Simula ing Monoli hic Ac i e Pixel Senso s:
A Technology-Independen App oach Using Gene ic Doping P o iles
H˚akan Wennl¨o a,∗
, Dominik Dannheimb, Manuel Del Rio Vie aa,1, Ka ha ina Do b,2, Do is Ecks eina, Finn Feind a,
Ing id-Ma ia G ego a, Lenna Hu ha, S ephan Lachni a,3, La issa Mendesa,1, Daniil Ras o gue a,4, Sa a Ruiz Dazaa,1,
Paul Sch¨u zea, Ad iana Simancasa,1, Wal e Snoeysb, Simon Spannagela, Ma cel S ani zkia, Alessand a Tomalc,
Anas asiia Velykaa, Gianpie o Vignolaa,1
aDeu sches Elek onen-Synch o on DESY, No kes . 85, 22607 Hambu g, Ge many
bCERN, Gene a, Swi ze land
cUni e si y o Campinas, Cidade Uni e si a ia Ze e ino Vaz, 13083-970, Campinas, B azil
Abs ac
The op imisa ion o he sensi i e egion o CMOS senso s wi h complex non-uni o m elec ic ields equi es p ecise
simula ions, and his can be achie ed by a combina ion o elec os a ic ield simula ions and Mon e Ca lo me hods. This
pape p esen s he guiding p inciples o such simula ions, using a CMOS pixel senso wi h a small collec ion elec ode and
a high- esis i i y epi axial laye as an example. The ull simula ion wo k low is desc ibed, along wi h possible pi alls and
how o a oid hem. Fo comme cial CMOS p ocesses, de ailed doping p o iles a e con iden ial, bu he p esen ed me hod
p o ides an op imisa ion ool ha is su icien ly accu a e o in es iga e senso beha iou and ade-o s o di e en senso
designs wi hou knowledge o p op ie a y in o ma ion.
The wo k low s a s wi h de ailed elec ic ield ini e elemen me hod simula ions in TCAD, using gene ic doping
p o iles. Examples o he e ec o a ying di e en pa ame e s o he simula ed senso a e shown, as well as he c ea ion
o weigh ing ields, and ansien pulse simula ions. The ields esul ing om TCAD simula ions can be impo ed in o he
Allpix2Mon e Ca lo simula ion amewo k, which enables high-s a is ics simula ions, including modelling o s ochas ic
luc ua ions om he unde lying physics p ocesses o pa icle in e ac ion. Example Mon e Ca lo simula ion se ups a e
p esen ed and he di e en pa s o a simula ion chain a e desc ibed.
Simula ion s udies om small collec ion elec ode CMOS senso s a e p esen ed, and example esul s a e shown o
bo h single senso s and mul iple senso s in a es beam elescope con igu a ion. The s udies shown a e hose ypically
pe o med on senso p o o ypes in es beam campaigns, and a compa ison is made o es beam da a, showing a
maximum de ia ion o 4% and demons a ing ha he app oach is iable o gene a ing ealis ic esul s.
Keywo ds: Shockley-Ramo, Simula ion, Mon e Ca lo, Silicon De ec o s, TCAD, D i -di usion, Gean 4, Allpix
Squa ed, Pixella ed de ec o s, Cha ged pa icle acking, Monoli hic ac i e pixel senso s, MAPS
Con en s
1 In oduc ion 2
2 Gene al layou and assump ions 2
2.1 Doping wells ................. 2
2.2 Con ac s and biasing ............. 3
2.3 Rec angula and hexagonal pixels ...... 3
3 Fini e elemen me hod simula ions 3
3.1 Simula ion wo k low ............. 4
∗Co esponding au ho
Email add ess: [email p o ec ed] (H˚akan Wennl¨o )
1Also a Uni e si y o Bonn, Ge many
2Also a Uni e si y o Giessen, Ge many
3Also a Uni e si y o Hambu g, Ge many
4Also a Uni e si y o Wuppe al, Ge many
3.2 Subs a e di usion simula ion ........ 4
3.3 Impac o senso doping concen a ion . . . 5
3.4 Impac o senso geome y .......... 5
3.5 Hexagonal pixel geome y simula ion . . . . 6
3.6 T ansien simula ions ............. 7
3.7 Gene a ing weigh ing po en ials ....... 7
4 Mon e Ca lo simula ions 8
4.1 Simula ion low ................ 8
4.2 Senso geome y and se up ......... 8
4.3 Impo ing esul s om TCAD simula ions . 9
4.4 Cha ge ca ie gene a ion .......... 10
4.5 Cha ge ca ie p opaga ion ......... 11
4.6 Cha ge ans e ................ 12
4.7 Signal digi isa ion .............. 12
4.8 T ansien simula ions ............. 12
4.9 E ec o dopan di usion in elec ic ields . 13
4.10 Simula ion pa ame e op imisa ion ..... 13
P ep in submi ed o NIM A Augus 2, 2024
a Xi :2408.00027 1 [physics.ins-de ] 31 Jul 2024
5 Senso pe o mance s udies 14
5.1 Clus e size and o al cha ge ........ 14
5.2 In-pixel s udies ................ 15
5.3 Impac o h eshold alue .......... 15
5.4 T ansien pulse s udies ............ 17
5.5 Mul i-senso s udies ............. 17
6 Compa isons o da a and p e ious simula-
ions 19
7 Summa y and ou look 19
1. In oduc ion
Monoli hic ac i e pixel senso s (MAPS) p oduced us-
ing comme cial CMOS imaging p ocesses a e a ac i e
in a pa icle physics con ex , as hey allow o a educed
ma e ial budge and educ ion o p oduc ion complexi y
compa ed o mos hyb id senso s. The use o comme cial
p ocesses enables ela i ely cheap la ge-scale p oduc ion
o senso s, bu i also means ha p ecise in o ma ion o
he manu ac u ing p ocess may no be publicly a ailable.
P edic ions o senso beha iou a e hus di icul o make,
as he de ailed elec ic ield con igu a ion in he sensi i e
ma e ial is highly dependen on he ex en and concen a-
ion o di e en doping egions in he silicon.
By u ilising a quad uple-well echnology (p o iding n-
wells, p-wells, and deep n-wells and p-wells) [1], MAPS can
be cons uc ed wi h a small collec ion elec ode, which e-
duces senso capaci ance and imp o es signal- o-noise a-
io while educing powe consump ion compa ed o sen-
so s wi h la ge collec ion elec odes. Howe e , designs
wi h a small collec ion elec ode lead o a highly non-linea
elec ic ield in he pixels, u he complica ing senso be-
ha iou p edic ion. As p o o ype senso submissions and
in es iga ions a e expensi e and ake a long ime, simu-
la ions o senso beha iou a e becoming mo e and mo e
impo an o gain insigh and speed up he design p ocess.
This pape aims o demons a e ha by making simple
assump ions and pe o ming simula ions based on he un-
damen al p inciples o silicon de ec o s and using gene ic
doping p o iles, pe o mance pa ame e s o MAPS can
be in e ed and compa ed o di e en senso geome ies.
This will be done in con ex o simula ions pe o med o
he Tange ine p ojec [2,3,4,5] and in collabo a ion wi h
he CERN EP R&D p og amme on echnologies o u-
u e expe imen s [6], bu he me hodology desc ibed is use-
ul o many di e en silicon senso simula ions. The de-
sc ibed me hod hus cons i u es a oolbox o pe o ming
simila simula ions, use ul in ex ac ing a ealis ic desc ip-
ion o senso beha iou wi hou knowledge o p op ie a y
in o ma ion. The e icacy o combining de ailed elec ic
ield simula ions wi h high-s a is ics Mon e Ca lo simula-
ions has been p e iously demons a ed o simila silicon
senso s [7,8], and he p ocess desc ibed in his pape is
gene al and applicable in mul iple di e en cases. Fo sen-
so s wi h non-linea elec ic ields, simula ions like hose
p esen ed he e a e use ul o gaining a deepe unde s and-
ing o he senso pe o mance. I is impo an o no e ha
he p esen ed simula ions by no means cap u e he in ica-
cies o CMOS imaging p ocesses, bu me ely desc ibe he
la ge ea u es o he senso equi ed o model an accu a e
signal esponse.
