Specific series resistance evaluation using photoluminescence signal of Si solar cells

Te-yuan Chung; Ying-Chang Chung; Sheng-Hui Chen

doi:10.1364/OE.20.00A822

1. Introduction

Series resistance is one of the most important parameters of solar cells. Several methods based on measuring the i-V curve of solar cells have been proposed to evaluate the effective series resistance [1]. Usually, these methods are slow due to their requirement of a complicated measurement system or procedure. Trupke et al. proposed using the photoluminescence (PL) method to evaluate specific series resistance in a unit of Ω⋅cm² of solar cells [3]. This method can resolve the spatial distribution for the specific series resistance and dramatically increase the speed of measurement. However, this method requires a calibration parameter on each position of the solar cell to obtain the correct corresponding voltage. PL images under different electrical and different illumination conditions are required for each sample. Kampwerth et al. proposed an advanced version of a similar measurement scheme [4]. In this paper, a more precise method and theoretical background are given. However, this method still demands complicated measurement steps. Moreover these two methods both demand highly uniform illumination to ensure the simplicity of the calculation [5].

Here, we propose a method to evaluate the spatial distribution of specific series resistance over a solar cell using a single frame of a PL image under the short circuit condition. The specific series resistance values obtained does not vary with the intensity of the illumination.

2. Theory

Figure 1 shows a solar cell equivalent circuit [6] which consists of a diode, a current source, a series resistor and a shunt resistor. R_s is the series resistance, R_sh is the shunt resistance, V_ph is the photovoltage, V_term is the terminal voltage, i_ph is the photocurrent and i_term is the terminal current. R_load is the external resistance load of the solar cell. For an efficient solar cell, the shunt resistance should be large enough to reduce the leakage yet the series resistance should be small to reduce the power consumption inside the solar cell. In the following discussion, the shunt resistance is assumed to be much larger than the series resistance and will be excluded.

Fig. 1 Equivalent circuit of a solar cell.

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The i-V relation of the diode part of the solar cell without light illumination can be written as the Shockley ideal diode Eq. or

i = i_{s} (e^{\frac{q \cdot V_{D}}{η_{i d e a l} \cdot k \cdot T}} - 1),

where i_s is the saturation current, q is the electron charge, V_D is the voltage across the ideal diode, k is the Boltzmann constant, T is the temperature and η_ideal is the ideality factor of the solar cell. The saturation current can be written as [7]

i_{s} = q \cdot A \cdot (\sqrt{\frac{D_{n p}}{τ}} \cdot p_{n 0} + \sqrt{\frac{D_{p n}}{τ}} \cdot n_{p 0}),

where A is the cross section area, p_n₀ is the hole number density in the n-type region, n_p₀ is the electron number density in the p-type region, D_np is the hole diffusivity in the n-type region, D_pn is the electron diffusivity in the p-type region, and τ is the excess carrier lifetime. The carrier diffusivities can be obtained when the corresponding doping concentrations are given. The excess carrier lifetime spatial distribution can also be obtained using the PL method [5, 8, 9]. Under illumination and with an external load, the terminal current can be represented as

i_{t e r m} = i_{p h} - i_{s} \cdot [e^{\frac{q \cdot (V_{t e r m} - i_{t e r m} \cdot R_{s})}{η \cdot k \cdot T}} - 1] - \frac{V_{t e r m} - i_{t e r m} \cdot R_{s}}{R_{s h}} .

With the large shunt resistance assumption, the second term of Eq. (3) may be neglected. Since a solar cell can be considered as many small solar cell nodes connected in parallel [3, 4], using the current density is more flexible in the following discussion. Therefore, the series resistance and the contributed current product on each node can than be written as

i_{t e r m} \cdot R_{s} = j_{t e r m} \cdot R_{s - s},

where j_term is the contributed current density on each solar cell node and R_s-s is known as the specific series resistance, which is in units of Ω⋅cm². Equation (1) and Eq. (2) can be rewritten as

j_{t e r m} = j_{s} \cdot (e^{\frac{V_{t e r m} - j_{t e r m} \cdot R_{s - s}}{η_{i d e a l} \cdot V_{T}}} - 1),

j_{s} = q \cdot (\sqrt{\frac{D_{n p}}{τ}} \cdot p_{n 0} + \sqrt{\frac{D_{p n}}{τ}} \cdot n_{p 0}),

where j_s is the saturation current of each node and is a function of position. Equation (4) can then be rewritten as

V_{t e r m} = \frac{1}{η_{i d e a l} \cdot V_{T}} \cdot \ln (\frac{j_{t e r m}}{j_{s}} + 1) + j_{t e r m} \cdot R_{s - s} .

