^{*}

A theoretical investigation of photoluminescence spectra for amorphous silicon quantum-dots (1 - 4 nm), at room temperature, were used to study the effect of both spatial and quantum confinements spontaneously via determination the energy peak of maximum intensity transition. The results show a continuous shifting toward low energy peak (red shift) and toward high-energy peak (blue shift), with the decreasing of quantum dot size, due to spatial and quantum confinements respectively. These results have leaded us to believe that such quantum dot size (1 nm) changes the nature of amorphous silicon optical band gap from indirect to direct transition material.

Silicon is a dominant material in micro and nanoelectronics technology, but because of its indirect band structure, it has not been possible to use silicon as optoelectronics devices such as light emitting diode [1-3]. The recent advancement in silicon technology has led to the fabrication of zero-dimensional structure, which is called silicon nanocrystals or silicon nanoparticle [

The light emission from silicon nanostructures is a research topic of current interest due to its potential for applications in silicon-based optoelectronic devices [

In this paper, we try to shed light upon a new proposal, which suggests the coexistence of both the spatial and the quantum confinement’s effects in amorphous silicon nanostructure material. The mechanism of transitions by considering the proposal of a-SiQDs model can make an understanding to the dynamic of these confinement effects.

It is possible to control the shape, dimensions, the structure of energy levels, and the number of confined carriers in this model, where it is adopted for a-Si because, silicon is very well known as indirect transition material. The key peculiarity of a-SiQDs emerges from the three dimensions confinement of charge carrier’s motion determined by shape and size [_{t}, which is estimated to be in the range (1 - 4) nm. This range represents the active quantum dots [

Dunstan and Boulitrop [_{C} [_{C}, may be viewed as the characteristic length explored by a photocreated electron-hole pair before its radiative recombination [_{C} must not be adopted randomly, an expression for the effective capture radius was derived that takes the form [19,21].

where N_{nr} is volume nonradiative recombination density (cm^{−3}), T_{O} is an experimentally determined constant and is the low temperature maximum quantum efficiency limit. R_{C} is considered as a characteristic length for this case. The consequence of spatial confinement effect is the localization of electrons and holes inside small size that leads to reflections or folding of phonons in k-space [

On the other hand, a characteristic length is considered the de Broglie wavelength for the quantum confinement case. According to the effective mass approximation, electrons and holes in a semiconductor are described as independent particles with an effective mass, and respectively. They can be determined by the convexity of the band edge structure. Both the electron states in the conduction band and the hole states in the valence band are quantized, where the 1s electron state increases in energy with decreasing the radius dot, R, while the 1 s hole state decreases [

where is the bulk band gap of the a-Si that takes the value (1.56 eV) obtained from the best fitting for the experimental data, which has been mentioned above. This value is very close to the values reported in literature for bulk a-Si [(1.5 - 1.6) eV] [_{t} is the radius of the dot and C is the confinement parameter which is estimated to be about (2.4 eV nm^{2}). Since the is included in the confinement parameter, where and are the electron and hole effective masses respectively. On contrast of assumption of Park et al., it was assumed [

The type of transitions in this study is allowed to be indirect transitions, as is normally done with bulk amorphous silicon. The absorption coefficient of amorphous silicon can be obtained from Taus expression in the photon energy range [

here is a constant related to the amorphous silicon material. takes the value of energy gap in bulk case for the spatial confinement, while it has the value of for the quantum confinement.

The chance that a tail state receives a carrier thermalizing from above (or below) is independent of its energy. The probability of thermalized carriers in one tail state is determined by the local configuration of other tail states, both above it and below it [

In this model, the same recombination process is active in both a-SiQDs and bulk a-Si [

In general, the recombination process depends on the tunneling type, tunneling distance and tunneling time. Tunneling can be either a radiative or a nonradiative process [^{8} sec^{−1}) for radiative process [18,24]. It may be useful to mention that, the radiative process occurs to carriers in the other band [^{12} - 10^{13}) sec^{−1}] [18,33], and it happens in deepest tail states of the same band [

In this equation is the tunneling attempt rate and is the effective Bohr radius. According to Street [^{−1}) depends exponentially on the pair separation [

An important consequence of quantum confinement is the increase in the band gap as the quantum dot size is decreased [34,35]. Since the wavefunction of carriers is strongly localized within the quantum dot [

Each size has a magnitude for the energy gap which results from the quantum treatment. It is noted that the energy gap magnitude (1.56 eV for bulk) increases with the decrease of the size. The cause of this behavior is that from the energy levels have moved to higher energies depending on the size. Thus, the energy gap of one size is different from the other. On the other hand, the spatial confinement effect increases with the decrease of quantum dot size. In fact, this is in agreement with the first suggestion of spatial position, but not by a random walk of steps. It is clarifying the principle of spatial confinement that is suggested by Tiedje et al. From this, it can be said that the spatial confinement may occur at any size with the quantum confinement (below the bulk-like size).

In the case of quantum confinement, the behavior of the absorption coefficient is different, as shown in

The absorption spectrum of a-SiQDs consists of a series of discrete lines as indicated in

fining electrons in a-SiQD produce a series of even sharper spikes which corresponds to a series of confinement quantum levels for these electrons, where the quantum dots absorb all wavelengths shorter than the absorption maximum (~E_{g}) [

The enhancement of radiative recombination results from a confinement effect, which is considered as one of the most effective means to convert an indirect optical transition into a direct [

where is defined in equation (4).

the size is reduced.

The mechanism of transitions in the present a-SiQDs model may be considered to understand the dynamic of confinement. It is well known, that in bulk crystal of indirect gap materials, the electron-hole recombination is possible only through phonon emission or phonon absorption. This because the wave-vector difference between the conduction-band bottom and the valence-band top must need to be compensated. However, in the quantum mechanics theory, the radiative transitions of electromagnetic waves with determination of their frequency, are due to Bohr's law, while the allowed transitions are due to the selection rule (n_{2} − n_{1} = 1) [

here is the frequency of vibration that is given by

, being the force constant. This energy represents the phonon energy and its frequency is equal to the frequency of the oscillator, which has the same mass, and the same force constant at allowed transitions. Consequently, the conduction band bottom is folded on Г point [

The dynamic of quantum confinement for present model is characterized by the shifting of energy levels of both conduction and valence bands. This shifting is proportional to the dot size, where the carriers are confined in three-dimensions, in the case of quantum dots. This leads to the overlap of the envelope wavefunctions, which is more than in the other nanostructures. However, the shifting of energy levels becomes more when the dot size decreases. This can be explained as following: As the dot size decreases the effective mass of electron (hole) decreases (increases), therefore; the energy levels are shifted to higher (lower) magnitude of conduction (valence) band, which results from the assumption of

. For this reason, the energy gap has different values depending on the dot size. By applying Bohr's law in this case, the photon energy for each size has unique value, where it increases as the dot size decreases (blue shifting).

In fact, the origin of quantum confinement is known to arise from the spatial confinement of carriers within the quantum dot boundary [

In conclusion, the optical energy gap transition of a-Si has been changed from indirect to direct transition by adopting the a-Si as quantum dot with very low radius

dimension (<1.8 nm). This is happened when spatial and quantum confinement’s effects are taken both in consideration. Also, a continuous shifting toward low energy (red shift) and toward high energy (blue shift) with the decreasing of the quantum dot size are due to the existence of both spatial and quantum confinements respectively.

We thank Dr. Ahlam H. Al-Mousawy, for her useful discussions.