Photovoltage in c-Si/mc-Si Solar Cells: Grain Size, Base Thickness, and Illumination Mode: A Mathcad-Based Analytical Model

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Eu opean Jou nal o Ad ances in Enginee ing and Technology, 2025, 12(10):37-46
Resea ch A icle
ISSN: 2394 - 658X
37
Pho o ol age in c-Si/mc-Si Sola Cells: G ain Size, Base Thickness,
and Illumina ion Mode: A Ma hcad-Based Analy ical Model
Youssou GNING1, Aly TOURE1, Dimi y DIASSY1, Mamadou Lamine SAMB1*, Moussa
TOURE1, Ahmed MOHAMED-YAHYA2
1Depa men o Physics and Chemis y, Uni e si y Iba De Thiam o Thies, Thies, Senegal
2Applied Resea ch Uni o Renewable Ene gies, Uni e si y o Nouakcho , Nouakcho , Mau i ania
*mlsamb@uni - hies.sn
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ABSTRACT
We p esen a h ee-dimensional di usion– ecombina ion model ha yields closed- o m exp essions o he
pho o ol age Vph o c ys alline-silicon (c-Si/mc-Si) sola cells unde on , ea , and dual illumina ion. The
amewo k sol es he mino i y-ca ie con inui y equa ion in (x,y,z) wi h ini e la e al dimensions (gx, gy),
explici ly e aining geome ic eigenmodes and bounda y condi ions, and pa ame e izes mic os uc u e h ough a
g ain size g and de ice geome y h ough he base hickness H. We in oduce ca ie -collec ion eloci y (CCV) a
he junc ion as he ope a ing-poin knob connec ing anspo o he diode ela ion o ol age. The analysis
cla i ies h ee egimes:
i. a low-CCV ex ac ion-limi ed egime (open-ci cui -like o cu en ),
ii. a ansi ion egime whe e he slope o Vph e sus CCV diminishes, and
iii. a high-CCV e icien -collec ion egime (sho -ci cui -like cu en pla eau), in which Vph is lowes o a
gi en gene a ion le el.
Pa ame ic esul s show ha Vph dec eases wi h inc easing CCV, consis en wi h he educ ion o s eady-s a e
excess densi y and quasi-Fe mi-le el spli ing. La ge g ains sys ema ically inc ease Vph by lowe ing g ain-
bounda y ecombina ion, while hinne bases sho en anspo pa hs and cu b bulk SRH losses. F on
illumina ion is weakly sensi i e o H (gene a ion nea he junc ion); ea illumina ion shows a s ong dependence
on H (longe di usion pa hs); and dual illumina ion aises he ol age baseline and mi iga es hickness penal ies
by sho ening a e age collec ion dis ances. A ixed CCV nea open-ci cui , he ol age gene ally inc eases
mono onically wi h g ain size ac oss illumina ion modes; in dual illumina ion we obse e “blockwise” g ouping
by H, consis en wi h enhanced gene a ion and educed pa h leng hs.
Design guidance ollows di ec ly: p io i ize la ge g ains (g≳0.01 cm), mode a e H (≈ 150 µm) unless supe io
passi a ion is a ailable, and dual illumina ion when applicable. Low-J0, low-SRV passi a ing con ac s li he
en i e Vph-CCV cha ac e is ic. The model’s closed- o m na u e enables apid explo a ion o geome y,
mic os uc u e, ope a ion couplings and p o ides physically anspa en a ge s o bi acial c-Si op imiza ion.
Keywo ds: bi acial pho o ol aics; pho o ol age; ca ie -collec ion eloci y (CCV); g ain size; base hickness;
analy ical modeling.
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INTRODUCTION
Bi acial c ys alline-silicon (c-Si/mc-Si) pho o ol aics ha e mo ed om niche o mains eam, d i en by hei
supe io ene gy yield and compe i i e LCOE when in eg a ed wi h mode n acke s and op imized si e albedo [1].
Accu a e p edic ion and in e p e a ion o de ice-le el obse ables, pa icula ly he pho o ol age unde on , ea ,
and dual illumina ion, equi e coupling op ical i adiance models wi h cha ge- anspo and ecombina ion physics.