Pape ou line
The pape aims o show guiding p inciples o pe o m-
ing de ailed Mon e Ca lo simula ions o silicon senso s,
using basic assump ions and es ima es. In Sec ion 2, a
gene al MAPS layou is desc ibed, and assump ions o
he geome y and doping ypes used in he simula ions
a e discussed. Then, doping concen a ion and elec ic
ield ini e-elemen simula ions using echnology compu e -
aided design (TCAD) a e p esen ed in Sec ion 3, wi h a
de ailed simula ion p ocedu e using gene ic doping p o iles
and assump ions based on he physics o a semiconduc-
o senso . Mon e Ca lo simula ions using he Allpix2[9]
amewo k wi h elec ic ields and doping p o iles om
TCAD a e desc ibed in de ail in Sec ion 4, going h ough
he simula ion se up s ep by s ep. Some example esul s
o he high-s a is ics Mon e Ca lo simula ions ca ied ou
in he Tange ine p ojec a e shown in Sec ion 5, including
in-pixel s udies, ansien cu en pulses, and simula ion
esul s om a mul i-senso se up. Finally, example com-
pa isons o simula ion esul s o da a a e shown in Sec-
ion 6.
2. Gene al layou and assump ions
The MAPS simula ed in his wo k consis o a high-
esis i i y p-doped epi axial laye g own on an elec onics-
g ade p-doped silicon subs a e, wi h implan ed doping
wells in he epi axial laye . The doping wells unc ion as
collec ion elec odes and/o shielding o he in-pixel elec-
onics. As he epi axial laye in he senso s is ela i ely
hin (o he o de o 10 µm), he senso hickness is domi-
na ed by he subs a e. Mos o he isible signal is gene -
a ed in he epi axial laye , and he subs a e is hus o en
hinned a e senso p oduc ion, down o a o al senso
hickness below 50 µm.
The doping concen a ions used in he simula ions p e-
sen ed he e a e no alues om a speci ic senso o ech-
nology, bu app oxima ions de i ed om p e ious s ud-
ies [10,11]. The subs a e is assumed o ha e a dop-
ing concen a ion o 1 ·1019 cm−3, he epi axial laye ap-
p oxima ely 3 ·1013 cm−3, and doping wells anging om
1·1015 cm−3 o 1·1019 cm−3, depending on hei pu pose.
2.1. Doping wells
In he cen e o he pixels, an n-doped well is loca ed.
This well is simula ed wi h a doping concen a ion o ap-
p oxima ely 1019 cm−3, is posi i ely biased, and se es as
he collec ion elec ode. The size o he well is o he o de
2
o 1 µm ac oss, and i has a squa e shape (when iewed
om abo e).
Su ounding he collec ion elec ode is an opening wi h-
ou wells, and hen a squa e deep p-well ( o squa e pixels).
This deep p-well is assumed o ha e a doping concen a-
ion o app oxima ely 1015 cm−3. While none o he dop-
ing wells in he simula ions con ain any in e nal s uc u e
o elec onics, he main pu pose o he deep p-well in a
physical senso is o con ain bo h NMOS ansis o s and
in e nal n-wells ha con ain PMOS ansis o s. In his
way, ull CMOS on -end elec onics a e possible in he
pixels. The deep p-well shields he elec onics om he
sensi i e egion, which ensu es ha he n-well collec ion
elec ode is he only node elec ons d i o. This also al-
lows o a highe bias ol age o be applied o he senso
bulk wi hou damaging he elec onics.
The ex en and shape o he wells can be used o shape
he elec ic ield, and may signi ican ly a ec he cha ge
collec ion p ope ies o a senso . Fo example, he e ec
o changing he size o he opening be ween he collec ion
elec ode and he p-well is explo ed in he wo k p esen ed
he e.
2.2. Con ac s and biasing
Ohmic con ac s a e essen ial o p o ide bias ol ages o
and ex ac signals om a senso , and hey a e achie ed
by ha ing a highly-doped egion in he silicon nex o he
me al con ac . In he senso s p esen ed he e he e a e
con ac s o he collec ion elec ode, he p-well, and he
senso subs a e. In a physical senso , he biasing o he
collec ion elec ode and he p-well a e done ia me al con-
ac s, and he subs a e is biased h ough su ace con ac s
ou side he pixel ma ix. In he simula ions howe e , he
subs a e is ins ead biased ia a con ac di ec ly on he
backside as gua d ings and senso edge s uc u es a e no
included.
The collec ion elec ode in he simula ions p esen ed
he e has a posi i e bias ol age o 1.2 V, whe eas he p-well
and subs a e ha e bias ol ages be ween 0 V and −6 V.
The p-well and he subs a e a e commonly biased o he
same ol age, bu can also be biased sepa a ely. The bias
ol age ha can be applied o he p-well in a physical sen-
so is limi ed by he beha iou o he NMOS ansis o s,
as hei cha ac e is ics will change and hei unc ion may
cease a high ol ages.
2.3. Rec angula and hexagonal pixels
Th ee main designs a e simula ed and es ed in his
wo k, labelled s anda d layou [12], n-blanke layou [13],
and n-gap layou [14]. The s anda d layou is simila o
wha is used in he ALPIDE senso [15], which is a MAPS
used in he ALICE expe imen since he ITS2 upg ade, de-
eloped in a 180 nm CMOS imaging p ocess. This layou
has a small n- ype collec ion elec ode in a p- ype epi axial
laye , and deple ion g ows in an app oxima ely sphe ical
shape om his pn-junc ion. This deple ed egion ends
o no ex end ully below he p-well. The n-blanke layou
in oduces a blanke laye o n-doped silicon in he p- ype
epi axial laye , which o ms a deep plana pn-junc ion, al-
lowing o ull deple ion o a pixel. This layou lea es
an elec ic ield minimum unde he p-well a pixel edges
and co ne s, howe e , leading o slow cha ge collec ion
and possible e iciency loss in hese egions. This can be
amended by in oducing a e ical pn-junc ion nea he
edge [14]. One way o achie e his is by lea ing a gap in
he blanke o n-doped silicon unde he p-well, which is
done in he n-gap layou . As a pn-junc ion is hus o med
nea he pixel edges, a la e al elec ic ield is o med he e,
pushing cha ges owa d he pixel cen e. The n-blanke
and n-gap layou modi ica ions we e o iginally de eloped
o a 180 nm CMOS imaging p ocess, bu simila de elop-
men s ha e been implemen ed in a 65 nm CMOS imaging
p ocess as well [16].
The h ee sensi i e olume designs desc ibed abo e a e
applied in bo h ec angula and hexagonal pixel geome-
ies in he wo k p esen ed he e. Using a hexagonal geome-
y dec eases he amoun o sha ed cha ge in pixel co ne s,
as a pixel only sha es a co ne wi h wo neighbou s a he
han h ee. I also educes he maximum dis ance o he
pixel bounda y om he cen e compa ed o ec angula
geome ies, while main aining he same a ea. Hexagonal
pixel shapes hus educe egions wi h low elec ic ields
a pixel edges. The maximum dis ance in a squa e g id
be ween he pixel co ne and he collec ion elec ode is
educed by 12% o he same pixel a ea on a hexagonal
g id [17]. As he pixel co ne egion and p-well edges ha e
a la ge opening angle, he elec ic ields he e di e sig-
ni ican ly compa ed o ec angula pixel geome ies. The
dis ance be ween collec ion elec odes is also he same o
all adjacen pixels in a hexagonal con igu a ion.
3. Fini e elemen me hod simula ions
Technology Compu e -Aided Design (TCAD) is a simu-
la ion ool ha uses ini e elemen me hods o model semi-
conduc o de ices in 2D and 3D. In each node o a c ea ed
mesh, calcula ions o he elec os a ic po en ial and o he
p ope ies a e ca ied ou by sol ing Poisson’s and ca -
ie con inui y equa ions. This wo k implemen s TCAD
simula ions wi h gene ic doping p o iles o s udy e ec s o
layou design on he elec ic ield o CMOS senso s, and
he p esen ed simula ions ha e been pe o med in 3D wi h
Sen au us TCAD om Synopsys [18].
The body o he senso is c ea ed ini ially om simple
geome ical shapes, which a e hen adjus ed o ep esen
he di e en s udied layou s. To ob ain insigh in o he
e ec s o he adjus men s, i e a ions o layou modi ica-
ions and simula ion e alua ions a e pe o med. To e ine
he simula ions, he ollowing p inciples a e aken in o ac-
coun , o ensu e a physically ealis ic and ope a ional sen-
so :
•The doping concen a ions in he in e aces be ween
3
di e en doping s uc u es (n- and p-wells, epi axial
laye /subs a e) should be di used o a oid unphysi-
cal e ec s, such as ab up changes in doping concen-
a ion and he co esponding elec ic ield.
