PRX ENERGY 3, 011001 (2024)
Pe spec i e
Induc i e and Capaci i e Hys e esis o Cu en -Vol age Cu es: Uni ied
S uc u al Dynamics in Sola Ene gy De ices, Mem is o s, Ionic T ansis o s, and
Bioelec onics
Juan Bisque *
Ins i u e o Ad anced Ma e ials (INAM), Uni e si a Jaume I, 12006 Cas elló, Spain
(Recei ed 25 Augus 2023; e ised 29 No embe 2023; published 8 Janua y 2024)
Hys e esis obse ed in he cu en - ol age cu es o bo h elec onic and ionic de ices is a phenomenon
whe e he cu e’s shape is al e ed on he basis o he measu emen speed. This effec is d i en by in e nal
p ocesses ha in oduce a ime delay in he esponse o an ex e nal s imulus, leading o measu emen s
being dependen on he his o y o he pas dis u bances. This hys e esis effec has posed challenges,
pa icula ly in solu ion-p ocessed pho o ol aic de ices such as halide pe o ski e sola cells, whe e i sig-
nifican ly complica es he e alua ion o pe o mance quali y. In o he de ices, such as mem is o s and
o ganic elec ochemical ansis o s o neu omo phic applica ions, hys e esis is an inhe en aspec o hei
unc ionali y, acili a ing ansi ions be ween diffe en conduc i i y s a es. Na u al and a ificial ionically
conduc ing channels also exhibi p onounced hys e esis, a c ucial componen o gene a ing ac ion po en-
ials in neu ons. In his s udy, we aim o ca ego ize a ious o ms o hys e esis by iden i ying sha ed
elemen s among di e se physical, chemical, and biological conduc ing sys ems. Ou me hod in ol es
examining hys e esis om mul iple angles, using simplified models ha cap u e essen ial esponse ypes.
We analyze sys em beha io using echniques such as linea sweep ol amme y and impedance spec-
oscopy and ansien cu en s esul ing om small ol age s eps. Ou in es iga ion e eals wo p ima y
hys e esis ypes based on how cu en esponds o apid sweep a es: capaci i e hys e esis and induc-
i e hys e esis. These e ms co espond o he dominan componen in he equi alen ci cui , de e mining
he ansien ime esponse. Rema kably, hese concep s p o ide insigh s in o as ly diffe en sys ems,
spanning sola cells, capaci o s, ansis o s, elec ofluidic nanopo es, and p o ein ion channels. The con-
sis ency in elec ical esponses ac oss he diffe en cases enables he iden ifica ion o he p ima y cause o
hys e esis. We also elucida e he equency dependence o hys e esis and he s epwise esponses o sola
cells, illus a ing how undamen al elaxa ions con ibu e o he o e all su plus o defici o cu en du ing
ex ensi e ol age sweeps ha define he cu en - ol age cu e.
DOI: 10.1103/PRXEne gy.3.011001
I. INTRODUCTION
Hys e esis in cu en - ol age cu es is a phenomenon
obse ed in a ious elec onic, ionic, and molecula
de ices, significan ly influencing hei ope a ional ai s
[1–9]. This effec leads o he cu en - ol age cu e’s
beha io being con ingen on he kine ic p ope ies o he
measu emen p ocess. This mani es a ion a ises om a
delay in he cu en esponse ela i e o ol age changes,
he eby al e ing he s eady-s a e cu en - ol age ela ion-
ship du ing ime- a ying pe u ba ions [10].
*[email p o ec ed]
Published by he Ame ican Physical Socie y unde he e ms o
he C ea i e Commons A ibu ion 4.0 In e na ional license. Fu -
he dis ibu ion o his wo k mus main ain a ibu ion o he
au ho (s) and he published a icle’s i le, jou nal ci a ion, and
DOI.
The main app oach o hys e esis is ob ained in he ol -
age sweep echnique wi h scan eloci y sused o measu e
he cu en - ol age cu e. The ol age u a ies wi h ime as
u= s .(1)
The measu emen is pe o med in a cycle o o wa d and
e e se di ec ions, o ice e sa, so as o e u n o he s a -
ing s a e, and i may be epea ed many imes o check he
s abili y o he esponse. This me hod is applied in many
kinds o de ices o ob ain he cu en - ol age cu e a a
low sweep eloci y and o p o ide he dynamic esponse
by measu ing a inc easing sweep a es.
In gene al, hys e esis indica es ha hose physical phe-
nomena ha p oduce he cu en ake some ime o espond
o he changes o applied ol age; hence, he cu en
esponse depends on he his o y o he pe u ba ion. The e-
o e one can apply, o example, a sinusoidal ol age
co e ing he ange (−u1,u1), i.e., u=u1sin(s ), ins ead
2768-5608/24/3(1)/011001(23) 011001-1 Published by he Ame ican Physical Socie y
JUAN BISQUERT PRX ENERGY 3, 011001 (2024)
o Eq. (1), and he hys e esis will be mani es as well.
These measu emen s ex end o e a wide ol age window
in highly nonlinea sys ems and ypically he esponse is
difficul o in e p e in e ms o he in e nal p ocesses.
The specifical physical mechanisms o hys e esis a e
analyzed in diffe en esea ch fields [1–7,9]. Howe e , he
ela ed s udies a e no able o e eal he gene al p inci-
ples o hys e esis ha occu s ac oss a wide a ie y o
conduc ing sys ems, om pho o ol aics o biology. Recen
findings ob ained in a a ie y o sys ems ha shows in ense
hys e esis, such as in mem is o s and halide pe o ski e
sola cells [11–16], enable us o p o ide a gene al clas-
sifica ion o he main hys e esis ypes, using e y sim-
ple models. As he phenomena ha lie behind he hys-
e esis espond in diffe en ypes o measu emen o a
ime-dependen pe u ba ion, his gi es us an oppo uni y
o p obe he sys em in diffe en ways o co ela e he
esponses and ob ain insigh in o he dominan phenom-
ena. This is he app oach we use in his wo k, in con as o
elabo a ing a mechanis ic model o each sepa a e physical
sys em.
The ad an age o his app oach is ha i combines he
measu emen s o diffe en expe imen al echniques, as he
ol age scan o e a wide window, impedance spec oscopy
on a se o s a iona y s a es, he ansien esponse o a se
o s ep pe u ba ions, and he equency dependence o he
cu en - ol age cu es. By me ging hese me hods on he
basis o simple models ha con ain jus a ew pa ame e s,
one can ob ain much insigh in o he dynamical p ope ies
o a sys em and one can p edic and s udy impo an ypes
o beha io . O cou se, hese me hods can be also ega ded
as a fi s s ep owa ds elabo a ed molecula o anspo
models ha implemen pa icula mechanisms, some imes
using la ge pa ame e se s [12,17–21]. These mo e com-
plex models mus finally obey he gene al ules ha will
be ou lined he e.
