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/Si 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 (-point) to the top of the valence band (-point). The photon wave vector is about three orders of magnitude less than that required for the transition between the - and -points. This difference in -space is , with 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 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 Si, 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 for holes and for electrons, respectively. Thus, the momentum conservation law is not violated in this case, which yields a nonzero probability of the 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 for the valence states and 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 function with 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 Section 2. 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 Figure 1. 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 Si. 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 to , where is the nanocrystal radius. Thus, the optical gap may be calculated as , where is the bandgap of bulk silicon, is some positive constant, and 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 , 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 matrix—Kim et al. [49] (empty triangles).

Furthermore, the blue-shift energy , determined from the experiments, does not obey the law following from the simplest quantum-mechanical model. This dependence is rather with or an even weaker dependence on 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 gap ) 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 other than . In addition, electron-hole Coulomb interaction contributes a further 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 times vary within a wide range of about 1 to1000 s 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 time ; and nonradiative, with typical time . The rate of the photoluminescence decay is the sum of the radiative and nonradiative recombination rates: . 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 . The quantum yield , which is proportional to the PL intensity, may be defined as the “weight” of the radiative channel in the recombination process: . If , then . Obviously, one can use two physically different ways to increase . 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, 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, , 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 , and lower triplet with about 2-3 orders-of-magnitude longer lifetime . The decay rate is then defined by where is the energy of the singlet-triplet splitting, is the Boltzmann constant, and 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 the decay of the singlet state is strongly suppressed, and the total lifetime coincides with the longer time . In the opposite case (when ), is close to . 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 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).

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).

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 ( 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 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 (Figure 4). 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 Here is the confining potential equal to zero inside and infinity outside the dot. The second part 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. is often referred to as a self-polarization term. It can be represented as where and 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 : Here, is the bulk Hamiltonian operator acting on the six-dimensional (6D) envelope-function vectors , and is the electron energy. The components of the 6D-vector are slowly varied expansion coefficients 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 the -point is , , , , , , where and are “up” and “down” spinors, respectively, and the Bloch functions , , belong to the irreducible representation . The Hamiltonian matrix in 4 is the sum of three parts: . Here is the isotropic part obtained as the average of the total matrix over all angles in -space. is the anisotropic part that can be represented by two equivalent blocks situated on the main diagonal of the total    matrix: where denotes , denotes , and denotes .

Finally, the term describes the spin-orbit interaction. We have introduced above the spin-orbit energy equal to 44 meV for silicon, the identity matrix , and the hole effective mass , where the numbers , , are dimensionless empirical parameters in the 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 operator , it is possible to classify its eigenstates similarly to atomic systems using common terminology such as -, -, and -type states, and so forth [92]. Accordingly, one may expand the components of the envelope-function vectors over these eigenstates as where stands for the -, -, -, states and are the expansion coefficients. For instance, the -, -, -, (in the following simply -, -, and -) and -type states are described by the following functions: where are the spherical Bessel functions of argument , and are the first roots of . Below, we restrict the basis of envelope functions with these states only, and determine the electron and hole spectra. Substitution of into (4) yields algebraic equations for : where denotes the energy of the state , stands for the Kronecker delta, and the Einstein convention has been used when summing over . 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 by , , , and 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 is triply degenerate (or sixfold, if spin is taken into account). It is the energy of the state. Hybridization of the -type states yields four different levels with the energies where . The energies of the - hybridized states are as follows: where . Finally, the energies of the - hybrids are written as 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 .

In Figure 5 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 -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 -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: -type level. Short dashed lines: - hybridized levels. Dots: - hybridized levels. Long dashed lines: - 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 different -points in the Brillouin zone [99]. We denote three pairs of these functions as , ; , ; , . Each pair belongs to the twofold degenerate irreducible representation of the corresponding -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 -functions as the Bloch basis allows us to describe correctly a dispersion-law nonparabolicity that originates mainly from the energy-branch crossing in the -points.

The Hamiltonian operator in the conduction band represents a matrix having the form Each element of the matrix is a block matrix defined by the following expressions [99]: Here is the distance in -space from any of the energy minima to the nearest -point, is the lattice constant of silicon. and are the “longitudinal” and “transverse” effective masses. The origin of the -axis coincides with the -point energy. Each of the small indices (, , ) runs over the values , , or, 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 - hybrids. Their energy levels are given by where are - and - type matrix elements of anisotropic part , , and are the energies of the - and -type states of the isotropic unperturbed Hamiltonian operator. The isotropic average electron effective mass is defined by . The - hybridization forms states with energies: Here . The next three levels are the result of the - hybridization: Finally, the - hybridized energies are obtained numerically: 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 -point energy. Solid line: - type levels. Short dashed lines: - hybridized levels. Dots: - hybridized levels. Long dashed lines: - 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 with - and -type envelope functions (see [92] for details). The “weight” of the -type envelope function is greater than that of the -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 between - and -points in bulk  eV: 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/Si 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 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 becomes weaker than , especially for the case of the discontinuous effective mass where the dependence is closer to . In the last case the gap dependence is well fitted by the function: with  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 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 . 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 .

