Comparative Study of Si-Ge-Sn Resonant Cavity Enhanced Heterojunction Bipolar Phototransistor (RCE-HPT) under Quantum Conned Stark Effect (QCSE) and Franz Keldysh Effect (FKE) at 1.55 µm

Group-IV and their alloy based Heterojunction Bipolar Phototransistors (HPTs) are of immense interest in recent day optical communication. In this paper first resonant cavity enhanced heterojunction bipolar phototransistor (RCE-HPT) with Ge 0.992 Sn 0.008 /Si 0.30 Ge 0.61 Sn 0.09 Quantum Well/barrier structure under Quantum Confined Stark Effect (QCSE) has been evaluated. Further the bulk GeSn absorption region has been considered instead of QW/barrier structure and estimated the Franz Keldysh Effect (FKE). Finally different RCE-HPT related parameters such as quantum efficiency-bandwidth product, responsivity, collector current and optical gain have been studied and compared under QCSE and FKE.


Introduction
Over last few decades, optical communication has attracted new data hungry generation due to its low loss, low attenuation and high speed than conventional electronic communication. Among different active optical devices, phtotodetector (PD) is an important building block. In last few years a large number of researches have been done on PDs with III-V materials and their alloy.
Although their performance is excellent in optical communication [1], but high fabrication cost has restricted their applications. Therefore, researchers have shifted their focus to alternative group-IV semiconductors. Si is an elementary group-IV semiconductor, but it is not compatible with modern optical communication window (1.3 μm and 1.55 μm), due to its bandgap energy i.e, 1.12eV [2]. This problem can be partially eradicated by using Ge, as its direct bandgap supports 1.55 μm at room temperature. But beyond 1.5 μm the optical response of Ge based devices falls drastically [3]. Therefore, entire optical telecommunication bands cannot be covered by them. This scenario has been changed dramatically in last few decades after successful growth of high quality Ge1-ySny films and SiGeSn alloys on Si substrate by practical chemical vapor deposition (CVD) [4,5] and molecular beam epitaxy (MBE) [6]. Beyond 8% Sn concentration the Ge1-ySny alloy becomes a direct bandgap material [7] and further enhancement of Sn concentration reduces the bandgap and not only increases the absorption coefficient but also shows a red shift [8]. Based on those notable features researchers have developed different types of optical devices such as, light emitting diode [9], LASER [10,11], p-i-n PD [12][13][14][15], metal-semiconductor-metal PD (MSM PD) [16], avalanche photodiode (APD) [17], waveguide PD [18], quantum well infrared photodetector (QWIP) [19][20][21][22] and heterojunction bipolar phototransistor (HPT) [8,[23][24][25][26][27].
Among all those aforesaid PDs, HPT is a potential device. It is a strong competitor of p-i-n and APD due to its inherent gain and absence of excess noise. A number of researchers have reported the characteristics of HPT based on III-V compounds and their alloys [28][29][30][31][32] and group-IV compounds [23][24][25][26][27]. Basu et al. [23] have reported their study on HPT based on GeSn, which shows that GeSn based HPT can compete with InGaAs based devices.
Fidaner et al. [33] and Basu and Das [34] have reported the characteristics of photoconductors exploiting Quantum Confined Stark Effect (QCSE) and Franz-Keldysh Effect (FKE) respectively. Chang and Chang [35] have reported a strain free, type-I structure of Ge0.992Sn0.008 quantum well (QW) with Si0.30Ge0.61Sn0.09 barrier. The lowest bandgap of QW is direct in nature reduces to 773.41 meV, which is lower than the bandgap of pure Ge. Chakraborty et al. [36] have shown resonant cavity enhanced (RCE) p-i-n PD, exploiting both FKE and QCSE. In our previous work [25] we have reported a comparison of HPT with and without resonant cavity.
The RCE-HPT shows approximately eight times better response than conventional HPT with equal thickness of absorption layer. The multi pass of the light wave inside the device increases the absorption in the active layer and thereby the optical response. These findings encourage the author for extending the study on RCE based HPT under QCSE and FKE and also comparing their performance at the recent telecommunication window i.e, 1.55 µm.
In this paper different RCE-HPT parameters exploiting FKE and QCSE such as Quantum Efficiency-Bandwidth Product (QEBWP), responsivity, terminal current, optical gain etc have been studied. The proposed structure exhibits in sec. 2, theoretical analysis part presents in sec.
3, followed by analytical results in sec. 4. Finally, the conclusion of this study presents in sec. 5.

Structure
In this study a N-p-n HPT with 100 µm 2 cross-sectional area has been considered with normal biasing technique. The transverse electric (TE or x-y) polarized light incident on the emitter-base junction, i.e., front side illumination has been considered. Fig. 1

Theoretical Modeling
In this section, the theoretical modeling of the proposed RCE-HPT, including the parameters related to absorption coefficient, quantum efficiency (QE), responsivity, bandwidth (BW), terminal current and optical gain are given. The parameters related to GeSn and SiGeSn alloys are obtained by linear interpolation of these of Si, Ge and Sn which can be found in Ref. [10].