We s a by e iewing some specific sys ems o in e es
o see wha gene al p ope ies we can in e om obse a-
ions o hys e esis. Since e y diffe en sys ems spanning
biological and a ificial ion channels, pe o ski e sola cells,
mem is o s, and elec ochemical ansis o s show p ope -
ies ha appea closely ela ed, he e mus be an unde lying
s uc u e o hys e esis effec s. We es ablish such s uc-
u e by analyzing he unde lying equi alen ci cui in he
equency domain [10,22]. We show ha his app oach
p o ides a s ong ool o he analysis o a he in ol ed
hys e esis ea u es in a a ie y o expe imen al sys ems.
II. SOME EXAMPLES OF HYSTERESIS
RESPONSE
Hys e esis has been a p ominen phenomenon in lead
halide pe o ski e sola cells, and i s significance has been
widely ecognized [1–3,22–26]. In pho o ol aic de ices,
hys e esis as shown in Fig. 1(A) is a majo p oblem
ha complica es he achie emen o a s a iona y cu en -
ol age cu e, which is essen ial o de e mine he powe
con e sion efficiency (PCE). This d awback makes i nec-
essa y o apply ime-consuming ad anced measu emen
p o ocols such as maximum powe poin acking [30–
35]. Howe e , in o he de ices, such as mem is o s, as
shown in Fig. 1(B) [27], he hys e esis loop is an in in-
sic phenomenon p o iding a unc ionali y ha needs o
be op imized. Indeed, he defining p ope y o mem is o s
[36–43] is he esis i e swi ching om a low- esis ance
s a e o a high- esis ance s a e in a se p ocess ha occu s
a some h eshold ol age. The low- esis ance s a e is
main ained in he e u n pa o he cycle, and a ese p o-
cess in he nega i e ol age side eco e s he ini ial high
esis ance. Also, in field-effec ansis o s and in ionic an-
sis o s o memo ies, synapsis, and neu ons [44–48], he
hys e esis is a main unc ional p ope y.
Capaci o s and elec ochemical supe capaci o s a e cen-
al de ices o elec onics and ene gy s o age. While hey
do no conduc di ec cu en , hey show hys e esis unde
ol age cycling, as p esen ed in Fig. 1(C), which also
shows a gene al cha ac e is ic o hys e esis in cu en -
ol age cu es: he effec becomes amplified when he
ol age scan eloci y inc eases.
The p e ious h ee sys ems ha e in common he p op-
e y ha he elec ical cu en is elec onic, al hough i
is s ongly influenced by ionic phenomena. Halide pe -
o ski es a e mixed ionic-elec onic conduc o s, whe e he
slow hys e esis esponse is a ibu ed o ionic eo ganiza-
ion in he sample [49–52]. In mem is o s, he esis i e
ansi ion is associa ed wi h he buildup o an ionic fila-
men [53,54]. The capaci i e esponse in Fig. 1(C) is due o
he ionic-elec onic double laye o a me al pla e in con ac
wi h a solu ion [28].
Hys e esis can be ound also in many sys ems ha ha e
exclusi ely ionic conduc ion, such as ionically conduc -
ing glasses [4] and elec okine ic anspo in nanopipe es
[5]. Ionic cu en ec ifica ion is a equen ly obse ed
occu ence in bo h p o ein ion channels ha o m na u-
ally and syn he ic nanopo es [55]. These phenomena a e
amply s udied because anspo h ough biological chan-
nels and po es plays a cen al ole in many physiological
p ocesses in li ing o ganisms [56]. Inwa dly ec i ying
po assium (KIR) channels in cell memb anes con ol he
passi e and ac i e elec ical p ope ies o cells, and hey
also link cellula me abolic s a e and memb ane exci abil-
i y in i o. These channels ope a e by in acellula di alen
ca ions and o he molecules blocking he asymme ic open
channel po es [57]. Figu e 1(D) shows he hys e esis in
he p ope y o inwa d ec ifica ion o he KIR channel a
diffe en po assium concen a ions [29]. The s eady-s a e
measu emen p oduces a much la ge cu en han he as
measu emen . This is ano he way o look a hys e esis:
o compa e he cu en be ween a e y slow measu emen
and a e y as one. The hys e esis o ion channels in he
011001-2
INDUCTIVE AND CAPACITIVE HYSTERESIS . . . PRX ENERGY 3, 011001 (2024)
(A)
(B) (C)
(D)
(a)
(a) (b)
(b)
(b)
(c)
(a)
FIG. 1. (A) (a) I-Vcu es measu ed in o wa d scan (FS) and e e se scan (RS) o a pe o ski e sola cell using CH3NH3PbI3.The
maximum powe poin (mpp) is indica ed. The ol age se ling ime was 200 ms and he ligh in ensi y was AM1.5 G 1 sun (100 mW
cm−2). (b),(c) Time-dependen pho ocu en esponse as a unc ion o ol age se ling ime in (b) o wa d scan and (c) e e se scan.
Rep oduced wi h pe mission om H.-S. Kim, N.-G. Pa k, J. Phys. Chem. Le . 5, 2927–2934 (2014). Licensed unde a C ea i e
Commons A ibu ion (CC BY 4.0) license. (B) (a) Cha ac e is ic cu en - ol age cu es o a ha nium oxide–based mem is i e de ice
swi ching om he high- esis ance s a e o he low- esis ance s a e and back. Vol age amps a h ee diffe en speeds: g ay, slow
(43.6 mV s−1); blue, medium (480 mV s−1); o ange, as (4.8 V s−1). The inse shows eadou I-Vcha ac e is ics o diffe en esis ance
s a es depending on he amp speed o he swi ching. A ske ch o he laye s ack o he mem is i e de ice is gi en on he igh side.
(b) Complex plane impedance plo s a e applica ion o 2.1 V o 1 and 30 s. Rep oduced wi h pe mission om R. Ma qua d , F.
Zaha i, J. Ca s ensen, G. Popki o , O. G onenbe g, G. Kolha ka , H. Kohls ed , M. Ziegle , Ad . Elec on. Ma e . 9, 2201227 (2023).
