1. Introduction

Transparent conductive films (TCFs) are characterized by their high transmission of light and simultaneously very high electrical DC (direct current) conductivity. TCFs can be applied as transparent electrodes, which are a key component of optoelectronic devices [1–3]. There are two kinds of transparent conductive films, doped oxide (DO) films [4] and dielectric metal dielectric (DMD) structured films [5,6]. DO films, for example, indium tin oxide (ITO) and fluorine-doped tin oxide (FTO), have been investigated and applied extensively due to their low resistivity and high light transmittance in the visible region [7–10]. However, these DO films are very costly because the indium for ITO is difficult to extract, and it is difficult to dope fluorine into tin oxide for FTO. DMD films can also be called oxides metal oxide (OMO) films and show better performance than DO films in terms of their sheet resistance, mechanical flexibility, and lower cost, but they are not as optically transparent.

There are two reasons for conductivity. One is the majority carrier density, and the other is the mobility [11–14]. For metals, the mobility is small, but the carrier density is very large. For semiconductors, the majority carrier density is very low, but the mobility is very high. However, a high mobility can offer carrier transport channels (electron transport layer or hole transport layer) and is widely used in the design of optoelectronic devices.

The good electrical conductivity of DO films is due to the large carrier density provided by the doped ions and the high mobility provided by the oxide semiconductor. Additionally, DMD films can provide good electrical conductivity. Because the metal layer in a DMD enables a large carrier density, the dielectric layers (oxide semiconductors) can provide a high mobility.

Here, we theoretically propose a new kind of TCF, called a quantum tunneling transparent conductive film (QTTCF). This kind of film is constructed of multiple subnanometer layers. There are three kind of materials in the film: a metal material, high refractive index material, and low refractive index material. The good electrical conductivity of the films produced with this structure results from the quantum tunneling phenomena. It is worth noting that the transparency of the QTTCF can be regulated by the thickness of the layers.

2. Principle

Quantum mechanical effects have drawn the interest of researchers in the field of nanophotonics over the past few decades [15]. A quantum-corrected model (QCM) has been established to incorporate electron tunneling effects into the local classical formalism [16]. The quantum relationship between an oscillating field and the current can be reproduced by assigning a local effective conductivity to the subnanometer gap. Following this work, refined treatments have been proposed based on the theory of laser-assisted tunneling in metal-insulator-metal junctions [17]. Based on these theoretical and experimental results, the thickness of the insulator or gap determines the fundamental physical processes. There are three types of physical processes: tunneling processes, nonlocal processes, and local processes. However, the first two processes belong to the quantum regime, and the third process belongs to the classical regime [15]. Large thicknesses correspond to the classical regime, for which the local Maxwell equations correctly describe the physical processes. As the thickness decreases to less than a few nanometers, the system enters the quantum regime, requiring a more detailed treatment. As the thickness continues to decrease to less than one nanometer, the electron tunneling effect gradually emerges. Therefore, this kind of multilayer film can obtain good electrical conductivity because of the subnanometer insulator layers between the metal layers.

A diagram of the QTTCF is shown in Fig. 1, where the insulator and metal materials are alternatively grown on a substrate medium. Generally, the substrate medium is glass, and the incidence medium is air. To enter the quantum regime and obtain good conductivity, the thickness of the insulator layers should be on the subnanometer scale. The number and thickness of the layers are determined by the design requirements for the transparency at the reference wavelength. In the following sections, we theoretically studied the optical characteristics and detailed design of QTTCFs and investigated the influence of the incoming QCM on the transparency

 

Fig. 1 Diagram of the QTTCF.

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3. Theory and results

Table 1 shows the detailed design of the QTTCF, where the metal material is Ag, the high refractive index material is Ta2O5, and the low refractive index material is SiO2. The columns of L mean layer number; the columns of M mean material; the columns of T mean layer thickness, and the unit is nm. The incidence medium is air, and the substrate medium is glass. The materials Ta2O5 and SiO2 are insulators with low dispersity but no absorption in the broadband spectrum. Therefore, we can change the thickness and number of layers to regulate the transparency at a given wavelength. The insulator layers and metal layers overlap one another. To approach the quantum tunneling regime, the thicknesses of the insulator layers should be 1 nanometer or smaller. In the following paragraph, the optical properties (transmittance, reflectance and admittance) are studied in detail.

Table 1. The detailed design of the QTTCF at a wavelength of 510 nm.

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The green dotted and red solid lines in Fig. 2(a) are the transmittance and reflectance over the wavelength range of 300 nm to 1300 nm, respectively. The incoming radiation is vertically incident on the QTTCF. In other words, the incident angle is zero degrees. The peak value of the transmittance corresponds to the zero reflectance point. Notably, in film optical theory, the optical admittance can be adjusted to reach the zero reflectance point [18]. Admittance loci technology has been widely used in the design, analysis and monitoring of optical film coatings [19–22]. As shown in Fig. 2(b), the admittance loci originate from the refractive index of the substrate material, and the thickness of each sublayer determines the subsequent trajectory. If the destination of the admittance loci is the index of the incidence material, the optical performance can reach zero reflectance. For a transparent film, the goal is a high transmittance over a wide range of wavelengths in the visible spectrum. However, a well-designed admittance loci at one wavelength does not necessarily work well at other wavelengths because of chromatic dispersion. Therefore, we designed the admittance loci at 510 nm, which is in the middle of the visible spectrum.

