ECS Journal of Solid State Science and Technology, 2 (11) P423-P431 (2013) P4232162-8769/2013/2(11)/P423/9/$31.00 © The Electrochemical Society

Resistive Switching: A Solid-State Electrochemical PhenomenonH. Nafe∗,z

Institut fur Materialwissenschaft, Universitat Stuttgart, 70569 Stuttgart, Germany

Resistive switching is the consequence of the electrochemical equilibrium between electrons, ions and the respective neutral speciesin a mixed ionic-electronic conductor that acts as electrolyte of a galvanic cell under ion-blocking mode conditions due to which anexternally applied voltage induces drastic variation of the n- and p-type electronic conductivity lagging behind the voltage.© 2013 The Electrochemical Society. [DOI: 10.1149/2.006311jss] All rights reserved.

Manuscript submitted May 14, 2013; revised manuscript received July 30, 2013. Published August 7, 2013.

Since Strukov et al.1 have announced the invention of a new pas-sive electronic circuit element, i.e. memristor, the number of papersin this field has literally exploded. This is particularly true with re-gard to a certain type of memristors, viz. resistive switchers, becauseof their possible application as non-volatile memory. The fact thatthere are much older reports in the literature on resistive switching(e.g.2) rightly made Prodomakis et al.3 state that memristor is not aninvention but a description of a basic phenomenon of nature previ-ously known. Recently, Chua4 has given mathematical relationshipsunderlying the very phenomenon in terms of current-voltage curves ofa model electrical equivalent circuit. As useful as these relationshipsmay be, they hardly reveal anything about the microscopic and macro-scopic physical and chemical processes and circumstances controllingthe cause-action relation for the functioning of a resistive switcher.This cause-action relation will be described in the following, as acomplex sum of purely solid-state electrochemical phenomena.

Role of Solid-State Electrochemistry

That electrochemistry may play a role for the topic has alreadybeen suggested by Strukov et al.1 insofar as they vagely emphasizedthe role of coupled electronic and ionic transport.5 Their opinion mighthave been triggered by the work of Aono et al.6–9 and Kozicki et al.10,11

who both regarded the switching effect as being conditional on thepresence of a solid electrolyte, i.e. an ion-conducting solid. In themanner these authors described the relevant phenomenon, namely ascaused by ionic charge carrier migration into the electrolyte materialand out of it, followed by precipitation or dissolution of the pertainingneutral species, the process more resembles the functioning of a solid-state electrochemical pump rather than that of a resistive switcher. It is,at any rate, only one type of switchers that is intended to be covered bythis electrochemistry-related attempt of interpretation, viz. the bipolarone with cation conductors. Because of the significance that has beenattached to the topic in the current literature, a critical discussiondedicated to the matter will be provided in a separate paragraph furtherbelow in the text.

In the most general sense a resistive switcher is a solid electrolytegalvanic cell under load.12 The material sandwiched between twoother prevalently electronically conducting materials, functioning aselectrodes, is a solid ionic conductor of primarily inorganic nature,to be more precise, a solid mixed ionic-electronic conductor (MIEC)with ionic and electronic conductivities of largely comparable mag-nitude. The MIEC extends along the x-axis from position x′ to x′′

with the distance L lying inbetween. For the sake of simplicity, theMIEC membrane is regarded to be unconfined in the space directionsperpendicular to x (Fig. 1).

Interrelationships Between Quantities Related to the Ionic andElectronic Charge Carriers

The flux of the charge carrier k through the solid from one electrodeside to the other is determined by the current density ik that is related

∗Electrochemical Society Active Member.zE-mail: naefe@imw.uni-stuttgart.de

to the conductivity of k, σk , and to the gradient of the electrochemicalpotential, ηk, throughout the conducting medium:

ik = − σk

zkFgrad ηk [1]

where F is the Faraday constant and zk is the charge number or valenceof k. The electrochemical potential of a charged species is definedas sum of the chemical potential μ and the electrical potential ϕ:ηk = μk + zk F ϕ, with the chemical potential of k being related tothe thermodynamic activity ak by the relation: μk = const. + RT ln ak

(R: gas constant; T: temperature). For the sake of simplicity, in thefollowing each gradient is supposed to be one-dimensional with anon-vanishing component in x-direction only.

Mobile charge carriers are either the cations or anions of the crystallattice of the MIEC and electrons as well as holes. Independent of thenature of k the conductivity σk is the product between the concentrationck and the mobility uk, with F and the absolute value of zk beingproportional constants:

σk = |zk| F uk ck [2]

Ionic conduction is caused by the existence of charged defects,e.g. vacancies or interstitials, due to intrinsic disorder in the solid.Because of electrical neutrality, the concentration of these defectscannot be changed independently as charge has to be compensated.Charge compensation may take place due to a change of the concen-tration of oppositely charged ionic defects or due to a change of theconcentration of the electronic charge carriers of the solid.

Figure 1. Schematic representation of a resistive switcher as solid-state gal-vanic cell with a mixed ionic-electronic conductor (MIEC) as electrolyte.(L = x′′-x′; thickness of the MIEC membrane).

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P424 ECS Journal of Solid State Science and Technology, 2 (11) P423-P431 (2013)

Figure 2. Logarithmic diagram of the concentration of defects as a functionof the activity of neutral species M, corresponding to the mobile ion M+ ina cation-conducting solid with Frenkel disorder (cf. eq. 5). The curves resultfrom mass action law considerations of the equilibria (3), (4) and (5) fordifferent limiting cases of the electro-neutrality conditions given on top of thediagram. ([D]: concentration of an arbitrary defect D; Ke: equilibrium constantof equilibrium (4); KF: equilibrium constant of equilibrium (5)).

The concentration of oppositely charged ionic defects may beaffected by shifting the position of the relevant defect equilibrium as aconsequence of various measures. For instance, chemically differentcounterions are partially substituted for the counterions of the hostlattice by means of doping. Charge compensation involving electroniccharge carriers results from the equilibrium between electrons, e′, andmobile ions, i 1), on the one hand, and neutral species Xξ existing inthe ambience of the solid and interacting with it, on the other hand:

i + zi e′ −−−⇀↽−−− 1

ξXξ [3]

The concentration of electrons in its turn is related to that of holes,h. by the intrinsic electronic equilibrium:

0 −−−⇀↽−−− e′ + h· [4]

X corresponds to the discharged mobile ion i. ξ is the number ofatoms associated in the standard state of pure substance of X.