Licensed unde a C ea i e Commons A ibu ion (CC BY 4.0) license. (C) (a) Expe imen al cyclic ol amme y cu es o an Al
elec ode in 0.01 MNa2SO4 eco ded a scan a es be ween 100 and 2000 mV s−1and (b) he capaci i e cu en as a unc ion o he
scan a e. Rep oduced wi h pe mission om O. Gha bi, M. T. T. T an, B. T ibolle , M. Tu mine, V. Vi ie , Elec ochim. Ac a 343,
136109 (2020). Copy igh 2020, Else ie . (D) Cu en - ol age ela ions o he egg cell memb ane a ou diffe en K concen a ions
(10, 25, 50, and 100 mM) in Na- ee medium. Con inuous lines, ins an aneous cu en ; b oken lines, s eady-s a e cu en . Rep oduced
wi h pe mission om S. Hagiwa a, S. Miyazaki, N. P. Rosen hal, J. Gen. Physiol. 67, 621–638 (1976). Copy igh 1976, Rocke elle
Uni e si y P ess.
011001-3
JUAN BISQUERT PRX ENERGY 3, 011001 (2024)
cell memb ane is a cen al p ope y o neu on unc ional-
i y. In he model o Hodgkin and Huxley o he dynamics
o ac ion po en ials in neu ons [58,59], he conce ed delay
o he sodium and po assium channels ha conduc oppo-
si e cu en s is esponsible o he p oduc ion o he ac ion
po en ials by which neu ons ansmi in o ma ion.
A ificial solid-s a e nanopo es and nanochannels can be
ega ded as mimics o p o ein ion channels and ha e also
been de eloped o sensing applica ions. The unde lying
physical cause o ec ifica ion a ises om he asymme y
in he geome y o he nanopo e and/o he dis ibu ion o
su ace cha ges and hei pola i ies along he sidewalls o
he nanopo e [55]. Conical nanopo es a e obse ed o dis-
play hys e esis by dec easing cu en a highe scan a es
[5,60–62].
III. DISTINCTION BETWEEN CAPACITIVE AND
INDUCTIVE RESPONSE
We b iefly analyze he physical na u e o hys e esis.
In he examples in Fig. 1we can obse e wo diffe en
kinds o hys e esis beha io wi h espec o he eloci y
o he ex e nal pe u ba ion. In he capaci o co espond-
ing o Fig. 1(C) he cu en inc eases wi h he scan a e.
The squa e shape becomes p og essi ely la ge . Howe e ,
in he ionic channel co esponding o Fig. 1(D) he as es
scan a es p oduce a dec easing slope and lowe o al
cu en , as ound also in conical nanopo es [5,60–62].
As we a e conce ned wi h he cu en - ol age cu e,
he ac impedance p o ides a na u al amewo k o analyze
he esponse. In gene al, a physical model o any kind o
de ice is highly nonlinea . Howe e , ope a ing a a s a ion-
a y poin o he cu en - ol age cu e, one can pe o m a
small signal measu emen a angula equency ωo he
ol age-cu en a io, and his is called he “impedance” Z.
Then an a bi a y complex physical sys em becomes a
linea model ha can be ep esen ed in e ms o esis-
ances, capaci o s, and induc o s [63–65]. This me hod
also se es o analyze s abili y and o he significan p ope -
ies o he nonlinea sys em [66–69]. In halide pe o ski e
sola cells, he connec ion be ween he impedance com-
ponen s and he hys e esis beha io is unde s ood [10,25,
51,70,71], wi h impo an applica ion o s abiliza ion o
he cu en - ol age cu es and de e mining he s a iona y
pe o mance [72].
One needs o keep in mind ha he elemen s o he
equi alen ci cui depend s ongly on he s a iona y ol -
age, so nonlinea p ope ies, as he hys e esis beha io ,
may change along he measu ed cu en - ol age cu e.
This ea u e is commen ed on in Sec. VII.
The cons i u i e equa ion o a cons an capaci o o
capaci ance Cis he cha ge Q o ol age V ela ion
Q=CV.(2)
The cu en unde ol age sweep (1) is gi en by
I=dQ
d =C s.(3)
Thus, he o wa d cu en is posi i e and he e e se cu -
en is nega i e; see Fig. 1(C)(a). In each case he cu en is
p opo ional o he scan a e. This ac is well known and i
is widely used o he cha ac e iza ion o elec ochemical
capaci o s, as shown in Fig. 1(C)(b) [28].
The impedance o he capaci o is ob ained by he
Laplace ans o m o Eqs. (2) and (3), and he esul is
Z=1
iωC.(4)
Now conside an induc o wi h induc ance L, which is
desc ibed by he equa ion
V=LdI
d ,(5)
wi h he co esponden impedance
Z=iωL.(6)
On he basis o Eq. (4) we can define a gene alized
complex capaci ance C∗as ollows:
C∗=1
iωZ.(7)
We ob ain he capaci ance o he induc o as [73]
C∗=− 1
ω2L.(8)
The e o e, he induc o elemen is associa ed wi h a “neg-
a i e capaci ance” [74,75].
We obse e ha a capaci o and an induc o p o ide e y
diffe en impedance esponses, due o hei in insically
diffe en ime delays. Nex we analyze he significance o
hese p ope ies o he in e p e a ion o hys e esis.
IV. LINEAR MODEL WITH MEMORY TO
EXPLAIN THE BASIC TYPES OF HYSTERESIS
As men ioned, ionic-elec onic de ices may equi e e y
complex models, p o iding in ica e hys e esis p ope ies.
Howe e , he e is an in insic s uc u e o many models
ha we can analyze on he basis o a simple model wi h wo
equa ions ha link he cu en , I o , he ol age, u,andan
addi ional in e nal s a e a iable, w. To explo e hese basic
011001-4
INDUCTIVE AND CAPACITIVE HYSTERESIS . . . PRX ENERGY 3, 011001 (2024)
p ope ies o hys e esis, we conside he linea equa ions
I o =u
Rb
+w+Cm
du
d ,(9)
τk
dw
d =u
Rw
−w
b, (10)
whe e Cmis a capaci ance, and Rband Rwa e cons an
esis ances. The o al cu en in Eq. (9) has h ee compo-
nen s: an ins an aneous cu en u/Rb, a capaci i e cu en ,
and a slow a iable cu en w ha is de e mined by he
elaxa ion equa ion (10), wi h cons an s τkand b.
