Characterization of weakly absorbing thin films by multiple linear regression analysis of absolute unwrapped phase in angle-resolved spectral reflectometry

Jingtao Dong; Rongsheng Lu

doi:10.1364/OE.26.012291

1. Introduction

Weakly absorbing thin films play an important role in the modern optoelectronic devices. The optical response of the weakly absorbing thin film is mainly determined by its thickness t and optical constants (n, κ). Characterization of these properties has been in wide demand and has been greatly improved in numerous research activities. Optical techniques that determine the thin film properties by measuring how the films interact with light are the preferred method since they are nondestructive, and require little or no sample preparation with a potential of real time measurement [1]. The two most common optical measurement types are the ellipsometry and the spectral reflectometry (SR). Ellipsometry with single wavelength and single angle of incidence cannot infer more than two parameters of thin films because only two data points, the amplitude ratio and the phase difference, are measured. Variable angle spectroscopic ellipsometry (VASE) is an extended technique of the ellipsometry providing two independent data points per wavelength over a wide spectral region and per angle of incidence. However, there are a number of reasons for the VASE to be a tough task for the simultaneous determination of t, n(λ) and κ(λ) of thin films: 1) Basic nonlinear regression analysis presents high correlation of fit parameters and thereby unknown reliability of fit parameters and probably unphysical results [2, 3]. 2) The dispersion relations such as Drude [4], Tauc-Lorentz [5], Forouhi-Bloomer [6], PJDOS [7] are needed to alleviate correlation concerns by reducing the free parameters, while the strong assumptions about the type of dispersion relation should be made before starting the fitting procedure. 3) The oblique angle of incidence limits the spot size to dozens of micrometers at least, thus preventing micro-scale area characterization.

SR has been a representative alternative for characterization of thin films by analyzing the reflectance fringe as a function of wavelength, assuming normal incidence in most cases. However, the problem of numerical aperture (NA) effect arises if one uses microscope objective or fiber probe to obtain high lateral resolutions. The objective and fiber probe introduce angles of incidence different from the default angle of normal incidence. Then, the polarization-dependent error is involved in the measured reflectance. Several approaches have been taken to compensate the NA effect of the objective and fiber probe, such as the effective angle of incidence obtained by the experimental tests [8] and the averaged angle of incidence with a weight factor according to different aperture angles [9, 10]. Nevertheless, the compensation methods are approximated and do not eliminate the error fundamentally. An alternative to reflectance fitting is the measurement of reflectance fringe as a function of angle of incidence [3, 11–14] (e.g., by using a high NA objective in the beam profile reflectometry). In contrast to SR, the NA effect is eliminated.

Either the reflectance as a function of wavelength or angle of incidence provides only one independent data point per wavelength or per angle of raw data. If one attempts to solve for too many parameters, more than one possible combination of parameter values may result in a calculated reflectance that matches the measured reflectance [1, 15, 16]. Similar to VASE, assumptions about dispersion relation are frequently made in SR to alleviate the problem of correlation. In addition, the methods of simultaneous determination of thin film parameters at present mainly focus on the nonlinear regression analysis by finding the best fit between the measured and calculated reflectance data, and parameters are inferred from the model that gives the best fit [3, 12–14]. While the nonlinear regression analysis of the reflectance is a complicated and tedious work, because there is no direct solution for minimizing the sum of squared errors (SSE). Thus, the iterative algorithms, along with the risk of falling into local optimal solutions, are commonly employed to systematically adjust the parameter estimates to reduce the SSE. Other methods based on the direct calculation are also available, such as counting fringe orders [17–19], calculating fringe envelopes [20–23] and measuring fringe extrema shift [11, 15, 24]. The problem is that the measurement accuracy is relatively low by analyzing only the typical points on the fringe.

In this work, we present an original method for simultaneous determination of weakly absorbing thin film parameters, which is free of correlation and no need of dispersion relation. In this method, a reflectance interferogram is recorded by the proposed angle-resolved spectral reflectometry in both angular and spectral domains in a single-shot measurement. The monotonicity of the merit function based on the absolute unwrapped phase with respect to t, n and κ is investigated. The unique solution of t, n and κ can be directly determined with precision from the extremum of the 3D data cube of the merit function for each wavelength by multiple linear regression. A sample of GaN thin film grown by metal organic chemical vapor deposition (MOCVD) on a polished sapphire substrate is tested. The experimental results of thickness and optical constants are compared with those of scanning electron microscopy (SEM) and other related works, respectively. The error sources and other concerns are also analyzed before the conclusion is made.

2. Method

2.1 General considerations

Although the method described in this work is materials and properties agnostic, some preconditions are required: 1) both the film and the substrate are assumed to be homogeneous and isotropic media with smooth, plane and parallel interfaces. 2) The thin film is weakly absorbing (0<κ<<n) and the reflectance of thin film under the incoherent illumination shows interference oscillations due to the coherent superposition of multiple reflections. 3) The substrate is assumed to be semi-infinity and treated incoherently, presenting no phase relation between waves reflecting off the front and back surfaces of the substrate.

The first precondition can be assumed because the proposed method generates a diffraction limited spot using a high NA objective. The thickness and interfaces within this micro-scale area can be considered as uniform, parallel and smooth with no scattering loss. For the last two preconditions, the optical path difference (OPD) between any two successive beams inside the thin film should be within the coherence length of the light source, while the OPD inside the substrate should be greatly larger than the coherence length [25].

