Performance evaluation of high-detectivity p-i-n infrared photodetector based on compressively-strained Ge0.964Sn0.036/Ge multiple quantum wells by quantum modelling

GeSn/Ge p-i-n photodetectors with practical Ge0.964Sn0.036 active layers are theoretically investigated. First, we calculated the electronic band parameters for the heterointerfaces between strained Ge1−xSnx and relaxed (001)-oriented Ge. The carrier transport in a p-i-n photodiode built on a ten-period Ge0.964Sn0.036/Ge multiple quantum well absorber was then analyzed and numerically simulated within the Tsu−Esaki formalism by self-consistently solving the Schrödinger and Poisson equations, coupled to the kinetic rate equations. Photodetection up to a 2.1 μm cut-off wavelength is achieved. High responsivities of 0.62 A W−1 and 0.71 A W−1 were obtained under a reverse bias voltage of −3 V at peak wavelengths of 1550 nm and 1781 nm, respectively. Even for this low Sn-fraction, it is found that the photodetector quantum efficiency (49%@1.55 μm) is higher than those of comparable pure-Ge devices at room temperature. Detectivity of 3.8 × 1010 cm Hz1/2 W−1 and 7.9 × 1010 cm Hz1/2 W−1 at −1 V and −0.5 V, respectively, is achievable at room temperature for a 1550 nm wavelength peak of responsivity. This work represents a step forward in developing GeSn/Ge based infrared photodetectors.


Introduction
In recent years, there has been increasing effort toward the development and realisation of group IV semiconductor optoelectronic devices, recently encouraged by their possible heterogeneous integration with CMOS technology on silicon on insulator (SOI) and germanium on insulator (GeOI) substrates. The progress made on key photonics components, including laser sources, optical modulators and photodetectors, has been demonstrated in detail [1]. Due to its relatively high absorption coefficient at 1.3-1.55 μm, Ge is regarded as the best candidate material for detection in this spectral range. However, the optical response of thin vertical Ge-based high frequency photodetectors at the telecommunication wavelength of 1.55 μm is limited [2]. An increase of the responsivity for these wavelengths can be achieved by intermixing Sn to Ge [3]. Thus, Ge 1−x Sn x alloy is an interesting alternative since its bandgap, which is smaller than that of Ge, may further improve the optical response at 1.55 μm and will push the spectral range to longer infrared wavelengths [4][5][6]. On the other hand, GeSn alloy has poor thermal stability [7,8], which prevents its application in optoelectronic devices. In addition, the large lattice mismatch with Ge (∼15%) and the extremely low (x < 1%) solid solubility of Sn in Ge makes it very difficult to grow high-quality defect-free GeSn [9]. Notwithstanding, GeSn alloys with a few per cent Sn content can only be grown under non-equilibrium conditions. Despite these difficulties, some effort has been made toward the epitaxial growth of GeSn alloys [7,8,10] and the fabrication of GeSn based devices [4]. Recently, there have been reports on GeSn p-i-n photodetectors with small Sn content for full telecommunication spectral range applications [11]. These results show that adding Sn in a p-i-n Ge photodiode matrix also increases the responsivity of the detector in the telecom wavelength range and extends the cut-off wavelength beyond 1.7 μm [12][13][14]. Su et al published the first GeSn p-i-n photodetector grown by solid-source molecular beam epitaxy with 3% Sn composition and 820 nm thick layers [14]. At 1640 nm, they measured strong reverse voltage dependence with a small optical responsivity of 50 mA W −1 under zero bias operation. Strangely, the Sn incorporation shifts the detector cut-off responsivity towards the infrared spectral range by 60 nm only. The detection wavelength of GeSn p-i-n photodetectors fabricated on Ge substrate with Sn contents up to 3.6% in the active layer has recently reached 1.95 μm [15]. Photodiodes built on lattice relaxed Ge 1−x Sn x layers [15,16] may exhibit large densities of defects, such as dislocations acting as recombination traps, resulting in an excessive dark current. Strained Ge 1−x Sn x /Ge multiple quantum wells (MQWs) may be used as an absorbing layer to adjust the absorption threshold and to reduce the dark current. However, owing to the serial resistance induced by the number of quantum barriers, a limited number of cells or coupled quantum wells in the superlattice regime should be used. In order to design such a device, we first determine the electronic properties of strained Ge 1−x Sn x on (001)-oriented Ge, such as bandgap energy and band discontinuities. Then, we propose a p-i-n infrared photodetector with an intrinsic absorbing layer formed by straincompensated Ge/Ge 0.964 Sn 0.036 /Ge multi-quantum wells. This heterostructure is modeled with a system of Schrödinger and kinetic equations self-consistently solved with the Poisson equation. The current components due to these processes contributing to the total dark current are described. The room temperature performance is finally presented and discussed in terms of responsivity and detectivity.

