1. Theory

Definition: The aluminium back-surface-field (Al-BSF) cell is the conventional industrial crystalline-silicon cell: a p-type boron-doped wafer (Czochralski or cast multicrystalline), a phosphorus emitter made by POCl $_{3}$ diffusion on a textured front, a silicon nitride coating, a screen-printed silver H-grid on the front and a full-area screen-printed aluminium layer on the rear, all fired in one belt-furnace step.
History: pv-manufacturing.org and Sinovoltaics call it the workhorse of the industry for decades; the industry changed to PERC from about 2013, and PERC needs three additional steps (a rear dielectric stack of aluminium oxide capped with silicon nitride, laser openings, and the removal of the light-induced degradation by a final step).
Wafer and emitter: Werner et al. use 156 mm Cz wafers of about 200 µm and 2 $Ω$ cm with an emitter of about 92 $Ω/ sq$ ; Xu et al. use multicrystalline wafers of about 180 µm and 0.5 to 3 $Ω$ cm with about 90 $Ω/ sq$ , a junction depth of 0.3 µm and a surface concentration of $4.5$ to $6.1 \times 1 0^{20}$ $c m^{- 3}$ .
Back-surface field: Above the eutectic temperature of 577 °C the aluminium reacts with the silicon; on cooling the melt regrows as a silicon layer doped with aluminium up to the solid solubility at the start of the regrowth, about 5 to 7 µm thick; the high-low junction repels the electrons from the rear and lowers the effective recombination velocity.
Rear properties: The effective velocity is 300 to 700 $cm/s$ (Herguth et al.) or about 1000 $cm/s$ (Liu et al.), the internal reflectance of the rear is about 65 % against 93 % for PERC, and the aluminium layer absorbs light parasitically.
Performance: Reported cells have $V_{oc}$ of 627 to 644 mV, $J_{sc}$ of 35.6 to 38.5 $mA/c m^{2}$ , a fill factor of 78 to 80 % and an efficiency of 17.5 to 19.7 %; a source quotes 17.0 % for a high-throughput process, over 19 % for a laboratory process and 20.1 % with a plated front; PERC cells are about 20 mV higher in $V_{oc}$ .
Mechanics: The thermal expansion of aluminium is about seven times that of silicon (23.1 against 3.36 per $1 0^{6}$ K), which bows the cell by 1.4 to 2.6 mm over 150 mm for 200 µm wafers, and the bow becomes a major problem for thinner wafers.
Light-induced degradation: Boron-doped Cz cells degrade by the boron-oxygen defect, which can regenerate at elevated temperature and illumination; the firing temperature changes the regeneration time constants.

2. Working Principle

Collection: The light is absorbed in the base, mostly within the 180 to 200 µm, the electrons diffuse to the emitter, which is thin so that the blue light is also collected, and the front grid carries the current.
Role of the field: Without the BSF the electrons that reach the rear would recombine at the metal; the barrier of the high-low junction (116 mV for the assumed doping ratio) keeps the minority-carrier density near the rear high and lowers $S_{e f f}$ .
Co-firing: One firing step fires the silver through the silicon nitride at the front and alloys the aluminium at the rear; a high peak temperature gives a stronger BSF and a higher $V_{oc}$ , which is why the cells are fired hot, but it also raises the bow.
Loss budget: The voltage is limited by the sum of the emitter, the base with the rear and the metal terms of the saturation current; the current by the shading, the reflection and the low rear reflectance; the fill factor by the series resistance of the emitter, the grid and the contacts.

3. Applications

Mass production: A low-cost and mature process for terrestrial modules, with a cell-to-module factor of 0.83 in the model of Liu et al.; it has been largely replaced by PERC in new lines.
Reference: The baseline for the comparison with PERC, TOPCon, heterojunction and back-contact cells on the following pages.
Research: The optimization of the pastes and the firing, including the claim of Schutt et al. that some local regions of an Al-BSF cell have a better rear velocity than PERC cells.