In the PL experiment, the total terminal current and terminal voltage can both be measured. However, each node of the solar cell offers individual voltage and current. The light absorbed by the solar cell generates excess carriers which eventually recombine. The recombination mechanism can be categorized into non-radiative recombination, radiative recombination and outflow current. Since silicon is well-known for its disappointingly low efficiency of band-to-band PL signal, we define the efficiency of excess as the PL signal from each node on the solar cell which is contributed by a small portion of the excess carriers generated. Under the assumption of the carrier diffusion length is much larger than the thickness of the solar cell, the PL quantum efficiency can be defined as

η_{P L} (x, y) = \frac{I_{P L_i n t} (x, y)}{I_{i l l u_i n t} (x, y)} \frac{h ν_{i l l u}}{h ν_{P L}} = \frac{I_{P L_i n t} (x, y)}{I_{i l l u_e x t} (x, y) \cdot [1 - R_{i l l u} (x, y)]} \frac{h ν_{i l l u}}{h ν_{P L}},

where the symbol I indicates the light intensity or exitance, ν indicates the frequency of the photon, R indicates the effective reflectivity of the light at the surface of the solar cell and h indicates the Plank’s constant. The subscripts _PL and _illu represent the PL signal and illumination, respectively; the subscripts _{_int} and _{_ext} indicate the quantity is inside and outside the solar cell, correspondingly. The illumination and PL intensities can be obtained experimentally. The observed PL signal by the CCD can be written as

I_{P L_e x t} (x, y) = I_{P L_i n t} (x, y) \cdot [1 - R_{P L} (x, y)] \cdot η_{o p t} (x, y),

where η_opt indicates the collecting efficiency of the entire optical system.

The solar cell can be operated under a short circuit or open circuit condition which will be indicated by a subscript labeled as _sc or _oc respectively in the following discussion. The total excess carrier density generation rate can be written as

{\dot{n}}_{t o t} (x, y) = G (x, y) = \frac{I_{i l l u_i n t} (x, y)}{h ν_{i l l u} \cdot W},

where W is the thickness of the solar cell. Since silicon is known for its poor band-to-band emission efficiency, the PL signal is only contributed by a small fraction of the excess carrier recombination. The PL signal can be derived and is proportional to the product of the local electron and hole number density for silicon [10, 11]. In other words, the outflow carriers reduce the local carrier number density and reduce the PL signal. Here we assume under short circuit condition, only a small fraction of carriers contribute to the current. In other words, different recombination channels remain having the same rates and the PL quantum efficiency defined can be applied. The excess carrier density recombination rate contributed to the PL signal can then be written as Eq. (11) under open circuit and short circuit conditions, respectively.

\begin{array}{l} {\dot{n}}_{P L, o c} (x, y) = η_{P L} (x, y) \cdot G (x, y) \\ {\dot{n}}_{P L, s c} (x, y) \approx η_{P L} (x, y) \cdot [G (x, y) - {\dot{n}}_{i} (x, y)] \end{array}

η_{P L}

is the efficiency of local excess carrier density conversion to PL signals. In general,

η_{P L}

is also a function of temperature, doping concentration, excess carrier lifetime and optical geometry;

{\dot{n}}_{i}

is the change rate of the excess carrier density which contributes to the outflow current. Equation (11) is proportional to the PL signal generated and can be written as

I_{P L_i n t} (x, y) ​ = {\dot{n}}_{P L} (x, y) \cdot h ν_{P L} \cdot W

Under identical illumination for both circuitry connection conditions, an outflow current efficiency can be defined and be derived using Eq. (9-12) as

η_{i} (x, y) \equiv \frac{{\dot{n}}_{i} (x, y)}{G (x, y)} = 1 - \frac{{\dot{n}}_{P L, s c} (x, y)}{{\dot{n}}_{P L, o c} (x, y)} = 1 - \frac{I_{P L_e x t, s c} (x, y)}{I_{P L_e x t, o c} (x, y)} .

Consequently, the short circuit current density on each node can be obtained as

j_{s c} (x, y) = \frac{i_{s c}}{A_{t o t}} \frac{η_{i} (x, y)}{\iint_{x, y} η_{i} (x, y) d x d y},

where i_sc is the total short circuit current of the entire solar cell and A_tot is the total illuminated area of the solar cell sample. Under the short circuit condition, the current provided by each node is the contributed terminal current. Meanwhile, V_term is 0, the saturation current density, excess carrier lifetime and the specific series resistance corresponding to each node can then be solved simultaneously using Eq. (7) and 2D Newton’s method to find the roots. Although the current provided by each node is different. Equation (13) does not contain

η_{P L}

which implies the illumination intensity uniformity will not influence the calculation.

