉=π2R3j0(πr/R),|pa〉=32πR3j1(μ1r/R)j0(μ1)xar,|dab〉=152πR3j2(μ2r/R)j1(μ2)xaxbr2,a≠b,|dx2−y2〉=158πR3j2(μ2r/R)j1(μ2)x2−y2r2,|d3z2−r2〉=158πR3j2(μ2r/R)j1(μ2)3z2−r2r2,|2s〉=π2R3j0(2πr/R), where jn(x) are the spherical Bessel functions of argument x,
and μn are the first roots of jn(x).
Below, we restrict the basis of envelope functions |α〉 with these states only, and determine the
electron and hole spectra. Substitution of Φj(r) into (4) yields algebraic equations for Cjα:(9)(E−Eα)Ciα=∑β〈α|Hij(1)+Hij(so)+Vsp(r)δij|β〉Cjβ, where Eα denotes the energy of the state |α〉, δij stands for the Kronecker delta, and the Einstein
convention has been used when summing over j.
Considering all operators in the right side of (9)) as perturbations [92], one can solve the equation and find
the energies in the valence band.Restricting the basis of the
envelope states|α〉 by 1s, 2s, 1p, and 1d functions, and
neglecting the spin-orbit interaction, we obtain the following “hierarchy” of
energies (for convenience, we write down the hole energies differing in the
sign from the electron energies, which are determined with (9))). The
ground-state energy(10)Eh(s)=ℏ2π22mhR2 is triply degenerate (or sixfold,
if spin is taken into account). It is the energy of the 1s state. Hybridization
of the p-type states yields four different levels with the energies(11)Eh1(p)=ℏ2μ122mhR2[1−3N+2L−2M5L+10M],Eh2(p)=ℏ2μ122mhR2[1−3N−4L+4M5L+10M],Eh3(p)=ℏ2μ122mhR2[1+3N−2L+2M5L+10M],Eh4(p)=ℏ2μ122mhR2[1+6N+4L−4M5L+10M], where μ1=4.4934.
The energies of the d-d hybridized states are as follows:(12)Eh1(d)=ℏ2μ222mhR2[1−6N7L+14M],Eh2(d)=ℏ2μ222mhR2[1−3N+4L−4M7L+14M],Eh3(d)=ℏ2μ222mhR2[1+3N−2L+2M7L+14M],Eh4(d)=ℏ2μ222mhR2[1+6N−4L+4M7L+14M], where μ2=5.7634.
Finally, the energies of the 1d-2s hybrids are written as(13)Eh1(2sd)=ℏ22mhR2[4π2+μ222+3μ2214N+2L−2ML+2M−1265πμ24π2−μ22NL+2M],Eh2(2sd)=ℏ2μ222mhR2,Eh3(2sd)=ℏ22mhR2[4π2+μ222+3μ2214N+2L−2ML+2M+1265πμ24π2−μ22NL+2M]. The contribution of the
self-polarization term from (10)–(13) has been
omitted since it represents, in fact, a common shift of all the levels by e2(1/εd−1/εs)/2R.In Figure5 the energy levels from (10)–(13) have been
plotted as functions of the NC radius. The two lowest hole levels have very
similar energies. The lowest level, being sixfold degenerate, corresponds to
six electron states with an s-type envelope function. These states have an
isotropic macroscopic (averaged over the unit-cell volume) charge distribution
inside the dot. The second level corresponds to six electron states with p-type
envelope functions. Correspondingly, the macroscopic charge has an anisotropic
distribution for these states (see [92] for details).Figure 5
Hole energy spectrum according to (10)–(13) in case of no spin-orbit interaction. Solid line: s-type level. Short dashed lines: p-p hybridized levels. Dots: d-d hybridized levels. Long dashed lines: 2s-d hybridized levels. All energies are counted deep into the valence band with respect to the valence band maximum. The degeneracy degree including spin (from bottom to top) is 6; 6; 6; 4; 6; 4; 6; 6; 2; 6; 2; 6.In the conduction
band the Bloch-state basis consists of six Bloch functions
of three differentX-points in the Brillouin zone
[99]. We denote three pairs of these
functions as |X〉, |X′〉; |Y〉, |Y′〉; |Z〉, |Z′〉.
Each pair belongs to the twofold degenerate irreducible representation X1 of the corresponding X-point. The unprimed
Bloch functions have a nonzero value at the lattice site, while the primed
functions are zero at those points. The choice of the X1-functions as the Bloch basis allows us to describe correctly a
dispersion-law nonparabolicity that originates mainly from the energy-branch
crossing in the X-points.Thek⋅p Hamiltonian operator in the conduction band
represents a 6×6 matrix having the form(14)H=(Hx000Hy000Hz). Each element of the matrix H is a block 2×2 matrix defined by the following expressions [99]:(15)Ha=(pa22ml+p2−pa22mt(1mt−1m0)pbpc+ip0mlpa(1mt−1m0)pbpc−ip0mlpapa22ml+p2−pa22mt). Here p0≈0.14(2πℏ/a0) is the distance in p-space from any of the energy minima to the nearest X-point, a0 is the lattice constant of silicon. ml=0.92m0 and mt=0.19m0 are the “longitudinal” and “transverse”
effective masses. The origin of the E-axis coincides with the X-point energy. Each of the
small indices (a, b, c) runs over the values x, y,
or, z and always differs from the others.Solving (4) for the conduction
band is done in the same way as for the valence band, as in (9)–(13) above. One
can then obtain the following groups of energies. The ground states, as well as
the second excited ones, are the 1s-1p hybrids. Their energy levels are given
by(16)Ee1(sp)=Es+Ep−2Hpp2−(Ep−Es−2Hpp2)2+Hsp2,Ee2(sp)=Es+Ep−2Hpp2+(Ep−Es−2Hpp2)2+Hsp2, where(17)Hsp=ip0ml〈s|pz|pz〉=2πℏμ1p0[3mlR(μ12−π2)],Hpp=112(1mt−1ml)〈pz|p2−3pz2|pz〉=ℏ2μ12(ml−mt)15mtmlR2 are s-p and p-p type matrix
elements of anisotropic part Ha, Es=ℏ2π2/2meR2,
and Ep=ℏ2μ12/2meR2 are the energies of the s- and p-type states
of the isotropic unperturbed Hamiltonian operator. The isotropic average
electron effective mass is defined by me−1=(2mt−1+ml−1)/3≈(0.26m0)−1. The p-p hybridization forms states with energies:(18)Ee1,2(p)=Ep+Hpp∓Hxy. Here Hxy=ℏ2μ12(m0−mt)/5mtm0R2.
The next three levels are the result of the d-d hybridization:(19)Ee1(d)=ℏ2μ222meR2(1−27ml−mt2ml+mt−67mlm0m0−mt2ml+mt),Ee2(d)=ℏ2μ222meR2(1+47ml−mt2ml+mt),Ee3(d)=ℏ2μ222meR2(1−27ml−mt2ml+mt+67mlm0m0−mt2ml+mt). Finally, the 2s-d hybridized
energies are obtained numerically:(20)Ee1(2sd)=ℏ22meR2(4π2+2μ223−0.477μ22),Ee2(2sd)=ℏ22meR2(4π2+2μ223−0.455μ22),Ee3(2sd)=ℏ22meR2(4π2+2μ223+0.932μ22). The results for the
single-particle electron spectrum in the conduction band are shown in Figure 6.Figure 6
The lowest energy levels in the conduction band as functions of the NC radius, from (16)–(20). The energies are counted from the X-point energy. Solid line: s-p type levels. Short dashed lines: p-p hybridized levels. Dots: d-d hybridized levels. Long dashed lines: 2s-d hybridized levels. All the levels are 12-fold degenerate including spin degeneracy.The wave function of the ground state turns out to be
a superposition of the terms withs- and p-type envelope functions (see [92] for details). The “weight” of the
s-type envelope function is greater than that of the p-type function. Therefore
the macroscopic charge for the ground state has a distribution close to
isotropic.From the above calculations, the optical gap of the NC
can be obtained as the sum of the lowest energies in both bands and the energy
difference betweenX- and Γ-points in bulk ΔXΓ=1.24 eV:(21)εg(R)=ΔXΓ+Eel(sp)+Eh(s). This energy somewhat overestimates value compared to
that obtained with the numerical methods, as indicated by the solid line in Figure 7.
This is a consequence of two main factors.Figure 7
NC gap versus reciprocal dot radius. Solid line: infinite barriers (21). Circles: finite barriers (3.2 eV for electrons and 4.3 eV for holes, which corresponds to Si/SiO2 interface) and constant effective mass. Disks—finite barriers and free electron mass outside the dot. Both of the latter are numerical results by EFA [92]. The dashed line is a guide to the eye.The first factor is that we have assumed infinitely
high potential barriers at the interface. The authors of  [91, 93, 95] calculated the carrier energies supposing the
potential barriers to be finite and equal to approximately 3 eV (or a
little more) for electrons and 4-5 eV for holes. Moreover, the discontinuity
of the effective mass has been taken into account. In Figure 7 we have plotted
the optical gap of silicon nanocrystal versus the confining parameter 1/R for three models differeing according the
nanocrystal’s surroundings. The solid line corresponds to infinitely high
potential barriers. Circles represent the finite barriers and constant
effective mass throughout the sample. Finally, in case of the mass discontinuity
with effective mass outside the dot coinciding with free electron mass, the
optical gap has been represented by disks. For the last two cases, the
dependence Δεg(R) becomes weaker than R−2,
especially for the case of the discontinuous effective mass where the
dependence is closer to 1/R.
In the last case the gap dependence is well fitted by the function:(22)εg(1)(R)=εg2+D1R2 with D1=4.8 eV2·nm2. In general, one may conclude that the calculated values of
the optical gap are reduced due to the barrier finiteness and mass
discontinuity, and agree better with those shown in Figure 4.The second factor, strongly influencing the dot gap,
is electron-hole Coulomb interaction. Its contribution to the optical gap will
be discussed in the next subsection.


## 3.1.3. EFA Calculations: Exciton Corrections

The electron-hole interaction reduces the energy of
the electron-hole pair. As a result, the exciton optical gap is also reduced.
In order to estimate the exciton correction to the total electron-hole
single-particle energy one must solve the two-particle problem. This has been
done for silicon NCs embedded in an infinitely wide-band-gap material [67, 90, 94]. The exciton corrections
were found to be about 0.4 eV [67], 0.25 eV [90], and 0.3 eV [94] for 2-nm-diameter NCs, with the
correction decreasing approximately as R−1 with increasing size.Meanwhile, as has been pointed out in the preceding section, the
finiteness of the potential barriers and effective mass discontinuity can
appreciably influence the electron and hole energies. Therefore, it is possible
to expect the same for the excitonic energies. Ferreyra and Proetto [91] have studied excitonic states for a
quantum dot in vacuum. Accordingly, they accepted the barrier heights equal to
the electron affinity of the corresponding bulk material for electrons and
infinity for holes, and the effective mass being m0.
They found an exciton correction of the order of 0.2 eV for a 2-nm-diameter
dot. As the nanocrystal size increases, the magnitude of this correction
decreased as R−0.7.The authors of some of the previous work (e.g., [67, 90, 91]) treated the Coulomb potential energy as VC=−e2/εsreh similar to bulk silicon. However, in a quantum
dot the electron and hole interact not only with each other but also with their
“images” due to the difference in the permittivity of the materials inside and
outside the dot. These are the so-called polarization fields arising because of
the charge polarization at the dot boundary. The interaction with the
polarization fields has been taken into account in [94]. Nevertheless, the
polarization fields did not lead to any significant corrections to the optical
gap. As mentioned above, the correction does not exceed 0.3 eV, which is
close to the values obtained by other authors.Evidently, the calculations carried out with the EFA [67, 90, 91, 94] provide sufficiently good coincidence with both experimental and
theoretical data. For comparison, we have depicted in Figure 4 with the
solid line the EFA single-particle gap according to (22). The gap values εg(1)(R) agree well with those computed by PP, TB, and
DFT methods. Obviously, a small exciton correction of about 0.3–0.1 eV
for 2 to 5 nm diameter crystallites, respectively, does not significantly
change the single-particle gap values.It should be noted, in conclusion, that all the authors employed the
perturbation theory considering the electron-hole interaction to be weak
compared to the typical quantum confinement energies in the dot. Such an
approach is justified if the exciton Bohr radius is significantly larger than
the dot radius. This requirement remains valid for small quantum dots with the
sizes of about 5-6 nm or less.


## 3.1.4. Interface States

We have already emphasized in the introduction that radiative
transitions between interface states are also often considered as the origin of
NIR, or even visible, emission from silicon nanocrystals. In a certain sense,
this point of view is alternative to the idea of quantum confinement.
Description of the interface states with EFA is difficult because the real
potential existing in the vicinity of the nanocrystal surface has in principle
a microscopic nature that manifests itself at a distance on the order of a bond
length. Meanwhile, the EFA is, in fact, a “macroscopic” method which is not
able to distinguish the spatial structure on scales smaller than the size of
the primitive cell. Therefore TB and DFT methods are essentially more suitable
for this goal.As a rule, interface states originate from various-type defects existing
at the surface of the crystallite. These include, mainly, dangling bonds (Pb centers) which can be neutral, positively, or negatively charged, and different
SiO bonds (backbonded and double bonded oxygen) arising at Si/SiO2 interface. Charged dangling bonds corresponding to zero or two electrons in the
bond level, respectively, produce deep levels inside the band gap of bulk
silicon. Their energies are approximately 0.3 eV below the conduction band
minimum for the negatively charged bond, and 0.3 eV above the valence band
maximum if the bond is positively charged [100]. Consequently, the effective gap
between these levels is about 0.5-0.6 eV [101]. For nanocrystals, the effective gap gradually rises as the nanocrystal size
decreases. In particular, for a 5-nm-diameter NC this shift equals
approximately 0.6 eV, while for a 2 nm crystallite it is 1.3-1.4 eV [100]. Thus, we see that the dangling bond levels are sensitive to the NC size.Nanocrystal oxidation has been studied theoretically by TB [8, 102], DFT [103, 104],
and GW+BSE [89]
methods. The role of the so-called
backbonded (Si–O–Si) and double bonded (Si=O) oxygen atoms has been examined
for the case where the spherical NC surface is passivated with hydrogen. In the
first case (Si–O–Si), oxygen forms two bonds with two different silicon atoms
which already have four bonds with other silicon or hydrogen atoms. In the
second case (Si=O), oxygen replaces two hydrogen atoms bonded with one silicon
atom. It is natural to suppose that the system perturbation in the second case
should be more significant because of the stronger distortions of the structure.
In fact, it was found that the presence of one Si=O bond at the NC surface
provides an optical gap almost independent of the dot size [8].
At the same time, the nanocrystal
with a Si–O–Si bond has a gap that gradually decreases with increasing size.
Nevertheless, this gap remains smaller, and varies more slowly than the gap of
perfect nanocrystal with hydrogen passivated surfaces. The
calculations carried out by different methods [89, 102, 103]
for oxidized crystallites show an essential difference
in the energy of the ground electron-hole transition in crystallites with
Si–O–Si and Si=O bonds. All the methods used exhibit the smallest gap for the
case of double-bond oxidation, while the crystallites with backbonded oxygen
have the greater gap. In turn, the latter is less than the gap of a perfect NC.
The authors of [89] have found that only the bridge-type bond (Si–O–Si) can
provide a satisfactory agreement with experiments on the photoluminescence in
nc-Si/SiO2 system. This fact may be explained by the considerably
different oscillator strengths of the ground transitions for
NCs with Si=O and Si–O–Si defects. As has been shown by Nishida [102], the oscillator strength for Si–O–Si
case is several orders of magnitude greater than that for Si=O case, especially
for NCs smaller than 1 nm in diameter. This is precisely the case studied
by Luppi et al. [89], where NCs of 0.5 nm and 0.9 nm in
diameter were considered. However, according to [102]
the difference between the
size-quantized energy levels and those of the Si=O and Si–O–Si defects, as well
as the difference in the oscillator strengths of the ground electron-hole
transitions, are essential only for NC diameters less than 2 to 2.5 nm [8, 102].
For larger sizes, the energies and
the transition intensities for perfect and oxidized crystallites almost
coincide.Finally, nonradiative trapping
rates are also critical in the light emission properties of silicon NCs. The
group of neutral or chargedPb centers at silicon dangling bonds is the most important nonradiative trap in
bulk silicon [105],
and acts also in the case of Si NCs [58]. In a detailed theoretical investigation, Lannoo et al.
[106] found that the carrier trapping at neutral Pb centers is nonradiative, whereas capture at charged
centers can lead to photon emission at energies smaller than the bandgap of
bulk silicon. The rate for carrier trapping at a neutral dangling bond defect
at the Si–SiO2 interface in a NC with a single Pb center was established theoretically as a function of
the NC size [106]:(23)wnr=σ0vV12π((E0ℏω)2+z2)−0.25×exp[−Scoth(ℏω2kT)+E02kT+((E0ℏω)2+z2)0.5−E0ℏωasinh(E0ℏωz)]. In (23), σ0 is the capture cross section given by c0/v where c0 is the capture coefficient and v is the thermal velocity equal to 8kt/πm*, E0 is the ionization energy of the defect
(approximately equal to the Franck-Condon energy plus the carrier confinement
energy), S is the Huang-Rhys factor (S=~15) [107], ℏω is the average phonon energy, V is the nanocrystal volume, and z=S/[sinh(ℏω/2kT)]. The capture rate increases strongly as a function of NC size due to the
decreasing defect ionization energy, until eventually the volume term begins to
dominate the trapping rate, which then begins to decrease again. Since, the
calculated rate is for a single isolated defect, in practice the rate should be
modified for the likelihood of multiple defects on larger clusters, as
discussed in Section 4.


