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Review ArticleElectrochemical Impedance Spectra of Dye-Sensitized SolarCells: Fundamentals and Spreadsheet Calculation

Subrata Sarker,1 A. J. Saleh Ahammad,2 Hyun Woo Seo,1 and Dong Min Kim1

1 Department of Materials Science and Engineering, Hongik University, Sejong 339-701, Republic of Korea2Department of Chemistry, Jagannath University, Dhaka 1100, Bangladesh

Correspondence should be addressed to Dong Min Kim; dmkim@hongik.ac.kr

Received 14 July 2014; Accepted 29 August 2014; Published 27 November 2014

Academic Editor: Rajaram S. Mane

Copyright 2014 Subrata Sarker et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Electrochemical impedance spectroscopy (EIS) is one of the most important tools to elucidate the charge transfer and transportprocesses in various electrochemical systems including dye-sensitized solar cells (DSSCs). Even though there are many books andreports on EIS, it is often very difficult to explain the EIS spectra of DSSCs. Understanding EIS through calculating EIS spectraon spreadsheet can be a powerful approach as the user, without having any programming knowledge, can go through each step ofcalculation on a spreadsheet and get instant feedback by visualizing the calculated results or plot on the same spreadsheet. Here, abrief account of the EIS of DSSCs is given with fundamental aspects and their spreadsheet calculation. The review should help oneto develop a basic understanding about EIS of DSSCs through interacting with spreadsheet.

1. Introduction

Impedance spectroscopy is a powerful method for charact-erizing the electrical properties of materials and their inter-faces [14]. When applied to an electrochemical system, itis often termed as electrochemical impedance spectroscopy(EIS); examples of such systems are electrochemical cellssuch as fuel cells, rechargeable batteries, corrosion, anddye-sensitized solar cells (DSSCs) [2, 3]. Recently, EIS hasbecome an essential tool for characterizing DSSCs [517].Typically, a dye-sensitized solar cell (DSSC) is composed ofa ruthenium dye loaded mesoporous film of nanocrystallineTiO2on fluorine-doped tin oxide (FTO) glass substrate as

photoelectrode (PE), an iodide/triiodide (I/I3

) based redoxelectrolyte solution, and a Pt coated FTO glass substrate ascounter electrode (CE) [14, 15, 1820]. Analysis of EIS spec-trum of a DSSC provides information about several impor-tant charge transport, transfer, and accumulation processesin the cell. These are (i) charge transport due to electrondiffusion through TiO

2and ionic diffusion in the electrolyte

solution; (ii) charge transfer due to electron back reaction atthe FTO/electrolyte interface and recombination at theTiO2/electrolyte interface and the regeneration of the redox

species at CE/electrolyte interfaces; and (iii) charging of the

capacitive elements in the cells including the interfaces, theconduction band, and surface states of the porous network ofTiO2[2, 7, 15, 2123]. Even though there are many books and

reports on EIS, it is often very difficult to explain the EIS spec-tra of DSSCs. Moreover, the details of EIS calculation alwaysremain under several layers of programming abstraction andthus cannot be accessed by the user.

Calculating EIS spectra on spreadsheet can be a powerfulapproach as the user, without having any programming know-ledge, can go through each step of calculation on a spread-sheet and get instant feedback by visualizing the calculatedresults or plot on the same spreadsheet. From our experienceof learning EIS of DSSCs from scratch, we found that it wasfar more easy and fun to learn EIS through spreadsheetcalculation than trying to decipher the abstract ideas of EISon books or papers.

Here, a brief account of the general aspects of EIS isgiven with mathematical expressions and their calculationon spreadsheet (see the interactiveMicrosoftExcel 2010 file inthe Supplementary Material available online at http://dx.doi.org/10.1155/2014/851705). Most importantly, we summarizethe fundamental charge transfer processes that take place inworking DSSCs and how those processes give rise to EISspectra.

