1. Theory

Definition: A III-V multi-junction cell is a stack of p-n subcells with different gaps, joined in series by tunnel junctions, with the widest gap on top; each cell absorbs the photons between its own gap and the gap above it and passes the rest on (Wikipedia). The standard lattice-matched stack is GaInP (about 1.86 to 1.88 eV), GaAs (1.42 eV) and germanium (0.65 to 0.67 eV) (Wikipedia, see the question on the gaps). The tunnel junctions must be transparent to the light for the cells below, so their gap must exceed that of the cell beneath (Wikipedia; my reading of its statement).
Series constraint: Voltages add and the current is that of the weakest subcell, so the gaps and thicknesses are chosen to make the subcell currents equal (Wikipedia). In a GaInP/GaAs/Ge stack the germanium cell would carry 29.03 mA/cm $^{2}$ against 17.99 and 14.06 mA/cm $^{2}$ in the two upper cells, so it wastes current (computed, gaps 1.86/1.42/0.66 eV).
Routes to other gap sets: Growth lattice-matched on germanium or GaAs (GaAs on germanium has a misfit of -0.07 %, computed); inverted metamorphic growth, in which the upper cells are grown first on GaAs and a graded buffer leads to lower-gap InGaAs cells before the substrate is removed (the six-junction cell has about 140 layers, Semiconductor Today); wafer bonding of two stacks (the five-junction cell of Spectrolab and the four-junction cell of Fraunhofer ISE and Soitec); mechanical stacking with four terminals, and growth on silicon (GaAs on silicon has a misfit of +4.09 %, computed).
One-sun records (efficiency tables, version 64): Green et al. list a five-junction bonded cell (2.17/1.68/1.40/1.06/0.73 eV) of 38.8 ± 1.2 % (1.021 cm $^{2}$ , 4.767 V, 9.564 mA/cm $^{2}$ , 85.2 %, Spectrolab, NREL 2013), a GaInP/GaAs/GaInAs triple of 37.9 ± 1.2 % (Sharp, 1.047 cm $^{2}$ , 3.065 V, 14.27 mA/cm $^{2}$ , 86.7 %, AIST 2013), a GaInP/GaAs double of 32.8 ± 1.4 % (LG, 2.568 V, 14.56 mA/cm $^{2}$ , 87.7 %, NREL 2017), a GaInP/GaAs/GaInAs triple of 37.8 ± 1.4 % (Microlink, NREL 2018), a GaInP/GaAs(quantum wells)/GaInAs inverted metamorphic triple of 39.5 ± 0.5 % (0.242 cm $^{2}$ , 2.997 V, 15.44 mA/cm $^{2}$ , 85.3 %, NREL 2021) and a six-junction inverted metamorphic cell (2.19/1.76/1.45/1.19/0.97/0.7 eV) of 39.2 ± 3.2 % (0.247 cm $^{2}$ , 5.549 V, 8.457 mA/cm $^{2}$ , 83.5 %, NREL 2018). The NREL release for the 39.5 % cell gives 34.2 % under the space spectrum at the beginning of life and 37.9 % (Sharp, 2013) as the previous one-sun record.
Stacks on silicon: A bonded GaInP/GaInAsP//Si triple reached 36.1 ± 1.3 % (3.987 cm $^{2}$ , 3.309 V, 12.70 mA/cm $^{2}$ , 86.0 %, Fraunhofer ISE 2023), a four-terminal GaInP/GaAs/Si stack 35.9 ± 0.5 % (2.52 and 0.681 V, 13.6 and 11.0 mA/cm $^{2}$ , 87.5 and 78.5 %, NREL 2017), a four-terminal GaAs/Si stack 32.8 ± 0.5 % (NREL 2016), a monolithic GaInP/GaAs/Si triple 25.9 ± 0.9 % (2.647 V, 12.21 mA/cm $^{2}$ , 80.2 %, Fraunhofer ISE 2020) and a GaAsP/Si double 23.4 ± 0.3 % (NREL 2020). A spectrally split minimodule (GaInP/GaInAs/Ge and silicon, 27.83 cm $^{2}$ ; UNSW, Azur, Trina) gave 34.5 ± 2.0 % (NREL 2016).
Modules: Sharp’s GaInP/GaAs/GaInAs module (40 cells, 965 cm $^{2}$ ) reached 32.65 ± 0.7 % (24.30 V, 1.520 A, 85.3 %, AIST 2022), a four-terminal GaInP/GaAs//silicon module (775 cm $^{2}$ ; Sharp and Toyota) 33.7 ± 0.7 % and a GaInP/GaAs//CIGS module (Sharp and Idemitsu) 31.2 ± 0.7 % (AIST 2023).
Commercial level: Wikipedia puts commercial GaInP/GaAs/Ge cells at about 30 % under one sun and about 40 % under concentration (see the question on commercial values).
Theoretical limits: Wikipedia lists 37 % for one junction, 50 % for two, 56 % for three, 72 % for 36 and 86.8 % for an infinite number. My calculation for the AM1.5G global spectrum gives 33.16, 44.97 and 50.85 % for one to three junctions with free gaps and 58.64 % for six (computed).

