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Abstract
Due to dependence of nanostructural material properties on the deposition conditions, the refractive index (RI) of a thin-metal alloy can vary by over $100 \times$. For a more accurate RI estimation of thin-film alloys under a specific deposition condition, a model of RI for untested alloys is derived by extension from the alloy’s main element properties. One of the key derivations is to simplify RI polynomial equations into monomial expressions through four assumptions. Thus, a ratio method can accurately predict the RI of thin-film alloys from the properties of their main metals under the same deposition condition. The model is valid in the red to infrared spectrum for thin-film alloys MX where M is one of copper, silver, or gold and X is one or more elements with total concentration ${\lt}{10}\%$.
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1. INTRODUCTION
New optical thin-film metal alloys are needed for many optical applications, yet properties such as the refractive index (RI) of the vast majority of thin-film alloys are unknown and not easily predicted from the thin-film alloy composition and experimental conditions. Thin-film alloy RI is difficult to predict accurately from prior models because the value can vary by more than 100 times depending on deposition conditions. Researchers typically either miss out on substantial material opportunities, or have to spend significant time and resources in physical experiments to find a thin-film alloy that is within a reasonable range of a desired RI so that further experiments can co-optimize material properties for device integration.
Thin-film alloys of Group-11 metals silver (Ag), gold (Au), and copper (Cu) are known to provide practical electrical and optical properties from the visible to infrared (IR) range, such as a refractive index (RI) that can be tuned to different controllable values and durability properties with the addition of alloying elements such as zinc (Zn), silicon (Si), carbon (C), etc. These materials are particularly useful in IR communication and night-vision IR optical devices. However, despite work such as the U.S. Materials Genome Initiative to simulate thin-film compositions and structures to avoid expensive experimentation [1], complex atomic-scale interactions prevent such approaches from achieving sufficient accuracy to be able to predict properties. Practical optical thin-film R&D still requires screening a wide range of new thin-film alloy compositions in a complex experimental deposition matrix, a necessarily slow and expensive approach.
There has been more than a hundred years of study in electrical conduction and optical response in bulk metals [2–4]. Drude [5] suggested the model of a gas of free-electrons traveling with the ions in the metal. Lorentz studied the dielectric properties through a dipole oscillator model. The Drude–Lorentz model describes the electronic response in a metal by two parameters, plasma frequency and electron collision time, which can effectively explain many electrical and optical properties in bulk metals [2]. This model is especially good at predicting the free-electron optical response of Ag, Au, and Cu in the infrared (IR) and near-infrared (NIR) regions [6]. Ding et al. previously studied the optical properties of Ag thin films, examining dependence on thickness and nanostructures [7].
However, modeling the RI of thin films is much more complex than that of bulk material [8,9]. This complexity arises from the fact that the optical properties of thin-film alloys depend on nanostructural factors, including grain size, defects, substrate material, and roughness, which can only be determined by experiment [10]. For example, the RI real component ($n$) of pure Ag thin films at 600 nm can range from 0.06 to 0.8, varying by ${\gt}{10}$ times. The well-known Thornton diagram shows part of the complexity of thin-film properties through a three-dimensional dependency graph [11].
This paper demonstrates a novel method to overcome the complexities of red to infrared optical thin-film parameter modeling by extending the main metal parameters using existing data from materials databases or literature. By simplifying the polynomial Drude–Lorentz equations into monomials though four assumptions, an alloy thin-film RI can be derived from the main element metal RI and the alloy resistivity through a ratio method that cancels out most of the factors from different deposition conditions. These simplified RI equations can accurately model properties of complex thin-film alloys and facilitate initial material screening through standard optical design simulations in many new product developments.
Fig. 1. (a) Experimental setup schematic drawing of the thin-film PVD tool, and (b) inside-chamber photo showing magnetron guns and rotating deposition stage.
We used silver–zinc (AgZn) alloys in this study to derive simplified RI models. The experimental RI closely corresponded with the modeled RI over the valid spectra wavelength range of 0.6–3.0 µm. This model is valid for alloys MX, where the main metal M is from Group-11 of the periodic table and X is one or more alloying elements up to a total of 10%.