2.2 Measurement principle

The proposed system mainly comprises three parts: the broadband light source, the measurement probe and the detection unit. The first two parts are shown in Fig. 1(a). When a linearly polarized monochromatic ray from the broadband light source is converged by the objective lens L_obj and incident on a thin film at an angle of incidence θ₀, two parallel rays reflected at the interfaces of air-film and film-substrate interferes with each other in the rear exit pupil (i.e., Fourier plane) of L_obj. In this plane, a reflectance pattern is thus formed by the coherent superposition of the reflected rays at various angles of incidence and wavelengths, and the relay lenses L₁ and L₂ then magnify and image the reflectance pattern upon the image plane. It is the reflectance pattern, which is produced by the interaction of the light with the thin film, that contains the information used to obtain the desired parameters [11–14]. The simulated reflectance pattern in the image plane apparently shows no interference oscillations, as shown in the left part of Fig. 1(b). In fact, it is composed of various fringe patterns of different wavelengths if we look into it in depth, as shown in the right part of Fig. 1(b). The cross-section profiles shown in Fig. 1(c) illustrate that the coherent superposition of the p-polarized rays with the angle of incidence and the wavelength ranging from −64° to 64° (objective NA = 0.9) and from 400 nm to 920 nm, respectively, produces a reflectance pattern with a Gaussian-like distribution, whilst the fringe patterns for the discrete wavelengths present different interference oscillations. This reflectance pattern in the image plane, seeming like incoherent, is not able to determine the thin film parameters directly. Further treatment is desired to extract the useful information.

Fig. 1 (a) Schematic of optical path of angle-resolved spectral interference. (b) Reflectance fringe simulated in the image plane. (c) Cross-section profiles of the reflectance patterns for different wavelengths along the direction of p-polarization. L₁, L₂: lenses; L_obj: objective lens; f₁, f₂, f_obj: focal lengths of L₁, L₂, L_obj; BS: beam splitter. Note that three discrete wavelengths of 400, 500, 700 nm are only shown for clarity in Fig. 1(b) and Fig. 1(c).

Download Full Size | PDF

Any cross-section through the center of the reflectance pattern in the image plane yields a plot of reflectance as a function of angle of incidence. If the incident light is s-polarized with respect to the x-z plane, then it is transformed into p-polarized with respect to the y-z plane. Thus, a slit is placed in the center of the image plane along the y-axis (i.e., p-polarization), as shown in Fig. 2(a). For each angle of incidence, the filtered light containing p-polarized reflectance over a broadband spectrum is dispersed and focused onto a 2D sensor by using an imaging spectrometer. Therefore, a reflectance interferogram is recorded in both angular and spectral domains simultaneously, as shown in Fig. 2(b). In the angular domain [y-axis, Fig. 2(c)], the light is incident at a continuum of angles ranging from -arcsin(NA) to arcsin(NA). The angle of incidence is nonlinearly related to the pixel coordinate j of the rows of the sensor by

θ_{0, j} = arc \tan \frac{d \cdot [j - (N + 1) / 2]}{f_{obj}},

where d is the pixel size of the sensor, N is the number of the pixels in the row of the sensor and f_obj is the focal length of the objective lens L_obj. In the spectral domain [z-axis, Fig. 2(d)], there is a linear correspondence between the wavelength and the pixel coordinate of the columns of the sensor calibrated by the imaging spectrometer. It is evident that the proposed system with 2M + 1 unknowns to be determined is able to obtain M × N (N>>2) independent data points in a single-shot measurement. Therefore, it is possible to eliminate the correlation without making assumptions about dispersion relation.

Fig. 2 (a) Principle of single-shot measurement of 2D angle-resolved spectral interferogram. (b) The reflectance interferogram simulated in both angular and spectral domains. The angle of incidence and the wavelength range from −64° to 64° and from 400 nm to 920 nm, respectively. Cross-section profiles of reflectance patterns as functions of (c) angle of incidence at wavelength of 670 nm and (d) wavelength at angle of incidence of 0°, respectively. (e) The wrapped and unwrapped phase maps recovered from the reflectance interferogram.

Download Full Size | PDF

2.3 Phase analysis

For a single layer film with the complex refractive index of $\tilde{n}$ ₁ = n₁-i·κ₁, the reflection coefficient of the p-polarized light is

\tilde{r} = \frac{{\tilde{r}}_{01} + {\tilde{r}}_{12} exp (- i {\tilde{γ}}_{1})}{1 + {\tilde{r}}_{01} {\tilde{r}}_{12} exp (- i {\tilde{γ}}_{1})},

where the phase change of the round-trip within a thin film layer is

{\tilde{γ}}_{1} = \frac{4 π t_{1}}{λ} {\tilde{n}}_{1} \cos {\tilde{θ}}_{1} .

The reflection coefficients for the air-film and film-substrate interfaces,

{\tilde{r}}_{01}

and

{\tilde{r}}_{12}

, respectively, are given by Fresnel equations

{\tilde{r}}_{01} = ρ_{01} \exp (i ψ_{01}) = \frac{{\tilde{n}}_{1} \cos θ_{0} - n_{0} \cos {\tilde{θ}}_{1}}{{\tilde{n}}_{1} \cos θ_{0} + n_{0} \cos {\tilde{θ}}_{1}} = \frac{{\tilde{n}}_{1}^{2} \cos θ_{0} - n_{0} {\tilde{n}}_{1} \cos {\tilde{θ}}_{1}}{{\tilde{n}}_{1}^{2} \cos θ_{0} + n_{0} {\tilde{n}}_{1} \cos {\tilde{θ}}_{1}},

{\tilde{r}}_{12} = ρ_{12} \exp (i ψ_{12}) = \frac{n_{2} \cos {\tilde{θ}}_{1} - {\tilde{n}}_{1} \cos θ_{2}}{n_{2} \cos {\tilde{θ}}_{1} + {\tilde{n}}_{1} \cos θ_{2}} = \frac{n_{2} {\tilde{n}}_{1} \cos {\tilde{θ}}_{1} - {\tilde{n}}_{1}^{2} \cos θ_{2}}{n_{2} {\tilde{n}}_{1} \cos {\tilde{θ}}_{1} + {\tilde{n}}_{1}^{2} \cos θ_{2}} .

