1
Time se ies s a is ical analysis: a powe ul ool o e alua e he
a iabili y o esis i e swi ching memo ies
J.B. Roldán1*, F. J. Alonso2, A. M. Aguile a2, D. Maldonado1, M. Lanza3
1Depa amen o de Elec ónica y Tecnología de Compu ado es. Uni e sidad de G anada.
Facul ad de Ciencias. A d. Fuen enue a s/n, 18071 GRANADA, Spain.
*Co esponding au ho Email: j oldan@ug .es
2Depa amen o de Es adís ica e In es igación Ope a i a. Uni e sidad de G anada.
Facul ad de Ciencias. A d. Fuen enue a s/n, 18071 GRANADA, Spain.
3Ins i u e o Func ional Nano and So Ma e ials (FUNSOM), Collabo a i e Inno a ion
Cen e o Suzhou Nanoscience & Technology, Soochow Uni e si y, 199 Ren-Ai Road,
Suzhou, 215123, China.
ABSTRACT
Time se ies s a is ical analyses (TSSA) ha e been employed o e alua e he
a iabili y o esis i e swi ching memo ies, and o model he se and ese ol ages o
modeling pu poses. The con en ional p ocedu es behind ime se ies heo y ha e been
used o ob ain au oco ela ion and pa ial au oco ela ion unc ions and de e mine he
simples analy ical models o o ecas he se and ese ol ages in long se ies o
esis i e swi ching p ocesses. To do so, and o he sake o gene ali y in ou s udy, a
wide ange o de ices ha e been ab ica ed and measu ed. Di e en oxides and
elec odes ha e been employed, including bilaye dielec ics in de ices such as:
2
Ni/H O2/Si-n+, Cu/H O2/Si-n+ and Au/Ti/TiO2/SiOx/Si-n+. The TSSA models ob ained
allowed o o ecas he ese and se ol ages in a se ies i p e ious alues we e known.
The s udy o au oco ela ion da a be ween di e en cycles in he se ies allows
es ima ing he ine ia be ween cycles in long esis i e swi ching se ies. O e all, TSSA
seems o be a e y p omising me hod o e alua e he in insic a iabili y o esis i e
swi ching memo ies.
Index Te ms—Resis i e swi ching memo y, RRAM, Conduc i e ilamen s,
Va iabili y, Time se ies modelling, Au oco a iance, S a iona y ime se ies.
1 - INTRODUCTION
Resis i e andom access memo ies (RRAM) ha e shown an ou s anding
po en ial o in o ma ion s o age, especially o in e ne o hings (IoT) and ela ed
applica ions, due o hei excellen scalabili y, low powe ope a ion, as swi ching
speed, easy ab ica ion and good compa ibili y wi h he well-es ablished complemen a y
me al-oxide-semiconduc o (CMOS) echnology [1, 2, 3, 4, 5, 6, 7]. The mos p omising
RRAM uni cells consis o ma ixes o wo- e minal me al/insula o /me al (MIM)
nanocells, in which he elec ical esis ance o he insula ing ilm can be swi ched
be ween a high esis i e s a e (HRS) and a low esis i e s a e (LRS) depending on he
elec ical impulses applied be ween he wo me allic elec odes. Howe e , a e mo e
han one decade o in ense esea ch, RRAM de ices s ill ha e no been mass p oduced
by he indus y, no en e ed in he ma ke o non- ola ile memo ies (NVM), mainly due
o a iabili y p oblems [1, 2, 5, 8, 9, 10]. In RRAM de ices he esis i e swi ching (RS)
3
is ela ed o he o ma ion and up u e o de ec - ich conduc i e ilamen s (CFs) wi hin
he dielec ic laye , which leads o a HRS- o-LRS (se ) and a LRS- o-HRS ( ese )
ansi ion ( espec i ely). Se and ese ansi ions a e ela ed o di usion, edox and
nuclea ion o di e en chemical species wi hin he MIM nanocells [1, 2, 5, 7, 10],
which ake place wi h a e y high deg ee o andomness om one cycle o ano he ,
leading o an in insic high a iabili y. Consequen ly, he elec ical cha ac e is ics
measu ed in a RRAM de ice e lec s he s ochas ici y o hese physical p ocesses, and
p oduces he so-called cycle- o-cycle a iabili y.
