Posi on beam analysis o s uc u ally o de ed po osi y in
mesopo ous silica hin ilms
A. an Veen
a,
*, R. Escoba Galindo
a
, H. Schu
a
, S.W.H. Eij
a
, C.V. Falub
a
,
A.R. Balkenende
b
, F.K. de Theije
b
a
De ec s in Ma e ials, In e acul y Reac o Ins i u e, Del Uni e si y o Technology, Mekelweg 15, NL-2629 JB Del , The Ne he lands
b
Philips Resea ch Labo a o ies, P o . Hols laan 4, NL-5656 AA, Eindho en, The Ne he lands
Abs ac
Posi on beam echniques ha e been employed o cha ac e ise low-kdielec ic silica based ilms, which ha e wo o h ee
dimensional s uc u es o nanome e size po es. Po e ac ions a y om 5 o 50%. Posi ons implan ed in he laye slow down and
e en ually annihila e wi h he elec ons om he laye . Howe e , in po es o he dielec ic ilms posi onium (Ps) is o med be o e
annihila ion akes place. The wo s a es o Ps (pa a-posi onium (p-Ps) and o ho-posi onium (o-Ps)) a e o med wi h a he
di e en li e imes o 125 ps and 142 ns, espec i ely. The beha iou o Ps in he po ous ma e ial can be desc ibed as ho pa icles
wi h 1 eV ini ial kine ic ene gy, which lose hei ene gy by equen collisions wi h he a oms o he po e walls. When he po es a e
in e connec ed o sepa a ed by hin walls allowing pe mea ion o he Ps some o he Ps will e use om he ilm in o he acuum.
The 2D-ACAR echnique enables one o moni o he eloci y dis ibu ion o he annihila ing p-Ps and shows an inc easing ac ion
o su ace emi ed p-Ps when posi ons a e implan ed close o he ilm su ace. Measu emen s o Dopple b oadening o 2-gamma
annihila ion and de ec ion o 3-gamma e en s (o-Ps) gi e insigh in o he equency o o he annihila ion e en s. Combining he
esul s, a comple e pic u e can be ob ained o he in e ac ions and anspo o he Ps pa icles in he ma e ial. By modelling he Ps
beha iou in o ma ion is ob ained on s uc u al pa ame e s o he po ous ma e ial. A anspo model based on mul i ene gy g oup
di usion o pa icles desc ibes he esul s well.
#2003 Else ie B.V. All igh s ese ed.
Keywo ds: Posi ons; Posi onium; Annihila ion; Silica; Po es and pe cola ion phenomena
1. In oduc ion
Ma e ials wi h a low dielec ic cons an (kB/2.2) a e
equi ed o u u e gene a ions o ICs, in o de o
subs i u e he p esen dielec ic laye s in ul a la ge scale
in eg a ed (ULSI) de ices. Candida es include, e.g.
o ganic and ino ganic polyme s, and nanopo ous silica.
A high po osi y o he ma e ial is necessa y o lowe he
dielec ic cons an su icien ly. Fu he mo e, po e sizes
should be small ( ypicallyB/5 nm). Cha ac e isa ion o
he po ous s uc u e by a numbe o posi on annihila-
ion echniques [1/4] has shown o be success ul. The
majo i y o posi ons a e implan a ion in he ma e ial
o m posi onium (Ps) in he po es. Since Ps can be
conside ed as a e y ligh ace ‘a om’ wi h he physical
size o a hyd ogen a om wande ing a ound in he po e
s uc u e i can gi e in o ma ion on po e size and
anspo o gas h ough he po ous s uc u e. In his
s udy he emphasis will be on a desc ip ion o he Ps
his o y in low-k hin ilms wi h well-de ined po es. The
expe imen al me hods a e dep h selec i e Dopple
b oadening and high esolu ion 2D-ACAR which mea-
su e momen um dis ibu ions o annihila ing elec on/
posi on pai s and Ps in he ilms. A companion a icle
desc ibes low-k ilms in which he composi ion o he
in e nal su ace is modi ied [5].
2. Sample p epa a ion and expe imen al p ocedu e
The mesopo ous silica samples we e p oduced using
solu ions con aining silicon alkoxides ( e ae hylo ho-
silica e, TEOS) and a su ac an (ce yl ime hylammo-
* Co esponding au ho . Tel.: /31-15-278-2801; ax: /31-15-278-
6422.
E-mail add ess: a [email p o ec ed] .nl (A. an Veen).
Ma e ials Science and Enginee ing B102 (2003) 2/7
www.else ie .com/loca e/mseb
0921-5107/03/$ - see on ma e #2003 Else ie B.V. All igh s ese ed.
doi:10.1016/S0921-5107(02)00752-3
nium b omide, CTAB) o Plu onic F127 (e hylene
oxide)
106
(p opylene oxide)
70
(e hylene oxide)
106
. Du ing
coa ing deposi ion he sol en s e apo a e, micelles a e
o med and he silicon alkoxide condenses o o m
(o ganically modi ied) silica a ound he micelles. A
well-o de ed po e s uc u e is ob ained once he micelles
a e emo ed om he coa ing. The TEOS/CTAB sys em
o ms a high po osi y a h ee dimensional (3D)
hexagonal phase, while he TEOS/F127 sys em esul s
a high po osi y in a wo dimensional (2D) hexagonal
packing o cylinde s in he plane o he su ace. The six
samples used in his s udy a e all hyd ophilic due o he
p esence o silanol g oups a he in e nal po e su aces.
The ilm p ope ies a e gi en in Table 1.
The PBA expe imen s we e pe o med wi h he
posi on beams a he Del Posi on Cen e [6,7]. The
posi ons we e injec ed in he samples wi h ene gies
uned be ween 100 eV and 30 keV. The maximum
implan a ion ene gy co esponds o a ypical implan a-
ion dep h o /2mm in ce amics o densi y 2.5 g cm
3
.
All expe imen s we e ca ied ou a oom empe a u e
unde a acuum o abou 10
6
Pa. The PBA esul s a e
desc ibed in e ms o h ee pa ame e s (S,Wand -Ps)
[8]. The Spa ame e indica es he ac ion o posi ons
ha annihila es wi h low momen um elec ons ( alence
o conduc ion elec ons). This pa ame e is ela ed o
he open olume de ec s p esen in he sample (e.g.
po es). The Wpa ame e indica es he ac ion o
posi ons ha annihila es wi h high momen um elec-
ons (co e elec ons). This pa ame e is ela ed o he
chemical en i onmen whe e he annihila ion akes
place. Bo h pa ame e s can be combined in S/W
maps whe e he di e en annihila ion si es (laye s) can
be dis inguished. The da a was analysed wi h he
VEPFIT
[7] p og am. The hi d pa ame e ( -Ps) ep esen s he
ac ion o implan ed posi ons ha is ela ed o o ho-
posi onium (o-Ps) sel -annihila ion and is obse ed by
3-gannihila ion. In he companion a icle he b anching
o he posi ons in o di e en s a es is p esen ed [5]. Fo
low-kcoa ings i e annihila ion s a es can be dis in-
guished: pa a-posi onium (p-Ps), o-Ps annihila ion ia
he po e wall, o-Ps sel -annihila ion, posi on annihila-
ion inside he walls sepa a ing he po es, and he
ac ion o Ps ha escapes om he ilm. The ac ion
-Ps is de ined such ha o highly po ous ma e ial wi h
la ge po es -Ps/100%.
The 2D-ACAR me hod [9] measu es he de ia ion
om collinea i y be ween he wo annihila ion pho ons.
This de ia ion is o he o de o a ew milli adians. A
high-in ensi y (10
8
e

s
1
) uneable keV posi on beam
POSH enables dep h-selec i e s udies in he (sub)-
mic ome e ange. 2D-ACAR is measu ing exclusi ely
2-gannihila ion e en s and is able o esol ing p-Ps
annihila ion and o de i e he momen um dis ibu ion
o he annihila ing p-Ps.
3. Resul s
In Fig. 1 he measu ed alues o S,W, and -Ps a e
plo ed e sus he posi on ene gy o he six samples. In
he ou h panel he S/W alues a e plo ed in a so
called S/Wmap. The esul s a e ypical o a hin ilm
on a silicon subs a e. Posi ons en e ing wi h a low
ene gy a e implan ed e y shallowly and p obe he ou e
su ace and, he e o e, yield di e en S,Wpa ame e s
han hose in he ene gy ange 1/4 keV co esponding
o implan a ion wi hin he ilms. Fo ene gies la ge
han 4 keV a g adual ansi ion akes place o annihila-
ion in he subs a e. Fi ing p ocedu es (
VEPFIT
[6])
ha e been applied o de e mine he hickness (by weigh )
o he ilm. This hickness ag ees well wi h he weigh
hickness de i ed om Table 1 (using hickness and
densi y). Cha ac e is ic alues de i ed o Sand Wo
all he ilms a e gi en in Table 1. They a e also indica ed
in he S/Wmap o Fig. 2 o six o he ilms. The Ps
ac ion a ies om e y low o abou 50% o he
in es iga ed hyd ophilic samples. Resul s o hyd opho-
bic ilms in [5] show e en highe alues o 85%. In
gene al, he Ps ac ion inc eases wi h po osi y bu he
ac ion is much highe o he high po osi y F127 wo
dimensional s uc u e wi h la ge diame e cylinde s han
o he CTAB h ee dimensional s uc u e wi h small
diame e po es.
A ypical esul ob ained wi h he 2D-ACAR se up
o he F127 (2c) ilm is shown in Fig. 2. The momen um
Table 1
Desc ip ion o he low-k ilms and measu ed posi on annihila ion pa ame e s
Numbe Desc ip ion Thickness (nm) Po osi y (%) Po e size (nm) Densi y (g cm
3
)S
el. laye
W
el. laye
Ps
1a TEOS/CTAB 0.01
a
R
b
184 10.9 1 (0.5) 2.0 0.954 1.64 0
1b TEOS/CTAB 0.05 R 267 31.0 1 (0.5) 1.5 0.992 1.46 0.12
1c TEOS/CTAB 0.1 3D 338 46.3 1.75 (0.25) 1.2 1.010 1.40 0.12
2a TEOS/ F127 0.0005 R 190 7.4 1 (0.5) 2.0 0.982 1.56 0
2b TEOS/ F127 0.002 R 285 35.0 1 (0.5) 1.4 0.994 1.40 0.08
2c TEOS/ F127 0.005 2D 457 44.7 4.5 (0.5) 1.2 0.994 1.48 0.48
a
Composi ion a io.
b
O de ing: R, andom; 2D, wo dimensional; 3D, h ee dimensional.
A. an Veen e al. / Ma e ials Science and Enginee ing B102 (2003) 2 /73
dis ibu ion o he ilm implan ed wi h 3.5 keV posi ons
shown in Fig. 2a e eals asymme y in he y-di ec ion
(co esponding o he di ec ion ou wa d di ec ed om
he su ace). Appa en ly, he dis ibu ion is composed o
an iso opic and an aniso opic componen . I appea s
ha he la e componen is a cosine momen um
dis ibu ion (Fig. 2b) which can be assigned o p-Ps
pa icles lying o he su ace, like a gas molecule
e using om a hole in a olume con aining a low
densi y Knudsen gas. The iso opic con ibu ion can be
decomposed in o a na ow and a b oad componen . The
na ow componen can be assigned o p-Ps ‘gas’ ha
annihila es inside he po es. In Fig. 3 c oss-sec ions o
he momen um dis ibu ion in xand ydi ec ion a e
displayed o he ilm. Analysis o he momen um
dis ibu ion shows ha he a e age ene gy o he Ps
amoun s o 0.9 eV o shallowly implan ed and o 0.5 eV
o 3.5 keV implan ed posi ons. An a e age momen um
Fig. 1. Posi on annihila ion pa ame e s S,W, and -Ps plo ed as a unc ion o he a e age posi on implan a ion dep h, and an S,Wmap showing
all he S,Wda a. Fo he CTAB-1c ilm he cha ac e is ic annihila ion s a es a e gi en: (1) su ace, (2) po ous ilm, (3) in e ace and (4) Si-subs a e.
Fig. 2. Momen um dis ibu ions measu ed o sample 127 2c.
A. an Veen e al. / Ma e ials Science and Enginee ing B102 (2003) 2 /74
o 10
3
m
o
c co esponds o an a e age Ps kine ic ene gy
o 0.128 eV.
4. Posi onium slowing down and anspo in po ous
SiO
2
The implan a ion p o ile o posi ons can be desc ibed
by he so called Makho p o ile (a de i a i eo a
Gaussian) which is also used o solids wi hou po es [8].
The po osi y is aken in o accoun by using he educed
densi y o he po ous silica in he Makho exp ession.
Posi ons do no slow down in he po es, only in he wall
ma e ial. Ps is o med by combining one o he elec ons
p oduced in he ack o he slowing down posi on wi h
he he malised posi on. The Ps pa icle is a ac ed by
he po es and is c ea ed wi h a kine ic ene gy o abou 1
eV, as ollows om Ps su ace emission o Ps om silica
and om ACAR measu emen s on Ps in xe ogel [10].
Once Ps is o med, wi h he wo s a es p-Ps and o-Ps in a
a io o 1/3, i will beha e as an a om in a Knudsen gas.
The e a e only in e ac ions wi h he walls o he
s uc u e bu no wi h o he Ps. The eason o his is
he sho li e ime o Ps a e age li e imes o sel -
annihila ion a e 125 ps o he p-Ps and 140 ns o o-
Ps, espec i ely, when sel -annihila ion occu s. The
annihila ion o p-Ps yields wo g- ays o abou 511
keV, he sel annihila ion o o-Ps yields h ee g- ays in
he ene gy egion 0/511 keV. Annihila ion o o-Ps is
accele a ed by a pick-o eac ion wi h he walls o he
po es; subsequen ly, annihila ion by wo g- ays occu s.
The anspo and cooling o he Ps can be desc ibed
by app oxima ing he Ps anspo by di usion heo y.
Le c(E,x, ) be he concen a ion o Ps a dep h xwi h
ene gy Ea ime .
A balance equa ion can hen be w i en as ollows:
dc(x;E)
d
D(E)d2c(x;E)
dx21
41
Vn(E) sOc(x;E)1
4
1
Vn(Eƒ) sOc(x;E)I(x)lPsc(x;E)
d
lpick-o c(x;E)
whe e he i s e m a he igh hand side desc ibes he
loss/gain by di usion ( he di usion coe icien Dis E
dependen ), he second e m he loss a e by sca e ing
(E0/E?), wi h inal ene gy E?/E/aa e one collision,
and he hi d e m he g ow h a e by sca e ing (Eƒ0
/
E) wi h ene gy Eƒ/aEa e one collision.
In a collision wi h a oms in he mal mo ion also some
ene gy will be ans e ed om he a om o he Ps;
inally, his will lead o he malisa ion o he Ps. This is
no ye included in he heo y. The ou h e m is he
implan a ion a e which is ze o o EB/E
0
, he i h e m
is he sel annihila ion a e and he six h e m he pick-
o a e (impo an only o o-Ps), wi h d he pick-o
laye in he po es wi h adius .
The di usion coe icien can be desc ibed by D(E)/
1/6l
2
n o 3D o de ed and 1/4l
2
n o 2D o de ed po e
s uc u es, whe e lis he jump dis ance/dis ance
be ween he wo cen es o neighbou ing ca i ies/
2

/l
w
, wi h l
w
he hickness o he wall sepa a ing he
po es. V is he a emp equency, in all di ec ions, o
one Ps p esen in a ca i y olume V/4/3/p/
3
, whe e V1/4
(1/V)n
Ps
O
Owhe e
Ps
is he eloci y o he Ps de i ed
Fig. 3. C oss-sec ions h ough momen um dis ibu ion measu ed by 2D-ACAR. (a) Momen um pe pendicula o he su ace, (b) momen um pa allel
o he su ace.
A. an Veen e al. / Ma e ials Science and Enginee ing B102 (2003) 2 /75
om he ene gy E,n
Ps
/V2eE/m
Ps
, and
O
is he
ac ion o Ps ha is pene a ing h ough he walls o
he ca i y gi en by
O
/O*/O,O* is he open o
anspa en pa o he wall,
s
is he ac ion ha
sca e s o he wall. The ac o 1/ais close o uni y
because m
Ps
is much smalle han he mass o he wall
a oms. I s lowes alue o maximum ene gy ans e , is
1/a/(1/m
Ps
/m
a
)
2
/(1/m
Ps
/m
a
)
2
/(1/4m
Ps
/m
a
). The
ene gy loss o Ps wi h ene gy Edu ing ime D equals
1/4(1/V)n(E)
s
O(collision equency)/D /E/a
The bounda y condi ions a e gi en by e lec ion o Ps
a he silicon subs a e in e ace a x/l:(dc/dx)
x1
/
0. I is assumed ha Ps a he acuum side is emi ed
in o he acuum: c
(x0)
/0. The emission a e is gi en
by D(dc/dx). The ime dependen anspo equa ion is
sol ed nume ically.
5. Discussion
The abo e model applied o an o de ed 2D po e
s uc u e (e.g. 127 2c) yields a wall hickness o 1.8 nm.
The e ec i ep-Ps di usi i y is calcula ed o be 2 cm
2
s
1
, and he di usion leng h is 158 nm i he po e
anspa ency pa ame e
O
/0.2 is aken. I ollows
om he alue o he di usion leng h (o he same o de
as he ilm hickness) ha a signi ican amoun o p-Ps
c ea ed h oughou he ilm will escape om he su ace
du ing i s limi ed li e ime. In [5] i is ound ha o a
sample po osi y /45%, p-Ps e usion is obse ed. I is
e y likely ha when walls ge hinne , i.e. a ew imes
he s uc u al po e size in silica, Ps, which has an
e ec i e, quan um mechanically de e mined, size o
abou 0.8 nm when i is in in e ac ion wi h he silica
s uc u e, will easily pe mea e. Nume ical esul s o
anspo o p-Ps implan ed in such a po ous ilm a e
gi en in Fig. 4. Time dependen ene gy dis ibu ions o
p-Ps a a dep h o 200 nm a e plo ed which show he
e olu ion o he ene gy om 1 eV a /0 o a b oad
s a iona y ene gy dis ibu ion when equilib ium is
es ablished be ween he implan a ion o posi ons and
he annihila ion o p-Ps (a e age li e ime 125 ps). The
bo om igu e shows he s a iona y ene gy dis ibu ions
a a ying dep h. A e age ene gies a e 0.35 eV o he
deep laye s and 0.41 eV o he shallow laye s. The
ene gy dis ibu ions de ia e om he Maxwell/Bol z-
mann dis ibu ion which applies o pa icles in he mal
equilib ium. An impo an eason is ha a any ime ho
Ps is in oduced in o he ilm. I seems ha he ene gy
dis ibu ions can be desc ibed a he well by he sum o
wo Maxwell/Bol zmann dis ibu ions, e.g. one wi h
kT
1
/0.11 eV, and kT
2
/0.6 eV o he dis ibu ion
wi h an a e age ene gy o 0.35 eV (no shown he e).
The abo e model can be applied and used o i he
measu ed ene gy and eloci y dis ibu ions, oge he
wi h he o he pa ame e s measu ed in he posi on
beam expe imen s. The model will yield ac ions o all
annihila ing modes which con ibu e o he Sand W
pa ame e s and he Ps ac ion.
6. Conclusions and inal ema ks
Dep h selec i e posi on annihila ion s udies o low-k
ilms show clea ly ha wi h inc easing po osi y o he
ilms p-Ps c ea ed in he ilm is eme ging om he ilm.
The eloci y dis ibu ion o he e using Ps and he Ps
emaining in he ilm can be de e mined. Cooling o he
Ps is slow and is de e mined by in e ac ion wi h he po e
walls. The p esence o o-Ps indica es ha po es ha e
g own beyond 4 nm size. Dopple b oadening pa a-
me e s and Ps ac ion can be used o de e mine he
Fig. 4. (a) Calcula ed ene gy dis ibu ions du ing slowing down o p-
Ps in mesopo ous silica wi h po es o 5 nm and densi y 1 g cm
3
The
Ps pa icles ha e an ini ial ene gy o 1 eV. The implan a ion o
posi ons is con inuous wi h an ene gy o 3.5 keV a an a e age dep h
o 300 nm. The ene gy dis ibu ion is plo ed a ime in e als o 37.5
ps. A e 500 ps a s a iona y s a e is a ained. (b) S a iona y ene gy
dis ibu ions o di e en dep hs in he ilm. A e age ene gies a e
indica ed. A Maxwell/Bol zmann dis ibu ion is shown wi h simila
a e age ene gy.
A. an Veen e al. / Ma e ials Science and Enginee ing B102 (2003) 2 /76

b anching in o all annihila ion channels which depend
on he po ous s uc u e o he ilm. The model p esen ed
he e will in he u u e be used o pe o m i s o he
b anching a ios.
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