Keeping in mind that the ionic and electronic defects are solutesof the solid solution MIEC and that Xξ is a species of a separate phaseambient to the solid, equilibrium (3) can actually be established onlyat the phase boundary between MIEC and the ambience, i.e. at bothelectrodes of the galvanic cell of Fig. 1. According to Wagner13 thesame equilibrium can, nevertheless, be assumed to exist as a virtualone throughout the solid. Therefore, the concentrations of the ionicand electronic defects of the MIEC are related to each other and to thechemical potential μXξ

, expressed as thermodynamic activity a Xξ.

Fig. 2 exemplarily illustrates such a relationship for a cation-conducting MIEC with Frenkel disorder in the sublattice of the mono-valent cation M+ 2):

MM + Vi −−−⇀↽−−− M·i + V′

M [5]

where MM and M·i designate M-cations on a normal cation lattice site

and on an interstitial site, respectively. V′M and Vi are vacancies in the

cation lattice and on interstitials. aM as a special case of aXξstands for

the thermodynamic activity of the neutral species M corresponding tothe mobile cation M+.

1)Throughout the text, i denotes either an arbitrary ionic defect or the current densityaccording to eq. 1. As a subscript, i is also used to characterize ion-related quantities orinterstitial defects.2)Kroger-Vink notation is preferred14.

M A M A

x0 MaAb +

1 x0( ) McAd

x1 MaAb +

1 x1( ) McAd

a zM = b zA

c zM = d zA

zA = 1+ zA

Figure 3. Variation of the macroscopic MIEC composition by change of theproportions of the two stoichiometric crystals MaAb and McA∗

d being theconstituents of the MIEC. (x0, x1: mole fractions; a, b, c, d: stoichiometricnumbers of the two crystalline compounds; α, β: M-to-A ratio).

The diagram of Fig. 2 characterizes the situation in an infinitely thinlayer out of the MIEC membrane at an arbitrary position x assumingthat this layer is fictively equilibrated with aM. It demonstrates theinterdependence of the concentrations of electrons, holes and ionicdefects with the thermodynamic activity aM as a consequence of theinterplay of equilibria (3), (4) and (5). At moderate values of aM theconcentration of ions in the solid is practically invariant as the equalityof the concentrations of the oppositely charged ionic defects M·

i andV′

M controls electro-neutrality. Consequently, equilibrium (3) requiresthe concentration of electrons to be increased if the rising activity aM

forces M to be built in the crystal lattice of the solid. The oppositehappens upon decrease of aM. Then ions tend to leave the solid andtransform to neutral M with the opposite consequences for the changeof the concentration of the electronic charge carriers. At extremelylow and high values of aM the same shift of equilibrium (3) takesplace, then, however, under different conditions with regard to theprevailing electro-neutrality conditions as they are given on top of thediagram of Fig. 2. As a result, the concentration of electronic chargecarriers responds less effectively to variation of aM.

By means of definition (2) the chemical potential dependence ofthe concentration of electronic charge carriers exemplarily depictedin Fig. 2 can be recast into a chemical potential dependence of theelectronic conductivities of n- and p-type, σn and σp:

σn = Kn ·a1/ξzi

Xξ

a1/zii

[6]

σp = Kp · a1/zii

a1/ξziXξ

[7]

where the abbreviations σn = σe′ and σp = σh. are used. ai is thethermodynamic activity of the mobile ion. Kn and Kp are temperature-dependent constants related to the equilibrium constants of equilibria(3) and (4).

As illustrated in Fig. 2, ai may either be practically constant withrespect to a Xξ

or may be reduced or increased by a shift of the con-centration of one sort of ionic defects relative to the concentration ofthe other one. The enrichment or depletion of ions implies a changein the macroscopic composition of the MIEC the extent of which isstrongly related to the level in the concentration of electrons or holes.Is the level of the electronic charge carrier concentration high enough,ionic defects otherwise present in the solid can be re-charged by trap-ping a part of the freely mobile electrons or holes. Such a change inthe overall MIEC stoichiometry is exemplarily illustrated in Fig. 3 byassuming that the valence of a definite proportion of the host anion ofa cation conductor changes by unity. Keeping in mind that the MIECserves as a solvent for all defects, the change in the MIEC compositionmay affect the magnitudes of the equilibrium constants of the defectequilibria involved, which may lead to a shift of the relative positionof Ke and KF in Fig. 2.

With the equilibrium between ionic, electronic and neutral speciesbeing the essence of electrochemistry, the phenomena resulting fromequilibrium (3) are electrochemical by their nature which is expressedby a coupling of the gradients along the x-direction of the neutral andcharged species Xξ, i and e ≡ e′:

1

ξgrad μXξ

= grad ηi + zi grad ηe [8]

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ECS Journal of Solid State Science and Technology, 2 (11) P423-P431 (2013) P425

By virtue of the foregoing interrelationship, a gradient in the chem-ical potential of Xξ may cause an electrical potential difference �ϕto be generated, i.e. a voltage, and, conversely, an externally appliedvoltage may induce a gradient in terms of μXξ

. The question as towhich of these quantities takes over the control over the other one isanswered by the operation mode and the operation conditions of thegalvanic cell under consideration.

Ion-Blocking Operation Mode

The most important prerequisite for that the voltage, U, appliedbetween the electrodes, controls the chemical potential gradient isthe presence of an ion-blocking electrode. What hereinafter shall becalled the ion-blocking operation mode of the galvanic cell was firstrealized in practice by Hebb15 and afterwards theoretically interpretedby Wagner.16 In such a cell the electrode at position x′ is completelyreversible in terms of the establishment of the equilibrium between iand Xξ according to eq. 3 so that the electrical potential ϕ‘ is fixedin a well-defined manner by μ′

Xξ. On the contrary, the electrode at

position x′′ is ideally polarizable in terms of the equilibrium betweeni and Xξ. In other words, the electrode totally blocks any ionic currentthrough the interface and, thus, through the solid. With the currentdensity ii being zero at steady-state, eq. 1 requires the gradient of ηi

to vanish:

grad ηi = 0 [9]

which together with eq. 8 provides:

U = ϕ′′ − ϕ′ = − RT

ziξ · F· ln

a′′Xξ

a′Xξ

[10]

In order to arrive at eq. 10, the conventional set-up of an electro-chemical cell is taken as a basis according to which each cell termi-nates in two identical metallic poles in contact with the electrodes. Asa consequence, the chemical potentials of the electrons of both polesare identical: μ′′

e = μ′e.