## 3.2. Shallow Impurities in Silicon Nanocrystals

There is one more type of defect which can play an
important role in the optical and transport properties of bulk semiconductors
and their low-dimensional counterparts (in particular, silicon). These are
impurity centers, of which some experimental results were discussed previously. Let
us now consider NCs doped with shallow impurities and discuss impurity states
in silicon quantum dots.Various aspects of the problem
have been explored. One of them—the central-cell effect [108–111]—causes splitting of the sixfold degenerate
ground energy level in bulk silicon into a singlet, doublet, and triplet with a
typical energy splitting of about 10–20 meV
for various donors [112]. According to a number of theoretical
studies [73, 113–119], in quantum dots doped with donors or acceptors the central-cell
potential strengthens the level splitting compared to (i) bulk silicon, and
(ii) undoped dots [66, 85, 120, 121]. Investigations of
the spatial charge distribution [122, 123], the formation of impurity centers
inside silicon nanocrystals from the energy point of view [114–117, 124, 125], intervalley scattering [126], hyperfine splitting and optical gap effects [127], and the screening of the point-charge field [123, 128–136] have been
performed. Below, we present EFA calculations of the electronic structure of
both donor- and acceptor-doped silicon quantum dots and examine the dependence
on the impurity position inside the dot within the framework of the hydrogenic-impurity
model. Also, we discuss the effect of the central-cell potential in screening
the point charge in NCs.


### 3.2.1. Screening in Quantum Dots

In order to solve accurately
the Schrodinger-like equation for the envelope functions in the dot in the
presence of an impurity ion, we need the correct expression for the Coulomb
potential energyVC.
Its determination in the quantum dot is not a simple task because of the
complicated character of screening in NCs. Some authors described the screening
properties in an NC with a modified dielectric constant ε(R) that depends on the dot radius [137–139]. Such a simple model, indeed, gives a moderate increase of the dielectric
properties due to the finite size of the crystallite but it does not reflect
correctly the local structure of the electric field.The correct description of such
a structure requires first-principles calculations [123, 129–136]. Nevertheless, the microscopic
picture permits a clear qualitative macroscopic interpretation using the local
dielectric function ε(r) depending on electron position
vector, r.
Taking into account the short-range field and charge polarization at the dot
boundary due to different static dielectric constants εs and εd of an NC and its surroundings,
one can in principle obtain ε(r) (Figure 8) [73].Figure 8
Dielectric function of silicon NCs in silicon dioxide matrix (εd=3) for (1)  R=1 nm; (2)  R=1.75 nm; (3)  R=2.5 nm.An overall decrease ofε(r) in a nanocrystal takes place compared to the
bulk value of εs=12. As was pointed out earlier [123, 128, 129], this decrease is mainly due to the polarization charges at the dot
boundary. The monotonic decrease of ε(r) extends to
the nanocrystal boundary, where the value of the dielectric function becomes
equal to εd. At the same time, the sharp reduction of ε(r) toward unity occurring at small r is exclusively due to the short-range
central-cell potential.


### 3.2.2. Donor States in Silicon Crystallite

First, we consider electronic
states in a silicon NC with a hydrogenlike donor placed in some arbitrary
position inside the dot. The total electron potential energy described by (2) transforms
into U(r,h)=U0(r)+VC where h stands for the impurity position vector, and VC is a Coulomb potential energy, including not
only the direct electron-ion interaction but also the interaction with
polarized charges arising at the dot boundary. Solving the eigenstate and
eigenvalue problem for the conduction band [140] yields the ground-state energy splitting shown in Figure 9 at nx=0.8, ny=0.5, and nz=0.33, where n=h/h,
and h stands for the absolute value of h.Figure 9
Fine structure of the energy spectrum with respect to the unperturbed sixfold degenerate energy level. Solid lines correspond to two states located in the Brillouin zone at theX-point (001). Dashed line: X-point (010). Dots: X-point (100). The direction of the donor's position vector is defined by  nx=0.8,  ny=0.5, and nz=0.33. The NC radius is indicated in the figure.In Figure10, the
envelope-function correction ΔΦ is shown for the ground state due to the
hydrogen-like donor, as a function of the angles θ and φ on the spherical surface r=h for the former values of na. Angles θ and φ are the spherical coordinates. The maximal
values of ΔΦ (light areas in the figure) are located around
the donor site marked with the cross. This is a natural result in that the
electron density shifts towards the donor site.Figure 10
Contour plot of the first-order correction to the envelope function for a 3-nm-diameter quantum dot. The value of the correction rises from dark to light. The cross indicates the donor position defined by:nx=0.8, ny=0.5, and nz=0.33. h/R=0.46.


### 3.2.3. Acceptor States in Silicon Crystallite

It is interesting to compare the fine structure of the
spectrum for donor- and acceptor-doped dots. Obviously, the fine structure in
the valence band occurs because of the spin-orbit interaction and the asymmetry
of the Coulomb field inside the dot, when the acceptor occupies some noncentral
position. The energy splitting has been presented in Figure11. Similarly to
the donor case, the splitting turns out to be large compared to the magnitudes
of chemical shifts in bulk silicon [141–144].Splitting of the two lower hole levels shown in Figure5 as function of the dimensionless acceptor displacement from the dot center in (a), and with respect to the NC radius in (b). The direction of the acceptor position-vector is defined by  nx=0.8, ny=0.5, and nz=0.33. Dashed and solid lines represent the energies originating from doublet and quadruplet at  h=0, respectively. All the energies are counted from the mean quadruplet energy at  h=0.
(a)(b)The Coulomb and spin-orbit interactions remove the sixfold degeneracy of
both lowest levels shown in Figure5, keeping only the double spin degeneracy
of each the levels. Furthermore the splitting increases with decreasing the
nanocrystal size. This is evidence in favor of a quantum confinement effect, as
was the case in the donor-doped dots.The spatial distribution of the
hole density in B-doped crystallite for all the six doublets is shown in Figure12 for h/R=0.1 (upper panels) and h/R=0.46 (lower panels). We have plotted the average of
the squared absolute value of the total wave function over the unit cell. In
this case the Bloch-function oscillations do not appear in the electron density,
which reduces to the “density of envelope function” ρenv(r)=∑j=16|Φj(r)|2,
where Φj(r) is the jth element of the 6D envelope function vector |Φ〉 as before.Density plot of the probability distributionρenv(r)   of all the six doublets at    h/R=0.1 (upper images), and h/R=0.46 (lower images) for NCs containing a hydrogenic acceptor. The acceptor position is indicated with a red point. (a): ground state; (b) to (f): first to fifth excited states. For each state the hole density is normalized to its maximum in the state, and rises from dark to light. The nanocrystal boundary is marked by the yellow circle, with R=1.5 nm.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)As the calculations show, for
all the six doublets the envelope-function density has an axial symmetry with
respect to the line drawn through the NC center and the acceptor. Thereforeρenv(r) has the same distribution in any dot
cross-section to which the acceptor position-vector belongs. Figure 12 shows such
a central cross-section of the electron density averaged over the unit cell for
all the doublets. Brighter areas in the density plots correspond to higher
values of ρenv(r).
The circle represents the nanocrystal boundary, and the bold point situated at
the vertical axis indicates the acceptor location.It should be noted that the general trends in the location of the
electron density under the action of the acceptor electric field are, in fact,
similar for all the doublets in both casesh/R=0.1 and h/R=0.46.
In particular, the ground-state electron density shifts to the acceptor site,
while for the excited states, as the energy of the state increases and the
electron density gradually moves into the areas free of the acceptor.


## 3.2.1. Screening in Quantum Dots

In order to solve accurately
the Schrodinger-like equation for the envelope functions in the dot in the
presence of an impurity ion, we need the correct expression for the Coulomb
potential energyVC.
Its determination in the quantum dot is not a simple task because of the
complicated character of screening in NCs. Some authors described the screening
properties in an NC with a modified dielectric constant ε(R) that depends on the dot radius [137–139]. Such a simple model, indeed, gives a moderate increase of the dielectric
properties due to the finite size of the crystallite but it does not reflect
correctly the local structure of the electric field.The correct description of such
a structure requires first-principles calculations [123, 129–136]. Nevertheless, the microscopic
picture permits a clear qualitative macroscopic interpretation using the local
dielectric function ε(r) depending on electron position
vector, r.
Taking into account the short-range field and charge polarization at the dot
boundary due to different static dielectric constants εs and εd of an NC and its surroundings,
one can in principle obtain ε(r) (Figure 8) [73].Figure 8
Dielectric function of silicon NCs in silicon dioxide matrix (εd=3) for (1)  R=1 nm; (2)  R=1.75 nm; (3)  R=2.5 nm.An overall decrease ofε(r) in a nanocrystal takes place compared to the
bulk value of εs=12. As was pointed out earlier [123, 128, 129], this decrease is mainly due to the polarization charges at the dot
boundary. The monotonic decrease of ε(r) extends to
the nanocrystal boundary, where the value of the dielectric function becomes
equal to εd. At the same time, the sharp reduction of ε(r) toward unity occurring at small r is exclusively due to the short-range
central-cell potential.


## 3.2.2. Donor States in Silicon Crystallite

First, we consider electronic
states in a silicon NC with a hydrogenlike donor placed in some arbitrary
position inside the dot. The total electron potential energy described by (2) transforms
into U(r,h)=U0(r)+VC where h stands for the impurity position vector, and VC is a Coulomb potential energy, including not
only the direct electron-ion interaction but also the interaction with
polarized charges arising at the dot boundary. Solving the eigenstate and
eigenvalue problem for the conduction band [140] yields the ground-state energy splitting shown in Figure 9 at nx=0.8, ny=0.5, and nz=0.33, where n=h/h,
and h stands for the absolute value of h.Figure 9
Fine structure of the energy spectrum with respect to the unperturbed sixfold degenerate energy level. Solid lines correspond to two states located in the Brillouin zone at theX-point (001). Dashed line: X-point (010). Dots: X-point (100). The direction of the donor's position vector is defined by  nx=0.8,  ny=0.5, and nz=0.33. The NC radius is indicated in the figure.In Figure10, the
envelope-function correction ΔΦ is shown for the ground state due to the
hydrogen-like donor, as a function of the angles θ and φ on the spherical surface r=h for the former values of na. Angles θ and φ are the spherical coordinates. The maximal
values of ΔΦ (light areas in the figure) are located around
the donor site marked with the cross. This is a natural result in that the
electron density shifts towards the donor site.Figure 10
Contour plot of the first-order correction to the envelope function for a 3-nm-diameter quantum dot. The value of the correction rises from dark to light. The cross indicates the donor position defined by:nx=0.8, ny=0.5, and nz=0.33. h/R=0.46.


## 3.2.3. Acceptor States in Silicon Crystallite

It is interesting to compare the fine structure of the
spectrum for donor- and acceptor-doped dots. Obviously, the fine structure in
the valence band occurs because of the spin-orbit interaction and the asymmetry
of the Coulomb field inside the dot, when the acceptor occupies some noncentral
position. The energy splitting has been presented in Figure11. Similarly to
the donor case, the splitting turns out to be large compared to the magnitudes
of chemical shifts in bulk silicon [141–144].Splitting of the two lower hole levels shown in Figure5 as function of the dimensionless acceptor displacement from the dot center in (a), and with respect to the NC radius in (b). The direction of the acceptor position-vector is defined by  nx=0.8, ny=0.5, and nz=0.33. Dashed and solid lines represent the energies originating from doublet and quadruplet at  h=0, respectively. All the energies are counted from the mean quadruplet energy at  h=0.
(a)(b)The Coulomb and spin-orbit interactions remove the sixfold degeneracy of
both lowest levels shown in Figure5, keeping only the double spin degeneracy
of each the levels. Furthermore the splitting increases with decreasing the
nanocrystal size. This is evidence in favor of a quantum confinement effect, as
was the case in the donor-doped dots.The spatial distribution of the
hole density in B-doped crystallite for all the six doublets is shown in Figure12 for h/R=0.1 (upper panels) and h/R=0.46 (lower panels). We have plotted the average of
the squared absolute value of the total wave function over the unit cell. In
this case the Bloch-function oscillations do not appear in the electron density,
which reduces to the “density of envelope function” ρenv(r)=∑j=16|Φj(r)|2,
where Φj(r) is the jth element of the 6D envelope function vector |Φ〉 as before.Density plot of the probability distributionρenv(r)   of all the six doublets at    h/R=0.1 (upper images), and h/R=0.46 (lower images) for NCs containing a hydrogenic acceptor. The acceptor position is indicated with a red point. (a): ground state; (b) to (f): first to fifth excited states. For each state the hole density is normalized to its maximum in the state, and rises from dark to light. The nanocrystal boundary is marked by the yellow circle, with R=1.5 nm.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)As the calculations show, for
all the six doublets the envelope-function density has an axial symmetry with
respect to the line drawn through the NC center and the acceptor. Thereforeρenv(r) has the same distribution in any dot
cross-section to which the acceptor position-vector belongs. Figure 12 shows such
a central cross-section of the electron density averaged over the unit cell for
all the doublets. Brighter areas in the density plots correspond to higher
values of ρenv(r).
The circle represents the nanocrystal boundary, and the bold point situated at
the vertical axis indicates the acceptor location.It should be noted that the general trends in the location of the
electron density under the action of the acceptor electric field are, in fact,
similar for all the doublets in both casesh/R=0.1 and h/R=0.46.
In particular, the ground-state electron density shifts to the acceptor site,
while for the excited states, as the energy of the state increases and the
electron density gradually moves into the areas free of the acceptor.


## 3.3. Interband Recombination in Silicon Nanocrystals

The PL intensity and quantum efficiency is defined by the
relative contributionτR−1/τPL−1 of a radiative recombination channel in an
interband transition. In turn, the total decay rate is defined by both
radiative and nonradiative recombination rates: 1/τPL=1/τR+1/τNR.
Consequently, in order to discuss possible means of increasing the quantum
efficiency of silicon crystallites, we should understand what factors influence
the radiative and nonradiative rates. For this purpose in this section we
analyze theoretically both radiative and nonradiative recombination processes.


