Reliable modeling of ultrathin alternative plasmonic materials using spectroscopic ellipsometry [Invited]

R. Secondo; D. Fomra; N. Izyumskaya; V. Avrutin; J. N Hilfiker; A. Martin; Ü. Özgür; N. Kinsey

doi:10.1364/OME.9.000760

1. Introduction

Plasmonics [1,2] is a wide-reaching field that has a variety of applications including near-field imaging [3,4], biological sensing [5], magnetic memory [6], data transfer [7–9], particle trapping [10], energy harvesting [11], metasurfaces [12–14] and more [15]. Traditional plasmonics often relies on noble metals for their low loss and large negative permittivity. These factors are necessary for enabling long propagation lengths and strongly confined modes. Yet, noble metals are limited by aspects such as a low melting temperature and difficulties in forming sub-nanometer smooth films that can be important in light confinement and high-power optics research.

These limitations of traditional plasmonic materials have spurred research into alternative plasmonic materials with optical properties that are similar to noble metals like silver and gold [16]. Titanium nitride (TiN) is one of the more promising alternatives due to its robustness [17], gold-like properties, and CMOS-compatibility [18–22]. Additionally, it is tunable with plasma frequency variations from less than 450 nm to over 600 nm with widely varying loss [23–27]. However, the optical material properties of such alternative plasmonic materials can vary widely based on the growth methods and the conditions employed. To understand the origins of such large variation and reliably tune the growth conditions to achieve the desired material characteristics, consistent and accurate measurement of optical properties is needed.

Many techniques exist to measure the optical properties of thin films, including spectroscopic ellipsometry (SE) and reflection-transmission (RT). Both measurements utilize known material properties and physically-based calculations to find unknown material properties from measured values. SE is a widely used method [28–31] due to its relative ease and self-referenced nature. It is often used for non-absorbing and semi-absorbing thin films that are easily determined from layers of unknown thickness. Absorbing films, such as metals, present a special difficulty to measurement of either RT or SE.

In this work we show examples of characterizing titanium nitride ultrathin films using SE with and without supplemental transmission intensity (T) data. Without T data, the resulting permittivity, which provides a good match to the SE data through modeling, shows a significant variance, of up to 3 × for a single film, characterized by the figure of merit (ε’/ε”). Combining SE and T data reduces the model ambiguity allowing the optical constants, film thickness and growth rate of films to all be determined. Through a case study of three TiN films on sapphire substrates using plasma enhanced atomic layer deposition, we show that the addition of T is an important step in the characterization of such materials.

2. Ellipsometry

Spectroscopic Ellipsometry is a powerful method for finding the refractive index and thickness of thin films. Spectroscopic Ellipsometry measures the reflected change of polarized light at oblique angles to characterize the material properties. Bulk materials can be characterized from the complex reflection coefficients for s and p polarized light, see Fig. 1 [32]. For thin films, reflections from the second surface interfere with the top-reflections to produce interference patterns in the measured SE spectra that can be used to calculate both film thickness and refractive index. SE data is characterized using two values, $Ψ$ and $Δ$ , which are calculated using the formula

\frac{{\tilde{r}}_{p}}{{\tilde{r}}_{s}} = \tan Ψ e^{i Δ}

where

{\tilde{r}}_{p}

and

{\tilde{r}}_{s}

are the complex reflection coefficient for p and s polarized light, respectively.

Fig. 1 Schematic of spectroscopic ellipsometry. Here, a light source with wavelength λ is configured to produce an arbitrary polarization state (E_i) by a variable polarizer. The incident beam is then used to interrogate the sample at an angle θ. The polarization state of the reflected beam (E_o) is altered by the interaction with the sample. After passing through an analyzer, which allows the polarization state of the reflected beam to be measured at two perpendicular states, the complex s and p reflection coefficients of the sample can be determined.

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The data represented by $Ψ$ and $Δ$ do not directly translate to thin film thickness and refractive index; rather, the data must be matched to a model-calculation based on best-fit sample properties. To reduce unknown parameters that describe material optical constants, a variety of dispersion equations are often implemented such as Cauchy, Sellmeier, general oscillators (e.g. Drude, Lorentz), and B-Spline [33].

