Direct Current Conductivity of Thin-Film Ionic Conductors from

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D.C. Conductivity of Thin-Film Ionic Conductors from Analysis of Dielectric Spectroscopy Measurements in Time and Frequency Domain Eleftherios John Kapetanakis, Paschalis Gkoupidenis, Vassilios Saltas, Antonios M. Douvas, Panagiotis Dimitrakis, Panagiotis Argitis, Konstantinos Beltsios, Stella Kennou, Christos Pandis, Apostolos Kyritsis, Polycarpos Pissis, and Pascal Normand J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b06979 • Publication Date (Web): 26 Aug 2016 Downloaded from http://pubs.acs.org on August 28, 2016

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The Journal of Physical Chemistry

D.C. Conductivity of Thin-Film Ionic Conductors from Analysis of Dielectric Spectroscopy Measurements in Time and Frequency Domain

Eleftherios Kapetanakis,*,1 Paschalis Gkoupidenis,2,3 Vassilios Saltas,1 Antonios M. Douvas,2 Panagiotis Dimitrakis,2 Panagiotis Argitis,2 Konstantinos Beltsios,3 Stella Kennou,4 Christos Pandis,5 Apostolos Kyritsis,5 Polycarpos Pissis,5 and Pascal Normand2

1

School of Applied Sciences, Technological Educational Institute of Crete, 73133 Chania,

Greece. 2

Institute of Nanoscience and Nanotechnology, National Center for Scientific Research

‘Demokritos’, 15310 Aghia Paraskevi, Athens, Greece. 3

Department of Materials Science and Engineering, University of Ioannina, 45110 Ioannina,

Greece. 4

Department of Chemical Engineering, University of Patras, 26500 Patras, Greece.

5

Department of Physics, National Technical University of Athens, Zografou Campus, 15780

Athens, Greece.

ACS Paragon Plus Environment

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ABSTRACT A method is developed for extracting the d.c. conductivity (σ ) of ion conducting materials from frequency- and time-domain dielectric spectroscopy measurements. This method exploits the electrode polarization effects arising from the charging of an ion-blocking capacitor and provides a useful way of obtaining σ for ionic conductors which do not exhibit frequency(time-) independent conductivity plateau; the latter absence of plateau is often encountered in the case of thin film materials. It allows, by proper design of the test cells, the estimation of σ independently of the specimen thickness as it is demonstrated herein for SiO2 blocking layers and electrolyte systems made of a polyoxometalate (POM) molecule embedded in poly(methyl methacrylate) (PMMA) polymeric matrices. For different post-preparation and measurement conditions, the σ values obtained for thick (8 µm) POM-PMMA layers are in good agreement not only with the observed conductivity plateaux but also with the values determined in the case of thin (270 nm) POM-PMMA layers for which no plateau is detected. The proposed method allows for the probing of a possible dependence of material properties on thickness and is of substantial interest for low-dimensional systems. The applicability and accuracy of the method are discussed and assessed in relation to the main methods currently used in the field.

1. INTRODUCTION Many solid-state ion conducting materials are being investigated owing to their fundamental scientific interest and increasing use in a wide range of applications such as advanced energy storage (batteries and supercapacitors), energy conversion (fuel cells and photoelectrochemical solar cells), electromechanical transduction devices (electroactive ionic


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actuators and sensors) and electronics (electrolyte gated transistors and memories).1-8 A central issue in the development of such materials is the identification and understanding of the dynamics of ionic motions for which many aspects of charge-transport mechanism on the microscopic scale still continue to present a scientific challenge.9-11 Ionic conduction in many disordered solids can be described by activated hopping of charge carriers in a randomly varying energy landscape and is theoretically well understood.12-14 The direct-current (d.c.) component of the ionic conductivity, , which represents the longrange diffusion of ions as they hop from site to site through the matrix, is an important intrinsic feature of the materials. provides information about ion motion in the bulk and its dependence on factors like temperature, chemical composition and moisture content of material; it relates9 to the ion charge, , the number of mobile ions, , and the ion mobility, , through the relation: = . and other electrical properties of ionic materials are usually obtained from impedance spectroscopy15 at frequencies typically in the mHz to MHz range (referred to as frequency domain, FD, measurements) or, for lowest frequencies, from measurements of time-dependent displacement currents16,17 (or absorption currents) resulting from the application (polarization) or removal (depolarization) of a step voltage (time domain, TD, measurements). In systems exhibiting a linear dielectric response to a perturbation (referred to as linear systems), the TD response is related to the FD response by means of Laplace or Fourier transforms. Representative conductivity spectra ( () and (), with the real part of the effective conductivity) that can be derived from FD and TD measurements of ionic materials are depicted in Figure 1. Two extreme cases are shown corresponding to () characteristics with a well-developed frequency-independent conductivity plateau and no conductivity plateau.18 These characteristics refer to specimens of the ion-conducting material used in the present study


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(see section 3) with film thicknesses, >> 1 µm (referred herein to as "thick" sample) and < 1 µm, ("thin" sample), respectively.

Figure 1. Idealized apparent conductivity σ' versus frequency (time) characteristics in two extreme cases related to the presence or not of a frequency-independent conductivity plateau. These characteristics refer to specimens of the ion-conducting material used in the present study. The dash line curve corresponds to a specimen with a film thickness of 8 µm ("thick" sample) while the solid line curve stands for a film thickness of 0.27 µm ("thin" sample).