We demonstrated modeling of the RI of AgZn and AgMn alloys at different film thicknesses and substrate conditions. Based on pragmatic assumptions, this approach can predict the RI of novel thin-film alloys, which can dramatically speed up the R&D of infrared optical products by avoiding expensive and tedious physical experimentation. The estimation accuracy of this model is typically on the order of 10%, plus the accuracy of the literature input data compared to experimental data. This will allow researchers to quickly find a starting material to fine-tune and integrate optical properties.
2. METHOD
Thin films of AgZn with varying Zn concentrations were deposited under vacuum in a physical vapor deposition (PVD) co-sputtering magnetron chamber. A PVD chamber with 3 magnetron guns (Fig. 1) was used to deposit 26–38 nm thin continuous smooth films on 3 mm thick glass substrates at room temperature. The purities of Ag and Zn targets were 99.99%. The target size was 66 mm in diameter, and the targets were produced by ACI Alloys Inc. The processing gas used was ultra-high purity 99.999% Argon.
The glass substrates were annealed low-iron from Industrial Glass Technologies and thoroughly cleaned using standard glass industry processes for silver coating: aqueous detergent clean, triple rinse, and cold air dry. The coating size was 50 mm in diameter, and the sample was 20 cm away from the sputtering targets. The substrate was rotated during deposition, and coated thickness uniformity was within 3% (five-point measurement). Substrate temperature during deposition was uncontrolled within the room temperature range of 20–25°C. The Zn sputtering power was 6 W for all alloy depositions, while the Ag sputtering power was 100, 150, or 200 W.
The background vacuum was ${2} \times {{10}^{- 7}}\;{\rm Torr}$ after overnight pumping by turbo pump backed by mechanical pump, while the deposition pressure was 2.5 mTorr using a continuous argon flow of 36 sccm. A throttle valve was used to maintain excellent vacuum background pressure to ensure water vapor was extremely low. Pre-sputtering was always used to clean the target surface, while a shutter covered the sample during this step; typically two cycles of 60 W sputtering for 300 s each.
After each deposition, to avoid thin-film corrosion/oxidation, the samples were measured within 2 h of removal from the chamber. An ellipsometer and spectrometer measured the RI of Ag and AgZn. The UV-VIS-IR spectrometer was a double-beam Shimadzu 3700 with three sensors to accurately measure 250–2500 nm wavelengths and an error bar of 0.2% or 0.1 nm. The spectroscopic ellipsometer was a Woollam VASE with a spectral range of 300–1700 nm and thickness accuracy of 0.1 nm. The ellipsometer was calibrated daily using the standard provided by Woollam with ${\lt}{0.1}\;{\rm nm}$ thickness error. The glass coating roughness during these experiments were typically within 1 nm, and the haze was typically smaller than 0.3%. Each coating sample was subjected to a roughness simulation test with the ellipsometer combined with the UV-VIS spectrometer, and the roughness was less than 1 nm.
A four-point probe with an accuracy of 0.04 ohm measured the resistivity of six areas of each sample, and the final sheet resistance was calculated as the average of those six measurements. All thin-film measurements were taken ${\lt}{2}\;{\rm h}$ after film exposure to air and showed negligible oxygen impact on film properties.
Ag volume concentration (Vol%) is calculated from Ag and AgZn alloy deposition rates, where Ag Vol% is equal to Ag deposition rate divided by AgZn deposition rate. All samples were processed in identical experimental conditions to minimize error, except for the negligible additional Zn sputter power during AgZn depositions. Zn Vol% is 100% minus Ag Vol%. The ellipsometer and UV-VIS spectrometer on film deposition thickness achieved high accuracies, within 0.1 nm error. The repeatability of the thickness measurement error for both Ag and AgZn films is 0.6%, so combining Ag and AgZn in the concentration calculation may result in an error of 1.2%.