We calcula e he impedance spec oscopy esponse o
his model o a small sinusoidal pe u ba ion o angula
equency ω. Since Eqs. (9) and (10) o m a linea sys em,
he Laplace ans o m o he a iable s=iωis
ˆ
I o =ˆu
Rb
+w+Cmsˆu, (11)
τksˆw=ˆu
Rw
−ˆw
b. (12)
He e he ci cumflex accen indica es a small pe u ba ion
o he a iable y. We ob ain he impedance as ollows:
Z(s)=ˆu
ˆ
I o
=Cms+Rb
−1+1
Ra+Las−1
, (13)
whe e he esis ance and he induc o a e defined as
Ra=Rw
b, (14)
La=Rwτk, (15)
and he induc o cha ac e is ic ime is gi en by
τL=La
Ra
=bτk. (16)
The equi alen ci cui ha ep esen s Eq. (13) is shown in
Fig. 2(a). This model indica es he h ee pa allel b anches
men ioned be o e: he capaci i e cha ging, he di ec con-
duc ion mode Rb, and he slow induc i e b anch (Ra,La).
The capaci i e elemen has been included in Eq. (9),as
mos de ices do ha e an in insic capaci ance. On he o he
hand, he induc i e elemen is due o he delay equa ion
(10) o he slow a iable. This is no a gene al p ope y
o de ices, bu i is mo e common ha one may suspec ,
especially when ionic-elec onic mixed effec s a e p esen .
Since he induc o is no based on elec omagne ism, i is
gene ally desc ibed as a chemical induc o [73,76].
When he induc o effec is negligible, he impedance is
o med by a posi i e RbCma c; see Fig. 2(b),a cA. I , how-
e e , he induc o pa ame e is la ge, he impedance aces
a loop in he ou h quad an ha ep esen s a “nega i e
capaci ance” ea u e, as shown by a c Bin Fig. 2(b).
Figu e 2(e) shows a mo e gene al equi alen ci cui [77]
wi h an RACAline ha p oduces a double-a c ea u e a
low equencies ha will be discussed la e . This ea u e,
co esponding o su ace pola iza ion o ion diffusion ha
is blocked a he con ac s, was o iginally obse ed in pe -
o ski e sola cells by ligh -modula ed echniques [78]and
by impedance me hods [79,80].
In he ollowing we a gue ha he combina ion o
he s anda d RbCmindica ed in Fig. 2(a) wi h he delay
mode ep esen ed by he RaLab anch explains he exis-
ence o bo h ypes o hys e esis men ioned ea lie in
cu en - ol age cu es o elec onic de ices as capaci o s
and mem is o s, and he wo ypes will be classified as
capaci i e and induc i e hys e esis [11–16,81,82].
The s a iona y cu en - ol age cha ac e is ic o he
model is gi en by
Idc =1
Rb
+1
Rau(17)
and he cu en will be modified when he ol age scan a e
sis fini e.
We now es ic he model o he ime dependence in
Eq. (1). We fi s discuss he case in which he slow cu -
en w esponds wi hou delay, o ocus on he capaci i e
cu en ; hence, w=u/Ra, and he o al cu en is gi en by
I o =u
Rb
+u
Ra
+Ic, (18)
whe e
Ic=Cm s. (19)
As shown in Fig. 2(c), unde a o wa d scan, a posi i e
capaci i e cu en is added o he s a iona y cu en . The
capaci i e cu en is p opo ional o he scan a e s,as
al eady s a ed in Eq. (3).
We now discuss he second componen o he ime-
dependen cu en , he slow a iable w ha esponds o he
applied ol age wi h cha ac e is ic ime τk.F omEq.(10),
we ob ain o a cons an sweep a e
τk s
dw
du =u
Rw
−w
b. (20)
This equa ion needs o be in eg a ed and inse ed in o
Eq. (9) o ob ain he cu en . Fo he ini ial condi ion
w(0)=0, he solu ion is
I o (u)=u
Rb
+u
Ra
+ sτL
Ra
(e−u/( sτL)−1). (21)
011001-5
JUAN BISQUERT PRX ENERGY 3, 011001 (2024)
(a)
(b)
(e)
(c)
(d)
FIG. 2. (a) Equi alen ci cui model. (b) Impedance spec a o Cm=1, Rb=1, Ra=1, and La=0.1 (a c A) o La=20 (a c B).
The ed poin is he dc esis ance. (c),(d) Cu en - ol age cu es a diffe en ol age sweep a es s,u= s . (c) The s a e a iable
esponds as , and he capaci i e cu en wi h capaci ance Cm=1 is added o he di ec cu en . The s eady-s a e cu en (slow)
is shown as he g ay line. (d) The capaci ance is emo ed, Cm=0, and he a iable wcon ibu es o he cu en . The pu ple line
co esponds o an infini ely as scan. The pa ame e s a e Rb=1, Ra=1, τk=0.01, and b=1. (e) A mo e gene al equi alen ci cui
model. The cu es in (b), (c), (d) a e gi en in a bi a y uni s o illus a e he gene al shapes caused by he indica ed pa ame e s.
The induc i e cu en IL=I o −Idc is nega i e in he
o wa d scan and posi i e in he e e se scan, as shown
in Fig. 2(d), co esponding o he nega i e capaci ance o
Eq. (8).I sis small, he ull cu en in Eq. (17) is ac i-
a ed, indica ed by he g ay line in Fig. 2(d). Howe e ,
i sτLu, he induc i e cu en h ough Ra anishes in
Eq. (21) and he emaining as componen
I as =u
Rb
(22)
is indica ed by he pu ple line in Fig. 2(d). This ep esen a-
ion p o ides an excellen accoun o he p ope ies o he
KIR channel obse ed in Fig. 1(D).
V. GENERAL CHARACTERISTICS OF
INDUCTIVE AND CAPACITIVE HYSTERESIS
To ob ain a gene al sys em ha displays bo h capaci i e
and induc i e hys e esis, we can w i e he dynamical equa-
ions wi h he same gene al s uc u e o Eqs. (9) and (10),
011001-6
INDUCTIVE AND CAPACITIVE HYSTERESIS . . . PRX ENERGY 3, 011001 (2024)
bu wi h a bi a y nonlinea unc ions o conduc i i y, ,
and ime delay o he in e nal a iable, g, co esponding o
he specific physical si ua ion:
I o =Cm
du
d + (u,w), (23)
τk
dw
d =g(u,w). (24)
The delay equa ion (24) is e y ypical o b oad kinds o
model, such as he Hodgkin-Huxley model in neu obiol-
ogy [58,59], he ela ed models o p o ein ion channels
wi h memo y effec s [83,84], and he gene ic model o a
mem is o wi h an in e nal s a e a iable [36,40]. I was
epo ed in e y old measu emen s ha ion channels in
neu ons con ain a la ge induc o componen [85–87]. In he
field o halide pe o ski e sola cells, his model ep esen s
he essen ial mechanism o he elec onic ecombina ion
con olled by ionic mo ion ha has been sugges ed in many
pape s o explain he nega i e capaci ance obse ed in he
de ice [15,21,79,88,89].
