photovoltaic solar cell technologies: analysing the state ...10.1038/s41578-019-0097-0... · 1...

Photovoltaic solar cell technologies: analysing the state of the artPabitra K. Nayak , Suhas Mahesh, Henry J. Snaith and David Cahen

https://doi.org/10.1038/s41578-019-0097-0

AnAlysis

Nature reviews | Materials

In format as provided by the authors

Supplementary InformatIon

https://doi.org/10.1038/s41578-019-0097-0

1

Supplementary Information

Photovoltaic solar cell technologies: analysing the state of the art

Pabitra K. Nayak, Suhas Mahesh, Henry J. Snaith and David Cahen

• Supplementary Figure 1


• Supplementary Table 1

• Derivation of the analytical expression for 𝑞𝑞𝑞𝑞OCSQ

• Relationship between 𝑞𝑞𝑞𝑞OCSQ and 𝑞𝑞𝑞𝑞OCRad

• Relationship between VMP and VOC


• Operational Loss





• Relationship between JMP/JSC and VOC

• The architecture of GaAs cells


• Passivation of c-Si with a-Si






• Urbach energy determination in OPVs



• Relationship between the fill factor and VOC

• Commercial solar cell module efficiencies

• Methods

2

Supplementary Figure 1: Current density versus voltage (J(V)) curve, JSC, JMP, VOC and VMP for a typical solar cell under illumination. The total (maximum) photocurrent is given by JSC where the ap-plied voltage is zero (i.e., short-circuit condition). The VOC of the cell is where the net current flowing across the device is zero. VOC of the device represents the maximum difference of electrical potential between two terminals of the cell. However, electrical power conversion is not possible either at open-circuit or short-circuit as J × V = 0 for both the cases. There exists a point on the J–V curve where the JV product is at a maximum, the maximum power point (MPP). The voltage at the MPP is termed VMP, and the current density is JMP. The power conversion efficiency of the solar cell under the global AM1.5 spectrum at 25 °C is given by (JMP∙VMP)/(100 mW cm–2).

Supplementary Figure 2: Measured external quantum efficiency (EQE) of a typical solar cell (black circles), its derivative (blue circles) as a distribution of Shockley–Queisser (SQ)-type bandgaps P(E). The photovoltaic (PV) gap (𝐸𝐸gPV) is determined from the distribution of the P(E). The S-Q step-function quantum efficiency for this sample cell is shown as the red line. The dashed vertical lines show the limits used for the integration (cf. equation 1 in the main text).

0.0 0.4 0.8 1.2 1.60

3

6

9

VMPVOC

JSC

Curr

ent d

ensi

ty (m

A/cm

2 )

Voltage (V)

J(V) curve

JMP

(VMP , JMP)

1.50 1.55 1.60 1.65 1.70

0.0

0.2

0.4

0.6

0.8

1.0

EQE

Energy (eV)

Exte

rnal

Qua

ntum

Effi

cien

cy (E

QE)

SQ- EQE

P(E)

0

1

2

3

4

d(EQ

E)/d

E (e

V-1)

3

Supplementary Table 1:Photovoltaic gap (𝐸𝐸gPV ) of the cells and the optical bandgap of the absorber, where available.

RT, room temperature.

Cell type (absorber) 𝐸𝐸gPV(eV) RT bandgap (eV)

c-Si 1.10 1.11(ref 1) GaAs 1.42 1.43 (ref 1) InP 1.38 1.35 (ref 2) GaInP 1.88 NA mc-Si 1.11 1.11(ref 1)

CdTe1–xSex 1.42 NA

CuInxGa1–xSe2 1.12 NA

Cu2ZnSnS4–ySey 1.18 NA

Cu2ZnSnS4 1.48 NA

ABX3 1.55 NA

a-Si:H 1.77 1.69 (ref 3)

OPV (Toshiba) 1.62 NA

QD 1.77 NA

4

Derivation of the analytical expression for 𝑞𝑞𝑞𝑞OCSQ

The photon flux from the sun which can be treated as a thermal emitter (black body) at temperature

𝑇𝑇𝑆𝑆 is be given by Planck’s law:

𝑛𝑛 (𝐸𝐸,𝑇𝑇𝑆𝑆) = 2π ℎ3𝑐𝑐2

E2

𝑒𝑒 𝐸𝐸𝑘𝑘𝑇𝑇𝑆𝑆

−1

(S1)

where c is the speed of light in vacuum, E is the photon energy, h is Planck’s constant and k is the Boltzmann

constant.

The total current that the photons, absorbed by the solar cell (on earth), can generate, is given by

∫ 2Ω𝑖𝑖𝑖𝑖 ℎ3𝑐𝑐2

a(E)E2


−1

∞0 𝑑𝑑(𝐸𝐸) (S2)

where Ω𝑖𝑖𝑖𝑖 is the solid angle subtended by the sun on the earth’s surface and a(E) is the absorptance of

the cell.

With a step function-like absorptance at 𝐸𝐸 = 𝐸𝐸𝑔𝑔, the above equation S2 becomes


E2


−1

∞𝐸𝐸𝑔𝑔

𝑑𝑑(𝐸𝐸) (S3)

Würfel’s generalised Planck’s law4 , which we show in equation S4 below, gives the photon flux, n

per energy interval d (E) per unit solid angle (Ω) for a semiconductor absorber at temperature 𝑇𝑇𝐴𝐴 with

chemical potential μ

𝑛𝑛 𝐸𝐸,Ω,𝑇𝑇𝐴𝐴, µ, a(E) = 2Ω𝑜𝑜𝑜𝑜𝑜𝑜 ℎ3𝑐𝑐2

a(E)E2

𝑒𝑒𝐸𝐸−µ𝑘𝑘𝑇𝑇𝐴𝐴

−1

(S4)

With a step function-like absorptance at 𝐸𝐸 = 𝐸𝐸𝑔𝑔, the total emitted photon is

∫ 2Ω𝑜𝑜𝑜𝑜𝑜𝑜 ℎ3𝑐𝑐2

E2

𝑒𝑒𝐸𝐸−µ𝑘𝑘𝑇𝑇𝐴𝐴

−1

∞𝐸𝐸𝑔𝑔

𝑑𝑑(𝐸𝐸) (S5)

In the detailed balance formalism, the photocurrent from the solar cell is the difference between the

number of photons absorbed and number of photons emitted. At open-circuit voltage, where = 𝑞𝑞𝑞𝑞𝑂𝑂𝑂𝑂 ,

the number of photons absorbed = number of photons emitted, which implies (equating S5 and S4).

Since we use all the assumptions that are required in in the S-Q formalism, here, 𝑞𝑞𝑞𝑞𝑂𝑂𝑂𝑂 = 𝑞𝑞𝑞𝑞𝑂𝑂𝑂𝑂𝑆𝑆𝑆𝑆


E2


−1

∞𝐸𝐸𝑔𝑔

𝑑𝑑(𝐸𝐸) = ∫ 2Ω𝑜𝑜𝑜𝑜𝑜𝑜ℎ3𝑐𝑐2

E2

𝑒𝑒𝐸𝐸−q𝑉𝑉𝑂𝑂𝑂𝑂

𝑆𝑆𝑆𝑆

𝑘𝑘𝑇𝑇𝐴𝐴

−1

𝑑𝑑(𝐸𝐸)∞𝐸𝐸𝑔𝑔

(S6)

5

For most solar cell materials 𝐸𝐸𝑔𝑔 > 1 eV, in which case the “–1” term in the denominators from both

sides can be neglected. As shown by Hirst et al.5 , this approximation has a negligible effect on the

calculation of open-circuit voltages, which implies


E2


∞𝐸𝐸𝑔𝑔

𝑑𝑑(𝐸𝐸) = ∫ 2Ω𝑜𝑜𝑜𝑜𝑜𝑜ℎ3𝑐𝑐2

E2

𝑒𝑒 𝐸𝐸𝑘𝑘𝑇𝑇𝐴𝐴

𝑑𝑑(𝐸𝐸)𝑒𝑒

q𝑉𝑉𝑂𝑂𝑂𝑂

𝑆𝑆𝑆𝑆

𝑘𝑘𝑇𝑇𝐴𝐴 ∞

𝐸𝐸𝑔𝑔 (S7)

⟹ 𝑒𝑒q𝑉𝑉𝑂𝑂𝑂𝑂

𝑆𝑆𝑆𝑆


=Ω𝑖𝑖𝑖𝑖 ∫ ∞

𝐸𝐸𝑔𝑔𝐸𝐸2𝑒𝑒

−𝐸𝐸𝑘𝑘𝑇𝑇𝑆𝑆

𝑑𝑑(𝐸𝐸)