3. THEORY
A. Free-Electron Models on RI Calculation
Free electrons are widely understood to be the source of both optical and electrical properties in relatively pure, highly conductive materials. The Drude–Lorentz model mathematically describes that the electronic response in a metal is governed by two parameters: plasma frequency (${\omega _{\textit{pe}}}$) and electron collision time ($\tau$). The thin-film dielectric constant (${\varepsilon _r}$) at a given frequency ($\omega$) is expressed as
The collision time τ is related to the resistivity $\rho$ as per the Drude model of $\frac{1}{\tau} = \frac{{{n_e}{e^2}\rho}}{{{m_e}}}$; and the plasma frequency is related to electron density by $\omega _{\textit{pe}}^2 = \frac{{{n_e}{e^2}}}{{{\varepsilon _0}{m_e}}}$, where ${n_e}$ is free-electron density, ${m_e}$ is the free-electron mass, $e$ is the free-electron charge, and ${\varepsilon _0}$ is vacuum permittivity.
The refractive index (RI) is expressed as
B. Simplify RI as Monomial Expressions through Assumptions
Mathematically, a ratio between monomials can cancel out common factors. In this study, we used assumptions to simplify the RI formula into monomial expressions from the polynomials expressed in Eqs. (1)–(4). Thus, the ratio between the alloy RI can allow for a greatly simplified model.
Assumption 1 is the Drude–Lorentz model of free electrons in pure metals determining thin-film optical properties.
Assumption 2 [7] is that the real component of the dielectric constant is much greater than the imaginary component:
Substitute Eq. (5) into Eqs. (3) and (4), and the RI become monomials of ${\varepsilon _1}$ and ${\varepsilon _2}$; note that ${\varepsilon _1}$ is a negative value number, with the absolute value much larger than that of ${\varepsilon _2}$:
Assumption 3 [7] is that frequency squared (${\omega ^2}$) times electron collision time squared (${\tau ^2}$) is much greater than 1:
Then, the dielectric constant Eq. (1) can be simplified as
Assumption 4 is that plasma frequency (${\omega _{\textit{pe}}}$) is much greater than the light frequency ($\omega$):
So dielectric constants ${\varepsilon _1}$ and ${\varepsilon _2}$ in Eq. (9) can be expressed as monomials instead of polynomials:
Thus, we can substitute ${\varepsilon _1}$ and ${\varepsilon _2}$ of Eqs. (10) and (11) into equations Eqs. (6) and (7) and combine the ${\omega _{\textit{pe}}}$ and $\tau$ relationship, leading to
The RI $n$ and $k$ in Eqs. (12) and (13) are dependent on electron density ${n_e}$, resistivity $\rho$, frequency $\omega$, electron mass ${m_e}$, charge $e$, vacuum permittivity ${\varepsilon _0}$, and extinction coefficient, as first derived in studying silver [7], and we now extend their silver application to any thin-film alloy MX.
For silver and alloy, the refractive index $n$ can be expressed as monomials:
C. Calculate Thin-film Alloy RI from its Main Metal RI
Taking the ratio between Eqs. (14) and (15) leads to Eq. (16):
Thus, after RI is simplified as a monomial, the RI $n$ ratio between the main element metal, such as Ag, and its alloy, such as AgZn, is only dependent on electron density and resistivity, with all other factors being canceled out.
Similarly, the extinction coefficient $k$ of an alloy can be expressed as in Eq. (17) when using ratios of Eq. (13) between Ag and AgZn:
As for the $n$ value, this approach allows for a silver alloy RI $k$ value to be predicted by the same two parameters: electron density and the main metal $k$ value.
This ratio method requires that metal and alloy deposition conditions be nearly identical so that nanostructure thickness, roughness, defects, etc. will be nearly identical. In this experiment, Ag and AgZn alloys were deposited in the same well-controlled PVD chamber with changes only from the small additional power used to co-sputter Zn for alloys.
D. Alloying Concentration and Free-Electron Density Ratio
The alloy free-electron density ratio in Eqs. (16) and (17) between the thin-film alloy and its primary element is associated with the alloy concentration, because the electron density is associated with the atom density when each atom can contribute a fixed number of electrons. Using AgZn as an example, the alloying concentration and free-electron density relationship is derived: we know that one Ag atom contributes one free electron, and there are two possible scenarios depending on whether Zn contributes two or zero free electrons.
