This paper presents modeling and simulation of a silicon-based group IV semiconductor injection laser diode in which the active region has a multiple quantum well structure formed with Ge0.9Sn0.1 quantum wells separated by Ge0.75Si0.1Sn0.15 barriers. These alloy compositions were chosen to satisfy three conditions simultaneously: a direct band gap for Ge0.9Sn0.1, type-I band alignment between Ge0.9Sn0.1 and Ge0.75Si0.1Sn0.15, and a lattice match between wells and barriers. This match ensures that the entire structure can be grown strain free upon a relaxed Ge0.75Si0.1Sn0.15 buffer on a silicon substrate – a CMOS compatible process. Detailed analysis is performed for the type I band offsets, carrier lifetime, optical confinement, and modal gain. The carrier lifetime is found to be dominated by the spontaneous radiative process rather than the Auger process. The modal gain has a rather sensitive dependence on the number of quantum wells in the active region. The proposed laser is predicted to operate at 2.3 μm in the mid infrared at room temperature.
©2010 Optical Society of America
This paper focuses on laser diodes (LDs) that utilize the new silicon-germanium-tin technology [1,2] capable of true monolithic integration on silicon or SOI in a CMOS Fab. These miniature waveguided LDs would be key “enablers” for chip-scale networks of group IV active and passive components. This paper addresses the issue of CW room-temperature operation at wavelengths beyond the 1.55 μm telecom band, wavelengths approaching the mid infrared. Our previous work on a 1.8 μm SiGeSn/GeSn/SiGeSn double heterostructure (DH) laser  showed that Auger recombination was a limiting process that constrained the LD to operate at temperatures around 200K. By contrast, the multiple-quantum-well (MQW) approach proposed in this paper offers suppression of Auger recombination to the extent that carrier lifetime is dominated by spontaneous radiative recombination. This MQW offers a lower density of states in the active layers, resulting in a lower electron-and-hole concentration required for population inversion. The MQW is significantly better than the DH because of its lower threshold and higher temperature of operation, including 300K.
This article presents design-and-simulation results on a waveguided electrically injected PIN MQW diode in which the active region has GeSn layers as QWs and SiGeSn ternary layers as barriers. This SiGeSn serves also as waveguide cladding regions. The compositions of GeSn and SiGeSn are chosen to provide type-I band alignment at the Γ-point as well as lattice matching. The laser would be situated on a relaxed buffer layer of SiGeSn-upon-silicon or SOI , a buffer whose lattice parameter is the same as that of the GeSn/SiGeSn laser; hence the entire laser structure is unstrained. The band-to-band MQW laser diode simulated in this paper has an emission wavelength of 2.3μm in the mid infrared. It advances applications of mid-IR semiconductor lasers in chemical and biological sensing, medical therapy, free-space communication, spectroscopy-on-a-chip and it aids the development on-chip laser radar transmitters. The results reported here compliment the previous simulations of a 1.55-μm GeSn-quantum-well laser , a strained GeSn/GeSiSn QW laser , and a Terahertz Ge/SiGeSn quantum cascade laser . Our strain-free MQW should be easier to implement than the strained prior-art QW lasers [5,6]. The analysis of carrier lifetime presented here includes recombination effects due to radiative and non-radiative Auger processes. Our result indicates that this MQW laser would operate at room temperature.
2. Band structure of GeSn/GeSiSn quantum wells
Energy-band theory  and FTIR absorption experiments  have indicated that the bandgap of unstrained crystalline GeSn makes a transition from indirect to direct as the percent of α-Sn is increased. Since the band offsets between ternary Sn-containing alloys and Si or Ge are not known experimentally, we follow the assumptions made in Ref . and calculate the conduction-band minima for the lattice-matched heterostructure consisting of Ge1-zSnz and a ternary Ge1- x - ySixSny. We used Jaros' band offset theory  which gives results in good agreement with experiment for many heterojunction systems. For example, this theory predicts an average valence band offset, eV for a Ge/Si hetero-interface (higher energy on the Ge side), close to the accepted value of eV. The basic ingredients of our band-alignment calculation are the average valence-band offset between the two materials (an average between heavy, light, and split-off hole bands) and the compositional dependence of the ternary-alloy’s band structure. For the Ge/-Sn interface. Jaros’ theory predicts eV (higher energy on the α-Sn side). Thus, relative to the average valence band of Ge, the average valence band position for Ge1- x - ySixSny is simply a linear interpolation
Similarly, with these spin-orbit splitting values eV eV, eV , the spin-orbit splitting for Ge1- x - ySixSny is
The top of the valence band for Ge1- x - ySixSny can then be determined as
The minima of the conduction band at points L and Γ can then be calculated by evaluating the compositional dependence of the band gaps of the ternary alloy as13,14]. These values at L and Γ points have been given in Table 1 .