It becomes apparent from eq. 10 that, with U and a′Xξ

being fixed,the magnitude of a′′

Xξis well defined and, for the reasons discussed

above, the same is true in terms of the position-dependent magnitude ofaXξ

(x) throughout the MIEC. The shape of this gradient can be exactlycalculated,17 at least for the region of moderate voltages where ai canroughly be taken as position-invariant implying that:

grad μi ≈ 0 [11]

As to whether the μXξ-gradient relative to the level of the reversible

electrode is rising or falling depends on the polarity of the voltage thatin its turn is related to the sign of the conducting ion, which becomesevident from eq. 10. In other words, the polarity of the voltage, thenature of the MIEC and the shape of the μXξ

(x)-gradient are correlatedwith each other, as it is schematically illustrated in Figs. 4a–4d.

According to eqs. 6 and 7, the MIEC under consideration exhibitspartial electronic conductivities that apart from being temperature-and material-dependent quantities are functions of the chemical sur-roundings expressed in terms of the magnitude of a Xξ

. As the chemicalpotential μXξ

is subject to a position-dependent distribution along thex-axis, the partial electronic conductivities become functions of x. Theunderlying reason is the polarization of the concentration of electroniccharge carriers, i.e. depletion or enrichment along the x-direction. Itfollows from eqs. 8, 9 and 11:

grad μe = 1

ziξgrad μXξ

[12]

where the chemical potential gradient of the electrons reflects theirconcentration gradient. Based on the possible variants for the chemicalpotential profiles depicted in Figs. 4a–4d, the schematic conductivityprofiles for the electronic majority charge carriers shown in Figs. 5a–5d can be derived depending on the polarity of the applied voltageand depending on whether an anion or cation conductor is underconsideration.

Figure 4. Polarity-dependent schematic profiles of the chemical potential ofthe virtual species Xξ throughout the MIEC along x-direction under ion-blocking conditions. (left-hand side: anion conductor as MIEC; right-handside: cation conductor as MIEC; x′: position of the reversible electrode; x′′:position of the ion-blocking electrode; +/–: polarity of the applied voltage).

By integrating eq. 1 over the spatial coordinate x the electroniccurrent I flowing through the cell can be determined for the conditionsspecified by eqs. 8 and 9, i.e. ion blocking. Together with eq. 10,the position-dependence of the partial conductivities σn(x) and σp(x)throughout the MIEC yields a voltage dependence of the current. Theresulting I-U relationship reads:16

I = RT

F· AMIEC

L·[σ′

p · (1 − exp(FU/RT)) + σ′n · (exp(−FU/RT) − 1)]

[13]where AMIEC is the area of the cross-section of the conducting mediumwhich, for simplification, is assumed to be invariant with respectto x. σn′ and σp′ are the n- and p-type electronic conductivities inthe outermost MIEC-layer close to the reversible electrode, i.e. atposition x′.

One has to keep in mind that both I and U may be positive ornegative depending on whether n- or p-type conduction in a cationor anion conductor becomes dominating with rising |U|. This is inaccordance with electrochemical terminology concerning the sign ofa cathodically or anodically oriented current.

Based on the I-U relationship the Ohmic resistance RMIEC can bequantified as a function of the applied voltage. This function is plottedin Fig. 6 for exemplary conditions, e.g. T = 300 K, L = 50 nm andAMIEC corresponding to a circular area of 100 nm in diameter. Theconductivities σ′

n and σ′p chosen for the computation are arbitrary. In

fact, the ratio between them may become crucial for some details ofthe curves, however, the key feature is not concerned. The steep decay

Figure 5. Polarity-dependent schematic profiles of the electronic conductivityof the majority charge carriers throughout the MIEC along x-direction underion-blocking conditions. (σn: n-type conductivity; σp: p-type conductivity;left-hand side: anion conductor; right-hand side: cation conductor; x′: positionof the reversible electrode; x′′: position of the ion-blocking electrode; +/–:polarity of the applied voltage).

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P426 ECS Journal of Solid State Science and Technology, 2 (11) P423-P431 (2013)

Figure 6. Variation of the MIEC resistance as a function of the amount of theapplied voltage in a galvanic cell under steady-state ion-blocking conditionsaccording to eq. 13 (Temperature: 300 K; MIEC-thickness: 50 nm; MIEC area:7.85 · 103 nm2). a) Top: Resistance drop due to voltage-induced increase of n-or p-type conductivity for σ′

n = σ′p = 10−5 S/cm. b) Bottom: Resistance drop

due to voltage-induced increase of n-type conductivity (curve 1) and p-typeconductivity (curve 2) for σ′

n = 10−5 S/cm; σ′p = 10−12 S/cm.

of the resistance over several orders of magnitude, characteristic ofthe curves, exclusively results from the voltage-induced change of theelectronic conductivities σn or σp relative to the ground levels σ′

n andσ′

p. That means, the resistance decay is a macroscopic phenomenonbased on the integral over the MIEC membrane under conditions con-trolled by the charge carrier flux equations and the resulting potentialprofiles.

In Fig. 6a the particular case is considered that, with σ′n = σ′

p, thevoltage causes the n-type conductivity of an anion or cation conductorto rise, which corresponds to the conductivity profiles of Figs. 5a andd. It follows from eq. 13 that the curve would look the same if p-type conductivity were concerned. Only the voltage would have beenapplied in the opposite direction. The symmetry in terms of n- orp-type behavior of the curves gets lost as soon as the parameters σ′

nand σ′

p differ from each other, which is demonstrated by Fig. 6b. Thefigure shows the voltage-induced decay of the overall resistance by theincrease of the n-type conductivity (curve 1) or p-type conductivity(curve 2) depending on the polarity of the voltage applied to one andthe same MIEC, i.e. a cation conductor, on condition that σ′

p � σ′n.

