Electrochemical strain microscopy of local electrochemicalprocesses in solids: mechanism of imaging and spectroscopyin the diffusion limit

Sergei V. Kalinin & Anna N. Morozovska

Received: 20 December 2012 /Accepted: 8 April 2013# Springer Science+Business Media New York 2013

Abstract Changes in ionic concentration and electrochemi-cal processes in solids are invariably associated with changesin molar volume. Correspondingly, materials with mobile ionsdevelop strain in response to applied electric bias. This elec-tromechanical coupling mediated by mobile ions lays thefoundation for the electrochemical strain microscopy (ESM)of energy storage and conversion materials. Here, we analyzethe imaging and spectroscopic mechanism in ESM in thediffusion limit and discuss the similarities between ESM andmacroscopic current-based electrochemical measurements.The theoretical challenges in ESM are formulated.

Keywords Scanning probemicroscopy . Chemicalexpansion . Electrochemical strainmicroscopy

1 Introduction

Progress of energy storage and conversion technologies andionic-based information storage and logic devices alike neces-sitates local probing electrochemical processes in solids. Whilesignificant progress was achieved using microfabricated elec-trodes combined with classical current-based electrochemicalstrategies [1–3], many solids are inhomogeneous at the level ofindividual grain, grain boundaries, and other microstructuralelements. This requires development of electrochemical

methods capable of exploring local electrochemical reactionsand ionic transport at these sub-10 nm scales.

Recently, it was proposed that electrochemical phenomenain solids can be explored on the nanometer scale by employingthe principle of bias-induced static (electrochemical dilatome-try) and dynamic strain detection. In static measurements,material expansion during electrochemical reaction is probedby an AFM tip or dilatometer [4–10]; alternatively, shape andsize change of particle or nanofabricated structure areascertained by AFM [11]. Dynamic detection is embodied inElectrochemical Strain Microscopy (ESM) [12–15], the peri-odic electric bias is applied to sharp scanning probe micro-scope (SPM) tip in contact with the surface. The applied biasresults in the ionic motion in the solid, either due to electrodepolarization or surface electrochemical process. The associatedchanges in molar volume result in oscillatory surface deforma-tion that can readily be detected by the SPM electronics at the2–3 pm level (for sufficiently high frequencies in the 10 kHz–1 MHz range that allow imaging at cantilever resonances).Combination of this high-frequency detection principle withslow large-amplitude waveforms allows a broad set of time-and voltage spectroscopies to be developed [12, 16–18].

Since its inception, ESM was applied to a variety of elec-trochemical systems, including Li-ion conductors [13, 19], Li-ion electrolytes [20–22], and oxygen electrolytes [16], oftendemonstrating sub-10 nm features. However, the relationshipbetween the ESM contrast and local electrochemical activityremains largely unexplored. Here, we analyze the image for-mation mechanism in ESM and establish the relationshipbetween high-frequency ESM contrast and electrochemicalcapacitance in the diffusion-limited regime. We further ana-lyze non-Vegard contributions to the ESM signal, brieflymention the complexities associated with the migrative con-tribution to the signal, and examine the ESM evolution as afunction of the state of charge, providing insight into ESMtime-and voltage spectroscopies.

S. V. Kalinin (*)Center for Nanophase Materials Science,Oak Ridge National Laboratory,Oak Ridge, TN 37831, USAe-mail: sergei2@ornl.gov

A. N. MorozovskaInstitute of Physics, National Academy of Sciences of Ukraine,46, pr. Nauki,03028 Kiev, Ukraine

J ElectroceramDOI 10.1007/s10832-013-9819-7

2 ESM signal in the general and high frequency limits

ESM is based on the detection of the response of the ionicsolid to the high-frequency electric field induced by theprobe [19]. Typically, the excitation frequency is selectedto be at the resonant frequency of the cantilever, providingthe advantage of mechanical amplification (by the factor of50–200 depending on the surface material) and minimizingcontribution of the 1/f noise [23, 24]. Furthermore, highfrequency operation allows for a broad range of spectro-scopic measurements. We note that for most cantilevers thecontact mode resonances are in the 50 kHz–1 MHz range.For typical tip-surface contact areas of the order of 3–30 nm,the characteristic diffusion times L2/D are of the order of~1 s. Hence, the ESM imaging is performed in the high-frequency limit at which transport length are significantlysmaller than the tip surface contact area. In comparison,hysteresis loop measurements in ESM and static electro-chemical strain measurements can be performed at frequen-cies comparable or well below diffusion frequencies.

2.1 General ESM formalism

The relationship between the ESM signal and the amount oftransferred ions can be analyzed in a straightforward fashion.We assume that the moving chemical species are generated atthe tip-surface junction and are transferred into the materialunder the combined effect of diffusion and electromigration.The change in local concentration δCk results in change in local

molar volume in accordance with Vegard law, duij ¼ bkij dCk ,

where bkij is the Vegard strain tensor [25–28], corresponding

full equation of state is uij ¼ bkij dCk þ sijklσkl . Surface dis-

placement can be then found from the convolution of the ionconcentration profile δCk and the Green’s tensorial function

GSij x1 � x1; x2 � x2; x3; x3ð Þ . Note that the analytical expres-

sions for Green functions are available for a film on a rigidsubstrate [29] and for elastic half space [30]. For more com-plex elastic symmetries and geometries numerical estimatescan be used.

