1. Theory

Definition: A solar cell is a diode, usually a p-n junction, that turns the energy of absorbed photons into electrical power; under illumination it behaves like a diode in the dark in parallel with a current source $J_{L}$ (the superposition of Lindholm, Fossum and Burgess, 1979, as cited by PVEducation).
Four numbers: The open-circuit voltage $V_{oc}$ , the short-circuit current density $J_{sc}$ , the fill factor $F F$ and the efficiency $η = V_{oc} J_{sc} F F / P_{in}$ describe the I-V curve; the standard test conditions are 1000 $W/ m^{2}$ , the AM1.5 spectrum and 25 °C.
Typical silicon values: PVEducation quotes $V_{oc}$ up to 764 mV (laboratory) and about 690 mV (commercial), $J_{sc}$ over 42 $mA/c m^{2}$ (laboratory) and 28 to 35 $mA/c m^{2}$ (commercial), and $F F$ of 0.85 and 0.83; the page is inconsistent on the commercial voltage (see the questions).
Recombination and ideality: The ideality factor $n$ is 1 for band-to-band or Shockley-Read-Hall recombination at low injection, 2 at high injection or for recombination in the depletion region, and 2/3 for Auger recombination.
Parasitics: A series resistance of 0.5 (laboratory) to 1.3 $Ω c m^{2}$ (commercial) mainly lowers the fill factor, and a shunt resistance from MΩ (laboratory) to about 1000 $Ω c m^{2}$ (commercial), caused by defects of the process, lowers it at low light.
Temperature and concentration: $V_{oc}$ falls by about 2 $mV/K$ in silicon, $J_{sc}$ rises by 0.06 %/K, and the power falls by 0.4 to 0.5 %/K; concentration raises $J_{sc}$ in proportion and $V_{oc}$ logarithmically, but the series loss grows with the square of the concentration.
Limit: With a step absorber and radiative recombination only, the efficiency of a single junction is limited by the photons below the gap, the thermalization of the others and the radiative saturation current; the maximum is 33.1 to 33.7 % at 1.34 eV for the AM1.5G spectrum (computed).

2. Working Principle

Generation and collection: A photon above the gap creates an electron-hole pair, the junction field separates the carriers that reach it, and at short circuit the current is $J_{sc}$ .
Forward bias by the light: When the terminals are open, the collected charge raises the voltage until the dark diode current equals $J_{L}$ , which gives $V_{oc} = (n k_{B} T / q) ln (J_{L} / J_{0} + 1)$ .
Operating point: The load selects a point of the curve $J (V)$ ; the power $J V$ is largest where $e^{x} (1 + x) = J_{L} / J_{0} + 1$ with $x = q V / n k_{B} T$ (my derivation), and the fill factor measures how square the curve is.
Resistive effects: The series resistance subtracts $J R_{s}$ from the junction voltage, and the shunt resistance diverts the current $V / R_{s h}$ ; the first leaves $V_{oc}$ unchanged, and the second lowers it at low light.
Thermodynamic floor: An ideal cell still emits photons because of its radiative recombination, at a rate that rises exponentially with the voltage; the balance with the absorbed sun gives the limit in the last section.

3. Applications

Rating and design: The four numbers, the series and shunt resistance and the temperature coefficients are the parameters of the data sheets and of the models used for the rating of cells and modules (background, the pages read give the definitions only).
Diagnosis: The slopes of the I-V curve at open circuit and at short circuit give $R_{s}$ and $R_{s h}$ , and the ideality factor shows the recombination path.
Concentrator cells: Cells for 10 to 1000 suns need a series resistance far below the one-sun values, as the numerical example shows (background for the range of concentrations).
Targets for new absorbers: The detailed-balance limit and the voltage deficit $E_{g} / q - V_{oc}$ set the goal for new materials such as perovskites, GaAs and thin films.

