Chemical Sensors: Simulation and Modeling Volume 2: Conductometric-Type Sensors

Type Sensors

• Phenomenological

• Approachesused

the surroundings

for chemical sensing.

About the e

Hisresearchactivities

MoldovanAcademies

CHEMICAL

Avolumeinthe

v

Contents

Preface xiiiabout the editor xviicontributors xix

1 numerical simulation of electrical resPonses to Gases in advanced structures 1

A. Šetkus

1 Introduction 1

2 Analytic and Numeric Modeling 32.1 Basic Equations 32.2 Analytical Approaches 72.3 Numerical Simulations 152.4 Verification of Models 22

3 Resistive Sensors 283.1 Introductory Remarks 283.2 Polycrystalline Films 293.3 Nanostructured Films 343.4 Conductive Polymer Layers 383.5 Molecular Structures 40

4 Concluding Comments 43

References 44

2 co-adsorPtion Processes and Quantum mechanical modelinG of Gas-sensinG effects 51

J.-J. Velasco-Vélez

1 Introduction 51

2 Solid–Gas Interaction 55

vi • contents

2.1 Adsorption 552.2 Chemisorption 572.3 Electronic Transitions in Chemisorption 652.4 Chemisorption in Equilibrium, “Wolkenstein” Isotherm 672.5 Reaction Time 712.6 Charge Transfer Model (CTM) 73

3 Co-adsorption 793.1 Quantum Model 793.2 Statistical Model 813.3 Adsorption Time 83

4 Discussion 84

5 Summary 90

6 Nomenclature 92

Dedication 93

Acknowledgment 93

References 94

3 nanosensors: a Platform to model the sensinG mechanisms in metal oxides 97

F. Hernandez-RamirezJ. D. PradesA. Cirera

1 Introduction 97

2 Toward a Better Description of Gas-Sensing Mechanisms in Metal Oxides: Oxygen Diffusion in Tin Dioxide Nanowires 100

2.1 Description of Oxygen Sensing Using Diffusion 1052.2 Summary 107

3 Toward a Systematic Understanding of Photo-Activated Gas Sensors 108

3.1 Experimental Background 1093.2 Theoretical Model of the Photo-Activated Response to

Oxidizing Gases (NO2) 1123.3 Comparison with Experiments 1163.4 Other Target Gases 1193.5 Summary 119

4 Conclusions 120

Acknowledgments 120

References 120

contents • vii

4 surface state models for conductance resPonse of metal oxide Gas sensors durinG thermal transients 127

A. FortM. MugnainiS. RocchiV. Vignoli

1 Introduction 127

2 Surface-State–Based Models of Resistive Chemical Sensors 1292.1 Depleted Surface 1352.2 Enhanced Surface 141

3 Building a Chemical-Physical Sensor Model: From the Chemistry to the Resistance Variations 143

3.1 The Mechanism of Conduction in the Film: Effect of the Film Structure 143

3.2 Selection of a Model for Surface Potential Barrier Height as a Function of the Surface Charge: Solution of the Poisson Equation 146

3.3 Selection of a Model for the Evolution of the Surface Charge as a Function of the Surface Chemical Reactions 151

4 Surface State–Based Models for Chemical Resistive Sensors: Different Assumptions and Points of View 155

5 Developing a Treatable Gray Model from the Physical-Chemical Model 156

5.1 The Intrinsic Model 1575.2 The Extrinsic Model: Contributions from Oxygen and

Reducing Gas 1595.3 Effects of Water Vapor 161

6 Conclusions 168

Nomenclature 169

References 170

5 conductance transient analyses of metal oxide Gas sensors on the examPle of sPinel ferrite Gas sensors 177

K. Mukherjee S. B. Majumder

1 Introduction 177

2 Salient Features of Gas–Solid Interaction during Gas Sensing 178

3 Experimental 181

viii • contents

4 Modeling the Conductance Transients during Response and Recovery 182

4.1 Derivation of Response and Recovery Conductance Transients Based on Langmuir Adsorption Isotherm 184

4.2 Nonlinear Fitting of Response and Recovery Transients 1874.3 Variation of Response and Recovery Time Constants with

Sensor Operating Temperature 1894.4 Variation of the Estimated Fitted Parameters with Test Gas

Concentration: Addressing the Selectivity Issue 1935 Characteristic Features Observed in Resistance Transients 196

5.1 Investigations on Irreversible and Reversible Gas Sensing in Oxide Gas Sensors 198

5.2 Periodic Undulation of the Resistance Transients during Response and Recovery 211

5.3 Spikelike Features in Resistance Transients 2186 Summary and Conclusions 222

7 Appendix 2237.1 Solution of Eq. (5.36) 2237.2 Solution of Eq. (5.42) 224

8 Nomenclature 226

Acknowledgment 228

References 228

6 model of thermal transient resPonse of semiconductor Gas sensors 233

Akira Fujimoto

1 Introduction 233

2 Improvement in Selectivity of the Semiconductor Gas Sensor Using Transient Response 234

3 Model of Thermal Transient Response of Semiconductor Gas Sensors 236

3.1 Transient Response of Semiconductor Gas Sensors 2363.2 Thermal Transient Response of Semiconductor Gas Sensors 2373.3 Physical and Chemical Processes in the Semiconductor

Gas Sensor Under Transient Response 2384 Modeling of Gas Sensor Processes 239

4.1 Heat Conduction Processes 2394.2 Chemical Reaction Processes 2424.3 Diffusion Processes 2434.4 Sensor Output 244

contents • ix

5 Calculation Methods 2455.1 Heat Conduction 2455.2 Gas Concentrations on the Sensor Surface 246

6 Calculated Transient Responses of Gas Sensors 2476.1 Temperature Change on the Sensor Surface 2476.2 Concentration Change of Substance in the Vicinity of the

Sensor Surface 2496.3 Comparison to Experimental Results 250

7 Application of the Model of Transient Response 2517.1 Transient Responses Under Heating with Various Waveforms 2517.2 Activation Energy Dependence of Transient Response 253

8 Conclusions 257

References 258

7 exPerimental investiGation and modelinG of Gas-sensinG effect in mixed metal oxide nanocomPosites 261

L. I. TrakhtenbergG. N. GerasimovV. F. GromovM. A. KozhushnerO. J. Ilegbusi

1 Introduction 261

2 Types of Mixed Metal Oxides 263

3 Synthesis of Metal Oxide Nanocomposites 264

4 Charge Transfer Processes and Conductivity 265

5 Conductivity Mechanism 267

6 Sensor Properties 269

7 Mechanism of Sensor Effect 2727.1 Sensors Based on Single Nanofibers 2727.2 Polycrystalline Sensors 275

8 Modeling of the Sensory Effect for Reduced Gases 2838.1 Qualitative Discussion of the Sensory Mechanism 2838.2 Equilibrium Electronic Characteristics Of SnO2 2848.3 Sensor Response 285

9 Conclusions 290

Acknowledgment 291

References 291

x • contents

8 the influence of Water vaPor on the Gas-sensinG Phenomenon of tin dioxide–based Gas sensors 297

R. G. Pavelko

1 Introduction 297

2 Direct Water Effects on Tin Dioxide–Based Gas Sensors 2992.1 Undoped SnO2 2992.2 Doped SnO2 307

3 Indirect Water Effects on Tin Dioxide–Based Gas Sensors 3103.1 Reducing Gases 3113.2 Oxidizing Gases 322

4 Phenomenological Model 323

5 Conclusions 330

Acknowledgments 330

References 330

9 comPutational desiGn of chemical nanosensors: transition metal–doPed sinGle-Walled carbon nanotubes 339

Duncan J. MowbrayJuan María García-LastraIker Larraza ArocenaÁngel RubioKristian S. ThygesenKarsten W. Jacobsen

1 Introduction 339

2 TM-Doped SWNTs as Nanosensors 342

3 Density Functional Theory 346

4 Kinetic Modeling 351

5 Nonequilibrium Green’s Function Methodology 3555.1 Divacancy II 3585.2 Divacancy I 3615.3 Monovacancy 3635.4 Target and Background Molecules 364

6 Sensing Property 369

7 Conclusions 372

Acknowledgments 373

References 373

contents • xi

10 al-doPed GraPhene for ultrasensitive Gas detection 379Z. M. AoQ. JiangS. Li

1 Emerging Graphene-Based Gas Sensors 3791.1 The Role of Aluminum Doping in Sensing Applications 380

2 Aluminum-Doped Graphene for CO Detection 3812.1 Sensitivity Enhancement of CO Detection in Aluminum-

Doped Graphene 3812.2 Effect of Electric Field on CO Detection 3872.3 Effect of Temperature on CO Detection 393

3 Aluminum-Doped Graphene for Formaldehyde Detection 3993.1 Adsorption Enhancement with Aluminum Doping 3993.2 Variation of Electronic Properties Induced by Adsorption 402

4 Aluminum-Doped Graphene for Detection of HF Molecules 4044.1 Adsorption Enhancement of Aluminum-Doped Graphene 4064.2 Adsorption Enhancement Mechanism 4104.3 Effect of Electric Field on Adsorption 410

5 Conclusion and Future Challenges 411

Acknowledgments 413

References 413

11 Physics-based modelinG of sno2 Gas sensors With field-effect transistor structure 419

P. Andrei L. L. Fields A. J. Soares R. J. Perry Y. Cheng P. Xiong J. P. Zheng

1 Introduction 419

2 Physics-Based Modeling of the Nanobelts 421

3 Model Calibration 425

4 Analytical Model for Nanobelt Sensors 4304.1 Case 1: Nanobelt with Ohmic Contacts in the Presence of

Hydrogen 4314.2 Case 2: Nanobelt with Ohmic Contacts in the Presence of

Oxygen 435

xii • contents

4.3 Case 3: Nanobelt with Schottky Contacts in the Presence of Oxygen 439

5 Conclusion 440

Appendix: Fabrication and Experimental Data 441

References 443

12 modelinG and simulation of nanoWire-based field-effect biosensors 447

S. BaumgartnerM. Vasicek C. Heitzinger

1 Introduction 447

2 Homogenization 450

3 The Biofunctionalized Boundary Layer 4523.1 The Site-Dissociation Model 4533.2 Screening and Biomolecules 4543.3 Summary 460

4 The Current Through the Nanowire Transducer 4614.1 The Drift-Diffusion-Poisson System 4614.2 Self-Consistent Simulations of Sensor Systems 462

5 Summary 464

Acknowledgment 465

References 465

index 471

xiii

PrefaCe

This series, Chemical Sensors: Simulation and Modeling, is the perfect comple-ment to Momentum Press’s six-volume reference series, Chemical Sensors: Fundamentals of Sensing Materials and Chemical Sensors: Comprehensive Sensor Technologies, which present detailed information about materials, technologies, fabrication, and applications of various devices for chemical sensing. Chemical sensors are integral to the automation of myriad industrial processes and every-day monitoring of such activities as public safety, engine performance, medical therapeutics, and many more.

Despite the large number of chemical sensors already on the market, selec-tion and design of a suitable sensor for a new application is a difficult task for the design engineer. Careful selection of the sensing material, sensor platform, technology of synthesis or deposition of sensitive materials, appropriate coatings and membranes, and the sampling system is very important, because those deci-sions can determine the specificity, sensitivity, response time, and stability of the final device. Selective functionalization of the sensor is also critical to achieving the required operating parameters. Therefore, in designing a chemical sensor, de-velopers have to answer the enormous questions related to properties of sensing materials and their functioning in various environments. This five-volume com-prehensive reference work analyzes approaches used for computer simulation and modeling in various fields of chemical sensing and discusses various phenomena important for chemical sensing, such as surface diffusion, adsorption, surface reactions, sintering, conductivity, mass transport, interphase inter actions, etc. In these volumes it is shown that theoretical modeling and simulation of the pro-cesses, being a basic for chemical sensor operation, can provide considerable assistance in choosing both optimal materials and optimal configurations of sensing elements for use in chemical sensors. The theoretical simulation and model ing of sensing material behavior during interactions with gases and liquid surroundings can promote understanding of the nature of effects responsible for high effectiveness of chemical sensors operation as well. Nevertheless, we have to understand that only very a few aspects of chemistry can be computed exactly.

xiv • preface

However, just as not all spectra are perfectly resolved, often a qualitative or ap-proximate computation can give useful insight into the chemistry of studied phe-nomena. For example, the modeling of surface-molecule interactions, which can lead to changes in the basic properties of sensing materials, can show how these steps are linked with the macroscopic parameters describing the sensor response. Using quantum mechanics calculations, it is possible to determine parameters of the energetic (electronic) levels of the surface, both inherent ones and those introduced by adsorbed species, adsorption complexes, the precursor state, etc. Statistical thermodynamics and kinetics can allow one to link those calculated surface parameters with surface coverage of adsorbed species corresponding to real experimental conditions (dependent on temperature, pressure, etc.). Finally, phenomenological modeling can tie together theoretically calculated characteris-tics with real sensor parameters. This modeling may include modeling of hot plat-forms, modern approaches to the study of sensing effects, modeling of processes responsible for chemical sensing, phenomenological modeling of operating char-acteristics of chemical sensors, etc.. In addition, it is necessary to recognize that in many cases researchers are in urgent need of theory, since many experimental observations, particularly in such fields as optical and electron spectroscopy, can hardly be interpreted correctly without applying detailed theoretical calculations.

Each modeling and simulation volume in the present series reviews model-ing principles and approaches particular to specific groups of materials and de-vices applied for chemical sensing. Volume 1: Microstructural Characterization and Modeling of Metal Oxides covers microstructural characterization using scanning electron microscopy (SEM), transmission electron spectroscopy (TEM), Raman spectroscopy, in-situ high-temperature SEM, and multiscale atomistic simulation and modeling of metal oxides, including surface state, stability, and metal oxide interactions with gas molecules, water, and metals. Volume 2: Conductometric-Type Sensors covers phenomenological modeling and computational design of conductometric chemical sensors based on nanostructured materials such as metal oxides, carbon nanotubes, and graphenes. This volume includes an over-view of the approaches used to quantitatively evaluate characteristics of sensitive structures in which electric charge transport depends on the interaction between the surfaces of the structures and chemical compounds in the surroundings. Volume 3: Solid-State Devices covers phenomenological and molecular model-ing of processes which control sensing characteristics and parameters of various solid-state chemical sensors, including surface acoustic wave, metal-insulator-semiconductor (MIS), microcantilever, thermoelectric-based devices, and sensor arrays intended for “electronic nose” design. Modeling of nanomaterials and nano-systems that show promise for solid-state chemical sensor design is analyzed as well. Volume 4: Optical Sensors covers approaches used for modeling and simu-lation of various types of optical sensors such as fiber optic, surface plasmon resonance, Fabry-Pérot interferometers, transmittance in the mid-infrared region,

preface • xv

luminescence-based devices, etc. Approaches used for design and optimization of optical systems aimed for both remote gas sensing and gas analysis cham-bers for the nondispersive infrared (NDIR) spectral range are discussed as well. A description of multiscale atomistic simulation of hierarchical nanostructured materials for optical chemical sensing is also included in this volume. Volume 5: Electrochemical Sensors covers modeling and simulation of electrochemical pro-cesses in both solid and liquid electrolytes, including charge separation and transport (gas diffusion, ion diffusion) in membranes, proton–electron transfers, electrode reactions, etc. Various models used to describe electrochemical sensors such as potentiometric, amperometric, conductometric, impedimetric, and ion-sensitive FET sensors are discussed as well.

