Local Voltage Mapping of Solar Cells in the Presence of Localized Radiative Defects

Abstract

Hyperspectral electroluminescence and photoluminescence imaging of photovoltaic materials and devices produces three-dimensional spatially and spectrally resolved luminescence data, which can be calibrated to an absolute scale, enabling the extraction of high resolution maps of quantities such as the local voltage (quasi-Fermi-level splitting). This extraction requires supplemental measurements of external quantum efficiency (EQE), but these do not have the same spatial resolution. Previously, assumptions have been made to overcome this limitation. In this work, we evaluate these assumptions for InGaAs solar cells with significant spatial variation in the luminescence spectrum shape due to small regions with elevated concentrations of radiative defects. Although appropriate for small variations in spectral shape, we find that with more significant variation, these assumptions can result in non-physical EQEs and too-low voltages. Combining multiple methods can help alleviate this, or a minimum voltage map can be extracted, which will be similar to the actual voltage when EQE is high.

Photoluminescence (PL) and Electroluminescence (EL) have been used extensively in both imaging and spectral forms for characterization of photovoltaic materials and devices. Luminescence imaging can provide spatial information, revealing cracks, defects, grain boundaries, and other types of non-homogeneity either inherent to the material or introduced from processing or aging^1–4. Luminescence spectra can provide information about the semiconductor band gap, radiative efficiencies, absorptivity, external quantum efficiencies (EQEs), internal voltages (or quasi-Fermi level splitting), and pseudo-IV curves, among other values^5–9.

Hyperspectral imaging allows for the collection of spatially and spectrally resolved EL and PL data. High spectral resolution combined with radiometric calibrations of the emission signal determine luminescence flux, regardless of material or luminescence spectrum. This makes it a powerful tool able to determine those properties, such as local voltage, which are accessible by luminescence spectra for individual locations on the sample, and visualize them across the sample, identifying spatial features and non-uniformity. Alternative methods for mapping local voltage, such as KPFM and AFM, or luminescence scanning tend to take longer, have less flexibility of spatial scale, require specific sample structures, or be highly sensitive to surface properties of the sample^10–12.

However, the usefulness of the spatial resolution of hyperspectral imaging can be compromised by the need for additional measurements, which generally do not produce spatially resolved data. This has led researchers to use hyperspectral imaging data from a single location or in a spatially averaged form for the calculation of local voltages, pseudo-IV and EQE curves^13–15. In other cases, assumptions are made, such as about spatial uniformity of externally measured EQE or absorptivity, for the calculation of a spatially resolved local voltage map^15–18.

Here we consider the applicability of these assumptions to cases where the semiconductor band-edge and the luminescence spectrum shape differ across the device area, focusing on determining the high-resolution local voltage of solar cells containing areas with high concentrations of radiative defects. The methods and considerations described will apply to any photovoltaic device with sufficient luminescence for EL/PL analysis and spatial variation in the luminescence spectrum. This includes devices with larger anomalous areas than seen in this work, and microphotovoltaic cells, where anomalous areas will have larger impact on overall device performance. In all cases, qualitative understanding of the behaviour and effect of such regions is improved.

Commercial 4 cm² InGaAs on Ge solar cells with a 2 % indium alloy fraction (estimated from optical modeling) were measured with a Photon ETC. Grand-EOS hyperspectral imaging system, with spectral range of 400 nm to 1600 nm with spectral resolution down to 2 nm in both EL and PL modes. EL measurements were taken with a 20 mm field of view, showing the full cell (Fig. 1a (Multimedia view)), and under a 10 X microscope objective (Fig. 1b (Multimedia view)). PL measurements were taken under a 10 X microscope objective with 532 nm laser illumination. Absolute calibration of the image cubes is performed as described in previous work^14,19. EQE measurements were taken using the differential spectral responsivity (DSR) method²⁰.

FIG. 1.

a) Absolute hyperspectral EL image cube taken with 20mm field of view and 100 mA injected current (Multimedia version) b) The same, taken with 10 X microscope objective and 25 mA injected current, centered on region with a high concentration of radiative defects. Original image has been cropped to allow larger printing of this region. (Multimedia version) c) Absolute EL spectra extracted from indicated spots in parts a) and b).