Back Surface Field and Rear Contact

Name	Formula	Related Formulae	Variables/Constants
Recombination velocity of the aluminium back-surface field	$S_{B S F} = \frac{N _{A}}{N _{A}^{+}} \frac{D ^{+}}{L ^{+}} e^{Δ E_{g}^{+} / k_{B} T} \frac{\frac{S _{0} L ^{+}}{D ^{+}} c o s h \frac{W ^{+}}{L ^{+}} + s i n h \frac{W ^{+}}{L ^{+}}}{\frac{S _{0} L ^{+}}{D ^{+}} s i n h \frac{W ^{+}}{L ^{+}} + c o s h \frac{W ^{+}}{L ^{+}}}$	$S_{B S F} \approx \frac{N _{A}}{N _{A}^{+}} \frac{D ^{+}}{W ^{+}} e^{Δ E_{g}^{+} / k_{B} T} (S_{0} \to \infty, W^{+} ≪ L^{+})$ $S_{B S F} \approx \frac{N _{A}}{N _{A}^{+}} \frac{D ^{+}}{L ^{+}} e^{Δ E_{g}^{+} / k_{B} T} coth \frac{W ^{+}}{L ^{+}}$	$S_{B S F} : e f f ec t i v e r eco mbina t i o n v e l oc i t y a t t h e e d g e o f t h e p + l a y er (see n b y t h e ba se)$ $N_{A}, N_{A}^{+} : a cce pt or co n ce n t r a t i o n s o f t h e ba se an d o f t h e B S F$ $D^{+}, L^{+} : min or i t y - c a r r i er (e l ec t r o n) d i f f u s i v i t y an d d i f f u s i o n l e n g t h in t h e p + l a y er$ $W^{+} : t hi c k n ess o f t h e p + l a y er$ $S_{0} : r eco mbina t i o n v e l oc i t y a t t h e r e a r co n t a c t, Δ E_{g}^{+} : ban d - g a p na r r o w in g o f t h e p + l a y er, t h e f or m i s t h e s t an d a r d hi g h - l o w - j u n c t i o n r es u l t (F er d ian s j ah e t a l . p r in t a s imi l a r e x p r ess i o n w i t h a g a r b l e d l a y o u t); i t i s ba c k g r o u n d k n o w l e d g e an d n o t co p i e d$ $k_{B} : B o l t z mann co n s t an t$ $(a ss u m e d N_{A} = 1.1 \times 1 0^{16} c m^{- 3}, N_{A}^{+} = 1 0^{18} c m^{- 3}, W^{+} = 6 μ m, D^{+} = 7 c m^{2} / s, τ^{+} = 1 μ s (L^{+} = 26 μ m) an d S_{0} \to \infty g i v e 131 c m / s, an d 345 c m / s f or Δ E_{g}^{+} = 25 m e V, co m p u t e d; H er g u t h e t a l . : 300 t o 700 c m / s, L i u e t a l . : ab o u t 1000 c m / s)$
Base doping, base resistance and rear barrier	$N_{A} = \frac{1}{q μ _{p} ρ}, R_{ba se} = ρ W$	$ψ_{B S F} = \frac{k _{B} T}{q} ln \frac{N _{A}^{+}}{N _{A}}$ $R_{s, ba se} = ρ W (Ω c m^{2})$	$N_{A} : ba se d o p in g$ $μ_{p} : h o l e m o bi l i t y$ $ρ : r es i s t i v i t y o f t h e w a f er$ $W : w a f er t hi c k n ess$ $R_{s, ba se} : v er t i c a l r es i s t an ce o f t h e ba se p er u ni t a r e a$ $ψ_{B S F} : p o t e n t ia l ba r r i er o f t h e hi g h - l o w j u n c t i o n t ha t r e p e l s e l ec t r o n s f r o m t h e r e a r (t h e so u r ce d escr ib es t h e f i e l d in w or d s)$ $k_{B} : B o l t z mann co n s t an t$ $(ρ = 1.3 Ω c m (H er g u t h) g i v es N_{A} = 1.1 \times 1 0^{16} c m^{- 3} an d 2 Ω c m (W er n er) 7.3 \times 1 0^{15} f or an a ss u m e d μ_{p} = 430 c m^{2} / V s; W = 180 μ m g i v es R_{s, ba se} = 0.023 Ω c m^{2}; N_{A}^{+} / N_{A} = 91 g i v es ψ_{B S F} = 116 mV, co m p u t e d)$