Cell Equations and Operating Point

Name	Formula	Related Formulae	Variables/Constants
Ideal solar-cell equation	$J = J_{L} - J_{0} [exp (\frac{q V}{n k _{B} T}) - 1]$	$J = J_{L} - J_{d a r k} (V)$ $J \approx J_{L} - J_{0} exp (\frac{q V}{n k _{B} T})$	$J : c u r r e n t d e n s i t y d e l i v er e d t o t h e l o a d (p o w er q u a d r an t p os i t i v e; P V E d u c a t i o n w r i t es t h e s am e e q u a t i o n w i t h t h e o pp os i t e s i g n an d in am p er es)$ $J_{L} : l i g h t - g e n er a t e d c u r r e n t d e n s i t y$ $J_{d a r k} : c u r r e n t d e n s i t y o f t h e d i o d e in t h e d a r k$ $J_{0} : d a r k s a t u r a t i o n c u r r e n t d e n s i t y$ $V : t er mina l v o l t a g e$ $n : i d e a l i t y f a c t or$ $k_{B} : B o l t z mann co n s t an t$ $T : t e m p er a t u r e$ $q : e l e m e n t a r y c ha r g e, t h e - 1 c an b e d r o pp e d ab o v e ab o u t 100 mV$ $(J_{L} = 40 mA/c m^{2}, J_{0} = 1 0^{- 12} A/c m^{2}, n = 1 an d T = 300 K (a ss u m e d e x am pl e v a l u es) g i v e k_{B} T / q = 25.852 mV an d V_{oc} = 0.6311 V, co m p u t e d)$
Open-circuit voltage	$V_{oc} = \frac{n k _{B} T}{q} ln (\frac{J _{L}}{J _{0}} + 1)$	$V_{oc, im pl i e d} = \frac{k _{B} T}{q} ln [\frac{( N _{A} + Δ n ) Δ n}{n _{i}^{2}}]$ $Δ V_{oc} = \frac{n k _{B} T}{q} ln 10 per decade of J_{0}$	$V_{oc} : o p e n - c i r c u i t v o l t a g e$ $J_{L} : l i g h t - g e n er a t e d c u r r e n t d e n s i t y$ $J_{0} : d a r k s a t u r a t i o n c u r r e n t d e n s i t y$ $N_{A} : d o p in g co n ce n t r a t i o n$ $Δ n : e x cess c a r r i er co n ce n t r a t i o n$ $n_{i} : in t r in s i c c a r r i er co n ce n t r a t i o n, t h e im pl i e d f or m g i v es t h e v o l t a g e t ha t t h e c a r r i er d e n s i t i es w o u l d s u pp or t w i t h o u t co n t a c t s$ $k_{B} : B o l t z mann co n s t an t$ $(n = 1 an d 300 K g i v e 59.5 mV p er d ec a d e o f J_{0} (co m p u t e d); P V E d u c a t i o n q u o t es u p t o 764 mV f or cr y s t a l l in e s i l i co n a t o n e s u n an d ab o u t 690 mV f or co mm er c ia l d e v i ces)$
Short-circuit current	$J_{sc} = q \int QE (λ) Φ (λ) d λ$	$J_{sc} = q G (L_{n} + L_{p})$ $J_{sc, ma x} (E_{g}) = q \int_{E_{g}}^{\infty} Φ_{s} (E) d E$	$J_{sc} : s h or t - c i r c u i t c u r r e n t d e n s i t y$ $QE : q u an t u m e f f i c i e n cy (t h e g e n er a l f or m w i t h t h e s p ec t r u m i s ba c k g r o u n d k n o w l e d g e, n o t w r i tt e n o n t h e P V E d u c a t i o n p a g e)$ $Φ : in c i d e n t p h o t o n f l ux d e n s i t y p er u ni t w a v e l e n g t h$ $G : g e n er a t i o n r a t e (u ni f or m)$ $L_{n}, L_{p} : d i f f u s i o n l e n g t h s o f e l ec t r o n s an d h o l es (t h e s im pl e f or m h o l d s f or a p er f ec tl y p a ss i v a t e d s u r f a ce)$ $Φ_{s} : p h o t o n f l ux d e n s i t y p er u ni t e n er g y in t h e A M 1.5 G s p ec t r u m$ $E_{g} : ban d g a p$ $(A M 1.5 G o f A S T M G 173 (1000.4 W / m^{2} f r o m 280 t o 4000 nm) g i v es J_{sc, ma x} = 43.8 m A / c m^{2} f or E_{g} = 1.12 e V an d 35.0 m A / c m^{2} f or 1.34 e V, co m p u t e d; P V E d u c a t i o n q u o t es 46 f or s i l i co n, o v er 42 f or l ab or a t or y ce l l s an d 28 t o 35 f or co mm er c ia l ce l l s)$
Fill factor and efficiency	$F F = \frac{J _{m p} V _{m p}}{J _{sc} V _{oc}}, η = \frac{V _{oc} J _{sc} F F}{P _{in}}$	$F F_{0} \approx \frac{v _{oc} - l n ( v _{oc} + 0.72 )}{v _{oc} + 1}, v_{oc} = \frac{q V _{oc}}{n k _{B} T}$ $P_{in} = G_{S T C} A, G_{S T C} = 1000 W/ m^{2}$	$F F : f i l l f a c t or$ $J_{m p}, V_{m p} : c u r r e n t d e n s i t y an d v o l t a g e a t t h e ma x im u m p o w er p o in t$ $η : p o w er co n v er s i o n e f f i c i e n cy$ $P_{in} : in c i d e n t p o w er$ $F F_{0} : e m p i r i c a l f i l l f a c t or w i t h o u t p a r a s i t i c r es i s t an ce$ $v_{oc} : n or ma l i z e d o p e n - c i r c u i t v o l t a g e$ $G_{S T C} : i r r a d ian ce a t s t an d a r d t es t co n d i t i o n s (A M 1.5 s p ec t r u m, 25 C)$ $A : ce l l a r e a$ $k_{B} : B o l t z mann co n s t an t$ $(V_{oc} = 764, 690 an d 644 mV w i t h n = 1 an d 300 K g i v e F F_{0} = 0.856, 0.844 an d 0.836 (co m p u t e d; P V E d u c a t i o n q u o t es 0.85 an d 0.83); a 156 mm s q u a r e ce l l r ece i v es 24.3 W; t h e e x am pl e ce l l o f t h e f i r s t r o w ha s F F = 0.8337 an d η = 21.0% a t 100 mW / c m^{2}, co m p u t e d)$
Maximum power point (Lambert W form)	$V_{m p} = \frac{n k _{B} T}{q} [W (e (\frac{J _{L}}{J _{0}} + 1)) - 1]$	$V_{m p} = V_{oc} - \frac{n k _{B} T}{q} ln (\frac{q V _{m p}}{n k _{B} T} + 1)$ $J_{m p} = J_{L} - J_{0} [exp (\frac{q V _{m p}}{n k _{B} T}) - 1]$	$V_{m p} : v o l t a g e a t ma x im u m p o w er$ $W : L amb er t W f u n c t i o n (t h e so l u t i o n o f W e^{W} = z)$ $e : E u l er n u mb er, t h e im pl i c i t f or m i s so l v e d b y i t er a t i o n f r o m V_{m p} = 0.9 V_{oc}, i t f o l l o w s f r o m d (J V) / d V = 0 (m y d er i v a t i o n); t h e p a g e p r in t s, a s e x t r a c t e d, n V_{t} W (J_{L} / J_{0}), w hi c h d i f f er s s l i g h tl y$ $k_{B} : B o l t z mann co n s t an t$ $(t h e e x am pl e ce l l g i v es V_{m p} = 0.5508 V an d J_{m p} = 38.21 m A / c m^{2}; t h e i t er a t i o n g i v es 0.5501, 0.5509 an d 0.55083 V a f t er o n e t o t h r ee p a sses, an d t h e e x t r a c t e d f or m g i v es 0.5520 V, co m p u t e d)$

