2.1.

Conceptual Design of Back-Contact Solar Cells

We present a conceptual illustration of the back-contact SC layout in Fig. 1.

Fig. 1

Simplified illustration of an interdigitated back-contact SC architecture.¹⁷ The polarity labels match those on the C60 SCs and indicate the direction of current flow when an external load is attached. Light incident from the top in this diagram creates electron–hole pairs in the monocrystalline silicon, and the resulting photocharge moves to back-side electrodes that reside under doped regions. The PN junction occurs between the electrodes and creates lateral electrical fields. A layer of doping on the top surface drives the electrons away from surface states and creates a local vertical electric field component. As we discuss in Sec. 3.5, the periodicity of the electrode/doping structure may give rise to regions where charge trapping can occur, degrading the effective quantum efficiency (EQE).

The device’s quantum efficiency (QE), the number of collected charge carriers per incident photon, is a function of incident photon wavelength, $\lambda$. As with CCDs, the index of refraction mismatch produces reflections, especially for $\lambda <400\mathrm{nm}$. At long wavelengths, the limitation is photon energy approaching the bandgap.

In our use of SCs as photometric calibrators, we care about their efficiency in turning incident photons into measurable electrical current, what we refer to as their EQE. This stands in contrast to the conventional application of SCs in generating renewable energy, where the figure of merit is the fraction of incident solar energy converted into useful electrical power or the energy efficiency. A single incident photon excites a single electron, but all energy of the incident photon in excess of the bandgap energy is lost as heat. Thus, a cell’s EQE is generally greater than its energy efficiency.

Note that a cell’s EQE is the product of its true QE and the fraction of the excited electrons that are successfully driven across the current-amplifier load. It is always true that $\mathrm{QE}>\mathrm{EQE}$.

In back-contact SCs, the electrical potential is periodic in the direction perpendicular to the electrodes, creating a series of regions where the perpendicular electric field is zero. This will reduce the QE of the device if the charge carriers dwell in these regions of null electric field long enough to recombine, or if these corrugations in the electrical potential trap photon-excited charges. We search for and discuss the spatial dependence of a cell’s EQE in Sec. 3.5.

Other internal loss mechanisms can also reduce the SC QE. These mechanisms include charge traps (local minima in the electrical potential energy), undesirable energy levels from contamination and defects, and electron–hole recombination. Manufacturers have worked hard to maximize the QE of silicon SCs by minimizing these effects and by applying an antireflection (AR) coating to the detecting surface to minimize reflections. The QE of contemporary SCs is in excess of 95% over much of the wavelength range of interest.^18,19 Detailed simulations of the charge collection efficiency as a function of electrode geometry and doping concentration²⁰ have been useful in guiding the design of these devices.

2.2.

Useful Equivalent Circuit

In analyzing the properties of a SC, we found it useful to model the cell’s electrical behavior using a simple equivalent circuit (EC). In this work, we use the standard single-diode EC²¹ shown in Fig. 2. A more robust analysis could generalize to a two-diode model^22,23 to account for distinct recombination timescales.

Fig. 2

(a) The single-diode EC and (b) current-source EC for the current amplifier that we use to characterize the C60 SCs. Our measurements suggest that every C60 SC has distinct characteristic values. Typically, $R_{\mathrm{ser}}\sim \mathcal{O}(0.01\text{to}0.1\mathrm{\Omega }),R_{\mathrm{sh}}^{1\mu \mathrm{A}}\sim \mathcal{O}(100\mathrm{to}1000\mathrm{\Omega })$, and $C_{\mathrm{sh}}\sim \mathcal{O}(1\text{to}10\mu \mathrm{F})$. The arrows of the current sources indicate the direction of current flow relative to the polarity designations (±) on the electrodes of the C60 SC.