I believe that this series will be of interest of all who work or plan to work in the field of chemical sensor design. The chapters in this series have been prepared by well-known persons with high qualification in their fields and therefore should be a significant and insightful source of valuable information for engineers and researchers who are either entering these fields for the first time, or who are al-ready conducting research in these areas but wish to extend their knowledge in the field of chemical sensors and computational chemistry. This series will also be interesting for university students, post-docs, and professors in material science, analytical chemistry, computational chemistry, physics of semiconductor devices, chemical engineering, etc. I believe that all of them will find useful information in these volumes.

G. Korotcenkov

xvii

about the editor

Ghenadii Korotcenkov received his Ph.D. in Physics and Technology of Semiconductor Materials and Devices in 1976, and his Habilitate Degree (Dr.Sci.) in Physics and Mathematics of Semiconductors and Dielectrics in 1990. For a long time he was a leader of the scientific Gas Sensor Group and manager of various national and international scientific and engineering projects carried out in the Laboratory of Micro- and Optoelectronics, Technical University of Moldova. Currently, Dr. Korotcenkov is a research professor at the Gwangju Institute of Science and Technology, Republic of Korea.

Specialists from the former Soviet Union know Dr. Korotcenkov’s research results in the field of study of Schottky barriers, MOS structures, native oxides, and photoreceivers based on Group III–V compounds very well. His current research interests include materials science and surface science, focused on nanostructured metal oxides and solid-state gas sensor design. Dr. Korotcenkov is the author or editor of 11 books and special issues, 11 invited review papers, 17 book chapters, and more than 190 peer-reviewed articles. He holds 18 patents, and he has presented more than 200 reports at national and international conferences.

Dr. Korotcenkov’s research activities have been honored by an Award of the Supreme Council of Science and Advanced Technology of the Republic of Moldova (2004), The Prize of the Presidents of the Ukrainian, Belarus, and Moldovan Academies of Sciences (2003), Senior Research Excellence Awards from the Technical University of Moldova (2001, 2003, 2005), a fellowship from the International Research Exchange Board (1998), and the National Youth Prize of the Republic of Moldova (1980), among others.

xix

Contributors

Arūnas Šetkus (Chapter 1)Department of Physical TechnologiesCenter for Physical Sciences and TechnologyVilnius LT01108, Lithuania

Juan-Jesús Velasco-Vélez (Chapter 2)Materials Sciences DivisionLarge Lawrence Berkeley National LaboratoryBerkeley, California 94720, USA

Francisco Hernandez-Ramirez (Chapter 3)Institut de Recerca en Energia de Catalunya (IREC)Barcelona, SpainandDepartament d’ElectrònicaUniversitat de BarcelonaBarcelona, Spain

J. Daniel Prades (Chapter 3)Departament d’ElectrònicaUniversitat de BarcelonaBarcelona, Spain

Albert Cirera (Chapter 3)Departament d’ElectrònicaUniversitat de BarcelonaBarcelona, Spain

Ada Fort (Chapter 4)Information Engineering DepartmentUniversity of Siena53100 Siena, Italy

xx • contributors

Marco Mugnaini (Chapter 4)Information Engineering DepartmentUniversity of Siena53100 Siena, Italy

Santina Rocchi (Chapter 4)Information Engineering DepartmentUniversity of Siena53100 Siena, Italy

Valerio Vignoli (Chapter 4)Information Engineering DepartmentUniversity of Siena53100 Siena, Italy

Kalisadhan Mukherjee (Chapter 5)Materials Science Centre Indian Institute of TechnologyKharagpur 721302, India

Subhasish Basu Majumder (Chapter 5)Materials Science Centre Indian Institute of TechnologyKharagpur 721302, India

Akira Fujimoto (Chapter 6)Wakayama National College of TechnologyNadacho, Gobo-shi 644-0023 Japan

Leonid I. Trakhtenberg (Chapter 7)Semenov Institute of Chemical PhysicsRussian Academia of SciencesMoscow 119991, Russia

Genrikh N. Gerasimov (Chapter 7)Semenov Institute of Chemical PhysicsRussian Academia of SciencesMoscow 119991, Russia

Vladimir F. Gromov (Chapter 7)Semenov Institute of Chemical PhysicsRussian Academia of SciencesMoscow 119991, Russia

contributors • xxi

Mortko A. Kozhushner (Chapter 7)Semenov Institute of Chemical PhysicsRussian Academia of SciencesMoscow 119991, Russia

Olusegun J. Ilegbusi (Chapter 7)University of Central FloridaOrlando, Florida 32816-2450, USA

Roman G. Pavelko (Chapter 8)Department of Energy and Material SciencesFaculty of Engineering SciencesKyushu UniversityKasuga-shi, Fukuoka 816-8580, Japan

Duncan J. Mowbray (Chapter 9)NanoBio Spectroscopy Group and ETSF Scientific Development CentreDepartamento de Física de MaterialesUniversidad del País Vasco UPV/EHU and DIPCE20018 San Sebastián, Spain

Juan María García-Lastra (Chapter 9)NanoBio Spectroscopy Group and ETSF Scientific Development CentreDepartamento de Física de Materiales, Centro de Física de Materiales CSICUPV/ EHUMPC and DIPCUniversidad del País Vasco UPV/EHUE20018 San Sebastián, SpainandCenter for Atomic-Scale Materials Design, Department of PhysicsTechnical University of DenmarkDK2800 Kgs. Lyngby, Denmark

Iker Larraza Arocena (Chapter 9)NanoBio Spectroscopy Group and ETSF Scientific Development CentreDepartamento de Física de MaterialesUniversidad del País Vasco UPV/EHUE20018 San Sebastián, Spain

Ángel Rubio (Chapter 9)NanoBio Spectroscopy Group and ETSF Scientific Development CentreDepartamento de Física de Materiales, Centro de Física de Materiales CSICUPV/ EHUMPC and DIPCUniversidad del País Vasco UPV/EHUE20018 San Sebastián, Spain

xxii • contributors

Kristian S. Thygesen (Chapter 9)Center for Atomic-Scale Materials Design, Department of PhysicsTechnical University of DenmarkDK2800 Kgs. Lyngby, Denmark

Karsten W. Jacobsen (Chapter 9)Center for Atomic-Scale Materials Design, Department of PhysicsTechnical University of DenmarkDK2800 Kgs. Lyngby, Denmark

Zhimin Ao (Chapter 10)School of Materials Science and EngineeringThe University of New South WalesSydney, New South Wales 2052, Australia

Qing Jiang (Chapter 10)Key Laboratory of Automobile Materials, Ministry of Education, and School of Materials Science and EngineeringJilin UniversityChangchun 130022, People’s Republic of China

Sean Li (Chapter 10)School of Materials Science and EngineeringThe University of New South WalesSydney, New South Wales 2052, Australia

Petru Andrei (Chapter 11)Department of Electric and Computer Engineering Florida A&M University—Florida State University College of EngineeringTallahassee, Florida 32310, USA

Leonard L. Fields (Chapter 11)Corning Inc.Optical Physics and Networks TechnologyCorning, New York 14831, USA

Antonio J. Soares (Chapter 11)Department of Electronic Engineering TechnologyFlorida A&M UniversityTallahassee, Florida 32301, USA

Reginald J. Perry (Chapter 11)Department of Electric and Computer EngineeringFlorida A&M University—Florida State University College of EngineeringTallahassee, Florida 32310, USA

contributors • xxiii

Yi Cheng (Chapter 11)Institute for Systems Research (ISR)University of MarylandCollege Park, Maryland 20742, USA

Peng Xiong (Chapter 11)Department of Physics and Integrative NanoScience Institute (INSI)Florida State UniversityTallahassee, Florida 32306, USA

Jianping Zheng (Chapter 11)Department of Electric and Computer EngineeringFlorida A&M University—Florida State University College of EngineeringTallahassee, Florida 32310, USA

Stefan Baumgartner (Chapter 12)Department of MathematicsUniversity of Vienna1010 Vienna, AustriaandAIT Austrian Institute of TechnologyVienna, Austria

Martin Vasicek (Chapter 12)Department of MathematicsUniversity of Vienna1010 Vienna, AustriaandWolfgang Pauli Institute c/o Department of MathematicsUniversity of Vienna1010 Vienna, Austria

Clemens Heitzinger (Chapter 12)Department of Applied Mathematics and Theoretical Physics (DAMTP)University of CambridgeCambridge CB2 1TN, United KingdomandDepartment of MathematicsUniversity of Vienna1010 Vienna, AustriaandAIT Austrian Institute of TechnologyVienna, Austria

1DOI: 10.5643/9781606503140/ch1

Chapter 1

Numerical SimulatioN of electrical reSpoNSeS to GaSeS iN advaNced

StructureS

a. Šetkus

1. IntroductIon

Research and development of gas sensors stands mainly on known technologies that implement fundamental principles of the conversion of chemical interaction into a change of physical properties. For most practical applications it is essential to produce well-functioning and stable devices, but detailed models seem hardly necessary, because empirical approaches allow one to accomplish this task suc-cessfully. However, it seems evident that understanding the key mechanisms of the response, and having a fundamental description of the processes involved, will make it possible to better define the targets of research and development work as well as to evaluate expectable progress in the modification, improvement, and optimization of gas sensors.

Depending on the varying physical properties, proposed devices can be divided into a few main classes, namely, electrical, optical, and mechanical gas-sensitive solid structures. Analysis of the processes and models of the mechanisms require different fundamental approaches and tools of description for these three classes of gas sensors, which exceeds the limits of this survey. This chapter is devoted only to sensors in which the electrical properties of sensitive materials depend on

2 • ChemiCal sensors – moDelinG anD simUlation: VolUme 2

the influence of the gas. Metal oxides are frequently used as the sensitive material in these sensors.

In general, metal oxide gas sensors should be assumed to be partly electronic conductors and partly ionic conductors. Depending on the dominating component of conductivity, the oxides are accepted as being typical ionic conductors (Y2O3, ZrO2, etc.) or typical electronic conductors (SnO2, In2O3, WO3, etc.). For certain ceramics that are typically based on transition metals, both the electronic and ionic conductivities have to be considered in the analysis of the electrical prop-erties. From the viewpoint of practical applications, ionic and mixed conductors are typically used in the development of oxygen gas sensors, while metal oxides with dominant electronic conductance are widely used in the development of vari-ous gas sensors for diverse odor-detection systems. Though general principles of simulation are analogous for all the metal oxides, in the present review, only the metal oxides with dominating electronic conductance are discussed. For readers who are interested in ionic conductors and applications of these oxides, it may be useful to start with recent publications such as those of Fergus (2008), Zhulykov (2008), Röder-Roith et al. (2009), Hubert et al. (2011), and Schonauer et al. (2011).

In metal oxides with dominating electronic conductance, the response to gas is determined by the changes in the electrical charge transport produced by the interactions between the surfaces and gases. In scientific publications, these sen-sors are typically called conductive, conductometric, and resistive sensors. In this chapter, this type of sensor is preferably called a resistive gas sensor.

For the classical resistive sensors, the response to gas is determined by a se-ries of interrelated processes occurring in the heterogeneous system that includes the gas medium, the interface region, and the semiconducting material. The com-plete model of the system must combine the descriptions of diverse mechanisms that are highly specific to the individual parts of the system. Well-defined and justified connections among these parts are crucial for development of simulation models for gas sensors. This chapter represents an attempt to overview the publi-cations containing suggestions about the simulation of electrical responses to gas and to arrange the known approaches in some overall picture that explains the fundamentals of functioning of these advanced gas sensors. In this chapter, we first discuss the basic equations defining the processes of sensor functioning. It is shown that using special simplifications, analytical solutions can be obtained for these equations and adapted for simulation of the sensor response. In more com-plicated situations the sensor response can be simulated by numerical methods. The ways used to verify the simulation models are also reviewed in this chapter. This chapter also includes several sections in which some specific aspects of the response simulations are discussed for the polycrystalline metal oxide sensors, nanostructured films, conductive polymers, and molecular sensors. Finally, some general concluding comments about possibilities to simulate both existing and emerging resistive gas sensors are presented.

nUmeriCal simUlation of eleCtriCal responses to Gases • 3

2. AnAlytIc And numerIc modelIng

2.1. Basic Equations

Quantitative description of electrical properties of solid-state chemical sensors requires deep understanding of a complete picture of the physical processes that determine the response of these sensors to external chemical influence. Details about diverse aspects of the response mechanism make it possible to develop the most acceptable model for analysis of experimental data. Theoretical description of the response model gives the basis for numerical evaluation of the properties under investigation. Consequently, one needs either to develop an original ap-proach or choose an already-known approach to theoretical modeling of the func-tioning of chemical sensors, aiming to reveal the most important factors in the sensor technology and the methods of application. This overview is focused on the electrical properties of semiconducting gas sensors, though general approaches to the numerical evaluation of the parameters may be acceptable for other classes of sensors.

It must be noted here that, to date, there were no detailed studies analyzing proportions between the ionic and electronic components in metal oxide gas sen-sors. On the other hand, the known experimental facts have been successfully explained and described by models based on only the electronic conduction com-ponent. This approach to charge transport is sufficiently good for the synthesized ceramic metal oxide sensors or even for thin-film gas sensors with thicknesses greater than about 100 nm. On the other hand, there is experimental data about the specific role of ionic transport in nanostructured materials, which, conse-quently, reduces the ability to rely on numerical simulation of electrical properties based on classical approaches in these nanostructured metal oxide gas sensors. These special aspects will be discussed later in this chapter.

Classical models of metal oxide gas sensors are based on the semiconduct-ing properties of nonstoichiometric metal oxides. The oxygen vacancies are typi-cally associated with donor-type point defects in these oxides. The shallow donor levels are completely oxidized at temperatures above 300 K and provide the free electrons in the conduction band in these materials. In general, in the presence of an electric field

E , the electrical charge transport is determined by the drift of carriers and the diffusion due to nonuniform distribution of carriers in semicon-ducting materials. Thus, the current density

j is equal to

µ= × × × + × ×Ñ

qj q n E q D n (1.1)

In Eq. (1.1), the first term describes the drift component of conductivity while the second one describes the diffusion component. These two components in-clude specific parameters: q, the electrical charge of a single carrier; n, the carrier


concentration; m, the mobility; and Dq, the diffusion coefficient. Typically, the diffusion term is omitted in the models describing the response mechanisms in metal oxide gas sensors. However, this term can be important at least for those sensors in which the layer controlled by the surface potential is comparatively thick with respect to the dimensions of the conductive channel. Therefore, the Eq. (1.1) can be considered in metal oxide sensors in which the Debye screen-ing length is comparable to the dimensions of structural elements (e.g., grains in polycrystalline oxides).

