1. Introduction

Transparent conductive layers (TCLs) are an inevitable component of various opto-electronic devices: liquid crystal and quantum dot displays, light emitting diodes, solar cells, touch sensors and smart windows [1–5]. Novel opto-electronic devices demand specific properties of TCLs such as flexibility, accessibility, ease of fabrication and low cost [6–10]. Recent studies demonstrate that nanopatterned metallic layers possess all the above mentioned properties, as well as opto-electronic performance exceeding indium-tin-oxide (ITO), which is the most used TCL today [11, 12]. Moreover, surface plasmons, which exist at the interface between metal and dielectric at certain wavelengths, can significantly improve the optical properties of the metallic nanopatterns and/or its surrounding material [13–18]. These facts demonstrate the attractiveness of the nanopatterned metallic layers for the next generation of TCLs.

Two common types of the nano-patterned layers, which are nano-pores (NPs) and nano-wires (NWs), were investigated for being used as TCLs [19–24]. Interpore/interwire distance, pore/wire diameter and thickness of the patterned layer can be modified to obtain various application specific transmittance and sheet resistance. For example, the transmittance of 97% and the sheet resistance of 3 Ohm/sq was demonstrated in case of silver NW TCLs [12]. However, it is still unclear which type of the nano-patterned layers can possess better opto-electronic performance. In this paper, the theoretical comparison of the opto-electronic properties between NP layers with square and hexagonal arrangement of the pores and uniformly arranged NW layers for aluminium and silver is demonstrated. We show that NW configuration exhibits higher average transmittance in visible wavelength range than square and hexagonal NP designs.

2. Methodology

Figure 1 shows the geometrical models for NP and NW layers on glass substrate. NPs were arranged squarely (NP_sq) and hexagonally (NP_hex) with interpore distance a and diameter of pore d [Figs. 1(a) and 1(b) respectively], while NWs were uniformly arranged with the interwire distance a and width of wires w. The simulation area was narrowed to the unit cell, which size along X and Y axes was set to the interpore/interwire distance a for the squarely arranged NP and NW layers. In case of the hexagonally arranged NP layers the unit cell was set to a and a × $\sqrt{3}$ along X and Y axes, respectively.

 

Fig. 1 Geometrical models of the following distributions: squarely (a) and hexagonally (b) arranged NPs with interpore distance a and diameter of pores d; (c) uniformly arranged NWs with the interwire distance a and the width of wires w. Red rectangles are the unit simulation cells, which are equal to a² for the NPs with square arrangement and NWs and a² × $\sqrt{3}$ for NPs with hexagonal arrangement, respectively.

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The optical properties were simulated using the finite-difference time-domain method (FDTD) which is commercially available within Lumerical software [25]. The incident light ranged from 300 to 900 nm was distributed along Z axis. The periodic boundary conditions were applied along X and Y axes to simulate a periodic structure in XY plane. While the periodic boundary condition was also applied along Z axis to simulate the “air/layer/glass” structure along Z axis and the perfectly matched layers were applied parallel to XY plane. Mesh size for metallic layer was set to 10, 10 and 5 nm in X, Y, and Z directions, respectively.

The sheet resistance is calculated by percolation model in accordance to [26–28], which is given by the following equation:

3. Results and discussion

The distinction of opto-electronic properties between uniformly and randomly arranged NP layers was previously studied by our group in [24]. It was shown that layers with uniform arrangement of NPs possess higher transmittance (up to 15%) and lower sheet resistance (down to 12 times) compared to random configuration of NPs. Therefore, the randomly arranged NP layers were excluded from the current paper. The thickness of NP and NW layers from 10 to 100 nm is typically implemented within experimental studies [20, 29–34]. We set the height (h) of the NWs to 60 nm. The shortest distance between edges of two nearest NPs and the width of NWs, which are equal to the distance w in Fig. 1, was set to 60 nm and was kept fixed for all simulations, while the interpore/interwire distance a was varied from 8 μm to 80 nm. Such interpore/interwire distance was chosen in order to analyze the opto-electronic properties for two regions: interpore/interwire distance less than (i) and larger than (ii) incoming wavelengths. The decrease of a, while keeping w fixed, results in decrement of open area. For instance, the open areas for NW layers with a of 8 μm and 80 nm are equal to 98.51 and 6.25%, respectively. The concentration of NPs and the concentration of NWs were calculated per area of 8 × 8 μm², where the minimum amount of NPs (one) and NWs (two: one along X and one along Y axes) is at a = 8 μm and the maximum amount of NPs (100) and NWs (200: 100 along X and 100 along Y axes) is at a = 80 nm.

Figure 2 shows the influence of the concentration of NPs (C_NP) and concentration of crossings of NWs (C_NWcrs) on the transmittance of Al NP and NW layers for the wavelength range from 300 to 900 nm. The average transmittance of squarely arranged Al NP layers in visible spectrum decreases from 75 to 2.25% when the concentration of NPs increases from 1 to 50, while the hexagonal arrangement of NPs possesses higher average transmittance, which decreases from 85 to 2.5% when C_NP increases from 1 to 50. Al NW layers outperform NP layers and demonstrate the average transmittance ranging from 95 to 5% when C_NWcrs increases from 1 to 50. The maximum difference of the transmittance between NP_sq and NP_hex against NW layers, which is 20 and 10% respectively, is observed at the concentration of NPs and NW_crs equal to one. This difference decreases with the increment of concentration of NPs and NW_crs and becomes insignificant at C_NPs = C_NWcrs = 50 (less than 2.5%), which is explained by the decrement of open area. Dips of the transmittance in range from 300 to 400 nm are related to localized surface plasmons (LSPs) which are oscillations of electrons inside NP/NW layers along the distance between edges of two pores and in crosswise direction to the individual NWs [11]. Another dip of the transmittance from 700 to 800 nm is due to interband electron transition in Al [35]. Continuous decrement in the transmittance from 450 to 850 nm is attributed to surface plasmon polaritons (SPPs), which excite and propagate along the NWs and metal tract between the neighboring rows of NPs [11]. Specific dip of the transmittance is observed only for NP layer in range from 560 to 620 nm, which can be explained by interference between LSPs traveling in opposite directions along the distance between edges of two pores.