The high-frequency (short-time) regime corresponds to bulk dielectric phenomena arising from back-and-forth ionic motions,9 which are independent of sample thickness (indicated as dielectric region in Fig. 1). Moving to intermediate frequencies, macroscopic (or long-range) charge transport leads to a conductivity plateau in the case of the thick sample, which is usually identified with the d.c. conductivity ( region in Fig.1). At lower frequencies, the conductivity dispersion results from charge “pile up” at the metallic electrodes (interfacial region in Fig. 1). Such charge accumulation, referred to as electrode polarization (EP),19,20 induces a timeincreasing back field leading to an apparent infinite decrease of () (or ()). Although EP is a direct result of the ion motion, it is a non-equilibrium, extrinsic feature that depends both upon the nature of the electrode interface and the thickness of the specimen.21 As depicted in Fig. 1,


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interfacial polarization in the case of the thin sample starts at higher frequency (shorter time) and no frequency-independent (time-independent) plateau is observed in () (or ()). In such situations, the determination of remains a challenging issue. The main objective of the present communication is to provide a method appropriate for the extraction of the d.c. conductivity of ion conducting materials from frequency and time domain dielectric spectroscopy; the method is particularly useful for specimens that do not exhibit a frequency-independent conductivity plateau, a feature generally observed in thin film materials. This method exploits the EP effect22 and allows, upon proper design of the test cells, for the estimation of σ independently on the specimen thickness. The latter is of particular interest in studying the possible dependence of material properties on scaling. How and to what extent these properties are affected by miniaturization is a topic of intense debate23,24. It is therefore important to develop approaches that provide pertinent information as miniaturization has become one of the key features of modern technology and many applications rely on the use of low-dimensional material systems. The applicability and accuracy of the proposed method are assessed in relation to the main methods currently used in the field.

2. EXPERIMENTAL The materials used in the present study are poly(methyl methacrylate) (PMMA) polymeric matrices doped with a Keggin-type polyoxometalate (POM) anion in the form of heteropolyacid (H3[PW12O40]). The methodology followed to prepare such organic-inorganic hybrid materials was as described in the literature.6,25 The mobile ions of the POM-PMMA electrolyte system are in the form of protons. FD and TD measurements were performed using


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two-terminal

cells

consisting

of

metal/POM-PMMA/metal

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(MEM)

and

metal/POM-

PMMA/SiO2/metal (MEOM) structures. Cell preparation was as follows. Heavily-doped n++-type (0.001-0.005 Ω-cm) 4 in. silicon wafers with and without SiO2 blocking layers (BLs) of nominal thickness in the 12 to 600 nm range were used as starting materials. A 500 nm-thick aluminum layer was deposited by e-beam evaporation on the back side of the wafers after removal of the corresponding (i.e., native or intentional) oxide layer by buffered HF treatment. The same procedure was repeated on the front-side of the wafers aiming at MEM structures. A POM-PMMA film was then formed on the Al (MEM) and oxide-coated Si (MEOM) surfaces by spin-coating of the corresponding polymeric solution (POM-PMMA 5/5% (w/w)). Spin coating was followed by a thermal treatment step at 120 ℃ for 1 hr in ambient atmosphere. Such a treatment removes the remaining casting solvent (PGMEA) and leads to free-volume reduction, whereas it does not introduce significant acid-induced chemical change to PMMA.6,25 The thickness of the resulting POMPMMA film was 270 nm. Thick POM-PMMA films (~8 µm) were formed by pouring a few drops of the above solution on the sample surface followed by thermal treatment at 60 ℃ for 6 hrs under vacuum and 120 ℃ for 1 hr in ambient atmosphere. Finally, Al circular top electrodes with diameter 560 µm and thickness 300 nm were deposited onto the POM-PMMA films by ebeam evaporation through a shadow mask. The samples were stored in ambient atmosphere (24 ℃–50% relative humidity) for 1 week before measurement. Broadband impedance spectroscopy (IS) measurements (10 mHz-1 MHz, 0.1V AC signal without DC bias) were performed with a Novocontrol Alpha analyzer that was connected to a shielded probe-station. The accuracy of the measurements at the used frequency range is 0.2 – 1%, depending on the measured impedance of the sample (103 – 1011 Ω). The measured complex


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impedance, ∗ , was obtained in the usual way, i.e., ∗ = + = ∗ ⁄ ∗ , where = √−1, ∗ and ∗ are the applied voltage and the resulting current, respectively. ∗ measurements allow derivation of the complex relative dielectric permittivity, ∗ = − = 1⁄ !" ∗ , or complex electric modulus, # ∗ = # + # = 1⁄ ∗ , where

denotes the radial frequency (

= 2%) and

!" is the geometric capacitance (= " &⁄" , with & the sample active area, " the distance between the electrodes, and " (= 8.854×10-14 F cm-1) the permittivity of free space). While the modulus representation26 is still matter of debate,27-31 it has the advantage of suppressing the signal intensity associated with electrode polarization and its imaginary part should be independent on the BL thickness; the latter feature was observed in our experiments. The complex electrical conductivity ∗ = + is related to ∗ by the relation: ∗ = " ∗ = (" ⁄!" )⁄ ∗ . Current-time, (), measurements were carried out in N2 atmosphere inside a shielded probe station connected to an HP4140A picoamperometer and a Keithley 230 voltage source. Instrument control and data acquisition were accomplished using a computer via an RS-232 connection and a custom program written in LabView. The smallest achievable acquisition time was 0.1 s and the current noise level was typically in the 10-20 fA range.