To de e mine he equi alen ci cui , we ob ain he small
pe u ba ion expansion:
ˆ
I o =Cmsˆu+ uˆu+ wˆw, (25)
τksˆw=guˆu+gwˆw, (26)
whe e he subindex indica es a pa ial de i a i e, u=
∂ /∂uand so o h.
Equa ions (25) and (26) ep oduce Eqs. (11) and (12)
and he impedance model is he same as Eq. (13).The
equi alen ci cui elemen s a e gi en by [69,73,77]
Rb=1
u
, (27)
Ra=− gw
wgu
, (28)
La=τk
wgu
. (29)
These elemen s may all be unc ions o he ol age acco d-
ing o he p ope ies o he unc ions and gp oducing
e olu ions o he hys e esis p ope ies ha a e discussed
below.
The induc o ime cons an is gi en by
τL=−τk
gw
. (30)
Conside he special ole o he cons an bin Eq. (10)
and mo e gene ally gwin Eq. (26) o he sys em’s s able
esponse. I b<0(gw>0), hen he induc o ime con-
s an τLis nega i e, which p oduces an exponen ial g ow h
in Eq. (24). Then he sys em is uns able [90]. Hence, we
equi e ha gw<0.
We can now gene alize he esul s o Fig. 2 o a non-
linea sys em o he ype o Eqs. (23) and (24). We apply a
ol age schedule wi h a ying sweep a es, as indica ed in
Fig. 3(a). In he fi s pa we ha e s>0 and in he second
pa we ha e s<0, so as o e u n o he ini ial ol age.
Simila ly, he ol age can be cycled wi h equency s.
(a)
(b) (c)
(d)(e)
FIG. 3. (a) Time- a ying ol age scan a e wi h a posi i e scan di ec ion pa (1) and a nega i e scan di ec ion pa (2) o he cycle.
(b)–(d) Cu en - ol age cu es. The hick line ep esen s he s a iona y cu en and he dashed line ep esen s he cu en unde he
a ying speed indica ed in (a). (c)–(e) Basic impedance spec a and he associa ed equi alen ci cui . The a ow indica es he di ec ion
o inc easing angula equency. (b),(c) Capaci i e sys em. (d),(e) Induc i e sys em.
011001-7
JUAN BISQUERT PRX ENERGY 3, 011001 (2024)
We assume fi s a sys em ha is domina ed by he capac-
i i e ansien cu en ; see Fig. 3(b). In he fi s pa o he
cycle i is Ic>0, while in he second pa i is Ic<0.
The e o e, he cu en will desc ibe a clockwise loop as
indica ed in Fig. 3(b). We e m his loop a “capaci i e
hys e esis”, and he associa ed impedance spec um is a
posi i e a c; see Fig. 3(c).
I , on he o he hand, he sys em is domina ed by he
slow cu en componen , hen we ha e IL<0in hefi s
pa and IL>0 in he second pa . The hys e esis loop
is coun e clockwise, and we e m i “induc i e hys e esis”;
see Fig. 3(d). The co esponden impedance spec um is a
nega i e a c as indica ed in Fig. 3(e).
F om Fig. 3we conclude ha o a capaci i e hys e esis
he o wa d cu en is la ge han he s a iona y cu e, and
he cu en in he e e se di ec ion is lowe . Howe e , o
sola cell de ices, he ep esen a ion is o en e e sed. This
is because he o al cu en is gi en by
I o =Ipho o −I ec, (31)
whe e Ipho o is he pho ocu en , aken o be posi i e, and
I ec is he ecombina ion cu en associa ed wi h Eq. (23),
which con ains he hys e esis effec s. Hence, o he o -
wa d scan he capaci i e hys e esis cu en appea s lowe
han he e e se scan cu en , as shown in Fig. 1(A) [22],
and he effec o capaci i e and induc i e hys e esis in he
sola cell cu en is opposi e ha in Fig. 3. In he field
o halide pe o ski e sola cells, he capaci i e hys e esis
is e med “ egula ” and he induc i e hys e esis is e med
“in e ed” [12,18,26,91].
A majo p oblem o de e mining he PCE in pe o ski e
sola cells wi h hys e esis, men ioned long ago [1], is ha
he pe o mance is diffe en in he o wa d and e u n
cycles, consequen ly necessi a ing equilib a ion me hods
by successi e slow scans [34,72]. The PCE is a p oduc
o cu en and ol age a he maximum powe poin . We
can see in Fig. 1(A) ha he capaci i e hys e esis p oduces
a la ge PCE in he e e se scan han in he o wa d cycle
[22], while o induc i e hys e esis he con e se is ue: he
PCE is la ge in he o wa d di ec ion [10,12]. These p op-
e ies a e independen o he sign con en ion adop ed o
plo he cu en .
In he li e a u e he e a e diffe en ways o quan i y he
capaci i e and induc i e hys e esis. A hys e esis index is
based on he in eg al o he hys e esis loop [8,92]:
A=V2
V1
IdV. (32)
This o equi alen in eg als (such as he hys e esis en ich-
men cha ge [5,62]) p o ide a posi i e a ea o induc i e
hys e esis and a nega i e a ea o a capaci i e hys e esis, o
ice e sa, acco ding o he sign con en ion o he cu en .
VI. INTERPRETATION OF SYSTEMS WITH
CAPACITIVE AND INDUCTIVE HYSTERESIS
S a ing om he e e ence linea sys em o med by
Eqs. (9) and (10) we pass o he mo e gene al model
o Eqs. (23) and (24) ha p oduces a gene al connec-
ion be ween he dominan elemen s o impedance spec-
oscopy and he ype o hys e esis loop ha will be
ob ained [14,16]. Many ma e ials and sys ems may equi e
s ill addi ional ea u es wi h espec o Eqs. (23) and (24),
depending on he mo phology, he numbe o in e nal p o-
cesses, and hei e olu ion wi h he applied ol age [11,
93]. Ne e heless, he basic dis inc ion o capaci i e and
induc i e hys e esis obse a ion p o ides a use ul diag-
nos ic echnique, as he equi alen ci cui ob ained om
he measu emen o small-pe u ba ion impedance spec-
oscopy can ell us he dominan hys e esis ype ha is
expec ed in he la ge pe u ba ion o linea sweep ol am-
me y, acco ding o he equency o eloci y o he mea-
su emen [13,15,22]. Fu he mo e, he obse a ion o he
p e alen ype o hys e esis can p oduce a guideline o
he basic s uc u al equa ions needed in a specific model
sys em.