### 3.3.1. Radiative Recombination Times in Undoped Si NCs

In bulk silicon, no-phonon radiative transitions between the
conduction-bandΔ-point and the valence-band Γ-point are forbidden because of indirect band-gap of silicon. In this
case, only phonon-assisted transitions may take place, as shown in Figure 13.
In a quantum dot no-phonon emission becomes possible but remains a low
probability process for reasons discussed below. Meanwhile, the phonon-assisted
radiative transitions have much greater rate. Here, we calculate both no-phonon
and phonon-assisted radiative lifetimes in silicon nanocrystal.Figure 13
Schematic representation of the silicon band structure with phonon-assisted radiative transitions from the conduction-band minimum to the valence-band maximum indicated by arrows.We first calculate the rateτ0−1 of the no-phonon radiative recombination.
Using Fermi's golden rule in the first-order perturbation theory, one can write
the decay rate for the transition between initial (I) and final (F) states in the conduction and valence bands, respectively, in the
following form:(24)τ0,IF−1=2πℏ∑Q,σ|WIF|2δ(εg(R)−ℏωσ(Q)). Here, ωσ(Q) and Q stand for the photon frequency and wave
vector. The Dirac delta-function reflects the energy conservation law for the
electron transition. The operator W describes the electron-photon interaction,
and has the form:(25)W=∑k,σ2πℏe2κ(εs;εd)m02ωσ(Q)V0εs(cQσ+cQσ+)eQσp, where p=−iℏ∇ is the
electron momentum operator, eQσ is the polarization vector, the operator cQσ annihilates and cQσ+ creates a photon with the wave vector Q and polarization σ,
and V0 stands for the volume of the electromagnetic
resonator. The function κ(εs;εd) is written as [145](26)κ(εs;εd)=εdεs(3εd2εd+εs)2. This function represents the
correction factor in the field magnitude due to the replacement of homogeneous
media with bulk permittivity εs by a spherical silicon nanocrystal surrounded
by silicon dioxide with a dielectric constant εd.The initial state in (24) corresponds
to an electron-hole pair being in its ground state, and an ensemble of photons
whose distribution over Q is described by the Bose-Einstein statistics.
In the final state, the valence band is completely occupied and the conduction
band is empty. The number of photons in the final state always increases by
one.As has already been mentioned
(see Section3.1.2), the ground spinless electron state in the conduction
band is sixfold degenerate. However, TB [85, 120], PP [66], and DFT [121] calculations revealed a weak
splitting of the ground state into the singlet, doublet, and triplet states due
to the tetrahedral symmetry of the spherical silicon nanocrystal. The authors
pointed out that as the NC size varies, A1-, E-, or T2-type states alternately become the ground level. Correspondingly, in
order to take into account tetrahedral symmetry of the quantum dot, we build
the six ground states in accordance with the symmetric transformations of the irreducible
representations A1 (singlet), E (doublet), and T2 (triplet) of the tetrahedral group. These functions
are defined as follows [73]:(27)ΨS=cos(λ)|A1〉|s〉+sin(λ)|X′〉|px〉+|Y′〉|py〉+|Z′〉|pz〉3,ΨD(1)=cos(λ)|E(1)〉|s〉+sin(λ)|X′〉|px〉−|Y′〉|py〉2,ΨD(2)=cos(λ)|E(2)〉|s〉+sin(λ)|X′〉|px〉+|Y′〉|py〉−2|Z′〉|pz〉6,ΨT(1)=cos(λ)|X′〉|s〉−sin(λ)2|A1〉+3|E(1)〉+|E(2)〉6|px〉,ΨT(2)=cos(λ)|Y′〉|s〉−sin(λ)2|A1〉−3|E(1)〉+|E(2)〉6|py〉,ΨT(3)=cos(λ)|Z′〉|s〉−sin(λ)|A1〉−2|E(2)〉3|pz〉. Here, the parameter λ is defined by the relationships [92](28)cos(2λ)=Ep−Es−2Hpp(Ep−Es−2Hpp)2+4Hsp2,sin(2λ)=2Hsp(Ep−Es−2Hpp)2+4Hsp2, where vectors |s〉 and |pa〉 stand for the s- and p-type envelope functions
as before. The Bloch states |A1〉=(|X〉+|Y〉+|Z〉)/3, |E(1)〉=(|X〉−|Y〉)/2, |E(2)〉=(|X〉+|Y〉−2|Z〉)/6,
and |T2〉=|X′〉, |Y′〉,
or |Z′〉 belong to the representations A1, E,
and T2 of the tetrahedral group. In what follows,
when computing the interband matrix elements of W we will choose one of the states given by (27)
as the initial one. For convenience, one can rewrite the wave functions of the
initial states in a general form:(29)ΨI=∑α,jBIαjFα(r)ψj(r), where the index α indicates the “values” of the s, px,
py, and pz, functions, and ψj(r) stands for the six Bloch functions |X〉, |Y〉, |Z〉, |X′〉, |Y′〉, |Z′〉 of the irreducible representation X1.
The s- and p-type envelope functions are denoted as F(r),
and Fx(r), Fy(r), Fz(r),
respectively. Selecting expansion coefficients BIαj,
we can construct any of the states defined by (27).The final state is one of the
three degenerate states (neglecting the spin-orbit coupling) in the valence
band [92] with the s-type envelope function:(30)ΨF=F(r)ψF(r), where ψF(r) stands for the Bloch state basis functions |YZ〉, |XZ〉,
or |XY〉 of the irreducible representation Γ25′.After some algebra, one can
obtain recombination rateτ0,IF−1 in the form:(31)τ0,IF−1=4e2εsκ(εs;εd)εg(R)3m02ℏ2c3|pIF|2, where pIF=〈ΨI|p|ΨF〉.
In order to compute the momentum matrix element, we expand the Bloch functions ψj(r) and ψF(r) over the reciprocal lattice vectors gn as follows:(32)ψj(r)=∑mCjmexp[2πi(gm+ej)r/a0],ψF(r)=∑nVFnexp[2πignr/a0], where a0 is the lattice constant of silicon, and unit
vectors ej define the X-point. We set e1,4=ex, e2,5=ey,
and e3,6=ez.
Expansion coefficients Cjm and VFn are determined with the local pseudopotential
method. They satisfy the normalizing condition ∑n|Cjn|2=∑n|VFn|2=1.
As a result, the momentum matrix element is given by(33)pIF=ℏ4πa0bIF(R)(a0R)4, where bIF(R)=∑α,jBIαjJα(R) with Jα(R) being oscillating vector functions [146]
of the dot radius. For s- and p-type subscripts, respectively, these functions
are(34)J(R)=∑n,mCjm∗VFndn+mdn−m4cos(2πdn−mRa0),Ja(R)=−i3μ1π∑n,mCjm∗VFndn+mdn−m(a)dn−m5sin(2πdn−mRa0). Here, a=x,y,z, dn±m=gn±gm±ej,
and dn−m(a) represents an ath component of dn−m.
The nonzero value of pIF is exclusively due to the Heisenberg
uncertainty relations which cause the wide distribution of the wave function in p-space.Averaging (31) over all
possible initial and final states yields(35)τ0−1=e2εsκ(εs;εd)εg(R)12π2m02c3a02(a0R)8〈|bIF(R)|2〉. In Figure 14 we show the
no-phonon recombination rate τ0−1 as function of the dot size. Although the
no-phonon decay rate has a nonzero value, its magnitude remains small (less
than 103 s−1) within the range of the NC sizes shown
in the figure.Figure 14
Recombination rates of no-phonon and phonon-assisted radiative transitions between the ground electron and hole states in silicon NCs, after averaging over all degenerate initial and final states.Next, we calculate the
phonon-assisted recombination rate [95, 147–149]. The total phonon-assisted
rate of the radiative electron-hole recombination is determined in the
second-order perturbation theory as(36)τR,IF−1=2πℏ∑Q,σ∑q,ℓ|∑aWFAUAI+UFAWAIεI−εA|2×[δ(εg(R)−ℏωσ(Q)−ℏνℓ(q))+δ(εg(R)−ℏωσ(Q)+ℏνℓ(q))]. Here the
matrix elements of the electron-photon (W) and electron-phonon (U) interaction operators are calculated between the
initial I,
the final F,
and an intermediate state A; εA and εI are the total energies of the
intermediate and initial states, respectively, including not only the energies
of the electrons (or holes) but also the energies of the photons and phonons.
The phonon frequency of ℓth polarization is denoted as νℓ(q) with q being the dimensionless phonon wave vector
taken in units of 2π/a0.The
electron-phonon interaction operator is treated within the framework of the
rigid-ion model and is given by(37)U=−∑q,ℓ∑n,sℏ2MNνℓ(q)∇Vns×(eqℓsexp{i2πqRna0}bqℓ+eqℓs*exp{−i2πqRna0}bqℓ+). Here, N is the number of primitive cells in the
crystal, M is the mass of a silicon atom, Vns=Vat(r−Rn−τs) is the atomic potential, where Rn stands for the position vector of the nth unit cell, and τs represents the position vectors of two atoms
within the unit cell: τ1=0,
and τ2=(1,1,1)×a0/4, bqℓ+ and bqℓ are the phonon creation and annihilation
operators, and the phonon polarization vectors are denoted as eqℓs.
Making the Fourier transformation of the atomic potential: Vat(r)=N−1∑pVpexp{ipr},
one can rewrite the operator U in the form (see also [150, 151]):(38)U=2πia0∑q,ℓ,s,mℏ2MNνℓ(q)(q+gm)×[eqℓs∗Vq+gm∗exp{−i2π(q+gm)(r−τs)a0}bqℓ+−eqℓsVq+gmexp{i2π(q+gm)(r−τs)a0}bqℓ]. Calculations
of the electron-photon and electron-phonon matrix elements yield(39)τR,IF−1=4π2ℏ2e2εsκ(εs;εd)εg(R)3Mm02c3a04(a0R)3×∑ℓPIF(ℓ)ℏνℓcoth[ℏνℓ2kT], where(40)PIF(ℓ)=BIαm(BIβn)∗xFm(ℓ)xFn(ℓ)2π2R3×∫drFα(r)Fβ(r)F2(r),xFm(ℓ)=∑A〈ψF|w|ψA〉〈ψA|u|ψm〉+〈ψF|u|ψA〉〈ψA|w|ψm〉εI−εA.ψA represents the Bloch functions of
the intermediate states and the operators w and u are directly proportional to W and U:
(41)W=w2πℏa02πℏe2κ(εs;εd)m02ω(Q)V0εs,U=u2πa0ℏ2MNνℓ(q). As has been
shown earlier, the electron subsystem in silicon nanocrystals interacts more
efficiently with TO and LO phonons [147, 148] similarly to bulk silicon [150, 151]. The contribution of TA phonons is
negligibly small. Therefore, in the sum over the phonon modes in (39), only TO
and LO modes may be taken into account.For
convenience, we have again averaged the calculated rates over all the possible
initial and final degenerate states and obtained the recombination rate in the
following form:(42)τR−1=π2ℏ2e2εg(R)εsκ(εs,εd)9Mm02c3a4(a0R)3×(7.527coth[ℏνLO/2kT]ℏνLO+32.768coth[ℏνTO/2kT]ℏνTO)×(4cos2λ+9.26). Here: ℏνLO=0.051eV and ℏνTO=0.0576eV represent the energies of LO and TO phonons
and the numbers 7.527, 32.768 come from numerical integration of
electron-phonon matrix elements. The size dependence of the recombination rate
has been presented in Figure 14. The dependence of τR−1 on the dot radius is close to R−3.
This is much slower than the rate obtained for no-phonon transitions (R−8).The results presented here agree well both qualitatively and
quantitatively with those obtained by other authors [95, 147, 148] for similar systems. In particular,
according to their results, the recombination rates for NCs from 2 to 6 nm in
diameter do not exceed 105 s−1 and are never
less than 5×102 s−1. It should be noted,
however, that the TB calculations [148] produce a considerably slower
decrease of the radiative decay rate than the EFA (as developed here and in
[95, 147, 149]) predicts.Finally, concluding this subsection, we would like to discuss briefly
the relation between our single-particle calculations and the singlet-triplet
model which gives a reduction of the radiative lifetime with increasing
temperature, as in (1). It is, of course, not applicable in the frame of a
single-particle treatment to obtain the state splitting caused by the exchange
electron-hole interaction. Therefore, the temperature dependence of τR−1 (42)) is defined exclusively by the phonons. As
a result, no sharp rise of the decay rate appears in our model as the
temperature increases up to the room values. At temperatures higher than about
150–200 K, both models nevertheless should give similar results because the
upper excitonic states become highly populated. Thus, the single-particle model
remains valid for T>200 K. As well, it should be noted that the separation of dark and bright
excitons is not always possible. The spin-orbit coupling transforms forbidden
exciton transitions into suppressed ones due to the entanglement of
single-particle states with different spin projections. The entanglement can be
crucially strengthened by imperfections of the crystallite structure such as
shape nonsphericity or, especially, by various point defects such as Pb centers, SiO bonds, and impurities. In the presence of the latter effects, all
the transitions can become allowed and consequently the single-particle model
is acceptable.


### 3.3.2. Radiative Recombination in Doped Dots

Experimental evidence of shallow impurity effects in
the PL of silicon NCs (see Section2.3) has no corresponding rigorous
theoretical grounds. Usually, when discussing various changes in optical
properties of the NCs due to doping, the nonradiative recombination channel is
considered more closely. For instance, doping may passivate dangling bonds, and,
as a consequence, essentially reduce the effectiveness of the nonradiative
channel. On the other hand, a large number of donors or acceptors in the
nanocrystal leads to the appearance of extra carriers that can recombine
through the Auger process. Thus, doping is an important factor influencing the
nonradiative recombination (a more detailed description of the nonradiative
recombination mechanisms can be found in the next subsection). The nonradiative
lifetime τNR can be varied by doping, which leads to the change
of the emission efficiency.The impact of doping on the radiative channel has not
yet reported. However, it is interesting to examine also the possibility of increasing
the quantum yield via enhancement of the radiative rateτR−1.
This is exclusively a quantum effect that can be explained by the
reconstruction of the wave functions in different symmetry (or asymmetry), and
changing selection rules for the interband transitions. We would like to touch
upon this question in the present subsection and compute the radiative
recombination rates in a silicon NC doped with a hydrogen-like donor and acceptor.In order to compute the decay rates for the doped dot
we follow (36). The main reason for the rate changes compared to the case of
the undoped dot lies in some corrections to the electronic wave
functions of both initial and final states (27), (30) due to the existence of a
shallow impurity in the dot. In the following, we discuss two different cases.The first case corresponds to an off-center position for the impurity
inside the dot. In the valence band such an impurity location creates a field
asymmetry in the system, which entangles the three spinlesss-type states (30))
with the three spinless p-type states [92](43)Ψp1=Fx(r)|XZ〉−Fy(r)|YZ〉2,Ψp2=Fz(r)|YZ〉−Fx(r)|XY〉2,Ψp3=Fy(r)|XY〉−Fz(r)|XZ〉2. These states (30) and (43) correspond to the closely spaced lowest triply
degenerate energy levels shown in Figure 5. As a result, the ground hole state becomes
a superposition of the states with s- and p-type envelope functions. In the
conduction band, the ground electron states are the s-p combinations even
without impurities, as seen from (27). Embedding the impurity into some
arbitrary position within the NC raises the number of the p-type envelope
functions in the ground states as well as in the valence band. However, PIF(ℓ) becomes smaller if the integrated envelope
functions are p-type. Consequently, the squared transition matrix element,
proportional to PIF(ℓ),
and the rate itself decrease relative to the case of an undoped NC, as shown in
Figure 15 for two different displacements of the impurity from the dot center.The rates of the phonon-assisted ground radiative transition as functions of the dot radius for a doped dot. Forh≠0, we have  nx=0.8, ny=0.5, and nz=0.33.
(a)(b)The ratioh/R=0.46 corresponds to the greatest possible mix of s-
and p-type states with maximal weight of the p-type envelope functions in the
ground states both in the conduction and valence bands. The strong mixing
results in maximal decrease of the decay rate independently of the impurity type.
When the impurity is situated near the nanocrystal boundary, h/R=0.9,
the weight of the p-type envelope functions reduces in the ground state of both
bands. As a consequence, the rate rises with respect to the case h/R=0.46,
but remains smaller than that for the undoped NC.The second case we consider is the case of the central-located impurity.
Ifh=0,
no s-p entanglement takes place in the valence band. The functions with the
s-type envelope (30)) describe the ground states of both donor- and
acceptor-doped NCs.The situation is different for the conduction band. Formally, the wave
functions of the six spinless ground states are described by (27) as before.
However, for doped dots, parameter λ has a form which differs from that given by (28).
In the presence of an impurity center, λ is defined as:(44)cos(2λ)=Ep−Es+δV−2Hpp(Ep−Es+δV−2Hpp)2+4Hsp2,sin(2λ)=2Hsp(Ep−Es+δV−2Hpp)2+4Hsp2, where δV=Vpp−Vss is the difference between the p-p and s-s type
matrix elements of the Coulomb potential energy VC.
The appearance of this term in the definition of λ is explained by the shift of the energy levels
of the s- and p-states Es and Ep,
respectively, due to the Coulomb interaction. The relative shift δV turns out to be positive for donors, and
negative for acceptors. Consequently, cos(λ) increases and sin(λ) decreases for donors, and vice versa for
acceptors.This implies that the weight of thes-type envelope function in the
electron ground state increases for the donor-doped dot and shrinks for the dot
doped with acceptor, with respect to its value in the undoped dot. Hence, the
coefficient PIF(ℓ) for the donor-doped dot is greater than that
for the undoped dot. In turn, the latter exceeds PIF(ℓ) for the acceptor-doped dot. Precisely the same
relationships take place for the recombination rates, as shown in Figure 15.Note in conclusion, that the central-symmetric caseh=0 qualitatively corresponds to some real
situation, where the NC has been highly doped with donors or acceptors. It is
natural to assume a homogeneous impurity distribution within the nanocrystal in
this case. Obviously, such a homogeneous density of charge induces a spherically
symmetric electric field that cannot mix the s- and p-type states. Moreover,
the relative Coulomb shift δV remains positive for donors and negative for
acceptors, as in the above case of a single impurity center in the dot [73]. Therefore, one can expect similar
behavior of τR−1 for highly doped NCs.