3. RESULTS AND DISCUSSION 3.1 Frequency domain measurements. Representative dielectric spectra of the MEM cells at room temperature (24 ℃) are shown in Figure 2. In the case of the thick POM-PMMA electrolyte sample, the real part of the conductivity vs. frequency curve (Fig. 2a) shows a plateau extending over a broad frequency range, which marks a d.c. conductivity, , of about 3.1×10-9 S/cm. In this case, the impedance Nyquist plot (− vs. ), shown in the left inset of Fig. 2a, exhibits a semicircular arc at high frequencies that corresponds to the bulk properties of


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the electrolyte, whereas the appearance of an almost vertical-line at low frequencies (right of the graph) is related to electrode polarization. Ion (i.e., proton in the present case) migration in the bulk can be approximately modeled by the simple equivalent electrical circuit shown in Figure 4b. The circuit consists of the high-frequency capacitance !' (= " ( &⁄, ( is the dimensionless high-frequency limiting value of the bulk dielectric constant) in parallel with a single resistance * (the frequency dependence of the bulk conductivity is here ignored), which is related to the material d.c. conductivity by: = ⁄(&* )

(1)

~ 3.1×10-9 S/cm for * ~ 105 MΩ, as extracted from the low-frequency intersection of the semicircle with the axis in the − vs. plot. The large interfacial capacitance (also referred to as ionic double layer capacitance), !+, , arising from electrode polarization, is connected in series with the bulk * !' parallel element (see Fig. 4b) and shows a noticeable effect only at low frequencies (vertical growth of − with , right of the − vs. plot). While in principle it is then possible to estimate simply from the complex impedance plane plots (− vs. ), this is not always the case in practice, especially for electrolyte systems with submicrometer film thicknesses (e.g., no semicircle exists for the 270 nm POM-PMMA case, right inset of Fig. 2a), where time constants of different elements might overlap. A quantitative description of electrode polarization (EP) has been recently proposed from which can be derived using the following expression:22,32 /⁄ = 2%" - ( . 01 )

(2)


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Figure 2. Typical dielectric spectral features of MEM cells with 8µm (open symbols) and 270 nm (closed symbols)-thick POM-PMMA electrolyte layers. (a), (b), (c) and (d) depict (), ′(), # (), and (), respectively. Insets of (a) and (d) show the − vs. and () 3 characteristics. . , 01 and 01 mark, respectively, the onset and the full development of electrode polarization, and the maximum loss in # . The horizontal dash line in (a) marks the value of () at about the middle of the conductivity plateau, which is identified with . where . and 01 mark the frequencies of the onset and full development of EP, corresponding to the minimum and maximum in () respectively, as depicted in Fig. 2b (thick sample case). - is the dimensionless static permittivity of the material in the bulk measured at 4 . . It is worth mentioning that the maxima observed in () and () occur at the same frequency, 01 (see Fig. 2d), which corresponds also to the low-frequency limit of the frequencyindependent plateau observed in (). Note also that . marks the start of a steep increase in () whereas at 01 a plateau in () starts to develop (“thick” sample case) as the frequency decreases (inset of Fig. 2d). At frequencies 5 01 , () is dominated by the interfacial


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capacitance, !+, , which leads to the apparently colossal value33 of = !+, ⁄!" 4 104 (!" is the geometric capacitance) in the case of 8 µm-thick sample. Such a result may be misinterpreted and particular care should be taken when extracting the static permittivity of the electrolyte system in the bulk. Using Eq. (2), it is clear that in the case of the thick sample, can be estimated using the parameters extracted from the EP effects, except - , Increase of - with decreasing frequency in the 4 . regime, expressed under the form, - = ( + ∆, which reaches a saturation plateau before the EP onset, is often observed in ionic conductors;29 ∆ is related to polarization owing to the long-range displacement of mobile ions. Due to the absence of a plateau-like shape in the () curve (inset of Fig. 2d) for the 4 . regime, it was not possible to reliably evaluate - for the POM-PMMA electrolyte system under study and therefore, to safely extract from Eq. (2). It should be noted here that on the basis of the experimental values of . , 01 and (3.1×10-9 S/cm as evaluated here above), an - value of ca. 16 is expected. Finally, it is obvious that in the case of the thin POM-PMMA layer, not only - but also . and 01 cannot be estimated with confidence (see, e.g., Fig. 2b); a trait which emphasizes the limitations of the above method for the extraction of . Figure 2c shows the frequency dependence of the imaginary part of the electric modulus (# ) for the two different thicknesses, , of the POM-PMMA layers (MEM cells). As predicted by the definition of # 34,35 (=

!" ′), the # vs. curves are independent of within

experimental error (note that for both the thick and thin POM-PMMA layers, ≅ " , the 3 distance between the electrodes). The single peak in # (), marked with 01 , relates to ion

conductivity relaxation time and appears near the ‘‘crossover’’ region of (); i.e., between the limiting low-frequency constant and the high-frequency power law behavior of (). The average conductivity relaxation time, 5 89 4, is the Maxwell-like relaxation time defined by the


10

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relation:30,35,36 5 89 4= * !' = " ( ⁄ ; 5 89 4 accounts for the usual finding of a distribution of relaxation times needed for describing conductivity relaxation (i.e., the long-range ionic diffusion process). The most probable conductivity relaxation time corresponds to the peak : frequency in the loss modulus (# ) spectrum and is given by:30,37 5 89 4 ≅ 1⁄2% 01 . Taking

into account the above, the following expression for can be derived: : = 2%" ( 01

(3)

