Improving Absorption in Single Silicon Nanowires by Symmetry-breaking Design from Square to Rectangular Cross-section

Light absorption in single nanowires (NWs) is one of the most crucial factors for photovoltaic applications. In this paper, we carried out a detailed investigation of light absorption in single rectangular NWs (RNWs). We show that the RNWs exhibit improved light absorption compared with the square NWs (SNWs), which can be attributed to the symmetry-breaking structure that can increase the light path length by increasing the vertical side and the enhanced leaky mode resonances (LMRs) by decreasing the horizontal side. We found that the light absorption in silicon RNWs can be enhanced by engineering the horizontal and vertical sides, the photocurrent is significantly increased 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. This work provides an effective way for designing high-efficiency single NW photovoltaic devices based on the symmetry breaking from the SNWs to RNWs.


Introduction
Single nanowire (NW) solar cells have attracted more and more attention in recent years serving as powering nanoscale devices [1][2][3][4][5][6][7][8]. Improving light absorption is an effective way to enhance the photoelectric conversion efficiency of single NW solar cell. It is well known that light absorption in single NWs is significantly enhanced due to the leaky mode resonances (LMRs) [9][10][11]. However, the light absorption in single NWs is still far below expectation.
It has been shown that the light absorption could be increased by engineering the size, geometry, and orientation of the NWs [12][13][14][15][16][17][18]. We have also shown that the light absorption could be further enhanced by introducing a single shell [19,20] or graded dual shells [21]. Recently, some new strategies have been performed to improve the light absorption in the NWs based on the symmetry breaking, such as opening crescent design [22,23], off-axial core-shell design [24,25], asymmetrical nanovoid design [14], partially capped design [26][27][28][29] and crescent nanohole design [30]. More recently, we investigated the symmetry-breaking structure from circular NWs to elliptical NWs [31]. However, the light absorption in single NWs based on the symmetry-broken structure from the square NWs (SNWs) to rectangular NWs (RNWs) has rarely been explored so far.
In this paper, we present a detailed study on the light absorption in single RNWs. We show that the light absorption is enhanced when the cross-section of NWs is changed from square to rectangle. The results show the light path length can be increased by increasing the vertical side and the LMR modes can be enhanced by decreasing the horizontal side. The 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 and Methods
In Figure 1, we show schematics of the cross-section of a RNW and two SNWs. The horizontal (x) and vertical (y) axes of the RNWs are labeled by a and b. The light illumination direction is perpendicular (or parallel) to the horizontal (or vertical) side, as presented using the colorful arrows. 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, and 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 values of a and b are chosen to be 100 and 300 nm as the typical representative nanoscale size, respectively. It is worth noting 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 due to the enhanced LMRs by decreasing a and the increased light path length by increasing b. The material is chosen to be silicon and its complex refractive index is taken from the ref. [32]. The electric field inside the RNWs is calculated by two dimensional (2D) FDTD simulation by solving Maxwell's equations [33][34][35], Also, the absorption mode profile, the absorption efficiency, the photogeneration rate, and the ultimate photocurrent can be obtained [21,31]. Representative values of the horizontal (x) side a and the vertical (y) side 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 typical semiconductor and the light illumination is normal to the horizontal side 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.

Results and Discussion
In Figure 2, we show λ-dependent Qabs spectra of the SNW1 with a = b = 300 nm, the RNW with a = 100 and b = 300 nm and the SNW2 with a = b = 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 increased and can be readily fixed by 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 ascribed to the fact that the absorption cross-section is bigger than the physical cross-section. Therefore, the RNWs have huge potential in improving light absorption owing to the NW's symmetry-breaking structure from the square to the rectangular cross-section. In the insets of the top right corner of Figure 2, we show the ultimate photocurrent Jph of the RNW and SNWs for TM, TE, and unpolarized light illumination, respectively. On one hand, the value of Jph of the RNW is much larger than those of the SNW1 and SNW2. Jph 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. Note here that the photocurrent enhancement is mainly attributed to the improved LMRs due to the decrease of a (here from 300 to 100 nm) in contrast with the SNW1. On the other hand, Jph can be dramatically increased due to the improved light path length by the increase of b (here 100 to 300 nm) compared with the SNW2. Jph is 70.5%, 121.8%, and 91.6% bigger than that of the SNW2 (8.84, 6.14, and 7.49 mA/cm 2 ) for TM, TE, and unpolarized light illumination, respectively.
In Figure 3, we present the normalized electric field (Er) profiles of the SNW1, RNW, and SNW2 for typical absorption peaks in Figure 2a nm) light illumination, respectively. On the one hand, there are similar features of the Er profiles between the RNW and SNWs. Firstly, the improved light absorption of the RNW is ascribed to the excitation of the LMRs, likewise in SNW [9,10]. For instance, 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 NWs illustrate much larger Er intensities in the long-than in the shortwavelength range. For example, the Er intensities for the RNW are much higher in Figure 3e(3-5) than Figure 3e(1-2) for TE light, likewise the elliptical NWs. On the other hand, the Er intensities of the resonant peaks in the short wavelength range are much bigger inside the whole RNW owing to the excitation of more complex LMRs (for example, Figure 3b(1) for TM light, indicating the stronger interaction between the incident light and the RNW, resulting in a stronger absorption in comparison with the SNWs. In Figure 4, we show the corresponding normalized absorption mode profiles (Pabs) of the SNW1, RNW, and SNW2 under TM and TE light illumination, respectively. Note that Figure 4a,d present the normalized Pabs of the SNW1 for TM and TE light, Figure 4b, exhibit those of the RNW for TM and TE light, while Figure 4c,f describe those of the SNW2, respectively. It is shown 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.  In Figure 5, we show the normalized photogeneration rate (G) profiles of the SNW1, RNW, and SNW2 for TM and TE light illumination, respectively. Note that Figure 5a,d illustrate the normalized G profiles of the SNW1 for TM and TE light, Figure 5b,e present those of the RNW, while Figure 5c,f exhibit those of the SNW2, respectively. It is shown 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. 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 reveal that the light absorption is significantly enhanced due to the LMRs by decreasing a compared to the SNW1 and enhanced light path length by increasing b. In Figure 6a-c, we show 2D Jph maps as a function of a (from 100 to 1000 nm) and b (from 100 to 1000 nm) of the RNWs for TM, TE, and unpolarized light illumination, respectively. It is shown that we show Jph sharply increases with decreasing a at 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 (6). 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 presented the improved light absorption in the NWs based on symmetrybreaking structure from the square to the rectangular cross-section. We found that the light absorption in RNWs could be significantly improved in comparison with the SNWs and the enhancement effect mainly resulted from the symmetry-broken structure, which can simultaneously realize the increase of the light path length by the vertical side and the enhanced LMRs by the horizontal side. The simulation results showed that the photocurrent was significantly enhanced by 276.5% or 82.9% compared 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, the RNWs can be employed to other semiconductors to increase light absorption and provides an exciting pathway for the future development of high-efficiency single NW solar cells.