Recen echno-economic and ield assessmen s quan i y bi acial gains and unde sco e he need o elec ical models
ha esol e on / ea gene a ion asymme y and ex ac ion bo lenecks [1,2]. On he op ical side, p ac ical and
compu a ionally e icien schemes o ea -side i adiance (now s anda d in bankable a ing me hods) enable obus
links be ween geome y, albedo, and backside lux [3,4].
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Elec ically, open-ci cui pho o ol age 𝑉𝑜𝑐 and i s e olu ion wi h ope a ing poin hinge on he balance be ween
pho ogene a ion and ecombina ion cu en s (𝐽0 e ms). P og ess in passi a ing con ac s has lowe ed 𝐽0 and
s abilized pe o mance unde ou doo s ess, di ec ly boos ing 𝑉𝑜𝑐 in bo h mono and bi acial cells [5–7]. A he
ma e ial scale, mode n iews o mc-Si mic os uc u e, g ain bounda ies, disloca ions, and impu i y complexes,
ela e s uc u al desc ip o s o ecombina ion ac i i y, in o ming g ain-enginee ing s a egies ha aise ca ie
li e ime and imp o e ol age a a gi en base hickness [8]. Finally, e ined in insic ecombina ion
pa ame e iza ions (Auge and adia i e) a e essen ial o a oid sys ema ic bias when ex apola ing 𝑉𝑜𝑐 ends ac oss
injec ion egimes and doping anges [9].
In his wo k we adop a 3D di usion– ecombina ion amewo k and de i e closed- o m pho o ol age exp essions
o on , ea , and dual illumina ion. We emphasize he coupled oles o la e al dimensions, g ain size, and base
hickness, and we benchma k quali a i e ends agains he con empo a y unde s anding summa ized abo e.
PHYSICAL MODEL
Desc ip ion o he Simula ed Sola Cell
We conside a 3D columna domain cen e ed on a single g ain o polyc ys alline silicon. The de ice is bi acial and
comp ises ou egions: a hin, highly doped n+ emi e (≈ 0.5 − 1 µ𝑚; 1017–1019 𝑐𝑚−3) con ac ed by a me al
g id; he space-cha ge egion (𝑝−𝑛 junc ion) ha sepa a es ca ie s; a ligh ly doped p- ype base (1015–1017 𝑐𝑚−3)
go e ning gene a ion, di usion and ecombina ion wi h hickness 𝐻∈[100,400] 𝜇𝑚; and a ea p+ BSF ha
edi ec s mino i y ca ie s owa d he junc ion o limi deep ecombina ion. This layou e lec s mode n c-Si/bi acial
p ac ice and passi a ing-con ac s acks o high collec ion e iciency [10,11].
Figu e 1: Simula ed sola cell model
Modeling assump ions:
he emi e ’s di ec con ibu ion o he pho ocu en is small compa ed wi h he base;
only he junc ion ield is e ained in he quasi-neu al egions. The junc ion is a 𝑧=0 (o igin o he (𝑥,𝑦) axes); 𝐻
is swep om 100 o 400 µ𝑚. Ca ie anspo and ecombina ion ollow he d i –di usion + SRH amewo k
[12,13].
The mino i y-ca ie con inui y equa ion in he base becomes
∂2
∂x2δ(x,y,z)+∂2
∂y2δ(x,y,z)+∂2
∂z2δ(x,y,z) − δ(x,y,z)
L2 =−G(z)
𝐷 (3)
wi h δ(x,y,z) he excess mino i y-ca ie densi y, G(z) he gene a ion a e D he di usion coe icien and 𝐿 he
di usion leng h.
We ep esen he dep h-dependen gene a ion by a compac mul i-exponen ial i o AM1.5:
G(z) = n∑aie−biz
3
i=1 (4)
whe e n is he concen a ion ac o (in “suns”) and {ai, 𝑏i} a e i coe icien s cap u ing he wa eleng h-in eg a ed
abso p ion [14].
