Mid-Infrared Emission in Ge/Ge_1-xSn_x/Ge Quantum Well Modeled Within 14-Band k.p Model

Abstract

Band structure and gain in a Ge/Ge_1-xSn_x/Ge quantum well are described theoretically using a 14-band k.p model. It has been shown that the quantum well width and the α-Sn concentration considerably modify the conduction and valence subband structure, and, as a result, the optical gain changes with the insertion of a very small concentration of α-Sn. In particular, we have determined the necessary injection carrier density Nj and the critical α-Sn concentration for elevated high gain lasing. It is found that for Nj = 1.5 × 10¹⁸ cm⁻³, we achieved a maximum peak gain for α-Sn concentration of the order 0.155. We can predict that Ge/Ge_1-xSn_x/Ge QWs should be manufactured with an α-Sn concentration less than 0.155 in devices for optoelectronics applications such as telecommunication and light emitting laser diodes.

1. Introduction

Ge, Sn group-IV-materials have drawn a remarkable amount of interest for both their electronic and optical properties [1,2,3,4]. The optical properties, such as laser and optical gain, can be enhanced when the Ge semiconductor is combined with the semi-metallic α-Sn. Recently, SiGeSn quantum well (QW) structures have shown enormous benefits in both theoretical and experimental studies [5,6,7,8,9,10]. However, significant progress in optoelectronic devices, such as infrared photodetectors in terahertz wavelength regions, has been reported by using GeSn/SiGeSn [11] or by optimization of different structural parameters of SiGeSn [12]. In addition, the optoelectronics properties of SiGeSn are greatly enhanced by the optimization of the α-Sn concentration, in particular by the transition from an indirect to direct band gap [13,14]. However, it is experimentally shown that the direct bandgap structure can be acquired by the incorporation of a small concentration of α-Sn [8], and that a high photoluminescence intensity can be also attained. Indeed, the elaboration and photoluminescence characterization of GeSn has significantly contributed to the growth of Si-based optoelectronics technology [9,13]. Further, several practical investigations demonstrated that the direct bandgap structure of Ge_1−xSn_x can be obtained with x = 0.12 [15]. In addition, thanks to changing the bandgap of GeSn, the ternary alloy SiGeSn also presents an analogous inter-band transition. Furthermore, several experimental and theoretical studies demonstrated that the ternary alloy SiGeSn has a higher range of bandgap and lattice variation compared with the binary one, therefore providing supplementary applications. An interesting question is related to the threshold carrier densities of GeSn QW laser light emission and the physical cause of low-temperature lasing of current. This has been studied in order to evaluate the threshold carrier densities of habitual III–V and II-VI QW lasers [16]. Indeed, the lasing has been investigated, and it has been shown that optical gain emission behavior analogous to II–VI and III–V QW has been found for GeSn QW by n-type doping of 6 × 10¹⁸ cm⁻³. In addition, a theoretical analysis of GeSn quantum dots for photodetection applications has also been conducted [17].

Up to now, the optical properties performances such as the optical cross section and the optical gain emission of GeSn/Ge QW have been considered with 6- and 8-band k.p Hamiltonian. To investigate the optimal injection carrier density and concentration of α-Sn with Ge, it is essential to apply the 14-band model in order to consider the effect of the far conduction (CB) and valence bands (VB) levels, which can considerably influence the inter-band-transitions and optical gain emission procedure. Compared to the 8-band k.p model, the 14-band k.p Hamiltonian is more appropriate in applications involving a sensible injection carrier density and doped concentration. This can be explained by the far levels bands intervening in the 14-band k.p model, such as the [(Γ_7C, Γ_8C)-Γ_6C, (Γ_7C, Γ_8C)-(Γ_7V, Γ_8V) and Γ_6C-(Γ_7V, Γ_8V)] interactions, which represent, respectively, the coupling between the two-type of CBs, the higher-CB with VB, and the coupling between the lower-CB and VB, whereas the 8-band k.p Hamiltonian is governed only by the lower-CB with VB interaction. Thus, in this paper, we theoretically consider the influence of the injection carrier density and the α-Sn concentration on the CB and VB-band structures and the wave functions as well as the optical gain emission through the framework of the 14-band k.p model, taking into account the strain effect and the interaction between VB and CB-bands. For the numerical step, we employ self-consistent iteration and variational methods [5,18].