Ω𝑜𝑜𝑜𝑜𝑜𝑜 ∫ 𝐸𝐸2𝑒𝑒 −𝐸𝐸𝑘𝑘𝑇𝑇𝐴𝐴

𝑑𝑑(𝐸𝐸)∞

𝐸𝐸𝑔𝑔

Integrating the numerator and denominator in the RHS, using integration by parts, gives

𝑒𝑒q𝑉𝑉𝑂𝑂𝑂𝑂

𝑆𝑆𝑆𝑆


= Ω𝑖𝑖𝑖𝑖 𝐸𝐸𝑔𝑔2.𝑘𝑘𝑇𝑇𝑠𝑠𝑒𝑒

−𝐸𝐸𝑔𝑔𝑘𝑘𝑇𝑇𝑆𝑆

+2𝐸𝐸𝑔𝑔𝑘𝑘2𝑇𝑇𝑆𝑆

2𝑒𝑒−𝐸𝐸𝑔𝑔𝑘𝑘𝑇𝑇𝑆𝑆

+2𝑘𝑘3𝑇𝑇𝑆𝑆

3𝑒𝑒−𝐸𝐸𝑔𝑔𝑘𝑘𝑇𝑇𝑆𝑆

Ω𝑜𝑜𝑜𝑜𝑜𝑜 𝐸𝐸𝑔𝑔2.𝑘𝑘𝑇𝑇𝐴𝐴𝑒𝑒−𝐸𝐸𝑔𝑔𝑘𝑘𝑇𝑇𝐴𝐴

+2𝐸𝐸𝑔𝑔𝑘𝑘2𝑇𝑇𝐴𝐴

2𝑒𝑒−𝐸𝐸𝑔𝑔𝑘𝑘𝑇𝑇𝐴𝐴

+2𝑘𝑘3𝑇𝑇𝐴𝐴

3 𝑒𝑒−𝐸𝐸𝑔𝑔𝑘𝑘𝑇𝑇𝐴𝐴

(S8)


𝑆𝑆𝑆𝑆


= Ω𝑖𝑖𝑖𝑖 𝐸𝐸𝑔𝑔2 + 2𝐸𝐸𝑔𝑔𝑘𝑘𝑇𝑇𝑆𝑆 + 2𝑘𝑘2𝑇𝑇𝑆𝑆

2𝑘𝑘𝑇𝑇𝑠𝑠𝑒𝑒−𝐸𝐸𝑔𝑔𝑘𝑘𝑇𝑇𝑆𝑆

Ω𝑜𝑜𝑜𝑜𝑜𝑜 𝐸𝐸𝑔𝑔2 + 2𝐸𝐸𝑔𝑔𝑘𝑘𝑇𝑇𝐴𝐴 + 2𝑘𝑘2𝑇𝑇𝐴𝐴 2𝑘𝑘𝑇𝑇𝐴𝐴𝑒𝑒

−𝐸𝐸𝑔𝑔𝑘𝑘𝑇𝑇𝐴𝐴


𝑆𝑆𝑆𝑆


= Ω𝑖𝑖𝑖𝑖Ω𝑜𝑜𝑜𝑜𝑜𝑜

𝐸𝐸𝑔𝑔2+2𝐸𝐸𝑔𝑔𝑘𝑘𝑇𝑇𝑆𝑆+2𝑘𝑘2𝑇𝑇𝑆𝑆 2𝑘𝑘𝑇𝑇𝑠𝑠

𝐸𝐸𝑔𝑔2+2𝐸𝐸𝑔𝑔𝑘𝑘𝑇𝑇𝐴𝐴+2𝑘𝑘2𝑇𝑇𝐴𝐴 2 𝑘𝑘𝑇𝑇𝐴𝐴

𝑒𝑒𝐸𝐸𝑔𝑔𝑘𝑘𝑇𝑇𝐴𝐴

−𝐸𝐸𝑔𝑔𝑘𝑘𝑇𝑇𝑆𝑆


𝑆𝑆𝑆𝑆


= Ω𝑖𝑖𝑖𝑖Ω𝑜𝑜𝑜𝑜𝑜𝑜

𝐸𝐸𝑔𝑔2+2𝐸𝐸𝑔𝑔𝑘𝑘𝑇𝑇𝑆𝑆+2𝑘𝑘2𝑇𝑇𝑆𝑆 2𝑘𝑘𝑇𝑇𝑠𝑠

𝐸𝐸𝑔𝑔2+2𝐸𝐸𝑔𝑔𝑘𝑘𝑇𝑇𝐴𝐴+2𝑘𝑘2𝑇𝑇𝐴𝐴 2 𝑘𝑘𝑇𝑇𝐴𝐴

𝑒𝑒𝐸𝐸𝑔𝑔𝑘𝑘𝑇𝑇𝐴𝐴

1−𝑇𝑇𝐴𝐴𝑇𝑇𝑆𝑆

⟹ q𝑞𝑞𝑂𝑂𝑂𝑂𝑆𝑆𝑆𝑆 = 𝑘𝑘𝑇𝑇𝐴𝐴𝑙𝑙𝑛𝑛

Ω𝑖𝑖𝑖𝑖Ω𝑜𝑜𝑜𝑜𝑜𝑜

+ 𝑘𝑘𝑇𝑇𝐴𝐴𝑙𝑙𝑛𝑛 𝑘𝑘𝑇𝑇𝑠𝑠𝐸𝐸𝑔𝑔2+2𝐸𝐸𝑔𝑔𝑘𝑘𝑇𝑇𝑆𝑆+2𝑘𝑘2𝑇𝑇𝑆𝑆

2 𝑘𝑘𝑇𝑇𝐴𝐴𝐸𝐸𝑔𝑔2+2𝐸𝐸𝑔𝑔𝑘𝑘𝑇𝑇𝐴𝐴+2𝑘𝑘2𝑇𝑇𝐴𝐴

2 + 𝐸𝐸𝑔𝑔 1 − 𝑇𝑇𝐴𝐴

𝑇𝑇𝑆𝑆

⟹ q𝑞𝑞𝑂𝑂𝑂𝑂𝑆𝑆𝑆𝑆 = 𝐸𝐸𝑔𝑔 1 − 𝑇𝑇𝐴𝐴

𝑇𝑇𝑆𝑆 − 𝑘𝑘𝑇𝑇𝐴𝐴𝑙𝑙𝑛𝑛

Ω𝑜𝑜𝑜𝑜𝑜𝑜Ω𝑖𝑖𝑖𝑖

+ 𝑘𝑘𝑇𝑇𝐴𝐴𝑙𝑙𝑛𝑛 𝑇𝑇𝑠𝑠𝐸𝐸𝑔𝑔2+2𝐸𝐸𝑔𝑔𝑘𝑘𝑇𝑇𝑆𝑆+2𝑘𝑘2𝑇𝑇𝑆𝑆

2 𝑇𝑇𝐴𝐴𝐸𝐸𝑔𝑔2+2𝐸𝐸𝑔𝑔𝑘𝑘𝑇𝑇𝐴𝐴+2𝑘𝑘2𝑇𝑇𝐴𝐴

2

q𝑞𝑞𝑂𝑂𝑂𝑂𝑆𝑆𝑆𝑆 = 𝐸𝐸𝑔𝑔 1 − 𝑇𝑇𝐴𝐴

𝑇𝑇𝑆𝑆 + 𝑘𝑘𝑇𝑇𝐴𝐴𝑙𝑙𝑛𝑛

𝛾𝛾𝐸𝐸𝑔𝑔,𝑇𝑇𝑠𝑠 𝛾𝛾𝐸𝐸𝑔𝑔,𝑇𝑇𝐴𝐴

− 𝑘𝑘𝑇𝑇𝐴𝐴𝑙𝑙𝑛𝑛 Ω𝑜𝑜𝑜𝑜𝑜𝑜Ω𝑖𝑖𝑖𝑖

(S9)

where 𝛾𝛾 (𝐸𝐸,𝑇𝑇) = 𝑇𝑇(𝐸𝐸2 + 2𝑘𝑘𝑇𝑇𝐸𝐸 + 2𝑘𝑘2𝑇𝑇2) .

