## Abstract

The effective medium approximation (EMA) has been widely applied to model the effect of a solid sample with surface roughness in spectroscopic ellipsometry (SE). There are two specific cases to utilize the EMA model. One is utilizing the EMA model to perform the inversion of the optical constants of solid samples from the SE measurements. Another is utilizing the EMA model to estimate the thickness of the rough layer at solid surface from the SE measurements under the condition in which the optical constants of samples are known. For the first case, the thickness of the rough layer is usually assumed to be the root mean square (rms) roughness as measured by atomic force microscopy (AFM). We theoretically investigate the error of the EMA model to estimate optical constants for different surface morphologies and materials. Because the EMA model only accounts for the height irregularities of rough surfaces but neglects the effect of the lateral irregularities on electromagnetic scattering from rough surfaces, it is difficult to obtain high-precision results for optical constants. Moreover, the inversion error of optical constants by using the EMA model is difficult to evaluate. In the second case, the thickness of the rough layer is estimated by using the EMA model from the SE measurements, called the EMA model roughness. We show that the EMA model roughness generally has a deviation from the rms roughness as measured by AFM. Some correlated relationships are established between the EMA model roughness and the morphological parameters of rough surfaces. It is found that these relationships have similar forms but not identical coefficients for different materials. The results from this work may facilitate a better understanding and utilization for the EMA model in SE.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

As a powerful tool, spectroscopic ellipsometry (SE) has been rapidly developed and widely applied to evaluate optical constants and thin film thickness of samples owning to its fast, precise, and non-destructive capabilities [1]. As is well known, data analysis of SE is mainly based on the Fresnel equation, which only describes the reflection properties for perfectly smooth surfaces. Moreover, SE is highly sensitive to surface structures [2], hence restricting its application generally to samples with a sharp planar surface. However, any practical surface inevitably exhibits some roughness no matter how carefully it is polished. Actually, rough surfaces can cause electromagnetic scattering of incident light, which results in a large deviation from the electromagnetic propagation of the Fresnel reflection. In data analysis of SE, the deviation yields pseudo optical constants [3]. Generally, the morphologies of randomly rough surfaces are characterized by the root mean square (rms) roughness *σ* and the correlation length *ζ*. The rms roughness represents the standard deviation of the height distribution and the correlation length denotes the distance above which the heights of two points are statistically uncorrelated, respectively. The correlation length is further defined by a lag length over which the peak value of the auto-covariance function drops by a factor of *e*^{−1} [4]. For Gaussian distributed randomly rough surfaces, the autocovariance function has the equivalent form [5]:

As a general modification, a sample with surface roughness is usually modeled by a flat layer with an effective dielectric function placed on a bulk layer. The effective medium approximation (EMA) has been applied widely to estimate the effective dielectric function of the flat surface roughness layer. The correspongding optical model is known as the EMA model. The EMA model provides other two parameters to characterize surface roughness: the volume fraction and the thickness of the rough layer. Provided the surface roughness layer is composed of the bulk material and air, the void volume fraction is generally assumed as a constant 50% to perform the analysis, which leaves the thickness of the rough layer as the sole unknown parameter [6]. With the EMA model, there are two ways to determine the thickness of the rough layer. One is to utilize the EMA model to perform the inversion of the optical constants of a solid sample from the SE measurements [2]. In this situation, the thickness of the rough layer is usually assumed to be the rms roughness as measured by atomic force microscopy (AFM). Another is to utilize the EMA model to estimate the thickness of the rough layer from the SE measurements under the condition in which the optical constants of the solid sample are known.