Absorption Coefficient Calculation
Due to the physical phenomenon QCSE, the absorption coefficient may be expressed as [36], where, the optical energy is ℏω, the excitonic peak energy is ℏΩ, Γ's are the line width For the bulk absorption region, the absorption coefficient due to FKE is given below [36], where, Eg, µr and E0 stand for direct band gap energy, reduced mass and subband energy respectively.
By using the approach of model solid theory [37], the band gap energy Eg is obtained by [36], where ac is the conduction band hydrostatic deformation potential and Edir denote the direct band gap of the absorption region.
The sub band energy (E0) and the absorption coefficient (αb) depend on the external electric field F, are given below [36], where, the momentum matrix is, Δ is the spin-orbit splitting and n is the refractive index (R.I.).

Photodetector parameters
For a conventional structure without any resonating cavity, the QE can be expressed as [25], where, α is the absorption coefficient, d is the active region thickness and R1 is the reflectivity of the anti-reflection (AR) coating for reducing light reflection from the top surface of the HPT.
The QE of the resonant cavity structure is given as [25], 1 − 2√R 1 R 2 e −αd cos(2βL + Ψ 1 + Ψ 2 ) + R 1 R 2 e −2αd (6) where, α(E,F) is the absorption coefficient at photon energy E and external electric field F, L is the cavity length, Ψ1 and Ψ2 are the phase shifts introduced by top and bottom mirror, and R1 and R2 are the front and back mirror reflectivity respectively. The propagation constant (β) is calculated by, β = 2πn λ , where λ is the wavelength and n is the R.I. As, the R.I. (n) changes to (n + Δn), due to the linear electro optic effect, so the propagation constant (β) and QE (η) change according to the external electric field.
The BW of the PD depends on both RC and transit time delay components. As the active region is very thin, in this analysis the RC time delay can be neglected. Therefore, the BW can be calculated by using the relation [25], where, v is the electron drift velocity [25], where μ is the mobility and vs is the electron saturation velocity. Since, till now the saturation velocity of GeSn alloy is unknown; therefore, in this analysis the saturation velocity of pure Ge i.e, 6×10 4 m/s has been considered [14]. As, GeSn alloy has higher carrier mobilities than pure Ge [38], the consideration of the value of saturation velocity of pure Ge may be true and reasonable.
The peak QE at resonant point, may be expressed as [39], The base-collector junction responsivity can be calculated by [14],

HPT Modeling
The common emitter current gain is obtained by [24], where, γ = γ 0 (ξ 1 ξ 2 ⁄ ) sinh(W B L nB ⁄ ) (12) and, The responsivity of HPT can be calculated by [40], where hFE is the common emitter current gain and ℜ is the base-collector junction responsivity.
The enhancement factor (EF), which indicates the enhancement of the responsivity of the RCE-HPT than conventional one can be calculated by [15], The collector current can be expressed as [25], where IC0 stands for the reverse saturation current, can be written as [25], where, D and L denote the diffusivity constant and diffusion length of the carrier, W is the thickness, A is the cross-sectional area, np0 and pn0 are the minority carrier concentrations in the emitter and base under thermal equilibrium condition. The subscripts p, n, E and B stand for hole, electron, emitter and base respectively.
Iph is the photocurrent, is obtained by [25], I ph = ℜ × P inc (18) where Pinc is the incident photon power.
The optical gain (G) can be defined as [25],

Results and Discussions
With the help of eqn. (1) the absorption spectra in the presence of QCSE phenomenon has been evaluated. The different parameters regarding this calculation are taken from the Ref. [41]. When a bulk Ge0.992Sn0.008 absorption layer has been considered instead of GeSn/SiGeSn well-barrier structure, then a strong FKE has been noticed in the presence of external electric field. The absorption coefficient under FKE has been calculated by eqn. (2). Those values of absorption coefficients are used for QE and responsivity calculation.
The QEBWP has been calculated by using eqns. (6) and (7). The velocity of the electron in eqn.
(7) is the drift velocity, which is a simplified assumption at high speed [25]. The responsivity of HPT has been estimated by eqn. (14). In fig. 2 and 3 we have plotted the QEBWP and HPT responsivity against optical frequency (wavelength). The variation of relative QE (Δη/η) as a function of optical frequency (wavelength), which has been evaluated from eqns. (6) and (9)

Conclusion
From this study it has been noticed that, when the absorption region is considered the GeSn/SiGeSn well-barrier structure, the RCE-HPT shows a strong QCSE under external electric field. Again, when this structure has been replaced by equivalent thickness of bulk GeSn, due to the direct band gap nature and high absorption it shows FKE in the presence of electric field. In this study first different characteristics of RCE-HPT exploiting strong QCSE and FKE have been compared. After that at 1.55 µm the optical and small signal current gains have been evaluated as a function of width and carrier concentration of the base and emitter region. Though the band gap of the GeSn does not match with 1.55 µm but nevertheless at that wavelength QCSE based device gives approximately three times better response than FKE based RCE-HPT. It requires further experimental and analytical study to get proper Sn composition in Ge for exploiting FKE and QCSE correctly at 1.55 µm.