Light Management in Single Rectangular Silicon Nanowires for Photovoltaic Applications

Light management in single nanowires (NWs) is of great importance for photovoltaic applications. However, square NWs (SNWs) can limit their light-trapping ability due to high geometrical symmetry. In this work, we present a detailed study of light management in single silicon NWs with a rectangular cross-section (RNWs). We demonstrate that the RNWs exhibit significantly enhanced light-harvesting compared with the SNWs, which can be attributed to the symmetry-broken structure that can orthogonalize the direction of light illumination and the leaky mode resonances (LMRs). That is, the rectangular cross-section can simultaneously increase the light path length by increasing the vertical side and reshape the LMR modes by decreasing the horizontal side. We found that the light absorption can be engineered via tuning the horizontal and vertical sides, the photocurrent is significantly enhanced by 276.5% or 82.9% in comparison with that of the SNWs with the same side length as the horizontal side of 100 nm or the vertical side of 1000 nm, respectively. This work advances our understanding of how to improve light-harvesting based on the symmetry breaking from the SNWs to RNWs and provides an effective way for designing highefficiency single NW photovoltaic devices.


Introduction
Single nanowire (NW) solar cells have increasingly attracted attention in recent years serving as powering nanoscale devices [1][2][3][4][5][6][7][8]. Light management in single NW solar cells is very important for ensuring both high absorption and little material [9][10][11][12][13][14]. Therefore, light management is an effective way to improve light absorption and enhance the photoelectric conversion efficiency of single NW solar cell. It is well known that the strong interaction between the incident light and a single NW has been applied to dramatically increase light trapping due to the leaky mode resonances (LMRs) [15][16][17]. However, the overall light-trapping performance of a single NW is still far below expectations owing to the narrow resonant peaks.
In this work, we carry out detailed investigations on the light-trapping effect of single RNWs. We demonstrate that the giant enhancement of the light absorption occurs when the RNWs replace the SNWs. The detailed analysis of the electric field, absorption mode profile and photogeneration rate shows that this enhancement is mainly attributed to the asymmetry breaking from the SNWs to RNWs. Specifically, the light path length can be increased by increasing the vertical side and the LMR modes can be reshaped by decreasing the horizontal side. Simulation results reveal that the photocurrent is significantly enhanced by 276.5% or 82.9% compared with that of the SNWs with the same side length as the horizontal side of 100 nm or the vertical side of 1000 nm, respectively.

Model
Figure 1 schematically shows the cross-sectional views of a RNW and two SNWs. The horizontal (x) and vertical (y) axes of the RNWs are denoted by a and b. The horizontal (or vertical) side is perpendicular (or parallel) to the light illumination direction, as presented using the colorful arrows in Figure 1. Note here that the unpolarized light (e.g., sunlight) can be expressed as the average of transverse electric (TE, electric field normal to the NW axis) and transverse magnetic (TM, magnetic field normal to the NW axis) light. The values of a and b are chosen to be 300 and 100 nm as the typical representative nanoscale size, respectively. It should be noted that the SNWs are also shown for comparison, where the side lengths of the SNWs are chosen to be 300 and 100 to investigate the improved light absorption performance due to the reshaped LMRs by decreasing a and the increased light path length by increasing b. Silicon is chosen as a typical semiconductor material and its wavelength-dependent refractive index is adopted from the experimental data [58]. Figure 1. Schematics of the cross-section of a rectangular nanowire (RNW) and two square nanowires SNWs). The geometrical metrics used to describe the geometry of the RNW, the horizontal (x) side a and the vertical (y) side b, are presented. A representative value of a and b of the RNW are chosen to be 100 and 300 nm, respectively. The side lengths of the SNWs are chosen to be 300 or 100 nm for comparison, the material of the RNW is set to be silicon as a representative semiconductor and the light illumination is perpendicular to the axis of the RNW from above. Note that the unpolarized light (e.g., sunlight) illumination can be regarded as the average of transverse electric (TE, electric field normal to the NW axis) and transverse magnetic (TM, magnetic field normal to the NW axis) light illumination.
3 of 13 co-workers for details [22][23][24]30]. In this simulation, the RNW is illuminated normally by sunlight from the top, the wavelength range of the incident light is from 300 to 1100 nm with a step size of 5 nm considering solar radiation and the bandgap of silicon, the perfectly matched layers (PML) boundary conditions are considered to avoid any non-physical reflection with the boundaries, the total-field scattered-field (TFSF) method was applied to ensure that a single NW interacts with an infinite plane wave. Also, the minimum cell size of the FDTD mesh is set to from 0.05 to 5 nm corresponding to b (from 5 to 1000 nm) to guarantee the accuracy of the simulation results.