In this model circuit, a pair of current sources drive current toward the output terminals. One source, $I_L$, increases monotonically with the incident light. The other, $I_B$, sources the bias current. With no external load, these currents flow through the system’s internal diode, shunt resistance, $R_{\mathrm{sh}}$, and shunt capacitance, $C_{\mathrm{sh}}$. When the SC is connected to an electrical load, such as a current amplifier, current is also driven across this load in series with the cell’s series resistance, $R_{\mathrm{ser}}$. Our tests in Secs. 3.2 and 3.4 find that this single diode circuit is a good model for the C60 SCs with one notable exception: when back biased, the cell’s current response deviates from the linear behavior expected from $R_{\mathrm{sh}}$, presumably due to quantum tunneling.²⁴

2.3.

Specific Example: C60 Solar Cells

We obtained a sample of Sunpower C60 monocrystalline back-contact silicon SCs (“C60 SCs” henceforth) from the Amazon online storefront.

The C60 SCs are 125-mm truncated square silicon elements with interdigitated anode and cathode electrodes. The front (illuminated) side has no electrical connection and the electrodes on the back (dark) side have solder tab connections. The vendor advertises an AR coating as having been applied to enhance photon conversion efficiency in the blue. The vendor claims a QE of $>0.95$ for $\lambda$ between 500 and 900 nm and a QE of $>0.9$ for $\lambda$ between 400 and 1050 nm, consistent with the modern standards for interdigitated back-contact SCs.^18,19

The vendor’s specifications are listed in Table 1. Front and back images of a C60 SC are shown in Fig. 3.

Table 1

Manufacturer’s characteristics of C60 SC.

Fig. 3

(a) Front and (b) back images of a C60 SC. The cells are 125 mm squares with truncated corners. The magnification of the back side shows the electrodes (silver bands) fused to the back of the silicon surface (dark bands) with a ruler for scale. Parallel electrodes sit alternately at positive and negative voltages. Parallel electrodes are 0.46 mm apart and parallel electrodes of matching electric potential are 0.92 mm apart.

3.1.

Preparing the Cells for Use

The C60 SCs do not come with any external wires for connection to other devices. The manufacturer recommends that the user directly solder a pair of specially designed “interconnect” tabs, one for the anode and one for the cathode, to the system’s solder points. However, we found that soldering the SCs directly caused their unbiased dark currents to increase by a factor of 5 to 100 or more. Any nonphotoelectric current introduces measurement error (both statistical and systematic) and ought to be minimized.

We soldered wire leads to the interconnect tabs and used electrically conductive copper tape, reinforced with nonconductive super glue, to affix the interconnect tabs to the SCs. This method produced stable connections at room temperature without degrading the dark current. However, we are concerned about these attachment’s multiyear reliability and their stability under large current loads. We suspect that a conducting adhesive would yield better long-term results.

3.2.

Unilluminated IV Curve of C60 Solar Cell

As we model in Fig. 2, a current amplifier attached to the output SC imposes a bias voltage, $V_b$, and carries a load impedance, $Z_L$. An ideal current amplifier has both $V_b=0$ and $Z_L=0$. When a current, $I_{\text{Out}}$, passes through the load, the voltage drop over the series resistance and the load will drive a current, $I_{\mathrm{sh}}$, through the shunt impedance, $Z_{\mathrm{sh}}$:

Current generated within the SC, $I_{\mathrm{SC}}$, will split between the shunt impedance and the load:

The fraction of the SC current that passes through the load is thus dependent on the applied bias voltage and the ratio of the impedances:

A value of $I_{\text{Out}}/I_{\mathrm{SC}}$ less than unity reduces the EQE of the detector over all photon wavelengths. Any variations in the resistances or in the bias voltage will produce a wavelength-independent, systematic change in the cell’s EQE and a corresponding error in flux measurements. Furthermore, because the shunt impedance of the SC is nonohmic [see Fig. 4(b)], this QE reduction depends on the incident light intensity and is a potential source of system nonlinearity. It is paramount that $r$ be kept as large as possible by minimizing the series resistance and by maximizing the shunt resistance. In practice, this means making good electrical connections and selecting SCs with the largest values of $R_{\mathrm{sh}}$.