Alloying of the aluminium contact	$N_{A}^{+} \approx N_{A l}^{so l} (T_{s t a r t})$	$T_{e u t} = 577^{\circ} C, T_{m e l t, A l} \approx 660^{\circ} C$ $W^{+} \approx 5 to 7 μ m (T_{p e ak} = 790 to 845^{\circ} C)$	$N_{A}^{+} : a cce pt or co n ce n t r a t i o n o f t h e r e g r o w n l a y er$ $N_{A l}^{so l} : so l i d so l u bi l i t y o f a l u mini u m in s i l i co n$ $T_{s t a r t} : t e m p er a t u r e a t w hi c h t h e r ecr y s t a l l i z a t i o n s t a r t s (t h e co n ce n t r a t i o n f o l l o w s t h e so l u bi l i t y a t t ha t t e m p er a t u r e, H er g u t h e t a l .)$ $T_{e u t} : e u t ec t i c t e m p er a t u r e a t w hi c h t h e r es t o f t h e m e l t so l i d i f i es$ $W^{+} : t hi c k n ess o f t h e B S F$ $T_{p e ak} : p e ak w a f er t e m p er a t u r e (C es a r e t a l .), b or o n in t h e p a s t e c an r ai se t h e d o p in g b ey o n d t h e so l u bi l i t y l imi t$ $(F er d ian s j ah e t a l . : so l u bi l i t y ab o u t 1 0^{18} c m^{- 3} a t a f i r in g t e m p er a t u r e o f ab o u t 750 C; r ec i p es : p e ak 797 C w i t h a r am p o f 3.8 s f r o m 450 t o 785 C, or 826 t o 846 C f or 30 s (A w aah e t a l .), s i l v er a t 715 C, f or min g - g a s ann e a l 20 min a t 400 C)$
Bow of the cell	$δ = \frac{κ L ^{2}}{8}, κ = \frac{6 ( α _{A l} - α _{S i} ) Δ T ( 1 + m ) ^{2}}{h [ 3 ( 1 + m ) ^{2} + ( 1 + mn ) ( m ^{2} + \frac{1}{mn} ) ]}$	$m = \frac{t _{A l}}{t _{S i}}, n = \frac{E _{A l}}{E _{S i}}, h = t_{A l} + t_{S i}, Δ T = T_{f} - T_{m}$ $δ \propto t_{S i}^{- 1.9} (elastic model, computed)$	$δ : b o w (d i s t an ce o f t h e ce n t r e f r o m t h e pl an e o f t h e e d g es)$ $κ : c u r v a t u r e o f t h e bim e t a l s t r i p (T im os h e nk o f or m, ba c k g r o u n d k n o w l e d g e : t h e e q u a t i o n s o f t h e so u r ces w er e g a r b l e d)$ $L : ce l l l e n g t h$ $α : t h er ma l e x p an s i o n coe f f i c i e n t s$ $E : Y o u n g m o d u l i$ $t : t hi c k n esses o f t h e a l u mini u m l a y er an d t h e s i l i co n, T_{f} = 577 C : t e m p er a t u r e a t w hi c h t h e co n t a c t f or m s$ $T_{m} : t e m p er a t u r e o f t h e m e a s u r e m e n t, t h e m o d e l t r e a t s t h e f i r e d l a y er a s o n e e l a s t i c l a y er$ $(m e a s u r e d : 1.4, 2.6 an d 1.8 mm o v er 150 mm f or t h r ee p a s t es o n 200 μ m w a f er s (E = 41, 44, 46 GP a, P o p o v i c h e t a l .), 2.13 mm f or 165 μ m M W T + P E R C ce l l s (P e n g e t a l .); w i t h E_{A l} = 43 GP a, E_{S i} = 162.5 GP a, α = 23.1 an d 3.36 \times 1 0^{- 6} K^{- 1}, Δ T = 552 K an d an a ss u m e d t_{A l} = 30 μ m t h e e l a s t i c m o d e l g i v es 35 mm, i . e . 14 t o 25 t im es m or e, co m p u t e d)$

