**4. Advanced understanding of silver-based low-E coatings: thin film silver properties**

Silver-based low-E window coating currently accounts for 90% of the market. The thin film silver properties are key in low-E coating technology. Thus, a separate section is used to focus on the thin film silver properties from practice to theory.

There is a hundred years of research history on electric conduction and optical response in metals [27–29]. The Drude model [30] describes two main parameters governing the electronic response: (i) the electron collision time *τ*, a statistical parameter describing the mean time between collisions, and (ii) the plasma frequency *ω<sup>p</sup>* , mainly determined by the concentration of carriers. Lorentz analyzed the electronic behavior using the dynamical theory of gases [31]. These theories gave a reasonable optical response of metals [32]. In addition, surface electronic scattering effects are extremely important, and the optical response of metal nanoparticles with the interfaces were reported [33–35].

The low-E industry history started with gold thin film then shifted to silver thin film, and the transition significantly improve the window color appearance, and the energy-saving efficiency, and the cost. Materials innovation is the key to low-E coating performance. The following figure clearly shows the silver benefit in optical absorption in comparison with those of gold and copper with 15 nm thin film, although they are all excellent conductors and are shown in **Figure 15** [11].

**Figure 15.** The spectral absorbance for 15 nm thin film of Ag, Au, and Cu.

Further, if the three blocks of single silver of **Figure 8** were put together [13], it would make a triple silver stack which is shown in **Figure 14**. The selectivity of IR and visiblity is the best in

**Figure 13.** The typical double silver spectra of transmittance and reflectance from film side and glass side [26].

today's market, however, the cost is higher.

422 Modern Technologies for Creating the Thin-film Systems and Coatings

**Figure 14.** A typical triple silver stack.

#### **4.1. Silver thickness effect in the electrical properties**

The thin film silver thickness effects on its electrical properties were studied in many literatures, and Ref. [12] gave a good comparison of the theoretical and experimental results. **Figure 16** shows experimental resistivity values (*ρ*) decreased as Ag films with thicknesses ranging from 3 to 74 nm and the theoretical model predictions [12]. The silver was either directly deposited on the glass or was deposited in a stack with seed and blocker layers. There are two models used in the fitting of the experiments:

**Figure 16.** Experimental electrical resistivity for Ag stacks (red dots), Ag on glass (blue dots), and calculated values using Fuchs-Sondheimer's theory for polycrystalline films (*p* = 0) and single crystal films (*p* = 0.5). The dotted lines show how roughness and intergrain scattering affect resistivity using the Rossnagel and Kuan formalism [12].

(i) First is the Fuchs-Sondheimer theory [31], considering electronic scattering by the interfaces, and the resistivity of metal thin films model used in the fitting is shown below:

$$\rho = \rho\_i \left[ 1 - \frac{3}{2\kappa} (1 - p) \left[ \frac{1}{t^3} - \frac{1}{t^5} \right) \frac{1 - e^{-wt}}{1 - p} dt \right]^{-1} \tag{2}$$

where *κ* = *d*/*l*, with *d* is the thin film thickness, *l* is the electronic mean free path, *ρ<sup>i</sup>* is the bulk resistivity, *t* is an integration parameter, and *p* is the probability that an electron will be specularly reflected upon scattering from one of the surfaces (*p*). Typical values for *p* are 0 for polycrystalline films and 0.5 for single crystal films. The electrical and optical properties of Ag thin films exhibit a marked dependence with thickness when this is comparable to the electronic mean free path.

(ii) The nonoptimum growth of Ag on glass leads to rougher films and agglomeration in the lower thickness limit. Mayadas et al. [36, 37] extended Fuchs-Sondheimer theory in Eq. (1) to consider electronic scattering by grain boundaries. Furthermore, Rossnagel and Kuan [38] revised this model and extended it to take into account surface roughness and grain size as 1 + \_ *<sup>d</sup>* <sup>+</sup> \_1.5 *Rl*

**4.1. Silver thickness effect in the electrical properties**

424 Modern Technologies for Creating the Thin-film Systems and Coatings

are two models used in the fitting of the experiments:

The thin film silver thickness effects on its electrical properties were studied in many literatures, and Ref. [12] gave a good comparison of the theoretical and experimental results. **Figure 16** shows experimental resistivity values (*ρ*) decreased as Ag films with thicknesses ranging from 3 to 74 nm and the theoretical model predictions [12]. The silver was either directly deposited on the glass or was deposited in a stack with seed and blocker layers. There

(i) First is the Fuchs-Sondheimer theory [31], considering electronic scattering by the inter-

**Figure 16.** Experimental electrical resistivity for Ag stacks (red dots), Ag on glass (blue dots), and calculated values using Fuchs-Sondheimer's theory for polycrystalline films (*p* = 0) and single crystal films (*p* = 0.5). The dotted lines show how