### 3.3.3. Nonradiative Recombination

In parallel with radiative transitions, nonradiative processes occur in
silicon nanocrystals. As a rule, nonradiative processes play a problematic role
in silicon crystallites, due to their high recombination rates which
substantially exceed the rates of the radiative transitions. Among the possible
nonradiative processes, capture on dangling bonds and Auger recombination are the
most efficient. Both processes are phonon-assisted, possibly even multi-phonon.
The first one takes place both at low and high pumping levels, while Auger
recombination becomes possible exclusively at high excitation power when more
than one excited electron (hole) exist in the conduction (valence) band of the
nanocrystal.The capture on a neutral dangling bond has been studied earlier by TB
method [100, 106]. It was shown that excited
electron-hole pairs can recombine on a neutral dangling bond in two stages. First,
the electron (hole) is trapped on a neutral Pb center making it
negatively (positively) charged. Then, the hole (electron) is trapped on the
charged dangling bond. According to estimations [100], the capture rate on the neutral
dangling bond for holes is always much greater than that for electrons. In
turn, the electron capture rate at room temperature increases sharply as the
nanocrystal gap decreases from 2.7 eV to 1.5 eV (see (23)). Within
the same range of energy gaps, the hole capture rate also increases but less
sharply. We should emphasize the presence of two main features of the capture
process. First, there is a very stron dependence of the capture rates (even for
holes) on the energy gap, that is, the NC size.
The second feature is the overall high capture rates, which are much
greater than the radiative decay rates. Evidently, the presence of a dangling
bond in silicon crystallite drastically reduces the quantum efficiency of
photon generation. Therefore the passivation of dangling bonds is required for
achievement of relatively efficient luminescence.Let us now touch briefly on the role of SiO defects in the
photoluminescence. The calculations of Nishida [102] showed that the oscillator strength
is almost independent of the existence of Si–O–Si bonds. In contrast, the Si=O
bonds reduce the oscillator strength by several orders of magnitude. Recently,
however, Sa'ar et al. [64] demonstrated experimentally that
both back-bonded and double-bonded oxygen atoms can enhance the PL intensity
due to formation of coupled states of electrons (holes) and SiO vibrations.
Such vibrations induce electric polarization fields generating long-lived
coupled states named “vibrons”. Usually, the surface vibrations scatter the
carriers; however, vibrons, being coupled states, do not scatter the carriers
and thereby exclude one of possible nonradiative channels from the
recombination process.For highly excited
nanocrystals, Auger recombination may take place. It is sufficient to have only
three particles (two electrons and a hole, or vice versa) for this process to
occur. It is well known that Auger recombination has high efficiency in bulk
silicon. Therefore, it is natural to expect a similar behavior in silicon
crystallites. Calculations of the Auger-recombination rate in crystallites
carried out with TB [58, 152] and EFA [153] techniques confirm, indeed, the fast character of this mechanism. In
particular, the characteristic times have been found to be of the order
0.1 to 10 ns−1 and gradually rise with decreasing
nanocrystal size. Thus, Auger recombination, as well as dangling bonds, can
strongly quench the luminescence in silicon crystallites, as confirmed
experimentally [152].


## 3.3.1. Radiative Recombination Times in Undoped Si NCs

In bulk silicon, no-phonon radiative transitions between the
conduction-bandΔ-point and the valence-band Γ-point are forbidden because of indirect band-gap of silicon. In this
case, only phonon-assisted transitions may take place, as shown in Figure 13.
In a quantum dot no-phonon emission becomes possible but remains a low
probability process for reasons discussed below. Meanwhile, the phonon-assisted
radiative transitions have much greater rate. Here, we calculate both no-phonon
and phonon-assisted radiative lifetimes in silicon nanocrystal.Figure 13
Schematic representation of the silicon band structure with phonon-assisted radiative transitions from the conduction-band minimum to the valence-band maximum indicated by arrows.We first calculate the rateτ0−1 of the no-phonon radiative recombination.
Using Fermi's golden rule in the first-order perturbation theory, one can write
the decay rate for the transition between initial (I) and final (F) states in the conduction and valence bands, respectively, in the
following form:(24)τ0,IF−1=2πℏ∑Q,σ|WIF|2δ(εg(R)−ℏωσ(Q)). Here, ωσ(Q) and Q stand for the photon frequency and wave
vector. The Dirac delta-function reflects the energy conservation law for the
electron transition. The operator W describes the electron-photon interaction,
and has the form:(25)W=∑k,σ2πℏe2κ(εs;εd)m02ωσ(Q)V0εs(cQσ+cQσ+)eQσp, where p=−iℏ∇ is the
electron momentum operator, eQσ is the polarization vector, the operator cQσ annihilates and cQσ+ creates a photon with the wave vector Q and polarization σ,
and V0 stands for the volume of the electromagnetic
resonator. The function κ(εs;εd) is written as [145](26)κ(εs;εd)=εdεs(3εd2εd+εs)2. This function represents the
correction factor in the field magnitude due to the replacement of homogeneous
media with bulk permittivity εs by a spherical silicon nanocrystal surrounded
by silicon dioxide with a dielectric constant εd.The initial state in (24) corresponds
to an electron-hole pair being in its ground state, and an ensemble of photons
whose distribution over Q is described by the Bose-Einstein statistics.
In the final state, the valence band is completely occupied and the conduction
band is empty. The number of photons in the final state always increases by
one.As has already been mentioned
(see Section3.1.2), the ground spinless electron state in the conduction
band is sixfold degenerate. However, TB [85, 120], PP [66], and DFT [121] calculations revealed a weak
splitting of the ground state into the singlet, doublet, and triplet states due
to the tetrahedral symmetry of the spherical silicon nanocrystal. The authors
pointed out that as the NC size varies, A1-, E-, or T2-type states alternately become the ground level. Correspondingly, in
order to take into account tetrahedral symmetry of the quantum dot, we build
the six ground states in accordance with the symmetric transformations of the irreducible
representations A1 (singlet), E (doublet), and T2 (triplet) of the tetrahedral group. These functions
are defined as follows [73]:(27)ΨS=cos(λ)|A1〉|s〉+sin(λ)|X′〉|px〉+|Y′〉|py〉+|Z′〉|pz〉3,ΨD(1)=cos(λ)|E(1)〉|s〉+sin(λ)|X′〉|px〉−|Y′〉|py〉2,ΨD(2)=cos(λ)|E(2)〉|s〉+sin(λ)|X′〉|px〉+|Y′〉|py〉−2|Z′〉|pz〉6,ΨT(1)=cos(λ)|X′〉|s〉−sin(λ)2|A1〉+3|E(1)〉+|E(2)〉6|px〉,ΨT(2)=cos(λ)|Y′〉|s〉−sin(λ)2|A1〉−3|E(1)〉+|E(2)〉6|py〉,ΨT(3)=cos(λ)|Z′〉|s〉−sin(λ)|A1〉−2|E(2)〉3|pz〉. Here, the parameter λ is defined by the relationships [92](28)cos(2λ)=Ep−Es−2Hpp(Ep−Es−2Hpp)2+4Hsp2,sin(2λ)=2Hsp(Ep−Es−2Hpp)2+4Hsp2, where vectors |s〉 and |pa〉 stand for the s- and p-type envelope functions
as before. The Bloch states |A1〉=(|X〉+|Y〉+|Z〉)/3, |E(1)〉=(|X〉−|Y〉)/2, |E(2)〉=(|X〉+|Y〉−2|Z〉)/6,
and |T2〉=|X′〉, |Y′〉,
or |Z′〉 belong to the representations A1, E,
and T2 of the tetrahedral group. In what follows,
when computing the interband matrix elements of W we will choose one of the states given by (27)
as the initial one. For convenience, one can rewrite the wave functions of the
initial states in a general form:(29)ΨI=∑α,jBIαjFα(r)ψj(r), where the index α indicates the “values” of the s, px,
py, and pz, functions, and ψj(r) stands for the six Bloch functions |X〉, |Y〉, |Z〉, |X′〉, |Y′〉, |Z′〉 of the irreducible representation X1.
The s- and p-type envelope functions are denoted as F(r),
and Fx(r), Fy(r), Fz(r),
respectively. Selecting expansion coefficients BIαj,
we can construct any of the states defined by (27).The final state is one of the
three degenerate states (neglecting the spin-orbit coupling) in the valence
band [92] with the s-type envelope function:(30)ΨF=F(r)ψF(r), where ψF(r) stands for the Bloch state basis functions |YZ〉, |XZ〉,
or |XY〉 of the irreducible representation Γ25′.After some algebra, one can
obtain recombination rateτ0,IF−1 in the form:(31)τ0,IF−1=4e2εsκ(εs;εd)εg(R)3m02ℏ2c3|pIF|2, where pIF=〈ΨI|p|ΨF〉.
In order to compute the momentum matrix element, we expand the Bloch functions ψj(r) and ψF(r) over the reciprocal lattice vectors gn as follows:(32)ψj(r)=∑mCjmexp[2πi(gm+ej)r/a0],ψF(r)=∑nVFnexp[2πignr/a0], where a0 is the lattice constant of silicon, and unit
vectors ej define the X-point. We set e1,4=ex, e2,5=ey,
and e3,6=ez.
Expansion coefficients Cjm and VFn are determined with the local pseudopotential
method. They satisfy the normalizing condition ∑n|Cjn|2=∑n|VFn|2=1.
As a result, the momentum matrix element is given by(33)pIF=ℏ4πa0bIF(R)(a0R)4, where bIF(R)=∑α,jBIαjJα(R) with Jα(R) being oscillating vector functions [146]
of the dot radius. For s- and p-type subscripts, respectively, these functions
are(34)J(R)=∑n,mCjm∗VFndn+mdn−m4cos(2πdn−mRa0),Ja(R)=−i3μ1π∑n,mCjm∗VFndn+mdn−m(a)dn−m5sin(2πdn−mRa0). Here, a=x,y,z, dn±m=gn±gm±ej,
and dn−m(a) represents an ath component of dn−m.
The nonzero value of pIF is exclusively due to the Heisenberg
uncertainty relations which cause the wide distribution of the wave function in p-space.Averaging (31) over all
possible initial and final states yields(35)τ0−1=e2εsκ(εs;εd)εg(R)12π2m02c3a02(a0R)8〈|bIF(R)|2〉. In Figure 14 we show the
no-phonon recombination rate τ0−1 as function of the dot size. Although the
no-phonon decay rate has a nonzero value, its magnitude remains small (less
than 103 s−1) within the range of the NC sizes shown
in the figure.Figure 14
Recombination rates of no-phonon and phonon-assisted radiative transitions between the ground electron and hole states in silicon NCs, after averaging over all degenerate initial and final states.Next, we calculate the
phonon-assisted recombination rate [95, 147–149]. The total phonon-assisted
rate of the radiative electron-hole recombination is determined in the
second-order perturbation theory as(36)τR,IF−1=2πℏ∑Q,σ∑q,ℓ|∑aWFAUAI+UFAWAIεI−εA|2×[δ(εg(R)−ℏωσ(Q)−ℏνℓ(q))+δ(εg(R)−ℏωσ(Q)+ℏνℓ(q))]. Here the
matrix elements of the electron-photon (W) and electron-phonon (U) interaction operators are calculated between the
initial I,
the final F,
and an intermediate state A; εA and εI are the total energies of the
intermediate and initial states, respectively, including not only the energies
of the electrons (or holes) but also the energies of the photons and phonons.
The phonon frequency of ℓth polarization is denoted as νℓ(q) with q being the dimensionless phonon wave vector
taken in units of 2π/a0.The
electron-phonon interaction operator is treated within the framework of the
rigid-ion model and is given by(37)U=−∑q,ℓ∑n,sℏ2MNνℓ(q)∇Vns×(eqℓsexp{i2πqRna0}bqℓ+eqℓs*exp{−i2πqRna0}bqℓ+). Here, N is the number of primitive cells in the
crystal, M is the mass of a silicon atom, Vns=Vat(r−Rn−τs) is the atomic potential, where Rn stands for the position vector of the nth unit cell, and τs represents the position vectors of two atoms
within the unit cell: τ1=0,
and τ2=(1,1,1)×a0/4, bqℓ+ and bqℓ are the phonon creation and annihilation
operators, and the phonon polarization vectors are denoted as eqℓs.
Making the Fourier transformation of the atomic potential: Vat(r)=N−1∑pVpexp{ipr},
one can rewrite the operator U in the form (see also [150, 151]):(38)U=2πia0∑q,ℓ,s,mℏ2MNνℓ(q)(q+gm)×[eqℓs∗Vq+gm∗exp{−i2π(q+gm)(r−τs)a0}bqℓ+−eqℓsVq+gmexp{i2π(q+gm)(r−τs)a0}bqℓ]. Calculations
of the electron-photon and electron-phonon matrix elements yield(39)τR,IF−1=4π2ℏ2e2εsκ(εs;εd)εg(R)3Mm02c3a04(a0R)3×∑ℓPIF(ℓ)ℏνℓcoth[ℏνℓ2kT], where(40)PIF(ℓ)=BIαm(BIβn)∗xFm(ℓ)xFn(ℓ)2π2R3×∫drFα(r)Fβ(r)F2(r),xFm(ℓ)=∑A〈ψF|w|ψA〉〈ψA|u|ψm〉+〈ψF|u|ψA〉〈ψA|w|ψm〉εI−εA.ψA represents the Bloch functions of
the intermediate states and the operators w and u are directly proportional to W and U:
(41)W=w2πℏa02πℏe2κ(εs;εd)m02ω(Q)V0εs,U=u2πa0ℏ2MNνℓ(q). As has been
shown earlier, the electron subsystem in silicon nanocrystals interacts more
efficiently with TO and LO phonons [147, 148] similarly to bulk silicon [150, 151]. The contribution of TA phonons is
negligibly small. Therefore, in the sum over the phonon modes in (39), only TO
and LO modes may be taken into account.For
convenience, we have again averaged the calculated rates over all the possible
initial and final degenerate states and obtained the recombination rate in the
following form:(42)τR−1=π2ℏ2e2εg(R)εsκ(εs,εd)9Mm02c3a4(a0R)3×(7.527coth[ℏνLO/2kT]ℏνLO+32.768coth[ℏνTO/2kT]ℏνTO)×(4cos2λ+9.26). Here: ℏνLO=0.051eV and ℏνTO=0.0576eV represent the energies of LO and TO phonons
and the numbers 7.527, 32.768 come from numerical integration of
electron-phonon matrix elements. The size dependence of the recombination rate
has been presented in Figure 14. The dependence of τR−1 on the dot radius is close to R−3.
This is much slower than the rate obtained for no-phonon transitions (R−8).The results presented here agree well both qualitatively and
quantitatively with those obtained by other authors [95, 147, 148] for similar systems. In particular,
according to their results, the recombination rates for NCs from 2 to 6 nm in
diameter do not exceed 105 s−1 and are never
less than 5×102 s−1. It should be noted,
however, that the TB calculations [148] produce a considerably slower
decrease of the radiative decay rate than the EFA (as developed here and in
[95, 147, 149]) predicts.Finally, concluding this subsection, we would like to discuss briefly
the relation between our single-particle calculations and the singlet-triplet
model which gives a reduction of the radiative lifetime with increasing
temperature, as in (1). It is, of course, not applicable in the frame of a
single-particle treatment to obtain the state splitting caused by the exchange
electron-hole interaction. Therefore, the temperature dependence of τR−1 (42)) is defined exclusively by the phonons. As
a result, no sharp rise of the decay rate appears in our model as the
temperature increases up to the room values. At temperatures higher than about
150–200 K, both models nevertheless should give similar results because the
upper excitonic states become highly populated. Thus, the single-particle model
remains valid for T>200 K. As well, it should be noted that the separation of dark and bright
excitons is not always possible. The spin-orbit coupling transforms forbidden
exciton transitions into suppressed ones due to the entanglement of
single-particle states with different spin projections. The entanglement can be
crucially strengthened by imperfections of the crystallite structure such as
shape nonsphericity or, especially, by various point defects such as Pb centers, SiO bonds, and impurities. In the presence of the latter effects, all
the transitions can become allowed and consequently the single-particle model
is acceptable.