•The p-well mus shield i s con en om he elec ic
ield in he ac i e senso a ea; he doping mus hus
be su icien o he deple ed egion o no pene a e
deep in o he well.
•The cha ge ca ie s gene a ed in he senso olume
ha e o each he collec ion elec ode.
•The e should be no conduc i e channel be ween di -
e en biased s uc u es, i.e. punch- h ough in he
senso should be a oided.
•The limi a ions on he ope a ing ol ages o he an-
sis o s in he eadou elec onics o a physical senso
should be espec ed.
I should be no ed ha no in e nal s uc u e o he dop-
ing wells is simula ed in his wo k, so no eadou elec on-
ics a e included. The basic p inciples needed o p o ec
hem (ou lined abo e) a e included, howe e , in o de o
ha e a ealis ic senso desc ip ion.
3.1. Simula ion wo k low
The simula ion p ocess s a s wi h de ining he sen-
so geome y using he Sen au us S uc u e Edi o ool.
The ma e ials o s uc u es a e de ined oge he wi h hei
shapes, and he ma e ials used in hese simula ions a e
aluminium o he elec odes, silicon oxide o he dielec-
ic ma e ial, and silicon o he senso bulk. In o de o
apply elec ical bounda y condi ions, i is necessa y o de-
ine in e ace egions called con ac s, which co espond o
physical con ac s be ween he elec odes and he silicon
bulk. Fo simplici y, only he op pa o he senso co -
esponding o he egion aken up by he epi axial laye
and i s in e ace o he subs a e is used in he TCAD
simula ions p esen ed he e.
In addi ion o he geome ical de ini ion o he senso ,
doping p o iles and meshing pa ame e s can be inco po-
a ed o di e en pa s o he s uc u e. This has been
done o he epi axial laye , he collec ion implan , and
he p-well. Re inemen /e alua ion windows a e de ined o
place he co esponding doping p o iles. Analy ical doping
p o iles a e used o emula e he well s uc u es; he wells
a e o med wi h an e o unc ion dis ibu ion in dep h and
a Gaussian dis ibu ion la e ally. This emula es a dopan
di usion egion wi h an ex en o 0.3−0.4µm in he dep h
di ec ion, and an ex en o 0.4−0.5µm in he la e al di-
ec ion. Fla doping concen a ions a e used ac oss he
ull well s uc u es, which is a simpli ying assump ion, bu
deemed su icien o unde s anding he physical beha iou
o he signal o ma ion in a senso . In he in e ace egions
be ween he silicon bulk and he elec odes, i is necessa y
o add a highly doped egion o c ea e an Ohmic con ac .
Once doping egions and p o iles ha e been de ined, he
e inemen pa ame e s o he mesh a e es ablished. This
includes minimum and maximum mesh sizes, and he e-
inemen unc ion. A ine mesh p o ides mo e accu a e e-
sul s. Howe e , since he numbe o calcula ion nodes can
be subs an ial, he simula ions ake longe and can make
subsequen calcula ions in ac able wi h a ine mesh. This
can be add essed by using an uns uc u ed mesh ha is
e ined only in ce ain egions. In his wo k, he mesh e-
inemen is a unc ion o he doping g adien , meaning ha
he mesh will be ine in places whe e he e a e signi ican
changes in he doping concen a ion, e.g. a he edges o
he well s uc u es.
When he geome y has been buil and he mesh is de-
ined, he de ice simula ions can be pe o med wi h he
Sen au us De ice ool. Simula ions ha e been pe o med
in bo h quasis a iona y mode, o ob ain elec ic ields,
and in ansien mode o ob ain he signal esponse o
a cha ged pa icle a e sing he senso . The g id ile c e-
a ed using he S uc u e Edi o is impo ed in o Sen au us
De ice and he con ac s a e iden i ied. Physical p ope ies
and sol e p ope ies a e de ined, as well as he bound-
a y condi ions o he simula ion. The esul s o he qua-
sis a iona y simula ions a e ol age-dependen cu es and
se e al elec ical p ope ies wi hin he 3D olume. The
TCAD simula ions p esen ed he e we e pe o med wi h a
collec ion elec ode bias ol age o 1.2 V and a p-well and
subs a e bias ol age o −5 V, unless o he wise s a ed.
P ope ies s udied in his wo k a e he elec ic ield mag-
ni ude, la e al elec ic ield, cha ge cu en densi y, and
deple ed olume, shown in Sec ions 3.3,3.4, and 3.5. The
esul s o ansien simula ions a e ime-dependen cu es
and snapsho s o he pe u bed elec ical p ope ies wi hin
he 3D olume, shown in Sec ion 3.6 and compa ed o e-
sul s o Mon e Ca lo simula ions in Sec ion 5.4.
3.2. Subs a e di usion simula ion
As he subs a e is no expec ed o di ec ly con ibu e
o he elec ic ield in he senso , he TCAD simula ions
only include he epi axial laye and he in e ace egion
be ween he epi axial laye and he subs a e. The only
possible in luence s ems om he di usion o p-dopan s
om he highly-doped subs a e o he lowe -doped epi-
axial laye . Du ing he p ocess o semiconduc o ab i-
ca ion, a high di e ence in doping concen a ion and he
high empe a u es he de ice is exposed o is expec ed o
p oduce a signi ican di usion egion a he in e ace o
he epi axial laye and he subs a e. To simula e he di -
usion o dopan s om he subs a e o he epi axial laye ,
simula ions o a senso p oduc ion p ocess we e pe o med
using he Sen au us P ocess ool o he Synopsys TCAD
amewo k. The simula ion includes 10 minu es o a chem-
ical apou deposi ion (CVD) p ocess on he subs a e
wi h a empe a u e o 1050 ◦C, which esul s in 10 µm o
epi axial silicon [19]. The assumed doping concen a ion
o he subs a e in his simula ion is 1 ·1019 cm−3, and
3·1013 cm−3 o he epi axial laye [10]. All he implan ed
4
s uc u es need o go h ough an annealing p ocedu e o
elec onically ac i a e he implan ed ions. The simula ed
ac i a ion p ocess in ol es hea ing o he s uc u e o a
empe a u e o 1100 ◦C o 240 min. The esul ing s uc-
u e is con e ed o a one-dimensional doping p o ile o
he epi axial laye , which can be seen in Figu e 1. This
p o ile is hen impo ed o he epi axial laye in u he
simula ions.
Figu e 1: Simula ed di usion om he subs a e o he epi axial
laye . Red indica es a highe doping concen a ion, and blue a lowe
one (by app oxima ely six o de s o magni ude).
3.3. Impac o senso doping concen a ion
Doping concen a ion is an impo an pa ame e in he
design o silicon senso s, especially o he s uc u es ha
cons i u e he junc ions ha shape he elec ic ield inside
he senso . S udies conduc ed on changing he assumed
doping concen a ions on he p- and n-wells desc ibed in
Sec ion 2 o he s anda d layou by an o de o magni-
ude did no signi ican ly change he elec ic ield o ming
in he senso . Th ough he s udies, a alue o he p-well
doping concen a ion o 5 ·1015 cm−3was selec ed as a
baseline assump ion o s udies o he n-blanke layou ,
whe e he pn-junc ion is la ge and he impac p esumed
g ea e . S udies o he impac o al e ing he doping con-
cen a ions o bo h he n-blanke and he p-well we e pe -
o med using his layou , while keeping he epi axial laye
doping concen a ion ixed a 3 ·1013 cm−3. The limi al-
ues o he s udy we e selec ed as he minimum alue ha
would p oduce an e ec on he deple ed olume and he
maximum alue ha would s a o ha e an ad e se e ec
on he su ounding doping s uc u es (e.g. deple ion deep
in o he p-well). The e ec s o al e ing he doping concen-
a ions a e s udied by obse ing plo s o he elec ic ield
magni ude and he deple ed olume.
The doping concen a ion o he n-blanke was s ud-
ied wi h a ixed alue o he p-well concen a ion o
5·1015 cm−3, a ying he n-blanke concen a ion be ween
1·1014 cm−3and 4·1015 cm−3, shown in Figu e 2a and Fig-
u e 2c espec i ely. The igu es show a c oss-sec ion o a
pixel, wi h hal a collec ion elec ode in he uppe co ne s
and he p-well in he cen e o he image. In oduc ion
o a low-doped n- ype blanke implan c ea es a la ge pn-
junc ion in he senso , bu a bulbous shape o he deple ion
egion below he collec ion elec odes is s ill p esen . The
highly-doped n- ype blanke implan is no ully deple ed,
which leads o a conduc i e pa h being p esen be ween
he wo collec ion elec odes, and hus a non- unc ioning
senso . This can be obse ed by he shape o he deple ion
line. The doping concen a ion selec ed o he n-blanke
o use in he inal simula ions was 9·1014 cm−3, shown in
Figu e 2b, as i p o ides a good deple ion a he bo om o
he senso , and he leas deple ion in usion in he p-well.