Scenario 1: If Zn contributes zero free electrons (only the Ag atoms contribute one free electron each), the electron density ratio of ${n_{e - {\rm alloy}}}/{n_{e - {\rm Ag}}}$ is associated with the Ag concentration. Thus, the RI in Eqs. (16) and (17) can be calculated as
Scenario 2: If Zn contributes free electrons to the thin-film alloy properties, then the electron density ratio would require a different formula. The ratio of Ag Vol% to Zn Vol% is 1.12., so if Zn becomes ${{\rm Zn}^{2 +}}$ by adding two free electrons per atom, then each Zn atom would contribute 2.24 free electrons in the volume of one Ag atom. Since each Ag atom contributes one valence electron, the replacement of Ag by Zn would add 1.24 (2.24–1) free electrons to the alloy.
The refractive index of the AgZn alloy could then be calculated as
4. RESULTS AND DISCUSSION
A. Model and Experiment Comparison
Alloys of AgZn were co-sputtered using two magnetron sputtering guns with Ag and Zn targets at ${\sim}{2.5}\;{\rm mTorr}$ in an argon ambient. Zn sputtering power was kept at 6 W for a low deposition rate, while Ag sputtering power varied between 100, 150, and 200 W to vary the Zn%.
Table 1 lists experimental conditions, major parameter measurements, and simple derived results. The deposition time was varied to maintain thicknesses in a range of 26–38 nm. The sheet resistances (R) were measured six times with a four-point probe, and the standard deviations of those Rs were listed; note that all standard deviation 1-sigma values were 1%–3%.
Table 1. AgZn Thin-film Co-sputtering Conditions, Major Parameter Measurements, and Simple Derived Results for each Experiment
Resistivity is calculated by multiplying sheet resistance by the film thickness. Alloy volume zinc concentration (Zn Vol%) is calculated through the Zn and Ag deposition rates. Electron density is calculated from the Ag atom number density by assuming that each Ag atom contributes one free electron. Collision time is calculated through the Drude model of $\frac{1}{\tau} = \frac{{{n_e}{e^2}\rho}}{{{m_e}}}$. Based on the collision time and the conditions of Eq. (8), the valid wavelength is ${\lt}{3.0}\;{\unicode{x00B5}{\rm m}}$. Plasma frequency is calculated through electron density by the formula of $\omega _{\textit{pe}}^2 = \frac{{{n_e}{e^2}}}{{{\varepsilon _0}{m_e}}}$. Based on the plasma frequency, the critical wavelength of the valid condition for Assumption 4, Eq. (10) was calculated, giving 30% leeway, so that the valid wavelength is more than 600 nm. We will discuss those valid conditions further in a later section.
We used Eq. (18) to calculate the RI $n$ values, and then plotted the result along with data from alloy depositions, as shown in Fig. 2. The modeled alloy $n$ values matched the experimental values well, especially at longer wavelengths.
We used Eq. (19) to calculate the alloy RI $k$ values, and we compared these results with the alloy data as shown in Fig. 3. The modeled alloy $k$ values matched those of the experimental ones.
B. Free-Electron Scenario Discussions
Using data from our AgZn alloy experiments, we can check the contribution of free electrons from Zn alloying atoms. We can modify Eqs. (18) and (20) into Eqs. (22) and (23) to clarify the two scenarios.
To compare the alloy RI from our experimental results, we calculate $\frac{{{n_{\rm{alloy}}}}}{{{n_{\rm{Ag}}}}}\big/\frac{{{\rho _{\rm{alloy}}}}}{{{\rho _{\rm{Ag}}}}}$ average value at spectra from 1000 to 1700 nm. This value is plotted versus AgZn alloy concentration in Fig. 4. The data aligns with the result derived from the formula that assumes Zn contributes no free electrons to AgZn thin-film properties. The error bar is less than ${+}{/} {-} 9\%$, with a 4% error bar attributed to the index ratio of ${n_{\rm{AgZn}}}$ and ${n_{\rm{Ag}}}$, and an additional 5% error bar attributed from the resistivity ratio experiments.