2. Working Principle

Spectrum splitting by gap: The top cell takes the blue and visible light and is transparent to photons below its gap, the next cell takes the next band, and so on. The thermalisation loss of each photon is limited to the distance to the gap of the cell that absorbs it, so every added junction reduces the loss that a single junction suffers at the high-energy end (my reading; background knowledge).
Voltage and current: With the gap set 1.88/1.42/1.0 eV the three cells deliver 17.46, 14.59 and 16.17 mA/cm $^{2}$ and the stack 14.46 mA/cm $^{2}$ at 3.186 V and 46.07 % in my model (computed). The same top two cells with a 1.2 eV third cell give only 26.78 %, since that cell carries 7.93 mA/cm $^{2}$ (computed); an unsuitable bottom gap therefore costs far more than a small voltage loss.
Why more junctions raise the voltage: The five-junction cell has 4.767 V and the six-junction cell 5.549 V, about 0.95 and 0.93 V per junction (computed); the voltage per junction stays close to that of a GaAs cell (1.127 V) while the photocurrent falls, which lowers the resistive loss at a given power (background knowledge).
Losses against the model: The five-junction cell reaches 0.760 of my ideal-split value and the six-junction cell 0.703, with voltage ratios of 0.862 and 0.862 and current ratios of 0.974 and 0.901 (computed; the model open-circuit values are 5.532 and 6.434 V and 9.82 and 9.39 mA/cm $^{2}$ ). The voltage deficit per junction is 0.455 and 0.452 V against 0.302 and 0.304 V in the model (computed), and the fill factors are 0.970 and 0.954 of the ideal values (computed).
Tunnel junctions and interfaces: Heavily doped tunnel diodes connect the cells with a small voltage drop at one sun (background knowledge), and their transparency requires a gap above that of the next cell (Wikipedia). Their resistance becomes important under concentration (background knowledge), which the next page treats.
Silicon as a subcell: Silicon absorbs up to about 1200 nm against about 1780 nm for the Fraunhofer four-junction cell (Fraunhofer ISE), and bonding or mechanical stacking avoids the +4.09 % misfit (computed). The monolithic stacks on silicon reach 0.921 and 0.894 of the ideal fill factor against 0.965 for the bonded one (computed).

3. Applications

Space: III-V multi-junction cells are the standard high-efficiency cells for satellites (background knowledge); the 39.5 % cell reaches 34.2 % under the space spectrum at the beginning of life (NREL).
Concentrators: Under concentration the same stacks reach 47.6 % (665 suns, four junctions) and 47.1 % (143 suns, six junctions) (Green et al.; Fraunhofer ISE; Semiconductor Today); the next page treats these cells.
Flat-plate modules: Sharp’s modules reach 32.65 % (three junctions) and 33.7 % (four-terminal on silicon) (Green et al.).
Tandems on silicon and CIGS: III-V top cells on silicon (36.1 % bonded, 35.9 % mechanical stack) and CIGS (31.2 % module) are routes to higher efficiency with a low-cost bottom cell (Green et al.).

