Analysis of threshold current of uniaxially tensile stressed bulk Ge and Ge/SiGe quantum well lasers

Jialin Jiang; Junqiang Sun; Jianfeng Gao; Ruiwen Zhang

doi:10.1364/OE.25.026714

1. Introduction

The Si-based light source has long been considered to be the “holy grail” of Silicon Photonics. Many efforts have been made to achieve light sources on Si platform, such as Si nanocrystals [1], erbium-doped Si [2], band-engineered Ge [3], GeSn alloy [4], Ge Zener-Emitter [5] and III-V material on Si [6]. Among those techniques, the band-engineered Ge has attracted great interest due to its Si-compatible fabrication process and ability to be electrically driven. The idea of band-engineered Ge was originally proposed by a MIT group [3], conceiving that the energy difference between the Γ and L points can be reduced by introducing tensile strain and the remaining electronic states can be further filled by extrinsic electrons from n-type doping so that more injected electrons can occupy the Γ valley where efficient radiative recombination occurs. Subsequently, the optically pumped [7] and electrically driven [8] Ge lasers have been demonstrated but with unsatisfying threshold current density as high as 280 kA/cm². In order to lower the threshold current density of Ge laser, the tensile strain needs to be enhanced and therefore a variety of strain techniques have been explored including SiN_x stressor [9] and suspended microstructures [10, 11]. The highest reported tensile strain values, to the best of our knowledge, are 4.9% [12] and 1.9% [11] in the case of uniaxially and biaxially tensile stressed, respectively.

On the other hand, it is widely accepted that the lasers using low dimensional quantum structures as gain media have lower threshold current density due to the modified density of state [13] and scaling law [14]. Indeed, the III-V lasers with quantum structures have achieved desirable performance, motivating a series of theoretical [15, 16] and experimental investigations [17–20] of the Ge/Si_1-xGe_x quantum structure. Compared with quantum wire and quantum dot, quantum well features easy epitaxial process and can be combined with tensile stress. It has been discussed in [21] that the unstrained Ge/Si_1-xGe_x quantum well underperforms bulk Ge since the quantum confinement of Γ valley is stronger than that of L valley, making the energy difference between Γ- and L-points even larger. Recently, uniaxial tensile stress and biaxial tensile stress have been realized in Ge/Si_0.19Ge_0.81 quantum well [22], signifying that it is possible to introduce high tensile strain in Ge/Si_1-xGe_x quantum well. Therefore, a comparison of threshold currents of lasers based on tensile strained bulk Ge and Ge/Si_1-xGe_x quantum well is in demand. We calculate the optical gain of uniaxially tensile stressed Ge/Si_1-xGe_x quantum well in [23]. Nevertheless, a laser structure based on uniaxially tensile stressed Ge/Si_1-xGe_x quantum well has rarely been proposed. In [24], we propose a Distributed Bragg Reflector (DBR) bulk Ge laser using the micro-bridge structure to introduce high uniaxial stress. The DBR laser has a low threshold current since it combines high tensile stressed Ge, low loss optical cavity and quasi-heterojunction. Similar structure can be utilized to obtain a uniaxially tensile stressed Ge/Si_1-xGe_x quantum well laser.

In this paper, we analyze and compare the threshold current densities of DBR lasers based on <100> uniaxially tensile stressed bulk Ge and Ge/Si_1-xGe_x quantum well growing along the <001> direction. The effects of strain and quantum confinement on the densities of state of L valley and carrier injection efficiency are analyzed. Differing from the laser structure in [24], the new design adopts a full etched ultra-broadband circular Bragg grating to enable larger fabrication tolerance of the gratings. Besides, a lateral p⁺-n-n⁺ junction is designed for the bulk Ge laser to obtain longer optical cavity so that the threshold current can be reduced. Through band structure calculation, optical and electrical transport simulations, we show that the threshold current density of the DBR laser based on the uniaxially tensile stressed Ge/Si_1-xGe_x quantum well is about two orders of magnitude higher than that of its bulk Ge counterpart. Therefore, we suggest that bulk Ge rather than Ge/SiGe quantum well should be used as the gain medium for Ge laser.