Saturation Current and Open-Circuit Voltage

Name	Formula	Related Formulae	Variables/Constants
Saturation-current budget of the cell	$j_{01} = j_{0 e} (1 - F M) + j_{0, m e t} F M + j_{0, ba se} + j_{0, r e a r}$	$V_{oc} = \frac{k _{B} T}{q} ln (\frac{j _{sc}}{j _{01}} + 1)$ $Δ V_{oc} \approx - \frac{k _{B} T}{q} \frac{Δ j _{01}}{j _{01}}$	$j_{01} : t o t a l s a t u r a t i o n c u r r e n t d e n s i t y o f t h e f i r s t d i o d e$ $j_{0 e} : e mi tt er t er m (p a ss i v a t e d a r e a)$ $j_{0, m e t} : m e t a l l i z e d a r e a u n d er t h e f r o n t co n t a c t s$ $F M : m e t a l l i z a t i o n f r a c t i o n$ $j_{0, ba se} : t er m o f t h e ba se$ $j_{0, r e a r} : t er m o f t h e r e a r (B S F), t h e s u m i s E q . 1 o f W er n er e t a l ., w hi c h t h e so u r ce d oes n o t r eso l v e f or A l - B S F ce l l s$ $k_{B} : B o l t z mann co n s t an t$ $(W er n er e t a l . s t a t e j_{01} \approx 850 f A / c m^{2} an d m e a s u r e 627 t o 628 mV; j_{sc} = 38.1 m A / c m^{2} an d 850 f A / c m^{2} g i v e 630.1 mV, an d 37.2 m A / c m^{2} g i v es 629.5 mV a t 298.15 K; 100 f A / c m^{2} m or e l o w er s V_{oc} b y 2.9 mV, co m p u t e d)$
Base and rear terms for a finite wafer	$j_{0, ba se + r e a r} = \frac{q n _{i}^{2} D}{N _{A} L} \frac{\frac{S L}{D} c o s h \frac{W}{L} + s i n h \frac{W}{L}}{\frac{S L}{D} s i n h \frac{W}{L} + c o s h \frac{W}{L}}$	$L = D τ, n_{i} (T) = 9.38 \times 1 0^{19} (\frac{T}{300})^{2} e^{- 6884/ T}$ $L_{e f f} (finite bulk, includes S and W)$	$j_{0, ba se + r e a r} : s a t u r a t i o n c u r r e n t d e n s i t y o f t h e ba se in c l u d in g t h e r e a r s u r f a ce t h r o ug h S$ $n_{i} : in t r in s i c co n ce n t r a t i o n$ $D : d i f f u s i v i t y o f e l ec t r o n s in t h e ba se$ $L : d i f f u s i o n l e n g t h$ $W : w a f er t hi c k n ess$ $S : e f f ec t i v e r e a r r eco mbina t i o n v e l oc i t y$ $τ : b u l k l i f e t im e, H er g u t h e t a l . d escr ib e t hi s w i t h an e f f ec t i v e d i f f u s i o n l e n g t h an d d o n o t p r in t a n u mb er; t h e f or m i s t h e s t an d a r d o n e (t h e s am e s ha p e a s t h e co l l ec t i o n p r o babi l i t y)$ $(a ss u m e d D = 27 c m^{2} / s, τ = 100 μ s (L = 520 μ m), N_{A} = 1.1 \times 1 0^{16} c m^{- 3}, W = 180 μ m : S = 25, 100, 500 an d 1000 c m / s g i v e 215, 283, 562 an d 788 f A / c m^{2}; w i t h j_{0 e} = 150 f A / c m^{2} (t h e v a l u e a ss u m e d b y H er g u t h e t a l .) an d j_{sc} = 38 m A / c m^{2}, V_{oc} = 652, 647, 635 an d 628 mV, co m p u t e d)$
Open-circuit voltage gain of a better rear	$Δ V_{oc} = \frac{k _{B} T}{q} ln \frac{j _{01, A l - B S F}}{j _{01, P E R C}}$	$j_{01} = j_{0 e} + j_{0, ba se + r e a r} (S)$ $S_{P E R C} \approx 100 cm/s, S_{A l - B S F} \approx 500 to 1000 cm/s$	$Δ V_{oc} : g ain o f t h e o p e n - c i r c u i t v o l t a g e w h e n t h e A l - B S F i s r e pl a ce d b y a d i e l ec t r i c - p a ss i v a t e d r e a r w i t h l oc a l co n t a c t s, t h e r e a r v e l oc i t i es a r e t h ose o f L i u e t a l . (ab o u t 100 c m / s f or P E R C an d ab o u t 1000 c m / s f or A l - B S F) an d o f H er g u t h e t a l . (300 t o 700 c m / s f or t h e B S F, 25 t o 50 c m / s u n d er P E R C co n t a c t s)$ $k_{B} : B o l t z mann co n s t an t$ $(w i t h t h e n u mb er s o f t h e p r e v i o u s r o w : 500 t o 100 c m / s g i v es 12.8 mV an d 1000 t o 100 c m / s 19.9 mV; H er g u t h e t a l . s t a t e t ha t P E R C ce l l s r e a c h 650 t o 660 mV, ab o u t 20 mV ab o v e A l - B S F, co m p u t e d)$

