Forecasting how organic solar cells age, from the first hours

Waiting for a cell to age

Single-junction organic solar cells now exceed 20% efficiency, but they still do not last long enough. Finding out whether a new blend is stable means running it under light for hundreds to thousands of hours. Stronger stress shortens the test but can switch on degradation routes that do not occur in real operation.

The alternative here is a forecast. Measure the performance Y₁ up to an early time t₁, fit the curve and predict the value Y₂ at a later time t₂, at the same, constant stress. The acceleration factor AF = t₂/t₁ says how far the forecast reaches beyond the data. An earlier deep-learning approach worked at AFs that varied from 5 to 20; the aim here is a fixed AF of 10 with a known error.

Schematic, after Fig. 1a. The curve is drawn with eqn (1) and illustrative parameters, not a measured device.

Hundreds of ageing curves from a robot

All training and test devices come from the robotic line AMANDA Line 1, which makes and measures cells automatically. They aged for 200 h under a metal halide lamp at room temperature, in nitrogen in a glovebox. Two campaigns are combined: one keeps PM6:Y6 fixed and varies more than 100 processing conditions; the other, previously unpublished, tests more than 40 donor–acceptor pairs with fixed processing.

One carrier holds 9 substrates with 6 pixels each, so 54 cells see the same lamp and the same compartment temperature. A flicker in either shows up in all 54 curves at once, an artifact the method has to live with. Curves that first improve ("light soaking") were left out, because they start from an ill-defined state.

Curve counts from Fig. 2 and the text. The 1000 h set, used only to test generalization, is the DeepDeg dataset (ref. 19 of the paper). Carrier drawing schematic.

Six curves stand for the whole set

Symbolic regression is costly, so it is not run on all 226 curves. Instead each curve is fitted with a sum of exponentials, and the number of exponentials needed for a residual-free fit (one, two or three) puts it into a shape class.

One, two and three representative curves are then taken from the three classes, six in total. They cover the main shapes: stable or mildly degrading, steady continuous loss, and a strong burn-in that later levels off. The authors stress that the classes describe shapes, not microscopic mechanisms.

Schematic, after Fig. 1b and Fig. S2: the curves show the three shape classes, not the selected devices.

Let an algorithm write the formula

Class symbolic regression, here the code PhySO, searches over mathematical expressions with reinforcement learning. It looks for the simplest formula that fits all six curves with shared structure, and the search was restricted to forms that decrease monotonically, as expected for ageing under inert illumination.

Each candidate has an accuracy (R²) and a complexity. Plotted against each other they form a Pareto front. Formulas with only 3 free parameters already reach R² > 0.95; the three-exponential classification model needs up to 6. Fewer parameters means less redundancy, and that matters when a fit has to be extrapolated.

Schematic, after Fig. 1c and Fig. S4: positions of the candidates are illustrative.

Linear at first, then dispersive

Among the accurate candidates, the forecasts from 10 h and 50 h windows were statistically indistinguishable for all but one, so the formula with the fewest parameters was chosen:

y = a − t·(2t + √(b+1))^c / b

Here y is the normalised metric, t the ageing time in hours, and a = 1 is the normalised starting value, so only b and c are fitted. At short times the loss grows linearly with t. Once 2t ≫ √(b+1) it grows as t^c+1, a dispersive decay; b sets when the change happens and c the long-time exponent. Because y keeps falling forever, the formula is meant for small to medium losses, not outright failure, and the authors call it a compact forecasting expression, not a universal law.

Formula and limits from eqn (1) and the text. The plot uses illustrative b and c on log–log axes, where the two regimes are straight lines.

Trained on current, it fits efficiency too

On the six training curves the normalised RMS error is 0.17–0.61%. On all 226 short-circuit-current (J_SC) curves of 200 h the median is 0.32% and 218 curves (98.2%) stay below 1%.

The formula was found from J_SC alone, yet it also fits the efficiency (PCE): 92.5% of 133 curves below 1%, median 0.43%. Fill factor and V_OC behave similarly (over 90% below 1%). On 246 J_SC curves of 1000 h, five times longer than training, 95.5% stay below 2%, median 1.20%. Coupled degradation processes give all the metrics a similar shape in time.

All values from the text (Fig. 2, Figs. S7–S8). The 2% threshold for the 1000 h set was chosen by the authors because longer runs are noisier.

How long must you measure?

Now the formula is fitted only to the data up to t₁ and extrapolated to the end of the run. The forecast error is the mean absolute deviation over the last four time points. The longer the test window, the smaller the error, until it reaches the measurement noise.

For 200 h of J_SC, the mean error falls below 3% once t₁ > 35 h (the median below 2% already after 10 h). For PCE the mean error drops below 4% after about 40 h; for the 1000 h set after about 450 h. The 3% mark follows the 1–3% uncertainty of certified efficiency measurements. Single-, bi- and stretched-exponential fits do worse at early times in both 200 h sets; for 1000 h the stretched exponential wins only in the 50 h window.

Crossing times from the text (Fig. 3, Figs. S10–S11). Curve shapes are schematic and omit the small bump the paper reports near 40 h.

A random forest corrects each forecast

Error propagation from the fit uncertainties assumes that only random noise is at play, and real curves carry short-time fluctuations that break that assumption. So the authors let a random forest learn the signed forecast deviation from earlier complete curves. Its inputs are the fitted b and c (log-transformed) and their correlation, the drop ΔY seen so far, the forecast Y₂ and the noise level.

Y₂ matters most, followed by c and b for the 200 h J_SC set; lower forecasts carry larger errors. Tested on held-out curves (ten 80/20 splits balanced by stability), the correction shrinks the mean absolute deviation from 3.37% to 1.52% (200 h J_SC), from 8.70% to 2.06% (200 h PCE) and from 9.37% to 1.25% (1000 h J_SC).

Values (mean ± spread over the splits) from the text (Fig. 4a,b; Fig. S18).

20 hours for a 200-hour verdict

The practical test: does about 90% of the forecasts fall within the error threshold? Without the correction this holds only for AF of 1.43–3.33. With it, 10% of the ageing time is enough. At AF 10, 91.96% of held-out 200 h J_SC forecasts are within 3%, and 90.00% (200 h PCE) and 91.60% (1000 h J_SC) within 4%. Turned into stable/unstable calls (final retention ≥ 0.90, or ≥ 0.80 for 1000 h), the decisions are right 96.7%, 88.9% and 92.6% of the time.

For a new lab the authors suggest running 20–30% of devices to the end at first to calibrate the forest, then 5–10%. All data were taken in nitrogen under constant light; oxygen, humidity, thermal cycling, UV, applied bias and outdoor conditions were not tested and may need retraining.

Values from the text (Fig. 4c–e, Figs. S19–S20, Fig. 5). Calibration fractions are the authors' heuristics.