For the extinction coefficient ($k$) we can modify Eqs. (19) and (21) into:
We then calculated the ${k_{\rm{alloy}}}/{k_{\rm{Ag}}}$ average value at spectra from 1000 to 1700 nm and plotted the experimental results of ${k_{\rm{alloy}}}/{k_{\rm{Ag}}}$ versus AgZn alloy concentration in Fig. 5. The error bar was less than ${+}{/} {-} {4}\%$. Like Fig. 4, this data also aligns with the result derived from the formula that assumes Zn contributes no free electrons to the AgZn thin-film properties.
These results show that the Zn valence electrons did not contribute to the AgZn alloy optical properties in these experiments. Thus, the real and imaginary RI of AgZn (concentration Zn ${\lt}{10}\;{\rm Vol}\%$) thin films can be expressed as Eqs. (18) and (19). A similar phenomena has been reported where free carrier densities of AgFe and AgCr thin-film alloys decrease with increasing Fe or Cr% up to 2%, which indicates that Fe and Cr did not contribute free electrons [12].
Fig. 2. AgZn alloy refractive index (RI) $n$ compared with calculated results at different alloy concentrations (a) 4.7%, (b) 6.8%, and (c) 8.7%, showing good overall match; the silver index at the corresponding conditions is shown for comparison.
Fig. 3. AgZn alloy refractive index $k$ comparison with calculation results at different alloy concentrations (a) 4.7%, (b) 6.8%, and (c) 8.7%, showing good overall match; the silver index $k$ values at the corresponding conditions are shown for comparison.
On the other hand, free carriers of AgGe thin film show that Ge contributes free electrons to the alloy, and Ge concentrations are at much higher concentration, 26%–37% [13].
In more general terms, for an alloy MX, where M is one of Cu, Ag, or Au and X is another material, we can discuss models at different scenarios.
Scenario 1 is true for many materials. There are some materials X that won’t contribute free electrons to the thin-film alloy MX; for example, if X is a non-conductive material. Also, even if X is a metal, there is still a possibility in some cases, especially at low concentration alloys, that X did not contribute free electrons to the alloy properties as in [12]. In this case, the model calculations are Eqs. (18) and (19).
Scenario 2, with X contributing free electrons, is true for many other materials. In such cases, the model calculations are Eqs. (20) and (21) (if each atom contributes two free-electrons).
In general, without digging out free-electron information, the simpler of Eqs. (18) and (19) can be the first-order approximation. The maximum error from the difference between two scenarios could be estimated; so according to the results in Fig. 4, the maximum difference error for the $k$ value could be around 10%, and the maximum difference error for $n$ value could reach up to ${\sim}{30}\%$ when the alloy concentration is 10%. However, this maximum difference error tends to decrease proportionally with lower concentrations. For example, the maximum difference error of $n$ accuracy is below 10% at 3% alloy concentrations. The model accuracy remains sufficient in initial material screenings for many infrared optical product developments, especially at lower concentration at a few percentages.
Fig. 4. Comparing the RIs from AgZn experimental depositions with those modeled by assuming that each Zn atom either contributes two (red triangles) or zero free electrons (orange circles) to the thin-film optical properties.
C. Model Constraints and Valid Ranges
The following constraints allow for this novel thin-film optical model to predict RI accurately:
Fig. 5. Comparing $k$ values from AgZn experimental depositions with those modeled by assuming that each Zn atom either contributes two (red triangles) or zero free electrons (orange circles) to the thin-film optical properties.
Fig. 6. Graph of common metals in the periodic table according to their $|{\varepsilon _1}/{\varepsilon _2}|$ ratio.
