5.2 Determining the electrical properties of a multicrystalline silicon wafer
5.2.1 Quantifying the luminescence spectrum
This Section follows from the theoretical model of the generalized Planck law presented in Chapter 2. The spatially-resolved hyperspectral data may be quantified using the gen- eralized Planck law [279, 346] as shown previously. The detected electroluminescence spectrum depends on the emitted spectrum and a wavelength (λ) dependent constant instrument functionFI(λ) containing the sensor quantum efficiencyQE(λ) and a trans- missionT(λ) accounting for the elements between the sample and sensor.
The instrument function FI(λ) may be characterized using a reference source of known intensity and spectrum. The detected luminescence spectral signature can be written using the minority carrier diffusion lengthLd as
jγd(λ, Ld) =FI(λ)jγ(λ, Ld) (5.2.1)
where jγ(λ, Ld) can be solved for radiative recombination across the indirect silicon bandgap.
The emitted luminescence is [276, 280, 346]
djγ(λ, Ld) = (1−Rf(θ)) Z Z Z
gγ(z, λ, Ld)A(z, λ, θ)dzdλdθ (5.2.2)
in the paraxial approximation whereA(z, λ, θ) may be determined from Equation 2.3.36 and the Fresnel Equations 2.2.4 and 2.2.5. for cell thickness d, silicon absorption coef- ficient α(λ), and reflection Rf /b from the front/back of the cell [346]. The reflections are non-trivial functions of the emission angleθ, however, the angular dependence is not considered here, similarly to other works [204].
Assuming a p-type wafer, radiative recombination between the excess carrier density
∆n and doping NA concentrations, and normalization constant K, the generation rate at depthz in the cell isgγ(z, λ, Ld) =Kα(λ)e−2¯hπc/kBT λNAne(z, Ld)/n2i [204]. Electro- luminescence boundary conditions give
ne(z, Ld)/n0e=f−ez/Ld+f+e−z/Ld (5.2.3)
where n0e = (n2i/NA)eqV /kBT, f± = r±/(r± +r∓e∓2d/Ld) and r± = 1±SLd/Dq, for rear surface recombination velocityS, diffusivity Dq, intrinsic carrier concentration ni, junction voltageV, electron charge q, Boltzmann constant kB and temperature T [204].
The main parameters which have been investigated by their effect on the lumines- cence spectra are the rear surface recombination velocity S, which is of interest for characterization, the diffusion length Ld of bulk minority carriers, and the voltage V which must be removed from affecting the luminescence spectrum [197, 204]. Deriving Equation 5.2.1 byκ∈ {V, S, Ld}gives the change in the spectral signature dependent on physical variables of interest including the junction voltageV, the surface recombination velocity at the rear of the solar cellS, and the bulk diffusion lengthLdof minority charge carriers on the silicon solar cell.
∂FI/∂κ= 0 meaning that the constant response of the instrument does not depend on the properties of the sample. This means that the constant instrument function must be measured, or the instrument calibrated to a reference variable. Inspection shows κ affects the spectral signature through the injection process through the excess carrier concentrationne, so that ∂j∂κγ ∝ ∂ne∂κ(z,κ) which involves three Equations, namely
∂ne(z, κ)
∂V =n0e e kBT
h
ez/Ldf−+e−z/Ldf+i
(5.2.4)
∂ne(z, κ)
∂S =n0e
ez/Ld∂f−
∂S +e−z/Ld∂f+
∂S
(5.2.5)
∂ne(z, κ)
∂Ld
=n0e
ez/Ld ∂f−
∂Ld
−∂f−z
∂L2d
+e−z/Ld ∂f+
∂Ld
+∂f+z
∂L2d
(5.2.6) The effect ofκ on the spectral signature may be determined as
∂jγd(λ, Ld)
∂κ =K0 Z Z d
0
∂ne(z, κ)
∂κ A(z, λ)dzdλ. (5.2.7) We note again that the angular dependence is ignored. However, the texture of the cell will lead to a larger pathlength of the luminescence in the solar cell, and thus reabsorption will also increase, as discussed in Section 2.3. This effect could be quantified in look-up
Figure 5.2.2: (Comparison of the (a) effective diffusion length image obtained from light beam induced current measurements of a silicon wafer solar cell (b) EL image of the same cell using an indium gallium arsenide camera (c) the spatially-resolved spectrum amplitude (the peak intensity of the spectrum), and (d) the integral of the spectrum evaluated over the entire luminescence spectrum detected. The cell was scanned left-to- right, resulting in the shadow and reflection seen from the contact assembly.
tables for various kinds of textured silicon. We will employ an empirical calibration of our instrument so the texture effect may be ignored.
The characteristic behavior of the spectrum to changes in the set of physical param- etersκ may be expressed as inequalities
∂jγd
∂Ld
6= ∂jγd
∂V 6= ∂jγd
∂S (5.2.8)
by inspection of Equations 5.2.4 to 5.2.6. This suggests the potential to characterize the setκ by detecting specific features of the luminescence spectrum, with the effect of cell parametersκonjγdbeing distinct. Separating the impact of various physical parameters on luminescence emission may enhance characterization of PV materials and devices by isolating them for independent determination [184].
For example, much work has focused on the bulk diffusion length characterization of
ingots since the back surface recombination is negligible in this case. It may be possible to extract the front surface recombination velocity using the spectrum of luminescence to complement other techniques such as internal quantum efficiency analysis [479]. Further research on the impact of the bulk diffusion lengths and surface recombination veloci- ties on the spectrum may thus allow the determination of both these parameters, which would be useful for inline inspection. Currently, most characterization methods mea- sure effective lifetimes or diffusion lengths of the charge carriers in a device [284, 347].
Hyperspectral imaging instruments thus have an advantage over luminescence intensity imaging instruments.
5.2.2 Developing characterization of diffusion length of minority charge car- riers from the luminescence spectrum
For example, diffusion length characterization was developed based on a specific feature of the EL spectrum. Changes in the voltage (V, Equation 5.2.4) induce an amplitude change ∆V /kbT, but changes in the diffusion length have an effect on local features of the luminescence spectrum since the minority carrier distribution modifies the distribution of photons which are reabsorbed in the silicon. The location of the spectrum peak, the slope of low wavelength luminescence, and the spectrum skew are effected by Ld. Comparatively, the effect ofSis located at the cells rear boundary, andzis not included in the integral over the cell depth in 5.2.5. The peak intensity of the spectrum occurs at a wavelength which may be evaluated from the spectrum, which we denote asλp. To some extent, the peak shift is not affected byS, depending on the magnitude ofLd, as seen with the ratio imaging method as well [204].
Characterization ofLdbased on the wavelength of the peak intensityλp is proposed.
An empiricalλp versusLdcalibration dependent on the cell thickness, optical properties like surface texture and reflectance of the back surface, and the rear surface recombina- tion velocity was performed. Determination of theλp versusLd relationship for various types of PV devices may be completed using look-up tables, or analytically by solving FI(λ).
The empirical calibration is advantageous because an analytical description of lu- minescence emission is approximate, particularly for textured cells where reabsorption depends on the pathlength enhancement and the angular dependence of ray propagation.
Additionally, owing to the sensor properties, an intensity calibration is inconclusive over a significant wavelength range where
jγd(λ)/QE(λ) =noise/0. (5.2.9)
The empirical correlation must be determined for a distinct device. Its accuracy depends on the spatial uniformity of parameters across the cell, their variation between cells, and the accuracy of the referenceLd map.