ACTAUNIVERSITATIS

UPSALIENSISUPPSALA

2020

Digital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 1980

Dynamic and quasi-stationaryelectrochromic response ofamorphous tungsten oxide thinfilms

In situ combined electrochemical and opticalmeasurements during lithium intercalation

EDGAR ALONSO ROJAS GONZÁLEZ

ISSN 1651-6214ISBN 978-91-513-1044-2urn:nbn:se:uu:diva-423191

Dissertation presented at Uppsala University to be publicly examined in Siegbahnsalen,Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, Friday, 11 December 2020 at 09:30 forthe degree of Doctor of Philosophy. The examination will be conducted in English. Facultyexaminer: Professor Aline Rougier (Université de Bordeaux).

AbstractRojas González, E. A. 2020. Dynamic and quasi-stationary electrochromic responseof amorphous tungsten oxide thin films. In situ combined electrochemical and opticalmeasurements during lithium intercalation. Digital Comprehensive Summaries of UppsalaDissertations from the Faculty of Science and Technology 1980. 93 pp. Uppsala: ActaUniversitatis Upsaliensis. ISBN 978-91-513-1044-2.

Electrochromic (EC) materials can adjust their optical properties, reversibly, by means of anexternal electrical stimulus. They have relevant technological applications; for example, energy-efficient smart windows, which can adapt dynamically—according to the given environmentalconditions—to control the heat and visible light fluxes between the interior and exterior ofa building. EC applications are currently available on the market. However, there are stillquestions concerning the fundamental processes responsible for the EC effects.

This thesis focuses on EC inorganic oxide materials in the thin film form; particularly,amorphous tungsten oxide. In this case, the electrochromism is induced by the intercalationof small ions (such as lithium ions) into the material and the insertion of electrons from anexternal circuit due to charge neutrality requirements within the film. These electrons are theones causing the optical changes. This work centers its attention to the quasi-equilibrium anddynamic EC processes. They were studied by in situ simultaneous electrochemical and opticalmeasurements at different conditions—that is, for a wide range of intercalation levels and biaspotentials.

The experimental results from quasi-equilibrium measurements were in accordance with aphenomenological optical absorption model for amorphous tungsten oxide that is based onelectronic transitions between states localized on neighboring tungsten atoms. In this case, theconsideration of W4+ sites in the model was needed to properly reproduce the experimentalresults.

The dynamic measurements used an experimental setup which can acquire simultaneouslythe frequency-dependent electrical and optical responses of the EC system. In the frequencydomain, different mechanisms with various characteristic times and responses can be isolated.Here, the coloration was mainly assigned to the ion and electron diffusion within the film.However, adsorption-related phenomena were also found to contribute to the coloration,especially at high bias potentials—corresponding to low intercalation levels. Interestingly,the dynamic optical response was in-phase with the electrical one at high bias potentials.Nevertheless, a delay between the former and the later was noticed as the bias potentialdecreased—that is, increasing intercalation level.

The methods and results from this thesis provide new perspectives into the fundamentalcoloration mechanisms in EC systems. In addition, studies like those presented here can bereadily extended to different materials at diverse conditions—for example, at various opticalwavelengths, material compositions, and film thicknesses.

Keywords: Electrochromism, Optical absorption, Amorphous materials, Tungsten oxide,Electrochemical impedance spectroscopy, Color impedance spectroscopy, Intercalation

Edgar Alonso Rojas González, Department of Materials Science and Engineering, Solid StatePhysics, Box 534, Uppsala University, SE-751 21 Uppsala, Sweden.

© Edgar Alonso Rojas González 2020

ISSN 1651-6214ISBN 978-91-513-1044-2urn:nbn:se:uu:diva-423191 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-423191)

Dedicado a María Fátima Rojas Vásquez ♥, nuestra angelita que está enel cielo. De parte de mamá y papá que siempre te amarán.

Dedicated to María Fátima Rojas Vásquez ♥, our little angel who is inheaven. From mom and dad who will always love you.

List of papers

This thesis is based on the following papers, which are referred to in the textby their Roman numerals.

I E. A. Rojas-González and G. A. Niklasson, Setup for simultaneouselectrochemical and color impedance measurements of electrochromicfilms: Theory, assessment, and test measurement, Review of ScientificInstruments 90, 085103 (2019)

II E. A. Rojas-González and G. A. Niklasson, Differential colorationefficiency of electrochromic amorphous tungsten oxide as a function ofintercalation level: Comparison between theory and experiment,Journal of Applied Physics 127, 205101 (2020)

III E. A. Rojas-González and G. A. Niklasson, Coloration of tungstenoxide electrochromic thin films at high bias potentials and lowintercalation levels, Materials Letters: X 7, 100048 (2020)

IV E. A. Rojas-González and G. A. Niklasson, Charge colorationdynamics of electrochromic amorphous tungsten oxide studied bysimultaneous electrochemical and color impedance measurements, inmanuscript

Reprints were made with permission from the publishers.

My contribution to the appended papers

I I participated in the conception of the idea, designed and performed allthe experiments, made the data analysis, developed the required theory,and did most of the writing.

II I participated in the conception of the idea, designed and performed allthe experiments, made the data analysis, developed the required theory,and did most of the writing.

III I performed all the experiments, made the data analysis, and did mostof the writing.

IV I participated in the conception of the idea, designed and performed allthe experiments, made the data analysis, developed the required theory,and did most of the writing.

Papers not included in this thesis

V E. A. Rojas-González, A. Borne, B. Boulanger, J. A. Levenson, and K.Bencheikh, Continuous-Variable Triple-Photon States Quantum Entanglement,Physical Review Letters 120, 043601 (2018)

VI M. A. Arvizu, H.-Y. Qu, U. Cindemir, Z. Qiu, E. A. Rojas-González,D. Primetzhofer, C. G. Granqvist, L. Österlund, and G. A. Niklasson,Electrochromic WO3 thin films attain unprecedented durability by potentiostaticpretreatment, Journal of Materials Chemistry A 7, 2908 (2019)

VII H.-Y. Qu, E. A. Rojas-González, C. G. Granqvist, and G. A. Niklasson,Potentiostatically pretreated electrochromic tungsten oxide films withenhanced durability: Electrochemical processes at interfaces of indium-tin oxide, Thin Solid Films 682, 163 (2019)

VIII I. Sorar, E. A. Rojas-González, I. B. Pehlivan, C. G. Granqvist, andG. A. Niklasson, Electrochromism of W-Ti Oxide Thin Films: CyclingDurability, Potentiostatic Rejuvenation, and Modelling of ElectrochemicalDegradation, Journal of The Electrochemical Society 166, H795 (2019)

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.1 Electrochromism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2 General scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Electrochromic amorphous tungsten oxide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1 Structure and composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Electronic density of states and ion intercalation . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.1 Schematic band diagram and general properties . . . . . . . . . . . 152.2.2 Introduction to the electrochemical density of states

(EDOS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.3 Details of the conduction band of amorphous tungsten

oxide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3 General optical response of non-intercalated and intercalated

amorphous tungsten oxide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.1 Linear systems, transfer functions, and frequency-dependent

variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2 Electrochemical impedance spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2.1 Circuit elements and basic connections . . . . . . . . . . . . . . . . . . . . . . . . . 293.2.2 Impedance of normal and anomalous diffusion . . . . . . . . . . . . . 323.2.3 Equivalent circuit for amorphous tungsten oxide . . . . . . . . . . 383.2.4 Mott-Schottky analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.3 Color Impedance Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.4 Optical absorption models in amorphous tungsten oxide . . . . . . . . . . . . 44

3.4.1 Intervalence-transfer and small polarons . . . . . . . . . . . . . . . . . . . . . . . 443.4.2 Optical density and differential coloration efficiency . . . 493.4.3 Site-saturation theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4 Electrode preparation and characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.1 Thin film deposition by reactive DC magnetron sputtering . . . . . . . . 554.2 Stylus profilometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.3 X-ray diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.4 Rutherford backscattering spectrometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5 Electrochemical and in situ optical measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.2 Electrochemical techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.2.1 Open circuit potential (OCP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.2.2 Linear sweep voltammetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.2.3 Cyclic voltammetry (CV) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.2.4 Chronopotentiometry (CP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6 Summary of important results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.1 Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.2 Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.3 Paper III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706.4 Paper IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

7 Concluding remarks and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

8 Sammanfattning på svenska . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

1. Introduction

Climate change presents a huge risk for humanity and ecosystems due to thecurrent trend of greenhouse gas emissions [1]. Finding and implementingways of reducing energy consumption is fundamental to solve this issue. Itis estimated that about 40% of the global energy consumption is designatedto cooling and heating in the building sector [2], and most of it is poweredby fossil fuel sources. Clearly, there is significant room for improvement inthis area, and the use of energy-efficient technologies is key for this. In thisregard, fenestration is an important aspect to consider. On one hand, it iscrucial for the comfort in the building—providing contact with the outdoorenvironment and natural sunlight—but on the other hand is associated withsignificant heat exchange with the surrounding [3]. The latter requires theuse of artificial cooling, and heating in warm, and cold climates, respectively.This could be mitigated by a good design of static windows optimized for aspecific situation. However, a significant reduction in energy consumption isbetter achieved by dynamic smart windows [4] because they are able to adjustaccording to the specific requirements at a certain moment—these can indeedchange throughout the year or even during the day.

The so-called chromogenic materials are able to adjust their optical proper-ties by external stimuli and are the basis of smart windows [5]. The chro-mogenic family includes, for example, thermochromic, photochromic, andelectrochromic (EC) materials. The latter kind shows tremendous potentialin energy efficient applications [4] and is the one studied in this thesis.

1.1 ElectrochromismEC materials adjust their optical properties, reversibly, by the application ofan external electrical voltage [6]. The term electrochromism was coined in1961 by Platt [7] related to the change of color of certain dyes by the applica-tion of an electric field. This work was based on previous studies during the1950s on the change in the absorption spectra of some dyes by polarizationeffects in the solvent [8, 9]. In 1969, Deb published a seminal paper concern-ing the electrochromism in tungsten oxide (WO3) thin films [10]. This workattracted the attention of the scientific community due to its potential applica-tion in display devices. However, the interest fade away in time because liquidcrystals became the technology of choice in this particular area. The workof Svensson and Granqvist in the mid 1980s [11, 12, 13] introduced the term

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Figure 1.1. Schematic of the coloration in an EC electrode based on an aWO3 thinfilm and a lithium-containing electrolyte. Four layers are depicted, from left to right,the electrolyte, the EC film (aWO3), the transparent conductive electrode (TCE), andthe substrate. A potential difference V is applied between the TCE and a reference,and it can induce a current I, which is carried by lithium ions Li+ and electrons e−.

"smart window" and together with the contribution from Lampert [14] initi-ated the use of EC coatings for this kind of energy-efficient technology. Sincethen, there has been a great and sustained interest in electrochromism up tothe present. Other EC applications are for instance spacecraft thermal control[15], tinted helmet visors [16], sunglasses [17], rear-view mirrors with vari-able reflectance for the automotive sector [18], indicator labels [19], read-outinterfaces for sensors [20], and color displays [21, 22].

In this thesis, we are mainly concerned with electrochromism due to ionintercalation (and deintercalation) in transition metal oxides, that is the basisof the most mature EC technology [23]. Particularly, we are focused on thestudy of amorphous tungsten oxide aWO3, which is almost ubiquitous in allcurrent practical EC applications [3]. The schematic of the coloration in ahalf cell using an EC electrode based on an aWO3 thin film in contact with alithium-containing electrolyte is depicted in Fig. 1.1 and described as follows.

Four layers are shown in Fig. 1.1. First, the electrolyte containing smallions (such as lithium ions or protons), it can be an inorganic solid, a poly-mer, or a liquid [24]—the latter is more suitable for research in the laboratory.Second, the EC film. Third, the transparent conductive electrode (TCE), pre-dominantly In2O3 : Sn (ITO) but other options are available [25]. Fourth, thesubstrate, that can be glass plates or transparent flexible foils [26]. Ideally, the

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electrolyte should be an ion conductor but an electron isolator, and the EC filmmust be a good mixed conductor (both for ions and electrons) [26]. At the topof Fig. 1.1, the aWO3 film starts in a transparent state with no intercalation.Then, a negative bias potential between the TCE and a reference is applied,and this generates a current that leads to lithium intercalation and electron in-sertion into the EC film due to charge neutrality requirements. This processgives rise to a coloration in the aWO3 film. The inserted charge-compensatingelectrons are the ones responsible for the optical modulation in the EC film[27], and the coloration state is governed by the electron density. The latter isusually described by the intercalation ratio defined as the number of interca-lated ions per formula unit of the host—note that if the cation is monovalentone electron is inserted per each intercalated ion. The EC film can be madetransparent again by applying a potential with an opposite polarity, which pro-vokes the deintercalation of the ions and the extraction of the electrons, asindicated in the bottom of Fig. 1.1. In principle, the process is reversible andcan attain several cycles between colored and bleached states.

It is important to remark that the situation explained above corresponds toaWO3, which in usual operational conditions colors upon ion intercalation.This is called cathodic coloration and is also the case for other materials likeoxides based on Ti, Nb, Mo, and Ta [24]. The opposite behavior, colorationdue to ion deintercalation, is called anodic, and can be observed for examplein oxides of Cr, Mn, Fe, Co, Ni, Rh, and Ir [24]. V oxides are both anodic andcathodic in different optical wavelength ranges.

The principle of an EC smart window is similar to that explained above inconnection with Fig. 1.1. Its generic design consists of two EC films (backedby TCEs) connected by an electrolyte [24]. The EC films must be comple-mentary to each other—that is, one should be cathodic and the other anodic.Thus, when a potential difference is applied between the TCEs (for example,in the direction of ion intercalation into the cathodic side) both EC films willcolor, and bleach when the polarity is inverted. Alternatively, the anodic filmcould be substituted by a neutral (non-EC) ion storage layer, but better results(related to the optical modulation) can be achieved in the former case.

1.2 General scopeBy now, it is clear that electrochromism is in essence complicated. This isbecause it involves different bulk phases—that is, the electrolyte, the EC film,and the TCE—and their respective boundaries. The coloration mechanismsand dynamics in aWO3 are not completely understood. Here, several pro-cesses take place, such as the intercalation of ions into the EC film, the in-sertion of electrons from the TCE, the ion and electron diffusion within theEC material, and the optical absorption phenomena occurring therein. In thisthesis, we focus on detailed in situ electrochemical and optical studies of elec-

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trochromic materials for an extended intercalation level range. Particularly,aWO3 was chosen as a case study due to its relevance in electrochromism. Inthis way, we aim to obtain new vistas for the analysis of the EC mechanismsby looking at the correlation between the electrical and optical processes. Insitu methods were chosen because we were interested in both the dynamic andquasi-stationary responses of the EC system during ion intercalation.

In Paper I, we present and characterize a setup for in situ dynamic simul-taneous electrochemical and optical measurements on EC materials based onfrequency-dependent techniques. In Paper II, we develop a concept for study-ing the correlation between the optical and electrical responses of an EC ma-terial, both in the dynamic and quasi-stationary regimes, and implement it tothe aWO3 case for a wide intercalation level range. In Paper III, we focuson the quasi-stationary coloration response of aWO3 at low intercalation lev-els. Paper IV shows detailed measurements on aWO3 at different intercalationlevels using the setup from Paper I. Here, the dynamic frequency-dependentelectrical and optical responses are compared.

The remainder of this thesis is organized as follows. Chapter 2 presents rel-evant properties of the main material studied here—that is, amorphous tung-sten oxide. In Chapter 3, the theoretical aspects related to the frequency-dependent techniques used in this work and the optical absorption in amor-phous tungsten oxide are developed. The details concerning the preparationand characterization of the amorphous tungsten oxide thin films are given inChapter 4. Chapter 5 describes the experimental setup for simultaneous elec-trochemical and optical measurements as well as the non frequency-dependentelectrochemical techniques employed in this work. A small summary of keyresults connected with the papers appended to this thesis is provided in Chap-ter 6. The conclusions and outlook are given in Chapter 7. Finally, a smallsummary in Swedish is included in Chapter 8.

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2. Electrochromic amorphous tungsten oxide

The color of tungsten oxides can vary upon the intercalation of small ions[24]. These changes were first reported by Berzelius in the beginning of the19th century [28].

In this chapter, the main structural, compositional, electronic, and opticalcharacteristics of amorphous tungsten oxide (aWO3) are presented. In addi-tion, the effects of intercalation and oxygen deficiency are discussed. Con-venient comparisons with the crystalline counterpart are made when this con-tributes to the understanding of the subject. Moreover, a special emphasis isgiven to previous works employing similar conditions than those used in thisthesis—that is, closely stoichiometric aWO3 thin film samples and lithiumintercalation.

2.1 Structure and compositionIt is instructive to begin this description with the properties of crystalline tung-sten oxides before turning to the amorphous case. Cubic crystalline WO3presents a ReO3 structure, which is a subset of the perovskite configurationABO3 with missing A cation [27]. In this case, the unit blocks correspond toWO6 octahedra with W atoms at the center and O atoms at the nodes [24].The ordered structure is formed by corner-sharing octahedra with each oxy-gen shared by two unit blocks, as depicted on the left-hand side of Fig. 2.1.In reality, there are deviations from a perfect ReO3 structure—that is, slightlyoff-centered tungsten atoms and rotations of the surrounding oxygen octahedra[24, 29]. Single WO3 crystals can present a certain symmetry—for example,monoclinic, triclinic, tetragonal, and orthorhombic—depending on its temper-ature [30, 31]. The phases observed close to room temperature are the triclinicP1 (δ -WO3) between about 230-300 K and monoclinic P21/n (γ-WO3) in therange of about 300-623 K [31]. The latter has a bulk density of 7.16 g/cm3

[24]. Crystalline substoichiometric tungsten oxide phases contain arrange-ments with edge-sharing octahedra like that depicted on the right-hand side ofFig. 2.1 [24].

As inferred from Fig. 2.1, the different tungsten oxide configurations thatcan be made out of corner- and edge-sharing octahedra arrangements havestructures with voids and tunnels which allow the insertion of external atoms[32]. In relation to this, the so-called tungsten bronzes refer to compoundsrepresented by the formula MxWO3, with M the foreign atom (for example,

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Figure 2.1. Corner- (left) and edge-sharing (right) arrangements of the octahedra thatconstitute the building blocks of crystalline tungsten oxides. Each octahedron has atungsten atom at its center surrounded by six oxygen atoms located at the corners.