6

Relationship between 𝒒𝒒𝒒𝒒𝐎𝐎𝐎𝐎𝐒𝐒𝐒𝐒 and 𝒒𝒒𝒒𝒒𝐎𝐎𝐎𝐎𝐑𝐑𝐑𝐑𝐑𝐑:

The current density of a solar cell at any applied voltage, J(V) can be expressed as6

𝐽𝐽(𝑞𝑞) = 𝐽𝐽𝑆𝑆𝑂𝑂 − 𝐽𝐽0(𝑒𝑒𝑞𝑞𝑉𝑉𝑖𝑖𝑘𝑘𝑇𝑇 − 1)

where 𝐽𝐽𝑆𝑆𝑂𝑂 is the photocurrent density at short circuit, 𝐽𝐽0 is the dark saturation current, n is the diode

ideality factor, T is the temperature (in Kelvin) and k is the Boltzmann constant.

At VOC,

𝐽𝐽𝑆𝑆𝑂𝑂 − 𝐽𝐽0(𝑒𝑒𝑞𝑞𝑉𝑉𝑂𝑂𝑂𝑂𝑖𝑖𝑘𝑘𝑇𝑇 − 1) = 0

⟹ 𝑞𝑞𝑞𝑞𝑂𝑂𝑂𝑂 = 𝑛𝑛𝑘𝑘𝑇𝑇ln 𝐽𝐽𝑆𝑆𝑂𝑂𝐽𝐽0

+ 1

Since 𝐽𝐽𝑆𝑆𝑂𝑂𝐽𝐽0

>>1

⟹ 𝑞𝑞𝑞𝑞𝑂𝑂𝑂𝑂 = 𝑛𝑛𝑘𝑘𝑇𝑇ln 𝐽𝐽𝑆𝑆𝑂𝑂𝐽𝐽0

In the radiative limit, the dark saturation current is 𝐽𝐽0𝑅𝑅𝑅𝑅𝑑𝑑 and 𝑛𝑛 = 1

where 𝐽𝐽0𝑅𝑅𝑅𝑅𝑑𝑑 = ∫ EQE(E)∅bb(E)dE∞0 , ∅bb is the radiation flux corresponding to a black body at ambi-

ent temperature (TA)

(𝑞𝑞OCRad is the open-circuit voltage in the radiative limit)

⟹ 𝑞𝑞𝑞𝑞𝑂𝑂𝑂𝑂𝑅𝑅𝑅𝑅𝑑𝑑 = 𝑘𝑘𝑇𝑇ln𝐽𝐽𝑆𝑆𝑂𝑂𝐽𝐽0𝑅𝑅𝑅𝑅𝑑𝑑

⟹ 𝑞𝑞𝑞𝑞𝑂𝑂𝑂𝑂𝑅𝑅𝑅𝑅𝑑𝑑 = 𝑘𝑘𝑇𝑇ln𝐽𝐽𝑆𝑆𝑂𝑂𝐽𝐽𝑆𝑆𝑂𝑂𝑆𝑆𝑆𝑆 ×

𝐽𝐽0𝑆𝑆𝑆𝑆

𝐽𝐽0𝑅𝑅𝑅𝑅𝑑𝑑×𝐽𝐽𝑆𝑆𝑂𝑂𝑆𝑆𝑆𝑆


where

𝐽𝐽0𝑆𝑆𝑆𝑆 = ∫ EQE(E)∅bb(E)dE∞

E𝐺𝐺𝑃𝑃𝑉𝑉 , 𝐽𝐽𝑆𝑆𝑂𝑂

𝑆𝑆𝑆𝑆 = ∫ EQE(E)∅𝑆𝑆𝑜𝑜𝑖𝑖(E)(E)dE∞E𝐺𝐺𝑃𝑃𝑉𝑉

∅bb and ∅Sun are the radiation flux corresponding to a blackbody at the operational temperature of

the cell and standard full sun illumination, respectively.

7

⟹ 𝑞𝑞𝑞𝑞𝑂𝑂𝑂𝑂𝑅𝑅𝑅𝑅𝑑𝑑 = 𝑘𝑘𝑇𝑇ln𝐽𝐽𝑆𝑆𝑂𝑂𝐽𝐽𝑆𝑆𝑂𝑂𝑆𝑆𝑆𝑆+ 𝑘𝑘𝑇𝑇ln


𝐽𝐽0𝑅𝑅𝑅𝑅𝑑𝑑+ 𝑘𝑘𝑇𝑇ln

𝐽𝐽𝑆𝑆𝑂𝑂𝑆𝑆𝑆𝑆


Since 𝑞𝑞𝑞𝑞𝑂𝑂𝑂𝑂𝑆𝑆𝑆𝑆 = 𝑘𝑘𝑇𝑇ln 𝐽𝐽𝑆𝑆𝑂𝑂

𝑆𝑆𝑆𝑆


⟹ 𝑞𝑞𝑞𝑞𝑂𝑂𝑂𝑂𝑅𝑅𝑅𝑅𝑑𝑑 = 𝑘𝑘𝑇𝑇ln𝐽𝐽𝑆𝑆𝑂𝑂𝐽𝐽𝑆𝑆𝑂𝑂𝑆𝑆𝑆𝑆+ 𝑘𝑘𝑇𝑇ln


𝐽𝐽0𝑅𝑅𝑅𝑅𝑑𝑑+ 𝑞𝑞𝑞𝑞𝑂𝑂𝑂𝑂

𝑆𝑆𝑆𝑆

⟹ 𝑞𝑞𝑞𝑞𝑂𝑂𝑂𝑂𝑅𝑅𝑅𝑅𝑑𝑑 = 𝑞𝑞𝑞𝑞𝑂𝑂𝑂𝑂𝑆𝑆𝑆𝑆 − 𝑘𝑘𝑇𝑇ln 𝐽𝐽𝑆𝑆𝑂𝑂

𝑆𝑆𝑆𝑆

𝐽𝐽𝑆𝑆𝑂𝑂 − 𝑘𝑘𝑇𝑇ln 𝐽𝐽0

𝑅𝑅𝑅𝑅𝑅𝑅


Relationship between VMP and VOC

From ref. 6

𝐽𝐽(𝑞𝑞) = 𝐽𝐽𝑆𝑆𝑂𝑂 − 𝐽𝐽0(𝑒𝑒𝑞𝑞𝑉𝑉𝑖𝑖𝑘𝑘𝑇𝑇 − 1) (S10)

At the maximum power point the product, 𝐽𝐽(𝑞𝑞) × 𝑞𝑞 is at a maximum, which means

𝑑𝑑 (𝐽𝐽(𝑞𝑞) × 𝑞𝑞 ) = 𝑑𝑑𝐽𝐽(𝑞𝑞)× 𝑞𝑞 + 𝐽𝐽(𝑞𝑞)𝑑𝑑𝑞𝑞 = 0

⟹ 𝑑𝑑 𝐽𝐽(𝑉𝑉)𝑑𝑑𝑉𝑉

𝑀𝑀𝑀𝑀

= − 𝐽𝐽(𝑉𝑉)𝑉𝑉𝑀𝑀𝑀𝑀

⟹ 𝑑𝑑 𝐽𝐽(𝑉𝑉)𝑑𝑑𝑉𝑉

𝑀𝑀𝑀𝑀

= −𝐽𝐽0𝑞𝑞

𝑖𝑖𝑘𝑘𝑇𝑇𝑒𝑒

𝑞𝑞𝑉𝑉𝑀𝑀𝑃𝑃𝑖𝑖𝑘𝑘𝑇𝑇 = − 𝐽𝐽𝑀𝑀𝑃𝑃

𝑉𝑉𝑀𝑀𝑃𝑃

⟹ 𝐽𝐽0𝑞𝑞

𝑖𝑖𝑘𝑘𝑇𝑇𝑒𝑒

𝑞𝑞𝑉𝑉𝑀𝑀𝑃𝑃𝑖𝑖𝑘𝑘𝑇𝑇 = 𝐽𝐽𝑀𝑀𝑃𝑃

𝑉𝑉𝑀𝑀𝑃𝑃 (S11)

At the maximum power point, equation S10 becomes

𝐽𝐽𝑀𝑀𝑀𝑀 = 𝐽𝐽𝑆𝑆𝑂𝑂 − 𝐽𝐽0 𝑒𝑒𝑞𝑞𝑉𝑉𝑀𝑀𝑃𝑃𝑖𝑖𝑘𝑘𝑇𝑇 − 1 (S12)

At open-circuit condition, 𝐽𝐽𝑂𝑂𝑂𝑂 = 0

⟹ 𝐽𝐽𝑆𝑆𝑂𝑂 = 𝐽𝐽0 𝑒𝑒𝑞𝑞𝑉𝑉𝑂𝑂𝑂𝑂𝑖𝑖𝑘𝑘𝑇𝑇 − 1 (S13)