3 By using the experimental values of ( = 4.4 (inset of Fig. 2d) and 01 = 2.4 kHz (8

µm sample thickness, Fig. 2c), is around of 5.9 ×10-9 S/cm; a value about twice in magnitude as compared to that obtained using Eq. (1), which corresponds to the conductivity plateau (3.1×10-9 S/cm, Fig. 2a). Despite this discrepancy, it should be noted that additional measurements of the thick POM-PMMA MEM and MEOM cells after post-fabrication thermal treatments reveal conductivity plateaus in close agreement with as derived with Eq. (3). This point will be discussed further below. As proposed by Richert et al,38 may also be evaluated using the following expression: = 2%" -

(4)

where is the frequency corresponding to the crossing point of () and "() in the 4 . regime (i.e., ( ) = "( ), () = - and ′() = ⁄2%" ). The real and imaginary part of the complex dielectric permittivity shown in Fig. 2d, are compared in Figure 3 for the two POM-PMMA thicknesses. In the case of the 8 µm POM-PMMA layer, the values of - and extracted from Fig. 3a are about 10.6 and 1.0 kHz, respectively, leading to a value of 5.9×109

S/cm by application of Eq. (4); the same value as derived from Eq. (3). For the 270 nm POM-


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Figure 3. Comparison of the complex dielectric function ( ∗) of MEM cells with (a) 8 µm and 3 (b) 270 nm-thick POM-PMMA electrolyte layers. , and 01 mark, respectively, the frequencies at which = ", and the maximum loss in # . PMMA case, no crossing point of () and "() is observed (Fig. 3b) and therefore, cannot be estimated using Eq. (4). On the basis of Figs. 2c and 3a, a further comment can be made on the so-called conductivity relaxation time, 89. 30,38,39 In the electric modulus representation, 89 stands for the time scale with which the electric field decays to zero at : constant dielectric displacement. This relaxation time (89 = 8> ≅ 1⁄2% 01 ) differs from the

time 8+ (≅ 1⁄2% , referred also to as retardation time) for polarization relaxation at constant electric field. 8> and 8+ scale as:26,30,40 8> ⁄8+ ~ ( ⁄- (exact equality applies only for the case of Debye-like relaxation) and can differ substantially in systems with high dielectric strengths. On the other hand, in systems with weak dielectric processes the two relaxation times, 8> and 8+ , are nearly equal.41


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Figure 4b shows a simplified electrical equivalent circuit (EC) based on a four-layer model (illustrated in Fig. 4a) of the conventional MEM cell. Like in the case of many metals, a thin passivation oxide layer forms on the aluminium surface when exposed to air.42 Such a native alumina layer, AlOx, of 3.5 nm in thickness as extracted from X-Ray Photoelectron Spectroscopy (XPS) measurements (not shown here), develops on the top of the Al substrate electrode and acts as an “unintentional” blocking layer (BL). The latter can be regarded as a capacitor of capacitance, !'@ , defined as: !'@ = " '@ &⁄'@ , where '@ and '@ are the thickness and the relative static permittivity of the AlOx layer. Assuming that Al electrodes and BL do not allow the transfer of ions into the external measuring circuit, formation of charge separation layers takes place at the AlOx /electrolyte and electrode/electrolyte interfaces for ensuring charge neutrality upon electrode polarization. This accumulation of charges is called electrical double layer (EDL), or more shortly double layer (DL) whose the effective thickness is often approximated by the Debye length, > . The latter can be derived from9 /> = " - A' B⁄ C / , where is the concentration of mobile ions (protons in the present case), A' the Boltzmann’s constant, T the absolute temperature and e the fundamental unit of charge. Assuming that is approximately equal to the estimated POM concentration (~ 2.8×1020 mol/cm3), > is about 0.3 nm. In the case of symmetric DLs (i.e., > of protons = > of the counter anions) the associated capacitance can be expressed as: !>@ = " - &⁄2> . Hence, the capacitance !+, (Fig. 4b) is equivalent to two capacitors in series and can then be expressed as: 1⁄!+, = 1⁄!'@ + 1⁄!>@ . !+, is connected in series with the electrolyte equivalent circuit, which has been modeled as a parallel * and !' combination for the sake of simplicity.


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Figure 4. (a) Schematic of a MEM cell. is the POM-PMMA thickness and " is the total thickness between the metallic electrodes (i.e., + '@ , where '@ is the thickness of the native alumina layer). (b) Simplified electrical equivalent circuit describing the low-field ac conductivity and permittivity spectra of POM-PMMA between blocking electrodes. The capacitor is represented by a parallel * and !' combination (the frequency dispersion of the bulk conductivity is ignored) in series with the interfacial capacitance !+, (which is equivalent to two capacitors in series, i.e., !'@ and !>@ , respectively). In the case of the MEM cell, !>@ ≫ !'@ (or equivalently, - ⁄2> ≫ '@ ⁄'@ ) and therefore, !+, ≅ !'@ . The low- and high-frequency limits of the net capacitance, !, of the cell (see Fig. 4b) are: ! → !'@ and ! → !'@ !' ⁄(!'@ + !' ) (≅ !' when !'@ ≫ !' ), for → 0 and → ∞, respectively.35 The characteristic relaxation time of electrode polarization, 8+, , is defined as:43 τ+, = (!+, + !' )* ≅ (!'@ + !' )* and can be extracted from the frequency at maximum dielectric loss (); i.e., τ+, ≅ 1⁄2π 01 . Combining the above with Eq. (1), can be expressed as follows:

σ = 2π 01 ε" (ε'@ ⁄'@ + ε( )

(5)

Taking '@ = 3.5 nm, = 8 µm, '@ = 5 − 944 and ( = 4.4, would range from 1.9 to 3.4 ×10-9 S/cm in the case of the thick POM-PMMA layer. This result is in good agreement with the


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value obtained using Eq. (1) and the − vs. plot (see here above) or directly estimated from the () curve (~3.1×10-9 S/cm, Fig. 2a). In the case of the 270 nm POM-PMMA layer, 01 (see Fig. 2d) and therefore, , cannot be safely extracted probably due to leakage currents through the POM-PMMA/BL stack. It should be also noted that Eq. (5) depends on the thickness and permittivity of the blocking layer. In the case of a native aluminium-oxide layer grown in ambient air, the latter properties cannot be easily controlled and determined with accuracy. While the thickness of the AlOx layer has been estimated herein by XPS and ellipsometry to be about 3.5 nm, it was not possible to determine its permittivity with confidence. This is the reason for which the above calculation of has been done using a range of permittivity values reported in the literature instead of giving an estimate based on a somewhat arbitrary value. The above highlight also the need of using a blocking layer with known and controllable properties as we report hereafter in the case of a 12 nm-thick SiO2. Representative dielectric spectra at room temperature of MEOM cells with various POMPMMA / SiO2 stacked layers are presented in Figure 5. In the case of the 8 µm POM-PMMA / 12 nm SiO2 stack, () exhibits a plateau extending over a broad frequency range (Fig. 5a), which marks a similar to that of the MEM cell discussed here above. As in the case of the MEM cell with a 270 nm POM-PMMA / AlOx stack, no plateau is detected in the () characteristics of the MEOM cells with a 270 nm POM-PMMA / SiO2 stack, regardless of the SiO2 BL thickness. Figure 5c shows the frequency dependence of the real part (! ) of the apparent capacitance, !, of the equivalent circuit reported in Fig. 4b. In the low-frequency saturation regime, ! corresponds to the electrode polarization capacitance, !+, . It is quasiindependent from the POM-PMMA thickness (see the 12nm-thick SiO2 case) and decreases almost linearly with increasing thickness of the blocking SiO2 layer (!+, ≅ !'@ ); it can be easily


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calculated using '@ = 3.9 (SiO2) and A = 0.246 mm2 for the three nominal oxide thicknesses, '@ = 12, 50 and 600 nm. For the more conventional MEM cell, the saturation value also matches !'@ as calculated with '@ = 3.5 nm (native alumina thickness) and an assumed value of 8 for '@. 44

Figure 5. (a) Real part of conductivity , (b) imaginary part of permittivity and (c) real part of capacitance ! versus frequency at 24 °C of the MEOM cells (inset of (b)) with various POMPMMA (indicated as PM) / SiO2 stacked layers. The square symbols refer to the 8 µm PM/ 12 nm SiO2 stack, while the other symbols refer to the stacks made of a 270 nm PM layer with a 12 nm (circles), 50 nm, (up triangles) or 600 nm (down triangles) SiO2 layer. (d) The "normalized" () spectra of (b) is depicted. The straight line curves in (a)-(c) refer to the 8 µm POMPMMA MEM cell. At high frequencies and in both cases of 12 and 50 nm SiO2, ! saturates at values very close to the high frequency capacitance, !' (= " ( &⁄). For this frequency regime, the ratio of saturation capacitances between the thin and thick POM-PMMA layers is very close to the


16

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thickness ratio of these layers. For a constant thickness of the POM-PMMA layer (i.e. for constant * ), the time constant 8+, (≅ 1⁄2π 01 ), expressing the dispersion associated with !+, , will occur at higher frequencies (shorter times) as the thickness of the blocking oxide increases (due to the decrease of !+, ), in agreement with the experimental results (see, Figs. 5b and d).

Table 1. Parameters extracted from IS measurements and average values of the dc conductivity, , calculated from Eqs [1, 3 and 5] for 8 µm and 270 nm thick POM-PMMA layers using MEM and MEOM two-terminal cells. Twoterminal cell

L (POMPMMA)

LBL

LBL

(Al2O3)

(SiO2)

(µm)

(nm)

(nm)

MEM

8

3.5 3.5

*

3 01

01

(MΩ)

(kHz)

105±5

2.4±0.05

QRS a

QRS b

QRS c

(Hz)

(nS/cm)

(nS/cm)

(nS/cm)

0.3±0.05

3.1

5.9

1.9-3.4

2.4±0.05

2.0±0.1

3.2

5.9

2.9

MEM

0.27

MEOM

8

12

9.5

MEOM

0.27

12

1.3±0.05

60±0.5

3.2

3.0

MEOM

0.27

50

2.8±0.05

220±5

6.9

3.1

MEOM

0.27

600

1.5±0.05

850±5

3.70

2.9

3.9±0.05 100±5

= ⁄(&* ), & is the sample active area (= 24.63 × 10 WX : = 2%" ∞ 01 , ∞ is the relative high-frequency dielectric permittivity of POM-PMMA(= 4.4). c σ = 2π " 01 (ε'@ ⁄'@ + ε∞ ), '@ is the relative static permittivity of AlOx (=5-9) or SiO2 (=3.9). a

b

UV

/ ).

