Semiconductor Physics, Photovoltaics & Tandem Solar Cell Engineering
Shockley-Queisser Bandgap Thermalization & Voc Loss Formula
Calculate Solar Cell Sub-Bandgap Transmission vs Hot-Carrier Phonon Thermalization Loss (eV), Detailed-Balance Open-Circuit Voltage Voc (V), Extracted Electrical Work Per Photon (eV), and Monochromatic Photon Conversion Efficiency.
See why single-junction solar cells cap near ~33.7%—and how perovskite-silicon tandems beat it!
Shockley and Queisser’s detailed-balance limit (~33.7% under AM1.5G sunlight at Eg ≈ 1.34 eV) arises from two competing quantum losses! First, any photon with energy E_ph < Eg passes straight through the absorber unabsorbed (Sub-Bandgap Transmission Loss -> 0 extracted work)! Second, every absorbed high-energy photon (E_ph >= Eg) immediately sheds its excess kinetic energy (E_ph - Eg) as lattice heat via picosecond hot-carrier phonon relaxation (Thermalization Loss), before radiative recombination and Carnot/etendue entropy subtract a voltage deficit ΔV_oc (~0.35–0.42 V) so q*V_oc = Eg - q*ΔV_oc and Extracted Electrical Work = (Eg - q*ΔV_oc) * FF!
Verify on a Green 554 nm Solar Photon (E_ph = 2.24 eV, B2) hitting a Crystalline Silicon Cell (Bandgap Eg = 1.12 eV, B3, Radiative/Entropy Voc Deficit = 0.40 V, B4, Fill Factor FF = 0.85, B5): Hot-Carrier Thermalization immediately wastes 2.24 - 1.12 = 1.12 eV (50% of photon energy! E2); Detailed-Balance Voc is 1.12 - 0.40 = 0.72 V (E3); Extracted Electrical Work is 0.72 * 0.85 = 0.6120 eV (E4); and Monochromatic Photon Conversion Efficiency is 0.6120 / 2.24 = 0.2732 (27.32% in E5)!
| Shockley-Queisser Photon Metric (E_ph = 2.24 eV, Si Eg = 1.12 eV, ΔVoc = 0.40 V, FF = 0.85) | Formula | Verified Output |
|---|---|---|
| Hot-Carrier Thermalization (eV) | IF(E_ph < Eg, 0, E_ph - Eg) | 1.12 eV (0.48 eV @ 1.76 eV Top) |
| Detailed-Balance Voc (V) | Eg_eV - Voc_Deficit_V | 0.72 V (1.36 V @ 1.76 eV Top) |
| Extracted Work / Photon (eV) | IF(E_ph < Eg, 0, (Eg - ΔVoc) * FF) | 0.6120 eV (1.1560 eV @ 1.76 eV) |
| Photon Conversion Efficiency | Extracted_Work_eV / Photon_Energy_eV | 0.2732 (0.5161 @ Perovskite Top) |
Detailed-Balance Photon Work & Conversion Efficiency Formula
Validate Photon_eV > 0, Bandgap_Eg > 0, 0 < Voc_Deficit < Bandgap_Eg, and Fill_Factor in (0, 1)!
=LET(ep_h, B2, eg_v, B3, dv_c, B4, ff_c, B5, IF(OR(ep_h <= 0, eg_v <= 0, dv_c <= 0, dv_c >= eg_v, ff_c <= 0, ff_c >= 1), "Invalid Bandgap Detailed Balance", IF(ep_h < eg_v, 0, ROUND(((eg_v - dv_c) * ff_c) / ep_h, 4))))1.76 eV Perovskite Tandem Top-Cell Harvest & Sub-Bandgap IR Transmission Checks
Intercept that 2.24 eV green photon first in a wide-bandgap 1.76 eV Perovskite Tandem Top Cell (B3 = 1.76): Thermalization Loss E2 plunges from 1.12 eV to 0.48 eV, nearly doubling Photon Conversion Efficiency E5 from 27.32% to 51.61% (0.5161)! Conversely, set B2 = 0.90 eV (sub-bandgap infrared < 1.12 eV) to confirm E4 = 0 and E5 = 0!
=IF(ISTEXT(E5), E5, IF(B2 < B3, 0, ROUND((B3 - B4) * B5, 4)))Try the example
Use the original values or change them to test your understanding.