Silicon photonics have diverse applications in optoelectronic integration, high-speed optical communications, and ultrafast signal processing. The key Si photonic components, including modulators, photodetectors, waveguides, filters, etc., have been demonstrated.^[1] But the indirect band-gap of group IV semiconductor challenges the realization of laser sources because of the very low probability of indirect radiative recombination. However, germanium has a slight energy difference (136  meV) between the direct band-gap and the indirect band-gap, which increases the probability of radiative recombination between the Γ valley and the valence band. It has been indicated that heavy n-type doping and tensile strain engineering can effectively provide population inversion in the direct band Γ valley of Ge, and a net gain of 400/cm can be obtained theoretically.^[2] Using this method, researchers have recently realized optically pumping^[3] and electrically pumping^[4] Ge lasers. However, the electrically pumping Ge laser needs heavy n-type doping and a high threshold current density (about 218  kA/cm²), which induces the intensive free-carrier absorption (FCA) and instantaneous breakdown of the metal contact. Such extreme conditions greatly hinder its practical applications. Adequate tensile-strain to reduce the energy difference between Γ valley and L valley plays a pivotal role in reducing the threshold current. However, it is very difficult to achieve high tensile-strain in Ge while keeping its high crystal quality and sufficient thickness.

Fortunately, the introduction of Sn into Ge is expected to diminish the direct band-gap to a larger extent than the indirect band-gap. High quality Ge_{1− x}Sn_x alloy epitaxy by MBE was reported from our previous work.^[5] Recently, Ge_{1− x}Sn_x p– i– n photodetectors for detecting all telecommunication bands (x = 3%)^[6] and short-wave infrared (x = 3.64%)^[7] have been reported. Moreover, many experiments and theories have indicated that a Ge_{1− x}Sn_x alloy with a Sn percentage of about 6%-10% is a kind of prospective direct band-gap material.^{[8– 11]} Several experiments have successfully been conducted to study the photoluminescences of Ge_{1− x}Sn_x materials. Mid-infrared electroluminescence from GeSn/Ge heterojunction on a Si substrate has also been reported.^{[12, 13]} In Ref.  [14], an electrically injected p– i– n SiGeSn/Ge_0.922Sn_0.078/SiGeSn double heterostructure (DHS) diode was designed and it was estimated that Ge_{1− x}Sn_x becomes a pseudo-direct band semiconductor when the Sn composition was about 6%, while the bowing parameter about direct band-gap used in the paper does not accord with the latest experimental result.^[11] In addition, the key factor of the FCA effect was not considered.

In the present paper, the optical gain characteristics of direct band Ge_{1− x}Sn_x alloys are systematically investigated. The effects of the L indirect band valley, the temperature, the injected current density, the proportions of Sn, the Auger process, and the free-carrier absorption effect are taken into consideration in our calculations. A double heterostructure Ge_0.554Si_0.289Sn_0.157/Ge_0.922Sn_0.078/Ge_0.554Si_0.289Sn_0.157 short-wave infrared laser diode is then designed to achieve high injection efficiency and low threshold current density. Finally, the device threshold current densities are simulated at different temperatures and wavelengths.

The physical parameters of Ge_{1− x}Sn_x or Si_xGe_{1− x − y}Sn_y can be obtained using the linear interpolation formulae, excluding the band-gap. There is a nonlinear relationship between alloy band-gap and the alloy composition x. As far as Ge_{1− x}Sn_x alloys are concerned, the band-gap is usually calculated by a parabolic function, which is listed as follows:

where x is the Sn percentage, y is the Si percentage, and b is bowing parameter.

For a Ge_{1− x}Sn_x alloy, b_L = 0.89^[8] and b_Γ = 2.1^[11] are cited from the latest result; for GeSi and SiSn, b_Γ = 0.21 and 13.2, ^[15] respectively. Figure  1(a) shows the direct band-gap E_{g, Γ} and indirect band-gap Eg, L of Ge_{1− x}Sn_x alloy, each as a function of Sn percentage at room temperature. The band parameters of Si, Ge, and Sn are cited from the data handbook.^[16] In order to obtain a more precise result the temperature-dependence of the Ge band-gap is taken into account (that of Sn is neglected because of its low percentage), as follows:^[17]

As figure  1(a) shows, when x (Sn) equals 7.8%, the direct and indirect band-gap reach the same value (about 0.56  eV). Based on this model, the band parameters of Ge_{1− x}Sn_x alloys, with x in a range between 0% and 15% (0%, 3%, 6%, 7.8%, 10%, 12%, and 15% are chosen in the calculation) are calculated and then utilized for the next optical simulations.

