# Off-Resonant Absorption Enhancement in Single Nanowires via Graded Dual-Shell Design

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model and Methods

#### 2.1. Model

_{1}of the outer shell and the thickness t

_{2}of the inner shell of the DSNW which varies from t

_{2}= 0 nm (i.e., OSNW) to t

_{2}= 180 nm (i.e., ISNW), the total shell thickness t = t

_{1}+ t

_{2}= 180 nm and the total radius R = r + t = 280 nm. Note here that the radius of the Si core is chosen to be 100 nm as a representative nanoscale size and the total shell thickness is chosen to be 180 nm as the DSNW shows the optimal absorption in this study and the OSNW (or ISNW) approaches the optimal absorption at this thickness in our previous study [31,32,33]. The incident light indicated by colorful arrows in the insets of Figure 1 is assumed to be illuminated perpendicularly to the axial from the top, the wavelength range is from 300 to 1100 nm with a step size of 5 nm considering solar radiation and the bandgap of Si. The wavelength-dependent complex refractive index of Si fitted with the experimental data [44] and that of the inner shell, the outer shell and the surrounding medium (air) are set to be 2.5, 1.5 and 1.0, respectively. Note here that for simplicity, we have neglected the wavelength dependence of the refractive indices to neatly determine the impact of their size on the absorption spectra [30,31,45], as discussed later; however, as long as the wavelength dependence is negligible, our results could apply to any other dielectric with similar refractive index, such as SiO

_{2}, Si

_{3}N

_{4}and so forth.

#### 2.2. Methods

#### 2.2.1. The Absorption Efficiency and Mode Profile

_{abs}of the Si core as [38,49,50,51]:

_{geo}is the geometric cross-section (i.e., the projected area of the Si core) and C

_{abs}is the absorption cross-section calculated by [38,49,50,51]

_{0}is the wave vector in air, ${{\epsilon}^{\u2033}}_{r}$ is the imaginary part of the relative permittivity, E

_{r}is the normalized electric field intensity, V is the volume of the Si core, I

_{0}is the solar incident light intensity and P

_{abs}describes the wavelength-dependent absorption mode profile calculated from Poynting theorem, which can be expressed as [38,49,50,51]

_{0}is the angular frequency, the speed of light and the electric field intensity of the solar incident light; ${\epsilon}_{0}$ and ${\epsilon}^{\u2033}$ are the permittivities in air and the imaginary part of the permittivity of Si; n and k are the real and imaginary part of the complex refractive index of Si (i.e., m = n + ik, m

^{2}= ${\epsilon}_{r}={{\epsilon}^{\prime}}_{r}+i{{\epsilon}^{\u2033}}_{r}$); E is the electric field intensity in the Si core, respectively.

#### 2.2.2. The Photogeneration Rate

#### 2.2.3. The Photocurrent Density

_{ph}by:

#### 2.2.4. The Photocurrent Enhancement Factor (PEF)

_{ph,DSNW}and J

_{ph,rNW}are the photocurrent density for the DSNW and the reference NWs (BNW, OSNW and ISNW), respectively.

## 3. Results and Discussion

#### 3.1. The Absorption Mechanism

_{ph}), the absorption efficiency (Q

_{abs}), the absorption mode profile (P

_{abs}) and the photogeneration rate (G), respectively. Note here that r = 100 nm, t = 180 nm, R = r + t = 180 nm, t

_{2}= 0 → 180 nm, where t = 0, t

_{2}= 0 and t

_{2}= 180 nm denote the cases of the BNW, OSNW and ISNW, respectively; m

_{3}is the complex refractive index of the Si core, m

_{2}= 2.0, m

_{1}= 1.5 and m

_{0}= 1.0, as shown in the insets of Figure 1.