## 3.3.2. Radiative Recombination in Doped Dots

Experimental evidence of shallow impurity effects in
the PL of silicon NCs (see Section2.3) has no corresponding rigorous
theoretical grounds. Usually, when discussing various changes in optical
properties of the NCs due to doping, the nonradiative recombination channel is
considered more closely. For instance, doping may passivate dangling bonds, and,
as a consequence, essentially reduce the effectiveness of the nonradiative
channel. On the other hand, a large number of donors or acceptors in the
nanocrystal leads to the appearance of extra carriers that can recombine
through the Auger process. Thus, doping is an important factor influencing the
nonradiative recombination (a more detailed description of the nonradiative
recombination mechanisms can be found in the next subsection). The nonradiative
lifetime τNR can be varied by doping, which leads to the change
of the emission efficiency.The impact of doping on the radiative channel has not
yet reported. However, it is interesting to examine also the possibility of increasing
the quantum yield via enhancement of the radiative rateτR−1.
This is exclusively a quantum effect that can be explained by the
reconstruction of the wave functions in different symmetry (or asymmetry), and
changing selection rules for the interband transitions. We would like to touch
upon this question in the present subsection and compute the radiative
recombination rates in a silicon NC doped with a hydrogen-like donor and acceptor.In order to compute the decay rates for the doped dot
we follow (36). The main reason for the rate changes compared to the case of
the undoped dot lies in some corrections to the electronic wave
functions of both initial and final states (27), (30) due to the existence of a
shallow impurity in the dot. In the following, we discuss two different cases.The first case corresponds to an off-center position for the impurity
inside the dot. In the valence band such an impurity location creates a field
asymmetry in the system, which entangles the three spinlesss-type states (30))
with the three spinless p-type states [92](43)Ψp1=Fx(r)|XZ〉−Fy(r)|YZ〉2,Ψp2=Fz(r)|YZ〉−Fx(r)|XY〉2,Ψp3=Fy(r)|XY〉−Fz(r)|XZ〉2. These states (30) and (43) correspond to the closely spaced lowest triply
degenerate energy levels shown in Figure 5. As a result, the ground hole state becomes
a superposition of the states with s- and p-type envelope functions. In the
conduction band, the ground electron states are the s-p combinations even
without impurities, as seen from (27). Embedding the impurity into some
arbitrary position within the NC raises the number of the p-type envelope
functions in the ground states as well as in the valence band. However, PIF(ℓ) becomes smaller if the integrated envelope
functions are p-type. Consequently, the squared transition matrix element,
proportional to PIF(ℓ),
and the rate itself decrease relative to the case of an undoped NC, as shown in
Figure 15 for two different displacements of the impurity from the dot center.The rates of the phonon-assisted ground radiative transition as functions of the dot radius for a doped dot. Forh≠0, we have  nx=0.8, ny=0.5, and nz=0.33.
(a)(b)The ratioh/R=0.46 corresponds to the greatest possible mix of s-
and p-type states with maximal weight of the p-type envelope functions in the
ground states both in the conduction and valence bands. The strong mixing
results in maximal decrease of the decay rate independently of the impurity type.
When the impurity is situated near the nanocrystal boundary, h/R=0.9,
the weight of the p-type envelope functions reduces in the ground state of both
bands. As a consequence, the rate rises with respect to the case h/R=0.46,
but remains smaller than that for the undoped NC.The second case we consider is the case of the central-located impurity.
Ifh=0,
no s-p entanglement takes place in the valence band. The functions with the
s-type envelope (30)) describe the ground states of both donor- and
acceptor-doped NCs.The situation is different for the conduction band. Formally, the wave
functions of the six spinless ground states are described by (27) as before.
However, for doped dots, parameter λ has a form which differs from that given by (28).
In the presence of an impurity center, λ is defined as:(44)cos(2λ)=Ep−Es+δV−2Hpp(Ep−Es+δV−2Hpp)2+4Hsp2,sin(2λ)=2Hsp(Ep−Es+δV−2Hpp)2+4Hsp2, where δV=Vpp−Vss is the difference between the p-p and s-s type
matrix elements of the Coulomb potential energy VC.
The appearance of this term in the definition of λ is explained by the shift of the energy levels
of the s- and p-states Es and Ep,
respectively, due to the Coulomb interaction. The relative shift δV turns out to be positive for donors, and
negative for acceptors. Consequently, cos(λ) increases and sin(λ) decreases for donors, and vice versa for
acceptors.This implies that the weight of thes-type envelope function in the
electron ground state increases for the donor-doped dot and shrinks for the dot
doped with acceptor, with respect to its value in the undoped dot. Hence, the
coefficient PIF(ℓ) for the donor-doped dot is greater than that
for the undoped dot. In turn, the latter exceeds PIF(ℓ) for the acceptor-doped dot. Precisely the same
relationships take place for the recombination rates, as shown in Figure 15.Note in conclusion, that the central-symmetric caseh=0 qualitatively corresponds to some real
situation, where the NC has been highly doped with donors or acceptors. It is
natural to assume a homogeneous impurity distribution within the nanocrystal in
this case. Obviously, such a homogeneous density of charge induces a spherically
symmetric electric field that cannot mix the s- and p-type states. Moreover,
the relative Coulomb shift δV remains positive for donors and negative for
acceptors, as in the above case of a single impurity center in the dot [73]. Therefore, one can expect similar
behavior of τR−1 for highly doped NCs.


## 3.3.3. Nonradiative Recombination

In parallel with radiative transitions, nonradiative processes occur in
silicon nanocrystals. As a rule, nonradiative processes play a problematic role
in silicon crystallites, due to their high recombination rates which
substantially exceed the rates of the radiative transitions. Among the possible
nonradiative processes, capture on dangling bonds and Auger recombination are the
most efficient. Both processes are phonon-assisted, possibly even multi-phonon.
The first one takes place both at low and high pumping levels, while Auger
recombination becomes possible exclusively at high excitation power when more
than one excited electron (hole) exist in the conduction (valence) band of the
nanocrystal.The capture on a neutral dangling bond has been studied earlier by TB
method [100, 106]. It was shown that excited
electron-hole pairs can recombine on a neutral dangling bond in two stages. First,
the electron (hole) is trapped on a neutral Pb center making it
negatively (positively) charged. Then, the hole (electron) is trapped on the
charged dangling bond. According to estimations [100], the capture rate on the neutral
dangling bond for holes is always much greater than that for electrons. In
turn, the electron capture rate at room temperature increases sharply as the
nanocrystal gap decreases from 2.7 eV to 1.5 eV (see (23)). Within
the same range of energy gaps, the hole capture rate also increases but less
sharply. We should emphasize the presence of two main features of the capture
process. First, there is a very stron dependence of the capture rates (even for
holes) on the energy gap, that is, the NC size.
The second feature is the overall high capture rates, which are much
greater than the radiative decay rates. Evidently, the presence of a dangling
bond in silicon crystallite drastically reduces the quantum efficiency of
photon generation. Therefore the passivation of dangling bonds is required for
achievement of relatively efficient luminescence.Let us now touch briefly on the role of SiO defects in the
photoluminescence. The calculations of Nishida [102] showed that the oscillator strength
is almost independent of the existence of Si–O–Si bonds. In contrast, the Si=O
bonds reduce the oscillator strength by several orders of magnitude. Recently,
however, Sa'ar et al. [64] demonstrated experimentally that
both back-bonded and double-bonded oxygen atoms can enhance the PL intensity
due to formation of coupled states of electrons (holes) and SiO vibrations.
Such vibrations induce electric polarization fields generating long-lived
coupled states named “vibrons”. Usually, the surface vibrations scatter the
carriers; however, vibrons, being coupled states, do not scatter the carriers
and thereby exclude one of possible nonradiative channels from the
recombination process.For highly excited
nanocrystals, Auger recombination may take place. It is sufficient to have only
three particles (two electrons and a hole, or vice versa) for this process to
occur. It is well known that Auger recombination has high efficiency in bulk
silicon. Therefore, it is natural to expect a similar behavior in silicon
crystallites. Calculations of the Auger-recombination rate in crystallites
carried out with TB [58, 152] and EFA [153] techniques confirm, indeed, the fast character of this mechanism. In
particular, the characteristic times have been found to be of the order
0.1 to 10 ns−1 and gradually rise with decreasing
nanocrystal size. Thus, Auger recombination, as well as dangling bonds, can
strongly quench the luminescence in silicon crystallites, as confirmed
experimentally [152].


## 4. Ensembles of Quantum Dots

### 4.1. Size Distributions

As discussed in detail in the
previous sections, the recombination energy for Si NCs is dependent on the NC
size. In order to achieve optical gain and stimulated emission, therefore, a
narrow size distribution would be beneficial in order to narrow the gain
profile. In ensembles of silicon nanocrystals grown by various thin film
methods or by ion implantation, it is so far not possible to obtain the narrow
size distributions that can be achieved with chemical methods, where
post-synthesis size selection methods are routinely employed [154]. However, there is hope for better size selectivity, as will be
discussed toward the end of this section.The nucleation and growth
behavior of nanocrystals in ion implanted systems, in particular, has been
extensively characterized [155, 156]. The lognormal size
distribution results in general when multiple microscopic processes govern the
nucleation and growth kinetics, as is typical in these systems. The lognormal
distribution is found even in cases where the “memory” of the specific initial
conditions is lost, for example, after annealing of ion implanted samples [156].
In such cases, the nanocrystal size
distribution is governed by the standard lognormal curve:(45)P(x)=1Sx2πexp[−(ln(x)−M)22S2], where the mean and variance can
be related to the M and S parameters by μ=exp(M+S2/2) and var=exp(S2+2M)[exp(S2)−1].For thin films also, a
lognormal distribution is predicted on the basis of random nucleation and
growth in a homogeneous medium, regardless of the growth temperature, specific
method of crystallization, and the mean grain size [157, 158], which is consistent with previous theories on cluster growth in metals
and ceramics [159]. Lognormal distributions in the case of silicon grains crystallizing in
amorphous silicon have been reported [157],
and the lognormal size distribution
has also been observed in silicon NCs over the range of annealing temperatures
from 400 to 1100°C [47, 160]. Figure 16 shows
the size distributions obtained from TEM data for two samples available in the
literature. In Figure  16(a), the data from
Vinciguerra et al. [161] are reproduced along with a Gaussian least squares fit (μradius=1.68 nm, σ=0.65 nm) which is similar to
the fit in the original publication (quoted as μ=1.7±0.6 nm), and a lognormal
fit with S=0.27 and M=0.54). The data look decidedly lognormal with a skew toward large radii; the lognormal fitting
matches the data better on both the high and low tails of the distribution, has
a better correlation (R2=0.94 versus 0.86), and is naturally equal to
zero for negative radii. Figure 16(b)
shows the original data from Glover and Meldrum [162], along with a
lognormal fit with S=0.29, M=0.22. This latter data and fit will be used in
the model in the following subsections. Obviously, the exact numbers obtained
for M and S depend on the histogram binning and on the quantity of the data.(a) Experimental size histogram from [161], with a Gaussian fit (blue line) and a lognormal fit (red line). Panel (b) shows the data from [162] with a lognormal fit.
(a)(b)Size distributions can
potentially be different than lognormal whenever the nucleation and growth
processes are not random but are controlled in some way, such as via
irradiation-induced nucleation [163], or through nonuniformities in the concentration of silicon, or when
size selection is built into in the process as, for example, in the formation
of silicon nanocrystals by laser pyrolysis [164], or when size selection is done after crystal formation in solution, or
when there is a mechanism for producing nonuniform (i.e., bimodal) distributions. Nevertheless, in numerous studies
of silicon nanocrystals a Gaussian distribution is assumed. It can be mentioned
that in most cases where a Gaussian fit is used to model the silicon NC size
distribution, there is in fact no theoretical basis for it. In fact, it can predict
unphysical results (e.g., the
Gaussian function does not go to zero when its argument—the radius or diameter—is negative).The lognormal function should,
therefore, be used in cases except those in which there are reasons
to expect a different distribution. Since there is a strong dependence of the
bandgap energy on the cluster size, accurate modeling of the behavior of
ensembles of Si NCs does imply that the size distributions should be well known
and correctly modeled. For modeling luminescent properties of ensembles of
silicon nanocrystals, as discussed below, the lognormal distribution will be
used in every case.


### 4.2. Pl Spectra and Dynamics

Silicon NCs are characterized
by a broad emission spectrum typically peaked between 800 and 900 nm, with a
full-width-at-half-maximum (FWHM) of as much as 200 nm, even in “monodispersed”
multilayered samples [165]. Although the size
distributions for the monodispersed samples in [165]
were not actually reported, it
seems likely that they would nevertheless be much wider than typically obtained
for CdSe NCs due to the lack of post-synthesis size selection methods. Therefore,
the broad emission band could simply be due to the range of confined carrier
energies in clusters of different sizes, or to the random nature of the
interface states that may affect the PL spectrum. Single quantum dots may show
homogeneous broadening up to ~150 meV at room temperature [19],
although this is still narrower
than the overall inhomogeneous broadening due to the size distribution. Below,
we will investigate the spectral effects of broadening and nonradiative defects
on the emission spectrum of an ensemble of Si NCs.In order to investigate the
theoretical luminescence spectrum and dynamics, the NC size distribution for
SiO films annealed at 1000°C from [47] (lognormal radius parameters S=0.21, M=0.74) was used along with the
theory developed in Sections 1–3. First,
the effect of inhomogeneous broadening was investigated by plotting one
spectrum in which the energies of 106 NCs produced with a lognormal
probability function were binned in histogram form. In this case, for
simplicity the NCs were treated as isolated, and energy transfer between them
was not permitted (purple curve, Figure 17). The effect of homogeneous broadening was
determined by assigning each cluster a Gaussian range of emission energies with
standard deviations ranging from 0 to 200 meV and adding all the resulting
curves (Figure  17). As the homogeneous
broadening increases, the emission spectrum became broader, effectively similar
to a “Gaussian smoothening” of the spectrum. Additionally, the spectrum was observably
wider even for small levels of homogeneous broadening, and the peak shifts to
slightly shorter wavelengths due to the overall shape of the lognormal function.
The spectrum maximizes at shorter wavelengths than actually observed for a real
specimen with this size distribution because NC-NC interactions were not yet
turned on in the simulation. At this stage, the intent is simply to demonstrate
the effect of homogeneous broadening on the observed PL spectrum. One may
furthermore propose that the term “monodisperse” may only be used rigorously in
situations in which the width of the PL spectra is governed primarily by
homogeneous rather than inhomogeneous spectral broadening.Figure 17
Effect of homogeneous broadening on the simulated (normalized) PL spectrum. The FWHM of the Gaussian homogeneous broadening is shown in the legend.Next, the effect of nonradiative
interface defects on the emission spectra and dynamics was estimated. Two size
distributions corresponding to those in [47] (S=0.21; M=0.74) and [162] (S=0.29; M=0.22) were used, and
for the present evaluation NC-NC energy transfer was not permitted. Several
different concentrations of nonradiative defects ranging from 0 to 1020 cm−3 were assigned to the clusters with probabilities weighted
proportionally to the surface areas. Clusters were checked in random order and
a defect was assigned to a cluster if the random number was smaller than ANC/Aall NCs.
The process was continued until all defects had been assigned, allowing for
multiple defects on a single particle (Figure 18). Every cluster was then given an
electron-hole pair with a lowest excited-state energy calculated according to (22).
The radiative and nonradiative rates were calculated from (42) and (23),
respectively. Using a simple Monte Carlo procedure, the probabilities of
radiative and nonradiative decay were calculated for each NC and a random
number generator used to determine whether either type of decay occurred. This
process continued until no excited carriers remained.Figure 18
Left: Plot of a simulation box, 4000 NCs, initial state. Dark clusters contain one or more defects. Right: multilayered sample in a time window between 50–100μs after the end of the pump pulse. Yellow NCs had a radiative recombination event during this time period and white NCs have no charge carriers remaining. Interaction effects were turned off for these simulations. The orange NC in the lower left corner of both images is 5 nm in diameter.The resulting PL spectra were
characterized by an asymmetric curve shape, tailing toward higher energies due
to the form of the equations leading up to (22) (Figure 19). The effect of various concentrations of
nonradiative surface defects is clearly observable in the simulated PL spectra.
In addition to decreasing the overall PL intensity, the effect of the nonradiative
centers is to blueshift the peak wavelength. The spectral blueshift occurs
because the larger clusters are statistically more likely to contain defects,
and the effect would be even stronger if defects were assigned with a volume
rather than surface-area weighting (e.g.,
for volume defects). If extended annealing periods can remove nonradiative
centers without increasing the NC size [166], then the effect would be to cause an increased intensity and a redshift
of the PL peak, exactly as reported experimentally [53]. This redshift is not due to
increasing cluster size, but is due to the removal of nonradiative traps.  Finally, the high-energy skewness of the
simulated spectra is due to a combination of effects, including the lognormal
size distribution, and the lack of carrier transfer from small to large
clusters.PL spectra as a function of surface defect concentration for non-interacting NCs. The spectra in the top panel correspond toμradius=2.1 nm, varradius=0.21 nm2; the bottom panel is for μradius=1.3 nm, σradius=0.38 nm. The legends show the defect concentration in cm−3.
(a)(b)As discussed in Section1, PL
decay dynamics in ensembles of Si NCs are characterized by the “stretched
exponential” function It/I0=exp[−(t/τ)β], with values of
the exponent β often
between 0.7-0.8. The stretched exponential has been suggested to be due to a hopping
mechanism in which excitons are temporarily trapped and delayed at sites in the
oxide matrix or at the cluster-oxide interfaces [32, 167, 168]. However, the
stretched exponential decay could also be due to the distribution of lifetimes
in NCs of different sizes, as has been indicated by a few groups [35, 169]. In the absence of
radiative centers, the recombination energy is given by (22), and the
size-dependent radiative rates were developed in (42)). Here we found that when
the defect concentration was set to zero (uppermost line in Figure 20), values of β=0.58 and τ=33 μs emerged naturally. The small lifetime is due to the small size of the
clusters used in this simulation. Also, since interactions were not enabled,
there is no energy transfer from small to large clusters which would also
affect the time constant (and the value of β). Including
defects changed these values considerably: with increasing defect
concentration, the decays are more precipitous (Figure  20). The stretched exponential function with
low β values can, therefore, be
obtained naturally as a result of the lifetime distribution. Approximately
100-microsecond-timescale carrier hopping, as suggested in previous work [168] as an explanation for the stretched
exponential decay, while not ruled out by this analysis, is not required in
order to generate the stretched exponential dynamics, a conclusion that is in
agreement with recent experimental results [169].Figure 20
Simulated PL decays for a specimen withμradius=1.3 nm, varradius=0.15 nm2. The lines correspond to defect concentrations of 0, 1019 cm−3, 1020 cm−3. Interactions were not permitted, which minizes the effect of defects.This is not meant to imply that
energy transfer does not occur—in fact, it certainly does occur in most
samples as discussed below—but that the shape of the decay curve is in
fact governed by the size distribution even in the presence of energy transfer.
However, energy transfer will also affect the shape of the decay curve—in the absence of nonradiative processes
shifting the decay to longer lifetimes and largerβ. In fact, in the
absence of nonradiative decay, it is in theory possible to work back from the
observed decay to the lifetime distribution using the inverse Laplace transform
for the stretched exponential function [170].
From the lifetime distribution one could in principle then work back to the
size distribution if τpl≈τrad
and the radiative rate model is accurate. If nonradiative processes are
included, the values of β and τ are sensitive to the size distribution,
filling fraction, and defect concentration. In the discussion below, the first
interaction simulation including Forster transfer and tunneling will be
performed and its effect on the PL spectrum investigated.