Table 2. AgZn Thin-film Alloy Resistivity and ${\rm RI}(n,k)$ at Different Thicknesses and Concentrations on Glass Conditions
Table 3. AgZn Thin-film Alloy Resistivity and ${\rm RI}(n,k)$ at Different Thicknesses and Concentrations within a “Stack” of Glass/10 nm ${{\rm TiO}_2}/10\;{\rm nm}$ ZnO/x nm AgZn/2 nm ${{\rm TiO}_2}/10\;{\rm nm}$ ZnO [7]
Table 4. AgMn Thin-film Alloy Resistivity and ${\rm RI} (n,k)$ at Different Thicknesses and Concentrations on Glass Substrates
Table 5. AgMn Thin-film Alloy Resistivity and ${\rm RI} ({\rm n},{\rm k})$ at Different Thickness and Concentrations within a “Stack” of Glass/10 nm ${{\rm TiO}_2}/10\;{\rm nm}$ ZnO/x nm AgMn/2 nm ${{\rm TiO}_2}/10\;{\rm nm}$ ZnO [7]
- 1. Upper wavelength of$\textbf{RI} \boldsymbol\lt \textbf{3.\bf 0}\;\textbf{\unicode{x00B5}}\textbf{m}$, based on (${\omega ^2} {\tau ^2} \gg {1}$) of Assumption 3 and the collision time (τ) calculated from the resistivities shown in Table 1 and literature values.
- 2. Lower wavelength of$\textbf{RI} \boldsymbol\gt \textbf{0. 6}\;\textbf{\unicode{x00B5}}\textbf{m}$, based on ($\omega _{\textit{pe}}^2 \gg {\omega ^2}$) of Assumption 4 and the following assumptions: ${\omega _{\textit{pe}}}$ for Cu, Ag, and Au as per literature are 115, 138, and 138 nm wavelengths, respectively, while Ag or Au alloy ${\omega _{\textit{pe}}}$ are typically 30% greater, which is ${\sim}{180}\;{\rm nm}$. If ${10} \times$ is needed for the condition of $\omega _{\textit{pe}}^2 \gg {{ \omega}^\textbf{2}}$, then we multiply 180 by $\sqrt {10}$ to get 570 or ${\sim}{600}\;{\rm nm}$. Also, an interband electronic transition peak near 560 nm for Cu constrains the lower-limit wavelengths as ${\gt}{0.6}\;{\unicode{x00B5}{\rm m}}$.
- 3. Validmaterials are thin-films alloy MX, where M is Ag, Au, or Cu and X is one or more alloying elements with total concentration$\boldsymbol{\lt}\textbf{10}\%$, based on ($|{\varepsilon _{1}}| \gg \;|{\varepsilon _2}|$) of Assumption 2. Reference [7] first raised the $|{\varepsilon _1}| \gg |{\varepsilon _2}|$ assumption for silver; we here extended the range of valid materials for this assumption. Exploring data from an elements database [14], we found that this model is also valid when M is an element from Group 1 or Group 11 elements (Fig. 6), yet Group 1 elements are too reactive in atmospheric conditions so that they are not typically used, and their RI databases are very limited. The elements in Group 11—including Ag, Au, and Cu—are less reactive and commonly used by industry. We further extend Group 11 elements to their alloys MX with the main element ${\rm M}\; \gt {90}\%$.
D. Method to Predict Thin-Film Alloy RI
Since Ag, Au, and Cu are common elements with rich data (e.g., RI and resistivity) under various conditions, the RI ($n$, $k$) of their alloy thin films under the same deposition conditions may be predicted as follows:
- 1. Calculate thin-film alloy $k$ at a specific deposition condition using the alloy concentration and RI database of the main element under the same deposition conditions, through Eq. (19).
- 2. Calculate the refractive index $n$ of the thin-film alloy from the alloy concentration, main metal refractive index $n$, and film resistivities through Eq. (18).
The data of alloy resistivities of MX alloys, where ${\rm M} = {\rm Ag}$, Au, and Cu, and X concentrations are ${\lt}{10}\%$ can be found in [15,16]. The resistivities of some alloys can be calculated using the linear relationship of alloying concentrations when the X concentration in MX alloys is ${\lt}{1}\%$. For the concentration X in the range of 1%–10%, second- or third-order correction factors might sometimes be required.
