Solar cell design using graphene-based hollow nano-pillars

In this paper, the full solar spectrum coverage with an absorption efficiency above 96% is attained by shell-shaped graphene-based hollow nano-pillars on top of the refractory metal substrate. The material choice guarantees the high thermal stability of the device along with its robustness against harsh environmental conditions. To design the structure, constitutive parameters of graphene material in the desired frequency range are investigated and its absorption capability is illustrated by calculating the attenuation constant of the electromagnetic wave. It is observed that broadband absorption is a consequence of wideband retrieved surface impedance matching with the free-space intrinsic impedance due to the tapered geometry. Moreover, the azimuthal and longitudinal cavity resonances with different orders are exhibited for a better understanding of the underlying wideband absorption mechanism. Importantly, the device can tolerate the oblique incidence in a wide span around 65°, regardless of the polarization. The proposed structure can be realized by large-area fabrication techniques.


Results and discussions
In the following sections, the design procedure and performance analysis of a graphene-based solar cell is investigated in detail. After exhibiting the absorption capability of graphene material in the solar spectrum, by wideband impedance matching and transmission elimination, a full-spectrum solar cell is proposed and its performance is discussed employing multiple simulations.
Material choice. Let us initially investigate the equivalent permittivity of graphene material in the solar spectrum. The solar radiation covers the wavelength range of 295-2500 nm (~ 100-1200 THz), comprised of ultraviolet, visible, and infrared light 9 . In the infrared regime, graphene is commonly modeled by its surface conductivity calculated based on Kubo formulas. The outstanding feature of graphene in the plasmonic state is being reconfigurable through its chemical potential and relaxation time 23 . In this band, graphene surface conductivity can also be converted to the equivalent bulk permittivity with the negative real part using the Ampere's low 24 . In the visible spectrum and beyond, graphene behaves as an ordinary dielectric. Thus, the Drude Lorentz model can be used to approximate the monolayer graphene dispersive permittivity using the measured data 25 . Hence 26 : where m = 3 , ε ∞ = 1.964 , �ε j = (6.99, 1.69, 1.53) , Ŵ j = (7.99, 2.01, 0.88) eV , ω gra = 6.02 eV , γ gra = 4.52 eV , � j = (3.14, 4.03, 4.59) eV . Figure 1a shows the real and imaginary parts of graphene permittivity in the solar spectrum, where relatively high real and imaginary parts are observed. In the next step, the attenuation constant of the illuminating wave to a graphene slab is studied to investigate whether the electromagnetic wave can penetrate it or not. The attenuation constant (Neper/meter) reads as 27 : The parameters a and b in (2) are defined as: Fig. 1b confirms, the graphene material has a large attenuation constant in the solar spectrum, making it suitable for the absorber design. Where prime and double prime respectively denote the real and imaginary parts of the permittivity (ε) and permeability (μ). Graphene is a non-magnetic material, therefore the real and imaginary parts of its permeability are respectively one and zero.
Proposing the device and performance analysis. The unit cell of our proposed absorber for the full coverage of the solar spectrum is illustrated in Fig. 2. This geometry is constructed by a refectory metal substrate, where its surface is covered by high aspect ratio graphene-based shell-shaped hollow nano-pillars. The height of the substrate and nano-pillars are considered h 1 = 100 nm and h 2 = 1600 nm, respectively. The dispersive permittivity of TiN refractory metal is extracted from 28,29 and that of the graphene is based on (1). The graphene nano-pillar is with a bottom radius of R 1 = 120 nm and a top radius of R 2 = 40 nm. Moreover, the periodicity www.nature.com/scientificreports/ is T = 2R 1 + p = 240 nm (p = 0) and the thickness of the graphene shell is t g = 10 nm. The dispersive permittivity introduced in (1) is used for the layered graphene as well 30 . Note that 3D nano-pillars can be fabricated by hole-mask colloidal lithography 31 . The method can also be used for the large-scale fabrication of shaped high index dielectric patterns 32 . Tape-assist transfer and spin coating can be used for wrapping graphene on curved surfaces [33][34][35] . Moreover, the core template can be removed by immersing it in hydrofluoric acid (HF) solution with stirring to reach ultrathin-shell graphene hollow particle 36 . As Fig. 3a shows, the achieved absorption rate is above 96% in the whole solar spectrum, being above 99% beyond 247 THz. Included in the figure are the reflectance and transmittance curves, showing a negligible amount of them. Moreover, the retrieved surface impedance is exhibited in Fig. 3b. The results are provided based on the simulated scattering parameters via 37 : Based on the figure, wideband impedance matching is observed in this spectrum. The results of the above figures show that the underlying mechanism of the absorption is wideband impedance matching and transmission blockage 38 . The geometrical parameters are comparable with the TiN-shell based nano-pillar solar cell 31 .
The spatial components of the electric field at three different frequencies (begin, middle, and end of the spectrum) are provided in Table 1. As the table indicates, azimuthal and longitudinal cavity resonances with  www.nature.com/scientificreports/ different orders are responsible for the high absorption rate. The field distribution at the lower frequency edge shows that the height of the cone can be tuned based on the minimum frequency requirement. Note that although high aspect ratio micro-nano structures are realizable 39 , the impact of the height on optical absorption is further discussed in the last section to reach geometrically compact devices for high-frequency operation. The sensitivity of the absorption to the incident angle of the radiating wave is shown in Fig. 4 for TE and TM waves. The device has a polarization-insensitive response due to its four-fold symmetry 40 and the absorption rate above 90% can be attained for angles up to 65°. Figure 5a shows the absorption rate of the proposed solar cell when the core is made of SiO 2 material with a refractive index of 1.45 9 . Based on the figure, the hollow nano-pillars have better low-frequency performance. Absorption rate enhancement by decreasing the core permittivity is expected since it is equivalent to the decrease Table 1. Spatial distribution of the electric field at three different frequencies.