Series Connection and Detailed Balance

Name	Formula	Related Formulae	Variables/Constants
Series connection of subcells	$V = \sum_{i = 1}^{N} V_{i}$	$J = min_{i} J_{i}$ $V_{oc} \approx \sum_{i = 1}^{N} V_{oc, i}$	$V : s t a c k v o l t a g e$ $V_{i} : v o l t a g e o f s u b ce l l i$ $N : n u mb er o f j u n c t i o n s$ $J : s t a c k c u r r e n t d e n s i t y, w hi c h c ann o t e x cee d t h e s ma l l es t s u b ce l l c u r r e n t, W ik i p e d ia s t a t es t ha t t h e s h or t - c i r c u i t c u r r e n t o f t h e ser i es s t a c k i s t h e s ma l l es t o f t h e s u b ce l l c u r r e n t s, t h e v o l t a g e s u m i s ba c k g r o u n d k n o w l e d g e$ $(co m p u t e d f r o m t h e e n t r i es o f G r ee n e t a l . a s V_{oc} / N : 4.767 V /5 = 0.953 V f or t h e f i v e - j u n c t i o n ce l l, 5.549/6 = 0.925 V f or t h e s i x - j u n c t i o n ce l l, 3.065/3 = 1.022 V f or t h e S ha r p t r i pl e, 2.997/3 = 0.999 V f or t h e 39.5 % t r i pl e an d 2.568/2 = 1.284 V f or t h e L G d o u b l e; f or t h e g a p se t 1.88/1.42/1.0 e V t h e i d e a l - s pl i t s u b ce l l p h o t oc u r r e n t s a r e 17.46, 14.59 an d 16.17 m A / c m^{2} an d t h e s t a c k d e l i v er s 14.46 m A / c m^{2} a t ma x im u m p o w er)$
Ideal-split photocurrent of a subcell	$J_{i} = q \int_{E_{g, i}}^{E_{g, i - 1}} Φ (E) d E$	$E_{g, 0} \to \infty, \sum_{i = 1}^{N} J_{i} = J_{sc} (E_{g, N})$	$J_{i} : p h o t oc u r r e n t o f s u b ce l l i$ $Φ (E) : p h o t o n f l ux p er u ni t e n er g y o f t h e A M 1.5 G s p ec t r u m$ $E_{g, i} : g a p o f s u b ce l l i in d esce n d in g or d er (E_{g, 1} i s t h e t o p ce l l), t h e s t e p ab sor pt an ce (e a c h ce l l t ak es a l l p h o t o n s b e tw ee n i t s o w n g a p an d t h e g a p ab o v e i t, w i t h n o r e f l ec t i o n) i s m y m o d e l a ss u m pt i o n$ $(co m p u t e d f r o m t h e p v l ib A M 1.5 G g l o ba l s p ec t r u m o f 1000.37 W / m^{2} : t h e p h o t oc u r r e n t ab o v e 1.0 e V i s 48.22 m A / c m^{2}, ab o v e 0.73 e V 58.97 m A / c m^{2} an d ab o v e 0.70 e V 60.70 m A / c m^{2}; t h e t o p ce l l o f 1.88 e V t ak es 17.46 m A / c m^{2}, an d in a G a I n P / G a A s / G e s t a c k o f 1.86/1.42/0.66 e V t h e g er mani u m ce l l w o u l d ha v e 29.03 m A / c m^{2} a g ain s t 17.99 an d 14.06 in t h e tw o u pp er ce l l s)$
Subcell voltage and stack power	$V_{i} (J) = \frac{k _{B} T}{q} ln (\frac{J _{i} - J}{J _{0, i}} + 1)$	$η = max_{J} \frac{J \sum _{i} V _{i} ( J )}{P _{in}}$ $J_{0} = \frac{4 π q}{h ^{3} c ^{2}} \int_{E_{g}}^{\infty} \frac{E ^{2} d E}{e ^{E / k_{B} T} - 1}$	$J_{0, i} : r a d ia t i v e d a r k c u r r e n t o f s u b ce l l i (e mi ss i o n f r o m b o t h f a ces, i . e . 2 π t im es t h e b l a c k b o d y f l ux 2 E^{2} / (h^{3} c^{2}) a s o n t h e o t h er p a g es, w hi c h i s w ha t t h e n u mb er s b e l o w u se; t h e p r e f a c t or w a s p r in t e d a s 2 π an d t h e l ab e l a s f r o n t o n l y b e f or e t h e a u d i t; n o l u min esce n t co u pl in g b e tw ee n t h e ce l l s, m y m o d e l a ss u m pt i o n), k_{B} T / q = 25.693 mV$ $P_{in} : 1000.37 W / m^{2} (g l o ba l) or 900.14 W / m^{2} (d i r ec t), t h e m e t h o d i s t h e o n e o f t h e S o l a r C e l l F u n d am e n t a l s p a g e e x t e n d e d t o a ser i es s t a c k (m y o w n c a l c u l a t i o n)$ $k_{B} : B o l t z mann co n s t an t$ $(co m p u t e d : o n e j u n c t i o n g i v es 33.15 % a t 1.34 e V an d 32.67 % a t 1.42 e V; t h e g a p se t 2.17/1.68/1.40/1.06/0.73 e V g i v es 51.05 %, V_{m p} = 5.228 V an d J_{m p} = 9.77 m A / c m^{2} (V_{oc} = 5.532 V, J_{sc} = 9.82 m A / c m^{2}), an d t h e se t 2.19/1.76/1.45/1.19/0.97/0.7 e V g i v es 55.75 %, V_{m p} = 5.974 V an d J_{m p} = 9.34 m A / c m^{2} (V_{oc} = 6.434 V, J_{sc} = 9.39 m A / c m^{2}); 1.88/1.42/1.0 e V g i v es 46.07 % an d 3.186 V, 1.88/1.42 e V g i v es 36.50 %, 1.86/1.42/0.66 e V g i v es 40.55 % an d 1.86/1.42/1.2 e V o n l y 26.78 % b ec a u se t h e 1.2 e V ce l l c a r r i es 7.93 m A / c m^{2})$
Efficiency limit against the number of junctions	$η_{N} = max_{E_{g, 1}, \dots, E_{g, N}} η$	$Δ η_{N} = η_{N} - η_{N - 1}$	$η_{N} : hi g h es t e f f i c i e n cy o f a ser i es s t a c k o f N j u n c t i o n s w i t h t h e g a p s c h ose n f r ee l y, W ik i p e d ia l i s t s 37, 50 an d 56 % f or o n e, tw o an d t h r ee j u n c t i o n s, 72 % f or 36 an d 86.8 % f or an in f ini t e n u mb er (t h e e x t r a c t g i v es n o s p ec t r u m or co n ce n t r a t i o n), t h e v a l u es b e l o w a r e f r o m m y m o d e l w i t h d i f f er e n t ia l e v o l u t i o n an d a l oc a l p o l i s h$ $(co m p u t e d, o n e s u n g l o ba l (N = 1 t o 6) : 33.16, 44.97, 50.85, 54.45, 56.78 an d 58.64 % w i t h t h e g a p s o f N = 3 a t 1.900/1.367/0.934 e V an d o f N = 6 a t 2.238/1.787/1.470/1.194/0.958/0.691 e V; 500 s u n s o f t h e d i r ec t s p ec t r u m : 39.37, 52.63, 59.61, 63.53, 66.01 an d 67.71 %; t h e g ain s p er a dd e d j u n c t i o n a r e 11.81, 5.88, 3.60, 2.33 an d 1.86 p o in t s a t o n e s u n an d 13.26, 6.98, 3.92, 2.48 an d 1.70 p o in t s a t 500 s u n s)$