A sepa a ed- a iables solu ion is sough unde g ain-bounda y (GB) ecombina ion a he la e al edges. The gene al
o m is: δ(x,y,z)=∑ ∑ Zj,k(z)cosCjxcosCky
kj (5)
wi h la e al wa enumbe s 𝐶𝑗, 𝐶𝑘 ixed by GB bounda y condi ions
W
H
O
z
y
x
Hea ily doped
p- egion ( )
Emi e ( )
Base (p)
Junc ion
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∂
∂xδ(x,y,z)]x=±gx
2=∓ Sg
2D∗δ(±gx
2,y,z) (6)
∂
∂yδ(x,y,z)]y=±gy
2=∓ Sg
2D∗δ(x,±gy
2,z) (7)
Whe e 𝑆𝑔 is he GB ecombina ion eloci y. These yield wo anscenden al ela ions
an(Cjgx
2)= Sg
2D Cj (8)
an(Ckgy
2)= Sg
2DCk (9)
sol ed g aphically o nume ically [13,15].
Subs i u ing (5) in o (3) gi es, o each mode (𝑗,𝑘),
𝑑2
𝑑z2Zj,k(z)−Zj,k(z)
(𝐿𝑗,𝑘)2=−G(z)
𝐷𝑗,𝑘 (10)
wi h e ec i e modal pa ame e s
1
Lj,k²=Cj2+Ck
2+1
L² (11)
1
Dj,k =16sin(Cjgx
2)sin(Ckgy
2)
D[Cjgx+sin(Cjgx)][Ckgy+sin(Ckgy)] (12)
so Lj,k and Dj,k ac as e ec i e di usion leng h and coe icien [13,15]. A pa icula solu ion based on (4) leads o
Zj,k(z)=Aj,ksinh z
Lj,k +Bj,kcosh z
Lj,k +∑kie−biz
3
i=1 (13)
wi h: ki=− n
Dj,k aiLj,k²
bi²Lj,k²−1 (14)
The cons an s Aj,k and Bj,k ollow om he on and ea su ace condi ions
𝐷∂
∂zδ(x,y,z)=S δ(x,y,z) pou z=0 (15)
D∂
∂zδ(x,y,z)=−Sbδ(x,y,z) pou z=H (16)
Whe e 𝑆𝑓=𝑆𝑓0 +𝑆𝑓𝑗 combines he in insic junc ion loss (e.g. shun - ela ed) and he ope a ing-poin -dependen
lux imposed by he ex e nal ci cui ; 𝑆𝑏 is he ea SRV. Mode n ex ac ions o 𝑆𝑅𝑉/𝐽0 mo i a e hese bounda y
e ms [16].
Exp ession o he pho o ol age (in V)
The pho o ol age unde illumina ion mode m ∈{F on , Rea , Dual} is compu ed om he junc ion-plane excess:
Vphm=VTln(1+ 1
nO∫ ∫ δm(x,y,0)dxdy
gy
2
−gy
2
gx
2
−gx
2) (17)
wi h VT=𝑘𝑇/𝑞 and nO=ni2/ Nb (in insic densi y 𝑛𝑖, base doping Nb) [12].
F on -side Illumina ion
Unde his illumina ion, abso p ion occu s p ima ily nea he uppe su ace, i.e., in he icini y o he junc ion,
which acili a es s ong ca ie collec ion. The co esponding pho o ol age is gi en by:
Vph on =VTln(1+ 1
nO∑ ∑ (B on j,k+∑ki
3
i=1 )
kj ×4sin(Cjgx
2)sin(Ck(B)gy
2)
CjCk(B) ) (18)
wi h:
𝐵 on ,j,k: mode ampli ude coe icien s o on -side illumina ion
𝑘𝑖 : cons an de i ed om he analy ical solu ion, associa ed wi h gene a ion and ecombina ion mechanisms.
Rea -side Illumina ion
When he de ice is illumina ed om he ea , ca ie s a e p oduced a he om he deple ion egion and mus
di use ac oss he base, which penalizes collec ion. The exp ession is gi en by:
Vph ea =VTln(1+ 1
nO∑ ∑ (Bea j,k +∑kie−biH
3
i=1 )
kj ×4sin(Cjgx
2)sin(Ck(B)gy
2)
CjCk(B) ) (19)
whe e:
𝐵 ea ,j,k: mode ampli ude coe icien s o ea -side illumina ion,
𝑏𝑖: cons an de i ed om he analy ical solu ion, associa ed wi h gene a ion and ecombina ion mechanisms.
𝑒−𝑏𝑖𝐻: exponen ial a enua ion e m ep esen ing ca ie ecombina ion in deepe egions.