3. Experimental setup

Figure 2 shows the setup for the PL experiment. The illumination light source is an 808 nm diode laser with a maximum output power of 50 W (Limo FP-A 0263). The laser output has passed a bandpass filter, Filter 1, to eliminate the long wavelength sideband of the diode laser and ensure only the laser wavelength is delivered to the following optical system. The laser light enters a rectangular lightpipe (Newport LPH-PIP-8) with a clear aperture of 8 × 8 mm² and 10 cm in length. The lightpipe is labeled as LP. A convex lens L1 with a focal length of 2.54 cm is placed near the exit facet of the lightpipe and is used to project the laser light onto the solar cell sample which is about 14.5 cm away from the lens. The normal direction of the solar cell sample has a 26.6° angle with respect to the optical axis of the incident laser light. The sample is in thermal contact with a temperature controlled aluminum plate which is labeled TC. The aluminum plate temperature is set to be 27°C. The sample electrodes are connected to a digital multimeter to measure the short circuit current of the sample. A CCD camera (Electrophysics Micron Viewer 7290A) is placed 42 cm away from the normal of the solar cell sample. A band pass filter, Filter 2, is placed on the CCD camera to ensure the observed signal is from the bandgap transition of the silicon which is 1.1 μm. The transmission spectrum of the filter is shown in Fig. 3 .

Fig. 2 Experimental setup.

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Fig. 3 The transmission spectrum of the filter for the CCD.

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The illumination light distribution is shown in Fig. 4 . The illumination pattern has an X-shaped dark pattern which is about 75% of the neighboring illumination intensity in average. It is caused by the chamfer of the light pipe. There is a more than 4-fold variation in the overall illumination intensity. However, the effect of non-uniform illumination will be cancelled in the calculation.

Fig. 4 (a) Laser illumination pattern on the sample. (b) The intensity distribution along the dashed line in (a). The intensity distribution over the illumination area has a variation as large as 40%.

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4. Results and discussion

With the knowledge of the illumination and PL signal distribution, the photovoltage distribution under the open circuit condition can be calculated as shown in Fig. 5(a) . Along the red dotted line, the voltage versus position can be plotted as in Fig. 5(b). The voltage varies with position with the period exactly matching the wire grids. In each period, the higher voltage part is located roughly in the middle of the position of two neighboring wires. Overall, the maximum voltage is about 0.697 V which is located at the position with higher light illumination intensity on the upper right part of the illumination area. Figure 6(a) shows the calculated spatial distribution of the specific series resistance of the sample under the average illumination intensity of 0.82 W/cm² using the method proposed here. The darker regions correspond to lower specific series resistance. The dark pattern along the vertical direction indicates the regions near the electrode grid lines where the specific series resistance is noticeably smaller. On the other hand, the specific series resistance is larger right in between the two neighboring grid lines since the current path is expected to be longer. The average specific series resistance is 1.42 Ω⋅cm² and the error is about 5.0 × 10⁻³ Ω⋅cm². Figure 6(b) shows the calculated spatial distribution of the specific series resistance under the average illumination intensity of 0.67 W/cm². The average specific series resistance is 1.45 Ω⋅cm². The average error is about 0.01 Ω⋅cm². Figure 6(c) shows the pixel-to-pixel fraction between Fig. 6(a) and 6(b). The average value of Fig. 6(c) is about 1.02 which indicates that the proposed method obtains well-agreed results under different levels of illumination.

Fig. 5 (a) Calculated photovoltage distribution of the solar cell sample, (b) The photovoltage profile along the red dashed line in (a).

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Fig. 6 Using the proposed method, the specific series resistance distribution of a solar cell can be calculated under an average illumination intensity of (a) 0.82 W/cm² and (b) 0.67 W/cm². (c) is the result of (a) divided by (b).

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5. Conclusion

A method for calculating the specific series resistance spatial distribution of solar cells is proposed; it uses the PL signal of the solar cell under short circuit and open circuit conditions along with the ideal diode Eq. and saturation current. The specific series resistance distribution obtained with this method is insensitive to the PL illumination intensity.

Acknowledgment

The author would like to specially thank Mr. Chia-Liang Yeh for his contribution. This work is financially supported by the National Science Council and the Ministry of Education of Taiwan under project numbers NSC100-2221-E-008-110, NSC101-2221-E-008-053, NSC 101-3113-E-008-001 and NCU-DEL-101-A-04.

References and links

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Specific series resistance evaluation using photoluminescence signal of Si solar cells

Abstract

1. Introduction

2. Theory

3. Experimental setup

4. Results and discussion

5. Conclusion

Acknowledgment

References and links

Cited By

Figures (6)

Equations (14)

Optics Express