To u he de e mine he significance o hese ques ions,
in he ollowing we p o ide a se o examples whe e he
connec ion be ween hys e esis and impedance illus a es
he kine ic and physical elemen s o he sys em.
Fi s we e u n o he ha nium oxide–based mem is-
o co esponding o Fig. 1(B) and we no e ha he
la ge induc i e (“in e ed”) hys e esis loop cha ac e is ic
o he se p ocess o he mem is o , in which he esis-
ance swi ches o a high-conduc ance s a e [94], e eals
he appea ance o he associa ed induc o in he complex
plane plo o he impedance, Fig. 1(B)(b), as p edic ed by
he gene al model. Equa ions (23) and (24) a e he s anda d
cons i u i e equa ions o a mem is o [36,40,95], and we
can ema k ha mem is o s, in he ol age ange o he se
p ocess, will display a la ge in e ed hys e esis loop in he
se cycle, whe e he cu en inc eases a a ce ain h esh-
old o ol age o cu en [12–14]. The associa ed chemical
induc o appea s na u ally in he impedance esponse, as
obse ed in Fig. 1(B)(b). A simila connec ion o in e ed
hys e esis and he induc i e loop has been epo ed o
halide pe o ski e mem is o s and sola cells [14,16,96],
and hese de ices a e discussed in Sec. VII.
To u he analyze he change o hys e esis ype, we
conside he p ope ies o conical ionic channels in an
elec oly e solu ion, shown in Fig. 4(a), ha show ec i y-
ing p ope ies due o he elec ical in e ac ion be ween he
unc ionalized cha ges on he conical po e su ace and he
nanoconfined ionic solu ion [55,98]. Nanopo e mem is-
o s ha e been s udied o hei s ong hys e esis p ope ies
[5,61,62], and induc i e impedance has been epo ed [60],
bu he co ela ion be ween he ype o hys e esis and he
induc i e elemen was no es ablished.
011001-8
INDUCTIVE AND CAPACITIVE HYSTERESIS . . . PRX ENERGY 3, 011001 (2024)
(a)
(b)
(c)
(d)
(e)
FIG. 4. (a) Cu en - ol age cu e measu ed a equency s=10 Hz o a mul ipo e memb ane in 100 mMKCl solu ion a neu-
al pH. The inse shows he elec ochemical cell wi h he memb ane. (b) Impedance spec a a diffe en e e se ol ages, wi h he
co esponding Bode plo o he imagina y pa o he impedance in (c). (d),(e) Impedance spec a and Bode plo s a o wa d ol age.
Adap ed om P. Rami ez, J. Ce e a, S. Nasi , M. Ali, W. Ensinge , S. Ma e, J. Colloid In e ace Sci. 655, 876–885 (2024) wi h
pe mission om Else ie .
We show in Fig. 4 esul s ob ained ecen ly by Rami ez
e al.[97]. On he posi i e ol age side o he cu en -
ol age cu e in Fig. 4(a), he mem is i e po es show a
s ong in e ed hys e esis, accompanied by he no o ious
induc i e impedance loop in Fig. 4(d). The o wa d and
e e se scans o he cu en - ol age cu e c oss a he o i-
gin, and i ollows ha he ype o hys e esis is changed.
On he nega i e side, he hys e esis is capaci i e, and he
impedance shows pu ely capaci i e spec a; see Fig. 4(b).
Rema kably, i has been shown ha he side o induc-
i e hys e esis o he po e esponse can be ansposed om
a posi i e po en ial o a nega i e po en ial by modula-
ion o he elec oly e composi ion [8], opening in e es ing
applica ions o nanofluidic neu omo phic ci cui s.
011001-9
JUAN BISQUERT PRX ENERGY 3, 011001 (2024)
capaci i e hys e esis in e ms o elemen a y elaxa ions,
we now apply a simila me hod o he linea model o
Eqs. (9) and (10), ep esen ed as he equi alen ci cui in
Fig. 2(a).
When a ol age s ep is imposed on he conduc ing sys-
em, i is impo an o conside he p esence o he se ies
esis ance, as indica ed in Fig. 2(a). Then he applied ol -
age Vapp is di ided be ween he ac i e zone ol age, u,and
he se ies esis ance, as ollows:
Vapp =RsI o +u. (42)
We apply wo consecu i e ol age s eps V,2Vo du a-
ion and measu e he cu en . The esul s o he cal-
cula ion a e shown in Fig. 8(B) o a sys em in which
he induc i e p ocess is e y as and can be igno ed, so
ha he ime esponse is domina ed by he capaci ance
and esis o s. In each diag am he g ay line is he s a-
iona y cu en ha can be expec ed a he gi en applied
ol age, i.e.,
Idc =Vapp
R o
, (43)
whe e
R o =Rs+1
Rb
+1
Ra−1
. (44)
We obse e ha when a ol age s ep is applied a 0in
Fig. 8(B)(a), he e is a capaci i e peak. This is because in
he ini ial ins an he capaci o is discha ged and all he
(a) (b)
(c) (d)
(e)( )
FIG. 9. S eady-s a e cu en measu emen s using o wa d and e e se s epwise ol age sweeps o wo me al halide pe o ski e sola
cells diffe ing in he con ac : (a),(b) spi o-OMeTAD and (c),(d) CuI-based. (e) Complex plane impedance plo s o he wo de ices, and
fi o he equi alen ci cui model. ( ) Va ia ion in low- equency esis ance and capaci ance in bo h de ices. Impedance measu emen s
we e pe o med unde cons an illumina ion. OC, open ci cui . Rep oduced wi h pe mission om G. A. Sepalage, S. Meye , A. Pascoe,
A. D. Scully, F. Huang, U. Bach, Y.-B. Cheng, L. Spiccia, Ad . Func . Ma e . 25, 5650–5661 (2015). Copy igh 2015, Wiley.
011001-16
INDUCTIVE AND CAPACITIVE HYSTERESIS . . . PRX ENERGY 3, 011001 (2024)
ol age goes o he se ies esis ance [121], so he ini ial
peak heigh is V/Rs. The ea e he capaci o cha ges
wi h a cha ac e is ic ime τC=RsCm, un il he equilib-
ium cu en Idc is eached. The nex inc ease V aises
he cu en again. In Fig. 8(B)(b) he capaci ance is 10
imes la ge , so he cha ging ime is 10 imes longe . The
cha ging canno be comple ed in each s ep, and a diffe ence
Iis accumula ed a he end poin . This is he posi i e
capaci i e hys e esis cu en .
