Dielectric Breakdown by Electric-Field Induced Phase Separation

Journal of the Electrochemical
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Dielectric breakdown by electric-field induced phase separation

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Journal of The Electrochemical Society
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Dielectric breakdown by electric-field induced phase
separation
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Journal: Journal of The Electrochemical Society
Manuscript ID JES-101542.R1
Manuscript Type: Research Paper
Date Submitted by the

07-Jul-2020
Author:
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Complete List of Authors: Fraggedakis, Dimitrios; Massachusetts Institute of Technology,
Mirzadeh, Mohammad; Massachusetts Institute of Technology
Zhou, Tingtao; Massachusetts Institute of Technology
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Bazant, Martin; Massachusetts Institute of Technology
Intercalation, Phase Separation, Memristors, Energy Storage,

Keywords:
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Information Storage, Dielectrics, Metal Insulator Transition
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https://mc04.manuscriptcentral.com/jes-ecs
Page 1 of 27 Journal of The Electrochemical Society
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3 Dielectric breakdown by electric-field induced phase separation
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6 Dimitrios Fraggedakis1 ,∗ Mohammad Mirzadeh1 , Tingtao Zhou2 , and Martin Z. Bazant1,3†
7
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8 Department of Chemical Engineering,
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10 Massachusetts Institute of Technology, Cambridge, MA 02139 USA
11 2
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Department of Physics, Massachusetts Institute
13 of Technology, Cambridge, MA 02139 USA and
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15 3
Department of Mathematics, Massachusetts Institute
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17 of Technology, Cambridge, MA 02139 USA
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19 (Dated: July 7, 2020)
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21 Abstract
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The control of the dielectric and conductive properties of device-level systems is important for
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25 increasing the efficiency of energy- and information-related technologies. In some cases, such as
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neuromorphic computing, it is desirable to increase the conductivity of an initially insulating

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28 medium by several orders of magnitude, resulting in effective dielectric breakdown. Here, we
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30 show that by tuning the value of the applied electric field in systems with variable permittivity
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32 and electric conductivity, e.g. ion intercalation materials, we can vary the device-level electrical
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34 conductivity by orders of magnitude. We attribute this behavior to the formation of filament-
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35
like conductive domains that percolate throughout the system, which form only when the electric
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37 conductivity depends on the concentration. We conclude by discussing the applicability of our
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39 results in neuromorphic computing devices and Li-ion batteries.

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∗
56 email: dimfraged@gmail.com
57 †
email: bazant@mit.edu
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60 https://mc04.manuscriptcentral.com/jes-ecs
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Journal of The Electrochemical Society Page 2 of 27
1
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3 I. INTRODUCTION
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Phase separating materials play a key role in several applications related to energy har-
8 vesting and storage, as well as information storage and processing. Some characteristic
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10 examples are Li-ion batteries [1], phase change [2] and redox [3] memristive devices [4], alloy
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12 catalysts [5] and self-organized surface nanoreactors [6]. In some cases, phase separation is
13
desirable as one can increase the efficiency of the system (e.g. increase catalytic activity [7],
14
15 change of electric [8, 9] and/or thermal [10] conductivity). However, there are other cases
16
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17 where phase separation degrades the performance of a device resulting in decreased life-
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19 time (e.g. fracture of secondary electrode particles in Li-ion batteries [11, 12] that decreases
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the available active material, delamination at electrode-electrolyte interface in all-solid-state
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22
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batteries [13] resulting in loss of contact of active material with the electrode). Hence, it
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24 is crucial to find ways that can actively control the occurrence and/or suppression of phase
25
26 separation, which can help us increase the efficiency of existing technologies, as well as open
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new possibilities on exploiting physical phenomena for new applications.
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Phase separation can be described as a form of instability. For example, a homogeneous

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31 binary mixture is thermodynamically unstable when its average concentration lies inside the
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33 spinodal region [14]. In this situation, any infinitesimal perturbation on the concentration
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35 field would evolve in time, making the system to form domains of the two phases [15]. In
36
general, there have been several efforts to control or induce the formation of instabilities in
37
38 equilibrium and non-equilibrium systems. Some examples are: 1) control of viscous finger-
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39
40 ing through the application of electric fields [16, 17], 2) stabilization of thermodynamically
41
42 unstable mixtures used in Li-ion batteries, such as LiFePO4 , using non-equilibrium driving
43
forces, galvanostatic conditions [1, 18–20] 3) destabilization of homogeneous polymeric, col-
44
45 loidal, electrolyte and glass mixtures under the application of electric field [21–28]. In the
46
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47 present work, we are interested in understanding the control of phase separation of mixtures
48
49 through electric fields and its impact on the transport properties of the system, e.g. change
50 of electric conductivity after phase separation occurs.
51
52
In many dielectric mixtures that phase separate under electric fields, e.g. colloids [29–32],
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54 polymers [33], amorphous solids [23], electrolytes [27], the electric permittivity depends on
55
56 the corresponding species concentration [34]. The effect of this dependence can be under-
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58 stood in the simple case of a binary mixture, where the applied electric field contributes
59
2
1
2
3 to the total chemical potential µ, i.e. µE ∼ ∂c ε |E|2 [35]. Therefore, a combination of a
4
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5 nonlinear concentration-dependent permittivity combined with high electric fields can alter
6
7 the free energy to change the miscibility gap and spinodal region.
8
9 The effect of the electric field has mainly to do with the thermodynamics stability, how-
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ever, the formed phase morphologies, e.g. filament-like structures [36, 37], can greatly affect
11
12 the transport properties on the macroscopic level, e.g. from low to high electric conductance
13
14 and vice versa, [9], leading to phenomena that resemble dielectric breakdown [38, 39]. Pre-
15
16 vious studies have focused identifying the conditions for dielectric breakdown due to phase
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17
separation solely based on thermodynamics [9, 23, 29–32], and few studies have discussed the
18
19 formation and dynamics of the conductive filaments [36], which is crucial for technological
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21 applications [2]. There are several solid-state materials that form conductive and insulating
22
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domains when they phase separate. For example, most of the commercial Li-ion interca-
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lation materials, i.e. Lix CoO2 [8, 40], Lix Ni1/3 Co1/3 Mn1/3 O2 [41], Li4+3x Ti5 O12 [42, 43],
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26 share this property as they undergo a metal-to-insulator transition (MIT), along with an
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28 ion concentration-dependent permittivity. This combination of characteristics makes them
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30 perfect candidates for studying the effect of electric fields on both the phase separation and
31 the electric conduction, which can lead to dielectric breakdown.
32
33 The goal of the present work is to develop a simple phenomenological theory that de-
34
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35 scribes dielectric breakdown due to electric-field induced phase separation. Based on a