2. Design of the DBR lasers

2.1 Micro-bridge for quantum well and quantum well material stack

Tensile strained Ge as the gain medium is indispensable for a low threshold Ge laser. SiN_x stressor [9] and suspended micro-structures [10, 11] are the two main techniques to introduce tensile strain in Ge, which can also be applied to Ge/Si_1-xGe_x quantum well. Compared with SiN_x stressor technique, the micro-bridge structure can incorporate low loss optical cavity and p-n junction for carrier injection, which is demonstrated in our previous work [24]. Therefore, we adopt the micro-bridge structure to introduce tensile strain. The high tensile strain is obtained by redistributing the thermal mismatch tensile strain in Ge-on-Si or Germanium on Insulator (GeOI) wafer using the suspended micro-bridge structure. The bulk Ge micro-bridge has been widely investigated while the Ge/Si_1-xGe_x quantum well micro-bridge is less straightforward but can be conceived analogously. In the annealing process of SiGe buffer, thermal expansion coefficients mismatch between SiGe alloy and Si also gives rise to residual tensile strain which depends on the annealing temperature and Ge composition of the buffer [25]. Therefore, the buffer layer can be used as a stressor layer. Figure 1(a) shows the micro-bridge structure for introducing tensile strain in Ge/SiGe quantum well. Note that the upper quantum well layer needs to be etched to form a ridge waveguide in order to enhance the tensile strain in the quantum well and confine the light. The micro-bridge is along the x-direction, i.e. the <100> crystal orientation. The thicknesses of buffer layer $H_{buffer}$ and multiple quantum wells (MQWs) layer $H_{r}$ are 300 nm and 450 nm, respectively. The width and length of the narrow bridge are 2 μm and 10 μm, respectively. The total length of the bridge is 740 μm and the taper angle θ is 30°. The width of the ridge is 600 nm. 3-D Finite Element Method (FEM) simulation is performed using COMSOL Multiphysics software to obtain the strain distribution of the Ge/Si_1-xGe_x quantum well micro-bridge. In the simulation, linear elastic deformation is assumed. The initial biaxial stress conditions in the buffer layer are $τ_{x x} = τ_{y y} = (C_{11} + C_{12} - 2 C_{12}^{2} / C_{11}) ε_{x x}$ and $τ_{z z} = 0$ with $ε_{x x} = 0.18 %$ . Fixed constraint is used for the bottom of the structure and symmetry condition is used in the mirror planes of the structure to reduce computational complexity. Other boundaries are considered to be free. Generally, strain-balanced quantum well stack is utilized to achieve high crystalline quality when there exists lattice mismatch between the quantum well and barrier. In the case of Ge/Si_1-xGe_x strain-balanced quantum well, the SiGe quantum barrier and Ge quantum well are tensile and compressive strained, respectively. To simplify the simulation of strain redistribution process, we assume that the effects of well and barrier counteract due to the strain balance mechanism. Namely, only the lower SiGe buffer layer is slightly biaxial tensile strained while the material above the buffer layer is considered to be unstrained. It should be noted that this assumption is only used in the FEM simulation. For the subsequent optical gain and electrical simulation, the strain of micro-bridge needs to be modified to obtain the exact strain of the well and barrier. Before the micro-bridge is fabricated, the in-plane lattice constants of the well and barrier are equal to that of the buffer layer which is $a_{b u f} (1 + ε_{b i})$ if the thermal mismatch strain $ε_{b i}$ is taken into account. After the micro-bridge forms, tensile stress is introduced in the narrow bridge and the lattice constant along the x-direction becomes $a_{b u f} (1 + ε_{x x}^{b r i})$ , where $ε_{x x}^{b r i}$ is the strain component of the micro-bridge, directly obtained from the FEM simulation. Therefore, the $ε_{x x}$ strain component of the well and barrier can be expressed as

ε_{x x} = \frac{a_{b u f}}{a} (1 + ε_{x x}^{b r i}) - 1

where

a_{b u f}

is the unstrained lattice constant of the buffer layer.

a

is the lattice constant of the well or barrier. Since the total strain contains the lattice mismatch strain and the strain induced by uniaxial stress,

ε_{y y}

and

ε_{z z}

can be divided into two parts and are described by

ε_{y y} = \frac{a_{b u f}}{a} (1 + ε_{b i}) - 1 - \frac{C_{12}}{C_{11} + C_{12}} (ε_{x x}^{b r i} - ε_{b i})

ε_{z z} = - 2 \frac{C_{12}}{C_{11}} (\frac{a_{b u f}}{a} (1 + ε_{b i}) - 1) - \frac{C_{12}}{C_{11} + C_{12}} (ε_{x x}^{b r i} - ε_{b i})

where

C_{11}

and

C_{12}

are the elastic constants.

Fig. 1 (a) Schematic diagram of the mirco-bridge structure for introducing uniaxial stress in Ge/SiGe quantum well (not to scale). (b) Distribution of the strain component $ε_{x x}^{b r i}$ calculated by 3-D FEM, when $H_{b u f f e r} = 300 n m$ , $H_{r} = 450 n m$ , $W_{r} = 600 n m$ , $θ = 30 °$ , $L_{1} = 10 μ m$ , $L_{2} = 740 μ m$ and the width of narrow bridge is 2μm.