#### 3.1.1. The Photocurrent Density (J_{ph})

_{ph}) obtained by Equation (7). In Figure 1, we show t

_{2}-dependent J

_{ph}under normally-incident TE, TM and unpolarized light illumination, respectively. It is observed that J

_{ph}increases rapidly first when initially increasing t

_{2}, reaches a peak at t

_{2}= 85 nm and then decreases when continuing to increase t

_{2}. More importantly, J

_{ph}of the DSNW is always bigger than that of the OSNW as long as the inner shell is adopted and higher than that of the ISNW in a broad inner thickness range of t

_{2}> 40 nm. For a direct comparison, we list the J

_{ph}values of the considered BNW, OSNW, ISNW and DSNW configurations under TE, TM and unpolarized illumination, as shown in Table 1. The maximum J

_{ph}values for TE and TM light are 14.85 and 15.50 mA/cm

^{2}, respectively. Under unpolarized light illumination (e.g., sunlight), the maximum J

_{ph}reaches 15.18 mA/cm

^{2}, which is 96.6%, 31.2% and 10.2% higher than that of the BNW (7.72 mA/cm

^{2}), OSNW (11.57 mA/cm

^{2}) and ISNW (13.77 mA/cm

^{2}), respectively. It is found that this photocurrent enhancement is mainly ascribed to the improvement of J

_{ph}under any polarized situations (especially TM light), indicating the potential of the DSNW in improving the light absorption of single NWs.

#### 3.1.2. The Absorption Efficiency (Q_{abs})

_{abs}as a function of t

_{2}under TE and TM light illumination, which is given by Equation (1). It is clear that the dual-shell design can lead to absorption enhancement in the whole spectrum range compared to the OSNW and almost the entire spectrum range (except several narrow peaks) compared to the ISNW under both TE and TM light illumination (especially at the off-resonant wavelengths), as discussed later. Moreover, Q

_{abs}can be divided into three regions using two vertical dashed lines by employing the characteristic wavelengths λ

_{c1}(~430 nm) and λ

_{c2}(~525 nm) for both TE and TM lights, as labeled in Figure 2a,b. Firstly, in the wavelength range of λ < λ

_{c1}, Q

_{abs}periodically changes with increasing t

_{2}. Note here that Q

_{abs}can be divided into two regions using a horizontal dashed line by employing the characteristic inner shell thickness t

_{2c}(~90 nm) for both TE and TM lights, as labeled in Figure 2a,b. Q

_{abs}reaches the maximum absorption near t

_{2}= 50 nm and t

_{2}= 130 nm for the first (t

_{2}< t

_{2c}) and second (t

_{2}> t

_{2c}) period, respectively. In other words, the excellent absorption can be obtained in the inner shell thickness range of 40 < t

_{2}< 60 nm and 110 < t

_{2}< 140 nm for two periods, respectively. Secondly, in the wavelength range of λ

_{c1}< λ < λ

_{c2}, Q

_{abs}reaches the maximum absorption near t

_{2}= 50 nm, that is, the superior absorption can be obtained in the inner shell thickness range of 60 < t

_{2}< 120 nm. Finally, in the wavelength range of λ > λ

_{c2}, Q

_{abs}appears to be comparable due to the trade-off between the suppression at the resonant wavelengths and the enhancement at the off-resonant wavelengths, resulting in little contribution to the photocurrent enhancement. Therefore, the photocurrent enhancement of the DSNW with t

_{2}< 60 nm and t

_{2}> 120 nm is attributed to the improved absorption in the wavelength range of λ < λ

_{c1}, while that of the DSNW with 60 < t

_{2}< 120 nm is mainly attributed to the improved absorption in the wavelength range of λ

_{c1}< λ < λ

_{c2}, which is due to the fact that there is a much higher solar radiation in the wavelength range of λ

_{c1}< λ < λ

_{c2}than λ < λ

_{c1}, leading to a more significant contribution to the photocurrent according to Equation (7).