36
37 concentration-dependent electric permittivity, we show that a homogeneous stable solution
38
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can phase separate in two (or more) phases after a critical electric field is applied. Addi-
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40 tionally, we consider the case where one of the phases is a metal and the other an insulator,
41
42 which translates in concentration-dependent electric conductivity. When the initial con-
43
44 centration is such that the material is insulating, after phase separation occurs we find the
45
system to conduct current like a metal. The transition from insulating to metallic behav-
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47 ior after electric field is applied corresponds to an effective dielectric breakdown, which is
48
49 attributed to the formation of filament-like structures that span the entire domain. This
50
51 phenomenon is related to the pioneering works by Goldhammer [44] & Herzfeld [45], that
52
link the changes in the electric permittivity with the species concentration to the metal-to-
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54 insulator transition. We relate our results to Li-ion intercalation materials, such as Lix CoO2
55
56 and Li4+3x Ti5 O12 , and we discuss the implications of our theory for resistive switching and
57
58 Li-ion battery applications.
59
3
1
2
3 II. THEORY
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6 We assume a phase separating dielectric medium placed between two blocking electrodes,
7
8 as shown in Fig. 1. Purple and yellow show the two phases, respectively, that can be formed
9
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10 either because the mixture is thermodynamically unstable or due to the application of an
11
12 electric field E that demixes the solid solution mixture. In the present picture, the two
13 phases have different electric permittivities, ε1 and ε2 , and electrical conductivities, σ1 and
14
15 σ2 . For simplicity we assume ε2 > ε1 and σ2 > σ1 , respectively. Representative examples of
16
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17 such system are ion intercalation materials [46–49], which are thermodynamically unstable
18
19 for a wide range of Li-ion fraction. Additionally, some of them undergo metal-to-insulator
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20 transition, e.g. Lix CoO2 [8, 40] (LCO), Li4+3x Ti5 O12 [42, 43] (LTO), and Lix TiO2 [50, 51],
21
22 where one of the phases has much larger electrical conductivity compared to the other - the
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24 difference can be up to six orders of magnitude [38]. In the following sections, we continue
25
26
with the thermodynamics and transport theory that describes the electric-field induced phase
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27 separation and consequently the dielectric breakdown of the medium along the direction of
28
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29 the electric field.
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32
33
34 A. Thermodynamics
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35
36
37 We are interested in modeling the dielectric phase separating medium shown in Fig. 1.
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The neutral species is described by the local fractional concentration c = n/nmax , where
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39
40
41
nmax is the maximum species concentration in the medium. Under constant temperature
42 and pressure, the Gibbs free energy of the system is [9, 15, 35, 52]
43
44 Z
45 1 1
46 G [c, φ, ρ] = dx gh (c) + κ |∇c|2 − ε(c) |∇φ|2 + ρφ (1)
V 2 2
ce
47
48
49 The first term is the homogeneous free energy of the system gh and the second term cor-
50
51 responds to the penalty gradient term [15, 53–55], which is used to describe the phase
52
separation of the material. The phenomenological parameter κ controls the thickness of the
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53
54 interface between the formed phases and is linked to their interfacial tension. The third and
55
56 fourth term describe the total electrostatic energy, where ε(c) is the electric permittivity of
57
58 the material as a function of the species concentration, ρ is the mobile charge density. For
59
4
1
2
3 the homogeneous free energy term gh we chose the regular solution model [18, 56, 57]
4
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5 kB T
6 gh = Ωc(1 − c) + (c ln c + (1 − c) ln(1 − c)) (2)
v
7
8 where Ω controls the interaction between the species particles - positive (negative) Ω corre-
9
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10 sponds to attractive (repulsive) interactions between the species. In the absence of particle-
11 particle interactions, v is the particle volume. Assuming local equilibrium, we define the
12
13 chemical potential of the neutral species as the variational derivative of the Gibbs free en-
14
15 ergy [14, 58]
16 δG 1 ∂ε
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17 µ= = µh (c) − κ∇2 c − |∇φ|2 (3)
δc 2 ∂c
18
19 where µh = ∂gh /∂c. Also, the chemical potential of the charged species that contribute to
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20
the charge density ρ reads
21
22
δG
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µρ = e= eφ (4)
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23 δρ
24 Regarding the dielectric model, we assume a simple monotonically increasing phenomeno-
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26 logical form
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28 ε = εf ε0 eγc (5)
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30 where γ controls both the change and the curvature of the permittivity. This form has been
31
32 previously used to model the electric-field induced phase separation of colloidal mixtures [32,
33
59].
34
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37 B. Transport
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40 We consider a phase separating material with both the permittivity and conductivity to
41
42 be functions of species concentration. The equations to model the process are [38, 60, 61]
43 ∂c
44 = −∇ · j (6a)
45 ∂t
46 ∂ρ
= −∇ · jρ (6b)
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47
∂t
48
49 δG
=ρ−∇·D=0 (6c)
50 δφ
51 where j and jρ are the species and electronic fluxes, respectively. Based on the assumptions
52
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53 of local equilibrium [14, 62] and microscopic reversibility [63, 64], the constitutive relation
54
55 for the fluxes and the dielectric displacement D are
56
57
D(c)c δG D(c)c
j=− ∇ =− ∇µ (7a)
58 kB T δc kB T
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5
1
2
3 δG
4 jρ = −σ(c)∇ = −σ(c)∇φ (7b)
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δρ
5
6 D = ε(c)E = −ε(c)∇φ (7c)
7
8
9 Eqs. 6(a)-(b) are the conservative descent [65] of the free energy G with respect to c and ρ,
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11
while eq. 6(c) is an equilibrium point with respect to φ.
12 In eq. 7(b), we consider the electric flux to be described by Ohm’s law [38] and we do
13
14 not specify neither the mechanism of electric conduction nor the identity of charge carriers.
15
16 This approach is similar to the model of leaky dielectrics that has been widely used in
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17
18
electrohydrodynamics [66–68], where the conductivity is assumed to originate from dissolved
19 ions. The model of eq. 7(b) is known to be a special limit of Poisson-Nerst-Planck type
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21 models under large applied electric fields and low carrier densities [69, 70].
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23 The functional form of the diffusivity D (c) is taken that of a lattice gas model, where the
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25 transition state is influenced by excluded volume effects leading to D (c) = D0 v (1 − c) [35].
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The permittivity is given in eq. 5. The conductivity is assumed to be linear in concentration