Dual- ace Illumina ion
Wi h dual-side incidence, he wo ca ie s eams o e lap wi hin he base; ecombina ion and la e al anspo
couple hei e ec s, so he pho o ol age depa s om a simple sum. The exp ession is:
Vphdual =VTln(1+ 1
nO∑ ∑ (Bdualj,k+∑ki(1+e−biH)
3
i=1 )
kj ×4sin(Cjgx
2)sin(Ck(B)gy
2)
CjCk(B) ) (20)
Whe e:
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𝐵dual,j,k: mode ampli ude coe icien s unde dual- ace illumina ion,
(1−𝑒−𝑏𝑖𝐻): con ibu ion o pho ogene a ed ca ie s h oughou he ull base hickness.
RESULTS AND DISCUSION
In luence o ca ie collec o eloci y 𝐂𝐂𝐕 on he pho o ol age o a sola cell illumina ed om he on , he
ea , and on bo h sides, o di e en g ain sizes
F on -side illumina ion.
Figu es 2 and 3 plo he pho o ol age Vph unde on -side illumina ion as a unc ion o he CCV, o se e al g ain
sizes and wo base hicknesses (H=150 μm and H=300 μm).
Figu e 1 : Pho o ol age o a 150μm (le ) and a 300μm ( igh ) PV cell unde on illumina ion as a unc ion o
ca ie collec ion eloci y o a ious g ain sizes
In all cases, Vph dec eases as CCV inc eases, consis en wi h he diode ela ion Vph ∝ln(1+Jph/J0): as e
ex ac ion lowe s he excess ca ie densi y a he junc ion and educes he quasi-Fe mi-le el spli ing [17]. The
sensi i i y o Vph o CCV also e lec s su ace ecombina ion and con ac quali y (which con ol J0 and e ec i e
SRV); imp o ed passi a ion/con ac s acks shi he cu es upwa d and delay he d op o Vph wi h CCV [11,16].
Rega ding g ain size, la ge g ains yield highe Vph owing o educed g ain-bounda y ecombina ion and longe
e ec i e di usion pa hs be o e loss [8,18]. In ou da a, he highes alues occu o g=0.015 cm, wi h a
maximum Vph ≈0.30 V a H=150 μm. Finally, he small gap be ween he H=150 and 300 μm cu es unde
on illumina ion is consis en wi h nea -junc ion gene a ion: once ca ie s a e c ea ed close o he deple ion egion,
inc easing H has only a modes in luence on Vph a a gi en CCV [17].
Rea -side illumina ion.
Figu es 3 plo he pho o ol age 𝑉𝑝ℎ unde ea -side illumina ion as a unc ion o he CCV, o se e al g ain sizes
and wo base hicknesses (𝐻=150 𝜇𝑚 and 𝐻=300 𝜇𝑚).
Figu e 2 : Pho o ol age o a 150μm and a 300μm PV cell unde ea illumina ion as a unc ion o ca ie collec ion
eloci y o a ious g ain sizes
101102103104105106107
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0,002 cm
0,005 cm
0,01 cm
0,012 cm
0,014 cm
0,015 cm
Ca ie collec o eloci y (cm/s)
Pho o ol age (V)
F on side illumina ion
Base hickness = 150 m
101102103104105106107
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0,002 cm
0,005 cm
0,01 cm
0,012 cm
0,014 cm
0,015 cm
Ca ie collec o eloci y (cm/s)
Pho o ol age (V)
F on side illumina ion
Base hickness = 300 m
101102103104105106107
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0,15
0,20
0,25
0,30
0,002 cm
0,005 cm
0,01 cm
0,012 cm
0,014 cm
0,015 cm
Pho o ol age (V)
Ca ie collec o eloci y (cm/s)
Rea side illumina ion
Base hickness = 150 m
101102103104105106107
0,00
0,05
0,10
0,15
0,20
0,25 0,002 cm
0,005 cm
0,01 cm
0,012 cm
0,014 cm
0,015 cm
Ca ie collec o eloci y (cm/s)
Pho o ol age (V)
Rea side illumina ion
Base hickness = 300 m
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As in he on -side case, Vph dec eases wi h inc easing CCV, e lec ing he educ ion o excess ca ie densi y a he
junc ion and he consequen d op in quasi-Fe mi-le el spli ing [17]. Howe e , he dependence on base hickness is
much s onge om he ea : ca ie s a e gene a ed a he om he junc ion and mus a e se he base, so a la ge
H inc eases ansi ime and exposu e o bulk SRH and g ain-bounda y ecombina ion, he eby lowe ing Vph a a
gi en CCV [16-18]. The g ain-size end emains obus : la ge g ains (lowe bounda y densi y) yield highe Vph
ac oss he CCV ange [18]. In ou da a, he uppe en elope eaches abou Vph≈0.30 V o g=0.015 cm a H=
150 μm. Finally, ea -su ace passi a ion/con ac s (which se SRV and J0) a e c i ical unde ea incidence;
imp o ed s acks shi he cu es upwa d and mi iga e he loss in hick bases [16,17]. These beha io s a e consis en
wi h ecen assessmen s o bi acial silicon ope a ion, whe e ea -side gene a ion accen ua es anspo -and-
ecombina ion limi a ions h ough he base [10].