As plasmonic materials are inherently lossy, general oscillator and B-Spline dispersion are used in this work. B-Spline dispersion equations provide an efficient and easy way to extract optical parameters, but some limitations exist such as an inability to extract physical meaning from the results [34]. General oscillators are a broad category of dispersion equations that use physically-derived functions, such as Drude, Lorentz, Tauc-Lorentz, and others, to model the permittivity [33] (see Methods). As an example of parameter extraction, the Drude equation has the unscreened plasma frequency $ω_{p}$ that can be calculated as $ω_{p} = \sqrt{\frac{N q^{2}}{ε_{0} m^{*}}} = \sqrt{\frac{1}{ρ_{0} ε_{0} τ}}$ , and from these, possible extracted values include the free electron density N, effective mass $m^{*}$ , mobility, relaxation time $τ$ , and resistivity $ρ_{0}$ if the other parameters are known.

B-Spline dispersion, on the other hand, has no physically-retrievable values and is purely a fitting function to accurately match the ellipsometric data. B-Spline uses a set number of nodes for basis-functions to describe the optical functions and find a best match for $Ψ$ and $Δ$ . B-Spline can be implemented with Kramers-Kronig consistency to keep the optical functions physically plausible, but it does not allow for the extraction of any physical parameters from the fit. This method is very easy to implement and can then be converted to general oscillators and vice versa.

The optical model is used to generate $Ψ$ and $Δ$ for a given sample structure using parameters such as thickness and refractive index for each film, along with the substrate optical functions. The model-generated data are compared to the measured $Ψ$ and $Δ$ spectral data using a single output value to quantify the fit quality. Mean squared error (MSE) is a typical representation of how well the modelled data matches the SE data (see Methods). This formalization provides a single value for how well the model fits the data, but we note that there is no ideal universal MSE value, as the value depends upon the assumptions, known quantities about the material, and measurement noise. A major limitation of the MSE is a tendency to define a “correct fit” based solely upon its minimum value. If correlation exists between multiple unknown fit-parameters, then many combinations of values can manifest in an acceptable MSE.

Dielectric films have negligible loss so the refractive index is purely real. For dielectric films of unknown thickness, the balance of variables is data-heavy to $Ψ$ and $Δ$ , see Fig. 2(a), as the parameters fit are refractive index $n (λ)$ and thickness $t$ while the measured values are $Ψ (λ)$ and $Δ (λ)$ . This data saturation allows for very nice data fits to determine both thickness and refractive index for many material systems.

Fig. 2 Comparison of data balance for a) transparent film, b) semi-absorbing film, and c) thin absorbing film. For transparent and semi-absorbing films, the ellipsometry measurement are sufficient to allow for unique retrieval of the refractive index and thickness. However, for absorbing films, the ellipsometry measurements alone are insufficient to allow for a unique retrieval of the optical properties and thickness, and additional information should be added to improve the confidence of the result.

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When loss is added to a system the refractive index becomes complex leading to two values, $n$ and $k$ where $\tilde{n} = n + i k$ , at each wavelength. For some materials, such as doped-semiconductors, a region exists where $k$ is negligible so the balance of variables continues to favor $Ψ$ and $Δ$ , see Fig. 2(b), as the thickness can be uniquely fit within the transparency window and fixed. However, in very thin films with absorption across the spectrum, the balance of data no longer favors the measured values, Fig. 2(c), which can lead to fit ambiguity. Multiple methods exist for overcoming the data discrepancy [35,36]: adding RT measurements, interference enhancement [35], multi-sample analysis, and in situ ellipsometry [32].

In this work, SE measurements are conducted on a single titanium nitride film of unknown thickness, using various metrics to analyze the results. Later, a case study comparison of three films, grown with the same conditions but of different unknown thickness, is presented. We demonstrate the combination of SE and T measurements to overcome the data content shortage. To facilitate T measurements, the absorbing thin film must be on a polished, transparent substrate. This can be counterproductive to the simplicity of ellipsometry, due to backside reflections from the substrate, but can generally be managed [37]. Transmission must also exist at some point in the material which limits the thickness of the absorbing film and the substrate type.

3. Characterization of titanium nitride films

Titanium nitride is metallic with a color similar to gold; therefore, when fitting with general oscillators, we use a Drude oscillator and two Lorentz oscillators to model the transitions [23]. The film absorbs across the spectrum and typically has a plasma frequency between 450 and 650 nm, which, along with quality, varies greatly by the method and parameters of growth. Due to this variation in materials, the data is presented with very few assumptions of expected thickness or permittivity values.