By omitting the diffusion term in Eq. (1.1), the simulation models are signifi-cantly simplified. As a result, the theoretical description acceptable for numerical evaluation of the parameters considers only the conductance s, which is defined by the factor from the first term of Eq. (1.1) as follows:

σ µ= × ×q n (1.2)

In fact, Eq. (1.2) is typically the basis of simulation models for gas sensors that have been presented in the literature to date. In Eq. (1.2), both the carrier con-centration and the mobility can be dependent on the surface properties of the semiconducting constructions. The exact form of the dependence is defined by the individual model of gas sensor. However, two notes can be added here. First, in the sensor models, the resistance R of gas-sensitive structures is frequently evaluated instead of the conductance (R = l/S·s–1, with length l and cross-sectional area S as the geometric factors). Second, gas sensor models do not include an explicit analy-sis of scattering mechanisms that determine the mobility in a homogeneous semi-conductor. For example, the mobility is inversely proportional to the concentration of ionized impurities (m ~ Nii

−1) if the ionized impurities are the dominant scattering mechanism in the transport of charge carriers. The effect of this scattering can be illustrated quantitatively by well-known facts about crystalline semiconductors. According to the experiments of Prince (1953), in p-Ge an increase in concentra-tion of ionized impurities from 1014 cm−3 to 1017 cm−3 results in a decrease in the electron mobility from about 3900 cm2 V−1 s−1 to about 1500 cm2 V−1 s−1. An analo-gous effect of the scattering on the mobility of charge carriers was also obtained for doped silicon (Prince 1954). Since the mobilities of the charge carriers are signifi-cantly less studied in metal oxides, it is not possible to evaluate quantitatively the influence of the scattering effects on the mobility in homogeneous parts of metal oxide gas sensors. Typically, it is assumed that the effect is negligible compared to other mechanisms determining an electrical response to gas.

In metal oxide gas sensors the surface potential defines both the charge trans-port and the spatial variation of free carriers as well as ionized point defects. Most simulation models includes Poisson’s equation, the solution of which describes a dependence of the potential on the space charge density. In one dimensional form, Poisson’s equation is written as


φ πρ

εε¶

=¶

2

20

4x

(1.3)

In Eq. (1.3), f is the electrostatic potential, r is the electrical charge density, and e and e0 are the relative permittivity of the material and the vacuum permit-tivity (e0 = 8.85 × 10−12 F/m), respectively. The exact definition of the charge r depends on the individual simulation model but, in general, the free carriers and ionized impurities have to be considered as follows:

( )ρ + -= × - - +1 1d aq N N n p (1.4)

where Nd and Na are the concentrations of donors and acceptors, respectively, while n and p are the concentration of free charges carriers, namely, electrons and holes, respectively. Simplifying the simulation of gas sensors, it is typically assumed that (1) only one type of carriers (n or p) is present and (2) the concentra-tion and distribution of ionized impurities is constant for all analyzed processes in the sensors. It is reasonable to expect that acceptability of these simplifications should be carefully considered because it can result in crucial deviations in the calculations of sensor characteristics.

In gas sensors at nonequilibrium conditions, the carrier densities within a given unit of volume of the material varies as a function of time due to the carrier transport, capture by the appearing surface states and release from the disappearing states. A change of the surface states is produced by adsorption– chemisorption, desorption, and chemical reactions on the surfaces. The changes in carrier densities with time are describe by the current continuity equations. Considering the electronic-type metal oxides, in this work only n is assumed to be important. Assuming analogy between optical generation–recombination pro-cesses and the trapping–releasing phenomena for the surface states, the continu-ity equation for the current density can be written as

¶¶

=- + -¶ ¶

nn n

jn d at x

(1.5)

In Eq. (1.5), n is the concentration of electrons in the conduction band, jn is the electron current, and dn and an are the rates of release (delocalization) and capture of electrons from/to the surface states, respectively. In general, it follows from the continuity equation that if charge is moving out of a differential volume (i.e., di-vergence of the current density is positive), then the amount of charge within that volume will decrease, so that the rate of change of the charge density is negative. Therefore the continuity equation amounts to a conservation of charge.


A change of free carriers in solid chemical sensors is typically related to the chemical interaction between the surfaces and the surrounding particles. For exam ple, the chemisorption of the atmosphere oxygen on metal oxide is associ-ated to the localization of conductive electrons at the chemisorption sites on the solid surface. Formation and the properties of the point surface defects depend on the materials, adsorbed particles, and specific processes. Therefore, the exact description of the rates for released dn and captured electrons an in Eq. (1.5) de-pends on the individual model. Some general considerations about the capture of free electrons by the surface chemisorption sites can be found in the study by Wolkenstein (1991) but, in simplified simulations, it can frequently be assumed that an is equal to the oxygen chemisorption rate and dn is equal to the sum of the oxygen desorption and surface reaction rates.

In general, the kinetics of the surface coverage by the chemisorbed species of gases can be described by a modified rate equation based on a Langmuir ap-proach as follows:

( )Φ ν ν Θæ ö¶ ÷ç= × × - - × × - - × ×÷ç ÷÷çè ø¶

desads 0 des OG G

0

expEN NS N N N

t kT N (1.6)

where N and N0 are the densities of the chemisorbed gas species and the chemi-sorption surface sites, respectively, F is the flux of particles hitting the solid sur-faces, Sads is the sticking probability of gas particles, ndes is the desorption rate, Edes is the desorption activation energy, nOG is the rate of chemical reaction be-tween chemisorbed particles and the gas particles hitting the surface, and QG is the flux of gas particles hitting the solid surface. The third term in Eq. (1.6) de-scribes a chemical reaction between two particles of different origin. One particle is chemisorbed on the surface, while another hits the surface area close to the chemisorption site. The Eq. (1.6) form without this third term is the most well known in the surface science models, and it describes the dynamics of the surface coverage determined by the adsorption and desorption processes.

The dynamic picture of adsorption and desorption for solid surfaces has been thoroughly studied in surface science for a number of years. Therefore, the simu-lation models for solid surface coverage with particles define influences of diverse processes (see, e.g., Kreuzer 1990; Zhdanov 2002). The difficulties in simulation of adsorption–desorption kinetics were discussed by Zhdanov (2001). A theory that incorporates the solid surface reconstruction phenomena occurring during adsorption and desorption was proposed in terms of the Langmuir approach in work by Cerofolini (2003). Based on statistical rate theory, simulation models for adsorption–desorption processes on heterogeneous surfaces were proposed by Rudzinski et al. (2005). Panczyk and Rudzinski (2004) showed that applica-tion of statistical rate theory to describe the kinetics of dissociative adsorption leads to very flexible expressions which may account for the variety of physical


situations found in these systems. The desorption kinetics and a variation of the solid surface coverage with gaseous species produced by desorption was explicitly described by Payne et al. (2006).

The basic equations are used to define the characteristics that have to be in-cluded in consistent simulation models, allowing one to calculate the responses of semiconducting sensors to gas and to evaluate the key characteristics of materi-als and processes producing the most significant influence on the parameters of sensors. Using special approaches, the equations can be modified and solved to provide explicit relationships among the characteristics of the sensors.

2.2. analytical approachEs

The simulation models that provide analytical descriptions of the response to gas are the most valuable results of studies on fundamental processes in gas sensors. Based on these models, a quantitative description of gas sensor characteristics can be obtained. However, in general, it is impossible to obtain an analytical solution for a set of the equations that defines the relationship between the gas–surface interaction and a change in the conductance of a solid–state construc-tion. Typically, a series of specific simplifications is made in the mechanism and processes, aiming to obtain an analytical form for description of parameters for gas sensors. These simplifications are dependent on the mechanisms selected to describe the conversion of the gas–surface interaction into the response and, consequently, are justified by the limitations that determine the acceptability of the simulation model.

The complete model of the response mechanisms in metal oxide gas sensors is still under discussion. In spite of this, the core aspects in the understanding of the response of metal oxides to gases are commonly accepted and have been used by sensor researchers and developers for more than 20 years. These classical ap-proaches have been nicely presented in previous papers (e.g., Barsan et al. 2001, 2007; Oprea et al. 2009). We will use this classical fundamental understanding of the response mechanisms in metal oxides for definition of simulation models resulting in analytical descriptions of the characteristics and a quantitative evalu-ation of the dependencies between the sensor parameters and the core factors in the technology as well as in functioning.

The classical approaches in the explanation of sensor functioning are based on the polycrystalline structure of typical gas sensors. Supposing grains similar to spheres with the same diameter, the sticking points between the grains are accepted as the contacts through which electrical current flows. The electrical charge on the surfaces of the grains can be changed by chemical interaction be-tween the surface and gas. A variation of the surface charge produces changes in the electron transport in the polycrystalline metal oxide. There are at least three


models explaining the relationship between the surface charge and the electron transport, depending on which Eqs. (1.1), (1.2), (1.3), and (1.4) can be presented in diverse forms acceptable for the simulation of the conductance. The most-exploited approach proposes the contacts between the grains being similar to the double Schottky junction. Thermally activated flow of electrons above the po-tential barrier of the grain boundary is the core assumption in this approach. Another approach includes an assumption about the coalescent grains. It follows from this approach that the electrical charge flows in a channel with varying cross section. The largest parts in the channel are equal to the diameter of grains, while the necks in the channel appear at the junctions between the grains. The width of the channel can be changed by the surface electrical charge. In this approach, the basic equations have to be used to define the relationship between the surface charge and the width of the conductive channel. The third approach rests on the assumption about the straightforward relationship between the free electron con-centration in the bulk of the semiconductor and the adsorbed gas species on the surfaces of sensor. Definition of this relationship leads to comparatively simple modification of the basic equations acceptable for quantitative description of the sensor characteristics.

We limit ourselves to the double Schottky barrier approach in this chapter because this approach is used more frequently in simulations than the other two. Without deep analysis, a few remarks can be added here about the choice of ap-proach. First, development of constructions with a continuous conductive channel (without potential barriers at the junctions) requires special growth technologies that allow one to obtain continuous metal oxide layers with the monocrystal-line structure. To date, most sensor technologies used have been acceptable for formation of polycrystalline materials. Second, supposing 1000 ppm for a gas, the maximum density of chemisorbed gas species is about 1014 cm−2. Assuming one extra free electron per each chemisorbed species, an increase in the free electron density nextra can be estimated as 1014 cm−3. The concentration of free electrons in gas-sensitive metal oxides n is typically estimated to be about 1017–1018 cm−3. Consequently, an electron concentration response to gas can be about nextra/n <10−3 and even less.

It must be also noted here that there are significant uncertainties that limit acceptability of an analytical solution for the simulation of sensor properties in the double Schottky barrier approach. There are ongoing discussions about the surface properties of metal oxides. It has been shown by various studies that the surface properties are dependent not only on the material but also on the struc-ture of the solid. The details about these dependencies can be found in recent publications, namely, for tin oxide in Batzill and Diebold (2005), for titanium oxide in Diebold (2003), and for indium oxide in King et al. (2009) and O’Neil et al. (2010). Summarizing the studies on the surface properties, one can conclude that there is no straightforward relationship between the density of surface oxygen and


an increase in the density of surface negative charge. Both depletion and accumu-lation layers can be obtained on the surfaces of metal oxides. The type of surface charge layer depends on the crystallographic planes of the oxides and the oxygen adsorption modes [see, e.g., for SnO2, Sensato et al. (2002), and for In2O3, Walsh (2011)]. Since the details of simulation depend on the exact model, classical prin-ciples in the depletion-layer approach are accepted as the basis for the analysis of the response of metal oxides to gas in this section.

In our approach, Eq. (1.1) must be modified to describe electron transport in conductive channels with potential barriers. Typically, it is supposed that the electric charge is transported only by the conduction-band electrons with energies higher than the height of the potential barrier. The thermally activated electron transport can be described by two analogous forms, however, whose origin is dif-ferent. The most typical approach is based on general understanding of electron distribution with energies. Using the Boltzmann distribution, the concentration of electrons with energies exceeding the height of the potential barrier Vb can be defined as

æ ö÷ç= ÷ç ÷÷çè ø0 exp b

ceV

n nkT

(1.7)

In Eq. (1.7), k is Boltzmann’s constant, T is the temperature of the sensor, and n0 is the free electron concentration in the conductance band in the bulk of the semiconductor. Neglecting the diffusion in Eq. (1.1), the density of current through the intergranular potential barrier can be defined by

σ σ µæ ö÷ç= × =÷ç ÷÷çè ø0 0 0exp beV

j E qnkT

(1.8)

In general, the characteristics of the junction barrier depend on various para meters such as the position relative to the junction, ionized impurities, sur-face charge, external electric field, and so on. Therefore, an analytical descrip-tion of Vb can be obtained for some simplified double Schottky model. Supposing a one-dimensional approach with the position of the junction at x = 0 and com-pletely ionized donor impurities outside the junction zone to the left (x < xL) and the right (x < xR), the charge density in the barrier region can be described by the following formula:

( ) ( ) ( ) ( )ρ θ θ δé ù= + - - -ë ûD L R Ix eN x x x x Q x (1.9)

where ND defines the donor concentration in the metal oxide bulk, θ(x) is the Heaviside step function, xL and xR are the lengths of the left and the right depletion


region, respectively, and d(x) is the Dirac d function, QI defines the density of sur-face charges and can be described by

( ) ( )¥

= òI

I I I

E

Q q dE N E f E (1.10)

In (1.10), NI(E ) and EI are the density and the lowest energy of the electronic traps in the gas chemisorption centers on the surface, respectively, and fI(E ) is the electron distribution function. Supposing flat bands outside the depletion region and constant concentration of ionized impurities in the region and replacing (1.4) with (1.9), it is possible to obtain an analytical solution for Eq. (1.3) similar to that of Blatter and Greuter (1986a, 1986b). The solution is as follows:

φφ

æ ö÷ç ÷= -ç ÷ç ÷çè ø

2

00

1 14b b

b

VV (1.11)

Here the barrier height without the external electrical field is

( )φεε

º = =2

0 0 02

Ib b

q D

QV VqN

(1.12)

V is the drop of external electrical field across the junction V = (xL + xR)E. The ap-plied voltage Vc = fb0 is often interpreted as the critical voltage for electrical break-down in the single junction.

It must be noted here that the simulation of the response in metal oxides is much easier if an external electric field E is assumed comparatively low, so that an influence on the barrier height Vb can be neglected. In the classical simulation model this assumption is justified by comparatively low measurement voltages (~10 V), long distances between the contacts (~10−3 m), and high number of the junctions between the contacts (~103). Assuming the junction thickness is about 10−7 m, the electric field across the single junction is about 105 V/m and practi-cally negligible. However, for higher fields, the influence of the applied external voltage on the properties of the gas sensor can be considered as in Varpula et al. (2008). More significant influence of the field can be detected in the nanostruc-tured metal oxide gas sensors. Some aspects of this effect will be discussed below.

Neglecting the influence of the external electric field on the characteristics of the junctions, Eq. (1.12) can be accepted as a definition of the relationship be-tween the surface charge and the barrier height that defines the electron trans-port in metal oxide gas sensors in (1.8). Here we can suppose that QI is determined only by the chemisorbed gas species on the surfaces of the sensor and QI = qNA,


with NA the density of the chemisorbed species. Using the description of the resis-tance R = L/(Sσ) for a sample with length L and the cross section S, from (1.8) and (1.12) one can obtain for the sensor resistance

εε

æ ö÷ç ÷= ç ÷ç ÷çè ø

22

00

exp8 A

d

qR R nkTN

(1.13)

The definition (1.13) is actually obtained for the static situation, after the transient processes are finished. Assuming the density of the chemisorbed gas species to be proportional to the gas partial pressure in atmosphere, NA ~ pg, a dependence of the resistance response on the amount of gas in the air follows from (1.13) if the adsorption is the core mechanism in the response. However, the resistance response to gas is typically determined by the surface chemical reac-tions between the chemisorbed oxygen species and the gas. Then, a relationship between the gas and the surface charge has to be obtained and included in (1.13) supposing that NA represents the density of chemisorbed oxygen species on the surfaces of the metal oxide gas sensor.