Dual-side illumina ion.
Figu e 4 show he pho o ol age Vph unde dual-side illumina ion as a unc ion o he CCV, o se e al g ain sizes
and wo base hicknesses (H=150 μm and H=300 μm).
Figu e 3: Pho o ol age o a 150μm and a 300μm PV cell unde dual illumina ion as a unc ion o ca ie collec ion
eloci y o a ious g ain sizes
As expec ed o bi acial ope a ion, illumina ing bo h aces inc eases he o e all gene a ion and shi s he Vph −CCV
cu es upwa d compa ed wi h single- ace cases [10,19]. A ixed i adiance and g ain size, Vph s ill dec eases wi h
inc easing CCV, consis en wi h he diode ela ion Vph ∝ln(1+Jph/J0): as e ex ac ion educes he s eady-s a e
excess ca ie densi y a he junc ion and hence he quasi-Fe mi-le el spli ing [17]. The absolu e Vph is, howe e ,
highe unde dual illumina ion due o he la ge Vph inpu ; his e ec is mo e p onounced when su ace passi a ion
and con ac s a e op imized (lowe SRV and J0), which li s he cu es and mi iga es losses ac oss he CCV ange
[11,16]. The g ain-size end emains obus : la ge g ains ( ewe bounda ies) educe ecombina ion and yield
highe Vph [18]. In ou da a, a maximum Vph ≈0.35 V is obse ed o g=0.015 cm a H=150 μm. The smalle
gap be ween he H=150 and H=300 μm cu es compa ed wi h ea -only illumina ion is consis en wi h sho e
a e age collec ion dis ances p o ided by dual incidence [10,19].
Complemen a y iew (g ain-size sweep a ixed CCV). The igu es below epo he a ia ion o Vph e sus g ain
size a CCV = 30.2 cm·s−1 o mul iple base hicknesses and illumina ion modes, illus a ing he mono onic ise
o Vph wi h g ain size and he addi ional bene i o dual illumina ion [10,18,19].
In luence o g ain size on he pho o ol age o a sola cell illumina ed om he on , he ea , and on bo h
sides, o di e en base hickness
Figu es 5, 6, and 7 display he pho o ol age 𝑉𝑝ℎ as a unc ion o g ain size 𝑔 o on , ea , and dual illumina ion,
espec i ely, a se e al base hicknesses 𝐻. The ca ie -collec ion eloci y is ixed a 𝐶𝐶𝑉 =30,2 𝑐𝑚.𝑠−1, i.e.,
nea open-ci cui , so hese ends closely e lec he open-ci cui ol age 𝑉𝑜𝑐 beha io ia Vph ≈Voc ∝ln(1+Jph/J0
) [17].
101102103104105106107
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0,15
0,20
0,25
0,30
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0,002 cm
0,005 cm
0,01 cm
0,012 cm
0,014 cm
0,015 cm
Ca ie collec o eloci y (cm/s)
Pho o ol age (V)
Dual illumina ion
Base hickness = 150 m
101102103104105106107
0,05
0,10
0,15
0,20
0,25
0,30
0,35
0,002 cm
0,005 cm
0,01 cm
0,012 cm
0,014 cm
0,015 cm
Ca ie collec o eloci y (cm/s)
Pho o ol age (V)
Dual illumina ion
Base hickness = 300 m

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Figu e 5: Pho o ol age as a unc ion o g ain size o a
PV cell wi h di e en base hicknesses unde on -side
illumina ion.