When fitting materials for both refractive index and thickness, a popular tool for checking the certainty of a value is to plot parameter uniqueness. Figure 3 is an example of thickness uniqueness plots for the analysis of SE data when describing the TiN layer using a general oscillator (GenOsc) (red) and B-Spline (blue) dispersion equation. A sharp ‘v’ shape in parameter uniqueness shows a high certainty of the fit to a specific thickness. A sharp ‘v’ is seen in the modeling of GenOsc for a fit of 48 nm with a reasonable MSE value near 4, but the B-Spline has an MSE below 3 for a wide range of thicknesses (from about 40 nm to well beyond 100 nm). At thicknesses of 90 nm for this material, the thin film is considered optically opaque, so the thickness no longer has a role in the fit-quality as all secondary reflections are quenched. The GenOsc preferentially fits with 48 nm layer thickness but a less prominent fit also occurs when the layer is over 100 nm. To properly characterize this material using either method, more must be known about the material.

Fig. 3 Mean Squared Error (MSE) vs Thickness for a thin TiN layer using general oscillator and B-Spline dispersion equations to fit only SE data. The general oscillator shows a distinct minimum at 48 nm and the B-Spline has a flat, low MSE fit from 20 to 120 nm leading to a large ambiguity in fit quality.

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Another way to check the uniqueness of fits is to calculate the correlation matrix, which shows the interdependence between fitting parameters. A value of ± 1 corresponds to a perfect correlation where parameters A and B can be interchanged and still achieve the same result. Correlation near zero is ideal for fitting parameters as no variation in a variable can be countered by another parameter to achieve the same result. A measured correlation matrix is shown in Table 1 where the correlation between resistivity and scattering time are shown to be inversely correlated (e.g. approximately −1). By examining the Drude formula used to model the film, this correlation is expected and does not affect the quality of fit; it simply means there is no information to separate these two values and only one is required to fit the data with the other fixed. To retrieve information about the film resistivity from SE measurements, we would need to separately know the accurate scattering time. A more troublesome correlation is between thickness and the dispersion model equations (here, both resistivity and scattering time). This correlation suggests that an arbitrary thickness increase can be compensated by a similar factor increase in resistivity. Figure 4 shows a two-dimensional uniqueness plot corresponding to thickness and resistivity. The trough representing the minimum MSE is elongated, demonstrating this correlation between thickness and resistivity.

Table 1. Correlation matrix between thickness, resistivity, and scattering time using GenOsc models. The correlation between thickness and resistivity or thickness and scattering time being close to ± 1 is non-ideal, meaning there is not uniqueness to the fit.

View Table | View all tables in this article

Fig. 4 Logarithmic surface plot of Mean Squared Error for various resistivity and thickness combinations. The elongated trough shows a strong correlation between thickness and resistivity which induces a large variability in material properties.

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B-Spline produces a similar issue as shown in the thickness uniqueness plot (Fig. 3). The basis functionality also works to produce a high correlation between each node, which results in a flat, low MSE, curve without a sharp ‘v’. To demonstrate the variety of material characteristics this can produce, the permittivities and figure of merit values are compared for TiN films fit from 30 nm to 70 nm in Fig. 5. We see a variation in figure of merit of almost 3 × from minimum to maximum and a similar spread is found when fitting with GenOsc (not shown).

Fig. 5 B-Spline fit for various thicknesses calculated real and imaginary permittivity a) and b) and figure of merit c). Despite each fit having a low Mean Squared Error fit, a large change in optical properties is seen between each fit with almost 3 × variation.

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In this case, neither the GenOsc nor the B-Spline fit the SE data confidently and more data is necessary to accurately determine the material properties. Accurate thickness measurements could be taken via methods such as transmission electron microscopy or step etching, thereby reducing the unknown variables. Another method to provide additional data content is to supplement SE with T data. When we add T data the balance of variables favors the measured values, see Fig. 6(a), as we have both SE and T data for each wavelength. Even when some spectral regions are completely opaque, only a few transmission data points are necessary to allow for film characterization.

Fig. 6 a) Balance of variables with transmission included. The balance favors measured values, which is ideal for fitting. b) Transmission measured and generated with B-Spline fits shown in Fig. 5. The 50 nm fit is closest to the actual measured transmission of the film.