We ema ked ega ding Eq. (31) ha he ep esen a ion
o he pho ocu en in a sola cell is in e ed wi h espec
o he da k cu en . Fo Fig. 8(B)(c), we apply Eq. (31) and
flip he capaci i e peaks, so ha Ibecomes nega i e. The
esul ing pa e n explains well he cu en esponse o ol -
age s eps in a o wa d scan in Figs. 1(A)(b) and 8(A)(b).
We can deduce om he model ha in a as e scan, he
s eps will be sho e in ime, and he accumula ed defici
o cu en will be la ge . This is he p ope y o capaci i e
hys e esis ha is obse ed in Fig. 8(A)(c).
Fo Fig. 9 he s ep p ocedu e is applied o wo pe -
o ski e de ices ha diffe in he hole ex ac ion con ac
[120]. In he fi s cell [Figs. 9(a) and 9(b)] he elaxa ion
is no comple ed in each s ep and he sample shows signi -
ican hys e esis. In he second case [Figs. 9(c) and 9)(d)]
he cha ging ime τCo he capaci o is much sho e , so
he hys e esis is educed. This was achie ed by dec ease
o bo h he low- equency esis ance and he capaci ance,
as shown in Fig. 9( ).
We now u n ou a en ion o he influence o he induc-
o in he s ep esponse o he cu en , shown in Fig. 10.
The ansien begins wi h a capaci i e peak, as in he
o me case, bu now addi ional pa allel esis ance Rais
ac i a ed when he induc o esponds in ime o o de τL
[121]. Hence, he cu en inc eases wi h ime un il he equi-
lib ium cu en is eached; see Fig. 10(a). In Fig. 10(b)
he induc o ime is longe , and he final cu en canno
be achie ed. This p oduces a o al nega i e defici , which
explains he p ope y o nega i e hys e esis in e ms o he
elemen a y s ep beha io . A ull classifica ion o he ime-
domain esponse when se e al induc i e and capaci i e
elemen s ac in conce was p esen ed in Re . [111].
In summa y, capaci i e and induc i e p ocesses p o-
ide opposi e ypes o ime ansien esponse. This is
closely ela ed o he diffe en equency-domain esponse
o bo h elemen s indica ed in Figs. 3(c) and 3(e).Ina
capaci i e p ocess he impedance dec eases a inc eas-
ing equency. Co espondingly, he cu en o capaci i e
cha ging dec eases wi h ime. Howe e , he impedance o
he induc o dec eases a lowe equency. Consequen ly,
he cu en o he induc i e p ocess inc eases wi h ime,
as shown in Fig. 10. The induc i e p ope y is an essen-
ial componen o he phenomenon o synapse po en ia ion
ha is necessa y o neu omo phic compu a ion elemen s
[8,112,121–123].
IX. THE FREQUENCY DEPENDENCE OF
HYSTERESIS
In his sec ion we commen on he mani es a ion o hys-
e esis as a unc ion o he equency so measu emen o
cu en - ol age cu es [114]. This is shown in Fig. 11(A)
o he po ous nanochannels o Fig. 4[8]. The ol age is
scanned by a sinusoidal wa e. A low equencies he ull
s a iona y cu en is obse ed, bu a highe equencies a
posi i e ol age he induc i e hys e esis becomes signifi-
can and he cu en emains lowe due o emo al o he
slow- esponse componen .
We conside an impo an esul shown in Fig. 11(B)
[72], which illus a es he co espondence o he hys e esis
effec and he equency o measu emen , acco ding o he
(a) (b)
FIG. 10. Cu en esponse o he linea model o wo consecu i e ol age s eps, om ime 0, o du a ion , he fi s wi h ampli ude
V=0.1 and he second wi h ampli ude V=0.2. The common pa ame e s a e Rb=1, Ra=0.1, Rs=0.05, 0=0.5, =1, and
Cm=1. The diffe en cases a e (a) τk=0.3 and (b) τk=3. The ed lines indica e he applied ol age s eps and he g ay lines indica e
he s eady-s a e cu en a he gi en ol age. The o ange a ows indica e he excess cu en a he end o he cycle wi h espec o he
s eady-s a e alue. The cu es a e gi en in a bi a y uni s o illus a e he gene al shapes caused by he indica ed pa ame e s.
011001-17
JUAN BISQUERT PRX ENERGY 3, 011001 (2024)
(A)
(C)
(B)
(a) (b) (c)
FIG. 11. (A) (a) I-Vcu es o a mul ipo e memb ane in 100 mMKCl solu ion a neu al pH, pa ame ically in he elec ical
po en ial scan a e, cha ac e ized by he signal equency =s/2π, ob ained wi h a ol age ampli ude o 2 V. The a ows indica e he
signal ime e olu ion. Rep oduced wi h pe mission om P. Rami ez, V. Gomez, J. Ce e a, S. Ma e, J. Bisque , J. Phys. Chem. Le .
14, 10930–10934 (2023). Copy igh 2023, Ame ican Chemical Socie y. (B) Complex plane plo o he impedance o a CH3NH3PbI3-
based mesoscopic sola cell, measu ed a 950 mV unde equi alen 1000 W m−2sola i adia ion. The Lissajous cu es (cu en
e sus ol age) co esponding o a sinusoidal pe u ba ion (V ms=22 mV) unde he same measu ing condi ions (ligh and dc bias)
a e shown a he op o he indica ed equencies. Rep oduced wi h pe mission om N. Pelle , F. Gio dano, M. I. Da , G. G ego i,
S. M. Zakee uddin, J. Maie , M. G ä zel, P og. Pho o ol . Res. Appl. 25, 942–950 (2017). Copy igh 2017, Wiley. (C) Simula ion
o halide pe o ski e sola cell hys e esis a diffe en scan eloci ies. (a) Measu emen p ocedu e o eco d he cu en - ol age cu es
and band diag ams s a ing a open-ci cui ol age (OC) and p og essing o sho -ci cui cu en (SC) and back. (b) Simula ed as -
hys e esis powe con e sion efficiency plo in o wa d and e e se scans and he cha ac e is ic efficiencies a slow, medium, and as
scan speeds. (c) Co esponding simula ed cu en - ol age cu es. Rep oduced wi h pe mission om V. M. Le Co e, J. Diekmann,
F. Peña-Cama go, J. Thiesb ummel, N. Tokmoldin, E. Gu ie ez-Pa ida, K. P. Pe e s, L. Pe digón-To o, M. H. Fu sche , F. Lang, J.