Figu e 6: Pho o ol age as a unc ion o g ain size o a
PV cell wi h di e en base hicknesses unde ea -side
illumina ion
Figu e 7: Pho o ol age as a unc ion o g ain size o a PV cell wi h di e en base hicknesses unde dual
Two le e s go e n he e olu ion:
i. any inc ease in Jph (e.g., dual illumina ion) o dec ease in J0 (be e passi a ion/con ac s) aises Vph
[11,16,17];
ii. la ge g ains educe g ain-bounda y ecombina ion, e ec i ely lowe ing ecombina ion losses and li ing
Vph ac oss H [8,18].
F on illumina ion (Fig. 5). The cu es o di e en H a e nea ly supe posed: gene a ion occu s close o he
junc ion, so hickness changes ha e li le impac on Vph a ixed CCV. The dominan end is he mono onic ise o
Vph wi h g ain size, d i en by he educed bounda y densi y [8,17,18].
Rea illumina ion (Fig. 6). The cu es sepa a e wi h hickness: a la ge H leng hens anspo pa hs and exposu e o
bulk SRH and g ain-bounda y ecombina ion, hus lowe ing Vph a a gi en g. Fo a ixed H, Vph inc eases wi h
g ain size, and he hinne base (e.g., H=150 μm) yields he highes ol ages, especially o g∈[0, 0.012] cm
[8,16-18].
Dual illumina ion (Fig. 7). The beha io is in e media e be ween on and ea : dual incidence boos s Jph, li ing all
cu es, while he dependence on H is weake han ea -only because a e age collec ion dis ances sho en when bo h
sides gene a e ca ie s. The g ain-size bene i emains clea and is enhanced by good su ace passi a ion and
passi a ing con ac s ha supp ess J0 [10,11,16].
In luence o CCV on pho o ol age o di e en base hicknesses and ixed g ain size
Figu es 8–9 show he pho o ol age 𝑉𝑝ℎ 𝐶𝐶𝑉 o on , ea , and dual illumina ion, espec i ely, a se e al base
hicknesses 𝐻, wi h a ixed g ain size 𝑔=0.01 𝑐𝑚.
0,004 0,008 0,012 0,016 0,020
0,22
0,24
0,26
0,28
0,30
0,32
0,34
0,36
G ain size (cm)
Pho o ol age (V)
F on side illumina ion
Ca ie collec ion eloci y 30,2 cm/s
100m
150m
200m
250m
300m
350m
400m
0,004 0,008 0,012 0,016 0,020
0,12
0,16
0,20
0,24
0,28
0,32
G ain size (cm)
Rea side illumina ion
Ca ie collec o eloci y 30,2 cm/s
100m
150m
200m
250m
300m
350m
400m
Pho o ol age (V)
0,004 0,008 0,012 0,016 0,020
0,22
0,24
0,26
0,28
0,30
0,32
0,34
0,36
G ain size (cm)
Pho o ol age (V)
Dual illumina ion
Ca ie collec o eloci y 30,2 cm/s
100m
150m
200m
250m
300m
350m
400m
Gning Y e al Eu o. J. Ad . Engg. Tech., 2025, 12(10):37-46
43
Figu e 8: Pho o ol age as a unc ion o ca ie
collec ion eloci y o PV cells o a ying base hickness
unde on illumina ion and a cons an g ain
Figu e 9: Pho o ol age as a unc ion o ca ie
collec ion eloci y o PV cells o a ying base
hickness unde ea illumina ion and a cons an g ain
size.
Figu e 10: Pho o ol age as a unc ion o ca ie collec ion eloci y o PV cells o a ying base hickness unde
dual illumina ion and a cons an g ain size.
As a eminde , low CCV co esponds o an ex ac ion-limi ed egime (open-ci cui -like o cu en ), whe eas high
CCV indica es e icien ex ac ion (sho -ci cui -like, wi h a cu en pla eau) [17]. Consis en wi h Vph ≈Voc ∝
ln(1+Jph/J0), Vph dec eases as CCV inc eases, because as e ex ac ion lowe s he s eady-s a e excess ca ie
densi y a he junc ion and he quasi-Fe mi-le el spli ing [17]. The magni ude and CCV sensi i i y o Vph a e se by
su ace ecombina ion and con ac quali y (SRV, J0), imp o ed passi a ion/con ac s acks gene ally li he cu es
and delay he d op wi h CCV [11,16].