For the isotropic and coordinate-independent Vegard tensorβ11 =β22 =β33 =β, elastic isotropy and a single species δC(x1,x2, x3, t), the displacement field can be written as [15, 31]:

u3 x1; x2; 0; tð Þ ¼ � 1þ nð Þp

b XXXx3>0

x3 dC x1; x2; x3; tð Þdx1dx2dx3x1 � x1ð Þ2 þ x2 � x2ð Þ2 þ x23

� �3 2=

ð1ÞHere ν is the Poisson coefficient. Equation (1) defines the

surface displacement at location (x1, x2, 0) induced by theredistribution of mobile ions as described by excess concen-tration δC(x1, x2, x3, t). Note that the concentration field

δC(x1, x2, x3, t), given in Refs. [14, 15], can be calculated forspecified boundary conditions and transport mechanism,and for a special case of purely diffusion transport drivenby reaction at the tip-surface junction.

We further assume axially symmetric electric potentialφ(ρ, z, 5) induced by the bias probe and resultant axiallysymmetric redistribution of ionic concentration, δC(ρ, z, 5).

Here,ρ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

pand δC(ρ, z,5) is the frequency spectrum

of the relative concentration deviation from the stoichiometricconcentration (i.e. Fourier transform of the concentrationfield). Then the linear surface displacement at the tip-surfacejunction z = 0 induced by the redistribution of mobile ions, asdetected by SPM electronics, is derived as [16]:

u3 0; ρ ¼ 0;wð Þ ¼ � 1þ nð Þp

bZh0

dz

Z10

d eρeρ � z � dC eρ; z;wð Þeρ2 þ z2� �3 2=

ð2ÞWhere 5 is the frequency of applied voltage, c33 ~ 1011Pa

is the tensor of elastic stiffness, h is the film thickness.

2.2 High-frequency ESM signal

We further analyze frequency regimes in ESM for the ionictransport. Relevant length scales that describe the system are thecharacteristic size, which in this case is the tip surface contact

area, R0, and the Debye length of the material, Rd ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi""0kBT q2n=

p, where ε0 is the universal dielectric constant, ε

is relative dielectric permittivity, kB is Boltzmann constant, T isthe absolute temperature, q = Ze is the charge species charge inthe electron charge units e, n is the average concentration of freeelectrons/holes. The diffusion length is frequency-dependent as

lD ¼ ffiffiffiffiffiffiffiffiffiD w=

p, with D as diffusion coefficient. In comparison,

electromigration length is directly proportional to the inversefrequency, lM ∝ 1/5.

Corresponding time scales are as following. The Maxwellrelaxation time is τM = ε0ε/qnη, where η is charge carriermobility. Using Nernst-Einstein relationship, η = qD/kBT, theMaxwell relaxation time can be rewritten in terms of diffusioncoefficient as tM ¼ R2

d D= . Diffusion time is tD ¼ R20 2D= . The

immediate sequence of the expressions is the relation betweenthe diffusion and Maxwell relaxation times:

tM tD= ¼ R2d R2

0

�: ð3Þ

We note that the scaling of diffusion and migrationlengths with frequencies suggests that the diffusion effectswill be more relevant at high frequencies. Practically, thisconclusion is valid only locally, and realistic picture is morecomplex and affected by the non-local details of the system,namely spatial extent of the chemical gradient vs. electricfields and presence of sources/sinks of species. For example,

J Electroceram

electric field distribution in the material at low frequencyis dominated by static screening and field is concentratedin the corresponding screening layer with the extent con-trolled by Debye length (small potential) or ioniccrowding (large potential). Correspondingly, diffusion isthe dominant transport mode in macroscopic systems inagreement with classical electrochemical arguments. How-ever, this picture becomes inapplicable at the scales of thescreening length.

Secondly, the relative contributions of diffusion andmigration effects cannot be separated from the issue ofthe availability of the moving species. Generally, gener-ation of ionic species requires interfacial reaction whichbecomes progressively less efficient with high frequencies(due to both kinetic effects and voltage divider effects atthe Stern/dense layer capacitances). Hence, at high fre-quencies the availability of diffusion species decreases.These countervailing effects of intrinsic physics and fieldextent/availability suggest that the picture of relative roleof diffusion and migration as a function of frequency isvery complex, and sensitively affected by the lengthscale of the problem (vs. screening length), and requiresmore detailed numerical studies. However, in all casesthe high frequencies suggest decrease of correspondingtransport length.