Wide field EL measurements showed several dark areas, visible near the center of Fig. 1a (Multimedia view), which we investigated further under the 10 X microscope objective (Fig. 1b (Multimedia view)). We see that these areas are centered on small regions that show a relatively strong luminescence peak at 932 nm. This indicates a radiative defect present in large concentration in this region, with the surrounding dark area due to carriers being drained from the bulk material. We will refer to this region as the defect region. Numerous radiative defects have been identified in GaAs^21,22. Based on additional temperature-dependent analysis which will be treated in a future publication, the peak seen here is attributed to a Ga antisite defect.

The significant difference in the luminescence spectrum shape in the defect region presents challenges in determining the local voltage using previously employed assumptions. We first consider the smaller variations in luminescence spectrum shape seen throughout the device, and what they can tell us about appropriate methods of analysis.

The local junction voltage ($V$) is related to the measured electroluminescent photon flux ($R$) at a given energy, $E$, by the reciprocity relationship²³ as:

$$R\left(E\right)=EQE\left(E\right)e^{qV/kT}B\left(E\right)$$ (1)

where $q$ is the elementary charge, $k$ is the Boltzmann constant, $T$ is the cell temperature, $EQE$ is the external quantum efficiency, defined as the probability that an incident photon creates a carrier which contributes to the light-generated current, and $B$ is the black body spectral photon flux, with

$$B\left(E\right)\cong 2\pi E^2{h}^{-3}{c}^{-2}\mathit{exp}\left(-E/kT\right)$$ (2)

where $h$ is Plank’s constant and $c$ is the speed of light in a vacuum. The primary challenge in using the reciprocity relation is the interdependence of $V$ and EQE, which introduces the need for additional external measurements.

The reciprocity relationship has also been expressed and used, primarily for determining overall cell voltage and I-V curves, as^13,14:

$$R_{tot}=<EQE>e^{qV/kT}{\int }_{E_1}^{\infty }B\left(E\right)dE$$ (3)

where $R_{tot}$ is the total photon flux out of the cell and $<EQE>$ is a weighted average of the EQE, defined as:

$$<EQE>=\frac{\int R\left(E\right)EQE\left(E\right)}{\int R\left(E\right)dE}$$ (4)

$E_1$ has previously been written as $E_g$ and described as the semiconductor band gap^13,14 However, Eqns 1–4 can be rearranged to find that:

$${\int }_{E_1}^{\infty }B\left(E\right)dE=\frac{R_{tot}^2}{\int R{\left(E\right)}^2/B\left(E\right)dE}$$ (5)

We do not need to calculate $E_1$ itself to determine the local voltage, which is advantageous as Eqn. 5 must be solved numerically for $E_1$. However, if we wish to map $E_1$ for a more intuitive interpretation of the data, a good linear fit of $ln({\int }_{E_1}^{\infty }B\left(E\right)dE)$ as a function of $E_1$ can be found over a moderate range (see Supplementary Material).

Eqn. 5 shows that the spatially resolved value of $E_1$ can be definitively determined from just the absolute hyperspectral EL image cube. No knowledge of EQE or V is required. Fig. 2 shows a map of $E_1$ for this cell, as well as the local EL luminescence spectrum, local relative EQE calculated with Eqn. 1, and $E_1$ calculated with Eqn. 5 for the two spots indicated (using 5 x 5 pixel areas). Notably, $E_1$ is non-uniform, potentially as a result of specific fabrication processes. It also does not correspond to the band gap determined from the EL peak or inflection point of the local (relative) EQE curve. $E_1$ is therefore better described as an energy level which is below most of the EL photon flux (approximately 85 % in this case), and which can be calculated precisely by Eqn. 5.

FIG. 2.

a) Map of local $E_1$ calculated from hyperspectral EL image cube. b) EL spectrum, extracted local relative EQE (scaled to DSR method EQE), and $E_1$ for two locations indicated in (a).

Although $E_1$ is determinable, finding $V$ using Eqn. 3 still requires knowledge of the EQE. Therefore, extracting the local voltage requires knowledge of the local EQE, which is not obtainable by the DSR method. Local relative EQE curves can be extracted by the reciprocity equation, as in Fig. 2b, but absolute values are dependent on the local voltage. One approach is to use Eqn. 1 and assume a spatially uniform EQE from the DSR measurement, which we will refer to as the “uniform EQE method” going forward. Another is to, also with Eqn. 1, assume a spatially uniform EQE only for a spectral region that is both well above the band gap, and contains noticeable luminescence^16–18, which we will refer to as the “uniform high-energy EQE method”. For these devices, we have used an average of the EQE over the 850 to 870 nm range. The second option is forgiving of local variations in the EQE near the band gap. Eqn. 3 also provides the option of assuming a spatially uniform $<EQE>$, with the locally calculated $E_1$, which we will refer to as the “uniform $<EQE>\left(E_1\right)$ method”. A similar method with the locally calculated $E_1$ replaced by a uniform E_g based on DSR EQE measurements will be referred to as the “uniform $<EQE>\left(E_g\right)$ method”.