In this simulation model we assume that a single type of oxygen species is dominant on the surfaces of the metal oxide and all allowable chemisorption sites can be occupied only by oxygen. Specific sites exist on these surfaces for adsorp-tion of gas that, however, do not create surface traps for electrons—i.e., the surface charge does not depend directly on adsorption of gas. The density of chemi sorbed oxygen is determined by the equilibrium of the chemisorption–desorption pro-cesses under constant conditions. This initial state of the sensor surfaces defines the electrical properties of the metal oxide gas sensor before exposure to gas as it was proposed by Šetkus (2002). The system “sensor–atmosphere” is accepted as being stable under equilibrium conditions.

Consider a response of a metal oxide sensor to a comparatively low amount of impurity gas (reducing) in the atmosphere. We suppose here that the partial pressure of the impurity gas Pgas << PO2 in the atmosphere and a steplike change in the atmosphere composition occurs. After the change in the composition of the atmosphere, the amount of oxygen remains unchanged, but a fixed amount of the target gas Pgas is mixed into the atmosphere. Heterogeneous catalytic reactions typically take place on the solid surfaces of the sensor. These reactions can be de-scribed as a sequence of a few elementary steps, namely, the adsorption (chemi-sorption), desorption, and Langmuir-Hinshelwood (LH) bimolecular interaction, including adsorbed species of oxygen and gas. A solution of the rate equations for these steps results in a relationship between the surface electrical charge and current in the metal oxide and, consequently, defines the simulation model of the sensor response to gas.

On real surfaces, these equations are typically complicated by terms related to adsorbate–adsorbate lateral interaction (including electrostatic interaction),


surface heterogeneity, spontaneous and adsorbate-induced changes in the sur-faces, and/or limited mobility of adsorbed species. In spite of this, the basic prop-erties can frequently be simulated by a model of an ideal adsorbed layer, when the surface is uniform and stable and there is no adsorbate–adsorbate lateral inter-action. These simplifications make it possible to obtain analytical solutions for the model equations.

Considering our simplifications, the surface reactions that produce a response by the sensor to gas can formally be split into the following steps:

- -+ «1 1gas adsO Oe (1.14)

«gas adsG G (1.15)

( )- -+ ® +1 1ads ads gasO G OG e (1.16)

Step 1 (1.14) represents the chemisorption of oxygen, while the interaction between the target gas (Ggas) and the surface is described in step 2 (1.15). Step 3 of the bimolecular interaction (1.16) includes the surface species of oxygen and gas. It is typically assumed that the product (OG)gas is removed from the surface to the atmosphere at once.

According to the common understanding of heterogeneous catalysis (see, e.g., Kreuzer 1990; Xu and Koel 1994; Carlsson and Madix 2001; Busse et al. 2001; Zhdanov 2002), the gaseous species are adsorbed on the surfaces prior to the LH reaction in step 3. Usually, as in Lantto and Romppainen (1987), Clifford and Tuma (1982/1983), Gardner (1990), Rantala et al. (1993), Simon et al. (2001), Sakai et al. (2001), and Nakata et al. (2002), it is supposed for the sensor response that the surface coverage with gases is determined by the adsorption–desorption equilibrium prior to step 3 (1.16). Based on this assumption, the simulations of the sensor response are crucially simplified, excluding a series of processes except for step 3 (1.16). Such simplifications allow one to limit the response description to a comparatively simple rate equation and to obtain a solution that typically is a pure exponential transient for the response signal.

In a more realistic approach, an injection of impurity gas is associated with changes in the coverage of the sensor surfaces by both the oxygen and the impu-rity gas species. Taking into account the steps of the surface chemical reactions in (1.14)–(1.16), two rate equations can be written for the oxygen coverage (QO) and the impurity gas coverage (QG) in a form analogous to that of Nakata et al. (2002), Carlsson and Madix (2001), Busse et al. (2001), and Kreuzer (1990). These rate equations are as follows:

( )

( ) ( ) ( ) ( )Θ

Θ β Θ ν Θ Θé ù= - - -ë ûO

O O O0 O O O OG O G1d t

F c S t t t tdt

(1.17)


( )

( ) ( ) ( ) ( )Θ

Θ β Θ ν Θ Θé ù= - - -ë ûG

G G G0 G G G OG O G1d t

F c S t t t tdt

(1.18)

In Eqs. (1.17) and (1.18) the constant parameters are as follows: F represents the flux of gas particles hitting a unit area of the surface from the atmosphere with constant P, where P is the partial pressure of gas particles in the atmosphere, c is the area of a single adsorption site, b is the probability for the desorption from a site per time unit, and nOG is the probability of step 3 (1.16) occurring at a site per time unit. The index O is for oxygen, while G is for the impurity gas. For general understanding, it is acceptable to replace F with P.

The parameters SO0 and SG0 represent the sticking probabilities for the oxygen and impurity gas on the clean surfaces, respectively. In general, the sticking prob-ability is not a constant but should be dependent on various specific factors such as the surface structure, parameters of chemisorption sites, the surface band bending, etc. According to the studies of co-adsorbed layers and multicomponent surfaces of Carlsson and Madix (2001), Busse et al. (2001), Kreuzer (1990), and Xu and Koel (1994), the sticking probability is influenced by the surface adatom modifiers because the modifier precursor state can be created on the solid sur-faces. In some studies, e.g., Zhdanov (2002) and Persson (1992), it was demon-strated that the different gas species can occupy the individual adsorption sites on the heterogeneous surfaces. In our presentation, simplifying the understand-ing of chemisorption, it is supposed that the oxygen is adsorbed on the surface sites with density NmaxO, while the target gas is adsorbed on some precursor states with constant density NmaxG.

In the rate equations (1.17) and (1.18) the first terms on the right-hand sides of the equations describe the adsorption rate, while the second terms represent the desorption rates. The third terms define the rates of the LH bimolecular step in which adsorbed species of oxygen and impurity gas are involved as defined in Eq. (1.15).

The rate equations (1.16) and (1.17) have to be solved numerically if accurate descriptions are desired for the fractional coverage QO and QG in the response simulation. However, an explicit definition of these parameters provides better understanding about influences of various factors on the sensor response. An analytical solution can be obtained for the rate equations (1.17) and (1.18) if the surface coverage QG with the impurity gas is supposed small with respect to the oxygen surface species (QG << QO). Based on this assumption, the total oxygen coverage QO is supposed to be represented by almost constant initial coverage QO0 and a small correction dQO produced by the surface chemical reaction between the impurity gas and oxygen. Therefore, the solution for the rate equation (1.17) can be written in the form

Θ Θ δΘ= -O O0 O (1.19)


In (1.19) the fractional coverage QO0 is defined by the equilibrium of the oxygen adsorption–desorption processes in clean air [step 1 (1.14)] and dQO is a correction to the coverage produced by the surface chemical reaction between the oxygen and the impurity gas in the LH bimolecular step (1.16). It is accepted for all condi-tions dQO << QO0.

For nondissociative chemisorption and simple bimolecular reaction on the sensor surfaces defined by (1.16), it is acceptable to suppose that dQO = QG. Consequently, only a solution of (1.18) is required for a simulation of the sensor response to gas defined in (1.13). For this, the coverage QO in (1.18) has to be re-placed by the following expression

( ) ( )Θ Θ Θ= -O O0 Gt t (1.20)

After the substitution, Eq. (1.18) is written in the following form:

( )Θν Θ β ν Θ Θ= - + + +2G

OG G G G G0 G OG O0 G G G G0d

F c S F c Sdt

(1.21)

The initial condition can accepted as QG(t ® 0) = 0 for the injection of impurity gas into the atmosphere. Depending on the parameters, the solution of Eq. (1.21) can be obtained in the following two forms:

( ) ( ) ( )Θ Φ ν γ Φ ν γ γ γ Φ νν

ì üé ùï ïï ïê ú= - × - × + <í ýï ïê úë ûï ïî þ

1 2 1 22 2 21G G OG G OG G OG

OG

1 4 4 if 42 2

tt tg (1.22)

( )( )

( ) ( )τ

Θ γ Φ ντ

-= × >

-2

2G G OG1 exp

if 41 exp

tt p

p q t (1.23)

where

Φ =G G G G0F c S (1.24)

γ Φ β ν Θ= + +G G OG O0 (1.25)

( ) ( )γ Φ ν γ ν --é ù= + - × ×ê úë û

1 2 12G OG OG1 1 4 2p (1.26)

( ) ( )γ Φ ν γ ν --é ù= - - × ×ê úë û

1 2 12G OG OG1 1 4 2q (1.27)

( )γ Φ ν γτ

-= × - ×1 22

G OG1 1 4 (1.28)


According to the definition, fractional coverage of the surfaces with oxygen QO represents a ratio between the density of chemisorbed oxygen NO and the density of the chemisorption sites for oxygen NOmax (QO = NO/NOmax). Replacing NA in (1.13) with NO and using (1.20), one can obtain a simulation formula for the resistance response of a metal oxide sensor to the injection of impurity gas as follows:

( ) ( ){ }Θ Θé ù= × - =ë û2

0 O0 Gexp 1,2,...iR t R Z t i (1.29)

where

εε

= =2 2

Omax max

08b

d

q N qVZ

kTN kT (1.30)

The formula (1.29) can be rewritten in more convenient form for the simula-tion of sensor response. To do this, a definition of R(0) = R(t ®0) can be obtained from (1.29) and, consequently, a normalized resistance response can be described as follows:

( )( )

( ){ }Θ Θ Θ-æ öé ù ÷ç= × - × - ÷ç ê ú ÷ç ë ûè ø22 1

O0 O0 Gexp 1 10

R tZ t

R (1.31)

The analytical expression (1.31) is acceptable for quantitative calculation of transient signal for the resistance response of a metal oxide sensor to a single im-purity gas in air. Time dependencies of the response after exposure of the sensor to gas are explicitly defined by the solutions (1.22) and (1.23). The magnitude of the resistance response can be obtained for the sensors from the saturated transient response signals evaluated under the assumption of extremely long exposure time using the limit t ® ¥ in (1.31). The acceptability of the response simulation using the analytical expression (1.31) was discussed by Šetkus (2002, 2004).

2.3. numErical simulations

Analytical solutions of gas sensor problems typically describe the dependencies between the response to gas and the factors important in the sensor technology, the measurement methodology, and the response mechanisms. A special form of the solution is typically generated in such studies that explicitly reveals the rela-tionships among various parameters. The analytical solutions actually represent a theoretical simulation model for the sensor response and the related pheno-mena in the sensors. This simulation model allows one to calculate the response


curves and fit these curves with experimental ones, which, consequently, gives a quantitative description of the parameters in the model. Such an approach gives quite detailed insight into the processes that determine the mechanisms and pa-rameters of the sensor. Analogous results can also be obtained without defining the analytical solutions. There are numerical methods acceptable for solving the basic equations of the simulation model. However, this approach makes the fit-ting to the experimental data rather complicated. Therefore, the approaches based on analytical solutions are much frequently used for the simulation of sensor re-sponses than those based on pure numerical methods.

Numerical representation of the analytical model makes it possible to use the transient responses for definition of a series highly important parameters. The dy-namics of a change in electrical charge transport produced by exposure of a sen-sor to gas contains specific information about both the physical and the chemical processes in the bulk and on the surfaces of semiconducting metal oxides. At least three independent methods have been proposed for obtaining the transients of the electrical signals in metal oxides exposed to gas. Techniques based on the controlled change in the sensor temperature, the bias voltage, and the composi-tion of gas atmosphere have been used. An interesting simulation model is pro-posed by Varpula et al. (2011) for the transients generated by small step changes in either temperature, bias voltage, or concentration of reducing gas. Varpula et al. (2011) showed that model-based analysis of experimental data allows calcula-tion of several important quantities, such as the time constant of the electronic trapping process, the height of the grain boundary (GB) potential barrier, the relative change of the occupied GB states, the resistance coefficient, and the ef-fective number of GBs between the electrodes of the sensor. The parameter values obtained can be used for further study of the underlying physical and chemi-cal phenomena in gas-sensitive metal oxides, including determining the electron transport over the grain boundaries.

Thermally activated processes of surface chemical reactions were analyzed by Ahlers et al. (2005) aiming to simulate the bell-shaped variation of the sensor response to gas with the sensor operation temperature T. In this study, it was supposed that, at typical sensor operation temperatures above 425 K, the prevail-ing adsorbate is the oxygen species O−. Adsorption and desorption of oxygen is defined by the transition

- -+ «2( ) ( )O 2 2Og Se (1.32)

An influence of temperature appears in this process because the chemisorp-tion requires two electrons to be thermally emitted from the Fermi energy EF across the surface barrier (EC + qVs − EF_bulk) to become trapped at a pair of adsorbed oxy-gen atoms O(S)

−. EC is the energy of the conduction band. Desorption of the sur-face oxygen ions requires re-emission of two electrons across an energy barrier of height (EC − EO_minus) with respect to the energy level EO_minus of chemisorbed oxygen


species O(S)−. If reducing analyte gas molecules are present in the atmosphere, the

reactions between the gas and surface oxygen take place. These reactions reduce the density of surface oxygen ions, NO. It is supposed that for this reaction a kine-tic barrier, represented by activation energy Ea, needs to be overcome. The reac-tion products then leave the surface and the electrons formerly trapped at surface oxygen ions are released to the conduction band.

Based on these ideas, the simulation model of Ahlers et al. (2005) includes the solution (1.12) of Poisson’s equation (1.3) for the intergranular junction with the double Schottky barrier and the solution of the rate equation for the surface chemical reaction (1.6) that was specifically modified by including the parameters of thermally activated processes as follows:

( ) ( )κ κ

é ù é ù+ - -ê ú ê ú= - - - -ê ú ê úê ú ê úë û ë û

_ bulk O _ minus2 2O0 O2 0 O

2 2exp expC S F C

f C r

E qV E E EdNp N N G

dt kT kT (1.33)

where pO2 is the oxygen partial pressure, κf0 and κr0 are the kinetic parameters for adsorption and desorption, and G is the term describing the reaction between the gas and chemisorbed oxygen as follows:

Θæ ö÷ç= - ÷ç ÷÷çè øO 0 exp aE

G N kkT

(1.34)

The term G includes a Langmuir relative coverage Q defined by

Θ =+gas

gas 0

pp p

(1.35)

In (1.35), pgas is the partial pressure of reducing gas and p0 is

æ ö÷ç= - ÷ç ÷÷çè ø

ads0 exp

Q

EkTpV kT

(1.36)

( )-=3 2*

gas 0 gasQV h M M kT (1.37)

Here Tgas is the gas temperature (e.g., 300 K), T is the sensor operation tempera-ture, Eads is the binding energy of the adsorbate, M0 is the atomic mass unit (1.67 × 10−27 kg), and Mgas is the relative atomic mass of the reducing gas (e.g., 28 for CO and 2 for H2). Considering the descriptions in (1.34)–(1.37) and simplifications analogous to that in Section 1.2, the solution for NO is obtained from (1.33) by Ahlers et al. (2005).