Whereas the resistance immediately falls with U if the polarity issuch that n-type conductivity is affected, the curve first passes a flatmaximum and only then falls, if the alternate voltage increases theproportion of the p-type conductivity. The reason for the maximumis that with growing σp the effective value of σn decreases as a resultof equilibrium (4) and as long as σn exceeds σp the decreasing n-type conductivity controls the situation before the p-type conductionbecomes dominant. Therefore, the resistance drop is asymmetricalwithin a definite voltage range.

Since all phenomena underlying the mathematical relationshipsplotted in Fig. 6 are reversible, the functional dependence is an in-verse one. In the same way as the resistance drops down with risingvoltage it also increases upon oppositely changing the voltage. Thus,the prerequisite is given for what characterizes the phenomenon ofresistive switching.

Concerning the principles underlying the functioning of a cell inion-blocking mode, the question might arise: Does a chemical poten-tial gradient of Xξ inside a solid body really exist or is its assumption

only a useful concept? The latter is true. It is a fact that in a solid-state oxygen-ion conductor molecular oxygen is not a species and,hence, μO2 is not at all defined.18 The same applies to the chemicalpotential of, for instance, metallic sodium in a solid-state sodium-ionconductor. Otherwise, it could not be excluded that a thermodynami-cally nonsensical sodium chemical potential higher than that of puremetallic sodium is generated throughout the sodium-ion conductor ifthe applied voltage was oriented as in Fig. 4d and if the reversibleelectrode was metallic sodium. The substitution of U for μXξ

result-ing in eq. 13 avoids the emergence of such a conflicting situation andestablishes a direct relationship between current and voltage. Never-theless, this does not detract from the heuristic value of the μXξ

-relatedconsideration.

There is another advantage connected with the substitution of Ufor μXξ

. Combination of eqs. 9 and 11 leads to the conclusion thatgrad ϕ ≈ 0. It means that there is no electrical potential drop insidethe MIEC as it is the key feature of a cell without any electronictransference, i.e. a cell with a totally pure ionic conductor acting aselectrolyte. In such a cell the electrical potential drop exclusively con-centrates on the electrochemical double layer at the electrodes. It is afact, however, that under ion-blocking conditions the MIEC becomespurely electronically conductive. As a consequence, the electrical po-tential drop expands from the electrodes into x-direction and spreadsover the whole MIEC. This can be understood as being consistentwith the relationship between U and μXξ

according to eq. 10. Thevoltage decay across the MIEC thickness corresponds to the virtualμXξ

distribution and vice versa.The above discussion sheds certain light on the ambivalence of

the electrochemical approach. On the one hand, this approach is inaccordance with a zero electrical potential gradient within a pure ionicconductor and, on the other hand, it is supposed to describe the volt-age distribution over a purely electronically conducting solid. Theapparent contrast originates from the limitation inherent to the elec-trochemical interpretation of what controls the behavior of a mixedconductor in the limiting case of pure electronic conduction. Never-theless, there is diversity of experimental observations proving thatthe concept under consideration provides an appropriate descriptionof reality. For more than half a century eq. 13 has been the basis ofall polarized-cell experiments aiming at the separate determination ofthe n- and p-type electronic conductivities of MIECs of all kind.

Resistive Switching

The conditions for which the curves of Fig. 6 have been calculatedare idealized. They do not take several additional effects into accountthat distinguish a cell in ion-blocking operation mode from the situa-tion prevailing in a resistive switcher. Therefore, the following aspectsdeserve particular attention as they, in synergy with the relationshipsdiscussed above, provide the prerequisites for the memristive proper-ties and for the spectrum of phenomena characteristic of the behaviorof a resistive switcher:

1. The defect equilibria underlying the curves of Fig. 6 are consid-ered to be in equilibrium position the establishment of which maynecessitate elevated temperatures and finite equilibration times assolid-state materials are involved.

2. The I-U relationship for a cell in ion-blocking mode takes steady-state conditions for granted. Each step of changing the appliedvoltage entails the establishment of a new steady-state. In practice,reaching the new state is time-consuming which is manifested bya transient change of the total current over times of up to 101. . . 2 hdepending on the temperature, the magnitude of the voltage step,the nature of MIEC and the amount of substance involved in theestablishment of the steady-state conditions.19

3. The I-U relationship for a cell in ion-blocking mode is based onthe validity of eq. 11 which is roughly fulfilled only in the centralregion of Fig. 2. The reality is that even under moderate voltages ameasurable amount of neutral species Xξ is reversibly exchangedwith the MIEC upon reaching steady-state. The amount of sub-stance involved increases with increasing voltage.20

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ECS Journal of Solid State Science and Technology, 2 (11) P423-P431 (2013) P427

4. The steady-state electronic current obeying eq. 13 is superim-posed by another electronic current that itself is subject to time-dependence.19,20 For quite a long period of time, this currentexceeds that of the steady-state and controls the electronic con-duction properties of the MIEC in each operation mode of the per-taining galvanic cell, which is especially effective after a changeof the dominating conditions in terms of U. This phenomenon isdescribed in the literature21–29 as chemical potential dependenceof the electronic conduction parameters, with these parametersrepresenting constants related to the equilibrium constants Kn

and Kp of eqs. 6 and 7.30 Thus, the reality demonstrates that theassumptions underlying eqs. 6 and 7 are oversimplified by ne-glecting the influence of the chemical potential μXξ

on the mag-nitude of the equilibrium constants Kn and Kp due to variation ofthe MIEC composition.

5. The realization of a totally polarizable and a totally unpolarizableelectrode in one and the same cell is a challenge and usuallyrequires special measures to take. Moreover, the polarizability ofone and the same electrode often strongly depends on the polarityand the magnitude of the applied voltage. An electrode with afinite capability of delivering ions through the interface need notremain reversible but may become blocking for the passage ofions due to changing conditions. Therefore, it is more probablethat, independent of the polarity of the applied voltage, a heavilyloaded electrode always acts fairly ion-blocking.

Remarks on topic (1) and (2).— It is obvious that the establish-ment of steady-state and equilibrium conditions is interrelated to eachother. Both of these phenomena together lay the foundations for atime delay between cause, i.e. voltage, and action, i.e. current andresistance, respectively. As a result, a hysteresis is generated which ischaracteristic of resistive switching. The voltage changes faster thanthe voltage-induced phenomena can follow.