By putting the expression of 𝐽𝐽𝑆𝑆𝑂𝑂 from S13 into equation S12,

𝐽𝐽𝑀𝑀𝑀𝑀 = 𝐽𝐽0 𝑒𝑒𝑞𝑞𝑉𝑉𝑂𝑂𝑂𝑂𝑖𝑖𝑘𝑘𝑇𝑇 − 1 − 𝐽𝐽0 (𝑒𝑒

𝑞𝑞𝑉𝑉𝑀𝑀𝑃𝑃𝑖𝑖𝑘𝑘𝑇𝑇 − 1

⟹ 𝐽𝐽𝑀𝑀𝑀𝑀 = 𝐽𝐽0 𝑒𝑒𝑞𝑞𝑉𝑉𝑂𝑂𝑂𝑂𝑖𝑖𝑘𝑘𝑇𝑇 − 𝑒𝑒

𝑞𝑞𝑉𝑉𝑀𝑀𝑃𝑃𝑖𝑖𝑘𝑘𝑇𝑇 (S14)

8

By putting the above expression for JMP into equation S11,

𝑞𝑞𝑖𝑖𝑘𝑘𝑇𝑇

𝑒𝑒𝑞𝑞𝑉𝑉𝑀𝑀𝑃𝑃𝑖𝑖𝑘𝑘𝑇𝑇 =

𝑒𝑒𝑞𝑞𝑉𝑉𝑂𝑂𝑂𝑂𝑖𝑖𝑘𝑘𝑇𝑇 −𝑒𝑒

𝑞𝑞𝑉𝑉𝑀𝑀𝑃𝑃𝑖𝑖𝑘𝑘𝑇𝑇

𝑉𝑉𝑀𝑀𝑃𝑃

⟹ 𝑞𝑞𝑉𝑉𝑀𝑀𝑃𝑃𝑖𝑖𝑘𝑘𝑇𝑇

𝑒𝑒𝑞𝑞𝑉𝑉𝑀𝑀𝑃𝑃𝑖𝑖𝑘𝑘𝑇𝑇 = 𝑒𝑒

𝑞𝑞𝑉𝑉𝑂𝑂𝑂𝑂𝑖𝑖𝑘𝑘𝑇𝑇 − 𝑒𝑒


⟹ 𝑒𝑒𝑞𝑞𝑉𝑉𝑂𝑂𝑂𝑂𝑖𝑖𝑘𝑘𝑇𝑇 = 𝑒𝑒


𝑞𝑞𝑞𝑞𝑀𝑀𝑀𝑀𝑛𝑛𝑘𝑘𝑇𝑇

+ 1

⟹ 𝑞𝑞𝑉𝑉𝑂𝑂𝑂𝑂𝑖𝑖𝑘𝑘𝑇𝑇

= 𝑞𝑞𝑉𝑉𝑀𝑀𝑃𝑃𝑖𝑖𝑘𝑘𝑇𝑇

+ ln 1 + 𝑞𝑞𝑉𝑉𝑀𝑀𝑃𝑃𝑖𝑖𝑘𝑘𝑇𝑇

⟹ 𝑞𝑞𝑉𝑉𝑀𝑀𝑃𝑃𝑖𝑖𝑘𝑘𝑇𝑇

= 𝑞𝑞𝑉𝑉𝑂𝑂𝑂𝑂𝑖𝑖𝑘𝑘𝑇𝑇

−ln(1 + 𝑞𝑞𝑉𝑉𝑀𝑀𝑃𝑃𝑖𝑖𝑘𝑘𝑇𝑇

)

If we define, 𝑞𝑞𝑉𝑉𝑀𝑀𝑃𝑃𝑖𝑖𝑘𝑘𝑇𝑇

= 𝑞𝑞𝑀𝑀𝑀𝑀′ and 𝑞𝑞𝑉𝑉𝑂𝑂𝑂𝑂𝑖𝑖𝑘𝑘𝑇𝑇

= 𝑞𝑞𝑂𝑂𝑂𝑂′

Then 𝑞𝑞𝑀𝑀𝑀𝑀′ = 𝑞𝑞𝑂𝑂𝑂𝑂′ − ln(𝑞𝑞𝑀𝑀𝑀𝑀′ + 1) (S15)

We now try to get an approximate explicit analytical function for 𝑞𝑞𝑀𝑀𝑀𝑀′ , because equation S15 is an

implicit one.

The logarithm weakly depends on its argument and 𝑞𝑞𝑀𝑀𝑀𝑀′ ≈ 𝑞𝑞𝑂𝑂𝑂𝑂′ , 𝑞𝑞𝑀𝑀𝑀𝑀′ can be substituted with 𝑞𝑞𝑂𝑂𝑂𝑂′ in

the right-hand side of equation S15, which yields

𝑞𝑞𝑀𝑀𝑀𝑀′ = 𝑞𝑞𝑂𝑂𝑂𝑂′ − ln (𝑞𝑞𝑂𝑂𝑂𝑂′ − ln (𝑞𝑞𝑂𝑂𝑂𝑂′ + 1) + 1) (S16)

To check the validity of equation S16, we took a series of 𝑞𝑞𝑀𝑀𝑀𝑀′ values and calculated 𝑞𝑞𝑂𝑂𝑂𝑂′ using the

following equation

𝑞𝑞𝑂𝑂𝑂𝑂′ = 𝑞𝑞𝑀𝑀𝑀𝑀′ + ln(𝑞𝑞𝑀𝑀𝑀𝑀′ + 1) (S17)

which is obtained by rearranging equation S15.

Then we used those values of 𝑞𝑞𝑂𝑂𝑂𝑂′ to regenerate 𝑞𝑞𝑀𝑀𝑀𝑀′ using equation S16. In Supplementary Figure 3a

we show the comparison of the original and regenerated values 𝑞𝑞𝑀𝑀𝑀𝑀′ .To show how much we can de-

viate from the original VMP values of the solar cells under consideration in this work, we show the dif-

ference in VMP values as a function of VOC/nkT in Supplementary Figure 3b.

9

Supplementary Figure 3: (a) Difference in 𝑞𝑞𝑀𝑀𝑀𝑀′ values due to the approximation in the equation

𝑞𝑞𝑀𝑀𝑀𝑀′ = 𝑞𝑞𝑂𝑂𝑂𝑂′ − ln (𝑞𝑞𝑂𝑂𝑂𝑂′ − ln (𝑞𝑞𝑂𝑂𝑂𝑂′ + 1) + 1). (b) The underestimation in 𝑞𝑞𝑀𝑀𝑀𝑀 = 𝑞𝑞𝑀𝑀𝑀𝑀′ .𝑛𝑛𝑘𝑘𝑇𝑇 values due to

the approximation for the values under consideration. We see that the underestimation will be less

than 0.2 mV.

Operational Loss

For a given 𝐸𝐸gPV, we calculate the 𝑞𝑞𝑞𝑞OCSQ from S9 (equation 2 in the main article) , the corresponding

𝑞𝑞𝑀𝑀𝑀𝑀′ from equation S16, and then 𝑞𝑞𝑀𝑀𝑀𝑀𝑆𝑆𝑆𝑆 as 𝑞𝑞𝑀𝑀𝑀𝑀′ .𝑘𝑘𝑇𝑇𝐴𝐴 and the operational loss in the S-Q limit as 𝐸𝐸𝑔𝑔𝑀𝑀𝑉𝑉-

𝑞𝑞𝑀𝑀𝑀𝑀𝑆𝑆𝑆𝑆 and the experimental operational loss (OL) is given by 𝐸𝐸𝑔𝑔𝑀𝑀𝑉𝑉 − 𝑞𝑞𝑞𝑞𝑀𝑀𝑀𝑀

To disentangle the factors for the experimental OL for comparison between cell types, we now use

rearranged S16 and express the OL as

OL ≈ 𝐸𝐸𝑔𝑔𝑀𝑀𝑉𝑉 − 𝑞𝑞𝑞𝑞𝑂𝑂𝑂𝑂𝑐𝑐𝑒𝑒𝑐𝑐𝑐𝑐 − 𝑛𝑛𝑘𝑘𝑇𝑇𝐴𝐴 ln 𝑞𝑞𝑉𝑉𝑂𝑂𝑂𝑂𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐

𝑖𝑖𝑘𝑘𝑇𝑇𝐴𝐴− ln( 𝑞𝑞𝑉𝑉𝑂𝑂𝑂𝑂

𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐

𝑖𝑖𝑘𝑘𝑇𝑇𝐴𝐴+ 1) + 1 (S18)

Using equation 3 from the main article .