The authors of some of the previous work (e.g., [67, 90, 91]) treated the Coulomb potential energy as 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 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 ( centers) which can be neutral, positively, or negatively charged, and different SiO bonds (backbonded and double bonded oxygen) arising at Si/Si 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/Si 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 charged 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 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–Si interface in a NC with a single center was established theoretically as a function of the NC size [106]: In (23), is the capture cross section given by where is the capture coefficient and is the thermal velocity equal to , is the ionization energy of the defect (approximately equal to the Franck-Condon energy plus the carrier confinement energy), is the Huang-Rhys factor () [107], is the average phonon energy, is the nanocrystal volume, and . 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 energy . 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 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 depending on electron position vector, . Taking into account the short-range field and charge polarization at the dot boundary due to different static dielectric constants and of an NC and its surroundings, one can in principle obtain (Figure 8) [73].

Figure 8: Dielectric function of silicon NCs in silicon dioxide matrix () for (1)  nm; (2)  nm; (3)  nm.

An overall decrease of in a nanocrystal takes place compared to the bulk value of . 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 extends to the nanocrystal boundary, where the value of the dielectric function becomes equal to . At the same time, the sharp reduction of toward unity occurring at small 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 where stands for the impurity position vector, and 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 , , and , where , and stands for the absolute value of .

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 the -point (001). Dashed line: -point (010). Dots: -point (100). The direction of the donor's position vector is defined by , , and . The NC radius is indicated in the figure.

In Figure 10, the envelope-function correction is shown for the ground state due to the hydrogen-like donor, as a function of the angles and on the spherical surface for the former values of . Angles and are the spherical coordinates. The maximal values of (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: , , and . .

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 Figure 11. Similarly to the donor case, the splitting turns out to be large compared to the magnitudes of chemical shifts in bulk silicon [141–144].

Figure 11: Splitting of the two lower hole levels shown in Figure 5 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 , , and . Dashed and solid lines represent the energies originating from doublet and quadruplet at , respectively. All the energies are counted from the mean quadruplet energy at .

The Coulomb and spin-orbit interactions remove the sixfold degeneracy of both lowest levels shown in Figure 5, 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 Figure 12 for (upper panels) and (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” , where is the th element of the 6D envelope function vector as before.

Figure 12: Density plot of the probability distribution of all the six doublets at (upper images), and (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  nm.

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. Therefore 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 . 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 cases and . 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 contribution of a radiative recombination channel in an interband transition. In turn, the total decay rate is defined by both radiative and nonradiative recombination rates: . 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-band -point and the valence-band -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 rate 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 () and final () states in the conduction and valence bands, respectively, in the following form: Here, and stand for the photon frequency and wave vector. The Dirac delta-function reflects the energy conservation law for the electron transition. The operator describes the electron-photon interaction, and has the form: where is the electron momentum operator, is the polarization vector, the operator annihilates and creates a photon with the wave vector and polarization , and stands for the volume of the electromagnetic resonator. The function is written as [145] This function represents the correction factor in the field magnitude due to the replacement of homogeneous media with bulk permittivity by a spherical silicon nanocrystal surrounded by silicon dioxide with a dielectric constant .

The initial state in (24) corresponds to an electron-hole pair being in its ground state, and an ensemble of photons whose distribution over 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 Section 3.1.2), 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, -, -, or -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 (singlet), (doublet), and (triplet) of the tetrahedral group. These functions are defined as follows [73]: Here, the parameter is defined by the relationships [92] where vectors and stand for the - and -type envelope functions as before. The Bloch states , , , and , , or belong to the representations , , and of the tetrahedral group. In what follows, when computing the interband matrix elements of 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: where the index indicates the “values” of the , , , and , functions, and stands for the six Bloch functions , , , , , of the irreducible representation . The - and -type envelope functions are denoted as , and , , , respectively. Selecting expansion coefficients , 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 -type envelope function: where stands for the Bloch state basis functions , , or of the irreducible representation .

After some algebra, one can obtain recombination rate in the form: where . In order to compute the momentum matrix element, we expand the Bloch functions and over the reciprocal lattice vectors as follows: where is the lattice constant of silicon, and unit vectors define the -point. We set , , and . Expansion coefficients and are determined with the local pseudopotential method. They satisfy the normalizing condition . As a result, the momentum matrix element is given by where with being oscillating vector functions [146] of the dot radius. For - and -type subscripts, respectively, these functions are Here, , , and represents an th component of . The nonzero value of is exclusively due to the Heisenberg uncertainty relations which cause the wide distribution of the wave function in -space.