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Figure 1(b) shows the strain component $ε_{x x}^{b r i}$ distribution of the micro-bridge simulated by 3-D FEM. As can be seen, both the buffer layer and MQWs layer are uniaxially tensile stressed with $ε_{x x}^{b r i} = 4.4 %$ . Moreover, $ε_{x x}^{b r i}$ reaches its maximum value 7% at the corner of the taper, which is still below the breakdown threshold [26]. It is worth notice that the achievable strain value depends on the crystalline quality of the material. Therefore, it is beneficial to use the SiGe on insulator (SiGeOI) wafer which is similar to the GOI wafer [27] so that high uniaxially tensile stressed quantum well can be attained.

The strain-balanced quantum well is designed using an elastic energy minimization method [28]. First, the average lattice constant of the MQWs region is calculated by

a_{| |}^{M Q W} = (a_{b} G_{b} h_{b} + a_{w} G_{w} h_{w}) / (G_{b} h_{b} + G_{w} h_{w})

G = 2 (C_{11} + 2 C_{12}) (1 - C_{12} / C_{11})

where

h_{b}

and

h_{w}

are the thicknesses of barrier and well, respectively. For a pseudomorphic interface, the lattice constants in the plane

a_{| |}

remain the same over the structure. In other words, the in-plane lattice of the buffer layer should match that of the MQWs region. In this way, Ge composition of the buffer layer can be determined. Note that the residual thermal strain should be included in the calculation of the lattice constant of SiGe buffer. The unstrained lattice constant of Si_1-xGe_x alloy is fitted from the experimental data [29].

a_{S i G e} = 0.0282 x^{2} + 0.1981 x + 5.4315

where

a_{S i G e}

is in the unit of

\overset{°}{A}

. For a laser diode, type-I quantum well and large band offset are required for strong quantum confinement and efficient carrier injection. The conduction band minima of unstrained Ge and SiGe alloy with high Ge composition are located at L-points. Nevertheless, tensile strain shifts the band edges of the quantum well and barrier and hence the optimal Ge composition of the barrier is changed. According to the model solid theory and deformation potential theory [28], we calculate the band edges of Γ-, L- and Δ-valleys and then estimate the optimal Ge composition of barrier for maximum conduction band offsets under various strain values. The energy shifts of valley i are described by [28]

Δ E_{c}^{i} = [Ξ_{d} \overset{\leftrightarrow}{1} + Ξ_{u} {{\hat{a}}_{i} {\hat{a}}_{i}}] : \overset{\leftrightarrow}{ε}

where

Ξ_{d}

and

Ξ_{u}

are the dilatation and uniaxial deformation potential, respectively.

\overset{\leftrightarrow}{1}

is the unit tensor and

{\hat{a}}_{i}

is the unit vector parallel to the k vector of valley i.

\overset{\leftrightarrow}{ε}

is the strain tensor.

{}

denotes dyadic product. The energy shift of Γ-valley is

Δ E_{c}^{Γ} = a_{c} (ε_{x x} + ε_{y y} + ε_{z z})

There are eight equivalent L-valleys and six equivalent Δ-valleys for unstrained Ge and SiGe. Uniaxial tensile stress along [100] direction breaks the degeneracy of Δ-valleys, while the L valleys remain degenerate. For L valleys, the energy shifts are

Δ E_{c}^{L} = a_{L} (ε_{x x} + ε_{y y} + ε_{z z})

For Δ-valleys along the [100] and $[\bar{1} 00]$ directions, the energy shifts are expressed as

Δ E_{c}^{Δ 2} = Ξ_{d}^{Δ} (ε_{x x} + ε_{y y} + ε_{zz}) + Ξ_{u}^{Δ} ε_{x x}

For Δ-valleys along the [010], [001], $[0 \bar{1} 0]$ and $[00 \bar{1}]$ directions, the energy shifts are

Δ E_{c}^{Δ 4} = Ξ_{d}^{Δ} (ε_{x x} + ε_{y y} + ε_{zz}) + Ξ_{u}^{Δ} ε_{y y}

The band offsets of Ge and SiGe alloy interface are determined using the model solid theory [28]. First, the unstrained band edges of conduction and valence band are determined by

E_{v} = E_{v . a v} + Δ_{s o} / 3

E_{c}^{i} = E_{v} + E_{g}^{i}

where

E_{v . a v}

and

Δ_{s o}

are the average energy of valence band and spin-orbit split off energy, respectively.