Voltage, Fill Factor and Lattice

Name	Formula	Related Formulae	Variables/Constants
Measured against ideal-split values	$r_{η} = \frac{η}{η _{i d e a l}}$	$r_{V} = \frac{V _{oc}}{V _{oc, i d e a l}}, r_{J} = \frac{J _{sc}}{J _{i d e a l}}$	$r : r a t i o o f t h e m e a s u r e d v a l u e t o t h e v a l u e o f m y m o d e l f or t h e s am e g a p se t, t h e g a p se t s a r e t h ose p r in t e d b y G r ee n e t a l . f or t h e f i v e - j u n c t i o n ce l l (2.17/1.68/1.40/1.06/0.73 e V) an d t h e s i x - j u n c t i o n ce l l (2.19/1.76/1.45/1.19/0.97/0.7 e V), an d b o t h a r e co m p a r e d o n t h e g l o ba l s p ec t r u m (a ss u m e d)$ $(co m p u t e d : f i v e j u n c t i o n s r_{η} = 0.760 (38.8/51.05), r_{V} = 0.862 (4.767/5.532) an d r_{J} = 0.974 (9.564/9.82); s i x j u n c t i o n s r_{η} = 0.703 (39.2/55.75), r_{V} = 0.862 (5.549/6.434) an d r_{J} = 0.901 (8.457/9.39); b e f or e t h e a u d i t r_{V} an d r_{J} w er e t ak e n a g ain s t t h e m o d e l^{'} s ma x im u m - p o w er v a l u es 5.228 V an d 9.77 m A / c m^{2} (an d 5.974 V, 9.34); t h e g a p se t s t h e m se l v es r e a c h 0.899 (51.05/56.78) an d 0.951 (55.75/58.64) o f t h e b es t i d e a l e f f i c i e n cy f or f i v e an d s i x j u n c t i o n s)$
Voltage deficit per junction	$Δ V_{N} = \frac{1}{N} (\sum_{i = 1}^{N} \frac{E _{g, i}}{q} - V_{oc})$	$Δ V_{1} = \frac{E _{g}}{q} - V_{oc}$	$Δ V_{N} : a v er a g e d e f i c i t p er j u n c t i o n$ $E_{g, i} : g a p o f s u b ce l l i$ $V_{oc} : v o l t a g e o f t h e s t a c k, t h e g a p se t s a r e t h ose p r in t e d b y G r ee n e t a l . f or t h e f i v e - an d s i x - j u n c t i o n ce l l s$ $(co m p u t e d : f i v e j u n c t i o n s ha v e a s u m o f g a p s o f 7.04 e V an d a d e f i c i t o f 0.455 V p er j u n c t i o n a g ain s t 0.302 V in m y m o d e l (m o d e l V_{oc} 5.532 V); s i x j u n c t i o n s 8.26 e V an d 0.452 V a g ain s t 0.304 V (6.434 V); t h e e x cess o v er t h e m o d e l i s 0.153 V an d 0.148 V p er j u n c t i o n; t h e G a A s r ecor d o f t h e p r e v i o u s p a g e ha s 0.293 V w i t h 1.42 e V a ss u m e d)$
Ideal fill factor of a stack	$F F_{0} = \frac{v - l n ( v + 0.72 )}{v + 1}$	$v = \frac{V _{oc}}{N k _{B} T / q}$	$F F_{0} : f i l l f a c t or w i t h o u t r es i s t i v e l osses$ $v : n or ma l i se d v o l t a g e p er j u n c t i o n$ $N : n u mb er o f j u n c t i o n s, t h e e m p i r i c a l e x p r ess i o n i s ba c k g r o u n d k n o w l e d g e (a s o n t h e H I T an d G a A s p a g es), e q u a l v o l t a g es o f t h e j u n c t i o n s an d an i d e a l i t y f a c t or o f 1 a r e m y a ss u m pt i o n s$ $k_{B} : B o l t z mann co n s t an t$ $(co m p u t e d w i t h k_{B} T / q = 25.693 mV : f i v e j u n c t i o n s F F_{0} = 0.878 an d 85.2 % i s 0.970 o f i t; S ha r p t r i pl e 0.885 an d 0.980; L G d o u b l e 0.903 an d 0.971; b o n d e d G a I n P / G a I n A s P // S i 0.891 an d 0.965; 39.5 % t r i pl e 0.883 an d 0.966; s i x j u n c t i o n s 0.876 an d 0.954; m o n o l i t hi c G a I n P / G a A s / S i 0.871 an d 0.921; G a A s P / S i 0.869 an d 0.894)$
Lattice mismatch	$f = \frac{a _{f i l m} - a _{s u b}}{a _{s u b}}$	$Δ a = a_{f i l m} - a_{s u b}$	$f : mi s f i t o f t h e f i l m o n t h e s u b s t r a t e$ $a : l a tt i ce co n s t an t, W ik i p e d ia l i s t s 0.5431 nm f or s i l i co n, 0.5653 nm f or G a A s, 0.5657 nm f or g er mani u m an d 0.5868 nm f or I n G a A s (n o co m p os i t i o n g i v e n), an d 0.5451 nm f or I n G a P, w hi c h i s d i sc u sse d in a q u es t i o n b e l o w$ $(co m p u t e d : G a A s o n g er mani u m - 0.07 %, G a A s o n s i l i co n + 4.09 %, t h e I n G a A s o f 0.5868 nm o n G a A s + 3.80 % an d o n g er mani u m + 3.73 %; a mi s f i t o f a f e w p er ce n t n ee d s a g r a d e d b u f f er (ba c k g r o u n d k n o w l e d g e))$