The Normalized Electric Field (Er)
The normalized electric field (Er) can be expressed as [37]: where E is the electric field of the RNWs, which is obtained by FDTD numerical simulation, and E0 is the electric field of the incident light, respectively.

The Absorption Mode Profile (Pabs)
The wavelength-dependent absorption mode profile (Pabs) calculated from the Poynting theorem can be regarded as [32,39,44,62]: where ω is the angular frequency of the incident light and ε" is the imaginary part of the permittivity of silicon, respectively.

The Absorption Efficiency (Qabs)
To qualify the light absorption performance of the RNWs, we define the absorption efficiency (Qabs) as [32,39,44,62]: where Cgeo is the projected area per unit length of the RNWs and Cabs is the absorption cross-section per unit length obtained by, where k0 is the wave vector in air, r   is the imaginary part of the relative permittivity of silicon, x and y are the coordinate axes shown in Figure 1, and I0 is the solar incident light intensity expressed as [32,39,44,62]: where c is the speed of light and ε0 is the permittivity in air, respectively.

The Photogeneration Rate (G)
The spatially dependent photogeneration rate (G) is readily calculated by [63,64]: where ħ is the reduced Planck's constant and λ is the wavelength of the incident light. It should be impressed that when using Equation (7), each photon absorbed in the RNW contributes to the photocurrent without considering recombination losses.

The Ultimate Photocurrent (Jph)
The overall light absorption performance is evaluated using the ultimate photocurrent (Jph) calculated by: where q is the elementary charge and Γ is the AM1.5G standard solar photon flux density spectrum. It should be noted here that 100% collection efficiency is assumed, which has been widely employed to evaluate the ultimate photocurrent [16,64].

The Photocurrent Enhancement Factor (PEF)
The photocurrent enhancement is evaluated by employing the photocurrent enhancement factor (PEF) using the relation: where Jph,RNWs and Jph,SNWs are the photocurrent density for the RNWs and SNWs, respectively.

Light absorption Mechanism in Single RNW
To understand the light absorption mechanism responsible for the improved photocurrent of the RNW, we investigate the absorption efficiency (Qabs), ultimate photocurrent (Jph), normalized electric field (Er), absorption mode profile (Pabs) and photogeneration rate (G), respectively. Note here that a = 100, b = 300 nm and the side lengths of the SNW are chosen to 300 (SNW1) and 100 (SNW2) nm for comparison, respectively.

The Absorption Efficiency (Qabs)
To quantitatively characterize the light absorption performance of the RNW compared with the SNW, we first examine the absorption spectra calculated by Equation (3). In Figure 2, we show λdependent Qabs spectra of the RNW with a = 100 and b = 300 nm and the RNWs with a = b = 300 and 100 nm under TM, TE and unpolarized light, respectively. Firstly, Qabs of the RNW is much bigger than that of the SNW2 almost the whole wavelength range, except for 560 < λ < 615 nm near the 3rd absorption peak of the SNW2 for TM light, 430 < λ < 455 nm near the 2nd absorption peak of the SNW2 for TE light and 560 < λ < 610 nm for near the 3rd absorption peak of the SNW2 for unpolarized light, which can lead to a significant photocurrent enhancement. Secondly, Qabs of the RNW is much bigger than that of the SNW1 in the short-wavelength range of λ < λcTM ~ 490, λ < λcTE ~ 425 or λ < λc ~ 485 nm for TM, TE or unpolarized light, which can result in a significant photocurrent enhancement. In contrast, the light absorption of the RNW seems to be comparable in the long-wavelength range of λ > λcTM, λ > λcTE or λ > λc, which can lead to a little contribution to the photocurrent enhancement. Note here that λcTM, λcTE and λc are the characteristic wavelengths for TM, TE and unpolarized light, below which the light absorption is always enhanced and can be readily determined for a fixed a and b. More importantly, the Qabs spectra exhibit two strong absorption peaks in the wavelength range of λ < λcTM or λ < λcTE for TM or TE light, respectively. Specifically, the Qabs value reaches 2.71 near λ = 440 nm for TM light and 2.14 near λ = 395 nm for TE light, which results in a dramatic photocurrent enhancement. Note here that some Qabs values exceed unity, which is attributed to the fact that the absorption cross-section is bigger than the physical cross-section. All in all, the absorption results indicate the great potential of the single RNW in improving light-harvesting due to the symmetrybroken structure from the SNW to RNW.