The final formula acceptable for the simulation of the response versus T was obtained by defining the relationship between NO and the width of the space charge layer WSCR in the double Schottky barrier and the resistance of the sensor. In the work of Ahlers et al. (2005), the analytical description of the simulation model was presented in the following form:

( )-

= - = --

SCR_gasairgas

gas SCR_air

, 1 1S

S

D WRS T c

R D W (1.38)

Here Rair and Rgas are the sensor resistances in clean air and air with gas, respec-tively, WSCR_air is the space-charge region width in clean air, WSCR_gas is the space-charge region width in the presence of reducing gas, and DS is the metal oxide layer thickness. The function S (T, cgas) in (1.38) defines the relationship between the normalized resistance response to gas and two parameters, temperature T and amount of tested gas cgas, and was used for the calculation of theoretical depen-dencies. This simulation model of Ahlers et al. (2005) reproduces the particular form of the relationship between the gas response and sensor temperature, with the sensitivity maximum SM occurring at the temperature TM. The solution also demonstrated the sublinear variation of S with the analyte gas concentration cgas. Details of the comparison between the simulation model and the experiment can be found in the work of Ahlers et al. (2005).

In studies with strictly defined conditions, the simulation models can be ex-tremely simplified using empirical descriptions of some parameters and limiting the analysis to specific tasks, as proposed by Vilanova et al. (1998). In this work, the influence of electrode position and geometry on the semiconductor gas sen-sor response was studied. It was assumed by Vilanova et al. (1998) that the dif-fusion of gas into a thick-layer sensor is the only process limiting the kinetics of the sensor response. Adsorption and chemical kinetics are assumed to be faster than physical diffusion of gas. At equilibrium, the relation between free (C ) and adsorbed gas concentration (N ) can be defined by the frequently used formula

γ=N BC (1.39)

where B is a constant. Assuming one-dimensional diffusion, the rate equation for the surface chemical reaction can be written as

¶ ¶ ¶- + + =

¶ ¶¶

2

2 0aC C ND KNt ty

(1.40)

Here K is the reaction rate constant and a is the reaction order, assumed to be a = 1 by Vilanova et al. (1998). The first two terms in (1.40) represent the diffusion of


gas into the thick sensor, while the terms with N define the variation of the density of the gas on the sensor surfaces. The problem can be solved by computational methods, and the carrier concentrations can be derived assuming proportionality between absorbed gas concentration and conduction electron concentration.

The numerical simulation model of Vilanova et al. (1998) demonstrated a cor-relation between the contacts and the sensor response. It was shown by this simulation that when the sensor has low catalytic activity for a gas, the variables electrode position, active-layer thickness (yO), and electrode gap (W ) have no rele-vance. For higher catalytic activity, when electrodes are placed on the bottom, the sensor sensitivity is lower, although it can be increased slightly using wider elec-trodes. For a sensor with a very wide electrode gap, the relative response to gas is obtained as being almost similar for all electrode positions in the sensor.

As follows from the above paragraph, the solutions of equations can be obtained in numerical form using special computational methods without the need to accept at least some of simplifications required for an analytical solution. Numerical methods frequently seem less restricting for sensor analysis than spe-cific requirements for the conditions of tests; however, these methods do not re-sult in explicit relationships between the response and probed factors. Numerical simulation of gas sensor characteristics is typically based on similar models that are used for obtaining analytical description of sensor parameters, but the results of numerical simulations are frequently limited to a quantitative representation of a relationship between a selected factor and the sensor response.

Rothschild and Komem (2003), proposed a computational method for numeri-cal calculations of the amount of chemisorbed species and the electrostatic po-tential barrier, which are crucial for simulation of the sensor response. In the proposed method, Wolkenstein’s suggestions about the relationship between the gas chemisorption on semiconductors and the filling of the surface electronic lev-els were used (Wolkenstein 1963). This proposition actually complicates the rate equations because the chemisorbed species are split into two groups: neutral che-misorbed species with empty electronic levels and ionized chemisorbed species with occupied electronic levels. Excluding the detailed analysis of the occupation mechanism, a simplified description of the occupation probability of the chemi-sorption-induced surface electronic states is proposed in this approach. The de-scription of the occupation is actually based on a trivial use of the Fermi-Dirac distribution function as follows:

--

-

é ùæ ö- ÷çê ú= + - ÷ç ÷÷çê úè ø+ ë û

1

0 1 exp F SSE ENkTN N

(1.41)

Here N0 is the number of neutral adsorbates per unit surface area, N− is the number of chemisorbed adions per unit surface area, and ESS is the energy of


chemisorption-induced electronic levels on the surfaces. Defining the coverage of the surfaces with the chemisorbed particles by

Θ- +

=0

partial*N N p

N (1.42)

where N* is the number of adsorption sites per unit surface area, one can use the rate equation (1.6) for the definition of the equilibrium problem with explicitly included factors defining the sensor properties and the surroundings. Thus, as-suming that the desorption of the neutral chemisorbed species is determined by specific desorption energy x0 and omitting the bimolecular interaction, it is pos-sible to obtain the equation for the equilibrium coverage of the surfaces produced by nondissociative chemisorption as in Rothschild and Komem (2003):

( )ξξΘ Θ ν Θ ν

π- -

æ öæ ö + - ÷÷ çç ÷÷- = - + -çç ÷÷ çç ÷÷ç ÷çè ø è ø

00* 0 0 *

0 1 exp exp2

bC SSE EpS N N

kT kTmkT (1.43)

Here S0 is the adsorption probability, n0 and n– are the desorption probabilities for neutral and ionized chemisorbed species, respectively, Q0 = N 0/N* and Q– = N −/N*, and EC

b is the conduction-band edge in the bulk of the sensor. Since the capturing of the electrons by the chemisorption-induced surface levels depend on the surface potential VS, Poisson’s equation has to be solved before calculating the surface coverage Q with the chemisorbed species.

Rothschild and Komem (2003) proposed a method for calculation of the equili brium coverage of chemisorbed species and the height of the chemisorption- induced potential barrier. This method actually enables numerical simulations of the sensor response to gases and quantitative evaluation of the effects of various parameters such as the operating temperature or the doping level on the sensor response. This method was tested by Rothschild and Komem (2003) by simulating n-type SnO2-based sensors.

Wolkenstein’s proposition (Wolkenstein 1963, 1991) about distinguishing be-tween weak (neutral chemisorbed species) and strong (ionized chemisorbed spe-cies) gas chemisorption states for the interactions between the surface and gas was successfully applied in a simulation model of the dynamic response of sen-sors to gas by Guerin et al. (2008). Following Wolkenstein’s approach, the chemi-sorption was split into two steps by Guerin et al. (2008). During the first step, the bond between the adsorbate and the substrate is weak and does not involve electronic transfer from the bulk to the surface or vice versa. The electrons of the atom or the molecule remain located in the vicinity of the adsorbate, involving a simple deformation of the orbitals. The binding energy of the adsorbate is Eads and corresponds to the loss of free energy of the system during the adsorption process.


This neutral chemisorption does not change the electrical properties of the mate-rial, but the perturbation created by the adsorbate induces a surface state ESS in the bandgap. This surface state acts as a trap for the electrons.

The second step (strong chemisorption) occurs when an electron of the con-duction band, whose energy is EC, is transferred from the semiconductor to the adsorbed species. The binding energy of the adsorbate is increased by ES = EC − ESS, that is, the loss of free energy of the system during the ionization process. This process involves the creation of a negative superficial charge and a chemisorption-induced surface potential barrier VS (VS < 0) that is defined by Poisson’s equation.

Considering these two steps, the evolution of the surface coverage with chem-isorbed gas Q is defined by the rate equation corresponding to that in (1.43), where we assumed dQ/dt = 0 at equilibrium. The rate equation includes both the weak and strong chemisorption states characterized by the ionized Q− and neutral Q0 coverage of the surface. These two parameters are related to the total coverage Q by the Fermi-Dirac statistics.

The simulation of sensor response requires simultaneous numerical solution of both the rate equation and Poisson’s equation. Moreover, some assumption must be added to obtain the relationship between the solution and the resist-ance response of the sensor to gas. For this, it is assumed that the mechanisms of adsorption–desorption at the grain surface are much slower than the genera-tion recombination effects in the semiconductor. For the free charge carriers it must be acceptable that ∂n/∂t = ∂p/∂t = 0. The continuity equation (1.5) and the transport equation (1.1) also have to be solved for the definite structure of sen-sor. Therefore, the grains in the polycrystalline sensor are assumed to be quasi-spherical, identical in size, and single-crystal. They are joined, coupled by a small contact surface that allows a high porosity. This simplified model allows one to easily determine the electrical features of a grain taking into account the mecha-nisms of the electrical conduction.

The numerical simulation of sensor response needs exact quantitative defini-tions of all the parameters included in the model. This requirement makes numer-ical simulation more difficult than the analytical approach. This is because only some estimated magnitudes are known for the required parameters. However, some parameters, such as the energy parameters, are practically unknown but are particularly critical for the simulation because a small change of these val-ues can produce large variations in the final results. Guerin et al. (2008) made a particular attempt to quantitatively define the parameters required for the simu-lation of WO3 sensors. Tabulated magnitudes of parameters can be found in this work. The response of gas sensors to ozone in dry air was calculated based on the adsorption/desorption mechanisms of the species O2, O2

−, O, and O− at the surface of the grains. The response kinetics based on the Wolkenstein adsorption theory were simulated numerically for comparatively low temperatures (less than


525 K) and nondissociative adsorption. The model was then extended to dissocia-tive adsorption at temperatures higher than 525 K.

It is evident from the reported studies that the numerical simulation of sen-sor response to gas based on any model, including the analytical models and the computational methods, aims at quantitative evaluation and fundamental under-standing of sensor functioning. Accurate description of the response becomes increasingly important for the comprehensive analysis of the complete physical-chemical picture of the conversion of the interaction between the surfaces and gases into the detectable signal and, consequently, for the intentional optimiza-tion of sensor characteristics and measurement methodologies. The numerical implementation of the model description allows one to compare theory and experi-ment by fitting the calculations according to the mathematical models with the measured quantities.

2.4. VErification of modEls

The barrier-limited conduction model accompanied by chemical rate equations has been very successful in simulating gas-sensing properties in metal oxide construction. Quantitative comparison between the outputs of simulation mod-els and experimental results allows one to evaluate both the acceptability of the model formalism and the correctness of understanding about the core factors in the analysis. Successful simulation typically results in close match between the calculated and measured parameters under a comparatively wide spectrum of test conditions. It is also possible to predict some specific features from the theoretical calculations using a simulation model of sensor response.

The analytical description of the response (1.30) with the solution (1.21) pre-dicts instability of the response of a metal oxide to gas under specific conditions defined by relations between the parameters in (1.21). It follows from (1.21) that a periodic variation of resistance should be detected for the metal oxide sensors in response to gas. The conditions under which the response oscillations may occur are dependent on a special combination of parameters, namely, the flux of particles F, the effective cross section of the chemisorption site c, the sticking probability S0, the desorption probability b, the rate of bimolecular interaction on the surfaces nOG, and the surface coverage with oxygen QO0. In addition, included in the simulation model is a series of measurable parameters that actually define the parameters explicitly (1.21). For example, the flux F is defined by the partial pressure of the corresponding gas pgas in air, the surrounding temperature T, and the mass of the gas molecule mG. The other parameters from (1.21) to (1.27) can also be defined by special relationships, including the measurable characteristics of sensor properties. Consequently, the theoretical calculations of the dependen-cies between the response and various factors in the response simulations are


extremely complicated because of lack of reliable information about the quantita-tive definitions of required parameters. Nevertheless, the simulation model allows one to reveal the key processes producing the oscillations of the sensor response to gas. Instability in the response can be obtained for special proportions between parameters defining the three steps (1.13)–(1.15) in the surface–gas interaction.

In general, the simulation model prediction is verified by a series of experimen-tal results. Galdikas et al. (2000), for instance, demonstrated experimentally that for SnO2-x-based sensors originally modified with metallic catalysts, oscillations can occur in the resistance response of these specific sensors to gas under specific conditions. It was found that, if an odor with a fixed amount of formaldehyde was tested, the oscillations of the resistance were typically detected by these sensors. The effect was verified by comparing the response to the odors of three analogous solutions, glucose–glycine, glucose–glycine–formaldehyde, and glucose– glycine–acetaldehyde. There were no other types of sensors with oscillations of the re-sponse, but the oscillations were reproduced in other similar sensors under the same conditions. It must be noted here that the oscillations of the response were detected only within a strictly limited interval of sensor temperatures.

The study of Galdikas et al. (2000) is not the only one that has reported about the oscillations of the resistance response. The oscillations of the sensor response are frequently detected for CO gas. It was reported by Nitta et al. (1978) and Nitta and Haradome (1979) that, in a ThO2-doped SnO2 sensor, a new self-oscillation phenomenon was found only when the sensor was exposed to CO gas. Based on experimental results, this phenomenon was related to the environ-mental CO gas concentration, substrate temperature, and applied voltage. It was demonstrated that the oscillation is extremely sensitive to the concentration of CO gas and can be detected especially in the region of 0.2–0.3%. The effect was also detected for a SnO2 gas sensor in a study by Nakata et al. (1996). The simul-taneous measurement of the temperature on the semiconductor surface and the conductance suggested that the temperature change was a key variable in the oscillatory phenomenon. As a preliminary theoretical model, a simulation was performed using the surface concentration of CO and the temperature as two independent variables.

In addition, an oscillation has also been detected in the response to H2 gas in air (for details, see Kanefusa et al. 1981). Sensors based on sintered SnO2 with mixture of additives Pd, Rh, and MgO were used in these tests. It was experimen-tally demonstrated that this phenomenon depends on the H2 gas concentration, the working device temperature, and the voltage applied to the device.

The simulation model (1.30) with the solution (1.22) results in the transient response defined by monotonically changing signal that approaches constant magnitude asymptotically as t ® ¥ . This result has been commonly verified by numerous experiments for various sensors tested under various conditions. Actually, it proves that the conditions for the parameters in (1.22) are typically


acceptable for most practical metal oxide sensors. In these cases, the formula (1.30) with definition (1.22) seems to be less advantageous compared to other simulation models proposed for comparatively simple forms of the response tran-sient in various studies. In general, any relationship containing an exponent can be acceptable for simulation of the sensor response under typical conditions. Compared to this, the most outstanding advantage of the simulation model based on (1.30) with (1.22) is in adequate description of several specific aspects of the response kinetics in metal oxide sensors.

The time constant t defined in (1.27) and included in (1.22) actually de-termines the response time for the metal oxide sensors (see, e.g., Šetkus 2002). Considering the commonly known relationship between the response time and the amount of gas, the most unusual conclusion follows from the simulation model (1.27) for the time constant in the interval of comparatively low amount of the impurity gas. The simulation based on (1.27) leads to a singular minimum in the dependency of the time constant t on the partial gas pressure pgas. This result can be obtained for comparatively low desorption rate bG and the high rate for the LH step nOG. The calculated dependencies were proved experimentally for a SnO2-based sensor exposed to H2 gas by Šetkus (2002). Šetkus et al. (2004, 2005) also demonstrated that the proportions between the rates can be changed by sur-face modifications with catalytic metals. These modifications results in significant change of the dependencies of the time constant t versus the partial pressure of the impurity gas pgas. The rates of the processes, namely chemisorption, desorp-tion, and bimolecular interaction, can be increased or decreased depending on the metal and the amount of it if the impurity only partially covers the surfaces of the basic oxide film. The comparison between the simulation model and experi-ment was performed for the SnO2-based sensors exposed to pure H2 and CO gases in air.

