1. Introduction

Transparent conductors are a key component of optoelectronic devices. Transparent conductors with high electrical conductivity and optical transmittance with good mechanical flexibility and durability are increasingly becoming more important for designing high performance optoelectronic devices including solar cells, photodetectors, light emitting diodes, and touch panels [1]. The widely used transparent conductor - indium-tin-oxide (ITO) suffers from high cost and lack of mechanical flexibility, a criterion that is becoming more important with the emergence of the flexible electronics era. In replacement of ITO, new materials for flexible transparent conductors have been extensively studied such as carbon-based materials (Graphene [2,3], CNT [4,5], conductive polymers [6]), or metal-based materials like a metal mesh [7,8], nanowires [9,10], or thin films [11–13]. Please refer to Table 1 for electrical and optical characteristics of ITO and emerging flexible transparent conductors on a flexible substrate. Carbon-based materials and conductive polymers offer excellent mechanical flexibility as a transparent conductor but suffer from low electrical conductivity, which can limit device performance when used for device applications [14–17]. Metal-based transparent conductors, like metal mesh or nanowires, have excellent optical transmittance and low sheet resistance but suffer from large optical haze due to light scattering and high cost from complex fabrication steps [10,18]. Among these candidates, thin metal film-based transparent conductors received great attention for their high electrical conductivity, optical transmittance with low haze, and excellent mechanical flexibility. Moreover, thin metal films exhibit unique advantages including simple fabrication steps, low-cost, and process compatibility suitable for large-area device applications [13].

The key to a thin metal film-based transparent conductor is in controlling its thickness. The thickness of the metal film should be kept well below its skin depth at optical frequency so that light can transmit through the film [19]. For example, the skin depth of silver (Ag) at visible wavelength is in the range of 20 nm [20]. Optimum thickness of the metal-film needs to be chosen so that its Figure-of-Merit (FoM) as a transparent conductor, a metric proportional to electrical conductivity and optical transmittance can be maximized. Although the actual FoM of a thin metal film may vary depending on the choice of material and its electrical or optical properties [1], almost all metal films’ FoM have a strong thickness dependence due to metal’s intrinsic inability to transmit light when the film becomes sufficiently thick. For this reason, the thickness range of interest for metal-film based transparent conductors is usually in the ultrathin regime (< 10nm). At this thickness, the metal film’s morphology dynamically change [21–23] due to de-wetting [24], which can impact its electrical and optical properties. Therefore, a good understanding of a metal-film’s electrical and optical properties at the ultrathin regime is crucial for designing the film for transparent conductor applications.

There are recent review papers that broadly discuss the development of metal-film-based flexible transparent conductors in terms of film growth, material preparation, and applications [13,17,25]. In this review paper, we aim to focus on optical, electrical, and mechanical properties of metal-film-based transparent conductors. First, we explore the evolution of an ultrathin metal film’s optical properties by associating a dielectric function with film morphology. We will then study how its optical properties can be correlated to electrical resistivity models, which could serve as a foundation for improving the film properties of an ultrathin metal film. Strategies to improve an ultrathin metal film’s intrinsic electrical and optical properties will be introduced through its figure-of-merit as a transparent conductor. Second, we investigate design considerations to enhance optical transmission by using anti-reflective coatings and introduce recent machine-learning approaches to further aid in the design of transparent conductors. Third, we will discuss requirements for mechanically robust flexible electrodes, comparison of emerging flexible electrodes, and the mechanism of a metal film-based transparent conductor's excellent durability against mechanical stress. Finally, we will touch upon key advantages in using metal-film based transparent conductors for optoelectronic applications.

Table 1. Comparison of electrical (sheet resistance Rs) and optical (Transmittance) characteristics of ITO and emerging flexible transparent conductors on a flexible substrate.

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2. Optical and electrical properties of ultrathin metal films

Understanding of a thin metal film’s dielectric properties is essential in designing it for various optoelectronic and plasmonic applications. As there are numerous literatures/reviews available that comprehensively discuss this topic [31–33], this paper will primarily focus on the evolution of the dielectric function at the ultrathin regime and associate it with the film’s electrical properties. The first part of this section discusses the dielectric function of an ultrathin metal film under the homogeneous and inhomogeneous film regime. The homogeneous film regime is when the thin metal film becomes continuous and free of any voids whereas the inhomogeneous film regime is when the film can be treated as a quasi-continuous film as a metal-void composite material. At the ultrathin film regime, these two regimes become less distinct and so it is important to understand how optical properties of the film evolve as film thickness is varied at this regime. The second part of the section focuses on studying various electrical scattering (or resistivity) models at each morphological regime. This is important as the film’s dielectric function is related to the electrical resistivity via scattering time in the classical Drude model. Therefore, a good understanding of the ultrathin metal film’s resistivity scaling due to scattering events can also give insight on the film’s optical properties as well as direction on how to improve film quality based on this understanding. The last part of this section will discuss dominant carrier scattering models at each morphological regime of the ultrathin metal film.

2.1 Dielectric function of homogeneous metal film: optical scattering time

The optical response of a homogeneous metal film can be ascribed to the frequency-dependent permittivity of metals, which can be modeled using the Lorentz harmonic oscillator model (for bound electrons) and the Drude model (for free electrons). For example, in Ag, UV and visible regions are dominated by inter-band transition described by the Lorentz oscillator or bulk plasmonic absorption, while response at infrared is determined by free electron which can be modeled using the Drude model [34]. The Drude contribution of the dielectric function for an infinite medium can be given as below [35]:

 

Fig. 1. (a) Experimental real (${\varepsilon _1}$) and imaginary (${\varepsilon _2}$) part of the dielectric constant along with model fitting results considering the Drude and Lorentz contributions for a 74 nm silver film.(b) Electron scattering time τ vs. thickness for Ag thin films. (Reprinted with permission from Ref. [35], AIP Publishing LLC). (c) A comparison of the resistivity determined by optical (squares and circles) and electrical (triangles) methods. The inset shows the areas for optical and electrical measurement (Reprinted with permission from Ref. [37], Elsevier B.V.).

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Optical scattering time extracted from the dielectric function of a metal film provides a link between the electrical and optical properties of the film. It is an important metric to gauge the quality of the metal film associated with its electronic properties. First-principles calculation of thin metal film’s scattering time can provide a theoretical limit on how good of a film quality can be achieved. For example, a first-principles calculation of Ag’s carrier relaxation time was shown as 30 fs [39] to 36.8 fs, [40] while optically extracted carrier relaxation time from various experimental Ag films were in the 10-20 fs range [35,41] including that from Palik’s data [42]. This indicates there is still room for improvement of the Ag film’s quality in practice for its electrical and optical properties, i.e. lower resistivity leading to a smaller value of the imaginary part of permittivity in the optical frequency range.

2.2 Dielectric function of inhomogeneous metal film

Metals commonly used as a transparent conductor like Ag, Au, and Cu are known to have low adhesion energy on oxide substrates in which the film growth would follow the Volmer-Weber mode [13]. In such a growth mode, initially formed metal islands at the nucleation stage will form a conductive network as deposition progresses. The film will be comprised of a metal-insulator composite medium until all insulating voids are filled by metal adatoms. In such case, the dielectric function of such inhomogeneous medium can be modeled using the percolation model or EMA models.

Hovel et al. studied the optical conductivity ${\sigma _1}$ and dielectric constant ${\varepsilon _1}$ of an evaporated Au film on a Si/SiO₂ substrate with thickness varied between 3 and 10 nm [22]. Optical conductivity ${\sigma _1}$ is the response of a material’s electrical conductivity under alternating electric field which can be calculated from the imaginary part of the dielectric function via ${\sigma _1}(\omega )= ({\omega /4\pi } )\cdot {\varepsilon _2}(\omega )$ [43]. First, low frequency ${\sigma _1}$ from the Drude model matches well to the directly measured DC conductivity added as dots in Fig. 2(a). Also, as shown in Fig. 2(a) and (b), the behavior of ${\sigma _1}$ and ${\varepsilon _1}$ can be clearly divided in two regimes: thickness above 7 nm and below 6 nm, where the percolation threshold is in between these two values. This is supported by the in-situ resistance measurement as a function of thickness as shown in Fig. 2(c). Above the percolation threshold thickness, the film behaves as a typical metallic film where ${\sigma _1}$ decreases with frequency with ${\varepsilon _1}$ as negative. This trend reverts for a film thickness below 6 nm near the metal-to-insulator transition regime where ${\sigma _1}$ vanishes and ${\varepsilon _1}$ is positive indicative of dielectric behavior. As a result, reflectivity of the film at infrared frequency increases. The metal-to-insulator transition was investigated using percolation theory to model the measured static ${\varepsilon _1}$ as shown Fig. 2(d). It is expected that ${\varepsilon _1}$ diverges at the percolation threshold where maximum ${\varepsilon _1}$ was fitted at a thickness of 6.4 nm. Also note that the zero crossing point of ${\varepsilon _1}$ occurs at ${d_0}$ = 6.7 nm, slightly higher than the percolation threshold, indicative of the metallicity of the film [43].

 

Fig. 2. (a) Optical conductivity and (b) dielectric constant of gold films at room temperature. The included DC values (solid squares) perfectly match to the extrapolated conductivity ${\sigma _1}$. (c) In situ DC-resistivity measurement showing the sharp transition at the percolation threshold written as ${d_c}$ in this figure (this is different from critical thickness used in the manuscript). (d) Divergence of the static dielectric constant as a function of nominal film thickness d for Au on Si/SiO₂. (Reprinted with permission from Ref. [22], APS Physics). (e) Real part ${\varepsilon _1}$ and (f) imaginary part ${\varepsilon _2}$ of Ag (Cu) film’s dielectric function for different film thickness ${d_{eff}}$. Black arrows indicate the curve change behavior as decreasing ${d_{eff}}$. (g) Ag (Cu) film’s ${\varepsilon _1}$ as a function of wavelength for different air fraction. Solid lines are the measured ${\varepsilon _1}$ for varying projected air fraction ${\phi _{air}}$. Dashed curves are ${\varepsilon _1}$ (real part of ${\varepsilon _{eff}}$) calculated from BEMA for varying volumetric air fraction ${f_{air}}$. For each curve, corresponding film thickness ${d_{eff}}$ are noted as well. (Reprinted with permission from Ref. [23], John Wiley & Sons, Inc.)