E_{g}^{i}

is the bandgap of the i conduction band. The bandgap of Γ-valley for

{Si}_{1-x} {Ge}_{x}

alloy at 300 K is [16]

E_{g}^{Γ} = 0.7985 x + 4.185 (1 - x) - 0.14 x (1 - x)

The bandgaps of L- and Δ-valleys for Si_1-xGe_x alloy at 300 K are fitted from the data of Braunstein [30]

E_{g}^{L} = 1.86 - 1.2 x

E_{g}^{Δ} = 1.087 - 0.487 x + 0.264 x^{2}

Then the conduction band edges and band offsets under strain are evaluated using Eqs. (7)-(11). The valence band edges are obtain by the 8-Band k∙p method. The material and band structure parameters of Ge and Si at 300 K are presented in Table 1. The parameters for SiGe alloy are calculated by linear interpolation. Figure 2(a) shows the optimal Ge compositions of barrier for maximum conduction band offsets at different strain values. It can be seen that the optimal Ge composition of barrier rises from 0.847 to 0.926 with the increase of $ε_{x x}^{w e l l}$ from 0 to 5%. Figure 2(b) depicts the band alignment of the Ge/Si_0.09Ge_0.91 which is the optimal configuration for $ε_{x x}^{w e l l} = 4 %$ . The conduction band minima of uniaxially tensile stressed Si_0.09Ge_0.91 are located at L points. Meanwhile, the Δ-points along the <100> and $< \bar{1} 00 >$ approximately overlap the L points due to the strain splitting effects. Therefore, choosing Ge composition lower than 0.91 will downshift the Δ-points, leading to the decrease of the conduction band offset.

Table 1. Material and band structure parameters at 300 K for Ge and Si^a

View Table

Fig. 2 (a) Optimal Ge composition of the barrier as a function of the strain component $ε_{x x}^{w e l l}$ . (b) Band alignment of the Ge/Si_0.09Ge_0.91 quantum well with $ε_{x x}^{w e l l} = 4 %$ .

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After obtaining the Ge composition of barrier, the SiGe buffer layer should be designed to make the quantum well stack under a strain balance status. Since we focus on the uniaxially tensile stressed quantum well with $ε_{x x}^{w e l l} = 4 %$ , Si_0.09Ge_0.91 is chosen to be the barrier. The widths of well and barrier are 17 nm and 21 nm, respectively. 0.18% biaxial tensile strain induced by thermal mismatch is considered in the buffer layer. Using Eqs. (4)-(6), the corresponding Ge composition of the buffer layer is calculated to be 0.91.

2.2 Optical cavity and optical gain spectra

In our previous work, we have designed a bulk Ge DBR laser utilizing the micro-bridge structure to introduce high uniaxial stress. Simulation results show that the laser has a low threshold current thanks to the high uniaxial stress, low loss optical cavity and quasi-heterojunction. However, 3-D Finite Different Time Domain (FDTD) simulation indicates that a 5 nm deviation of the grating period can result in a 20 nm wavelength shift of the reflection peak, leading to the misalignment of the two reflection peaks. Therefore, the original design requires high fabrication accuracy. To relax the fabrication requirement, we propose a new optical cavity design which involves a full-etched circular grating with high reflectivity over ultra-broad band and a normal waveguide grating as the output mirror, as depicted in Figs. 3(a) and 3(b). Figure 3(c) illustrates the ridge waveguide used in the lasers. For the bulk Ge laser, the waveguide only supports fundamental TE-mode. For the quantum well laser, owing to the restriction of the waveguide dimensions, fundamental TE- and TM-mode are supported in the waveguide. The single-mode operation and two-mode operation of the waveguides indicate that the photons are well confined in the active region for both bulk Ge and Ge/SiGe quantum well lasers. The ultra-broad band high reflectivity of the circular grating results from the considerable refractive index contrast between the air gap and grating blade. However, the air gap can evoke intense diffraction loss if deployed in a normal straight waveguide. In order to reduce the diffraction loss, Wang [32] has proposed and demonstrated a circular grating where the wave front of the diffraction wave matches the grating blade. We adopt a similar design with a different two-section taper region, as shown in Fig. 3(d). The first taper with small taper angle acts as a transition region that can reduce the refractive index difference between the straight waveguide and the second taper. The second taper with large angle serves as a diffraction region that allows the wave to form a circular wave front. 3-D FDTD simulation signifies that there will exist ripples in the reflectance spectrum without the first taper. It should be noted that the first trench of the circular grating is designed to be narrower to avoid intense vertical diffraction loss. For the bulk Ge laser, the sidewall grating can be used as the output mirror on the other side of the optical cavity in consideration of its easy fabrication process. However, for the Ge/SiGe quantum well laser, sidewall grating fails to realize reflectivity higher than 90%. Therefore, surface grating is employed as the output mirror of the quantum well laser.

Fig. 3 Schematic diagrams of the uniaxially tensile stressed (a) bulk Ge laser and (b) Ge/SiGe quantum well laser. (c) Cross section view of the straight waveguide. For bulk Ge laser, $W_{r} = 600 n m$ , $H_{1} = 100 n m$ and $H_{2} = 200 n m$ . For Ge/SiGe quanum well laser, $W_{r} = 600 n m$ , $H_{1} = 450 n m$ and $H_{2} = 300 n m$ . (d) Top view of the circular grating. Reflection coefficient spectra of the fundamental TE-mode for the gratings of (e) bulk Ge laser and (f) Ge/SiGe quantum well laser.

