## Abstract

We propose and develop a theoretical gain model for an n-doped, tensile-strained Ge-Si* _{x}*Ge

*Sn*

_{y}_{1-x-y}quantum-well laser. Tensile strain and n doping in Ge active layers can help achieve population inversion in the direct conduction band and provide optical gain. We show our theoretical model for the bandgap structure, the polarization-dependent optical gain spectrum, and the free-carrier absorption of the n-type doped, tensile-strained Ge quantum-well laser. Despite the free-carrier absorption due to the n-type doping, a significant net gain can be obtained from the direct transition. We also present our waveguide design and calculate the optical confinement factors to estimate the modal gain and predict the threshold carrier density.

©2009 Optical Society of America

## 1. Introduction

An efficient silicon-based laser is *the missing component* for integrated silicon-based photonics. The main difficulty is that group-IV semiconductors, such as silicon (Si) and germanium (Ge), are indirect bandgap materials. The first-order band-edge electron-hole recombination process which involves a single photon does not conserve momentum, because photons have a small momentum compared with that of electrons. Therefore, electron-hole radiative recombination can only occur with the assistance of a phonon to conserve momentum. However, such a second-order process in group-IV semiconductors is very inefficient compared with the direct transition in III–V semiconductors. As a result, group-IV semiconductors are usually poor light emitters. Still, a silicon-based laser remains the last unrealized building block of silicon photonics. One of the main driving forces for silicon photonics today is silicon-based optoelectronic integrated circuits (OEICs), that is, integrating optoelectronic components, such as lasers, modulators, multiplexers, and detectors, into a single silicon chip. By transporting information optically rather than electrically, the data-transfer rate and heat dissipation can be enormously improved. Therefore, Si-photonics holds promise for next-generation telecommunications. To be useful for telecommunications or as intra-chip interconnections, it is desirable for a silicon-based laser to satisfy the following key ingredients: (1) the silicon-based laser can be monolithicly grown on silicon substrates, which cost much less than any other semiconductor material; (2) it is compatible with complementary metal-oxide-semiconductor (CMOS) processing, which allows for the integration of the laser with other electronic or optoelectronic components; (3) it emits light around the telecommunication wavelengths of 1300 nm or 1550 nm; and (4) it is desirable to be electrically-pumped, so it requires no additional pump light source. Although lots of efforts have been made to improve the light radiation efficiency of group-IV semiconductors, for example, silicon Raman laser [1], porous silicon [2], and erbium-doped silicon [3], no solution was found that satisfied all the four key ingredients until now.

Among group-IV semiconductors, germanium is a promising material for an efficient light emitter. Germanium is a quasi-direct bandgap material, that is, the direct-conduction band edge lies only 134.5 meV above the lowest indirect-conduction band edge at room temperature. Therefore, it is possible to convert the quasi-direct bandgap of Ge into a direct bandgap through proper engineering of the bandgap allowing to serve it as a gain medium. Another particular characteristic of Ge is that its direct bandgap is about 0.8 eV, corresponding to an emission wavelength of 1550 nm, the most popular wavelength in telecommunications. Those advantages make Ge a potential candidate for an efficient silicon-based laser for telecommunications. Recently, the SiGeSn material system has attracted attention with its optoelectronic applications. New ideas to produce a direct-bandgap material based on group-IV semiconductors is to grow tensile-strained Ge layers by SiGeSn technology [4
5, 6]. The lattice constant of *α*-Sn is larger than that of Ge, so the growth of tensile-strained Ge layers on the SiGeSn material system becomes possible. Such tensile-strained Ge layers grown on Si via a GeSn buffer layer have been demonstrated recently, and a tensile strain of 0.45% has been observed experimentally [7]. In addition, SiGeSn technology is fully compatible with Si-based CMOS processing [8], which allows large-scale integration with other electronic and optoelectronic components on one single silicon chip. Based on the above benefits, the SiGeSn material system may be an ideal platform for silicon photonics. On the other hand, the use of n doping and tensile strain in bulk Ge has been proposed to achieve population inversion [9]. It has been predicted that a net gain of about 400 cm^{-1} is achievable for a bulk Ge laser with an n-type doping concentration of 7.6×10^{19} cm^{-3}. Furthermore, a strong photoluminescence at wavelengths around 1550 nm from n-doped Si/Ge superlattices has been observed experimentally [10].