Finally, for the indirect conduction band minimum near the X-point, Weber and Alonso find14]. On the other hand, the empirical pseudopotential calculations of Chelikovsky and Cohen place this minimum at 0.90 eV in α-Sn, virtually the same as its value in pure Ge . We thus assume that the position of this minimum in ternary Ge1- x - ySixSny alloys is independent of the Sn concentration y, and thus is also given by Eq. (5). Obviously, the calculation of band structures outlined above is an approximation that is subject to experimental corrections as more measurements become available. This implies that the compositions of Ge1-zSnz and Ge1- x - ySixSny are necessarily adjusted in the QW structure to arrive at the band structure that is being proposed here. But it should be pointed out that the laser behavior depends only on the band structure, and that the results obtained in this design should be valid albeit at slightly different binary and ternary compositions, and possibly at slightly different lasing wavelengths.
The α-Sn composition dependence of the conduction band gaps for Ge1-zSnz at the three valleys , Γ, and X is first calculated using Eqs. (4) and (5) to establish the crossing point where the Γ-point band gap drops below that of the L-point. Figure 1(a) shows that for α-Sn composition greater than , the Ge1-zSnz gap becomes direct. We thus choose Ge0.9Sn0.1 to be the QW layer with a direct band gap of eV. Fixing at this Ge0.9Sn0.1 composition, we then looked for a lattice matched Ge1- x - ySixSny that can be used as barriers that form type-I band alignment with Ge0.9Sn0.1. Such a simultaneous requirement for lattice parameter and band alignment can be satisfied by the additional degree of freedom in the Ge1- x - ySixSny where both and can be tuned. Using Vegard’s law for the lattice constant of Ge1- x - ySixSny, the lattice constant of Ge1- x - ySixSny is11], respectively, and we can vary the Si and α-Sn compositions in GeSiSn simultaneously to yield exactly the lattice constant of Ge0.9Sn0.1. Adding the band gaps to the top of the valence band Eq. (3), we obtain the band alignment between at the Γ-point as shown in Fig. 1(b).
There is a wide range of Sn composition over which the ternary Ge1- x - ySixSny forms type-I confinement with Ge0.9Sn0.1, i.e., both elections and holes are confined in the Ge0.9Sn0.1 QWs by the Ge1- x - ySixSny barriers. In particular, we choose barriers with the composition Ge0.75Si0.1Sn0.15 that gives the largest conduction band offset of meV, and offset of the valence band meV as shown in Fig. 1(b). The laser device shall be grown on a relaxed Ge0.75Si0.1Sn0.15 buffer on a Si substrate to ensure that the entire structure is strain free as illustrated in Fig. 2 .
3. Carrier lifetime
The proposed laser device has MQWs in its active region. Since the compositions of Sn and Si are relatively small in comparison with that of Ge for either QWs or barrier layers, we shall use Γ-point Ge parameters in the following calculations. The quantum confinement leads to energy subbands in both conduction and valence bands. The energy levels of these subbands can be calculated by solving the one-dimensional Schrödinger equation following the envelope function approximation . The laser device under consideration has a forward-biased PIN structure where the active MQW region is undoped. The band-to-band lasing transitions occur as stimulated emissions triggered by recombination of electron-hole pairs that are injected into this region. The analysis proceeds as follows. For a given carrier density, we can derive quasi Fermi levels at a specific temperature (T) for electrons in the conduction band () and for light holes (LHs) and heavy holes (HHs) in the valence band (). However, only the electron-HH pair recombination contributes to lasing transitions since the ground-state HH subband lies lower-in-valence energy than that of the LH subband. The structure that we have calculated was chosen to have 20nm Ge0.9Sn0.1 QWs that are separated by 20nm Ge0.75Si0.1Sn0.15 barriers. The energy separation between the ground-state electron and HH subbands has been determined to be eV, which is 36meV larger than the Ge0.9Sn0.1 band gap (0.505eV) due to the quantum confinement.
In order to estimate the carrier lifetime, it is necessary to calculate the radiative as well as the nonradiative Auger recombination rate in the MQW active region. The radiative process is spontaneous consisting of electron-heavy hole (e-hh) as well as electron-light hole (e-lh) recombination. The spontaneous emission rate per unit area in the energy interval due to the e-hh process can be calculated as ,16]. The occupation probabilities at the states that are separated by a photon energy E with the same k in the reciprocal space are
The radiative lifetime can then be obtained by where n is the area carrier density. The result for the area carrier density of /cm2, corresponding to a carrier concentration of 1018/cm2 in QW layers for the well thickness of d = 20nm, is shown in Fig. 3 for a range of temperature. It can be seen that the radiative lifetime for a fixed carrier density is rather insensitive to temperature change, showing a slight increase with the temperature.
This radiative carrier lifetime can be shown to be much shorter than that of the Auger process where the recombination of an electron-hole pair takes place by transferring energy and momentum to a third particle which could be either an electron or a hole. For comparison, we have also estimated the Auger lifetime by following the calculation procedure outlined in Ref . The result for the same area carrier density of /cm2 determined by the Auger process is shown in Fig. 3. Clearly the Auger lifetime decreases rapidly with the increase of temperature, but even at T = 300K, it remains longer than that of the radiative process. We therefore conclude that the spontaneous radiative recombination is the dominant process in determining the carrier lifetime. In comparison with the DH laser that we have simulated earlier , this represents a significant improvement as a result of reduced density of states in QWs relative to that of bulk material, which leads to its potential room temperature operation.