In he pas ew yea s, he a iabili y o RRAM de ices has been s a is ically
analysed in mos cases using he Weibull dis ibu ion (WD) [1, 11, 12]. The WD comes
ou in he ield o eliabili y physics [13] and i s use makes sense o RRAMs unde
ilamen a y conduc ion since i is a weakes -link ype dis ibu ion, i.e. he ailu e o he
whole is domina ed by he deg ada ion a e o he weakes elemen . Howe e , his
me hod does no desc ibe all he inhe en s a is ical pa icula i ies o RRAM de ices:
and al hough i has been p e iously employed o deepen on he expe imen al da a
cha ac e is ic ob ained o di e en echnologies [1, 11, 12], i does no en i ely cap u e
he essence o he RS p ocess. In ac , classical eliabili y analyses wi h he Weibull
dis ibu ion assume ha imes o ailu e, in ou case se ol age (VSET) and/o ese
ol age (VRESET), a e independen wi hin a RS se ies. This assump ion may no be alid
in he case o s ochas ic p ocesses associa ed wi h RRAMs because successi e
obse a ions could be highly dependen (in ac , a CF is o med making use o b oken
pa s o p e ious ones). I is wo h highligh ing ha o he ma hema ical app oaches
ha e been p oposed o ackle di e en ace s o he s a is ical s udy o a iabili y in
RRAMs and, in a mo e gene al scope, hin dielec ics. Among hem, he use o a
clus e ing s a is ical app oach complemen ing he use o he WD [14] can be coun ed; in
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line wi h his, con olu ion-based modelling is also in e es ing being no iced [15].
S a egies making use o comple ely di e en dis ibu ion unc ions ha e also been
epo ed; o example, he employmen o phase- ype dis ibu ion unc ions o ce ain
de ices led o in e es ing esul s [16]. Ma ko models ha e also been employed in he
analysis o hese de ices [17, 18]. Kine ic Mon e Ca lo (KMC) and ela ed simula ions
can also be conside ed as s a is ical ools o analyse RRAM a iabili y, as hey allow
modi ying e y speci ic physical pa ame e s o he de ices (i.e. concen a ion o a omic
acancies/dopan s, insula o hickness luc ua ions) wi hin a easonable ange and
analyse he de ia ion o he elec ical cha ac e is ics. A b oad numbe o con ibu ions
ha e been p esen ed in his espec [10, 19-25]. The only d awback o KMC simula ions
is he longe compu a ional ime, which obliga es esea che s o make assump ions ha
simpli y he calcula ions, leading in some cases o a loss o accu acy.
I is clea ha a iabili y is s ill an un esol ed p oblem in RRAM de ices bo h
om he echnological and he modelling iewpoin , and de eloping new analy ical
me hods o shed ligh in o his p oblem is highly necessa y. Times se ies s a is ical
analyses (TSSA) a e powe ul nume ical me hods ha ha e been success ully applied
o decades in he ields o economics and sociology, and mo e ecen ly hey ha e been
also spo adically used in he ield o enginee ing and eliabili y o elec onic de ices
[26-27]. TSSA may be use ul o analyse he a iabili y o RRAM de ices because: i)
he da a (VSET and VRESET) a e collec ed in a con inuous manne o e he ime (cycle- o-
cycle) o a long RS se ies [28-29]; ii) TSSA is app op ia e o physical p ocesses ha
exhibi any kind o ine ia in some o hei pa icula ea u es [30, 31]. In his espec , in
RS cycling, he CF is o med (se p ocess) making use o he emnan s o he CFs
up u ed in he p e ious cycle ( ese p ocess). The e o e, om a s a is ical poin o iew
i is ele an o analyse any nume ical ela ions be ween neighbou ing cycles and assess
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he sys em “memo y” in a long RS se ies ( o pu i in o mal wo ds, he pa ame e s ha
cha ac e ize consecu i e cycles in a RS se ies can be co ela ed and, he e o e, he e m
au oco ela ion comes up na u ally). And iii) unde ce ain ma hema ical condi ions (i.e.