The E– k relation between electron and hole are assumed to be parabolic, as follows:

And the photon energy hν emitting through Γ – h transition is given as follows:

According to Eq.  (5), the electron energy in the Γ valley E_e and hole energy E_h are described by

In a steady state, electrons and holes obey quasi-Fermi distribution, which is described by electron quasi-Fermi level E_fc and hole quasi-Fermi level E_fv. When the energy difference between an electron (E_e) and a hole (E_h) with the same k in the reciprocal space is equal to photon energy E = hν , the occupation probabilities of an electron and a heavy hole are described, respectively, by

Quasi-Fermi level values (E_fc and E_fv) corresponding to injected carrier density N_inj = P_inj and doping density N_D under a degenerate condition (with heavy doping and high injection) can be obtained by the calculation method from the second chapter of Ref.  [18] through the following relationship:

The percentage of electrons occupying Γ band valley can then be obtained as

Compared with the direct Γ band valley, the indirect L band valley of Ge or Ge_{1− x}Sn_x can accommodate a majority of electrons because of its lower energy valley, high degeneracy (s = 4), and the bigger effective mass of the electron. Even worse, the electron in the indirect L band valley has a very low emission transition probability and high contribution of free-carrier absorption. While the key point for the laser process is to achieve considerable stimulated radiation from a direct Γ valley, the indirect L band valley has a serious negative influence on this material luminescence quantum efficiency.

Figure  1(b) shows the percentage of electrons occupying the Γ band valley of Ge_{1− x}Sn_x with x value being between 0% and 15% (n-doping density: 1 × 10¹⁸/cm³, injected carrier density is 1 × 10¹⁹  cm^{− 3}; room temperature: 300  K). It is indicated that the percentage of electrons occupying the Γ band valley of Ge_0.88Sn_0.12 is 1000 times as high as that of Ge; however, that of pseudo-direct band Ge_0.922Sn_0.078 alloy is still only about 1%, as shown in Fig.  1(b).

Fig.  1.

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	Fig.  1. (a) The L and Γ conduction band band-gaps of Ge1− xSnx alloy each as a function of Sn percentage at room temperature. (b) The variation of the percentage of electrons in Γ conduction valley of Ge1− xSnx with x in a range between 0% and 15% at room temperature.

After the calculation of band structure and carrier distribution, the optical gain spectrum (g_{Γ − hh}) and the spontaneous emission spectrum (R_{spΓ − hh}) through Γ – hh radiative recombination can be obtained; the formulae are given as follows:^[19]

where e is the electron charge, m₀ is the free electron mass, ɛ ₀ is the permittivity of vacuum, h is the Planck constant (ħ = h/2π ), c is the speed of light in vacuum, n₀ is the index of refraction, m_cΓ is the effective mass of electron in Γ band valley, m_hh is the heavy hole mass, m_lh is the light hole mass, E_g is the band gap, and is the average matrix element for the Bloch states.

The optical gain spectrum g_{Γ − lh} and the spontaneous emission spectrum R_{spΓ − lh} through Γ – lh radiative recombination can be obtained in a similar way, i.e., just replacing m_hh in Eqs.  (10) and (11) with m_lh. Thus, the total optical gain spectrum G (hν ) and the spontaneous emission rate R_sp are given as follows:

The total optical gain spectra of Ge_{1− x}Sn_x with different Sn percentages and with a same injected carrier density (5 × 10¹⁸/cm³) and doping density (5 × 10¹⁸/cm³) at room temperature are then calculated and shown in Fig.  2(a). The optical gain significantly increases with the increase of Sn percentage. The maximal gain coefficient of Ge_{1− x}Sn_x alloy increases from 200/cm (x = 7.8%) to 1000/cm (x = 15%), which indicates that the introduction of Sn into Ge can promote optical gain evidently and may reduce threshold carrier density. The decrease of threshold carrier density also reduces the FCA and Auger recombination process, which will further lower the threshold current density.