Hence, in the high frequency regime (or under appli-cation of short voltage pulses) and assuming interfacialreaction driven-transport, the redistribution of mobile spe-cies can be expected only in the close vicinity of the tip.In this case, once the axially symmetric electric potentialφ(ρ, z, 5) is applied, the concentration variation can beapproximated as [15]

dC ρ; z; tð Þ � dC z; tð Þθ R0 � ρð Þ: ð4Þ

Note, that the assumption made in Eq. (4) is valid whenthe diffusion length is smaller than tip radius and we are inthe diffusion controlled regime, i.e. the regime, where thespatial–temporal evolution is defined by diffusion length.The concentration field (4) is uniform in ρ ≤ R0 in the tip-surface contact area R0, zero outside tip-surface contact, anda function of z only at ρ ≤ R0, which vanishes when z >> R0

(θ(x>0) = 1 and θ(x<0) = 0 is the unit-step function). Aleading order approximation of Eq. (2) is then:

u3ðz ¼ 0; ρ;wÞ � � 1þ nð Þp

bθ R0 � ρð ÞZh0

dC z;wð Þdz ð5Þ

As derived from Eq. (4), Eq. (5) is valid under theconditions:

lD < R0; R0 < Rd ; wtD >> 1 ð6Þ

Equation (5) can be expressed via the average concentra-

tion variation dCh i ¼ 1h

Rh0dC ρ; z;wð Þdz and the total charge

Q ¼ qRh0dC ρ; z;wð Þdz (q = Ze is the ionic charge) as

u3 0;wð Þ / � 1þ nð Þp

bh dCh i � � 1þ nð Þp

bqQ wð Þ ð7Þ

This equation establishes the general relationship be-tween high-frequency ESM signal and the transferredcharge, i.e. resistance/differential capacitance. Hence, theconcentration variation can be approximated, if the filmthickness h is known, from the high-frequency ESM signal

as dCh i / � pu3 0;wð Þhb 1þnð Þ . Note, that when deriving Eqs. (5–7) no

assumptions are made on the relation between the concen-tration field penetration depth in z direction and tip radius.So, the z-integration limit in Eqs. (5) and (7) is formally {0,h}. In many cases, however, the lower limit of integrationcan be taken as corresponding screening length.

The corresponding Faradaic current frequency spectrumwill be

J wð Þ ¼ �iwQ wð Þ / iwq pu3 0;wð Þb 1þnð Þ ) JðtÞ / �pq

b 1þnð Þ@u3 0;tð Þ

@t

ð8ÞNote, that relations (7) and (8) are independent of the nature

of material and transport process. Additionally nature of thecurrent depends on conduction mechanisms, i.e. Eq. (8) de-fines the magnitude of the impedance but not the phase angle,and hence can define resistance or differential capacitance.However, the relationship between transferred Faradaiccharge and surface displacement holds.

3 Contributions of the non-Vegard terms

It is further instructive to briefly consider the contribution ofthe other interactions to the measured signal. In an SPMexperiment, the measured response comprises the sum oflong range electrostatic interactions acting on the tip andcantilever, and mechanical surface displacement. Electro-static contributions are well-analyzed in the context ofpiezoresponse force microscopy [32–34], and generally arelinear in the tip-surface potential difference. Hence, thesecan be readily identified based on the bias dependence of theresponse. Furthermore, (often dominant) electrostatic canti-lever–surface interactions scale reciprocally with cantileverspring constant and can generally be ignored for sufficientlystiff cantilevers (see detailed analysis in Ref. [32]).

The electromechanical response of the surface is signifi-cantly more complex, and in addition to Vegard strain cancomprise a number of additional contributions [35, 36].

J Electroceram

These include (a) piezoelectric, (b) flexoelectric, (c) defor-mation potential, (d) space charge, and (e) surface electro-chemistry contributions.

The piezoelectric contribution describes the intrinsic cou-pling between the strain and electric field, common for non-centrosymmetric materials. The tip-induced surface defor-mation for piezoelectric coupling was explored in detail inthe context of piezoresponse force microscopy, and corre-sponding coupling coefficient is comparable to the dij of thematerials, of the order of 10–300 pm/V for ferroelectric andstrong piezoelectric materials. However, for centrosymmet-ric material this response is zero.

The flexoelectric contribution [37–40] describes the cou-pling between the strain gradient with polarization and is non-zero for both centrosymmetric and non-centrosymmetric ma-terials. For the simplest 1D case, the flexoelectric contributionrenormalized Vegard coefficient in Eqs. (1–7) as [33]:

b ! b � 2q

""0f33

� ð9Þ

Here the relative dielectric permittivity is ε, ε0 is theuniversal dielectric constant. Flexoelectric strain tensor com-ponent f33 has been measured experimentally for several sub-stances and it was found to vary by several orders ofmagnitude [41]. Reliable values of the flexoelectric couplingare f33 ~ (1 − 4) V, these comes from recent experimental data[42] andmicroscopic estimations [36, 37] (see Table 1). So thevalue of the flexoelectric contribution 2qf33/εε0 is an order ofmagnitude smaller than the Vegard contribution β ~ (5 −50)×10−30 m3 [28, 43–46]. It is worth to underline that therenormalization Eq. (9) is formally valid for arbitrary filmthickness h. In this case the flexoelectric effect impact canbe important for nanosized, as well as mesoscopic films. The

unexpected “global” renormalization originated from thePoisson equation for electric potential ε0ε(∂2φ/∂z2) = −qδCis already included in the flexoelectric strain as δu33 /f33(∂2φ/∂z2)/ − f33qδC/ε0ε. Hence, this effect is likely to bealready encompassed in Vegard strain tensor values measuredmacroscopically.