 

Fig. 2 The optical properties of the QTTCF: (a) transmittance and reflectance and (b) admittance.

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Then, the transmittance of the designed QTTCF in Table 1 was studied in detail for oblique incident angles, at which the incident light can be divided into two types of polarized lights (s-polarized light and p-polarized light). First, the mean values of the s- and p-polarized cases are analyzed, as shown in Fig. 3, where the color map denotes the transmittance. Figure 3(a) shows the three-dimensional plot for changes in the transmittance (mean value), incident angle and wavelength. Figure 3(b) shows the two-dimensional plot for changes in the incident angle and wavelength. Figure 3(c) shows the two-dimensional plot for changes in the transmittance and wavelength. Figure 3(d) shows the two-dimensional plot changes in the transmittance and incident angle. However, the visible spectrum ranges from 400 nm to 700 nm. The results in Fig. 3 show high transmittance in the visible spectrum for large incident angles.

 

Fig. 3 Three-dimensional plot for changes in the transmittance (mean value of the s- and p-polarized cases), incident angle and wavelength.

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Second, the p-polarized and s-polarized cases were separately studied, as shown in Fig. 4 and Fig. 5, respectively. These results explicitly show high transmittance in the visible spectrum for large incident angles in the p-polarized and s-polarized cases.

 

Fig. 4 Three-dimensional plot for changes in the transmittance (p-polarized case), incident angle and wavelength.

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Fig. 5 The three-dimensional plot for the changes in transmittance (s-polarized situation), incident angle and wavelength.

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Finally, the conductivity of the QTTCF is due to the quantum tunneling effect, and thus, the influence of the incoming QCM on the transparency was also studied. However, only order-of-magnitude differences in the conductivity were studied because the dispersion of the conductivity can be ignored for the quantum tunneling effect. To simplify the problem, we obtain

 

Fig. 6 Transmittance and reflectance properties for order-of-magnitude differences in the electrical conductivity$\sigma$.

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Furthermore, we have also studied the whole conductivity of the QTTCF. The effects of quantum tunneling and nonlocal process affect the properties of the insulator materials, and it is embodied in the imaginary part that affects the dielectric constant. The pioneer works [15–17] have proposed theory and method to obtain the electrical conductivity of the subnanometer insulator material. Here we would use the QCM electrical conductivity of the subnanometer insulator material to obtain the whole conductivity of the QTTCF. In general, there are two kinds of resistance for a film. One resistance R_p is the current direction parallel to film surface, and the other resistance R_v is the current direction vertical to the film surface, as shown in Fig. 7(a), where the currents are i_p and i_v, respectively. These two resistances have the following expressions

 

Fig. 7 (a) Sketch diagram of the parallel current and vertical current on a thin film. (b) Current distribution of the QTTCF. (c) Resistance topological structure of QTTCF.

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In the study of TCF, the conductivity property along the direction parallel to the film surface is more important than that along the direction vertical to the film surface due to the requirements of the applications in transparent electrodes.

As shown in Fig. 7(b), the current i to be studied is along the direction parallel to the QTTCF surface, which is indicated by black arrow. Then, the detailed current distribution is described by the red arrows in Fig. 7(b). The dotted red arrows represent the quantum tunneling current along the direction vertical to the insulator layer surface, and the solid red arrows represent the current along the direction parallel to the metal layer surface. The resistance topological structure of QTTCF can be obtained from the current distribution, as shown in Fig. 7(c). The resistance R_m is the parallel resistance of the metal layer, and the resistance R_i is the vertical resistance of the insulator layer.

We can use Eqs. (2) and (3) to calculate the values of R_m and R_i. However, in order to study the conductivity, we must use the electrical conductivity of the material at low frequency, and this is different from the optical situation which is at high frequency. Take the quantum tunneling effect and the subnanometer thickness into account, the vertical resistance R_i is much less than parallel resistance R_m. Hence, the final equivalent resistance is the parallel resistance of all metal layers, and thereby is the resistance of a metal film with equal thickness of all metal layers in QTTCF, which is easy to prove. However, the parallel resistance of insulator layer is neglected in the above discussion.

The total thickness of QTTCF in Table 1 is 55.01nm, and the sum thickness of Ag layers is 7.74nm. Therefore, the conductivity of QTTCF in Table 1 is equivalent to a 7.74nm Ag film. Though the total thickness of QTTCF is thicker than one-layer metal film, QTTCF has better optical tunability. Generally, the thickness of DMD, ITO and FTO are about 80nm, 500nm and 500nm, respectively. The total thickness of QTTCF is much less than ITO and FTO, while QTTCF and DMD are about the same thickness. However, QTTCF has better optical tunability than DMD because of the much more layers.

4. Conclusion

Here, we theoretically propose a new kind of transparent conductive film, called a quantum tunneling transparent conductive film (QTTCF). The good electrical conductivity of the films utilizing this structure results from the quantum tunneling phenomena because the thicknesses of the insulator layers are subnanometer. Notably, the transparency can be regulated by adjusting the thicknesses of the layers. Lastly, the conductivity of QTTCF is due to the quantum tunneling effect, and thus, the influence of the incoming QCM on the transparency is also studied. The electrical conductivity $\sigma$ of 10³ S/m as influenced by the QCM can present good conductivity and good transparency in the visible spectrum.