Fig.  2.

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	Fig.  2. (a) Total optical gains of Ge1− xSnx (x = 0, 3, 6, 7.8, 10, 12, 15% respectively) with a same injected carrier density (5 × 1018/cm3) and doping level (5 × 1018/cm3) at room temperature. (b) Optical gain spectra of alloy with 7.8% Sn at different temperatures.

Optical gain spectra of alloy with 7.8% Sn (injected carrier density: 5 × 10¹⁸/cm³, n-doping level: 3 × 10¹⁸/cm³) at different temperatures are simulated. As figure  2(b) indicates, it is hard for Ge_0.922Sn_0.078 to realize optical amplification at 300  K under this carrier density condition, but a low temperature can improve optical gain effectively and the maximal gain can reach 500  cm^{− 1} at 100  K. The blue shift of maximal gain is due to the band-gap broadening at low temperatures.

The optical gain spectra of a Ge_0.922Sn_0.078 alloy with four different injected carrier densities are also studied (n-doping concentration: 5 × 10¹⁸/cm³, and temperature: 300  K), as shown in Fig.  3(a). Although Ge_0.922Sn_0.078 has become a direct band-gap semiconductor, a high injection density, such as 6 × 10¹⁸/cm³, is needed to achieve population inversion in direct Γ band valley and provide significant positive optical gain. In addition, a comparison between Ge_0.922Sn_0.078 and Ge with an identical injected carrier density of 9 × 10¹⁸/cm³ is made. It is indicated by the dashed line and solid line in Fig.  3(a) that the maximum gain of Ge_0.922Sn_0.078 can reach 500  cm^{− 1}, while the optical gain of Ge is still negative.

Fig.  3.

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Fig.  3. (a) Optical gain spectra of Ge0.922Sn0.078 alloy with four different injected carrier densities and a comparison with Ge. (b) The optical gain and FCA spectra of Ge0.922Sn0.078 alloy simulated with the injected carrier densities of 5 × 1018, 9 × 1018, 2 × 1019  cm− 3, respectively. (c) Comparison between optical gain and FCA of Ge0.922Sn0.078 alloy and that of Ge0.9Sn0.1 on an identical injection and n-doping level at room temperature.

It is obvious that high injection could contribute to the achievement of high optical gain; however, the FCA effect is considerable in high doping or at a high injection level, which is bad for optical gain. The FCA effect, consisting of the absorptions of L-conduction valleys, Γ -conduction valley, heavy-hole band, and light-hole band, can be described by Drude– Lorentz equation^[20] as follows:

where λ is the free space wavelength; n_Γ and n_L are the electron densities in the Γ - and L-conduction valleys, respectively; p_hh and p_lh are the hole densities in the heavy and light valence band, respectively; m_cΓ and m_cL are the electron effective mass values of the Γ - and L-conduction valleys, respectively; m_hh and m_lh are the heavy and light hole effective masses; μ _L and μ _Γ are the electron mobilities in the L- and Γ -conduction valleys, respectively; and, μ _P is the hole mobility in the valence band.

Moreover, the Ge_{1− x}Sn_x carrier mobility (μ _L, μ _Γ , and μ _P) dependences of temperature and carrier density could be estimated by that of Ge for simplicity. The motilities in undoped Ge at the values of temperature T (100  K– 300  K) are given by^[16]

In addition, the mobility is also carrier-concentration-dependent, and the electron mobility of L-conduction valley and hole mobility of Ge (electron density of L-conduction valley: n_L, and hole density: n_p) at 300  K is given by^{[21, 22]}

The above equations are extended to other temperatures reasonably, and equation  (14) is substituted into Eq.  (15) to obtain μ _L(n, T) and μ _p(p, T) each as a function of carrier density n and p (in cm^{− 3}) at T (in K). However, to date there are no experimental data about the mobility of electrons the in Ge Γ -conduction valley. The relationship like Eq.  (16) and Eq.  (17) is then assumed to be