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28 σ (c) = σ0 + σ1 c, where for σ1 > 0 increasing charge carriers correspond to increasing
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30 conduction [38]. While being simple, this specific form of σ does not alter our conclusions.
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When phase separation occurs, we assume the two formed phases to have conductivities
33 which differ by several orders of magnitude (metal-insulator/semi-conductor contact and
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35 vice versa [8]). This large difference in combination with the phase separation due to electric
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37 fields leads to dielectric breakdown in our system. Finally, we close the description of the
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system using the following boundary conditions: 1) for the species number density we assume
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40 blocking electrodes, which translates into j · n = 0 along all domain boundaries; 2) for the
41
42 potential φ, we consider φ (0) = V and φ (L) = 0 to simulate the voltage drop across the
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44 cell, while on all the other boundaries we assume n · ∇φ = 0; 3) we assume a contact angle
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46
of π/2 at any triple contact point between the formed phases and the boundaries of the
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47 system, n · ∇c = 0. In all cases, n is the outward normal vector.

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49
50
51 C. Characteristic Scales and Material Parameters
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54 The characteristic scales we consider herein are the following: (i) time tch = L2 /Dmax ,
55
56 (ii) length Lch = L, (iii) voltage φch = kB T /e, (iv) conductivity σch = σmax , (v) charge
57
58 density ρch = enmax NA , (vi) volumetric energy kB T /v, where NA is the Avogadro number.
59
6
1
2
3 Substituting these scales in the transport equations we arrive at the following dimensionless
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5 forms
6 ∂c D(c)
7 =∇· ∇µ (8a)
∂τ Dmax
8
9 ∂q σ(c)
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10 =∇· Φ ∇φ (8b)
∂τ σmax
11
12 1
13 − ∇ · [ε(c)∇φ] = q (8c)
λ2D
14
15 where Φ = σmax kB T
, λ−2
εf ε0 kB T
Dmax nmax NA e2 D = nmax NA L2 e2
. The dimensionless number Φ quantifies the
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17 ratio between the electronic and the species mobilities. An interesting limit is that of
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19 Φ 1, where the electron/hole redistribution in the domain occurs much faster compared to
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species diffusion, and can be thought as the continuum equivalent of the Born-Oppenheimer
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approximation [71]. In that case, eq. 8(b) is always at quasi-equilibrium. This is not the
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24 case though when the material enters the insulating regime. Finally, the homogeneous free
25
26 energy combined with the electric energy reads
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28 gv Ωv
c (1 − c) + c ln c + (1 − c) ln(1 − c)
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=
29 kB T kB T
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(9)
30 εf ε0 vkB T γc
31 − 2 2
e |∇φ|2
32
2e L
33 Before we dive further into analyzing the implications of the applied electric field on the
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35 thermodynamics stability of mixing and the dynamics of phase separation, it is instructive