2. Theoretical Method for Optical Laser-Gain of Ge/Ge_1−xSn_x/Ge QWs

Investigations of optical gain properties, such as injection carrier density and α-Sn concentration, require accurate knowledge of the subband structure. However, both the CB and VB subbands of Ge/Ge_1−xSn_x/Ge QW will be investigated by the 14-band k.p Hamiltonian H [5,18]:

$$H=H_{kp}+H_{strain}+V_{VBO}(z)+V_{CBO}(z)$$

(1)

where $H_{kp}$ represent the k.p Hamiltonian, $H_{strain}$ represents the Hamiltonian of strain, and $V_{VBO}(z)$ and $V_{CBO}(z)$ are, respectively, the VB and CB potential offset. The bands alignment scheme and CB and VB parameters used in 14-band k.p model of the Ge/Ge_1−xSn_x/Ge QWs are represented in Figure 1. The $H_{kp}$, matrix, and the $\left|J,M_J\right.〉$ base are given in Appendix A.

For the optical gain $g(\omega )$ calculation, it is well known that $g(\omega )$ for QW depends on the subband structure and is related to the density of states ρ_2D, the wave functions overlap, the intra-band relaxation time ${\tau }_{in}$, the quasi-Fermi levels $f_n^c$ and $f_m^v$, and on the momentum matrix element for the optical transitions [19]:

$$\begin{array}{ll}g(\omega )=& {\displaystyle \frac{e^2}{m_0^2{n}_r c {\epsilon }_0 \omega }}{\displaystyle \frac{1}{L_{QW}}}{\displaystyle \sum _{\sigma =\uparrow ,\downarrow }{{\displaystyle \sum _{n m}{J}_{cv}^{n m}{\displaystyle \underset{0}{\overset{+\infty }{\int }}{\rho }_{2D}}\left|M_{cv}\right|}}^2\left[f\left(E_{n,k}^c-E_F^c\right)-f\left(E_F^v-E_{m,k}^v\right)\right]\times }\\ & {\displaystyle \frac{\hslash /{\tau }_{in}}{{\left(\hslash \omega -(E_{n,k}^c-E_{m,k}^v)-E\right)}^2+{\left(\hslash /{\tau }_{in}\right)}^2}}dE\end{array}$$

(2)

where, m₀ is free electron mass, L_Qw is well width, n_r is refractive index, e is the electron charge, ${\epsilon }_0$ is the permittivity of free space, c is the light-velocity, and τ_in is the intraband relaxation time, which is equal to 10⁻¹³ s [19].

3. Results and Discussions

To study the effect of the QW-width and the α-Sn concentration on the subband structure and optical gain, we calculated the relation dispersion of both CB and VB of the Ge/Ge_1−xSn_x/Ge QW for α-Sn concentration and QW-width. The necessary parameters of Ge and α-Sn used in the present numerical calculation are given in Table 1 of reference [18]. The well and barrier width are around (8, 10, 12, 14) nm and 24 nm, respectively, and the profound for the well is about 130 meV for the heavy HHn and light LHn-holes [20]. Note that the unit of the energy is meV and the energy reference is chosen at the highest of the unstrained HHn-well. All the numerical parameters of the bulk Ge and α-Sn used in this work are taken from Refs. [18,20].

Figure 2 plots the subband structure of the CB and VB of a 12 nm single QW with various Sn concentration.

The HH_n, LHn, and e_n represent the confined hole- and electron-subband-states in CB and VB, respectively, for various α-Sn concentrations. For the different behaviors, one should note the strong non-parabolicities resulting from the admixture of (HHn, LHn) in the VB. Indeed, strong coupling has been shown between the HHn, and LHn. This reality suggests that the VB-states are highly dependent between them and with the two CBs in the Ge/Ge_1−xSn_x/Ge QW. One can notice that the number of confined states and the difference between the HHn and LHn subbands increase with α-Sn concentration. On the other hand, from Figure 2, we notice that both the HH₁- and HH₂-states are shifted down, whereas the LH₁-states are pressed up when the concentration of Sn increases. This can be attributed to the influence of α-Sn concentration on the band gap energy [11], which can explain the higher energy level as shown in Figure 2a. In addition, the HH₁-e₁-transition energy is very dependent on the α-Sn concentration, and it is shifted to the low-energies. This comportment will considerably influence the variation of the optical gain. Optical gain emission for different QW-width (8, 10, 12, and 14 nm) and α-Sn (0.1 0.15, 0.155, and 0.159) concentrations of the Ge/Ge_1−xSn_x/Ge QW are represented in Figure 3a–d.