H or alkali metals such as Li, Na, and K). Transitions between crystallinephases in tungsten bronzes occur with increasing x value [29]. Dependingon the conditions, tungsten bronzes can present orthorhombic, monoclinic,tetragonal, hexagonal, and cubic symmetries (or a mixture of them). The cubicconfiguration is the one expected at x values close to 1.

As a first approximation, it can be assumed that the amorphous case isformed by a random network of corner- and edge-sharing distorted octahe-dra—that is, the same building blocks as in crystalline tungsten oxide but witha distribution of bond lengths and angles (in contrast to the regular octahedra).However, a detailed study combining experimental extended X-ray absorptionfine structure (EXAFS) data and state-of-the-art calculations suggested thatthis is not the real case [33]. In this work, the structure of sputtered stoichio-metric aWO3 was observed to be made out of a distribution of distorted WO6,WO5, and WO4 polyhedra building blocks—instead of only the WO6 octahe-dra, as depicted in Fig. 2.1. In addition, it was concluded that some of thesepolyhedra formed chains and rings which resulted in a vast amount of sitesand channels adequate for ion insertion, diffusion, and storage. This is consis-tent with the fact that aWO3 can attain high intercalation ratios. For example,using electrochemical methods lithium can be reversibly intercalated up to anintercalation ratio (x=Li/W) of about 0.65 [34], and in the irreversible regimehigh x values can be achieved—for example, see Ref. [35] that reports data upto x = 1.8. The particularities in the amorphous case most likely depend onthe preparation condition of the material. However, the previously commentedstudy sheds some light on important aspects that characterize the structure ofaWO3.

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Figure 2.2. Schematic energy band diagram of WO3 (left panel) and WO2 (rightpanel) after Refs. [36] and [37]. Reproduced from Ref. [39] with permission fromThe Royal Society of Chemistry. The filled states at 0 K are shadowed. The 5d (egand t2g) orbitals of W and the 2s as well as the 2p orbitals of O are indicated. TheFermi level (εF) is depicted by the arrows in each panel. For each band, the availablenumber of electrons (e−) per formula unit is specified.

2.2 Electronic density of states and ion intercalation2.2.1 Schematic band diagram and general propertiesThe schematic representation of the energy diagrams of WO3 and WO2 aredepicted in Fig. 2.2 [36, 37]. For both cases, it can be seen that the valenceband (VB) is mainly formed by O 2p orbitals, whereas the conduction band(CB) consists predominantly of W 5d orbitals. This can be qualitatively un-derstood by looking at the first ionization energies of W, and O, which areabout 7.86 eV, and 13.62 eV, respectively [38].

As depicted on the left-hand side of Fig. 2.2, WO3 is a semiconductorwith the Fermi level positioned in the band gap. On the other hand, WO2is metallic with the Fermi level positioned in the conduction band (see theright-hand side of Fig. 2.2), and this is also the case for substoichiometriccrystalline phases WOy with 2 < y < 3. Conceptually speaking, we can thinkthat W has six electrons to release and each O can receive two electrons. Thus,for stoichiometric WO3 (with three O atoms per one W) all the electrons fromthe W can be accommodated in the valence band, which will be filled. In thesubstoichiometric case, the valence band would not be able to receive all theavailable W electrons. Thus, some of them will populate the conduction band.

15

Thin films of aWO3, prepared with similar conditions than those used inthis thesis, present an optical band gap of about 3.2 eV [40]. The values forsputtered aWO3 are usually in the range of about 3.0−3.4 eV [24].

When lithium is intercalated into a WO3 host the accompanying charge-balancing electron occupies the lowest available energy state in the CB ofWO3. As discussed by Hjelm and coworkers [41], the conceptual reason is thefollowing. The first ionization energy of Li is of about 5.39 eV [38], smallerthan that of W (7.86 eV). Thus, it is energetically favorable for the extraelectron to occupy the conduction band of WO3 rather than staying in a bandformed by the intercalated Li, which remains basically ionized.

The EC properties of aWO3 are governed by the inserted electrons in thelower part of its conduction band. As a result, the characteristics of the lat-ter are specially relevant in the present case. The lack of long-range order inamorphous materials does not allow to apply the Bloch formalism for the de-scription of the system. Thus, band structure representations—that is, energyeigenstates as a function of the wave vector—are not adequate. Instead, a goodway of describing the electronic structure in amorphous materials is the den-sity of states [42]—that is, the number of energy eigenstates per unit volumeper unit energy. For the same reason commented above, ab initio calculationsof the density of states in amorphous materials are complicated [33, 43]. Inthe following, a method for probing the density of states in the bottom of theconduction band of aWO3 is introduced.

2.2.2 Introduction to the electrochemical density of states(EDOS)

Information of the electronic density of states (DOS) of intercalation mate-rials has been obtained by a simple and convenient electrochemical method,the so-called intercalation spectroscopy technique [44]. As explained abovefor WO3, when lithium is intercalated into the host the charge-balancing elec-trons start to populate the lowest available energy levels in the CB. Also, thecharacteristics of the top of the valence band can be probed in cases whenelectrons get extracted from this one upon ion deintercalation. Nevertheless,we are going to focus the discussion below on the particularities of WO3.

Strong similarities between the electronic DOS—obtained by density-functionalcomputations of the band structure in crystalline cases—and the results us-ing this method have been observed, for instance, in CeO2 [45], IrO2 [46],V2O5 [47], SnO2 [48], as well as in sputtered amorphous TiO2 [49] andaWO3 [49, 50]. The latter is relevant to this thesis. Similar approacheshave been employed for the study of nanostructured NiO2 [51] and TiO2[52, 53]—including a review of related works on nanostructured TiO2 andorganic hole conductors by Bisquert [54].

16

The typical experimental configuration used to this end is a three-electrodesetup. Here, the material of interest is the working electrode (WE) and, inthe case of lithium intercalation, metallic Li pieces can be used both as thereference (RE) and counter (CE) electrodes. The electrochemical potential Vof the WE measured with respect to the RE takes the form [55]

V =−(ze)−1(μ −μ refLi ), (2.1)

with μ the chemical potential of the WE, μ refLi the chemical potential of

metallic lithium (the reference), z the valence of the intercalated ion (z = 1for lithium and this value will be used hereinafter in this section), and e theelementary charge.

The electrochemical density of states EDOS is defined as [49]

EDOS =−dx/dV, (2.2)

with x = Li/W the number of intercalated lithium atoms (in this case, equalto the number of inserted electrons) per formula unit (f.u.) of tungsten oxide.Note that EDOS corresponds to the number of inserted electrons per potentialunit per f.u. of tungsten oxide. The quantity EDOS is expected to approximatethe electronic DOS of the CB. This is the case provided that the measurementsare performed as close as possible to equilibrium, the rigid band approxima-tion (RBA) is valid (meaning no or small changes of the electronic DOS dueto intercalation), no phase changes occur (if the material is crystalline), andthe changes of V upon intercalation are mainly determined by the variation ofthe Fermi level εF of the inserted electrons (which accommodate in the lowestavailable state in the CB) [50].

Three techniques are usually employed in intercalation spectroscopy. Thoseare chronopotentiometry (CP) [56], galvanostatic intermittent titration tech-nique (GITT) [57], and electrochemical impedance spectroscopy (EIS) [56].In CP, lithium ions are intercalated by applying a small constant current to theWE and the corresponding change of potential is monitored. GITT is similarto CP, but alternates between small steps of ion intercalation and a subsequentrelaxation. That is, the current is applied during a small time interval, and itis later interrupted during a certain period to let the WE stabilize, so that thequasi-stationary condition is assured. EIS uses frequency-dependent measure-ments to obtain information about the DOS.

In principle, GITT should be better in terms of avoiding kinetics effectsaffecting the measurement. However, it gives noisy data and is significantlytime-consuming. The latter may even induce irreversible changes in the WE.Despite CP can be somehow affected by kinetic effects, it has given goodagreements between the EDOS and theoretical DOS calculations (of crys-talline phases). Furthermore, it is more convenient in terms of the durationof the experiment. The EDOS data obtained from EIS is qualitatively similar

17

to that from CP, but in general gives smaller values [44]. In this thesis, CP wasused for the study of the EDOS.

The RBA is indeed a big assumption in intercalation spectroscopy thatwould require theoretical and experimental verification in each specific case.There are two aspects to consider here. First, modifications in the bottom ofthe CB (the one of interest in this thesis) due to lithium intercalation. Thisshould be more relevant at high rather than at low intercalation levels. Thus,the validity of the RBA is expected to be more applicable in the latter case.Second, the creation of an impurity band—due to the intercalated ions—whichalters the DOS in the region of interest of the CB. Regarding this point, thereis an advantage with the particular choice of material and ion. At least forWO3 in the cubic crystalline phase, calculations have shown that the states in-troduced by Li form a band centered at about 6.6 eV above the CB band edge[41]—well above of the energy region of interest in this thesis.

Looking at Eq. (2.1), it can be seen that the potential V is directly relatedto μ—which contains other contributions apart from the electronic one. In anintercalation material, as an approximation, μ can be expressed as [55]

μ = μLi +μe, (2.3)

with μLi, and μe the chemical potentials of the intercalated ions, and theinserted electrons, respectively. The ionic part can be modeled by the lat-tice gas model [55, 58]. Here, it is assumed that the ions arrange randomlyinto energetically equivalent sites, and the ion-ion interaction is treated in themean-field approximation. In this case, μLi can be expressed as

μLi =−e{V0 + kT ln[(1− xr)/xr]−Wxr}, (2.4)

with V0 the ion site potential, k the Boltzmann constant, xr the occupiedfraction of the available sites for ions, T the temperature, and W the total meaninteraction potential that an ion in the host would perceive if all the remainingion sites were full. The second term of the right-hand side of Eq. (2.4) isrelated to the entropy of the random ion distribution in the fraction xr of theequivalent sites.

Equation (2.4) describes μLi. However, the quantity of interest to the DOSis μe—which, in general, can contain entropy and interaction contributionsapart from that of the energy of the electronic states. Here, we associate μewith the Fermi level εF of the electrons in the CB.

In the case of a metallic material with extended electronic states, μe wouldremain close to constant upon intercalation (at low x) according to the Friedelscreening model [59]. This theory suggests that the electrons screen the inter-calated ions such that the absolute energy level of the CB translates downward.Here, the variation of the Fermi level accounts only for the portion of the ioniccharge that is not screened. The left and center panels of Fig. 2.3 depict asituation where the ions are completely screened and the Fermi level remains

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Figure 2.3. Schematic representation of the conduction band (CB) diagram. A caseat a certain intercalation level x = x1 is shown on the left-hand side. Upon furtherintercalation up to x = x2 > x1, the situations with strong screening (center) and with-out screening (right-hand side) are depicted. The Fermi level εF is indicated in eachcase and the filled states are shaded. With strong screening (center) the Fermi leveldoes not change with intercalation. On the other hand, without screening (right) theintercalation produces a variation ΔεF in the Fermi level with respect to that of the leftpanel (which presents a lower intercalation level).

unchanged after intercalation. In this extreme case, any information of theDOS would be lost and the total chemical potential would only be related tothe ions. That is, Eq. (2.2) would not give any details about the electronicstructure of the conduction band. Hence, a situation without strong screeningand with a Fermi level that changes with the intercalation would be desirable,like that presented on the right-hand side of Fig. 2.3. Ideally, the variation inV should be the same as that of the Fermi level. In reality, the situation maybe in between the latter and the case with constant Fermi level.

It is worth mentioning that the Friedel screening model was derived formetals, which present a high electron density. The situation is different inaWO3, where the concentration of the intercalated ions (the impurity) and theinserted electrons would be equal—at least for monovalent ions such as Li+.A detailed modeling of the screening in this instance may require a differentapproach from that used in the Friedel screening model, and this is out of thescope of this thesis.

It has been suggested that, when the bottom of the CB consists of localizedstates, at low intercalation levels the screening can not be completely achievedand, in this case, the changes in the total chemical potential would be dom-

19

inated by the electronic contributions. Thus, the EDOS in Eq. (2.2) wouldresemble the DOS of the CB [50]. Indeed, it has been shown that disorderinduces a strong localization of the electrons in the bottom of the CB of aWO3[45]. The estimated radius of the electron wave function was smaller thanabout 1 Å, and 4 Å for intercalation ratios below about 0.3, and 0.55, respec-tively. Therefore, we assume that intercalation spectroscopy can be used forprobing the electronic structure of the CB of aWO3—at least at low interca-lation levels, and probably at higher values as suggested by the work in Ref.[49]. Particularly, it is inferred that this method can provide valuable detailsof the localized band gap states below the CB of aWO3.

Although amorphous samples appear to be more convenient for intercala-tion spectroscopy [44], in some cases, the EDOS measured on nanocrystallinematerials [46, 48] has shown comparable features to calculations of the elec-tronic DOS. Here, the states should be considerably more extended than inamorphous cases. Thus, the validity of EDOS measurements may not be nec-essarily circumscribed to situations with localized states. Instead, amorphousand crystalline materials may present different degrees of screening that deter-mine the extent up to which the information of the electronic structure can beretrieved from intercalation spectroscopy.

A further issue of intercalation spectroscopy is that it does not give theposition of the CB edge. Consequently, this information must be obtained bycomplementary methods such as Mott-Schottky analysis, as described later inSection 3.2.4.

2.2.3 Details of the conduction band of amorphous tungstenoxide

Illustrative examples of the DOS in the bottom of the CB of aWO3 and crys-talline WO3 are depicted in Fig 2.4. The EDOS of aWO3 (using CP) [49] andamorphous WO2.93 (using GITT) [60] thin films as well as the calculated DOSof room temperature monoclinic WO3 (γ-WO3) [61] and aWO3 (in combina-tion with experimental EXAFS data) [33] are shown in Fig 2.4.

The calculated DOS of aWO3 is relatively similar to that of γ-WO3 butwith smoothed details. Interestingly, the former does not display a band tail.The experimental EDOS values from CP are lower than those of the calcu-lated DOS of γ-WO3. However, the former seems to follow qualitatively thedetails of the latter—that is, from low to high energies, an elbow followedby a local minimum and a subsequent increase. For the probed energies, theEDOS obtained from GITT displays higher values than that from CP, which isexpected because the former is closer to the equilibrium condition. Variationsin the shape between the CP and GITT cases shown here may also be due todifferences in the preparation conditions. Note that both experimental datasets show a band tail. In general, aWO3 films are not identical to each other

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Figure 2.4. Experimental electrochemical density of states −dx/dV of an amorphousWO2.93 thin film using GITT taken from Ref. [60] (black solid line) and that of anaWO3 thin film using CP from Ref. [49] (blue dash-dotted line). In both cases, themeasured initial open circuit potential was chosen as the zero of energy. The calcu-lated density of states of aWO3 (red dashed line) from Ref. [33] and room temperaturemonoclinic WO3 (γ-WO3; green dotted line) from Ref. [61] are also presented. Thecalculated data sets, as shown here, account for the occupation of two electrons withopposite spin per state. In each case, the CB edge was taken as the zero of energy.Here, only the bottom of the conduction band is depicted.

and their experimental EDOS values are expected to be highly dependent ofthe preparation conditions and the stoichiometry—the latter can be observedin Ref. [60]. An issue that makes difficult the comparison between the experi-mental EDOS and the calculated DOS is the location of the CB edge, which isnot given by the intercalation spectroscopy technique—that is, the position ofthe experimental data with respect to the absolute energy scale. In Paper III,an attempt to estimate the positions of the CB edge of an aWO3 thin film in alithium-containing electrolyte is made.

2.3 General optical response of non-intercalated andintercalated amorphous tungsten oxide

Optical changes in tungsten oxide occur both due to ion intercalation and oxy-gen deficiency. In the crystalline case, substoichiometric tungsten oxide ofthe form WOy is bluish for y values slightly lower than 3 and change color

21

with decreasing y down to y = 2 (WO2), which presents a brownish tonality[62]. Moreover, bulk crystals of sodium tungsten bronzes NaxWO3 start witha greenish color at x = 0. With increasing x value, they change through darkgrey, dark blue, royal blue, purple, brick red, orange, and reach a yellowishappearance at x = 1 [29].

A representative example of the changes in the optical spectra of an amor-phous tungsten oxide thin film upon lithium intercalation is depicted in Fig.2.5. This particular instance is relevant to this thesis because corresponds to asputtered amorphous tungsten oxide thin film (WO2.89) close to the stoichio-metric condition. In addition, it contains data at high intercalation levels (thatis, x� 0.5), which are difficult to find in literature.

The transmittance begins at a high level at x = 0 and decreases significantlyas x increases up to 0.30, see Fig. 2.5(a). The curve at x = 0.45 presents onlysmall changes with respect to that at x = 0.30. Notably, at higher intercalationlevels the transmittance reverses its direction of change, and starts increasingwith the intercalation level, see Fig. 2.5(c). Another experiment showing dataat high intercalation levels for aWO3 is consistent with this picture [63].

It is important to remark that during the whole wide interval of x (from 0up to 1.8) the reflectance shows only minor modifications, see Figs. 2.5(b)and 2.5(d). The most noticeable being a variation in the amplitude of theinterference fringes, which decrease with the absorption coefficient. Still, thereflectance curves are hardly affected by the intercalation in comparison withthose of the transmittance. Hence, the optical changes upon intercalation aremainly associated with an optical absorption.

The dependence of the absorption coefficient with respect to the intercala-tion level is opposite to that of the transmittance, see Fig. 2.5(e). This is betterobserved in Fig. 2.5(f), which presents the data sets without the contributionof the non-intercalated case (x= 0). Here, a predominant absorption band cen-tered at about 1.4 eV can be noticed. Another peak centered at about 3.4 eVdevelops as the intercalation level increases. There is probably an extra peakcentered between about 2.45 and 2.70 eV [35].

The abrupt increase of the absorption coefficient at high energies in Fig.2.5(e) is linked to interband transitions between the VB and the CB of tungstenoxide. In addition, in Fig. 2.5(e), the small values for the curve at x = 0 at lowenergies are related to the absorption in the ITO layer. It is interesting tomention that the optical spectra of lithium-intercalated and oxygen-deficientamorphous tungsten oxide thin films are similar, at least at low intercalationand deficiency levels [40].