In this paper, we propose and develop a theoretical model for an n-doped, tensile-strained Ge/Si* _{x}*Ge

*Sn*

_{y}_{1-x-y}multiple-quantum-well (MQW) laser. The use of tensile strain can effectively reduce the energy difference between the direct and indirect conduction band edges of the Ge wells. By employing extrinsic electrons from n doping to fill the L-valley conduction subbands up to the onset of the Γ-valley conduction subband, injected electrons via bias current are allowed to populate the direct Γ-conduction subband, achieving population inversion and providing significant optical gain. In addition, by using quantum-well structures as the active medium rather than bulk materials, there are two advantages: (1) the n doping requirement is moderate in contrast to that of bulk materials leading to less free-carrier loss, and (2) the transition energy can be flexibly adjusted by changing the well width to serve the telecommunication applications. Even in the presence of the free-carrier loss, our calculations show that a significant net gain is achievable. Threshold condition analysis indicates that the modal gain can reach the threshold condition to make lasing action possible.

## 2. Theory of electronic band structure and optical gain

#### 2.1. Band structure

Figure 1 shows our proposed n^{+}-Ge/Si_{0.2}Ge_{0.7}Sn_{0.1} MQW structure. An n-doped Si_{0.24}Ge_{0.66}Sn_{0.1} layer which serves as the bottom contact is grown on a (100)-oriented silicon substrate via a fully strain-relaxed Ge_{0.965}Sn_{0.035} buffer layer. Then five pairs of n-doped Ge/Si_{0.2}Ge_{0.7}Sn_{0.1} QWs, that is, five Ge wells and six Si_{0.2}Ge_{0.7}Sn_{0.1} barriers, are grown on the n-contact. Finally, the whole structure is capped by a p-doped Si_{0.2}Ge_{0.7}Sn_{0.1} layer. Because the buffer layer is fully strain-relaxed, the lattice constants of the Ge wells and the Si_{0.2}Ge_{0.7}Sn_{0.1} barriers will be forced to follow that of the buffer layer. Therefore, the Ge wells are subjected to a biaxially tensile strain of 0.514% while the Si_{0.2}Ge_{0.7}Sn_{0.1} barriers are subjected to a biaxially compressive strain of 0.154%, respectively. In this calculation, we assume that the width of the Ge quantum well is 70 Å. In order to balance the strain between the tensile-strained Ge wells and the compressive-strained Si_{0.2}Ge_{0.7}Sn_{0.1} barriers, the barrier width is set to 200 Å, so that the whole structure is approximately strain-balanced.

We adopt the model-solid theory to calculate the potential profiles of various bands [11] and the effective mass theory to calculate various subbands of the strained Ge/SiGeSn MQW structure [12]. The procedures and parameters we use here can be found in our previous work [6], except for the deformation potentials of *α*-Sn (*a _{c}*=-6.00 eV,

*a*

_{L}=-2.14 eV,

*a*

_{v}=1.58 eV) [13, 14]. In this calculation, most parameters for the Si

*Ge*

_{x}*Sn*

_{y}_{1-x-y}material system are obtained by linear interpolation among those of Si, Ge, and

*α*-Sn, except for the unstrained direct bandgap from the experimental data [15]

$$-{b}^{\mathrm{SiGe}}\mathrm{xy}-{b}^{\mathrm{GeSn}}y\left(1-x-y\right)-{b}^{\mathrm{SiSn}}x\left(1-x-y\right),$$

where the direct bandgaps are *E*
^{Si}
_{g,Γ}=4.185 eV, *E*
^{Ge}
_{g,Γ}=0.7985 eV, and *E*
^{Sn}
_{g,Γ}=-0.413 eV, respectively, and the bowing parameters are *b*
^{SiGe}=0.21 eV, *b*
^{GeSn}=1.94 eV, and *b*
^{SiSn}=17.5 eV, respectively. Notice that SiSn alloys have a giant bowing parameter, leading to a tunable direct bandgap of SiGeSn alloys ranging from 0.8 eV to 1.2 eV, which is sufficient to provide suitable energy barriers for the Ge well. On the other hand, since there is a lack of experiments on the indirect-bandgap, we use linear interpolation to calculate the indirect bandgap of the Si* _{x}*GeySn

_{1-x-y}materials.