We notice that the optical gain curves are very sensitive to the α-Sn concentration and to the QW-width. The injected carrier density has been chosen to overcome the free carrier absorption losses [6] and it is varied from $N_j=1\times {10}^{18}\mathrm{c}{\mathrm{m}}^{-3}$ to $2\times {10}^{18}\mathrm{c}{\mathrm{m}}^{-3}$. It is important to note that within the 8-band k.p Hamiltonian, the injected carrier density has been varied from $N_j=2\times {10}^{18}\mathrm{c}{\mathrm{m}}^{-3}$ to $6\times {10}^{18}\mathrm{c}{\mathrm{m}}^{-3}$ in references [7,16]. One can observe that for the value of x between 0.1 and 0.159, the gain peak moves to a higher wavelength when we increase the Sn concentration, i.e., we reached an emission wavelength of the order of 1.96 μm for x = 0.1, whereas for x = 0.155, an emission value in the order of 2.61 μm was obtained. These results confirm that mid-infrared laser structure operating at room temperature can be obtained by Ge/Ge_1−xSn_x/Ge QW structure. Otherwise, for x = 0.159, we remarked the strong reduction for optical gain. We reached an intensity lesser than 50 cm⁻¹, whereas for x = 0.155 an intensity in the order of 929 cm⁻¹ has been obtained. Therefore, this variance does not exist in the calculations provided by the 8-band model [2,7]. Furthermore, the 14-band k.p Hamiltonian 0.155 represents the critical concentration of a-Sn, which leads to the strong decrease in the intensity of the optical gain as presented in Figure 4. In order to highlight the evolution of wavelength and the peak gain maximum versus α-Sn concentration, we plotted (in Figure 4) the wavelength extracted from the peak gain maximum as a function of the insertion of the α-Sn contents.

It is observed that the increase of α-Sn concentration from 0.1 to 0.155 increases both the wavelength and the gain maximum. Beyond these concentrations, one can notice the strong decrease of optical gain and the wavelength. This indicates that 0.155 represents the critical α-Sn concentration for elevated high gain lasing for the Ge/Ge_1−xSn_x/Ge QW structure. By comparing the 8-band and 14-band models, the maximum injected carrier density used to produce gain is less than 2 × 10¹⁸ cm⁻³ for the 14-band model while it is greater than 6 × 10¹⁸ cm⁻³ for the 8-band model [2,7,16]. This difference in the injected carrier density justifies the importance of the application of the 14-band model for the GeSn structures. On the other hand, in refs [7,16], the authors indicate that the enlargement of the QW-width can considerably enhance the optical gain. Thus, we have reviewed the optical gain dependence on QW-width. Figure 3a displays the optical gain of Ge/Ge_1−xSn_x/Ge QW calculated for x = 0.150 for a typical injected density of about 1.7 × 10¹⁸ cm⁻³ and for different QW-widths (8, 10, 12, and 14 nm). The significant influence of the increasing of QW-width is noticeable. This can be explained by the confinement of particles, as shown in Figure 2. One can observe that the gain spectrum has been shifted slightly for the higher wavelength with the increase of the QW-width. The QW-width can improve the gain intensity, but when the concentration of α-Sn is increased, the gain spectrum and the intensity decrease quickly to give a very weak peak, as shown in Figure 3c. To remedy this anomaly, we have inserted a very small quantity of Sn in the barrier with Ge. Figure 3d shows the optical gain of Ge_0.99Sn_0.01/Ge_0.835Sn_0.165/Ge_0.99Sn_0.01 QW as a function of photon energy and for different QW-widths (8, 10, 12, and 14 nm). One can observe the significant difference when incorporating the small quantity of the Sn in the barrier. Indeed, the obtained results of the insertion of the α-Sn in the barrier will be helpful the experimental step for the engineering of Ge_1-xSn_x/Ge_1-xSn_x/Ge_1-xSn_x QWs. Consequentially, it is important to conclude that for the potential fabrication of the Ge_1-xSn_x/Ge_1-xSn_x/Ge_1-xSn_x QWs, the insertion of α-Sn should not exceed 0.155 to ensure high optical gain in mid-infrared emission. In addition, 2 × 10¹⁸ cm⁻³ is sufficient to produce gain as an injected carrier density for Ge_1-xSn_x/Ge_1-xSn_x/Ge_1-xSn_x QWs. These sophisticated precision measurements for the α-Sn concentration and injected carrier density are performed in relation to the practice of the 14-band k.p model; they are not relevant for the 8-band k.p Hamiltonian. This is due to the second CB$\left({\Gamma }_{8C}-{\Gamma }_{6C}\right)$ and VB$\left({\Gamma }_{8C}-{\Gamma }_{8V}\right)$ interaction, which performed the modification noted by the two models.

4. Conclusions

Using 14-band k.p Hamiltonian, we have calculated the electron and hole-subband-states and optical gain of Ge/Ge_1−xSn_x/Ge QW for different α-Sn concentrations and QW-widths. Our calculation shows that a 1.5 × 10¹⁸ cm⁻³ injection carrier density was sufficient to produce high gain intensity. For the optimization of the α-Sn, our results predicted that Ge/Ge_1-xSn_x/Ge QWs should be realized with a concentration less than 0.155 in device applications involving optical gain in mid-infrared emission at room temperature. Finally, our results indicated that instead of extending the QW-width to enhance the optical gain emission, it is advisable to insert a small quantity of α-Sn into the barrier.