36
37 to consider a specific material for the parameters of our model: εr , γ, κ, Ω, v, nmax , σmax ,
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Dmax and L. An interesting example with practical implications in Li-ion batteries and
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40 neuromorphic computing is Li4+3x Ti5 O12 [9, 72]. Based on the density and the molecular
41
42 weight of Li7 Ti5 O12 , we know that nmax ' 0.72 × 104 mol/m3 as well as v ' 1 nm3 .
43
44 Additionally, previous phase field modeling [72] has reported NA Ωv ' 8.612 kJ/mol and
45
NA κv ' 8.61×10−15 J m2 /mol. The electric permittivity is known to increasing as a function
46
ce
47 of average Li-ion fraction x. In particular, for x → 0, ε ' 1.5ε0 , while for the fully lithiated
48
49 state, x → 1, the permittivity becomes ε ' 50ε0 [73]. This behavior can be approximated by
50
51 choosing εr ' 1.5 and γ ' 3.5. In terms of tracer diffusivity, NMR studies [74–76] measured
52
D0 ' 4 × 10−16 m2 /s, while electrochemical measurements found that the conductivity
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54 changes from σ(x → 0) ' 10−5 S/m (insulating) to σ(x → 1) ' 102 S/m (metallic) [42].
55
56 Finally, we consider a thin-film device with dimensions around L ' 100 nm, which is kept
57
58 at constant temperature T = 298 K.
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7
1
2
3 The discussed material and system parameters result in Φ ' 107 and λ−1 −4
D ' 10 . Given
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5 this value for Φ, it is clear that when the material is metallic, we can assume eq. 8(b) to
6
7 be in quasi-equilibrium. The small value of λ−1
D ' 10
−4
corresponds to double layers on the
8
9
scale of 10−1 Å, which is a reasonable value for a perfect metal. The RC timescale τC =
q
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10 e2 nmax NA εr ε0 L2
for charging the formed double layers after the electric field is applied [77]
σ 2 kB T
11
12 is of the order of 5 × 10−10 s for the conductive phase and 10−2 s for the insulating one,
13
14 values much lower than the diffusive timescale of the neutral species τD = L2 /Dmax ∼ 102 s.
15
Due to numerical stability, however, when we solve eqs. 8 we use a re-scaled value for λ−1
16 D
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17 that matches the interface thickness.
18
19 Finally, we discretize the set of eqs. 8 using finite elements [78], and more specifically, we
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21 use continuous linear basis functions for approximating all unknowns. Additionally, we solve
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the system of equations in a monolithic fashion, while for the time integration a second order
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24 scheme is used (BDF2) [79]. The non-linear system of equation is solved using Newton’s
25
26 method and for the inversion of the resulting linear system we use LU decomposition.
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30 III. INSTABILITY OF A HOMOGENEOUS STATE

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33 A. Thermodynamics Stability
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36 According to phase equilibria, cs,1 and cs,2 are the two spinodal points that indicate the
37 change in the curvature of the homogeneous free energy. Both of these values are solutions
38
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39 of [14, 52]
40 ∂ 2 gh ∂µh
41 2
= =0 (10)
42
∂c ∂c
43 In the spinodal region, the homogeneous solution is unstable (gh00 < 0) and tends to phase
44
45 separate in two immiscible phases. The concentration in each phase is determined by the
46
common tangent construction
ce
47
48
49 µ(ceq,1 , |E| = 0) =µ(ceq,2 , |E| = 0) =
50
gh (ceq,1 ) − gh (ceq,2 ) (11)
51
52 ceq,1 − ceq,2
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54 Here, we are interested in studying the electric-field induced phase separation. When the
55
56 electric field is uniform across medium, E ' −∆V ex (∆V is the dimensionless voltage drop,
57
58 scaled with the thermal voltage kB T /e), we can see from eq. 3 that the value of the chemical
59
8
1
2
3 potential will change. The effect of the electric field on the thermodynamics stability of the
4
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5 homogeneous mixture, however, depends on the functional form of ε. For example, when
6
7 ε is a linear function of c, then the spinodal region is not affected. Therefore, for having
8
electric field-induced phase separation we require that ε00 (c) 6= 0. The equation for finding
9
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10 the spinodal points reads [9, 23, 34]
11
12
13
14 ∂ 2 gh (c) 1 ∂ 2ε
15 = |∆V |2 (12)
16 ∂c2 2 ∂c2
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17
18
19
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20 Fig. 2(a) shows the dimensionless free energy as a function of the species fraction c for
21
22 four different values of the applied electric field |∆V |. In this example we set Ω = 0, as
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24 we are interested to understand the implications of the electric field on the de-mixing of a
25
26 homogeneous solution. For |∆V | = 0, the energy is convex, and the mixture remains in
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27 the solid solution regime. With increasing |∆V |, however, the energy landscape becomes
28
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29 distorted, shifting the minimum energy toward c ' 0.8. For |∆V | = 400, the electric
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30
31 field is strong enough to change the convexity of g (dashed region), making phase separation
32
33 thermodynamically favorable. At this point, the electrostatic energy becomes comparable to
34 the entropy of mixing due to alignment of the microscopic dipoles in the medium, Figs. 2(a) &
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36 (b). As it will be shown in more detail in later sections, after phase separation is completed
37
38 the system will consist of domains with low and high permittivity, respectively. Further
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39
40 increase of the E-field increases the distance between the binodal points (red dots) which
41 leads to further increase in the dielectric mismatch between the phase separated domains,
42
43 Fig. 2(a).
44
45
These observations can be summarized in the phase diagram of Fig. 2(c), which is con-
46
ce
47 structed in terms of the applied electric field |∆V | and the system fractional concentration
48
49 c. The binodal region, which is thermodynamically metastable, is shown with blue, while
50
51 the spinodal region with light brown. It is clear that there exist a critical electric field
52
(around |∆V |c ∼ 380 for the parameters used herein) above which the convexity of the free
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53
54 energy changes and the homogeneous state of the material becomes thermodynamically un-
55
56 stable. The implications of this phase diagram on the dielectric breakdown of the material
57
58 are discussed in Section IV.
59
9
1
2
3 B. Linear Stability
4
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5
6 The standard way to understand the dynamics during the onset of phase separation is
7
8 through linear stability analysis. In particular, given the physical parameters of our model,
9
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10 we can identify the critical wavelength that can be induced by random fluctuations that lead
11
to de-mixing of a homogeneous state. To do so, we assume an infinitesimal perturbation of
12
13 the form δy = δeik·x+ωt , where yT = (c, q, φ), δ is an infinitesimal vector, kT = (kx , ky , kz )
14
15 is the wavenumber, and ω is the growth rate of the instability. The base state around which
16
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17 we linearize eqs. 8 is y0T = (c0 , 0, −∆V (1 − x)). The dispersion relation is found by solving
18 the secular equation [80]
19
Fo
20 det |J − ωe1 e1 | = 0 (13)
21
22
an
where J is the Jacobian matrix defined as J = δy f - the components of J are given in the
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23
24 appendix - and e1 corresponds to the unit vector along the ‘concentration’ axis. Assuming
25
26 Φ 1, the growth rate becomes
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27
28 ω = − ε |∆V |2 (∂c ln σ) kx2 − D kx2 + kr2 ×
dM
29 (14)
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30 − (1/2) |∆V |2 ∂c2 ε + ∂c2 gh + κ kx2 + kr2