### 4.3. Interactions

In order to achieve better size
control, several studies have formed silicon nanocrystals using a thin film
multilayering approach. Instead of depositing a single thick layer of SiOx,
alternating thin layers of SiOx and SiO2 were used. In
this way, the thin SiOx layers are isolated from one another by the
SiO2 buffer layers. Upon annealing, silicon clusters crystallize in
the SiOx layers, but the cluster growth occurs only by diffusion
within the layer as opposed to three-dimensionally. Although the resulting
cluster sizes should clearly be smaller than for three-dimensional diffusion in a thick layer of the same composition,
it is not immediately clear without more detailed modeling whether the
distribution should, in theory, be more narrowly distributed with respect to
the mean value. However, narrower distributions have been reported in many
cases [171]. Nevertheless, even in apparently monodispersed multilayered samples [165] the PL peak is nearly 200 nm wide
at the half maximum.Through investigating the
effect of multilayering of the Si NCs on the luminescence spectrum, it has been
observed that the PL spectrum and dynamics from silicon nanocrystals depends on
the thickness of both the SiO2 buffer layers and the active
nanocrystal layers, even if the size distributions remain approximately
constant [23, 32, 162]. This is evidence of an interaction
mechanism operating between the clusters: as the NCs become more isolated—in one direction at least—by the buffer layers, the PL peak shifts to
shorter wavelengths and there is a change in the decay dynamics as well, with a
possible trend toward higher values of the exponent β. As already discussed, this has been widely interpreted in terms of a
kind of excitonic hopping or migration (e.g., see [23]) between clusters, although only
more recently have the mechanisms for carrier transfer among Si NCs been more
clearly elucidated [30, 160].There are at least two
fundamental means by which closely spaced nanocrystals can exchange charge
carriers. The first mechanism is via tunneling of individual electrons or holes
from one cluster to another. If the two clusters are approximately the same
size (or if the higher levels of a larger cluster overlap with lower levels of
a smaller one), then the tunneling is resonant, it is equally probable in
either direction, and its rate depends on the negative exponent of the
separation distance. Tunneling is well known to occur in silicon clusters and
forms the basis of silicon nanocrystal memories [172, 173]. Accurate calculation of the tunneling rate in ensembles of nanocrystals
with different energies and spacings is a difficult problem, however. The
tunneling rate can be approximated by the Wentzel-Kramers-Brouillin (WKB)
approximation, as has been established for the case of double quantum wells [174–176]:(46)wt=1ttunnel=(12dp)(2Emp)1/216E(V−E)(mp/mb)[V+(mp/mb−1)E]2×exp[−2db(2mb(V−E)ℏ2)1/2], where mb and mp are the carrier effective masses in the barrier and the particle, dp and db
are the particle diameter and barrier thickness, V is the barrier height, and E is the carrier energy. This equation
(using the corrected form shown in [160]) has previously been employed to
estimate tunneling rates between asymmetric quantum wells [176], despite the fact that (46) is
strictly applicable only to the resonant case.If the energy levels of
adjacent clusters are mismatched, then tunneling must be “assisted” by emission
or absorption of phonons. The tunneling rate dependence on the energy gap,ΔE, and the phonon energies (within the clusters
and in the SiO2 matrix) for these mechanisms is not well known and
may need to be estimated experimentally if sufficiently narrow size
distributions can be obtained. Recently, computer simulations showed [160] that a large Stokes shift between
absorption and emission in the specimen as a whole can occur as a result of
tunneling between nanocrystals although, as for the quantum well case, (46) was
used without accounting for phonon-assisted tunneling.The second fundamental
interaction mechanism is resonance energy transfer, or “Forster transfer” [177]. In this mechanism, electron-hole pairs may “migrate”
from particle to particle by a dipole-dipole or higher-order multipole coupling,
that is, the Forster process—which has already been established for the
case of CdSe nanocrystals [178]. Others have shown
that it applies to silicon nanocrystals also [30],
although due to the indirect gap
the rates can be much different (typically smaller) than for CdSe. Forster
transfer is the basis for fluorescence resonance energy transfer (FRET)
microscopy, and has been widely investigated in the case of interactions
amongst the rare earth ions [179]. The distance dependence of the transition dipole transfer rate, wd, is given by wd=wPL(R0/r)6,
where wPL
is the PL rate, R0 is the distance at which
the transfer rate and the PL rate are equal, and r is the distance between the edges of the particles. R0 itself depends on the
spectral overlap integral between the donor (D) and acceptor (A) clusters, and
is, therefore, technically different for any pair of nanocrystals undergoing
this process. Previous work in CdSe nanocrystals has assumed that the energy
levels must be resonant for the Forster transfer to occur [178], however this is not strictly true:
phonon-assisted Forster transfer has been calculated for the rare earth ions
and should apply to nanocrystals as well. In the Miyakawa-Dexter theory [180], the resonant
Forster rate is modified by considering the electron-phonon coupling parameters
in the density-of-states function for the donor and activator ions. In this
way, one obtains [180]:(47)wdip=wpl(R0r)6exp(−βΔE), where β depends on the electron-phonon coupling strength,
and its value can be extracted from a plot of the transfer rate versus ΔE in silicate glass. From the data provided in
[181], a rough estimate gives α=0.0018 for rare earth ions
embedded in SiO2, where α indicates internal transitions
and is about twice the value of β [180]. Unfortunately, the value of β is much more difficult to obtain
experimentally for silicon NCs, as a result of the size distributions (i.e., every NC is slightly different;
whereas all rare earth ions of the same type have identical properties).The Miyakawa-Dexter model has
been widely discussed and applied extensively in the case of transfer between
different rare earth ions [182–184]. In the present case, one may assume that the energy difference is lost
to phonons in the surrounding silicate glass, as would be the case for isolated
“point” dipoles such as the rare earth ions. As the energy gap grows, the
transfer rate decreases exponentially, and experimentally, for rare earth ions
at least, this trend shows little variation despite the many simplifications in
the Miyakawa-Dexter model [185]. The effective Forster distance, R0,
is more difficult to obtain, since it is not practical to calculate the overlap
integral (including phonon sidebands) for every possible pair of interacting
particles in a sample. Therefore, an experimental approach is taken to
approximate a single R0 value
for all pairs of particles. By varying the distance between layers of
interacting particles and using the geometrical constraints imposed by
interacting layers (as opposed to two point dipoles) [186], an estimated value of R0=5 nm  was used in the simulations. Obviously, this is a
significant simplification and it must be stressed that the results remain
approximate until phonon-assisted multipole transfer between all possible pairs
of NCs can be calculated in a reasonable way.The simulation setup was done
exactly as described previously: every particle was populated with a single
electron-hole pair at timet=0. This time, the effect of nonradiative traps
was not included in order to show only the energy migration effects. Next, for
every nanocluster the probabilities associated with each of the four processes
described above (radiative decay, nonradiative decay, tunneling, and Forster
transfer) were calculated in random order (the NCs themselves are also checked
randomly) and compared to a flat-probability random number between 0 and 1 to
determine whether one of these four events occurred. When the calculation
started, the simulation timestep was 1×10−12 s. After iterating
through all the particles, the time increments by one picosecond and the
process repeats. If ten complete iterations through all the clusters (in
randomized order each time) produce no events of any kind, the timestep is
multiplied by a factor of ten. In this way, it is possible to overcome the
otherwise impossible computational limits associated with the vastly different
rates for the different processes. The simulation ends when there are no
electron-hole pairs left. When a
particle radiates, the energy of the outgoing photon is stored for the spectral
output function, and the time is recorded in order to plot the decay of the
luminescence. Finally, a set of tests were performed to find the minimum sample
size for which edge effects (due to the finite box size) became unobservable. A
simulation of 5,000 particles was found to provide sufficiently short
computation time (~12 hours on a 2.6 GHz CPU) and no observable edge effects.
Auger effects were not simulated, which effectively approximates the behavior
of a lightly pumped ensemble.Figure21 shows the PL spectrum
with and without the Forster and tunneling “energy migration” interactions. In
the first few nanoseconds, numerous tunneling events and dipole-dipole
interactions occurred. The rates for these processes for adjacent nanocrystals
are many orders of magnitude faster than the radiative decay processes. Some nonradiative
trapping can also occur early on in the simulation. Even after only a few picoseconds, there was
already a small fraction of charged nanocrystals due to electron or hole
tunneling events. Many nanocrystals had no remaining charge carriers, and many
others—predominantly the larger ones—had multiple electron-hole pairs. After 500 ns, many thousands of tunneling and dipole events occurred and most of the
smaller clusters were devoid of charge carriers. In the microsecond timescale,
radiative decay joins the longer-range energy transfer events, and by 100
microseconds only a few excited nanocrystals remained. The long wavelength of
the emitting states (compared to absorption)—as well as a decrease in the overall skewness
of the PL peak—in dense ensembles of Si NCs can originate
predominantly from energy migration, indicating that the large Stokes shift in
ensembles of Si NCs can be attributed, at least in part, to these effects. We
have also found that the degree of “stretching” of the PL decay depends on the
resulting ensemble of particles with electron-hole pairs; in the absence of nonradiative
effects the PL lifetime is longer but if defects are introduced the PL lifetime
can be dramatically shorter in the presence of energy migration, due to the
greater statistical probability for defects to occur on larger NCs. Based on
the initial results presented here, it would seem that the combination of
theory with computational simulations may be one of the best tools to obtain a
more thorough understanding of the complicated and controversial luminescence
behavior of ensembles of silicon clusters.Figure 21
Effect of carrier migration on the PL histogram of a silicon NC ensemble with a lognormal distributionS=0.21, M=0.74, and volume fill = 35%. The PL spectrum is strongly redshifted as a result of tunneling and Forster transfer in this densely packed ensemble. For this simulation, there were no non-radiative defects and infinte barriers were assumed. The “after interactions” result agrees reasonably well with the observed CW PL from a specimen with the same size distribution and volume fill [47].At this point, although
instructive, the simulations have obvious limitations that we are currently
working to overcome. For example, we have not incorporated the effects of Si=O
surface bonds, which again may cause the simulated spectra to peak at wavelengths
slightly too short, especially for smaller NCs. An obvious limitation of the
model is the estimate of a single value ofR0 for all Forster interactions. The fact that phonon-assisted Forster processes
among silicon NCs remain beyond theoretical scope [30], required a further assumption that the phonon-assisted rate had a
similar dependence on ΔE as for the case for rare-earth ions in silicate glass. Higher-order
multipole interactions were also ignored. Finally, unlike the case in which infinite barriers were assumed [160], here the resulting spectra are narrower and redden than observed experimentally.Nevertheless, the implications
of these simulations are clear. First, the two energy transfer mechanisms
dominate the PL spectrum when the clusters are separated by less than 2 nm, and
can still be significant in the case of Forster transfer over distances of 5 nm
or more. Therefore, in most of the experimental work reported to date, NC-NC
interactions strongly affect the PL spectrum. Multilayer simulations such as
that shown in Figure18 have also been
performed, and the results are consistent with the experimentally observed
spectral blueshift and increased lifetime in multilayered samples. The decay
exponent β shows a more complicated behavior
however; its value depends on the distribution of lifetimes which in turn
depends on the size distribution and the nonradiative defect concentration.
Next, in most samples the largest clusters act as depressions in the energy
landscape and can attract numerous charge carriers. Under high pump rates,
therefore, the Auger problem can become significant when the clusters are
reasonably closely separated. Next, the simulations can easily be extended to
model the sensitizing action of the NCs on erbium ions; this work is currently
ongoing and uses the Forster mechanism of energy transfer. The results seem
consistent with the reports of optical gain (ASE) in very short timescales
before significant amount of energy transfer can occur (as long as the size
distribution is sufficiently narrow to have a narrow enough gain profile) but
for CW stimulated emission the NCs must be sufficiently well separated to
prevent carrier migration and buildup on the largest clusters. The largest NCs
are also the ones most likely to contain a defect, so their effects on the
emission spectrum and gain profile are extremely strong in most samples where
carrier interactions occur. Therefore, in the final section of this paper we briefly
discuss the controversial observations of optical gain in silicon nanocrystals,
and the implications of the theoretical results in Sections 1–3 and the
simulations of Section 4 for the development of photonic devices relating to
stimulated emission in silicon nanocrystals.


### 4.4. Stimulated Emission in Si NCs

There have been numerous recent
reports of optical gain in silicon nanocrystals [165, 187–189],
and stimulated emission was
reported in [190], although to our knowledge the results are not fully understood. Many
of these studies deduced the presence of gain by analyzing data from the variable
strip length (VSL) technique [191]. This method has been discussed in great detail in the literature, owing
to the many possible sources of error. In general, a planar waveguide film
containing silicon nanocrystals is optically pumped from above (parallel to the
specimen surface normal). The pump laser beam is first incident on a
variable-length slit, such that only a diffraction-limited strip of the sample
is pumped. The waveguided PL signal is then collected from the edge of the
specimen in the direction parallel to the axis of the excitation strip. The
signature of optical gain is a superlinear increase in the PL intensity (marked
by a distinct transition) for constant pump flux as the strip is lengthened.
This is due to the stimulated emission from excited nanocrystals interacting
with photons originally emitted at the farther end of the excitation strip.
This method, along with some corrections, has been one of the most widely used methods
to support optical gain in Si NCs. The other method that has been used,
although somewhat less extensively, is the standard pump-probe technique. In
that case, the transmission of a probe beam is measured as a function of the
probe power, with the transition from loss to gain marked by a shift from
positive to negative absorption of the pump. This method has been used both for
Si NC waveguides and in single-pass measurements.The presence (or absence) of SE
in Si NCs is a subject of much debate. After the initial report [5], which used both the VSL and the
single-pass pump-probe methods to report a net modal gain for a silicon
nanocrystal film of 100 cm−1, a net material gain of 104 cm−1, and a gain cross section per NC of ~10−16 cm2,
numerous groups have searched for optical gain in ensembles of Si NCs. The
results of these investigations have not been entirely consistent. Currently,
there seems to be some possible evidence in favor of SE in an Auger-limited
time window of a few ns after intense pulsed excitation, with gain cross
sections per NC of ~10−16 cm2 in the extreme short
wavelength side of the broad emission band [189]. However, in comparison to the case
for CdSe NCs, where there is clear and unequivocal observations of SE (e.g., marked by a sudden superlinear increase of PL intensity with pumping
power in a narrow wavelength range), in the case of Si NCs the signatures of SE
have been extracted from fairly elaborate and complicated experiments, and the
results have shown significant specimen-to-specimen variation. A
brief examination of the limiting factors and key specimen requirements for SE
in ensembles of silicon NCs would be of use with the ultimate objective
demonstrating, unequivocally, stimulated emission in nanocrystalline silicon.Gain Profile
The calculations discussed in
Sections1–3 indicate several key factors relating to optical gain in Si NCs.
First, for size distributions produced by standard physical vapor deposition or
ion implantation methods (including multi-layering) the gain profile becomes
extremely wide. This reduces the gain coefficient and the gain cross section
per NC in the sample. Dal Negro et al. [187], for example, reported a gain
coefficient of 1600 ± 300 cm−1 at λ=750 nm, with a
gain cross section at this wavelength of 3×10−16 cm2 per
NC. Since the overall PL spectra generally peak at longer wavelengths, only a
small fraction of the NCs may actually emit within the reported gain window.
Most of the nanocrystals, therefore, may not contribute to SE and are in fact,
detrimental as a result of confined carrier absorption, scattering, and other
loss mechanisms they can cause. If Si NCs could be prepared with a size
distribution similar to that for CdSe (e.g., 5% standard deviation), the
theoretical gain coefficient per nanocrystal in the specimen as a whole could
increase by a fairly large factor, simply because a greater number of NCs could
contribute to the overall gain at the peak wavelength. Therefore, true
monodispersivity (i.e., overall
emission profile controlled by homogeneous broadening) is likely to be a
requirement for producing SE in ensembles of Si NCs.
Secondly, the size distribution
presents an inherent problem for laser excitation of the specimen. With the
exception of erbium doped waveguides that used LED excitation [192],
SE or gain is reported as a result of pumping the sample with a laser. Laser
excitation implies a narrow excitation energy window, and only NCs that happen
to have a transition within that window will be absorbing. In a given sample
with a wide distribution, therefore, many of the NCs may not absorb and cannot
contribute to the overall specimen gain. Furthermore, these “unexcited” NCs can
enhance the energy migration or other undesired effects discussed above.
The cluster-cluster energy
transfer mechanisms found to be important in the overall luminescence spectrum
also have implications for the presence of SE in ensembles of Si NCs. Generally, a high-pump rate is necessary to
observe the transition from loss to gain. Under such high rates, one can assume
a relatively large number of excitons per NC (we hesitate somewhat to suggest a
number as high as 100) [5] but at least 1 exciton per NC is readily
possible. In this case, charge carriers will be rapidly redistributed through
the tunneling and Forster mechanisms, on the sub-nanosecond time scale for a typical
volume filling fraction of NCs (the volume fraction ultimately controls the average
NC-NC spacing). Therefore, in a short time after the initial pump pulse, one
has the accumulation of charge carriers on the largest NCs as a result of
tunneling and Forster interactions. This drastically decreases the number of
NCs that can contribute to the gain spectrum and can lead to enhanced nonradiative
trapping and undesirable Auger interactions. This may be one reason that gain
may only occur in very short timescales, as suggested in several recent
experimental works [189], and only in certain specimens but not in others. In such cases, gain may
only build up before the energy migration processes have occurred, and will be
very susceptible to the average NC-NC spacing (and size distribution, as above)
from specimen to specimen. The fact that the gain, if present, occurs on the
short-wavelength side of the PL band is also consistent with the idea that the
gain occurs before the charge carriers transfer to the larger and “redder”
nanoclusters. By increasing the NC-NC spacing it should be possible
to limit the energy migration problem, due to the strong distance dependence of
the main processes. This may be accomplished by, for example, freeing the Si
NCs via an HF etch and redepositing them at a known (low) filling fraction, an
idea we are currently exploring. Other methods are also possible.
Additionally, the level
degeneracies calculated in Section3 imply a further key limiting factor for SE
in silicon nanocrystals. With a two-fold degeneracy or more, more than 1 exciton
per NC must be present, on average, in order to switch from loss to gain, which
in turn raises the problem of the Auger interactions. This number becomes
higher as the degeneracy of the electron and hole levels increases. In the case
of CdSe NCs, Klimov et al. [193]
have shown that one way to avoid this problem is to separate electrons and
holes in a core-shell nanostructure, thereby creating a large crystal-field
shift of the absorption spectrum. Such problems (and solutions) may well apply
to Si NCs also.