Parasitics, Ideality, Temperature and Concentration

Name	Formula	Related Formulae	Variables/Constants
Series resistance	$J = J_{L} - J_{0} [exp (\frac{q ( V + J R _{s} )}{n k _{B} T}) - 1]$	$R_{C H} = \frac{V _{oc}}{J _{sc}}, r_{s} = \frac{R _{s}}{R _{C H}}$ $F F_{s} \approx F F_{0} (1 - 1.1 r_{s}) + \frac{r _{s}^{2}}{5.4}$ $p_{R s} = J^{2} R_{s}$	$R_{s} : ser i es r es i s t an ce (p er u ni t a r e a)$ $R_{C H} : c ha r a c t er i s t i c r es i s t an ce$ $r_{s} : n or ma l i z e d ser i es r es i s t an ce$ $F F_{s} : f i l l f a c t or w i t h ser i es r es i s t an ce$ $p_{R s} : p o w er d e n s i t y l os t in R_{s}, t h e e m p i r i c a l f or m o f F F_{s} i s f r o m m e m or y b ec a u se t h e p a g e s h o w s i t a s an ima g e, an d t h e p a g e g i v es i t s r an g e a s r_{s} < 0.4 an d v_{oc} > 10; R_{s} i s 0.5 Ω c m^{2} (l ab or a t or y) t o 1.3 Ω c m^{2} (co mm er c ia l), an d i t d oes n o t c han g e V_{oc}$ $k_{B} : B o l t z mann co n s t an t$ $(R_{C H} = 15.8 Ω c m^{2} f or t h e e x am pl e ce l l; R_{s} = 0.5 g i v es r_{s} = 0.032 an d F F_{s} = 0.8049 (n u m er i c a l so l u t i o n 0.8049), an d R_{s} = 1.3 g i v es 0.7594 (n u m er i c a l 0.7592); t h e l oss J_{m p}^{2} R_{s} i s 0.73 an d 1.90 mW / c m^{2}, 3.5% an d 9.0% o f P_{m p}, co m p u t e d)$
Shunt resistance	$J = J_{L} - J_{0} [exp (\frac{q V}{n k _{B} T}) - 1] - \frac{V}{R _{s h}}$	$P_{m p}^{'} = P_{m p} (1 - \frac{1}{r _{s h}}), r_{s h} = \frac{R _{s h}}{R _{C H}}$ $F F_{s h} \approx F F_{0} {1 - \frac{v _{oc} + 0.7}{v _{oc}} \frac{F F _{0}}{r _{s h}}}$	$R_{s h} : s h u n t r es i s t an ce (p er u ni t a r e a)$ $r_{s h} : n or ma l i z e d s h u n t r es i s t an ce$ $P_{m p}^{'} : ma x im u m p o w er w i t h t h e s h u n t$ $F F_{s h} : f i l l f a c t or w i t h t h e s h u n t (t h e e m p i r i c a l f or m i s f r o m m e m or y; t h e p a g e g i v es i t s r an g e a s r_{s h} > 0.4 an d s h o w s i t a s an ima g e), R_{s h} i s in t h e M Ω c m^{2} r an g e (l ab or a t or y) an d ab o u t 1000 Ω c m^{2} (co mm er c ia l), an d t h e e f f ec t i s s t r o n g es t a t l o w l i g h t$ $k_{B} : B o l t z mann co n s t an t$ $(R_{s h} = 1000 Ω c m^{2} (r_{s h} = 63) g i v es P_{m p}^{'} / P_{m p} = 0.9856 (n u m er i c a l) a g ain s t 0.9842 f r o m 1 - 1/ r_{s h}, an d F F = 0.8217 a g ain s t 0.8225; 100 Ω c m^{2} g i v es 0.715 a g ain s t 0.721; 20 Ω c m^{2} g i v es 0.317 a g ain s t 0.270, co m p u t e d)$
Two-diode model and ideality factor	$J = J_{L} - J_{01} [e^{\frac{q ( V + J R _{s} )}{k _{B} T}} - 1] - J_{02} [e^{\frac{q ( V + J R _{s} )}{2 k _{B} T}} - 1] - \frac{V + J R _{s}}{R _{s h}}$	$n = \frac{q}{k _{B} T} (\frac{d l n J _{d a r k}}{d V})^{- 1}$ $n_{S R H, l o w} = 1, n_{S R H, hi g h} = 2, n_{A ug er} = \frac{2}{3}, n_{d e pl e t i o n} = 2$	$J_{01} : s a t u r a t i o n c u r r e n t d e n s i t y o f t h e i d e a l i t y - 1 (d i f f u s i o n) d i o d e$ $J_{02} : s