Remarks on topic (3).— Topic (3) will become the more relevant,the higher the voltage amplitude is. Note that the establishment ofthe ion concentration gradient may be the result of a net flux throughat least one of the electrodes inwards or outwards the MIEC butneed not be, since an internal redistribution of the ions within theMIEC is less hindered and, thus, is comparatively faster. In a cell withhighly polarizable electrodes, the internal ion redistribution will bethe only possible process related to ion motion. It is suspected that thephenomenon called “electroforming of a switcher” is nothing else thanthe accelerated establishment of an optimum level in the concentrationof mobile ions by pumping ions into or out of the MIEC depending onthe initial ion concentration. Optimum in this respect means that theconcentration of ions is high enough in order to maintain a finite ionicconductivity of the MIEC and is low enough in order to ensure thations need not any longer leave the MIEC for reaching a time-invariantaverage ion concentration upon changing the voltage polarity.

Ion flux implies transport of matter, which is proportional to thevolume involved upon establishment of the ion concentration gradient.Hence, the effect under consideration is an amount-of-substance effectstrongly related to geometrical parameters. Primarily, it is not a nano-size effect. Moreover, the transport of ions is relatively slow so thatthe effects induced by the ion motion always lag behind the voltage,which amplifies the hysteresis discussed above.

To which extent topic (3) contributes to the voltage-induced re-sistance variation is difficult to estimate. Two kinds of influence areconceivable. First, the change of the stoichiometric composition ofthe MIEC may affect the concentration polarization of the electroniccharge carriers and thus the electronic conductivities for which ev-idences really exist (cf. topic (4)) and, second, the change in theconcentration of mobile ions may reduce or increase the ionic con-ductivity. The latter effect is too small as has been proved at least in thecentral region of Fig. 2, so that the voltage-dependent, i.e. position-dependent, change in the ionic conductivity σi(x) will only contributeto the change of the total resistance RMIEC on a negligibly small scale.

Remarks on topic (4).— As already discussed above, the phe-nomenon described under topic (4) is strongly related to topic (3).It demonstrates that an even minor voltage-induced re-distribution ofionic charge carriers affects the electronic structure of the solid insuch a way that the voltage-induced resistance change due to elec-tronic conduction is amplified. With its time- and voltage-dependencethe phenomenon may represent a key factor in terms of kinetics.

Remarks on topic (5).— As to the electrode behavior, a variety offactors plays a role. It is easier to polarize an electrode if the presenceof species Xξ corresponding to the mobile ion can be basically ex-cluded in the surroundings of the solid. Therefore, minimum effort isnecessary in order to block the electrode of a cation conductor. How-ever, in this case the suppression of the interfacial ionic exchange iseffective only if the metal corresponding to the mobile ion does notform a stable solution with the electronically conducting phase of theelectrode or reacts with it to an intermetallic compound. Otherwise,the electrode stabilizes the chemical potential of Xξ due to which theelectrode that initially might have been easily polarizable turns into amore reversible one as long as the solid solution or the intermetalliccompound exists.

On the contrary, gaseous oxygen is ubiquitous so that in the caseof an oxygen ion conductor, at least at elevated temperatures, it re-quires significant efforts in order to prevent oxygen from passing theelectrode. Nevertheless, the transition from O2 to O2−, i.e. from a bi-atomic neutral molecule to a negatively charged oxygen ion that canenter the solid, is a complex electrochemical process with a numberof possible rate-limiting steps.31 The obstacles represented by thesesteps add up to an overall polarization resistance that is a complexfunction of the polarization conditions.32 In general, the resistance isthe higher, the lower the temperature and the higher the polarizationcurrent, implying the voltage. Experience shows that the polarizationresistance of an interface between an oxygen-ion conducting solid andan oxygen-containing gas atmosphere is higher than that between thesame solid and an otherwise oxygen-ion bearing medium representedby a second oxide. In this case a considerable reservoir of oxygen ionsis provided which reduces the polarization resistance and may stabilizethe oxygen chemical potential at the respective interface. The samemay happen if a nominal metal electrode is in contact with an oxygenion conductor and the oxidation potential of the metal is covered uponvoltage change. Then the metal oxidizes and the oxide formed as athin film at the interface buffers the oxygen chemical potential μO2 ata level defined by the standard Gibbs free energy of formation of themetal oxide. The behavior will additionally become dependent on theoxygen mobility of the film and on other film properties. As a result,the chemical and electrical potential are temporarily or permanentlypinned which increases the degree of asymmetry in the cause-actionrelation.

For the sake of completeness it is mentioned that the oppositetendency may also play a role. In particular cases the polarizationresistance need not continue to increase beyond a threshold voltagebut may suddenly drop. The reason is an electro-catalytic effect of thehighly polarized electrode at which the elevated electron concentrationfacilitates the conversion from neutral to charged species and starts toact depolarizing.33

Summarizing the preceding discussion, it becomes apparent thatthe conductivity gradients depicted in Figs. 5a–5d represent model sit-uations from which the cause-action relation controlling the behaviorof a real resistive switcher can be derived phenomenologically. Real-ity differs the more, the more the electrodes deviate from the model.Provided that both electrodes act equally ion-blocking, the voltageonce will favor n-type conductivity and in the opposite direction itwill favor p-type conductivity, at one and the same electrode. As theprofile of one type of electronic conductivity is always the mirrorimage of the profile of the other type of conductivity with respectto the x-axis, the profile alternately flips up and down upon voltagevariation. The counter electrode behaves just the other way round. Asa result, a conductivity profile is generated throughout the MIEC withone half of the profile being fairly symmetrical with respect to a 180◦

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Figure 7. Current vs. voltage curve of the unipolar resistive switcher Pt|NiO|Ptaccording to experimental data of Inoue et al.34 (Inserts: Schematic shape ofthe conductivity profile through the MIEC; the attribution of the conductivityprofiles to either n- or p-type conductivity depends on the specific voltagepolarity).

rotation of the second one. Assuming that there is no process that pinsthe ordinate height of the rotation center, the conductivity profile ini-tially floats as a whole during the “electroformation process” until itstabilizes and becomes approximately symmetrical with respect to theextent of σn and σp. Then the symmetry point will better than beforecoincide with the intrinsic point of the n- and p-type conductivity atwhich σn ≈ σp so that the voltage-induced resistance drop becomesindependent of whether it is due to n- or p-type conductivity as it isexemplarily demonstrated in Fig. 6a. In other words, the resistancedrop is practically independent of the voltage polarity. Such kind ofmirror-image conductivity profiles are expected to determine the lowresistance states, i.e. the on-states, of a unipolar resistive switcher atthe voltage extremes as it is illustrated in the inserts of Fig. 7.

