Modeling interdiffusion in semiconductor distributed Bragg reflectors: An analytical approach

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Abstract

The effect of interdiffusion on some important optical properties of semiconductor distributed Bragg reflectors (DBRs) has been successfully modeled by simple analytical expressions. An analytical expression for the coupling coefficient κ was derived as a function of diffusion length. This expression allows the reflectivity, stop bandwidth, and penetration depth to be modeled for diffused DBR structures. A 19-period GaAs/AlAs DBR centered at 950 nm was used to test and validate the model. It has been shown that the obtained results agree very will with those reported earlier using the transfer matrix method. As so, the model may provide a simple, versatile, and rapid technique for the analysis of diffused DBRs composed of any material system.

Highlights

► Interdiffusion in DBR has been modeled using the CMT. ► Important DBR parameters are calculated analytically. ► The results compared very well with those obtained by TMM.

Introduction

Semiconductor based quarter-wave distributed Bragg reflectors (QW-DBRs) consist of periodically grown heterointerfaces serve as the key element(s) in various semiconductor optoelectronic devices, such as, the vertical cavity surface emitting lasers (VCSELs) [1], resonance-cavity-enhanced photo-diodes [2], and semiconductor saturable absorber mirrors (SESAMs) [3]. Interdiffusion has been reported to affect the electrical as well as the optical properties of DBRs in several material systems. Indeed, change in the optical properties of DBR has been attributed to interdiffusion in ion implanted DBRs [4], the doping type and level [5], and the grading and doping schemes [6]. In some devices, such as in VCSELs, the requirements for high reflectivity (>99%) and low series resistance are contradictory. Such high reflectivity require some or combination of abrupt heterointerfaces, maximum refractive index contrast, and large number of layers. On the other hand, low series resistance require graded heterointerfaces, minimum refractive index contrast and small number of layers. To meet these requirements, which are partially in conflict, and for device design, optimization and processing, several compositional grading schemes including interdiffusion have been employed [7]. Although interdiffusion of atoms across DBR heterointerfaces has been used and shown to affect their optical properties, very little has been done to theoretically model its effect.

There are several methods available for the analysis of DBRs, including, the transfer matrix method [8], the coupled mode theory (CMT) [9], hyperbolic tangent, tanh substitution technique (TST) [10], and others. The most widely used among them is the TMM. When the refractive index profile in the DBR stack is piecewise constant, the TMM can be used by characterizing each layer by a 2×2 transmission matrix. The matrices characterizing the DBR stack are then multiplied to obtain the input/output relationship from which all the optical parameters of the DBR can then be calculated. The TST can be used to calculate the overall reflection coefficient by a simple sum of transformed layer reflectivity. TST has a disadvantage that it only works at resonance or anti-resonance frequencies.

The simple approach based on layers having abrupt high and low refractive index profile cannot be always used for practical cases. The majority of real devices constitute DBRs with graded interfaces obtained by changing the DBR compositional profile smoothly leading to smooth continuous change of the refractive index. The computation of the optical properties of DBRs having such a profile is not as straight forward as in the case of abrupt one. TMM and TST can still be used for the computation by the discretization of the graded region into a number of very thin layers with constant refractive index, however, this approach is inefficient, time consuming, and impractical. The couple- mode theory can be employed to circumvent such problems, however, it is suitable only if the variation of the refractive index across the DBR is small compared to the average index.

In this paper, results are presented showing that, analytical expressions based on the coupled mode theory can be used to model and study the effect of interdiffusion on the optical properties of semiconductor DBRs. To validate the present method, the reflectivity, penetration depth and stop-bandwidth of a GaAs/AlAs DBR are calculated as a function of diffusion length. The results are found to agree very well with those of Floyd [13] using similar structure analyzed by the transfer matrix method.

Section snippets

Theoretical basis

To construct, test, and validate the mathematical model presented in this article a GaAs/AlAs DBR is considered. An initial square GaAs/AlAs refractive index profile will be modified due to AlGa interdiffusion across the heterointerfaces in the DBR stack leading to the formation of the ternary AlxGa1-xAs. Ficks second law of diffusion (Eq. (1)) is initially solved to determine the Al composition from which the refractive index profile can later be calculated.CAlt=DAl-Ga2CAlz2Assuming that

Results and discussion

The model presented in the previous section will be evaluated based on the DBR structure of Floyd et al. [13]. The DBR is a 19-period GaAs/AlAs centered at 950 nm grown on GaAs substrate. The as grown GaAs/AlAs refractive indices were calculated using Eq. (3) for Al mole fraction values of zero and unity giving nGaAs=3.54 and nAlAs=2.96. The quarter-wave layer thicknesses were calculated accordingly for Bragg condition and found to be dGaAs=67nm and dAlAs=80nm.

Fig. 1 shows the calculated Al

Conclusions

It has been shown that the effect of interdiffusion on important optical properties such as reflectance, stop bandwidth, and penetration depth of DBRs can be modeled using simple analytical expressions. These simple analytical expressions have been derived from a a modified coupled-mode theory which takes into account the refractive indices of the incident and exit interfaces to a DBR structure. An expression of the coupling coefficient κ as a function of diffusion length Ld has been derived

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