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3-D FDTD simulation is performed to simulate the reflection spectra and optimize the structure parameters. For the bulk Ge laser, the ridge width W_r, the ridge height H₁ and the total height of the ridge waveguide are 600 nm, 100 nm and 300 nm, respectively. The taper angles θ₁ and θ₂ are chosen to be 20° and 70°, respectively. The lengths of the first taper and the second taper are 3 μm and 4 μm, respectively. The width of the first trench is 100 nm. The widths of the other trenches and blades are 200 nm and 318 nm. The number of the trenches is 10. The etched width of the sidewall grating is 100 nm. The grating period is 420 nm with a duty cycle of 50% and a total number of 120. For the quantum well laser, the dimensions of the waveguide are the same as those used in the strain simulation. At the same time, the second taper angle changes to 60°. The widths of the other trenches and blades are both 250 nm. The etched depth of the surface grating is 200 nm with a grating period of 327 nm. The duty cycle is 50% and the number of the periods is 350.

Figures 3(e) and 3(f) show the reflection coefficient spectra of the fundamental TE-mode for the gratings of bulk Ge laser and quantum well laser, respectively. As can be seen, the bandwidths of the circular gratings of bulk Ge laser and quantum well laser with reflection coefficients above 95% are ~500 nm, which allows for relatively large fabrication tolerance. The peak reflection coefficients of the sidewall grating and the surface grating are 97% and 95%, located at 2524 nm and 2294 nm, respectively. The reflection peaks of the output gratings are determined by estimating the net gain peak positions of the gain media when the lasers are near threshold. Models based on 8-Band k∙p method are employed to calculate the net gain spectra of the uniaxially tensile stressed bulk Ge and Ge/SiGe quantum well, which can be found in our previous works [23, 24]. Figures 4(a) and 4(b) depict the net optical gain coefficient spectra of the gain media of the bulk Ge laser and the Ge/Si_0.09Ge_0.91 quantum well laser when the lasers perform at threshold. As can be seen, the peak gains of the TE-polarized light for the bulk Ge and Ge/SiGe quantum well are 180 cm⁻¹ and 600 cm⁻¹, located at 2524 nm and 2294 nm, respectively, which correspond to the reflection peaks of the output gratings. Furthermore, simulation results indicate that the optical gain of TE-polarized light is higher than that of the TM-polarized light. As a result, only TE-mode can lase in the quantum well laser even though the ridge waveguide supports two modes.

Fig. 4 (a) Net gain coefficient spectrum of the TE-polarized light for bulk Ge with a strain of 4%, a doping concentration of $7 \times 10^{18} c m^{- 3}$ and an injected carrier density of $2.3 \times 10^{18} c m^{- 3}$ . (b) Net gain coefficient spectra of the undoped Ge/Si_0.09Ge_0.91 quantum well with a strain of 4% and an injected carrier density of $2.1 \times 10^{19} c m^{- 3}$ .

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2.3 Electrical structure

Electrical structure for carrier injection is essential for an electrically driven laser. For the bulk Ge micro-bridge structure, the quasi-heterojunctions along the bridge utilizing the band offsets induced by strain distribution has been proposed [24]. However, the longitudinal heterojunction design limits the length of the optical cavity since the carrier densities experience an attenuation along the optical cavity. Therefore, longer optical cavity for lower threshold current cannot be deployed. To address this concern, a lateral Ge homo-junction is designed, which is shown in Fig. 5(a). To enhance the confinement of the carriers, the doping concentrations of the p⁺-region, n-region and n⁺-region are $1.5 \times 10^{19} c m^{- 3}$ , $7 \times 10^{18} c m^{- 3}$ and $1.5 \times 10^{19} c m^{- 3}$ , respectively. Figure 5(b) shows the p⁺-i-n⁺ junction of the Ge/SiGe quantum well laser. Since we target at the uniaxially tensile stressed quantum well with $ε_{x x}^{w e l l} = 4 %$ , the structure has a 400nm-thick Si_0.09Ge_0.91 buffer layer, followed by 5 pairs of Ge/Si_0.09Ge_0.91 quantum wells and a 140nm-thick n-doped Si_0.09Ge_0.91 layer. The doping concentrations of the p⁺- and n⁺-region are $1.5 \times 10^{19} c m^{- 3}$ . The MQWs region and part of the buffer layer are undoped in order to reduce the free carrier absorption (FCA). It should be pointed out that only the narrow bridge region is doped for both the bulk Ge and quantum well laser to avoid the FCA loss in the gratings and passive waveguide.

Fig. 5 Cross section of (a) the p⁺-n-n⁺ junction of bulk Ge laser and (b) the p⁺-i-n⁺ junction of Ge/Si_0.09Ge_0.91 quantum well laser. Electron concentration profiles of the (c) p⁺-n-n⁺ junction and (d) p⁺-i-n⁺ junction at a bias voltage of 0.6V.