Figure 2 shows the scheme of employing tensile strain and n doping to achieve population inversion in the Γ-conduction band of the Ge well. Figure 2(a) shows the band structure of an unstrained Ge well. The energy difference between the direct- and indirect-conduction band edges of the Ge well is only 134.5 meV at room temperature. By introducing a biaxially tensile strain of 0.514% into the Ge well, the indirect and direct bandgaps are reduced to 0.6157 eV and 0.7072 eV, respectively, as shown in Fig. 2(b). The energy difference between the two conduction band edges is thus reduced to 91.5 meV. Then, by building in an electron concentration of 2.14×10^{19}cm^{-3} via n doping into the L-valley conduction subbands, the electron quasi-Fermi level in the conduction band, *F _{c}*, will rise to the lowest Γ-valley conduction subband to compensate the remaining energy difference of 91.5 meV, as shown in Fig. 2(c). Since the L-conduction subbands, which are lower than the Γ-conduction subband, are mostly occupied by extrinsic electrons, injected electrons by bias current will begin to populate the Γ-conduction subband and thus achieve population inversion. Therefore, electron-hole radiative recombination through the direct transition can bring significant optical gain, as shown in Fig. 2(d).

#### 2.2. Carrier occupation in the conduction and valence bands

The quasi-Fermi levels *F*
_{c} and *F*
_{v} can be determined by the electron concentration *n* and hole concentration *p*, which satisfy the charge neutrality condition

where *N*
^{+}
_{D} and *N*
^{-}
_{A} are the ionized donor and acceptor concentrations, respectively. Since the Ge wells are heavily n-doped, we assume *N*
^{+}
_{D}≫*N*
^{-}
_{A}. For a given injected carrier density *N*
_{inj}, the electron quasi-Fermi level *F*
_{c} in the conduction band can be determined by using

where *m*
_{n,t} is the in-plane effective mass of the *n*-th subband in the Γ-conduction band; *E _{n}* is the subband energy of the

*n*-th subband in the Γ-conduction band;

*T*is the temperature;

*L*

_{QW}is the well width;

*f*

_{L,l}(

**k**

_{t}) is the Fermi occupation number of the

*l*-th subband in a certain L-conduction valley; and

*E*

^{L}

*is the subband energy of the*

_{l}*l*-th subband in the L-conduction band. Notice that the factor of eight in the summation for the L conduction subbands in Eq. (3) comes from the spin degeneracy of two and four equivalent L-conduction valleys. Similarly, the quasi-Fermi level in the valence band,

*F*

_{v}, can be determined by using

where U and L refers to the upper and lower blocks of the “*J*=*4*” Luttinger-Kohn Hamiltonian of the valence band, respectively; *f ^{σ}_{m}* (

*k*) is the electron Fermi occupation number in the valence band; and

_{t}*E*is the subband energy of the

^{σ}_{m,kt}*m*-th subband in the valence band. The two quasi-Fermi levels help determine the electron and hole distributions in the Γ- and L-conduction subbands as well as those in the valence subbands. For a given n-type doping concentration

*N*and an injected carrier density

_{D}*N*

_{inj}, the two quasi-Fermi levels can be calculated self-consistently.

#### 2.3. Transverse electric and transverse magnetic polarized optical gain

To calculate the transverse electric (TE) and transverse magnetic (TM) optical gain spectra, the TE and TM momentum matrix elements have to be determined first. For quantum wells, the optical momentum matrix elements are polarization-dependent. By averaging the TE and TM momentum matrix elements in the plane of quantum wells, the average squared TE momentum matrix elements are given by

where *ϕ _{n}*(

*z*) is the wavefunction of the

*n*-th state in the Γ-conduction band;

*g*

^{(1)}

*and*

_{m}*g*

^{(2)}

*(*

_{m}*g*

^{(3)}

*and*

_{m}*g*

^{(4)}

*) are eigenfunctions of the upper (lower) block of the “*

_{m}*J=4*” Luttinger-Kohn Hamiltonian of the valence band;

*M*

_{b}is the bulk momentum matrix element; and

*E*

_{p}is the optical energy parameter. Here a spin degeneracy of two has been included in the expressions of the average squared momentum matrix elements. Similarly, the average squared TM momentum matrix elements can be written as

Note that the TE momentum matrix elements depend on both the heavy-hole (HH) components, *g*
^{(1)}
* _{m}* (