Hindawi Publishing CorporationInternational Journal of PhotoenergyVolume 2014, Article ID 851705, 17 pageshttp://dx.doi.org/10.1155/2014/851705

2 International Journal of Photoenergy

2. Electrochemical Impedance Spectroscopy

2.1. Fundamentals. Let us begin with the notion of an idealresistor having resistance . According to Ohms law, current()flowing through the resistor and voltage () across the twoterminals of the resistor is expressed by the following relation:

= . (1)

On the other hand, impedance is amore general concept thanresistance because it involves phase difference [4]. Duringimpedancemeasurement, a small-amplitudemodulated volt-age (, ) is applied over a wide range of frequency ( =/2) and the corresponding currents (, ) are recorded,or vice versa. The resultant impedance () of the system iscalculated as [1, 2, 4]

() = (, )

(, )(2)

provided that (, ) is small enough to be linear with respectto (, ), or vice versa. At a certain frequency , (, )may have different amplitude and phase than that of (, )depending on the nature of the charge transfer processes inthe system that results in impedance of the correspondingcharge transfer process. When the frequency of the appliedperturbation is very low, the system is said to be driven withdc current and the impedance of the system coincides withits dc resistance (dc), that is, impedance with zero phasedifference [2, 10]:

(0) = (0)

(0)= dc. (3)

It is to be noted here that there are other response quantitiesrelated to impedance such as admittance (), modulusfunction (), and complex dielectric constant or dielectricpermittivity () [2, 4].

In complex number, a small-amplitude AC voltage canbe described as (, ) =

0exp() and response to this

potential is the AC current (, ) = 0exp{( )}, where

is the phase difference between (, ) and (, ) and =1. Therefore, (2) can be written as [1, 4]

() =0

0

exp () . (4)

Again, (4) can be rewritten in terms of magnitude (0) as

[1, 4]

() = 0exp () . (5)

Applying Eulers relationship and replacing 0with ||, (5)

can be expressed as [1, 4]

() = || (cos + sin ) . (6)

In general, impedance is expressed as [1, 4]

() = Re + Im (7)

or more simply as [1, 4]

= + , (8)

where Re = = || cos and Im =

= || sin are thereal and the imaginary parts of the impedance, respectively.The real and imaginary parts of the impedance are related tothe phase angle as

= tan1 (

) (9)

and the magnitude || as

|| = 2 + 2. (10)

EIS data can be displayed in differentways. In the complexplane, is plotted against . The complex plane plots areoften termed as Nyquist plots [4]. In Bode plot, both log ||and are plotted against log. Sometimes, it is helpful to plotlog against log [1, 2].

In the frequency domain, current-voltage relations canbe rearranged as (2). If a purely sinusoidal voltage (, ) =0sin() is applied across a resistor with resistance then

the current that flows through the resistor will be (, ) =(, )/ =

0sin()/, which can be written as (, ) =

0sin(). So, the impedance of the resistor,

(), is [1]

() =

(, )

(, )= . (11)

In this case, the applied voltage and the resultant current arein phase. If the voltage is applied to a capacitor having capaci-tance then the resultant current is (, ) = (, )/ =0cos(), where = / and = . The above

expression for the current passing through the capacitor canbe written as (, ) =

0cos( /2) or (, ) =

0sin(), where

0=

0. The impedance of the capacitor,

(), is thus [1]

() =

(, )

(, )=

1

, (12)

where 1/ or in complex notation 1/ is the reactanceof a capacitor and /2 is the phase difference. According tothe above description, reactance for any electrical element canbe deduced using fundamental relation between current andvoltage for that element as summarized in Table 1 [2, 4].

Analysis of EIS data is central to the study of EIS of anelectrochemical system. An overview of the system of interestfacilitates the translation of the charge transfer, transport, andaccumulation processes in the system to an electrical circuitcomposed of a lump of series and parallel combination ofresistors, capacitors, inductors, and so forth. The equivalentmodel is used to deduce the physically meaningful propertiesof the system. Any equivalent circuit model can be con-structed using Kirchoff s rules [1, 2]. For example, if two ele-ments are in series then the current passing through them arethe same and if two elements are in parallel then the voltagesacross them are the same.

International Journal of Photoenergy 3

Table 1: Basic electrical elements and their current-voltage relation.

Component Symbol Fundamental relation Impedance, ()

Resistor =

Capacitor =

1

Constant phase element

=

1

()

Inductor =

Figure 1: Screenshots of the spreadsheet calculation of impedance of a capacitor (dl) with capacitance of 100F at frequencies 10mHz and100 kHz showing formulas and corresponding results in MS Excel.

In spreadsheet, a complex number can be constructedusing built-in function and the number can be operated withall the basic mathematical operators available in the spread-sheet as functions for complex numbers. Figure 1 showssuch calculation implemented for impedance of a capac