a t u r a t i o n c u r r e n t d e n s i t y o f t h e i d e a l i t y - 2 (d e pl e t i o n - r e g i o n) d i o d e$ $n : i d e a l i t y f a c t or (l oc a l s l o p e f or m i s m y d e f ini t i o n), t h e l ec t u r e s l i d es w r i t e t h e tw o e x p o n e n t ia l s w i t h o u t t h e - 1 t er m s an d t h e t e x t i s g a r b l e d$ $S R H : S h oc k l ey - R e a d - H a l l r eco mbina t i o n, t h e t ab l e o f n i s f r o m P V E d u c a t i o n$ $k_{B} : B o l t z mann co n s t an t$ $(w i t h n = 2 t h e g ain o f V_{oc} f or a d o u b l e d l i g h t in t e n s i t y i s 35.8 mV in s t e a d o f 17.9 mV f or n = 1, a t 300 K, co m p u t e d)$
Temperature dependence	$n_{i} (T) = 9.38 \times 1 0^{19} (\frac{T}{300})^{2} e^{- 6884/ T} c m^{- 3}$	$J_{0} = A n_{i}^{2}$ $\frac{d V _{oc}}{d T} = - \frac{E _{g 0} / q - V _{oc} + γ k _{B} T / q}{T} \approx - 0.0043 + \frac{V _{oc}}{300} V/K$ $\frac{1}{I _{sc}} \frac{d I _{sc}}{d T} \approx + 0.06%/ K, \frac{1}{P _{m}} \frac{d P _{m}}{d T} \approx - (0.4 t o 0.5) %/ K$	$n_{i} : in t r in s i c c a r r i er co n ce n t r a t i o n o f s i l i co n (S p r o u l - G r ee n a pp r o x ima t i o n)$ $A : t e m p er a t u r e - in d e p e n d e n t r eco mbina t i o n co n s t an t$ $E_{g 0} : ban d g a p e x t r a p o l a t e d t o 0 K (1.2 e V f or s i l i co n)$ $γ : co n s t an t t ha t co v er s o t h er t e m p er a t u r e e f f ec t s (3)$ $I_{sc} : s h or t - c i r c u i t c u r r e n t$ $P_{m} : ma x im u m p o w er, t h e f i r s t e q u a l i t y o f d V_{oc} / d T i s t h e p a g e^{'} s an d t h e seco n d i s i t s s im pl i f i e d f or m a t 300 K$ $k_{B} : B o l t z mann co n s t an t$ $(n_{i} = 1.015 \times 1 0^{10} c m^{- 3} a t 300 K an d 8.70 \times 1 0^{9} a t 25 C; n_{i}^{2} r i ses b y 5.01 f r o m 300 t o 310 K an d b y 5.1 f r o m 25 t o 35 C; d V_{oc} / d T = - 2.26 mV / K f or V_{oc} = 0.6 V, - 1.96 f or 0.69 V an d - 1.71 f or 0.764 V, co m p u t e d)$
Light concentration	$J_{sc} (X) = X J_{sc} (1), V_{oc} (X) = V_{oc} (1) + \frac{n k _{B} T}{q} ln X$	$P_{R s} (X) = X^{2} p_{R s} (1)$ $η (X) = \frac{V _{oc} ( X ) J _{sc} ( X ) F F ( X )}{X P _{in}}$	$X : co n ce n t r a t i o n (n u mb er o f s u n s)$ $J_{sc} (1), V_{oc} (1) : v a l u es a t o n e s u n$ $P_{R s} : p o w er l os t in t h e ser i es r es i s t an ce p er u ni t a r e a$ $p_{R s} (1) : t h e s am e a t o n e s u n, t h e g ain o f e f f i c i e n cy co m es f r o m t h e l o g a r i t hmi c r i se o f V_{oc} an d i s c u t b y t h e ser i es r es i s t an ce an d b y t h e hi g h er t e m p er a t u r e$ $k_{B} : B o l t z mann co n s t an t$ $(n = 1 an d 300 K g i v e + 17.9 mV f or X = 2, + 59.5 mV f or X = 10 an d + 119.1 mV f or X = 100 (P V E d u c a t i o n : 18 mV f or X = 2); t h e e x am pl e ce l l w i t h R_{s} = 0 ha s η = 21.1, 23.3 an d 25.6% a t X = 1, 10 an d 100, an d w i t h R_{s} = 0.5 Ω c m^{2} 20.3, 16.3 an d 2.8%, co m p u t e d)$

