## Abstract

SiGe alloys present a large Infra-Red transparency window and a full compatibility with the standard Complementary Metal Oxide Semiconductor processing making them suitable for applications in integrated optics. In this paper we report on Mlines characterization of Si_{1-x}Ge_{x} graded index waveguides at 2.15 µm. First, a law giving the refractive index of a Si_{1-x}Ge_{x} alloy as a function of the Ge content x: n = 1.342x^{2} + 0.295x + 3.451, has been experimentally established in the 0 < x < 0.4 range. Then, we have demonstrated that our methodology based on Mlines measurements can be used as short-loop non-destructive technique to provide feedback for sample growth.

©2013 Optical Society of America

## 1. Introduction

The mid-infrared (MIR) spectral region offers promising prospects for a wide range of applications, such as gas sensing (many gases of interest present strong absorptions in this spectral region) or astronomy [1]. The development of efficient MIR waveguides and Integrated Optics devices has become a new challenge for many research groups. To address this issue, many material stacks have been proposed and tested. They are either based on semi-conductors: Silicon-On-Saphir [2], Silicon-On-Insulator [3], or different alloys as detailed by Soref *et.al* [4]. Infrared glasses, like chalcogenide and silver halide glasses have also been successfully used as explained by Labadie *et.al* [1]. Among those solutions, SiGe alloys present an interesting transparency window (2-9µm) and a full compatibility with standard Complementary Metal Oxide Semiconductor processing. Since SiGe is more and more used nowadays in mainstream microelectronics, processing of this material has become usual in Integrated Circuit production lines, opening the way for a number of applications in integrated optics.

It is noticeable that in the literature, Si_{1-x}Ge_{x} layers with constant Ge compositions have been thoroughly studied [5,6]. It should be emphasized that these studies are limited to the visible spectrum and the telecommunication wavelengths (i.e. 1.3 and 1.55 µm). Theoretical or experimental demonstrations have led to various formulations of the law linking the refractive index n, to the Germanium composition x [7,8]. But, as reviewed and demonstrated by Trita in her PhD dissertation [9], there are important discrepancies between those laws. Thus, it appeared clear to us that the development of Si_{1-x}Ge_{x} gradient waveguides for the MIR spectral region had to be accompanied by a precise characterization of their refractive index.

The purpose of this article is to report on the characterization of the refractive index profiles of these waveguides at a wavelength of 2.15 µm. This characterization procedure has been established to provide a reliable and non-destructive method to measure the optical index profile of the guiding layer and to infer from it the Ge concentration depth profile. Providing a reliable feedback loop to SiGe epitaxial growers and assessing the run-to-run reproducibility are then feasible. We will first of all describe the different experimental setups developed to accomplish this task. Then we will present the experimental results and the methodology used to analyze them. Finally, we will show that this experimental method can be used to follow quantitatively the drifts of the waveguide fabrication process.

## 2. Experimental material

The SiGe waveguides studied here were grown by reduced pressure chemical vapor deposition (RP-CVD) on bulk Si(001) and on Silicon-On-Insulator wafers. High temperatures (850°C-900°C) were used to grow the stacks, in order to promote the glide of misfit dislocations generated by the lattice mismatch between Si and SiGe. The density of defects and thus the optical losses are then minimized. To set up a robust fabrication flowchart, we chose to realize a fully epitaxial SiGe graded structure alloy capped with an epitaxial Si layer. Figure 1 shows a schematic of the SiGe stack. A detailed description of the design, fabrication and optical properties of those waveguides will be presented in a future paper [10]. The waveguide presents a Ge gradient profile obtained by varying the germane mass-flow during growth, the dichlorosilane mass-flow staying the same. A typical Ge profile measured by Secondary Ions Mass Spectrometry is presented in Fig. 1: the Ge content increases linearly from 0 to 0.4 within 1.5 µm and then decreases back to 0 symmetrically. As will be explained later, some samples presenting a constant Ge composition over their entire thickness were also fabricated.

Two experimental methods were used to characterize the samples. The first one is a commercially available dynamic SIMS. With this method, we measured the depth profile of Ge over the entire thickness of the sample. The typical precision obtained for the depth and the Ge content is about a few percents.

The second experimental setup specifically developed here is a Mlines setup. It measures directly the effective indexes of the modes of a waveguide. The working principle is schematized in Fig. 2.