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Park et al. observed change in the dielectric function of Cu-seeded Ag film for thickness ranging from 2 to 17 nm as shown in Fig. 2(e) and (f). ${\varepsilon _1}$ at the near-infrared region remains relatively unchanged or slight increased for thickness of 17 nm down to 5 nm [23]. Below 5 nm, the ${\varepsilon _1}$ curve at infrared rapidly increases toward zero value and crosses the zero point at an effective thickness ${d_{eff}}$ of around 2.6 nm. A similar trend was observed for ${\varepsilon _2}$ where the curve remains unchanged for film thickness down to 5 nm but then increases for thickness below 5 nm which indicates the film becoming lossy. The rapid change in dielectric function was attributed to the increase in inhomogeneity of the film below a certain critical thickness. They used Bruggeman’s effective medium approximation (BEMA) to model the real part of dielectric function ${\varepsilon _1}$ as a metal-insulator composite medium. As shown in Fig. 2(g), air fraction calculated from the BEMA matched the measured void fraction extracted from TEM results which optically verifies inhomogeneity of the film. One major limitation of BEMA is that this model is absent of any plasmonic effect that may occur in an actual sample. As will be discussed later, electrical scattering time of Cu-seeded Ag film rapidly decreases as film thickness reduces below a certain thickness, so called critical thickness ${d_c}$, in which optical scattering time can be expected to trend the same.

2.3 Size effect theory: surface and grain boundary scattering

For a bulk material, a sample’s boundary effect can be ignored as the material dimension is much larger than electron’s mean free path (${l_0}$). This no longer is the case for thin metal films when its thickness (or grain size) becomes comparable to ${l_0}$. In such case, scattering at the boundaries can contribute significantly to the increase in total resistivity. As a result, resistivity of the thin metal film becomes a strong function of the film’s dimension (thickness or grain size) which is known as the size effect theory. This size effect can be taken into consideration in the dielectric function by replacing optical scattering time as ${\tau _{size}}$ in Eq. (1). The two fundamental scattering mechanisms are surface scattering and grain boundary scattering which are the basis for various existing size effect theories.

The first comprehensive analysis of size effect in a metal thin film was suggested by Fuchs, and later known as Fuchs-Sondheimer (F-S) model which takes into account resistivity increase due to surface scattering as a film’s thickness becomes comparable to the electron mean free path [44]. The resistivity of a thin metal film described by this model is calculated as:

Soffer’s model assumes that the scattering at the surface is only attributed to the roughness. The limitation of this model is that the contribution of roughness to resistivity is minute unless roughness ${r_1}$ or ${r_2}$ is close to ${\lambda _F}$ [45].

Later, Namba represented 1-dimentional geometrical roughness using a Fourier series and proposed the inclusion of the geometrical film cross section due to roughness into the F-S model’s expression for resistivity [48]. This leads to a stronger increase in resistivity which better estimates experimental resistivity compared to other existing surface scattering models especially at the low thickness regime of 40 nm [49,50]. The resulting expression for the roughness incorporated thickness $d(x )$ is given as:

A typical metal film deposited via commonly used physical vapor deposition will be polycrystalline and have a finite grain size. The effect of grain boundary scattering on the resistivity is well described in the Mayadas-Shatzkes (M-S) model [51]. The M-S model treats electrons as waves traversing the grain boundary as a periodic potential well, and the probability that an electron wave is reflected at the grain boundary is represented by the reflection coefficient R. The expression of grain-boundary scattering limited resistivity of thin film is ${\rho _{MS}}$:

Mayadas and Shatzkes further derived the resistivity model for the case where there is interplay between the grain boundary and surface scattering [56]. This model, so called Mayadas Shatzkes Surface (MSS) model takes into account the grain boundary scattering as well as those electrons scattered at this boundary subsequently redirected to the film surface and get scattered there. The expression of MSS model is:

In principle, the MSS model is a violation of the Matthiessen’s rule, which states that each of the scattering sources are independent of each other. It is only at a low temperature regime where the thermal energy is low enough to depopulate the phonon density at the film surface that the mechanism in the MSS model will likely to come into play.

In practice, it is more likely that surface (or roughness) scattering and grain boundary scattering both take role in decreasing the conductivity, which is determined by the rate limiting step. To further quantify the contribution of each scattering event, Matthiessen’s rule can give a good approximation for various combinations of surface (roughness) scattering models and grain boundary models [53,57] :

The total resistivity is the summation of ${\rho _{Surf}}$, resistivity contribution determined by the surface scattering models and ${\rho _{GB}}$, that from the grain boundary scattering model, minus intrinsic resistivity of a bulk material ${\rho _i}$ (as this portion was each included in ${\rho _{Surf}}$ and ${\rho _{GB}}$).

2.4 Inhomogeneous film theory: percolation, effective medium approximation (EMA), general effective medium (GEM)

The size effect theory gives a good approximation of resistivity scaling as long as the film is homogeneous which is usually the case for sufficiently thick films. However, when a metal film’s thickness is further reduced to a regime where percolation effect starts to play a role, the film can no longer be treated as homogeneous medium and so size effect models alone are insufficient to describe the resistivity scaling. Additional models need to be considered to account for the resistivity increase due to morphological inhomogeneity of the film. For a metal-film as a transparent conductor, the thickness of interest is typically in the ultrathin regime below 10 nm in which the resistivity of the film gets impacted by both size effect theory and the inhomogeneity of the film.

At the lower thickness limit of the metal film near percolation threshold, the resistivity scaling encounters a phase transition problem, often described as the percolation effect. In such an extreme limit of reduced thickness where the film is no longer continuous but instead behaves as a random network of conductors (so called metal-insulator composites), rapid change in resistivity occurs in the vicinity of percolation threshold following a power-law behavior. An electrical percolation model has been widely used to describe the resistivity scaling in metal-insulator composite materials [58,59].

For the percolation model or the GEM model, the percolation threshold fraction ${\phi _c}$ and critical exponent of conductivity t need to be determined. This can be achieved by fitting data to the power law [68]. For a given shape and type of network system, universal values of ${\phi _c}$ and t can be mathematically calculated [22,60,69]. For example, bond percolation for a disk shape network which resembles a typical metal cluster has a ${\phi _c}$ of 0.6763 [70]. Also a metal-insulator composite is known to follow a three-dimensional network with theoretical universal value of $t\; $= 2 [10,11], although deviation from this value is seen in empirical results [71]. With rigorous calculation of ${\phi _c}$ and t, resistivity scaling of a inhomogeneous metal-film near percolation threshold can be approximated. Summary of all models discussed in this paper to describe resistivity scaling in homogeneous and inhomogeneous metal film is shown in Fig. 3.

 

Fig. 3. Summary schematic of all electrical models discussed in this section that are used to describe resistivity scaling of metal films in homogeneous film (size effect theory) and inhomogeneous film (GEM). Extended GEM model incorporates size effect theory into general effective medium theory to describe the resistivity scaling in the entire range of metal film thickness from continuous film down to percolation threshold.

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2.5 Resistivity modeling in homogeneous film regime

For homogenous metal films, there has been debate on whether surface or grain boundary scattering plays a dominant role in thin films. The answer to this may vary from film to film as the physical morphology of the film which determines the resistivity can vary by numerous factors. Here, we will introduce several studies aiming to provide insight into scattering events of metal films at the ultrathin regime for the interest of transparent conductor applications.

First, few studies emphasized the significance of surface roughness scattering in thin metal films. Li et al. conducted in-situ resistivity measurement of ultrathin cobalt (Co) films with varying monolayer thickness. When using various classical size effect models, the Namba model that accounts for surface roughness gives the best approximation for resistivity even down to a few monolayers [72]. Similar observation was reported by Liu et al. where they studied thickness dependent copper (Cu) film's resistivity below 40 nm. Using surface roughness measured by atomic force microscopy (AFM) to find h as 3.5 nm in the Namba model in Eq. (3a) provided the closest approximation of resistivity scaling of Cu film from 10–40 nm among different size effect models [50]. Ke et al. used ab initio to theoretically study the impact of electron scattering of the Cu film as a function of surface roughness [73]. Unlike using empirical models, this work provides quantitative analysis that the surface scattering contribution can be large even when surface roughness is small. On the other hand, Zhang et al. reported that grain boundary is the dominant scattering mechanism in a polycrystalline Cu film with reflection coefficient $R$ = 0.46 [74]. Also, Sun et al. showed that the grain boundary scattering model is the dominant contributor for a tantalum/silicon dioxide (Ta/SiO₂) encapsulated Cu thin film [53,75]. Also, Lin et al. showed iridium films with average thickness of 0.6–7 nm where they attributed resistivity scaling to strong reflection ($R$ > 0.94) at columnar grain boundary [76,77].

In practice, scattering from surface (or surface roughness) and grain boundaries co-exist and should be considered on the same footing in the study. Decoupling these two mechanisms becomes a challenge especially in models like MSS where grain boundary and surface scattering effects are convoluted in a single model. Chawla et al. investigated the scattering effect of Cu thin films and wires for different grain sizes, and provides a systematic method to deconvolute the electron scattering contributions from grain boundaries and from top, bottom, and side surfaces [57]. They used electron backscatter diffraction (EBSD) to find the grain size histogram of polycrystalline Cu(111) thin films while controlling the size by annealing with the top and bottom capping layers. Supported by their ab initio transport calculations [78] to consider the limiting case of Cu interfaces (Cu-air, Cu-Ta, and Cu-MgO) as complete diffusive scattering (${p_1}$=0, ${p_2}$=0), they were able to calculate the relative contribution of resistivity by phonons, phonons and surface, or phonons and grain boundaries for different grain size films. Also, in their 30-50-nm-thick polycrystalline Cu sample, a 10-15% decrease in resistivity is expected when increasing the average grain size by a factor of 1.8.