The deformation potential eIC;Vij contributes to the equa-

tion of state as uij ¼ sijklσkl � eICij dn� eIV

ij dp, for electrons

and holes concentration variations δn and δp [33], similar to

the Vegard equation for ions/vacancies, duij ¼ bkij dCk. In the

case when the electrons/holes are proper carriers, straininduced conduction (valence) band edge shift is proportion-

al to the strain and deformation potential tensor eIC;Vij in the

linear approximation [47–49]. Deformation potential leadsto the electron contribution to the surface displacement,

u3 0; ρ ¼ 0;wð Þ / � 1þ nð Þp

IC33

Zh0

dz

Z10

d eρeρ � z � dn eρ; z;wð Þeρ2 þ z2� �3 2=

;

ð10ÞThis is functionally similar to Eq. (2).Finally, a possible modality for strain generation is the

electrostriction, leading to the quadratic in field contributionis [50]

u3 0; 0ð Þ / � 1þ nð Þ2p

1

2

Zh0

dzeQ33kBT

""0E23 ; ð11Þ

where eQ33 is the corresponding electrostriction coefficientand electric field distribution E3 = −∂φ(z)/∂z. Hence, thegeneralization of Eq. (7) to include these mechanisms can bewritten down as:

u3ðtÞ / � 1þ nð Þp

h b � 2q

""0f33

� dCðtÞh i þ IC

33 dnðtÞh i þ IV33 dpðtÞh i þ

eQ33kBT

2""0E23

� !ð12Þ

Typical values for these chemo-electromechanical cou-pling coefficients are given in the Table 1. Note that theVegard terms dominate over the flexoelectric contribution,although not critically and in certain cases the two arecomparable for “generic material”, e.g. |β|/|2qf33/εε0|.

Electrostriction term becomes important only at high ap-plied fields. Numerical estimates then require a full solution ofNerst-Plank Einstein relations to yield complete concentrationfields. These are available for special cases [51]. Note howeverthat for non-hysteretic system the electrostriction contributes at

Table 1 Typical values of chemo-electromechanical couplingcoefficients

Physical quantity Numerical value References

Vegard coefficient eb � 5� 50ð Þ eV [26, 28, 50]b � 5� 50ð Þ � 10�30m3

Deformation potential eI � 1� 5ð Þ eV [69, 70]I � 1� 5ð Þ � 10�30m3

Flexoelectric coefficient f33 ~ (1−4) V [38–42]2q""0

f33~ 1� 4ð Þ � 10�30m3

J Electroceram

the double of the modulation frequency, and hence are notdetectable in the ESM experiment.

When electronic and electrostriction contributions aremuch smaller than ionic effects, electric current becomes

JionðtÞ / �q1þ nð Þp

b � f332q

""0

� �1@u3 0; tð Þ

@tð13Þ

Finally, another relevant impact on the ESM signal is surfaceelectrochemical effect coupled with bulk electrostriction re-sponse [52, 53]. In this case, the tip induces surface chargingthough ionic process, e.g. formation of protons or hydroxyls[54–58]. Thus induced charges establish an electric field in thematerial that results in the electrostrictive response. The mag-nitude of this effect is difficult to estimate given the uncertaintyof the surface electrochemistry in oxides, and requires dedicatedstudies of surface ionic charge dynamics.

4 Spectroscopy in ESM

We now aim to analyze strain and ESM hysteresis loops inthe diffusion limit. Unlike the general theory developed inSection 1, this analysis requires definitive assumption ofreaction mechanisms. Therefore we call on the associatedmechanism in the classical diffusion approximation, follow-ing the development by Cheng et al [59], for the analysis ofthe potential intermittent titration (PITT) measurements.

Consider a system described by uniquely defined state ofcharge–bias curve, U(x), as depicted in Fig. 1.

Following the analysis of Cheng et al. [59], we assumethat the surface reaction rate is described by classical Butler-Volmer (BV) equation

Jion ¼ c0 exp1� gð ÞF8

RT

� � exp � gF8

RT

� � ð14Þ

Where g is symmetry factor, χ0 is exchange currentdensity, F is the Faraday constant, R is universal gas

constant, and T is temperature. Overpotential φ is relatedto the applied bias U as φ(x, y) = U(x, y) − US, where thevalue US is the initial equilibrium potential value to reach aneffective steady state. All equilibrium values are furtherdenoted by a subscript S.