## 4.1. Size Distributions

As discussed in detail in the
previous sections, the recombination energy for Si NCs is dependent on the NC
size. In order to achieve optical gain and stimulated emission, therefore, a
narrow size distribution would be beneficial in order to narrow the gain
profile. In ensembles of silicon nanocrystals grown by various thin film
methods or by ion implantation, it is so far not possible to obtain the narrow
size distributions that can be achieved with chemical methods, where
post-synthesis size selection methods are routinely employed [154]. However, there is hope for better size selectivity, as will be
discussed toward the end of this section.The nucleation and growth
behavior of nanocrystals in ion implanted systems, in particular, has been
extensively characterized [155, 156]. The lognormal size
distribution results in general when multiple microscopic processes govern the
nucleation and growth kinetics, as is typical in these systems. The lognormal
distribution is found even in cases where the “memory” of the specific initial
conditions is lost, for example, after annealing of ion implanted samples [156].
In such cases, the nanocrystal size
distribution is governed by the standard lognormal curve:(45)P(x)=1Sx2πexp[−(ln(x)−M)22S2], where the mean and variance can
be related to the M and S parameters by μ=exp(M+S2/2) and var=exp(S2+2M)[exp(S2)−1].For thin films also, a
lognormal distribution is predicted on the basis of random nucleation and
growth in a homogeneous medium, regardless of the growth temperature, specific
method of crystallization, and the mean grain size [157, 158], which is consistent with previous theories on cluster growth in metals
and ceramics [159]. Lognormal distributions in the case of silicon grains crystallizing in
amorphous silicon have been reported [157],
and the lognormal size distribution
has also been observed in silicon NCs over the range of annealing temperatures
from 400 to 1100°C [47, 160]. Figure 16 shows
the size distributions obtained from TEM data for two samples available in the
literature. In Figure  16(a), the data from
Vinciguerra et al. [161] are reproduced along with a Gaussian least squares fit (μradius=1.68 nm, σ=0.65 nm) which is similar to
the fit in the original publication (quoted as μ=1.7±0.6 nm), and a lognormal
fit with S=0.27 and M=0.54). The data look decidedly lognormal with a skew toward large radii; the lognormal fitting
matches the data better on both the high and low tails of the distribution, has
a better correlation (R2=0.94 versus 0.86), and is naturally equal to
zero for negative radii. Figure 16(b)
shows the original data from Glover and Meldrum [162], along with a
lognormal fit with S=0.29, M=0.22. This latter data and fit will be used in
the model in the following subsections. Obviously, the exact numbers obtained
for M and S depend on the histogram binning and on the quantity of the data.(a) Experimental size histogram from [161], with a Gaussian fit (blue line) and a lognormal fit (red line). Panel (b) shows the data from [162] with a lognormal fit.
(a)(b)Size distributions can
potentially be different than lognormal whenever the nucleation and growth
processes are not random but are controlled in some way, such as via
irradiation-induced nucleation [163], or through nonuniformities in the concentration of silicon, or when
size selection is built into in the process as, for example, in the formation
of silicon nanocrystals by laser pyrolysis [164], or when size selection is done after crystal formation in solution, or
when there is a mechanism for producing nonuniform (i.e., bimodal) distributions. Nevertheless, in numerous studies
of silicon nanocrystals a Gaussian distribution is assumed. It can be mentioned
that in most cases where a Gaussian fit is used to model the silicon NC size
distribution, there is in fact no theoretical basis for it. In fact, it can predict
unphysical results (e.g., the
Gaussian function does not go to zero when its argument—the radius or diameter—is negative).The lognormal function should,
therefore, be used in cases except those in which there are reasons
to expect a different distribution. Since there is a strong dependence of the
bandgap energy on the cluster size, accurate modeling of the behavior of
ensembles of Si NCs does imply that the size distributions should be well known
and correctly modeled. For modeling luminescent properties of ensembles of
silicon nanocrystals, as discussed below, the lognormal distribution will be
used in every case.


## 4.2. Pl Spectra and Dynamics

Silicon NCs are characterized
by a broad emission spectrum typically peaked between 800 and 900 nm, with a
full-width-at-half-maximum (FWHM) of as much as 200 nm, even in “monodispersed”
multilayered samples [165]. Although the size
distributions for the monodispersed samples in [165]
were not actually reported, it
seems likely that they would nevertheless be much wider than typically obtained
for CdSe NCs due to the lack of post-synthesis size selection methods. Therefore,
the broad emission band could simply be due to the range of confined carrier
energies in clusters of different sizes, or to the random nature of the
interface states that may affect the PL spectrum. Single quantum dots may show
homogeneous broadening up to ~150 meV at room temperature [19],
although this is still narrower
than the overall inhomogeneous broadening due to the size distribution. Below,
we will investigate the spectral effects of broadening and nonradiative defects
on the emission spectrum of an ensemble of Si NCs.In order to investigate the
theoretical luminescence spectrum and dynamics, the NC size distribution for
SiO films annealed at 1000°C from [47] (lognormal radius parameters S=0.21, M=0.74) was used along with the
theory developed in Sections 1–3. First,
the effect of inhomogeneous broadening was investigated by plotting one
spectrum in which the energies of 106 NCs produced with a lognormal
probability function were binned in histogram form. In this case, for
simplicity the NCs were treated as isolated, and energy transfer between them
was not permitted (purple curve, Figure 17). The effect of homogeneous broadening was
determined by assigning each cluster a Gaussian range of emission energies with
standard deviations ranging from 0 to 200 meV and adding all the resulting
curves (Figure  17). As the homogeneous
broadening increases, the emission spectrum became broader, effectively similar
to a “Gaussian smoothening” of the spectrum. Additionally, the spectrum was observably
wider even for small levels of homogeneous broadening, and the peak shifts to
slightly shorter wavelengths due to the overall shape of the lognormal function.
The spectrum maximizes at shorter wavelengths than actually observed for a real
specimen with this size distribution because NC-NC interactions were not yet
turned on in the simulation. At this stage, the intent is simply to demonstrate
the effect of homogeneous broadening on the observed PL spectrum. One may
furthermore propose that the term “monodisperse” may only be used rigorously in
situations in which the width of the PL spectra is governed primarily by
homogeneous rather than inhomogeneous spectral broadening.Figure 17
Effect of homogeneous broadening on the simulated (normalized) PL spectrum. The FWHM of the Gaussian homogeneous broadening is shown in the legend.Next, the effect of nonradiative
interface defects on the emission spectra and dynamics was estimated. Two size
distributions corresponding to those in [47] (S=0.21; M=0.74) and [162] (S=0.29; M=0.22) were used, and
for the present evaluation NC-NC energy transfer was not permitted. Several
different concentrations of nonradiative defects ranging from 0 to 1020 cm−3 were assigned to the clusters with probabilities weighted
proportionally to the surface areas. Clusters were checked in random order and
a defect was assigned to a cluster if the random number was smaller than ANC/Aall NCs.
The process was continued until all defects had been assigned, allowing for
multiple defects on a single particle (Figure 18). Every cluster was then given an
electron-hole pair with a lowest excited-state energy calculated according to (22).
The radiative and nonradiative rates were calculated from (42) and (23),
respectively. Using a simple Monte Carlo procedure, the probabilities of
radiative and nonradiative decay were calculated for each NC and a random
number generator used to determine whether either type of decay occurred. This
process continued until no excited carriers remained.Figure 18
Left: Plot of a simulation box, 4000 NCs, initial state. Dark clusters contain one or more defects. Right: multilayered sample in a time window between 50–100μs after the end of the pump pulse. Yellow NCs had a radiative recombination event during this time period and white NCs have no charge carriers remaining. Interaction effects were turned off for these simulations. The orange NC in the lower left corner of both images is 5 nm in diameter.The resulting PL spectra were
characterized by an asymmetric curve shape, tailing toward higher energies due
to the form of the equations leading up to (22) (Figure 19). The effect of various concentrations of
nonradiative surface defects is clearly observable in the simulated PL spectra.
In addition to decreasing the overall PL intensity, the effect of the nonradiative
centers is to blueshift the peak wavelength. The spectral blueshift occurs
because the larger clusters are statistically more likely to contain defects,
and the effect would be even stronger if defects were assigned with a volume
rather than surface-area weighting (e.g.,
for volume defects). If extended annealing periods can remove nonradiative
centers without increasing the NC size [166], then the effect would be to cause an increased intensity and a redshift
of the PL peak, exactly as reported experimentally [53]. This redshift is not due to
increasing cluster size, but is due to the removal of nonradiative traps.  Finally, the high-energy skewness of the
simulated spectra is due to a combination of effects, including the lognormal
size distribution, and the lack of carrier transfer from small to large
clusters.PL spectra as a function of surface defect concentration for non-interacting NCs. The spectra in the top panel correspond toμradius=2.1 nm, varradius=0.21 nm2; the bottom panel is for μradius=1.3 nm, σradius=0.38 nm. The legends show the defect concentration in cm−3.
(a)(b)As discussed in Section1, PL
decay dynamics in ensembles of Si NCs are characterized by the “stretched
exponential” function It/I0=exp[−(t/τ)β], with values of
the exponent β often
between 0.7-0.8. The stretched exponential has been suggested to be due to a hopping
mechanism in which excitons are temporarily trapped and delayed at sites in the
oxide matrix or at the cluster-oxide interfaces [32, 167, 168]. However, the
stretched exponential decay could also be due to the distribution of lifetimes
in NCs of different sizes, as has been indicated by a few groups [35, 169]. In the absence of
radiative centers, the recombination energy is given by (22), and the
size-dependent radiative rates were developed in (42)). Here we found that when
the defect concentration was set to zero (uppermost line in Figure 20), values of β=0.58 and τ=33 μs emerged naturally. The small lifetime is due to the small size of the
clusters used in this simulation. Also, since interactions were not enabled,
there is no energy transfer from small to large clusters which would also
affect the time constant (and the value of β). Including
defects changed these values considerably: with increasing defect
concentration, the decays are more precipitous (Figure  20). The stretched exponential function with
low β values can, therefore, be
obtained naturally as a result of the lifetime distribution. Approximately
100-microsecond-timescale carrier hopping, as suggested in previous work [168] as an explanation for the stretched
exponential decay, while not ruled out by this analysis, is not required in
order to generate the stretched exponential dynamics, a conclusion that is in
agreement with recent experimental results [169].Figure 20
Simulated PL decays for a specimen withμradius=1.3 nm, varradius=0.15 nm2. The lines correspond to defect concentrations of 0, 1019 cm−3, 1020 cm−3. Interactions were not permitted, which minizes the effect of defects.This is not meant to imply that
energy transfer does not occur—in fact, it certainly does occur in most
samples as discussed below—but that the shape of the decay curve is in
fact governed by the size distribution even in the presence of energy transfer.
However, energy transfer will also affect the shape of the decay curve—in the absence of nonradiative processes
shifting the decay to longer lifetimes and largerβ. In fact, in the
absence of nonradiative decay, it is in theory possible to work back from the
observed decay to the lifetime distribution using the inverse Laplace transform
for the stretched exponential function [170].
From the lifetime distribution one could in principle then work back to the
size distribution if τpl≈τrad
and the radiative rate model is accurate. If nonradiative processes are
included, the values of β and τ are sensitive to the size distribution,
filling fraction, and defect concentration. In the discussion below, the first
interaction simulation including Forster transfer and tunneling will be
performed and its effect on the PL spectrum investigated.