Efficiency and Wavelength Checks

Name	Formula	Related Formulae	Variables/Constants
Efficiency from the reported parameters	$η = \frac{V _{oc} J _{sc} F F}{P _{in}}, P_{in} = 100 mW/c m^{2}$	$η_{c h ec k} = \frac{V _{oc} J _{sc} F F}{1000}$ $Δ η = η_{p r in t e d} - η_{c h ec k}$	$η : e f f i c i e n cy, V_{oc} in mV, J_{sc} in m A / c m^{2}, F F a s a f r a c t i o n (η_{c h ec k} i s in p er ce n t : t h e 1000 t u r n s mV in t o V, an d 1 mW / c m^{2} i s 1 p er ce n t o f P_{in}), t h e e n t r i es a r e f r o m t h e e f f i c i e n cy t ab l es (v er s i o n 64) o f G r ee n e t a l . an d a r e o n e - s u n ce l l s w i t h tw o co n t a c t s$ $(co m p u t e d : f i v e j u n c t i o n s 4767, 9.564, 85.2 % g i v e 38.84 % (38.8); S ha r p t r i pl e 3065, 14.27, 86.7 % g i v e 37.92 % (37.9); L G d o u b l e 2568, 14.56, 87.7 % g i v e 32.79 % (32.8); b o n d e d G a I n P / G a I n A s P // S i 3309, 12.70, 86.0 % g i v e 36.14 % (36.1); m o n o l i t hi c G a I n P / G a A s / S i 2647, 12.21, 80.2 % g i v e 25.92 % (25.9); G a A s P / S i 1732, 17.34, 77.7 % g i v e 23.34 % (23.4); G a I n P / G a A s w i t h q u an t u m w e l l s 2500, 15.36, 85.7 % g i v e 32.91 % (32.9); M i cr o l ink t r i pl e 3013, 14.60, 85.8 % g i v e 37.74 % (37.8); 39.5 % t r i pl e 2997, 15.44, 85.3 % g i v e 39.47 % (39.5); s i x j u n c t i o n s 5549, 8.457, 83.5 % g i v e 39.18 % (39.2))$
Efficiency of a four-terminal stack	$η_{4 T} = \frac{\sum _{k} V _{k} J _{k} F F _{k}}{P _{in}}$	$P_{ma x, k} = V_{k} J_{k} F F_{k}$	$η_{4 T} : e f f i c i e n cy o f a m ec hani c a l s t a c k w h ose t o p an d b o tt o m d e v i ces a r e w i r e d in d e p e n d e n tl y$ $V_{k}, J_{k}, F F_{k} : v o l t a g e, c u r r e n t d e n s i t y an d f i l l f a c t or o f d e v i ce k (h er e V_{k} i s t h e p r in t e d v o l t a g e), t h e e n t r i es a r e f r o m G r ee n e t a l .$ $(co m p u t e d : t h e G a I n P / G a A s / S i s t a c k g i v es 2.52 \times 13.6 \times 0.875 = 29.99 mW / c m^{2} f r o m t h e t o p an d 0.681 \times 11.0 \times 0.785 = 5.88 mW / c m^{2} f r o m t h e s i l i co n, 35.87 % (35.9); t h e G a A s / S i s t a c k g i v es 26.78 + 6.00 = 32.78 % (32.8))$
Module efficiency from voltage, current and area	$η_{m o d} = \frac{V _{oc} I _{sc} F F}{G A}$	$η_{m o d, ma x} = \frac{V _{oc} I _{sc}}{G A}$	$η_{m o d} : m o d u l e e f f i c i e n cy$ $V_{oc}, I_{sc} : m o d u l e v o l t a g e an d c u r r e n t$ $F F : f i l l f a c t or$ $G : 1000 W / m^{2}$ $A : d es i g na t e d i l l u mina t i o n a r e a, t h e e n t r i es a r e t h e S ha r p m o d u l es o f G r ee n e t a l . (40 ce l l s, e i g h t in ser i es, i s t h e r e a d in g o f a t oo l s u mma r y)$ $(co m p u t e d : 24.30 V, 1.520 A, 85.3 % o n 965 c m^{2} g i v e 31.50 W an d 32.65 % (32.65); 24.30/8 = 3.04 V p er ce l l a g ain s t 3.065 V o f t h e ce l l e n t r y (m y r e a d in g); t h e f o u r - t er mina l m o d u l e w i t h 20.3 V, 1.25 A an d 2.83 V, 1.93 A o n 775 c m^{2} ha s an u pp er b o u n d o f 39.8 % an d t h e p r in t e d 33.7 % im pl i es an a v er a g e f i l l f a c t or o f 0.847)$
Wavelength of an absorption edge	$λ = \frac{h c}{E} = \frac{1239.84 eV nm}{E}$	$E = \frac{1239.84 eV nm}{λ}$	$λ : w a v e l e n g t h$ $E : p h o t o n e n er g y or g a p, h c = 1239.84 e V nm, t h e F r a u nh o f er I S E p r ess r e l e a se s t a t es t ha t t h e f o u r - j u n c t i o n co n ce n t r a t or ce l l ab sor b s u p t o ab o u t 1780 nm a g ain s t ab o u t 1200 nm f or s i l i co n$ $(co m p u t e d : 1780 nm i s 0.697 e V an d 1200 nm i s 1.033 e V; t h e g a p s 2.17, 0.73 an d 0.7 e V o f t h e f i v e - an d s i x - j u n c t i o n ce l l s cor r es p o n d t o 571, 1698 an d 1771 nm)$