Material parameters
The band offsets between binary α-Sn containing alloys and Ge are not known experimentally [17]. In order to calculate the conduction and valence-band discontinuities between a Ge 1−x Sn x strained layer and (001)-oriented relaxed Ge substrate, we have adopted the calculation procedure of Van de Walle et al that we outlined in [18] using the parameters taken from [19][20][21] and the parameters listed in table 1. The following (1)−(4) approximated analytical laws of the conduction and valence-band discontinuities between strain split L and Γ valleys were extracted versus Sn composition: with energies given in eV. Ge 1−x Sn x /Ge heterointerfaces have type I alignments at Γ-Γ, L-L and Γ-L respective critical points, meaning both electrons, heavy and light holes are confined into Ge x Sn x alloys. The relaxed bandgaps of Γ and L valleys in the conduction band (CB) relative to the top of the valence band (VB) are given by: The resulting plots of the energy gaps for compressively strained and relaxed alloys including experimental data values are summarized in figure 1. It is found that pseudomorphic Ge 1−x Sn x /Ge (001) alloys may have a direct bandgap at Sn fractions higher than 10.5% at room temperature. Regarding Table 1. Summary of the lattice constants (in angstrom), elastic stiffness constants (in 10 11 dyn cm −2 ), conduction and valence band deformation potentials, spin-orbit splitting and bulk bandgaps (in eV) of Ge and α-Sn. All symbols are used in their conventional meanings (see [19][20][21]). the unstrained Ge 1−x Sn x alloys, indirect-to-direct crossing occurs at a Sn fraction of 5.3% at room temperature. The latter values are consistent with the experimental crossovers found in [22] at about 17 and 6.3% respectively for temperatures around 10 K. Additional experimental evidence of the indirect −direct bandgap transition for pseudomorphic Ge 1−x Sn x /Ge (001) would be of fundamental and practical importance.

Studied structure and modeling
The structure under investigation consists of 10 periods of intrinsic Ge 0.964 Sn 0.036 /Ge QW cell. Both compressively strained Ge 0.964 Sn 0.036 and relaxed Ge barriers are 10 nm thick. The device is sketched in figure 2. The material parameters used are taken from [23][24][25] and summarized in table 2. The compressive lattice mismatch between Ge 0.964 Sn 0.036 and Ge is 0.52% while the average mismatch of the 200 nm-thick stacks is 0.26%. Thus the chosen thicknesses are reasonable values regarding the critical thickness of Ge 0.964 Sn 0.036 layers and strain accumulation within the stack. However, thicker MQWs for better absorption would imply strain compensation through the use of Si 0.13 Ge 0.87 barriers under tensile strain. The active region is embedded between 70 nm-thick n and p-doped Ge contact layers doped at N A = N D = 1 × 10 18 cm −3 . For the sake of simplicity, we assume that there are no intrinsic impurities or defects in the host lattice. The theoretical model and numerical methods used in this paper are based on the self-consistent solving of the onedimensional Schrödinger equation in the effective mass approximation [26][27][28][29] and Poisson equations for electrons and holes. Numerically, the problem was treated using the finite differential method. The electron and hole densities n(z) and p(z) in the contact regions are calculated according to the Boltzmann statistic. Figure 3 shows the conduction-and valence-band profiles for this ten-period Ge 0.964 Sn 0.036 /Ge MQW at zero bias. The ground level is drawn for electron, light-hole and heavy-hole with the associated wave function. Electrons fill all available states in the active part of the device where V bias is the applied bias. The formula after Tsu and Esaki generalized for the case of several conduction valleys and anisotropy of effective mass is then used for calculation of the tunneling current. For parabolic dispersion relations with integration over the transverse mass component only, the current density from extended and localized states is written as [30]: where the transmission coefficient tr(E Z ) is deduced from the transfer matrix formalism within the step approximation as for the potential [31]; T is the absolute temperature and k B and ħ are the Boltzmann and reduced Planck constants, respectively. For an analysis of the thermally activated carrier transport, the current density can be described using the rates g g ,  bandgaps, respectively. Symbols correspond to available experimental data [22] and E R Γ and E L R are obtained by the empirical pseudo-potential method (EPM). where n i (p i ) is the electron (hole) concentration in the ith mesh point. This rate, defined within the thermally activated transport theory, includes the image-force effect in the following way [33]: where hz i is the step of the mesh, ϕ i is the electrostatic potential in the ith mesh (obtained by solving the Poisson equation), ε i is the dielectric constant in the ith mesh, and 0 ω is the frequency of the local carrier oscillations along the multiple quantum well. The system of kinetic equations has been solved for the stationary case.