 

Fig. 2 Dependence of the transmittance of Al NP and NW layers on the concentration of NPs with square (a) and hexagonal (b) arrangement and the concentration of crossings of NWs (c) for wavelength range from 300 to 900 nm. Note: one NW_cr requires two NWs, two NW_crs require four NWs, etc.

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Figure 3 shows the influence of the concentration of NPs and concentration of crossings of NWs on the transmittance of Ag NP and NW layers for the wavelength range from 300 to900 nm. The average transmittance of squarely arranged Ag NP layers in visible spectrum decreases from 75 to 20% when the concentration of NPs increases from 1 to 50, while the hexagonal arrangement of NPs possesses higher average transmittance, which decreases from 86 to 23.2% when C_NP increases from 1 to 50. Ag NW layers outperform NP layers and demonstrate the average transmittance ranging from 95 to 32.7% when C_NWcrs increases from 1 to 50. The maximum difference of the transmittance between NP_sq and NP_hex against NW layers is observed at the concentration of NPs and NW_crs equal to one, which is same as in case of Al layers (see Fig. 2) and is 20 and 10% respectively. This difference decreases with the increment of concentration of NPs and NW_crs and becomes equal to 12.7 and 9.5% between NP_sq and NP_hex against NW, respectively, which is up to 27% higher in comparison to Al layers. This can be explained by higher quality factor of surface plasmons for Ag NP/NW layers, which increases the momentum matching between wavevectors of photons and SPPs, resulting in enhanced transmittance. Dips of transmittance for Ag NP/NW layers can be explained in the same manner as for Al NP/NW layers: (i) dips of the transmittance in range from 300 to 450 nm are related to LSPs; (ii) continuous decrement in the transmittance from 450 to 850 nm is attributed to SPPs; (iii) specific dip of the transmittance in range from 560 to 620 nm related to the interference between LSPs traveling in opposite directions along the distance between edges of two pores. There is no dip of the transmittance from 700 to 800 which is related to interband transition only for Al.

 

Fig. 3 Dependence of the transmittance of Ag NP and NW layers on the concentration of NPs with square (a) and hexagonal (b) arrangement and the concentration of crossings of NWs (c) for wavelength range from 300 to 900 nm. Note: one NW_cr requires two NWs, two NW_crs require four NWs, etc.

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Figure 4 illustrates the dependence of the sheet resistance against the average transmittance in visible wavelength spectrum for Al (left) and Ag (right) NP/NW layers. Significant difference between Al NP and NW layers becomes evident since the sheet resistance equals to 3 Ohm/sq. In case of Ag NP and NW layers this difference takes place at around 1.5 Ohm/sq, which is explained by higher bulk conductivity σ₀ of Ag: 6.3 × 10⁷ S/m versus 3.5 × 10⁷ S/m for Al. NW layer possesses the transmittance higher by 10 and 20% than NPs layers with hexagonal and square arrangement, respectively. For instance, at the sheet resistance of 5 Ohm/sq the transmittance of NW layer is 90%, while hexagonally and squarely NP layer have the transmittance of only 80 and 70%, respectively. It can be explained by the larger open area of NW layers in comparison with NP arrangement, which results in higher influence of LSPs and SPPs on the transmittance rather than the absorbance and reflectance of NW layers. Interestingly, the values of the transmittance are almost similar for both Al and Ag at the sheet resistance larger than 20 Ohm/sq. This effect takes place when interpore/interwire distance becomes longer than the wavelength of the incoming light (in our case ≥ 800 nm), resulting in insufficient influence of LSPs and SPPs on the transmittance. Another behavior is observed at the interpore/interwire distance ≤ 800 nm: the transmittance of Ag layers is higher by ~20 and ~30% in comparison with Al layers for NP and NW, respectively. Thus, metals with higher quality factor of the surface plasmons possess better transmittance for the nanoscale interpore/interwire distance of NP/NW layers.

 

Fig. 4 Sheet resistance versus average transmittance in visible wavelength spectrum for Al (left) and Ag (right) NPs/NWs layers.

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4. Conclusion

The theoretical comparison of the optical and the electronic properties between the squarely and hexagonally arranged NP layers and the uniformly arranged NW layers of Al and Ag was presented. NW configuration possesses 20 and 10% higher average transmittance in the visible spectrum in comparison to the square and hexagonal NP designs, respectively. The difference of the transmittance for Al and Ag NP/NW layers is insignificant at the interpore/interwire distance larger than wavelengths of incoming light. This difference becomes considerable at the interpore/interwire distance of subwavelength range, resulting in up to 27% higher transmittance of Ag NP/NW layers in comparison to Al ones, which is due to the stronger quality factor of the surface plasmons for Ag. The given results grant the opportunities for more detailed analysis of the type and material of the nano-patterned transparent conductive layers chosen for different optoelectronic applications, such as displays and solar cells.