31
32 where kr2 = ky2 + kz2 , and D, σ and κ are in their dimensionless form. From eq. 14, it is clear
33
34
that the direction of the E-field affects critical wavelength in the x-direction. We can further
On
35 analyze the result by determining the set of (kx , kr ) that maximize ω. Solving ∂ω/∂k = 0,
36
37 we find r !
38 Q
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(kx , kr ) = 0, (15a)
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40
4κ
41  s 
2
42 DQ − 2ε |∆V | ∂c ln σ 
43 (kx , kr ) = ± ,0 (15b)
4κD
44
45
where Q = |∆V |2 ∂c2 ε − 2∂c2 gh . From the first set of solution, it is clear that for kr to be
46
ce
47 physical it has to be positive. This is true for |∆V |2 ∂c2 ε − 2∂c2 gh > 0, which is equivalent to
48
49 the thermodynamics stability condition we discuss in Sec. IIIA. On the contrary, the second
50
51 locus of solutions is physical for D |∆V |2 ∂c2 ε − 2∂c2 gh > 2ε |∆V |2 ∂c ln σ, which shows that
52
conductivity variations can suppress phase separation in the direction of electric field when
Ac
53
54 ∂c ln σ > 0.
55
56 Fig. 3(a) demonstrates the stability diagram for a mixture with average concentration
57
58 c = 0.8 in terms of the wavenumber set (kx , kr ). The lines denote the isocontour ω = 0
59
10
1
2
3 for different applied voltages across the domain. Inside the formed envelopes lies the region
4
pt
5 where ω > 0, which corresponds to the long-wave modes for which de-mixing occurs, while
6
7 short-waves are damped by the action of surface tension. As shown by eq. 15(b), when
8
kr = 0 there is a critical applied voltage - |∆V | . 700 below which perturbation in the x
9
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10 direction are suppressed.
11
12
13
14
C. Phase Separation Dynamics
15
16
us
17
In order to test the predictions of the theory and understand the time evolution of de-
18
19 mixing, we perform two-dimensional simulations of an initially homogeneous mixture with
Fo
20
21 concentration c = 0.8. As a representative example, we consider the case where the value
22
an
of the applied electric field across the domain is |∆V | = 600. According to the phase
rR
23
24
diagram of fig. 2(c), we expect the two formed phases to have concentrations cb,1 ' 0.12 and
25
26 cb,2 ' 0.998, respectively, where cb is the binodal point concentration.
ev
27
28 Fig. 3(b) shows the temporal evolution of the concentration field after the electric field
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29
iew
30 is applied. The light yellow color represents the rich phase, while the dark purple the low
31
32
concentration one. At time t = 0, the homogeneous profile is perturbed with zero-mean white
33 noise. After some time, these perturbations grow exponentially in time as predicted by the
34
On
35 linear stability analysis of Sec. III B. The exponential evolution of the instability stops right
36
37 after the two phases begin to form, i.e. for t = 0.1. At this moment, filament-like patterns,
38
ly
which consist of the poor and rich phase, span the entire domain. It is noticeable that the
pte
39
40 formed filaments align with the direction of the applied electric field, an observation that
41
42 connects with the phenomenon of dielectric breakdown, which is discussed in more detail in
43
44 the next section (Sec. IV). At later times, t = 1.0, the initially formed filaments with the
45
lowest concentration break into smaller islands. Due to the existence of multiple interfaces
46
ce
47 this state is not energetically favorable. As a result, the system undergoes further coarsening
48
49 (Ostwald ripening [81]) making the islands to merge into larger filamentary domains, leading
50
51 to the non-equilibrium steady state shown for t = 20.0. At this time, three large domains
52
that consist of the high concentration phase are formed, while the low concentration ones
Ac
53
54 occupy smaller fraction of the total volume. This is in agreement with the theoretically
55
56 predicted phase diagram, fig. 2(c), where the lever rule predicts that the phase with the
57
58 lowest concentration occupies ∼ 22% of the total system. Finally, when the electric field
59
11
1
2
3 is removed, the homogeneous free energy becomes convex again, leading to mixing of the
4
pt
5 two formed phases. Therefore, the recovery of a solid solution after the electric field bias is
6
7 removed corresponds to volatile behavior [2].
8
9
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10
11 IV. DIELECTRIC BREAKDOWN DUE TO FILAMENT FORMATION
12
13
14 The de-mixing of an initially solid-solution system demonstrates the basic principle of the
15
16 electric-field induced phase separation. Although studying these materials is informative,
us
17 many materials of practical relevance phase separate at room temperature, even in the
18
19 absence of E-field. Such examples are Lix CoO2 , Li4+3x Ti5 O12 , and Lix TiO2 where during
Fo
20
21 phase separation undergo metal-to-insulator (and vice versa) transition. Here, we focus
22
an
on the case of an initially phase separated material and show that, by applying electric
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23
24 field, it is possible to control the orientation of the phase boundaries and, consequently, the
25
26 current-voltage response of the material.
ev
27
28 For the ease of computations and without altering the final conclusions, we choose a
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29
iew
30 system with NA Ωv ' 5.601 kJ/mol, and electrical conductivity of the form σ = σ0 eσ1 c ,
31 where σ1 = 10−7 and σ1 = 16.11. All the other properties are the same as discussed in
32
33 Sec. II C. The reason for changing the functional form of σ is to replicate the abrupt change
34
On
35 in the electrical conductivity during the insulator-to-metal transition that take place in
36
37
materials like Li4+3x Ti5 O12 [42, 74].
38 Fig. 4(a) demonstrates the temporal evolution of the resulting current for three different
ly
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39
40 applied ∆V . The current is defined as the surface integral of the electric current density
41 R
42 across one of the electrodes, I = − J · ndA. For all the applied voltages, the initial phase
43
44 morphology corresponds to the earliest time instant shown in the inset of Fig. 4(a), while
45 the average concentration is set to c = 0.5.
46
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47 For ∆V = 100, the resulting current is always of the order of 10−3 , which can be under-
48
49 stood in terms of an equivalent circuit. Given the functional form for σ, we know that one
50
51 of the phases is insulating. Also, the phase morphology does not change after the voltage is
52 applied. Therefore, the equivalent circuit consists of two resistances in series, one of which
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53
54 corresponds to an insulator with resistance R = ∆V /I ∼ O (105 ).
55
56 For larger applied voltages, ∆V & 200, the temporal evolution of the current is quali-
57
58 tatively different. More specifically, we focus on the phase evolution for ∆V = 200. It is
59
12
1
2
3 apparent from the inset images that the applied electric field across the domain is able to
4
pt
5 change the morphology completely. At early times, tc < 0.27, the electric field forces the
6
7 binodal concentration to change, as shown in the phase diagram of fig. 4(b). Due to this
8
change, the system is perturbed and the interface between the two phases becomes unsta-
9
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10 ble forming tips in the direction of the electric field. At around tc ' 0.27, the instability
11
12 grows abruptly leading to the formation of highly conductive filaments. By the time these
13
14 filaments ‘touch’ the second electrode, the electric current increases by three orders of mag-
15
nitude, from I ' 0.01 to I ' 7. After this critical event, a dendrite-like structure is formed,
16
us
17 t ' 1, which evolves in four filaments for t & 5 - two consisting of the highly conductive
18
19 phase and two of the insulating one. The insulating domains within the dendrite-like struc-
Fo
20
21 ture demonstrate a specific angle that can be related to the Taylor cone behavior [82]. The
22
an
final configuration yields a steady state current around I ' 19, which exceeds by almost four
rR
23
24 orders of magnitude the value obtained when ∆V = 100 is applied. Therefore, we conclude
25
26 that the formation of highly electric conductive filaments after a critical voltage is applied
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27
28 leads to the phenomenon of dielectric breakdown.
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iew
30
The question that arises, though, is why voltages below ∆V ' 200 do not alter the
31 phase morphology into filamentary ones. The answer becomes clear when analyzing the
32
33 phase diagram of fig. 4(b). When no E-field is applied, the system is separated in two
34
On
35 immiscible phases with concentration cb,1 ' 0.21 and cb,2 ' 0.78, respectively. When a
36
voltage drop of ∆V = 100 is applied, these values lie the binodal region, which is known to
37
38 be metastable. As a result, the species are redistributed between the two phases and a new
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39
40 steady state is reached without altering the morphology of the existing interface. However,
41
42 when ∆V = 200 is applied, the initially stable state becomes thermodynamically unstable,
43
as the concentration of the rich phase lies inside the spinodal region - the spinodal points
44
45 are cs,1 ' 0.3 and cs,2 ' 0.8. Hence, any infinitesimal perturbation tends to destabilize the
46
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47 system from its initial state, which, as analyzed in fig. 4(a) leads to the formation of highly
48
49 conductive filaments that align with the direction of the applied electric field.
50
51
52
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53 V. DISCUSSION
54
55
56 Our findings on dielectric breakdown, as a result of filament formation due to the applied
57
58 electric field, are relevant to resistive switching and memristor devices [2, 4, 83–88]. One of
59
13
1
2
3 the desired properties of memristors is the ability to change the electrical conductivity of
4
pt
5 the device by orders of magnitude under an external stimulus, e.g. voltage or temperature.
6
7 Here, we show that through de-mixing of a solid solution mixture or change in the phase
8
morphology of an already phase separated material, we can tune the device-level resistance
9
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10 by orders of magnitude, fig. 4(a). This idea has been recently demonstrated experimentally
11
12 in Li4+3x Ti5 O12 , for x → 0 and x → 1, a material proposed as a potential candidate
13
14 for memristive devices [9]. It is known that Li4+3x Ti5 O12 undergoes an insulator-to-metal
15
transition [42], resulting in a change in the electric resistance of the memristive device by
16
us
17 orders of magnitude. Here, by using the properties of LTO, we show that a dimensionless
18
19 voltage of ∼ O (102 ) is required to induce de-mixing, and therefore change in the conductivity
Fo
20
21 of the device. This value corresponds to approximately 3 − 4 V, which is on the same order
22
an
of magnitude with the experimentally observed value for dielectric breakdown.
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23
24 Fig. 5 shows a schematic on the device current-voltage response related to the phenomena
25
26 presented in Secs. III-IV. When the system is initially prepared as solid solution, de-mixing
ev
27
28 occurs at a critical voltage ∆Vc , figs. 2(b) & 4(b). Two scenarios are possible: i) the mixture
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iew
30 shows solid solution behavior for all values of c, ii) the mixture is phase separating for a
31 wide range of concentrations. In the former situation, when the applied voltage is close to
32
33 critical value the two phases have very similar concentration, and thus, similar conductivi-
34
On
35 ties. Therefore, for resistive switching applications a voltage much larger than the critical
36
37
value needs to be applied in order to establish phases with very large mismatch in their con-
38 ductivity. For the latter case, when the critical voltage is applied and de-mixing occurs, the
ly
pte
39
40 current changes abruptly due to the large difference on the electric resistance between the
41
42 formed phases, fig. 4(a). Although dielectric breakdown is necessary for resistive switching,
43
44
memristors have the additional requirement of non-volatile operation [2]. When the electric
45 field bias is lifted, the de-mixed mixture is going to return to its initial homogeneous state
46
ce
47 under diffusive timescales. In particular, for a device with L = 100 nm [9], and a material
48
49 with maximum diffusion of ∼ 10−16 m2 /s [72], the system relaxes back to equilibrium within
50
51
∼ 100 seconds. This timescale is very short for any application that requires non-volatile
52 operation, such as neuromorphic computing.
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53
54 On the contrary, mixtures that are thermodynamically unstable under no applied electric
55
56 field are non-volatile due to the persistence of the filaments after we stop applying a voltage
57
58 bias. A schematic of the representative current-voltage curve is shown with the orange line
59
14
1
2
3 in fig. 5. At first, the system is prepared in a state where the interface between the two
4
pt
5 phases is aligned perpendicular to the direction of the electric field. As discussed earlier,
6
7 this configuration corresponds to a circuit with two resistors in series, where the one with
8
the lowest conductivity governs the I − V response of the device. At large enough voltages,
9
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10 though, the phase separated mixture becomes thermodynamically unstable and conductive
11
12 filaments are formed that connect the two electrodes. At this point, the macroscopic resis-
13
14 tance of the device drops by orders of magnitude causing an effective dielectric breakdown
15
- from a high resistance state (HRS) to a low resistance one (LRS) [2, 4]. After filament
16
us
17 formation, decreasing the voltage drop does not affect the phase morphology. Hence, the
18
19 persistence of the filamentary state, even after electric fields are not active, demonstrate the
Fo
20
21 potential application of such systems in neuromorphic computing.
22
an
Even though our model is able to predict the correct order of magnitude for the critical
rR
23
24 voltage that causes dielectric breakdown in LTO, it is greatly simplified. First, the functional
25
26 forms for both the electric permittivity and conductivity are purely empirical and do not
ev
27
28 describe an actual material. Therefore, a more complete theory for both ε and σ, which takes
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29
iew
30
into account information from first-principles calculations and/or experiments, is needed.
31 A more complete picture would identify the charged species, e.g. bound and/or free
32
33 electrons or ions, that can contribute to the electric conductivity of the medium. For ex-
34
On
35 ample, in the case of quasi-particles such as polaron-ion pairs, if the applied electric field is
36
37
strong enough, e.g. near electrode/bulk interfaces [89] or phase boundaries [27], the pairs
38 can split into its components, i.e. localized electrons and ions. Each of the newly generated
ly
pte
39
40 species can have its own conductivity [90, 91] where its diffusive/conductive motion would
41
42 be described by the corresponding conservation law.
43
44 Another effect we have neglected, which plays an important role when the system is at its
45 LRS, is Joule heating [92–97]. For nanometer scale phase change memristors, Joule heating
46
ce
47 is known to control resistive switching. Hence, the change of the local temperature due to
48
49 dissipation phenomena, such as electric conduction, is expected to affect the thermodynamics
50
51 and, consequently, the phase separation dynamics after the electric field is applied.
52
Most memristive devices are solid state in nature [2, 4, 92]. Thus, it is expected that elas-
Ac
53
54 tic and/or inelastic deformation, as well as the existence of grain boundaries and dislocations
55
56 to influence the dynamics of conductive filament formation. Additionally, phase-separating
57
58 intercalation materials are known to exhibit misfit strains which affect the morphology of
59
15
1
2
3 the formed interfaces [49, 98]. Therefore, there will be a competition between the interface
4
pt
5 orientation defined by the minimum elastic energy state and the one induced by imposed
6
7 electric field.
8
9
Finally, in our model we assumed a closed system where species concentration does not
cri
10
11 change. However, in ion intercalation materials one can change the total number of ions.
12
13 This is known to have a large impact on the phase morphology [18, 49] as well as on the
14
15 electronic conductivity of the material [8], e.g. metal-to-insulator transition. Therefore,
16
us
it would be interesting to explore the effects of species insertion/extraction on the phase
17
18 morphologies at the same time electric fields are applied. All these phenomena have to
19
Fo
20 be examined in greater detail for establishing qualitative design principles for memristive
21
22 devices that are based on the phenomenon of the electric-field induced phase separation.
an
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23
24
25
26
ev
27
28
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29
iew
30
31 VI. SUMMARY
32
33
34
On
35 In summary, we showed that when electric field is applied in a material with concentration-
36
37 dependent permittivity and electric conductivity, phase separation occurs and dielectric
38
ly
breakdown is observed. Through thermodynamics stability analysis we derived phase di-