It is convenient to set

{\tilde{n}}_{1} \cos {\tilde{θ}}_{1} = u_{1} - i v_{1},

where u₁ and v₁ are real and given by

u_{1} = (1 / \sqrt{2}) {(n_{1}^{2} - κ_{1}^{2} - n_{0}^{2} \sin^{2} θ_{0}) + {[{(n_{1}^{2} - κ_{1}^{2} - n_{0}^{2} \sin^{2} θ_{0})}^{2} + 4 n_{1}^{2} κ_{1}^{2}]}^{1 / 2}}^{1 / 2},

v_{1} = (1 / \sqrt{2}) {- (n_{1}^{2} - κ_{1}^{2} - n_{0}^{2} \sin^{2} θ_{0}) + {[{(n_{1}^{2} - κ_{1}^{2} - n_{0}^{2} \sin^{2} θ_{0})}^{2} + 4 n_{1}^{2} κ_{1}^{2}]}^{1 / 2}}^{1 / 2} .

From Eqs. (3)-(6) we have after straightforward calculation

{\tilde{γ}}_{1} = η (u_{1} - i v_{1}), η = \frac{4 π t_{1}}{λ},

ρ_{01} = {\frac{{[(n_{1}^{2} - κ_{1}^{2}) \cos θ_{0} - n_{0} u_{1}]}^{2} + {(2 n_{1} k_{1} \cos θ_{0} - n_{0} v_{1})}^{2}}{{[(n_{1}^{2} - κ_{1}^{2}) \cos θ_{0} + n_{0} u_{1}]}^{2} + {(2 n_{1} k_{1} \cos θ_{0} + n_{0} v_{1})}^{2}}}^{1 / 2},

ψ_{01} = arc \tan \frac{2 n_{0} \cos θ_{0} [v_{1} (n_{1}^{2} - κ_{1}^{2}) - 2 n_{1} κ_{1} u_{1}]}{{(n_{1}^{2} + κ_{1}^{2})}^{2} \cos^{2} θ_{0} - n_{0}^{2} (u_{1}^{2} + v_{1}^{2})},

ρ_{12} = {\frac{{[(n_{1}^{2} - κ_{1}^{2}) \cos θ_{2} - n_{2} u_{1}]}^{2} + {(2 n_{1} κ_{1} \cos θ_{2} - n_{2} v_{1})}^{2}}{{[(n_{1}^{2} - κ_{1}^{2}) \cos θ_{2} + n_{2} u_{1}]}^{2} + {(2 n_{1} κ_{1} \cos θ_{2} + n_{2} v_{1})}^{2}}}^{1 / 2},

ψ_{12} = arc tan \frac{2 n_{2} \cos θ_{2} [v_{1} (n_{1}^{2} - κ_{1}^{2}) - 2 n_{1} κ_{1} u_{1}]}{{(n_{1}^{2} + κ_{1}^{2})}^{2} \cos^{2} θ_{2} - n_{2}^{2} (u_{1}^{2} + v_{1}^{2})} .

The Eq. (2) now becomes

\tilde{r} = ρ \exp (i ψ) = \frac{ρ_{01} \exp (i ψ_{01}) + ρ_{12} exp (- η v_{1}) exp [i (ψ_{12} - η u_{1})]}{1 + ρ_{01} ρ_{12} exp (- η v_{1}) exp [i (ψ_{01} + ψ_{12} - η u_{1})]},

where the reflectance and the phase of the reflection coefficient are

R = {| ρ |}^{2} = \frac{ρ_{01}^{2} \exp (η v_{1}) + ρ_{12}^{2} \exp (- η v_{1}) + 2 ρ_{01} ρ_{12} \cos (ψ_{12} - ψ_{01} - η u_{1})}{\exp (η v_{1}) + ρ_{01}^{2} ρ_{12}^{2} \exp (- η v_{1}) + 2 ρ_{01} ρ_{12} \cos (ψ_{12} + ψ_{01} - η u_{1})} .

ψ = arc \tan \frac{ρ_{12} (1 - ρ_{01}^{2}) \sin (ψ_{12} - η u_{1}) + ρ_{01} [\exp (η v_{1}) - ρ_{12}^{2} \exp (- η v_{1})] \sin ψ_{01}}{ρ_{12} (1 + ρ_{01}^{2}) \cos (ψ_{12} - η u_{1}) + ρ_{01} [\exp (η v_{1}) + ρ_{12}^{2} \exp (- η v_{1})] \cos ψ_{01}} .

The phase values calculated with the arctangent function are wrapped to the interval [-π, π), the 2D phase unwrapping algorithm [26] can then be applied to remove the jumps, as shown in Fig. 2(e). Since the refractive indices of air and substrate, n₀ and n₂, respectively, are supposed to be known, and the ranges of the angle of incident θ₀ and the wavelength λ can be obtained from the experiment setup, the reflectance and the phase expressed by Eqs. (15) and (16) depend only on the parameters t₁, n₁ and κ₁ of the weakly absorbing thin film.

Traditionally, as in the methods of SR and beam profile reflectometry [12, 14], the reverse engineering is employed to infer the thin film parameters by minimizing the merit function established with the reflectance. However, the variations of the reflectance as functions of t₁ and n₁, as plotted in Fig. 3(a) and 3(b), respectively, show the periodic ambiguity, which gives rise to more than one possible combination of parameters generating a calculated reflectance that matches the measured reflectance. The unique solution of thin film parameters may be determined through the nonlinear iterative algorithms (e.g., Levenberg-Marquardt algorithm), where the initial parameter estimates should be carefully made in case the solution falls into a local minimum.