OL ≈ 𝐸𝐸𝑔𝑔𝑀𝑀𝑉𝑉 − 𝑞𝑞𝑞𝑞𝑂𝑂𝑂𝑂𝑅𝑅𝑅𝑅𝑑𝑑 − 𝑘𝑘𝑇𝑇𝐴𝐴ln|ln(𝜂𝜂𝑒𝑒𝑒𝑒𝑜𝑜)| − 𝑛𝑛𝑘𝑘𝑇𝑇𝐴𝐴 ln 𝑞𝑞𝑉𝑉𝑂𝑂𝑂𝑂𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐




𝜂𝜂𝑒𝑒𝑒𝑒𝑜𝑜 can be expressed as7,8

𝜂𝜂𝑒𝑒𝑒𝑒𝑜𝑜 = 𝜂𝜂𝑖𝑖𝑖𝑖𝑜𝑜.𝑀𝑀𝑐𝑐𝑠𝑠𝑐𝑐1−𝜂𝜂𝑖𝑖𝑖𝑖𝑜𝑜.𝑀𝑀𝑟𝑟𝑐𝑐𝑅𝑅𝑟𝑟𝑠𝑠

(S20)

where 𝜂𝜂𝑖𝑖𝑖𝑖𝑜𝑜 is the internal luminescence efficiency, 𝑃𝑃𝑒𝑒𝑠𝑠𝑐𝑐 and 𝑃𝑃𝑟𝑟𝑒𝑒𝑅𝑅𝑟𝑟𝑠𝑠 are the probabilities of photon es-

cape and re-absorption, respectively. 𝑃𝑃𝑟𝑟𝑒𝑒𝑅𝑅𝑟𝑟𝑠𝑠 depends on the 𝑃𝑃𝑒𝑒𝑠𝑠𝑐𝑐 and the probability of parasitic ab-

sorbance, 𝑃𝑃𝑝𝑝𝑅𝑅𝑟𝑟𝑅𝑅𝑠𝑠𝑖𝑖𝑜𝑜𝑖𝑖𝑐𝑐, as:

𝑃𝑃𝑟𝑟𝑒𝑒𝑅𝑅𝑟𝑟𝑠𝑠 = 1 − (𝑃𝑃𝑒𝑒𝑠𝑠𝑐𝑐 + 𝑃𝑃𝑝𝑝𝑅𝑅𝑟𝑟𝑅𝑅𝑠𝑠𝑖𝑖𝑜𝑜𝑖𝑖𝑐𝑐) (S21)

20 30 40 50 600.0

1.0x10-4

2.0x10-4

Diffe

renc

e (V

)

VOC/nkT0 10 20 30 40 50 60

0

20

40

60

V MP/n

kT

VOC/nkT

Original Regenerated

(a) (b)

10

and 𝜂𝜂𝑖𝑖𝑖𝑖𝑜𝑜 = 𝑘𝑘𝑟𝑟𝑘𝑘𝑖𝑖𝑟𝑟+𝑘𝑘𝑟𝑟

(S22)

where 𝑘𝑘𝑟𝑟 and 𝑘𝑘𝑖𝑖𝑟𝑟 are the rates of radiative and non-radiative recombination, respectively

Now we can write equation S19 as

OL ≈ 𝐸𝐸𝑔𝑔𝑀𝑀𝑉𝑉 − 𝑞𝑞𝑞𝑞𝑂𝑂𝑂𝑂𝑅𝑅𝑅𝑅𝑑𝑑 − 𝑘𝑘𝑇𝑇𝐴𝐴ln 𝜂𝜂𝑖𝑖𝑖𝑖𝑜𝑜.𝑀𝑀𝑐𝑐𝑠𝑠𝑐𝑐1−𝜂𝜂𝑖𝑖𝑖𝑖𝑜𝑜.𝑀𝑀𝑟𝑟𝑐𝑐𝑅𝑅𝑟𝑟𝑠𝑠

− 𝑛𝑛𝑘𝑘𝑇𝑇𝐴𝐴 ln 𝑞𝑞𝑉𝑉𝑂𝑂𝑂𝑂𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐




By substituting the expression for 𝑞𝑞𝑞𝑞OCRad ( that is 𝑞𝑞𝑞𝑞𝑂𝑂𝑂𝑂𝑅𝑅𝑅𝑅𝑑𝑑 = 𝑞𝑞𝑞𝑞𝑂𝑂𝑂𝑂𝑆𝑆𝑆𝑆 − 𝑘𝑘𝑇𝑇ln𝐽𝐽𝑆𝑆𝑂𝑂

𝑆𝑆𝑆𝑆

𝐽𝐽𝑆𝑆𝑂𝑂 − 𝑘𝑘𝑇𝑇ln𝐽𝐽0


𝐽𝐽0𝑆𝑆𝑆𝑆 )

into equation S23, we get

OL ≈ 𝐸𝐸𝑔𝑔𝑀𝑀𝑉𝑉 − 𝑞𝑞𝑞𝑞𝑂𝑂𝑂𝑂𝑆𝑆𝑆𝑆 − 𝑘𝑘𝑇𝑇𝐴𝐴 ln 𝐽𝐽𝑆𝑆𝑂𝑂

𝑆𝑆𝑆𝑆

𝐽𝐽𝑆𝑆𝑂𝑂 − 𝑘𝑘𝑇𝑇𝐴𝐴 ln 𝐽𝐽0


𝐽𝐽0𝑆𝑆𝑆𝑆 − 𝑘𝑘𝑇𝑇𝐴𝐴ln 𝜂𝜂𝑖𝑖𝑖𝑖𝑜𝑜.𝑀𝑀𝑐𝑐𝑠𝑠𝑐𝑐

1−𝜂𝜂𝑖𝑖𝑖𝑖𝑜𝑜.𝑀𝑀𝑟𝑟𝑐𝑐𝑅𝑅𝑟𝑟𝑠𝑠 − 𝑛𝑛𝑘𝑘𝑇𝑇𝐴𝐴 ln 𝑞𝑞𝑉𝑉𝑂𝑂𝑂𝑂


𝑖𝑖𝑘𝑘𝑇𝑇𝐴𝐴−

ln( 𝑞𝑞𝑉𝑉𝑂𝑂𝑂𝑂𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐

𝑖𝑖𝑘𝑘𝑇𝑇𝐴𝐴+ 1) + 1 .. (S24)

By substituting the expression for 𝑞𝑞𝑞𝑞𝑂𝑂𝑂𝑂𝑆𝑆𝑆𝑆 from equation S9 and rearranging equation S24

𝑂𝑂𝑂𝑂 ≈ 𝐸𝐸𝑔𝑔𝑀𝑀𝑉𝑉𝑇𝑇𝐴𝐴𝑇𝑇𝑆𝑆 − 𝑘𝑘𝑇𝑇𝐴𝐴ln 𝛾𝛾𝐸𝐸𝑔𝑔

𝑃𝑃𝑉𝑉,𝑇𝑇𝑆𝑆𝛾𝛾𝐸𝐸𝑔𝑔𝑃𝑃𝑉𝑉,𝑇𝑇𝐴𝐴

+ 𝑘𝑘𝑇𝑇𝐴𝐴ln Ω𝑜𝑜𝑜𝑜𝑜𝑜Ω𝑖𝑖𝑖𝑖

+ 𝑘𝑘𝑇𝑇𝐴𝐴 ln 𝐽𝐽𝑆𝑆𝑂𝑂𝑆𝑆𝑆𝑆

𝐽𝐽𝑆𝑆𝑂𝑂 + 𝑘𝑘𝑇𝑇𝐴𝐴 ln 𝐽𝐽0


𝐽𝐽0𝑆𝑆𝑆𝑆 + 𝑘𝑘𝑇𝑇𝐴𝐴 ln|(𝑃𝑃𝑒𝑒𝑠𝑠𝑐𝑐)|

+𝑘𝑘𝑇𝑇𝐴𝐴 ln 𝜂𝜂𝑖𝑖𝑖𝑖𝑜𝑜1−𝜂𝜂𝑖𝑖𝑖𝑖𝑜𝑜.𝑀𝑀𝑟𝑟𝑐𝑐𝑅𝑅𝑟𝑟𝑠𝑠

+ 𝑛𝑛𝑘𝑘𝑇𝑇𝐴𝐴 ln 𝑞𝑞𝑉𝑉𝑂𝑂𝑂𝑂𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐

𝑖𝑖𝑘𝑘𝑇𝑇𝐴𝐴− ln 𝑞𝑞𝑉𝑉𝑂𝑂𝑂𝑂


𝑖𝑖𝑘𝑘𝑇𝑇𝐴𝐴+ 1 + 1

In the case of non-ideal contacts, the energy loss due to the non-zero contact resistance from the con-