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Electrical transport simulation is performed using drift-diffusion model. The parameters of the strained material, such as bandgap, electron affinity, mobility and effective mass are determined using the method described in [24]. Figures 5(c) and 5(d) depict the calculated electron concentration profiles of the lateral p⁺-n-n⁺ junction and the vertical p⁺-i-n⁺ junction under a bias voltage of 0.6 V, respectively. For the bulk Ge laser, the active region has a high electron concentration varying from $1.1 \times 10^{19} c m^{- 3}$ to $1.2 \times 10^{19} c m^{- 3}$ with a homogeneous distribution along the z direction. For the quantum well laser, the electrons are confined in the quantum well with a concentration of $6 \times 10^{18} c m^{- 3}$ . These simulation results indicate that the designed electrical structures for the bulk Ge and Ge/SiGe quantum well laser can effectively inject carriers into the active region where radiative recombination mainly occurs.

3. Analysis of densities of state, carrier injection efficiencies and threshold current densities

The success of III-V quantum well in low threshold laser is the main drive for the investigation of Ge/SiGe quantum well. Theoretically, there are two widely accepted reasons for the low threshold of quantum well laser. First, owing to the step-like density of state, the injected carrier density of the quantum well is lower than that of the conventional bulk material when the peak gain of them are the same, making contribution to reducing the threshold current [13]. On the other hand, the scaling laws can also explain why lasers with quantum structures have lower threshold current. The scaling equation for the quantum well laser is described below [14]

I_{t h} (\frac{η γ τ}{e}) = N_{2 D}^{t r} (W L) + \frac{W d}{g_{2 D}} \ln R^{- 1} + \frac{W L d}{g_{2 D}} α_{s c a t}

where

η

is the carrier injection efficiency.

γ = 1 - b / g_{2 D}

, b is the coefficient for FCA with

α_{f c a} = b N_{2 D} / L_{z}

and

g_{2 D}

is the areal density of state.

N_{2 D}

is the areal density of carrier,

L_{z}

is the width of quantum well.

τ

is the carrier lifetime.

N_{2 D}^{t r}

is the transparency carrier density. W, L and d are the width, length and thickness of the active region, respectively. Assuming the facet reflectivity R approaches 1 and the scattering loss

α_{s c a t}

can be neglected, the minimum threshold current density can be expressed as

J_{t h} = N_{2 D}^{t h} e / η γ τ

. As a consequence, the scaling law predicts a threshold current density as low as 60 A/cm² for the III-V quantum well [14]. However, as far as the GeSi material system is concerned, those two mechanisms should be treated differently. Firstly, it has been demonstrated for untrained Ge/SiGe quantum well that the energy difference between the Γ- and L-points become larger than that of bulk Ge due to the stronger quantum confinement effect of Γ-valley [21]. Consequently, more electrons are needed to fill the states between the Γ- and L-points which raises the threshold current. To obtain better understanding and extend to the case of strained quantum well, we compare the density of state of L-valley and the number of states between the Γ- and L-points of the strained Ge/SiGe quantum well with their bulk Ge counterpart. Since the effective mass approximation is adopted to calculate the band structure of L-valleys, the strain is considered to have little effect on the density of state of L-valley. The volumetric density of state of L-valley for the quantum well is calculated numerically by

g_{3 D}^{Q W} = \frac{2}{π^{2} L_{z} Δ E} \sum_{n_{L}} \int H e a v i s i d e (E_{c} + Δ E - E_{n_{L}}^{L} (k_{t})) d k_{x} d k_{y}

where

H e a v i s i d e ()

is the unit step function.

E_{c}

and

Δ E

are the band edge of L-valley and the energy range, respectively. Note that the area density of state is divided by the length of quantum well

L_{z}

to obtain the volumetric density of state which can be compared with its bulk Ge counterpart. For the bulk Ge, the average density of state of L-valleys in the energy range of

Δ E

can be expressed as

g_{3 D}^{B u l k} = 8 π {(2 m_{n}^{*})}^{3 / 2} \sqrt{Δ E} / 3 h^{3}

, where

m_{n}^{*} = 0.55

is the density of state effective mass. When the energy range

Δ E

is chosen to be

k_{B} T

,

g_{3 D}^{B u l k}

is calculated to be

2.97 \times 10^{20} c m^{- 3} e V^{- 1}

while

g_{3 D}^{Q W}

is calculated to be

3.6 \times 10^{20} c m^{- 3} e V^{- 1}

which is larger than

g_{3 D}^{B u l k}

, indicating that the Ge/SiGe quantum well has no advantage over bulk Ge in terms of the density of state. Figure 6 shows the number of states between the Γ- and L-points for the bulk Ge and Ge/SiGe quantum well under various tensile strain. The Ge compositions of barrier are chosen to be the optimized values that maximize the conduction band offsets. As can be seen, bulk Ge has less states between the Γ- and L-points than Ge/SiGe quantum well when

ε_{x x}^{w e l l} > 0.5 %

. Even though Ge/GeSi quantum well with

ε_{x x}^{w e l l} = 0

has less states to fill than unstrained bulk Ge, positive net gain cannot be achieved due to the large number of states between the Γ- and L-points.