Fig. 4

The measured shunt and series resistances ($R_{\mathrm{sh}}$ and $R_{\mathrm{ser}}$) of 37 unmodified C60 SCs (a) and the full unilluminated $IV$ curve of the AX SC after wiring (b). In (a), each point is a different SC with a sequentially assigned label and the line with arrows shows increasing values of the figure of merit, $r=R_{\mathrm{sh}}/(R_{\mathrm{ser}}+R_L)$ when the load resistance, $R_L$, is negligible. Those cells with the largest value of $r$, in this case cells AN, AZ, and BN, are the best candidates for use in a precision photometric calibrator. In the right plot, we measured the $IV$ curve of the AX SC. We fit those points for applied voltages above 500 mV (crosses) to measure a best fit value of $R_{\mathrm{ser}}=0.192\mathrm{\Omega }$ (blue solid line) and we fit those points with applied voltages between $\pm 100\mathrm{mV}$ (red dots) to determine a best fit value of $R_{\mathrm{sh}}=1010\mathrm{\Omega }$ (red dashed line). Comparing these results to the measured resistances before the cell was wired [the point labeled “AX” in (a)], the shunt resistance is about 120% larger and the series resistance is about 1300% larger. The increased series resistance is likely due to the resistance of the affixed wiring and underscores the importance of good electrical contacts and short electrical leads. The increase in shunt resistance is somewhat surprising and something that we do not presently understand.

To better understand the device-to-device variation in $r$, we measured the shunt and series resistances of 37 C60 SCs before affixing the conducting tabs.

According to the EC of Sec. 2.2, $I_{\text{Out}}$ for an unilluminated SC with forward-bias voltage, $V_{\mathrm{fb}}$, is determined by Ohm’s law:

When the applied voltage biases the diode into conduction, $R_{\mathrm{sh}}$ is shorted and has a negligible effect on the cell’s internal resistance. The dc behavior of the cell in this regime is determined entirely by the summed impedances of the diode and the series resistance:

In Fig. 4(a), we show the measured values of $R_{\mathrm{sh}}$ and $R_{\mathrm{ser}}$, along with contours for the $r$ figure of merit assuming a shorted load.

Opting to preserve those cells with the largest values of $r$ for future projects, we measured the full, unilluminated room-temperature $IV$ curve of the C60 SC “AX” after affixing electrical leads by the process described in Sec. 3.1. We show this curve in Fig. 4(b). To measure the ohmic behavior of the cell for small dc bias voltages, we fit a line to those points for which the magnitude of the applied voltage is below 100 mV [red points in Fig. 4(b)] and took the inverse of the fitted slope as the shunt resistance, $R_{\mathrm{sh}}$. To measure the cell’s series resistance, $R_{\mathrm{ser}}$, we fit the $IV$ curve for points with forward-bias voltages above 0.5 V [blue points in Fig. 4(b)] using Eq. (5). These two slopes imply a $R_{\mathrm{sh}}$ and $R_{\mathrm{ser}}$ of 1010 and $0.192\mathrm{\Omega }$, respectively. By fitting the current of the cell for applied voltages $<3\mathrm{mV}$ in magnitude, we measured the cell’s intrinsic dark current, $I_B$, to be 2 nA.

Although these cells partially conform to the behavior predicted by the EC of Fig. 2, they are nonohmic. For example, cell AX has a nonlinear reverse current response to increasing back-bias voltages [negative voltages in Fig. 4(b)]. A back-bias voltage of 400 mV drives 3.4 times more reverse current through the SC than the EC predicts, with the disparity growing more extreme as the back-bias voltage grows.

3.3.