> 1 ∞ ( \_1 *<sup>t</sup>*<sup>3</sup> <sup>−</sup> \_<sup>1</sup>

resistivity, *t* is an integration parameter, and *p* is the probability that an electron will be specularly reflected upon scattering from one of the surfaces (*p*). Typical values for *p* are 0 for polycrystalline films and 0.5 for single crystal films. The electrical and optical properties of Ag thin films exhibit a marked dependence with thickness when this is comparable to the electronic

*<sup>t</sup>*<sup>5</sup>) <sup>1</sup> <sup>−</sup> *<sup>e</sup>* <sup>−</sup>*κ<sup>t</sup>* \_\_\_\_\_\_ <sup>1</sup> <sup>−</sup> *<sup>p</sup> <sup>e</sup>* <sup>−</sup>*κ<sup>t</sup> dt*

] −1

(2)

is the bulk

faces, and the resistivity of metal thin films model used in the fitting is shown below:

roughness and intergrain scattering affect resistivity using the Rossnagel and Kuan formalism [12].

<sup>2</sup>*κ*(1 − *p* ) ∫

where *κ* = *d*/*l*, with *d* is the thin film thickness, *l* is the electronic mean free path, *ρ<sup>i</sup>*

*<sup>ρ</sup>* <sup>=</sup> *<sup>ρ</sup><sup>i</sup>* [<sup>1</sup> <sup>−</sup> \_\_\_<sup>3</sup>

mean free path.

$$\rho = \rho\_i \left[ 1 + \frac{0.375(1 - p)Sl}{d} + \frac{1.5 \text{ R}l}{(1 - R) \text{ g}} \right] \tag{3}$$

where *S* is the roughness parameter that equals 1 in perfect, atomically flat interfaces, and increases as roughness does so. *R* is the scattering coefficient, illustrating the scattering of electrons at the grain boundaries. Finally, *g* is the average grain size, which is shown in **Figure 17**. These parameters clearly illustrate how thickness, grain size, and surface roughness affect the films reflectivity.

**Figure 17.** Ag grain size measured for the case of films deposited on glass and in stack configuration [12].

#### **4.2. Silver thickness effect in the optical properties**

Ref. [10] provided a good model and experiments on the silver thickness effect in the optical properties. **Figure 18(a)** shows the refractive index (*n*) and extinction coefficient (*k*) for Ag thin films deposited on glass with thickness ranging from 15.3 to 74.3 nm [12]. A progressive reduction in *n* is observed as the Ag films get thicker, achieving values close to bulk for thicknesses around 74.3 nm [36]. On the other hand, the extinction coefficient (*k*) was found to remain almost identical for all the thicknesses.

**Figure 18.** (a) Refractive index (*n*) and extinction coefficient (*k*) and (b) real and imaginary parts of the dielectric function for Ag thin films deposited on glass [12].

Using the ellipsometry method, the electron scattering times *τ* for the Ag films can be calculated. It is interesting to find that the product *τ* × *ρ* also remains constant, as shown in **Figure 19**, and the theory behind it have been discussed in the literature [12]. Further, a new model was developed to predict the silver refractive index as shown in **Figure 20**.

$$\eta = \frac{\rho}{2\varepsilon\_0 (\rho \tau)^2 \omega^3} = \frac{\rho}{2\varepsilon\_0 \Gamma^2 \omega^3} = \frac{\rho}{k}$$

where *C* is the product *ρ* × *τ*, with the value 59 ± 2 µΩ cm fs [10], independent of the wavelength, and *k* is nearly a constant for the silver film.

**Figure 19.** The product *τ* × *ρ* remains constant within the thicknesses range studied with silver directly deposited on glass, and with silver in the stack between seed and blocker layer [12].

Using the ellipsometry method, the electron scattering times *τ* for the Ag films can be calculated. It is interesting to find that the product *τ* × *ρ* also remains constant, as shown in **Figure 19**, and the theory behind it have been discussed in the literature [12]. Further, a new model was developed to predict the silver refractive index as shown in

**Figure 18.** (a) Refractive index (*n*) and extinction coefficient (*k*) and (b) real and imaginary parts of the dielectric function

where *C* is the product *ρ* × *τ*, with the value 59 ± 2 µΩ cm fs [10], independent of the wave-

<sup>2</sup> *<sup>ε</sup>*<sup>0</sup> *<sup>C</sup>*<sup>2</sup> *<sup>ω</sup>*<sup>3</sup> *<sup>k</sup>* (4)

*<sup>n</sup>* <sup>≈</sup> *<sup>ρ</sup>* \_\_\_\_\_\_\_\_\_\_\_ <sup>2</sup> *<sup>ε</sup>*<sup>0</sup> (*ρτ* )<sup>2</sup> *<sup>ω</sup>*<sup>3</sup> *<sup>k</sup>* <sup>=</sup> *<sup>ρ</sup>* \_\_\_\_\_\_\_\_\_\_\_

length, and *k* is nearly a constant for the silver film.