Fig. 6 Number of states between the Γ- and L- points for bulk Ge and Ge/SiGe quantum well as a function of the strain component $ε_{x x}^{w e l l}$ .

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Another factor that should be carefully treated is the carrier injection efficiency in the scaling equation. A higher carrier injection efficiency reflects a better confinement of the injected carrier. Most of the theoretical investigations assume a high carrier injection efficiency, indicating an ideal carrier confinement [14, 21, 33]. However, this is not true in the case of Ge/SiGe quantum well. Electrical transport simulation using drift-diffusion model is performed to obtain the relationship between the current density J and carrier density N. The carrier injection efficiency is derived by $η = e L_{z} (A N + B N^{2} + C N^{3}) / J$ , where $A = 1 / τ_{S R H}$ is the Shockley–Reed–Hall (SRH) recombination coefficient. B is the radiative recombination coefficient and C is the Auger recombination coefficient. The defect limited SRH life time is set to 3 ns according to the experimental data of a GOI sample [34]. The radiative recombination coefficient for uniaxially tensile stressed Ge/SiGe quantum well with $ε_{x x}^{w e l l} = 4 %$ is calculated to be $3.3 \times 10^{- 13} c m^{3} s^{- 1}$ . The Auger recombination coefficient C of tensile strained Ge is chosen to be $1.6 \times 10^{- 30} c m^{6} s^{- 1}$ [35]. It should be noted that the strain effects on the parameters used in the electrical simulation should be taken into account. The models for the strain-dependent parameters are described in our previous work [24]. A uniaxial tensile stress with $ε_{x x}^{w e l l} = 4 %$ is considered in our simulation. Figure 7(a) shows the calculated injected carrier density as a function of the current density for the strained quantum well p⁺-i-n⁺ junction. The relationship between the carrier injection efficiency and injected carrier density is shown by the red curve in Fig. 7(b). It can be seen that the carrier injection efficiency of the strained quantum well p⁺-i-n⁺ junction is below 1%, indicating that the carrier confinement is weak. Moreover, the injection efficiency deteriorates with the increase of injected carrier density. This is because the quasi-Fermi level of electron raises with the increase of carrier density, making it easier for the electrons to escape from the quantum well. However, an injected carrier density of ~ $1 \times 10^{19} c m^{- 3}$ is required for the quantum well to produce net gain. At such high injection level, the carrier injection efficiency is less than $10^{- 3}$ . To get insight into the cause of the low carrier injection efficiency, we change the parameters of the electrical simulation and find that the effective mass and band offset are two crucial factors. According to the 8-Band k∙p band structure calculation, the effective mass of hole for Ge decreases from 0.33 to 0.16. If an unstrained effective mass is used in the simulation, the carrier injection efficiency turns out to be higher especially in the high injection level, as shown by the blue curve in Fig. 7(b). On the other hand, the small conduction band offset also results in a low carrier injection efficiency. In the simulation for comparison, the conduction band offset is set to 0.15 eV instead of the original value 0.0284 eV. The simulation results, shown by the purple curve in Fig. 7(b), signify that the carrier injection efficiency is improved by two orders of magnitude. To summarize, the actual carrier injection efficiency for the Ge/SiGe quantum well is several orders of magnitude lower than the ideal 100% which is assumed in the scaling law and a number of theoretical models. As a result, the threshold current of Ge/SiGe quantum well laser can be much higher than envisioned.

Fig. 7 (a) Injected carrier density as a function of current density for the p⁺-i-n⁺ junction of Ge/Si_0.09Ge_0.91 quantum well laser. (b) Carrier injection efficiency as a function of injected carrier density for the p⁺-i-n⁺ junction of Ge/Si_0.09Ge_0.91 quantum well laser. The red curve and blue curve represent simulation results using strained effective mass and unstrained effective mass, respectively. The purple curve represents simulation results using unstrained effective mass as well as a larger conduction band offset of 0.15 eV.

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Finally, we compare the threshold current densities of uniaxially tensile stressed bulk Ge laser and Ge/SiGe quantum well laser. The threshold conditions are expressed below.