Mlines is based on the prism coupling method, which was first proposed by Tien and Ulrich [11,12] and which is commonly used to characterize waveguides in the visible, near-infrared range or at 10.6 µm [13,14]. A linearly polarized (either TE or TM polarizations can be selected) collimated laser beam (λ = 2.15 µm) is incident on the base of a Germanium prism. The prism is mounted on a rotation stage to allow selection of the angle of incidence θ of the beam with respect to the normal to the prism base. At this interface, for small angles, a part of the beam is transmitted, while for angles higher than the angle of total reflection, as defined by the Fresnel laws, the beam is totally reflected. When approaching the 2D wave-guiding-layer in close vicinity to this interface, for some particular angles the evanescent field is coupled in the waveguide modes. These coupling will lead to strong absorption peaks in the reflection spectrum, the angles of which can be precisely measured by a detector placed on another rotation stage.

Typical reflectivity curves measured for a SiGe waveguide are presented in Fig. 2. The mathematical condition for coupling the incident beam to a waveguide mode, defined by its propagation constant β, is: k_{0}n_{p}sin(θ) = β; where k_{0} is the wave number of the incident beam in vacuum and n_{p} is the index of refraction of the prism. In other terms, the coupling occurs when the tangential projection of the wave number equals the propagation constant. Then the effective index of the mode is defined by: k_{0}n_{eff} = β, which leads us to the equation: n_{eff} = n_{p}sin(θ). Finally, we can see that the measurement of the mode coupling angles allows the calculation of the effective indexes n_{eff} of the waveguide modes. The precision of measurement of our setup is about Δθ = 0.01° which corresponds to a Δn_{eff} = 5.10^{−4} for coupling angles around 60°.

Finally, a modeling program was developed to compare experimental results to theory. This software computes the effective indexes of the modes of a waveguide presenting any arbitrary refractive index profile. The graded index profile is modeled, using a matrix method [15,16], by a multi-layer stack, where each layer has a constant refractive index. The optimal thickness of each individual layer depends on the local slope of the index profile; the profile presented in Fig. 1 is modeled by 10 nm thick layers. The basic principle of the matrix method is to calculate the transmission and reflection of each layer and then to multiply these coefficients, through the stack, using a matrix formalism. From these coefficients, one can solve the Maxwell equations; boundary conditions for guiding solutions lead to a transcendental equation the solutions of which are the effective indexes of the waveguide. A complete waveguide theory can be found in Snyder *et.al* [17].

As a summary, we have two experimental tools which measure the Ge depth profile and the effective indexes of the modes of our SiGe waveguides, and a modeling program which calculates the effective indexes of the modes of any arbitrary waveguide. We are now going to describe the methodology used to analyze our experimental results with our modeling program.

## 3. Experimental results

The first step was to determine the law linking the index of refraction to the Ge content: n = f(x). To determine this relationship, we focused first on Si_{1-x}Ge_{x} layers having a constant Ge concentration. For this purpose, nine 3 µm thick layers of Si_{1-x}Ge_{x} having different Ge concentrations varying from x = 0.14 to x = 0.45 were epitaxially grown on Si. These layers were then characterized by SIMS. The obtained concentration profiles are shown in Fig. 3. To make the figure readable, only six of them are reported. One can see in Fig. 3 that each layer presents a constant Ge concentration over its entire thickness. Nevertheless, a clear overshoot can be observed at the Si/SiGe interface. This corresponds to an artifact usually observed when crossing a chemically abrupt interface. Indeed, when performing SIMS measurements, a material-dependent correction factor (called the “Relative Sensitivity Factor”) is used to convert the raw signal into an atomic concentration. When the ion beam reaches the Si/Ge interface, this correction factor becomes inaccurate. The discrepancy increases with the Ge content and so does the intensity of the overshoot. Thus, the overshoots observed in Fig. 3 are measurement artifacts and are not physically present at the edge of the guiding layers. This point is noteworthy since the guided modes are very sensitive to the refractive index step at the edge of guiding layers.

Finally, for each SIMS measurements, we extract a thickness and a mean Ge content; the results are reported in Table 1. The thickness is taken as the width at half maximum of the depth curves.

The nine samples were also characterized by Mlines in order to determine their refractive indexes and thicknesses (n, t). To explain the method followed, we will focus on the sample having a Ge content of 0.376. The coupling angles were measured for both polarizations and the corresponding effective refractive index, n_{eff_exp}, were deduced. Experimental curves are shown in Fig. 2.