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Figure 2.5. Transmittance [(a) and (c)], reflectance [(b) and (d)], and absorption co-efficients [(e) and (f)] for a LixWO2.89 (slightly substoichiometric) amorphous thinfilm sputtered onto glass pre-coated with ITO. The results shown here belong to thesame sample. Lithium was intercalated electrochemically in between the optical mea-surements, which were performed ex-situ. The x values range from 0 to 1.8 and areindicated in each plot. Panel (f) corresponds to the curves in panel (e) minus the dataat x = 0 (sample with no intercalation). Reprinted from Ref. [35] with permission ofAIP Publishing.

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3. Theory

The theoretical aspects of frequency-dependent measurements and optical ab-sorption models of aWO3 are presented in this chapter. As far as possible, thesame definitions and conventions employed in the appended papers are usedhere. In addition, references to literature related to aWO3 are preferred.

3.1 Linear systems, transfer functions, andfrequency-dependent variables

Let us assume that there is a linear system with a physical property of interestdescribed by a complex frequency-dependent transfer function S(ω)—withω = 2π f the circular frequency linked to the linear frequency f . In the generalimpedance context [64], this transfer function can be thought as a relationbetween a complex frequency-dependent input (or excitation) A(ω) and anoutput (or response) B(ω), as depicted in Fig. 3.1.

For simplicity, the time-dependent input A(t) can be a sinusoidal signal ofthe form

A(t) = 〈A〉+AA(ω)sin[ωt +φA(ω)], (3.1)

with 〈A〉 a steady state component of the input. Also, AA(ω), and φA(ω)are the amplitude, and phase of the sinusoidal excitation, respectively. Theamplitude AA(ω) is a real number equal or greater than zero. The connectionwith the frequency dependent quantity is given by A(ω) = AA(ω)eiφA(ω).

In the linear regime, the response B(ω) should correspond to a sinusoidalsignal with the same frequency ω as the excitation, but it can be dephased withrespect to this one. Its respective time-dependent signal B(t) is expressed as

B(t) = 〈B〉+BA(ω)sin[ωt +φB(ω)], (3.2)

with analogous definitions to those of the input case in Eq. (3.2) and withB(ω) = BA(ω)eiφB(ω).

The transfer function S(ω) of the system can be given by

S(ω) =B(ω)

A(ω)= |S(ω)|eiφS(ω), (3.3)

with |S(ω)|= BA(ω)/AA(ω) and φS(ω) = φB(ω)−φA(ω).

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Figure 3.1. Schematic of a general transfer function S(ω) associated with an inputA(ω) and an output B(ω).

Hence, the transfer function S(ω) can be built by measuring the input andoutput over a desired frequency range. Here, it is assumed that the system isstable so that S(ω) does not vary with time, at least during the period whenthe inputs and outputs are measured. S(ω) is a complex quantity whose real,and imaginary parts are related to the components of the output which arein-phase, and out-of-phase with respect to the input, respectively.

In the present thesis, the experimental configuration for electrochemicalmeasurements was a three-electrode setup. Here, the potential of the workingelectrode (WE) is measured with respect to the reference electrode (RE), andthe current through the system is regulated by the counter electrode (CE). Inthis case, the excitation was the potential of the WE, with time-dependent formV (t) given by

V (t) = 〈V 〉+VAsin(ωt), (3.4)

with an amplitude VA independent of the frequency (unless specified other-wise) and a phase set to zero for convenience. Thus, its frequency-dependentform is simply V (ω) =VA (constant in frequency).

Both the electrical and optical responses are of interest in an electrochromicmaterial. As a result, the analyzed responses are the current through the circuit(I), and the optical transmittance T with time-dependent quantities denotedrespectively by

I(t) = 〈I〉+ IA(ω)sin[ωt +φI(ω)], (3.5)

T (t) = 〈T 〉+TA(ω)sin[ωt +φop(ω)]. (3.6)

The frequency-dependent expressions for the current and the optical trans-mittance are given respectively by

25

I(ω) = IA(ω)eiφI(ω), (3.7)

T (ω) = TA(ω)eiφop(ω), (3.8)

The oscillating charge Q(t) can be obtained from the time integration of Eq.(3.5) as follows

Q(t) =∫ t

0dt ′[I(t ′)−〈I〉] = QA(ω)sin[ωt +φC(ω)], (3.9)

with the amplitude QA(ω) = IA(ω)/ω and phase φC(ω) = φI(ω)− π/2.Then, the frequency-dependent charge is defined as

Q(ω) = QA(ω)eiφC(ω). (3.10)

A summary of the transfer functions of interest to this thesis and their as-sociated excitation and response quantities is depicted in Table 3.1. Thesefunctions will be explained in detail in the following sections. The respec-tive technique for obtaining each quantity is indicated in Table 3.1. That is,electrochemical impedance spectroscopy (EIS), color impedance spectroscopy(CIS), and simultaneous electrochemical and color impedance spectroscopy(SECIS). Note that the charge was considered as the excitation in the differ-ential coloration efficiency transfer function. In addition, the charge was nor-malized by the active area of the WE, A. In the linear regime, the potential andthe current (or charge) can swap their roles of excitation and response with-out changing the values of their related transfer functions. The assignments inTable 3.1 are merely related to the experimental conditions used in this thesis.

Table 3.1. Summary of the transfer functions used in this thesis including the relatedmethod as well as the excitation and response quantities.

Method Transfer function Excitation Response

EIS

Impedance Potential Current

Z(ω) VA I(ω)

Capacitance Potential Charge

C(ω) VA Q(ω)/A

CISOptical capacitance Potential Optical change

Gop(ω) VA T (ω)/〈T 〉

SECISDifferential coloration efficiency Charge Optical change

K(ω) Q(ω)/A T (ω)/〈T 〉

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3.2 Electrochemical impedance spectroscopyThe impedance is a complex quantity defined as [65]

Z(ω) = V (ω)/I(ω). (3.11)

This is a generalization of the Ohm’s law V =RI, with R a resistance—whichis a real number that describes how much a system opposes to the flow of acurrent upon the application of a voltage difference. The real, and imaginaryparts of the impedance are associated with the resistance, and capacitance of acircuit, respectively. The existence of the latter causes a delay between voltageand current changes.

A complex frequency-dependent capacitance (per unit area) C(ω) can beexpressed as

C(ω) = A−1/[iωZ(ω)] = A−1Q(ω)/V (ω), (3.12)

which is analogous to a differential capacitance given by the derivative ofthe charge with respect to the voltage.

The impedance response of a system of interest can be measured experi-mentally over a desired frequency range. The information in the frequencyspace is useful because it allows the separation of processes with differentcharacteristic frequencies, which would be otherwise convoluted in the timespace.

Furthermore, physical properties of the system can be obtained by compar-isons between the experiment to either theoretical models or sensible equiva-lent circuits. In the latter case, the system is modeled using a combination ofcircuit elements which, ideally, can be associated with its physical properties.This approach has to be done with care because, usually, the choice of theparticular equivalent circuit configuration is not unique.

Electrochemical impedance spectroscopy (EIS) measurements are gener-ally nondestructive. Thus, EIS is a technique suitable for studying some prop-erties of interest at different conditions—for example, at various bias poten-tials of the WE.

Concerning the use of EIS in the context of intercalation EC systems, Ho etal. developed an equivalent circuit model in 1980 for the study of the chem-ical diffusion of the ion and the accompanying electron in aWO3 thin films[66]. It corresponds to a Randles circuit [67] adapted for finite diffusion in-stead of the semi-infinite case described by a Warburg element [68]—that is,solving the equations for normal diffusion (Fick’s laws) considering a reflec-tive boundary condition at the interface between the EC film and the TCE.Here, the fitting of the model to the experimental data can provide informationabout the chemical diffusion coefficient of the electron-ion couple within theEC film. In addition, properties of the interface between the electrolyte and theEC film can assessed. Also in 1980, Glarum and Marshall employed a similar

27

approach to that from Ho et al. [66] for the study of iridium oxide films withEIS [69].

Then, Franceschetti and Macdonald presented an extension to the modelfrom Ho et al. [66] adding an intermediate adsorption step before the interca-lation. This is conceptually consistent with the two-step intercalation modelproposed later by Bruce and Saidi [70]. Here, the ion reaches the electrolyte-EC film interface and combines with an electron from the film before the inter-calation into the host. However, the intermediate adsorption model has beenhardly used in EC studies, see for example the work on vanadium oxide byBae and Pyun [71]. The reason is that the extra step would generate an addi-tional semicircle in the complex plane apart from that related to a combinationbetween the double layer capacitance and the charge transfer resistance. Gen-erally, only one clear feature is observed. This would mean that either theexpected adsorption step is not present, the features overlap, or one of themlies in a frequency range that can not be resolved in the experiment.

The previous approaches considered normal diffusion. The determinationof the impedance of anomalous diffusion—including also the reflecting andabsorbing boundary condition cases at the EC film-TCE interface—was de-veloped by Bisquert and Compte [72]. This was done by solving generalizedFick’s laws in the small amplitude (linear) oscillating regime with appropri-ate boundary conditions. Diffusion throughout a lattice containing differentkinds of sites has also been treated [73, 74, 75]. Here, shallow sites allow fastdiffusion and deep sites act as traps with slower filling and release times thanthe former ones. This may generate a dispersion of waiting times between thedifferent sites which can be associated with the anomalous diffusion, as men-tioned in Ref. [73]. In addition, the impedance of diffusion coupled with asolid-state reaction has been studied by Fabregat-Santiago et al. [76]. In thiscase, the diffusion is considered as a preliminary step before the incorporationof the ion into a lattice site by means of the so-called solid-state reaction.

EIS measurements have been performed on aWO3 thin films prepared bydifferent methods. For example, thermal evaporation [66, 77, 78, 79, 80, 81,82, 83, 84, 85, 86], electron-beam evaporation [87, 88], sol-gel deposition[76, 89], and sputtering [90, 91, 92, 93, 94]. In general, the experimental spec-tra have been analyzed using an equivalent circuit similar to that from Ho et al.[66], or modifications and extensions of this one. The particular results varywith the method and preparation conditions as well as with the stoichiometryand the intercalation level. Reported values of the chemical diffusion coeffi-cient of the lithium-electron pair range from 10−13 to 10−8 cm2 s−1.

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3.2.1 Circuit elements and basic connectionsAn equivalent circuit consists of an arrangement of circuit elements. The basicpassive circuit elements are depicted in Table 3.2 together with their symbol,impedance, and Nyquist plot representation.

Table 3.2. Basic circuit elements. In each case, the symbol (used in this thesis),impedance formula, and schematic Nyquist plot representation are depicted. The ar-rows show the direction of increasing frequency.

Element Symbol Impedance Z(ω) Nyquist plot

Resistor R

Capacitor [iωC]−1

Inductor iωL

Constantphase element

(CPE)[Q(iω)n]−1

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The impedance of a resistor with resistance R [Ω] is expressed as

Z(ω) = R. (3.13)

It is a real number independent of the frequency, and in the Nyquist plot itis depicted by a point on the real axis.

The impedance of a capacitor with capacitance C [F] is given by

Z(ω) = [iωC]−1. (3.14)

It draws a line along the negative imaginary axis in the Nyquist plot whichtends to −∞, and 0 for ω → 0, and ω → ∞, respectively.

An inductor with inductance L [H] presents an impedance of the form

Z(ω) = iωL. (3.15)

It lies on the positive imaginary axis in the Nyquist plot. The impedance ofan inductor vanishes at ω = 0, has a real part equal to zero, and presents animaginary part proportional to ω .

The constant phase element (CPE) is usually introduced empirically to ac-count for the distributed nature of some characteristics of a system [65]; forexample, surface roughness of an electrode which results in an "imperfect"capacitor. Its impedance form is given by

ZCPE(ω) = [Q(iω)n]−1. (3.16)

It gives a tilted line in the Nyquist plot, see Table 3.2 for the cases with0< n< 1 and −1< n< 0. The CPE can take a limiting behavior of a capacitor(n = 1), an inductor (n =−1), and a resistor (n = 0).

Two arbitrary elements with impedances Z1 and Z2 can be arranged in seriesor in parallel, as depicted in Fig. 3.2. The total equivalent impedance Zeq of aseries connection has the form

Zeq = Z1 + Z2. (3.17)

Figure 3.2. Series (a) and parallel (b) connections of two elements with arbitraryimpedances Z1 and Z2.

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On the other hand, the total equivalent impedance Zeq of a parallel connec-tion is given by

Zeq = (Z−11 + Z−1

2 )−1. (3.18)

The situation is the opposite in the capacitance representation. Let us as-sume that we have two arbitrary elements with complex capacitances C1 andC2. The equivalent complex capacitance Ceq of their series connection is ex-pressed as

Ceq = (C−11 +C−1

2 )−1, (3.19)

and that of their parallel connection is given by

Ceq = C1 +C2. (3.20)

An illustrative example of an equivalent circuit is depicted in Fig. 3.3(a).It consists of a resistor R1 in series with a parallel combination of a resistorR2 and a capacitor C2. Its impedance is depicted in the Nyquist plot by thesolid line in Fig. 3.3(c). At ω → ∞, and ω → 0 the impedance tends to R1,and R2, respectively. Note that this response corresponds to a semicircle in thecomplex plane with diameter R2 and centered at the coordinate (R1+R2/2,0).The apex of the semicircle is located at the characteristic frequency ω0 of theR2C2 parallel combination given by

ω0 = (R2C2)−1. (3.21)

If the capacitor is swapped for a CPE element with impedance ZCPE2(ω) =[Q2(iω)n2 ]−1 and exponent 0 < n2 < 1, as depicted in Fig. 3.3(b), the responseremains similar. However, the semicircle becomes depressed, and the level ofdistortion increases with decreasing n2. Note that with n2 = 1 the "perfect"semicircle is recovered. Here, the apex of the arc occurs at the characteristicfrequency ω0 of the form

ω0 = (R2Q2)−1/n2 . (3.22)

The factor Q2 does not have the units of a capacitance. For a parallel com-bination like that shown in Fig. 3.2(b), an effective capacitance (per unit area)Ceff associated with the CPE can be approximated by [95]

Ceff = A−1Q1/n22 R(1−n2)/n2

2 , (3.23)

which corresponds to the formula given by Hsu and Mansfeld [96]. Alterna-tively, a formula for an effective capacitance (per unit area) Ceff also containingthe series resistance R1 was presented by Brug et al. [97] and it is given by

Ceff = A−1Q1/n22 [R1R2/(R1 +R2)]

(1−n2)/n2 . (3.24)

31

Figure 3.3. (a) Equivalent circuit of a resistor R1 in series with a parallel R2C2 combi-nation. (b) Same as in (a) but with a CPE with impedance ZCPE2(ω) = [Q2(iω)n2 ]−1

instead of the capacitor C2—an exponent value of n2 = 0.75 was used in this example.(c) Nyquist plot with the impedance response of the equivalent circuits depicted in (a)[solid line] and (b) [dashed line]. The frequencies at the apex of each arc are indicatedas well as the direction of increasing frequency.

The Hsu and Mansfeld formula given in Eq. (3.23) was used in this thesiswhen an effective capacitance estimation (related to a RCPE parallel combi-nation) was required.

3.2.2 Impedance of normal and anomalous diffusionThe schematic of the geometry of the problem is depicted in Fig. 3.4. It isassumed that the conductivity of the electrons is much higher than that of theions [66, 98, 99]. In this case, the electric field in the interior of the EC filmcan be neglected, and the ion transport within the film (in the form of neu-tral coupled ion-electron pairs) is dominated by diffusion due to concentrationgradients.

The Brownian motion—denoted here as normal diffusion (ND)— is char-acterized by a mean square displacement 〈r2(t)〉 proportional to the elapsedtime, so that 〈r2(t)〉 ∝ t [100]. The anomalous diffusion deviates from thislinear behavior, and in some cases can be described by a power law of theform 〈r2(t)〉 ∝ tγ . Here, the subdiffusion (γ < 1), and superdiffusion (γ > 1)regimes can be distinguished.

Returning to the case depicted in Fig. 3.4, because of symmetry consid-erations, the problem can be regarded as one-dimensional along the directionperpendicular to the surface of the film. The origin is placed at the bound-

32

Figure 3.4. Schematic of the geometry of the diffusion problem. The x-axis is per-pendicular to the surface of the film. The origin (x = 0) is set at the electrolyte-ECfilm interface, and x = d at the boundary between the EC film and the transparentconductive electrode (TCE). After the ions are intercalated from the electrolyte andthe electrons inserted from the TCE, a neutral Li+-e− couple forms at x = 0 and startsdiffusing throughout the EC film.

ary between the electrolyte and the EC film, and the EC film-TCE interface islocated at x = d, with d the film thickness.

Let c(x, t) be the number density of the ions and J(x, t) their flux along thex direction. The continuity equation reads

∂c(x, t)/∂ t =−∂J(x, t)/∂x, (3.25)

and the constitutive equation is given by Fick’s first law

J(x, t) =−Dch∂c(x, t)/∂x, (3.26)

with Dch the chemical diffusion coefficient of the ion-electron pair.Equations (3.25) and (3.26) govern the ND case and their combination gives

Fick’s second law

∂c(x, t)/∂ t = Dch ∂ 2c(x, t)/∂x2. (3.27)

Assuming that the ions can not penetrate the TCE, the reflective boundarycondition applies at x = d and takes the form [66]

∂c(x, t)/∂x|x=d = 0. (3.28)

The condition in Eq. (3.28) is used in the remainder of this thesis.Let the system be at equilibrium at t = 0 with the film at an equilibrium

potential η0, which corresponds to a constant equilibrium ion concentrationc0 in the film, so that

33

c(x,0) = c0. (3.29)

For a small potential variation δη(t) at x = 0, the concentration changes asmall δc(x, t). In this case, the boundary condition of c(x, t) at x = 0 can beexpressed as [66]

δc(x, t)|x=0 = (dη0/dc)−1c0

δη(t), (3.30)

with (dη0/dc)c0the derivative of the equilibrium potential with respect to

the concentration around the equilibrium condition determined by c0. Thatis, an overpotential at the surface (x = 0) creates a concentration gradient thatdrives the diffusion.

In addition, at x = 0 the diffusion current Id(t) and the flux J(x, t) are con-nected by

Id(t) = zeAJ(0, t), (3.31)

with z the charge of the ion, z = 1 for lithium.The solution to Eq. (3.27)—with the conditions and considerations given

in Eqs. 3.28, 3.29, 3.30, and 3.31—was obtained by Ho et al. [66], and theimpedance related to the normal diffusion case ZND(ω) can be expressed as[72]

ZND(ω) = RW[ωD/(iω)]1/2coth[(iω/ωD)1/2], (3.32)

with the multiplicative factor RW, and ωD the characteristic frequency ofdiffusion given by

ωD = Dch/d2 = (ARWCch)−1, (3.33)

with Cch a diffusion capacitance [101] (presented here per unit area) whichcan be considered a chemical capacitance [102].