_{ph}in Figure 1. In Figure 2c,d, we show λ-dependent Q

_{abs}of the DSNW with t

_{2}= 85 nm for TE and TM light, where the results of the BNW, OSNW and ISNW are also included for comparison. It is shown that Q

_{abs}of the DSNW is much higher than that of the BNW and OSNW in the wavelength range of λ < λ

_{c2}and that of the ISNW in the wavelength range of λ < λ

_{c2}(except for several narrow peaks, for example, λ = 470 for TE light) for both TE and TM lights, resulting in a significant photocurrent enhancement. In contrast, although Q

_{abs}of the DSNW is weaker at the resonant wavelengths, higher at the off-resonant wavelengths than that of all the other three NW structures in the wavelength range of λ > λ

_{c2}, leading to a similar contribution to the photocurrent, as discussed above. It is worth noting that Q

_{abs}can be substantially enhanced at the off-resonant wavelengths over the whole wavelength range for both TE and TM lights, especially for TM light (e.g., near λ = 470 nm), which results in the more prominent photocurrent enhancement for TM than TE light. It should also be noted that the match between the absorption efficiency and the solar spectrum becomes another essential factor in evaluating the photocurrent according to Equation (7). For instance, although Q

_{abs}of the DSNW for TE light is much higher than that for TM light in the wavelength range of λ < λ

_{c1}, solar radiation is much lower, which leads to a less photocurrent enhancement, while Q

_{abs}for TM light is much higher than that for TE light in the wavelength range of 450 < λ < 650 nm (except the narrow wavelength range of 490 < λ < 505 nm), as shown in the inset of Figure 2d and solar radiation is much higher at the same time, which results in a more significant contribution to the photocurrent.

#### 3.1.3. The Absorption Mode Profile (P_{abs})

_{abs}) calculated by Equation (3) [22,24,26,50,52]. In Figure 3, we examine the normalized absorption mode profiles inside the Si core corresponding to the wavelengths in Figure 2c,d under TE and TM light illumination (these profiles from left to right columns are related to the evolution of the structure from BNW to OSNW and then to ISNW and finally to DSNW). Figure 3a,c show the off-resonant absorption mode profiles for TE and TM light, while Figure 3b,d show the corresponding resonant absorption mode profiles. Note here that the resonant (or off-resonant) absorption for all the four NW configurations may occur at different wavelengths due to the difference of the thickness and the refractive index of the dielectric shells, that is, t

_{1}(or t

_{2}) or m

_{1}(or m

_{2})-driven shift [30,31], as shown in Figure 2a,b. It is observed that the absorption enhancement is attributed to the excitation of the LMRs, likewise in BNW [17,18], which can capture light by multiple total internal reflections at the Si core/inner shell interface when the wavelength of the incident light matches one of the LMRs supported by the Si core. The LMRs can be noted as TE

_{ml}or TM

_{ml}, where m and l are the azimuthal mode number and the radial order of the resonances, respectively. Figure 3b,d show that the resonant absorption mode profiles of the DSNW are different from that of the BNW, similar to that of the INSW due to the fact that the LMRs of the Si core occur at the Si core/inner shell interface. Specifically, the modes of the BNW, OSNW, ISNW and DSNW are TE

_{12}, TE

_{31}, TE

_{31}and TE

_{31}at λ = 495, 470, 470 and 470 nm for TE light and TM

_{12}, TM

_{12}, TM

_{41}and TM

_{41}at λ = 495, 500, 465 and 470 nm for TM light, respectively. The absorption of the DSNW is indeed enhanced compared to the BNW and OSNW for both TE and TM lights and slightly suppressed for TE light and enhanced for TM light compared to the ISNW. Figure 3a,c show that the off-resonant absorption mode profiles of the DSNW exhibit a transition mode referred to the LMRs, such transition modes are very close to the corresponding LMRs, which is attributed to the fact that the presence of the graded dual shells makes more light couple into the Si core, leading to a more significant absorption enhancement compared to all the other three NWs.

