A quantum cascade laser (QCLs) is a specific type of semiconductor laser that operates through principles of quantum mechanics. Already theoretically predicted in 1971 [1], QCLs had not been realized until 1994 at Bell Laboratories [2]. They have many advantages over other types of semiconductors’ lasers. Some of these advantages include precise tuning from one wavelength to another, higher optical power, continuous wave operation and the ability to produce light in the terahertz range of the spectrum [3-6].

From the physical point of view, the unipolarity of a QC laser indicates that electrons are solely responsible for releasing energy in the form of photons. These electrons transition from one quantum energy state to another within a layer, or group of layers, of semiconductor material releasing energy in the form of photons during their descent. The binding energy necessary to pull these electrons away from the Coulombic force of the nucleus in the atom is related to the extremely thin semiconductor layers. A property of quantum mechanics known as quantum confinement occurs when the electrons are trapped within a thin semiconductor quantum well layer. These electrons can freely move in only two directions within the plane of the thin layer. In this case quantum confinement, which leads to discrete energy levels that electrons can occupy in a material smaller than the de Broglie wavelength, occurs in only one dimension due to the quantum well structure [7]. Unlike the earliest form of semiconductor lasers where the energy bandgap determines the wavelength of the light emitted, with QC lasers the thickness of the layers determines the wavelength. This is a critically important property of QC lasers because it allows them to be tuned to a desired frequency through bandgap engineering [8]. Technologically speaking, this laser type is grown by epitaxial method such as molecular beam epitaxy (MBE) [9]. Layers of different semiconductor materials each only a few atomic layers thin are deposited onto a thin slice of a semiconductor crystal. In order to optimize the electronic wave functions with respect to energy and probability distribution, we have to choose the sequence of the layers, their width and materials.

It was established that these unipolar intersubband lasers might be realized not only in III-V semiconductors [10-13] but also in IV-IV structures [14-16]. Among semiconductors, the covalent semiconductors Si and Ge have been studied extensively both theoretically and experimentally [17,18]. Group IV semiconductors alloys like Si-Ge, have the immense potential for technological applications whose include the optoelectronic devices [19-21]. By using intersubband transitions within the same band, one can circumvent the main obstacle to silicon-based lasers, the indirect band gap. The large band offsets in the valence band of pseudomorphic SiGe layers on Si substrates imply a quantum cascade scheme with hole subbands.

In our previous publication [22], we were interested to the investigation of the structural, electronic and optical properties of Si, Ge, and Si_1–xGe_x for different compositions using the full-potential linear muffin-tin orbital (FP-LMTO) method augmented by a plane-wave basis (PLW), implemented in Lmtar code [23-25]. All the obtained results showed that the weakly strained G-rich SiGe layers possess very promising properties for both electronic and optical applications.

The aim of the present work is to provide a consistent and complete set of electronic and optical parameters of the Si/SiGe quantum well. The obtained results are going to be of use to a good understanding of the quantum phenomena of these devices. The second objective concerns the way which allows us to optimize the intrinsic parameters of the Si/SiGe quantum cascade structure.

Much theoretical work has been done to accompany the rapid experimental developments of QCLs as well as to better explain the design considerations of intersubband lasers. These include Monte Carlo simulations [26-29], self-consistent rate equations [30,31], as well as the nonequilibrium Green’s function formalism [32,33]. As was shown above, the quantum cascade lasers (QCL’s) are fabricated by stacking up alternating layers of semiconducting material with nanoscale thicknesses. This heterostructure of layers forms a series of conductionband quantum wells in the z direction which trap the electrons into subband states [2] (Figure 1). The calculation procedures described here follows the envelope function approach based on the effective-mass approximation [34,35]. This approximation was found to be much more computationally efficient than atomistic methods, making it more suitable as a design tool for QCLs [36].

The eigenstate of an electron in the unperturbed Hamiltonian of a QCL is the product of the Bloch envelope function B(x, y, z), the free electron wavefunction in the x and y direction, and the bound quantum-well eigen-functions ψ_n(z) in the z direction.

The Bloch function factor contains the effects on the electron state due to the non-uniform nature of the crystal potential on the atomic scale.

We assume the semiconductor layer widths are large compared to the atoms, so we make the approximation that the Bloch function factor is negligible. Each electron is pseudo-free in the x and y dimensions because the material is uniform in those dimensions. Even though each electron is bound to the crystal in these dimensions, we can treat each as free if we use the effective mass of the electron. The bound-state z component wave functions ψ_n(z) are found by numerically solving the one-dimensional Schrödinger equation when the potential profile is known. The potential profile is a combination of the

conduction-band edge quantum well profile of the material layers, the bias voltage, and the built-in potential which accounts for the effects of space charge. This minimum energy can be calculated as one of the eigen values of the Schrödinger equation along the growth direction z,

where h is the Planck constant and E_i is the minimum energy of subband i in a QW structure. By solving this equation, one obtains the energy E_i (eigenvalues) and the wave function ψn (eingenfunctions) of the n electron state. In this method, we have taken into account the boundary conditions for the wavefunction.

An analytical solution provided by solving the Schrö- dinger equation for the conduction minimum energy of subband i has the following form Bas du formulaire

Haut du formulaire where L_z represent the depth of the quantum well.

Bas du formulaire.

For a given temperature T, the population n_i for each subband i is expressed by

Two simplified expressions can be established for the population n_i:

For the subband situated below the Fermi level , the population is

.

For the subband situated above the Fermi level

, the population becomes

Haut du formulaire The effective states density according to x concentration for the Si/SiGe quantum well is given by

where ΔE_cb is the limit energy of the conduction band which is differentiated in two terms E_c(2) and E_c(4). E_c(2) is the value linked to both identical directions [001] and [001] while E_c(4) is the value linked to the four other identical directions [010], [010], [100] et [100].

Bas du formulaire We have then

with

The term e_T(x) is the difference between the strain tensors e_zz – e_xx, according to directions zz and xx. These strain tensors depend of the silicon lattice parameter as well as the SiGe bowing parameter.

To design a desired QW structure such as the quantum cascade laser and improve a device performance, a numerical simulation is needed to compute the energy levels and for different electrons states, the corresponding envelope functions, the intersubband transition dipole moments, carrier densities, relaxation times and other parameters. Then, subband formation and energy dispersion are described in the framework of envelope functions with the effective-mass approximation for both conduction and valence band. In Figure 2, we displayed the profiles of the envelope function for different electron states.

In Figure 3, we have illustrated the calculated electronic energy-band structure (a) and total DOS (b) of Si_1–xGe_x alloy for x = 0.5. For the calculations, we have used the full-potential linear muffin-tin orbital (FP-LMTO) method augmented by a plane-wave basis (PLW), implemented in Lmtar code [23-25]. The effects of the approximations to the exchange-correlation energy were treated by the local density approximation (LDA).

Figure 4 shows the variation of the confinement energy with respect to different width wells. This energy is carried out by solving the Schrodinger equation for the Si/SiGe quantum cascade structure. It is clear from this

results that for the low width wells the confinement energy is very important. This leads favorably to the intersubband transition. Hence, the emitted wavelength of the QCL only depends on the thicknesses of the layers. One can notice that the use of semiconductor with small effective mass excites well the confinement effects what is not the case for the material SiGe of which its effective mass is important compared to those of GaAs, InAs, equal to 0.067 m₀ and 0.023 m₀, respectively.

Figure 5 illustrated the variation of the effective states density with respect to germanium concentration x. We can see that the effective states density of the barrier material is very important to that of the well. Then we shall have continued and a strong carrier’s injection in the well to minimize the losses and increase the laser gain.