Important Questions

Ques: Which is the one-sun record: 39.5, 39.2 or 38.8 %?

Green et al. list 39.5 ± 0.5 % for the GaInP/GaAs(quantum wells)/GaInAs inverted metamorphic triple (0.242 $c m^{2}$ , NREL 2021), 39.2 ± 3.2 % for the six-junction cell (0.247 $c m^{2}$ , NREL 2018) and 38.8 ± 1.2 % for the five-junction bonded cell (1.021 $c m^{2}$ , NREL 2013). The products of voltage, current and fill factor are 39.47, 39.18 and 38.84 % (computed), so the printed values are consistent. The NREL release calls 39.5 % the record and names 37.9 % (Sharp, 2013) as the previous one, but the table also holds the 39.2 % cell, and the uncertainty of 3.2 points makes the three values indistinguishable. 39.5 % is the more likely highest value and the only one with a small uncertainty. The five-junction cell has four times the area of the other two (1.021 against about 0.24 $c m^{2}$ ), which may matter for current uniformity (my reading). Reference: Green, Dunlop, Yoshita, Kopidakis, Bothe, Siefer, Hinken, Rauer, Hohl-Ebinger and Hao, ‘Solar cell efficiency tables (Version 64)’, Prog. Photovolt. 32(7), 425-441 (2024); NREL news release on the 39.5 % triple-junction cell (2022; Steiner, France, Geisz et al., Joule, May 2022); Semiconductor Today, report on the six-junction inverted metamorphic cell (14 April 2020; Geisz et al., Nature Energy 2020).

Ques: Is the concentrator record 47.6 or 47.1 %, and how do they compare with the one-sun value?

Green et al. list 47.6 ± 2.6 % at 665 suns for the four-junction bonded cell (0.0452 $c m^{2}$ , Fraunhofer ISE, May 2022) and 47.1 ± 2.6 % at 143 suns for the six-junction cell (0.099 $c m^{2}$ , NREL, 2019); Fraunhofer ISE names 46.1 % (Soitec, 2016) as the previous record, and Semiconductor Today says that more than 50 % is achievable. The difference of 0.5 points lies inside the uncertainty of 2.6 points, so the order is not settled, although 47.6 % is the higher printed value. Both numbers are for the direct spectrum under concentration and cannot be compared directly with the global one-sun 39.5 %. My model gives 65.09 % for the six-junction gap set at 143 suns, so the cell reaches 0.72 of it (assumed, on the same power basis), and the next page treats the basis. Reference: Green, Dunlop, Yoshita, Kopidakis, Bothe, Siefer, Hinken, Rauer, Hohl-Ebinger and Hao, ‘Solar cell efficiency tables (Version 64)’, Prog. Photovolt. 32(7), 425-441 (2024); Fraunhofer ISE press release on the 47.6 % concentrator cell (30 May 2022); Semiconductor Today, report on the six-junction inverted metamorphic cell (14 April 2020; Geisz et al., Nature Energy 2020).