Fig. 3 Variations of reflectance as functions of (a) t₁, (b) n₁ and (c) κ₁, respectively. Variations of wrapped phase as functions of (d) t₁, (e) n₁ and (f) κ₁, respectively. Variations of unwrapped phase as functions of (g) t₁, (h) n₁ and (i) κ₁, respectively. In the simulation, the angle of incidence and wavelength are arbitrary chosen to be θ₀ = 52° and λ = 690 nm, respectively. The nominal values of the parameters are t₁ = 1000 nm, n₁ = 2.3, κ₁ = 0.01, n₀ = 1 and n₂ = 4.9.

Download Full Size | PDF

An alternative proposed in this work is to analyze the phase information contained in the reflectance interferogram. From Eq. (16), it is apparent that the phase of the reflection coefficient includes not only the phase change on reflection at each boundary of thin film layer, ψ₀₁ and ψ₁₂, but also the phase change due to the optical path length of the layer thickness, ${\tilde{γ}}_{1}$ , which is composed of the real part ηu₁ and the imaginary part ην₁. Both the phase changes result in the interference oscillations in the reflectance interferogram for different angles of incidence and wavelengths, as illustrated by Eq. (15). Therefore, the phase retrieved from the interferogram is directly connected to the phase of the reflection coefficient. The variations of the phase as functions of t₁ and n₁, as plotted in Fig. 3(d), 3(e), 3(g) and 3(h), respectively, reveal that although the phase show the periodic ambiguity as the reflectance, the unwrapped phase is monotonically related to t₁ and n₁ in a wide range of variation. For the weakly absorbing thin films, both the reflectance and phase are monotonically related to κ₁, as shown in Fig. 3(c), 3(f) and 3(i). Therefore, the monotonical interval of the unwrapped phase with respect to t₁, n₁ and κ₁ allows the problem of the nonlinear regression analysis of the merit function based on the reflectance to be alleviated by the multiple linear regression analysis of the merit function based on the phase.

However, the phase values retrieved from the interferogram by the Fourier carrier frequency method [27] are not equal to the phase of the reflection coefficient obtained by Eq. (16), because both of them are known with the problem of 2πp ambiguity (p is an integer) after the phase unwrapping. In order to remove the 2π ambiguity of the unwrapped phase, the extended Kramers-Kronig (KK) relation [28] is employed in this work, which is classically based on a causality principle that connects the absolute spectral phase and the observed spectral reflectance. Since the unwrapped phase map is a continuous surface, the 2π ambiguities of the phase values at all the angles of incidence and wavelengths can be removed provided that at least one absolute phase value is calculated from the reflectance data by the KK relation. This can be done by the numerical integration of the reflectance data within only a finite spectral range, as illustrated by Eq. (25) in the Ref [28]. The result of removing the 2π ambiguity of the unwrapped phase map is illustrated in Fig. 4. An absolute phase value at the angle of incidence of 0°and the wavelength of 720 nm is calculated by the KK relation. Since the ambiguity p is an integer, a small error (no more than π/30) on the calculated phase value has no effect on the determination of p.

Fig. 4 (a) Removal of the 2π ambiguity of the unwrapped phase map. Cross-section plots of the phase values along (b) the spectral domain and (c) the spectral domain. The green dot in Fig. 4(a) and 4(b) is the absolute phase value calculated by the extended KK relation at the angle of incidence of 0°and the wavelength of 720 nm.

Download Full Size | PDF

We did not retrieve the absolute unwrapped phase map directly by the KK relation in this work for a number of reasons: 1) The existence of zeros in the spectral reflectance may result in branch points and branch cuts in the phase calculation [28]. 2) The error on the phase becomes larger at oblique incidence. 3) The well-established Fourier method [27] has higher efficiency and accuracy in the phase calculation.

2.4 Multiple linear regression of the merit function

The merit function is established based on the absolute unwrapped phase and given by

χ^{2} = \sum_{q = 1, j = 1}^{M, N} {ψ_{th} [t_{1}, n_{1} (λ), κ_{1} (λ), n_{0}, n_{2}, θ_{0, j}, λ_{q}] - ψ_{ex} (n_{0}, n_{2}, θ_{0, j}, λ_{q})}^{2} .

In order to minimize the merit function, we first estimate the variation ranges of t₁, n₁ and κ₁ for a thin film and take the estimates into ψ_th expressed by Eq. (16). Then the phase values are unwrapped and the 2π ambiguities are removed to obtain the theoretical absolute unwrapped phase values. The experimental absolute unwrapped phase values ψ_ex are retrieved from the interferogram. Since we need to determine the dispersion of n₁ and κ₁, the dimensions of Eq. (17) can then be reduced by decomposing the merit function into low-dimensional merit function at each wavelength, which is expressed as

χ_{λ}^{2} = \sum_{j = 1}^{N} {[ψ_{th} (t_{1}, n_{1}, κ_{1}, n_{0}, n_{2}, θ_{0, j}) - ψ_{ex} (n_{0}, n_{2}, θ_{0, j})]}^{2} | \begin{matrix} λ \end{matrix} .