*k*

_{t},

*z*) and

*g*

^{(4)}

*(*

_{m}*k*

_{t},

*z*), and the light-hole (LH) components,

*g*

^{(2)}

*(*

_{m}*k*

_{t},

*z*) and

*g*

^{(3)}

*(*

_{m}*k*

_{t},

*z*), while the TM momentum matrix elements depend on only the LH components. Therefore the TE gain will be dominant if the highest valence band is HH-like, while the TM gain will be dominant if the highest valence band is LH-like. With the TE and TM momentum matrix elements determined, we can calculate the corresponding TE and TM material gain spectra based on the spontaneous emission transformation method, taking into account the homogenous broadening by the Lorentzian function with a full-width-at-half-maximum (FWHM) linewidth, Γ [16, 17]

where *e* is the elementary charge, *c* is the speed of light in free space, *ε*
_{0} is the permittivity of free space, Δ*F*=*F*
_{c}-*F*
_{v} is the quasi-Fermi level separation, *h̄ω* is the photon energy, and *n*
_{r} is the background refractive index.

## 3. Free-carrier absorption and optical confinement factor

#### 3.1. Free-carrier absorption

Because the Ge wells are heavily n-doped, the free-carrier absorption may be significant, and it has to be taken into account to predict the threshold of this MQW laser. The free-carrier absorption, including the contribution form the L-conduction valleys, can be described by Drude-Lorentz equation [18]

where *n*
_{Γ} and *n*
_{L} are the electron densities in the Γ- and L-conduction valleys, respectively; *m*
^{*}
_{c} and *m*
^{*}
_{L} are the electron effective masses of the Γ- and L-conduction valleys, respectively; *m*
^{*}
_{h} is the hole effective mass of the valance band; *µ*
_{L} and *µ*
_{Γ} are the electron mobilities in the L- and Γ-conduction valleys, respectively; *µ*
_{h} is the hole mobility in the valence band; and λ is the free space wavelength. Notice that the mobility is carrier-concentration-dependent. The electron mobility in the L-conduction valleys of Ge at room temperature is given by [19]

where *µ*
_{L0}=3900 cm^{2}V^{-1}s^{-1} [20] and *n*
_{L} is in the unit of cm^{-3}. The electron mobility in the Γ-conduction valley of Ge is still unavailable in the literature. However, the carrier mobility is related to the carrier effective mass *m*
^{*} and the scattering time *τ* by *µ*=*qτ/m*
^{*}. If we assume the scattering times *τ* for the electrons in the L- and Γ- conduction valleys are the same, we can simply estimate the electron mobility in the Γ-conduction valley from that in the L-conduction valleys. For the holes in the valence band, we obtain the hole mobility of Ge in the range of 10^{18}-10^{20}cm^{-3} at room temperature by fitting the experimental data [21]

where *µ*
_{p0}=1900 cm^{2}V^{-1}s^{-1} [20] and *p* is in the unit of cm^{-3}. Based on Eqs. (14), (15), and (16), several important characteristics for the free-carrier absorption should be pointed out. First, the free-carrier absorption is proportional to *λ*
^{2} [22], so it could be significant at long wavelengths. Second, at a high doping concentration, the free-carrier absorption is proportional to *n*
^{3/2}, which agrees well with the experimental results [23]. Last, the free-carrier absorption is inversely proportional to the effective mass squared, indicating that the free-carrier absorption will be significant if the carrier has a small effective mass.

#### 3.2. Modal gain, optical confinement factor, and threshold lasing condition

The modal gain and threshold modal gain are two key parameters which characterize laser performance. The former describes the available optical gain for guided modes in a laser cavity, while the later stands for the required modal gain to overcome the background losses and radiation losses for lasing action to take place. We also need to convert the material gain to the modal gain using the optical confinement factor, since different modes experience different amount of optical gain. In addition, owing to the anisotropic optical gain, we need two optical confinement factors for the modal gain to properly specify the contributions from the TE and TM material gains. The general expression of the modal gain *G*
_{mod} for a MQW laser with *N*
_{w} wells can be expressed as [24, 25, 26]

where Γ^{TE}
_{w} and Γ^{TM}
_{w} are the TE and TM optical confinement factors per well of guided modes, respectively; and g^{TE}
_{w} and g^{TM}
_{w} are the TE and TM material gains per well, respectively. If a guided mode propagates in the *x* direction and the crystal growth direction is along the *z* direction, the optical confinement factors for the well are

where *n*
_{w} is the refractive index; and ${\eta}_{0}=\sqrt{{\mu}_{0}\u2044{\epsilon}_{0}}$ is the intrinsic impedance. On the other hand, since the material loss attributed from the free-carrier absorption is isotropic, we need only one optical confinement factor Γ* _{i}*=Γ