ime se ies s a iona i y, an assump ion ha ou da a dis ibu ions ul il) a comp ehensi e
analysis can be pe o med h ough a ime se ies analysis app oach [30, 31]. Howe e ,
despi e his s ong pa allelism, o he bes o ou knowledge, TSSA ha e ne e been
employed o e alua e he a iabili y o RRAM de ices. In his wo k we p esen he i s
a iabili y s udy o h ee di e en ypes o RRAM de ices using he TSSA, and obse e
ha in all cases essen ial RS pa ame e s, such as VSET and VRESET, can be easonably
o ecas ed making use o ma hema ical models and he in o ma ion o hese pa ame e s
in p e ious cycles. In addi ion, in long RS se ies he dependence o o hcoming cycles
on p e ious ones can be co ec ly s udied by means o co ela ion and au oco ela ion
analyses, cha ac e izing in his manne he ine ia o RS ope a ion in RRAMs o
di e en echnologies.
The manusc ip is o ganized as ollows: in Sec ion II he new model is desc ibed
in dep h, in Sec ion III he ab ica ed de ices and measu emen p ocess de ails a e
gi en, in Sec ion IV he new s a is ical analysis is explained, and in Sec ion V he main
esul s and discussion a e p esen ed. Finally, he main conclusions a e d awn in Sec ion
VI.

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2 – MODEL DESCRIPTION
Back in 1927, G.U. Yule in oduced mode n TSSA o mula ing a model o a
pendulum dynamic mo emen ime dependency [32]. A e ea angemen s in he
co esponding equa ion desc ibing he pendulum mo emen he came ou wi h a second
o de au o eg essi e ime se ies model whe e he pendulum displacemen (z ) om he
equilib ium posi ion was eg essed on he wo p e ious obse a ions (z -1) and (z -2) —
he physics go e ning he pendulum is linked o a second o de di e en ial equa ion—.
In ou s udy, we model he alues o VSET and VRESET o di e en RRAM de ices o e
long se ies o RS cycles by conside ing he alues o p e ious cycles using TSSA [30,
31, 33]. One o he main di icul ies is o ind he o de o he model, i.e. how many
VSET o VRESET alues om p e ious cycles we need o o ecas he cu en cycle ( o a
gene al model p e ious cycles a e usually conside ed, see Equa ion 1). In addi ion, i is
also necessa y o ind he weigh s (Φ1…Φp) o he au o eg essi e model we a e seeking
(see Equa ion 1).
𝑉𝑟𝑒𝑠𝑒𝑡𝑡= Φ1𝑉𝑟𝑒𝑠𝑒𝑡𝑡−1 + Φ2𝑉𝑟𝑒𝑠𝑒𝑡𝑡−2 + ⋯ + Φ𝑝𝑉𝑟𝑒𝑠𝑒𝑡𝑡−𝑝 + 𝜀𝑡
(Eq. 1)
The o de o he model (p) depends on he physics go e ning RS, bu we will no
assume any knowledge o i and we will only make use o he in o ma ion wi hin he
expe imen al da a because he unde lying physics and he echnological de ails o he
ab ica ion p ocess a e “hidden” in he RS da a collec ed. The e o e, he models
ob ained wi h his app oach a e empi ical and he TSSA ou pu will consis o he o de
o he model (p) and he weigh s se (Φ1, ..., Φp). Some imes a model such as he one
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desc ibed in Equa ion 1 wo ks be e o he cen ed a iables; e.g., 𝑉𝑅𝐸𝑆𝐸𝑇 − 𝜇, whe e µ
s ands o VRESET mean in he RS se ies ( his o mula ion is equi alen o include a
cons an e m Φ0 in he model). The e m ε , as usually employed in ime se ies s udies,
s ands o a esidual ha accoun s o he model e o ( he di e ence be ween he
measu ed alue and he modelled alue). This e m is ob ained in TSSA heo y by
gene a ing andom numbe s wi h a no mal dis ibu ion whose a iance co esponds o
he one calcula ed om he measu ed da a. Ne e heless, o he sake o simplici y, his
e m will no be included in he models de eloped he e, ollowing he con en ional
no a ion in he enginee ing con ex . In his app oach, he algeb aic equa ions a e
employed in he o m usually seen in compac modelling, i.e., he cu en alue o ese
o se ol ages a e gi en as a unc ion o a iables al eady known, such as ese and se
ol ages o p e ious cycles. The exis ence o a di e ence be ween he measu ed and
modelled alues is assumed. Equa ion 1 shows wha is called an au o eg essi e (AR)
model [30]; howe e , no always such an easy model can be ob ained. Occasionally
mo e complex models a e needed; i his is he case, an au o eg essi e mo ing a e age
(ARMA) model ha includes AR and mo ing a e age (MA) pa s a e conside ed [30].