Ques: Is the upper tandem of the four-junction cell GaInP/AlGaAs or GaInP/GaInAs?

Green et al. label the 47.6 % cell as GaInP/GaInAs and GaInAsP/GaInAs, but the Fraunhofer ISE release says that the upper tandem is GaInP/AlGaAs and the lower tandem GaInAsP/GaInAs, bonded by Soitec. The two labels differ in the second junction. Neither extract lists the gaps, so I cannot test them. The press release is the description of the group that made the cell and is more likely to describe the real layers, whereas the table label is a short form that may have been simplified (my reading). A second junction of AlGaAs would have a gap above 1.42 eV, which would suit the current balance of a four-junction stack (my reading). My model needs the gap set, so the 60.03 % at 665 suns for 1.88/1.42/1.05/0.70 eV is only an assumed case. Reference: Green, Dunlop, Yoshita, Kopidakis, Bothe, Siefer, Hinken, Rauer, Hohl-Ebinger and Hao, ‘Solar cell efficiency tables (Version 64)’, Prog. Photovolt. 32(7), 425-441 (2024); Fraunhofer ISE press release on the 47.6 % concentrator cell (30 May 2022).

Ques: What are the gaps of the lattice-matched triple, and is the InGaAs gap 1.4 or 1.2 eV?

Wikipedia gives GaInP as 1.86 eV, GaAs as 1.42 in one place and 1.4 eV in another, germanium as 0.66 to 0.67 and elsewhere 0.65 eV, and InGaAs as 1.4 eV in one place and 1.2 eV in another. The extract does not show which cell each passage means. In my model the GaInP/GaAs/Ge stack of 1.86/1.42/0.66 eV gives 40.55 % and 0.65 eV gives 40.43 % (computed), so the germanium value hardly matters. An InGaAs cell of 1.2 eV below GaAs would give only 26.78 % because it carries 7.93 $mA/c m^{2}$ , while a gap near 1.0 eV gives 46.07 % (computed). I take 1.42, 1.86 and 0.66 eV as the more likely nominal values, 1.2 eV as unsuitable for a bottom cell, and 1.0 eV for the metamorphic bottom cell is background knowledge, not a value from the sources. Reference: Wikipedia, ‘Multi-junction solar cell’; Solar Cell Fundamentals page of this vault.

Ques: Is the single-junction limit 33.16, about 34 or 37 %?

Wikipedia’s multi-junction article gives 33.16 % in one passage, about 34 % in another and 37 % as the first entry of its list by number of junctions. My detailed-balance calculation for the AM1.5G global spectrum at 298.15 K gives 33.15 % at 1.34 eV and a limit of 33.16 % at 1.336 eV (computed), so the first value agrees with an unconcentrated cell. The direct spectrum without concentration gives 31.98 % at 1.42 eV (computed), and 500 suns give 39.37 % at 1.122 eV (computed). The 37 % of the list lies between my one-sun and 500-sun values, and the extract does not state the spectrum or concentration (my reading). 33.16 % is the more likely one-sun limit, 37 % belongs to a different assumption, and about 34 % is probably a rounding on an uncertain basis (my reading). Reference: Wikipedia, ‘Multi-junction solar cell’; Solar Cell Fundamentals page of this vault.

Ques: Is the two-gap optimum 0.77/1.70 eV, and are the limits 50 and 56 % for two and three junctions?

Wikipedia gives an optimum pair of 0.77 and 1.70 eV for two junctions and limits of 50 % for two and 56 % for three. My model for the global one-sun spectrum finds 44.97 % at 1.634 and 0.961 eV, and for 500 suns of the direct spectrum 52.63 % at 1.569 and 0.934 eV, with 50.85 and 59.61 % for three junctions (computed). The pair 0.77/1.70 eV gives only 39.6 % at one sun in my model (computed). At 46 200 suns the optimum falls to 1.439 and 0.699 eV, so the lower gap does fall with concentration, but not to 0.77 eV (computed). The Wikipedia values sit between my one-sun and 500-sun figures, which suggests a different spectrum or concentration (my reading). My values apply to the stated assumptions, and the extract does not allow a firmer choice. Reference: Wikipedia, ‘Multi-junction solar cell’; Solar Cell Fundamentals page of this vault.

Ques: Do commercial multi-junction cells reach 30 or 40 %, or the record of 39.5 %?