^{TE}

*+Γ*

_{i}^{TM}

*, which is the sum of the TE and TM optical confinement factors, for layer i to calculate the modal loss. If we assume that the free-carrier absorption due to the total electron density*

_{i}*n*occurs in the well region, the modal loss

*α*(

*n*) taking into account the absorption losses in the n- and p-contacts can be expressed as

where *α*
_{w}(*n*) is the absorption loss for one well; *α*
_{n} and *α*
_{p} are the absorption losses for the n-contact and p-contact regions, respectively; and Γ_{n} and Γ_{p} are the optical confinement factors of the n-contact and p-contact regions, respectively.

With the modal loss defined, it is sufficient to determine the threshold modal gain. For a Fabry-Pérot laser with a cavity length of *L*, the required modal gain at the threshold lasing condition is

where *R*
_{1} and *R*
_{2} are the reflectivities at the two facets of the cavity. Once the modal gain in Eq. (17) reaches the threshold modal gain in Eq. (22), lasing action can take place.

## 4. Theoretical results and discussions

Figure 3(a) shows the the potential profiles of various bands of the type-I Ge/Si_{0.2}Ge_{0.7}Sn_{0.1} quantum-well structure. The zero energy is set at the top of the LH band of the well. For the Ge well, the band edge of the LH band is shifted upward while that of the HH band is shifted downward due to the tensile strain. Meanwhile, for the Si_{0.2}Ge_{0.7}Sn_{0.1} barriers, the shift directions of the LH and HH bands are opposite to those of the Ge well due to the compressive strain. As a result, the highest valence subband in the Ge well is LH-like and there is no bounded HH-like state, because the tensile strain lowers the HH band edge so that the potential profile in the barrier region is higher than that of the Ge QW. Correspondingly, Fig. 3(b) shows the dispersion relation of the valence subband for the Ge well under the axial approximation. Only one quantized LH subband exists. On the other hand, in the conduction band, the tensile strain in the Ge well lowers the Γ-conduction band edge as well as the L-conduction band edge. However, the magnitude of the deformation potential of the Γ-conduction valley is much larger than that of the L-conduction valley. Therefore, the Γ-conduction band edge will lower more rapidly than the L-conduction band edge under biaxially tensile strain. Thus, the use of biaxially tensile strain on Ge can effectively reduce the energy difference between its direct and indirect conduction band edges. Theoretically, Ge becomes a direct-bandgap material under a biaxially tensile strain of 1.61% [6].

Figure 4(a) shows the injected surface carrier densities presenting in the L- and Γ-conduction valleys as a function of the total injected surface carrier density *N ^{s}*

_{inj}, which is related to the total injected volume carrier density

*N*

_{inj}by

*N*

^{s}_{inj}=

*N*

_{inj}LQW. Because of the larger electron effective mass and four equivalent valleys of the L-conduction valleys, the carrier occupation of the L-conduction valleys outnumbers that of the Γ-conduction valley. Since those carriers in the L-conduction valleys do not contribute to the optical gain through the direct transition, the effect of carrier leakage becomes important to this MQW laser. Figure 4(b) shows the TE and TM squared normalized momentum matrix elements

*M*

^{σ}^{TE}

*/*

_{nm}*M*

^{2}_{b,Ge}and

*M*

^{σ}^{TM}

*/*

_{nm}*M*

^{2}

_{b,Ge}as a function of the wave vector

*k*

_{t}, where

*M*

_{b,Ge}is the bulk momentum matrix element of Ge. Because only one LH subband exists, the TM magnitude is about four times larger than the TE magnitude. When the wave vector

*k*

_{t}increases, the TM component always decreases due to less wave function overlap. On the other hand, the TE magnitude slightly increases at a high wave vector

*k*

_{t}due to the valence-band-mixing effect.