A more commonly used method to consider both scattering mechanisms is to use Matthiessen’s rule as stated in Eq. (8). Camacho et.al, studied the electrical resistivity $\rho $ of thermally evaporated aluminum (Al), gold (Au), and Cu films for thickness range of 3–100 nm [52]. Calculation of resistivity contribution of F-S and M-S models can be done by sweeping the range of fitting parameters ($p$ and $R$) for each model. The result indicates that the grain boundary model is the major contribution to the experimental $\rho $ values for all three metals which becomes more prominent at sub-10 nm thickness. A more systematic approach can be done by calculating sum of squared errors (SSE) to fit empirical data by sweeping fitting parameters like p and R or other terms depending on the model used. Sun et al. conducted comprehensive analysis of comparing various classical size effect models to study which combination of models can best explain the best scattering behavior in nanometric Cu films cladded by SiO₂ or Ta deposited on a Si(100) substrate using DC sputtering [53,75]. For each set of models that represent either surface scattering, grain boundary scattering, or a combination of both, fitting parameters were obtained by minimizing the SSE. Since different sets of models can have a different number of fitting parameters where goodness-of-fit is expected to be better for models with more fitting parameters, a Bayesian information criterion (BIC) was calculated for each model set to penalize those having a larger number of fitting parameters allowing for a fair comparison among models. Models with a good fit will have smaller negative BIC values. From their analysis, using Matthiessen’s rule to combine resistivity contribution by the F-S and M-S model with fitting parameters $p$ = 0.52 and $R$ = 0.43, respectively showed lowest SSE and BIC values among models with same number of parameters for their sputtered Cu-film. For their sputtered Cu-film with thickness ranging from 27 to 158 nm, the partition of the resistivity contribution showed an average of 27% from surface scattering and 73% from grain boundary scattering at room temperature signifying the dominant grain boundary contribution in the Cu film. Dutta et al. also discussed the relative contribution of surface and grain boundary scattering in the MSS model for Cu and platinum-group metals (PGM) like Ru, Pt, or Ir ultrathin films with thickness ranging from 3 to 30 nm [79]. In Fig. 4, ${\sigma _{SS,GB}}({\; = 1/{\rho_{SS,GB}}} )$ is conductivity describing combined effects of surface and grain boundary scattering and ${\sigma _{GB}}({\; = 1/{\rho_{GB}}} )$ is conductivity contribution from grain boundaries. Their results of sweeping p and R parameters to calculate the SSE for Cu and Ru films show a weak contribution of surface scattering which is manifested as the SSE minima elongated along the $p$-axis as shown in Fig. 4. Also, the grain boundary coefficient for PGM materials like Ru was higher ($R$ ∼ 0.4 to 0.6) than for Cu ($R$ = 0.22). The reason for dominant grain boundary scattering in PGM metals even more so than for Cu was attributed to having a much smaller mean free path. More recently, Park et al. used Matthiessen’s rule to sweep fitting parameters of various models using Eq. (1)–(6) and calculate the SSE of Cu-seeded ultrathin Ag film’s resistivity scaling for thickness range from 5 to 40 nm [23]. Among various sets of models, selecting Namba model with $p$ = 0.1 for ${\rho _{Surf}}$ and M-S model with $R$ = 0.2 for ${\rho _{GB}}$ was shown to best describe Cu-seeded Ag film with thickness ranging from 5 to 40 nm. Also, similar to results by Dutta et al., the ultrathin Ag film’s SSE calculation showed strong dependence on the grain boundary coefficient R, indicating grain boundary contribution is strong compared to surface roughness scattering.

 

Fig. 4. ${\sigma _{SS,GB}}/{\sigma _{GB}}$ as a function of the surface scattering parameter p for (annealed) stacks as indicated for film thicknesses of 5 nm (a) and 20 nm (b), respectively. (c)–(e) Sum of squared errors (SSE) of fits to the experimental data vs. p and R fitting parameters for (c) TaN/Cu/TaN, (d) Ru/SiO₂, and (e) TaN/Ru/TaN, all after post-deposition annealing at 420°C. The full color scales correspond to the range between 1× and 4× the minimum SSE for all graphs. The white crosses represent the positions of minimum SSE. (Reprinted with permission from Ref. [79], AIP Publishing LLC).

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The approach of mapping SSE across the entire span of fitting parameters has a limitation in that the set of fitting parameters may not be uniquely defined but different sets of values may be equally well fit the experimental data. Strictly speaking, temperature-dependent resistivity measurements are needed to accurately reflect which contribution is the dominant factor [55,80,81]. This is because surface scattering is known to show a stronger interaction with phonon scattering than with grain boundary scattering [53]. As temperature reduces, less phonon scattering events will redirect electron momentum toward the surface hence a decrease in surface scattering. The detail on this topic is beyond the scope of this review paper and can be found elsewhere [53,80,82,83]. Nevertheless, the above discussion as well as various other works [84] clearly indicate stronger scaling behavior of resistivity due to the grain boundary scattering model as thickness is reduced. In practice, metal film growth is a complex problem where its physical properties like the metal property (mean free path), grain (size, shape, orientation) or surface roughness are affected by various factors like the type of metal, its interfacial energy with the substrate, and the deposition conditions making it difficult to generalize the dominant scattering mechanism. However, for ultrathin metal films of our interest for transparent conductor application where film’ surface roughness is kept small to guarantee low optical loss, surface roughness scattering is not expected to play a major role even with the Namba model which is known to provide strong scaling behavior at lower thicknesses [50] with consistent results observed elsewhere [23]. Therefore, reducing the grain boundary scattering portion via tuning film growth or applying post treatment can be effective in obtaining high quality ultrathin metal film for transparent conductor application.

2.6 Resistivity modeling in inhomogeneous regime and mechanism transition

In a metal film within the ultrathin or extremely thin regime where its thickness decreases down to near percolation threshold, the size effect theory alone is insufficient to describe the scaling behavior [23,85]. In this regime, the film’s resistivity is governed not only by the size effect theory but also by EMA/percolation effect arising from inhomogeneity of the film which both give rise to the total resistivity. This is manifested as a form of rapid increase in resistivity below certain thickness or alternatively a double-sloped behavior in logarithmic resistivity-thickness plot indicative of two different governing mechanisms. Such bimodal behavior was commonly observed in various metal films like Ag [23,86–88], Au [89], Cu [50,90], Pt [21], or NiCr [91]. Despite numerous studies discussed in detail about electrical resistivity modeling of the metal film at the metal-insulator transition regime (percolation theory [92–95], EMA [61,96,97], or GEM [63]), relatively less research was done in conjunction with the size effect model. This is partly because the region of interest for inhomogeneous metal films is at the vicinity of the metal-to-insulator transition which is usually at a much smaller thickness regime than the thickness of interest dealt by size effect theories. Also, for resistivity models of inhomogeneous metal films, resistivity is a function of metal fraction $\phi $ and not a function of thickness, which makes it less straightforward to combine the model with size effect models. Recent advancement in nanotechnology has enabled wider use of sub-10 nm ultrathin metal films for various optoelectronic applications, where at this thickness regime, the transition of the film’s morphology from inhomogeneous to homogenous is less distinct necessitating both mechanisms to be considered in the resistivity modeling.

Vancea et al. studied resistivity scaling of various metals like Cu, Ag, Al, Ni, Au, Pt on a glass substrate with in-situ resistivity measurements [45]. They demonstrated that their experimental resistivity data can be described by the Fuch-Namba model down to the onset thickness below which film becomes island-like morphology where the model no longer agrees with empirical results. Despite the absence of the percolation effect in the model they used, their studies of resistivity at various distinct thickness regimes provide insight on the implication of transition of the transport mechanism governed by inhomogeneity of the medium. Maaroof elegantly elaborated Pt and Ni film’s change in electrical conduction with respect to ${t_{min}}$, the thickness at which $R{t^2}$ becomes minimum where R and t are film’s resistance and thickness, respectively [21]. Below ${t_{min}}$, metallic island nuclei grow to form a conducting metallic network where the percolation theory was used to model a rapid drop in R as t is increased until ${t_{min}}$. Above ${t_{min}}$, the film is considered quasi-continuous film where its resistivity can be modeled using the classical size effect theory. This work provides valuable insight on how the conduction mechanism evolves at different stages of film growth. One limitation though is that the piece-wise representation of resistivity at each growth regime may be inaccurate as it should be a continuous transition from the percolation theory to size effect theory. Also, the percolation theory applies only to resistivity scaling near the conductor-insulator transition region in which the model becomes inaccurate at thicknesses near the quasi-continuous film regime [60]. More recent results on studying a metal film’s resistivity scaling at the ultrathin regime attempt to conceptually explain the conduction mechanism. For example, Zhang et al. described a zinc-oxide(ZnO)/Ag multilayered film’s resistivity showing two different slopes where conduction at the thinner region was attributed to a tunneling mechanism [98]. As the resistivity values calculated from the tunneling models (typically used for discontinuous metal films [49,99,100]) are at least a few orders of magnitude higher than what was reported in the study, it is not certain whether tunneling is responsible for conduction at this regime. Formica et al. observed the rapid decrease in electrical resistivity of ultrathin Ag and Cu-seeded Ag film indicating the coalescence of islands above which film resistivity can be expressed by the size effect theory using Matthiessen’s rule [101]. More recently, Park et al. used an extended-GEM model (${\rho _{ext.\; GEM}}$) to combine the GEM model with the classical size effect theory (Fig. 3). Instead of representing the resistivity transition from an inhomogeneous film to a homogeneous film regime in a piece-wise manner, ${\rho _{size}}$ is incorporated into the GEM as a varying function of effective thickness ${d_{eff}}$ (or metal fraction $\phi $) to describe resistivity scaling of Cu-seeded Ag film down to the percolation threshold [23]. As discussed in section 2.5, the GEM model combines both aspect of the percolation theory and the effective medium theory thereby giving more accurate approximation of resistivity even when the metal fraction is near unity (homogeneous film). Among various models that are used to describe resistivity of inhomogeneous metal film, the extended GEM model was found to give the best fitting result compared to other models like the percolation model, Landauer’s EMA, or conventional GEM model [23]. As shown in Fig. 5(a)-(b), the governing transport mechanism rapidly transitions from a GEM-dominated to a size-effect dominated regime with respect to the critical thickness ${d_c}$ = 5 nm marked as vertical dotted-line in Fig. 5(a). This critical thickness ${d_c}$ was defined as the intersection point in thickness at which the double-slope intersects in log-log plot of resistivity-thickness plot. Taking a closer look starting from the initial film growth stage is as follows: at the vicinity of percolation threshold thickness ${d_{eff}}({{\phi_c}} )$ of 2.4 nm for a Cu-seeded Ag film, transport is dominated by the percolation theory where a rapid decrease in resistivity is governed by a power-law as thickness is increased (or $\phi $ is increased). As thickness further increases and metal clusters coalesces to fill voids, transport is determined by the effective medium theory until ${d_c}$ of 5 nm. Above ${d_c}$, resistivity contribution due to the GEM model rapidly diminishes (Fig. 5(b)) and transport is dominated by the size effect theory ${\rho _{size}}$ where the film can be considered as a quasi-continuous film. Electron mean free path calculated from the extended GEM model shows that it is below ${d_c}$ in which the mean free path values rapidly diverges from the size effect linearity due to the increase in metal-insulator boundary that acts as additional scattering sites as shown in Fig. 5(c). In this work, authors also showed that ${d_c}$ also coincides with the ${t_{min}}$ value that yields minimum $\rho \times t$ or $R \times {t^2}$ as discussed by Vancea [45] or Maroof [21]. Furthermore, validity of the extended GEM model was tested which works well in describing not only an Ag(Cu) film but also a bare Ag film for the entire thickness range as shown in Fig. 5(d).

 

Fig. 5. (a) Ag (Cu) film’s $\rho $ versus ${d_{eff}}$ approximated by using size effect model only (${\rho _{size}}$) is plotted in blue dashed line and that by extended GEM model (${\rho _{ext.GEM}}$) is plotted in red dashed-dot line. Measured experimental resistivity values are plotted in open symbols for reference. (b) Ag (Cu) film’s total resistivity ${\rho _{total}}$ (= ${\rho _{ext.\; GEM}}$, dashed black line) on the left axis and contribution of size effect model over total resistivity, ${\rho _{size}}/{\rho _{total}}$ (dashed-dot red line) on the right axis both as a function of film thickness ${d_{eff}}$. Symbols on left axis are measured experimental resistivity values. Transport mechanism above and below critical thickness ${d_c}$ (= 5nm) are each governed by size effect theory and GEM, respectively. (c) Ag(Cu) film’s predicted effective mean free path l (symbol) as a function of ${d_{eff}}$ calculated from ${\rho _{ext.\; GEM}}$. Proportionality relation of grain size D with ${d_{eff}}$ is plotted (red dashed line) as well to illustrate the size effect theory. Ag (Cu) film’s critical thickness ${d_c}$ (= 5nm) is indicated as a vertical dotted line. (d) $\rho $ versus ${d_{eff}}$ approximated by ${\rho _{ext.GEM}}$ for Ag (Cu) film and bare Ag film. Bare Ag film data is newly added to the plot from the original figure (Reprinted with permission from Ref. [23], John Wiley & Sons, Inc.).