Parametric analysis. The optical absorption of the proposed device with changes in different geometrical
parameters is studied in this section. Initially, let us consider the impact of the thickness of the graphene shell on the absorption rate. The reported absorption manipulation by the increase in the number of wrapped graphene layers around a spherical resonator exhibit different performance for solid silver and dielectric-metal core-shell resonators, respectively increasing and decreasing for thicker shells 30 . As Fig. 6a shows, the mono/few-layer graphene material has a low absorption rate at low frequencies. Thus, the presence of around 10 graphene layers, each with a thickness of 1 nm, is essential for the full spectrum coverage in the proposed device. Note that graphene is essentially a 0.34 nm monolayer of carbon atoms, but inhomogeneities during the fabrication process may result in thicker graphene sheets. For CVD-grown graphene material, treating the monolayer graphene as 1 nm thick effective material is a reasonable choice 46 . Using few-layer graphene films in the optical design, cost, scalability, and ultra-broadband absorption features can be attained but the effective transport of the photothermal energy generated on the surface and the possibility of performance manipulation become limited 11 . Moreover, Fig. 6b investigates the influence of inter-particle distance in the performance and shows that nearfield coupling plays a crucial role in the absorption enhancement at lower frequencies since by increasing the distance between the nano-pillars the low-frequency performance is highly affected. The absorption rate of the proposed solar cell for different cone heights is investigated in Fig. 7. For covering the lower frequencies, the height of the cone should be increased. Since higher frequencies have more energy, the height can be shortened by partially covering the higher frequencies to reach a compact device. With the height of 500 nm (substrate thickness 100 nm), a 90% absorption rate for the frequencies beyond 300 THz can be achieved. The same performance is attained with metal-coated moth-eye films with the same geometry. The cone height and substrate thickness are respectively around 200 nm and 5 μm in this device 47 .
Finally, the morphology-dependent behavior of the proposed structure is investigated in Fig. 8. Notably, tapering of individual nano-objects is regarded as a promising strategy to engineer the far-field and near-field optical response of nanostructured metamaterials [48][49][50][51] . However, since the nanowire system is relatively easy to realize and control, two simpler geometries are analyzed in the following. Specifically, in Fig. 8a the pyramidal nano-pillar is converted to a cylindrical cavity with the same height and shell thickness and the radius of the cavity is varied from 40 to 120 nm. The absorption rate is large at low (high) frequencies for large (small) radii, as expected. Thus, the absorption rate can be sacrificed to attain a simpler geometry. For the intermediate radius of 80 nm, the absorption rate above 90% in the solar spectrum is attained. Moreover, illustrated in Fig. 8b is  www.nature.com/scientificreports/ the absorption rate of the same structure by converting it to a hollow tube. It is observed that by increasing the tube radius, the absorption increases as well, being above 95% in the solar spectrum for the radius of 120 nm.
To further illustrate the priority of the pyramidal geometry, the absorption rate of the pyramidal geometry is compared with those of cylindrical cavity (with the radius 80 nm) and cylindrical tube (with the radius 120 nm) in Fig. 8c. Moreover, the substrate thickness of the pyramidal geometry is varied in Fig. 8d, to confirm that it is chosen properly. Increasing the substrate height beyond 40 nm has a negligible impact on the absorption rate.

Conclusions
With the combined use of shell-shaped graphene-based hollow nano-pillars and a refractory metal substrate, a full-spectrum solar cell is designed. These two sections are respectively responsible for impedance matching and transmission blockage, leading to efficient energy absorption. Moreover, the performance is further explained by exhibiting the azimuthal and longitudinal cavity resonances with different orders in the spectrum. The polarization in-sensitive absorption of the solar cell is maintained up to 65°. Moreover, it is observed that tip truncation and using hollow particles are two key factors for the efficient absorption of the low-frequency waves. The proposed device is heat tolerant and environmentally robust and can be possibly realized by the current fabrication technology.

Methods
The unit cell analysis of CST software is used for simulation 52 . The Floquet ports with two prorogating modes are considered during the simulations. The absorption rate (A) of the structure is calculated using the simulated reflectance (R) and transmittance (T) as 37 :