pte
39
40 agrams in terms of the species concentration and the applied voltage drop between the
41
42 operating electrodes, and we demonstrated that one can de-mix a solid solution mixture.
43
44 Additionally, by performing simulations we predicted that once the system is thermodynam-
45
ically unstable, filament-like structures are formed. These structures percolate across the
46
ce
47 domain and are responsible for the dielectric breakdown by allowing electrons to conduct
48
49 through the metallic phase. Furthermore, we demonstrated the predictions of the theory
50
51 to be in agreement with recent experiments on Li4+3x Ti5 O12 . Finally, we discussed the
52
implications of our results on resistive switching, which can be useful in applications like
Ac
53
54 neuromorphic computing. In particular, we showed that phase separating materials can
55
56 exhibit the desired non-volatile behavior while solid solution materials do not, as they relax
57
58 back to their equilibrium state after the electric field is turned off.
59
16
1
2
3 ACKNOWLEDGMENT
4
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5
6 The authors would like to thank Tao Gao, Neel Nadkarni, Juan Carlos Gonzalez-Rosillo,
7
8 Moran Balaish, and Jennifer L. M. Rupp for insightful discussions.
9
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10
11
12 APPENDIX
13
14
15 Jacobian Matrix
16
us
17
18 The components of the Jacobian matrix discussed in Sec. III B are presented here. More
19 specifically,
Fo
20  
21
22
J1,1 0 J1,3 
 