The semi-infinite normal diffusion case can be obtained from Eq. (3.32)by letting d → ∞. The resulting expression is the Warburg impedance [68]ZW(ω) which is given by

ZW(ω) = AWω1/2 − iAWω1/2, (3.34)

with AW a coefficient that can be related to the diffusion coefficient [65].In the following, two special cases of anomalous diffusion are introduced.

Those are the AD1a and AD1b models presented in Ref. [72]. In these cases,either the continuity [Eq. (3.25)] or the constitutive [Eq. (3.26)] equationsare converted to generalized fractional equations [103] and this leads to theanomalous diffusion behavior.

The AD1a model generalizes the continuity equation in Eq. (3.25) to theform

34

∂ γc(x, t)/∂ tγ =−∂J(x, t)/∂x, (3.35)

with γ < 1 the anomalous diffusion exponent. This corresponds to a non-conservation of the number of ions participating in the diffusion. CombiningEqs. 3.26 and 3.35, the resulting diffusion equation becomes

∂ γc(x, t)/∂ tγ = D∂ 2c(x, t)/∂x2. (3.36)

Note that the coefficient D in Eq. 3.36 does not have the units of the typicaldiffusion coefficient (cm2 s−1).

The diffusion impedance obtained from the AD1a model ZAD1a(ω) can beexpressed as [72]

ZAD1a(ω) = RW[ωD/(iω)]γ/2coth[(iω/ωD)γ/2]. (3.37)

The Ad1b model generalizes the constitutive equation in Eq. 3.26 to

J(x, t) =−D∂ 1−γ

∂ t1−γ∂c(x, t)

∂x, (3.38)

with γ < 1. Here, a continuous-time random walk scenario is described[72, 103]. It considers a distribution of waiting times between jumps.

The diffusion impedance from the AD1b model ZAD1b(ω) is given by [72]

ZAD1b(ω) = RWωγ−1D [ωD/(iω)]1−γ/2coth[(iω/ωD)

γ/2]. (3.39)

For both the AD1a and AD1b models, the diffusion coefficients presentedin Ref. [72] do not have the proper dimensions of a diffusion coefficient.An effective chemical diffusion coefficient Dch in these cases can be obtainedbased on the characteristic frequency ωD by Dch = ωD d2 [46, 94], as in Eq.3.33.

Equations (3.32), (3.37), and (3.39)—corresponding respectively to the ND,AD1a, and AD1b diffusion impedances—can be expressed in the general form[72]

Z(ω) = (ζmχm)1/2coth[d(χm/ζm)

1/2], (3.40)

which is mathematically equivalent to the impedance of the transmissionline of length d depicted in Fig. 3.5(a). The open-circuit termination at x = dgives the reflecting boundary condition from Eq. (3.28). The elements χm[Ωm−1] and ζm [Ωm] take a particular form in each case.

For the normal diffusion

χm = rm, (3.41)ζm = (iω cm)

−1, (3.42)

35

Figure 3.5. (a) Transmission line of length d with the same impedance as that of Eq.(3.40). (b) Schematic Nyquist plot representation of the impedance obtained from theND [Eq. (3.32)], AD1a [Eq. (3.37)], and AD1b [Eq.(3.39)] models. The dotted linecorresponds to the Warburg (W) impedance [Eq.(3.34)], which describes the semi-infinite normal diffusion case—that is, with d → ∞ in the ND model from Eq. (3.32).The parameters used in the simulated spectra were RW = 1 Ω ([Ωsγ−1] in the AD1bmodel), ω = 2π rads−1, and γ = 0.75. The positions in the curves corresponding tothe frequency ω = 6ωD are indicated by dots. The direction of increasing frequencyis depicted by an arrow.

with

rm = RW/d, (3.43)χm = ACch/d, (3.44)

with RW from Eq. (3.32) and Cch from Eq. (3.33). Thus, χm represents aresistor and ζm a capacitor.

For the AD1a model, χm is a resistor and ζm a CPE expressed as

χm = rm, (3.45)ζm = [Qm(iω)nm ]−1, (3.46)

with

rm = RW/d, (3.47)Qm = (RW ωγ

D d)−1, (3.48)nm = γ, (3.49)

with RW, ωD, and γ from Eq. (3.37).In addition, in terms of the transmission line elements, the characteristic fre-

quency of diffusion and the effective diffusion coefficient in the AD1a modelcan be written respectively as

36

ωD = (d2Qmrm)−1/nm , (3.50)

Dch = d2(d2Qmrm)−1/nm , . (3.51)

For the AD1b model, χm is a CPE and ζm a capacitor given by

χm = [Qm(iω)nm ]−1, (3.52)ζm = (iω cm)

−1, (3.53)

with

Qm = d/RW, (3.54)cm = (RW ωγ

D d)−1, (3.55)nm = 1− γ, (3.56)

with RW, ωD, and γ from Eq. (3.39). In the AD1b model case, an expressionfor the chemical capacitance Cch (per unit area) is given by [104]

Cch = A−1 cm d. (3.57)

Moreover, in the AD1b model case, the relation between the transmissionline elements and ωD as well as Dch can be expressed respectively as

ωD = (d2cm/Qm)−1/(1−nm), (3.58)

Dch = d2(d2cm/Qm)−1/(1−nm), . (3.59)

A detailed correspondence between the parameters in the models—that is,Eqs (3.32), (3.37), and (3.39)—and those describing the transmission line rep-resentations was given here because the fittings to the experimental spectrawere done using the latter. That is, the parameters defined in Eqs. (3.47),(3.48), and (3.49) for the AD1a model, and those defined in Eqs. (3.54), (3.55),and (3.56) for the AD1b model. Note that both cases (the AD1a and AD1bmodels) reduce to the normal diffusion ND situation when γ = 1.

A comparison between the different diffusion impedance models is depictedin Fig. 3.5(b). The semi-infinite case, described by the Warburg impedance,presents a characteristic straight line with a slope of 45◦ with respect to thereal axis. The ND starts following the dotted line at high frequencies, but itbends over toward a capacitive response (slope of 90◦) at lower frequencies,the transition starts at about ω = 6ωD. The high-frequency slope for the AD1aand AD1b models deviates from 45◦, and their low-frequency response alsodiffers from a straight vertical line.

37

3.2.3 Equivalent circuit for amorphous tungsten oxideFigure 3.6 depicts the equivalent circuit used for the fitting of the EIS spectraof aWO3 thin films. In the complete version, see Fig. 3.6(a), it comprisesa high-frequency impedance Zhf(ω) connected in series with the impedancerelated to the electrolyte-EC film interface.

The form of Zhf(ω) is presented in Fig. 3.6(c), and it consists of an in-ductive CPE ZCPEL(ω) = [QL(iω)nL ]−1 connected in series with the high-frequency resistance Rhf. The element ZCPEL(ω) describes inductive effectsmostly due to connection cables and Rhf contains the resistive contributionsfrom the TCE, the electrolyte, and also the electrical connections.

The electrolyte-film interface is modeled by a capacitive double layer CPEZCPEdl(ω) = [Qdl(iω)ndl ]−1 in parallel with a charge transfer resistance Rctconnected in series with a combination that accounts for the contribution fromadsorption effects and the chemical diffusion. The latter array is composedof a capacitive adsorption CPE ZCPEad(ω) = [Qad(iω)nad ]−1 connected in par-allel with an adsorption resistance Rad in series with the diffusion impedanceZAD1b(ω).

The diffusion element corresponding to the AD1b model [Eq. (3.39)] wasused here because, for the experimental EIS spectra discussed in this thesis,it case gave the best results. Constant phase elements were used instead ofperfect capacitors and inductors to consider the distributed nature of the ex-perimental configuration.

The equivalent circuit in Fig. 3.6(a) has a similar structure to that pre-sented by Franceschetti and Macdonald [99] for the study of the impedanceresponse of electrochromic thin films (considering an intermediate adsorptionstep before the intercalation), which is actually described in more detail intheir previous work in Ref. [98].

The following description adopts the conceptual approach from the previ-ous works. The diffuse double layer has an associated capacitance describedby ZCPEdl(ω). The resistance Rct is related to the barrier that an ion feels whenit transverses the diffuse double layer to get adsorbed at the surface of the film.An electron, that entered the film from the TCE, combines with the ion at thesurface to form a neutral ion-electron pair. The barrier to the insertion of thecouple into the interior of the film is associated with Rad. Moreover, the ca-pacitance linked to the formation of the ion-electron couple at the surface isdescribed by ZCPEad(ω). In reality, as shown in Ref. [98], Rct and the adsorp-tion parameters (ZCPEad(ω) and Rad) are coupled. The latter could be thoughtas a "correction" of the charge transfer impedance due to a bottleneck effectat the surface, which regulates the insertion of the ion-electron couple into thebulk of the film.

38

=(c)

(a)

(b)

Figure 3.6. (a) Equivalent circuit employed for modeling the EIS spectra of the aWO3thin films. The total sinusoidal excitation voltage with amplitude VA is measuredbetween the WE and RE. A high-frequency impedance Zhf(ω) is put in series witha combination corresponding to the impedance of the electrolyte-EC film interface(described in detail in the main text). The voltage drop over Zhf(ω) is Vhf(ω). (b)Simplified version of (a) appropriate when the contribution of the double layer tothe experimental impedance is negligible—here, the elements Rct and ZCPEdl(ω) areignored. (c) Zhf(ω) consists of an inductive CPE ZCPEL(ω) in series with Rhf (a high-frequency resistance).

39

Using Eq. (3.23), the effective double layer Cdl, and adsorption Cad capaci-tances (per unit area) can be estimated respectively by

Cdl = A−1Q1/ndldl R(1−ndl)/ndl

dl , (3.60)

Cad = A−1Q1/nadad R(1−nad)/nad

ad . (3.61)

The characteristic frequencies of the double layer ωdl and adsorption ωadprocesses can be obtained by means of Eq. (3.22), and their respective expres-sions are

ωdl = (RctQdl)−1/ndl , (3.62)

ωad = (RadQad)−1/nad . (3.63)

Figure 3.6(b) depicts a reduced case of the equivalent circuit shown inFig. 3.6(a). The former does not contain the double layer elements—thatis, ZCPEdl(ω) and Rct. It can be employed in situations with small Rct orhigh ωdl (outside the experimental range or at frequencies dominated by high-frequency inductive effects). In these cases, it would not be possible to dis-tinguish the contribution of the double layer impedance or it could be simplyneglected.

Returning to the parallel combination containing the adsorption and diffu-sion elements in Figs. 3.6(a) and 3.6(b), the branches with ZCPEad(ω), and theseries Rad-ZAD1b(ω) array are going to be denoted here as the adsorption, andintercalation branches, respectively.

The complex capacitance (per unit area) associated with the adsorptionbranch CCPEad(ω) is given by

CCPEad(ω) = A−1[iωZCPEad(ω)]−1, (3.64)

and the complex capacitance (per unit area) of the intercalation branchCint(ω) is expressed as

Cint(ω) = A−1{iω[Rad + ZAD1b(ω)]}−1. (3.65)

If the impedance contribution of the double layer is negligible—that is, thecase portrayed in Fig. 3.6(b)—the complex Faradaic capacitance CF(ω) (re-lated to the voltage drop at the interface between the electrolyte and the ECfilm) can take the form

CF(ω) = CCPEad(ω)+Cint(ω), (3.66)

assuming that the Faradaic capacitance is composed of the contributionsfrom both branches.

40

3.2.4 Mott-Schottky analysisIt is possible to estimate the flat band potential Vfb of the aWO3 WE from theMott-Schottky relation [105]

C−2dl = [2/(εε0eN)](V −Vfb − kT/e), (3.67)

with the dielectric constant of aWO3 given by ε , the vacuum permittivityε0, the carrier concentration N, the Boltzmann constant k, and the temperatureT .

Here, the total double layer effective capacitance Cdl from Eq. (3.60) (ob-tained from EIS measurements) was used in Eq. (3.67). In reality, Cdl iscomposed of a series connection of the Helmholtz layer capacitance CH (inthe electrolyte) and the semiconductor capacitance CSC [106], so that

1Cdl

=1

CSC+

1CH

. (3.68)

The semiconductor capacitance CSC is the one that should be consideredin the Mott-Schottky relation instead of the total capacitance at the interfaceCdl. The form of Eq. (3.67) is a valid approximation when CH >>CSC, whichwould give Cdl ≈ CSC. CH is a geometrical capacitance which, presumably,does not vary with the potential. For a semiconductor, it presents typical valuesbetween about 10 and 20 μFcm−2 [106]. A potential range where the previousapproximation is valid and the Mott-Schottky plot (C−2

dl against V ) gives alinear dependence is expected to exist at potentials higher than the conductionband edge of aWO3. In this case, Vfb can be obtained from the intercept of thestraight line with the potential axis V |y=0 as follows

Vfb =V |y=0 − kT/e. (3.69)

Moreover, the carrier concentration N can be retrieved from the slope of theMott-Schottky plot mMS by

N = 2/(εε0emMS), (3.70)

provided that ε is known.In the non-degenerate case (for which the Fermi level lies below the con-

duction band edge), for a n-doped semiconductor (such as aWO3) the relationbetween Vfb and the conduction band edge potential VC is given by [107]

VC =Vfb − (kT/e)ln(NC/N), (3.71)

with NC the effective electronic density of states at the bottom of the CB,which can be expressed as

NC = 2(2π m∗ekT/h2)3/2, (3.72)

41

with m∗e the electron effective mass in WO3 and h denoting the Planck con-

stant.There are complications in the flat band determination by the Mott-Schottky

relation using EIS [106]. For example, the Helmholtz layer capacitance CHcould influence the value of the obtained Vfb and N. Moreover, at high carrierconcentrations, the potential drop in the Helmholtz layer can be comparableto that in the semiconductor. In this case, the assumption that V −Vfb merelyaccounts for the potential drop in the semiconductor would not be completelyvalid. The estimated N in Paper III was about 3× 1018 cm−3, which couldbe considered low having in mind that the obtained NC was about 1020 cm−3.However, the upper limit in the carrier concentration at which the previousassumption is not longer valid should, ideally, be verified by performing themeasurements at different carrier concentrations. On top of this, the estima-tion of an effective capacitance from constant phase elements, such as in Eq.(3.60), may add errors to the process. Furthermore, Eqs. (3.67) and (3.71)contain quantities like ε and m∗

e , which, in this work, were obtained from liter-ature. This could also add additional sources of uncertainty in the calculationof VC. For the reasons mentioned above, the obtained value of VC in Paper IIIshould be regarded as an estimation.

The flat band potential in WO3 has been obtained before in protic elec-trolytes, giving values of, for example, 3.27 V vs. Li/Li+ (pH = 0) [108].However, these can not be directly compared to those obtained using aproticelectrolytes [108, 109] (such as lithium-containing electrolytes). A flat bandpotential value of about 3.21 V vs. Li/Li+ (assuming kT/e = 0.025 V) wasreported in literature [110] for an aWO3 thin film (prepared by electrodepo-sition) measured with the same electrolyte than that used in this thesis (1 MLiClO4 dissolved in propylene carbonate). In this case, the Mott-Schottkyrelation was obtained by single-frequency EIS measurements at 1.1 kHz.

3.3 Color Impedance SpectroscopyAs explained in Section 3.2, EIS analyzes the relation between the oscillat-ing potential and the current. In an EC system, an oscillating optical re-sponse is also generated in relation to the previous quantities. The frequency-dependent measurement of the optical response in combination with the elec-trical signals has been denoted as color impedance spectroscopy (CIS) [111].It was introduced for probing the adsorption kinetics at the interface betweenan electrolyte and an electrode [112]. Afterward, this technique has beenused for studying different kinds of materials. For example, polyaniline [113,114, 115], polypyrrole [111, 116, 117, 118], nickel oxide [119], nile blueA and hemin [120], cytochrome c proteins [121, 122, 123, 124, 125], Prus-sian blue [126, 127, 128, 129], Alexa 488 fluorochromes [130], poly(3,4-ethylenedioxythiophene methanol) and poly(3,4-ethylenedioxythiophene) [131],

42

graphite [132], pyridine-capped CdSe nanocrystals [133], niobium oxide [134],and tungsten oxide [64, 82, 83, 134, 135, 136].

By analogy with the complex capacitance C(ω) defined in Eq. (3.12),Kalaji and Peter introduced the concept of complex optical capacitance [115],which is denoted here by Gop(ω) and can be expressed as follows

Gop(ω) = 〈T 〉−1T (ω)/VA = |Gop(ω)|exp[iφop(ω)], (3.73)

with |Gop(ω)|= 〈T 〉−1TA(ω)/VA and φop(ω) the amplitude and phase (withrespect to the sinusoidal excitation voltage) of Gop(ω), respectively. The quan-tities T (ω), 〈T 〉, TA(ω), and VA are defined in Eqs. (3.4), (3.6), and (3.8).

In connection with the analogy between Gop(ω) and C(ω), the term 〈T 〉−1T (ω)in Eq. (3.73) can be regarded as an "optical charge." It can be shown that it isproportional to small changes in the optical density (optical absorption coeffi-cient times the film thickness) when the variations in the reflectance are muchsmaller than those in the transmittance (as seen in Paper II).

Actually, if the optical absorption in an EC system were only associatedwith an unique redox reaction, and if the optical change were perfectly syn-chronized with the charge related to that Faradaic reaction, Gop(ω) would bedirectly proportional to the part of C(ω) connected with the Faradaic chargeparticipating in the optical absorption process [115].

A detailed study of the comparison between the complex optical and elec-trical capacitances—for an aWO3 thin film WE at different bias potentials—ispresented in Paper IV. In the case of WO3 thin films, disagreements betweenthese quantities have been assigned, for instance, to leakage currents [82, 136],the presence of various kinds of intercalation sites [64, 135], non-Faradaic ef-fects at high frequencies [82], and an optically inactive diffusion of the ionbefore it is trapped at a site where the optical absorption can occur [134, 136].