In the absence of the optical excitation G G 0, ni pi = = it is worth comparing the generalized formula of the dark current density described by equations (10)- (12) with the classical expression given by continuity equations [27] to account for additional current contributions:   (13), neither the three carrier Auger mechanisms, nor the band-to-band tunneling mechanism that are valid assumptions for low concentrations of photo-carriers and low reverse voltages of a p-i-n photodiode. The two sets of equations (10)- (12) or (13) thus account for same current contributions.
In the particular case of a GeSn absorbing layer (not MQWs), e.g. E 0 i C V , Δ = and no image-force effect, if one considers a first-order finite development on the electrostatic potential difference within a mesh, the transfer rate given in equation (12) [22] and thus decreases the dark current around 3.1 10 −7 A for the same −1 V reverse voltage. In order to understand the evolution of the dark current with the bias voltage, it is necessary to determine the respective influence of each component originating from the different considered mechanisms that include the conduction, diffusion, recombination and thermally activated tunnel currents. The calculation of each current component has been carried out at room temperature using the approach described in the previous section. Results are illustrated in figure 5. It is shown that the thermal diffusion and the well describe the forward bias characteristic while the behavior of the total dark current under reverse polarization thoroughly agree with the sum of the tunneling and recombination current contributions. The diffusion component remains the main mechanism for low (<0.5 V) direct bias voltages.

Results and discussion
Next we have investigated the influence of α-Sn concentration on the I−V dark characteristics. The results of our simulations of the dark current−voltage characteristics at room temperature for a pure bulk Ge p-i-n photodiode and two devices based on ten-period strained Ge 1−x Sn x /Ge MQWs with different α-Sn compositions, are illustrated in figure 6.
The dark current density not only indicates the material quality but also determines the optical receiver sensitivity. A direct comparison with the experimental results pointed out by recent investigations [15] proves that our proposed  structure allows improvement of the performance of the photodiode. The optical responsivity is obtained using the following expression: where η i is the internal quantum efficiency, Γ is the confinement factor of the optical mode within the multiple quantum wells, V ( , ) α λ is the absorption coefficient at wavelength λ and bias V, L is the waveguide length, and Ω is the coupling efficiency, including mode size mismatch and reflection [35]. i α accounts for all the propagation losses that do not generate photocurrent. We used η i = 1, 0.38, Γ = 0.25, Ω = 120 i α = cm −1 and V ( , ) α λ was theoretically calculated according to the formula outlined in our previous work [23].
The room temperature responsivity for three reverse bias voltages (0 V, −1 V and −3 V) are given in figure 7(a). This responsivity versus wavelength (0.9-2.3 μm) shows two peaks. These peaks corresponding to the hh 1 −cΓ 1 transition energy and the strained direct bandgap of Ge 0.964 Sn 0.036 /Ge QWs are observed around 1550 nm and 1781 nm, respectively. Similarly, we can see that the responsivity decreases when the reverse bias voltage increases. Under a reverse bias voltage of −3 V, the responsivities of 0.62 A W −1 at 1550 nm and 0.71 A W −1 at 1781 nm are obtained at room temperature.