Consider a system response to the rapid change in externalpotential applied to the probe, as illustrated in Fig. 1. Thelinearized form of Eq. (14) becomes Jion = χ0φF/RT. The

exchange current density can be represented as c0 ¼ c0S �CS � Cð Þ @j0 @C=ð Þ��

Sand potential as U = US − (CS − C)

(∂U/∂C)|S. Substitution in the linearized BV equation yieldsreaction current as

Jion ¼ � F

RTc0S

@U

@C

����S

C � CSð Þ: ð15Þ

The ionic current is balanced by the diffusion fluxthrough the interface z=0 defined as Jd = −DF ∂C/∂z, whereD is diffusion constant. This yields a boundary condition forthe diffusion problem

@C

@zþ c0S

DRT

@U

@C

����S

C � CSð Þ ¼ 0: ð16aÞ

In Eq. (16a) we can further regard that C − CS = δC,where δC is the same as introduced in the Section 2. Then,for convenience, the boundary condition can be written inthe dimensionless form as

@

@zdC þ B

R0dC ¼ 0; ð16bÞ

where B ¼ R0 DRTð Þ=ð Þc0s @U @C= jSð Þ is a dimensionlessBiot number and R0 is tip-surface contact radius that definesthe characteristic length scale of the problem.

The solution for the diffusion equation with inhomoge-neous boundary conditions l∂δC/∂z − ηδC = V0 similar toEq. (16b) for the disc electrode was previously derived inRef. [15]. The equivalence between the solutions in Ref.[15] and the present analysis is given by relations, l ¼DRT c0s

�, η = ∂US/∂C and B/R0 = −η/l. This analysis allows

strain response during the ESM measurements to be calcu-lated if the function U(x, y) is known.

4.1 Linear high-frequency response

Using the similarity in the boundary conditions given by Eq.(16) with those used in Ref. [15], linear ESM response canbe calculated using results [15] by

u3 0;wð Þ ¼ c0sDRT

Z10

dk�2 1þ nð ÞbU wð Þ � R0J1 kR0ð Þ

k þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 þ iw D=

p� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 þ iw D=

p � B l=ð Þ� �

ð17ÞFig. 1 Schematic depiction of potential-composition curve for anelectrochemical system and system response to potential change

J Electroceram

Here J1(kR0) is Bessel function of the first order, U(5) isthe bias applied to the probe. Approximate analyticalexpression for Eq. (17) is

u3 0;wð Þ ¼ c0sDRT

� 1þ nð ÞbU wð Þffiffiffiffiffiffiffiffiffiffiffiffi2iwtD

p � BR0 l=ð Þ �R20

iwtD

� exp � 1þ ið Þ ffiffiffiffiffiffiffiffiwtD

pð Þ � 1þ 1þ ið Þ ffiffiffiffiffiffiffiffiwtD

pð Þð18Þ

Here tD ¼ R20 2D= . At very high frequencies 5τD >> 1,

u3 0;wð Þ / c0sDRT

� 1þ nð ÞbU wð Þ2

� R20

iwtD: ð19Þ

The comparison of exact (14) and approximate (15)expressions is shown in Fig. 2 for typical material parame-ters. Note that the approximation (15) works with adequateaccuracy and contains the main features of the linear ESMresponse frequency spectrum. ESM response monotonicallydecreases with the frequency of applied field increase. Athigh frequency the response becomes negligible and thephase changes its sign.

4.2 ESM response for non-ideal solid solution

In this section we compare the static responses for the ideal andnon-ideal solid solution, for which the composition-potentialcurve isU =US − (RT/F)(ln(δ/(1 − δ)) − K(δ − 0.5)), where 0 <δ < 1 is a dimensionless concentration, δ = (C −CS)/CS = δC/CS

[57]. The exchange constant K is zero for the ideal solidsolution and nonzero in other cases. In accordance with lineartheory, results of which are summarized in the Sections 2 and 3,the vertical surface displacement measured by ESM, u3(ρ, z=0), is proportional to the linear integro-differential operator ofthe concentration variation, u3(ρ,5)/ L[C −CS]. For example,in 1D-case the rigorous expression is

u3 0; 0ð Þ ¼ �Zh0

dz0ebd z0ð Þ � �heb dh i; ð20Þ

Where eb ¼ b33 � 2s12b11s11þs12

� �the deviation δ(z, t) is

regarded small and weakly coordinate-dependent for the lastapproximate equality. Moreover, the variation δ(z′, t) islinearly proportional to the applied voltage U, again in thelinear theory. In the ideal case, K=0, the relation U = US −(RT/F)ln(δ/(1 − δ)) gives

u3 0; 0ð Þ ¼ �heb exp θð Þ1þexp θð Þ K ¼ 0ð Þ ð21Þ

Where θ = (U − US)F/RT is the dimensionlessoverpotential. Figure 3(a) illustrates that ESM response doesnot exhibit any hysteresis or bi-stable behavior when K=0.

Since dh i ¼ �u3 heb.in accordance with Eq. (20), for the

case of non-ideal solid solution (K≠0), the equation

θ ¼ ln � u3

heb 1þ u3

heb� �1

!� K � u3

heb � 0:5

� ð22Þ

can be solved graphically as θ = θ(u3) in order to obtain theresponse u3 dependence on the overpotential θ. FromFig. 3(a), the bi-stable behavior is possible for sufficientlylarge positive K only.