The high-frequency impedance Zhf(ω) is not expected to participate in thecoloration. Thus, assuming the situation shown in Fig. 3.6(b), the Faradaiccapacitance CF(ω) defined in Eq. (3.66), and not the total capacitance C(ω)[Eq. (3.12)], is the one that should be compared to the complex optical capac-itance. Moreover, the expression in Eq. (3.73) contains the total amplitude ofthe oscillating excitation voltage. A corrected expression, that considers onlythe EC film-electrolyte interface, is defined as the complex Faradaic opticalcapacitance GF

op(ω) and is given by

GFop(ω) = 〈T 〉−1T (ω)/[VA −Vhf(ω)] = Gop(ω)[1− Zhf(ω)/Zfit(ω)]−1,

(3.74)with Zfit(ω) the fitting of the experimental data to the equivalent circuit in

Fig. 3.6(b).Simultaneous electrochemical and color impedance spectroscopy (SECIS)

measurements allow to obtain a direct comparison between Gop(ω) [Eq. (3.73)]and C(ω) [Eq. (3.12)]. The ratio between the previous quantities can be

43

defined as a complex frequency-dependent differential coloration efficiencyK(ω), which is expressed as

K(ω) = Gop(ω)/C(ω) = A〈T 〉−1[T (ω)/Q(ω)] (3.75)

= |K(ω)|eiφK(ω),

with φK(ω) = φop(ω)−φC(ω), and |K(ω)|= A〈T 〉−1[TA(ω)/QA(ω)], thephase, and amplitude of K(ω), respectively. The definitions of T (ω), andQ(ω) and are given in Eqs. (3.8), and (3.10), respectively. Note that

Gop(ω)/C(ω) = GFop(ω)/CF(ω). (3.76)

The concept of a frequency-dependent differential coloration efficiency (ora similar one) has been used, for example, elsewhere [82, 83, 119, 127, 129,134, 137]. Paper II contains a comprehensive analysis of K(ω), including itsexperimental spectra for an aWO3 thin film electrode at various bias poten-tials. Particularly, the phase φK(ω) provides details on the coloration process.For instance, according to the conventions used in this thesis, the sinusoidaltransmittance lags, and leads the sinusoidal charge for −π < φK(ω) < 0, and0 < φK(ω) < π , respectively. Furthermore, the sinusoidal transmittance andcharge are fully synchronized at φK(ω) = 0 and φK(ω) = ±π . The formercorresponds to a perfectly correlated state and the latter to an anticorrelatedone. In addition, the coloration is cathodic, and anodic for Re{K(ω)} > 0,Re{K(ω)}< 0, respectively.

3.4 Optical absorption models in amorphous tungstenoxide

3.4.1 Intervalence-transfer and small polaronsIn aWO3, the optical absorption—associated with photon energies smallerthan the required ones to promote interband absorption—can be explained interms of electron transitions between states localized on neighboring tungstensites [27]. This can be conceptualized by the intervalence-transfer absorptiontheory [138]. Figure 3.7 depicts a schematic representation of this processbased on a two-site model. Here, two adjacent potential wells with an energydifference between their bottoms of E0 and centered at the configuration co-ordinates qi and qj are shown. They correspond respectively to tungsten sitesdenoted by Wi and Wj. The electron is initially localized on the Wi potentialwell, and makes a transition to that belonging to Wj by absorbing a photonwith energy Eop. The Franck-Condon principle is assumed in the formulationof the intervalence transition model [138]—that is, the electron excitation oc-curs without changes in the configuration coordinate. Thereafter, the electron

44

Figure 3.7. Illustrative diagram of the intervalence-transfer absorption model. Theparabolas represent the potential wells linked to the tungsten sites Wi and Wj, whichare centered at the configuration coordinates qi and qj, respectively. The energy dif-ference between the bottom of the potential well Wj with respect to that of Wi is givenby E0. The electron (initially at the Wi site) can get excited to the Wj potential wellby absorbing a photon with associated energy Eop. Ea corresponds to the activationenergy related to thermal hopping.

relaxes to the bottom of the Wj potential well by a reconfiguration of the latticeassisted by phonons.

In general, in the context of aWO3, the following transitions involving dif-ferent tungsten valence states can be considered [35]

W5+i +W6+

j + hω → W6+i +W5+

j , (3.77)

W4+i +W5+

j + hω → W5+i +W4+

j , (3.78)

W4+i +W6+

j + hω → W6+i +W4+

j , (3.79)

with hω representing a photon. The transitions outlined in Eqs. (3.77),(3.78), and (3.79) are hereinafter referred to as W5+ → W6+, W4+ → W5+,and W4+ → W6+, respectively.

An analogous picture to the intervalence-transfer absorption theory is givenby small-polaron absorption models [139, 140, 141]. A polaron is a quasipar-ticle comprising a charge carrier and the perturbation that it produces in thelattice [142, 143], as depicted in Fig. 3.8 for an electron acting as the carrier.A polaron is called large or small depending on the spatial extent associatedwith the wave function of the carrier. In the latter case, it is of the order or lessthan the separation between atoms in the lattice.

45

Figure 3.8. Schematic representation of a small-polaron consisting of an electron e−trapped by its self-induced potential, which is generated by the displacement of theions around the electron charge. The dashed contours correspond to the equilibriumposition of the ions in the lattice in the absence of the electron.

The complex electrical conductivity σ(ω) and dielectric function ε(ω) arerelated by

σ(ω) = −iωε0ε(ω)

= −iωε0[Re{ε(ω)}+ iIm{ε(ω)}]= ωε0Im{ε(ω)}− iωε0Re{ε(ω)}, (3.80)

with ε0 the vacuum permittivity. As a result,

Re{σ(ω)}= ωε0Im{ε(ω)}. (3.81)

Thus, Re{σ(ω)} is directly linked to the optical absorption. The real partof the complex conductivity Re{σ(ω)} is hereinafter referred to as the opti-cal conductivity. In fact, it is proportional to the absorption coefficient α , asshown by the expression

α = Re{σ}/(ncε0), (3.82)

with n the refractive index of the material and c the speed of light in vacuum.In the context of the intervalence-transfer absorption, Hush gives a descrip-

tion of the frequency-dependence of the optical absorption coefficient in thetwo-site approximation [138], depicted in Fig. 3.7. In a simplified case, anelectron interacts with two potential wells described by harmonic oscillatorswith identical characteristic frequency—that is, their potential energy can bedescribed by the same parabolic function Cq2, with both the potential energyand q measured from the bottom of the potential well. Given these condi-tions, and using the terminology from Fig. 3.7, the optical conductivity can beapproximated by

46

Re{σ(ω)}= A(ω)exp{−(Ei +E0 − hω)2

4EiEvib

}, (3.83)

with A(ω) a frequency-dependent multiplicative factor, Ei = C(qi − qj)2,

and the characteristic vibrational energy of the atoms Evib given by

Evib =hωlo

2coth

(hωlo

2kT

), (3.84)

with ωlo the longitudinal optical (LO) phonon frequency. At high temper-atures (kT >> hωlo/2), Evib ≈ kT . On the other hand, at low temperatures,(kT << hωlo/2), Evib ≈ hωlo/2. A more detailed expression of the opticalconductivity in the two-site model (including the prefactor) is given, for ex-ample, by Calvani [144].

The line-shape of the optical conductivity Re{σ(ω)} given in Eq. (3.83)is mainly determined by the Gaussian expression. The latter presents a peakpositioned at hω = Ei +E0 with a full width at half maximum (FWHM) of4√

ln2√

EiEvib. For the special case of E0 = 0, it can be shown (using geo-metrical considerations) that Ei = 4Ea. In this case, the peak of the opticalconductivity would be located at hω ≈ 4Ea and the FWHM would be given by8√

ln2√

EaEvib.The frequency dependence of intraband optical absorption in disordered

materials presenting a strong electron-phonon coupling has been treated byBryksin [145]—specifically, the associated optical conductivity. Here, thecase of absorption due to electron hopping between localized states near theFermi level was considered—for example, in aWO3, transitions of d electronsbetween tungsten cations presenting unsaturated covalent bonds. In principle,the carrier localization at the sites may be either due to Anderson localization[146], polaron effects, or a combination of them.

The derivation in Ref. [145] starts with an electron-phonon system Hamil-tonian with three main terms—that is, the ones describing the phonon field,the electrons, and their interaction.

The Kubo formula is used to obtain the linear response of the optical con-ductivity due to a time-dependent sinusoidal electric field perturbation (theillumination). The obtained expression is not valid for ω → 0 in the case ofdisordered systems, but starts to hold in the beginning of the infrared regionof the electromagnetic spectrum. The general form of the optical conductivity,presented in equation 10 from Ref. [145], is given by

Re{σ(ω)}= e2Ns

h3ωTA sinh

(hω2kT

)∫dεdε ′

P(ε)P(ε ′)K(ε − ε ′)

4cosh( ε−εF

2kT

)cosh

(ε ′−εF2kT

) ,(3.85)

47

with Ns the concentration of the sites (where localization can occur), TA aparameter related to the average coupling between two sites (specifically, theaverage over every pair of sites of the square of the hopping integral times thesquare of the distance between them), T the temperature, ε the energy, εF theFermi level, and P(ε) the normalized energy distribution of the atomic levels(proportional to the DOS of the CB). The term K(ε −ε ′) is a time integral thatdepends on the energy difference between two given sites ε − ε ′, the tempera-ture, as well as the phonon frequencies and electron-phonon coupling for thedifferent wave vectors that describe the phonon field. Both ε and εF are givenwith respect to the maximum of the DOS. Equation (3.85) can be rearrangedin the form

Re{σ(ω)} =e2Ns

h3ωTA sinh

(hω2kT

)

×∫

dεdε ′ P(ε) f (ε)︸ ︷︷ ︸occupied

P(ε ′)[1− f (ε ′)]︸ ︷︷ ︸unoccupied

K(ε − ε ′)e(ε−ε ′)/2kT ,

(3.86)

with f (ε) = {1+ exp[(ε − εF)/kT ]}−1 the Fermi-Dirac distribution. Thus,the argument in the integral in Eq. (3.86) contains the product between theoccupied [P(ε) f (ε)] and the unoccupied {P(ε ′)[1 − f (ε ′)]} states. This isactually the core of the phenomenological model for the optical absorptionintroduced in Section 3.4.3—that is, the site-saturation theory.

Next, a simplified expression for K(ε) was obtained in Ref. [145]. In ad-dition, for convenience, P(ε) was approximated by a Gaussian function withthe form

P(ε) =1

Γ√

2πexp(−ε2/2Γ2), (3.87)

with Γ a quantity that describes the band width. The choice of P(ε) in Eq.(3.87) is of course advantageous for pursuing analytical calculations. How-ever, it could be a rough estimation of the DOS shape, and it would be inter-esting to use instead a DOS more closely connected with the real case—thatis, obtained either from experimental or theoretical methods. This is out of thescope of the present thesis but it would be compelling to try this approach.

Bryksin presents a solution to Eq. (3.85) giving a function that acquires acharacteristic form in three different frequency regions [145]. In this case, theexpression for the optical conductivity reads

48

Re{σ(ω)}=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

A1exp[− ε2

FΓ2 − (hω−4Ea)

2

16EaEvib

],

for hω < 4Ea(1−2εFEvib/Γ2)

A2exp[− ε2

F2Γ2 − (hω−4Ea−εF)

2

16EaEvib+2Γ2

],

for 4Ea(1−2εFEvib/Γ2)< hω < 4Ea(1+2εFEvib/Γ2)+2εF

A3exp[− (hω−4Ea)

2

16EaEvib+4Γ2

],

for 4Ea(1+2εFEvib/Γ2)+2εF < hω(3.88)

with A1, A2, and A3 multiplying factors. Also, Ea and Evib have the samemeaning as above. The high-temperature approximation is used in Ref. [145];that is, Evib ≈ kT . Close to room temperature kT ≈ 0.025 eV and a value ofhωlo ≈ 0.123 eV for polycrystalline WO3 is reported elsewhere [147]. Thus,in this case, hωlo > kT and the approximation Evib ≈ hωlo/2 seems more ap-propriate, as done in Ref. [40].

Equation (3.88) gives a line-shape with its maximum at hω ≈ 4Ea + εF,which is higher (by εF) than that of the two-site approximation [Eq. (3.83)]for E0 = 0. The Bryksin model [Eq. (3.88)] has provided good agreementswith the experimental optical conductivity of lithium-intercalated aWO3 [40].

3.4.2 Optical density and differential coloration efficiencyThe optical density OD(λ ,x)—as a function of the optical wavelength λ andthe intercalation level x—is defined as

OD(λ ,x) = dα(λ ,x), (3.89)

with d the film thickness and α(λ ,x) the absorption coefficient.The coloration efficiency is a highly relevant quantity in electrochromism

[24]. It corresponds to the optical density change per inserted charge q (givenhere per unit area) in an EC material or device. It is usually reported in liter-ature in two forms. First, as ΔOD/Δq [Fig. 3.9(a)], in terms of the changesof the optical density ΔOD and the inserted charge (per unit area) Δq betweena bleached state and a colored one—see, for example, Ref. [148]. Second,as the initial slope of the optical density as a function of inserted charge (perunit area) [Fig. 3.9(b)]—see, for instance, Ref. [149]. Alternatively, the dif-ferential coloration efficiency K(λ ,x), as depicted in Fig. 3.9(c), is definedas

K(λ ,x) = dOD(λ ,x)/dq, (3.90)

49

Figure 3.9. Coloration efficiency obtained by the difference between a bleached and acolored state (a), the initial slope (b), and the derivative (c) of the optical density ODas a function of the inserted charge q (expressed here per unit area).

which can also be expressed in the form

K(λ ,x) = (edNW)−1dOD(λ ,x)/dx, (3.91)

with the inserted charge per unit area q given by

q = edNWx, (3.92)

with NW the number density [cm−3] of the tungsten atoms in the aWO3 thinfilm.

The quantity K(λ ,x) can provide valuable information during the interca-lation process. However, to our knowledge, it has been used only in few ECstudies—see, for example Refs. [150, 151, 152, 153] in relation to aWO3. InPaper II, the differential coloration efficiency is used for studying the opticalabsorption in aWO3 for an extended intercalation range.

3.4.3 Site-saturation theoriesLet the optical density OD(λ ,x) be expressed as the linear superposition of thecontributions from different absorption processes belonging to a set B, thus

OD(λ ,x) = Σ j∈BOD j(λ ,x), (3.93)

with j an element of B, and OD j(λ ,x) its associated optical density.In the following, we will consider that the optical absorption in aWO3—for

photon energies small enough to disregard interband processes—is assignedmainly to intraband absorption due to electronic transitions between localizedstates (associated with tungsten sites), as explained in Section 3.4.1. In thisframework, Denesuk and Uhlmann [150] developed the site-saturation (SS)theory, which provides a phenomenological relation between the optical den-sity and the intercalation level. In this thesis, the case of lithium intercalation,

50

with x = Li/W, is used. For an equilibrium condition determined by the in-tercalation level, the SS theory assumes that the optical density is proportionalto the probability of transition at a random tungsten site. In this case, onlythe W5+ → W6+ transition is considered, which involves a site occupied byone electron (W5+) and an empty one (W6+). Thus, the SS theory accountsfor a maximum occupancy of one electron per tungsten site. As a result, itis mathematically defined up to x = 1. The optical density given by the SStheory ODSS(λ ,x) can be expressed as

ODSS(λ ,x) = ASS(λ )PSS(x), (3.94)

with ASS(λ ) a wavelength-dependent absorption strength, and PSS(x) theprobability of a transition at a random tungsten site, which has the form

PSS(x) = x(1− x) = x− x2. (3.95)

In this context, for a random site, the probability of being occupied withan electron is x, and that of a neighboring site to be empty is (1− x). Themultiplication of the last two expressions gives the transition probability inEq. (3.95).

A conceptual visualization of the SS theory in aWO3 is depicted in Fig.3.10(a). Here, the aWO3 film is represented by a collection of boxes, whichcan be either empty (W6+) or full (W5+). At x = 0, and x = 1 all the boxes areempty, and full, respectively. At low x, there are plenty of empty sites that canreceive an electron, but few electrons to perform a transition. The situation isthe opposite at x close to 1. Thus, a maximum in PSS(x) (and consequently inODSS(λ ,x) as well) is expected at a value in between these two cases. Thisis depicted in Fig. 3.10(b), which shows that PSS(x) peaks at x = 0.5 andvanishes at x = 0 and x = 1.

The extended site-saturation (ES) theory presented by Berggren et al. [35]takes into account W4+ sites in addition to the W5+ and W6+ ones. Accord-ingly, a maximum occupancy of two electrons per tungsten site is allowed,and the theory is mathematically defined up to x = 2. In this case, threetransitions could participate in the optical absorption. That is, W5+ → W6+,W4+ → W5+, and W4+ → W6+. They present probabilities of transitions at arandom tungsten site given respectively by [35]

PES56 (x) = x(1− x/2)3, (3.96)

PES45 (x) = (x3/4)(1− x/2), (3.97)

PES46 (x) = (x2/4)(1− x/2)2, (3.98)

51

Figure 3.10. (a) Schematic representation of aWO3 in the SS theory context. Thetungsten sites are depicted as either empty (white) or full (dark gray) boxes, whichcorrespond respectively to W6+ (having no electrons) and W5+ (containing one elec-tron). Probabilities of transition at a random tungsten site as a function of intercalationlevel derived from the SS (b) and the ES (c) theories. The latter includes W4+ sitesapart from those corresponding to W5+ and W6+. In (b), the curves x and (1− x) areoutlined in the plot as a reference, they correspond respectively to the probability of arandom site to be full and empty (in the SS theory).

whose dependence with the intercalation level is depicted in Fig. 3.10(c).All the curves vanish at x = 0 and x = 2. PES

56 (x) is an asymmetric functionwhich presents a maximum at x = 0.5—just like PSS(x). PES

45 (x) peaks atx = 1.5 and corresponds to PES

56 (x) mirrored around x = 1. Finally, PES46 (x) is

symmetric with respect to x = 1 and presents its maximum value at this point.According to Eq. (3.93), the optical density in the ES case can be expressed

as

ODES(λ ,x) = AES56 (λ )P

ES56 (x)+AES

45 (λ )PES45 (x)+AES

46 (λ )PES46 (x), (3.99)

with AES56 (λ ), AES

45 (λ ), and AES46 (λ ) the wavelength-dependent absorption

strengths of the respective transitions.The differential coloration efficiency related to the SS theory KSS(λ ,x) can

be calculated using the expression of the optical density from Eq. (3.94) inEq. (3.91), which gives

KSS(λ ,x) =CSS(λ )dPSS(x)/dx, (3.100)

with the prefactor CSS(λ ) = (edNW)−1ASS(λ ) and

52

dPSS(x)/dx = 1−2x. (3.101)

In a similar way, by combining Eqs. (3.99) and (3.91), the differential col-oration efficiency connected with the ES theory KES(λ ,x) can be expressedas

KES(λ ,x) = CES56 (λ )dPES

56 (x)/dx+CES45 (λ )dPES

45 (x)/dx

+CES46 (λ )dPES

46 (x)/dx, (3.102)

with

dPES56 (x)/dx = (1/2)(x−2)2(1/2− x), (3.103)

dPES45 (x)/dx = (1/2)x2(3/2− x), (3.104)

dPES46 (x)/dx = (1/4)x(x−1)(x−2), (3.105)

and the respective weighting factors of the form CESi j (λ )= (edNW)−1AES

i j (λ ),with {i j}= {56,45,46}.