(Γ_{a c t} g_{b u l k} - Γ_{p} α_{p} - Γ_{n} α_{n}) L_{g} = \ln (\frac{1}{R_{1} R_{2}})

(Γ_{Q W} g_{Q W} - Γ_{b u f f e r} α_{b u f f e r} - Γ_{b a r r i e r} α_{b a r r i e r} - Γ_{n} α_{n}) L_{g} = \ln (\frac{1}{R_{1} R_{2}})

where g and α represent the optical gain and FCA loss, respectively. The overlap integration between the optical field and the active region of the bulk Ge laser

Γ_{a c t}

is 75%, calculated by the Finite Difference Method (FDM). For the quantum well laser, the overlap integration

Γ_{Q W}

is 8.5%. L_g denotes the length of the gain region rather than the total length of the optical cavity and is 10 μm. The reflection coefficients of the circular mirrors and output mirrors are 97% and 95%, respectively. The calculation procedure of the FCA loss can be found in our previous work [23, 24]. The threshold gain for the bulk Ge laser is calculated to be 180 cm⁻¹. Using the optical gain model described in [24], the threshold carrier density for uniaxially tensile stressed bulk Ge with

ε_{x x} = 4 %

and a n-doping concentration of

7 \times 10^{18} c m^{- 3}

is

2.3 \times 10^{18} c m^{- 3}

. Utilizing the relationship of carrier density and current density obtained by the electrical transport simulation, the threshold current density is 37 kA/cm². For the Ge/SiGe quantum well laser, assuming the FCA loss of barrier and buffer layer can be neglected, the threshold gain is calculated to be 600 cm⁻¹, corresponding to a threshold carrier density of

2.1 \times 10^{19} c m^{- 3}

. Such a carrier density requires a current density as high as

1 \times 10^{4}

kA/cm². Moreover, the confinement of electron becomes weak due to the rise of the quasi Fermi level of electron, giving rise to a high carrier density of the barrier and intrinsic buffer region. As a result, the FCA loss of barrier and intrinsic buffer region should be taken into account. Therefore, the actual threshold current density for the quantum well laser is even higher, signifying that the laser using uniaxially tensile stressed Ge/SiGe quantum well as the gain medium is inferior to its bulk Ge counterpart in a realistic strain range. Nevertheless, we suggest that future works toward group IV quantum well laser can be devoted to GeSn/SiGeSn material system since direct bandgap gain medium as well as larger band offsets can be achieved [36, 37].

4. Conclusion

The electrically driven uniaxially tensile stressed bulk Ge laser and Ge/SiGe quantum well laser are designed. Full etched circular gratings with high reflectivity bandwidths of ~500 nm are utilized to relax the fabrication requirement. Lateral p⁺-n-n⁺ homo-junction is designed for the bulk Ge laser, allowing for longer optical cavity compared with the quasi-heterojunction along the optical cavity. As a consequence, the threshold current can be reduced as long as a longer micro-bridge can be realized experimentally. For the Ge/SiGe quantum well laser, the Ge compositions of barrier under different strain levels are optimized to attain maximum band offsets. The quantum effects on the density of state and number of states between the Γ- and L- points are analyzed, indicating that Ge/SiGe quantum well has no advantages over bulk Ge in terms of those two factors. Additionally, electrical transport simulations show that the carrier injection efficiency of the quantum well laser is lower than $10^{- 3}$ , resulting from the reduced effective mass, small conduction band offset and high injection level. The threshold current density of uniaxially tensile stressed bulk Ge laser with a realistic $ε_{x x}$ value of 4% is calculated to be 37 kA/cm². However, the Ge/SiGe quantum well laser at the same strain level has a threshold current density higher than $1 \times 10^{4}$ kA/cm², revealing that the Ge/SiGe quantum well underperforms even though combined with tensile strain. Therefore, we suggest that future works of group IV quantum well laser aim at GeSn/SiGeSn material system.

Funding

National Natural Science Foundation of China (NSFC) (Grant No. 61435004).

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Parameters	Symbol	Ge	Si
Deformation potentials (eV)
	$a_{c}$	−8.24	1.98
	$a_{L}$	−1.54	−0.66
	$Ξ_{d}^{Δ}$	−0.59 ^b	1.127 ^b
	$Ξ_{u}^{Δ}$	9.42 ^b	9.16 ^b
Average valence band energy (eV)	$E_{a v}$	−6.35 ^c	−7.03 ^c
Spin-orbit split off energy (eV)	$Δ$	0.29	0.044
Elastic stiff constants (10Gpa)
	$C_{11}$	12.853	16.577
	$C_{12}$	4.826	6.393

Analysis of threshold current of uniaxially tensile stressed bulk Ge and Ge/SiGe quantum well lasers

Abstract

Corrections

1. Introduction

2. Design of the DBR lasers

2.1 Micro-bridge for quantum well and quantum well material stack

2.2 Optical cavity and optical gain spectra

2.3 Electrical structure

3. Analysis of densities of state, carrier injection efficiencies and threshold current densities

4. Conclusion

Funding

References and links

Cited By

Figures (7)

Tables (1)

Equations (20)

Optics Express