## 4.3. Interactions

In order to achieve better size
control, several studies have formed silicon nanocrystals using a thin film
multilayering approach. Instead of depositing a single thick layer of SiOx,
alternating thin layers of SiOx and SiO2 were used. In
this way, the thin SiOx layers are isolated from one another by the
SiO2 buffer layers. Upon annealing, silicon clusters crystallize in
the SiOx layers, but the cluster growth occurs only by diffusion
within the layer as opposed to three-dimensionally. Although the resulting
cluster sizes should clearly be smaller than for three-dimensional diffusion in a thick layer of the same composition,
it is not immediately clear without more detailed modeling whether the
distribution should, in theory, be more narrowly distributed with respect to
the mean value. However, narrower distributions have been reported in many
cases [171]. Nevertheless, even in apparently monodispersed multilayered samples [165] the PL peak is nearly 200 nm wide
at the half maximum.Through investigating the
effect of multilayering of the Si NCs on the luminescence spectrum, it has been
observed that the PL spectrum and dynamics from silicon nanocrystals depends on
the thickness of both the SiO2 buffer layers and the active
nanocrystal layers, even if the size distributions remain approximately
constant [23, 32, 162]. This is evidence of an interaction
mechanism operating between the clusters: as the NCs become more isolated—in one direction at least—by the buffer layers, the PL peak shifts to
shorter wavelengths and there is a change in the decay dynamics as well, with a
possible trend toward higher values of the exponent β. As already discussed, this has been widely interpreted in terms of a
kind of excitonic hopping or migration (e.g., see [23]) between clusters, although only
more recently have the mechanisms for carrier transfer among Si NCs been more
clearly elucidated [30, 160].There are at least two
fundamental means by which closely spaced nanocrystals can exchange charge
carriers. The first mechanism is via tunneling of individual electrons or holes
from one cluster to another. If the two clusters are approximately the same
size (or if the higher levels of a larger cluster overlap with lower levels of
a smaller one), then the tunneling is resonant, it is equally probable in
either direction, and its rate depends on the negative exponent of the
separation distance. Tunneling is well known to occur in silicon clusters and
forms the basis of silicon nanocrystal memories [172, 173]. Accurate calculation of the tunneling rate in ensembles of nanocrystals
with different energies and spacings is a difficult problem, however. The
tunneling rate can be approximated by the Wentzel-Kramers-Brouillin (WKB)
approximation, as has been established for the case of double quantum wells [174–176]:(46)wt=1ttunnel=(12dp)(2Emp)1/216E(V−E)(mp/mb)[V+(mp/mb−1)E]2×exp[−2db(2mb(V−E)ℏ2)1/2], where mb and mp are the carrier effective masses in the barrier and the particle, dp and db
are the particle diameter and barrier thickness, V is the barrier height, and E is the carrier energy. This equation
(using the corrected form shown in [160]) has previously been employed to
estimate tunneling rates between asymmetric quantum wells [176], despite the fact that (46) is
strictly applicable only to the resonant case.If the energy levels of
adjacent clusters are mismatched, then tunneling must be “assisted” by emission
or absorption of phonons. The tunneling rate dependence on the energy gap,ΔE, and the phonon energies (within the clusters
and in the SiO2 matrix) for these mechanisms is not well known and
may need to be estimated experimentally if sufficiently narrow size
distributions can be obtained. Recently, computer simulations showed [160] that a large Stokes shift between
absorption and emission in the specimen as a whole can occur as a result of
tunneling between nanocrystals although, as for the quantum well case, (46) was
used without accounting for phonon-assisted tunneling.The second fundamental
interaction mechanism is resonance energy transfer, or “Forster transfer” [177]. In this mechanism, electron-hole pairs may “migrate”
from particle to particle by a dipole-dipole or higher-order multipole coupling,
that is, the Forster process—which has already been established for the
case of CdSe nanocrystals [178]. Others have shown
that it applies to silicon nanocrystals also [30],
although due to the indirect gap
the rates can be much different (typically smaller) than for CdSe. Forster
transfer is the basis for fluorescence resonance energy transfer (FRET)
microscopy, and has been widely investigated in the case of interactions
amongst the rare earth ions [179]. The distance dependence of the transition dipole transfer rate, wd, is given by wd=wPL(R0/r)6,
where wPL
is the PL rate, R0 is the distance at which
the transfer rate and the PL rate are equal, and r is the distance between the edges of the particles. R0 itself depends on the
spectral overlap integral between the donor (D) and acceptor (A) clusters, and
is, therefore, technically different for any pair of nanocrystals undergoing
this process. Previous work in CdSe nanocrystals has assumed that the energy
levels must be resonant for the Forster transfer to occur [178], however this is not strictly true:
phonon-assisted Forster transfer has been calculated for the rare earth ions
and should apply to nanocrystals as well. In the Miyakawa-Dexter theory [180], the resonant
Forster rate is modified by considering the electron-phonon coupling parameters
in the density-of-states function for the donor and activator ions. In this
way, one obtains [180]:(47)wdip=wpl(R0r)6exp(−βΔE), where β depends on the electron-phonon coupling strength,
and its value can be extracted from a plot of the transfer rate versus ΔE in silicate glass. From the data provided in
[181], a rough estimate gives α=0.0018 for rare earth ions
embedded in SiO2, where α indicates internal transitions
and is about twice the value of β [180]. Unfortunately, the value of β is much more difficult to obtain
experimentally for silicon NCs, as a result of the size distributions (i.e., every NC is slightly different;
whereas all rare earth ions of the same type have identical properties).The Miyakawa-Dexter model has
been widely discussed and applied extensively in the case of transfer between
different rare earth ions [182–184]. In the present case, one may assume that the energy difference is lost
to phonons in the surrounding silicate glass, as would be the case for isolated
“point” dipoles such as the rare earth ions. As the energy gap grows, the
transfer rate decreases exponentially, and experimentally, for rare earth ions
at least, this trend shows little variation despite the many simplifications in
the Miyakawa-Dexter model [185]. The effective Forster distance, R0,
is more difficult to obtain, since it is not practical to calculate the overlap
integral (including phonon sidebands) for every possible pair of interacting
particles in a sample. Therefore, an experimental approach is taken to
approximate a single R0 value
for all pairs of particles. By varying the distance between layers of
interacting particles and using the geometrical constraints imposed by
interacting layers (as opposed to two point dipoles) [186], an estimated value of R0=5 nm  was used in the simulations. Obviously, this is a
significant simplification and it must be stressed that the results remain
approximate until phonon-assisted multipole transfer between all possible pairs
of NCs can be calculated in a reasonable way.The simulation setup was done
exactly as described previously: every particle was populated with a single
electron-hole pair at timet=0. This time, the effect of nonradiative traps
was not included in order to show only the energy migration effects. Next, for
every nanocluster the probabilities associated with each of the four processes
described above (radiative decay, nonradiative decay, tunneling, and Forster
transfer) were calculated in random order (the NCs themselves are also checked
randomly) and compared to a flat-probability random number between 0 and 1 to
determine whether one of these four events occurred. When the calculation
started, the simulation timestep was 1×10−12 s. After iterating
through all the particles, the time increments by one picosecond and the
process repeats. If ten complete iterations through all the clusters (in
randomized order each time) produce no events of any kind, the timestep is
multiplied by a factor of ten. In this way, it is possible to overcome the
otherwise impossible computational limits associated with the vastly different
rates for the different processes. The simulation ends when there are no
electron-hole pairs left. When a
particle radiates, the energy of the outgoing photon is stored for the spectral
output function, and the time is recorded in order to plot the decay of the
luminescence. Finally, a set of tests were performed to find the minimum sample
size for which edge effects (due to the finite box size) became unobservable. A
simulation of 5,000 particles was found to provide sufficiently short
computation time (~12 hours on a 2.6 GHz CPU) and no observable edge effects.
Auger effects were not simulated, which effectively approximates the behavior
of a lightly pumped ensemble.Figure21 shows the PL spectrum
with and without the Forster and tunneling “energy migration” interactions. In
the first few nanoseconds, numerous tunneling events and dipole-dipole
interactions occurred. The rates for these processes for adjacent nanocrystals
are many orders of magnitude faster than the radiative decay processes. Some nonradiative
trapping can also occur early on in the simulation. Even after only a few picoseconds, there was
already a small fraction of charged nanocrystals due to electron or hole
tunneling events. Many nanocrystals had no remaining charge carriers, and many
others—predominantly the larger ones—had multiple electron-hole pairs. After 500 ns, many thousands of tunneling and dipole events occurred and most of the
smaller clusters were devoid of charge carriers. In the microsecond timescale,
radiative decay joins the longer-range energy transfer events, and by 100
microseconds only a few excited nanocrystals remained. The long wavelength of
the emitting states (compared to absorption)—as well as a decrease in the overall skewness
of the PL peak—in dense ensembles of Si NCs can originate
predominantly from energy migration, indicating that the large Stokes shift in
ensembles of Si NCs can be attributed, at least in part, to these effects. We
have also found that the degree of “stretching” of the PL decay depends on the
resulting ensemble of particles with electron-hole pairs; in the absence of nonradiative
effects the PL lifetime is longer but if defects are introduced the PL lifetime
can be dramatically shorter in the presence of energy migration, due to the
greater statistical probability for defects to occur on larger NCs. Based on
the initial results presented here, it would seem that the combination of
theory with computational simulations may be one of the best tools to obtain a
more thorough understanding of the complicated and controversial luminescence
behavior of ensembles of silicon clusters.Figure 21
Effect of carrier migration on the PL histogram of a silicon NC ensemble with a lognormal distributionS=0.21, M=0.74, and volume fill = 35%. The PL spectrum is strongly redshifted as a result of tunneling and Forster transfer in this densely packed ensemble. For this simulation, there were no non-radiative defects and infinte barriers were assumed. The “after interactions” result agrees reasonably well with the observed CW PL from a specimen with the same size distribution and volume fill [47].At this point, although
instructive, the simulations have obvious limitations that we are currently
working to overcome. For example, we have not incorporated the effects of Si=O
surface bonds, which again may cause the simulated spectra to peak at wavelengths
slightly too short, especially for smaller NCs. An obvious limitation of the
model is the estimate of a single value ofR0 for all Forster interactions. The fact that phonon-assisted Forster processes
among silicon NCs remain beyond theoretical scope [30], required a further assumption that the phonon-assisted rate had a
similar dependence on ΔE as for the case for rare-earth ions in silicate glass. Higher-order
multipole interactions were also ignored. Finally, unlike the case in which infinite barriers were assumed [160], here the resulting spectra are narrower and redden than observed experimentally.Nevertheless, the implications
of these simulations are clear. First, the two energy transfer mechanisms
dominate the PL spectrum when the clusters are separated by less than 2 nm, and
can still be significant in the case of Forster transfer over distances of 5 nm
or more. Therefore, in most of the experimental work reported to date, NC-NC
interactions strongly affect the PL spectrum. Multilayer simulations such as
that shown in Figure18 have also been
performed, and the results are consistent with the experimentally observed
spectral blueshift and increased lifetime in multilayered samples. The decay
exponent β shows a more complicated behavior
however; its value depends on the distribution of lifetimes which in turn
depends on the size distribution and the nonradiative defect concentration.
Next, in most samples the largest clusters act as depressions in the energy
landscape and can attract numerous charge carriers. Under high pump rates,
therefore, the Auger problem can become significant when the clusters are
reasonably closely separated. Next, the simulations can easily be extended to
model the sensitizing action of the NCs on erbium ions; this work is currently
ongoing and uses the Forster mechanism of energy transfer. The results seem
consistent with the reports of optical gain (ASE) in very short timescales
before significant amount of energy transfer can occur (as long as the size
distribution is sufficiently narrow to have a narrow enough gain profile) but
for CW stimulated emission the NCs must be sufficiently well separated to
prevent carrier migration and buildup on the largest clusters. The largest NCs
are also the ones most likely to contain a defect, so their effects on the
emission spectrum and gain profile are extremely strong in most samples where
carrier interactions occur. Therefore, in the final section of this paper we briefly
discuss the controversial observations of optical gain in silicon nanocrystals,
and the implications of the theoretical results in Sections 1–3 and the
simulations of Section 4 for the development of photonic devices relating to
stimulated emission in silicon nanocrystals.


## 4.4. Stimulated Emission in Si NCs

There have been numerous recent
reports of optical gain in silicon nanocrystals [165, 187–189],
and stimulated emission was
reported in [190], although to our knowledge the results are not fully understood. Many
of these studies deduced the presence of gain by analyzing data from the variable
strip length (VSL) technique [191]. This method has been discussed in great detail in the literature, owing
to the many possible sources of error. In general, a planar waveguide film
containing silicon nanocrystals is optically pumped from above (parallel to the
specimen surface normal). The pump laser beam is first incident on a
variable-length slit, such that only a diffraction-limited strip of the sample
is pumped. The waveguided PL signal is then collected from the edge of the
specimen in the direction parallel to the axis of the excitation strip. The
signature of optical gain is a superlinear increase in the PL intensity (marked
by a distinct transition) for constant pump flux as the strip is lengthened.
This is due to the stimulated emission from excited nanocrystals interacting
with photons originally emitted at the farther end of the excitation strip.
This method, along with some corrections, has been one of the most widely used methods
to support optical gain in Si NCs. The other method that has been used,
although somewhat less extensively, is the standard pump-probe technique. In
that case, the transmission of a probe beam is measured as a function of the
probe power, with the transition from loss to gain marked by a shift from
positive to negative absorption of the pump. This method has been used both for
Si NC waveguides and in single-pass measurements.The presence (or absence) of SE
in Si NCs is a subject of much debate. After the initial report [5], which used both the VSL and the
single-pass pump-probe methods to report a net modal gain for a silicon
nanocrystal film of 100 cm−1, a net material gain of 104 cm−1, and a gain cross section per NC of ~10−16 cm2,
numerous groups have searched for optical gain in ensembles of Si NCs. The
results of these investigations have not been entirely consistent. Currently,
there seems to be some possible evidence in favor of SE in an Auger-limited
time window of a few ns after intense pulsed excitation, with gain cross
sections per NC of ~10−16 cm2 in the extreme short
wavelength side of the broad emission band [189]. However, in comparison to the case
for CdSe NCs, where there is clear and unequivocal observations of SE (e.g., marked by a sudden superlinear increase of PL intensity with pumping
power in a narrow wavelength range), in the case of Si NCs the signatures of SE
have been extracted from fairly elaborate and complicated experiments, and the
results have shown significant specimen-to-specimen variation. A
brief examination of the limiting factors and key specimen requirements for SE
in ensembles of silicon NCs would be of use with the ultimate objective
demonstrating, unequivocally, stimulated emission in nanocrystalline silicon.Gain Profile
The calculations discussed in
Sections1–3 indicate several key factors relating to optical gain in Si NCs.
First, for size distributions produced by standard physical vapor deposition or
ion implantation methods (including multi-layering) the gain profile becomes
extremely wide. This reduces the gain coefficient and the gain cross section
per NC in the sample. Dal Negro et al. [187], for example, reported a gain
coefficient of 1600 ± 300 cm−1 at λ=750 nm, with a
gain cross section at this wavelength of 3×10−16 cm2 per
NC. Since the overall PL spectra generally peak at longer wavelengths, only a
small fraction of the NCs may actually emit within the reported gain window.
Most of the nanocrystals, therefore, may not contribute to SE and are in fact,
detrimental as a result of confined carrier absorption, scattering, and other
loss mechanisms they can cause. If Si NCs could be prepared with a size
distribution similar to that for CdSe (e.g., 5% standard deviation), the
theoretical gain coefficient per nanocrystal in the specimen as a whole could
increase by a fairly large factor, simply because a greater number of NCs could
contribute to the overall gain at the peak wavelength. Therefore, true
monodispersivity (i.e., overall
emission profile controlled by homogeneous broadening) is likely to be a
requirement for producing SE in ensembles of Si NCs.
Secondly, the size distribution
presents an inherent problem for laser excitation of the specimen. With the
exception of erbium doped waveguides that used LED excitation [192],
SE or gain is reported as a result of pumping the sample with a laser. Laser
excitation implies a narrow excitation energy window, and only NCs that happen
to have a transition within that window will be absorbing. In a given sample
with a wide distribution, therefore, many of the NCs may not absorb and cannot
contribute to the overall specimen gain. Furthermore, these “unexcited” NCs can
enhance the energy migration or other undesired effects discussed above.
The cluster-cluster energy
transfer mechanisms found to be important in the overall luminescence spectrum
also have implications for the presence of SE in ensembles of Si NCs. Generally, a high-pump rate is necessary to
observe the transition from loss to gain. Under such high rates, one can assume
a relatively large number of excitons per NC (we hesitate somewhat to suggest a
number as high as 100) [5] but at least 1 exciton per NC is readily
possible. In this case, charge carriers will be rapidly redistributed through
the tunneling and Forster mechanisms, on the sub-nanosecond time scale for a typical
volume filling fraction of NCs (the volume fraction ultimately controls the average
NC-NC spacing). Therefore, in a short time after the initial pump pulse, one
has the accumulation of charge carriers on the largest NCs as a result of
tunneling and Forster interactions. This drastically decreases the number of
NCs that can contribute to the gain spectrum and can lead to enhanced nonradiative
trapping and undesirable Auger interactions. This may be one reason that gain
may only occur in very short timescales, as suggested in several recent
experimental works [189], and only in certain specimens but not in others. In such cases, gain may
only build up before the energy migration processes have occurred, and will be
very susceptible to the average NC-NC spacing (and size distribution, as above)
from specimen to specimen. The fact that the gain, if present, occurs on the
short-wavelength side of the PL band is also consistent with the idea that the
gain occurs before the charge carriers transfer to the larger and “redder”
nanoclusters. By increasing the NC-NC spacing it should be possible
to limit the energy migration problem, due to the strong distance dependence of
the main processes. This may be accomplished by, for example, freeing the Si
NCs via an HF etch and redepositing them at a known (low) filling fraction, an
idea we are currently exploring. Other methods are also possible.
Additionally, the level
degeneracies calculated in Section3 imply a further key limiting factor for SE
in silicon nanocrystals. With a two-fold degeneracy or more, more than 1 exciton
per NC must be present, on average, in order to switch from loss to gain, which
in turn raises the problem of the Auger interactions. This number becomes
higher as the degeneracy of the electron and hole levels increases. In the case
of CdSe NCs, Klimov et al. [193]
have shown that one way to avoid this problem is to separate electrons and
holes in a core-shell nanostructure, thereby creating a large crystal-field
shift of the absorption spectrum. Such problems (and solutions) may well apply
to Si NCs also.


## 5. Conclusion

There is now an enormous wealth
of data on ensembles of silicon nanoclusters. The  difficulties in
understanding the optical properties of these materials are caused by the
indirect gap of silicon, the large homogeneous, and inhomogeneous broadening,
the possible effects of radiative centers and nonradiative traps, the cluster
size dependence of the lifetimes, and energy transfer. All these processes must
be considered when evaluating a luminescence spectrum; it is insufficient to
study the luminescence as a function of cluster size only, although size
undoubtedly plays a role. Unfortunately, owing to the complexity of the
materials, there are many contradictory observations and much uncertainty in
the literature. We hope that, with this paper, we have been able to summarize
some of the main theoretical and computational results that should help to work
out the problems and issues with our understanding of ensembles of silicon NCs.
We  hope that the combination of theory and simulation, new single particle
spectroscopies [20], and new synthesis methods [194]
will eventually lead to unequivocal and readily reproducible reports of
stimulated emission in Si NCs.



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*Source: 279502-2008-06-29.xml*