Table 1 gathers together the values of the d.c. conductivity, , as calculated from Eqs (1), (3) and (5) for 8 µm and 270 nm thick POM-PMMA layers using MEM and MEOM cells. Eq. (5) gives very similar values in all cases except for the MEM cell with a 270 nm POMPMMA layer where safe extraction of 01 was not possible as discussed here above. These results reveal that: (1) is not dependent on the thickness of the POM-PMMA layer (at least in the 8 µm - 270 nm range), (2) can be accurately estimated given that the thickness and the relative permittivity of the blocking layer are known and 01 can be extracted from the () characteristics, (3) the high-frequency limiting dielectric constant, ( , of the POM-PMMA


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Page 18 of 35

materials should be accounted for when !'@ is comparable to or smaller than !' as in the present case of the 270 nm POM-PMMA / 50 nm (or 600 nm) SiO2 stacks.

Figure 6. as derived from Eqs (1), (3) and (5) after IS measurements at different temperatures for a 8 µm-POM-PMMA / 12 nm-SiO2 MEOM cell following a post-fabrication thermal treatment at 118 ℃ for 30 min. Typical spectra at 70 ℃ are shown in the down right inset (# (), () and ()) and the upper left inset ( ()).The horizontal dash lines mark the value of obtained from Eqs. (3) and (5). In order to gain more insights on the discrepancy of derived from Eq. (3), IS measurements were performed following the application of thermal treatments to the MEM and MEOM cells. Typical results are herein presented in the case of a thermal treatment at 118 ℃ for 30 min in an atmospheric oven applied to a MEOM cell with an 8 µm POM-PMMA / 12 nm SiO2 stack. IS measurements were carried out at temperatures ranging from 55 to 85 ℃ in order


18

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to be able to extract 01 and therefore, to calculate from Eq. (5). As shown in Figure 6, the values of derived from Eqs (1, 3, 5) exhibit small (and constant) deviations but remain in good agreement at each measurement temperature. The above thermal treatment not only reduces the measured ion conductivity probably as a result of a substantial decrease in the moisture content of the POM-PMMA materials,45 but also may minimize or eliminate additional dispersive contributions originating from water-related relaxation processes46 (e.g., surfaceconductivity and dipolar relaxation). It is assumed herein that the latter water-related relaxation : processes affect the peak location, 01 , of the non thermally treated cells and probably the

overall distribution of the # () characteristics and thereby, the correctness of Eq. (3).

3.2 Time domain measurements. The a.c.-impedance measurements are usually carried out at frequencies ≥ 10–2 Hz. For lower frequencies they become quite time consuming and in this regard, measurements in the time domain (TD) are more appropriate. Hereafter, we report on TD measurements of electrolyte systems with low ionic conductivity and further extraction of σ . For this purpose we have selected the case of MEOM cells with 8 µm or 270 nm POMPMMA layers having undergone a post-fabrication thermal treatment at 113 °C for 1h on a hot plate; which considerably reduces the measured ion conductivity. The blocking layer is a 12 nm SiO2 film. Figure 7a shows the time-dependent current responses, IZ (t), of the MEOM cells to a polarization step voltage, VZ , of 1.0 V. In the case of the 8 µm POM-PMMA layer, IZ (t) exhibits a time independent plateau from which σ may be estimated (see hereafter), while no plateau is detected for the 270 nm POM-PMMA MEOM cell. Considering the simple RC circuit of Fig. 4b, IZ (t) can be expressed as: , () = ' ()⁄* + !' (]' ()⁄]) = !+, (]+, ()⁄])

(6)


19


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Page 20 of 35

Figure 7. (a) Measured polarization current, , (), as a function of time following the application of 1.0V step voltage ( , ), to the 270 nm and 8 µm POM-PMMA MEOM cells. Inset: The product of the polarization current and the time, , (), for both cells. 01 indicates the peak location of the , () curves. (b) Effective conductivity, ^__ (), of both cells as derived from: ^__ () = `a ()" bc`a &b. The d.c. conductivities, , calculated in the time and frequency domain (TD and FD) are also indicated; ≅ 0.7 and 0.6 pS/cm in TD and FD for both cells. (c) The , () curves as derived from the measured (solid lines) and fitted (Eq. (8), symbols) , () responses for both cells. The corresponding imaginary parts of the Fourier transforms, dB"(, ) calculated from the fitted , () responses are also represented. Note that the product of dB"(, ) by (" ⁄(" &, )) gives the imaginary part of the susceptibility, e"(). 5 8 f> 4 and 5 8g> 4 are the average relaxation times corresponding to 8+, in TD and FD. Inset: the measured (solid lines) and fitted (Eq. (8), symbols) , () responses. (d) The , () and the dB"(, ()) curves derived from the , () response of the 270 nm POM-PMMA MEOM cell as fitted from Eq. (8) (sum of exponentials, ∑ Cij, symbols) and Eq. (9) (KWW function, solid lines). The average relaxation times in the TD (5 8 f> 4klmn = 8opp = 01 ) and FD (5 8 f> 4klmn ~ 5 8 (q) 4rss ) are the peak locations of the corresponding distributions. where ' () and +, () are the potential differences across the parallel * !' circuit and the equivalent capacitor, !+, (= (!'@ !>@ )⁄(!'@ + !>@ )), respectively. Using ' () + +, () = , , Eq. (6) leads to +, () = , `1 − Cij(−⁄8)b, where 8 (= * (!+, + !' )) is the relaxation