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Adding T data, we can simply calculate the expected T from the previous B-Spline results at various thicknesses and compare to the measured T. Figure 6(b) shows that the results calculated from the 50 nm fit results produce the closest match to T. This visual check may be enough information to consider the film to be about 50 nm thick and use this to optimize our permittivity. However, it is best to consider the SE and T data simultaneously as the thickness and permittivity are regressed to the best-fit. The standard approach considers the single T data to be “weighted” as a single measurement, compared to the multiple-angles from SE measurements. To increase the importance of the T, when three angles of $Ψ (λ)$ and $Δ (λ)$ where measured, we increase the T weighting to 600% such that T and SE curves are more or less equivalent. As seen in Fig. 7(a), when T is added, the uniqueness plot produces a sharp ‘v’. This ‘v’ is sharper for higher T weighting, including tests with T weighted at 2400%, but the MSE also increases. This gives a confident thickness of 47 nm for the film. T data also breaks correlation as can be seen in Table 2 and Fig. 7(b). As a result, the previously elongated trough in the two-dimensional parameter uniqueness has been condensed such that a large change in thickness cannot be compensated by a change in resistivity.

Fig. 7 a) Mean Squared Error vs thickness for various weightings of transmission. Fits with transmission included have a sharp ‘v’ providing a unique fit for thickness at 47 nm. b) Two dimensional parameter uniqueness comparing resistivity and thickness. The conical trough is ideal, showing a unique fit between the thickness and resistivity parameters.

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Table 2. Correlation matrix with transmission included in the calculation. The correlation between thickness and resistivity or thickness and scattering time are no longer close to ± 1 so the correlation is broken

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Finally, we consider three films; film A, which has been the subject of the previous sections, along with films B and C. The growth method is plasma-enhanced atomic layer deposition, a technique with a consistent, but unknown, growth-rate when other deposition parameters are kept constant. The three films were grown to different unknown thickness to verify whether thickness of ultrathin films impacts the optical properties. Film A should be the thinnest with film C being the thickest based on deposition cycle numbers (See Methods for growth conditions). First the films were characterized with SE only (without T) using a GenOsc with one Drude and two Lorentz oscillators. Film A produced best fits at 48 nm and 105 nm, film B at 105 nm, and film C at 99 nm). All three sample optical properties are plotted in Fig. 8(a); the optical properties of the three films are shown to vary only for the lowest thickness in film A but the assumed thicknesses are clearly incorrect.

Fig. 8 a) Real (left axis) and Imaginary (right axis) permittivities of samples A, B, and C when using SE data alone fit with a Drude and two Lorentz oscillators (GenOsc). The fit thicknesses have clear deviations from a linear growth. b) Real (left axis) and Imaginary (right axis) permittivities of samples A, B, and C with transmission measurements fit with a Drude and two Lorentz oscillators(GenOsc). The trend is now seen that material quality does not strongly depend on the thickness of the film and a linear growth rate is found. c) AFM measurement of a step in the TiN films verifying the thicknesses obtained. The curvature seen in C is attributed to etching under the mask, more prevalent in C due to additional time of etch. d) AFM measurement of sample A showing <1nm surface roughness. Other samples illustrate similar roughness.

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However, the large discrepancy between expected film thickness and fit thickness led to the addition of T spectra for the three samples. After fitting both SE and T, we find that the actual thicknesses are 47 nm, 58 nm, and 80 nm, which is a linear growth rate of 0.07Å/ cycle between samples. The results of this correct modelling are shown in Fig. 8(b) and there is no significant change in optical properties between films. For parity, the thicknesses of the films were also measured using atomic force microscopy across a step profile of the film fabricated through photolithography and wet etching. The resulting thickness values are found to be in agreement with the ellipsometry measurements, Fig. 8(c), within the ± 3 nm error of the measurement. Surface roughness was measured using AFM to find <1 nm RMS, shown in Fig. 8(d).

4. Conclusions

In this paper, it has been shown that supplementing SE data with T significantly increases confidence in our extracted material parameters. The extracted parameters from “good fit” characterizations without T can significantly alter the expectations for ultrathin films. For example, a single film fit to various fits with an MSE below 3 showed single-interface plasmon propagation lengths varying from 23 to 51 $μ m$ . When T is included and the confidence in the fit is increased, a propagation length of 35 $μ m$ is found. The presented case study of films A, B, and C shows that fitting T can also prevent misunderstanding of film growth trends such as growth rate and how optical properties relate to the thickness of the ultrathin film.