The Ultimate Photocurrent (Jph)
To evaluate the light-harvesting of the RNW for photovoltaic applications, we then calculated the ultimate photocurrent (Jph) according to Equation (8). For a direct comparison, in the insets of the upper right corner of Figure 2, we show Jph of the RNW and SNWs corresponding to Qabs for TM, TE and unpolarized light illumination, respectively. It is observed that Jph of the RNW is much bigger than that of the SNW1 and SNW2. Jph for TM, TE and unpolarized light illumination reaches 15.07, 13.62 and 14.35 mA/cm 2 , which is 17.00%, 20.42% and 18.69% higher than that of the SNW1 (12.88, 11.31 and 12.09 mA/cm 2 ), respectively. It is worth noting here that the photocurrent enhancement is mainly attributed to the reshaped LMRs caused by decreasing a (here from 300 to 100 nm) compared to the SNW1, as discussed later. Moreover, Jph can be dramatically enhanced due to the increased light path length by increasing b (here 100 to 300 nm) compared with the SNW2. Jph is 70.5%, 121.8% and 91.6% higher than that of the SNW2 (8.84, 6.14 and 7.49 mA/cm 2 ) for TM, TE and unpolarized light illumination, respectively. The photocurrent results further indicate the huge potential of lightharvesting in single RNWs for photovoltaic applications.

The Normalized Electric Field (Er)
To understand the mechanism of light-harvesting in single RNWs, we first examine the normalized electric field (Er) calculated by Equation (1). In Figure 3, we present the Er profiles of the SNW1, RNW and SNW2 corresponding to the positions denoted by numerals in Figure 2a  It is observed that there are common characteristics of the Er profiles between the RNW and SNWs. Firstly, the improved light-harvesting of the RNW is ascribed to the excitation of the LMRs, likewise in SNW [15,16], which can confine light by multiple total internal reflections at the RNW/air interface when the wavelength of the incident light matches one of the LMRs supported by the RNW. The LMRs can be termed as TMml or TEml, where m and l describe the azimuthal mode number and the radial order of the resonances, respectively. For example, the Er profiles of the RNW in Figure  3b(3) and Figure 3e (5) show more characteristics of the TM12 and TE31 modes of the RNW, respectively. Secondly, both NW configurations show much higher Er intensities in the long-than in the short-wavelength range. For example, the Er intensities for the RNW are much higher in Figure  3e  long-than short-wavelength range, resulting in a stronger resonance. Moreover, the Er intensities of the resonant peak in the short wavelength range are much bigger inside the whole RNW owing to the excitation of more complex LMRs (for example, Figure 3c(1) for TM light, indicating the stronger interaction of incident light with the RNW, leading to a more significant light-harvesting in comparison with the SNWs.

The Absorption Mode Profile (Pabs)
To further understand the physics behind light-harvesting in single RNWs, we then examine the absorption mode profile (Pabs) calculated by Equation (2). In Figure 4, we present the normalized Pabs of the SNW1, RNW and SNW2 corresponding to the same positions denoted by numerals in Figures  2 and 3 under TM and TE light illumination, respectively. Note that Figure 4a,d show the normalized Pabs of the SNW1 for TM and TE light, Figure 4b,e show those of the RNW for TM and TE light, while Figure 4c,f show those of the SNW2, respectively. It is observed that the light absorption of the RNW in the short-wavelength range is much higher than that of the SNW2 (for example, Figure 3b,e(1)), leading to a significant photocurrent enhancement. Meanwhile, although the Pabs intensities of the RNW in the short-wavelength range is comparable with that of the SNW2, the light path length of the RNW is three times as much as that of the SNWs, resulting in a dramatic photocurrent enhancement.
It is worth noting that the match between r   and Er becomes another essential factor in evaluating the absorption in the specific wavelength according to Equation (2). For instance, although the Er intensities of the RNW at λ = λ3 = 630 nm for TM light is much larger, the corresponding r   is the smallest, which still leads to a lower absorption, while although those of the RNW at λ = λ1 = 440 nm for TM light is the smallest, the corresponding r   is much larger, which results in a more significant absorption.

