## Abstract

Threshold carrier densities of GeSn quantum well (QW) lasers and the physical reason of low-temperature lasing of current GeSn laser are investigated through the comparison of threshold carrier densities of conventional III-V QW lasers. Electrons distributed over L-band is the main cause of decreased gain for GeSn QWs. To increase the gain (and improve the laser characteristics), a modulation-doped GeSn QW is proposed and the material gain is analyzed based on many-body theory for both qualitative and quantitative simulation. Significant gain increase can be expected for n-type modulation doping QWs. The doping condition for elevated temperature lasing is discussed and it was found that material gain curve similar to III-V QW is obtained for GeSn QW with n-type modulation doping of 6 × 10^{18} cm^{−3}. It was also found that unlike III-V QW lasers, n-type modulation doping is more effective for high-speed operation in terms of differential gain than p-type modulation doping.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Optical devices using GeSn materials have been intensively studied recently since GeSn can be grown on Ge/Si substrate and becomes direct-gap material in mid-infrared (mid-IR) range by incorporating small amount of Sn into Ge [1]. So far, various active optical devices such as photo diodes, optical modulators, and lasers have been investigated [2–7]. Among them, the GeSn laser is the most important device since the monolithically integrable lasers on Si is a missing component in Si photonics. After the first report of GeSn laser [4], significant effort has been made to improve the performance of the GeSn laser [5–7]. Especially, increasing the operating temperature is critically important for integrated device application. So far, the improvement of the operating temperature has been mainly achieved by following methods. (1) By increasing Sn, GeSn becomes more direct-gap material and achieving population inversion becomes easier. Currently, the incorporation of Sn up to 17% [5,6] was realized. Using GeSn buffer layer helped to increase the composition of Sn. (2) Using doublehetero or quantum well (QW) structures [7] leads to stronger confinement of carriers. Especially, using QW is effective to reduce the density of states in the active region. (3) A microdisk structure can be used for strain engineering [5,7]. (4) Improved crystal quality with reduced misfit dislocation density [5–7]. Thanks to these efforts, the operation temperature is increased to 180 K [5,6] for the mid-IR wavelength range (2 to 3 μm). However, the value is still cryogenic and further improvement is highly desired.

In this paper, we theoretically discuss the operating temperature of GeSn QW laser through the comparison of threshold carrier density (*N*_{th}) of GeSn and conventional III-V lasers. Many-body theory (MBT) formulated for group IV material [8,9] is used for calculating material gain due to its strong predictability. It is shown that *N*_{th} of GeSn laser is three times larger than that of III-V laser for telecommunication [10] at room temperature due to large electron distribution in L-band. To reduce *N*_{th} of GeSn laser, modulation doping (MD) structure [11,12] is proposed as an alternative way for improving the operating temperature and the material gain of MD GeSn QW is analyzed. Significant gain increase can be expected for n-type MD and the technique is useful for increasing the operating temperature of GeSn laser. The doping condition for elevated temperature lasing is discussed and it was found that material gain curve similar to III-V QW is obtained for GeSn QW with n-type MD of 6 × 10^{18} cm^{−3}. It was also found that unlike III-V QW lasers, n-type MD is more effective for high-speed operation in terms of differential gain than p-type MD.

## 2. Device structure and theory

We consider Fabry-Perot laser based on the ridge waveguide structure [13,14], whose cross section is shown in the left part of Fig. 1. It consists of upper and lower p- and n-claddings (Si_{0.14}Ge_{0.7}Sn_{0.16}), 7QW region, and GeSn buffer layer on Si substrate. For QW, we consider Ge_{0.84}Sn_{0.16}/Si_{0.1}Ge_{0.76}Sn_{0.14} QWs on Ge_{0.88}Sn_{0.12} buffer layer. SiGeSn is used to make the bandgap of the barrier region large. The composition is determined to maximize the bandgap of SiGeSn with the condition of no strain in the barrier [8]. The well and barrier thicknesses are 10 nm and the strain of the well is −0.55% (compressive strain). The barrier bandgap wavelength is 2.2 μm and the barrier layer is lattice matched to the buffer layer. The bandgap wavelength of QW (bandgap between first conduction and valence subbands) is 2.9 μm at 298 K. This is the optimized structure for increasing the material gain for Ge_{0.88}Sn_{0.12} buffer presented in [8]. It should be noted that Ge_{0.88}Sn_{0.12} buffer was experimentally realized in [6]. The doping concentrations of upper p- and lower n-cladding are assumed to be 5 × 10^{18} and 1 × 10^{18} cm^{−3}. The free carrier absorption (FCA) loss coefficients for these doping densities are α_{f,p} = 28 and α_{f,n} = 15 cm^{−1} [15]. The quasi-TE mode field distribution calculated by finite-element method [16] is shown in the right part of Fig. 1. The waveguide width, the layer thicknesses are denoted in the left part of Fig. 1. The calculated confinement factors of the TE mode in the well, p-cladding, and n-cladding are Γ_{well} = 0.06, Γ_{p} = 0.27, and Γ_{n} = 0.41. We assume that the facets are cleaved and the total mirror loss is α_{m} = 20 cm^{−1} for the cavity length of 500 μm. *N*_{th} of the laser can be calculated by

*g*

_{th}is the threshold gain. α

_{f,well}is FCA loss coefficient in the well and depends on the injected carrier and MD densities. α

_{f,well}is calculated by Drude model [13] and given by

*e*is the electron charge, λ is the wavelength,

*c*is the speed of light in vacuum, ε

_{0}is the free-space permittivity, and

*n*

_{r}is the refractive index of the material.