Determining Nonlinearity of QE

We define the nonlinearity of a SC as the dependence of its QE on the incident photon flux, $\varphi$. An ideal SC is perfectly linear, meaning its QE is entirely independent of $\varphi$. Our method for determining the nonlinearity of the C60 SCs combines the simultaneous illumination of a reference and uncharacterized detector²⁶ with the use of a lock-in (LI) amplifier.²⁷

We used a Stanford Research Systems (SRS) DS345 function generator to drive a set of Thorlabs LEDs, listed in Table 2, with a 13 Hz 0 to 2 V square wave. We connected the LEDs to a 1.5-mm-diameter optical fiber, which illuminated one port of a Thorlabs 2” IS through a logarithmically varying neutral density (ND) filter and an iris. A Thorlabs SM1PD2A PD occupied another IS port, and the SC stood several inches away from a third IS port. The IS ports for the fiber, the PD, and the SC were mutually orthogonal. The SC’s collecting area was overfilled by the output of the IS. BNC cables connected both the SC and the PD to a BNC switch box, the output of which we connected to an SRS SR570 current preamplifier. We fed the output of the preamplifier into the input of an SRS SR2124 LI that was synchronized to the frequency of the function generator. The BNC switch box toggled between the PD and SC signals, both subject to the same loading effects of the downstream instruments. By moving the position of the variable ND filter with a translation stage, we changed the illumination intensity of the IS.

Table 2

The Thorlabs LEDsa used to study the linearity of the C60 SCs.

In Fig. 5, we measure the cell’s linearity by plotting the normalized ratio of the SC- and PD-modulated currents as a function of SC current densities assuming homogeneous illumination. This ratio is constant to within 1% over the range of monitored fluxes, suggesting that the C60 SCs are linear to within 1% for photocurrent densities between $\sim 100\mathrm{pA}/{\mathrm{cm}}^2$ and $\sim 100\mathrm{nA}/{\mathrm{cm}}^2$ for the wavelengths tested. The LSST CBP remains in development.¹⁴ Internal estimates predict current densities of $\mathcal{O}(10\mathrm{to}100{\mathrm{pA}/\mathrm{cm}}^2)$ for a monochromatic beam with adjustable wavelengths between 300 and 1100 nm. The linearity tests reported here demonstrate 1% linearity over a portion of this expected CBP parameter space, but further verification may be necessary when the design is finalized.

Fig. 5

C60 SC nonlinearity measured using an LI amplifier for the LEDs listed in Table 2 measured relative to a Thorlabs SM1PD2A PD. On the $x$ axis, we plot SC current density assuming that the photocurrent is generated uniformly across the cell surface. On the $y$ axis, we plot the ratio of SC and photodiode currents, both measured using the same LI amplifier. The ratios have been normalized to unity, and the gray shadings show the $\pm 1\%$ linearity bounds. Based on these results, the SC is linear to within 1% for photocurrent densities between $\sim 100\mathrm{pA}/{\mathrm{cm}}^2$ and $\sim 100\mathrm{nA}/{\mathrm{cm}}^2$ for incident photon wavelengths between 365 and 625 nm.

3.4.

Rise Time/Frequency Response

We are interested in applications where the calibration light is modulated at some modest frequency to discriminate calibration light from ambient light. To determine a SC’s viability as a calibrator for such a modulated light source, we must determine two semidistinct frequency dependencies.

Both of these effects could reduce the amplifier signal as the frequency of modulation increases.

We measured the shunt impedance of the SC, $Z_{\mathrm{SC}}(f)$, by measuring the voltage across a series capacitance of $1\mu \mathrm{F}$ given an applied 50-mV peak-to-peak sinusoidal voltage of varying frequency. The output voltage, $V_{\text{Out}}$, is determined by $Z_{\mathrm{SC}}(f)$ and by the impedance of the reference capacitor, $Z_{\mathrm{ref}}(f)=-i{(2\pi f{C}_{\mathrm{ref}})}^{-1}$, according to

In Fig. 6, we show both this measurement scheme and the results of applying this measurement to SC “AX.” We found that the EC model of Fig. 2 effectively describes the shunt impedance of the SC up to an input frequency of at least 3 kHz for voltages $\le 50\mathrm{mV}$.

Fig. 6

(a) Circuit configuration for measuring frequency dependence of C60’s parallel impedance and (b) the results of that measurement. We fit the measured voltage ratio values (points) to the voltage ratio predicted in Eq. (6) with the extracted best-fit values of $R_{\mathrm{sh}}=1020\mathrm{\Omega }$ and $C_{\mathrm{sh}}=1.36\mu \mathrm{F}$ (line). The $\sim 1\%$ disparity between this measurement of $R_{\mathrm{sh}}$ and the measurement of $R_{\mathrm{sh}}$ derived from the unilluminated $IV$ curve (Fig. 4) is likely due to the cell’s non-ohmic $IV$ response.