By illumina ion mode, he ends ma ch he anspo dis ance and ecombina ion exposu e.
F on illumina ion (Fig. 8): he cu es a e nea ly supe posed ac oss H, since gene a ion occu s nea he junc ion;
hickness plays a mino ole a ixed CCV.
Rea illumina ion (Fig. 9): cu es sepa a e wi h H, o H=100 −150 μm he beha io emains close, bu hicke
bases show a clea Vph educ ion a he same CCV due o longe pa hs and highe bulk/GB ecombina ion
[17,18,20].
Dual illumina ion (Fig. 10): we obse e blockwise supe posi ion: a i s g oup o H=100 − 200 μm wi h Vph ≈
0.28 V and a second o H=250 − 400 μm wi h Vph ≈0.325 V, e lec ing he highe Jph and sho e a e age
collec ion dis ances o bi acial ope a ion [10,20].
Absolu e a ia ion o pho o ol age o e he g ain-size ange as a unc ion o base hickness
Figu es 11, 12, and 13 epo he absolu e a ia ion o pho o ol age o e he g ain-size ange,
ΔVph =Vph(gmax )−Vph(gmin ) (21)
unde on , ea , and dual illumina ion, espec i ely, as a unc ion o base hickness 𝐻. The junc ion 𝐶𝐶𝑉 is ixed a
30.2 𝑐𝑚 𝑠−1, i.e., nea an open-ci cui -like egime o he ol age (slow ex ac ion, la ge excess ca ie densi y)
[1].
101102103104105106107
0,10
0,15
0,20
0,25
0,30
0,35
Ca ie collec o eloci y (cm/s)
Pho o ol age (V)
150m
200m
250m
300m
350m
400m
F on side illumina ion
g = 0,01 cm
101102103104105106107
0,00
0,05
0,10
0,15
0,20
0,25
0,30
Ca ie collec o eloci y (cm/s)
Pho o ol age (V)
100m
150m
200m
250m
300m
350m
400m
Rea side illumina ion
g = 0,01 cm
101102103104105106107
0,10
0,15
0,20
0,25
0,30
Ca ie collec o eloci y (cm/s)
Pho o ol age (V)
Dual illumina ion
g = 0,01 cm
100m
150m
200m
250m
300m
350m
400m
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44
Figu e 11: Absolu e a ia ion o he Pho o ol age o e
he g ain size ange unde on illumina ion as a
unc ion o base hickness
Figu e12: Absolu e a ia ion o he Pho o ol age o e
he g ain size ange unde ea illumina ion as a
unc ion o base hickness
Figu e 43: Absolu e a ia ion o he Pho o ol age o e he g ain size ange unde dual illumina ion as a unc ion o
base hickness
Physical eading.
• Role o 𝐻: Inc easing 𝐻 leng hens anspo pa hs and aises exposu e o bulk SRH and g ain-bounda y
ecombina ion. Since 𝑉𝑝ℎ ≈Voc ∝ln(1+Jph/J0), any hickness-induced inc ease o J0 ( ia highe SRV o
deepe ecombina ion) comp esses he ol age sp ead be ween la ge- and small-g ain cases, ending o educe
𝛥𝑉𝑝ℎ [11,16,17].
• F on illumina ion (Fig. 11). Gene a ion close o he junc ion makes 𝑉𝑝ℎ only weakly dependen on 𝐻; esidual
su ace/con ac losses domina e. In his egime, 𝛥𝑉𝑝ℎ o en dec eases wi h 𝐻 as added hickness b ings li le
bene i o la ge g ains bu inc eases pa asi ic losses [17].
• Rea illumina ion (Fig. 12). Ca ie s o igina e a om he junc ion; hicke bases penalize small g ains mo e
(highe bounda y densi y), which can inc ease 𝛥𝑉𝑝ℎ o e pa o he ange (con as e ec ) be o e e y la ge 𝐻
supp esses bo h endpoin s [8,17,18].
• Dual illumina ion (Fig. 13). Bi acial gene a ion sho ens a e age collec ion dis ances and mi iga es hickness
penal ies; acco dingly, 𝛥𝑉𝑝ℎ ypically dec eases wi h 𝐻, bu mo e slowly han unde ea -only ope a ion
[10,16].