Figure 7 shows the example of the I-U curve of a unipolar switcherbased on the symmetrical cell Pt|NiO|Pt.34 In such a switcher both un-der positive and negative voltage a switching process occurs from lowto high and from high to low resistance. The passage from low to highresistance is caused by the intermediate state between the positivelyand negatively sloped conductivity profile, which is expected to bereached at around zero voltage. This state is characterized by a flatconductivity profile, as it is shown by the inserts in the middle ofFig. 7, implying that there is no conductivity increase at all, neither byn-type nor p-type conductivity. As a result, the resistance is high. Dueto the lag in the cause-action relation the onset of the high resistance isdelayed and takes place during both decreasing and increasing voltageafter polarity change.

In case that one of the electrodes of a resistive switcher is morereversible than the other one, an asymmetrical behavior of the cell isthe consequence. Even in a nominally symmetrical cell this may inci-dentally happen due to an asymmetrical polarization treatment duringthe “electroforming process”. Then the Xξ-level at one electrode sideis fixed or to some extent stabilized. As a result, upon polarizing in onedirection the voltage induces a drastic increase of the electronic con-ductivity whereas the alternate voltage does hardly change anything,as it is demonstrated in Fig. 6b for the case of a cation conductor. Theconsequence is a transition from high to low resistance for one volt-age polarity and from low to high resistance for the opposite voltage,which is characteristic of bipolar switching.

An example of bipolar switching is given in Fig. 8 for the cell ar-rangement Ag|Glassy Ag+-conductor|Pt.11 Here, it is the presence ofthe Ag electrode that fixes the n-type electronic conductivity through-out the initially unpolarized MIEC and ensures that the situation isclose to the one underlying eq. 13 with the parameters σ′

n and σ′p dif-

fering by orders of magnitude. In general, the difference between σ′n

and σ′p depends on the Xξ-level defined by the reversible electrode

and on the characteristic conduction properties of the MIEC underconsideration. Note that this difference corresponds to the differencein the charge carrier concentration between holes and electrons at adefinite aM value in Fig. 2. The condition σ′

n � σ′p implies that the

Figure 8. Current vs. voltage curve of the bipolar resistive switcher Ag|GlassyAg+-conductor|Pt according to experimental data of Kozicki et al.11 (Inserts:Schematic shape of the n- and p-type conductivity profile through the MIEC).

general level of the electron concentration throughout the MIEC isfairly high but still low enough to keep the MIEC resistance high. Then-type electronic conductivity causes the resistance to drop not untilthe voltage rises with the Ag electrode being positive (cf. σn-profileon the right-hand side of Fig. 8). Consequently, the switcher is set on.Qualitatively, this state lasts until the voltage alternates polarity. Withopposite polarity, i.e. in the left-hand half of the diagram of Fig. 8,the hole rather than the electron concentration is increased. However,this does not manifest itself in an increase of the total electronic con-ductivity. Since the change of the hole concentration is necessarilyaccompanied by the opposite change of the electron concentration(cf. eq. 4), the total conductivity even falls or remains tolerably con-stant, as long as the electrons are the majority charge carriers. Thep-type conductivity profile proves to be quite flat and roughly stayson the same ordinate level as defined by the Ag electrode with a smallincrease in the area close to the Pt electrode. This is indicated by theinsert in the left-hand part of Fig. 8. As the total electronic conductivitydoes not change remarkably, the resistance that is low at the extremeof the positive voltage and that increases upon the voltage becomingmore negative remains high and does not drop. The prerequisite isthat the negative voltage does not exceed a finite span. Consequently,the switcher is set off. Between the two extremes the resistance lagsbehind the voltage in each direction, which leads to the typical shapeof the I-U curve of Fig. 8.

With regard to categorizing the switching phenomena recently athird type in addition to unipolar and bipolar switching has gainedgrowing interest, i.e. complementary switching, which could actuallybe regarded as a special bipolar variant. A complementary switcheris the combination of two back-to-back stacked bipolar switchers35–37

for which the electrochemical cell TiN|HfO2|Hf|HfO2|TiN35 is an in-teresting example. Characteristic of the cell is that the more reversibleelectrode is in the middle of the symmetrical arrangement and thatthe two terminal electrodes are highly polarizable with respect tothe mobile ion of the MIEC. Since metallic Hf has an extremelyhigh affinity to oxygen, it partially becomes oxidized by reducing theoxygen content of the surrounding HfO2 and converts its ambienceinto sub-stoichiometric HfO2-δ. Therefore, another terminology for thesame cell is also appropriate, viz. TiN|HfO2-δ|TiN.36 Independent ofwhether hafnium exists in metallic form and equilibrates with HfO2-δ

or two different sub-stoichiometric hafnias do coexist with each otherand represent the phase equilibrium HfO2−δ1/HfO2−δ2 , in both casesquite a stable oxygen potential is maintained with comparatively highreversibility with respect to μO2 and, according to eq. 12, also withrespect to μe. In other words, the electron concentration of this MIECarea is pinned and, correspondingly, the electronic conductivity of therespective MIEC layers is relatively stable against the influence of theapplied voltage. The absolute level of the ordinate height in terms of

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ECS Journal of Solid State Science and Technology, 2 (11) P423-P431 (2013) P429

Figure 9. Current vs. voltage curve of the complementary resistive switcherTiN|HfO2|Hf|HfO2|TiN according to experimental data of Wouters et al.35

(Inserts: Schematic shape of the n-type and p-type conductivity profile throughthe two half-cells).

electron concentration determines as to whether the relation σ′n � σ′

p

or σ′n � σ′

p is valid, which ultimately controls the behavior of themore polarizable MIEC regions close to the terminal electrodes uponvoltage application.