426 Modern Technologies for Creating the Thin-film Systems and Coatings

**Figure 20**.

for Ag thin films deposited on glass [12].

**Figure 20.** Experimental refractive index spectra for Ag films (a) 29 nm and (b) 74 nm thick along with calculations using three values of the constant *C*, i.e., 59 µΩ cm fs as middle point, and 57 and 61 µΩ cm fs as lower and upper limits, respectively [12].

#### **4.3. Emissivity dependence with silver thickness**

Low-emissivity coatings have important applications in energy-efficient windows and thermal coatings. **Figure 21** shows experimental emissivity values for Ag thin films with thicknesses ranging from 3 to 40 nm [12]. A progressive increase in emissivity is observed as the films get thinner, which means their ability to reflect infrared radiation decreases.

**Figure 21.** Emissivity versus thickness for Ag thin films. Calculations correspond to bulk Ag optical constants considering no dependence with thickness (blue line), considering the evolution of the electron collision time predicted for polycrystalline films *p* = 0 (red line) and single crystal films *p* = 0.5 (green line) [12].

The optical model could calculate the reflectivity by using silver's optical constants with different silver film thicknesses, which can be used for simulating silver emissivity [12]

teret sliver film thickness, which can be used for simulating silver emissivity [12]

$$\mathbf{e} = \frac{\sum\_{i=1}^{n} (1 - R\_{li}) \, E\_{li} \, \Lambda \, \lambda\_{i}}{\sum\_{i=1}^{n} E\_{li} \, \Lambda \, \lambda\_{i}} \tag{5}$$

where *Rλ* is the reflectivity at wavelength *λ* and *Ebλ* is the radiation emitted by blackbody at wavelength *λ*, and *R<sup>λ</sup>* can be calculated from the refractive index *n*, *k*, which can be calculated from the modeling for polycrystalline and single crystal. These results are compared with experimental results in **Figure 21**, which implied that the most experimental silver film results are agreed with the polycrystalline model.

#### **5. Conclusion**

Stronger legislations are helping to improve the energy efficiency of new buildings, so low-E technology, especially silver-based low-E technology, has developed very fast in the last 30 years. Silver-based low-E technology is reviewed on its application background, history, and on how technically to generate high-quality, silver thin films; further on how the silver thin film's electrical, optical, and emissivity properties are influenced by their microstructure, thickness, and by the materials on neighboring layers through a theoretical and an experimental perspective. Low-E window coating is one of the fastest growing sectors in the glass industry, and sputtered silver-based low-E will continue to grow in the near future globally.

## **Acknowledgement**

The authors would like to thank low-E group members' support at Intermolecular Inc.

### **Author details**

**4.3. Emissivity dependence with silver thickness**

428 Modern Technologies for Creating the Thin-film Systems and Coatings

Low-emissivity coatings have important applications in energy-efficient windows and thermal coatings. **Figure 21** shows experimental emissivity values for Ag thin films with thicknesses ranging from 3 to 40 nm [12]. A progressive increase in emissivity is observed as the

The optical model could calculate the reflectivity by using silver's optical constants with dif-

**Figure 21.** Emissivity versus thickness for Ag thin films. Calculations correspond to bulk Ag optical constants considering no dependence with thickness (blue line), considering the evolution of the electron collision time predicted

where *Rλ* is the reflectivity at wavelength *λ* and *Ebλ* is the radiation emitted by blackbody at wavelength *λ*, and *R<sup>λ</sup>* can be calculated from the refractive index *n*, *k*, which can be calculated from the modeling for polycrystalline and single crystal. These results are compared with experimental results in **Figure 21**, which implied that the most experimental silver film results

Stronger legislations are helping to improve the energy efficiency of new buildings, so low-E technology, especially silver-based low-E technology, has developed very fast in the last 30

*<sup>m</sup>* (1 − *Rλ<sup>i</sup>* ) *Ebi* Δ *λ<sup>i</sup>* \_\_\_\_\_\_\_\_\_\_\_\_\_\_ <sup>∑</sup>*<sup>i</sup>*=<sup>1</sup> *<sup>m</sup> Ebi* Δ *λ<sup>i</sup>*

(5)

ferent silver film thicknesses, which can be used for simulating silver emissivity [12]

for polycrystalline films *p* = 0 (red line) and single crystal films *p* = 0.5 (green line) [12].

<sup>ϵ</sup> <sup>=</sup> <sup>∑</sup>*<sup>i</sup>*=<sup>1</sup>

are agreed with the polycrystalline model.

**5. Conclusion**

films get thinner, which means their ability to reflect infrared radiation decreases.

Guowen Ding\* and César Clavero

\*Address all correspondence to: gding@intermolecular.com; dingguowen@yahoo.com

Intermolecular, Inc., San Jose, CA, USA

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