Finally, Eq. (22) allows for an evident extension of thedynamic case [60], with the limitation that ion dynamics is

reduced to the continuity equation,q @@t dC þ div J dCð Þð Þ ¼ 0.

Where the ionic flux J(δC) = −σgrad[μ(δC)], with μ(δC)being the corresponding electrochemical potential. Using avery raw approximation, for the 1D case, after the spatial

averaging of 1D equation @@t dh i ¼ @2

@z2 μ dð ÞD E

we get the

estimate @@t dh i ¼ σh�2 μ dð Þh i , where the electrochemical

potential μ(δ) includes the nonlinear configuration entropyand exchange terms, μ(δ) / q(U − US) + kBT(ln(δ/(1 − δ)) +K(δ − 0.5)) and δ = δC/CS. Finally it leads the twowell-knownfirst-order time dependent Ising-type equation for:

� 1

t@ dh i@t

þ lndh i

1� dh i�

� K dh i � 0:5ð Þ ¼ θ ð23Þ

Where τ is corresponding relaxation time. Using the

linear proportionality, dh i ¼ �u3 heb., corresponding ESM

ω ω

Fig. 2 Frequency dependenceof electrochemical strainu3(0,ω) real and imaginaryparts (a) and absolute value andphase (b). Parameters:D=10−12 m/s2, R0=100 nm,l/η=10−8 m. Solid and dashedcurves represent exact (14) andapproximate (15) expressionsrespectively

J Electroceram

loops u3(θ) are shown in the Fig. 3(b–d) with increasingfrequency 5Γ. As expected, hysteresis loops broaden asfrequency increases.

5 Summary

To date, electrochemical strain microscopy and associatedtime- and voltage spectroscopies have been demonstrated fora variety of energy storage and conversion materials, oftendemonstrating sub-10 nm resolution. The microscope re-sponses can often be calibrated [61–63], allowing for thequantitative measurements of bias-induced surface displace-ments [in pm/V]. This in turn necessitates relating bias-induced surface displacement to a material’s functionality. Inthis manuscript, the image and spectroscopy formation mech-anisms in the electrochemical strain microscopy in diffusionapproximation are analyzed. In this limit, the problem isequivalent to the well-known diffusion problem from the discelectrode and convolution of corresponding solutions with

appropriate Green’s function. The contributions of non-Vegard terms to the response are briefly discussed.

We further note that of significant interest is the ESMimaging under the general conditions of diffusion-migration transport, as e.g. was analyzed by Bazant [64,65] and Riess [66–68]. Of interest are then the associatedionic concentration profiles (for solids) or electrostaticforces mediated by ionic redistribution processes (for liq-uid electrolytes). Integration of these analytical solutionswith the quantitative data offered by modern SPM instru-mentation holds the promise of quantitative probing ofelectrochemical reactivity in solid and liquid environmentson the nanometer scale.

Acknowledgments This work was supported as part of the FluidInterface, Reactions, Structures and Transport (FIRST) Center, anEnergy Frontier Research Center funded by the U.S. Department ofEnergy, Office of Science, Office of Basic Energy Sciences. A.N.M.gratefully acknowledges multiple discussions and critical remarks fromEugene Eliseev (NAS Ukraine). The authors are grateful to A.Belianinov (ORNL) for valuable advice.

ωτ ωτ

ωτ ωτ

θ

θ θ

θ

Fig. 3 (a) Static ESM responsecalculated from Eq. (21).Different curves correspond todifferent factors K=0, 10, 30,−30 as indicated near thecurves. (b–d) Dynamic ESMresponse calculated from Eq.(21) at the frequency 5τ=0.05(b), 1 (c) and 5 (d). Differentloops correspond to differentfactors K=0, 5, 10 as indicatednear the curves

J Electroceram

References

1. F.S. Baumann, J. Fleig, H.U. Habermeier, J. Maier, Solid StateIonics 177(11–12), 1071–1081 (2006)

2. V. Brichzin, J. Fleig, H.U. Habermeier, G. Cristiani, J. Maier, SolidState Ionics 152, 499–507 (2002)

3. G.J. la O, S.J. Ahn, E. Crumlin, Y. Orikasa, M.D. Biegalski, H.M.Christen, Y. Shao-Horn, Angew. Chem. Int. Ed. 49(31), 5344–5347(2010)

4. M.M. Hantel, V. Presser, J.K. McDonough, G. Feng, P.T. Cummings,Y. Gogotsi, R. Kotz, J. Electrochem. Soc. 159(11), A1897–A1903(2012)

5. M. Hahn, H. Buqa, P.W. Ruch, D. Goers, M.E. Spahr, J. Ufheil, P.Novak, R. Kotz, Electrochem. Solid-State Lett. 11(9), A151–A154(2008)

6. S. Park, T. Kim, S.M. Oh, Electrochem. Solid-State Lett. 10(6),A142–A145 (2007)

7. M. Hahn, O. Barbieri, R. Gallay, R. Kotz, Carbon 44(12), 2523–2533 (2006)

8. M. Winter, G.H. Wrodnigg, J.O. Besenhard, W. Biberacher, P.Novak, J. Electrochem. Soc. 147(7), 2427–2431 (2000)

9. S.B. Adler, J. Am. Ceram. Soc. 84(9), 2117–2119 (2001)10. S.R. Bishop, K.L. Duncan, E.D. Wachsman, J. Electrochem. Soc.