tacts adds to the above expression for operational loss, which implies

𝑂𝑂𝑂𝑂 ≈ 𝐸𝐸𝑔𝑔𝑀𝑀𝑉𝑉𝑇𝑇𝐴𝐴𝑇𝑇𝑆𝑆 − 𝑘𝑘𝑇𝑇𝐴𝐴ln 𝛾𝛾𝐸𝐸𝑔𝑔

𝑃𝑃𝑉𝑉,𝑇𝑇𝑆𝑆𝛾𝛾𝐸𝐸𝑔𝑔𝑃𝑃𝑉𝑉,𝑇𝑇𝐴𝐴

+ 𝑘𝑘𝑇𝑇𝐴𝐴ln Ω𝑜𝑜𝑜𝑜𝑜𝑜Ω𝑖𝑖𝑖𝑖

+ 𝑘𝑘𝑇𝑇𝐴𝐴 ln 𝐽𝐽𝑆𝑆𝑂𝑂𝑆𝑆𝑆𝑆

𝐽𝐽𝑆𝑆𝑂𝑂 + 𝑘𝑘𝑇𝑇𝐴𝐴 ln 𝐽𝐽0


𝐽𝐽0𝑆𝑆𝑆𝑆 + 𝑘𝑘𝑇𝑇𝐴𝐴 ln|(𝑃𝑃𝑒𝑒𝑠𝑠𝑐𝑐)|

+𝑘𝑘𝑇𝑇𝐴𝐴 ln 𝜂𝜂𝑖𝑖𝑖𝑖𝑜𝑜1−𝜂𝜂𝑖𝑖𝑖𝑖𝑜𝑜.𝑀𝑀𝑟𝑟𝑐𝑐𝑅𝑅𝑟𝑟𝑠𝑠

+ 𝑛𝑛𝑘𝑘𝑇𝑇𝐴𝐴 ln 𝑞𝑞𝑉𝑉𝑂𝑂𝑂𝑂𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐

𝑖𝑖𝑘𝑘𝑇𝑇𝐴𝐴− ln 𝑞𝑞𝑉𝑉𝑂𝑂𝑂𝑂


𝑖𝑖𝑘𝑘𝑇𝑇𝐴𝐴+ 1 + 1 + loss due to contact resistance

(S25)

11

Supplementary Figure 4: Net loss in energy at VOC (𝐸𝐸gPV − 𝑞𝑞𝑞𝑞OCSQ). The net loss has a linear relation-

ship with 𝐸𝐸gPV in the range of interest (that is, from 1–2.5 eV).

1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.40.22

0.24

0.26

0.28

0.30

0.32

Net Loss = 0.171 eV +0.059*EPVg

EPVg (TA/TS) - kTAln(γ(E

PVg ,TS)/γ(E

PVg ,TA))+kTAln (Ωout/Ωin)

Linear Fit

Net L

oss

in E

nerg

y at

VOC

(eV)

EPVg

12

Supplementary Table 2: Efficiency, VOC, JSC, fill factor and area of the cells included in the analysis.

ap = aperture area, da = designated area c = single-crystalline, mc= multi-crystalline, a = amorphous; CIGS = CuInxGa1–xSe2, CZTSS = Cu2ZnSnS4–ySey, CZTS = Cu2ZnSnS4, ABX3 = metal halide perov-skite, QD = quantum dot. c References for the solar cell parameters and area of the cell.

Cell type Efficiency (%)

VOC (V)

JSC [mA/cm2)

Fill Factor (%)

Area (cm2)

Ref.c

Single-crystalline materials

c-Si

GaAs

InP a GaInP

26.7

29.1

24.2

21.4

0.74

1.13

0.94

1.46

42.6

29.8

31.1

16.3

84.9

86.7

82.6

87.7

79 (da)

0.998 (ap)

1.008 (ap)

0.25 (ap)

9 9 9 9

Polycrystalline materials

mc-Si

CdTe1–xSex

CIGS

CZTSS

CZTS

ABX3

22.3

21.0

22.9

11.3

10

20.9

0.67

0.88

0.74

0.53

0.71

1.12

41.1

30.2

38.8

33.6

21.8

24.9

80.5

79.4

79.5

63.0

65.1

74.5

3.923 (ap)

1.062(ap)

1.041 (da)

1.176 (da)

1.113 da)

0.991 (da)

9 9 9 9 9 9

Other materials

a-Si 10.2 0.90 16.4 69.8 1.001 (da) 9

OPV (Toshiba)

11.2 0.78 19.3 74.2 0.992 (da) 9

aQD 13.4 1.16 15.2 76.63 0.058(ap) 10

13

Supplementary Figure 5: ηext, calculated from equation 7 in the main text, vs. the measured ηext for

cells reported in the literature: GaInP (blue circles), metal halide perovskite (red circles), c-Si (wine

red circle ), CIGS (green circle) and GaAs ( pink circle) . The ηext value of the GaAs cell (prepared at

NREL, USA) was measured at Oxford University.

In Supplementary Table 3, we provide the 𝐸𝐸gPV values determined from the reported EQE of the cell

(or reported bandgap values), experimentally observed qVOC and ηext.

10-7 10-6 10-5 10-4 10-3 10-2 10-1 10010-7

10-6

10-5

10-4

10-3

10-2

10-1

100

GaAsPerovskiteGaInP

c-Si

Calculated ηExt

M

easu

red η Ex

t

CIGS

14

Supplementary Table 3: 𝐸𝐸gPV and qVoc values used to calculate ηext., measured ηext and references for

the reported work. ηext for GaAs cell (PCE = 27.7 %) is from an in-house measurement.

Technology 𝐸𝐸gPV (eV) qVoc (eV) Calculated ηext Measured ηext Reference

ABX3 1.61 1.08 5.1E–5 1.2E–4 11

ABX3 1.59 1.10 2.3E–4 5E–4 12

ABX3 1.60 1.18 3.6E–3 3E–3 13

ABX3 1.60 1.17 2.4E–3 5E–3 13

ABX3 1.63 1.24 1.2E–2 1.4E–2 14

GaInP 1.81a 1.458 8.2E–2 8.7E–2 15

GaInP 1.81 a 1.455 7.3E–2 7.6E–2 15

GaInP 1.81 a 1.392 6.3E–3 7.5E–3 15

GaInP 1.84 a 1.413 4.8E–3 5.3E–3 15

GaInP 1.84 a 1.406 3.6E–3 3.2E–3 15

c-Si 1.09 0.687 2E–3 1.3E–3 16

CIGS 1.2 0.747 4E–4 3E–4 16

GaAs 1.42 1.1 1.1E–1 6E–2 This work a EQE data are available only for the GaInP cell with 1.455 V Voc, which gave 𝐸𝐸gPV ≈ 1.81. For other

GaInP cells, reported bandgap values in ref 15 are used in lieu of 𝐸𝐸gPV calculated from EQE.

15

Relationship between JMP/JSC and VOC

Rearrangement of equation (S12) gives

𝐽𝐽𝑀𝑀𝑃𝑃𝐽𝐽𝑆𝑆𝑂𝑂

= 1 − 𝐽𝐽0(𝑒𝑒𝑞𝑞𝑉𝑉𝑀𝑀𝑃𝑃𝑖𝑖𝑘𝑘𝑇𝑇 −1)𝐽𝐽𝑆𝑆𝑂𝑂

(S26)

By putting the expression 𝐽𝐽𝑆𝑆𝑂𝑂 from S13 in the RHS of equation S26

𝐽𝐽𝑀𝑀𝑃𝑃𝐽𝐽𝑆𝑆𝑂𝑂

= 1 − 𝐽𝐽0(𝑒𝑒𝑞𝑞𝑉𝑉𝑀𝑀𝑃𝑃𝑖𝑖𝑘𝑘𝑇𝑇 −1)

𝐽𝐽0(𝑒𝑒𝑞𝑞𝑉𝑉𝑂𝑂𝑂𝑂𝑖𝑖𝑘𝑘𝑇𝑇 −1)

(S27)

⟹ 𝐽𝐽𝑀𝑀𝑃𝑃𝐽𝐽𝑆𝑆𝑂𝑂

= 1 − (𝑒𝑒𝑞𝑞𝑉𝑉𝑀𝑀𝑃𝑃𝑖𝑖𝑘𝑘𝑇𝑇 −1)

(𝑒𝑒𝑞𝑞𝑉𝑉𝑂𝑂𝑂𝑂𝑖𝑖𝑘𝑘𝑇𝑇 −1)

(S28)

Since 𝑒𝑒𝑞𝑞𝑉𝑉𝑂𝑂𝑂𝑂𝑖𝑖𝑘𝑘𝑇𝑇 and 𝑒𝑒

𝑞𝑞𝑉𝑉𝑀𝑀𝑃𝑃𝑖𝑖𝑘𝑘𝑇𝑇 >> 1, for reasonable solar cell performances, where VMP >100 mV, we can

ignore the subtracted '1's in the 2nd term on the R.H.S of equation S28.