Series Resistance, Shading and Efficiency

Name	Formula	Related Formulae	Variables/Constants
Series resistance of the cell	$R_{s} = R_{s, ba se} + R_{s, e mi tt er} + R_{s, f in g er} + R_{s, co n t a c t} + R_{s, b u s ba r} + R_{s, r e a r}$	$R_{s, e mi tt er} = \frac{ρ _{□} S ^{2}}{12}$ $R_{s, ba se} = ρ W$	$R_{s} : a r e a - n or ma l i z e d ser i es r es i s t an ce, t h e t er m s : ba se, e mi tt er (b e tw ee n t h e f in g er s), f in g er s, m e t a l - s i l i co n co n t a c t, b u s ba r s an d t h e r e a r A l l a y er$ $ρ_{□} : s h ee t r es i s t an ce o f t h e e mi tt er$ $S : f in g er s p a c in g, t h e e mi tt er, f in g er an d co n t a c t t er m s a r e d er i v e d o n t h e p a g e o n l oss m ec hani s m s, t h e r e a r t er m i s s ma l l an d i s n o t q u an t i f i e d b y an y so u r ce t ha t I r e a d$ $(W er n er e t a l . m e a s u r e R_{s} = 0.54 t o 0.66 Ω c m^{2} an d A w aah e t a l . 0.55 t o 1.30 Ω c m^{2}; w i t h an a ss u m e d s p a c in g o f 2 mm, 92 Ω/ s q g i v es 0.307 an d 640 Ω/ s q g i v es 2.13 Ω c m^{2} f or t h e e mi tt er a l o n e, an d t h e ba se o f 0.023 Ω c m^{2}, co m p u t e d)$
Metallization and shading	$Δ J_{sc} \approx \frac{Δ F M}{1 - F M} J_{sc}$	$F M = F M_{f in g er s} + F M_{b u s ba r s}$ $Δ η \approx Δ J_{sc} \frac{V _{oc} F F}{P _{in}}$	$Δ J_{sc} : c u r r e n t g ain f r o m l ess s ha d in g$ $F M : m e t a l l i z a t i o n f r a c t i o n o f t h e f r o n t s i d e$ $J_{sc} : s h or t - c i r c u i t c u r r e n t d e n s i t y a f t er t h e c han g e, t h e r e l a t i o n i s m y f i r s t - or d er es t ima t e t ha t n e g l ec t s t h e l i g h t r e f l ec t e d f r o m t h e m e t a l an d t h e p a ss i v a t i o n e f f ec t s$ $P_{in} : in c i d e n t p o w er$ $(W er n er e t a l . : F M f a l l s f r o m 6.3% t o 4.4% w i t h t a p er e d b u s ba r s, t h e s i l v er p a s t e f r o m 140 t o 110 m g, an d J_{sc} r i ses f r o m 37.2 t o 38.1 m A / c m^{2} (u p t o 0.9); t h e f or m u l a g i v es 0.76 m A / c m^{2}, an d 0.9 m A / c m^{2} g i v es 0.5 p o in t s o f e f f i c i e n cy a t 627 mV an d F F = 0.79, co m p u t e d)$
Efficiency from the reported parameters	$η = \frac{V _{oc} J _{sc} F F}{P _{in}}, P_{in} = 100 mW/c m^{2}$	$η_{c h ec k} = \frac{V _{oc} J _{sc} F F}{1000}$ $Δ η = η_{p r in t e d} - η_{c h ec k}$	$η : e f f i c i e n cy, V_{oc} in mV, J_{sc} in m A / c m^{2}, F F a s a f r a c t i o n (η_{c h ec k} i s in p er ce n t : t h e 1000 t u r n s mV in t o V, an d 1 mW / c m^{2} i s 1 p er ce n t o f P_{in}), P_{in} i s t h e s t an d a r d t es t co n d i t i o n, t h e v a l u es a r e m e d ian s or b es t ce l l s a s l ab e l l e d b y t h e so u r ces$ $(W er n er e t a l . : 627, 37.2, 78.8% g i v e 18.38% (p r in t e d 18.4); 627, 38.1, 78.9% g i v e 18.85% (18.9); 628, 38.1, 79.7% g i v e 19.07% (19.1); X u e t a l . : 630, 35.55, 78.9% g i v e 17.67% (17.65); A w aah e t a l . b es t ce l l : 644, 38.5, 78% g i v e 19.34% a g ain s t t h e p r in t e d 19.7%, co m p u t e d)$

Important Questions

Ques: How thick is the aluminium back-surface field: 5 to 7 µm or about 50 µm?

