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A GexSn1-x/Ge quantum well structure is studied by the 8-band k.p method. The band structures of both direct Γ valley and indirect L valley were calculated, and the relations between temperature and the spontaneous emission rate spectrum (SERS) were investigated. The results show an abnormal temperature dependent SERS phenomenon in the so-called pseudo-direct gap semiconductor. This can be explained by taking consideration of the contribution of electrons in the indirect L valley injecting into direct Γ valley under a higher temperature. Cases of higher Sn composition accompanied with larger compressive strain were investigated using the same model. The significant compressive strain effect compensates the dragging down of energy gap by the induced Sn atoms, which makes it difficult to achieve the transition from indirect band-gap material to direct band-gap material in quantum well devices.
© 2017 Optical Society of America
1. Introduction
Group IV semiconductors such as silicon (Si) and Germanium (Ge) have shown their benefits in mass production of electronic devices. However, due to such feature as the indirect band gap semiconductor, their poor performance of optical properties has limited their application in optoelectronic area. Ge has a direct band gap of 0.8 eV at the Γ point [1], while its L conduction band edge is just below the Γ conduction band edge 0.136 eV [2]. Therefore, Ge is also known as a pseudo-direct band gap semiconductor [3]. Recently, lasing from the direct-bandgap transition in the GeSn bulk material grown on Si substrate has been reported [4]. The indirect-to-direct transition was achieved by incorporate Sn atoms into germanium lattice to adjust the Γ conduction band edge below L conduction band edge [5].
In this paper, we studied the GeSn/Ge quantum well with different Sn composition from 5% up to 24%. By calculating the band structure using the multiple-band k.p method, we can theoretically reveal the relation between spontaneous emission rate and various band parameters. The contribution of the very close indirect L valley [6] is taken into consideration. The conduction band minimum (CBM) of the existing L valley significantly influences the spontaneous emission rate. The abnormal temperature dependent SERS [7] for indirect band-gap material is observed, which can be explained and simulated by using our model too.
To achieve the transition from indirect band-gap material to direct band-gap, cases of higher Sn composition are studied. For the quantum well (QW), the band gap is defined by the difference of the first quantized level in the conduction band and the first quantized level in the valence band. As a result, the CBM of QWs should be determined by the lowest energy level that may lie in either indirect Γ or direct L valley. The relatively larger effective mass of L valley makes the position of energy levels much closer comparing to Γ valley. The significant difference of effective mass m* between Γ and L valleys causes it hard to realize direct band-gap even though the CBM in Γ valley of bulk material is lower than that in L valley. Meanwhile the higher Sn composition results in the strong compressive strain and this strain effect will push CBM of both valleys to move up. CBM of Γ valley however is much more sensitive to the strain, so it moves faster which makes direct transition even harder.
2. Calculation method
The 8 band k.p model rather than the commonly used 6 band k.p model [8] is applied to the relatively narrow band gap semiconductor so that the interaction between the conduction band and valence band is included in the calculation. The details of the 8 band k.p method can be found in our previous work [9].
The Hamiltonian of L conduction subbands can be written as [8]:
where ml,L and mt,L stand for the longitudinal and transverse effective masses along the [111] direction, k1 and k2 stand are wave vectors along [110] and [-110] direction, a is the lattice constant of GexSn1-x which is assumed linearly proportional to the material composition given by [10]:V[111](z) is the energy band offset without inducing any strain, and Vε[111](z) is the strain-involved energy shift for L valley, εxx and εyy are the in-plane strain, εzz is the strain in Z direction.The expression of direct band spontaneous emission rate (SER) can be written as [11]:
E is the photon energy, ε0 is the free-space dielectric constant, n is the refractive index, c is the light speed, Qncnv is the squared optical transition matrix element, Eeh is the transition energy, and τ is the intraband relaxation time.In order to study the temperature dependence of the spontaneous emission rate, Varshni’s equation is applied into the expressions of energy gap of both direct Γ valley and indirect L valley. For the Γ and L conduction bands, the relation between band gap and temperature obeys Varshni’s equation [12]:
where E0 is the band gap at 0 K. α and β are the two constants. For the indirect band gap of Ge, E0 is 0.7412 eV, and for the direct band gap of Ge, E0 is 0.8893 eV. The two parameters α and β in the equation are 4.461x10−4 and 210 for indirect energy gap, 6.842x10−4 and 398 for direct energy gap. These two coefficients for GeSn have been measured by experimentally curve fitting [13–15]. However, a consistent set of the coefficients cannot be collected from the references. In order to avoid the uncertainty, we assume the two coefficients α and β are the same for GeSn and Ge in order to get a self-consistent result.The half-width at half-maximum (HWHM) of SERS as a function of temperature is a fit to the expression [16]:
where Γ0 equals to 2.3meV which for the inhomogeneous broadenings such as scattering by interface roughness and alloy fluctuations. The second term stands for the homogeneous broadening due to GeSn longitudinal optical (LO) phonon scattering using at 300k [8], and of Ge for approximation.3. Results and discussions
The QWs are grown on Germanium substrate where the Ge barrier is fully relaxed. The well width of GexSn1-x/Ge quantum well is fixed at 60 Å and barrier width is fixed at 200 Å. All the parameters used are taken from the reference [9] and [17]. Due to a larger lattice constant of Sn, the natural compressive strain exists in the GeSn QW [18]. When calculating energy gaps of Γ and L conduction bands, bowing effect is taken into consideration. The energy gaps of GexSn1-x at Γ and L valley is given by
and coefficients bΓGeSn and bLGeSn are 2.55 and 0.89, respectively. The bowing coefficient for GeSn is hard to determine because it may change with Sn component. The values used in our cases are from Ref. 17, which agree very well with experimental results.In the first part, we studied the case with smaller Sn composition to make sure it is still indirect material at the well region. The calculated band structure of Ge0.95Sn0.05/Ge before strain is shown Fig. 1(a) at room temperature. The energy reference zero point is set at the average energy over the three uppermost valence bands i.e. heavy hole, light hole and split-off hole. The negativity of Sn’s band gap has pulled down the conduction band minimum (CBM) to form a quantum well. Figure 1(b) shows the temperature effect on the band structure. A lower temperature enlarges the band gap of both Γ and L valleys according to Varshni’s experimental equation. There are two sets of coefficients for Γ and L which indicates that these two valleys react differently towards the changing temperature. Figure 1(c) is showing the dispersion curve in Γ valley in well region. The unit for the X axis is, where equals to the sum of well width and barrier width. The lowest transition energy is 782 meV from E1 to HH1. Partially relaxed bulk GeSn alloy grown virtual Ge substrate has already been reported to successfully change from indirect band-gap material to direct band-gap material by adding 12.6% Sn using the specialized chemical vapor deposition method [4]. Without the strain relaxation, the strain in quantum well will cancel out the inversion of two valleys as shown in Fig. 1(d).
Fig. 1 (a). Energy band structure of Ge0.95Sn0.05/Ge at room temperature. 1(b). The shift of energy of Ge0.95Sn0.05/Ge quantum well as a result of rising temperature. 1(c). Electron and hole energy dispersion curves of Ge0.95Sn0.05/Ge quantum well at 200k. 1(d). Strain effect on GeSn0.126/Ge Quantum well at room temperature.
The temperature and doping level dependence of spontaneous rate at direct Γ point are shown in Fig. 2. At 200 K, the first peak of the SERS locates at about 780 meV (see Fig. 2(a)) which is mostly attributed to the direct transition from E1 to heavy hole 1 (HH1) and partly to the transition from E1 to light hole 1 (LH1) at Γ point corresponding to the lowest to lowest transition energy in E-k dispersion graph shown above. The second peak locates at 830 meV corresponding to the third lowest transition energy from E2 to HH2. It demonstrates a normal temperature dependent SERS for direct band-gap material. A higher temperature results in a larger proportion of free electrons at CBM getting excited onto upper energy states which is the reason for the SERS decreasing. A blue shift is observed when keep lowering the temperature. This is due to the negative correlation between temperature and energy gap where E1 is pushed up when temperature goes down. Figure 2(b) is showing the effect of doping concentration on SERS. The temperature and injection electron-hole pair concentration are fixed at 200 K and 2x10−18 cm−3 while the doping level varies from 0 to 1x1019 cm−3. Transition of the higher levels is highly enhanced by doping comparing to the intrinsic one. The electrons of the donors first occupy the lowest E1 level when doping is started. And after E1 is fully occupied, electrons begin to occupy the higher E2 levels when the doping level rises from 5x1018 cm−3 to 1x1019 cm−3.
Fig. 2 (a). Temperature dependence of TE spontaneous emission rate spectra with no doping and 2(b). doping concentration dependence of TE spontaneous emission rate spectra at 200 K where only direct Γ valley is considered in GeSn0.05 quantum well.
The temperature and doping dependence of spontaneous emission rate with indirect L valley included are shown in Fig. 3. At 200 K, the peak of the spontaneous rate is observed at 780 meV corresponding to the direct transition from E1 to HH1 but the strength reduces to 1/10000 of the peak in previous condition where only direct Γ valley is considered. In this case, the minimum of conduction band is not at Γ point but at L point where most electrons stay. At higher temperature and/or doping concentration, more electrons will occupy the energy levels in Γ valley. This explains why the higher temperature and doping concentration can boost the SERS.