*n*

_{Γ}and

*n*

_{L}are the electron densities in Γ- and L-band.

*p*is the hole density in the valence band. μ

_{Γ}and μ

_{L}are the mobilities of electrons in Γ- and L-band. μ

_{p}is the hole mobility. These mobilities are the function of injected and doping densities.

For calculating material gain, we use MBT [8,9]. The microscopic polarization *p _{kt}* is calculated by solving Semiconductor Bloch equation given by

*f*and

_{c}*f*are the Fermi distributions for electrons and holes. The definitions of other terms can be found in [8]. By solving (3), the microscopic polarization is calculated and a macroscopic polarization,

_{v}*P*, is obtained by summing the microscopic polarization over all the states. The macroscopic polarization is related to material gain through Maxwell’s equations as

*V*is the volume,

*μ*is the dipole matrix element, and

_{kt}*E*

_{0}is the electric field of light. Since MBT can exclude artificial fitting parameters used in conventional approach, one can grasp inherent optical properties of QW both qualitatively and quantitatively. In [9] and [10], it was shown that calculated optical properties of QW are in good agreement with the measured ones for group IV and III-V materials.

## 3. Material gain and threshold characteristics

#### 3.1 GeSn/SiGeSn QWs without MD

Solid lines in Fig. 2 show the calculated peak material gain of GeSn QW as a function of injected carrier density for different temperature without MD. Dashed line shows the material gain of 1.3-μm InAlGaAs QW at 298 K presented in [10]. An open circle on the dashed line shows *N*_{th} of this laser and is 1.58 × 10^{18} cm^{−3} (*g*_{th} = 350 cm^{−1}) [10]. These values are used for representative ones of direct-gap III-V QW lasers. Of course, this is not perfectly fair comparison (since the laser using this InAlGaAs QW can lase at even 368 K [17], it is severe for GeSn QWs for room temperature comparison), we use these values to grasp inherent reason of cryogenic lasing of current GeSn laser. The red line in Fig. 2 shows the gain curve of GeSn QW laser at 298 K. The transparent density is significantly increased and the curve is shifted toward large density side. If we assume *g*_{th} of GeSn QW is similar (the assumption is valid as shown later), *N*_{th} of GeSn QW laser is about 4 × 10^{18} cm^{−3} at 298 K and almost 3 times larger than that of III-V QW laser. The increased *N*_{th} originates from the existence of L-band. Figure 3 shows the conduction Γ- and L-band structures of GeSn QW. Two band structures are plotted in the same graph for comparison. The effective mass of L-bandedge for [110] direction is very large, leading to the large density of states. Since the injected carriers are filled from lower energy side, the majority of injected electrons is distributed in L-band. For this QW (which is optimized to maximize the gain for Ge_{0.88}Sn_{0.12} buffer), the ratio of injected carrier density distributed in Γ-band is only 10% (In this optimized QW [8], the energies of lowest subbands of Γ- and L-band are almost the same). The increased carrier results in increased non-radiative recombination and generating heat in the active region. Especially, the effect is significant for Auger recombination since it is proportional to *N*^{3}. Increased temperature in the active region reduces the material gain and increases the carrier loss, leading to non-lasing. For low temperature, the carrier density for obtaining the same material gain is reduced. Green and blue lines in Fig. 2 show the gain curves of GeSn QW for 200 and 150 K. For lower temperature, the gain curve shifted toward small density side because it is easy to achieve population inversion (*f*_{c} + *f*_{v}-1 term in Eq. (3) becomes positive with reduced density). At 150 K, the gain curve of GeSn QW is very similar to that of III-V laser for the region of material gain < 1500 cm^{−1}. It means that similar material gain required for the lasing in III-V QW laser can be obtained for GeSn QW with similar carrier density. It is very interesting that this temperature is comparable to the maximum operating temperature of current GeSn laser (~180 K). From these results, for achieving high-temperature operation, *N*_{th} should be reduced for GeSn laser.