In the second case, to distinguish the rms roughness as measured by AFM, the thickness of the rough layer estimated by the EMA model from the SE measurements is called the EMA model roughness ${d}_{EMA}$. The interest of researchers is focused on the establishment of the relationship between the thickness ${d}_{EMA}$ and the morphological parameters, including the rms roughness *σ* and the correlation length *ζ*. Koh et al. [7] got roughness layer thicknesses deduced from real-time SE measurements and rms roughness from AFM of amorphous semiconductor thin films, and they found that ${d}_{EMA}$ and *σ* closely obey a linear relationship with a slope of 1.5. Petrik et al. [8] also reported the existence of a linear relationship on chemical vapor deposition deposited poly-Si and poly-Si-on-oxide. Fujiwara et al. [9] obtained similar results in microcrystalline silicon thin films, but the linear parameter changes to 0.88. Later, Franta et al. [10] showed that it is impossible to find a universal formula between just ${d}_{EMA}$ and *σ*. In the past few years, the linear relationship has evolved to a more complicated form including more factors influencing the optical response, such as the correlation length and roughness exponent (*α*). Yanguas-Gil et al. [11] studied the influence of surface morphology on the optical response of self-affine surfaces of a-Si:H by employing Rayleigh-Rice scattering theory. Moreover, the relationship ${d}_{EMA}$ ∼${\sigma}^{2}/{\zeta}^{\alpha}$ has been proposed in calculations [11]. Franta et al. [10] have also shown a similar quadratic relation in Gaussian distributed randomly micro-rough surfaces. Table 1 lists several correlations between the EMA model roughness and the morphological parameters of rough surfaces. As shown in Table 1, even for the same material, different growth conditions can generate various correlations and coefficients between the EMA model roughness and morphological parameters.

As an effective and new approach, the finite-difference time-domain (FDTD) method, a method employing first-principles calculations of Maxwell’s equations, was recently utilized for calculating the ellipsometric response from rough surfaces [12,13]. In our recent work, we have performed an experimental verification of the FDTD method on a two-dimensional SiO_{2} rough surface, and the simulated results agree well with the experimental results [12]. In the past, first-principles calculations of electromagnetic scattering by rough surfaces were challenging owing to computer memory limitations and long calculation times. With the remarkable progress of computer technology today, this approach is now capable of becoming a crucial tool in the data analysis of SE for rough surfaces.

In this work, aiming at the first case of using the EMA model, we demonstrate theoretically the error of the EMA model in estimating the optical constants of two-dimensional Gaussian distributed randomly micro-rough surfaces with different morphological parameters and materials. The FDTD method was utilized to simulate the ellipsometric response from Gaussian distributed randomly micro-rough surfaces. The FDTD simulations are regarded as the “measured” data. The optical constants are deduced from the FDTD simulations by using the EMA model. The results calculated from the EMA model are compared with the true optical constants to investigate the precision of the EMA model. For the second case, we then deduce the EMA model roughness from the SE and establish relationships between the EMA model roughness and the morphological parameters measured by AFM for different materials.

## 2. Theoretical background and methodology

#### 2.1 SE parameters

In SE measurements, a beam of linearly polarized light is emitted onto the surface of a solid sample at an oblique angle. SE can measure the amplitude ratio and the phase difference between *p*- and *s*-polarizations, from which the optical constants can be deduced. The measured values (Ψ, Δ) are known as the SE parameters and are defined by [1]

*ρ*is the ellipsometric ratio; Ψ and Δ denote the amplitude ratio and the phase difference, respectively;

*r*and

_{p}*r*are the Fresnel reflection coefficients in

_{s}*p*- and

*s*-polarizations;

*E*and

_{rp}*E*denote the reflected and the incident electric field intensity of

_{ip}*p*-polarizations, respectively; similarly,

*E*and

_{rs}*E*represent the reflected and the incident electric field intensity of

_{is}*s*-polarizations, respectively. It is worth noting that the SE parameters depend on the amplitude ratio and the phase difference between

*p*- and

*s*-polarizations at the same time, resulting in the fact that the SE parameters are very sensitive to the surface roughness.