MA models a e a linea combina ion o pas esiduals [30-32]. The gene al exp ession
o an ARMA model is desc ibed in he Supplemen a y In o ma ion (see Equa ion A1).
We ha e employed TSSA o s udy and model he da a ob ained om RRAMs
made o di e en ma e ials. Th ee ypes o de ices we e conside ed, wo o hem
including H O2 as he dielec ic, sandwiched by di e en elec ode ma e ials (Ni and
Cu), and ano he one based on a bilaye (TiO2/SiOX) insula ing s ack.
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3 – DEVICE FABRICATION AND MEASUREMENT
The H O2-based RRAM de ices consis ed o Ni/H O2/Si-n+ and Cu/H O2/Si-n+
(20nm hick dielec ic laye s we e used) s acks [28]. The ac i e a ea o he MIM cells
was 5 µm×5 µm, and he de ices we e pa e ned by pho oli hog aphy. A HP-4155B
semiconduc o pa ame e analyse was used in he measu emen p ocess, which
consis ed o collec ing long sequences (se ies) o cu en s. ol age (I-V) cu es by
applying amped ol age s ess (RVS). The Si-n+ subs a e (bo om) elec ode was
g ounded and a nega i e ol age was applied o he Ni (0.3V/s amps) o Cu (0.5V/s
amps) ( op) elec ode, al hough o simplici y we ha e assumed he absolu e alue o
he applied ol age hence o h [28]. The RS phenomenon obse ed o bo h ype o
de ices was unipola , as displayed by he I-V cu es (see Figu es 1a, 1b) [5].
A RS se ies o 2800 cycles was ob ained o he Ni/H O2/Si-n+ RRAMs, and a
se ies o 280 cycles o he Cu/H O2/Si-n+ RRAMs. The alues o VSET and VRESET we e
ex ac ed om he I-V cu es as epo ed in [1, 28, 34], and plo ed in Figu es 1b-1 .
The a iabili y o VSET and VRESET om one cycle o ano he can be clea ly obse ed.
9
Figu e 1. Typical I-V cu es obse ed in (a) Ni/H O2/Si-n+ and (d) Cu/H O2/Si-n+
RRAMs. Expe imen al alues o VSET (b, e) and VRESET (c, ) e sus cycle numbe o a
se ies o con inuous RS cycles unde RVS o RRAMs based on Ni/H O2/Si-n+ and
Cu/H O2/Si-n+ s acks.
The hi d ype o RRAM de ices was ab ica ed using an Au/Ti/TiO2/SiOX/Si-n+
s uc u e. A 2nm TiO2 ilm was g own by a omic laye deposi ion (ALD), on Si-n+
wa e wi h a ~1.5 nm hick na i e SiOX laye . The de ice a ea was 5 µm x 5 µm and he
16
See ha he sample PACF o Cu/H O2/Si-n+ RRAMs has only wo alues ( he
i s and second lagged cycles) abo e he h eshold bound (Equa ion A9), so ha an AR
(2) model could be conside ed. Howe e , aking in o conside a ion ha he second
componen is close o he h eshold bound an AR (1) can be easonable. A e he AR
(1) is adjus ed all he alida ion conside a ions a e sa is ied. The VRESET model o hese
de ices is gi en in Equa ion 5:
𝑉𝑅𝐸𝑆𝐸𝑇𝑡(𝑉) = 0.8577 + 0.5711 𝑉𝑅𝐸𝑆𝐸𝑇𝑡−1.