Finally, the FCA spectra of Ge_{1− x}Sn_x alloys as a function of carrier density and temperature can be obtained through Eqs.  (13)– (17). For the optimization of injection density, the optical gain and FCA spectra of the Ge_0.922Sn_0.078 alloy are simulated simultaneously with injected carrier densities of 5 × 10¹⁸, 9 × 10¹⁸, and 2 × 10¹⁹  cm^{− 3}, respectively. The net gain is defined to simplify the description as follows:

As shown by the three curves b, d, f in Fig.  3(b), α _FCA is inversely proportional to (hν )² and increases with the increase of carrier density. Moreover, g_net (hν ) is negative for the photon energy between 0.55  eV and 0.90  eV when the injected carrier density is 5 × 10¹⁸  cm^{− 3}, as shown by curve e and curve f in Fig.  3(b); when the density goes to 9 × 10¹⁸  cm^{− 3}, g_net(hν ) is positive within a certain value range of photon energy, as shown by curve c and curve d; when the injected carrier density reaches 2 × 10¹⁹  cm^{− 3}, g_net(hν ) is negative again, as shown by curve a and curve b. So it is concluded that the injected carrier density is not better higher, there should be an optimal injection to reach a maximal net gain for a special wavelength. In addition, as shown in Fig.  3(c), the comparisons of G(hν ) and α _FCA between Ge_0.922Sn_0.078 and Ge_0.9Sn_0.1 with the same injection level indicate that the FCAs of both alloys almost coincide but the maximal optical gain value of Ge_0.9Sn_0.1 alloy is much higher than that of Ge_0.922Sn_0.078.

In a real device simulation, when the G (hν ) and α _FCA(hν ) of the active layer are obtained, the device threshold carrier density n_th can be obtain from the following equation at a given wavelength (hν ):

where Γ is the confinement factor, α _i is the internal loss, mainly including FCA (α _i ∼ α _FCA), α _m is the mirror loss, L is the cavity length, R₁ and R₂ are the reflectivities of the front and back mirror, respectively. When the net optical gain

is positive, which means that the optical gain can overcome the loss, the lasing can occur.

In this real device simulation, Ge_0.922Sn_0.078 is chosen as an active region because it is difficult to achieve the high Sn percentage alloy by the epitaxy technique.^[23] In order to obtain high injection efficiency and low threshold current density, a double heterojunction (DHS) diode is designed. Figure  4(a) shows the band-gap and band-offset of Ge_{1− x − y}Si_xSn_y cladding layer (lattice-matched to Ge_0.922Sn_0.078 active region), each as a function of Sn percentage y. For well confining the carriers and light, the cladding layer of Ge_0.554Si_0.289Sn_0.157 (y = 0.157) can have the largest band-gap difference with Ge_0.922Sn_0.078 layer: Δ E_g = 215  meV, valence-band-offset Δ E_v = 95  meV and conduction-band-offset Δ E_c = 120  meV. The DHS laser device as shown in Fig.  4(b) is designed. In the next device simulation, α _m is assumed to be 10  cm^{− 1} and Γ is assumed to be 1.

Fig.  4.

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	Fig.  4. (a) Band-gap and band offset of Ge0.922Sn0.078 active region and lattice-matched Ge1− x − ySixSny cladding layer each as a function of Sn percentage (y). (b) The designed Si-based double heterojunction laser device.

Fig.  5.

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	Fig.  5. Net gains versus injected carrier density (lasing wavelengths are 2000  nm and 2050  nm at 200  K, and 2100  nm at 300  K).