The derivatives of the probabilities of transitions given in Eqs. (3.101),(3.103), (3.104), and (3.105) are depicted in Fig. 3.11 as a function of inter-calation level. dPSS(x)/dx is a straight line, with a slope equal to −2, thattakes the value of 1 at x = 0 and is equal to zero at x = 0.5. dPES

56 (x)/dx isalso equal to 1 at x = 0 and vanishes at x = 0.5. In addition, it decreaseswith increasing x up to x = 1, where it presents a minimum. Thereafter, it in-creases with increasing x until it vanishes at x = 2. dPES

45 (x)/dx is equal to thedPES

56 (x)/dx curve mirrored around x = 1 and the horizontal axis. dPES46 (x)/dx

is antisymmetric with respect to x = 1, and vanishes at x equal to 0, 1, and 2.Moreover, it is positive, and negative for 0 < x < 1, and 1 < x < 2, respec-tively. It is worth mentioning that dPSS(x)/dx and dPES

56 (x)/dx describe thesame transition—that is, W5+ → W6+. Actually, they are relatively similar inthe range 0 < x < 0.5—note that both change sign at x = 0.5. However, theydiffer considerably from each other for x > 0.5.

53

Figure 3.11. Derivatives of the probabilities of transitions at a random tungsten site asa function of intercalation level. The expressions derived from the SS [Eq. (3.101)]and the ES [Eqs. (3.103), (3.104), and (3.105)] theories are shown in the plot. Repro-duced from Paper II, with the permission of AIP Publishing.

54

4. Electrode preparation and characterization

In this chapter, the synthesis and characterization of the aWO3 thin films aredescribed.

4.1 Thin film deposition by reactive DC magnetronsputtering

The aWO3 thin films were prepared by reactive DC magnetron sputtering[154], which is a physical vapor deposition technique. This was done em-ploying a Balzers UTT 400 unit. The schematic of the configuration used inthis work is depicted in Fig. 4.1. The process takes place in a vacuum cham-ber initially evacuated at about 10−7 Torr (after a baking step for 10 h at about112 ◦C). The general principle is outlined in the following.

An Ar flow is injected into the vacuum chamber. A high negative voltagedifference is applied between the target (the cathode) and the anode. By doingthis, electrons are emitted from the cathode and these ionize the Ar atoms,generating a plasma formed by Ar+ ions, the secondary electrons producedfrom the ionization, and those from the cathode. An applied magnetic fieldconfines the plasma in the vicinity of the target, see Fig. 4.1(b). The previouslymentioned voltage difference accelerates the Ar+ ions toward the cathode,and they knock out target atoms, which will be deposited onto the substrates.When a reactive gas, like O2, is injected into the chamber during the depositionprocess, it can react with the target atoms. In this way, thin oxide films can becoated onto the substrates.

The target consisted of a 5-cm-diameter W disc (with a purity of 99.95%).It is positioned—with its normal tilted by an angle of 50◦ with respect to thenegative vertical axis—at a distance of 13 cm from the substrate holder (whichlies below the target), as depicted in Fig. 4.1. In order to remove surface con-tamination from the target, it was initially pre-sputtered in an Ar atmosphere(99.9997%-pure) for a period of about 5 min at a constant discharge power of240 W.

The substrates for opto-electrochemical measurements were ∼ 1× 2 cm2

glass pieces (1-mm-thick) pre-coated with In2O3:Sn (ITO; sheet resistance of15 Ω/sq and thickness of ∼ 150 nm) and those for Rutherford backscatteringspectrometry analysis were 1× 1 cm2 glassy carbon plates. These were pre-viously cleaned by ultrasonication in acetone (10 min), ethanol (10 min), anddistilled water (10 min) and, subsequently, they were dried by nitrogen gas.

55

Figure 4.1. Schematic of the deposition system (a) and close-up of the target depictingthe sputtering process (b). Magnetic field lines are portrayed in (b) as curved arrows.

During the deposition, a region along one of the borders of each substrate wasmasked by a Kapton tape. The purpose of this is twofold. First, to create a stepbetween the covered and uncovered portions of the substrate for measuring thethickness of the deposited coating. Second, to leave an exposed area on theITO to be used as an electrical contact in the electrochemical measurements.

The deposition was performed using a constant discharge power of 240 Win a mixed atmosphere of Ar (99.9997%-pure) and O2 (99.998%-pure) at30 mTorr. Ar and O2 were injected into the vacuum chamber at a flow of 50and 22 mL/min, respectively. Throughout this process, no intentional heatingwas applied to the substrates, and the holder was rotated to ensure the filmthickness homogeneity.

The schematic of the resulting aWO3 WE is depicted in Fig. 4.2.

Figure 4.2. Schematic geometry of the aWO3 WE depicted for the cross-sectional (a)and top view (b). In (b), the points labelled by 1 (left-hand side), 2 (center), and 3(right-hand side) correspond to the approximate places at which the thickness of theaWO3 thin films was measured.

56

4.2 Stylus profilometryThe thickness of each thin film was measured by a Bruker DetakXT profilome-ter with vertical resolution of 4 Å. It employs a mechanical stylus with a di-amond tip which contacts the sample surface. By a linear scanning betweentwo points on the surface, the information about its topography can be re-trieved. The step (thin film thickness) between the uncovered ITO and theaWO3 film can be obtained by setting a scan that covers these two regions.Thin film thickness d values of about 300 nm were obtained. These resultedfrom the average of three measurements at different positions on the edge ofthe aWO3 film, see Fig. 4.2(b)—that is, at the left-hand side, the center, andthe right-hand side of the edge.

4.3 X-ray diffractionX-ray diffraction (XRD) is a physical characterization technique suitable forstudying the structures of thin films [155]. Crystals can be thought of as an ar-ray of parallel planes with spacing distance a, as depicted in Fig. 4.3. Let θ bethe incident angle (measured with respect to the planes) of a collimated beamof X-rays with wavelength λxr. Thus, for each family of planes, a constructiveinterference occurs for an incident angle given by Bragg’s law [156]

mλxr = 2asinθm, (4.1)

with m an integer. A X-ray diffraction pattern can be obtained by scanningthrough different incident angles and measuring the intensity of the reflectedX-rays. It shows intense peaks at the places that satisfy the constructive in-terference condition given in Eq. (4.1). Conversely, no clear sharp peaks areobserved in amorphous materials (without long-range order).

Figure 4.3. Schematic of X-ray diffraction from planes of atoms with spacing a.

57

In this work, XRD spectra of the aWO3 WEs were measured at grazingincidence angle using a Siemens D5000 diffractometer employing Cu Kα ra-diation with a wavelength of 1.5406 Å. A typical pattern is depicted in Fig.4.4. No sharp reflections were observed apart from those related to the ITOlayer. Thus, the aWO3 thin films were considered as X-ray amorphous.

Figure 4.4. Typical X-ray pattern of an as-deposited amorphous tungsten oxide work-ing electrode (aWO3/ITO). The spectrum corresponding to the glass pre-coated withITO is shown as a reference.

4.4 Rutherford backscattering spectrometryRutherford backscattering spectrometry (RBS) is an ion beam technique thatcan provide, for example, information about the elemental composition anddepth profile in thin films [157]. A schematic of this technique is depicted inFig. 4.5. A incident beam of light ions (like 4He+ at an energy of 2 MeV)impinges on the sample (the target) and these ions are elastically scattered bythe atomic nuclei of the target. By setting a detector at a fixed scattering angleΘ, a RBS spectrum can be obtained—that is, the number of scattered ions(yield) as a function of their energy. Let E0 be the energy of the incident ions,and, for simplicity, it will be assumed that the thin film portrayed in Fig. 4.5presents an uniform composition consisting of two elements with masses M1and M2 (with M1 > M2). In addition, let the substrate be composed of onlyone element with mass Ms (with Ms < M2 < M1).

58

Figure 4.5. Schematic of the RBS technique applied to a sample (target) consisting ofa thin film—with homogeneous distribution of elements with masses M1 and M2 (withM1 > M2)—on top of a monoatomic substrate with associated atomic mass Ms (withMs < M2 < M1). An incident beam of ions, with energy E0, impinges on the targetand the detector is positioned at the scattering angle Θ. In this way, a RBS spectrumcan be obtained (that is, a yield vs. energy plot). The features associated with eachmass are depicted in the plot, including the position of their upper energy edge (K1E0,K2E0, and KsE0). For M1 and M2, the width of their signals ΔE is associated with thethin film thickness d. For each element, the energies corresponding to the scatteringevents taking place at the surface and at the interface between the thin film and thesubstrate are indicated by the arrows.

For each kind of atom in the target, the highest possible energy of its relatedsignal corresponds to a scattering at the surface and this energy edge is locatedat KjE0—with Kj the kinematic factor of the element j, which satisfies theinequality Kj < 1 and increases with the atomic mass. Thus, the higher theatomic mass, the higher the energy edge. This can be used to identify orseparate the signal corresponding to each element in the RBS spectrum.

The height of each signal is proportional to the areal atomic concentrationand the scattering cross-section. For RBS measurements, the latter is generallyclose or almost equal to the Rutherford cross-section—that is, proportionalto the square of the atomic number of the corresponding target atom. Thiscan provide quantitative information about the atomic composition. When theincident ions enter the target, they experience a frictional force characterizedby the stopping power in the sample. As a result, a scattering event originated

59

from a certain depth of the target will be detected at a smaller energy thanthat of the energy edge KjE0 of the respective target atom. This can giveinformation about the atomic depth profile in the sample. Thus, the widthΔE of the energy features—associated with the elements in the thin film—isrelated to the thin film thickness d.

The height of each signal is proportional to the areal atomic concentrationand the scattering cross-section. For RBS measurements, the latter is generallyclose or almost equal to the Rutherford cross-section—that is, proportionalto the square of the atomic number of the corresponding target atom. Thiscan provide quantitative information about the atomic composition. When theincident ions enter the target, they experience a frictional force characterizedby the stopping power in the sample. As a result, a scattering event originatedfrom a certain depth of the target will be detected at a smaller energy thanthat of the energy edge KjE0 of the respective target atom. This can giveinformation about the atomic depth profile in the sample. Thus, the widthΔE of the energy features—associated with the elements in the thin film—isrelated to the thin film thickness d.

The RBS measurements were done at the Tandem Laboratory at UppsalaUniversity employing 2 MeV 4He ions, which were detected at a backscatter-ing angle of 170◦. The elemental areal densities of W (ΓW) and O (ΓO) wereobtained—using the SIMNRA program [158]—from the fitting of the differ-ent RBS spectra to a model of the aWO3/C system (accounting for the thinfilm and the glassy carbon substrate). The tungsten NW, and oxygen NO num-ber densities were respectively calculated by NW = ΓW/d, and NO = ΓO/d. Ineach case, the film density ρ was obtained using the equation

ρ =1

NA(MWNW +MONO), (4.2)

with NA the Avogadro constant, and the molar masses of tungsten as wellas oxygen given respectively by MW and MO. In Eq. (4.2), only the main ele-ments of the aWO3 thin films are considered—that is, W and O. A more de-tailed analysis (or other techniques) would be needed in order to provide accu-rate information about adventitious contamination from lighter elements—like,H and C—or from very small concentrations of elements with atomic masshigher than that of O. However, this is out of the scope of this work becausethe main quantities of interest in this thesis are NW (the number density of thetungsten atoms) and the O/W ratio. A typical RBS spectrum of the aWO3 thinfilms is depicted in Fig. 4.6. The results for different batches used in this thesisare summarized in Table 4.4. The corresponding samples were slightly over-stoichiometric (WOy with y > 3). As mentioned before, further studies wouldbe needed to determine how much of this oxygen belongs either to the tung-sten oxide network or to unintentional contamination. The latter could consistof either water, organic compounds, or OH groups (primarily at external andpore surfaces of the aWO3 thin films).

60

Figure 4.6. Typical experimental (red circles) and simulated (solid line) RBS spectraof the aWO3 thin films. The features corresponding to W, O, and C are indicated

Table 4.1. Elemental density and ratio as well as the density of the aWO3 thin films.The data were obtained from a combination of film thickness and RBS measurements.

Batch NW (cm−3) NO (cm−3) O/W ρ (gcm−3)

Batch-1 (1.242±0.002)×1022 (3.86±0.04)×1022 3.11±0.03 4.81±0.01

Batch-2 (1.218±0.002)×1022 (3.81±0.04)×1022 3.13±0.03 4.73±0.01

Batch-3 (1.170±0.002)×1022 (3.80±0.04)×1022 3.24±0.04 4.58±0.01

In Paper II, samples from batch-1 were used in the CP and SECIS-1 exper-iments, and a sample from batch-2 was used in the SECIS-2 experiment. InPaper III, a sample from batch-3 was used in the Mott-Schottky analysis, andthe CP experiment was the same as that presented in Paper II (with a samplefrom batch-1). In Paper IV, the data used in the analyses were obtained fromthe SECIS-1 experiment in Paper II (which employed a sample from batch-1).

61

5. Electrochemical and in situ opticalmeasurements

In this chapter, the experimental setup and the electrochemical techniques usedduring the combined electrochemical and optical measurements are described.Note that the principles of EIS and CIS were already introduced in Sections3.2 and 3.3, respectively. Thus, the latter topics are not covered in this chapter.

5.1 Experimental setupThe schematic of the experimental setup used for electrochemical measure-ments—including the simultaneous electrochemical and color impedance spec-troscopy (SECIS) technique—is depicted in Fig. 5.1(a). It is thoroughly de-scribed in Paper I. The electrochemical measurements were performed em-ploying a three-electrode setup in an argon-filled glove box (H2O level <0.6 ppm). As mentioned before, unless specified otherwise, an aWO3 sam-ple acted as the working electrode WE and lithium foils were used both as thereference (RE) and counter (CE) electrodes. The electrodes were immersedin an electrolyte, which was 1 M LiClO4 dissolved in propylene carbonate.The latter was contained in a quartz cuvette. The combination comprising theelectrodes (WE, CE, and RE), the electrolyte, and the quartz cuvette is here-inafter denoted as the electrochemical cell. The potential difference betweenthe WE and RE (V ), as well as the current flow between the CE and WE (I)were measured by an electrochemical interface (SI-1286, Solartron).

The SI-1286 can control either the current of the electrochemical cell orthe potential of the WE. The former was used in the chronopotentiometrytechnique. The latter was employed in SECIS measurements as well as inother electrochemical techniques—that is, linear sweep voltammetry, poten-tiostatic treatment, and cyclic voltammetry. The previously mentioned tech-niques, apart from SECIS, are described in Section 5.2.

A frequency response analyzer (FRA; SI-1260, Solartron) was used to-gether with the electrochemical interface during the SECIS measurements. Inthese cases, the FRA provided the sinusoidal excitation voltage to the SI-1286.In addition, the V and I signals (measured by the SI-1286) were fed into theFRA inputs.

The light source was a fiber-coupled light-emitting diode (LED), which wascontrolled by a LED driver (DC4100, Thorlabs) in constant current mode.LEDs with peak wavelengths at 470 nm (M470F3, Thorlabs), 530 nm (M530F2,Thorlabs), and 810 nm (M810F2, Thorlabs) were utilized in this thesis. Theoptical fiber (M28L01, Thorlabs)—with a core diameter of 400 μm and nu-merical aperture of 0.39—was coupled to a collimating system (74-ACH,Ocean Optics), which was based on a pair of achromatic lenses (74-ACR,Ocean Optics).

62

Figure 5.1. Schematics of the combined electrochemical and optical experimen-tal setups. The configuration shown in (a) was employed for the experimental se-quences—that is, a series of measurements using electrochemical techniques—thatincluded simultaneous electrochemical and color impedance (SECIS) measurements,and the arrangement depicted in (b) was used otherwise. In each case, the elementswithin the red dashed contour were placed inside an argon-filled glove box (H2O level< 0.6 ppm).

63

Unless specified otherwise, the collimated light beam crossed the cuvette,the WE, and the electrolyte before reaching a photodetector (PDA-100A-EC,Thorlabs). The light beam arrived at the WE surface at a normal angle of in-cidence. The photodetector gave an output voltage signal V pd(t) proportionalto the intensity of the measured transmitted light. It was monitored during thewhole experimental sequence by a digital multimeter (34401A, HP/Agilent).The transmittance T (t) was calculated as follows

T (t) =V pd(t)/V pdB , (5.1)

with V pdB the measured value at a bleached state of the WE (for example,

at zero intercalation level in an aWO3 WE), which was chosen as the 100%-level of transmittance. During the SECIS measurements, V pd(t) was fed intothe FRA current input via a 150 Ω resistor—a prior calibration procedure wasdone for converting the FRA current output to the quantities associated withV pd(t). Throughout the frequency-dependent measurements, V pd(t) can beexpressed as

V pd(t) = 〈V pd〉+V pdA (ω)sin[ωt +φop(ω)], (5.2)

with φop(ω) defined in Eq. (3.6). In addition, 〈V pd〉, and V pdA (ω) corre-

spond to the stationary equilibrium value, and the amplitude of the oscillatorypart of V pd(t), respectively. Thus, the quantities 〈T 〉, and TA(ω) from Eq.(3.6) can be calculated, respectively, by

〈T 〉 = 〈V pd〉/V pdB , (5.3)

TA(ω) = V pdA (ω)/V pd

B . (5.4)

The FRA gives the amplitude and phase (with respect to a reference) of theoscillatory parts of the voltage, current, and transmittance. However, it doesnot provide the stationary equilibrium values of these signals. As a result,the additional monitoring of V pd(t) with the digital multimeter was necessaryto have access to the information of 〈V pd〉. Finally, a computer was used tocontrol the experiments, as well as for the acquisition, display, and analysis ofthe data.