20

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time for electrode polarization, 8+, , as discussed here above. Following the application of the step voltage, the displaced charge, t, (), as a function of time is given by: t, () = v

u" , () ] = !+, +, (), which reaches t,f = !+, , when time tends to infinity. As previously discussed, !>@ ≫ !'@ , and therefore, t,f ≅ !'@ , , which is nothing else than the total charge that can accumulate on the blocking-layer capacitor, i.e. the SiO2 layer. For a 12 nmthick SiO2 layer and an electrode diameter of 560 µm, t,f is about 0.71 nC; a value close to those extracted from integration of the , () curves of Fig. 7a (6.8 and 6.2 nC for the 0.27 and 8 µm-thick POM-PMMA layers, respectively). For the 8 µm POM-PMMA MEOM cell, the relaxation time, 8 ≅ * !'@ , can be estimated from the plateau, (~2.2 pA), of the , () curve. In that case, * ≅ 0.45 TΩ leading to a 8 value of about 320 s (!'@ = 0.71 nF). Such a value can also be obtained from the product of the polarization current and the time, , (), which exhibits a maximum, 01 , at = 8 (see inset of Fig. 7a). This information is particularly useful in the case of the 270 nm POM-PMMA MEOM cell where no plateau is detected in the corresponding , () characteristics. Knowledge of 8 allows for the estimation of the d.c. conductivity through the relation: ≅ (w" ⁄8)(w'@ ⁄'@ ) which is equivalent to Eq. (5) for w'@ ⁄'@ ≫ w( . Using 01 as extracted from the , () curves, a value of about 0.7 pS/cm is obtained for both the 8 µm and 270 nm POM-PMMA MEOM cells; 01 ~11.2 s in the latter case. As shown in Fig. 7b, is in good agreement with the time-independent plateau of the effective

conductivity,

^__ (),

of

the

8

µm

POM-PMMA

cell

expressed

by:

^__ () = (, ()" )⁄(, &). It is worth mentioning that the displaced charge from application of the step voltage to 01 can be expressed by: t, ( 01 ) = !+, +, (8) ≅ (!'@ , (C − 1)/C), which represents the accumulated charge on the BL capacitor until 01 . The so derived value of t, ( 01 ) is about


21


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Page 22 of 35

0.45 nC (= 0.71(C − 1)/C); a value close to those extracted from integration of the , () curves from = 0 to 01 : 0.40 and 0.41 nC for the 8 µm and 270 nm POM-PMMA MEOM cells, respectively. Note that in the case of a depolarization response, t> ( 01 ) = (!'@ , /C), which represents the amount of remaining charge at 01 after removal of the step voltage. According to the above model the polarization current follows a simple exponential decay: , () ≅ (!'@ , ⁄8) `Cij(−⁄8)b

(7)

from which the electrical susceptibility in the frequency domain, e(), can be calculated by means of Fourier transform (FT). The imaginary part of the susceptibility, e"(), and thereby, "(), exhibits a maximum, 01 , at = (1⁄2% 8) with 8 = 01 , thus leading to the same d.c. conductivity in the time and frequency domains by application of Eq. (5). It is important to keep in mind that the above , () function expresses the charging current of the blocking layer capacitor and applies in the time regime where electrode polarization occurs. While under the assumption of the validity of Eq. (5), fitting of the polarization currents over the entire range of time is not necessary per se for calculating , it is interesting to examine the case of a nonexponential current decay in the electrode polarization regime and its consequences on the relaxation time, 8+, , as extracted in the time and frequency domain representations. A practical way to accurately fit the , () characteristics is to use a discrete sum of exponential relaxations, which may be associated to a series array of parallel RC elements with different time constants similar to that of Fig. 4b. In this case, , () can be expressed as:

, () = ∑. ,(.) = ∑. z(.) Cij`−⁄8(.) b

(8)


22

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where z(.) = !+,(.) , ⁄8(.) with !+,(.) ≅ !'@(.) as discussed here above; note also that t,f ≅ , ∑ !'@(.) = , !'@ . For each MEOM cell, accurate fits of , () can be obtained in the region of interest (i.e., around the maximum ( 01 ) of the corresponding , () function) using two exponential decay functions ( = 2) and over the entire range of time for = 5 as shown in the inset of Fig. 7c. The Fourier transform of , () and thereby, "(), display a maximum, 01 (see Fig. 7c), which differs substantially from 1/(2% 01 ). We will not consider this issue here and we assume simply that 01 = 5 8 f> 4 and (1⁄2% 01 ) = 5 8g> 4, where 5 8 f> 4 and 5 8g> 4 are the average relaxation times corresponding to 8+, in the time and frequency domains (TD and FD), respectively. Except the particular case of an exponential decay where 5 8 f> 4 = 5 8g> 4= 01 , application of Eq. (5) leads in the case of a superposition of exponential responses to reduced ionic d.c.-conductivities in the frequency domain compared to those calculated in the time domain, as the ratio 5 8 f> 4⁄5 8g> 4 is less than unity. Such a ratio was 0.86 and 0.83 for the 270 nm and 8 µm POM-PMMA MEOM cells, respectively, leading to a value in the frequency domain of about 0.6 pS/cm for both cells. This result, reported in Fig. 7b, is consistent with those previously derived herein from impedance spectroscopy measurements in the sense that the values calculated from Eq. (5) were very close but systematically slightly lower than those corresponding to the conductivity plateau of the 8 µm POM-PMMA MEOM cells (see, e.g., Fig. 6); a trait which favors the TD representation. Finally, it is of interest to examine the case where the , () curve in the time region corresponding to electrode polarization (i.e., to the charging of the blocking-layer capacitor) can be represented by the Kohlrausch–Williams–Watts (KWW) function47-49 (also called stretched