(a) 1 ·1014 cm−3(b) 9 ·1014 cm−3(c) 4 ·1015 cm−3
Figu e 2: Di e en doping concen a ions o he n-blanke o a
10 µm pi ch senso in he n-blanke layou , ep esen ed by di e en
colou s. The colou scale co esponds o he o al doping concen a-
ion (wi h he highes p-doped egions being blue and he highes
n-doped egions ed), he b own line indica es he loca ion o a pn-
junc ion, and he whi e lines delimi he deple ed olume.
Wi h he alue o he n-blanke doping concen a ion
ixed o 9 ·1014 cm−3, he p-well doping concen a ion
was a ied. The s udy s a ed wi h he alue employed in
he n-blanke concen a ion s udy. Figu e 3displays he
elec ic ield magni ude a a close-up o he edge o he
p-well, wi h di e en es ed doping concen a ions. Fo
a doping concen a ion o 5 ·1015 cm−3, he e is a ela-
i ely la ge olume o he p-well ha has a non-ze o elec-
ic ield, which is undesi able as ha may in luence he
in-pixel elec onics con ained he e in a physical senso . A
highe doping concen a ion han he one in he p e ious
s udy should allow o a be e shielding o he p-well, bu
a oo high doping concen a ion p oduces a deepe s uc-
u e han wha is desi ed. Fu he mo e, Figu e 3b shows a
mo e uni o m elec ic ield ou side he p-well, when com-
pa ed o he simula ions using he uppe and lowe es ed
limi alues shown in Figu es 3a and 3c. The doping con-
cen a ion o 1 ·1016 cm−3was hus chosen as he alue
o he p-well o use in he inal simula ions.
3.4. Impac o senso geome y
Modi ica ions o he senso layou can ha e a signi i-
can impac on he s eng h and ex en o he elec ic ield
inside he senso , and on he deple ed olume. To in es i-
ga e his impac , s udies ha e been pe o med on he size
o he p-well opening, which co esponds o he dis ance
5
(a) 5 ·1015 cm−3(b) 1 ·1016 cm−3(c) 5 ·1016 cm−3
Figu e 3: Elec ic ield magni ude o h ee di e en doping concen-
a ions o he p-well, o a 10 µm pi ch senso . Close-up o he co ne
o he p-well. The b own line indica es he loca ion o a pn-junc ion
and he whi e line delimi s he deple ed olume. The elec ic ield
magni ude is gi en by he colou scale, whe e a da k blue colou in-
dica es an elec ic ield magni ude o ze o.
be ween he edge o he collec ion implan and he edge o
he p-well, and on he gap size in he n-gap layou . These
pu ely geome ical ea u es a e de ined by he mask de-
sign used in senso p oduc ion. The e ec s a e s udied by
obse ing plo s o he elec ic ield magni ude, he la e al
elec ic ield s eng h, and he deple ed olume. In he ig-
u es, he o me wo a e ep esen ed in colou scale, while
he la e one is delimi ed by a whi e line.
3.4.1. P-well opening
The p-well opening ex en was a ied om 1 µm o 4 µm,
a a pixel size o 20 ×20 µm2. I was obse ed ha in-
c easing he p-well opening c ea es a s onge la e al elec-
ic ield and inc eases he deple ed olume in he s anda d
layou , on he o de o µm in bo h wid h and dep h. A
la ge deple ed olume allows o mo e cha ge collec ion
h ough d i , while a s onge la e al elec ic ield p o ides
a highe d i eloci y o he ee cha ge ca ie s p oduced
in he edges o he pixels. Howe e , inc easing he p-well
opening means dec easing he p-well size and hence he
a ailable space o on -end elec onics. The s udy was
also pe o med using he n-blanke layou . He e, i was
obse ed ha he inc ease in la e al elec ic ield s eng h
and deple ed olume was no as signi ican as in he s an-
da d layou case. A la ge p-well opening he e leads o a
la ge undeple ed egion a ound he collec ion elec ode,
howe e , which has a nega i e impac on he senso capac-
i ance and cha ge collec ion beha iou .
A compa ison be ween he la e al elec ic ield and de-
ple ion bounda y o p-well openings o 1 µm and 4 µm
is shown in Figu e 4 o bo h he s anda d and he n-
blanke layou s. The egion wi h a s ong la e al elec ic
ield isibly inc eases in bo h layou s as he p-well open-
ing inc eases. A la ge p-well opening is expec ed o also
di ec ly a ec he senso capaci ance, bu s udies o his
ha e no been ca ied ou . An opening size o 2 µm is se-
lec ed o use in u he s udies, as a balance be ween he
inc eased deple ed egion and he o al p-well size.
3.4.2. Gap size in he n-gap layou
A la e al elec ic ield is obse ed o appea unde he
p-well once he e ical junc ions o he n-gap layou a e
added o he senso , as can be seen in Figu e 5. The gap
(a) S anda d layou , 1 µm
opening
(b) S anda d layou , 4 µm
opening
(c) N-blanke layou , 1 µm
opening
(d) N-blanke layou , 4 µm
opening
Figu e 4: La e al elec ic ield o wo p-well openings o 20 µm
pi ch senso s in he s anda d and n-blanke layou s. The b own
line indica es he loca ion o a pn-junc ion and he whi e line he
bounda ies o he deple ed olume.
size was a ied om 1 µm o 4 µm, and i was ound ha
he s eng h o he la e al elec ic ield is inc eased wi h
inc easing gap size, as he wo e ical junc ions mo e u -
he apa . When he junc ions a e close, he egions o
dopan di usion will o e lap, leading o a smalle la e al
ield. As can be seen in Figu e 5 he la e al ield di ec ions
o he wo e ical junc ions a e opposi e, implying ha
hey cancel ou in he cen e when he dis ance is small,
so a su icien ly la ge gap size he la e al ield s eng h
eaches a maximum. Howe e , when he gap is inc eased,
he e ical pn-junc ion as well as he la e al elec ic ields
a e shi ed away om he pixel edges, hus educing hei
use ulness in imp o ing cha ge collec ion a om he col-
lec ion elec ode and lea ing an elec ic ield minimum in
he gap. A gap size o 2.5 µm is su icien ly la ge o max-
imise he la e al ield s eng h, while keeping he junc ion
close o he pixel edge.
3.5. Hexagonal pixel geome y simula ion
De ailed in es iga ions o hexagonal pixel designs e-
qui e cus om ield maps om TCAD o hexagonal geome-
ies. One ull pixel cell wi h he collec ion elec ode in he
cen e is used in hese simula ions, and he p-well and sub-
s a e a e biased wi h a ol age o −1.2 V. In Figu e 6, a
pixel cell o a simula ion o a senso in he s anda d lay-
ou is shown, wi h he colou indica ing he doping le el.
The plane indica ed as C1 ep esen s a c oss-sec ion, along
which he elec ic ield magni ude is shown in Figu e 7 o
bo h he s anda d and n-gap layou s. The egions s ick-
ing ou om he op o he senso a e he me al biasing
con ac s o he collec ion elec ode and he p-well.
6
(a) 1 µm gap (b) 2.5 µm gap
(c) 4 µm gap
Figu e 5: La e al elec ic ield o h ee n-gap sizes o a 20 µm pi ch
senso in he n-gap layou . The b own line indica es he loca ion o
he pn-junc ion and he whi e line delimi s he deple ed olume.
I can be seen ha he deple ion egion is small o
he s anda d layou , only ex ending below he opening
be ween he collec ion elec ode and he p-well. Fo he
n-gap layou , howe e , i ex ends ac oss he ull pixel.
3.6. T ansien simula ions
T ansien simula ions we e pe o med o es ima e he
shape, ampli ude, and du a ion o a signal gene a ed by
a minimum ionising pa icle a e sing he senso . Fo
hese simula ions, he p-well and subs a e a e biased wi h
a ol age o −1.4 V. The pa icle a e sing he senso is
ep esen ed by linea cha ge deposi ion along he pa icle
ack wi h Gaussian la e al smea ing o 0.5 µm using he
“Hea y Ion” cha ge deposi ion model. The mesh is ad-
jus ed o he ansien simula ions o ha e a ine cell size
a ound he a eas wi h a doping concen a ion g adien and
he ack o he a e sing pa icle. Fo he hin senso s
used in he simula ions, a deposi ion o 63 elec on-hole
pai s pe mic ome e is assumed [20]. A ma ix o 3 ×3
pixels is simula ed in 3D, in o de o a oid edge e ec s.
The ime s ep o he ansien simula ion is adap ed o
he expec ed shape o he signal. Two inciden posi ions
we e simula ed; in he cen e and in he co ne o a squa e
pixel, o he h ee di e en layou s. The simula ed ack
o he a e sing pa icle is pe pendicula ly inciden on he
senso o all he simula ions.
The absolu e elec on cu en densi y o he s anda d
layou o he incidence o a MIP in he cen e o he pixel
is shown in Figu e 8. Th ee adjacen pixels a e shown.
The deple ion olume in he s anda d layou is limi ed,
and he layou allows o signi ican cha ge sha ing due o
di usion. The n-blanke and n-gap layou s we e de eloped
in o de o imp o e he cha ge collec ion e iciency o he
Figu e 6: Simula ed hexagonal pixel cell in TCAD. The colou s co -
espond o doping le el, and he plane ma ked C1 is a cu o display
pu poses. The elec ic ield magni ude o his cu is shown in Fig-
u e 7.
senso in inciden posi ions u he om he eadou im-
plan [13,14]. In Figu e 9 he signals o he cen e and
co ne inciden posi ions o he n-gap layou a e shown.
The du a ion o he signal is dependen on he MIP in-
ciden posi ion; he as e cha ge collec ion is obse ed in
he cen e o he pixel, due o he immedia e p oximi y
o he eadou implan . The cha ges a e deposi ed 0.5 ns
a e he s a o he simula ion, leading o he ising edge
o he pulse.
Figu e 10 shows he signals o co ne inciden posi ions,
o all h ee layou s. The n-blanke has a la ge deple ion
egion compa ed o he s anda d layou , and he n-gap has
an addi ional a ea wi h a s onge la e al elec ic ield com-
ponen , which imp o es he cha ge collec ion a om he
pixel cen e. As he s anda d layou is undeple ed in he
pixel co ne s, cha ges o med he e mo e slowly by di u-
sion, and he cha ge collec ion hus akes a compa a i ely
long ime.
3.7. Gene a ing weigh ing po en ials
T ansien simula ions using TCAD a e compu a ionally
in ensi e, and i can hus be bene icial o pe o m an-
sien simula ions using e.g. Allpix2ins ead. A weigh ing
po en ial is equi ed o be able o pe o m ansien sim-
ula ions using he Shockley-Ramo heo em [21,22]. The
po en ial can be calcula ed by aking he di e ence o he
elec os a ic po en ials a ising om applying wo sligh ly
di e en bias ol ages o one collec ion elec ode in a sen-
so , keeping he o he collec ion elec odes a a cons an
bias ol age. The wo equi ed elec os a ic po en ials can
be simula ed using TCAD, using he same senso geom-
e y wi h a di e ence o 0.01 V in he bias ol age o a
single collec ion elec ode. By calcula ing he di e ence
be ween he wo po en ials in each mesh poin , and di id-
ing he di e ence by he di e ence in collec ion elec ode
bias ol age, he weigh ing po en ial is acqui ed. Be o e
7
(a) S anda d layou
(b) N-gap layou
Figu e 7: Elec ic ield magni ude as ou pu om a TCAD simula ion
o a hexagonal pixel in he s anda d and n-gap layou s. The whi e
lines deno e deple ion bounda ies, and he colou scale deno es he
magni ude o he elec ic ield.
Figu e 8: Absolu e elec on cu en densi y. Th ee adjacen pixels
a e shown, and he inciden posi ion is in he cen e o he middle
pixel eadou implan . S anda d layou . The dashed lines indica e
pixel edges.
u ilising his weigh ing po en ial o simula ions, he al-
ues should be cons ained o be be ween 0 and 1, as his
is he physical ange o a weigh ing po en ial. La ge and
smalle alues may occu in he calcula ion due o nume -
ical e o s.
4. Mon e Ca lo simula ions
Simula ion o senso esponse o inciden pa icles can
be pe o med using TCAD, bu s udies wi h high s a is i-
cal signi icance aking s ochas ic luc ua ions in o accoun
a e no easible due o he long simula ion ime equi ed
pe pa icle hi . By combining he doping concen a ions,
elec ic ields, and weigh ing po en ials gene a ed using
TCAD wi h he Allpix2Mon e Ca lo simula ion ame-
wo k howe e , high-s a is ics simula ions can be ca ied
ou [9,23]. This sec ion demons a es how such simu-
la ions can be pe o med o he monoli hic senso s de-
sc ibed ea lie , using Allpix2 e sion 3.0 [24].
4.1. Simula ion low
Allpix2is buil on he concep o exchangeable modules,
making i possible o lexibly change simula ion aspec s
such as pa icle sou ce and cha ge p opaga ion me hod.
0 0.5 1 1.5 2 2.5 3
Time [ns]
0
100
200
300
400
500
600
Cu en [nA]
TCAD simula ion
Pixel cen e incidence
Pixel co ne incidence
, n-gap layou
2
mµSqua e pixels, 20x20
Figu e 9: Signal in he n-gap layou senso in he cen e and co ne
inciden posi ions.
0 5 10 15 20 25
Time [ns]
0
5
10
15
20
25
30
35
40
Cu en [nA]
TCAD simula ion
S anda d layou
N-blanke layou
N-gap layou
, co ne incidence
2
mµSqua e pixels, 20x20
Figu e 10: Signal o he senso wi h he inciden posi ion in he
co ne o he pixel, o he s anda d,n-blanke ,n-gap layou s.
The modules also cons i u e di e en s eps aken in he
simula ion p ocess, and pa ame e s o each module can be
con olled by con igu a ion iles wi h keywo d- alue pai s.
When p o iding alues o keywo ds ha ep esen physical
quan i ies in module con igu a ions, i is impo an o also
p o ide a uni in o de o a oid unexpec ed beha iou .
4.2. Senso geome y and se up
A de ec o model in Allpix2is de ined in a con igu a-
ion ile, and an example can be seen in Lis ing 4.1. The
example shows a monoli hic senso assembly wi h squa e
pixels, wi h a pixel size o 20 ×20 µm2and a o al senso
hickness o 50 µm. The senso excess consis s o senso
ma e ial wi hou pixels, and is an impo an pa ame e o
keep in mind when calcula ing senso e iciency om simu-
la ion esul s, as pa icles hi ing he senso excess should
no be coun ed as pa icles ha should p oduce a signal
in he senso .
8
ype = monoli hic
geome y = pixel
senso _ma e ial = silicon
numbe _o _pixels = 20 20
pixel_size = 20um 20um
senso _ hickness = 50um
senso _excess_ igh = 200um
[implan ]
ype = on side
shape = ec angle
size = 2.2um 2.2um 0.8um
Lis ing 4.1: De ec o model con igu a ion example.
The geome y pa ame e can be used o selec ec angu-
la , adial s ip, o hexagonal pixel geome ies. Fo hexag-
onal geome ies, he pixel size is de ined om co ne o
co ne along axes 60◦apa , i.e. he maximum dis ance
ac oss a hexagon.
The [implan ] sec ion de ines he x-, y-, and z-ex en
o he collec ion elec ode o he senso . In Allpix2, his
de ines he olume in which cha ge p opaga ion s ops, and
cha ges a e coun ed as “collec ed”. A small di e ence in
his pa ame e can ha e a sizeable e ec on he inal e-
sul s.
A geome y con igu a ion ile de ines he ull simula ed
geome y, and can con ain se e al senso s and passi e ol-
umes. An example con igu a ion o a single-senso simu-
la ion is shown in Lis ing 4.2. Fo each senso o passi e
[du ]
ype ="de ec o Model"
posi ion = 0mm 0mm 0mm
o ien a ion = 0deg 0deg 0deg
Lis ing 4.2: Geome y con igu a ion example, o a single senso
wi hou andom misalignmen .
olume, a posi ion and o ien a ion has o be de ined, along
wi h an alignmen p ecision. In he gi en example, he
name o he de ec o is “du ”, loca ed a he cen e o he
global coo dina e sys em. The alignmen p ecision is mo e
impo an o include when se e al senso s a e in ol ed.
The de ec o ype in he example is de ec o Model, which
is he name o a de ec o model con igu a ion ile such as
he example shown in Lis ing 4.1.
The global coo dina e sys em is de ined by he simula ed
wo ld olume, and posi ions o componen s a e de ined in
his sys em. Each de ec o placed in he wo ld has a local
coo dina e sys em, wi h an o igin de ined in he cen e o
he lowe le pixel o he senso pixel ma ix.
4.2.1. Cons uc ing he geome y o use wi h Gean 4
The [Geome yBuilde Gean 4] module cons uc s he
geome y o use wi h Gean 4 [25,26,27], which allows
o de ailed pa icle in e ac ion simula ions. To isu-
alise he geome y cons uc ed by he module, he mod-
ule [Visualiza ionGean 4] can be used. This opens a
Gean 4 g aphical use in e ace window, wi h he possi-
bili y o also s a ing a Gean 4 e minal, gi ing access o
bo h a isualisa ion o he se up and Gean 4 commands.
By using he / un/beamOn Gean 4 command, i is possi-
ble o see whe e pa icles om a de ined sou ce will hi
he se up. This is use ul o making su e ha he sou ce
is aligned wi h he de ec o s in he desi ed way, bu he
command canno be used o pe o m a p ope simula ion.
4.3. Impo ing esul s om TCAD simula ions
Elec ic ields and doping concen a ions can be im-
po ed om TCAD simula ions, which gi es access o mo e
de ailed ields han he buil -in pa ame ic models. To be
usable in Allpix2 he TCAD mesh has o be adap ed in o
a egula ly-spaced g id, and his p ocess is pe o med us-
ing he Mesh con e e ool. The ool ei he pe o ms
a ba ycen ic in e pola ion o alues o each poin in he
new egula ly-spaced g id o uses he alue o he closes
TCAD mesh poin wi hou in e pola ion. The second case
is pa icula ly use ul o con e sion o la ge ield maps.
An example o a con igu a ion ile o he mesh con-
e e is shown in Lis ing 4.3. This ile is used o con e -
ing he elec ic ield o he egion named “epi axial” o a
20 ×20 µm2pixel. The model keywo d de ines he ou pu
o ma , and he uni s o he con e ed obse able ha e o
be p o ided; o elec ic ields, i is ypically V/cm, and
o doping concen a ions cm−3(w i en /cm/cm/cm in he
Allpix2con igu a ion iles).
model ="APF"
egion ="epi axial"
obse able = Elec icField
obse able_uni s ="V/cm"
di isions = 300 300 100
xyz =xy-z
Lis ing 4.3: Mesh con e e con igu a ion example, o he elec ic
ield in he “epi axial” egion o a senso .
The di isions pa ame e de ines he numbe o poin s
used in he egula ly-spaced g id, and hus he g anula i y
o he ield map when impo ed in o he amewo k. The
numbe o g id poin s used can ha e a signi ican impac
9
0 5 10 15 20 25 30 35 40 45 50
m]µ x [
0
5
10
15
20
25
30
35
40
45
50
m]µ y [
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
E iciency, 4 pixels, 200 elec on h eshold
(a) S anda d layou
0 5 10 15 20 25 30 35 40 45 50
m]µ x [
0
5
10
15
20
25
30
35
40
45
50
m]µ y [
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
E iciency, 4 pixels, 200 elec on h eshold
(b) N-gap layou
Figu e 18: In-pixel e iciency o ou adjacen pixels wi h a pixel
size o 25 ×25 µm2, a a bias ol age o −4.8 V and a h eshold o
200 elec ons.
0 100 200 300 400 500 600 700
Th eshold [e]
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
Clus e size
S anda d layou
N-blanke layou
N-gap layou
Mean clus e size s h eshold
Figu e 19: Th eshold dependency o he mean clus e size, a a bias
ol age o −4.8 V o a senso wi h a 25 ×25 µm2pixel size.
pixels. The clus e size o he n-gap layou is signi ican ly
lowe han o he o he wo layou s o e all es ed h esh-
old alues, as i has a smalle amoun o cha ge sha ing
be ween pixels. The educ ion in clus e size as he h esh-
old inc eases o he s anda d layou is a ec ed by he loss
o e iciency in his layou a highe h esholds. The e -
iciency educ ion is la ges a pixel co ne s and edges,
which a e he main pa icle incidence a eas ha lead o a
highe clus e size, as can be seen in Figu e 17.
The dependence o he de ec ion e iciency on he
h eshold alue is shown in Figu e 20, whe e he mean
e iciency alue o he senso is plo ed o he h ee lay-
ou s. I can be obse ed ha he n-blanke and n-gap
layou s main ain e iciency o e a la ge h eshold ange
han he s anda d layou , which is consis en wi h wha
is shown in Figu e 18. The n-gap layou hus enables he
la ges e icien ope a ing ma gin. This end is consis en
o all es ed bias ol ages. Fo a pixel size o 15 ×15 µm2
he e icien h eshold ange is la ge o he s anda d and
n-blanke layou s, compa ed o he la ge pixel size o
25 ×25 µm2. This can be seen in Figu e 21, whe e e-
0 100 200 300 400 500 600 700
Th eshold [e]
0
20
40
60
80
100
E iciency [%]
Mean e iciency s h eshold
S anda d layou
N-blanke layou
N-gap layou
Mean e iciency s h eshold
Figu e 20: Th eshold dependency o he de ec ion e iciency, a a
bias ol age o −4.8 V o a senso wi h a 25 ×25 µm2pixel size.
sul s a e shown o a bias ol age o −1.2 V. Fo he n-gap
layou , he e icien h eshold ange is sligh ly smalle han
o he la ge pixel size a his bias ol age. This is due o
a educed e iciency in he gap in he n-blanke be ween
pixels; in a smalle pixel his egion akes up a la ge ac-
ional olume, as he gap size emains he same. The end
be ween he di e en layou s emains he same, wi h he
n-blanke layou mo e e icien han he s anda d layou ,
and he n-gap layou mo e e icien han he n-blanke lay-
ou .
0 100 200 300 400 500 600 700
Th eshold [e]
0
20
40
60
80
100
E iciency [%]
Mean e iciency s h eshold
mµS anda d, 25x25 mµN-blanke , 25x25
mµN-gap, 25x25 mµS anda d, 15x15 mµN-blanke , 15x15
mµN-gap, 15x15
Mean e iciency s h eshold
Figu e 21: Th eshold dependency o he de ec ion e iciency, o sen-
so s wi h wo di e en pixel sizes a a bias ol age o −1.2 V.
Figu e 22 shows he mean esolu ion o he senso in he
x-di ec ion e sus he h eshold, o he same simula ion
se up as be o e. As he pixels a e squa e and symme ic,
he esolu ion is iden ical in he y-di ec ion. The eso-
lu ion is de ined as he oo mean squa e o he cen al
3σ(99.73%) o he esidual dis ibu ion, i.e. he dis i-
bu ion o he di e ence o econs uc ed pa icle posi ion
and Mon e Ca lo u h posi ion o each e en . The e-
16
cons uc ed posi ion is aken as a cha ge-weigh ed mean
posi ion o all pixel hi s in a clus e . In hese simula ions,
he ull cha ge in o ma ion is used, a he han he alue
om a cha ge- o-digi al con e e wi h limi ed esolu ion.
0 100 200 300 400 500 600 700
Th eshold [e]
1
2
3
4
5
6
7
m]µResolu ion in x [
Residual RMS in x s h eshold
S anda d layou
N-blanke layou
N-gap layou
Residual RMS in x s h eshold
Figu e 22: Th eshold dependency o he spa ial esolu ion, a a bias
ol age o −4.8 V o a senso wi h a 25 ×25 µm2pixel size.
The esolu ion de e io a es as he h eshold inc eases,
and o a la ge ange o h eshold alues he s anda d lay-
ou has he bes (lowes ) esolu ion. A low h esholds,
his is due o he la ge amoun o cha ge sha ing com-
pa ed o he o he wo layou s. The econs uc ed posi-
ion is mo e accu a e when he clus e size is la ge , due
o he cha ge-weigh ed posi ion econs uc ion occu ing
be ween mo e pixels. A high h esholds, he esolu ion
o he s anda d layou dec eases as h eshold inc eases.
This is an e ec o he educ ion o he e iciency, as can
be seen in Figu es 18 and 21. As e iciency is educed a
pixel edges when he h eshold inc eases, only pa icle hi s
close o he pixel cen e can be econs uc ed, and hus he
e ec i e pixel size is educed.
The smalle clus e sizes o he n-blanke and n-gap lay-
ou s de e io a e hei esolu ions, bu as can be seen in
Figu es 20 and 18, hei e iciency is imp o ed. The n-gap
layou has he highes e iciency, bu he la ges in insic
esolu ion.
5.4. T ansien pulse s udies
Pe o ming ansien simula ions as desc ibed in Sec-
ion 4.8 can conside ably educe simula ion ime in com-
pa ison o ansien simula ions using TCAD [8]. Using
Mon e Ca lo simula ions also allows inclusion o s ochas ic
e ec s, such as Landau luc ua ions and seconda y pa i-
cles. A alida ion be ween bo h app oaches has been pe -
o med, u ilising he same pa ame e s and se up, o ensu e
ha using Allpix2mimics he esul s o TCAD ansien
simula ions as desc ibed in Sec ion 3.6. In hese alida ion
s udies, only an epi axial laye wi h a hickness o 10 µm
was simula ed. The mobili y model pa ame e alues in
Allpix2we e changed o ma ch he ex ended Canali mo-
bili y model used in TCAD [37].
The simula ed geome y consis ed o a ma ix o
3×3 pixels, wi h a pixel size o 20 ×20 µm2. Cha ges
we e injec ed along a s aigh line in he co ne be ween
ou pixels using he [Deposi ionPoin Cha ge] module,
wi h 63 elec on-hole pai s deposi ed pe µm. Elec ic
ields, doping concen a ions, and weigh ing po en ials
om TCAD we e impo ed in o Allpix2using hei e-
spec i e module eade s.
Using he [T ansien P opaga ion] module as de-
sc ibed in Sec ion 4.8, an in eg a ion ime o 40 ns was
used o all simula ions. A coa se alue o he imes ep
pa ame e may lead o smalle pulses han expec ed, so a
imes ep o 15 ps was used in he p esen ed esul s. As
he cha ge is injec ed in he co ne , pulses a e expec ed o
be induced in all ou pixels sha ing he co ne . The o al
pulses we e calcula ed as he a e age o he induced pulses
in he ou collec ion elec odes o each e en .
Figu e 23 shows he esul ing pulses o bo h TCAD and
Allpix2simula ions, o he s anda d and n-blanke lay-
ou s. The Allpix2pulses a e he a e age o 10 000 e en s,
whe eas he TCAD pulses come om single e en s. The
plo s show ha he wo me hods ag ee in e ms o pulse
heigh and peaking ime, which indica es ha he Allpix2
me hod la gely yields compa ible esul s wi h he TCAD
me hod. A he alling edge o he pulses, he e is a small
di e ence be ween he app oaches, howe e . The peak
s uc u e in he TCAD pulse o he n-blanke layou be-
ween 0 and 1 ns is an a e ac o he TCAD simula ions
om he ini ial cha ge deposi ion, and i s in eg al is ze o
and does no a ec he es o he pulse.
A no iceable di e ence in he pulse ise ime and du a-
ion is p esen be ween he shown s anda d and n-blanke
layou s, wi h he n-blanke pulse being as e , which is ex-
pec ed due o he la ge deple ed egion and hus mo e
cha ge collec ion by d i . This also inc eases he cha ge
collec ion e iciency pe pixel and esul s in a highe peak
and highe in eg a ed cha ge o he n-blanke layou .
5.5. Mul i-senso s udies
Se e al senso s can be simula ed simul aneously in
Allpix2, in o example a beam elescope se up. By us-
ing he [Co y eckanW i e ] module, he esul s o he
Allpix2simula ion can be expo ed in a o ma sui able o
he Co y eckan es beam econs uc ion amewo k [38].
This amewo k can hen be used o ex ac pa ame e s
such as elescope esolu ion a di e en posi ions o he
se up. The simula ion o mul i-senso se ups enables s ud-
ies o he acking pe o mance o di e en se ups and sen-
so designs, and cons uc ion o a beam elescope ep e-
sen s a possible use case o he senso s desc ibed in his
wo k.
Simula ions we e ca ied ou wi h a six-plane beam ele-
scope su ounded by ai , using senso s in he h ee di e -
en layou s wi h a pixel size o 20 ×20 µm2. A beam o
17
0 5 10 15 20 25 30 35 40
Time [ns]
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Cu en [nA]
TCAD simula ion
simula ion
2
Allpix
(a) S anda d layou
0 5 10 15 20 25 30 35 40
Time [ns]
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Cu en [nA]
TCAD simula ion
simula ion
2
Allpix
(b) N-blanke layou
Figu e 23: Compa ison be ween pulses ob ained wi h TCAD (blue)
and Allpix2using TCAD ields ( ed). A TCAD pulse co esponds
o a single e en , while he Allpix2pulse is he a e age o 10 000
e en s.
elec ons was i ed a he se up, wi h a single elec on pe
e en . The dis ance be ween elescope planes was a ied,
and he spa ial esolu ion a he de ice-unde - es (DUT)
posi ion (in he middle o he se up) ex ac ed. The es-
olu ion was de e mined om he dis ibu ion o he di -
e ence o ack in e cep loca ions and he ue pa i-
cle posi ions a he DUT, whe e he acks we e econ-
s uc ed using he elescope plane hi s and he gene al
b oken lines me hod [39]. Figu e 24a shows he esul -
ing elescope acking esolu ion a he DUT posi ion o
di e en dis ances be ween elescope planes, o he h ee
senso layou s. The dis ance is he same be ween any wo
adjacen planes, and he p esen ed esul s a e a a h esh-
old o 200 elec ons o each o he six senso s.
An es ima e o he elescope acking e iciency is shown
in Figu e 24b. The calcula ion is pe o med by di iding
he numbe o econs uc ed acks by he o al numbe
0 20 40 60 80 100 120 140 160
dz [mm]
1
2
3
4
5
6
m]µ
Telescope esolu ion in x [
N-gap layou
N-blanke layou
S anda d layou
(a) Resolu ion a he DUT posi ion
0 20 40 60 80 100 120 140 160
dz [mm]
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
T acking e iciency
N-gap layou
N-blanke layou
S anda d layou
(b) T acking e iciency es ima e
Figu e 24: Telescope esolu ion a he DUT posi ion, and acking
e iciency es ima e, as a unc ion o he dis ance be ween elescope
planes o he h ee di e en layou s a a h eshold o 200 elec ons.
o simula ed e en s o each da a poin . A leas i e o
he six elescope planes ha e o egis e hi s o an e en
o be conside ed o ack econs uc ion, and no all such
e en s will ha e a ack success ully econs uc ed due o
sca e ing leading o hi s ou side o he spa ial cu . A
spa ial cu equal o he pixel size (20 µm) is used in he
inal ack econs uc ion.
F om he elescope acking esolu ion, i can be seen
ha he s anda d layou p o ides he smalles esolu ion,
while he esolu ion o he n-blanke and n-gap layou s is
sligh ly la ge . This ag ees quali a i ely wi h he esul s
shown o single-senso s in Figu e 22, bu he di e ence
is smalle han indica ed he e. As he dis ance be ween
elescope planes inc eases, he acking esolu ion de e io-
a es. Bo h o hese e ec s ag ee well wi h expec a ions;
as mul iple senso s a e used, he esolu ion o he ull sys-
em is be e han ha o an indi idual senso , and when
he dis ance be ween planes inc eases he sca e ing in ai
inc eases along wi h he unce ain y in de lec ion angle.
The elescope acking e iciency es ima ion quali a i ely
18
ag ees wi h he single-senso esul s shown in Figu e 20; a
a h eshold o 200 elec ons, he acking e iciency is low
o he s anda d layou due o he low e iciency o each in-
di idual senso . The n-blanke layou shows a signi ican ly
highe acking e iciency, and he n-gap layou is he mos
e icien . The acking e iciency es ima e has a weak de-
pendence on he dis ance be ween elescope planes, wi h a
dec ease due o he inc eased sca e ing in ai as he dis-
ance be ween planes inc eases. In conclusion, he spa ial
esolu ion o a beam elescope consis ing o he in es i-
ga ed senso s is compa able o ha o he EUDET- ype
beam elescopes [40]. While he acking e iciency is low
o he s anda d layou , i can be imp o ed wi hou sig-
ni ican loss o acking esolu ion by u ilising one o he
o he layou s.
6. Compa isons o da a and p e ious simula ions
Compa isons o he ou lined simula ion p ocedu e o
p e iously published da a a e pe o med, using esul s
om a es beam ca ied ou in he amewo k o he
CLICdp Collabo a ion on he Towe Jazz In es iga o 1
senso , wi h a pixel size o 28 ×28 µm2[7]. This sen-
so is designed in he s anda d layou in a 180 nm CMOS
imaging p ocess wi h an epi axial laye hickness o ap-
p oxima ely 25 µm, and he s udies a e made a a bias
ol age o −6 V. The senso in es iga ed he e is hus di -
e en om wha was p e iously used as an example in
de eloping he simula ion p ocedu e, demons a ing he
e sa ili y o he app oach.
Figu e 25 shows compa isons be ween da a aken wi h
he senso and esul s using he simula ion p ocedu e ou -
lined in his pape . Figu e 25a shows he clus e cha ge
a a h eshold o 120 elec ons, and Figu e 25b shows he
clus e size e sus h eshold.
In he igu es, da a a e shown in blue and he esul s
o simula ions using he gene alised p ocedu e ou lined in
his pape a e shown in ed. Figu e 25a indica es ha he
simula ion esul clus e cha ge is shi ed sligh ly highe
compa ed o he da a, while he ising and alling slopes o
he dis ibu ions ma ch in shape. A i is pe o med using
a con olu ion o a Gaussian and Landau unc ion, which
gi es a mos p obable clus e cha ge alue o 1.47 kiloelec-
ons o he simula ion esul s. Fo he da a, he alue is
1.42 kiloelec ons [7]. The wid h o he Gaussian pa is
0.22 kiloelec ons in he simula ion esul s, and 0.21 kilo-
elec ons in he da a.
In Figu e 25b he simula ions and da a ma ch ac oss he
ull in es iga ed h eshold ange, wi h a sligh de ia ion a
h esholds smalle han 300 elec ons. The e o s shown
a e pu ely s a is ical o bo h da a and simula ions, and
he maximum de ia ion be ween he da a and simula ions
is 4%, a a h eshold o 120 elec ons.
Compa a i e s udies ha e also been ca ied ou in he
ame o he Tange ine p ojec , using es beam da a o
senso s in a 65 nm CMOS imaging p ocess [5]. These
0 500 1000 1500 2000 2500 3000 3500 4000
Clus e cha ge [e]
0
0.005
0.01
0.015
0.02
0.025
0.03
E en s (no m.)
Clus e cha ge
Da a
Simula ions
(a) Clus e cha ge dis ibu ion a a h eshold o 120 elec ons
0 100 200 300 400 500 600 700
Th eshold [e]
1
1.5
2
2.5
3
3.5
Mean clus e size
Mean clus e size
Da a
Simula ions
(b) Th eshold dependency o he clus e size
Figu e 25: Compa ison be ween simula ion esul s ob ained using
he me hod desc ibed in his pape and es beam da a [7].
s udies show an ag eemen be ween da a and simula ions
wi hin 1% o he n-gap layou .
In conclusion, he simula ions using he me hod p e-
sen ed in his pape ma ch da a well. The e is a maxi-
mum de ia ion o app oxima ely 4% in bo h he cha ge
dis ibu ion and he clus e size. The quali a i e ends
ag ee and o a le el su icien o d aw conclusions conce n-
ing senso pe o mance and i s o igins wi hou use o any
p op ie a y in o ma ion. The esul s a e also compa ible
wi h simula ions ca ied ou a CERN using mo e ealis ic
ields om TCAD, which ha e been compa ed o he same
da a [7].
7. Summa y and ou look
In his pape , a simula ion p ocedu e o silicon sen-
so s wi h complex non-uni o m elec ic ields has been
desc ibed, s a ing om i s p inciples o a simple pn-
junc ion, and going o high-s a is ics simula ions o a
mul i-senso beam elescope. Th ee-dimensional elec o-
s a ic TCAD simula ions we e p oduced, based on gene ic
doping p o iles and i s p inciples o senso ope a ion,
19
wi hou knowledge o p op ie a y in o ma ion. S udies
o he impac o a ying di e en senso pa ame e s ha e
been ca ied ou , obse ing hei impac on he elec ic
ields. Th ee di e en senso layou s ha e been es ed and
compa ed, in se e al di e en pixel sizes o bo h ec angu-
la and hexagonal pixel geome ies. The geome ies used
only desc ibe he la ge- ea u e geome y o he senso s,
and do no a emp o mimic he in icacies o a CMOS
imaging p ocess, bu hey a e su icien o modelling a
signal esponse desc ibing obse ed senso beha iou o
an accu acy wi hin a ew pe cen o key obse ables.
By impo ing he TCAD ields and doping p o iles
in o Allpix2, as and comple e simula ions o pa icle in-
e ac ions and cha ge anspo can be pe o med, ak-
ing s ochas ic luc ua ions s emming om he unde lying
physics p ocesses in o accoun . Th ough his p ocess, sen-
so pe o mance obse ables such as e iciency, clus e size,
and esolu ion can be ex ac ed. Example esul s o such
simula ions ha e been p esen ed, and ag ee well wi h ex-
pec a ions and s udies o simila senso s.
T ansien simula ions ha e been ca ied ou in bo h
TCAD and Allpix2, and he esul s ma ch well. Using
he induced cha ge gi en by ansien simula ions is mo e
accu a e han using he no ion o “collec ed cha ge”, and
when cha ge pulses a e a ailable mo e sophis ica ed digi-
isa ion simula ion can be pe o med o ex ac accu a e
alues o ime-o -a i al and ime-o e - h eshold. This
wo k is o eseen o con inue in he nea u u e, also in-
cluding mo e accu a e simula ion o he senso on -end
esponse.
The desc ibed simula ion p ocedu e is applicable in mul-
iple di e en cases, and cons i u es a gene ic oolbox o
pe o ming simila s udies wi hou using p op ie a y in-
o ma ion. These simula ions a e able o p o ide accu a e
p edic ions o senso beha iou and ade-o s wi h di e -
en designs, and can hus be used o in o m decisions aken
o u u e senso designs.
Acknowledgemen s
The p esen ed simula ion s udies ha e been pe o med
in he ame o he Tange ine p ojec , and in collabo a ion
wi h he CERN EP R&D p og amme.
CRediT au ho ship s a emen
Dominik Dannheim: Concep ualisa ion, Resou ces,
W i ing - Re iew & Edi ing. Manuel Del Rio Vie a:
Fo mal analysis, In es iga ion, W i ing - O iginal D a ,
Visualisa ion. Ka ha ina Do : Me hodology, Re-
sou ces, W i ing - Re iew & Edi ing. Do is Ecks ein:
Resou ces, Funding acquisi ion. Finn Feind : W i ing -
Re iew & Edi ing. Ing id-Ma ia G ego : Resou ces,
Funding acquisi ion. Lenna Hu h: Me hodology.
S ephan Lachni : So wa e. La issa Mendes: Fo mal
analysis, In es iga ion, W i ing - O iginal D a , Visuali-
sa ion. Daniil Ras o gue : W i ing - Re iew & Edi ing.
Sa a Ruiz Daza: Fo mal analysis, In es iga ion, W i ing
- O iginal D a , Visualisa ion. Paul Sch¨u ze: Me hod-
ology, So wa e. Ad iana Simancas: Fo mal analysis,
In es iga ion, W i ing - O iginal D a , W i ing - Re iew
& Edi ing, Visualisa ion. Wal e Snoeys: Concep u-
alisa ion, W i ing - Re iew & Edi ing. Simon Span-
nagel: Concep ualisa ion, Me hodology, So wa e, W i -
ing - Re iew & Edi ing, P ojec adminis a ion. Ma cel
S ani zki: Funding acquisi ion. Alessand a Tomal:
Supe ision. Anas asiia Velyka: Me hodology, So -
wa e, Fo mal analysis, In es iga ion, W i ing - O iginal
D a , W i ing - Re iew & Edi ing, Visualisa ion. Gian-
pie o Vignola: W i ing - Re iew & Edi ing. H˚akan
Wennl¨o : Me hodology, So wa e, Fo mal analysis, In-
es iga ion, W i ing - O iginal D a , W i ing - Re iew &
Edi ing, Visualisa ion.
Decla a ion o compe ing in e es
The au ho s decla e ha hey ha e no known compe ing
inancial in e es s o pe sonal ela ionships ha could ha e
appea ed o in luence he wo k epo ed in his pape .
Funding
This wo k has been ca ied ou wi hin Tange ine, a
Helmhol z Inno a ion Pool P ojec .
This p ojec has ecei ed unding om he Eu opean
Union’s Ho izon 2020 Resea ch and Inno a ion p og amme
unde G an Ag eemen No 101004761.
Pa o his wo k has been sponso ed by he Wol gang
Gen ne P og amme o he Ge man Fede al Minis y o
Educa ion and Resea ch (g an no. 13E18CHA).
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