First-principle calculation of solar cell efficiency under incoherent illumination

Michaël Sarrazin; Aline Herman; Olivier Deparis

doi:10.1364/OE.21.00A616

Optics Express
Vol. 21,
Issue S4,
pp. A616-A630
(2013)
•https://doi.org/10.1364/OE.21.00A616

First-principle calculation of solar cell efficiency under incoherent illumination

Michaël Sarrazin, Aline Herman, and Olivier Deparis

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Michaël Sarrazin, Aline Herman, and Olivier Deparis, "First-principle calculation of solar cell efficiency under incoherent illumination," Opt. Express 21, A616-A630 (2013)

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Abstract

Because of the temporal incoherence of sunlight, solar cells efficiency should depend on the degree of coherence of the incident light. However, numerical computation methods, which are used to optimize these devices, fundamentally consider fully coherent light. Hereafter, we show that the incoherent efficiency of solar cells can be easily analytically calculated. The incoherent efficiency is simply derived from the coherent one thanks to a convolution product with a function characterizing the incoherent light. Our approach is neither heuristic nor empiric but is deduced from first-principle, i.e. Maxwell’s equations. Usually, in order to reproduce the incoherent behavior, statistical methods requiring a high number of numerical simulations are used. With our method, such approaches are not required. Our results are compared with those from previous works and good agreement is found.

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Fig. 1 (a) Sketch of a light trapping diffracting structure (Z_I < z < Z_III) with many diffraction orders. Domains I and III are respectively the incident and emergence media. (b) Model of field amplitude F_in(t) for the incoherent monochromatic incident light. m(t) is the corresponding modulation which characterizes the random phase switching at different times τ_i. τ_c is the coherence time, equal to the average value <τ_i>.

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Fig. 2 Simulation of the absorption spectra of planar and corrugated 500 nm-thick c-Si slabs. The coherent spectra were obtained using RCWA and the incoherent ones using our convolution formula, Eq. (7).(a) Absorption spectra of the planar slab according to various coherence times. (b) Absorption spectra of the corrugated slab according to various coherence times. Inset: corrugated structure ; p = 500 nm, t = 500 nm, h = 300 nm, D = 450 nm, d = 320 nm. The structure follows a super-Gaussian profile with m = 3 (see Ref. [8]).

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Fig. 3 Comparison between the coherent spectra obtained with the RCWA method (blue lines) and incoherent spectra with various coherence times, for planar and grating structures, defined in [26]. (a) Reflectance spectrum of an unpatterned c-Si layer (225 nm) deposited on a 75 nm-thick Au film on a glass substrate. (b) Reflectance spectrum of the whole grating structure.

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Tables (1)

Table 1 Computed photocurrents related to the corrugated device of Fig. 2(b) for various coherence times. Photocurrent was integrated over the spectral range: 300 nm – 1200 nm.

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Equations (66)

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J = \frac{e}{h c} \int A (λ) S_{G} (λ) λ d λ

A_{coh} (ω) = - \frac{ε_{0} ω}{2 P_{in}} \int_{V} ε^{″} (ω, r) {| E (ω, r) |}^{2} d^{3} r

| F_{in} (t) 〉 = m (t) e^{- i ω_{c} t} | F_{in}^{(0)} 〉 .

D (ω) = {| m (ω) |}^{2}

I (ω) = \frac{D (ω)}{\int D (ω) d ω}

I (ω) = τ_{c} \sqrt{\frac{ln 2}{π^{3}}} e^{- \frac{ln 2}{π^{2}} τ_{c}^{2} ω^{2}}

A_{incoh} (ω) = I (ω) ★ A_{coh} (ω) .

| F_{scat} 〉 = [\begin{matrix} {\bar{N}}_{III}^{+} \\ {\bar{X}}_{III}^{+} \\ {\bar{N}}_{I}^{-} \\ {\bar{X}}_{I}^{-} \end{matrix}], | F_{in} 〉 = [\begin{matrix} {\bar{N}}_{I}^{+} \\ {\bar{X}}_{I}^{+} \\ {\bar{N}}_{III}^{-} \\ {\bar{X}}_{III}^{-} \end{matrix}] .

S (ω) | F_{in} (ω) 〉 = | F_{scat} (ω) 〉 .

J_{in} = σ \frac{1}{2} ε_{0} c \sqrt{ε_{I}} cos θ 〈 F_{in} (ω) | F_{in} (ω) 〉 .