(Eq. 5 )
Fo he Ni/H O2/Si-n+ de ices, he VRESET model wo ks simila ly o wha was
de e mined o he se ol age modeling (ACF and PACF dec ease). Then an ARMA
(1,1) model (Equa ion 5) holds o he VRESET ime se ies o Ni/H O2/Si-n+ de ices:
𝑉𝑅𝐸𝑆𝐸𝑇𝑡(𝑉) = 0.2377 + 0.8573𝑉𝑅𝐸𝑆𝐸𝑇𝑡−1 + 0.6523𝜀𝑡−1.
(Eq. 6)
The esiduals o he VRESET models depic ed in Equa ions 5 and 6 ha e whi e
noise beha io again, and con i m he app op ia eness o he modeling p ocedu e.
Fo he Au/Ti/TiO2/SiOx/Si-n+ de ices he ACF and PACF a e gi en in Figu e 6.
In his case, he in luence o o he lagged cycles is mo e impo an han in p e ious
cases; he e o e, he ese ol age can be gi en by a linea combina ion o ese ol ages
ob ained in he p e ious cycles. The model is a ac o ized AR (6) ype.

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Figu e 6. (a) ACF and (b) PACF e sus cycle lag o he ese ol age o he
Au/Ti/TiO2/SiOx/Si-n+ de ices unde s udy and he RS se ies desc ibed in sec ion II.
The ACF and PACF minimum h eshold bounds a e 0.2 espec i ely, shown in blue
lines.
The analy ical exp ession o he TSSA desc ip ion o he se ol age in
Au/Ti/TiO2/SiOx/Si-n+ de ices is gi en by Equa ion 7.
𝑉𝑅𝐸𝑆𝐸𝑇𝑡(𝑉)= 0.3228 + 0.3198 𝑉𝑅𝐸𝑆𝐸𝑇𝑡−1(𝑉)+ 0.2197 𝑉𝑅𝐸𝑆𝐸𝑇
𝑡−5(𝑉)
− 0.0703 𝑉𝑅𝐸𝑆𝐸𝑇𝑡−6(𝑉)
(Eq. 7)
The esiduals o he VRESET models depic ed in Equa ion 7 ha e whi e noise
beha io again and consequen ly he modeling p ocedu e is co ec . No e ha he
coe icien o he ( -6) componen is he p oduc o he componen s ( -5) and ( -1)
because o he cha ac e is ics o he pa ame e calcula ion p ocedu e.
The h ee echnologies unde s udy he e show s a iona i y in he se and ese
ol ages se ies. Ne e heless, o o he echnologies a d i in he mean and a iance
shows up. In hese cases, s a iona i y does no hold, he e o e s a iona i y is no a
gene al ule. I we a e aced wi h a nons a iona y da a se ies, he me hodology desc ibed
in he Supplemen a y In o ma ion would no be app op ia e and no models can be
ex ac ed. In his si ua ions he e can be o he op ions since he TSSA heo y p oposes
18
changes o a iables ha lead he newly de i ed se ies o ul ill he s a iona y
equi emen s ha a e needed p io o he modeling p ocess. Au o eg essi e in eg a ed
mo ing a e age (ARIMA) app oaches can be employed ins ead o he AR o ARMA
modeling schemes explained abo e, Re [30, 31].
5 - RESULTS AND DISCUSSION
In o de o es he accu acy o he models p e iously de eloped we supe posed
in he same g aphic he measu ed VSET and VRESET wi h he modeled ones o he de ices
unde conside a ion in his manusc ip . The modeling is a o ecas o he ac ual alue
conside ing p e ious measu ed alues, as i is con en ionally done in TSSA. These
esul s o VSET (Figu e 7) and VRESET (Figu e 8) a e plo ed aking in o conside a ion
Ni/H O2/Si-n+ and Cu/H O2/Si-n+ de ices. As can be seen, he VSET mean gene al end
is desc ibed easonably well by Equa ions 2 and 3. The main dependencies ha e been
co ec ly analyzed and inco po a ed wi h ou p ocedu e; hence, wi hin he ime se ies
con ex , he model wo ks well. We ha e alida ed his poin by s udying he esiduals
co ela ion, and we did no ob ain any signi ican co ela ion be ween he esiduals o
he cu en cycle and hose o he lagged ones ( his cons i u es he alida ion s ep, as
explained in he Supplemen a y In o ma ion). Consequen ly, no mo e dependencies
ha e o be inco po a ed o he models, since no in o ma ion is s a is ically “hidden”.