Asymmetry in the rise and the fall of the resistance response to gas also is given by the simulation model (1.30) with (1.22). The response time of the sensor is determined by the three steps of the gas–surface interaction during injection of impurity gas into the atmosphere. The time constant t (1.27) depends on the adsorption flux FG, desorption rate bG, and the high rate for the LH step nOG. After restoration of clean air conditions, the time constant that defines the time of the response fall depends only on the desorption rate bG and the high rate for the LH step nOG. Consequently, the transient signals are defined by individual time con-stants for the rise and fall of the response of the sensor to gas.

In general, a good match between simulation of the response and experimen-tal results can be obtained only if similar conditions are applied to the simulation and the experimental measurements. Consequently, it is acceptable to fit the cal-culations of the response defined by (1.30) and (1.22) only if a steplike change in gas composition is accomplished during the experiments. However, sometimes it is possible to modify the simulation model by simple substitution of parameters,


aiming to evaluate the influence of specific conditions. Based on the formalism in Section 2.2.2, the resistance response of metal oxide sensors can be obtained for varying amounts of impurity gas, as demonstrated in Šetkus (2004). In this study, the transient signals of the response were obtained for periodically varying amounts of impurity gas in air. In general, various special functions can be used for the simulation of the response of sensor to varying amounts of impurity gas by the formalism based on (1.30) and (1.22).

Discrepancies between the theoretical simulation of the sensor response and the measured results typically occur due to mismatch between the conditions of theory and those of experiment, and due to lack of specific aspects in the model. Though the origin of the discrepancies is not always clear, the situation can fre-quently be improved by adding some details in the theoretical description of the response that solve the problem of the mismatch.

An attempt to improve the commonly known approach of thermally stimulated electron transport in metal oxide sensors with intergranular junctions was made by introducing the tunneling effect. Lee (2006) assumed that if a gas sensor is highly doped, up to degeneracy, then the Fermi level lies above the bottom of the conduction band. Because of heavy doping, the depletion region is very thin and at low temperatures free electrons with energy close to the Fermi level can tunnel across the double Schottki barrier between the grains. Though the proposition was not verified by reliable results, it was suggested that the tunneling effect can be important at comparatively low temperatures. At practically important temper-atures, however, thermoemission transport over the barrier should be dominant.

A much more successful attempt to introduce the tunneling effect in the simu-lation of sensor resistance response was presented by Malagu et al. (2009). In this study, it was suggested that after oxygen adsorption, the barrier height and the depletion width become larger and, as a consequence, the sample resistance increases. Different factors (such as type of defects, morphology, and additives) contribute to the electrical response of the gas sensor. The barrier height VS of the intergranular junction was related to the characteristics of the depletion region by the following solution of Poisson’s equation:

Λ Λε

-æ ö÷ç= - ÷ç ÷çè ø

22 121

2 3d

Sq N

qV R (1.44)

Here R is the radius of the particle, L is the depletion-layer width, and Nd is the donor density. After thermal treatment of metal oxide sensors, the donor concen-tration Nd defined by the oxygen vacancies in the bulk typically decreases, which can lead to an enlargement of the depletion layer (increase in L). These effects pro-duce an increase of the grain boundary resistance. In all the temperature range for such grain boundary conditions, the tunneling current can be more than an


order of magnitude larger than the thermionic current. Consequently, the electric current in the metal oxide sensor must be calculated according to the two-term definition as follows:

( ) ( )æ ö÷ç= + - ÷ç ÷÷çè øò 2

0

expSV

SVATJ f E P E dE ATk kT

(1.45)

In the definition of current, the first term corresponds to the tunneling cur-rent and the second to the thermionic current. The parameters A and k are the Richardson and Boltzmann constants, and f (E ) is the Fermi-Dirac distribution. P (E ) is the transmission probability for a reverse-biased Schottky barrier, and it is defined by a special function by Malagu et al. (2009). The contribution of the tunneling current was verified by experimental results of impedance spectroscopy in the study of Malagu et al. (2009).

In metal oxide sensors the electronic processes can also be accompanied by ionic drift–diffusion processes caused by the drift–diffusion of the particles ad-sorbed on the surface, as suggested by Liess (2002) and Simakov et al. (2006). Simulation of the electrical current in the sensor in this approach includes varia-tion of the surface ions with the position between the electrical contacts of the sensor. Simakov et al. (2006) proposed modifying the rate equation for the che-misorbed particles [similar to (1.16) and (1.17)] by including the ion migration on the surfaces. Simplifying the analysis by the limit of the equilibrium state, the rate equation was proposed in the following form:

( ) ( )γννν -æ öæ ö ÷÷ çç ÷÷- - - - =çç ÷÷ çç ÷÷ç ç+ +è ø è ø

1 0A ad A a ia S A

a a

N nN n dIN N qw

n n n n dx (1.46)

Here νa is the gas particle adsorption effective frequency, νd = 1/τl is the desorption effective frequency, and νr is the effective frequency of the reaction between the oxidizing and reducing gas at the surface. NA is the surface density of the particles absorbed from the gas, and NS is the maximum (saturated) density at the anode of the sensor [NS = NA(x = L), with L being the length of the sensor]. The ion current is defined by the relationship

µ-=i A iI qN Ew (1.47)

with mobility of ions mi, external electric field E, elementary electrical charge q, contact width w, density of the chemisorbed gas ions NA

−= Nan/(n + na). n is the concentration of the free bulk electrons and na is the electron concentration cor-responding to the Fermi level, equal to the energy of the surface acceptor level Ea. A solution of (1.46) was obtained for the distribution of electrical charge between


the contacts of a sensor by Simakov et al. (2006). Unfortunately, the simulation of the electrical current in the sensor was not supported by adequate experimen-tal results. The measured dependence of current versus applied voltage (the I–V characteristic) does not directly represent the distribution of the surface electrical charge (or chemisorbed gas ions) in the sensor. Therefore, an improvement of the simulation model of the electrical response of sensor to gas by migration of che-misorbed ions on the surfaces of sensors still needs more detailed studies about the simulation model and much more reliable verification of this model performed under adequate experimental conditions.

Varying density of chemisorbed gas particles and possible migration of these particles on the surfaces suggested an original modification of the simulation model for resistive gas sensors. Varpula et al. (2008) assumed that oxygen ion-ization on the surfaces of sensors due to chemical reactions is equivalent to the phenomenon of electron trapping in the surface states. Based on this, it was ac-cepted that free electrons in the gas sensors can be trapped on the surfaces of the grains or grain boundaries in two kinds of surface states: intrinsic and extrinsic surface states. The intrinsic states are composed of the states which are created by the existence of the surface itself and the states which are created by surface impurities, doping, and surface defects. The extrinsic surface states are created by adsorbed gas atoms or molecules at the surface. If the electron is trapped in an adsorbed atom or molecule (extrinsic state), creating an ion, this process is called ionization. In other words, electron trapping and oxygen ionization is the same phenomenon. The simulation model of sensor conductance of Varpula et al. (2008) suggests an influence of external electric field on trapping of electrons into the extrinsic states due to a change in the grain boundary characteristics produced by the applied electric field. Poisson’s equation is solved for the sensor without and with an applied electric field, assuming that the complete length of the depletion region (xd1 + xd2) at the contact between adjacent grains with the double Schottky barrier is constant and independent of the field. By solving the continuity equation (1.1), the current density through the potential barrier is ob-tained in the following form:

µæ ö æ ö÷ ÷ç ç= - ×÷ ÷ç ç÷ ÷÷ ÷ç çè ø è ø

max 0 bar0 exp sinh

2B

dqV qU

J q N EkT kT

(1.48)

where Nd is the donor concentration, E0max is the maximum applied electric field

producing the potential shift equal to the barrier height, and Ubar is the applied voltage. An initial grain boundary potential VB0 is defined by the following solution of Poisson’s equation:

ε

=2

0 8B

Bd

qNVN

(1.49)


where NB is the charge density on the extrinsic states on the grain boundary. Based on the same equations, the density of trapped electrons on the grain

boundary is defined by Varpula et al. (2008) by

æ ö æ ö÷ ÷ç ç= - ×÷ ÷ç ç÷ ÷÷ ÷ç çè ø è ø

0 barexp cosh2

BB d

qV qUn N

kT kT (1.50)

The simulation model proposed by Varpula et al. (2008) suggests two regions in the dependence of the response versus applied voltage: a linear regime at low voltages and a superlinear regime at high voltages. Consequently, the extension of the conductive model used for simulation of barrier-limited electron transport in gas sensors allows one to combine the conduction and surface-state models and to obtain a description of nonlinear I –V characteristics in which the linear, sublinear, and superlinear current regimes are detected. The model simulation was applied and verified by response tests on WO3-based sensors in the study of Varpula et al. (2008). A good fit between experimental data and the proposed model was obtained. According to the model, sensitivity degradation occurs only in the regime where traps are almost already filled in thermodynamic equilibrium in clean air. In the opposite regime, where the traps are emptier, the model suggests constant or slightly increasing sensitivity with the bias voltage. The generality of the proposed model allows investigation of the fundamental physical-chemical phenomenon, the trapping process, in all resistive metal oxide sensors.

3. resIstIve sensors

3.1. introductory rEmarks

Resistive gas sensors are still the most-exploited type of sensors for both research projects and practical applications. Simplicity of the resistance measurements is the most significant advantage of these sensors over the other ones, especially in commercial devices. Moreover, it has been demonstrated by various studies that comparatively simple models of electric charge transport can adequately explain the most essential features of sensor functioning and the most promising develop-ments in sensor technology. Since these models can typically be described by standard fundamental equations, it would seem that simulation of the response would be quite an easy task. However, there are at least two aspects in the simula-tion models that must be carefully considered.

First, resistive gas sensors are typically based on nonhomogeneous materials. In general, such sensors can be described as some complicated network of semi-conducting elements each of which can be individually analyzed in terms of the basic equations describing the electronic properties. Therefore, the effect of the


complicated structure of the material is typically excluded from sensor simula-tion models by limiting the analysis to the single most essential part of sensor— assuming that this selected part completely determines the functioning of the sensor. This limitation must be sufficiently justified for the sensor simulation model to be accepted as reliable.

Second, simulation models of the resistance response are frequently verified by comparing the calculated data with experimental results obtained for sensors with two electrodes. These can be highly affected by the electrode resistances caus-ing a significant mismatch between the calculated and measured responses, be-cause the simulation models typically do not include the effects of the electrodes.

The most systematic studies are currently required for development of re-sponse simulation models for new resistive gas sensors based on nanostructured and organic materials. Attempts have been made to adapt the classical models of electrical properties to describe the processes in constructions with nanometer dimensions. However, simulation of the resistance response in nanosensors and molecular sensors is still challenging for sensor researchers, and highly innova-tive approaches are probably necessary. In this section, an overview of existing approaches to response simulation is presented for several of the most popular types of resistive sensors.

3.2. polycrystallinE films

To date, gas sensors based on metal oxide films with polycrystalline structure have been the most popular type of sensors to be investigated in various studies and applied in practical devices. Extensive research and development of these sensors has been highly stimulated by the commercial successes of Figaro, Inc. (Japan) since 1969. Basic understanding has been deepened and simulation models of sensor responses have been extended to explain various aspects of the response mechanisms in metal oxide gas sensors, as has already been presented in Section 2.2 of this chapter. It is commonly recognized that the structure of the sensor is important in the development of simulation models of the response. In addition, it is already proved that some processes determining the response can be related only to the polycrystalline structure of sensors. One such process is gas diffusion in and out of the pores in polycrystalline materials. A model for simulation of the response, including the influence of gas diffusion, has been developed and thor-oughly analyzed in a series of investigations (Sakai et al. 2001; Matsunaga et al. 2002, 2003).

The basis of the model rests on the understanding that, at equilibrium, the gas concentration inside the porous sensing layer will decrease with increasing depth due to diffusion-limited access. As a result, the gas concentration profile inside the film must depend on the rates of gas diffusion and the surface reaction.


In fact, the diffusion and the surface reaction are competing processes, the first of which results in an increase in the quantity of gas amount inside the film, while the second consumes the gas and, in general, limits the diffusion into the film.

Assuming that the diffusion of gas is effective only into the depth of the film, the process can be accepted as a one-dimensional problem with the main axis directed toward the film depth. Then, the access of gas to the surfaces of grains in a polycrystalline sensitive film can be characterized by the rate equation

( ) ( )

( )¶ ¶

= -¶ ¶

2

2

, ,,

C x t C x tD kC x t

t t (1.51)

Here C (x, t ) is the target gas concentration, D is the diffusion coefficient, k is the reaction constant, x is the depth from the surface, and t is time. The first term in the right side of Eq. (1.51) represents diffusion of gas, while the second defines the surface chemical reaction.

The partial differential equation (1.51) can be solved only if the boundary and initial conditions are defined. For this, several details about the model must be known. In the studies of Sakai et al. (2001), Matsunaga et al. (2002), and Matsunaga et al. (2003), it was supposed that the sensor is like a plate with thick-ness 2L and the target gas can access both surfaces of the plate. Before the test, there are no gases inside the sensor and, consequently, C (x, 0) = 0 is the initial condition for this problem. After injection of the target gas into the atmosphere, the amount of gas is CS at both surfaces of the plate representing the sensor. The target gas is supplied continually at the same rate, so the amount CS is constant in this analysis. Based on this assumption, the boundary conditions are defined as C (0, t ) = C (2L, t ) = CS. Using these initial and boundary conditions it is pos-sible to obtain the solution for Eq. (1.51). Details about how to obtain the solution are precisely described by, e.g., Matsunaga et al. (2003). Here we present only the final solution, which describes the changes in the target gas concentration profile in a platelike polycrystalline sensor after exposure to a constant amount of gas CS. The gas concentration profile is defined by the following solution of Eq. (1.51):

( ) ( ) πωì üï ïï ïé ù= - × - + ×í ýê úë ûï ïï ïî þ

2, 1 exp sin2S n

n xC x t C A t BL

(1.52)

( )

π

¥

=

- -= å

1

1 12n

n

An

(1.53)

( ) ( )

( )ω ω

ω ω

é ù- + × -ê úë û=é ù+ - -ê úë û

2 2

2 2

1 1 exp

1 exp

n n

n n

k t tB

k t (1.54)


πω = 1 2

2nn D

L (1.55)

The model developed in the series of studies by Sakai et al. (2001), Matsunaga et al. (2002), and Matsunaga et al. (2003) gives sophisticated variation of the depth profile of the amount of gas in the porous film, as shown by Eqs. (1.52)–(1.55). The profile of the amount of gas C (x, t ) is represented by a monotonic function with the film depth at some periods of time, but there are time intervals when the profile is defined by a unique shape. For the definite time interval, the profile of the amount of gas has a maximum at some x in the depth of the film. It was proposed that the maximum appears as a result of contribution by the surface reaction.

Analogous to (1.52), a profile C*(x, t ) can be obtained for the sensor after the injection of gas into the atmosphere is stopped. The details can be found in the cited works by Sakai et al. (2001), Matsunaga et al. (2002), and Matsunaga et al. (2003).