J_{X} = 〈 F_{scat} (ω) | C_{X}^{†} (ω) C_{X} (ω) | F_{scat} (ω) 〉,

C_{T} (ω) = [\begin{matrix} Q_{III} (ω) & 0 & 0 & 0 \\ 0 & Q_{III} (ω) & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}], C_{R} (ω) = [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & Q_{I} (ω) & 0 \\ 0 & 0 & 0 & Q_{I} (ω) \end{matrix}],

Q_{I (III)} (ω) = [\begin{matrix} \dots & 0 & 0 \\ 0 & \sqrt{\frac{σ}{2 μ_{0} ω} Re {k_{I (III), g, z}}} & 0 \\ 0 & 0 & \dots \end{matrix}] .

J_{X} = 〈 F_{X} (ω) | F_{X} (ω) 〉,

| F_{X} (ω) 〉 = S_{X} (ω) | F_{in} (ω) 〉,

S_{X} (ω) = C_{X} (ω) S (ω) .

| F_{in} (t) 〉 = | F_{in}^{(0)} 〉 e^{- i ω_{c} t}

| F_{X} (t) 〉 = S_{X} (t) ★ | F_{in} (t) 〉

\begin{array}{l} | F_{X} (t) 〉 & = & \int S_{X} (t - t^{'}) e^{- i ω_{c} t^{'}} d t^{'} | F_{in}^{(0)} 〉 \\ = & e^{- i ω_{c} t} S_{X} (ω_{c}) | F_{in}^{(0)} 〉, \end{array}

S_{X} (ω_{c}) = \int S_{X} (t^{'}) e^{i ω_{c} t^{'}} d t^{'}

| F_{X} (t) 〉 = | F_{X}^{(0)} 〉 e^{- i ω_{c} t},

| F_{X}^{(0)} 〉 = S_{X} (ω_{c}) | F_{in}^{(0)} 〉 .

R_{coh} (ω_{c}) = \frac{J_{R}}{J_{in}} = \frac{〈 F_{R}^{(0)} | F_{R}^{(0)} 〉}{J_{in}} = \frac{〈 F_{i n}^{(0)} | S_{R}^{†} (ω_{c}) S_{R} (ω_{c}) | F_{in}^{(0)} 〉}{J_{in}},

T_{coh} (ω_{c}) = \frac{J_{T}}{J_{in}} = \frac{〈 F_{T}^{(0)} | F_{T}^{(0)} 〉}{J_{in}} = \frac{〈 F_{in}^{(0)} | S_{T}^{†} (ω_{c}) S_{T} (ω_{c}) | F_{in}^{(0)} 〉}{J_{in}} .

\begin{array}{l} J_{in} (t) & = & σ \frac{1}{2} ε_{0} c \sqrt{ε_{I}} cos θ 〈 F_{in} (t) | F_{in} (t) 〉 \\ = & σ \frac{1}{2} ε_{0} c \sqrt{ε_{I}} cos θ 〈 F_{in}^{(0)} | F_{in}^{(0)} 〉, \end{array}

J_{in} = \frac{1}{T} \int_{- T / 2}^{T / 2} J_{in} (t) d t = σ \frac{1}{2} ε_{0} c \sqrt{ε_{I}} cos θ 〈 F_{in}^{(0)} | F_{in}^{(0)} 〉 \equiv J_{in} (t) .

| F_{in} (t) 〉 = | F_{in}^{(0)} 〉 m (t) e^{- i ω_{c} t} .

D (ω) = {| m (ω) |}^{2},

m (ω) = \int m (t) e^{i ω t} d t

\begin{array}{l} | F_{X} (t) 〉 & = & S_{X} (t) ★ | F_{in} (t) 〉 \\ = & | F_{X}^{(0)} (t) 〉 e^{- i ω_{c} t} \end{array}

| F_{X}^{(0)} (t) 〉 = U_{X} (t) | F_{in}^{(0)} 〉,

U_{X} (t) = m (t) ★ S_{X} (t) e^{i ω_{c} t} .

J_{X, incoh} = \frac{1}{T_{c}} \int_{0}^{T_{c}} J_{X} (t) d t .

J_{X, incoh} = \int_{- \infty}^{\infty} J_{X} (ω) (\frac{1}{2 π} \frac{sin (ω T_{c} / 2)}{ω T_{c} / 2}) d ω .

lim_{T_{c} \to + \infty} \frac{T_{c}}{2 π} \frac{sin (ω T_{c} / 2)}{ω T_{c} / 2} = δ (ω),

J_{X, incoh} \sim \frac{1}{T_{c}} J_{X} (ω = 0) .