We repurposed the configuration used to measure the SC linearity (Sec. 3.3) to measure the SC’s response to modulated light. Our results are dominated by the gain bandwidth product of the current amplifier and by the upper frequency limit of the LED driver. We determined that the SC has an acceptable time response out to at least 200 Hz.

3.5.

Spatial Uniformity of QE

The electrodes along the back of the C60 SCs (see Fig. 3) produce a corrugated electric potential that may adversely affect the QE. To determine the strength of this effect, if it exists at all, we illuminated a $30\text{-}\mu \mathrm{m}$ pinhole with the M415F3 Thorlabs LED (see Table 2). We used a pair of convex lenses to image the pinhole onto the surface of SC “AX,” forming a $150\text{-}\mu \mathrm{m}$ diameter spot. We used a translation stage to scan the pinhole image across the cell parallel to the electrodes, perpendicular to the electrodes, and diagonally with respect to the electrodes. We show the results of these three scans in Fig. 7.

Fig. 7

C60 spatial response to scans of a $150\text{-}\mu \mathrm{m}$-diameter spot (lower-left column) and to scans of a 30-mm-diameter spot (lower right column) across the front of the cell as the spot is moved perpendicular to the electrodes (black), nearly parallel to the electrodes (blue), and offset 10 deg from parallel to the electrodes (orange). We show the approximate orientation of the scans with respect to the electrodes in the included image of the back side of the cell. For the small spot moving perpendicular to the electrodes, the SC signal is characterized by a series of sharp “troughs” in which the response drops by between 1% and 17%. These troughs are 1.2 mm apart. The considerable structure within the troughs of the small spot’s diagonal and parallel scans indicates that the SC response is highly nonuniform along the troughs. Some $\le 0.04\%$ variations at the $\mathcal{O}(1\mathrm{mm})$ spatial scale are apparent in the SC response to both the the perpendicular and parallel scans of the 30-mm spot.

The response of the SC as the $150\mu \mathrm{m}$ spot is moved parallel to the electrodes is characterized by a series of sharp “troughs” of varying depths. Fourier analysis indicates that the spacing between these troughs is 1.2 mm, $\sim 2.5$ times larger than the spacing between adjacent electrodes of opposite polarity. It is unclear if this spatial variation is caused by the corrugations in the electric field produced by the electrodes or due to some other regularly structured spatial inhomogeneity in the SC.

In the parallel and diagonal scans, the spot obliquely traversed the electrodes and produced a series of much broader troughs in the measured current. These broader troughs contain considerable substructure, suggesting that the QE response along the electrodes is not homogeneous.

This spatial dependence of the cell’s QE introduces a source of systematic error that will diminish as the cross section of the incident light grows. To assess the effect of this systematic error on large optical beams, we scanned a 30-mm-diameter spot across the cell parallel and perpendicular to the electrodes. The results of these scans are shown in Fig. 7. Spatial variations in induced current on the scale of the electrode width are visible in both the perpendicular and parallel scan but are no bigger than 0.04%. These scans also exhibit a $\mathcal{O}$ (10 to 100 mm) large-scale spatial inhomogeneity in the cell’s QE that is as large as 4%. When using these devices as precision calibration tools, the location of the incident beam should be kept stable to minimize the systematic error introduced by such large-scale spatial inhomogeneities.

3.6.

Statistical Uncertainty

We are primarily concerned with the statistical uncertainty due to Johnson noise³¹ in the resistances and due to shot noise in the photocurrent, $I_L$. If we assume that the photocurrent is substantially larger than both the bias current and the current driven over the shunt resistance due to the load’s bias voltage, we can estimate the statistical uncertainty in the load current, ${\sigma }_{I,\text{Out}}$, by adding the shot noise and the Johnson noise in quadrature: