J. Appl. Phys. 110, 043705 (2011); h ps://doi.o g/10.1063/1.3622615 110, 043705
© 2011 Ame ican Ins i u e o Physics.
In e p e a ion o ap-limi ed mobili y in
space-cha ge limi ed cu en in o ganic
laye s wi h exponen ial densi y o aps
Ci e as: J. Appl. Phys. 110, 043705 (2011); h ps://doi.o g/10.1063/1.3622615
Submi ed: 24 Ma ch 2011 . Accep ed: 29 June 2011 . Published Online: 17 Augus 2011
José M. Mon e o, and Juan Bisque
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In e p e a ion o ap-limi ed mobili y in space-cha ge limi ed cu en
in o ganic laye s wi h exponen ial densi y o aps
Jose´ M. Mon e o and Juan Bisque
a)
Pho o ol aics and Op oelec onic De ices G oup, Depa amen de Fı´sica, Uni e si a Jaume I, 12071
Cas ello´, Spain
(Recei ed 24 Ma ch 2011; accep ed 29 June 2011; published online 17 Augus 2011)
Cha ge ca ie anspo in diso de ed o ganic semiconduc o s, pe o med in elec onic de ices
such as op oelec onic and pho o ol aic ones, is usually a ec ed by an exponen ial dis ibu ion o
localized s a es in he band-gap ( aps) unde space-cha ge limi ed cu en . In his pape , we
p o ide a ull analysis o he ap-con olled anspo o he single-ca ie de ice in he equency
domain. T ap-limi ed mobili y is in e p e ed in e ms o he classical mul iple- apping pic u e wi h
one anspo s a e and he apping-de apping dynamics o he exponen ial densi y o aps. This
allows us o p o ide a sui able explana ion o he usual expe imen al ea u es o he mobili y
dependence on ol age as along wi h he capaci ance spec a. V
C2011 Ame ican Ins i u e o
Physics. [doi:10.1063/1.3622615]
I. INTRODUCTION
O e he las wo decades, he applica ion o ligh -emi -
ing diodes (LEDs) o he echnology o daily li e has c ea ed
an inc easing in e es , going beyond he adi ional ole o
ligh indica o s and displays o home appliances and gene al
illumina ion.
1,2
A he same ime, an exhaus i e echnologi-
cal esea ch e o was igge ed by he po en ial applica ions
o he new ongoing gene a ion o hese de ices based on o -
ganic semiconduc o s.
3
Low-cos manu ac u ing p ocessabil-
i y and la ge a ea and lexible de ices may be achie ed by
polyme -based o ganic ligh -emi ing diodes.
4,5
Ne e he-
less, he pe o mance o o ganic elec onic de ices s ongly
depends upon he cha ge anspo p ocess ca ied ou , he e-
o e, u he unde s anding o he physical beha io o such
ma e ials is needed. Fo ins ance, he desc ip ion o cha ge
anspo in o ganic laye s by space-cha ge limi ed cu en
(SCLC)
6–9
equi es an in e p e a ion o mobili y by di e en
semiempi ical models (wi h ield- o densi y-dependence),
which is he c ucial pa ame e go e ning he anspo in he
bulk.
10
Recen ly,
11
we ha e ea ed he ela ionship be ween
he di e en models and in his pape we p o ide a ull anal-
ysis o he in e p e a ion o he capaci ance and conduc ance
spec a, along wi h a compa ison wi h he obse ed expe i-
men al ea u es.
In he 1990s, ield-dependen mobili y models we e p o-
posed by Ba¨ssle , as a esul o assuming hopping anspo
in a Gaussian densi y o s a es (DOS).
12
Field-dependen
mobili y in o ganic laye s, such as in sandwiched ilms com-
posed o ei he poly(p-phenylene inylene) (PPV) de i a-
i es o aluminum hyd oxyquinoline (Alq3), became widely
accep ed. Howe e , Tanase e al. p esen ed a compa ison o
mobili y alues o wo solu ion-p ocessed o ganic poly-
me s: poly(2-me hoxy-5-(30,70-dime hyloc yloxy)-p-phenyl-
ene inylene) (OC
1
C
10
-PPV) and poly(3-hexyl hiophene)
(P3HT), pe o med in wo di e en con igu a ions, i.e.,
ield-e ec ansis o s and hole-only diodes.
13
The i s
s uc u e displayed mobili y esul s up o h ee o de s o
magni ude highe han he la e con igu a ion. These obse -
a ions endo sed he densi y-dependen mobili y model, p o-
posed by Vissenbe g and Ma e s in amo phous o ganic
ansis o s, ha s ems om hopping pe cola ion in an expo-
nen ial DOS.
14
As demons a ed by A khipo e al., hopping
anspo in diso de ed ma e ials can be educed o a ap-
con olled anspo composed o an e ec i e anspo le el
and a b oad dis ibu ion o localized s a es ( aps) ha only
e ain mobile cha ges.
15
Cu en ly, se e al au ho s a e con-
side ing his amewo k: anspo ia an ex ended s a e
unde he in luence o an exponen ial densi y o aps.
11,16–18
In he p esen pape , we implemen his assump ion in SCLC
o analyze i s implica ions o cha ge ca ie mobili y.
F om an expe imen al poin o iew, he de e mina ion
o mobili y is commonly gi en h ough he s udy o ansi
imes (i.e., ime needed o ca ie s o c oss he sample
elec ode- o-elec ode) in he wide ange o he me hods
a ailable in he li e a u e: ime-o - ligh , ansien elec olu-
minescence, da k injec ion, and impedance spec oscopy
(IS), among o he s.
19,20
The IS echnique will ocus ou com-
pu a ional calcula ions o p o ide physical insigh s on he
expe imen al measu emen s o capaci ance spec a. As dem-
ons a ed in an ea lie heo e ical wo k o a single-ca ie de-
ice in SCLC wi h only a single ap,
21
he e is a s ong
co ela ion be ween he shape o he capaci ance spec a and
he na u e o aps lying in he band-gap. In pa icula , a clas-
si ica ion o hem was es ablished using i s ene gy dep h and
i s dynamic ac i i y o cap u e and elease cha ge ca ie s
(i.e., shallow: as and slow, and deep aps). Meanwhile
as -shallow aps we e esponsible o he delay o ansi
imes (i.e., shi ed capaci ance s ep-ups), slow-shallow aps
we e o low- equency capaci ance inc eases. The aim o
his pape is o ex end hese ideas o a wide se o aps,
om a single o an exponen ial densi y in he band-gap,
11
in
o de o u he es he heo e ical amewo k wi h expe i-
men al capaci ance da a: a single-ca ie de ice composed
o N,N0-diphenyl-N,N0-bis(1-naph ylphenyl)-1,10-biphenyl-4,40-
diamine (a-NPD).
a)
Elec onic mail: [email p o ec ed].
0021-8979/2011/110(4)/043705/6/$30.00 V
C2011 Ame ican Ins i u e o Physics110, 043705-1
JOURNAL OF APPLIED PHYSICS 110, 043705 (2011)
This pape is s uc u ed as ollows: i s , we discuss he
exponen ial-densi y- ap model; second, he esul s and dis-
cussion sec ion; and inally, he conclusions.
II. THE EXPONENTIAL-DENSITY-TRAP MODEL
The SCLC egime o elec on anspo in diso de ed
o ganic semiconduc o s comp ises wo classes o ene gy
s a es: one anspo le el wi h densi y, nc, ha d i s in he
elec ic ield, F, and an exponen ial dis ibu ion o local-
ized s a es wi h densi y, n , ha co esponds o he immo-
bilized apped cha ge. The ma hema ical desc ip ion is
well-known in he li e a u e and en ails he con inui y
equa ion, he d i -cu en equa ion, and he Poisson equa-
ion, espec i ely,
22
dJ
dx ¼0;(1)
J¼ql0nncFþe e0
@F
@ ;(2)
dF
dx ¼q
e e0
ncþn
ðÞ:(3)
Fu he mo e, he ap dynamics equa ion is conside ed o
e e y ene gy le el, E , in he exponen ial dis ibu ion o aps
wi h occupancy, ðE Þ, along he band-gap as,
23
@
@ ¼cnc1
½e :(4)
He e, qis he elemen a y cha ge, l0nis he ap- ee mobil-
i y, e e0is he dielec ic cons an , and cand ea e he coe i-
cien s o elec on cap u e and elease, espec i ely. The
po en ial can be calcula ed by in eg a ing he elec ic ield
along he hickness, L,
V¼ðL
0
Fdx:(5)
In ac , he same sys em o equa ions s ands o he hole
anspo jus by swapping he spa ial and ene gy scales o i-
gins a once. The popula ion o he ex ended s a es a he
ene gy le el, Ec, o a non-degene a e semiconduc o , ela es
o he Fe mi le el, EF,as
ncðEFÞ¼NceðEFEcÞ=kBT;(6)
whe e Ncis an e ec i e densi y o s a es in he anspo
le el.
The apped popula ion is he o e all densi y o cha ge
ca ie s loca ed by he exponen ially dis ibu ed aps along
he band-gap,
n ðEFÞ¼ðEc
EV
g ðE Þ ðE ;EFÞdE ;(7)
g ðE Þ¼ N
kBT
e
E Ec
kBT ;(8)
whe e N
is an e ec i e densi y o aps and T
is he cha ac-
e is ic ap empe a u e.
Assuming ha e e y ap ene gy le el, E , eaches equi-
lib ium wi h he ex ended s a e (wi h he same Fe mi le el),
he ap occupancy is gi en by,
ðE ;EFÞ¼ 1
1þeðE EFÞ=kBT:(9)
In s eady s a e, Eq. (4) yields,
¼1
1þe=ðcncÞ:(10)
The e o e, he de ailed balance condi ion p o ides he ol-
lowing ela ionship o he ap emission and cap u e coe i-
cien s, Eqs. (6) and (9), in o Eq. (10) yields,
e¼cNceðE EcÞ=kBT:(11)
Le us deno e he s eady-s a e by 
xand a small pe u ba ion
by ^
xapplied a a ce ain angula equency, x. Hence, e e y
elec ical a iable can be exp essed as x¼
xþ^
x o linea ize
he whole sys em o equa ions up o he i s o de .
24
As
shown in Re . 25, by sol ing Eq. (4) o a small pe u ba ion,
we ob ain,
^
ðE Þ¼1

nc

ð1
Þ
1þix=x
^
nc:(12)
This e m gi es he con ibu ion o he spec a o he capaci-
ance and conduc ance o e e y ap le el. E e y ap e-
quency is de ined as
x ðE Þ¼ e
1
:(13)
This is he maximum equency ha he ap is ac ing as
such, since a highe equencies he ap canno ollow he
ac pe u ba ion.
21
Inse ing Eq. (11) in o Eq. (13), we ind
he dependence o x on he ap ene gy and occupa ion, as
x ¼cNceðE EcÞ=kBT
1
:(14)
I should be no ed ha in he SCLC egime, 
is posi ion-
dependen along he o ganic laye . The impedance is de ined
as he quo ien o he po en ial o he cu en densi y,
ZðxÞ¼
^
VðxÞ
^
JðxÞ:(15)
The quan i y, ^
VðxÞ, is de e mined by he spa ial in eg a ion
o ^
FðxÞ om he solu ion o he p eceding model. The
bounda y condi ions a he injec ing con ac used o sol e he
elec ical a iables along he hickness in dc and ac condi-
ions a e,
26,27

ncðx¼0Þ¼Ncand ^
Fðx¼0Þ¼0:(16)
The capaci ance and conduc ance a e de ined as ollows:
CðxÞ¼Re 1
ixZðxÞ

;(17)
043705-2 J. M. Mon e o and J. Bisque J. Appl. Phys. 110, 043705 (2011)
gðxÞ¼Re 1
ZðxÞ

:(18)
The dielec ic capaci ance o he o ganic laye is deno ed
as Cg.
The p esen physical model leads o a i s -o de di e -
en ial equa ion. The nume ical algo i hm used o achie e he
solu ion is based on a double disc e iza ion; one o he
hickness L(i.e., spa ial coo dina e x), and he o he one o
he ene gy bandgap (i.e., ap ene gy E ). On he one hand,
he s eady-s a e is sol ed om Eqs. (1)–(3) and Eqs. (5)–(9)
by pinning he mobile ca ie s as he e ec i e densi y o
s a es a he injec ing con ac in Eq. (16). On he o he hand,
he ime-dependen egime s ems om he applica ion o an
addi ional small ac ol age o e he dc one, he e o e, he
inclusion o he ime-dependen apping ac ion is equi ed,
as shown in Eq. (4) and Eqs. (10)–(14). The ac bounda y
condi ion is se o ze o elec ic ield a he injec ing con ac
[Eq. (16)] and he equency- esponse solu ion [Eq. (15)]is
ob ained.
III. RESULTS AND DISCUSSION
A. Theo e ical amewo k
1. S eady-s a e cha ac e is ics o o ganic laye s wi h
an exponen ial densi y o aps
Expe imen al measu emen s o J-V cu es a e usually
analyzed o in e p e he pe o mance beha io and cha ge
anspo o a wide ange o elec onic de ices, such as o -
ganic ligh -emi ing diodes,
28
ansi o s,
13
and sola cells.
29
In ou case, o single-ca ie o ganic laye s, we calcula e he
s eady-s a e solu ion om he p eceding model and he
esul s a e displayed in Fig. 1. Inpu da a a e shown in Table
I. A low ol ages, mos o he ohmically injec ed cha ges
a e apped (n n, in he Poisson equa ion) and he ep e-
sen a ion app oaches he o mula,
16,17
J¼el0nNc
e
eN

ll
lþ1

l2lþ1
lþ1

lþ1Vlþ1
L2lþ1;(19)
whe eas a high ol ages, n n, he Mo -Gou ney squa e
law modi ied by shallow aps is ollowed,
J¼9
8ehl0n
V2
L3;(20)
whe e h1ð¼ 1þn
hi
=nc
hiÞ
is a ca ie -densi y dependen
ac o o apped and ee cha ge loca ed by he shallow
aps.
30
I should be ema ked ha ap dynamics [Eq. (4)]
ha e no been included in he p esen calcula ions and he
mobili y pa ame e is he eby de ined as l0n, which is inde-
penden o ol age.
2. Impedance esponse o o ganic laye s wi h an
exponen ial densi y o aps
Expe imen al measu emen s o he capaci ance spec a
ex ac ed om he impedance esponse [Eq. (17)] a e consid-
e ed a powe ul ool o de e mine he cha ge anspo pa-
ame e s such as ca ie mobili y o holes and
elec ons.
19,31,32
The model desc ibed in Sec. II was al eady analy ically
sol ed by Dascalu wi h he app oxima ion o J/V2and
wo sugges ed apping coe icien s independen o occupa-
ion (dand W). Howe e , his app oach does no ake in o
accoun he ol age dependence o mobili y i ex ended o
any ol age ange.
33–35
In con as , ou compu a ional esul s
o he capaci ance wi h he exponen ial densi y o aps ha e
no es ic ions and co e he whole model p e iously
exposed. Figu e 2(a) shows capaci ance calcula ions wi h he
ma e ial pa ame e s o Table I. De ia ion om he well-
known ap- ee spec um, i.e., a s ep up om 0.75C
g
o C
g
a a ce ain equency,
8,36
is displayed as a e e ence. Low-
equency capaci ance exhibi s an inc eased alue a mo e
han he adi ional 0.75C
g
and his beha io is a ibu ed o
he slow-shallow aps wi hin he dis ibu ion ha canno
achie e he quasi-equilib ium wi h he anspo le el. A
highe equencies app oaching he ansi ime, he ap-
ping-de apping dynamics o as -shallow aps in e sec
wi h he ansi o cha ge ca ie s om he injec ing o he
collec ing con ac s, hus causing a ime delay, as shown by
peaks (a ows) in Fig. 2(b). T ap-limi ed mobili y is in e -
p e ed in e ms o his ap-con olled anspo . Bo h oles o
shallow aps (slow and as ) now occu a once wi h he ex-
ponen ial densi y o aps, unlike in ou p e ious wo k o a
single- ap.
21
The deep aps con ibu ion is also sligh ly
obse ed in he i s peak o he capaci ance. As a mino
commen , o he capaci ance simula ions wi h a slowe
FIG. 1. Simula ions o cu en -densi y- ol age cha ac e is ics (line and sca -
e plo ) pic u ed oge he wi h he analy ical o mulas: Eq. (19) (solid line)
and Child’s law, Eq. (20) wi h h¼1 (dashed line).
TABLE I. Ma e ial pa ame e s implemen ed o simula ions.
Pa ame e Value
Thickness, L80 nm
T anspo e ec i e densi y o s a es, N
c
10
19
cm
3
Rela i e dielec ic cons an , e
3
T ap- ee mobili y, l
0n
510
7
cm
2
/(Vs)
Tempe a u e, T300 K
Band-gap, E
c
E
2.4 eV
T ap e ec i e densi y o s a es, N
510
17
cm
3
Cha ac e is ic ap empe a u e, T
1500 K
T apping cap u e coe icien , c710
14
cm
3
/s
043705-3 J. M. Mon e o and J. Bisque J. Appl. Phys. 110, 043705 (2011)
cap u e ap a e yielded sligh ly lowe alues han 0.75C
g
in
he icini y o he ansi ime equency.
Since we a e dealing wi h an inhomogeneous sys em,
he Fe mi le el and occupa ion a y along he hickness
[Fig. 3(a)], especially ab up ly close o he injec ion egion
and wi h a smoo he unc ional dependence in he wide ange
o he sample hicknesses. The e o e, an a e age Fe mi le el
hE
F
ican be gi en as an app op ia e app oxima ion o sim-
pli y he heo e ical analysis. Thus, he a e age occupancy
becomes,

ðE ;EFÞ


ðE ;EF
hiÞ
;(21)
and Fig. 3(b) shows he a e age occupied densi y o aps
and he emp y ones unde he p eceding app oach. P esum-
ably, he mos shallow and emp y aps will beha e as as ,
whe eas he less shallow and ull ones will be slow [see Eq.
(14)]. Ne e heless, a u he app oach o he apping a es
along he ene gy dis ibu ion in he band-gap is o p ime im-
po ance o disen angle he di e en oles o e e y ap
ene gy s a e in he con ibu ion o he capaci ance spec a
(see Fig. 4). The cap u e a e, c, is de ined as ene gy inde-
penden wi h a highe alue han he no malized emission
a e, e/N
c
, gi en in Eq. (11). Howe e , he de ini i e disc im-
inan pa ame e s used o classi y whe he he shallow ene gy
s a es a e pu ely as o no , a e: he cap u e a e, c, and he
c i ical cap u e coe icien , c
c
,de ined as,
ccðE Þ3pl0nV
2L2
1
ðE ;EF
hiÞ
NceðE EcÞ=kBT;(22)
and i s discussion was also published in Re . 21.I cis la ge
han c
c
along a ce ain ene gy in e al in he band-gap, hese
shallow ene ge ic le els beha e as as -shallow aps esul -
ing in a la ge ansi ime and he eby, in a ap-limi ed mo-
bili y. This ene gy egion below he anspo le el
ðEcE ELÞcan be calcula ed as, ccðELÞffic, whe e EL
is in e p e ed as he lowes ene gy le el ac ing as pu ely as ,
EL¼kBTLn 3pl0V
cNcL2

exp ðEcEF
hiÞ
kBT

1

þEF
hi
;
(23)
which is dependen upon he ol age, V, d opped along he
o ganic laye in space-cha ge, he Fe mi le el hE
F
i, and he
de ice speci ica ions. Fo shallow le els below EL
FIG. 3. (Colo online) (a) Simula ions a 3 V o he Fe mi le el along he
hickness (solid line) and a e age occupancy (dashed line) ha s ems om
he a e age Fe mi le el, hE
F
i. (b) Colo ed a ea is displayed o he a e age
occupied ap densi y o s a es (DOS) whe eas shadowed a ea co esponds
o he emp y apping densi y o s a es. A e age occupancy is also p in ed
(pink dashed line) oge he wi h a e age Fe mi le el, hE
F
i, and anspo
ene gy le el, E
c
, as e e ences.
FIG. 2. (Colo online) Model simula ions a 3 V o (a) capaci ance, and (b)
di e en ial suscep ance spec a [see Eq. (24)], wi h an exponen ial densi y
o aps (o ange dashed line) in compa ison o he ap- ee spec um (black
solid line). F equencies a e no malized o ansi ime, s
dc
, [see Eq. (26)].
043705-4 J. M. Mon e o and J. Bisque J. Appl. Phys. 110, 043705 (2011)

ðEL>E >EF
hiÞ
, he ap dynamics s a o g adually
change om as o slow, as a as he deepe s a es wi h
which we a e dealing. Tha makes i di icul o quan i y he
educed ac o , h, ha commonly ela es he ap-limi ed mo-
bili y o he ap- ee one.
30
In o de o expe imen ally es he model,
37
capaci ance
spec a a e gene a ed a di e en ol ages in Fig. 5.By
inc easing he ol age d opped in he space-cha ge, he
Fe mi le el mo es up owa d he anspo le el, co e ing
mo e apping s a es in he band-gap. As a consequence, he
ange o he ene gy dis ibu ion ac ing as pu ely as
ðEcE ELÞbecomes na owed since EL[Eq. (23)] also
mo es up. This indica es ha he e a e less unoccupied ap-
ping si es ha can ap and elease cha ges quickly while
hey a e d i ed elec ode- o-elec ode by he local elec ic
ield. In o he wo ds, ap limi a ion o mobili y becomes
lowe he mo e ol age is applied since less as -shallow
aps a e ac ing as such. As ega ds he low- equency beha -
io wi h he ol age, he same easoning can be gi en, since
he ene gy egion ðEL>E >EF
hiÞ
con aining he slow-
shallow aps, also becomes mo e educed. Thus, less low-
equency inc ease is expec ed he mo e ol age is applied.
B. Expe imen al analysis
Le us now apply he model and heo e ical amewo k o
in e p e he capaci ance spec a o he hole-only de ice based
on he a-NPD o Nguyen e al.
38
In ha publica ion, he au ho s
model he da a by conside ing d i -di usion anspo wi h a
Gaussian densi y o aps and ield-dependen mobili y, how-
e e , he low- equency capaci ance exhibi s a a he sha p
beha io in con as o he expe imen s. This ea u e may be be -
e desc ibed by an exponen ial dis ibu ion o aps ha would
esul in a smoo he capaci ance a ia ion a low- equency.
We show he expe imen al capaci ance spec a a di e en
ol ages o a hick hole-only de ice e ie ed om Re . 38 in
Fig. 6. The capaci ance beha io and shape ag ees well wi h
ou model a e e y equency ange, howe e , no he magni-
udes, mainly due o a no iceable di e ence o hickness, L,
om simula ions (Fig. 5), in con as o he expe imen al da a
showninFig.6. The low- equency (LF) capaci ance pa dis-
plays an inc ease, which is di ec ly modula ed by he apping
dis ibu ion, and conc e ely by he slow-shallow aps. The LF
capaci ance inc ease is mo e no iceable he less he ol age is
applied, i.e., o he lowe se o ol ages (3–5 V) mo e han
o he highe ones (9–10 V), as expec ed. The in e media e
equency (IF) ange is cha ac e ized by he p esence o a min-
imum alue o he capaci ance spec a and i s posi ion shi s
depending on he ol age. Ou model p edic s all o he min-
ima achie ing he alue o 0.75 o he geome ical capaci-
ance, C
g
, o he o ganic laye , howe e , he expe imen al da a
lies mo e o e his limi he less he ol age is applied. This
ea u e seems o be be e desc ibed by Nguyen’s model. In
he IF egion i is belie ed ha he a e age ansi ime o he
ca ie s, s , can be ex ac ed by means o he IS echnique
wi h he ep esen a ion o he suscep ance (¼Im(Y)), con-
c e ely, he nega i e di e en ial suscep ance,
DBðxÞ¼xðCðxÞCgÞ:(24)
The posi ion o he maxima de ine he peaks co esponding
o ac ansi imes a di e en ol ages, he e o e,
39
s ac 0:72  1
max;(25)
and he mobili y alues can be ex ac ed by using he dc
ansi ime exp ession,
FIG. 4. (Colo online) T apping coe icien s a e displayed o he 3 V model
simula ion: no malized emission a e (e/N
c
) and apping a e (c) a e ep e-
sen ed by colo ed solid lines. C i ical cap u e coe icien is also gi en by a
dash-do ed line. Re e ence ene gy le els a e: anspo E
c
, limi E
L
, and a -
e age Fe mi hE
F
ile els.
FIG. 5. (Colo online) Model ep esen a ion o capaci ance spec a a ol-
ages anging om 3 o 7 V. Simula ion pa ame e s a e shown in Table I.
FIG. 6. (Colo online) Expe imen al capaci ance spec a o a hole-only
a-NPD de ice a di e en ol ages. Con ac a ea, A, is 0.235 cm
2
.
043705-5 J. M. Mon e o and J. Bisque J. Appl. Phys. 110, 043705 (2011)
s dc ¼4
3
L2
lðVbias VbiÞ:(26)
Inse ing Eq. (25) in o Eq. (26), i holds,
l¼4
3
L2 max
0:72 ðVbias VbiÞ:(27)
Calcula ions o mobili y yielded mobili y enhancemen he
mo e he ol age is applied in he bulk. In pa icula , he i -
ing o he classical ield-dependen mobili y exp ession,
l¼l0exp cffiffiffi
F
p

;(28)
wi h he app oxima ion o ffiffiffi
F
pffiffiffiffiffiffiffiffiffi
V=L
p, p o ided expe i-
men al mobili y alues o l0¼2:9104cm2=ðVsÞand
c¼2:6103ðcm=VÞ1=2. This mobili y dependence upon
ol age, measu ed by means o IS, is in e p e ed in he p esen
pape as a ap-limi ed mobili y go e ned by he dynamics o
he as -shallow aps in he band-gap. The ield-dependen
mobili y is induced by he educ ion o he apping ac ion as
a as mo e ol age co e s mo e apping si es in he exponen-
ial dis ibu ion o localized-s a es. As ega ds he high e-
quency ange (HF); i is composed o wo di e en pa s: a
ela i ely wide pla eau lying a he capaci ance alue o C
g
,
and a sha p dec ease om a equency cu o onwa ds. The
la e beha io o he HF is domina ed by he se ies esis ance
o he whole de ice s uc u e causing he capaci ance d op.
In summa y, he shape o he capaci ance spec a wi h
an exponen ial densi y o aps is s ongly de e mined by he
bias- ol age, p o iding: (1) a de ia ion o ansi imes ans-
la ed in o a ield-dependen mobili y, and (2) a low- e-
quency capaci ance inc ease o e he adi ional 0.75C
g
o
ap- ee ma e ials.
IV. CONCLUSIONS
We ha e co obo a ed he heo e ical amewo k o he
mul iple- apping pic u e in o ganic laye s ha comp ises an
exponen ial densi y o apping s a es unde SCLC. The anal-
ysis o he capaci ance spec a acili a es he in e p e a ion o
he ol age dependence o he ap-limi ed mobili y and he
low- equency capaci ance beha io . The p esence o pu e
as -shallow aps de e mines he limi a ion o he cha ge
anspo mobili y, whe eas slow-shallow aps cause he
low- equency capaci ance o inc ease. Bo h ea u es a e
modula ed by wo espec i e ol age-dependen ene gy
egions in he band-gap.
ACKNOWLEDGMENTS
We a e hank ul o inancial suppo om MEC o
Spain unde p og am Consolide -Ingenio 2010 (P ojec
HOPE CSD2007-00007), and Gene ali a Valenciana unde
P ojec PROMETEO/2009/058.
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