23


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Page 24 of 35

exponential relaxation function), d() = Cij`−(⁄8opp ){ b. In that case, , () is proportional to ]d()⁄] and can be expressed as: , () = z (| ⁄8opp )(⁄8opp ){Uq Cij`−(⁄8opp ){ b

(9)

where 0 5 | ≤ 1 (| = 1 for an exponential decay) and z ≅ !'@ , . Here, the , () function exhibits a maximum at 8opp = 01 . The Fourier transform of , () and thereby, calculation of e(), can be performed by using the Fast Fourier Transform (FFT) algorithm as there is no analytic solution for arbitrary β. The 01 value is then extracted from the e"() curves. The so obtained values of 5 8g> 4 = (1⁄2% 01 ) for both MEOM cells under consideration were in excellent agreement with the average relaxation time derived from the first moment associated to KWW relaxation and expressed as:50 5 8 (q) 4 = (8opp ⁄| ) Γ(1⁄| )

(10)

where Γ refers to the gamma function (see Fig. 7d for the 270 nm POM-PMMA layer case). If indeed, 5 8g> 4 = 5 8 (q) 4, thus the average relaxation time in the frequency domain can be estimated from the parameters 8opp and | without any Fourier transformation. Curve fitting for | extraction should be done in a region around 01 for limiting the distortion effects caused by other processes not associated to KWW relaxation.50 The extracted | values from fitting of the , () characteristics of both MEOM cells were close to those given by: | = (C⁄z ) 01 , ( 01 )

(11)


24

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derived from Eq. (9) at = 8opp . Eq. (11) represents the peak amplitude of the , () function multiplied by the factor (C⁄z ). A similar method was proposed by Wagner and Richert51 in their study of the electric-field relaxation of ionic conductors by measurements of the modulus in the time domain. Using 5 8 (q) 4 and Eq. (11), the 5 8 f> 4⁄5 8g> 4 ratio is given by (| ⁄Γ(1⁄| )), and takes the values of ~0.84 (| ≅ 0.75) and 0.90 (| ≅ 0.82) for the 270 nm and 8 µm POMPMMA MEOM cells, respectively (z ≅ !'@ , = 0.71 nC). These values agree fairly well with those extracted in the case of the sum of discrete exponentials discussed here above. It is believed that the small difference in | detected between the two MEOM cells would be due to experimental "error" (e.g., measurement precision, instrument calibration, etc.) since there is no reason to expect different | values for the same material systems. At last, it should be noted that for a KWW-type relaxation and the RC circuit of Fig. 4b, the potential difference across the equivalent capacitor, !+, , would be given by: +, () = , 1 − Cij`−(⁄8opp ){ b and according to Eq. (6), 8opp = |* (!'@ + !' ), as !>@ ≫ !'@ . As a consequence, the d.c.conductivity, , derived from Eq. (1) would be reduced by a factor | in comparison to the case of the exponential decay discussed here above. This result is interesting but obviously does not fit the d.c. plateau of the effective conductivity of the 8 µm POM-PMMA MEOM cell (see Fig. 7b). Note also that in this case the values corresponding to the frequency domain should be also reduced by a factor | as they are derived herein using the same model.

4. CONCLUSIONS The main purpose of this work was to develop a method appropriate for the extraction, from dielectric spectroscopy measurements, of the d.c. conductivity ( ) of ion conducting


25


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Page 26 of 35

materials that do not exhibit a frequency-independent conductivity plateau, a feature generally observed in thin film materials. This method is independent of the specimen thickness and is of particular interest for examining the possible dependence of on the thickness of thin films. The method exploits the parameters associated to electrode polarization effects arising from the charging of an ion-blocking capacitor located between the ionic materials and the metal electrode. By proper design of this added capacitor (thickness and materials) in the fabrication of the test cells, can be accurately estimated through the use of Eq. (5) as it has been demonstrated herein for POM-PMMA electrolyte systems and SiO2 blocking layers. The value obtained for thick (8 µm) POM-PMMA layers is in excellent agreement not only with the observed frequency- (time-) independent conductivity plateau but also with the value determined in the case of thin (270 nm) POM-PMMA layers for which no plateau is detected. While coherent results have been also obtained for a blocking layer made of native alumina in the case of thick POM-PMMA layers, it is important to use a blocking layer (SiO2 herein) that eliminates (or reduces as much as possible) the leakage currents and has controllable properties, especially in terms of relative permittivity and thickness for application of Eq. (5). The latter equation requires also knowledge of the characteristic relaxation time for electrode polarization (τ+, ), which can be estimated from the peak location of the () and , () spectra for measurements carried out in the frequency and time domain, respectively. As discussed herein, the τ+, value remains unchanged in the FD and TD representations for a Debye-like relaxation while it appears overestimated in the case of a.c. impedance measurements for non-exponential type relaxations. In this regard, it would be interesting to perform FD and TD measurements, which both allow application of Eq. (5), on specimens with same history. Finally, as derived from the simple RC model used herein, Eq. (5) is dependent on the high-


26

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frequency limiting dielectric constant, ( , of the ion conducting materials. Such a parameter should be accounted for when the blocking layer capacitance is comparable to or smaller than the capacitance of the ion conducting materials.

AUTHOR INFORMATION Corresponding Author *E-mail: [email protected]; Phone number: +30 28210 23056, Fax: +30 28210 23003. Notes The authors declare no competing financial interest.

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Direct Current Conductivity of Thin-Film Ionic Conductors from

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