#### 2.2 Effective medium approximation

The optical constants of materials are well known as the dielectric function *ε* and the complex refractive index *N*. The dielectric function can be correlated with the complex refractive index as $\epsilon =N{\text{\hspace{0.05em}}}^{2}$ for nonmagnetic materials. The complex refractive index *N* is defined as$N\text{\hspace{0.05em}}\text{\hspace{0.17em}}\equiv \text{\hspace{0.05em}}\text{\hspace{0.17em}}n-i\text{\hspace{0.05em}}k$, where *n* is the refractive index and *k* is the extinction coefficient. In general, in SE analysis using the EMA model, the surface roughness layer is replaced by a single homogeneous and flat layer with an effective dielectric function (${\epsilon}_{eff}$). Bruggeman’s EMA theory has been widely employed to calculate the effective dielectric function of the surface roughness layer and it takes the form [14]:

*f*and ${\epsilon}_{n}$ represent the volume fraction and dielectric function of the measured sample, respectively; ${\epsilon}_{v}$ is the dielectric function of void spaces. In this work, we assume

*f*= 50% to perform the analysis. For the inversion of optical constants by using the EMA model, the known parameters are the incident angle, the measured (Ψ, Δ), ${\epsilon}_{v}$, and

*f*. In this case, because the thickness of the EMA model is assumed to be the rms roughness as mentioned before, the only unknown parameter is ${\epsilon}_{n}$. In this case, by mathematical inversion, the measured (Ψ, Δ) can be converted directly to the optical constants

*n*and

*k*[15].

## 3. Results and discussion

#### 3.1 Case one: Using the EMA model in the inversion of the optical constants

### 3.1.1 Effects of correlation length and relative roughness on SE parameters

This section discusses the effects of the morphological parameters on the precision of the EMA model to perform the inversion of the optical constants of a sample from SE measurements. In fact, using the EMA model to estimate the optical constants from the SE parameters is an inverse problem. Conversely, when the optical constants are known, utilizing the EMA model to calculate the SE parameters is a direct problem. Actually, solving the inverse problem is more complex than solving the direct problem. Consequently, we first investigate the precision of the EMA model solving the direct problem, which is more convenient for discussing the effects of the morphological parameters on the precision of the EMA model. Firstly, the effect of the correlation length is discussed. Figure 1 shows the FDTD simulations of the SE parameters (Ψ, Δ) of Si compared with the EMA model calculations. The morphological parameters of Gaussian randomly micro-rough surfaces are as follows: *σ* = 0.05 μm and *ζ* = 0.1, 0.2, and 0.4 μm, respectively. The incident angle of 65° is selected. The discussed wavelength range of light is 4 ≤ *λ* ≤ 10 μm because the optical constants of Si are relatively unchanged within the wavelength range, so that we can exclude the influence of optical constants of dispersive materials on the precision of the EMA model. The optical constants of Si come from the literature [16]. As shown in Fig. 1, when *σ* is fixed, a change of *ζ* can result in a variation of (Ψ, Δ). This is because *ζ* is an important factor that influences electromagnetic scattering from rough surfaces. However, the EMA model cannot capture the influence of *ζ* on the ellipsometric response from rough surfaces. This derives from the assumption that the thickness of the rough layer is equal to the rms roughness in the inversion process of optical constants. This assumption only accounts for the height irregularities but not the effect of the lateral characteristic dimensions on electromagnetic scattering from rough surfaces. As a result, the EMA model calculations cannot fit the FDTD simulations well unless *σ* and *ζ* are appropriately chosen. For instance, the EMA model can just fit Δ well when *ζ* = 0.2 μm in Fig. 1. Obviously, the results reflect the fact that the EMA model also suffers the significant disadvantage of neglecting the lateral characteristic parameter *ζ* in the inversion of optical constants from the SE parameters.