an
J = J2,1 0 J2,3  (16)
rR
23  
24
J3,1 J3,2 J3,3
25
26
ev
where
27
28 2 2 2 1 2 2
J1,1 = −k Dc ∂c gh + κk − ∂c ε |∇φ| (17a)
dM
29 2
iew
30
31 J1,3 = i∂c εk 2 k · ∇φ (17b)
32
33 J2,1 = i∂c σk · ∇φ (17c)
34
On
35
J2,3 = −σk 2 (17d)
36
37
38 J3,1 = −i∂c εk · ∇φ (17e)
ly
pte
39
40 J3,2 = λ2D (17f)
41
42
43
J3,3 = εk 2 (17g)
44
45
46
ce
47
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41
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curves correspond to different values of the applied electric field. The thermodynamics model is
rate ω = 0 for different values of the wave numbers kx and kr = ky2 + kz2 . Different colored
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phase separated state as shown in the inset for the earliest time instance. The average concentration
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is c = 0.5. At low voltage drop, e.g. |∆V | = 100, the initial morphology remains intact. For |∆V |
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the blue region is the binodal region, where the solution is metastable. The critical voltage drop
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FIG. 5. Schematic of the device-level current-voltage response for a mixture with electric permit-
tivity that depends on species concentration ε (c). Both curves correspond to the parameters used
in Secs. III-IV. The blue curve corresponds to a system initially prepared as a solid solution mix-
ture, while the orange line is that for a phase separated system. The arrows indicate the direction
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of the voltage sweep. In both cases, there is a critical applied voltage above which filaments are
formed between the two electrodes and the resistance of the device drops by orders of magnitude.
This phenomenon corresponds to resistive switching, which is the key operation for the operation
of memristors.
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27