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3. Critical thickness of ultrathin metal film and its figure-of-merit (${\phi _{TC}}$)

3.1 Implication of critical thickness

As discussed previously, electron transport of ultrathin metal film experiences a transition point at a critical thickness (either ${t_{min}}$ in [21] or ${d_c}$ in [23]). Due to the nature of rapid increase in metal-film’s electrical resistivity below this critical thickness, the film below this thickness becomes less ideal to be used as a transparent conductor. The term critical thickness ${t_{min}}$ or ${d_c}$ should be distinguished from the percolation threshold which is typically smaller than ${t_{min}}$ or ${d_c}$, despite several literatures mistakenly mixing up the two terms. This critical thickness has important implication in terms of the optical properties as well. Above critical thickness where theultrathin metal film is still homogeneous, optical transmission of the film increases with reduced thickness as more light can transmit [23,87]. This trend continues until the critical thickness point, below which there is no benefit in optical transmission despite reduction of film thickness. Below the critical thickness, plasmonic resonance absorption increases caused by island-like morphology of the inhomogeneous metal-insulator medium as shown in Fig. 6(a). What this implies for metal films as a transparent conductor is that its Haake’s Figure-of-Merit ${\phi _{TC}}$ (= ${T^{10}}/{R_s}$) [102] reaches a maximum near this critical thickness [23,103] where T is transmission at 550 nm wavelength and ${R_s}$ is the sheet resistance. Figure 6(b) shows measured and modeled ${\phi _{TC}}$ of Ag(Cu) and bare Ag film where ${\phi _{TC}}$ reaches a maximum at around critical thickness of ${d_C}$ = 5 nm [23] (bare Ag’s ${\phi _{TC}}$ was calculated and plotted to the original figure). It is desirable for metal film to be as thin as possible to maximize optical transmission but not too thin to a point at which both electrical and optical properties start to deteriorate. This implies that the thickness of metal film will need to be at or above ${d_C}$ to maximize optical and electrical properties for the transparent conductor application. Remarkably, Ag(Cu) with ${d_C}$ = 5 nm on a glass substrate alone can achieve $T$ = 80.7% and ${R_s}$ = 21.3 ohms/sq even without any anti-reflective coating signifying the implication of ${d_C}$ [104]. Also, it is below this critical point at which the electrical resistivity not only rapidly increases but also shows large sample-to-sample variation due to the randomness of quasi-continuous film. This could raise a practical concern on the uniformity of the sample as it would mean that resistivity becomes uncontrollable as film gets thinner. In total, critical thickness ${d_C}$ can serve as an important engineering metric for designing ultrathin metal film as a transparent conductor.

 

Fig. 6. (a) Ag (Cu) film’s measured average transmission (T_AVE) and absorption (A_AVE) over visible wavelength (380–780 nm) plotted as a function of ${d_{eff}}$. Critical thickness ${d_c}$ (= 5nm) is indicated as a vertical solid line (Reprinted with permission from Ref. [23], John Wiley & Sons, Inc.) (b) Haacke’s figure-of-merit ${\phi _{TC}}$ (= ${T^{10}}/{R_s}$) of Ag (Cu) and bare Ag as a function of ${d_{eff}}$ where T is transmission at 550 nm wavelength and ${R_s}$ is sheet resistance. Symbols and dotted line are experimental and modeled ${\phi _{TC}}$, respectively. For modeling of ${\phi _{TC}}$ for Ag (Cu) and bare Ag, where ${R_s}$ was calculated from using ${\rho _{ext.\; GEM}}$ and T is calculated from measured optical constants of each film using transfer-matrix method. Vertical dotted-lines indicate ${d_c}$ for each sample. Ideal Ag’s ${\phi _{TC}}$ (blue) is also plotted where the data extracted from Fig. 3 of Ref. [104]. (Reprinted with permission from Ref. [104], American Association for the Advancement of Science)

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3.2 Enhancing ${\phi _{\textrm{TC}}}$ by percolation threshold reduction

An ultrathin metal film’s ${\phi _{TC}}$ can be enhanced by reducing the critical thickness of the ultrathin metal film. As critical thickness is associated with the film’s transition from the inhomogeneous to homogeneous regime, reducing the percolation threshold during film growth can be effective in achieving high ${\phi _{TC}}$. As metal films widely used for transparent conductor applications (e.g. Au, Ag, or Cu) have de-wetting properties causing island-like Volmer-Weber growth, various methods have been proposed to promote the wetting of these metal films to lower its percolation threshold. As this topic was extensively covered in other recent review papers [13,17], it will be briefly discussed in this paper. Commonly practiced methods include using inorganic wetting-layers like germanium (Ge) [105,106], Cu [101], titanium (Ti) [107], chromium (Cr) [108] or polymeric surfactants [89,109] prior to the deposition. Modulation of the substrate surface energy [88,110] or using underlying dielectric layers [111,112] to promote wetting of metal films have been demonstrated as well. Alternatively, introduction of dopants [87,113,114] during the metal film deposition or even partial oxidation of the metal film during the initial stage of deposition [115] are known to be effective in reducing the percolation threshold of the film. All these methods have been demonstrated to be effective in making an ultrathin metal film with an ultrasmooth surface and good optoelectronic properties by inhibiting Volmer-Weber growth and lowering the percolation threshold thickness, thereby achieving enhanced ${\phi _{TC}}$ compared to that of bare metal film [104,109,115] (Fig. 6(b) shown as an example). Such ultrathin metal films with film growth methods to reduce percolation threshold show excellent electrical and optical properties and so are considered as promising candidates for emerging flexible transparent conductor [104,114].

3.3 Enhancing electrical property by film morphology engineering

Metal-film based transparent conductor’s ${\phi _{TC}}$ can be further enhanced by improving the electrical resistivity of the film. Blue-dotted line in Fig. 6(b) is the data plot extracted from [104] for an ideal Ag film’s ${\phi _{TC}}$ when only the surface scattering F-S model ($p$ = 0.9) is considered showing potential improvement in ${\phi _{TC}}$ when resistivity due to grain boundary scattering can be neglected. Considering such an ideal Ag film case, its ${\phi _{TC}}$ can even exceed that of the state-of-the-art ITO [104]. As already discussed, grain boundary scattering plays a significant role in poly-crystalline ultrathin metal films [23,53]. The grain boundary scattering portion becomes significant especially for the case of metal films where wetting/adhesion layers or dopants are used to achieve lower a percolation threshold as these strategies are known to render smaller grain sizes [101,113] in return for a smoother surface. Smaller grain size will result in more frequent scattering events at grain boundaries which leads to higher electrical resistivity and larger optical loss. If grain boundary scattering can be minimized in an ultrathin metal film, electrical and optical properties of the film are expected to improve followed by higher ${\phi _{TC}}$.

One of the strategies to reduce grain boundary contribution is via thermal annealing. Annealing the metal film will allow grain growth and removes defects inside the film thereby reducing the damping rate of electron scattering [116]. Fortunately, studies show that adhesion layer or dopants in metal film makes the metal film thermally stable [106,114,117] thereby preventing the film from forming agglomerates upon thermal stress [94]. Adhesion layers like Ti are known to be effective in improving the thermal stability of Ag [118] or Au films [107] where the thickness of the adhesion layer plays a critical role in enhancing the thermal stability. For example, Abbott, et al. firstly demonstrate that Ti adhesion layer with sub-nanometer thickness can significantly increase the thermal stability of the 50 nm Au film [107]. According to their study, too thick of an adhesion layer results in diffusion of Ti through Au grain boundaries causing Ti to reach the surface and oxidize, leading to volume expansion and compressive stress within the film.

Several studies show improvement in the ultrathin metal-film’s electrical and optical performance upon annealing. For example, Chen et al. showed that a 1-nm of Ge wetting layer was effective in promoting lateral grain growth of a Ag thin film without increase in the surface roughness of the film, which was not achievable for the Ag film without the wetting layer [106]. As a result of annealing, the Ag thin film with a 1-nm Ge wetting layer exhibits reduced damping rate and lower resistivity owing to the reduced grain boundary scattering of the film as shown in Fig. 7. It is also important to mention that the annealing temperature and time needs to be carefully controlled for a film with given thickness as excess thermal stress can exacerbate the optical/electrical properties of the film. Similarly for the case of a co-sputtered film, Gu et al. reported that Al-doping in a doped Ag film curbs the enlarging of the grain size [113]. Zhang et al. reported that an ultrathin Al-doped Ag film (7 nm) exhibited higher optical loss than a thicker pure Ag film (30 nm) owing to various factors including Al impurities, reduced thickness, and increased scattering due to fine grains. In this study, they also observed improved optical property of 7 nm Al-doped Ag film after annealing at 500°C under nitrogen environment for 10 seconds where imaginary part of permittivity was significantly reduced over near infrared range [114].

 

Fig. 7. (a) Experimentally retrieved damping rate (${\gamma _p}$) curves and (b) Measured resistivity ($\rho $) of silver films at different thicknesses on a 1 nm Ge layer after annealing at different temperatures for 3 minutes. (c) Linear relation between the ${\gamma _p}$ and $\rho $ in the silver films. (Reprinted with permission from Ref. [106], AIP Publishing LLC).

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Aside from annealing, tuning the deposition conditions can also result in better electrical property of the film. While details of the deposition conditions may depend on the specific deposition system or conditions, McPeak et al. provides a comprehensive guideline in obtaining high quality metal films (Al, Cu, Au, Ag) with optimal optical properties [119]. In specific, factors like residual gas (water vapor or oxygen) can adsorb on the film and pin grain boundaries, which reduce the average grain size in the film [120]. Substrate temperature can also affect the grain growth as low substrate temperature with minimal surface diffusion can produce small grains. Deposition rate also needs to be carefully chosen so that the balance between grain growth and grain-boundary pinning can be obtained. Figure 8 shows the effect of the deposition rate on the Ag film morphology. It is seen that the number of voids reaches a minimum at a deposition rate of 2 Å/s, with more voids at both higher and lower rates [121]. Considering these aspects, high quality metal film can be obtained at sufficiently low vacuum where impact of residual gas is minimal, with deposition rate sufficient to prevent grain-boundary pinning (due to gas trapping or void formation) while promoting grain growth where temperature heating can be used to excite surface diffusion but not so much to increase the roughness of the film. While tuning deposition kinetics to improve the film properties can be a quick solution to improve the quality of the film, exploring new techniques to obtain epitaxial metal film can also be a solution to improve film quality by obtaining grain-free metal films. Deposition methods like chemical synthesis, electrodepositions, molecular beam epitaxy, or atomic layer deposition with the aid of substrate engineering techniques have been successful in obtaining epitaxial or crystal growth of metal films like Au [122,123], Ag [124–126], Cu [127,128], or Al [129]. These deposition techniques can be further developed and utilized for achieving grain-free (or large grainsize) ultrathin metal films, where significant enhancement in ${\phi _{TC}}$ is expected for its application as a transparent conductor upon improvement in electrical/optical performance.