Detailed-Balance Limit

Name	Formula	Related Formulae	Variables/Constants
Detailed-balance current-voltage characteristic	$J (V) = J_{sc} - J_{0} (e^{q V / k_{B} T} - 1)$	$J_{sc} = q \int_{E_{g}}^{\infty} Φ_{s} (E) d E, J_{0} = q π \int_{E_{g}}^{\infty} Φ_{bb} (E) d E$ $Φ_{bb} (E) = \frac{2 E ^{2}}{h ^{3} c ^{2}} \frac{1}{e ^{E / k_{B} T} - 1}$ $J_{0} = \frac{q}{k _{B}} \frac{15 σ}{π ^{4}} T^{3} \int_{u}^{\infty} \frac{x ^{2}}{e ^{x} - 1} d x, u = \frac{E _{g}}{k _{B} T}$	$J_{sc} : p h o t o g e n er a t e d c u r r e n t d e n s i t y (o n e e l ec t r o n p er p h o t o n ab o v e t h e g a p, s t e p ab sor pt i v i t y)$ $J_{0} : r a d ia t i v e s a t u r a t i o n c u r r e n t d e n s i t y o f a ce l l a t t e m p er a t u r e T t ha t e mi t s in t o o n e h e mi s p h er e (t h e f a c t or π i s n o t e x pl ain e d b y t h e so u r ce)$ $Φ_{s} : p h o t o n f l ux d e n s i t y o f t h e s u n$ $Φ_{bb} : b l a c k b o d y p h o t o n f l ux d e n s i t y$ $h : P l an c k co n s t an t$ $c : s p ee d o f l i g h t$ $σ : S t e f an - B o l t z mann co n s t an t, t h e l a s t f or m i s t h e minim u m J_{0} o f P V E d u c a t i o n; w i t h o u t a ba c k r e f l ec t or J_{0} d o u b l es$ $k_{B} : B o l t z mann co n s t an t$ $(E_{g} = 1.12 e V an d T = 298.15 K g i v e J_{0} = 6.25 \times 1 0^{- 17} A / c m^{2} (t h e P V E d u c a t i o n in t e g r a l g i v es t h e s am e) an d V_{oc} = 0.878 V, F F = 0.8705, η = 33.5%; w i t h o u t t h e r e f l ec t or J_{0} = 1.25 \times 1 0^{- 16}, V_{oc} = 0.861 V, η = 32.7%, co m p u t e d)$
Limiting open-circuit voltage and non-radiative penalty	$q V_{oc} = E_{g} (1 - \frac{T _{D}}{T _{S}}) - k_{B} T_{D} ln \frac{Ω _{D}}{Ω _{S}} - k_{B} T_{D} ln \frac{γ _{S Q} ( T _{D} )}{γ _{S Q} ( T _{S} )} - k_{B} T_{D} ln \frac{1}{η _{e x t}}$	$γ_{S Q} (T) = \frac{2 k _{B} T}{c ^{2} h ^{3}} (E_{g}^{2} + 2 k_{B} T E_{g} + 2 k_{B}^{2} T^{2})$ $Δ V_{oc} = \frac{k _{B} T _{D}}{q} ln \frac{1}{η _{e x t}}$	$T_{S}, T_{D} : t e m p er a t u r es o f t h e s u n (6000 K) an d o f t h e d e v i ce (300 K)$ $Ω_{S}, Ω_{D} : so l i d an g l es o f t h e s u n (6.8 \times 1 0^{- 5} sr) an d o f t h e e mi ss i o n o f t h e d e v i ce (2 π)$ $γ_{S Q} : f u n c t i o n o f t h e b l a c k b o d y p h o t o n f l ux ab o v e t h e g a p in t h e B o l t z mann a pp r o x ima t i o n$ $η_{e x t} : e x t er na l r a d ia t i v e (f l u or esce n ce) e f f i c i e n cy$ $Δ V_{oc} : n o n - r a d ia t i v e l oss o f v o l t a g e; a f u r t h er s ma l l cor r ec t i o n t er m o f t h e so u r ce i s n o t w r i tt e n h er e$ $k_{B} : B o l t z mann co n s t an t$ $(E_{g} = 1.12 e V an d η_{e x t} = 1 g i v e 0.8667 V (d i r ec t in t e g r a t i o n w i t h t h e s am e b l a c k b o d y : 0.8678 V); η_{e x t} = 0.57% g i v es Δ V_{oc} = 129 mV a s p r in t e d in t h e so u r ce, w hi c h m y f or m u l a r e p r o d u ces o n l y a t ab o u t 290 K, co m p u t e d)$
Ultimate efficiency	$u (x_{g}) = \frac{x _{g} \int _{x_{g}}^{\infty} \frac{x ^{2}}{e ^{x} - 1} d x}{\int _{0}^{\infty} \frac{x ^{3}}{e ^{x} - 1} d x}, x_{g} = \frac{E _{g}}{k _{B} T _{S}}$	$\int_{0}^{\infty} \frac{x ^{3}}{e ^{x} - 1} d x = \frac{π ^{4}}{15}$ $P_{in} = \frac{2 π ^{4}}{15} \frac{Ω _{S} ( k _{B} T _{S} ) ^{4}}{c ^{2} h ^{3}}$	$u : u l t ima t e e f f i c i e n cy (e v er y p h o t o n ab o v e t h e g a p g i v es t h e e n er g y E_{g}, n o o t h er l oss)$ $x_{g} : n or ma l i z e d ban d g a p$ $T_{S} : t e m p er a t u r e o f t h e s u n$ $P_{in} : p o w er d e n s i t y o f a b l a c k b o d y s u n o f so l i d an g l e Ω_{S}; K han e t a l . p r in t t h e coe f f i c i e n t 12 f or t h e e x a c t 2 π^{4} /15 = 12.99$ $k_{B} : B o l t z mann co n s t an t$ $(6000 K g i v es u = 43.9% a t 1.1 an d a t 1.12 e V (t h e ma x im u m, a g ain s t 2.2 k_{B} T_{S} = 1.137 e V), an d t h e d e t ai l e d - ba l an ce e f f i c i e n cy a t 1.1 e V i s 30.0% w i t h a r e f l ec t or; P_{in} = 1591 W / m^{2} f or 6.8 \times 1 0^{- 5} sr an d 1470 W / m^{2} w i t h t h e coe f f i c i e n t 12, co m p u t e d)$
Power budget of an ideal cell on AM1.5G	$P_{in} = P_{s u b} + P_{t h} + P_{V oc} + P_{F F} + P_{o u t}$	$P_{s u b} = \int_{λ_{g}}^{\infty} G (λ) d λ, P_{t h} = \int_{0}^{λ_{g}} G (λ) d λ - E_{g} N_{p h}$ $P_{V oc} = (\frac{E _{g}}{q} - V_{oc}) J_{sc}, P_{F F} = V_{oc} J_{sc} (1 - F F)$	$P_{s u b} : p o w er o f t h e p h o t o n s b e l o w t h e g a p$ $P_{t h} : t h er ma l i z a t i o n l oss$ $P_{V oc} : v o l t a g e d e f i c i t$ $P_{F F} : f i l l - f a c t or l oss$ $P_{o u t} : e l ec t r i c a l o u tp u t$ $G : s p ec t r a l i r r a d ian ce$ $λ_{g} : w a v e l e n g t h o f t h e g a p$ $N_{p h} : n u mb er o f p h o t o n s p er seco n d an d a r e a ab o v e t h e g a p, t h e s pl i t o f P_{V oc} in t o r a d ia t i v e an d o t h er p a r t s i s n o t ma d e h er e$ $(E_{g} = 1.12 e V, w i t h a ba c k r e f l ec t or an d 298.15 K : 19.2% b e l o w t h e g a p, 31.7% t h er ma l i z a t i o n, 10.6% v o l t a g e d e f i c i t, 5.0% f i l l f a c t or an d 33.5% o u tp u t; E_{g} = 1.34 e V : 29.9, 23.2, 9.0, 4.2 an d 33.7%, co m p u t e d)$