The reciprocal of the low-dimensional merit function at each wavelength $χ_{λ}^{2}$ performs a 3D data cube with variables of t₁, n₁ and κ₁, as shown in Fig. 5(a). The minimization of the low-dimensional merit function is to locate the peak of the 3D data cube. Although t₁, n₁ and κ₁ vary in wide ranges, the monotonicity of the absolute unwrapped phase forces the low-dimensional merit function to rapidly converge to the unique solution, indicated by the arrow, by the multiple linear regression analysis. In contrast, the merit functions based on the wrapped phase and the reflectance show multiple solutions indicated by the double-headed arrows in Fig. 5(b) and 5(c). The multiple linear regression implemented on the low-dimensional merit function returns t₁, n₁ and κ₁ at each wavelength. Since all the wavelengths should share the common thickness, the unique solution of t₁ can then be obtained directly by averaging over all the thickness values in the spectral range, t₁(λ). The dimensions of the low-dimensional merit function are then reduced further to 2. The optimal solution of n₁ and κ₁ can be finally determined by locating the peak of the 2D data at each wavelength without the assumptions of dispersion relation. In accordance with the analysis above, the major procedure of determination of thin film parameters is summarized in Fig. 6. Note that if the optimal solution occurs at the boundaries of the 3D data cube, the estimates of the ranges of t₁, n₁ and κ₁ probably do not include the true solution and should be re-estimated.

Fig. 5 3D data cube slices of the reciprocal of the low-dimensional merit function based on (a) the absolute unwrapped phase, (b) the wrapped phase and (c) the reflectance, respectively. In the simulation, the nominal values of the parameters are t₁ = 1000 nm, n₁ = 2.3, κ₁ = 0.01, n₀ = 1 and n₂ = 4.9. The ranges of t₁, n₁ and κ₁ are [800, 1200] nm, [1.6, 3.0] and [0, 0.1], respectively. The angle of incidence θ₀ is in the range of [-64°, 64°] and the wavelength λ is 690 nm. The values of the 3D data cube slices are displayed in the logarithmic scale for clarity.

Download Full Size | PDF

Fig. 6 Flow chart of determination of thin film parameters.

Download Full Size | PDF

3. Results

3.1 Experiment description

The proposed method is used for determining the thickness and the optical constants of epitaxial GaN thin film grown by MOCVD on a polished sapphire (0001) substrate (hexagonal GaN). The root mean square (RMS) surface roughness of the GaN was evaluated to be 2 Å within a micro-scale area of 5 × 5 μm² by atomic force microscopy (AFM). The averaged thickness of the GaN layer was estimated to be 1.12 μm by SEM. The collimated beam from a high-power tungsten-halogen light source (coherence length close to 2.2 μm) focuses on the sample through a high NA objective (M Plan Apochromat, 60X, NA = 0.9). Following the Rayleigh criterion, the diffraction limited spot size is in the order of 1.08 μm for a central wavelength of 800 nm. The multiple reflected light from the interfaces of the GaN layer form the interference fringe at the Fourier plane of the objective, which is exactly 19.1 mm from the flange of the objective. The fringe pattern is magnified by the relay optics to a diameter slightly less than 5 mm and is then filtered by the slit (5 mm × 25 μm) of the imaging spectrometer (400~920 nm, spectral resolution of 1 nm) along the direction of p-polarization located at the image plane as shown in Fig. 2(a). A reference sapphire substrate is employed to calculate the absolute reflectance interferogram R_film following the relation of

R_{film} = (I_{film} / I_{ref}) \cdot R_{ref} .

where I_film and I_ref are the measured intensity of the reflected light of the thin film layer and the reference substrate, respectively. R_ref is the theoretical reflectance of the reference substrate that can be calculated from the Fresnel equations directly. The nonlinearity of the angle of incidence of the interferogram is corrected related to the pixel coordinate of the rows of the sensor in accordance with Eq. (1).

The corrected absolute interferogram is shown in Fig. 7(a). The interferogram shows mainly the noises due to the 2D sensor such as shot noise, dark current and electronic interference, but it shows little speckle noise and ghost fringe patterns as appeared in the traditional beam profile reflectometry that uses a laser source [11–14]. The wrapped phase map is recovered from the interferogram by the Fourier method after low-pass filtering and smoothing and then unwrapped, and the 2π ambiguity is removed, as shown in Fig. 7(b) and 7(c), respectively.

Fig. 7 Experimental results of (a) the absolute interferogram formed by the GaN layer, (b) the wrapped phase map and (c) the absolute unwrapped phase map. The angle of incidence and the wavelength range from −64.16° to 64.16° and from 400 to 920 nm, respectively.

Download Full Size | PDF

3.2 Determination of t, n and κ

As discussed in Section 2.3, the phase map in Fig. 7(c) varies monotonically after it is unwrapped with respect to the initial estimated ranges of t₁, n₁ and κ₁. In the experiment, we estimated t₁ = [900, 1200] nm, n₁ = [2.0, 3.0] and κ₁ = [0.01, 0.15] for the GaN layer in the spectral range from 400 to 920 nm. For each wavelength, the unique solution of t₁, n₁ and κ₁ can be obtained by locating the peak value from the 3D data cube of the reciprocal of the low-dimensional merit function, as indicated in Fig. 5(a). The thicknesses t₁ for all the wavelengths are plotted in Fig. 8(a). Since all the wavelengths should share the common thickness, t₁ is averaged to be 1.117 μm. The measurement accuracy, in theory, can be as high as possible by digital subdivision during multiple linear regression, but the noisy raw data determines the uncertainty. The thickness is thus estimated to be 1.117 ± 0.002 μm (3σ) with a level of confidence greater than 99%, indicating good agreement with the result (1.12 μm) of SEM.

Fig. 8 Experimental results of (a) t₁, (b) n₁ and (c) κ₁ of the GaN layer in the wavelength range from 400 to 920 nm.

Download Full Size | PDF

After the averaged thickness t₁ is obtained, the dimensions of the merit function is reduced to 2. Since the merit function is free of periodic ambiguity, the unique solution of n₁ and κ₁ can then be directly determined through the peak finding from the 2D data distribution at each wavelength, as shown in Fig. 8(b) and 8(c). The film dispersion in this spectral range is a normal dispersion. The measured refractive index dispersion n₁(λ) in Fig. 8(b) are analyzed using the first-order Sellmeier equation

n^{2} (λ) = A + \frac{B λ^{2}}{λ^{2} - C},

where A, B and C are the fitting parameters and λ is in μm. The solid line in Fig. 8(b) shows the fitted results of Eq. (20) to the measured n₁(λ). The values of the fitting parameters are A = 2.751, B = 2.099 and C = 0.06768. The goodness of fit (i.e. R-sqaure) and the root mean squared error (RMSE) is 0.9984 and 0.0028, respectively, indicating that the experimental results obey, as a good approximation, the first-order Sellmeier equation.