= 1 − 𝑒𝑒𝑞𝑞𝑉𝑉𝑀𝑀𝑃𝑃𝑖𝑖𝑘𝑘𝑇𝑇

𝑒𝑒𝑞𝑞𝑉𝑉𝑂𝑂𝑂𝑂𝑖𝑖𝑘𝑘𝑇𝑇

(S29)


= 1 − 𝑒𝑒(𝑞𝑞𝑉𝑉𝑀𝑀𝑃𝑃𝑖𝑖𝑘𝑘𝑇𝑇 −𝑞𝑞𝑉𝑉𝑂𝑂𝑂𝑂𝑖𝑖𝑘𝑘𝑇𝑇 )


= 1 − 𝑒𝑒(𝑉𝑉𝑀𝑀𝑃𝑃′ −𝑉𝑉𝑂𝑂𝑂𝑂

′ )

where 𝑞𝑞𝑉𝑉𝑀𝑀𝑃𝑃𝑖𝑖𝑘𝑘𝑇𝑇

= 𝑞𝑞𝑀𝑀𝑀𝑀′ and 𝑞𝑞𝑉𝑉𝑂𝑂𝑂𝑂𝑖𝑖𝑘𝑘𝑇𝑇

= 𝑞𝑞𝑂𝑂𝑂𝑂′


= 1 − 𝑒𝑒(𝑉𝑉𝑀𝑀𝑃𝑃′ −𝑉𝑉𝑂𝑂𝑂𝑂

′ ) (S30)

Since 𝑞𝑞𝑀𝑀𝑀𝑀′ = 𝑞𝑞𝑂𝑂𝑂𝑂′ − ln (𝑞𝑞𝑂𝑂𝑂𝑂′ − ln (𝑞𝑞𝑂𝑂𝑂𝑂′ + 1) + 1) (from equation S16)


= 1 − 𝑒𝑒(−ln (𝑉𝑉𝑂𝑂𝑂𝑂′ −ln (𝑉𝑉𝑂𝑂𝑂𝑂

′ +1)+1)) (S31)


= 1 − 1𝑉𝑉𝑂𝑂𝑂𝑂′ −ln (𝑉𝑉𝑂𝑂𝑂𝑂

′ +1)+1


= 𝑉𝑉𝑂𝑂𝑂𝑂′ −ln (𝑉𝑉𝑂𝑂𝑂𝑂

′ +1)𝑉𝑉𝑂𝑂𝑂𝑂′ −ln (𝑉𝑉𝑂𝑂𝑂𝑂

′ +1)+1 (S32)

16

The architecture of GaAs cells

In GaAs, the electrons have 20 times higher mobility than that of the holes. To compensate the mis-

match, the cell structure has a thinner p layer on the top, whereas the base is n-type.

Supplementary Figure 6: (a) Upright and inverted configurations for the GaInP cells18. [2019] IEEE.

Reprinted with permission, from Steiner, M. A., Geisz, J. F., Reedy, R. C. & Kurtz, S. A direct

comparison of inverted and non-inverted growths of GaInP solar cells. 2008 33rd IEEE Photovolatic

Specialists Conference 1–6 (IEEE, 2008). (b) External quantum efficiency (EQE) for the champion

GaInP cells reported over the last few years. The data for the 17.5% cell is not certified and is from

ref.19.

Passivation of c-Si with a-Si

a-Si is made up of the same element as c-Si (no difference in electronegativity) and can be deposited

on to c-Si in a controlled manner without creating defects on the substrate. Moreover, a-Si deposition

eliminates (curing) the existing dangling bond on the c-Si surfaces.

300 400 500 600 700

0

20

40

60

80

100

EQE

(%)

Wavelength (nm)

21.4% 20.8% (rescaled) 17.5%

(a) (b)

Upright configuration Inverted configuration

17

Supplementary Figure 7: External quantum efficiency (EQE) for 26.7% c-Si (SHJ–IBC) cell. The

EQE is rescaled to match the reported current density in ref20. The shaded region shows the filtering

effect from a-Si layer. Adapted with permission from ref. 20, Wiley-VCH.

Supplementary Figure 8: Schematic device architectures for (a) Poly-crystalline CIGS, (b) Poly-crystalline

CdTe, (c) Halide Perovskite (ABX3) and (d) OPV.

400 600 800 1000 12000

20

40

60

80

100

c-Si

EQE

(%)

Wavelength (nm)

Glass (substrate)

Mo

Poly-crystalline CIGS (absorber)

CdS/ZnSZnO window AR coating

Front contact

Back contact

Glass (substrate)

TCO

Metal halide perovskite(absorber)

Electron selective contact

Metal contactHole selective contact

Front contact

Back contact

Glass (substrate) TCO layer CdS(Se)

Poly-crystalline CdTe1-xSex(absorber)

Metal contact

AR coating

Back contact

Front contact

Organic donor:acceptor blend(absorber)

Glass (substrate)

TCO

Metal contact

Front contact

Back contact

(a) (b)

(c) (d)

18

Supplementary Figure 9: External quantum efficiency (EQE) for the champion CZTS and CZTSS cells report-

ed over the last few years. 7.5% and 9.5% CZTS cells have an area <1 cm2. 9.5% and 10% CZTS cells have

buffer layers of Zn1–xCdxS instead of CdS. The dashed vertical line is the bandgap of CdS.

400 600 800 1000 12000

20

40

60

80

100

EQE

(%)

Wavelength (nm)

CZTS* (7%) CZTS*(9.5%) CZTS (10%) CZTSS (9.8%) CZTSS (11.3%)

19

1.2 1.3 1.4 1.5 1.60.00.20.40.60.81.01.21.41.6

qVSQOC - qVOC = 0.35 eV

dEQE/dE EL

Energy (eV)

dEQ

E/dE

(eV-1

) PIPCP:PCBM

EPVg

0.0

0.2

0.4

0.6

0.8

1.0

Inte

nsity

(nor

m.)

1.2 1.3 1.4 1.5 1.6 1.7 1.80

1

2

3

4

5


dEQE/dE EL

Energy (eV)

dEQ

E/dE

(eV-1

)EPV

g

0.0

0.2

0.4

0.6

0.8

1.0

Inte

nsity

(nor

m.)

PffBT4t-2DT:FBR

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.80.00

0.05

0.10

0.15

0.20

0.25


dEQE/dE EL

Energy (eV)

dEQE

/dE

(eV-1

)

EPVg

MDMO-PPV:PCBM

0.0

0.2

0.4

0.6

0.8

1.0

Inte

nsity

(nor

m.)

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.00

1

2

3

4

5

6

7


dEQE/dE EL

Energy (eV)

dEQ

E/dE

(eV-1

)

EPVg

0.0

0.2

0.4

0.6

0.8

1.0

Inte

nsity

(nor

m.)

PBDTTPD:PCBM(a)

(b)

(c)

(d)

(e)

(f)

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.90

2

4

6

8

10

12 dEQE/dE EL

dE

QE/

dE (e

V-1)

Energy (eV)

0.0

0.2

0.4

0.6

0.8

1.0

PDCBT-2F:IT-M

Inte

nsity

(nor

m.)

EPVg


1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.80

2

4

6

8

10

12

dEQE/dE EL

dE

QE/

dE (e

V-1)

Energy (eV)

0.0

0.2

0.4

0.6

0.8

1.0

PTB7-Th:PCBM

Inte

nsity

(nor

m.)