 

Fig. 8. SEM images of an 11-nm-thick Ag layer on 30-nm-thick MoO3 layer with various Ag deposition rates of (a) 0.5 Å/s, (b) 1 Å/s, (c) 2 Å/s, (d) 3 Å/s, and (e) 2 Å/s at a low magnification level, and (f) a 2µm ${\times} $ 2µm atomic force microscopy (AFM) image of the M30A11M30 (MoO₃/Ag/MoO₃) multilayer structure with an Ag deposition rate of 2 Å/s. The root-mean-square (RMS) roughness of the M30A11M30 was 0.67 nm (Reprinted with permission from Ref. [121], IEEE).

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4. Optical design consideration and extending transparency to the NIR range

Due to the highly reflective nature of thin metal films over optical frequencies, light transmission through the film is still limited. Even when the thickness of metal film is reduced to an extremely thin regime, light reflected at the metal-air/metal-substrate interface can be significant, especially at longer wavelengths. A widely practiced method to further boost transmittance by salvaging reflected portion of light is using anti-reflective (AR) dielectric coatings in each side of metal film [11,12,114].

It is illustrative to examine the analytical and quantitative design principle used by Ji et al. [12] to optimize the transmittance of the dielectric-metal-dielectric (DMD) based transparent conductor. Figure 9(a) shows the DMD structure on top of a transparent substrate, where in total there are three reflection coefficient components that are significant: ${r_1}$, ${r_2}$ and ${r_3}$. Consider normally incident light, ${r_1}$ is the reflection coefficient of incident light reflected at the air/dielectric interface, ${r_2}$ is the reflection coefficient when light transmits through dielectric layer and gets reflected back at the dielectric/metal interface and ${r_3}$ is the reflection coefficient when light transmits through metal layer and gets reflected back, including reflecting from the metal/dielectric 2 and dielectric 2/substrate interfaces. For simplicity, all three components do not consider multiple reflections between different interfaces, and they are complex numbers with amplitude and phases (including the propagation phase inside medium and reflection phase at each interface). A DMD structure should be optimized such that the sum of ${r_1},\; {r_2},\; {r_3}$ equals zero, leading to the minimum reflection and maximized transmission. Compared to ${r_2}$, ${r_1}$ and ${r_3}$ (decaying due to propagating in metal layer) have relatively low amplitude. Therefore, destructive interference happens when ${r_1} + {r_3} = {r_2}$ (as illustrated in Fig. 9(b)), which will completely suppress the overall reflection. Following this principle, the designed structure achieves an average of 88.4% transmission in visible band (shown in Fig. 9(c) and 9(d)).

 

Fig. 9. (a) The design principle formulated by Ji et al. [12]., where the complex reflection ratio of ${r_1},{\; }{r_3}$ need to sum up to give destructively interference with ${r_2}$ such that the reflection is minimized. (b) Phase diagram of reflected waves. (c) The simulated and measured transmission of designed structure: Polyethylene Terephtalate (PET) substrate/24 nm ZnO/6.5 nm Cu-doped Ag/56 nm Al2O3 (aluminum oxide) structure. (d) A photograph of fabricated AR-TC. (Reprinted with permission from Ref. [12], NPG).

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In some applications (e.g. photovoltaics), we want the transparent conductors to have high transmission of light not only in the visible band but also in near infrared (NIR) spectrum range to harvest the broad solar spectrum. However, the structure reported in [12] does not give a sufficiently high transmittance when extending to NIR range. Figure 10(c) shows the simulated transmission, reflection and absorption of the same structure in Fig. 9(c) from 400-1100nm, where the transmittance starts to decrease significantly at NIR. The decreased transmittance attributes from the increased extinction coefficient (imaginary part of refractive index) of Cu-doped Ag, which leads to the failure of the destructive interference where ${r_1} + {r_3} \ne {r_2}$. To see this, Fig. 10(d) and (e) show the relative amplitude and net phase shift of these three complex reflection coefficients. At the visible range, ${r_1}$ and ${r_3}$ are mostly in phase and both are out-of-phase with ${r_2}$, leading to almost complete destructive interference. However, at NIR, because of the high extinction when propagating through the metal layer, the amplitude of ${r_3}$ decreases considerably due to absorption; the abrupt phase shift at metal/dielectric 2 interface also makes ${r_3}$ not in phase with ${r_1}$ anymore. Both effects result in ${r_1} + {r_3} \ne {r_2}$. This is the limitation of the simple three-layer DMD structure which has a limited number of interfaces and propagation mediums, where both can contribute to modulate the reflection coefficients. We suggest by adding extra material layers below or above the metallic layer, it is possible to introduce extra components in ${r_1}$, ${r_2}$ or ${r_3}$ to further suppress the reflection through destructive interference in NIR. As an example, Zhang et al. used an additional layer of MgF₂ on top of the TiO₂/Al-doped Ag/TiO₂ structure and achieved averaged transmittance of 92.4% on a glass substrate [114].

 

Fig. 10. Demonstration of adding additional layers in DMD structure to extend the high transmittance to NIR. Schematic representation of the simulated DMD structure (a) and DDMDD structure (b). The DMD structure is the same as the structure shown in Fig. 1(c). Compared to DMD, the DDMDD structure in (b) is modified by adding two additional dielectric layers: MgO and TiO₂, above and below the middle metallic layer, respectively. The thickness of each layer is optimized to achieve a high averaged transmittance. The simulated transmission, reflection and absorption in 400-1100nm for the DMD (c) and DDMDD (f) structure. DMD structure shows a decrease in transmittance at NIR, where DDMDD structure shows a transmittance improvement from 81.2% to 85.4%. The amplitude of three refection coefficients in DMD (d) and DDMDD (g). The net phase shift of three refection coefficients in DMD (e) and DDMDD (h). Compared to DMD, in NIR, DDMDD structures gives a higher amplitude in ${r_1}$ and ${r_3}$ (illustrated in red arrow in (g)), as well as a closer phase relationship between ${r_1}$ and ${r_3}$ (illustrated in red arrow in (h)), which contribute together to give destructive interference and lead to lower reflections in NIR.

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Considering different metals and substrate used in Ref. [12] and Ref. [114], here we show the performance improvement in NIR in a more illustrative and comparable way when using additional layers, in the form of dielectric/dielectric/metal/dielectric/dielectric (DDMDD) (structure diagram is shown in Fig. 10(b)). Compared to the structure in Fig. 10(a), extra MgO and TiO₂ layers are added between the Al₂O₃/Cu-doped Ag and the Cu-doped Ag/ZnO, respectively. The thickness of each layer is optimized to achieve a higher averaged transmittance. Figure 10(f) shows the simulated transmission, reflection and absorption from 400-1100nm, which exhibits an improved transmission from 81.2% to 85.4%. Because of extra components in each reflection coefficient at the additional material interfaces (as illustrated in Fig. 10(b)), at NIR, the amplitude of ${r_1}$ and ${r_3}$ are increased (as illustrated in red arrow in Fig. 10(g)) and the net phase shift between ${r_1}$ and ${r_3}$ is no longer completely out-of-phase (as illustrated in red arrow in Fig. 10(h)). Therefore, the destructive interference of ${r_1} + {r_3} = {r_2}$ is still satisfied, leading to the anti-reflection effect that extends to NIR. In order to further suppress the reflection to maximize the transmission, it is anticipated that more dielectric layers can be added to boost the performance.

Although adding dielectric antireflection layers on both sides of metal layers is widely used to decrease the reflection, it is still unknown if there exist other types of structures that can surpass the performance of such a DMD-type (or extended DDMDD-type) structure. Recently, many researchers are using deep learning to deal with the photonic inverse design and demonstrate excellent performance in many complicated design problems [130–136]. Specifically, by learning through the interaction with the physical environment, reinforcement learning (RL) is good at exploring the whole design space [137], which can potentially find new designs with better performance than exiting known structures. For example, Wang el. al [136] proposed a RL algorithm called optical multi-layer proximal policy optimization (OMLPPO) that can sequentially generate and design for multilayer thin film structures. The authors applied this method to two optical design tasks: ultra-wideband absorbers and incandescent light bulb filters, and both discovered optimal designs with performance much better than existing results. Currently no research has been done by using RL to design for transparent conductors. We would like to highlight the great potential and possibly new revolution in transparent conductor design when considering the RL algorithm for design.

5. Mechanical properties of the metal-film based transparent conductor

5.1 Improving mechanical properties on DMD structure

There has been tremendous effort to advance electronic materials and structures with better bendability and flexibility to open a new era of flexible devices [138–148]. In the case of the flexible transparent conductor, electrical failure can occur as a consequence of large mechanical deformation along with catastrophic crack propagation across the conductor under repeated bending or large straining. Interestingly the dielectric/metal/dielectric (DMD) multilayered sandwich structures introduced in the previous section not only improves the light transmission [12,13,149], but shows excellent endurance against mechanical stress in bending tests [138]. It enables their application in flexible optoelectronic devices and helps with better fracture and fatigue resiliency. These advantages make the multiple layer electrodes becomes more reliable and robust transparent conductive electrode materials by deflecting rapid crack propagation under large deformation. Since the first report by Fan et al. in 1974 [150], DMD multilayered structures have been studied extensively to achieve highly transparent and conductive electrodes [151–153], because they offer higher transmittance with low sheet resistance [113,154–157]. From another perspective, inserting a ductile metallic layer between two dielectric layers can benefit mechanical durability. The superior flexibility of DMD based transparent electrode is mainly attributed to the ductile metal interlayer that provides effective electrical conductivity even after the dielectric layer is far beyond its fracture strain [158,159]. While these structures have been investigated in detail for improved conductivity electrodes for displays, their utility as mechanically robust transparent conductive electrode has not been thoroughly studied due to experimental difficulties at the nanometer scale, and there have been only few attempts to address this important issue [160–162]. Nanoscale fracture mechanics are substantially different from those on the macroscopic scale, hence fracture mechanics confront new challenges at the nanoscale since only a limited number of atoms exist in the vicinity of the crack tip [163,164]. In this section, the improved bending performance of DMD over various ITO-replacement electrodes with regards to conductivity and flexibility will be reviewed.