156(10), B1242–B1248 (2009)11. Y. Tian, A. Timmons, J.R. Dahn, J. Electrochem. Soc. 156(3),

A187–A191 (2009)12. S. Jesse, N. Balke, E. Eliseev, A. Tselev, N.J. Dudney, A.N.

Morozovska, S.V. Kalinin, ACS Nano 5(12), 9682–9695 (2011)13. N. Balke, S. Jesse, Y. Kim, L. Adamczyk, A. Tselev, I.N. Ivanov,

N.J. Dudney, S.V. Kalinin, Nano Lett. 10(9), 3420–3425 (2010)14. A.N. Morozovska, E.A. Eliseev, S.V. Kalinin, Appl. Phys. Lett.

96(22) (2010)15. A.N. Morozovska, E.A. Eliseev, N. Balke, S.V. Kalinin, J. Appl.

Phys. 108(5), 053712 (2010)16. A. Kumar, F. Ciucci, A.N. Morozovska, S.V. Kalinin, S. Jesse,

Nat. Chem. 3(9), 707–713 (2011)17. S. Guo, S. Jesse, S. Kalnaus, N. Balke, C. Daniel, S.V. Kalinin, J.

Electrochem. Soc. 158(8), A982–A990 (2011)18. N. Balke, S. Jesse, Y. Kim, L. Adamczyk, I.N. Ivanov, N.J.

Dudney, S.V. Kalinin, ACS Nano 4(12), 7349–7357 (2010)19. N. Balke, S. Jesse, A.N. Morozovska, E. Eliseev, D.W. Chung, Y.

Kim, L. Adamczyk, R.E. Garcia, N. Dudney, S.V. Kalinin, Nat.Nanotechnol. 5(10), 749–754 (2010)

20. S. Jesse, A. Kumar, T.M. Arruda, Y. Kim, S.V. Kalinin, F. Ciucci,MRS Bull. 37(7), 651–658 (2012)

21. T.M. Arruda, A. Kumar, S.V. Kalinin, S. Jesse, Nano Lett. 11(10),4161–4167 (2011)

22. T.M. Arruda, A. Kumar, S.V. Kalinin, S. Jesse, Nanotechnology23(32), 325402 (2012)

23. S. Jesse, S.V. Kalinin, J. Phys. D. Appl. Phys. 44(46), 464006 (2011)24. S. Jesse, S.V. Kalinin, R. Proksch, A.P. Baddorf, B.J. Rodriguez,

Nanotechnology 18(43), 435503 (2007)25. Y.T. Cheng, M.W. Verbrugge, J. Power Sources 190(2), 453–460

(2009)26. X.C. Zhang, A.M. Sastry, W. Shyy, J. Electrochem. Soc. 155(7),

A542–A552 (2008)27. Y. Kim, A.S. Disa, T.E. Babakol, J.D. Brock, Appl. Phys. Lett.

96(25) (2010)28. D.A. Freedman, D. Roundy, T.A. Arias, Phys. Rev. B 80(6) (2009)29. A.N. Morozovska, E.A. Eliseev, S.V. Kalinin, J. Appl. Phys.

102(7) (2007)30. A.N. Morozovska, S.V. Svechnikov, E.A. Eliseev, S.V. Kalinin,

Phys. Rev. B 76(5) (2007)

31. A.N. Morozovska, E.A. Eliseev, S.V. Kalinin, J. Appl. Phys.111(1) (2012)

32. S. Jesse, A.P. Baddorf, S.V. Kalinin, Nanotechnology 17(6), 1615–1628 (2006)

33. C. Harnagea, M. Alexe, D. Hesse, A. Pignolet, Appl. Phys. Lett.83(2), 338–340 (2003)

34. B.D. Huey, C. Ramanujan, M. Bobji, J. Blendell, G. White, R.Szoszkiewicz, A. Kulik, J. Electroceram. 13(1–3), 287–291 (2004)

35. A.N. Morozovska, E.A. Eliseev, A.K. Tagantsev, S.L. Bravina,L.Q. Chen, S.V. Kalinin, Phys. Rev. B 83(19) (2011)

36. A.N. Morozovska, E.A. Eliseev, G.S. Svechnikov, S.V. Kalinin,Phys. Rev. B 84(4), 045402 (2011)

37. V.S. Mashkevich, K.B. Tolpygo, Sov. Phys. JETP-USSR 5(3),435–439 (1957)

38. S.M. Kogan, Sov. Phys.-Solid State 5(10), 2069–2070 (1964)39. J.W. Hong, D. Vanderbilt, Phys. Rev. B 84(18), 180101 (2011)40. A.K. Tagantsev, Phys. Rev. B 34(8), 5883–5889 (1986)41. A.K. Tagantsev, V. Meunier, P. Sharma, MRS Bull. 34(9), 643–647

(2009)42. P. Zubko, G. Catalan, A. Buckley, P.R.L. Welche, J.F. Scott, Phys.