Al hough he scales and he modeling s a egies a e di e en o VSET, i.e. AR (1)
o Cu/H O2/Si-n+ de ices and ARMA (1,1) o Ni/H O2/Si-n+ de ices, he accu acy is
simila ly easonable, as Figu e 7 shows. The model e o (ε ), as highligh ed be o e,
p esen s a whi e noise s uc u e. We ha e also checked ha he model accu acy is
main ained i he numbe in e al is changed.
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Figu e 7. VSET e sus cycle numbe o he RS se ies unde conside a ion. The
measu ed alues a e shown in blue lines and he modelled ones in ed. (a) Ni/H O2/Si-
n+ RRAMs, cycles 1000-1200, (b) Ni/H O2/Si-n+ RRAMs, cycles 1800-2000 (c)
Cu/H O2/Si-n+ RRAMs, cycles 50-150, (d) Cu/H O2/Si-n+ RRAMs, cycles 50-250.
The esul s o VRESET a e in line wi h hose o VSET. In his case, an AR (1)
model is used o he Cu/H O2/Si-n+ de ices and ARMA(1,1) o Ni/H O2/Si-n+
de ices. Again, he model ep oduces accu a ely he VRESET mean e olu ion o all he
cycle numbe in e als conside ed, as displayed in Figu e 8.
1000 1050 1100 1150 1200
2
3
4
1800 1850 1900 1950 2000
2
3
4
50 100 150
1
2
3
4
5
6
7
8
50 100 150 200 250
1
2
3
4
5
6
7
8
(a) Ni
VSET (V)
Cycle numbe
Measu ed
Modeled
(b) Ni
VSET (V)
Cycle numbe
Measu ed
Modeled
Measu ed
Modeled
(c) Cu
VSET (V)
Cycle numbe
VSET (V)
Cycle numbe
Measu ed
Modeled
Cu
(d)
20
Figu e 8. VRESET e sus cycle numbe o he RS se ies unde conside a ion. The
measu ed alues a e shown in blue lines and he modeled ones in ed. (a) Ni/H O2/Si-n+
RRAMs, cycles 1000-1200, (b) Ni/H O2/Si-n+ RRAMs, cycles 1500-1700 (c)
Cu/H O2/Si-n+ RRAMs, cycles 50-150, (d) Cu/H O2/Si-n+ RRAMs, cycles 50-250.
Fo he Au/Ti/TiO2/SiOx/Si-n+ de ices he compa ison be ween measu ed and
modelled da a is shown in Figu e 9. The p edic ion is also easonable al hough mo e
alues o se (Equa ion 4) and ese (Equa ion 7) ol ages o p e ious cycles a e
conside ed.
1000 1050 1100 1150 1200
1
2
3
1500 1550 1600 1650 1700
1
2
3
50 100 150
0
1
2
3
4
50 100 150 200 250
0
1
2
3
4
Ni
VRESET (V)
Cycle numbe
Measu ed
Modeled
(a) Ni
VRESET (V)
Cycle numbe
Measu ed
Modeled
(b)
Cu
VRESET (V)
Cycle numbe
Measu ed
Modeled
(c) Cu
VRESET (V)
Cycle numbe
Measu ed
Modeled
(d)
21
Figu e 9. (a) Se (b) Rese ol age e sus cycle numbe o he RS se ies unde
conside a ion. The measu ed alues a e shown in blue lines and he modeled ones in
ed. Au/Ti/TiO2/SiOx/Si-n+ de ices a e conside ed he e.
I can be obse ed in he igu es abo e ha in ce ain cases he i is smalle han cu en
alues. This e ec is seen when alues much di e en han he mean show up.