Wikipedia puts commercial GaInP/GaAs/Ge cells at about 30 % under one sun and about 40 % concentrated. Green et al. list 39.5 % for a laboratory triple, 37.9 % for Sharp’s triple, 32.8 % for the LG double and 32.65 % for the Sharp module. My detailed-balance value for the 1.86/1.42/0.66 eV stack is 40.55 % at one sun and 47.95 % at 500 suns (computed), so 30 % is 0.74 of the one-sun limit and 40 % is 0.83 of the concentrated one (computed). The germanium cell is the cause of the poor limit, since it has a current surplus (computed). The lab cells with metamorphic cells are higher because their bottom gap is closer to the optimum. The commercial values are probably older and refer to the standard Ge-based product; the extract gives no date (my reading). Reference: Wikipedia, ‘Multi-junction solar cell’; Green, Dunlop, Yoshita, Kopidakis, Bothe, Siefer, Hinken, Rauer, Hohl-Ebinger and Hao, ‘Solar cell efficiency tables (Version 64)’, Prog. Photovolt. 32(7), 425-441 (2024).

Ques: Which III-V cell on silicon is best: 36.1, 35.9 or 25.9 %?

Green et al. list a bonded GaInP/GaInAsP//Si triple of 36.1 ± 1.3 %, a four-terminal GaInP/GaAs/Si stack of 35.9 ± 0.5 %, a four-terminal GaAs/Si stack of 32.8 ± 0.5 %, a monolithic GaInP/GaAs/Si triple of 25.9 ± 0.9 % and a GaAsP/Si double of 23.4 ± 0.3 %. The first two are within each other’s uncertainty, so their order is open. The monolithic cells have a voltage of 0.88 V per junction against 1.10 V for the bonded cell and fill factors of 0.921 and 0.894 of the ideal against 0.965 (computed). The misfit of GaAs on silicon is +4.09 % (computed), and dislocations are the likely cause of the lower voltage and fill factor (my reading). Bonding and mechanical stacking avoid the misfit, and 36.1 % is the more likely best value. Reference: Green, Dunlop, Yoshita, Kopidakis, Bothe, Siefer, Hinken, Rauer, Hohl-Ebinger and Hao, ‘Solar cell efficiency tables (Version 64)’, Prog. Photovolt. 32(7), 425-441 (2024); Wikipedia, ‘Multi-junction solar cell’.

Ques: How close are the five- and six-junction cells to their limits?

With the printed gap sets my model gives 51.05 % for five junctions and 55.75 % for six (computed), so the cells of 38.8 and 39.2 % reach 0.760 and 0.703 of them. The voltage ratios are 0.862 and 0.862 and the current ratios 0.974 and 0.901 against the model’s open-circuit values (computed). The voltage deficit per junction is 0.455 and 0.452 V, against 0.302 and 0.304 V in the model (computed), so the excess is 0.153 and 0.148 V. The six-junction cell loses more current, probably through the thin subcells and reflection (my reading), and its uncertainty is 3.2 points. The model has no reflection, no series resistance and perfect step absorption, so a ratio of 1 cannot be reached. The gap sets themselves reach 0.899 and 0.951 of the best N-junction values (computed). Reference: Green, Dunlop, Yoshita, Kopidakis, Bothe, Siefer, Hinken, Rauer, Hohl-Ebinger and Hao, ‘Solar cell efficiency tables (Version 64)’, Prog. Photovolt. 32(7), 425-441 (2024); Solar Cell Fundamentals page of this vault.

Ques: Do more junctions help: 3, 5 or 6?

Green et al. list 39.5 % for three junctions, 38.8 % for five and 39.2 % for six at one sun, so the extra junctions have not raised the record. In my model the best efficiency rises from 50.85 % for three junctions to 56.78 % for five and 58.64 % for six, so the fourth, fifth and sixth junctions add 3.60, 2.33 and 1.86 points (computed). The measured cells reach 0.760 and 0.703 of the ideal values of their gap sets (computed), so each added junction brings a new layer with its own losses. Against the best ideal values of 50.85, 56.78 and 58.64 % the three records are 0.777, 0.683 and 0.668 (computed). More junctions would help only if each added cell kept the voltage per junction, which the table does not yet show (my reading). Reference: Green, Dunlop, Yoshita, Kopidakis, Bothe, Siefer, Hinken, Rauer, Hohl-Ebinger and Hao, ‘Solar cell efficiency tables (Version 64)’, Prog. Photovolt. 32(7), 425-441 (2024); Solar Cell Fundamentals page of this vault.

Ques: Is the lattice constant of InGaP 0.5451 nm?

My extract of Wikipedia lists 0.5451 nm for InGaP. That is the value I know for the binary gallium phosphide (background knowledge), and a GaInP layer that is lattice-matched to GaAs must have almost the lattice constant of GaAs, 0.5653 nm. Taken literally, 0.5451 nm would give a misfit to GaAs of -3.57 % (computed), which would make the top cell of the standard triple a metamorphic layer, and that is not how the lattice-matched stack is described. The silicon value of 0.5431 nm gives a misfit of 0.37 % to 0.5451 nm (computed), a coincidence that is part of the reason GaP is used on silicon (background knowledge). The more likely reading is that the table entry is the GaP value or an error, and the lattice-matched GaInP has 0.5653 nm; the extract does not give the composition. Reference: Wikipedia, ‘Multi-junction solar cell’.

Quartz 5

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III-V Multi-Junction Solar Cell