J_{X} (t) = 〈 F_{X}^{(0)} (t) | F_{X}^{(0)} (t) 〉 = 〈 F_{in}^{(0)} | U_{X}^{†} (t) U_{X} (t) | F_{in}^{(0)} 〉,

J_{X, incoh} = \frac{1}{T_{c}} 〈 F_{in}^{(0)} | I_{X} (ω = 0) | F_{in}^{(0)} 〉

\begin{array}{l} I_{X} (ω) & = & \int U_{X}^{†} (t) U_{X} (t) e^{i ω t} d t \\ = & \frac{1}{2 π} U_{X}^{†} (ω) ★ U_{X} (ω) . \end{array}

I_{X} (ω) = \frac{1}{2 π} m (ω) S_{X}^{†} (ω - ω_{c}) ★ m (ω) S_{X} (ω + ω_{c}),

I_{X} (ω = 0) = \frac{1}{2 π} \int_{- \infty}^{\infty} m (- ω^{'}) S_{X}^{t} (- ω_{c} - ω^{'}) m (ω^{'}) S_{X} (ω^{'} + ω_{c}) d ω^{'} .

I_{X} (ω = 0) = \frac{1}{2 π} \int_{- \infty}^{\infty} {| m (ω_{c} - ω^{'}) |}^{2} S_{X}^{†} (ω^{'}) S_{X} (ω^{'}) d ω^{'} .

J_{X, incoh} = \frac{1}{2 π T_{c}} \int_{- \infty}^{\infty} D (ω_{c} - ω^{'}) 〈 F_{in}^{(0)} | S_{X}^{†} (ω^{'}) S_{X} (ω^{'}) | F_{in}^{(0)} 〉 d ω^{'} .

\begin{array}{l} J_{in, incoh} (t) & = & σ \frac{1}{2} ε_{0} c \sqrt{ε_{I}} cos θ 〈 F_{in} (t) | F_{in} (t) 〉 \\ = & σ \frac{1}{2} ε_{0} c \sqrt{ε_{I}} cos θ {| m (t) |}^{2} 〈 F_{in}^{(0)} | F_{in}^{(0)} 〉 . \end{array}

\begin{array}{l} J_{in, incoh} & = & \frac{1}{T_{c}} \int_{0}^{T_{c}} J_{in, incoh} (t) d t \\ = & J_{in} \frac{1}{T_{c}} \int_{0}^{T_{c}} {| m (t) |}^{2} d t \end{array}

\begin{array}{l} \frac{1}{T_{c}} \int_{0}^{T_{c}} {| m (t) |}^{2} d t & = & \frac{1}{T_{c}} \frac{1}{2 π} m^{*} (ω) ★ m (ω) |_{ω = 0} \\ = & \frac{1}{2 π T_{c}} \int_{- \infty}^{\infty} D (ω) d ω . \end{array}

J_{in, incoh} = J_{in} \frac{1}{2 π T_{c}} \int_{- \infty}^{\infty} D (ω) d ω .

X_{incoh} (ω_{c}) = \frac{1}{\int_{- \infty}^{\infty} D (ω) d ω} \int D (ω_{c} - ω^{'}) \frac{〈 F_{in}^{(0)} | S_{X}^{†} (ω^{'}) S_{X} (ω^{'}) | F_{in}^{(0)} 〉}{J_{in}} d ω^{'},

X_{coh} (ω^{'}) = \frac{〈 F_{in}^{(0)} | S_{X}^{†} (ω^{'}) S_{X} (ω^{'}) | F_{in}^{(0)} 〉}{J_{in}} .

\begin{array}{l} X_{incoh} (ω_{c}) & = & \int_{- \infty}^{\infty} I (ω_{c} - ω^{'}) X_{coh} (ω^{'}) d ω^{'} \\ = & I (ω_{c}) ★ X_{coh} (ω_{c}) \end{array}

I (ω) = \frac{D (ω)}{\int_{- \infty}^{\infty} D (ω) d ω}

ε (ρ, ω) = \sum_{g} ε_{g} (ω) e^{i g \cdot ρ} .

[\begin{matrix} E \\ H \end{matrix}] = \sum_{g} [\begin{matrix} E_{g} (z) \\ H_{g} (z) \end{matrix}] e^{i (k + g) \cdot ρ} e^{- i ω t} .