Important Questions

Ques: Is the Shockley-Queisser optimum 1.1 eV with 30 %, or 1.34 eV with 33 %?

Both, for different assumptions. For a 6000 K blackbody sun of $6.8 \times 1 0^{- 5}$ sr (1591 $W/ m^{2}$ ) and a rear reflector, I compute 30.0 % at 1.1 eV and about 31 % near 1.3 eV; the 1.1 eV of the original is the peak of the ultimate efficiency $u$ , which I compute at 1.12 eV (43.9 %). For the AM1.5G spectrum the optimum is 1.34 eV, with 33.1 % for emission from both faces and 33.7 % with a reflector, at 298.15 K (computed). Wikipedia pairs 33.16 % with the mirror, but my numbers pair it with no mirror, so its label is probably swapped; that is my reading, as its equations were stripped. Khan et al. give about 1.35 eV, and the $π$ in $J_{0}$ of arXiv:1705.07762 is the mirror case. Reference: Wikipedia, ‘Shockley-Queisser limit’ (equations stripped in the text that the tool returned); Khan, Bermel and Alam, arXiv:1401.2886 (title not shown to me); ‘Exceeding the Shockley-Queisser limit within the detailed balance framework’, arXiv:1705.07762 (authors not shown to me); ASTM G173 reference spectra as distributed in pvlib 0.16.1 (pvlib.spectrum.get_reference_spectra()), integrated by me.

Ques: Does the saturation current double for every 10 °C, as PVEducation says?

Not for the diffusion current of its own model. The page gives $J_{0} = A n_{i}^{2}$ and $n_{i} = 9.38 \times 1 0^{19} (T /300)^{2} e^{- 6884/ T}$ , which yields $n_{i} = 1.015 \times 1 0^{10}$ $c m^{- 3}$ at 300 K and a ratio of 5.01 for $n_{i}^{2}$ from 300 to 310 K, and 5.1 from 25 to 35 °C (computed). A doubling per 10 °C is the ratio of $n_{i}$ itself, 2.24 (computed), which is the dependence of a current set by recombination in the depletion region ( $n = 2$ ); that is my reading, the page does not say it. For the ideal diode the factor of five is the more likely, and it is what drives $V_{oc}$ down. The page’s $n_{i}$ at 25 °C, $8.6 \times 1 0^{9}$ , is 1 % below my $8.70 \times 1 0^{9}$ . Reference: PVEducation, pveducation.org/pvcdrom/solar-cell-operation/effect-of-temperature.

Ques: Is the Lambert W expression for the maximum power point exact?

The exact condition is $e^{x} (1 + x) = J_{L} / J_{0} + 1$ with $x = q V / n k_{B} T$ , which gives $V_{m p} = (n k_{B} T / q) [W (e (J_{L} / J_{0} + 1)) - 1]$ (my derivation). The fill-factor page, as extracted, prints $n V_{t} W (J_{L} / J_{0})$ , which differs. For $J_{L} = 40$ $mA/c m^{2}$ , $J_{0} = 1 0^{- 12}$ $A/c m^{2}$ and 300 K the exact value is 0.5508 V and the printed form gives 0.5520 V, 1.1 mV (0.2 %) higher (computed). The implicit form started at $0.9 V_{oc}$ gives 0.5501, 0.5509 and 0.55083 V after one to three passes, so the page’s error limits of below 1 % and below 0.01 % hold. The extraction probably lost part of the argument, but I did not see the page itself, so the exact form should be used. Reference: PVEducation, pveducation.org/pvcdrom/solar-cell-operation/fill-factor.

Ques: Is the voltage of a commercial silicon cell 690 mV or 644 mV, and what fill factor follows?