We now extract and compare local voltage maps of this device using the options discussed above, with the results shown in the top row of Fig. 3, and differences between the methods shown in the bottom row. Fig. 3a shows the voltage map extracted with the uniform high-energy EQE method, the most commonly accepted method in previous work^16–18. The effect of error in the EQE measurement is shown in Fig. 3b, with a 10 % error in the EQE corresponding to less than 3 mV error in voltage. Fig. 3c shows the voltage map extracted with the uniform EQE method, and Fig. 3f shows the error compared to the uniform high-energy EQE version.

FIG. 3.

Voltage maps extracted using a) uniform high-energy EQE, c) uniform EQE, d) uniform $<EQE>\left(E_g\right)$ and e) uniform $<EQE>\left(E_1\right)$ methods. b) Shows the effect that an error in EQE has on the extracted voltage, and f-h) show the difference between the voltage maps above and that shown in part (a).

Fig. 3d and e show the voltage map calculated by the uniform $<EQE>\left(E_g\right)$ and uniform $<EQE>\left(E_1\right)$ methods, respectively. Fig. 3g and h show the error using these methods compared to the uniform high-energy EQE method. The uniform $<EQE>\left(E_1\right)$ is, as expected, consistent with the uniform high-energy EQE results, while the uniform $<EQE>\left(E_g\right)$ results in tens of mV of non-spatially uniform error. Therefore, the qualitative understanding of the local voltage pattern is adversely affected, along with the quantitative values. The uniform EQE method results in smaller errors (Fig. 3f), but, due to the band-edge variation present in this device (Fig. 2), also introduces spatial variation in local voltage which is not present with the uniform high-energy EQE method. Additionally, this method can be influenced by the selection of the integration limits applied to Eqn. 1, particularly the low-energy limit if the DSR-measured EQE does not go strictly to zero below the band gap. As the DSR method is least accurate when EQE is low, errors in this range are expected¹⁵. Based on these results, the most reliable methods for determining the local voltage in the presence of small band-edge fluctuations are the uniform high-energy EQE and uniform $<EQE>\left(E_1\right)$ methods.

We now consider more significant variations in the luminescence spectrum, in this case, regions with a high concentrations of radiative defects resulting in an EL peak at 932 nm.

Using the uniform EQE method is particularly problematic here. Doing so produces the voltage map seen in Fig. 4a, with an apparent higher local voltage in the immediate defect region than in the surrounding area. However, this would not result in the draining of carriers from the surrounding region that is indicated by the dark area in the EL image (Fig. 1). The strong luminescence from the radiative defect might naively be expected to indicate a high voltage in this region, but this fails to take into account the rapid increase in $B(E)$ with decreasing energy.

FIG. 4.

a) Local voltage map extracted from 10 X EL hyperspectral image cube with the uniform EQE method. Arrow points to apparent high voltage in defect region. b) Local relative EQEs extracted from defect and bulk regions, scaled to DSR-EQE in high-energy region (left axis) and effect that the luminescence wavelength has on the extracted voltage (right axis, dashed line). c) Areas for which assuming spatially uniform high-energy EQE or uniform $<EQE>$ result in non-physical spectral EQE. d) Voltage map extracted using maximum of uniform high-energy EQE and uniform $<EQE>\left(E_1\right)$ methods. e) Minimum possible voltage map. f) difference between (d) and (e).

$B(E)$ is 12 times as high at 932 nm as it is at 890 nm, resulting in a 60mV lower $V$ for the same luminescence flux measured at 932 nm compared to that measured at 890 nm, assuming a constant EQE (Fig. 4b). Therefore, the strong luminescence peak at 932 nm in the defect region does not necessarily indicate a higher voltage in this region. We note that this effect will also be important in applying previous luminescence imaging methods, such as lifetime mapping, which implicitly rely on the spatial homogeneity of the luminescence spectrum¹, to cells with radiative defect regions or other significant spatial variation in luminescence spectrum.