Rev. Lett. 99(16), 167601 (2007)43. X.Y. Chen, J.S. Yu, S.B. Adler, Chem. Mater. 17(17), 4537–4546

(2005)44. E.V. Tsipis, E.N. Naumovich, M.V. Patrakeev, A.A. Yaremchenko,

I.P. Marozau, A.V. Kovalevsky, J.C. Waerenborgh, V.V. Kharton,Solid State Ion. 192(1), 42–48 (2011)

45. A.N. Petrov, V.A. Cherepanov, A.Y. Zuev, J. Solid State Electrochem.10(8), 517–537 (2006)

46. J.F. Mitchell, D.N. Argyriou, C.D. Potter, D.G. Hinks, J.D.Jorgensen, S.D. Bader, Phys. Rev. B 54(9), 6172–6183 (1996)

47. N.D. Mermin, N.W. Ashcroft, Solid State Physics (Holt, Rinehartand Winston, New York, 1976)

48. S.M. Sze, Physics of Semiconductor Devices, 2nd edn. (Wiley-Interscience, New York, 1981)

49. A.I. Anselm, Introduction to Semiconductor Theory (Prentice-Hall, Englewood Cliffs, 1981)

50. A.N. Morozovska, E.A. Eliseev, S.L. Bravina, F. Ciucci, G.S.Svechnikov, L.Q. Chen, S.V. Kalinin, J. Appl. Phys. 111(1) (2012)

51. B.W. Sheldon, V.B. Shenoy, Phys. Rev. Lett. 106(21), 216104(2011)

52. C.W. Bark, P. Sharma, Y. Wang, S.H. Baek, S. Lee, S. Ryu, C.M.Folkman, T.R. Paudel, A. Kumar, S.V. Kalinin, A. Sokolov, E.Y.Tsymbal, M.S. Rzchowski, A. Gruverman, C.B. Eom, Nano Lett.12(4), 1765–1771 (2012)

53. A. Kumar, T.M. Arruda, Y. Kim, I.N. Ivanov, S. Jesse, C.W. Bark,N.C. Bristowe, E. Artacho, P.B. Littlewood, C.B. Eom, S.V. Kalinin,ACS Nano 6(5), 3841–3852 (2012)

54. F. Bi, D.F. Bogorin, C. Cen, C.W. Bark, J.W. Park, C.B. Eom, J.Levy, Appl. Phys. Lett. 97(17) (2010)

55. S.V. Kalinin, C.Y. Johnson, D.A. Bonnell, J. Appl. Phys. 91(6),3816–3823 (2002)

56. S.V. Kalinin, D.A. Bonnell, Nano Lett. 4(4), 555–560 (2004)57. S. Cunningham, I.A. Larkin, J.H. Davis, Appl. Phys. Lett. 73(1),

123–125 (1998)58. D.Y. He, L.J. Qiao, A.A. Volinsky, J. Appl. Phys. 110(7), 074104

(2011)59. J.C. Li, X.C. Xiao, F.Q. Yang, M.W. Verbrugge, Y.T. Cheng, J.

Phys. Chem. C 116(1), 1472–1478 (2012)60. C. Brissot, M. Rosso, J.N. Chazalviel, S. Lascaud, J. Electrochem.

Soc. 146(12), 4393–4400 (1999)61. H.J. Butt, B. Cappella, M. Kappl, Surf. Sci. Rep. 59(1–6), 1–152

(2005)62. R.J. Cannara, M.J. Brukman, R.W. Carpick, Rev. Sci. Instrum.

76(5) (2005)

J Electroceram

63. D.F. Ogletree, R.W. Carpick, M. Salmeron, Rev. Sci. Instrum.67(9), 3298–3306 (1996)

64. M.Z. Bazant, K.T. Chu, B.J. Bayly, SIAM J. Appl. Math. 65(5),1463–1484 (2005)

65. M.Z. Bazant, K. Thornton, A. Ajdari, Phys. Rev. E 70(2), 021506(2004)

66. Y. Gil, O.M. Umurhan, I. Riess, Solid State Ion. 178(1–2), 1–12 (2007)

67. I. Riess, J. Electroceram. 17(2–4), 247–253 (2006)68. Z. Rosenstock, I. Feldman, Y. Gil, I. Riess, J. Electroceram. 14(3),

205–212 (2005)69. Y. Sun, S.E. Thompson, T. Nishida, J. Appl. Phys. 101(10),

104503 (2007)70. H.F. Tian, J.R. Sun, H.B. Lu, K.J. Jin, H.X. Yang, H.C. Yu, J.Q. Li,

Appl. Phys. Lett. 87(16), 164102 (2005)

J Electroceram