Howe e , we would like o highligh ha he modeling me hodology we a e p esen ing
deals well wi h he p edic ion o he se and ese ol age mean. The cu en ol age
alues a e also p edic ed well in mos cases by using he s a is ical in o ma ion o
p e ious cycles. The e o e, aking in o accoun ha cu en models in he li e a u e do
no ha e his in o ma ion, and ha p edic ion o a iabili y is in e es ing in de ices
(RRAMs) ha show inhe en s ochas ici y, we belie e ha , al hough i is no a modeling
inal solu ion, his echnique is a s ep o wa d ha can be wo hwhile o cha ac e ize he
de ice physics and help wi h a iabili y modeling.
I is impo an o highligh ha Figu e 3 and Figu e 5 (in addi ion o he da a
needed o model building) p o ide in o ma ion abou he RS p ocesses o he H O2-
based de ices analysed. In Figu e 3a, we can see ha he co ela ion be ween cycles is
highe o he Cu/H O2/Si-n+ de ices o he i s lag cycles wi h espec o he se
ol age. In Figu e 5a, a simila end can be obse ed o he ese ol age se ies. So, in
gene al, he in luence o p e ious cycles in Cu/H O2/Si-n+ de ices is highe ; i.e., he CF
emnan s om ese s p ocesses in luence mo e he ollowing se cycles, and ha is why
020 40 60 80 100
0.5
1.0
1.5
020 40 60 80 100
-1.0
-0.8
-0.6
-0.4
-0.2
(a)
VSET (V)
Cycle numbe
Measu ed
Modeled
(b)
VRESET (V)
Cycle numbe
Measu ed
Modeled

22
a highe co ela ion comes ou o he se ol age alues when se e al lag cycles a e
conside ed.
The quali a i e explana ion o his beha iou could be in he na u e o RS in
hese de ices. I is known ha de ices wi h Cu and Ag elec odes a e employed in
conduc i e-b idge RAMs [2, 36, 37] because o he capaci y o Cu and Ag ca ions o
di use in he dielec ic and o m, a e a educ ion p ocess, a me allic-like conduc i e
ilamen . In hese de ices, he educ ion and oxida ion po en ial a e desc ibed by a
he mally ac i a ed p ocess whose ac i a ion ene gy depends on he numbe o a oms
su ounding he one which is aken in o conside a ion [1, 23, 24]. In his manne , he
o ma ion o he pe cola ion pa h in oduces “ine ia” as he educed a om clus e s g ow
dense in he dielec ic since hey end o main ain hei shape hinde ing he oxida ion
p ocesses o hei a oms. This beha iou could explain he highe RS “ine ia” shown by
Cu/H O2/Si-n+ de ices since hey main ain be e he CF o m and size ha de e mine
VSET and VRESET alues. These e ec s would be e lec ed in he ACF plo wi h highe
au oco ela ion alues o cycles no dis an away in he se ies.
The lowe au oco ela ion (Figu es 3a and 5a) o he Ni/H O2/Si-n+ de ices
e lec s a lowe RS “ine ia”, as highligh ed abo e. Tha could also be linked o a
mix u e o RS phenomena, since oxygen acancies could be also in ol ed in hei
esis i e swi ching ope a ion, as sugges ed in Re . [38]. In his espec , he e ec s linked
o ac i a ion ene gy lowe ing o oxida ion o clus e ed me al a oms om he elec ode
would be mi iga ed, p oducing less co ela ion be ween he se and ese alues o
consecu i e cycles.
In bo h cases, Ni/H O2/Si-n+ and Cu/H O2/Si-n+ de ices, he mos signi ican
dependency (co ela ion) is wi h he p e ious alue ( i s lag). In line wi h hese la e
issues, he esul s o Au/Ti/TiO2/SiOx/Si-n+ de ices can be analysed. In Figu es 4 and 6,
23
he cycles co ela ed in he ACF and PACF (mos ly o he ese ol age) is highe han
o he echnologies s udied p e iously. In pa icula , o he ese ol age, he
co ela ion o cycles 9, 11, 13 is high wi h espec o he p e ious cases. This ac shows
again an impo an ine ia in he RS ea u es o his echnology in compa ison o he
de ices based on H O2. The na u e o CFs in TiO2 o he ilamen a y cu en
componen and he p esence o a olume cu en componen ( his componen is linked
o he ion dis ibu ion ha is sp ead ou in he dielec ic and ha can a ec se e al
consecu i e cycles in a RS se ies, as i was shown in de ices o his kind [29]), is key o
explain he highe co ela ion highligh ed abo e.