We now consider the uniform high-energy EQE method. Fig. 4b shows a relative EQE curve from a 5 by 5 pixel square at the center of the defect region, scaled to match DSR-EQE between 850 nm and 870 nm. Not only does the shape of the EQE differ greatly from that measured by DSR or extracted from other regions of the device, this scaling results in an EQE over 100 % between 912 and 934 nm. Therefore, the uniform high-energy EQE method would result in underestimation of the voltage in the defect region. The areas where such an assumption results in a non-physical EQE are shown in red in Fig. 4c.

If we use the uniform $<EQE>\left(E_1\right)$ method, with $<EQE>$ calculated from the DSR-measured EQE and the spatially averaged EL spectrum, the associated EQE in the middle of the defect region remains nominally below 100 %. However, around the edges of this region, where both the 890 nm and 932 nm peaks are present in similar intensity, the associated EQE(E) becomes greater than 100 %. This is due to the dependence of $B$ on $E$ described above. When calculating $<EQE>$, both peaks contribute relatively equally to the denominator in Eqn. 4, but the lower energy peak contributes relatively little to the numerator, as the $\mathit{EQE}(E)$ term is proportional to $1/B(E)$. Therefore $<EQE>$ in areas with two peaks of similar intensity will be lower than in areas with only one significant peak. The areas for which the maximum physical $<EQE>$ is less than 95 % of our uniform value are shown in blue in Fig. 4c, and have little overlap with the regions for which the uniform high-energy EQE is non-physical. As using a higher EQE results in extracting a lower voltage (Eqn. 1), we eliminate the non-physical (too-high EQE) region by taking the maximum voltage from the uniform high-energy EQE and uniform $<EQE>$ methods, to find the voltage map shown in Fig. 4d. Using each method alone results in voltages up to 22 meV lower in or near the defect region of this sample.

Having noted that we are limited by the physical requirement that EQE not be above 100 %, we can also calculate a local minimum voltage map, which is the voltage map if we use Eqn. 1 and determine the EQE of each pixel by scaling the relative EQE such that its spectral maximum is 100 %. We do this using the average value between 850 nm and 870 nm (to reduce noise where luminescence is low) and the maximum value between 920 nm and 940 nm, and find the map shown in Fig. 4e. Fig. 4f shows the difference between the method described in the previous paragraph, and the minimum voltage. In the bulk of the device, where the higher energy peak dominates, the two are very similar due to the high EQE of the device. Either of these methods therefore provides a reasonable map of the local voltage in such a situation. If the device EQE is known to be significantly lower than 100 %, the first method may be more appropriate, or the minimum voltage can be calculated by scaling for a different maximum EQE(E) in each pixel.

The methods above are also applicable to PL measurements, using the generalized Planck equation, which follows the form of Eqn. 1, with EQE replaced by the absorptivity, A (the probability that a photon incident on the surface will be absorbed). It has been previously stated that, for solar cells, A can be taken to be 1 at energies above the band gap and 0 at energies below the band gap^16,24. PL hyperspectral imaging of this device (Fig. 5a), however, shows that inside the defect region, A is higher around 930 nm than in the aboveband-gap range (Fig. 5b). Therefore, we scale A such that the spectral maximum is equal to 1, producing a minimum voltage map as in Fig. 5c as a good approximation for the local voltage in this case. At this illumination condition, the voltage in the defect area is about 100 mV lower than in bulk locations a similar distance from the metal finger, compared to the 60 meV difference in luminescence peak energies and local $E_1$. This could indicate that the high concentration of defects in this region also increases non-radiative recombination.

FIG. 5.

a) Two frames of a hyperspectral PL image cube, taken with a 10 X microscope objective and an illumination intensity of 230 mW/cm². b) Local relative A extracted from two spots. c) Minimum voltage map extracted from PL hyperspectral image cube.

The determination of high-resolution spatially resolved local voltage in photovoltaic devices and materials from absolutely calibrated hyperspectral EL and PL image cubes can be a valuable characterization tool, but requires the application of certain assumptions as to device EQE or absorptivity. If local variations are small, an assumption of a uniform high-energy EQE or A may be valid, but when hyperspectral imaging reveals small regions with a high concentration of radiative defects, or other significant variations in spectral luminescence shape, this assumption cannot be used uncritically. Attention must be paid in particular to ensure that scaling EQE or A by one spectral region does not result in non-physical values of EQE or A in another spectral region. The correct use of an alternate form of the reciprocity or generalized Planck equation can extend the spatial region for which a valid scaling can be achieved. In extreme cases, only a local minimum voltage map may be extractable, but even this can provide valuable information.

DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available from the corresponding author upon reasonable request