_{c2}, as shown in Figure 2, the resonant absorption is greatly enhanced due to the constructive interference with the reemitted light of the weaker LMRs of the Si core. However, for stronger LMRs in the wavelength of λ > λ

_{c2}, the resonant absorption is suppressed due to the destructive interference with the reemitted light of the stronger LMRs of the Si core. In contrast, the off-resonant absorption over the whole spectrum is greatly enhanced due to the constructive interference with the reemitted light of the weaker transition modes of the Si core. In a word, the off-resonant (or weaker resonant) absorption is dramatically enhanced owing to an improved coupling between the reemitted light of weaker transition modes (or weaker LMRs) of the Si core and the nanofocusing light from the graded dual shells at the core/inner shell interface [38,39].

#### 3.1.4. The Photogeneration Rate Profile (G)

#### 3.2. The Optimization of the Light-Harvesting Performance

_{2}, all the other structural details of the DSNW are consistent with that shown in the insets of Figure 1. In Figure 4a, we show 2D J

_{ph}as a function of t

_{2}and m

_{2}of the DSNW and the optimal t

_{2}as a function of m

_{2}. Figure 4a shows J

_{ph}sharply increases with increasing t

_{2}at a fixed m

_{2}, reaches its maximum and then decreases when continuing to increase t

_{2}. More importantly, J

_{ph}of the DSNW is always much larger than that of the OSNW at any t

_{2}values and higher than that of the ISNW in a broad inner shell thickness range of t

_{2}> 40 nm for m

_{2}< 3.5 and t

_{2}> 60 nm for 3.5 < m

_{2}< 4.0. It is observed that the maximum values of J

_{ph}can be obtained in the inner shell thickness range of 90 < t

_{2}< 110 nm for 3.0 < m

_{2}< 3.5. In Figure 5b, we show m

_{2}-dependent J

_{ph}of the DSNW (corresponding to the optimal t

_{2}in Figure 5a), together with that of the BNW, OSNW and ISNW for comparison. Also, in Figure 5c, we show the photocurrent enhancement factors (PEFs) defined by Equation (8). It is readily observed that J

_{ph}of the DSNW is much larger than all the other three NWs. In particular, the maximum J

_{ph}reaches 18.10 mA/cm

^{2}at t

_{2}= 100 nm for m

_{2}= 3.25, which is 134.4%, 56.4% and 12.4% much larger than that of the BNW (7.72 mA/cm

^{2}), OSNW (11.57 mA/cm

^{2}) and ISNW (16.10 mA/cm

^{2}), respectively.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**J

_{ph}versus t

_{2}of the dual-shell coated nanowire (DSNW) for TE (Transverse electric field to the nanowire axis), TM (Transverse Magnetic field to the nanowire axis) and unpolarized light illumination. The insets are the cross-sectional views of the NW only coated with the outer shell (OSNW) (left), DSNW (middle) and the NW only coated with the outer shell (ISNW) (right), where J

_{ph}is the photocurrent density; t

_{1}, t

_{2}, t = t

_{1}+ t

_{2}= 180 nm, r = 100 nm and R = r + t = 280 nm are the outer shell thickness, the inner shell thickness, the total shell thickness, the core radius and the total radius; m

_{3}, m

_{2}, m

_{1}and m

_{0}are the complex refractive indices of the Si core, the inner shell, the outer shell and air, respectively.

**Figure 2.**Q

_{abs}versus λ and t

_{2}of the DSNW for (

**a**) TE and (

**b**) TM light illumination; Q

_{abs}versus λ of the DSNW (t

_{2}= 85 nm) under (

**c**) TE and (

**d**) TM light illumination, together with the BNW (t = 0), OSNW (t

_{2}= 0) and ISNW (t

_{2}= 180 nm) as references. The inset in (

**d**): Q

_{abs}versus λ (450–650 nm) for TE and TM light, where Q

_{abs}and λ are the absorption efficiency and the wavelength of the incident light, respectively. Note that λ

_{c1}(~430 nm) and λ

_{c2}(~525 nm) are the characteristic wavelengths denoted by two vertical dashed lines and t

_{2c}(~90 nm) is the characteristic inner shell thickness in the wavelength range of λ < λ

_{c1}denoted by a horizontal dashed line.