The two pages of PVEducation disagree. The voltage page gives up to 764 mV for crystalline silicon and about 690 mV for commercial devices, whereas the fill-factor page speaks of a difference of about 120 mV and fill factors of 0.85 and 0.83. With $n = 1$ and 300 K the ideal fill factor is 0.856 for 764 mV, 0.844 for 690 mV and 0.836 for 644 mV (computed); 764 minus 120 is 644, and 0.856 and 0.836 become the quoted 0.85 and 0.83 if truncated, whereas 690 mV would give 0.84. So the fill-factor page needs 644 mV and the voltage page gives 690 mV; the pages do not say which is typical, and the numbers cannot settle it. The pair 690 mV and 0.844 is self-consistent. Reference: PVEducation, pveducation.org/pvcdrom/solar-cell-operation/open-circuit-voltage; PVEducation, pveducation.org/pvcdrom/solar-cell-operation/fill-factor.

Ques: Is the maximum short-circuit current of silicon 46 or 43.8 $mA/c m^{2}$ ?

PVEducation states that silicon has a maximum of 46 $mA/c m^{2}$ under AM1.5, laboratory cells over 42 and commercial cells 28 to 35. Integrating the global spectrum of ASTM G173 (1000.4 $W/ m^{2}$ from 280 to 4000 nm) with one electron per photon above the gap, I get 43.8 $mA/c m^{2}$ for 1.12 eV (cutoff 1107 nm) and 44.2 for 1.1 eV; 46 needs a cutoff of 1190 nm, i.e. 1.04 eV (computed). So 46 comes from an older spectrum, a longer cutoff or a loose figure; the page does not say which. The value of about 44 is more likely the bound for a sharp edge at 1.1 eV, and it is consistent with laboratory cells over 42. Reference: PVEducation, pveducation.org/pvcdrom/solar-cell-operation/short-circuit-current; ASTM G173 reference spectra as distributed in pvlib 0.16.1 (pvlib.spectrum.get_reference_spectra()), integrated by me.

Ques: Are the empirical fill-factor formulas for the series and shunt resistance accurate?

Only in their stated ranges. They are from memory, as the pages show them as images, and I checked them against a numerical solution of the example cell ( $R_{C H} = 15.8$ $Ω c m^{2}$ ). For the series resistance, $F F_{s}$ is 0.8049 (formula) and 0.8049 (numerical) at 0.5 $Ω c m^{2}$ , and 0.7594 and 0.7592 at 1.3. For the shunt, 1000 $Ω c m^{2}$ gives 0.8225 and 0.8217, 100 gives 0.721 and 0.715, and 20 gives 0.270 and 0.317 although $r_{s h} = 1.27$ lies above the stated limit of 0.4 (computed). The simple $1 - 1/ r_{s h}$ is 0.984 against 0.9856 for 1000 $Ω c m^{2}$ . A strongly shunted cell should be solved numerically. Reference: PVEducation, pveducation.org/pvcdrom/solar-cell-operation/series-resistance; PVEducation, pveducation.org/pvcdrom/solar-cell-operation/shunt-resistance.

Ques: Is the sign of the term with the open-circuit voltage in the temperature coefficient correct?

The simplified form on the page, $- 0.0043 + V_{oc 1} /300$ V/K, is correct: differentiating $V_{oc} = (k_{B} T / q) ln (J_{L} / A n_{i}^{2})$ with $n_{i}^{2} \propto T^{4} e^{- 13768/ T}$ gives $V_{oc} / T - (k_{B} / q) (4 + 13768/ T)$ , and the second term is $- 4.3$ $mV/K$ at 300 K (computed). The full expression as extracted ends with $- V_{oc} / T_{1}$ , which has the opposite sign and disagrees with the simplified one on the same page; I take it as an extraction error, which is my reading. The result is $- 2.26$ $mV/K$ at 0.6 V, $- 1.96$ at 0.69 V and $- 1.71$ at 0.764 V (computed), so the page’s 2.2 $mV/K$ holds for about 0.6 V; better cells lose less. The result also depends on the assumed $E_{g 0} = 1.2$ eV and $γ = 3$ , so it is an estimate for design. Reference: PVEducation, pveducation.org/pvcdrom/solar-cell-operation/effect-of-temperature.

Ques: What does concentration do to the efficiency when the series resistance is not zero?

The voltage gain is small, $(n k_{B} T / q) ln X$ : 17.9 mV at $X = 2$ (the page says 18 mV), 59.5 mV at 10 and 119 mV at 100 (computed). The series loss grows with $X^{2}$ . For the example cell ( $J_{L} = 40$ $mA/c m^{2}$ per sun, $J_{0} = 1 0^{- 12}$ $A/c m^{2}$ ) the efficiency with $R_{s} = 0$ is 21.1, 23.3 and 25.6 % at 1, 10 and 100 suns, with $R_{s} = 0.5$ $Ω c m^{2}$ it is 20.3, 16.3 and 2.8 %, and with 1.3 it is 19.2, 8.5 and 1.1 % (computed). So the one-sun resistances are of no use at 10 suns, and at 10 suns a resistance of about 0.05 $Ω c m^{2}$ keeps $J_{m p}^{2} R_{s}$ near 3 % of the output (computed). Reference: PVEducation, pveducation.org/pvcdrom/solar-cell-operation/effect-of-light-intensity; PVEducation, pveducation.org/pvcdrom/solar-cell-operation/series-resistance.

Ques: Does Table I of Khan et al. follow their own formula at 300 K?