The sources differ by a factor of ten. Cesar et al. measured a BSF thickness of 5 to 7 µm for peak temperatures of 790 and 845 °C and call it comparable to the state of the art of full Al-BSF cells. Awaah et al. write that the BSF of about 50 µm forms during the firing of the screen-printed aluminium. The model of Popovich et al. has separate layers, an Al-14 % Si eutectic layer, the BSF layer and the particle layer, so that 50 µm probably includes the eutectic or alloyed region and not only the doped silicon; that is my reading, as neither paper compares them. The 5 to 7 µm value is more likely for the p+ layer that sets the rear recombination, and the doping and the 6 µm thickness enter the formula of the first row. Reference: Cesar, Granneman, Vermont, Khatri, Kerp, Shaikh, Manshanden, Mewe, Romijn and Weeber, ‘Industrial Application of Uncapped Al2O3 and Firing-Through Al-BSF In Open Rear Passivated Solar Cells’, ECN (publications.tno.nl/publication/34631367); Awaah, Awaah, Apalangya and Das, ‘Investigation and Characterization of Al paste as back surface field (BSF) for High Efficiency Si Solar cell’, 2021 IEEE PES/IAS PowerAfrica; Popovich, Janssen, Richardson, van Amstel and Bennett, ‘Microstructure and Mechanical Properties of Aluminium Back Contact Layers’, 24th EU PVSEC, Hamburg (2009).

Ques: Is the rear recombination velocity of an Al-BSF 300 to 700 cm/s or about 1000 cm/s?

Herguth et al. give typical effective rear velocities of 300 to 700 $cm/s$ , depending on the peak firing temperature, the paste and the doping of the substrate, and use 500 $cm/s$ as a reference; Liu et al. use about 1000 $cm/s$ in a thin-silicon model, with PERC at about 100. With assumed parameters (base $1.1 \times 1 0^{16}$ $c m^{- 3}$ , BSF $1 0^{18}$ $c m^{- 3}$ and 6 µm, band-gap narrowing of 25 meV) the high-low-junction formula gives 345 $cm/s$ (computed), inside the first range. A measured $j_{01}$ of 850 $fA/c m^{2}$ (Werner et al.) corresponds in my model to about 800 $cm/s$ , because the metal and the emitter add to it. So 300 to 700 $cm/s$ is more likely for a well-fired BSF, and 1000 $cm/s$ is a conservative figure for a poorly fired one. Reference: Herguth, Horbelt, Wilking, Job and Hahn, ‘Comparison of BO Regeneration dynamics in PERC and Al-BSF solar cells’, Energy Procedia 77, 75-82 (2015); Liu, Sofia, Laine, Woodhouse, Wieghold, Peters and Buonassisi, ‘Revisiting Thin Silicon for Photovoltaics: A Technoeconomic Perspective’, arXiv:1906.06770; Werner, Lohmuller, Maier, Kimmerle, Spribille, Wasmer, Clement and Wolf, ‘Process Optimization for the Front Side of p-Type Silicon Solar Cells’, 29th EU PVSEC, Amsterdam (2014).

Ques: Does the open-circuit voltage advantage of PERC follow from the rear velocities?

Roughly. Herguth et al. state that PERC cells reach 650 to 660 mV, which is about 20 mV higher than Al-BSF cells. In my model (assumed 100 µs lifetime, 180 µm, $j_{0 e}$ of 150 $fA/c m^{2}$ , 38 $mA/c m^{2}$ ) a rear velocity of 100 $cm/s$ for PERC against 500 $cm/s$ for the BSF gives only 12.8 mV, and against 1000 $cm/s$ it gives 19.9 mV (computed). The full 20 mV needs a BSF velocity near 1000 $cm/s$ or additional gains, such as a lower metal term and a longer lifetime in the PERC process, which is my reading. The rear velocity alone explains between two thirds and all of the gap, and Cesar et al. estimate 0.5 % absolute efficiency for a PERC option. Reference: Herguth, Horbelt, Wilking, Job and Hahn, ‘Comparison of BO Regeneration dynamics in PERC and Al-BSF solar cells’, Energy Procedia 77, 75-82 (2015); Liu, Sofia, Laine, Woodhouse, Wieghold, Peters and Buonassisi, ‘Revisiting Thin Silicon for Photovoltaics: A Technoeconomic Perspective’, arXiv:1906.06770; Cesar, Granneman, Vermont, Khatri, Kerp, Shaikh, Manshanden, Mewe, Romijn and Weeber, ‘Industrial Application of Uncapped Al2O3 and Firing-Through Al-BSF In Open Rear Passivated Solar Cells’, ECN (publications.tno.nl/publication/34631367); Solar Cell Fundamentals page of this vault.

Ques: Do the reported open-circuit voltage, current, fill factor and efficiency multiply?