For the sake of comparison with the results of other works, we reproduced in Fig. 8(b) and 8(c) the experimental data of n(λ) and κ(λ) of GaN film from Kawashima et al [29] and EI-Naggar et al [30], respectively. The consistency between the results of Ref [29], and our work is noticeable in the available spectral region of comparison (400-920 nm). The deviation of our results from Ref [30] may be attributed to differences in surface roughness and crystal quality.

4. Discussions

4.1 Error analysis

In accordance with Eqs. (15) and (16), the relative measurement accuracy of the thickness and optical constants is mainly limited by the following error sources: 1) the calibration errors of the wavelength Δλ and the angle of incidence Δθ₀, 2) the noisy raw data of reflectance ΔR and 3) the phase error Δψ in the phase retrieval and phase unwrapping. The common way of error analysis is to take the partial derivative of each error source. However, it is complicated to cope with Eqs. (2)-(16) analytically. The numerical evaluation is thus carried out for each error source while keeping others as the nominal values. The values of the error sources, taking the GaN film as an example, are estimated as follows.

The wavelength drift of the light source can be neglected for the single-shot measurement of the proposed method. The calibration accuracy of wavelength of the imaging spectrometer is claimed to be ± 1 nm in the visible range. The alignment of the angle of incident on the thin film sample can be implemented by analyzing the left and right symmetry (i.e. in the angular domain) of the interferogram. The angle of incidence ranges from −64.16° to 64.16°, which corresponds to almost 640 pixels of the sensor. The uncertainty of the angle of incidence is thus estimated to be ± 0.2°.

The signal-to-noise ratio (SNR) of the reflectance interferogram depends on, besides the noises of the 2D sensor as mentioned in Section 3.1, the alignment of the Fourier plane of the objective and the relay optics [11,14]. After the optimization of the optical path, the SNR better than 13 dB is obtained when the Fourier plane is located at 19.1 mm from the flange of the objective. If the integration of the 2D sensor is implemented for a couple of seconds, the SNR better than 20 dB can be obtained.

The phase error occurs due to the phase retrieval and phase unwrapping. We simulated both the ideal interferogram and the phase map in accordance with Eqs. (15) and (16), respectively, and compared the phase map recovered from the interferogram with the ideal phase map. The relative error is estimated to be around ± 1.4%.

Taking the above estimated errors into Eqs. (2)-(16), we can roughly solve the numerical solutions of the relative errors of t₁, n₁ and κ₁ with respect to each error source, respectively, as listed in Table 1. First, the relative error of κ₁ is about two orders of magnitude larger than those of t₁ and n₁, indicating that κ₁ has to change much more in order to produce the same error due to t₁ and n₁. In other words, κ₁ has a much weaker influence on the reflectance and phase than t₁ and n₁. Second, the relative errors of t₁, n₁ and κ₁ due to ΔR are much larger than those due to Δψ, showing that the measurement accuracy is relatively low for the direct calculation methods [11, 15, 17–24] by analyzing only the typical points on the reflectance fringe, as described in Section 1, while the proposed method based on the phase analysis maintains high accuracy on the determination of thin film parameters.

Table 1. Error analysis of t₁, n₁ and κ₁ with respect to Δλ, Δθ₀, ΔR and Δψ

View Table

4.2 Thickness limits

The proposed method gives reliable results in the presence of interference oscillations in the reflectance. Therefore, the upper limit of the layer thickness that can be determined mainly depends on the coherence length of the light source, as described in Section 2.1. The lower limit of the layer thickness depends on a couple of factors. First, the Fourier method [27] used in this work may have difficulties in the retrieval of the phase map if the interferogram does not contain high enough carrier frequency to produce a frequency modulated signal in the spectral domain. In Fig. 9, taking the GaN film as an example, the frequency spectrum along the dashed line in the Fourier image of the interferogram is plotted for different thicknesses. It is difficult to separate the carrier frequency f₀ from the dc component if the thickness is less than 100 nm. At larger thicknesses the carrier frequency f₀ is sufficiently far from the dc component so it can be separated easily. It should be noted that the minimum thickness that can be determined will vary for the refractive indices of different thin film materials. Second, if the axial chromatic aberration of the objective is not properly corrected, the SNR of the reflectance in the spectral domain will degrade for the thin thickness layer. An apochromatic objective with a large NA is thus employed to provide a corrected chromatic aberration in a narrow depth of focus less than 200 nm. Therefore, the minimum thickness that can be determined is estimated to be larger than roughly (λ/n)/2 for weakly absorbing (κ<<n) thin films in accordance with the specification of the objective and the carrier frequency in the interferogram.

Fig. 9 (a) Fourier image of the interferogram. (b) Frequency spectrum in the spectral domain for different layer thicknesses.