Fig. 3 (a). Temperature dependence of TE spontaneous emission rate spectra with no doping and (3b). doping concentration dependence of TE spontaneous emission rate spectra at 200 K where the contribution of indirect L valley is considered in GeSn0.05 quantum well.
Contrary to the direct gap III-V materials such as GaAs, indirect gap materials such as Ge, the strength of SERS increases with temperature as shown in Fig. 3(a). This agrees to the result of tensile strained Ge-on-Si reported by Sun at el [7] and Wirth at el [4]. The very close band edges between L point and Γ point due to the induced Sn atom make electrons in L valley to be easily exited into the states in Γ valley. This consequence enhances the direct transition from the conduction band to the valence band in Γ valley.
The lowest electron levels in both valleys with the change of Sn composition are shown in Fig. 4(a). Although the rising Sn composition drags down the E1 of Γ point, but it is not sufficient yet to make it below the E1 of L point. This situation is significantly different from partially relaxed bulk material where the Γ and L electron energy level cross at Sn composition of 12.6%. The reason is the strain effect caused by the mismatch of lattice constant between Ge substrate and GeSn thin film as shown in Fig. 4(b). Without strain, the CBM of Γ point is lowered dramatically by the increasing Sn composition which is the case of bulk material. But when considering quantum well, the energy offset induced by the strain is also increased dramatically with increasing Sn composition. The consequence of these two effects is the almost unmoving CBM, the E1 in Γ valley. The after-strain CBM is the summation of the before-strain CBM and the strain induced offset. This restriction by QW has limited the possibility of GeSn to realize the transformation from indirect band-gap material in direct band-gap material.
Fig. 4 (a). The lowest electron energy levels at L and Γ points of the GeSn QW with different Sn composition at room temperature. Figure (4b). The strain effect of different Sn composition on the CBM at room temperature.
Engineering of other relevent parameters can be useful to minimize the negative effect caused by strain. The main purpose of this engineering is to lower down the E1 in Γ valley, while E1 in L valley remains unchanged. Figure 5 is showing one possible way to achieve this which is increasing the well width. Because electrons of two valleys have very different effective masses, this method can greatly changes the CBM of Γ point meanwhile leave tiny influences on the CBM of L point. Other methods such as partly relaxing the compressive strain are also worth trying to solve this problem. The detailed influence of different well width is clearly shown in the recent work reported by Maczko et al [19]. They also reported achieving successful indirect to direct band gap transferring in GeSn/Ge QWs with Sn concentration reaching 12% and above which is quite different from ours. The reason could be the effective mass of electron in Sn for their work is a positive value which is 0.058m0. In our case, the effective mass is −0.058m0 as Sn is a semimetal whose effective mass should be set as negative, see Moontragoon et al [20]. As a result, the induced Sn atoms will negatively contribute to the effective mass of electrons in well region, which cause the smaller electron effective mass than that in Ref [19]. After finishing this paper, an optically pumped 2.5 μm GeSn laser on Si operating at 110 K was reported by Al-Kabi at el in Ref [21]. The work indicates the promising potential application of GeSn for the integrated photonics in the future.
Fig. 5 The lowest level E1 of Γ point in Ge0.8Sn0.2/Ge QW with different well width at room temperature.
4. Conclusion
We developed a model for the simulation of Ge1-xSnx/Ge QW. The contribution of the L valley is included. Based on this model, we calculated E-k dispersion curves of both direct Γ and indirect L valleys, and spontaneous emission rate spectra of Ge1-xSnx/Ge QW. We observed and explained the abnormal temperature dependent photoluminescence for the pseudo-direct gap material when the amount of Sn is smaller. Our model indicates that the transformation of this material from indirect band-gap to direct band-gap material is hard to achieve in the quantum well. The resulting compressive strain from the GeSn alloy will significantly compensate the contribution of Sn’s negative band gap, so the strained band structure almost remains unchanged. The quantized energy levels in indirect valley are always below the ones in the direct valley which makes the transition from indirect to direct band gap difficult to occur. This theoretical work can be the guidance of experiment based on GeSn QW.
Funding
We would like to acknowledge the support from Ministry of Education (RG 182/14 and RG86/13), A*Star (1220703063), Economic Development Board (NRF2013SAS-SRP001-019) and Asian Office of Aerospace Research and Development (FA2386-14-1-0013).
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