EPVg

qVSQOC - qVOC = 0.58eV

Supplementary Figure 10: Distribution of bandgaps and electroluminescence (EL) spectra of, (a) PBDTTPD:PCBM21, (b) MDMO-PPV:PCBM22 , (c) PTB7-Th:PCBM23 (d) PIPCP:PCBM24, (e) PffBT4t-2DT:FBR25 and (f) PDCBT-2F:ITM23. 𝑞𝑞𝑞𝑞OC

SQ of the cells are calculated using equation 6 in the main article. FBR, (5Z,5′Z)-5,5′-(9,9-dioctyl-9H-fluorene-2,7-diyl)bis[2,1,3-benzothiadiazole-7,4-diyl(Z)methylylidene]bis(3-ethyl-2-thioxo-1,3-thiazolidin-4-one); IT-M, 3-(1,1-dicyanome thylene)-1-methyl-indanone)-5,5,11,11-tetrakis(4-hexylphenyl)-dithieno[2,3-d:2′,3′-d′]-s-indaceno [1,2-b:5,6-b′]-dithiophene; MDMO-PPV, poly[2-methoxy-5-(3′,7′-dimethyloctyloxy)-1,4-phenylenevinylene]; P3HT, poly(3-hexylthiophene-2,5-diyl); P3TEA, a donor polymer; PBDTTPD, poly(benzo[1,2-b:4,5-b′]dithiophene–alt–thieno[3,4-c]pyrrole-4,6-dione); PCBM, [6,6]-phenyl-C61-butyric acid methyl ester; PffBT4t-2DT, a difluoro-benzothiadiazole donor polymer; PIPCP, a donor polymer; PTB7-Th, poly([2,6′-4,8-di(5-ethylhexylthienyl)benzo[1,2-b;3,3-b]dithiophene]3-fluoro-2[(2-ethylhexyl)carbonyl]thieno[3,4-b]thiophenediyl); SF-PDI2, a small-molecule acceptor.

20

Supplementary Table 4: Urbach energy (EU) associated with solar cell materials.

Technology

c-Si 26

GaAs 26

MAPbI3

26

InP 2

aGaInP 27

bCdTe 28

cCIGS 29

da-Si 3

bCZTS 30

CZTSS 31

EU (meV) 11 7.5 15 9.4 9 10 16 43 56.5 54

a From absorbance spectra. bFrom single crystal absorbance. cFrom the EQE data. dFrom champion a-

Si cell.

21

Urbach energy determination in OPVs

Since there is contribution of CT states and energetic disorder, in the spectrum in the sub 𝐸𝐸gPV region,

we fitted that region as a combination of a Gaussian (for CT states) and exponential (for band tailing)

function.

𝑓𝑓(𝐸𝐸) = 𝐴𝐴𝑒𝑒−𝐸𝐸−𝐸𝐸′𝐸𝐸𝑈𝑈

+ 𝐵𝐵𝑒𝑒−𝐸𝐸−𝐸𝐸′′2𝜎𝜎 + 𝐶𝐶

In Supplementary Figure 11 we show one such example.

Supplementary Figure 11: Urbach tail in PBDTTPD: PCBM system.

Supplementary Figure 12: Experimental EQE from a CIGS solar cell with 19 meV Urbach energy

and the simulated EQE with same 𝐸𝐸gPV and Urbach energy.

1.3 1.4 1.5 1.6 1.7

10-7

10-6

10-5

10-4

10-3

10-2

10-1

EQE Fitted lineGaussian Exponential

EQE

Energy (eV)

EU = 36 meV

PBDTTPD:PCBM

0.0 0.5 1.0 1.5 2.0 2.5 3.010-22

10-17

10-12

10-7

10-2

EQE_experimental EQE_generated

EQE

(nor

m.)

Energy (eV)

EPVG

Eu = 19 meV

22

Relationship between the fill factor and VOC

Fill factor (FF) of a solar cell is given by

𝐹𝐹𝐹𝐹 = 𝑉𝑉𝑀𝑀𝑃𝑃×𝐽𝐽𝑀𝑀𝑃𝑃𝑉𝑉𝑂𝑂𝑂𝑂×𝐽𝐽𝑆𝑆𝑂𝑂

(S33)

⟹ 𝐹𝐹𝐹𝐹 =𝑞𝑞𝑉𝑉𝑀𝑀𝑃𝑃𝑖𝑖𝑘𝑘𝑇𝑇 ×𝐽𝐽𝑀𝑀𝑃𝑃𝑞𝑞𝑉𝑉𝑂𝑂𝑂𝑂𝑖𝑖𝑘𝑘𝑇𝑇 ×𝐽𝐽𝑆𝑆𝑂𝑂

⟹ 𝐹𝐹𝐹𝐹 = 𝑉𝑉𝑀𝑀𝑃𝑃′ ×𝐽𝐽𝑀𝑀𝑃𝑃𝑉𝑉𝑂𝑂𝑂𝑂′ ×𝐽𝐽𝑆𝑆𝑂𝑂

By putting the expression of 𝑞𝑞𝑀𝑀𝑀𝑀′ from equation S16 and 𝐽𝐽𝑀𝑀𝑃𝑃𝐽𝐽𝑆𝑆𝑂𝑂

from equation S32

FF = 𝑉𝑉𝑂𝑂𝑂𝑂′ −ln (𝑉𝑉𝑂𝑂𝑂𝑂

′ −ln (𝑉𝑉𝑂𝑂𝑂𝑂′ +1)+1)

𝑉𝑉𝑂𝑂𝑂𝑂′ × 𝑉𝑉𝑂𝑂𝑂𝑂

′ −ln (𝑉𝑉𝑂𝑂𝑂𝑂′ +1)

𝑉𝑉𝑂𝑂𝑂𝑂′ −ln (𝑉𝑉𝑂𝑂𝑂𝑂

′ +1)+1 (S34)

Commercial solar cell module efficiencies

We gathered information regarding the efficiency of the best commercial modules, which is available

on the websites of the manufacturing companies. We then compared the efficiencies of commercial

modules to those of the best laboratory modules and standard and mini cells to assess the drop in per-

formance during the transition.

SunPower (21%) (Model no. X21-345-COM)

https://us.sunpower.com/sites/sunpower/files/uploads/resources/sp-x21-345com-327com-ds-en-ltr-

mc4comp-505700.pdf

Panasonic (19.7%) (HIT technology, Model no. HIT® N330)

https://eu-solar.panasonic.net/cps/rde/xbcr/solar_en/2018_Panasonic_HIT_Catalogue_EN.pdf

LG (21.1%) (IBC technology , Model No. NeONR)

http://www.lg.com/global/business/solar/business-resources/download

Solar Frontier (13.8%) (CI(G)S thin film technology, Model: SF 150-170S)

http://www.solar-frontier.com/eng/solutions/products/index.html

First Solar (17%) (CdTe thin film technology, Model: First Solar Series 4™)

23

http://www.firstsolar.com/-/media/First-Solar/Technical-Documents/Series-4-Datasheets/Series-4V3-

Module-Datasheet.ashx

Methods

We obtained high-quality published figures from the scientific literature and used Graph Grabber

software to digitize the graphs. On many occasions the authors of the original work provided their raw

data to us.

Wherever the EQE is presented in a normalized scale, we rescaled it to match the reported Jsc value

of the cell under AM 1.5G illumination (ASTM G‐173‐03 global, sourced from ref.17).

To handle noisy derivatives of EQE, we used the Savitzky–Golay method to smooth the data (using

Origin 8.5). Whenever we used the Savitzky–Golay method, we interpolated the EQE data to an in-

terval of 1 meV. We used Python codes (with help of the package Numpy) to simulate the EQEs with

different Urbach energies and their effect on 𝑞𝑞OCRad using equation S35.

The EQE is modelled as a step function at E = Eg with a tail that decays with an Urbach energy EU.

The Urbach tail is a feature seen in the absorption coefficient. However, we assume a tail of the same

form in the EQE to make our calculations independent of the precise optical properties and thickness

of the absorber. There is a slight deviation in the simulated EQE spectrum from the actual EQE close

to the 𝐸𝐸gPV. If anything, this deviation implies that our calculated VOC is slightly underestimated, but

this does not affect the conclusions drawn here. We show simulated EQE and normalised EQE of a

CIGS solar cell in Supplementary Figure 12.

We integrate the EQE over the solar AM1.5 spectrum ∅𝐴𝐴𝑀𝑀 1.5(𝐸𝐸) and the Planck’s Black Body spec-

trum ∅𝑟𝑟𝑟𝑟 (𝐸𝐸) at 298K and solve for the open-circuit voltage using the following equation

𝑞𝑞OCRad = kTln ∫𝐸𝐸𝑆𝑆𝐸𝐸(𝐸𝐸)∅𝐴𝐴𝑀𝑀 1.5(𝐸𝐸)(𝑑𝑑𝐸𝐸)∞

0

∫ 𝐸𝐸𝑆𝑆𝐸𝐸(𝐸𝐸)∅𝐵𝐵𝐵𝐵(𝐸𝐸)(𝑑𝑑𝐸𝐸)∞0

(S35)

The integration is performed numerically between 280 and 1,200 nm using scipy.integrate. This is

repeated for as many combinations of Urbach Energies and bandgaps as desired to generate the plot of

qVOC vs EU.

24

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