5. Supplementary materials

5.1 Optical models

Based on the expected properties of the material being examined, the most applicable dispersion equations can be selected. For example, the Cauchy equation does not include absorption whereas general oscillator models are often derived directly from the absorption of the material.

These dispersion models are often physically derived and result in Kramers-Kronig consistent optical functions. Being physically derived models, they can often provide meaning in the interpretation of results. For example, in the Drude-Lorentz equation which we use in this work,

ε = ε_{\infty} + \sum_{j} \frac{f_{j}}{ω_{0 j}^{2} - ω^{2} - i Γ ω} - \frac{ω_{p}^{2}}{ω^{2} + i γ ω}

the Lorentz portion has a value of

ω_{0 j}

, which corresponds to absorption lines in a material with a strength

f_{j}

and sharpness

Γ_{j}

. The Drude portion has the unscreened plasma frequency

ω_{p}

and the dampening

γ

which can be calculated as

ω_{p} = \sqrt{\frac{N q^{2}}{ε_{0} m^{*}}} = \sqrt{\frac{1}{ρ_{0} ε_{0} τ}}

and

γ = \frac{1}{τ}

from these possible extracted values include the free electron density N, effective mass

m^{*}

, mobility, relaxation time

τ

, and resistivity

ρ_{0}

. However, care must be taken as these values can be highly correlated and one could arbitrarily double N and double

m^{*}

to arrive at the same plasma frequency. For this reason, additional measurements may be necessary to extract physical values from the model, such as a Hall Effect measurement to determine resistivity.

5.2 MSE

The MSE is calculated using the formula

M S E = \sqrt{\frac{1}{3 n - m} \sum_{i}^{n} [{[\frac{N_{E_{i}} - N_{G_{i}}}{.001}]}^{2} + {[\frac{C_{E_{i}} - C_{G_{i}}}{.001}]}^{2} + {[\frac{S_{E_{i}} - S_{G_{i}}}{.001}]}^{2}]}

Where

N = \cos (2 Ψ)

,

C = \sin (2 Ψ) \cos (Δ)

, and

S = \sin (2 Ψ) \sin (Δ)

summed over

n

wavelengths for

m

varied fit-parameters. The subscripts

E

and refer to experimentally measured and model-generated values, respectively.

G

5.3 Plasma enhanced atomic layer deposition growth

Titanium nitride films were grown using Plasma-Enhanced Atomic Layer Deposition (PE-ALD) on c-plane sapphire. The precursor used was tetrakis (dimethylamido)titanium (TDMAT) at a temperature of 375 $° C$ . The plasma exposure time was 10 seconds per cycle. Film A, B, and C, were grown for 600, 775, and 1050 cycles respectively. Surface roughness was measured using atomic force microscopy, Fig. 8(d) and found to be < 1 nm.

Funding

Virginia Microelectronics Consortium

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	Thickness (nm)	Resistivity (Ω·cm)	Scat. Time (fs)
Thickness (nm)	1	0.934	−0.931
Resistivity (Ω·cm)	0.934	1	−0.992
Scat. Time (fs)	−0.931	−0.992	1

	Thickness (nm)	Resistivity (Ω·cm)	Scattering Time (fs)
Thickness (nm)	1	0.324	−0.237
Resistivity (Ω·cm)	0.324	1	−0.949
Scattering Time (fs)	−0.237	−0.949	1

	Thickness (nm)	Resistivity (Ω·cm)	Scat. Time (fs)
Thickness (nm)	1	0.934	−0.931
Resistivity (Ω·cm)	0.934	1	−0.992
Scat. Time (fs)	−0.931	−0.992	1

	Thickness (nm)	Resistivity (Ω·cm)	Scattering Time (fs)
Thickness (nm)	1	0.324	−0.237
Resistivity (Ω·cm)	0.324	1	−0.949
Scattering Time (fs)	−0.237	−0.949	1

Reliable modeling of ultrathin alternative plasmonic materials using spectroscopic ellipsometry [Invited]

Abstract

1. Introduction

2. Ellipsometry

3. Characterization of titanium nitride films

4. Conclusions

5. Supplementary materials

5.1 Optical models

5.2 MSE

5.3 Plasma enhanced atomic layer deposition growth

Funding

References

Cited By

Figures (8)

Tables (2)

Equations (3)

Optical Materials Express