Mostly. Werner et al. print 18.4, 18.9 and 19.1 % for the three layouts, and 627 mV, 37.2 $mA/c m^{2}$ and 78.8 % give 18.38 %, 627 mV, 38.1 $mA/c m^{2}$ and 78.9 % give 18.85 %, and 628 mV, 38.1 $mA/c m^{2}$ and 79.7 % give 19.07 % (computed). Xu et al. have 630 mV, 35.55 $mA/c m^{2}$ and 78.9 %, which give 17.67 % against 17.65 % printed. The best cell of Awaah et al. does not: 644 mV, 38.5 $mA/c m^{2}$ and 78 % give 19.34 %, but 19.7 % is printed. The printed efficiency requires a fill factor of 79.5 % (computed), so the 78 % is probably the average of the series and not that of the best cell; that is my reading, as the paper does not say it. Reference: Werner, Lohmuller, Maier, Kimmerle, Spribille, Wasmer, Clement and Wolf, ‘Process Optimization for the Front Side of p-Type Silicon Solar Cells’, 29th EU PVSEC, Amsterdam (2014); Xu, Wu and Wang, ‘Efficiency Improvement of Industrial Silicon Solar Cells by the POCl3 Diffusion Process’, Materials 16(5), 1824 (2023); Awaah, Awaah, Apalangya and Das, ‘Investigation and Characterization of Al paste as back surface field (BSF) for High Efficiency Si Solar cell’, 2021 IEEE PES/IAS PowerAfrica.

Ques: Is the Voc of 627 mV consistent with the saturation current that Werner et al. quote?

Yes, to 3 mV. The paper gives an overall $j_{01}$ of about 850 $fA/c m^{2}$ for its cells and the median open-circuit voltages are 627 to 628 mV. The ideal-diode equation with $j_{sc} = 38.1$ $mA/c m^{2}$ and 850 $fA/c m^{2}$ gives 630.1 mV at 298.15 K, and 629.5 mV for 37.2 $mA/c m^{2}$ (computed); the cells were measured after 36 hours at 0.2 suns, and a small difference of temperature or of the second diode could explain the rest. The paper concludes that because $j_{01}$ is so high, a lower metallization fraction has a negligible effect on $V_{oc}$ : 100 $fA/c m^{2}$ more costs 2.9 mV and the metal term is a small part of the total (computed). Reference: Werner, Lohmuller, Maier, Kimmerle, Spribille, Wasmer, Clement and Wolf, ‘Process Optimization for the Front Side of p-Type Silicon Solar Cells’, 29th EU PVSEC, Amsterdam (2014).

Ques: Is the current gain of the tapered busbars consistent with the metallization fraction?

It is, within the scatter. Werner et al. report that the metallization fraction falls from about 6.3 % to about 4.4 %, the silver from about 140 mg to 110 mg, and the current density rises by up to 0.9 $mA/c m^{2}$ , from the median 37.2 to 38.1 $mA/c m^{2}$ . The first-order estimate $Δ F M / (1 - F M) \cdot J_{sc}$ gives 0.76 $mA/c m^{2}$ (computed), 0.14 below the reported value; light that is reflected from the metal and the larger spread of single cells can account for it, which is my reading. With 627 mV and a fill factor of 0.79, 0.9 $mA/c m^{2}$ is 0.5 points of efficiency, which agrees with the reported gain from 18.4 to 18.9 %. Reference: Werner, Lohmuller, Maier, Kimmerle, Spribille, Wasmer, Clement and Wolf, ‘Process Optimization for the Front Side of p-Type Silicon Solar Cells’, 29th EU PVSEC, Amsterdam (2014).

Ques: Is a sheet resistance of 640 ohm/sq after the POCl3 diffusion plausible?

Not for these cells. Awaah et al. quote about 640 $Ω/ sq$ after the diffusion, but Werner et al. have about 92 and Xu et al. about 90 $Ω/ sq$ for industrial emitters. The emitter term $ρ_{□} S^{2} /12$ for a 2 mm spacing is 0.307 $Ω c m^{2}$ at 92 $Ω/ sq$ and 2.13 $Ω c m^{2}$ at 640 $Ω/ sq$ (computed, spacing assumed), and the latter exceeds the whole series resistance of 0.55 to 1.3 $Ω c m^{2}$ of the paper. Their fill factor of 78 % and voltage of 640 mV are normal for a standard emitter. So 640 is probably a misprint of 64 or of 90, which is my reading; the paper does not give the value again. Reference: Awaah, Awaah, Apalangya and Das, ‘Investigation and Characterization of Al paste as back surface field (BSF) for High Efficiency Si Solar cell’, 2021 IEEE PES/IAS PowerAfrica; Werner, Lohmuller, Maier, Kimmerle, Spribille, Wasmer, Clement and Wolf, ‘Process Optimization for the Front Side of p-Type Silicon Solar Cells’, 29th EU PVSEC, Amsterdam (2014); Xu, Wu and Wang, ‘Efficiency Improvement of Industrial Silicon Solar Cells by the POCl3 Diffusion Process’, Materials 16(5), 1824 (2023).