In parallel, through simulation, we calculated the theoretical effective index n_{eff_simu} of the modes of a waveguide defined by its refractive index and thickness (n, t). Then we tried to adjust the parameters (n, t) in order to minimize the difference |n_{eff_exp} - n_{eff_simu}| for each mode. To better visualize the result, we plotted a map of this difference over a range of thicknesses and refractive indexes. In Fig. 4(a), we present the cartography obtained for the three TE modes. To make Fig. 4(a) readable Δn = |n_{eff_exp} - n_{eff_simu}| is plotted in false colors for each mode, and the plot is limited to Δn values smaller than 0.0015, which corresponds to 3σ_{exp} where σ_{exp} is the Mlines measurement accuracy. We can distinguish in Fig. 4(a) three broad lines with different slopes, each one being related to a TE mode. The central part, in cyan, of each line corresponds to a locus where |n_{eff_exp} - n_{eff_simu}| < σ_{exp}, i.e. simulation matches experiment.

The three lines intersect at (n, t) = (3.611, 2965nm) where |n_{eff_exp} - n_{eff_simu}| < σ_{exp} holds for the three modes. It is interesting to note that, when considering a single mode, an infinite number of solutions can be found; a unique solution can be calculated only when two or more coupling modes are measurable. This is a limitation of the Mlines technique which requires more than one mode to univocally determine thickness and refractive index. Then, the same curves were plotted for the TM modes (not presented here). The results are very close to the previous one: the three TM lines intersect at (n, t) = (3.610, 2975nm). This means that the layer might present a slight anisotropy which cannot be detected by this technique. Finally we have plotted in Fig. 4(b) a cartography of the mean value of |n_{eff_exp} - n_{eff_simu}| obtained for the six modes. The cyan ellipse defines the solutions for which |n_{eff_exp} - n_{eff_simu}| < σ_{exp}. The centre and size of this ellipse gives the precision (σ_{n}, σ_{t}) on the determination of (n, t) considering the experimental accuracy σ_{exp}: n = 3.6105 +/− 10^{−3} and t = 2970 +/− 20nm. Then the same method was applied to the other eight samples. The thicknesses and refractive indexes deduced are listed in Table 1. For all the samples, the precisions obtained are: σ_{n} = 10^{−3}, σ_{t} = 20 nm. It is noteworthy that the thicknesses measured by SIMS and by Mlines are very close, the maximum difference being less than 80nm i.e. 2.5% of the thickness. This value is comparable to the precision of measurement given for the SIMS: +/− 2% of total thickness. On Table 1, we intentionally added two columns which correspond to the refractive indexes of bulk Si and bulk Ge. Then from Table 1, one can plot the function n = f(x). The result obtained is presented in Fig. 5 together with a quadratic fit.

Finally, from this set of measurements, we were able to determine the law giving the refractive index at 2.15 µm of a Si_{1-x}Ge_{x} alloy as a function of the Ge composition x: n = 0.342x^{2} + 0.295x + 3.451.

#### 3.1 Characterization of Si_{1-x}Ge_{x} gradient waveguides

After determining the n = f(x) law, the second step was to characterize layers with a gradient in Ge concentration. A Si_{1-x}Ge_{x} thick layer with a triangular Ge concentration profile was deposited on a (Si / SiO_{2}(2000nm) / Si(1200nm)) SOI substrate. The Ge concentration profile z = f(x) of this layer was then measured by SIMS (cf. Fig. 1). Then, by coupling the n = f(x) and z = f(x) laws, we deduced the index profile of the layer: n = f(z). Finally, through simulation, we calculated the theoretical effective indexes n_{eff_simu} of the modes of our stack: Si / SiO_{2}(2000nm) / Si(1200nm) / Si_{1-x}Ge_{x}. The sample was then characterized by Mlines. Four coupling angles were detected for each polarization and the corresponding effective refractive indexes, n_{eff_exp}, were determined. From this point, we compared n_{eff_simu} to n_{eff_exp}. As before, we decided to use a cartographical representation of the difference Δn = |n_{eff_exp} - n_{eff_simu}|. Nevertheless, since we now deal with gradient index layers, the (n, t) parameters cannot be used anymore for a graphical representation. For this purpose, we defined the parameters x_{max} and t_{max} as homothetic coefficients applied to, respectively, the concentration and the thicknesses coordinates of the SIMS plot presented in Fig. 1. Thus, if (x_{max}, t_{max}) is the set of parameters used to plot the cartography of |n_{eff_exp} - n_{eff_simu}|; the point (1, 1) corresponds to the real SIMS profile. In Fig. 6(a), we present the cartography obtained for the four TE modes. For each mode, Δn = |n_{eff_exp} - n_{eff_simu}| is plotted in false colors using the same color map as in Fig. 4. We can distinguish four broad lines with different slopes, each one being related to a TE mode. The central part, in cyan, of each line corresponds to a locus where |n_{eff_exp} - n_{eff_simu}| < σ_{exp}, i.e. where modeling matches the experimental data within the measurements errors. We see that the four lines intersect around the point (1, 1).