Provided that the case σ′n � σ′

p is assumed to characterize reality inthe cell TiN|HfO2|Hf|HfO2|TiN, the following situation is encountered(cf. Fig. 9): By applying an increasing voltage with the positive poleon the left TiN-electrode, the p-type electronic conductivity close tothis electrode increases, however, without remarkable effect on thetotal conductivity of the MIEC of the left half-cell. The situation isthe same as described by curve 2 of Fig. 6b and is also the sameas in the left-hand part of Fig. 8. Simultaneously, with increasingvoltage the extent of n-type conductivity increases exponentially inthe right-hand half-cell according to curve 1 of Fig. 6b. As a result,the resistance of this half-cell and, consequently, the resistance ofthe whole cell decreases and the current flowing through both half-cells gets up, as it is demonstrated by the current jump upwards atabout U ≈ 0.5 V in Fig. 9. Inasmuch as the current rises, Ohm’slaw requires that the voltage drop across the left half-cell increasesconsiderably. Keep in mind that the left half-cell is still characterizedby a rather high resistance. If, in a circuit with two serial resistances,about hundred percent of the applied voltage drops across the firstresistance, there is no voltage drop any longer across the secondresistance. With this voltage approaching zero, the electrical potentialgradient that has maintained the σn-gradient throughout the secondhalf-cell vanishes. Subsequently, the resistance of the second half-cellgoes up and so does the resistance of the whole cell, which ultimatelyresults in a decay of the total current flowing through the cell. Beforethe same process can be repeated again, the applied voltage is reducedand approaches zero. Then the voltage changes its polarity and theright electrode becomes positive implying that the same situation asdescribed above proceeds anew. Ultimately, the mirror-inverted I-Ucurve of Fig. 9 is generated. For the sake of schematic illustration,characteristic conductivity profiles are given as inserts in this figure.

From the discussion above it is not surprising that cell struc-tures like Pt|Nb2O5-δ|Pt37 and Pt|Ta2O5-δ|Pt38 are also complementaryswitchers even though, from their appearance they seem to be moresimilar to symmetrical cells with unipolar behavior. The difference isthat in oxides with an extended stoichiometry region a pinning effectin terms of the potential distribution occurs and in others it does notor it does to a much smaller extent or on a different level in terms ofthe relative position of σ′

n compared to σ′p.

The present solid-state electrochemical concept allows both unipo-lar, bipolar and complementary switching to be explained and this isindependent of whether anion or cation conductors are concerned,which underlines the general character of the concept and helps toreveal that all of these types of resistive switchers are due to oneand the same phenomenon with just different characteristics. Sincequantitative features affect qualitative behavior, the type of switchercan be influenced to some extent by controlling the prevailing condi-tions. For instance, it should be possible to convert a bipolar switcherinto a unipolar one provided that the dominating asymmetry in thepolarization behavior is compensated by appropriate measures. Thisis consistent with real observations.34 It may be suspected that sucha conversion is the more likely, the higher the voltage amplitude is.Furthermore, it is conceivable that in a complementary cell the di-vision into two tolerably equal half-cells is not justified for differentreasons. As a consequence, the symmetry of the I-U curve can getlost. From that point of view, the cell structure Pt|TiO2-δ|Pt that isconsidered to exhibit bipolar switching39 could readily be interpretedas representing a complementary switcher with a highly asymmetricalI-U curve.

Among the materials known to exhibit the phenomenon of resis-tive switching there are oxides with extremely low ionic and electronicconductivities in their pure state, e.g. SiO2. This raises the questionas to whether the conduction properties of the solid are sufficient tointerpret the currents experimentally observed. Two aspects may beimportant. First, the degree of disorder prevailing in thin films maybe higher than that in bulk materials due to the technology of materialpreparation. Second, very small amounts of cations may be dissolvedin highly insulating oxides and may turn them into cation-conductingsolids40 with conduction properties different from those of the purematerial. In silica, for instance, such a process is connected with sta-bilization of the energetically unfavorable cristobalite modification.41

Previous Attempt of Electrochemical Interpretation

As outlined at the beginning, in the literature electrochemistry hasalready been considered as possible interpretation of resistive switch-ing which is due mainly to Aono et al.6–9 Hereinafter, the questionwill be answered as to how the present solid-state electrochemicalinterpretation differs from that kind of understanding especially sinceAono’s view is intensively propagated in the literature. This viewmainly goes back to the generalization of incidental observationsmade upon atomic nano-structuring by means of the tip of a scanningtunneling microscope (STM).6 If a voltage is appropriately appliedbetween the platinum tip of an STM and the vacuum-exposed sur-face of a solid cation conductor, grown as a layer on top of a metalthat embodies the neutral counterpart species of the mobile cation,an atomic cluster of the same metal precipitates out of the surface ofthe ion conductor and grows into the gap between the respective sur-face and the platinum tip3). This cluster may act as a nanometer-scaleelectrical bridge for the gap.6,7 Building and destroying the bridge asa result of electrochemical metal precipitation or dissolution due tochange of the voltage polarity which is accompanied by extendingor shortening the length of the formed nanowire may be thought asaccomplishing a switch function. This is the basic idea that the authorsadopted in order to explain the behavior of a real resistive switcherfor non-volatile memory application, which distinguishes itself fromthe STM configuration first of all by a gapless intimate contact be-tween the ionic conductor and both of the metallic electrodes. Inother words, a gap that could be bridged by the nanowire does notexist in a resistive switcher. Nevertheless, the authors erroneously as-sumed that the switching phenomenon has the same electrochemicalorigin as the above-described process of nano-structuring and, thus,the term atomic switch or nanoionics switch has been coined.42 Theauthors were even misled to speak of a quantized conductance atomic

3)Occasionally, the reverse configuration is considered with the metal/MIEC combina-tion being the STM tip and platinum being the substrate. The character of the relevantphenomena is the same, however.

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P430 ECS Journal of Solid State Science and Technology, 2 (11) P423-P431 (2013)

switch.43 The fact is, however, that the above kind of electrochem-ical nanowire formation has nothing to do with the phenomenon ofresistive switching.