Figure 5 shows the TE and TM material gain spectra under varying injected surface carrier densities *N*
^{s}
_{inj}. The FWHM linewidth of the Lorentzian function is set to 20 meV. Because the magnitude of the TM momentum matrix element is about four times larger than that of the TE one, the magnitude of the TM gain is also about four times larger than that of the TE gain. It means that the TM-polarized light is dominant in this MQW laser. As the number of injected carriers increases, both the TE and TM gains increase but tend to saturate. The peak gain is close to 0.8 eV, corresponding to an emission wavelength of 1.55 *µ*m. In this calculation, the bandgap shrinkage due to the heavily n doping is not included. However, the amounts of the indirect- and direct-bandgap shrinkages due to n doping at room temperature are almost the same [27], so it does not qualitatively influence the calculation results except for the red shift of the gain spectra. However, this red shift can be compensated by adjusting the well width. By decreasing the well width, the gain spectra will be blue-shifted to cancel the influence of the bandgap shrinkage effects caused by the heavy n doping to maintain the peak gain around 0.8 eV to serve telecommunication purposes.

Another important concern is that the high refractive index of *α*-Sn (*n*
_{Sn}=4.885) [28] compared with those of Si (*n*
_{Si}=3.4784) [29] and Ge (*n*
_{Ge}=4.275) [30] at *λ*=1550 nm. The high composition of *α*-Sn in Si_{x}GeySn_{1}-x-y compounds leads to the situation that the refractive index of the Ge well is slightly larger than that of the p-Si_{0.2}Ge_{0.7}Sn_{0.1} and n-Si_{0.24}Ge_{0.66}Sn_{0.1} cladding layers. It means that the n- and p-contacts cannot provide proper optical confinement for the active region. To confine light properly, we design a silica ridge waveguide structure, which has a smaller refractive index *n*
_{SiO2}=1.45, for index guidance, as shown in Fig. 6(a). The width of the ridge structure is set to 2 *µ*m. We use the finite element method (FEM) to calculate the field distributions of the guided modes. Since the TM-polarized gain is dominant in this MQW laser, we are more interested in the quasi-TM fundamental mode, whose guided power distribution is shown in Fig. 6(b). It is clear to see the silica ridge structure provides the upper cladding while the silicon substrate acts as a lower cladding for the waveguide. As a result, Table 1 tabulates the doping concentrations, the optical confinement factors, and the absorption losses in the n+-well, n-contact, and p-contact regions, respectively. The TM optical confinement factors are much larger than the TE components for the quasi-TM fundamental mode due to the weak waveguiding. Meanwhile, because of the proper waveguide design, the TM optical confinement factor in the Ge wells is as high as 5.91%, leading to an improved stimulated emission process in the active region. Although the optical confinement factors of n- and p-contacts are much larger than that of the Ge wells, the absorption-losses in the n- and p-contacts are much smaller in contrast to the material gain generated by the Ge wells, resulting in very little modal losses in the n- and p-contact regions.

Figure 7(a) shows the TE and TM material gains per quantum well from the direct transition and the free-carrier absorption (an n-doped, tensile strained Ge-well structure with a doping concentration *N*
_{D}=2.14×10^{19} cm^{-3}) as a function of the injected surface carrier density *N*
^{s}
_{inj} at λ=1550 nm. As mentioned earlier, the free-carrier absorption is a function of the carrier concentration and the carrier effective mass. In the conduction band, due to the less electron occupation and the larger electron mobility of the Γ-conduction subbands than those of the L-conduction subbands, the free-carrier absorption from electrons in the Γ-conduction subbands is much smaller than that from the L-conduction subbands. In the valence band, because only one bounded LH state exists, and its effective mass is very small (*m*
_{lh}=0.043 *m*
_{0} for Ge), the free-carrier absorption for holes in the valance band becomes significant at a high injected carrier density. At an injected surface carrier density of about 3.23×10^{12} cm^{-2}, the Ge well becomes transparent due to the cancelation between the TM optical gain and the free-carrier absorption loss. A net TM material gain of about 7474 cm^{-1} at *λ*=1550 nm is obtainable at an injected surface carrier density of 10^{13}cm^{-2}. We then estimate the threshold carrier density by using the threshold condition in Eq. (23), which is one of the most important parameters for a working device. Figure 7(b) shows the modal gain, the modal loss, and the threshold modal gain as a function of the injected surface carrier density *N*
^{s}
_{inj}, where the reflectivities *R*
_{1}=*R*
_{2}=0.37 are obtained form the waveguide analysis; and the mirror loss is 19.89 cm^{-1} if we assume the cavity length is 500 *µ*m. The intersection of the modal gain and the threshold modal gain defines the threshold injected surface carrier density *N*
^{th}
_{inj}. The threshold lasing condition is reached at a threshold injected surface carrier density *N*
^{th}
_{inj}=3.86×10^{12}cm^{-2}, which is still a reasonable value compared with that of typical III-V MQW lasers. A further increase in the injected surface density above this threshold injected surface density can lead to lasing action.