To get a better estimation, we zoomed on this area and plotted, in Fig. 6(b), a cartography of the mean value of |n_{eff_exp} - n_{eff_simu}| obtained for the eight modes (TE and TM). The result obtained is a series of concentric ellipses with a centre located at (x_{max}, t_{max}) = (1.005, 0.99). The smallest ellipse, in cyan, corresponds to a locus where |n_{eff_exp} - n_{eff_simu}| < σ_{exp}, where modeling matches the experimental data within the Mlines measurements errors. If we moreover add the SIMS error, of typically a few percent, one can definitely conclude that the two techniques, Mlines and SIMS, give the same results and that Mlines measurements associated with the modeling tool developed can be successfully used to evaluate the concentration profile of Ge in SiGe alloys. It should also be emphasized that these experimental results give us confidence in the validity of the empirical law n = f(x).

#### 3.2 Application to quantitative process monitoring

Finally, we decided to show that the methodology and the results presented above could be used to provide a feedback loop to the waveguide fabrication process. Using Mlines measurements, one can indeed follow the slow drift of the epitaxy process in a quantitative way, and correct it. The Ge depth profile obtained is for instance homothetic of the nominal profile when slow temperature drifts occur (long term coating of the quartz walls of the epitaxy reactor). In this case, Mlines measurements can be used to determine qualitatively and quantitatively the homothetic transformation to be applied to recover the nominal conditions. This approach was validated after a corrective maintenance of the epitaxy tool which is known to change slightly the deposition rates. The method followed was quite straightforward: first, we used the nominal deposition recipe on a standard SOI wafer. The SiGe wave-guiding-layer obtained was characterized by Mlines: four TE and TM modes were measured and the effective indexes were deduced n_{eff_exp}. Then from the nominal SIMS profile (presented in Fig. 1) we calculated the simulated effective indexes n_{eff_simu}. Following the procedure explained previously, we subsequently compared the experimental measurements and simulations by drawing a cartography of the average value of |n_{eff_exp} - n_{eff_simu}| obtained for the eight modes (TE and TM). The result is presented in Fig. 7.

A good correlation between experiment and simulation is obtained for the coordinates (x_{max}, t_{max}) = (0.8, 1.2). This means that, *a priori*, the nominal profile of the waveguide is missed by 20% for both coordinates: it is 20% too thick and the maximal Ge concentration is 20% too low.

Thus we modified the nominal process recipe using these correction factors and made a new cycle: process + Mlines characterization. A total of four cycles were made before stabilizing our process to a point considered satisfactory for our application. Then, afterwards, the four trial samples were characterized by SIMS. Results are summarized in Fig. 8.

The four graphics correspond to the four trial samples. For each graphic, we plot the nominal SIMS profile as in Fig. 1, the *a priori* profile (which corresponds to the best homothetic fit obtained from the Mlines measurement) and the real SIMS profile measured afterwards on the sample. On these graphics we can see that, for the four trial samples, the Mlines guessed profile (blue curves) is close to the SIMS real profile (red curves). The discrepancy, which can be estimated by measuring the homothetic difference between the SIMS and Mlines profiles, is less than a few percents. To further validate our approach, we measured, for each sample, the thickness of the SiGe alloy by cross-sectional Scanning Electron Microscopy (SEM). We obtained, once again, a very good correlation between the SEM measurements and the SIMS/Mlines measurements.

As a conclusion, we have demonstrated that our methodology based on Mlines measurements can be used as a short-loop non-destructive technique to provide feedback for sample growth. It should be emphasized that, since the method is based on the comparison of an estimated profile to a nominal profile, knowing the general shape and dimensions of this nominal profile (triangular, trapezoidal, square…) is a necessity to obtain a valid matching solution. This is however not an issue since these parameters (general shape and dimension) are dictated by the epitaxy recipe used.

## 4. Conclusion

We have experimentally established a law giving the refractive index of a Si_{1-x}Ge_{x} alloy as a function of the Ge content, x: n = 0.342x^{2} + 0.295x + 3.451. It is the first time to our knowledge that this law is determined for this range of Ge content, 0 < x < 0.4, and a wavelength of 2.15 µm. The authors are currently working on the experimental determination of similar laws, using the same method, for higher wavelengths.

Moreover, we have developed a short-loop non-destructive technique based on the Mlines measurement to estimate the Ge content in a gradient SiGe alloy waveguide. In our technique the effective index of each mode coupled in the waveguide is used to calculate the refractive index profile and the Ge depth profile. The comparison with the expected values can provide a quantitative feedback for process optimization.

## Acknowledgments

This work was partially funded by European projects Clarity (Grant agreement No 288304) and Doggies (Grant agreement No 285446).

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