I is impo an o highligh ha he au oco ela ion and pa ial au oco ela ion unc ions
employed o in he analysis p esen ed abo e (Figu es 3-6) can be used sepa a ely om
he TSSA modeling. They e lec he co ela ion o se and ese ol ages be ween he
di e en cycles, his means ex ac ing he dependencies o he cu en cycle on he
p e ious ones. This in o ma ion is use ul e en i no TSSA modeling is pe o med since
i shows ha he alues s udied as independen da a, a e, in ac , dependen , and his ac
has implica ions in he s udy o cycle- o-cycle a iabili y. I is impo an o highligh
ha ACF and PACF analysis could be easily pe o med on RRAM measu ed da a o
assess he co ela ion be ween he cha ac e is ics o successi e RS cycles.
F om he modeling iewpoin , TSSA models could be implemen ed in ci cui simula o s
wi h Ve ilog-A compile s. TSSA models could be embedded in p e ious models o
accoun o he RRAM s ochas ic beha io and o he co ela ion o ce ain pa ame e s
such as se and ese ol ages in long RS se ies, as explained abo e. We ha e done so
making use o he RRAM S and o d model [40]. In ou case, a log ile o p e ious se
and ese alues o he RRAMs has been employed in a model buil upon he TSSA
24
app oach making use o eg essed alues. The ile was w i en wi h he cu en alues o
se and ese ol ages e e y ime a se o ese was pe o med. P e iously, in o de o
simula e he RVS case we desc ibed expe imen ally abo e, we implemen ed a
modi ica ion o he local enhancemen ac o (γ) [41] o be able o ob ain he
co esponding se and ese ol ages, since hese la e pa ame e s a e no model
pa ame e s.
Finally, we would like o commen on he ac ha he ope a ion egime o hese de ices
would be cha ac e ized by pulses i hey a e used as s o age-class memo y in di e en
chips o spikes o a a ie y o shapes i employed o mimic synapses in neu omo phic
ci cui s. The de ice conduc ance depends on he pulse numbe o each signal
ampli ude, because o he di e en he mal ine ia ha is p oduced in each ope a ion
egime [39]. In his espec , we ha e chosen RVS measu emen s o show a pa icula
applica ion o he TSSA in a well-known cha ac e iza ion app oach. This app oach
could be conside ed as he DC ace o a classical compac model, u he de elopmen s
could be needed o deal wi h ansien e en s in a mo e gene al model whe e he mal
and capaci i e e ec s would be needed.
VI.-CONCLUSIONS
Times se ies s a is ical analyses (TSSA) ha e been used o s udy long se ies o
esis i e swi ching p ocesses. The expe imen al da a analyzed he e we e measu ed in
esis i e andom access memo ies wi h Ni/H O2/Si-n+, Cu/H O2/Si-n+,
Au/Ti/TiO2/SiOx/Si-n+ s uc u es. The con en ional ime se ies echniques we e applied
25
o model he VSET and VRESET o hese de ices; o do so, au oco ela ion unc ions and
pa ial au oco ela ion unc ions we e ob ained o all he ypes o RRAMs.
Au o eg essi e models we e ob ained o Cu/H O2/Si-n+ de ices and he au oco ela ion
unc ion be ween cycles was high, showing an impo an ine ia be ween esis i e
swi ching cycles. The be e di usion o Cu ions in he dielec ic is behind his
beha iou . Fo Ni/H O2/Si-n+ de ices mo e complex models a e needed and
au oco ela ion da a show less ine ia be ween esis i e swi ching cycles. Finally, o
Au/Ti/TiO2/SiOx/Si-n+ de ices a signi ican co ela ion can be obse ed o mo e dis an
cycles in he ese ol age desc ip ion. A physical explana ion has been de eloped in
connec ion wi h he co ela ion esul s ound. The models ob ained can be used o
o ecas he alues o se and ese ol ages in a esis i e swi ching se ies i p e ious
alues a e known. The in o ma ion ob ained in his con ex can be employed in
modelling and in he cha ac e iza ion o RRAM a iabili y.
32
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