Ques: Why do the contact resistivities of the silver on the emitter differ so much?

Werner et al. give a specific contact resistivity of at most 3.8 $mΩ c m^{2}$ for their p-type Cz cells with the emitter of 92 $Ω/ sq$ , and Xu et al. give 18.9 and 19.1 $mΩ c m^{2}$ for their multicrystalline cells with 90 $Ω/ sq$ and a surface concentration of 4.5 to $6.1 \times 1 0^{20}$ $c m^{- 3}$ . The ratio is five or more. The sources do not compare them, and differ in the paste, the firing, the surface and the way of measuring, so I cannot say which is typical. The value of Werner et al. is an upper bound, and the higher one is common for an unoptimized process, which is my reading. A resistivity of 19 $mΩ c m^{2}$ gives a contact term near 0.1 to 0.2 $Ω c m^{2}$ in a grid like the one of the loss-mechanisms page. Reference: Werner, Lohmuller, Maier, Kimmerle, Spribille, Wasmer, Clement and Wolf, ‘Process Optimization for the Front Side of p-Type Silicon Solar Cells’, 29th EU PVSEC, Amsterdam (2014); Xu, Wu and Wang, ‘Efficiency Improvement of Industrial Silicon Solar Cells by the POCl3 Diffusion Process’, Materials 16(5), 1824 (2023); Solar Cell Loss Mechanisms and Characterization page of this vault.

Ques: Can a bimetal model explain the bow of 1.4 to 2.6 mm?

Not with the elastic constants of the sources. Popovich et al. measured 1.4, 2.6 and 1.8 mm over 150 mm for three pastes on 200 µm wafers and derived a modulus of about 43 GPa for the fired layer, while Peng et al. use 25.5 GPa for the paste and 162.5 GPa for silicon. The Timoshenko bimetal formula with 43 GPa, expansion coefficients of 23.1 and 3.36 per $1 0^{6}$ K, a temperature step of 552 K and an assumed aluminium thickness of 30 µm gives 35 mm, 14 to 25 times more (computed). Their own equations were garbled, so I cannot see what differs; stress relaxation of the soft aluminium is my guess. The same model gives a bow proportional to $t_{S i}^{- 1.9}$ , which agrees with the thin-wafer warning. Reference: Popovich, Janssen, Richardson, van Amstel and Bennett, ‘Microstructure and Mechanical Properties of Aluminium Back Contact Layers’, 24th EU PVSEC, Hamburg (2009); Peng, Xi, Shao, Liu, Li, Lu, Xu, Huang and Zhang, ‘Study on the bowing of monocrystalline silicon MWT+PERC solar cells with different laser-ablation condition’, Optoelectron. Adv. Mater. Rapid Commun. 15(7-8), 373-380 (2021).

Ques: What is the right peak firing temperature for the Al-BSF?

The sources use different temperature scales, so they cannot be compared directly. Cesar et al. give peak wafer temperatures of 790 to 845 °C, measured by thermocouples, and expect a thicker BSF at the higher end. Awaah et al. print pyrometer values of 826, 836 and 846 °C for 30 s for the baseline and 797 °C with a ramp of 3.8 s for another paste, and the silver is fired at 715 °C. Herguth et al. use relative set points from 800 °C up by 60 K and conclude that it is beneficial to fire Al-BSF cells rather hot, since the voltage rises with the set temperature. All agree on a hot, short firing of 790 to 850 °C; the exact value depends on the paste, the belt speed and the furnace, and none of the sources gives an optimum. Reference: Cesar, Granneman, Vermont, Khatri, Kerp, Shaikh, Manshanden, Mewe, Romijn and Weeber, ‘Industrial Application of Uncapped Al2O3 and Firing-Through Al-BSF In Open Rear Passivated Solar Cells’, ECN (publications.tno.nl/publication/34631367); Awaah, Awaah, Apalangya and Das, ‘Investigation and Characterization of Al paste as back surface field (BSF) for High Efficiency Si Solar cell’, 2021 IEEE PES/IAS PowerAfrica; Herguth, Horbelt, Wilking, Job and Hahn, ‘Comparison of BO Regeneration dynamics in PERC and Al-BSF solar cells’, Energy Procedia 77, 75-82 (2015).

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Aluminium Back-Surface-Field (Al-BSF) Silicon Cell