Aiming to simulate the sensor response that is influenced by gas diffusion, one can use a commonly known empirical formula for the sensor electrical con-ductance s, which can be written in the following form:

σ σ γ= +0mC (1.56)

Here s0 is the clean-air conductance; g and m are the variable constants. The amount of gas C should be replaced by (1.52)–(1.55). It must be noted here that it is complicated to use the model developed by Sakai et al. (2001), Matsunaga et al. (2002), and Matsunaga et al. (2003) for the simulation of sensor response because the definition (1.56) is acceptable only for a thin piece of sensor plate defined by dx. To obtain the sensor resistance, integration of parallel slices of the sensor dx over the thickness of the sensor is required. Quantitative evaluation of gas diffusion on the sensor response was illustrated by the numerically simulated detection of sensor signal by Boeker et al. (2002). Supposing a series of repeating short pulses for the injection of target gas, Boeker et al. (2002) demonstrated that the sensor signal shows a “pseudo-drift” over long time compared to the short pe-riod of a single pulse of injected gas. The simulation proved that the drift resulted from the analytes remaining within the film. The period of drift was significantly shorter for a comparatively large amount of the target gas than it was for lower concentrations, when only very slow drift of the response signal was detected in the numerical experiment by Boeker et al. (2002).

The structure of polycrystalline gas sensors can be important not only for gas access to the surfaces of the sensor, but also to the electron transport that determines the electrical response of the sensor to the gas. Numerous funda-mental studies have proposed simulation models for the electrical properties in non homogeneous semiconductors that are based on the theory of solid-state


electronics. However, models for the simulation of metal oxide gas sensors typi-cally include only some specific aspects of the electrical properties of polycrystal-line films. An influence of proportions between the length of the depletion region and the grain size is the most frequently discussed problem for the sensor re-sponse models. An attempt to include this effect in the simulation of sensor re-sponse was described in recent reports by Yamazoe and Shimanoe (2010, 2011).

Yamazoe and Shimanoe (2010, 2011) assumed that the occupation of the electronic states localized on the surfaces of grains is determined by the charac-teristics of the depletion region if the length of the region defined by the Debye screening length LD is comparable to the radius of the crystals, a. Since the den-sity of the surface states depends on the amount and type of chemisorbed gas species, an influence of the grain size on the sensor response to gas can be in-cluded in the simulation model. For this, however, a few general assumptions are implicitly accepted, simplifying the theoretical definition of the model called “receptor action of grain” by Yamazoe and Shimanoe (2010, 2011). These simpli-fications extremely restrict the acceptability of the simulation model for analysis of the resistance response of metal oxide polycrystalline sensors to both reducing and oxidizing gases.

The first assumption introduces an imaginary flat zone approach that omits the fundamental origin of surface electronic levels. It is also assumed that, in an almost insulating grain, the trapping of electrons into the surface levels created by chemisorption can be simply described by statistics based on the Fermi distri-bution, which allow calculation of the occupation of electron levels in the bulk of semiconductors. Based on this, a relationship between the grain size and electron transport is derived in the following form (Yamazoe and Shimanoe 2010, 2011):

[ ]

∆ æ öæ ö ÷ç÷ç ÷= = × × ç÷ç ÷÷ ç÷ç ÷çè ø è ø

2

0 2exp exp6

d F

DS

N E aR Re kT L

(1.57)

where Nd is the concentration of donors, [e]S is the density of conduction electrons on the surfaces of grains, R0 is the resistance of the sensor corresponding to the flat-band assumption, LD is the Debye length, and DEF is the Fermi energy shift due to the band bending at the surface.

The second assumption excludes adsorption and desorption of reducing gas from the response mechanism. It is assumed that the adsorption and desorption determines only the coverage of the grain surfaces with oxygen species, while the density of reducing gas particles reacting with the surface oxygen is simply pro-portional to the partial pressure of the gas in the atmosphere. Therefore, only the rate equation for oxygen is included in the model in the following form (Yamazoe and Shimanoe 2010, 2011):


[ ]-

- --

é ùê úë û é ù é ù= - -ê ú ê úë û ë û22

1 O2 1 2 H2

OO OS

dk P e k k P

dt (1.58)

Here [O−] is the density of the chemisorbed oxygen species O−, PO2 and PH2 are the partial pressures of oxygen and reducing gas in the atmosphere, respectively, k1 and k−1 are the rate constants of the forward and reverse interactions between the oxygen and the grain surfaces, respectively, and k2 is the rate constant of the reaction between the reducing gas (H2 in this case) and the surface oxygen. This simplification allows excluding from the model the coupling between the two rate equations for both the oxygen and the reducing gas that was described by Eqs. (1.17) and (1.18) in this chapter.

The third general assumption of Yamazoe and Shimanoe (2010, 2011) defines the interaction between the grain surfaces and the two types of oxidizing gases. In the original study, oxygen and NO2 gases were considered. It was assumed that the total surface electrical charge QSC was the same in both atmospheres, that is, the clean air with only [O−] and the air with NO2 gas. It was also assumed that the proportion between the amounts of the two chemisorbed gas species was mainly determined only by the adsorption rates. Therefore, the exposure to NO2 in air produced a change of the surface coverage with the species [O−] and [NO2

−] that can be described by the following rate equations:

[ ]-

--

é ùê úë û é ù= - ê úë û2

2 NO2 2 2

NONOS

dk P e k

dt (1.59)

[ ]-

--

é ùê úë û é ù= - ê úë û2

1 O2 1

OOS

dk P e k

dt (1.60)

Here k2 and k−2 are the rate constants of adsorption and desorption of the oxidiz-ing gas (NO2), respectively, and PNO2 is the partial pressure of NO2 gas in air.

Details about deriving the expressions that are acceptable for simulation of rate transients of sensor response to both reducing and oxidizing gases are thoroughly described in the reports of Yamazoe and Shimanoe (2010, 2011). In these studies, problems were solved for the responses of a semiconductor gas sensor to particular gases, that is, O2, H2, and NO2, however, the simulation model is acceptable for sensors exposed to other analogous gases. In each case analyzed, the rate con-stant (reciprocal of time constant) is determined as a combination of three terms, which reflect the rate of surface reactions, the semiconductor properties of the constituent crystals, and the size (and shape) of the crystals. One of the terms in


the description of the simulation model displays the influence of the depletion re-gion on the sensor response. Yamazoe and Shimanoe (2010, 2011) found that the rate of response is significantly reduced if the proportion a/LD increases beyond 5 for spherical crystals. In addition, in Yamazoe and Shimanoe (2011) there are detailed instructions about how to derive an explicit equation for the response to reducing gases (represented by H2). In that study, the simulation model was used to reveal explicitly how various physicochemical parameters contribute to the gas response as well as to find a way to proper analyses of response data.

It must be noted here that the model of Yamazoe and Shimanoe (2010, 2011) contains a large number of original parameters. These parameters are hardly re-placeable by the characteristics commonly used for description of surface pro-cesses and electronic properties in fields such as surface science. Therefore, the simulation of sensor responses is actually dependent on a large number of vari-ables that must be independently defined and that can hardly be quantitatively evaluated using the results of these specialized studies.

The most recent experimental investigations, namely, those of Šetkus et al. (2009) and Bukauskas et al. (2010), demonstrated that a decrease in the grain size down to nanometer scale creates additional problems in simulation of the sensor resistance response of metal oxide gas sensors. It was found that the structure of the films can be significantly altered by a comparatively low external voltage applied to the film. For SnO2-x and In2O3 films with thickness < 30 nm and nanometer-size grains (about 10–20 nm), the grains are physically rearranged by applied voltages exceeding some critical magnitude [about 1 V according to Šetkus et al. (2009)]. The structural changes are dependent on the surrounding atmosphere and are comparatively larger in air with CO than in clean air, as was found by Bukauskas et al. (2010). In metal oxide nanosystems, the electron transport is accompanied by permanent changes in the surface properties and the intergranular junctions that seem to originate from lattice oxygens hopping through the vacancies along the external electric field. Therefore, the classical approaches for improving the re-sistance simulations models for nanosized metal oxides can hardly be successful. Novel methods must be used for these metal oxide–based nanosystems.

3.3. nanostructurEd films

Nanostructured films can be obtained from polycrystalline metal oxides in which the grain size is drastically reduced to the dimensions comparable to the nano-meter scale. It is expected that the properties of the nanostructured materials may be much more acceptable than those used previously for the development of gas sensors. According to Di Francia et al. (2009), reducing sensors to nanome-ter size may be a way to make use of specific low-dimension properties such as quantum effects, high surface-to-volume ratio, controlled structure with a specific


surface termination, nanoparticle doping, morphology, aggregation, and nanoma-terial agglomeration state. However, the technology of chemical nanosensors is still being developed; consequently, the properties of these sensors are not usually reproducible in practice, and the quantitative definition of the characteristics is not very reliable. As a result, effort is still necessary to develop simulation models for calculation of the response. In this respect, the models proposed so far seem to be acceptablre as ideas for consideration in further studies.

Straightforward transfer of ideas about gas-sensing mechanisms in metal ox-ides to nanosensors is the most typical approach in the development of response simulation models. This approach seems most justified for films with nano-metric grains, as proposed by Malagu et al. (2008). Highly reduced dimensions of grains are comparable with the length of the depletion region at the surfaces of the nanograins, similar to the ideas discussed in Section 2.3.2. Malagu et al. (2008) proposed including both the thermionic and the tunneling current com-ponents in the simulation of electron transport in films with junctions between the nanograins and the corresponding potential barriers. Accepting the Wentzel-Kramers-Brillouin approximation to a double parabolic barrier at the junction between the nanograins (Bender and Orszag 1978; Sakurai 1993), the electrical current in the films is defined as

= +1 2I I I (1.61)

where I1 is the thermionic current component,

( )α

α αé ùæ ö× ÷çê ú÷= - × +ç ÷ê úç ÷÷çè øê úë û

ò1

100

0

2expS S kT yeV eV

I dkT kT E

(1.62)

and I2 is the tunneling current component,

æ ö÷ç= - ÷ç ÷÷çè ø2 exp SeV

IkT

(1.63)

where

( ) ( )( )α

α α αα

é ù+ -ê ú= - - × ê úê úë û

0.50.5

0.5

1 11 lny (1.64)

α =S

EeV

(1.65)


π ε¥

æ ö÷ç= × ÷ç ÷÷çè ø

1 2

*4dNehE

m (1.66)

In these definitions, E is the energy of an electron in the conduction band. The two-component current model allows one to prove numerically that the

electrical current in films with nanograins is lower than with comparatively large grains, because the increase in thermionic current across the junction between the nanograins is practically compensated by the significant decrease in the tun-neling current in films with nanograins. Assuming that the barrier height VS de-pends on the gas chemisorption, the response of nanograin-based films to gas can be simulated by an electrical current model analogous to that defined by Eq. (1.61).

An exchange of electrons between the bulk and the surface states that results in variation of the characteristics of the surface depletion layer is the core ef-fect in most of the simulation models proposed for nanostructured gas-sensitive materials. In a series of studies, by Lupan et al. (2010), Hongsith et al. (2010), and Dmitriev et al. (2007), the simulation of the response of nanowires to gas was based on variation of the characteristics of the depletion region and corre-sponding changes in the conductive electrons. Lupan et al. (2010) proposed two components for description of ZnO nanowire response to H2 gas: an electron con-centration term and a term of geometric factors. Accepting

π µ= ×2

0 0rG qnl

(1.67)

for the conductance of the nanowire, the change in electrical conductance of the nanowire exposed to gas (the electron concentration term) was determined by Lupan et al. (2010) as a change in concentration of charge carriers Dn as follows:

∆∆

= S

g g

nGG n

(1.68)

Here DnS = ng − n0, ng and n0 are the concentrations of charge carriers correspond-ing to gas and clean air, respectively, and r and l are the radius and length of the nanowire, respectively.

The gas response term due to the changes in geometric factors includes the length of the depletion area and, according to Lupan et al. (2010), can be obtained from the following expression:

( ) ( )εε∆λ λ

æ ö- ÷ç ÷= = × - = × -ç ÷ç ÷çè ø

1 21 2 1 20

0

2 2g aDDa Dg Sa Sg

g g

G GG V VG G r r qN

(1.69)


Here lDa and lDg are the Debye radius in clean air and gas, respectively, and VSa and VSg are the band bending at the surfaces of nanowire in the clean air and gas, respectively.

In general, the response simulation model for a nanowire sensor of Lupan et al. (2010) was supported by the study of Hongsith et al. (2010), where the re-sponse of a ZnO nanowire sensor to ethanol gas was described by the model. It must be noted here that only one term was used in the model by Hongsith et al. (2010), and it contained a specific time constant because the rate equations of the chemical reactions were used for development of the simulation model. In addi-tion, it was suggested that the expression of the nanowire sensor response to gas can be used not only for simulation of ZnO-based nanosensors but also for other sensitive metal oxide nanostructures.

The effect of narrowing of the channel for the electron transport in nanow-ires can be enhanced by intentional modulation of the nanowire diameter, as proposed by Dmitriev et al. (2007). In contrast to the work of Lupan et al. (2010) and Hongsith et al. (2010), the nanowire in the Dmitriev et al. (2007) study was assumed to be constructed from two segments with equal length L but different cross sections, namely, a thick segment with radius rD and a thin segment with radius r. The electrical resistance of the nanowire with two segments and the con-tacts connected in series is simply defined by

ρ βπ

æ ö÷ç ÷= × + +ç ÷ç ÷çè ø

2 2

seg 2 221

DD

L r rRrr r

(1.70)

Here r is the initial resistivity of the segments and b is determined by a geometric factor a that is independent of the radius of the nanowires (b = pa/2rL). Accepting an influence of the ionosorbed oxygen on thickness W of the depletion layer and, consequently, on the effective diameter of the thin nanowire, r = r0 + W, the resis-tance response of the segmented nanowire to gas can be calculated according to the following expression:

∆ β

-æ ö æ ö÷ ÷ç ç÷ ÷= » - × +ç ç÷ ÷ç ç÷ ÷÷ çç è øè ø

12 2seg

seg 2seg 0

21 1D

R r rSR rr

(1.71)

This simulation model was applied to SnO2-based segmented nanowires by Dmitriev et al. (2007). In the calculations of Dmitriev et al. (2007), the thick seg-ments were accepted as being rD = 500 nm, while diameter of thin segments r was varied from 10 to 60 nm. The simulation by Dmitriev et al. (2007) proved that the response of segmented nanowires to gas is significantly higher than that of straight (without the “necks”) nanowires.


Much more sophisticated methods can also be used for simulation of the resistance response of nanowires to adsorption of gas. A nonequilibrium Green’s function formalism was used for analysis of electron-transport properties in atomic-chain-scaled Si nanowires with two gold electrodes by Zhang et al. (2009). The current–voltage and conductance–voltage dependencies were calculated for the nanowires without and with adsorbed gas molecules. The simulation demon-strated that the conductance of Si nanowire depends on the distance between the adsorption site and the electrode, though the overall transport properties are hardly dependent on the gas adsorption. In fact, the simulation of Zhang et al. (2009) suggests that the ionosorption-dependent depletion-layer approach may not be acceptable for models of the response mechanism in comparatively very short nanowires. The outcome of Zhang et al. (2009) was supported by analo-gous results for single-walled carbon nanotubes (SWCNTs) by Hsieh et al. (2011). Similar dependence of the electronic conductance on the distance between the gas adsorption location and the electrode was obtained for SWCNTs by Hsieh et al. (2011). The results were obtained from simulation of the conductance carried out using the self-consistent nonequilibrium Green’s function method for the system Al electrode–SWCNT–Al electrode and a single NO2 molecule.

In spite of promising results, the simulation models of the response of nano-structured materials to gas seem hardly acceptable as reliable at the present state of investigations. Attempts to adapt the well-known approaches approved for comparatively large sensors still have to be justified theoretically. Various investi-gations focused on nanosystems suggest that unique and unexpected phenomena may exist in the systems at nanometer scale. These phenomena have to be con-sidered in the development of simulation models for the nanosensors.