The effect of the relative roughness *σ*/*λ* is then investigated. Figure 2 illustrates the relative mean square error (RMSE) of the SE parameters calculated by using the EMA model as a function of the relative roughness. In this work, the RMSE is defined as

*FDTD*’ and ‘

*EMA*’ represent the FDTD simulations and the EMA model calculations, respectively. The incident angle and the wavelength range remain unchanged. Three materials with the same refractive indices

*n*and different extinction coefficients

*k*are considered. The values of

*n*are set as 3.4294 at 4 μm, 3.4242 at 6 μm, 3.4224 at 8 μm, and 3.4215 at 10 μm, respectively. The

*k*values of the three materials are chosen as 0, 2, and 5 in the whole discussed wavelength range, respectively. Here,

*k*and

*n*are chosen as uniform also for the purpose of excluding the influence of the optical constants of dispersive materials. Two kinds of rough surfaces are investigated:

*σ*= 0.05 μm,

*σ*/

*ζ*= 0.75 in Fig. 2(a) and

*σ*= 0.05 μm,

*σ*/

*ζ*= 0.125 in Fig. 2(b). The result shows that the RMSE of the EMA model calculations decreases as the relative roughness

*σ*/

*λ*decreases. This implies that the EMA is more applicable when the wavelength of incident light is much greater than characteristic sizes of the surface roughness.

### 3.1.2 Precision of the EMA model in the inversion of optical constants

By combining the Fresnel equations of smooth surfaces and Eq. (2), ellipsometry data can be transformed into the optical constants

*θ*denotes the incident angle of light. This section shows the error of utilizing the Fresnel equations to obtain pseudo optical constants. The precision of the EMA model is also presented in the inversion of optical constants from the SE parameters for rough surfaces with different morphologies and materials. Here, we define the relative error of the complex refractive index aswhere

*N*denotes the complex refractive index obtained from the EMA model or the Fresnel equations;

*N** represents the true complex refractive index of materials.

There are two prominent morphological factors that affect electromagnetic scattering from rough surfaces: the rms roughness and the average local slope. For Gaussian distributed randomly micro-rough surfaces, the average local slope of the surfaces can be characterized by the parameter *σ*/*ζ* [10]. Figures 3 and 4 show the relative errors of the Fresnel equations and the EMA model to obtain the complex refractive index as a function of the parameter *σ*/*ζ* for different materials. Six kinds of rough surfaces are considered. The relative roughness *σ*/*λ* is fixed as 0.0125 and the incident angle of light is 65°. The results illustrate that the pseudo optical constants have a large deviation from the true optical constants, which again indicates that the surface roughness has a significant effect on the ellipsometric measurement. As the average local slope decreases, the pseudo optical constants become closer to the true values. This is because the surface gets smoother as the average local slope decreases. For the EMA model, its inversion precision varies with the change of the average local slope, which is consistent with the previous results that the EMA model cannot capture the influence of *ζ* on the ellipsometric response. Even so, the EMA model can obtain better results than the pseudo optical constants in most cases. However, when the average local slope decreases to a certain value, such as the case of *σ*/*ζ* ≤ 0.1 considered here, the results calculated by using the EMA model are worse than the pseudo optical constants. This is because EMA theory can only describe the electromagnetic interaction between the incident light and rough surfaces with relatively small lateral dimensions. Furthermore, it can be concluded that the inversion precision of the EMA model generally decreases with *n* or *k* increasing.

In Figs. 3 and 4, the relative roughness is held constant. Actually, the relative roughness can significantly affect the inversion precision of the Fresnel equations and the EMA model. Table 2 shows the relative errors of the inversion of optical constants of the Fresnel equations and the EMA model for Si with different morphologies of rough surfaces. The wavelength of light is chosen as 400 nm, where the refractive index of Si is 5.57 and the extinction coefficient is 0.387 [16]. It can be seen that, as the relative roughness increases, with *σ*/*ζ* being held constant, the pseudo optical constants will have a larger deviation from the true optical constants. Moreover, the inversion precision of the EMA model also decreases with increasing the relative roughness. For the same relative roughness, the influence of the average local slope on the inversion precision of the Fresnel equations and the EMA model is consistent with the aforementioned results. By combining Fig. 3 and Table 2, it can be concluded that the inversion precision of the EMA model is affected significantly by the morphologies and materials of rough surfaces. The precision of the EMA model, however, is difficult to evaluate.