4. Optical gain
In contrast to their advantage of having longer carrier lifetime, single-QW lasers typically have a very small optical confinement factor compared to that of DH lasers because their active regions are too thin relative to the lasing wavelength. Fortunately, the MQW structure offers a practical solution to the mode-overlap problem by increasing the effective thickness of the active region. We calculate the confinement factor by treating the MQW active region as having an index of refraction that is averaged between the QW index and the barrier layers index. Figure 4 shows the optical confinement factor for the fundamental TE mode of the Ge0.9Sn0.1 /Ge0.75Si0.1Sn0.15 QW laser for a range of QW numbers whose active regions consist of 20nm Ge0.9Sn0.1 QWs that are separated by 20nm Ge0.75Si0.1Sn0.15 barriers, and where the active region is cladded above and below by thick Ge0.75Si0.1Sn0.15 layers. The confinement factor Γ, defined as the spatial overlap integral of the TEo mode profile with the gain profile, increases from 0.003 for a single QW to 0.90 for 35 QWs.
For a given injected area carrier density , we can calculate the optical gain at a photon energy E due to e-hh recombination per period as Fig. 5 at T = 300K for the Ge0.9Sn0.1/Ge0.75Si0.1Sn0.15 QW laser with different numbers of QWs, , but all of the lasers have 20nm QWs and 20nm barriers.
Initially, the modal gain increases rapidly with pumping current as the injected carriers start to establish the population inversion between the ground-state electron and HH subbands and start to produce the optical gain at the photon energy that is equal to the energy separation between the bottom of the ground-state electron subband and the top of the ground-state HH subband. As the pumping current continues to increase, the occupation status of electrons and HHs at those extreme locations of the involved subbands will no longer change, i.e. and in Eq. (12) indicting that the maximum population inversion has been established. For different number of QWs, the modal gain reaches different saturation values. This is a direct result of the mode confinement factor as shown in Fig. 4. The modal gain of a laser must be sufficient to compensate for the various losses in the device such as the free carrier absorption and imperfect mirror reflectivity. Figure 5 shows that an adequate number of QWs must be designed to overcome a certain level of losses. For instance, 20 QWs are needed to provide modal gain of just over 100/cm with a current density of 3kA/cm2. In general, active regions consisting of a larger number of QWs are more capable of providing higher modal gains. This obviously creates strict demands upon the structural growth that must offer very fine control of layer thicknesses as well as uniformity of the layer thickness and alloy compositions. Fortunately, the recent result on epitaxial techniques exhibiting very fine control of GeSiSn alloy layers has indeed opened a pathway to developing GeSn QW lasers . The proposed laser design utilizes a lattice matched structure that conveniently avoids the situation of strain development as more QWs are deposited – a welcome factor for the device growth.
It may be desired to increase the laser’s emission wavelength into the 3 to 5 μm band (the atmospheric transmission window). Then it is necessary to increase the Sn content of the QWs beyond 10% and to change the barrier composition to lattice-match the new QWs. Having done this for λ = 3.5 μm, we found that the well/barrier conduction-band offset decreased to about 20 meV, a value not sufficient for good confinement of electrons. However, this offset problem may be solvable by employing SiGeSn QWs along with ternary barriers.
We propose a simple group-IV laser made of a PIN-diode Ge0.94Sn0.06 /Ge0.75Si0.15Sn0.1 MQW active region wherein the direct-gap Ge0.9Sn0.1 QWs are confined by Ge0.75Si0.1Sn0.15 barrier layers. The optical channel-waveguide confinement is provided by the same ternary Ge0.75Si0.1Sn0.15 cladding layers of large thickness. The compositions of both the QW and the barrier/cladding layers are determined to yield lattice matching and type-I band alignment between them. The laser structure would be grown on a relaxed Ge0.75Si0.1Sn0.15 buffer on either Si or SOI, hence the device is strain free. Both radiative and nonradiative Auger processes are included in the simulation of the carrier recombination rate. It is shown that the carrier lifetime is determined by the radiative process rather than the Auger process. In particular, we have analyzed a MQW laser with an active region consisting of 20nm Ge0.94Sn0.06 wells and 20nm Ge0.75Si0.15Sn0.1 barriers that are optically confined by thick Ge0.75Si0.1Sn0.15 cladding layers. The quantum confinement leads to an energy separation between the ground-state electron and HH subbands equal to 0.541eV, which yields a lasing wavelength of 2.3μm. Optical confinement varies over a wide range depending on the number of QWs employed in the active region. Modal gain is calculated as a function of injection current density for a range of QW numbers at room temperature. For a laser with 20 QWs, the optical confinement factor reaches 0.74 and the modal gain can exceed 100/cm for a pumping current density of 3kA/cm2, which is sufficient to compensate for losses in mid IR semiconductor lasers. It can be concluded that the implementation of this laser will lead to the first electrically injected group-IV near/mid-IR laser capable of operating at room temperature, a laser that is integrated on Si or SOI.
This work was supported in part by the Air Force Office of Scientific Research, Dr. Gernot Pomrenke, Program Manager.
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