Dielectric Breakdown by Electric-Field Induced Phase Separation

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Journal of the Electrochemical

Dielectric breakdown by electric-field induced phase separation

Manuscript version: Accepted Manuscript

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This content was downloaded from IP address 134.117.10.200 on 17/07/2020 at 02:22

Manuscript Type: Research Paper

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Intercalation, Phase Separation, Memristors, Energy Storage,

neuromorphic computing, it is desirable to increase the conductivity of an initially insulating

39 results in neuromorphic computing devices and Li-ion batteries.

Phase separation can be described as a form of instability. For example, a homogeneous

35 scribes dielectric breakdown due to electric-field induced phase separation. Based on a

The permittivity is given in eq. 5. The conductivity is assumed to be linear in concentration

47 system, n · ∇c = 0. In all cases, n is the outward normal vector.

35 thermodynamics stability of mixing and the dynamics of phase separation, it is instructive

30 III. INSTABILITY OF A HOMOGENEOUS STATE

30 − (1/2) |∆V |2 ∂c2 ε + ∂c2 gh + κ kx2 + kr2

breakdown is observed. Through thermodynamics stability analysis we derived phase di-

27 in Materials Science and Engineering 24, 035020 (2016).

30 67, 283 (2015).

27 [33] C. Liedel, C. W. Pester, M. Ruppel, V. S. Urban, and A. Böker, Macromolecular Chemistry

30 [34] J. Tomlinson, Journal of Electronic Materials 1, 357 (1972).

30 [52] P. M. Chaikin, T. C. Lubensky, and T. A. Witten, Principles of condensed matter physics,

39 [56] J. H. Hildebrand, , and R. Scott, Solubility of Non-electrolytes (Rheinhold Publishing Co.,

30 [75] W. Schmidt, P. Bottke, M. Sternad, P. Gollob, V. Hennige, and M. Wilkening, Chemistry of

39 [78] D. Fraggedakis, J. Papaioannou, Y. Dimakopoulos, and J. Tsamopoulos, Journal of Compu-

27 [94] J. P. Strachan, D. B. Strukov, J. Borghetti, J. J. Yang, G. Medeiros-Ribeiro, and R. S.

39 [98] D. A. Cogswell and M. Z. Bazant, ACS nano 6, 2215 (2012).

"1 <latexit sha1_base64="(null)">(null)</latexit>

FIG. 1. Schematic representation of a phase separating dielectric medium of length L placed

align with the applied electric field.

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