However, when the thin-film alloy resistivity for a specific thickness cannot be found, there is a practical two-step method to calculate it. First, find the resistivity of bulk alloy material from a handbook, database, or literature, including the concentration fitting calculations just discussed. Second, convert the bulk resistivity to the resistivity at the desired film thickness assuming ratios of resistivity versus film thickness reported in the literature [7,17].
Using this two-step method, we modeled silver alloys on glass or within a “stack” of glass/10 nm ${{\rm TiO}_2}/10\;{\rm nm}$ ZnO/Ag-alloy/2 nm ${{\rm TiO}_2}/10\;{\rm nm}$ ZnO [7] for different Ag-alloy film thickness with an adjustment of 1–2 nm.
Table 2 shows thin-film Ag, AgZn 5%, and 10% ${\rm RI}(n,k)$ on glass, where the data in blue boxes are from the literature or this study, data in yellow boxes are from experiment, and the data in the white boxes are modeled results. The differences between the model and experiment are ${\lt}{10}\%$ and ${\lt}{20}\%$ for alloy concentrations of 5% and 10%, respectively, as shown in Table 2. Such accuracies are sufficient for model estimation in the initial screening of ${\rm RI}(n,k)$ for new materials in typical optical product development.
Table 3 compares experimental and modeled results for AgZn alloys in the thin-film stack, glass/10 nm ${{\rm TiO}_2}/10\;{\rm nm}$ ZnO/Ag-alloy/2 nm ${{\rm TiO}_2}/10\;{\rm nm}$ ZnOfor different film thicknesses, where such an integrated stack of materials would typically be required for advanced infrared optical devices. Using the silver resistivity versus thickness trend and bulk resistivity values of 5% and 10% AgZn from literature, the alloy thin-film resistivity and ${\rm RI}(n,k)$ at different thicknesses can be modeled. The data from Tables 2 and 3 can be plotted in Fig. 7 to make it easier to observe the comparison of modeling results for AgZn thin film alloy ${\rm RI}(n,k)$ under different conditions.
Using this same two-step ${\rm RI} (n,k)$ estimation method for AgMn 5% and 10% alloys, Tables 4 and 5 shows data from the literature or this study in the blue boxes, and data from modeled results in the white boxes. Table 4 shows AgMn thin-film properties on glass substrates.
Table 5 shows the same AgMn film properties when integrated into an infrared optical materials stack of glass/10 nm ${{\rm TiO}_2}/10\;{\rm nm}$ ZnO/x nm AgMn/2 nm ${{\rm TiO}_2}/10\;{\rm nm}$ ZnO. Such a highly-p durable stack of thin films could be used for low-emissivity (Low-e) coatings on architectural window glass. The data from Tables 4 and 5 can be plotted in Fig. 8 to make it easier to observe the comparison of modeling results for AgMn thin film alloy ${\rm RI}(n,k)$ under different conditions.
5. CONCLUSION
Thin films of new metal alloys are needed for the research and development of advanced infrared optical devices. The RI is a critical parameter; yet the RI of novel thin-film alloys cannot be predicted by theory due to the complexities of nanostructures formed under different deposition conditions. This study shows a practical way to extend known RI of the main element metal to be able to estimate previously unknown thin-film alloys in the red to infrared spectra under the same deposition conditions.
This simplified model—valid over the 600–3000 nm spectrum as detailed in Section 4.C—is based on the four assumptions and a ratio method detailed in Section 3.B that allows RI to be predicted for previously unknown silver, gold, and copper thin-film alloys. The derived Eqs. (18) and (19) for AgZn may be extended to any MX alloy, where M is a Group 11 metal and X is one or more alloying elements with total concentration (by volume) ${\lt}{10}\%$:
Since the rich resistivity data for thin films of Cu, Ag, Au, and their alloys can be found or easily obtained from an experiment under a specific condition, Eqs. (26) and (27) allow ${\rm RI} (n, k)$ values of thin-film alloys to be estimated under controlled deposition conditions.
Acknowledgment
The author would like to thank Qi Lin and Ed Korczynski for their discussions of the paper.
Disclosures
The authors declare no conflicts of interest.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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