5.2 Requirements for robust flexible electrodes

The requirement for robust conductive electrodes that can bend is growing with the interest in flexible electronics. First a clarification of terminology is in order. The terms bending and flexing are often used mutually even though flexing is commonly intended to be elastic while bending often involves plastic deformation. Fracture strain indicates how much an electrode can be deformed plastically before breaking. Flexibility refers to the capability of bending without any breaking of chemical bonds in an electrode material. Stiffness is the extent to which a sample resists deformation in response to an applied load. Its complementary value is flexibility: the more flexible an object is, the less stiff it is. The flexibility of a structure is of principal importance in many engineering applications, so the elastic modulus is often one of the primary properties considered when selecting a material. A high modulus of elasticity is sought when deflection is undesirable, while a low modulus of elasticity is required when flexibility is needed. In addition, physical dimension factors, e.g., thickness, length, and the moment of inertia can also affect the electrode's ability to resist bending [165–168]. If the thickness of the sample increases, the stiffness will increase at least by a power-law dependence. It is important to reduce the mechanical stress and strain by making substrates thin to improve flexibility. Various types of flexible transparent conductive electrodes are evaluated under various types of stress, including tensile, compression, twinning and bending [169]. In a typical bending process, the two sides of a bent substrate experience different types of strain as shown in Fig. 11. Tensile strain is present on the convex side, while compressive strain on the concave side [147,170].

 

Fig. 11. Schematic illustration of the stresses in a thin film during bending deformation. Linear elastic deformation is governed by Euler and Bernoulli equation, which states inner layer has compressive stress whereas outer layer has tensile stress when subjected to bending. In the middle layer (black dashed line), there is a neutral plane.

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These opposites of nature may lead to different failure modes. Cracking of a film or channeling crack will occur when tensile stresses develop; buckling-driven delamination can occur if stresses are compressive [171,172]. Interfaces are intrinsic to these systems, and they are susceptible to interface debonding or delamination. Isolated surface cracks can propagate and channel across the film. If the substrate is brittle, the crack may penetrate into the substrate even though it is essentially unstressed. The interface may debond, or a debond crack may drive into substrate producing delamination through cracking, parallel to the interface in the substrate. Combined film cracking and interface debonding can also take place if the combination of film toughness and interface toughness lies within certain limits [173]. In a homogeneous structure, these peak strains can be calculated using a simple equation, (ε = t / R), or with more accurate formulae that invoke fewer simplifying assumptions [174]. From these expressions, therefore thinner substrates experience lower strain at a given bending radius, and components mounted on these thinner substrates tend to tolerate smaller bending radii before failing. Moreover, through the bulk of a bent substrate, the strain varies between the compressive and tensile extremes on either surface. The variation is linear in a homogeneous film and it may be more complex in a layered structure, but regardless, a mechanically neutral plane exists in the sample where neither tensile nor compressive strain is present. Positioning fragile materials along this neutral plane can maximize the overall flexibility of the system [175,176].

5.3 Various emerging alternative materials for flexible electronics

There are two different types of failures in bending operations. One is fracture, which is a one-time static bending. When materials are subjected to a large strain, they fail by fracture. Another is fatigue, which is weakening of a material caused by repeatedly applied loads. Bending-mode tests are widely used in both the academic and industry fields to evaluate the strain tolerance of flexible electronics. Over the last few decades, extensive efforts have been made on utilizing advanced materials and structures to improve cyclic bending performance of electrodes. ITO has been commonly used for transparent conductive materials despite the nature of its brittleness and cost [177–179]. Among the various emerging alternative materials to ITO, metallic and carbon-based nanomaterials have received much interest due to their outstanding optical-electrical properties, low cost, ease of manufacturing, flexibility, and a wide range of application. These involve metal nanowire percolating networks, lithographically patterned metal grids, carbon-based nanostructures, e.g., carbon nanotubes (CNTs) and graphene, emerging liquid metal, and conducting polymers [180–182]. Metal nanowire-based transparent electrode technologies have been significantly developed over the last decade to become a prominent low-cost alternative to ITO [183]. Copper based materials have been attracting significant attention due to their electric, mechanical, and thermal properties [184]. Copper nanowires (CuNWs) are typically prepared via hydrothermal methods and are widely used in flexible electronic devices due to their high conductivity and oxidation resistant characteristics. Both Zhang et al. and Tang et al. show that copper nanowire maintains very good stability after 1,000 cycles [185,186]. However, compared to silver nanowires (AgNW), CuNWs are much more vulnerable to chemical degradation by the formation of oxides (CuO, and Cu₂O) from reacting with oxygen easily under ambient conditions, aggravated due to its high surface-to-volume ratio [184,187,188]. On the other hand, AgNWs have already reached a certain level of technological maturity, and numerous commercialized AgNW products are available commercially. The good electrical conductivity of silver, coupled with the excellent ductility and bendability exhibited by the wires make them suitable for flexible electrodes [189–191]. Lee et al. developed a highly stretchable electrode via the solution-processing of very long AgNW and subsequent percolation network formation. The stretchable metal electrode from very long metal nanowires demonstrated high electrical conductivity (∼9 Ω/sq) and mechanical compliance (strain > 460%) at the same time [192]. Han et al. introduced a facile and large area silver nanowire electrode which exhibits excellent durability during cyclic bending (up to 10,000 cycles) and stretching (50% strain) [193]. Other than metallic nanowires, Rodriguez et al. introduced an aluminum nanoparticles on PET structure which shows an impressive mechanical resistance after 10,000 bending cycles [194]. In addition to metal-based nanowires, metal grids are also considered to be the best candidates because of their inherently high electrical conductivity, optical transparency, mechanical robustness and, more importantly, cost-competitiveness [7]. A nanoscale metal grid transparent conductor was reported back in 2007 and used as flexible anode for an OLED [195] and later for a organic photovoltaic [196]. Nanoscale metal grids allow one to exploit some special effects such as polarization selection and plasmonic excitation. Kang et al. took this advantage to demonstrate a plasmonic enhanced organic solar cell [9] and Park et al. explored colored organic photovoltaics [197]. Metal grid transparent electrodes optimized for low sheet resistances (8 Ω/sq at a relative transmittance of 94%) as well as optimized for high transmittance (97% at a sheet resistance of 20 Ω/sq) are reported by Schneider et al. which can be further improved on demand for use in various applications [198]. The finding of low-dimensional carbon nanostructures, such as graphene and CNTs, have attracted extensive interest and demonstrated great promise in flexible electronics areas [199]. Graphene holds great potential for application in flexible and wearable electronics. Nam and his colleagues successfully demonstrated that robust thin-film electrodes can be fabricated by inserting an atomically thin 2-D interlayer between the metal thin film and the flexible substrate [200]. Hyun et al. demonstrated the great bending stability of printed graphene [201] and Lan et al. produced good durability for bending over 5,000 cycles [202]. Carbon nanotubes (CNTs) are expected to obtain outstanding properties of the constituent material, especially superior tensile strength and elastic modulus of individual CNTs. However, CNT has not shown its advantage as next-generation flexible electrode due to the large contact resistance between CNTs, as reflected by its low conductivity [203–205]. Zou et al. coated CNT fiber with a metal layer, and it has become an effective solution to address this problem. The composite fiber can sustain up to 10,000 bending cycles [206]. Both CNTs and graphene exhibit superior mechanical stability, but they still suffer from a poor tradeoff between transparency and conductivity [144,203]. The high conductivity and fluidity of Gallium (Ga) based liquid metal makes it a promising candidate as an electrical conductor directly dispensable at room temperature. Unlike conventional metals in the solid state, the intrinsic nature of liquid metal is naturally soft and can be easily dispensed, patterned, deformed, and even stretched to form the desired structure [207–209]. In particular, Ga-based liquid metals have shown intrinsically high electrical conductivity and thermal conductivity, while their low toxicity, benign biocompatibility, and low viscosity (near that of water) can be widely used in the fields of soft and stretchable electronics, self-healing devices, and biology. Hu et al. showed excellent conductivity based on a polyvinyl alcohol and a liquid metal printable ink [210]. Park and colleagues developed a printable ink containing polyelectrolyte-attached liquid metal film, and it shows reliable and stable response under the application of repeated strain (over 10,000 cycles) without electrical failure [211]. Conducting polymers are extensively studied due to their outstanding properties, including tunable electrical properties, high optical and mechanical properties, easy synthesis, and fabrication with environmental balance over conventional inorganic materials [212]. Ko et al. presented a micropatterning process for high cyclic bending performance of PEDOT:PSS [213]. However, there are disadvantages in using PEDOT:PSS, which include acidity, and batch-to-batch variation in electrical and physical properties [214]. In theory, all these materials can be used as an effective conductive medium for flexible devices, but each conductor will produce different cyclic bending performance respectively when subjected to a large deformation. Many researchers and industries have been developing different bending test setups to evaluate the durability of flexible electronics, the measured results vary due to not only material constituents or structures but also different bending test set ups [215,216]. It is important to design an appropriate bending test setup that can confirm a good electric conductivity under bending conditions. Both the characteristics of individual components and integrated systems must be considered, and electrical properties should be measured during or after the bending operation of the sample through electrical connection at the grips.

5.4 Cyclic bending performance results and analysis

Long term durability analyses of electrodes are of high interest when determining the risk of failure in engineering applications [191,217]. Lee et al. conducted long-term reliability of various non-ITO conductors during bending fatigue under the same conditions of strain, as shown in Fig. 12. The structures and materials included: ITO, metal grids, silver nanowire networks (AgNW), DMD with a ITO/CuAg/ITO structure, and DMD with a Al₂O₃/CuAg/Al₂O₃, on PET substrates of similar thickness around 120 µm. Resistance variance (ΔR/R₀) was monitored during cyclic bending. Each electrode exhibited significantly different electrical fatigue trends during the cyclic bending process. Importantly the authors observed that even after more than 10,000 cycles bending test, the conductivity of the DMD films has almost not changed. This flexibility was attributed to the highly ductile behavior of the metallic CuAg interlayer between the two ITO layers. In sharp contrast, the resistance of pure ITO films with similar thickness to that of the DMD abruptly increased at a bending radius of 5 mm within a few cycles and rapidly increased with further decrease of bending radius, due to crack formation and propagation in the top surface of ITO layer. Once it passes over its critical tensile stress, crack generation under large straining is inevitable, the ITO thin film starts to develop cracks rapidly that lead to complete electrical and mechanical failure eventually. A simple model can describe the increasing resistance in cracked ITO electrode in terms of small volume of conducting material within each crack [218]. The fatigue response for the AgNWs, metal grids, and DMD with the Al₂O₃/CuAg/Al₂O₃ structure are found to be significantly worse than that of the DMD structure. In case of the AgNWs, the change in resistance under bending strain is closely related to the failure of each individual nanowire. Moreover, the AgNW shows the unique deformation characteristics with a size dependent ductile brittle transition, the brittle fracture can occur even under small deformation [219]. The metal grid is prone to fatigue damage, since stress will be concentrated in the narrow conductive path during cycling, making it difficult to resist any fracture and necking. Depending on the pattern of the mesh grid, the resistance will be displayed differently [220]. The DMD with a Al₂O₃/CuAg/Al₂O₃ multilayer induces an earlier failure than the DMD with a ITO/CuAg/ITO structure, attributed to a weak toughness of Al₂O₃. Mechanical properties of interest typically include elastic modulus, yield strain, crack onset strain, and fracture toughness. Among these, the crucial and simple reliability parameter when characterizing brittle thin films on polymeric substrates is the fracture toughness. Fracture toughness is a clear indication of the amount of stress required to propagate a preexisting flaw; in other words, toughness is the capability of material to resist fracture, and this is a critical mechanical property of interest because it determines the structural integrity and reliability of flexible electronics. Young's modulus measures the resistance of a material to elastic, i.e., recoverable deformation under load. A stiff material has a higher Young's modulus and changes its shape only slightly under elastic region. the elastic modulus of Al₂O₃ (200 GPa) is higher than that of ITO structure (150 GPa). However, Al₂O₃ has a weak toughness (0.24 to 1.20 MPa√m) compared to ITO structure (2.50 to 2.70 MPa√m) [221,222]. For that reason, it was concluded ITO/CuAg/ITO architecture can absorb fracture energy and plastically deform without fracturing, and consequently exhibits a superior crack tolerance. This phenomenal response suggests that the ductility of the CuAg layer plays a crucial role in the fatigue-resilient electrical performance under cyclic bending and provides an electrically conductive path even the ITO was strained beyond its failure limit.