If an experimental sequence did not include SECIS, the setup depicted inFig. 5.1(b) was employed instead of that shown in Fig. 5.1(a). The only differ-ences of the array outlined in Fig. 5.1(b) with respect to the configuration por-trayed in Fig. 5.1(a) are the following. A BioLogic SP-200 potentiostat wasused instead of the SI-1286, no extra FRA was utilized, and the output signalof the photodetector was directly fed into one of the analog inputs available inthe SP-200 potentiostat.

64

5.2 Electrochemical techniquesThe non-frequency-dependent electrochemical techniques used in this thesisare schematically described in Table 5.2—a more detailed explanation can befound, for example, in Ref. [56]. In all cases, the optical transmittance of theWE can act as a response signal that could be measured during the experi-ments. Furthermore, the expressions of the quantities derived from chronopo-tentiometry (CP) measurements are given—that is, the intercalation ratio, thedifferential capacitance, and the differential coloration efficiency.

Table 5.1. Schematic description of the (non-frequency-dependent) electrochemicaltechniques used in this thesis.

Technique Excitation ResponseOpen-circuit

potential (OCP) None (free WE) Potential, andtransmittance

Potentiostatictreatment

Current andtransmittance

Linear sweepvoltammetry

Current andtransmittance

Cyclic voltammetry(CV)

Current andtransmittance

Chronopotentiometry(CP)

Potential andtransmittance

65

5.2.1 Open circuit potential (OCP)The open-circuit potential (OCP) technique measures, during a certain periodof time, the potential of the free WE—that is, with neither current flowingthrough the electrochemical cell nor applied potential to the WE.

5.2.2 Linear sweep voltammetryThe linear sweep voltammetry technique applies a linear time variation to thepotential of the WE between two given potential values. The absolute value ofthe rate of change of the potential (|ΔV/Δt|) is set during the experiment.

5.2.3 Cyclic voltammetry (CV)In the cyclic voltammetry (CV) technique, a symmetrical triangular potentialwave is applied to the WE for a determined number of cycles and the resultingcurrent is measured. The main parameters in a CV measurement are the upperand lower bounds of the potential wave as well as the absolute value of therate of change of the potential (|ΔV/Δt|). Other parameters that are importantto consider in a CV are the initial and final potential values in addition to thedirection of the initial linear scan.

The typical optical transmittance and current density responses of the firstthree cycles of a CV measurement on an aWO3 WE are depicted in Fig. 5.2.In this case, negative, and positive current density values are associated withlithium intercalation, and deintercalation, respectively. For all the samplesstudied in this thesis, the first CV cycle differed considerably from the subse-quent ones. On the other hand, the second and third CV cycles were nearlyidentical to each other.

5.2.4 Chronopotentiometry (CP)In the chronopotentiometry (CP) technique, a constant current flow (betweenthe CE and the WE) is applied during a certain time interval and the potentialresponse is measured.

Assuming that the WE is at zero intercalation level at the beginning of theCP measurement (defined as the time t = 0) and that the current is applied insuch a way that the relevant ions are intercalated into the WE, the intercalationlevel x(t) at time t can be obtained by

x(t) =−(zedANW)−1∫ t

0dt ′I(t ′), (5.5)

with z the valence of the ion participating in the intercalation (for lithium,z = 1), d the film thickness, e the elementary charge, A the active area of

66

Figure 5.2. Typical current density (black curves) and optical transmittance (redcurves) responses from a CV measurement at 10 mV/s for an aWO3 WE. Cycles1 (solid lines), 2 (dashed lines), and 3 (dotted lines) are shown. The lower and up-per potential bounds were 2 and 4 V vs. Li/Li+, respectively. The CV measurementstarted and finished at the initial OCP value of the WE (3.81 V vs. Li/Li+)—that is,the one recorded when the WE was just immersed into the electrolyte. Cycle 1 startedjust after the measurement of the initial OCP, and it began with a scan toward the2 V vs. Li/Li+ edge. The optical transmittance was measured at the optical wave-length of 810 nm. The scanning directions are indicated by arrows.

the electrode, NW the tungsten number density (see Section 4.4), and I(t) thecurrent applied during the CP measurement. The minus sing on the right-hand side of Eq. (5.5) comes from the sign convention used throughout thisthesis—that is, I(t)< 0 for a net ion intercalation. The formulas related to thecalculation of x(t) in Paper II as well as in the supplementary information ofPaper III (without this minus sign) are valid when just the magnitude of theintercalation current is used.

The differential capacitance Cd of the aWO3 WE (obtained from CP) isgiven by Cd = −dq/dV , with q corresponding to the total integrated chargedivided by the active area of the WE. It can be expressed as q(t) = (edNW)x(t).Thus, using Eq. (2.2), we can write

Cd =−(edNW)dxdV

= (edNW)EDOS. (5.6)

67

The quantity Cd was employed in Paper III.The optical density OD(λ ,x)—defined in Eq. (3.89)—can be approximated

(from experimental quantities) by [159]

OD(λ ,x) = ln(

1−Rt(λ ,x)Tt(λ ,x)

), (5.7)

with Rt(λ ,x), and Tt(λ ,x) the total optical reflectance, and transmittance (atoptical wavelength λ and intercalation level x), respectively.

Equation (5.7) is intended for strong absorbing films. However, as pointedout in Ref. [159], it can also be used for weak absorbing films in cases pre-senting refractive indexes of the film, and substrate within the ranges of 1.35-2.5, and 1.5-1.7, respectively. Note that, for similar samples to those used inthis thesis, the refractive index (at λ = 810 nm) of the glass substrate was ofabout 1.6 and that of the ITO layer was of about 1.9 [160]. In addition, fromthe experimental real and imaginary parts of the dielectric function of aWO3presented in Ref. [40] (also employing similar samples to those used in thiswork), an associated refractive index value (at λ = 810 nm) of about 2 can beobtained (at intercalation levels from 0 up to 0.27), which is similar to that ofthe ITO layer.

Combining Eqs. (3.90) and (5.7), the differencial coloration efficiencyK(λ ,x) can be expressed as

K(λ ,x) =− 1Tt(λ ,x)

dTt(λ ,x)dq

− 1[1−Rt(λ ,x)]

dRt(λ ,x)dq

. (5.8)

For a case in which the change of the transmittance with the intercalation ismuch higher than that of the reflectance, K(λ ,x) can be written as

K(λ ,x) =− 1Tt(λ ,x)

dTt(λ ,x)dq

, (5.9)

In terms of the intercalation level x, Eq. (5.9) takes the form

K(λ ,x) =−(edNW)−1 1Tt(λ ,x)

dTt(λ ,x)dx

. (5.10)

Note that

dTt(λ ,x)Tt(λ ,x)

=dT (λ ,x)T (λ ,x)

, (5.11)

with the definition of T (λ ,x) given in Eq. (5.1). Thus, if convenient,T (λ ,x) could be used in Eqs. (5.9) and (5.10) instead of Tt(λ ,x).

68

6. Summary of important results

In this chapter, a small summary of the important results from the appendedpapers is given. In addition, some unpublished results related to Paper III arepresented

6.1 Paper IIn this work, an experimental setup for simultaneous electrochemical and colorimpedance spectroscopy (SECIS) measurements on EC thin films was pre-sented and characterized. Moreover, we dealt with the theoretical aspects con-cerning these techniques as well as with the different experimental quantitiesthat can be derived from them—that is, the complex impedance, capacitance,optical capacitance, and the differential coloration efficiency. An interestingfeature of this work is that the frequency-dependent background noise of theoptical signal was estimated, which was considered as the limiting factor inthe SECIS measurements. To our knowledge, this was the first time this wasdone in the context of color impedance spectroscopy (CIS).

In addition, illustrative test measurements were made on an amorphoustungsten oxide thin film (aWO3) at the bias potential value of 2.6 V vs. Li/Li+.Three different optical wavelengths were used here—that is, 470 nm, 530 nm,and 810 nm. In this case, the respective complex optical capacitance spec-tra presented the same shape (within experimental error) and only differed inamplitude.

Note that, for aWO3, the signal-to-noise ratio (SNR) is low at high fre-quencies. With the knowledge of the background noise, we estimated the fre-quency limit at which the optical signal could be retrieved. In order to do this,we only accepted data points with SNR of the optical signal greater than 5,and the previously mentioned limit was chosen as the highest frequency at andbelow which all the data points satisfied this requirement. Finally, a strategyfor shifting this limit toward higher frequencies was introduced—that is, theso-called variable-amplitude method.

6.2 Paper IIThis paper is focused on the differential coloration efficiency (DCE) of aWO3thin films. It was obtained both from quasi-equilibrium and dynamic mea-surements, which were acquired using chronopotentiometry (CP) and SECIS,

69

respectively. In both cases, a wide range of intercalation levels was probed(with their corresponding bias potentials). An optical wavelength of 810 nmwas used in this work.

Apart from a region at high potentials (low intercalation levels), the quasi-equilibrium DCE followed in an excellent manner the extended site-saturation(ES) model introduced in Section 3.4.3—which accounts for transitions in-volving W4+ sites, contrary to the simpler site-saturation (SS) model that con-siders only W5+ and W6+ sites. In this case, the optical response was dom-inated by the W5+ → W6+ transition. Even though the contributions fromthe W4+ → W5+ and W4+ → W6+ transitions were not directly observed inthe quasi-equilibrium DCE, the influence of the W4+ sites was clear. This isbecause the simpler SS model could not reproduce properly the experimentalresponse.

The dynamic DCE followed a trend similar to that of the quasi-equilibriumone. However, the former presented marked differences in comparison withthe latter. For example, the dynamic DCE did not show a change of sign ata critical intercalation level as observed in the quasi-equilibrium case. Thereason is not clear at the moment.

Finally, at low bias potentials (below about 2 V vs. Li/Li+), deviationsfrom the steady-state condition were clearly noticed during the SECIS mea-surements. This could affect both the optical and electrical responses. It couldbe related to irreversible trapping of Li+ or to the formation of lithium oxides.Nevertheless, more studies are required to clarify these aspects.

6.3 Paper IIIThis paper deals with the details of the quasi-equilibrium DCE of an aWO3thin film at low intercalation levels (or high bias potentials). To this end, theDCE and the differential capacitance obtained from CP were analyzed. Notethat the differential capacitance—defined in Eq. (5.6)—is proportional to theelectrochemical density of states. In addition, by means of Mott-Schottky(MS) analysis, employing EIS spectra from an aWO3 thin film, the conduc-tion band (CB) edge of aWO3 was estimated to be at about 3.11 V vs. Li/Li+.The electronic states below and above this bias potential were assigned to theCB and the band gap, respectively. A band tail was identified in the differentialcapacitance at bias potentials higher than 3.11 V vs. Li/Li+. It is worth men-tioning that if a Helmholtz layer capacitance CH of 20 μFcm−2 (10 μFcm−2)were assumed (see Section 3.2.4), the corrected CB edge would be located atabout 3.15 V vs. Li/Li+ (3.17 V vs. Li/Li+).

Interestingly, it was observed that the ES model could reproduce appropri-ately the DCE for CB states. On the other hand, there were clear differencesbetween the experimental DCE and the ES model for band gap states. Thus, itwas hypothesized that the average distribution of occupied and available sites

70

(as discussed in the ES theory) governs the quasi-equilibrium optical responseof aWO3 for CB states. This would mean that the other quantities relatedto small-polaron absorption (see Section 3.4.1) are relatively constant for CBstates. Conversely, they may vary significantly for band gap states.

For completeness, the Nyquist plot of the experimental complex electricalcapacitance C(ω) spectra used in the MS analysis are depicted in Fig. 6.1.In general, the higher the bias potential, the smaller the complex capacitancevalues. Each spectrum consists of relaxation contributions from the doublelayer, adsorption, and diffusion effects.

At high bias potentials (at and above 3.4 V vs. Li/Li+) [see Fig. 6.1(d)],the double layer and adsorption relaxations are merged in the small semicircle,and the diffusion one is observed at low frequencies. The respective semicir-cles (or arcs) corresponding to these processes tend to increase their size withdecreasing bias potential. However, the adsorption and diffusion relaxationsgrow faster than the double layer one, which presumably reaches a saturation(that is, it becomes nearly constant below a certain bias potential). In fact,the double layer arc gets hidden beneath the other contributions at and belowabout 3.3 V vs. Li/Li+. In this bias potential region, the parameters relatedto the double layer were difficult to obtain from EIS fitting. Actually, for biaspotentials below 3.1 V vs. Li/Li+, Qdl and ndl were assumed to be equal totheir respective values at 3.1 V vs. Li/Li+ (see Table 6.1). Having the previ-ous assumption in mind (which may introduce uncertainties), the fitting errorsof Rct were small in these cases (below about 6%).

71

Figure 6.1. Panel (a) depicts the Nyquist representation of the experimental complexcapacitance spectra used in the Mott-Schottky analysis in Paper III. They were mea-sured on an aWO3 thin film at bias potential values (with respect to Li/Li+) from 3.7down to 2.8 V. Panels (b), (c), and (d) are successive magnifications of panel (a). Asshown in panel (a), the closer a curve is to the origin, the higher its corresponding biaspotential is. The direction of increasing frequencies ω is outlined in the plots.

72

Table 6.1 presents the parameters obtained from the fittings of the equiv-alent circuit shown in Fig. 3.6(a) to the EIS spectra used in Fig. 6.1. Here,the high-frequency impedance from Fig. 3.6(a) consisted only of a simpleresistance. Rhf remained almost constant around 100 Ω. Qdl, Qad, Qm, andcm (in general) increased with decreasing bias potential. As explained before,the low bias potential values of Qdl and ndl corresponded to an estimation.Rct oscillated between about 39 and 59 Ω down to about 3.3 V vs. Li/Li+.Subsequently, it decreased significantly for lower bias potentials—presentingvalues of the order of 1 Ω, whose uncertainty was high due to the small sizeof the double layer feature in comparison with the adsorption and diffusionones. At large, Rad decreased with decreasing bias potential. ndl as well as nadwere close to 1 at high bias potentials and decreased down to about 0.75 atlow bias potentials. nm showed values between about 0.3 and 0.7. The param-eters that describe the diffusion are Qm, cm, and nm. Even though the fittingat 3.7 V vs. Li/Li+ was highly satisfactory (see Fig. 6.2), the fitting errors ofthe diffusion parameters Qm and cm were extremely high in this case—that is,above 600%. However, for lower bias potentials, the fitting errors associatedwith Qm, cm, and nm were always low—that is, smaller than about 10%.

In addition to the complex capacitance plots, the Nyquist representation ofthe experimental EIS spectra used in the MS analysis (including their fittings)are depicted in Figs. 6.2-6.11. In general, the fittings to the experimentalspectra were good.

73

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74

Figure 6.2. Panel (a) shows the Nyquist plot representation of the impedance spectrummeasured on the aWO3 thin film at the bias potential of 3.7 V (with respect to Li/Li+).It was employed in the MS analysis in Paper III. The symbols and the solid line corre-spond to the experimental data and the fitting (using the parameters presented in Table6.1), respectively. For illustrative purposes, the RadZCPEad(ω) (blue dashed curve) andRdlZCPEdl(ω) (red dash-dotted curve) parallel combinations are depicted. ZCPEad(ω)and ZCPEdl(ω) are defined in Section 3.2.3. Panels (b), (c) portray magnifications ofpanel (a). The frequency ω increases at the direction indicated in the plots.

Figure 6.3. Panel (a) shows the Nyquist plot representation of the impedance spectrummeasured on the aWO3 thin film at the bias potential of 3.6 V (with respect to Li/Li+).It was employed in the MS analysis in Paper III. The symbols and the solid line corre-spond to the experimental data and the fitting (using the parameters presented in Table6.1), respectively. For illustrative purposes, the RadZCPEad(ω) (blue dashed curve) andRdlZCPEdl(ω) (red dash-dotted curve) parallel combinations are depicted. ZCPEad(ω)and ZCPEdl(ω) are defined in Section 3.2.3. Panels (b), (c) portray magnifications ofpanel (a). The frequency ω increases at the direction indicated in the plots.

75

Figure 6.4. Panel (a) shows the Nyquist plot representation of the impedance spectrummeasured on the aWO3 thin film at the bias potential of 3.5 V (with respect to Li/Li+).It was employed in the MS analysis in Paper III. The symbols and the solid line corre-spond to the experimental data and the fitting (using the parameters presented in Table6.1), respectively. For illustrative purposes, the RadZCPEad(ω) (blue dashed curve) andRdlZCPEdl(ω) (red dash-dotted curve) parallel combinations are depicted. ZCPEad(ω)and ZCPEdl(ω) are defined in Section 3.2.3. Panels (b), (c) portray magnifications ofpanel (a). The frequency ω increases at the direction indicated in the plots.

Figure 6.5. Panel (a) shows the Nyquist plot representation of the impedance spectrummeasured on the aWO3 thin film at the bias potential of 3.4 V (with respect to Li/Li+).It was employed in the MS analysis in Paper III. The symbols and the solid line corre-spond to the experimental data and the fitting (using the parameters presented in Table6.1), respectively. For illustrative purposes, the RadZCPEad(ω) (blue dashed curve) andRdlZCPEdl(ω) (red dash-dotted curve) parallel combinations are depicted. ZCPEad(ω)and ZCPEdl(ω) are defined in Section 3.2.3. Panels (b), (c) portray magnifications ofpanel (a). The frequency ω increases at the direction indicated in the plots.

76

Figure 6.6. Panel (a) shows the Nyquist plot representation of the impedance spectrummeasured on the aWO3 thin film at the bias potential of 3.3 V (with respect to Li/Li+).It was employed in the MS analysis in Paper III. The symbols and the solid line corre-spond to the experimental data and the fitting (using the parameters presented in Table6.1), respectively. For illustrative purposes, the RadZCPEad(ω) (blue dashed curve) andRdlZCPEdl(ω) (red dash-dotted curve) parallel combinations are depicted. ZCPEad(ω)and ZCPEdl(ω) are defined in Section 3.2.3. Panel (b) portrays a magnification of panel(a). The frequency ω increases at the direction indicated in the plots.