**Figure 3.**The representative normalized absorption mode profiles inside the Si core corresponding to the wavelengths in Figure 2c,d [from left to right columns are associated with the BNW, OSNW, ISNW and DSNW, respectively]: (

**a**,

**b**) for TE and (

**c**,

**d**) for TM light illumination; (

**a**,

**c**) for off-resonant and (

**b**,

**d**) for resonant wavelengths. Note that the numbers in square brackets are the wavelengths and the modes of the the leaky mode resonances (LMRs) at the resonant wavelengths are also labeled in the left of square brackets in Figure 3b,d.

**Figure 4.**The normalized generation rate profiles of the DSNW for (

**a**) TE and (

**b**) TM light, together with the BNW, OSNW and ISNW for comparison. Note that the regions labeled by circles and triangles denote that the absorption of the DSNW is significantly enhanced for TE and TM light and the region labeled by the square denotes that the absorption of the DSNW is slightly decreased for TE light, respectively.

**Figure 5.**(

**a**) J

_{ph}versus t

_{2}and m

_{2}of the DSNW. The white dashed line represents the position of the maximum J

_{ph}at various m

_{2}values. (

**b**) J

_{ph}versus m

_{2}of the DSNW, corresponding to the optimal t

_{2}. Also, J

_{ph}versus m

_{2}of the BNW, OSNW and ISNW are included for comparison. (

**c**) photocurrent enhancement factor (PEF) versus m

_{2}of the DSNW compared to the BNW, OSNW and ISNW, respectively. Note that the PEF is the photocurrent enhancement factor compared to the reference NWs.

**Table 1.**Photocurrent densities (in mA/cm

^{2}) of the four typical configurations (in nm) of the bare NW (BNW), OSNW, ISNW and DSNW under TE (Transverse electric field to the nanowire axis), TM (Transverse Magnetic field to the nanowire axis) and unpolarized light illumination, where r, t

_{2}, and t are the core radius, the inner shell thickness and the total shell thickness, respectively.

Configuration | Parameter | ${\mathit{J}}_{\mathbf{ph}}^{\mathbf{TE}}$ | ${\mathit{J}}_{\mathbf{ph}}^{\mathbf{TM}}$ | ${\mathit{J}}_{\mathbf{ph}}$ |
---|---|---|---|---|

BNW | r = 100, t_{2} = 0, t = 0 | 7.24 | 8.20 | 7.72 |

OSNW | r = 100, t_{2} = 0, t = 180 | 11.27 | 11.87 | 11.57 |

ISNW | r = 100, t_{2} = 180, t = 180 | 13.94 | 13.59 | 13.77 |

DSNW | r = 100, t_{2} = 85, t = 180 | 14.85 | 15.50 | 15.18 |

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## Share and Cite

**MDPI and ACS Style**

Liu, W.; Guo, X.; Xing, S.; Yao, H.; Wang, Y.; Bai, L.; Wang, Q.; Zhang, L.; Wu, D.; Zhang, Y.;
et al. Off-Resonant Absorption Enhancement in Single Nanowires via Graded Dual-Shell Design. *Nanomaterials* **2020**, *10*, 1740.
https://doi.org/10.3390/nano10091740

**AMA Style**

Liu W, Guo X, Xing S, Yao H, Wang Y, Bai L, Wang Q, Zhang L, Wu D, Zhang Y,
et al. Off-Resonant Absorption Enhancement in Single Nanowires via Graded Dual-Shell Design. *Nanomaterials*. 2020; 10(9):1740.
https://doi.org/10.3390/nano10091740

**Chicago/Turabian Style**

Liu, Wenfu, Xiaolei Guo, Shule Xing, Haizi Yao, Yinling Wang, Liuyang Bai, Qi Wang, Liang Zhang, Dachuan Wu, Yuxiao Zhang,
and et al. 2020. "Off-Resonant Absorption Enhancement in Single Nanowires via Graded Dual-Shell Design" *Nanomaterials* 10, no. 9: 1740.
https://doi.org/10.3390/nano10091740