 

Fig. 12. Dynamic outer bending fatigue test results of the optimized DMD multilayer with increasing bending cycles at a fixed outer bending radius of 5 mm. (a) Comparison of flexible electrode cyclic performance results, clearly very stable repeated oscillations were observed. Based on our bending test results, AgNW and Metal grid sample (with Ag nanoparticle) failed rapidly comparing to DMD sample. These promising electrode exhibits brittle fracture due to the localized necking and leading to the abrupt increase of electrical resistance in flexible electrodes. (b) DMD multilayered structure resistance versus repeated oscillating bending events (at R = 5 mm radius). Inset: a zoomed-in fragment of the main curve showing perfectly repeatable and stable sheet resistance oscillations, DMD exhibits very stable resistance value even after over ten thousand bending cycle number. (c, d) Fatigue comparison results between tensile and compressive at bending radius 4mm. Compression shows better cyclic performance compared to that of tensile stress case. Inset figures indicates uneven surface was created due to the compressive loading. Channeling crack occurs when tensile stresses. Scale bar = 1 µm. Adapted from Ref. [124] by permission of copyright 2022 American Chemical Society.

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Based on fatigue experiments, DMD films on the flexible PET substrate exhibit substantial differences in the value of ΔR/R₀ between the inward and outward bending (Fig. 12(c) and (d)), and tensile stress causes early failure. The difference in the behavior is fully attributed to different fracture mechanisms [215,223–225]: under tensile strain generated in the outward bending, fracture can directly cause film rupture and forming channel cracks, impeding the current conduction; under compressive strain induced by inward bending, the film may first delaminate from the PET substrate and then buckle before the crack initiates. The applied strain energy was released through the formation of delamination, buckling and cracks in the inward bending test. In-situ bending test of DMD films on the flexible PET substrate was carried out to verify the assumption. As can be seen in Fig. 12(d) inset figures, the failure mechanism of the DMD film under tensile strain is directly related to cracking resulting in rapid electrical disconnect, while that of the DMD film under compressive strain experiences delamination and uneven surface, resulting in delayed failure.

5.5 Unique crack deflection in DMD multilayer when subjected to bending

At a bending radius of R = 3 mm, comparisons between ITO and DMD surfaces when subjected to bending are presented in Fig. 13(a) and (b). Scanning electron microscopy (SEM) image confirms many cracks cut through ITO sample. As opposed to ITO electrode, in the DMD multilayer structure, it was found that when the crack tip approaches the CuAg layer, the crack is abruptly deflected to the lateral direction and continues to propagate along the metallic and ITO interface, as shown by the SEM in Fig. 13(a), creating a very unusual step-like crack. Such a unique deflection in crack trajectory can be interpreted as the competition between the direction of maximum mechanical driving force and the weakest structural pathway [226]. It is assumed that the ductile layer of the thin film metal can greatly improve the mechanical properties of the whole DMD structure due to internal interfaces. Specifically, from a mechanical perspective, cracks in nominally brittle materials follow the path of maximum strain energy release rate. In the crack propagation process, the middle CuAg layer plays an important role in preventing rapid crack propagation as it absorbs energy through large shear deformations and strain energy transfers to plastic deformation. This results in the step-like fracture can be seen only in the DMD multilayered structure. Both He and Hutchinson’s crack-deflection mechanics solution [227–229] and FEA simulation results further illustrate this phenomenon.

 

Fig. 13. SEM images were taken at bending radius (R = 5 mm) (a) The path of the cracks meanders alternately along with CuAg interfaces, before cutting through bottom ITO layer (b) In ITO sample, there is only penetrated crack when subjected to bending. Scale bar = 500 nm. (c) a crack impinging onto a bi-material interface (d) and (e) He and Hutchinson provide a quantitative analysis tool to predict whether an impinging crack will penetrate through whole layer or deflect along with a linear elastic bi-material interface. Schematics of different crack propagation modes on a bare ITO structure (d), and on a DMD structure on the flexible PET substrate (e). FEA results compare between the multilayer structure with specific metallic layers, and pure ITO structures when subjected to bending (f) Pure ITO structure under bending radius, R = 5 mm. Strain level was increased due to the conventional beam bending theory (0.53% to 1%). (g) However, if we insert metal layer in between, strain level is decreased compared to pure ITO case. At bending radius, R = 5 mm, strain level of outer layer is 0.82%. Reproduced from Ref. [124] by permission of copyright 2022 American Chemical Society.

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The outcome fully depends on the elastic mismatch across the interface and the ratio of fracture toughness of the interface [230–232]. Deflected crack is the more robust mechanism to prevent rapid electrical failure compared to penetrated crack. From an electrical perspective, a vertically penetrated crack serves as the worst-case scenario as it severely damages the conducting material, which leads to abrupt electrical disconnection due to rapid crack propagation. In the presence of a thin metal layer, the crack diverts its propagation direction so that an overall conduction path can still connect, and therefore can effectively prevent catastrophic conduction failure in the DMD stack structure. The toughening mechanism of DMD structure is mainly associated with two-dimensional deflection of cracks that bypasses the thin metallic CuAg layer. The thin metal layer serves to dissipate strain energy from the occurrence of plastic deformation, greatly decelerating crack growth. In the metallic layer, bond rupture plays a crucial role in resisting crack growth, because fracture energy is highly associated with plastic deformation near the crack tip [233,234]. Moreover, the transverse sectional plane of the metal layer largely deformed which resulted in transferring of the bending stress to the shear stress [235–237]. As displayed in Fig. 13(f), even though the DMD and the pure ITO have identical thickness, the properties of the strain levels differ significantly. When the pure ITO structure was subject to bending, the plane cross section for the entire stack remained planar and it showed only a single neutral plane after deformation. In contrast, as presented in Fig. 13(g), the multiple neutral planes appear by inserting a ductile metallic layer in between the two ITO layers, and the ITO layers bent individually during bending are not coupled rigidly because the transverse sectional plane of the metallic layer deforms to accommodate the sheer stress. Multiple neutral planes allow the transfer of strain energy to the middle layer such that the maximum bending strain could be reduced with increased compliance of the structure. The reduced strain also reduces the probability of crack initiation based on the analysis presented above.

6. Applications in optoelectronic devices

6.1 Use of DMD in organic optoelectronic devices

As mentioned previously, the DMD structure serves as a good transparent conductor capable of replacing ITO in many optoelectronic devices. It has been widely studied and shown to function as well as ITO when used in organic light emitting diodes (OLEDs) [238–240], organic solar cells (OSCs) [241,242] and perovskite solar cells (PSCs) [243]. Some of the benefits that the DMD structure provides include: dual roles of transparent conduction and electron injection in OLEDs, waveguide mode elimination in OLEDs, and improved encapsulation for perovskite solar cells. Each of these will be discussed in the coming sections.

6.2 DMD dual role in OLEDs

DMD structures have widely been used as cathodes for top emitting, or transparent OLEDs. Typically, the dielectric materials used are ZnS and WO₃[239], however, in some cases a material can be chosen to not only fulfill the role of the dielectric, but also act as an important electrical component of the device. One such example is with Cs₂CO₃, which has been shown to be a good dielectric as well as electron injection layer [240]. In this study, an OLED with the structure of N,N'-bis(naphthalen-1-yl)-N,N'-bis(phenyl)-benzidine (NPB) as the hole transporting layer and tris(8-hydroxy-quinolinato) aluminum (Alq₃) on an ITO substrate was used. The device had various cathodes, one with Ag deposited directly on top of the Alq₃, one with Cs₂O₃ deposited between the Alq₃ and Ag, and one with Cs₂O₃, Ag and a ZnS capping layer. We can see in Fig. 14 below, that with the Cs₂O₃ between the Ag and Alq₃ the transparency is higher and sheet resistance is lower, respectively, when compared to the device without the Alq₃. This is due to the improved morphology of the Ag film as grown on the Cs₂O₃. Additionally, the benefit of Cs₂O₃ is shown in Fig. 14 below, where we see that the turn on voltage for the device is significantly lower with the Cs₂O₃. This is because the Cs₂O₃ acts as an EIL which provides much more efficient electron injection than with Ag grown directly on Alq₃. Lastly, the full effect of the DMD structure is shown as well, where the top emission has higher luminance with the presence of the ZnS acting as the top dielectric capping layer. Therefore, the DMD structure allows for the use of a bottom dielectric as an EIL and more compatible Ag growth layer, as well as a top dielectric to improve transparency, without sacrificing electrical performance.

 

Fig. 14. (a) Transmittance of a device with the structure of ITO/ NPB (50nm)/ Alq3 (50nm)/ x/ Ag (9, 12, and 15 nm) with x being either Cs2CO3 (dash) or nothing (line). (b) The scanning electron microscopy (SEM) images for the top surface of Ag electrodes in OLEDs shown in the inset in Fig. 5 either with or without Cs₂CO₃ for two different Ag thicknesses of 9 nm and 12 nm. (Reprinted with permission from Ref. [240] OPG).