According to the present solid-state electrochemical interpreta-tion, the configurations described by Aono et al. are examples ofbipolar switchers without a nano-scale gap like in an STM experi-ment. For the purpose of transferring the insight gained from the STMstudies about nano-bridging into the understanding of the switchingbehavior, Aono et al. had to locate the nano-wire formation, that isdeemed to cause the conductivity change, inside the bulk of the ionconductor. Therefore, they postulated the formation and annihilationof conducting channels of precipitated metal species throughout themembrane of the MIEC. They even speak of electroplated metal fila-ments spreading between the electrodes that sandwich the MIEC.9,42

Such an interpretation contradicts fundamental relationships aboutthe local distribution of electrochemically active species in a MIECand, thus, about the region for possible enrichment or depletion ofions and neutral particles. Provided that, for the sake of simplicity,parallel electrodes and an infinitely extended MIEC in between areconsidered, lines of equal potential are parallel to the electrodes or,in a more practical case, they are largely parallel, and so are the linesof equal thermodynamic activities of the ions, electrons and neutralspecies. On the contrary, filaments would require these lines to runperpendicular to the electrodes. Furthermore, from a thermodynamicpoint of view the formation of metallic filaments would imply phaseseparation, i.e. the simultaneous establishment of a unity activity ofthe ionic and pertaining electronic charge carriers. In reality, the ac-tivities of these species are very far away from being unity since thematerial under consideration is a non-metallic one. Therefore, pre-cipitation and dissolution of a metal phase in a reversible manner areexcluded on principle.

There are additional arguments speaking against metallic filamentsas a reason for the resistance switch. Provided that Aono’s electro-chemical interpretation of the STM experiments were correct, theprocess of nanowire growth should obey Faraday’s law implying aroughly linear relationship between the current and the rate of lengthchange of the nanowire. The exponential relationship observed inpractice42 significantly differs from that expectation which means thatthe wire growth does hardly require a current to flow through theset-up. More important is that the same relationship does not at alldepend on the bias voltage within the investigated interval of 1.5 . . .3 V while the switching time, i.e. the time period in which the set-up resistance decreases by a fixed amount, becomes exponentiallyshorter if the bias voltage increases by some tenth of a volt only.44

For the sake of resolving this inconsistency, the authors introducedthe impact of a finite activation energy,44 which, however, is not atall compatible with the prerequisites for the validity of Faraday’s law.Moreover, this does not in the least answer the question as to why,on the one hand, a large variation of the voltage does not affect thegrowth rate of the nanowire and, on the other hand, a very smallvoltage variation causes the switching time to change tremendously.Actually, according to Aono’s interpretation the nanowire growth rateand the switching time should be directly related to each other in thesense that the switching time becomes smaller, the higher the growthrate.

In contrast, the present solid-state electrochemical understandingprovides a conclusive explanation of the phenomena discussed above.The exponential relationship between switching time and applied volt-age is an immediate consequence of eq. 13. If the resistance variesexponentially with the voltage, the time within which a fixed resis-tance interval is covered becomes exponentially shorter the higher theapplied voltage is, provided that the kinetics of the underlying processremains the same.

Likewise, an explanation can be given for Aono’s STM observa-tion that the nanowire grows nearly without charge carrier flow. First,it must be remembered what precipitation of neutral species meansin the present concept of a MIEC cell under ion-blocking conditions.It means that the electrode at which the metal precipitates loses itsion-blocking character. Various reasons could be responsible for that.

One of such reasons is mentioned in the foregoing discussion of pos-sible polarization phenomena. For instance, it is conceivable that theelectrode working as an ion-blocking one may become depolarizedat comparatively high voltages. This is exactly what characterizesAono’s observations. While their switching experiments were per-formed in the voltage region 0 . . . 0.4 V, the experiments on thenanowire growth exclusively referred to voltages between 1.5 and 3V. As long as the MIEC surface operates as an ion-blocking one, thefunctionality as a resistive switcher is given. It can be assumed thatthis is true at comparatively low voltages. The functionality gets lostinasmuch as the voltage is increased up to such a high value thatthe surface becomes again reversible with respect to the exchangeof charged species due to electro-catalysis. The consequence is thatmetal atoms precipitate which leads to the formation of a nanowire inan STM configuration. The electrons necessary for the electrochemi-cal reduction of the cations do not even need to be delivered throughthe external electrical circuit as they are still accumulated at the MIECsurface from its functioning as ion-blocking electrode. Keep in mindthat this electron accumulation causes the conductivity of the MIECto increase (cf. Fig. 5d). Therefore, the current measurable in the ex-ternal electrical circuit of an STM configuration is much smaller thanrequired by Faraday’s law.

The above discussion once again proves that nanowire formationis not the prerequisite for resistive switching which is why the termi-nology introduced by Aono et al. is less appropriate for characterizingthe relevant phenomena. Insofar as Aono’s view has nothing to dowith a conclusive interpretation of resistive switching, the present ex-planation is the only valid one based on solid-state electrochemistry.

Quite recently, observations about stepwise changes in the con-ductivity of a switcher during holding a certain polarization state andsuperimposing voltage pulses have been regarded as giving evidenceof conductance quantization.45,46 In the light of the present solid-stateelectrochemical concept this phenomenon that might really be observ-able under STM conditions cannot be regarded as generally valid andseriously proven apart from STM conditions since (i) it would be im-possible in an anion-conducting MIEC; (ii) it would only be consistentwith an electrochemically unrealistic filament structure provided thata cation conductor is considered; (iii) it could likewise be explainedin a quite conventional manner not at all related to a quantizationeffect. With regard to the latter aspect it should be kept in mind thatthe high complexity of the interplay of electrical polarization anddepolarization, of the superposition of enrichment and depletion in-volving neutral species and charge carriers of different nature, yieldsa multi-variable complicated functional dependence for the conduc-tivity which offers a variety of possibilities to interpret conductivitysteps beyond conductance quantization.

Conclusions

Resistive switching is a complex process based on the electrochem-ical equilibrium between electrons, ions and the respective neutralspecies in a mixed ionic-electronic conductor that acts as electrolyteof a galvanic cell operated under variable conditions close to the ion-blocking mode. Different from a previous attempt of electrochemicalinterpretation the resistance of the solid conductor is understood tovary due to a voltage-induced change of the n- and p-type electronicconductivity throughout its cross-section, which is interfered by per-manent disturbance of the steady-state conditions and by additionalion redistribution within the solid. These phenomena give rise to thecharacteristic time delay between cause, i.e. voltage, and action, i.e.resistance.

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