Download Full Size | PDF

4.3 Potential for multilayer films

The potential of the proposed method for the determination of multilayer film parameters is preliminarily investigated by simulating a two-layer film. The top layer is a SiO₂ film and the second layer is a GaN film. The thicknesses of both the layers are 500 nm. The other parameters are the same as those in Fig. 3. The simulation results are plotted in Fig. 10. The variations of the reflectance, wrapped phase and unwrapped phase as functions of κ₁ and κ₂ are not shown here because they are monotonical for the weakly absorbing thin films, as illustrated in Fig. 3(c), 3(f) and 3(i). The unwrapped phase shows monotonicity with respect to t₁, n₁ of the SiO₂ film and t₂, n₂ of the GaN film. Therefore, the 6 unknowns (t₁, n₁, κ₁, t₂, n₂, κ₂) of the two-layer film at each wavelength can be determined by locating the peak of the 6D data generated by the merit function. After the unique solutions of t₁ and t₂ are obtained by averaging over all the thickness values in the spectral range, t₁(λ) and t₂(λ), respectively, the optimal solution of n₁, κ₁, n₂, κ₂ can then be determined by locating the peak of the remaining 4D data.

Fig. 10 Variations of reflectance, wrapped phase and unwrapped phase as functions of t₁, n₁, t₂, n₂, respectively. The nominal values of the parameters in the simulation are t₁ = 500 nm, n₁ = 1.455, t₂ = 500 nm, n₂ = 2.3 and n₃ = 4.9, θ₀ = 52°, λ = 690 nm.

Download Full Size | PDF

Since there are 3 unknowns (t, n, κ) to be determined for each layer at each wavelength, the multiple linear regression can only be implemented on an X-layer film, where X≤N/3 (N is the number of the data points in the angular domain). This is because solving for 3X unknowns with only N (3X>N) equations is highly underdetermined, resulting in multiple solutions.

5. Conclusion

Based on the developed angle-resolved spectral reflectometry, we presented an alternative way to determine the thickness and the optical constants of weakly absorbing thin films from the absolute unwrapped phase with respect to t, n and κ. We abandoned the assumptions about the dispersion relation of the thin film material and the nonlinear regression analysis of the traditional merit function based on the reflectance. Instead, we used the phase information recovered from the angle-resolved spectral interferogram, and investigated the phase variations as functions of t, n and κ. Monotonicity of the unwrapped phase with respect to t, n and κ is found and the 2π ambiguity of the unwrapped phase is removed. The merit function based on the absolute unwrapped phase at each wavelength performs a 3D data cube with variables of t, n and κ. One can determine the unique solution of t, n(λ) and κ(λ) from the peaks of the 3D data cubes directly by the multiple linear regression analysis with no iterative process. The measurement results of a hexagonal GaN thin film layer grown on a polished sapphire substrate is used to confirm the proposed method experimentally. The error analysis indicates that the proposed method is more robust against the errors from the phase. The thickness range that can be determined is also discussed and should be limited by the coherence length of the light source, the specification of the objective and the carrier frequency in the interferogram. The simulation of a two-layer film shows the potential of the proposed method for the determination of multilayer film parameters. Although several preconditions on the properties and structures of thin films are made, the proposed method is suitable for a wide variety of weakly absorbing thin films.

Funding

National Natural Science Foundation of China (50875074), National Major Scientific Instruments and Equipment Development Project of the Ministry of Science and Technology of China (2013YQ220749), Science and Technology Major Project of Anhui Province (15czz02012) and Fundamental Research Funds for the Central Universities.

References and links

1. M. Quinten, A Practical Guide to Optical Metrology for Thin Films (John Wiley & Sons, 2012).

2. P. Nestler and C. A. Helm, “Determination of refractive index and layer thickness of nm-thin films via ellipsometry,” Opt. Express 25(22), 27077–27085 (2017). [CrossRef] [PubMed]

3. R. A. DeCrescent, S. J. Brown, R. A. Schlitz, M. L. Chabinyc, and J. A. Schuller, “Model-blind characterization of thin-film optical constants with momentum-resolved reflectometry,” Opt. Express 24(25), 28842–28857 (2016). [CrossRef] [PubMed]

4. L. Fricke, T. Böntgen, J. Lorbeer, C. Bundesmann, R. Schmidt-Grund, and M. Grundmann, “An extended Drude model for the in-situ spectroscopic ellipsometry analysis of ZnO thin layers and surface modifications,” Thin Solid Films 571, 437–441 (2014). [CrossRef]

5. J. Jyothi, A. Biswas, P. Sarkar, A. Soum-Glaude, H. Nagaraja, and H. C. Barshilia, “Optical properties of TiAlC/TiAlCN/TiAlSiCN/TiAlSiCO/TiAlSiO tandem absorber coatings by phase-modulated spectroscopic ellipsometry,” Appl. Phys., A Mater. Sci. Process. 123(7), 496 (2017). [CrossRef]

6. M. Girtan, L. Hrostea, M. Boclinca, and B. Negulescu, “Study of oxide/metal/oxide thin films for transparent electronics and solar cells applications by spectroscopic ellipsometry,” AIMS Materials Science 4(3), 594–613 (2017). [CrossRef]

7. D. Nečas, J. Vodák, I. Ohlídal, M. Ohlídal, A. Majumdar, and L. Zajíčková, “Simultaneous determination of dispersion model parameters and local thickness of thin films by imaging spectrophotometry,” Appl. Surf. Sci. 350, 149–155 (2015). [CrossRef]

8. M. Quinten, F. Houta, and T. Fries, “Problems in thin film thickness measurement resolved: improvements of the fast Fourier transform analysis and consideration of the numerical aperture of microscope headers and collimators,” Proc. SPIE 9526, 95260R (2015). [CrossRef]

9. N. Saigal, A. Mukherjee, V. Sugunakar, and S. Ghosh, “Angle of incidence averaging in reflectance measurements with optical microscopes for studying layered two-dimensional materials,” Rev. Sci. Instrum. 85(7), 073105 (2014). [CrossRef] [PubMed]

10. K. Kim, S. Kim, S. Kwon, and H. J. Pahk, “Volumetric thin film thickness measurement using spectroscopic imaging reflectometer and compensation of reflectance modeling error,” Int. J. Precis. Eng. Manuf. 15(9), 1817–1822 (2014). [CrossRef]