The G_net(hν ) spectrum of Ge_0.922Sn_0.078 alloy (doping density: 5 × 10¹⁸  cm^{− 3}) as a function of injection level is calculated at a given wavelength of λ , the threshold carrier density n_th can then be obtained where G_net(hν ) equals zero. The net gain spectra of three different wavelengths, each as a function of injected carrier density, are shown in Fig.  5, and the arrow-marked threshold carrier densities are about 8 × 10¹⁸  cm^{− 3} (temperature: 300  K, and λ = 2100  nm), 2.2 × 10¹⁸  cm^{− 3} (temperature: 200  K, and λ = 2050  nm), and 3.3 × 10¹⁸  cm^{− 3} (temperature: 200  K, and λ = 2000  nm), respectively. G_net(hν ) at room temperature is much smaller than at 200  K. In addition, the maximal values of G_net(hν ) of these three conditions all appear at about n ∼ 9 × 10¹⁸  cm^{− 3}.

The the injected threshold current density J_th corresponding to threshold carrier density n_th can then be estimated according to the following formula:

where η _i is the injection efficiency, and assumed to be 0.8; d is the depth of active region, and assumed to be 10  μ m: R_sp is the spontaneous emission rate (as Eq.  (12)), and R_Aug is the Auger recombination rate, which is neglected at 200  K when considering that the Auger recombination coefficient of Ge_0.922Sn_0.078 is small at low temperatures (there is no report at present). However, the R_Aug of Ge_0.922Sn_0.078 at room temperature is estimated through using Auger recombination coefficients of Ge, as shown in Eq.  (22), which are C_ppn = 7.0 × 10^{− 32}  cm⁶/s^[24] and C_nnp = 3.0 × 10^{− 32}  cm⁶/s.^[25]

The corresponding values of J_th are then obtained through Eqs.  (12), (21), and (22), respectively, as 6.47  kA/cm² (temperature: 200  K, and λ = 2050  nm), 10.75  kA/cm² (temperature: 200  K, and λ = 2000  nm), and 23.12  kA/cm² (temperature: 300  K, and λ = 2100  nm). All of the above simulation results are shown in Table  1. These results indicate the possibility of obtaining a GeSn short-wave infrared laser at room temperature.

Table 1.

Table 1.

Table 1. Threshold carrier densities, spontaneous emission rates, Auger recombination rates, and threshold current densities at different temperatures and wavelengths. Temperature/K Wavelength/nm nth/cm− 3 Rsp/cm− 3· s− 1 RAug/cm− 3· s− 1 Jth/(kA/cm− 2) 200 2000 3.3 × 1018 5.37 × 1025 neglected 10.75 200 2050 2.2 × 1018 3.23 × 1025 neglected 6.47 300 2100 8.0 × 1018 1.14 × 1026 1.12 × 1024 23.12

Table 1. Threshold carrier densities, spontaneous emission rates, Auger recombination rates, and threshold current densities at different temperatures and wavelengths.

In this paper the optical gain characteristics of Ge_{1− x}Sn_x are systematically stimulated. Ge_{1− x}Sn_x can become a direct band material when the Sn concentration is larger than 7.8%. Owing to the reduction of the energy difference between direct and indirect conduction band valley with the increase of Sn concentration, the percentage of the direct band electrons increases evidently, which contributes to the realization of population inversion in direct Γ band valley and significant optical gain. The maximal gain value of Ge_0.922Sn_0.078 reaches 500/cm, while that of Ge is still negative at room temperature (the doping carrier density and injected carrier density both are 5 × 10¹⁸  cm^{− 3}). The simulation results indicate that there should be an optimal injection to reach a maximal net gain when the FCA effect is considered. To achieve a high injection efficiency and low threshold current density, a double heterojunction Ge_0.554Si_0.289Sn_0.157/Ge_0.922Sn_0.078/Ge_0.554Si_0.289Sn_0.157 short-wave infrared laser diode is designed and simulated. Through calculating the spontaneous emission rate and Auger recombination rate in Ge_0.922Sn_0.078, the injected threshold current densities J_th are obtained, which are about 6.47  kA/cm² (temperature: 200  K, and λ = 2050  nm), 10.75  kA/cm² (temperature: 200  K, and λ = 2000  nm), and 23.12  kA/cm² (temperature: 300  K, and λ = 2100  nm). They are of the same order of magnitude as III– V devices and indicate the possibility of obtaining a GeSn short-wave infrared laser at room temperature. Therefore, GeSn alloy is a promising candidate material, not only for all telecommunication bands in optoelectronics but also for shortwave infrared applications.