Solar-cell band-gap calculation

# 7. Solar cells have a band-gap limit

**Verdict: Proved**

*Working test · not peer reviewed · AI-assisted · no clinical or safety advice*

## The hypothesis, as published

> Single-junction solar efficiency is capped by the ratio of band gap to photon
> energy, producing an optimum gap rather than a monotonic improvement with
> absorption.

> Prediction: efficiency should peak at an intermediate gap.
> Refute: efficiency increasing with absorption would remove the trade-off.

## What was tested

This is a calculation, not a dataset. The Shockley–Queisser detailed-balance
model balances absorbed solar photons against radiative emission at the cell
temperature, with no concentration and no non-radiative losses. The solar source
is a 5778 K blackbody diluted to 1000 W/m²; using a blackbody rather than the
AM1.5G spectrum makes the low-gap arm optimistic, because the blackbody carries
more infrared, so the test is conservative about the *height* of the peak but
neutral about its *existence*.

## Results

| Quantity | Value |
|---|---|
| Optimal band gap | 1.27 eV |
| Efficiency at the optimum | 31.5% |
| Open-circuit voltage at the optimum | 1.01 V |
| Short-circuit current at the optimum | 35.1 mA/cm² |
| Fill factor at the optimum | 0.88 |
| Efficiency at 0.70 eV | 22.5% |
| Efficiency at 2.40 eV | 17.0% |
| Interior maximum? | True |

The efficiency curve turns over on both sides of the optimum. Below it, photons
are wasted as heat in the lattice; above it, photons pass through unabsorbed.
Neither loss can be removed by better material, which is what makes this a
limit rather than an engineering target.

Measured records sit below the cap and cluster near the peak region:

| Material | Gap (eV) | Detailed-balance limit (%) | Record cell (%) | Fraction of limit |
|---|---|---|---|---|
| c-Si | 1.12 | 30.9 | 27.6 | 89% |
| GaAs thin film | 1.42 | 31.0 | 29.1 | 94% |
| CdTe | 1.45 | 30.8 | 22.4 | 73% |
| CIGS | 1.15 | 31.1 | 23.6 | 76% |
| Perovskite | 1.48 | 30.6 | 26.1 | 85% |
| a-Si:H | 1.75 | 27.3 | 14.0 | 51% |

The mean record cell reaches 78% of its
material's detailed-balance limit. The gap is real: non-radiative recombination,
series resistance and imperfect absorption are all absent from the model.

## Sensitivity

- Cell temperature 300 K vs 350 K shifts the optimum by a few hundredths of an
  eV and lowers the peak efficiency by about a point.
- A 6000 K blackbody moves the optimum upward by about 0.05 eV; a 5778 K source
  is the conventional choice.
- Truncating the solar spectrum at 1.8 eV (an extreme atmospheric cutoff) moves
  the optimum to roughly 1.5 eV, still an interior maximum.

## What this test cannot see

The record-cell comparison is indicative, not a survey: the values are widely
reproduced records, not a systematic extraction from Green's tables, and the
band gaps are nominal rather than measured for each device. The model assumes
unity absorption above the gap and a step-function absorptivity, so it does not
describe any real material's absorption edge. The result is a bound, and bounds
are not predictions of what a cell will do.

## Files

| File | What it is |
|---|---|
| `hypothesis.md` | the published entry, quoted verbatim |
| `run.py` | the calculation, runnable |
| `data.csv` | the efficiency curve |
| `results.json` | every number quoted above |
| `figures/fig1.png` | the efficiency curve and the current–voltage trade-off |
| `SOURCE.md` | constants and method |

**Source.** Calculation. No measured dataset was used for the main result.