## 5. Conclusion

We have developed a theoretical gain model for an n-doped, tensile-strained Ge/Si* _{x}*Ge

*Sn*

_{y}_{1-x-y}multiple-quantum-well laser. The use of tensile strain and n doping can effectively achieve population inversion for the direct transition and provide significant optical gain. Due to the tensile strain, the highest valance band is LH-like in nature, which leads to a dominant TM-polarized optical gain in this device. Our calculations show that the material gain can overcome the free-carrier loss and provide a significant net gain. We have also shown our waveguide design and calculated the optical confinement factors in the various regions to estimate the modal gain and predict the threshold carrier density. Based on our results, we conclude that a sufficient modal gain is achievable to reach the threshold lasing condition using this n-doped, tensile-strained Ge/Si

*Ge*

_{x}*Sn*

_{y}_{1-x-y}MQW laser.

## Acknowledgments

This work at the University of Illinois at Urbana-Champaign was supported by AFOSR under the MURI program. Guo-En Chang was on leave from the Institute of Applied Mechanics, National Taiwan University, Taipei, Taiwan (R.O.C) and he thanks the support from the National Science Council of the Republic of China under the contract number NSC 096-2917-I-002-039. The authors would like to thank Prof. J. Kouvetakis and Prof. J. Menéndez at the Arizona State University, and Dr. Richard Soref at Hanscom AFB for useful discussions.

## References and links

**1. **O. Boyraz and B. Jalali, “Demonstration of a silicon Raman laser,” Opt. Express **12**, 5269–5273 (2004).
[CrossRef] [PubMed]

**2. **L. T. Canham, “Silicon quantum wire array fabrication by electrochemical and chemical dissolution of wafers,” Appl. Phys. Lett. **57**, 1046–1048 (1990).
[CrossRef]

**3. **H. Ennen, J. Schneider, G. Pomrenke, and A. Axmann, “1.54-µm luminescence of erbium-implanted III-V semiconductors and silicon,” Appl. Phys. Lett. **43**, 943–945 (1983).
[CrossRef]

**4. **R. A. Soref and L. Friedman, “Direct-gap Ge/GeSn/Si and GeSn/Ge/Si heterostructure,” Superlattices Microstruct. **14**, 189–193 (1993).
[CrossRef]

**5. **J. Menéndeza and J. Kouvetakis, “Type-I Ge/Ge_{1-x-y}Si_{x}Sn_{y} strained-layer heterostructures with a direct Ge bandgap,” Appl. Phys. Lett. **85**, 1175–1177 (2004).
[CrossRef]

**6. **S. W. Chang and S. L. Chuang, “Theory of optical gain of Ge-Si_{x}Ge_{y}Sn_{1-x-y} quantum-well lasers,” IEEE J. Quantum Electron. **43**, 249–256 (2007).
[CrossRef]

**7. **J. Kouvetakis, J. Tolle, J. Menéndeza, and V. R. D’Costa, “Advances in Si-Ge-Sn materials science and technology,” in *Proceedings of IEEE 4th International Conference on Group IV Photonics* (Institute of Electrical and Electronics Engineers, Tokyo, Japan, 2007), pp. 1–3.

**8. **J. Kouvetakis and A. V. G. Chizmeshya, “New classes of Si-based photonic materials and device architectures via designer molecular routes,” J. Mater. Chem. **17**, 1649–1655 (2007).
[CrossRef]

**9. **J. Liu, X. Sun, D. Pan, X. Wang, L. C. Kimerling, T. L. Koch, and J. Michel, “Tensile-strained, n-type Ge as a gain medium for monolithic laser integration on Si,” Opt. Express **15**, 11272–11277 (2007).
[CrossRef] [PubMed]

**10. **N. D. Zakharov, V. G. Talalaev, P. Werner, A. A. Tonkikh, and G. E. Cirlin, “Room-temperature light emission from a highly strained Si/Ge superlattice,” Appl. Phys. Lett. **83**, 3084–3086 (2003).
[CrossRef]

**11. **C. G. Van de Walle, “Band lineups and deformation potentials in the model-solid theory,” Phys. Rev. B **39**, 1871–1883 (1989).
[CrossRef]

**12. **S. L. Chuang, *Physics of Photonic Devices*, 2nd Ed. (Wiley, New York, 2009).

**13. **Y. H. Li, X. G. Gong, and S. H. Wei, “Ab initio all-electron calculation of absolute volume deformation potentials of IV-IV, III–V,and II–VI semiconductors: The chemical trends,” Phys. Rev. B **73**, 245206 (2006).
[CrossRef]

**14. **T. Brudevoll, D. S. Citrin, M. Cardona, and N. E. Christensen, “Electronic structure of α-Sn and its dependence on hydrostatic strain,” Phys. Rev. B **48**, 8629–8635 (1993).
[CrossRef]

**15. **V. R. D’Costa, Y. Y. Fang, J. Tolleb, J. Kouvetakis, and J. Menéndeza, “Direct absorption edge in GeSiSn alloys,” International Conference on the Physics of Semiconductors, Rio de Janeiro, Brazil, 2008.

**16. **T. Keating, S. H. Park, J. Minch, X. Jin, S. L. Chuang, and T. Tanbun-Ek, “Optical gain measurements based on fundamental properties and comparison with many-body theory,” J. Appl. Phys. **86**, 2945–2952 (1999).
[CrossRef]

**17. **J. Minch, S. H. Park, T. Keating, and S. L. Chuang, “Theory and experiment of In_{1-x}Ga_{x}As_{y}P_{1-y} and In_{1-x-y}Ga_{x}Al_{y}As long-wavelength strained quantum-well lasers,” IEEE J. Quantum Electron. **35**, 771–782 (1999).
[CrossRef]

**18. **R. A. Soref and J. P. Lorenzo, “All-silicon active and passive guided-wave components for λ=1.3 and λ=1.6µm,” IEEE J. Quantum Electron. **QE-22**, 873–879 (1986).
[CrossRef]

**19. **C. Hilsum, “Simple empirical relationship between mobility and carrier concentration,” Electron. Lett. **10**, 259–260 (1974).
[CrossRef]

**20. **B.G. Streetman, *Solid State Electronic Devices*, 4th Ed. (Prentice-Hall, New Jersey, 1995).

**21. **S. M. Sze and J. C. Irvin, “Resistivity, mobility and impurity levels in GaAs, Ge, and Si at 300°K,” Solid State Electron. **11**, 599–602 (1968).
[CrossRef]

**22. **J. I. Pankove, “Optical absorption by degenerate germanium,” Phys. Rev. Lett. **4**, 454–455 (1960).
[CrossRef]

**23. **J. I. Pankove, “Properties of heavily doped germanium,” *Prog. Semicond*. , **9**, 48 (1965).

**24. **T. D. Visser, H. Blok, B. Demeulenaere, and D. Lenstra, “Confinement factors and gain in optical amplifiers,” IEEE J. Quantum Electron. **33**, 1763–1766 (1997).
[CrossRef]

**25. **A. V. Maslov and C. Z. Ning, “Modal gain in a semiconductor nanowire laser with anisotropic bandstructure,” IEEE J. Quantum Electron. **40**, 1389–1397 (2004).
[CrossRef]

**26. **S.W. Chang and S. L. Chuang, “Fundamental formulation for plasmonic nanolasers,” IEEE J. Quantum Electron. (in press).

**27. **C. Hass, “Infrared absorption in heavily doped n-type germanium,” Phys. Rev. **125**, 1965–1971 (1962).
[CrossRef]

**28. **R. E. Lindquist and A. W. Ewald, “Optical constants of single-crystal gray tin in the infrared,” Phys. Rev. **135**, A191–A194 (1964).
[CrossRef]

**29. **D. F. Edwards, “Silicon (Si),” in *Handbook of Optical Constants of Solids*,
E.D. Palik, ed. (Academic, Orlando, Florida, 1985), pp. 547–569.

**30. **R. F. Potter, “Germanium (Ge),” in *Handbook of Optical Constants of Solids*,
'E.D. Palik, ed. (Academic, Orlando, Florida, 1985), pp. 465–478.