#### 3.2 Case two: Using the EMA model to estimate the thickness of rough layer

This section will illustrate the use of the EMA model to estimate the thickness of the rough layer from SE measurements when the optical constants of a solid sample are known. The EMA model roughness is transformed from the ellipsometric response simulated by using the FDTD method. Table 3 shows the EMA model roughness *d _{EMA}* corresponding to the same eighteen cases in Fig. 3. It can be seen that the value of

*d*varies with different correlation lengths and optical constants. This implies that the parameter

_{EMA}*σ*alone cannot describe the effect of surface roughness on spectroscopic ellipsometry in a wide range of experimental systems. Combining the results of Fig. 3 and Table 3 demonstrates that, as the value of

*d*gets closer to that of

_{EMA}*σ*, the precision of the EMA model can greatly improve in the inversion of the optical constants. For example, in the case of

*σ*= 0.05 μm,

*σ*/

*ζ*= 0.375 and

*k*= 2 in Fig. 3(b), the EMA model has the best inversion precision because the corresponding value of

*d*is 0.053 μm, which is near the

_{EMA}*σ*value of 0.05 μm.

The results in Table 3 suggest that a linear dependence between *d _{EMA}* and

*σ*is insufficient to describe the effect of surface roughness on SE. A more general correlation should be established between

*d*and morphological parameters, including

_{EMA}*σ*and

*ζ*. It is interesting that all the data presented in Table 3 coalesce into lines when

*d*is regarded as a function of a single parameter

_{EMA}*σ*

^{2}/

*ζ*as shown in Fig. 5. However, the cases in Fig. 5 are not general for different materials because the value of

*n*is fixed. The cases in Fig. 4 are replenished for different values of

*n*, as shown in Fig. 6. It can be seen that the linear relationship is still valid for different values of

*n*. All fitting results are given in Table 4. The parameter

*σ*

^{2}/

*ζ*can be interpreted as the product of

*σ*and

*σ*/

*ζ*. Therefore, the parameter

*σ*

^{2}/

*ζ*includes prominent information on the rms roughness and the average local slope that affect electromagnetic scattering from rough surfaces. In a similar way, corresponding to the cases in Table 2, the EMA model roughness values are shown in Fig. 7. The fitting results are found as a unified form:

*d*= (1.54 ± 0.12)

_{EMA}*σ*

^{2}/

*ζ +*(3.01 ± 0.83). It is worth noting that, although all the above mathematical relationships have unified forms, the coefficients are not totally identical for different materials.

## 4. Conclusions

In this work, we investigated theoretically the two most common cases of using the EMA to model the effect of surface roughness on SE. For the first case of utilizing the EMA model to perform the inversion of optical constants from SE, the thickness of the rough layer is assumed to be the rms roughness. The ellipsometric response from rough surfaces is simulated by FDTD. Compared with the FDTD simulations, the RMSE of the EMA model calculating the SE parameters is explored; the results imply that the EMA model cannot capture the influence of the correlation length on the ellipsometric response from rough surfaces. This is because the EMA model only considers the height irregularities but not the effect of the lateral characteristic dimensions on electromagnetic scattering from rough surfaces. Then the relative error of the EMA model to obtain the optical constants is presented. It is found that the EMA theory is more applicable to describe the electromagnetic interaction between light and rough surfaces with relatively small lateral dimensions. For the second case of using the EMA model to estimate the thickness of the rough layer from SE, we established the correlations between the EMA model roughness and the parameter *σ*^{2}/*ζ*, which includes the two most prominent factors affecting electromagnetic scattering from rough surfaces: the rms roughness and the average local slope. A linear relationship is found, but the coefficients are not totally identical for different materials. Conclusively, the results from this work may facilitate a better understanding for utilizing for the EMA model in ellipsometry. In addition, a novel inversion method, based on first-principles calculations of electromagnetic scattering, should be developed to obtain optical constants from the SE parameters for solid materials with randomly micro-rough surfaces.

## Funding

National Natural Science Foundation of China (51306043, 51336002) and the Open Foundation of State Key Lab of Digital Manufacturing Equipment and Technology (DMETKF2018015).

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