Enhanced external quantum efficiency in rectangular singlenanowire solar cells (supplementary material)–modeling method

Xiaofeng Li1* and Yaohui Zhan2

1Institute of Modern Optical Technologies, Key Lab of Advanced Optical Manufacturing Technologies of Jiangsu Province and Key Lab of Modern Optical Technologies of Education Ministry of China, Soochow University, Suzhou 215006, Jiangsu, PR China.

2Center for Composite Materials, Harbin Institute of Technology, Harbin, 150001, China.

*Email address: (X LI)

In this paper, the optical absorption and electrical response of the gallium arsenide (GaAs) single-nanowire solar cells (SNSCs) have been calculated through seamlessly linking electromagnetic and carrier transport behaviors in frequency and spatial domains. The device model is numerically solved using finite-element method (FEM). We next introduce this modeling schemefrom optical and electrical perspectives, respectively.

Electromagnetic (EM) response of SNSCs under external solar illumination(AM1.5 [1])is solved using scattered-field formulation. The computational domain is terminated by perfectly matched layers (PMLs) to avoid non-physical reflection (see Fig. S1). The solar incidence (under transverse electric or magnetic polarization) is specified as a background field (Eb) for all non-PML regions; therefore the incident and scattered fields (Escat) can be distinguished easily. The wave equation governing the photonic response of the solar cell is

, (S.1)

where ε0(εr) is the vacuum (relative) permittivity, μ0(μr) the vacuum (relative) permeability, k0 the vacuum wave vector, ω the angular frequency, and σ the electrical conductivity of the material. The total electric field (E) with the presence of the scatterer can then be obtained from E = Escat + Eb.

FIG. S1. (Color online) Schematic cross-sectional diagram of EM filed calculation. ASNSC composed of active layers (red circle: n region; blue square: p region) and SiO2 nanoshell (green square) is lying in vacuumwithout substrate (substrate can be included straightforwardly by adding another semi-infinite layer beneath the SNSC). The absorption and scattering electric field in near field are depicted visually in blue and green arrows.

The absorption efficiency (percentage)of the SNSCs, Qabs,can be obtained by [2,3]

, (S.2)

where v is the volume of photoactive absorber andε" the imaginary part of dielectric function of GaAs[4].Pin is the incident areal power density from the sun within the cross-sectional area (Ap) confined by thephotoactive region.

To link the optical and electrical domains, the detailed carrier generation profile (G)is calculated from Eq. (S.3) and coupled synchronously into semiconductorsolar cell model, which is composed of carrier transport[i.e., Eq. (S.4) for electrons and Eq.(S.5) for holes]and Poisson’s equation[i.e., Eq. (S.6)][1, 5, 6]

, (S.3)

, (S.4)

, (S.5)

, (S.6)

whereħis the reduced Planck constant,n (p) the electron (hole) concentration, Dn = µnKBT/q (Dp= µpKBT/q) the electron (hole) diffusion coefficient, µn (µp) the electron (hole) mobility,KB the Boltzmann’s constant, T (= 300 K) the operating temperature,qthe electron charge, Ф the electrostatic potential (electrostatic field F = –Ф),λ the wavelength, and U the carrier recombination rate including contributions from Shockley-Read-Hall (SRH), radiative, and Auger recombinations.ε is the material permittivity and C = ND – NAis the impurity concentration defined as the sum of the concentrations of ionized donors ND and acceptors NA, including the signs of the compensated charges.

At the outer boundary, surface recombination plays an important role and has to be carefully treated in the carrier transport modules. The minority surface recombination conditions at the exterior surfaces of the SNSCs are[6]

, (S.7)

where Sn (Sp) is the electron (hole) surface recombination velocity.

The forwardly biased (voltage of Va) SNSCs can be modeled by controlling the boundary conditions of the electrostatic module, i.e., the value of Ф at the center of n-type region is unchanged, while at the opposite end increased by Va from initial

. (S.8)

FIG. S2. (Color online) The effective domains and boundaries in electricalcalculation for (a) rectangular and (b) circular SNSCs, wherethe cylindrical coordinate (r, φ) is adopted andis the normal direction to the exteriorboundary .

The solutions of Eqs. (S.4) – (S.6) enable the examination of the electrical response of the SCs. The corresponding short-circuit current density jsc(λ) is given by the spatially averaged photocurrent at the surface of the SCs (device schematic refers to Fig. S2)

, (S.9)

where jn (jp) is the photocurrent contributed from electrons (holes). Eq. (S.9) allows the calculation of the frequency-dependent EQE

. (S.10)

where IAM1.5is the solar incident photon flux.

References

1J. Nelson. The Physics of Solar Cells (Imperial College Press, London, 2003).

2Y. Zhan, J. Zhao, C. Zhou, M. Alemayehu, Y. Li, and Y. Li, Opt.Express20, 11506 (2012).

3C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (Wiley, 1983).

4E. D. Palik, Handbook of Optical Constants of Solids, (Academic Press, 1985).

5X. Li, N. Hylton, V. Giannini, K.-H. Lee, N. J. Ekins-Daukes, and S. A. Maier, Opt. Express19, A888 (2011).

6X. Li, N. Hylton, V. Giannini, K.-H. Lee, N. J. Ekins-Daukes, and S. A. Maier, Prog. Photovolt.: Res. Appl. (2012),doi: 10.1002/pip.2159.