11. R. Casquel, J. A. Soler, M. Holgado, A. López, A. Lavín, J. de Vicente, F. J. Sanza, M. F. Laguna, M. J. Bañuls, and R. Puchades, “Sub-micrometric reflectometry for localized label-free biosensing,” Opt. Express 23(10), 12544–12554 (2015). [CrossRef] [PubMed]

12. W. D. Joo, J. You, Y. S. Ghim, and S. W. Kim, “Angle-resolved reflectometer for thickness measurement of multi-layered thin-film structures,” Proc. SPIE 7063, 70630Q (2008). [CrossRef]

13. J. M. Leng, J. J. Sidorowich, Y. D. Yoon, J. Opsal, B. H. Lee, G. Cha, J. Moon, and S. I. Lee, “Simultaneous measurement of six layers in a silicon on insulator film stack using spectrophotometry and beam profile reflectometry,” J. Appl. Phys. 81(8), 3570–3578 (1997). [CrossRef]

14. J. T. Fanton, J. Opsal, D. L. Willenborg, S. M. Kelso, and A. Rosencwaig, “Multiparameter measurements of thin films using beam - profile reflectometry,” J. Appl. Phys. 73(11), 7035–7040 (1993). [CrossRef]

15. J. E. Park, J. Kim, and M. Cha, “Measurement of thickness profiles of glass plates by analyzing Haidinger fringes,” Appl. Opt. 56(7), 1855–1860 (2017). [CrossRef] [PubMed]

16. C. Lee, H. Choi, J. Jin, and M. Cha, “Measurement of refractive index dispersion of a fused silica plate using Fabry-Perot interference,” Appl. Opt. 55(23), 6285–6291 (2016). [CrossRef] [PubMed]

17. R. P. Shukla, D. V. Udupa, N. C. Das, and M. V. Mantravadi, “Non-destructive thickness measurement of dichromated gelatin films deposited on glass plates,” Opt. Laser Technol. 38(7), 552–557 (2006). [CrossRef]

18. E. A. de Oliveira and J. Frejlich, “Thickness and refractive index dispersion measurement in a thin film using the Haidinger interferometer,” Appl. Opt. 28(7), 1382–1386 (1989). [CrossRef] [PubMed]

19. J. A. Pradeep and P. Agarwal, “Determination of thickness, refractive index, and spectral scattering of an inhomogeneous thin film with rough interfaces,” J. Appl. Phys. 108(4), 043515 (2010). [CrossRef]

20. S. T. Yen and P. K. Chung, “Extraction of optical constants from maxima of fringing reflectance spectra,” Appl. Opt. 54(4), 663–668 (2015). [CrossRef] [PubMed]

21. P. K. Chung and S. T. Yen, “Extraction of infrared optical constants from fringing reflectance spectra,” J. Appl. Phys. 116(15), 153101 (2014). [CrossRef]

22. J. Lunácek, P. Hlubina, and M. Lunácková, “Simple method for determination of the thickness of a nonabsorbing thin film using spectral reflectance measurement,” Appl. Opt. 48(5), 985–989 (2009). [CrossRef] [PubMed]

23. S. Humphrey, “Direct calculation of the optical constants for a thin film using a midpoint envelope,” Appl. Opt. 46(21), 4660–4666 (2007). [CrossRef] [PubMed]

24. B. Šantić, “Measurement of the refractive index and thickness of a transparent film from the shift of the interference pattern due to the sample rotation,” Thin Solid Films 518(14), 3619–3624 (2010). [CrossRef]

25. O. Stenzel, The physics of thin film optical spectra (Springer, 2005).

26. D. C. Ghiglia and M. D. Pritt, Two-dimensional phase unwrapping: theory, algorithms, and software (Wiley New York, 1998).

27. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982). [CrossRef]

28. B. Gralak, M. Lequime, M. Zerrad, and C. Amra, “Phase retrieval of reflection and transmission coefficients from Kramers-Kronig relations,” J. Opt. Soc. Am. A 32(3), 456–462 (2015). [CrossRef] [PubMed]

29. T. Kawashima, H. Yoshikawa, S. Adachi, S. Fuke, and K. Ohtsuka, “Optical properties of hexagonal GaN,” J. Appl. Phys. 82(7), 3528–3535 (1997). [CrossRef]

30. A. M. El-Naggar, S. Y. El-Zaiat, and S. M. Hassan, “Optical parameters of epitaxial GaN thin film on Si substrate from the reflection spectrum,” Opt. Laser Technol. 41(3), 334–338 (2009). [CrossRef]

	t ₁	n ₁	κ ₁	Descriptions
Δλ	0.18%	0.13%	4.7%	λ ranges from 400 to 920 nm, Δλ = ± 1 nm.
Δθ₀	0.13%	0.074%	3.5%	θ₀ ranges from −64° to 64°, Δθ₀ = ± 0.2°.
ΔR	0.23%	0.15%	7.9%	R is given by Eq. (15), ΔR = ± 1% (relative error).
Δψ	0.004%	0.0043%	0.13%	ψ is given by Eq. (16), Δψ = ± 1.4% (relative error).

Characterization of weakly absorbing thin films by multiple linear regression analysis of absolute unwrapped phase in angle-resolved spectral reflectometry

Abstract

1. Introduction

2. Method

2.1 General considerations

2.2 Measurement principle

2.3 Phase analysis

2.4 Multiple linear regression of the merit function

3. Results

3.1 Experiment description

3.2 Determination of t, n and κ

4. Discussions

4.1 Error analysis

4.2 Thickness limits

4.3 Potential for multilayer films

5. Conclusion

Funding

References and links

Cited By

Figures (10)

Tables (1)

Equations (20)

Optics Express