Wa by, H. J. Snai h, D. Nehe , M. S ol e oh , Sola RRL 6, 2100772 (2022). Copy igh 2022, Wiley.
ype o impedance esponse ha p e ails a he equency
chosen. By changing he cycling o he cu en - ol age
cu e (wi h a small ampli ude o 22 mV), we change he
poin o measu emen in he complex plane plo o he
impedance. I is obse ed ha when he imagina y pa o
he impedance becomes la ge, co esponding o a capac-
i i e esponse a his equency, he hys e esis inc eases,
as shown in he Lissajous cu es. In Fig. 11(B) he wo
componen s in oducing ime lag in he espec i e a cs
(one a low and ano he one a high equency) a e capac-
i i e, co esponding o he model o Figs. 2(e) and 5(c).
The low- equency induc o esponse is no p esen ed in
Fig. 11(B), al hough i could exis , bu he impedance is
no epo ed up o e y low equency. I we measu e up o
such a low equency, and find he induc o componen , we
can in e ha he induc i e ea u e o he hys e esis could
appea , causing he Lissajous figu e o spin in he opposi e
di ec ion, as shown in Fig. 3.
011001-18
INDUCTIVE AND CAPACITIVE HYSTERESIS . . . PRX ENERGY 3, 011001 (2024)
Figu e 11(C) shows a s udy o he hys e esis ai s
o d i -diffusion simula ed halide pe o ski e ope a ion,
which co esponds well o he expe imen al obse a ions
in he same wo k [113]. The sola cell is s abilized a open-
ci cui ol age and hen measu ed in e e se and o wa d
cycles, as shown in Fig. 11(C)(a). The PCE is la ge in he
e e se scan, so his model con ains only capaci i e hys-
e esis, as discussed in Sec. V, and he impedance pic u e
mus be he same as in Fig. 11(B). The end o hys e e-
sis as a unc ion o equency shown Fig. 11(C)(b) is he
same as ha in Fig. 11(B): he sys em su moun s a capac-
i i e a c cen e ed a a scan a e o 10 V s−1. A a high
equency o he scan, he sys em becomes a pu e esis-
ance, and he o al PCE inc eases sligh ly. This is because
he low- equency a c in Fig. 11(B) has been emo ed,
and i means ha he low- equency a c is dele e ious o
he pe o mance, because i is a ecombina ion loss a he
con ac s, caused by he ionic dis ibu ion, ha is a oided
in he as measu emen . In he s udy in Re . [114], he
efficiency and hys e esis inc ease a inc easing equency
as in Fig. 11(C), bu hen s a o dec ease a he high-
es equencies. This means ha he scan is affec ing he
high- equency a c ha holds he PCE o he sola cell.
Howe e , he p e ious conside a ions should be ega ded
as p elimina y in ui i e explana ions. In he ol age scan
he sys em changes o e a wide ol age ange ha co -
e s a la ge a ia ion o he impedance pa ame e s [63,124],
and a de ailed quan i a i e ans o ma ion be ween he ime
domain and he equency domain becomes necessa y [11].
We ema k ha one can ob ain a ansi ion o he kind
o hys e esis (capaci i e o induc i e o ice e sa) in wo
diffe en ways: by changing he ol age modi ying he
dominan low- equency componen , as shown in Fig. 6,
o by ope a ing a a fixed ol age, by changing he e-
quency o he measu emen o he cu en - ol age cu e,
as in Fig. 11.
X. CONCLUSION
Ou comp ehensi e explo a ion has un eiled he in i-
ca e na u e o hys e esis ac oss a spec um o elec-
onic, ionic, and mixed ionic-elec onic de ices. Th ough
a mul idimensional app oach, we ha e delinea ed wo
undamen al hys e esis a che ypes—capaci i e and induc-
i e— oo ed in dis inc i e ansien esponses. These
a che ypes, e ealed h ough di e se analy ical me hods,
p o ide a uni ying amewo k o deciphe hys e esis in
sys ems as dispa a e as sola cells, capaci o s, ansis o s,
and ion channels. Fu he mo e, he use o complemen-
a y me hods acili a es he p edic ion and classifica ion o
mo e conc e e physical and molecula models ha accoun
o he complica ed e olu ion o he cu en wi h ime in
highly he e ogeneous and nonlinea sys ems.
Ou analysis p o ides a c i e ion o making hese
ad anced physical models, such as hose elabo a ed by
d i -diffusion equa ions and pola iza ion assump ions.
F om he knowledge o basic s uc u al condi ions o
equa ions ha p oduce capaci i e and induc i e hys e esis
and hei combina ions, we ha e a ya ds ick o measu e
ad anced physical models ha con ain specific physical
effec s. We can fi s de e mine he e olu ion o hys e esis
in an expe imen al sys em using he equency analysis and
gene al desc ip i e models, and can hen de elop conc e e
specific explana ions ha sa is y he o e all beha io . We
emphasize ha he ac ha a model p o iding some ype
o desc ip ion o hys e esis is no enough o alida e such
a model. Many equi alen models can be made, p o ided
ha hey sa is y he basic s uc u al condi ions ha gi e he
igh e olu ion o he equi alen ci cui . A mo e s ingen
es o models and heo ies is needed, a leas conside ing
a a ie y o expe imen al esponse me hods.
By elucida ing he unde lying mechanisms d i ing
hys e esis, we inc ease ou unde s anding o i s pe a-
si e p esence in bo h na u al and enginee ed con ex s,
con ibu ing o he b oade unde s anding o dynamic
esponses in complex sys ems.
ACKNOWLEDGMENTS
This wo k was unded by he Eu opean Resea ch Coun-
cil ia Ad anced G an No. 101097688 (Pe oSpike ). I am
g a e ul o discussions wi h Agus ín Bou, An onio Gue -
e o, Ced ic Gonzales, En ique He nández-Balague a, and
Pa icio Rami ez.
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