The Photogeneration Rate (G)
To further confirm the physical mechanism discussed above, we show the photogeneration rate (G) calculated by Equation (6). In Figure 5, we present the normalized G profiles of the SNW1, RNW and SNW2 for TM and TE light illumination. Note that Figure 5a,d show the normalized G profiles of the SNW1 for TM and TE light, Figure 5b,e show those of the RNW, while Figure 5c,f show those of the SNW2, respectively. It is observed that the G intensities of the RNW are much greater in the whole NW than those of the SNW1 for both TM and TE light, and more absorption sites appear and fill in the whole RNW, leading to a giant photocurrent enhancement. Note that although the G intensities of the RNW are slightly smaller than those of the SNW2 for both TM and TE light, the light path length is three times as much as that of the SNW2, which results in a giant photocurrent enhancement. These results further demonstrate that this enhancement arises mainly from the excitation of more LMR modes caused by decreasing a compared to the SNW1 and enhanced light path length by increasing b. In other words, the RNW can better interact with the incident light compared to the SNW1 and SNW2, leading to a significant contribution to the photocurrent.

The ultimate photocurrent of Single RNWs
To verify that the improved light absorption is not just specific for the dimension discussed above, we calculate Jph of the RNWs with different a (from 100 to 1000 nm) and b (from 100 to 1000 nm). In Figure 6a-c, we show 2D Jph maps as a function of a and b of the RNWs for TM, TE and unpolarized light illumination, respectively. It is shown that we show Jph sharply increases with decreasing a at a fixed b, reaches its maximum at a = 100 nm, and Jph also dramatically increases with increasing b at a fixed a, reaches its maximum at b = 1000 nm. More importantly, Jph of the RNWs is always much larger than that of the SNWs at any a (< b) values. It is observed that the maximum values of Jph of the RNWs can be obtained in the length range of 100 < a < 300 and 450 < b < 1000 nm.
In Figure 6d,e, we show Jph as a function of a of the RNWs with b = 1000 nm and Jph as a function of b of the RNWs with a = 100 nm for TM, TE and unpolarized light. We then in Figure 6f,g show the photocurrent enhancement factors (PEFs) defined by Equation (9). It is observed that Jph of the RNWs increases with decreasing a at a fixed b = 1000 nm and increasing b at a fixed a = 100 nm for all polarized lights, reaches 26.90, 29.50 and 28.20 mA/cm 2 at a = 100 and b = 1000 nm, which is 81.0 %, 84.6% and 82.9% much larger than that of the SNW with a = b = 1000 nm (14.86, 15.98 and 15.42 mA/cm 2 ) and 204.3%, 380.5% and 276.5% much larger than that of the SNW with a = b = 100 (8.84, 6.14 and 7.49 mA/cm 2 ) for TM, TE and unpolarized light, respectively.
Finally, we show in Figure 6h,i the normalized G profiles of two SNWs and six RNWs for TM and TE light illumination, respectively. Note here that a =1000, 500, 200 and 100 nm at fixed b = 1000 nm and b = 1000, 500, 200 and 100 nm at a fixed a = 100 nm, respectively. As shown in this figure, with the symmetry breaking from SNWs to RNWs (a = 1000 → 500 → 200 → 100 nm) with b = 1000 nm, the LMR modes are reshaped due to the size decrease of the horizontal side, and the absorption enhancement sites better fill in the whole RNWs for both TM and TE lights, especially TE light. Meanwhile, with the decreased light path length from RNWs to SNWs (b = 1000 → 500 → 200 → 100 nm) with a = 100 nm, the photocurrent is decreased.

Conclusions
In summary, we demonstrated the effective light management from the SNWs to RNWs. The influences of the geometrical parameters of the RNWs on the light-harvesting performance were numerically investigated. It was found that the rectangular cross-section can lead to significantly improved light absorption. The examination of the spatial profiles of the electric field, absorption mode and photogeneration rate revealed that the enhancement effect resulted from the symmetrybroken structure, which can simultaneously realize the increase of the light path length by the vertical side and the reshaped LMRs by the horizontal side. The simulation results showed that the photocurrent was significantly enhanced by 276.5% or 82.9% in comparison with that of the SNW with the same side length as the horizontal side of 100 nm or the vertical side of 1000 nm, respectively. Therefore, such a RNW can be applied to various semiconductors to improve light-harvesting and provides a promising approach for the future development of high-efficiency single NW solar cells.