#### 3.2 GeSn/SiGeSn QWs with MD

To reduce *N*_{th}, we consider MD QW. MD QW is composed of highly doped barriers and nondoped well layers. For example, for n-type doping, electrons are supplied to the well layers from ionized donor impurities in barriers. For p-type doping, holes are supplied to the well layers from barriers. By using MD, achieving the population inversion in QW becomes easier because carriers in the well is increased for the same injection level, leading to increased material gain. The effect of MD on the laser performance was experimentally confirmed in [12]. Here, we consider n- or p-type MD QW. The MD density is *N _{MD}*. For calculating the material gain of MD QW, the quasi-Fermi levels for conduction and valence bands are calculated by following equations. For the n-type doping, they are given by

*N*and

*P*are total electron and hole densities in the well.

*N*

_{inj}and

*P*

_{inj}are the injected carrier densities.

*N*

_{Γ}and

*N*

_{L}are the electron densities in Γ and

*L*conduction band. The summation extends over the subbands of QWs.

*L*

_{w}is the well thickness and

*k*

_{t},

*k*

_{1}, and

*k*

_{2}denote the transverse wavenumbers.

*f*

_{c}and

*f*

_{v}are Fermi-Dirac distribution functions for electrons and holes.

*E*

_{Γ}and

*E*

_{L}are the band structures of the QW for Γ and L bands calculated by

**•**

*k***theory [18]. For the n-type doping, the quasi-Fermi level of the conduction band is affected. For the p-type doping, the quasi-Fermi levels for conduction and valence bands are calculated by following equations.**

*p**N*at 298 K. Dashed line is for III-V QW [same as Fig. 2]. By increasing

_{MD}*N*, the material gain is increased significantly, and for high MD density, the carrier density for obtaining the same material gain is reduced. This is because due to n-type MD, L-band is somewhat filled with electrons without injection and the population inversion occurs with reduced carrier density. For

_{MD}*N*= 6 × 10

_{MD}^{18}cm

^{−3}, the gain curve of GeSn QW is similar to that of III-V QW. Figure 5 shows the calculated peak material gain of p-type MD GeSn QW as a function of injected carrier density for different

*N*at 298 K. As in the n-type doping, the gain is increased for p-type doping, however, the increase is smaller for the same doping density than that of n-type doping. In conventional direct-gap QW laser, the effective mass of the valence band is large compared with that of the conduction band. Therefore, p-type MD is more useful to achieve the population inversion. However, for group IV material considered here, the effective mass of the L-band is large and the carriers distributed in the L-band cannot be used for light emission. Therefore, n-type doping is more useful for GeSn QW.

_{MD}Figure 6 shows *N _{th}* as a function of

*N*. The horizontal dashed line shows the

_{MD}*N*of III-V QW laser [10].

_{th}*N*is decreased with

_{th}*N*. The reduction is larger for n-type MD as expected by the gain curves shown in Figs. 4 and 5. For n-type MD, at

_{MD}*N*= 7 × 10

_{MD}^{18}cm

^{−3}, the value of

*N*is similar to that of III-V QW laser. Figure 7 shows the differential gain at the threshold density as a function of

_{th}*N*. While the differential gain for n-type MD is increased with

_{MD}*N*, it is almost constant for p-type MD. Although for III-V QW, it is known that p-type MD is useful for high-speed operation [11,12], the tendency is opposite for GeSn QW. Therefore, for GeSn QW, n-type MD is useful for both reducing

_{MD}*N*and increasing the differential gain. Large differential gain is useful for a high-speed operation [17].

_{th}Although we used FCA loss calculated by Drude model for estimating the threshold, the value may be increased due to various reasons (fabrication imperfection, degraded crystal quality, etc). Also, there are a lot of uncertainties in this material system in terms of loss, such as waveguide sidewall loss, etc. To investigate the effect of increased loss for MD QW, we intentionally increased the FCA loss and calculate the threshold. Blue line in Fig. 8 shows *g*_{th} as a function of *N _{MD}* for n-type MD QW. All the results in Fig. 8 are for 298K.

*g*

_{th}is increased for large

*N*because MD increases FCA loss of the active region. The value is several hundreds of cm

_{MD}^{−1}and comparable to III-V laser. Red line in Fig. 8 shows

*N*

_{th}as a function of

*N*calculated by Eq. (1) [same as Fig. 6]. Green, orange, and purple lines show

_{MD}*N*

_{th}for intentionally increased loss (1.5, 2, and 3 times larger than

*g*

_{th}). Up to 2

*g*

_{th},

*N*

_{th}comparable to III-V laser can be obtained for

*N*< 10

_{MD}^{19}cm

^{−3}. For 3

*g*

_{th}, although it seems to be difficult to achieve

*N*

_{th}comparable to III-V laser, the threshold reduction is possible by using MD QW. Reduced

*N*leads to reduced nonradiative recombination and high-temperature operation is expected.

_{th}## 4. Conclusion

Material gain of MD GeSn QW is investigated by MBT. Through the comparison of threshold carrier density with conventional direct-gap III-V QW laser, the reason for preventing GeSn laser from high-temperature lasing is discussed. By using n-type MD QW, the threshold carrier density can be reduced to the level of III-V QW laser. Although the fabrication of MD QW may be challenging task at this stage, this is a promising technique for increasing the operating temperature of GeSn laser.

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