Afterwards, we investigated the effect of the concentration of α-Sn on the optical responsivity for the Ge/Ge 1−x Sn x / Ge p-i-n photodetector at zero bias. Figure 7(b) shows that an increase of the optical responsivity for higher wavelengths can be observed with increasing α-Sn content at 0 V. We found that the Ge 0.98 Sn 0.02 /Ge MQW photodetector is achieved with a responsivity of 240 mA W −1 at the telecommunication wavelength of 1.55 μm. The response of the photodetector with Sn 2% is significantly higher than that reported in [36]. In addition, a rise of 3.6% of the α-Sn amount leads to an increase greater than an order of magnitude on the responsivity. Indeed, the optical responsivity curve of the Ge 0.964 Sn 0.036 /Ge p-i-n MQW is shifted ∼500 nm to higher wavelengths compared with pure Ge. At λ = 1.781 μm, an optical responsivity of ∼306 mA W −1 is observed for the Ge 1-x Sn x /Ge p-i-n photodetector with 3.6% α-Sn, whereas the Ge reference photodetector without α-Sn shows an optical responsivity of 0.17 A W −1 . Compared to Ge photodetector, responsivity is improved in the whole wavelength range from 0.9 μm to 2.3 μm. Gassenq et al [37] found that an increase of S n content enhances the responsivity of the detector in the telecom wavelength range and extends the cut-off wavelength beyond 2.4 μm.
Besides the dark current, we can test the MQW's performance by its detectivity D* which depends on the wavelength of incident light λ. The detectivity is given by the B L Sharma simplified formula: where λ is the operating wavelength and J 0 is the dark current density.
Using this expression in the region of zero bias voltage of the p−n junction, we have calculated room-temperature specific detectivity at a 1550 nm wavelength peak of responsivity. D* 7.9 10 10 cm Hz 1/2 W −1 and ∼ 3.8 10 10 cm Hz 1/2 W −1 have been achieved under reverse bias voltage of −0.5 V and −1 V, respectively, at 300 K. We have not investigated the temperature effect. Conley et al [38] found that the detectivity of GeSn photoconductors as expected increases with Sn increasing content when temperature decreases. But their device exhibits a poor value of detectivity of about 1 109 cm Figure 5. Total dark current at 300 K as a function of bias voltage from −5 V to 1 V is shown with four distinct regimes and different dominant mechanisms, the conduction current density, diffusion current density, recombination current density and tunneling current density. Hz 1/2 W −1 at 1.55 μm, as compared to the results of Zhang et al [15].
GeSn/Ge p-i-n photodetectors operating in the infrared region (1.3−1.55 μm) are finding extensive applications in long haul and high bit rate optical communication systems and local area networks [39][40][41].
In addition to optical communication, these devices are also useful for sensing applications as they have superior electro-optical characteristics, namely low dark current, high quantum efficiency, greater sensitivity and high speed of response.

Conclusion
In summary, we have theoretically studied the group-IV semiconductor in which the GeSn relaxed alloys have a direct bandgap for 5.3% α-Sn. Our investigation predicts, at room temperature, a direct bandgap for strained Ge 1−x Sn x /Ge (001) structures at a α-Sn fraction higher than 10.5%. We have theoretically simulated the characteristics of GeSn/Ge p-i-n photodetector with Sn contents up to 3.6%. The electrical and optical properties are compared with a similar Ge p-i-n reference photodetector without Sn. An increase of Sn content shifts the optical responsivity curve to higher wavelengths compared with pure Ge. The simulation of the responsivity (0.9-2.3 μm) shows that the detectors have a photo-response up to 2.1 μm, covering the entire telecommunication range. In particular, considering the absorption of the strained direct bandgap of the Ge 0.964 Sn 0.036 /Ge quantum wells, good responsivities of 0.62 A W −1 at 1550 nm and 0.71 A W −1 at 1781 nm were obtained at a reverse bias voltage of −3 V. Consequently, Ge 1−x Sn x /Ge heterostructures are a promising candidate material not only for all telecommunication bands in optoelectronics but also for near infrared applications.