Design implica ion. To p ese e a sizable 𝛥𝑉𝑝ℎ nea open-ci cui :
i. limi 𝐻 (e.g., 150 𝜇𝑚 s. 300−400 𝜇𝑚) o on /dual ope a ion;
ii. in ea ope a ion, no e ha hicke bases may empo a ily widen 𝛥𝑉𝑝ℎ (con as ) bu ul ima ely lowe absolu e
ol ages; and
iii. combine la ge g ains ( educed GB ecombina ion) wi h low-𝐽0, low-SRV passi a ing con ac s o li bo h
𝑉𝑝ℎ(𝑔𝑚𝑎𝑥 ) and he o e all cu e [10,11,16].
150 200 250 300 350 400
0,1090
0,1092
0,1094
0,1096
0,1098
0,1100
Base hickness (m)
F on side illumina ion
Ca ie collec o eloci y 30,2 cm/s
Absolu e a ia ion o he pho o ol age
o e he g ain size ange (A/cm²)
150 200 250 300 350 400
0,116
0,118
0,120
0,122
0,124
Absolu e a ia ion o he pho o ol age
o e he g ain size ange (A/cm²)
Rea side illumina ion
Ca ie collec o eloci y 30,2 cm/s
Base hickness (m)
150 200 250 300 350 400
0,1090
0,1095
0,1100
0,1105
0,1110 Rea side illumina ion
Ca ie collec o eloci y 30,2 cm/s
Base hickness (m)
Absolu e a ia ion o he pho o ol age
o e he g ain size ange (A/cm²)
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45
CONCLUSION
This wo k es ablished closed- o m exp essions o he pho o ol age unde on , ea , and dual illumina ion in a 3D
di usion– ecombina ion amewo k. Ac oss ope a ing condi ions, he ca ie -collec ion eloci y (𝐶𝐶𝑉) eme ges as
he p ima y knob con olling he quasi-Fe mi-le el spli ing a he junc ion: 𝑉𝑝ℎ dec eases as 𝐶𝐶𝑉 inc eases,
ansi ioning om an ex ac ion-limi ed (open-ci cui -like) o an e icien -collec ion (sho -ci cui -like) egime.
Mic os uc u e and geome y co-go e n he ol age: la ge g ains (𝑔) educe g ain-bounda y ecombina ion and
consis en ly aise 𝑉𝑝ℎ, while hinne bases (𝐻) sho en anspo pa hs and cu b bulk losses. Illumina ion mode se s
he anspo dis ance and hus he sensi i i y o 𝐻: on illumina ion is weakly hickness-dependen , ea
illumina ion is s ongly hickness-dependen , and dual illumina ion li s he ol age baseline while mi iga ing he
hickness penal y.
Design guidance.
• Fa o la ge g ains (𝑔≳0.01 𝑐𝑚) o supp ess g ain-bounda y ecombina ion.
• Use mode a e 𝐻 (e.g., 150 µ𝑚) o limi bulk SRH losses; hicke bases demand supe io passi a ion.
• Implemen low-𝐽0, low-SRV passi a ing con ac s; imp o emen s he e li all 𝑉𝑝ℎ 𝐶𝐶𝑉 cu es.
• P e e dual illumina ion when a ailable; i aises 𝑉𝑝ℎ and educes hickness sensi i i y, especially a low 𝐶𝐶𝑉.
Scope & ou look. The analysis assumes uni o m ma e ial p ope ies, 𝑔𝑥=𝑔𝑦 domains, and cons an empe a u e.
Ex ending he model o include la e al non-uni o mi ies, ealis ic op ics/albedo o bi acial ope a ion, se ies
esis ance, and empe a u e-dependen ecombina ion would enable di ec p edic ion o ull 𝐽−𝑉 cha ac e is ics
unde ield condi ions.
ABBREVIATIONS
• c-Si: monoc ys alline silicon
• mc-Si: mul ic ys alline (polyc ys alline) silicon
• PV: pho o ol aic
• BSF: Back Su ace Field
• SRH: Shockley–Read–Hall ( ap-assis ed ecombina ion)
• SRV: Su ace Recombina ion Veloci y
• CCV: Ca ie -Collec ion Veloci y (a he junc ion)
• AM1.5: Ai Mass 1.5 sola spec um
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