Figure 6.7. Panel (a) shows the Nyquist plot representation of the impedance spectrummeasured on the aWO3 thin film at the bias potential of 3.2 V (with respect to Li/Li+).It was employed in the MS analysis in Paper III. The symbols and the solid line corre-spond to the experimental data and the fitting (using the parameters presented in Table6.1), respectively. For illustrative purposes, the RadZCPEad(ω) (blue dashed curve) andRdlZCPEdl(ω) (red dash-dotted curve) parallel combinations are depicted. ZCPEad(ω)and ZCPEdl(ω) are defined in Section 3.2.3. Panels (b), (c) portray magnifications ofpanel (a). The frequency ω increases at the direction indicated in the plots.

77

Figure 6.8. Panel (a) shows the Nyquist plot representation of the impedance spectrummeasured on the aWO3 thin film at the bias potential of 3.1 V (with respect to Li/Li+).It was employed in the MS analysis in Paper III. The symbols and the solid line corre-spond to the experimental data and the fitting (using the parameters presented in Table6.1), respectively. For illustrative purposes, the RadZCPEad(ω) (blue dashed curve) andRdlZCPEdl(ω) (red dash-dotted curve) parallel combinations are depicted. ZCPEad(ω)and ZCPEdl(ω) are defined in Section 3.2.3. Panels (b), (c) portray magnifications ofpanel (a). The frequency ω increases at the direction indicated in the plots.

Figure 6.9. Panel (a) shows the Nyquist plot representation of the impedance spectrummeasured on the aWO3 thin film at the bias potential of 3.0 V (with respect to Li/Li+).It was employed in the MS analysis in Paper III. The symbols and the solid line corre-spond to the experimental data and the fitting (using the parameters presented in Table6.1), respectively. For illustrative purposes, the RadZCPEad(ω) (blue dashed curve) andRdlZCPEdl(ω) (red dash-dotted curve) parallel combinations are depicted. ZCPEad(ω)and ZCPEdl(ω) are defined in Section 3.2.3. Panels (b), (c) portray magnifications ofpanel (a). The frequency ω increases at the direction indicated in the plots.

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Figure 6.10. Panel (a) shows the Nyquist plot representation of the impedance spec-trum measured on the aWO3 thin film at the bias potential of 2.9 V (with respect toLi/Li+). It was employed in the MS analysis in Paper III. The symbols and the solidline correspond to the experimental data and the fitting (using the parameters pre-sented in Table 6.1), respectively. For illustrative purposes, the RadZCPEad(ω) (bluedashed curve) and RdlZCPEdl(ω) (red dash-dotted curve) parallel combinations are de-picted. ZCPEad(ω) and ZCPEdl(ω) are defined in Section 3.2.3. Panels (b), (c) portraymagnifications of panel (a). The frequency ω increases at the direction indicated inthe plots.

Figure 6.11. Panel (a) shows the Nyquist plot representation of the impedance spec-trum measured on the aWO3 thin film at the bias potential of 2.8 V (with respect toLi/Li+). It was employed in the MS analysis in Paper III. The symbols and the solidline correspond to the experimental data and the fitting (using the parameters pre-sented in Table 6.1), respectively. For illustrative purposes, the RadZCPEad(ω) (bluedashed curve) and RdlZCPEdl(ω) (red dash-dotted curve) parallel combinations are de-picted. ZCPEad(ω) and ZCPEdl(ω) are defined in Section 3.2.3. Panels (b), (c) portraymagnifications of panel (a). The frequency ω increases at the direction indicated inthe plots.

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The fitting parameters shown in Table 6.1 were used to calculate the re-lated effective double layer, adsorption, and chemical capacitances (per unitarea)—which are denoted by Cdl [Eq. (3.60)], Cad [Eq. (3.61)], and Cch [3.57],respectively. The results are presented in Table 6.2, which also includes theassociated chemical diffusion coefficients—designated as Dch and defined inEq. (3.59).

Cdl increased with decreasing bias potential down to 3.2 V vs. Li/Li+,and a subsequent decrease and stabilization was observed at lower bias po-tentials. Cad remained relatively stable down to about 3.4 V vs. Li/Li+. Be-low this bias potential, Cad increased abruptly down to 3.2 V vs. Li/Li+ andstayed somewhat constant at lower bias potentials. Apart from the value at3.7 V vs. Li/Li+, Cch increased as the bias potential decreased. Dch pre-sented values between about 10−13 and 10−9 s−1 cm−2 from 3.7 down to3.4 V vs. Li/Li+. For lower bias potentials, Dch remained of the order of10−10 s−1 cm−2.

The figures of the quantities Cad, Cch, Dch shown in Table 6.2 are consistentwith those given in Table II of Paper IV. That is, they are of the same orderof magnitude at similar bias potentials. The biggest difference is observed inDch at 3.15 V vs. Li/Li+. The latter was about 10−11 s−1 cm−2 in Paper IV,which is one order of magnitude smaller than Dch at 3.2 and 3.1 V vs. Li/Li+

in Table 6.2. Note that Cdl was not considered in Paper IV.

Table 6.2. Effective double layer Cdl, adsorption Cad, and chemical Cch capacitances(per unit area) as well as the chemical diffusion coefficient Dch at the different biaspotentials. These quantities were calculated using the fitting parameters from Table6.1, which were employed in the Mott-Schottky analysis presented in Paper III.

Biaspotential

(Vvs.Li/Li+)

Cdl(μFcm−2)

Cad(μFcm−2)

Cch(μFcm−2)

Dch(s−1 cm−2)

3.7 1.25 2.38 25.0 3.46×10−13

3.6 1.26 2.42 14.5 1.36×10−11

3.5 1.55 1.20 40.1 2.90×10−12

3.4 1.99 0.89 93.4 1.16×10−9

3.3 3.15 13.5 519 1.71×10−10

3.2 19.5 417 748 2.32×10−10

3.1 11.2 431 1844 2.82×10−10

3.0 8.19 399 3257 3.74×10−10

2.9 8.70 388 4702 4.72×10−10

2.8 8.94 396 6413 3.68×10−10

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6.4 Paper IVThis paper presents an analysis of the dispersion of the complex electricaland optical capacitances (see Section 3.3) measured on an aWO3 thin filmat various bias potentials (each of them associated with a given intercalationlevel). Particularly, the complex Faradaic optical and electrical capacitanceswere compared to each other—these quantities are denoted by GF

op(ω) [Eq.(3.74)] and CF(ω) [Eq. (3.66)], respectively. Note that the latter comprisesadsorption and diffusion effects.

At high bias potentials (low intercalation levels), CF(ω) and GFop(ω) were

almost identical up to a multiplicative constant—only small discrepancies werenoticed at high frequencies. In this case, the complex Faradaic optical capac-itance seemed to contain appreciable contributions from both the adsorptionand the diffusion processes.

At low bias potentials (high intercalation levels) CF(ω) and GFop(ω) differed

significantly from each other. In addition, only the diffusion process appearedto give an important contribution to the complex Faradaic optical capacitance.Moreover, at low bias potentials, the relaxation of the complex Faradaic op-tical capacitance was delayed with respect to that of the complex Faradaicelectrical capacitance. This could be due to the difficulty of finding a site withan available neighbor (for an electronic transition) at high intercalation levels.

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7. Concluding remarks and outlook

This thesis provided a systematic study of the electrochromic (EC) responseof amorphous tungsten oxide (aWO3) thin films for a wide intercalation levelrange, which is difficult to find elsewhere. In situ simultaneous electrochemi-cal and optical measurements present challenges in terms of the experimentalsetup. However, they can give valuable insights into the dynamic and quasi-equilibrium effects in EC systems.

As observed for aWO3, the dynamic EC response consists of a combina-tion of various processes. In this context, frequency-dependent techniques canbe employed to single out the different contributions in order to better under-stand their relevance at diverse conditions. Nevertheless, this kind of studiesrequires a careful and detailed data analysis. In addition, important assump-tions have to be made regarding the nature of the relevant processes, and theyrequire further validation. Concerning the dynamic EC effects in aWO3, itseems that the electrical and optical responses are to a large extent diffusionlimited, but there is an extra delay in the optical response with respect to theelectrical one—which is more noticeable at high intercalation levels.

Regarding the quasi-equilibrium response of aWO3, it was interesting toobserve that a relatively simple phenomenological model—namely, the ex-tended site-saturation theory—could reproduce appropriately the experimen-tal EC response of aWO3 for an extended intercalation level range. This re-sult strengthens the notion that, in this case, the coloration is mainly due toelectronic transitions between localized states, which is also connected withthe intervalence-transfer and polaron absorption theories. Furthermore, it wasconcluded that the W4+ sites play a role in the coloration of aWO3. In ad-dition, it was observed that the optical response due to electrons in band gapstates differed significantly from that assigned to electrons in conduction bandstates—contrary to the latter, the former could not be successfully modeled bythe extended site-saturation theory.

This work developed new methodologies (and extended already existingones) for the study of EC systems. The approach presented here can be em-ployed to obtain interesting and valuable information about the influence ofdifferent experimental parameters in the response of EC systems; for exam-ple, optical wavelength, film thickness, and material composition. It is partic-ularly appealing to perform studies similar to those presented in this thesis inthe following situations. First, employing other intercalation ions apart fromlithium. Second, studying various EC materials; for example, oxides basedon Ni, V, Ti, Mo, and Ir. Moreover, frequency-dependent techniques can helpto separate surface from bulk effects, and this characteristic makes these tech-niques especially helpful for improving our understanding of the colorationmechanisms in EC systems.

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8. Sammanfattning på svenska

Denna avhandling behandlar elektrokromism i tunna filmer av amorf vol-framoxid.

Vad är elektrokromism och varför är det viktigt?

Elektrokromism är förmågan hos vissa material att genomgå reversibla ochjusterbara förändringar av sina optiska egenskaper på grund av en pålagdelektrisk spänning. Denna effekt kan användas i energieffektiv teknik. Ettrelevant exempel på detta är smarta fönster som kan justera sina egenskaperdynamiskt för att kontrollera instrålningen av synligt ljus och värmeflödenberoende på klimatförhållandena. Detta kan spara mycket energi och relat-erade kostnader för uppvärmning i kalla klimat och kylning i varma klimat.Dessutom kan smarta fönster hjälpa till att förbättra komforten i en byggnadgenom att blockera oönskad solinstrålning. Som sagt är elektrokromism ettspeciellt forskningsområde som kombinerar grundläggande egenskaper hosmaterial och tekniska tillämpningar med positiv inverkan på miljön och vårtsamhälle i allmänhet.

Varför studerades amorfa volframoxidfilmer?

Volframoxid har haft en speciell plats i forskning om elektrokromism allt ifrånbörjan. Det är det mest studerade elektrokroma materialet och finns i nästanalla kommersiella tillämpningar som för närvarande finns tillgängliga. Dennaavhandling syftar till att utveckla nya metoder för studier av elektrokroma ma-terial. Således valdes amorf volframoxid för denna fallstudie för att tillämpaoch testa dessa nya metoder, eftersom dess elektrokroma egenskaper karak-teriserats i stor utsträckning. Trots detta upptäcktes nya resultat om infärgn-ingsmekanismerna i amorf volframoxid i denna avhandling. Varför amorf?Detta beror på att amorf volframoxid uppvisar en avsevärd optisk moduleringför synliga optiska våglängder (synligt ljus), i motsats till den kristallina vol-framoxiden. För tillämpningar i fönster är vi intresserade av det första fallet.

Vad är känt om elektrokromismen i amorf volframoxid?

En elektrokrom elektrod framställs i allmänhet genom att belägga en tunnfilm av det elektrokroma materialet på en transparent ledare (kom ihåg attvi skulle vilja se genom fönstret). Filmens framsida sätts i kontakt med enelektrolyt innehållande små joner (litiumjoner användes i denna avhandling)och elektroden är ansluten till en yttre krets. Således har filmen på ena sidanen jonreservoar (elektrolyten) och på den andra sidan en elektronreservoar

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(den genomskinliga ledaren). Från den yttre kretsen kan en spänningsskill-nad appliceras på ett sådant sätt att både jonerna och elektronerna vandrarin i filmen. Observera att när en jon införs (interkaleras) måste en elektronockså komma in i det elektrokroma materialet eftersom laddningsneutralitetenmåste bibehållas i filmen. Dessa elektroner är de som ger upphov till de op-tiska förändringarna. Under normala förhållanden färgas amorf volframoxidnär jonerna är interkalerade och bleknar när jonerna vandrar ut ur filmen. Detfinns material som beter sig på motsatt sätt. Genom att kontrollera spännin-gen i den externa kretsen, kan den elektrokroma elektroden justeras så att denuppnår en given infärgningsnivå, och den kan till och med cyklas mellan ettfärgat och ett blekt (mer transparent) tillstånd.

Som nämnts tidigare består mekanismen för elektrokromism av två huvu-daspekter, en elektrisk excitation och den resulterande optiska förändringen.Med tanke på att elektrokromismen i amorf volframoxid beror på inblandningav litiumjoner och införande av externa elektroner i filmen uppstår följandefrågor, och denna avhandling försöker ge några svar på dem.

• Vad händer med de optiska förändringarna när vi fortsätter att interkaleralitiumjoner över den normalt använda nivån?

• Är den optiska förändringen fördröjd med avseende på den elektriskaexcitationen, och hur beror detta på interkaleringsnivån?

• Verkar infärgningsprocesserna endast i filmens inre, eller sker de ocksåpå ytan?

Hur undersöktes dessa frågor?

I detta arbete mättes de elektriska och optiska egenskaperna hos den elek-trokroma elektroden samtidigt. På detta sätt kunde de optiska förändringarnavid litiuminterkalering studeras. Dessutom utformades och karakteriseradesen experimentell uppställning för att mäta de optiska och elektriska frekvenss-varen för elektrokroma elektroder samtidigt. Med hjälp av dessa frekvens-beroende tekniker undersöktes fördröjningen mellan de elektriska och optiskasvaren på den pålagda spänningen. Man kunde också separera bidrag till in-färgningen från processer vid filmens yta och dess inre.

Vad fann vi?

Som förväntat ökade den optiska absorptionen vid låga interkaleringsnivåerav litium . Efter en kritisk interkaleringsnivå observerades emellertid det mot-satta beteendet - filmen blev mer transparent när mer litium interkalerades.Intressant nog kunde det observerade elektrokroma beteendet hos den amorfavolframoxidfilmen reproduceras med en enkel modell som beaktade hoppningav elektroner mellan fyllda och tomma tillstånd lokaliserade på angränsande

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volframatomer. Vid låga interkaleringsnivåer var den optiska förändringennästan fullständigt synkroniserad med den elektriska excitationen. Emellertidkom de optiska förändringarna senare än den elektriska signalen då interkaler-ingsnivån ökade. Dessa resultat betonar att även om de optiska och elektriskaaspekterna av de fysikaliska fenomenen är nära förbundna så har de olika ur-sprung. Färgningen ägde rum både från processer vid filmens yta och dessinre. Vid låga interkaleringsnivåer observerades båda fallen tydligt. Å andrasidan följde färgningen vid höga interkaleringsnivåer huvudsakligen processeri filmens inre.

Vad är nästa steg?

Metoden som presenteras i denna avhandling kan användas för att studerainfärgningsmekanismerna för andra relevanta material. Dessutom kan deraselektrokroma egenskaper undersökas i detalj genom att testa olika experi-mentella förhållanden såsom olika interkalerade joner, olika optiska våglängder(färger), andra materialsammansättningar och så vidare. Elektrokrom teknikkan bidra till att förbättra byggnaders energieffektivitet. Även om det redanfinns kommersiella produkter baserade på elektrokromism, finns det fortfarandegrundläggande frågor att ta itu med avseende mekanismerna för infärgning.Förhoppningsvis kan mer hållbara och effektiva elektrokroma produkter ut-formas genom att förbättra vår förståelse för dessa fenomen.

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Acknowledgments

I am deeply grateful to my supervisor Gunnar Niklasson for his support,valuable pieces of advice, guidance, patience, and for giving me the opportu-nity to pursuit my doctoral studies under his supervision. In addition to yourscientific sharpness, I really admire and appreciate your kindness. I thank myco-supervisors Tomas Edvinsson and Lars Österlund for interesting conver-sations and for being an example of how to become a good researcher. I alsothank Claes-Göran Granqvist for his sharp comments and pieces of advice,there is always something to learn from such a remarkable researcher like you.

Daniel Primetzhofer, Marcos Moro, Mauricio Sortica, and the staff atthe Tandem Laboratory at Uppsala University are acknowledged for assistancewith RBS measurements.

The Anna Maria Lundin foundation in Smålands Nation in Uppsala isacknowledged for financing my participations in conferences abroad.

I would like to thank the members at the Division of Solid State Physics forproviding me with a pleasant atmosphere during my PhD studies. Thank youIlknur Pehlivan for your team spirit and José Montero Amenedo for beingsuch a good office mate and friend.

I am grateful for the scholarship received from the University of Costa

Rica to complete the PhD program at Uppsala University. I am sincerelyindebted to William Vargas Castro for believing in me, and for guiding mein the decision of coming to Uppsala University, thank you very much. I alsothank Esteban Avendaño Soto for the help at the beginning of this process.

I would like to express my sincere and deep gratitude to my mentors dur-ing my Masters (and further): Paul Urbach, Aurèle Adam, and Florian

Bociort from TU Delft (The Netherlands); Nicolas Dubreuil from the In-stitut d’Optique Graduate School (France); as well as Juan Ariel Levenson

and Kamel Bencheikh from the Laboratoire de Photonique et de Nanostruc-tures (CNRS, France)—now at C2N (CNRS/Université Paris-Saclay, France).Thank you very much.

I am extremely grateful to my friends Diana Campos, Oscar Mario Cor-

rales, Pablo Lizano, Roberto Vargas, and Esteban Pérez who helped mewhen I needed them the most.

Throughout this journey I met wonderful people whom I will always keep ina especial place in my heart. This list would be extremely large and I honestlydo not want to risk missing anyone. Thank you all of you who stood by meboth in the good and difficult times.

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To my family. Mín♥ (Yasmín), my love, we did this together. I havehad the huge blessing of being by your side and it has been much better thanwhat I could have ever dreamed of. Take my hand and let us keep walkingtogether. Te amo mi amor. Gracias por amarme siempre. Mami, papi ySilvia, los amo. Muchas gracias por apoyarme siempre, esto también es deustedes. Muchas gracias a la familia de Yasmín (y mía) por todo su apoyo,cariño, y amor—gracias doña Xenia, don Jorge, Adrián y Susana.

Lo más importante, gracias a Dios por todo, sobre todo por darnos la fort-aleza para seguir adelante aún en los momentos difíciles.

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