Not at 300 K. The penalty is $Δ V_{oc} = (k_{B} T / q) ln (1/ η_{e x t})$ . The printed values are 129 mV for $η_{e x t} = 0.57$ %, 130 for 0.56 %, 37 for 22.5 %, 109 for 1.26 % and 187 for 0.057 %. At 300 K I get 133.6, 134.0, 38.6, 113.1 and 193.1 mV, and at 290 K 129.1, 129.6, 37.3, 109.3 and 186.7 (computed); the implied temperature is 288 to 291 K for all five rows. The paper’s formulas take $T_{D} = 300$ K. The table was thus computed at about 290 K, or the temperature is a choice for the measured cells that the tool did not report; the difference of 4 to 6 mV does not change any conclusion. Reference: Khan, Bermel and Alam, arXiv:1401.2886 (title not shown to me).

Ques: Is the coefficient of the incident power in Khan et al. 12 or 12.99?

The paper prints $P_{in} = 12 Ω_{S} (k_{B} T_{S})^{4} / (c^{2} h^{3})$ , but the integral $\int_{0}^{\infty} x^{3} / (e^{x} - 1) d x = π^{4} /15$ gives the coefficient $2 π^{4} /15 = 12.99$ (computed). For $Ω_{S} = 6.8 \times 1 0^{- 5}$ sr and 6000 K, 12 gives 1470 $W/ m^{2}$ and 12.99 gives 1591 $W/ m^{2}$ , 8 % more (computed), and 1591 is what the Stefan-Boltzmann law gives for that solid angle. So the exact coefficient is the more likely one, and 12 is a truncation by the authors or an error of the text that I could read only through a fetch tool. Neither value is the 1000 $W/ m^{2}$ of AM1.5G, so a blackbody-sun result cannot be set beside the spectrum result without scaling. Reference: Khan, Bermel and Alam, arXiv:1401.2886 (title not shown to me).

Ques: Where does the sunlight go in an ideal silicon cell?

For a gap of 1.12 eV, a back reflector and 298.15 K, I find on the AM1.5G spectrum (computed): 19.2 % is below the gap, 31.7 % is lost by thermalization, 10.6 % is the deficit between $E_{g} / q$ and $V_{oc}$ (0.878 V, 78 % of the gap), 5.0 % is lost in the fill factor, and 33.5 % is delivered. Without the reflector, $V_{oc}$ is 0.861 V and the output 32.7 %. Wikipedia gives about 19 % below the gap and about 33 % of thermalization, close to my 19.2 and 31.7 %. At 1.34 eV the shares are 29.9, 23.2, 9.0, 4.2 and 33.7 %. The unavoidable terms are the first two; the others come from the radiative recombination. Reference: Wikipedia, ‘Shockley-Queisser limit’ (equations stripped in the text that the tool returned); ASTM G173 reference spectra as distributed in pvlib 0.16.1 (pvlib.spectrum.get_reference_spectra()), integrated by me.

Quartz 5

Explorer

Solar Cell Fundamentals

1. Theory

2. Working Principle

3. Applications

Cell Equations and Operating Point

Parasitics, Ideality, Temperature and Concentration

Detailed-Balance Limit

Important Questions

Ques: Is the Shockley-Queisser optimum 1.1 eV with 30 %, or 1.34 eV with 33 %?

Ques: Does the saturation current double for every 10 °C, as PVEducation says?

Ques: Is the Lambert W expression for the maximum power point exact?

Ques: Is the voltage of a commercial silicon cell 690 mV or 644 mV, and what fill factor follows?

Ques: Is the maximum short-circuit current of silicon 46 or 43.8 $mA/c m^{2}$ ?

Ques: Are the empirical fill-factor formulas for the series and shunt resistance accurate?

Ques: Is the sign of the term with the open-circuit voltage in the temperature coefficient correct?

Ques: What does concentration do to the efficiency when the series resistance is not zero?

Ques: Does Table I of Khan et al. follow their own formula at 300 K?

Ques: Is the coefficient of the incident power in Khan et al. 12 or 12.99?

Ques: Where does the sunlight go in an ideal silicon cell?

Graph View

Table of Contents

Backlinks

Quartz 5

Explorer

Solar Cell Fundamentals

1. Theory

2. Working Principle

3. Applications

Cell Equations and Operating Point

Parasitics, Ideality, Temperature and Concentration

Detailed-Balance Limit

Important Questions

Ques: Is the Shockley-Queisser optimum 1.1 eV with 30 %, or 1.34 eV with 33 %?

Ques: Does the saturation current double for every 10 °C, as PVEducation says?

Ques: Is the Lambert W expression for the maximum power point exact?

Ques: Is the voltage of a commercial silicon cell 690 mV or 644 mV, and what fill factor follows?

Ques: Is the maximum short-circuit current of silicon 46 or 43.8 mA/cm2?

Ques: Are the empirical fill-factor formulas for the series and shunt resistance accurate?

Ques: Is the sign of the term with the open-circuit voltage in the temperature coefficient correct?

Ques: What does concentration do to the efficiency when the series resistance is not zero?

Ques: Does Table I of Khan et al. follow their own formula at 300 K?

Ques: Is the coefficient of the incident power in Khan et al. 12 or 12.99?

Ques: Where does the sunlight go in an ideal silicon cell?

Graph View

Table of Contents

Backlinks

Ques: Is the maximum short-circuit current of silicon 46 or 43.8 $mA/c m^{2}$ ?