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6.3 Waveguide mode elimination

Along with the capability of acting as a transparent electrode to replace ITO, the DMD structure can also be used to increase the efficiency in OLEDS. In OLEDs, internal quantum efficiencies (IQE) of 100% have been achieved, but the external quantum efficiency (EQE) has still been limited due to the difficulty in outcoupling the light generated within the device into air [244]. The limiting factors are the different modes that the emitted light couples into within the device: the surface plasmon polariton (SPP) mode, the substrate mode, and the waveguide mode. Light trapping in the SPP and substrate modes can be improved by using a thick electron transport layer (ETL) [245] and a microlens array [246], or other light scattering method [247], respectively. The waveguide mode, however, is much more difficult to eliminate. The waveguide mode comes from the organic layers and transparent anode of the device acting as a waveguide, which traps the light inside, and is inherent to the device structure [244,248]. There have been efforts to extract the light from the waveguide mode, such as using patterning or grid structures, but these not only add complexity and fabrication costs, but also often protrude into the organic layers negatively affecting the surface smoothness important for OLEDs and their operation lifetime [247,249].

In order to better solve this issue, ultra-thin Ag based transparent conductors have been used to replace ITO in a polymer OLED that showed higher efficiency [154]. More recently a DMD structure as a transparent anode has also been shown to eliminate the waveguide mode in an archetypical OLED structure [250,251]. This is illustrated in upper panel of Fig. 15, with the device structure (Fig. 15(a)) and the effective refractive index (n_eff) of the organic layer stack including the transparent conductor as a function of the Ag metal thickness (Fig. 15(b)). In the device structure, a DMD layer made of ITO as the dielectric, and a Cu-seeded Ag layer as the metal layer is used. The DMD structure at the right metal thickness does not support the waveguide mode, whereas the ITO always supports it (shown in lower panel of Fig. 15(a) and (b) for ITO device and DMD device, respectively). This is due to the high refractive index present in the ITO, while the low-loss Ag film has the real part of its refractive index close to zero. This means that the n_eff of the device can be reduced to match that of the substrate, thus eliminating the waveguide mode.

 

Fig. 15. (upper panel) Modal analysis of an OLED with DMD. (a) Schematics of a DMD based OLED. (b) Effective index of TE₀ mode as a function of the Cu-Ag thickness with varied permittivity, with top and bottom ITO to be both 40 nm. ITO and Air in (b) indicate DMD with ITO and air permittivity instead of Cu-Ag. (lower panel) Spectral power distribution of an ITO or DMD OLED. Spectral power distribution in (a) ITO or (b) DMD device. Each waveguide mode is denoted, and k|| or k_EML is the wavenumber in propagation or transverse direction, respectively (Reprinted with permission from Ref. [250,251], Society for Information Display).

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This effect is demonstrated more clearly using Fig. 16, where the spectral power distribution of an ITO anode and Cu-Ag anode OLED device were calculated using Dyadic Green’s function [104]. While the Cu-Ag anode is not the same as the DMD structure, the thin Ag film eliminating the waveguide mode works the same in both. The device structure simulated consists of 5 nm molybdenum trioxide (MoO₃) / 40nm di-(4-(N,N -di-p-tolyl-amino)-phenyl) cyclohexane (TAPC) / 20 nm EML / 75 nm 1,3,5-Tris(1-phenyl-1Hbenzimidazol-2-yl)benzene (TPBi), where CBP was used as the emission layer (EML) calculation. We can see that the eliminated waveguide mode in the DMD OLED is funneled into the substrate mode, which is beneficial as it can be easily extracted using a microlens array or index matching fluid.

 

Fig. 16. Spectral power distribution of each mode in (A) ITO and (B) Cu-Ag OLEDs. a.u., arbitrary units. Intensity at 530 nm is the highest as it is the peak emission wavelength of Ir(ppy)2acac. Fractions of Air, Subs, W/G, and SPP are referred to the energy transferred to the air, glass, waveguide, and SPP modes, respectively. Waveguide mode was removed in the Cu-Ag device, showing zero intensity. Power distribution of each mode as functions of ETL thickness in (C) ITO and (D) Cu-Ag OLEDs. Extractable light portion (Air + Subs) increases in the Cu-Ag device due to waveguide mode elimination. Waveguide mode starts to appear in the Cu-Ag OLED at thick ETL. (Reprinted with permission from Ref. [104], American Association for the Advancement of Science)

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In addition to the simulation, this effect has been demonstrated experimentally as shown in Fig. 17 [250,251]. We can see that both devices have almost identical external quantum efficiency when no outcoupling method is used. However, when an index matching fluid is used to extract the substrate mode, the DMD structure sees a much greater increase than the ITO based device due to the light normally going to the waveguide mode being converted to the substrate mode and extracted. Additionally, the DMD OLED was made on a flexible PVA substrate, which was capable of folding over a razor blade, while in operation, which shows another huge benefit of the DMD anode over the ITO anode.

 

Fig. 17. Figure 4: Performances of ITO and DMD based OLEDs. (a) EQE – J or (b) J – V relation of ITO and DMD devices, where the substrate mode is extracted by IMF.(c) Ultrathin OLED lit up even when folded against a razor blade (Reprinted with permission from Ref. [250,251], Society for Information Display).

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6.4 DMD as encapsulation layer for perovskite solar cells

Along with OLEDs, the DMD structure has been used with both organic solar cells [241] and perovskite solar cells [243] as the top electrode. The reflective nature of thin metals in the long wavelength range help to improve the light absorption in the NIR range and the OSC efficiency [252]. Adding a dielectric layer under the thin metal electrode can function as external spacer to fine tune and optimize the light absorption by the organic semiconductor layer, further benefiting the power conversion efficiency [149]. In addition to their use as just a replacement for ITO as a transparent conductor, DMD structures have also been shown to possess additional benefits. One such case is their use as both a flexible transparent conductor, as well as an encapsulation layer for perovskite solar cells [243]. The device structure used to demonstrate this is shown below in Fig. 18 where MoO_x/Ag/MoO_x was used as the DMD. The sheet resistance of the DMD structure and short circuit current of the solar cell were tested over several day span in ambient air conditions, and it was found that the DMD structure exhibited little to no degradation when compared to a control sample of Ag grown on a layer of MoO_x and Au grown on the PSC. Further, placing the Ag layer between the two MoO_x layers helped prevent degradation due to a few reasons: First, the top MoO_x layer served as a good encapsulation layer to prevent the Ag from oxidizing in air. Second, the bottom MoO_x layer retards the corrosive reaction of Ag with the halide ions of the perovskite layer. Third, the MoO_x helps absorb UV light, which prevents photocatalytic decomposition of the perovskite material. As a result, the DMD structure is capable of not only acting as a transparent conductor, but also as an encapsulation layer allowing for the use of Ag with perovskites under ambient air conditions for extended periods of time.

 

Fig. 18. (a) Illustration of multi-functional effect of MoOx/Ag/MoOx electrode on top of perovskite solar cells, (b) a change in the sheet resistance of MoOx/Ag/MoOx and Ag/MoOx as a function of age under ambient conditions (20 ± 3 °C, 45 ± 5% relative humidity), (c) stability test of PSCs with MoOx/Ag/MoOx electrode and Ag continuing to operate at 0.8 V for top (DMD) and bottom (Au) illumination under atmosphere conditions; photograph of PSC with MoOx/Ag/MoOx and Au after operating for 24 h (upper, DMD; lower, Ag electrode). (Reprinted with permission from Ref. [243], Elsevier).

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The protection the dielectric provides in preventing the Ag from degrading the perovskite is an especially important benefit of the DMD structure. Without the DMD structure, a less reactive metal, such as Au, would typically need to be used as the electrode for the perovskite in order to prevent the electrode from forming a redox couple with the perovskite material and degrading the device [253]. The issue is that this increases the cost of fabrication, and so the use of the DMD structure allows for cheaper metals, like Ag, to be used instead for the electrode.

7. Conclusion and outlook

Ultrathin metal films have attracted intensive research interests as a potential candidate for flexible transparent conductors. Their excellent optoelectrical properties with robust mechanical flexibility when used in conjunction with dielectric coatings provide unique benefits that makes them stand out among other emerging transparent conductor candidates. Good understanding of the metal film's intrinsic optoelectrical properties at the ultrathin regime and its association with physical morphology of the film is the prerequisite for designing the film as a high-performance transparent conductor. Also, combining the film with dielectric coatings can enhance not only optical transmission but also enhance robustness in mechanical flexibility.

This review paper summarizes key design considerations of metal film-based transparent conductor technology from an optical, electrical, and mechanical perspective. First, we reviewed how optoelectrical properties of metal film evolves at the ultrathin regime. Theoretical models were introduced to explain the scaling behavior of dielectric function and electrical resistivity of the metal film near the critical thickness, also a thickness below which the film transitions from a homogeneous to inhomogeneous film. Below this critical thickness, the film’s optoelectrical properties significantly deteriorate in which one needs to carefully choose the film thickness to be at or above this thickness to ensure its high figure-of-merit as a transparent conductor. Surface engineering methods to reduce the percolation threshold of the metal film, hence critical thickness, was also introduced as an approach to further enhance the figure-of-merit. Also, findings from resistivity modeling studies have shown that grain boundary scattering is the dominant mechanism to deteriorate optoelectrical properties of the intrinsic metal films. Methods to reduce the metal film’s grain boundary contribution were discussed in terms of the film’s morphology engineering by using annealing method or novel growth methods. Yet the empirically obtained electrical or optical scattering time of silver thin films is usually lower than theoretical value calculated from first-principle methods. This states that there is still ample opportunity and room for improvement to enhance intrinsic property of metal film via post-treatment or novel growth methods to develop in the future. In addition to improve the intrinsic optoelectrical performance of the metal film itself, anti-reflective coating strategies to further enhance optical transmission of the metal-film stack were also discussed for its application as a transparent conductor. Optical design considerations of this optical stack were not limited to transparency at visible wavelength but was extended to the NIR range which is important for solar energy applications. We also briefly highlighted the recent research interest in deep learning to solve challenging optical and photonic inverse design problem, which can be a new research opportunity if applied to solve optical problems in metal-film-based transparent conductor design.

We also reviewed the mechanical flexibility and stability of thin metal film based transparent conductor, which represents an ever-increasing area of application. Because metallic thin is malleable, it makes the crack propagation more difficult as compared with the traditional ITO conductor. Outstanding cyclic bending results obtained for the DMD structure indicated that the thin metallic layer deposition between ITO layers can partially absorb the bending deformation energy by shear deformation of metal layer. This hierarchical multilayer structure can alter the direction of crack propagation and delay electrode degradation. The crack deflection mechanism in the DMD structure by sandwiching a thin metallic layer between two brittle dielectric layers suggests potential strategies for overcoming the relatively low fracture toughness of fragile materials through brittle/ductile hybrid structure. We hope the methods and analysis reviewed in this paper can expedite finding new paths to develop next-generation flexible electronics. Finally, we briefly discussed unique benefits that metal-film-based transparent conductor provides for optoelectronic device applications including eliminating the waveguide mode in OLEDs which is unachievable by conventional transparent conductors like ITO. In conclusion, further improvement in intrinsic optoelectrical properties of the metal film or thin film stack design improvement to tailor light transmission or enhance robustness of mechanical flexibility can motivate the research community to exploit and develop metal-film-based transparent conductors for various optoelectronic device or photonic applications.