Exact comprehensive equations for the photon management properties of silicon nanowire

Unique photon management (PM) properties of silicon nanowire (SiNW) make it an attractive building block for a host of nanowire photonic devices including photodetectors, chemical and gas sensors, waveguides, optical switches, solar cells, and lasers. However, the lack of efficient equations for the quantitative estimation of the SiNW’s PM properties limits the rational design of such devices. Herein, we establish comprehensive equations to evaluate several important performance features for the PM properties of SiNW, based on theoretical simulations. Firstly, the relationships between the resonant wavelengths (RW), where SiNW can harvest light most effectively, and the size of SiNW are formulized. Then, equations for the light-harvesting efficiency at RW, which determines the single-frequency performance limit of SiNW-based photonic devices, are established. Finally, equations for the light-harvesting efficiency of SiNW in full-spectrum, which are of great significance in photovoltaics, are established. Furthermore, using these equations, we have derived four extra formulas to estimate the optimal size of SiNW in light-harvesting. These equations can reproduce majority of the reported experimental and theoretical results with only ~5% error deviations. Our study fills up a gap in quantitatively predicting the SiNW’s PM properties, which will contribute significantly to its practical applications.

n is the refractive index of silicon, and λ is the wavelength of light. Taking the SiNW with diameter 100 nm as example, the obtained cut-off wavelengths are ~530 nm for TE 01 , TM 01 and HE 21 , ~415 nm for EH 11 , HE 12 and HE 21 , and ~380 nm for EH 21 and HE 41 .
Main peak has the longest wavelength thus it should correspond to the lowest-order modes.
Therefore, it is no doubt to assign Main peak to the degeneracy of TE 01 , TM 01 and HE 21 . This assignment can be confirmed by the fact that the electric field of the main peak follows the distribution of mode LP 11 (a degenerate mode of TE 01 , TM 01 and HE 21 ), as given in the inset of Figure 1c.  Table S1. Cut-off parameters of the first few guided modes Peak3 can't be assigned to mode LP 21 since its wavelength, 517 nm, is much longer than the cut-off wavelength of mode LP 21 , 415 nm. Besides, i) its electric field distribution shows obvious longitudinal oscillation pattern Figure 1c; ii) it becomes weaker until disappears as the length of SiNW increases, as shown in Figure S1, which reflects the variation of the extinction and absorption curves for SiNWs with increased length. What's shown in the insets are Peak3, appearing when the diameters greater than 90 nm. It can be seen that, with the length increases, Peak3 become more and more unobvious, even vanish for the SiNW with diameter 90 and 100 nm.
Taking into account it can only occurs when the diameter of the SiNW greater than 90 nm, this phenomenon indicates that Peak3 may come from the limitation on the length direction, and should be greatly related to the draw ratio of the SiNW.

Relationship between RW and the cut-off wavelength of SiNW:
From the waveguide theory, the cut-off wavelength is the critical wavelength where the mode can be leaked out.
At the cut-off wavelength, the light propagating in the nanowire starts to be leaked out (very little light can be leaked out), to couple with the light surround the nanowire (very little light can be coupled within the NW). With the wavelength increases, more light can be leaked out, and correspondingly more light can be coupled within the Si NW, till a peak value which corresponds to the resonance wavelength. While, when the wavelength is much longer than the cut-off wavelength, the leaky mode cannot be supported again. Therefore, the actual resonance wavelength should be some greater than the cut-off wavelength.
We have compared RW of SiNWs (with fixed length, 1μm, and various diameter 40, 60, 80, 100, 120 and 140 nm) calculated by the DDA simulations and those by the equation The values calculated by the equation exactly coincide with the leftmost wavelength of the resonance peaks. This effectively verified the above explanations.
The fitting pictures for RW of peak1 and peak3.

Regression process of the equations for the integrated extinction and absorption intensities
in full-spectrum with unit light intensity. The integrated extinction and absorption efficiencies are signified by Q ext-int and Q abs-int , respectively. Figure S6a shows that both of them firstly increase and then decrease with increased diameter. However, when we try to fit the data of the integrated extinction efficiencies, we found that using one cubic function of the diameter is better than using two separated quadratic functions, in describing total variation in the data. Therefore, the practical function used is a polynomial owning cubic term of the diameter. Figure S6b shows that both of Q ext-int and Q abs-int , show well linearity with length.
By regression method, the final equation to describe the integrated extinction intensity ( Figure   S7a) of SiNW is with correlation index R-squares=0.9989, which reflects the good fitness.
For the integrated absorption intensity ( Figure S7b), two functions can give very good fitting results to the original data. We attribute the dependent rules of Q abs-int before and after the points to Ext.
Abs. be quadratic and cubic, respectively. The obtained equations are