3.4. conductiVE polymEr layErs

In numerous studies (see reviews, e.g., Lange et al. 2008; Lu et al. 2011; Nambiar and Yeow 2011), conductive polymers, have been shown to be acceptable for ap-plication in chemical sensors and biosensors because of their highly attractive properties. In spite of extensive investigations and promising application tests, however, fundamental understanding of the processes and mechanisms in these materials must still be significantly improved. Theoretical descriptions of electri-cal properties and electrical responses to chemical interactions are only rarely proposed in scientific publications. Published theoretical studies frequently rep-resent attempts to test separated ideas of diverse approaches. In general, a com-paratively large variety of mechanisms have been proposed in different studies for the description of electrical properties in the conductive polymers.

Lei et al. (2007) proposed a mechanism based on varying distances be-tween the conductive carbon particles to describe the response to gas in a sensor


consisting of insulating polymers and dispersed carbon black particles. Based on this mechanism, the mathematical description of the response of the sensor to gas (chemical vapors) combines the dependence of electrical resistance on the concentration of carbon black with the dependence of the volume fraction of com-ponents in the composite on the vapor pressure of the vapor.

The resistivity of the sensor, defined as the resistivity of the composite mate-rial, rm, can be derived from the following equation obtained on the basis of gene-ral effective media:

( ) ( ) ( )ρ ρ ρ ρ

ρ ρ ρ ρ

- - - -

- - - -

× - - × -+ =

- -

1 1 1 1

1 1 1 1

10

k k k kC m C mk k k k

C m C m

f f

B B (1.72)

where

-

=1 C

C

fB

f (1.73)

rC and rP are the resistivities of the conductive component [carbon black in Lei et al. (2007)] and the polymer, respectively, k is an exponent, f is the volume fraction of the conductive component in the sensor, and fC is the critical volume fraction of this conductive component at the percolation threshold. The volume fraction of the other components in the sensor is defined by (1 − f ), which, in general, includes both the polymer and the vapor particles and consequently de-pends on the surrounding atmosphere. Supposing that the sensor response is produced by only one target gas, the volume fraction of the nonconductive com-ponents (1 − f ) is defined by the following relationship:

( )

-=

+ -0

01CCA

A p CC

f fff f f f

(1.74)

Here fA and fp are the volume fractions of the absorbed vapor particles and the polymer in the sensor; f0CC is the volume fraction of the conductive component in the sensor in the clean air. It follows from (1.74) that (1 − f ) = fp in the sensor in clean air (fA = 0).

Lei et al. (2007), defined the dependence of absorption of the target gas (vapor) by the sensor material on the vapor pressure in terms of thermodynamic theory. According to the definition, the partial pressure of the target gas is related to the volume fraction of the absorbed gas in the composite structure of the sensor. Lei et al. (2007) write this relationship as follows:

χæ ö æ ö÷ç ÷ç÷ = + + + ×ç ÷ç÷ç ÷ç÷ç è øè ø

2sat

1

1ln ln 1p A AP f f f

mP (1.75)


Here P and Psat are the partial pressure and the saturated partial pressure of the target gas, respectively, c is the gas–polymer interaction parameter, m = vp/vA, and vp and vA are the molar volumes of the polymer and the gas, respectively.

According to the simulation model of Lei et al. (2007), the resistance response (rAm − r0m) of a conductive polymer sensor to gas with partial pressure P can be obtained by calculating r0m and rAm from (1.72) with (1.73)–(1.75) for the clean air and the air with the target gas, respectively. This resistance response simulation model is acceptable for evaluation of the sensor responses to the vapor at various amounts of the vapor.

Absorption of analytes (vapor) from gaseous surroundings can directly change the electrical properties of conducting polymer sensors, as a result of localiza-tion/delocalization of electrons at the absorption sites, similar to some extent to the chemisorption effects in metal oxide–based gas sensors. Transfer of electrons between the adsorbed analyte and the conductive polymer can change the resis-tance of the sensor because the doping technology in most conductive polymers is based on the changes produced by redox reactions. In spite of existing qualita-tive interpretations of the effect, the mechanisms determining conversion of the analyte–polymer interaction into a response signal still have to be described in more detail, the absence of which is crucial for development of simulation models of sensor response. The state of the art of studies of conductive polymer sensors was reviewed recently by Bai and Shi (2007).

It is commonly accepted now that conductive polymers can be used success-fully for development of chemical sensors, producing a resistance response to volatile chemical compounds. However, more information and better understand-ing is required for development of strictly defined theoretical descriptions of the conversion of the interaction between the conductive polymers and analytes into the response signal.

3.5. molEcular structurEs

Molecular-scale electronic devices typically are assumed to be practical imple-mentations of understanding about the processes and mechanisms specific to systems with single molecules. In these systems, the molecules themselves can emulate the properties of well-known solid-state devices and can stand for novel nanosystems with unique properties. Fundamental studies of charge transport through an interface between metals or semiconductors and organic or inorganic molecules not only opened various ways for innovative developments of molecular electronics but also revealed serious problems defining new challenges. Research reviews on this topic include, among others, those of Weiss et al. (2007) and Heath (2009). It is recognised by researchers that the ultimate aim of studies on electron transport in single molecules is to define the influence of metal electrodes on the


energy spectrum of the molecule and to understand how the electron-transport properties of the molecule depend on the strength of the electronic coupling be-tween it and the electrodes. A variety of phenomena are observed to influence the charge transport in these systems, including those acceptable for recognition of chemical compounds and chemical interactions between the molecular system and the surroundings.

Depending on the mechanism determining the charge transport through the molecular systems, it is possible to classify the emerging approaches in the re-sponse of electrical parameters to chemical surrounding as the bridging-effect–recognizing systems, interaction-dependent tunneling transport devices, and organic molecule detectors.

The molecular bridging system reviewed by Lindsay et al. (2010) consists of two metal electrodes separated by a gap L. The electric current through the gap can be originated by the electrons in the existing highest occupied energy state close to the Fermi energy, EF. The potential barrier that retains electrons within the metal is V, so the work function, φ, is given by φ = V − EF. In the absence of a state close to EF, in the gap the tunnel conductance is given approximately by

( ) ( )ϕ β» - × = -0 0exp 1.02 expG G L G L (1.76)

where G0 is the quantum of conductance, φ is the work function in electron volts, L is the tunnel gap in angstroms, and b is the inverse electronic decay length. A real molecule spanning the gap will consist of molecular orbitals yn, often well described in terms of linear combinations of the atomic orbitals of the constituent atoms and the hopping matrix elements between adjacent orbitals. An electron is propagated from an electrode to the opposite one through all the paths made of interacting orbitals and connecting both electrodes. Mathematically, the trans-mission is calculated with a Green’s function,

( )Ψ Ψ

εµ

- +å,0

| |n nL R

nn

L RG E

E E i (1.77)

Here, L and R represent states at the energy E on the left and right electrodes, E0n

is the eigenenergy of the nth eigenstate of the system, and ε is an infinitesimal. The presence of G (E ) in the transmission includes the energy levels of the mol-ecule and naturally introduces the possibility of resonances and multiple reflec-tions. Therefore, the electron current through the molecular bridge between the metal electrodes is individual to the molecule type. This current appears only if the molecule binds to both electrodes. Based on this simulation model, a possibil-ity exists to identify molecules and specific molecular structures, as suggested by Lindsay et al. (2010).


A simulation model of a molecular wire containing a redox system was pro-posed by Crus et al. (2010). In this study, interaction of the redox system with a classical solvent was analyzed, assuming that the solvent state was represented by a solvent coordinate q in the spirit of the Marcus theory (Marcus 1956). Using the wide-band approximation, an exact expression for the quantum conductance of a chain of arbitrary length was derived in matrix form:

Tr aL Rg G Gγ Γ Γé ù= ê úë û (1.78)

Here the subscripts L and R refer to the left and right reservoirs (electrodes), G denotes the imaginary part of the self-energy, G is the Green’s function obtained from a special Hamiltonian defining the model system. The index r represents the redox species on the molecular wire, while a means the chain of atoms. Based on (1.78), an influence of electrochemical environment on the molecular wire with the redox system was demonstrated by Crus et al. (2010) by means of explicit calcula-tions performed for a chain of three atoms.

A classical approach in analysis of a one-dimensional conductivity problem (molecular wire) was proposed by Koslowski and Wilkening (2010). In this work, the solution of the problem is based on Kirchhoff’s second law applied to a resis-tor network with the nodes corresponding to individual molecules. The nodes are connected by channels characterized by local conductance G. It is assumed that an individual node can only be occupied by a limited number of charge carriers. Electron transfer between the nodes is defined by a classical Master equation that relates the occupation probability of individual node pi and the transfer rates be-tween the adjacent nodes, wij.

In general, the equations can be solved only numerically, but analytical ex-pressions can be obtained for illustration in a highly simplified system. This sim-plified system contains two sites connected to two leads. Conductance between the two sites is accepted to be G12 = G, and the conductances between sites and adjacent electrodes are G1e and G2e, respectively. Supposing p1 and p2 as the occu-pation probabilities of an individual site, V1 and V2 are the site voltages, while V1e and V2e are the external voltages applied to the system electrodes. For this simpli-fied system of Koslowski (2010), the current is defined by the following analytical expression:

( )

( ) ( )-

=+ + -

21 2 1 1

21 2 1 1 2 1

11

e e

e e e e

G G Gp pI

G G G p G G p (1.79)

This expression gives a general idea about the parameters that are important for the simulation model of a molecular wire. It also follows from this expression that the current depends on the structure and the components of the wire and, consequently, it can be used for identification of the molecule in a single-molecule


wire. Moreover, supposing that the conductivity of the components and occupa-tion of the sites can be influenced by an interaction between the molecular wire and the surrounding media, the theoretical model proposed by Koslowski and Wilkening (2010) can be adapted for simulation of an electrical response of a mo-lecular wire to the chemical surroundings.

4. concludIng comments

This chapter has provided an overview of simulation models of resistive gas sen-sors reported in the literature over a considerable period of research and appli-cation activities. The systematic search for publications with suggestions about sensor response simulations has revealed that theoretical calculations of the re-sponse are frequently based on empirical formulas aiming to quantitatively define some specific characteristics of sensor response to gas. The detailed description of gas sensor properties acceptable for the basic simulation of sensor responses to gas actually requires specific combinations of approaches from diverse sci-entific areas such as state electronics, semiconductor physics, surface states, heterogeneous catalysis, adsorption, transport in gases, etc. This considerably complicates the simulation of the sensor response and, possibly, explains the relative lack of reports about such studies in the literature. It must be noted here that various aspects of the problems have already been analyzed in specific stud-ies focused on surface chemical interactions, properties of the surface electronic states, electrical properties of nanostructured semiconductors, etc., and reported in numerous specialized scientific journals. Justified usage of proposed ideas and integration of the diverse problem descriptions into a systematic model seem to be the main difficulties in the development of a well-organized, reliable simulation model of gas sensor response.

It is obvious from the literature that the significant reduction of sensors down to the nanometer scale and to the molecule size creates new challenges of gas sensor modeling by introducing unexpected and unique features of the nano-systems. The methods and the physical theories of the nanosystems that are intensively discussed in the dedicated publications are hardly reflected in the gas sensor studies. However, the idea is increasingly supported that the pro-cesses in naonsystems can hardly be simulated by classical physical models with simply reduced dimensions. It seems that the processes and the mechanisms in solid nanosystems are frequently highly specific and cannot be predicted by these classical models, as demonstrated, for instance, for TiO2 based nanosystems by Strukov et al. (2008). These fundamental problems in understanding the response mechanisms may be the root cause of unexpectedly slow progress in obtaining considerable improvements in gas-sensing characteristics using the promising technologies of nanostructured sensors.


The existing simulation models are mainly acceptable for quantitative descrip-tion of some fundamental aspects in the sensor response to gas and are useful in explaining the basic mechanisms of sensor functioning. However, the influence of the basic parameters of sensing materials on the sensor response to gas is in-sufficiently defined. Attempts to improve the simulation models of the resistance response to gas are frequently aimed at enhancing the description of the elec-tronic processes at the surfaces in the semiconducting materials of the sensors. A distinct lack of explicit description of the relationship between the sensor char-acteristics and the properties of both the materials and the constructions is likely discouraging developers from using numerical tests in research and development projects. Therefore, fundamental studies focused on enhancing the theory of gas sensors and response simulation models seems to be highly important in accel-erating further progress in gas sensor technology and applications. It is hoped that the overview in this chapter about the present state of response simulation models will be interesting for gas sensor developers and persuade them to carry out much more systematic research on sensor technologies and functioning.

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Momentum Press is proud to bring to you Chemical Sensors: Simulation and Modeling Volume 2: Conductometric-Type Sensors, edited by Ghenadii Korotcenkov. This is the second of a new five-volume comprehensive refer-ence work that provides computer simulation and modeling techniques in various fields of chemical sensing and the important applications for chemical sensing such as bulk and surface diffusion, adsorption, surface reactions, sintering, conductivity, mass transport, and interphase interactions. In this second volume, you will find background and guidance on:

• Phenomenologicalmodelingandcomputationaldesignof conductometric chemical sensors,basedonnanostructured materials such as metal oxides, carbon nanotubes, and graphenes

• Approachesusedtoquantitativelyevaluatecharacteristicsofsensitivestructuresinwhichelectricchargetransport depends on the interaction between the surfaces of the structures and chemical compounds in the surroundings

Chemical sensors are integral to the automation of myriad industrial processes and everyday monitoring of such activities as public safety, engine performance, medical therapeutics, and many more. This five-volume reference work covering simulation and modeling will serve as the perfect complement to Momentum Press’s 6-volume reference work, Chemical Sensors: Fundamentals of Sensing Materials and Chemical Sensors: Compre-hensive Sensor Technologies, which present detailed information related to materials, technologies, construction, and application of various devices for chemical sensing. Each simulation and modeling volume in the present series reviews modeling principles and approaches peculiar to specific groups of materials and devices applied for chemical sensing.

About the editorGhenadii Korotcenkov received his Ph.D. in Physics and Technology of Semiconductor Materials and De-vices in 1976, and his Habilitate Degree (Dr.Sci.) in Physics and Mathematics of Semiconductors and Dielec-trics in 1990. For many years, he was a leader of the Gas Sensor Group, and manager of various national and international scientific and engineering projects carried out in the Laboratory of Micro- and Optoelectronics, Technical University of Moldova. Currently, Dr. Korotcenkov is a research Professor at the Gwangju Institute of Science and Technology, Republic of Korea. His research has included significant work on Schottky barri-ers, MOS structures, native oxides, and photo receivers on the base of III-Vs compounds. He continues with research in various aspects of materials sciences and surface science, with a particular focus on nanostructured metal oxides and solid state gas sensor design. Dr. Korotcenkov is the author or editor of eleven books and spe-cial issues, eleven invited review papers, seventeen book chapters, and more than 190 peer-reviewed articles. HisresearchactivitieshavebeenhonoredwiththeAwardoftheSupremeCouncilofScienceandAdvancedTechnology of the Republic of Moldova (2004) and The Prize of the Presidents of the Ukrainian, Belarus and MoldovanAcademiesofSciences(2003),amongmanyothers.

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CHEMICAL SENSORS VoLuME 2: ConduCtoMEtrIC-typE SEnSorS Edited by Ghenadii Korotcenkov, ph.d., dr. Sci.

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