\frac{d}{d z} [\begin{matrix} {\bar{E}}_{/ /} (z) \\ {\bar{H}}_{/ /} (z) \end{matrix}] = [\begin{matrix} 0 & A \\ \tilde{A} & 0 \end{matrix}] [\begin{matrix} {\bar{E}}_{/ /} (z) \\ {\bar{H}}_{/ /} (z) \end{matrix}],

[\begin{matrix} {\bar{E}}_{/ /} (z_{I}) \\ {\bar{H}}_{/ /} (z_{I}) \end{matrix}] = exp {[\begin{matrix} 0 & A \\ \tilde{A} & 0 \end{matrix}] (z_{I} - z_{III})} [\begin{matrix} {\bar{E}}_{/ /} (z_{III}) \\ {\bar{H}}_{/ /} (z_{III}) \end{matrix}] .

μ_{I, g} = \frac{k_{I, g, z}}{\sqrt{ε_{I}} \frac{ω}{c}} \frac{k + g}{| k + g |},

η_{g} = \frac{k + g}{| k + g |} \times e_{z},

χ_{I, g}^{\pm} = \mp μ_{I, g} + \frac{| k + g |}{\sqrt{ε_{I}} \frac{ω}{c}} e_{z} .

\begin{array}{l} E_{I} (ρ, z) & = & \sum_{g} [N_{I, g}^{+} η_{g} e^{i k_{I, g, z} (z - z_{I})} \\ + N_{I, g}^{-} η_{g} e^{- i k_{I, g, z} (z - z_{I})} \\ + X_{I, g}^{+} χ_{I, g}^{+} e^{i k_{I, g, z} (z - z_{I})} \\ + X_{I, g}^{-} χ_{I, g}^{-} e^{- i k_{I, g, z} (z - z_{I})}] e^{i (k + g) \cdot ρ} \end{array}

\begin{array}{l} H_{I} (ρ, z) & = & \frac{\sqrt{ε_{I}}}{c μ_{0}} \sum_{g} [- N_{I, g}^{+} χ_{I, g}^{+} e^{i k_{I, g, z} (z - z_{I})} \\ - N_{I, g}^{-} χ_{I, g}^{-} e^{- i k_{I, g, z} (z - z_{I})} \\ + X_{I, g}^{+} η_{g} e^{i k_{I, g, z} (z - z_{I})} \\ + X_{I, g}^{-} η_{g} e^{- i k_{I, g, z} (z - z_{I})}] e^{i (k + g) \cdot ρ} . \end{array}

[\begin{matrix} {\bar{N}}_{I}^{+} \\ {\bar{X}}_{I}^{+} \\ {\bar{N}}_{I}^{-} \\ {\bar{X}}_{I}^{-} \end{matrix}] = [\begin{matrix} T^{+ +} & T^{+ -} \\ T^{- +} & T^{- -} \end{matrix}] [\begin{matrix} {\bar{N}}_{III}^{+} \\ {\bar{X}}_{III}^{+} \\ {\bar{N}}_{III}^{-} \\ {\bar{X}}_{III}^{-} \end{matrix}] .

[\begin{matrix} {\bar{N}}_{III}^{+} \\ {\bar{X}}_{III}^{+} \\ {\bar{N}}_{I}^{-} \\ {\bar{X}}_{I}^{-} \end{matrix}] = [\begin{matrix} S^{+ +} & S^{+ -} \\ S^{- +} & S^{- -} \end{matrix}] [\begin{matrix} {\bar{N}}_{I}^{+} \\ {\bar{X}}_{I}^{+} \\ {\bar{N}}_{III}^{-} \\ {\bar{X}}_{III}^{-} \end{matrix}] .

J = \int_{σ} \frac{1}{2} Re (\vec{E} \times {\vec{H}}^{★}) \cdot {\vec{e}}_{z} d S .

J_{I}^{+} = \frac{σ}{2 μ_{0} ω} \sum_{g} k_{I, g, z} [{| N_{I, g}^{+} |}^{2} + {| X_{I, g}^{+} |}^{2}] Θ (ε_{I} (ω) \frac{ω^{2}}{c^{2}} - {| k + g |}^{2})

J_{III}^{+} = \frac{σ}{2 μ_{0} ω} \sum_{g} k_{III, g, z} [{| N_{III, g}^{+} |}^{2} + {| X_{III, g}^{+} |}^{2}] Θ (ε_{III} (ω) \frac{ω^{2}}{c^{2}} - {| k + g |}^{2})

J_{I}^{-} = - \frac{σ}{2 μ_{0} ω} \sum_{g} k_{I, g, z} [{| N_{I, g}^{-} |}^{2} + {| X_{I, g}^{-} |}^{2}] Θ (ε_{I} (ω) \frac{ω^{2}}{c^{2}} - {| k + g |}^{2})

τ_c(fs)	j_sc(mA/cm²)
coherent	20.69
95	20.70
41	20.75
20	20.81
10	20.93
5	21.16
3	21.14
2.5	21.01

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