Handbook of infrared spectroscopy of ultrathin films

By understanding optical theory for stratified media, it is also possible todistinguish optical effects artifacts, which present in each IR spectrum of anultrathin film, and, hence, to avo

HANDBOOK OF

INFRARED SPECTROSCOPY

OF ULTRATHIN FILMS

HANDBOOK OF

INFRARED SPECTROSCOPY

OF ULTRATHIN FILMS

Valeri P Tolstoy Irina V Chernyshova Valeri A Skryshevsky

A JOHN WILEY & SONS, INC., PUBLICATION

Copyright  2003 by John Wiley & Sons, Inc All rights reserved.

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Published simultaneously in Canada.

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Library of Congress Cataloging-in-Publication Data:

Tolstoy, V P (Valeri P.)

Handbook of infrared spectroscopy of ultrathin films / V P.

Tolstoy, I V Chernyshova, V A Skryshevsky.

p cm.

ISBN 0-471-35404-X (alk paper)

1 Thin films — Optical properties 2 Infrared spectroscopy.

I Chernyshova, I V (Irina V.) II Skryshevsky, V A (Valeri

A.) III Title.

1.1 Macroscopic Theory of Propagation of Electromagnetic

Waves in Infinite Medium / 2

1.2 Modeling Optical Properties of a Material / 10

1.3 Classical Dispersion Models of Absorption / 13

1.4 Propagation of IR Radiation through Planar Interface

between Two Isotropic Media / 24

1.4.1 Transparent Media / 26

1.4.2 General Case / 29

1.5 Reflection of Radiation at Planar Interface Covered by

Single Layer / 31

1.6 Transmission of Layer Located at Interface between Two

Isotropic Semi-infinite Media / 39

1.7 System of Plane–Parallel Layers: Matrix Method / 43

1.8 Energy Absorption in Layered Media / 49

1.8.1 External Reflection: Transparent Substrates / 50

1.8.2 External Reflection: Metallic Substrates / 52

1.8.3 ATR / 55

1.9 Effective Medium Theory / 60

1.10 Diffuse Reflection and Transmission / 65

Appendix / 68

References / 70

v

2 Optimum Conditions for Recording Infrared Spectra of

2.3.3 Buried Metal Layer Substrates

2.6 Choosing Appropriate IR Spectroscopic Method for

Layer on Flat Surface / 118

2.7 Coatings on Powders, Fibers, and Matte Surfaces / 120

2.7.1 Transmission / 120

2.7.2 Diffuse Transmittance and Diffuse

Reflectance / 122

2.7.4 Comparison of IR Spectroscopic Methods for

Studying Ultrathin Films on Powders / 130References / 133

3 Interpretation of IR Spectra of Ultrathin Films 140

3.1 Dependence of Transmission, ATR, and IRRAS Spectra of

Ultrathin Films on Polarization (Berreman Effect) / 141

3.2 Theory of Berreman Effect / 146

3.2.1 Surface Modes / 147

3.2.2 Modes in Ultrathin Films / 151

3.2.3 Identification of Berreman Effect in IR Spectra of

Ultrathin Films / 1573.3 Optical Effect: Film Thickness, Angle of Incidence, and

Immersion / 159

CONTENTS vii

3.3.1 Effect in “Metallic” IRRAS / 159

3.3.2 Effect in “Transparent” IRRAS / 164

3.3.3 Effect in ATR Spectra / 167

3.3.4 Effect in Transmission Spectra / 169

3.4 Optical Effect: Band Shapes in IRRAS as Function of OpticalProperties of Substrate / 171

3.5 Optical Property Gradients at Substrate–Layer Interface: Effect

on Band Intensities in IRRAS / 175

3.9 IR Spectra of Inhomogeneous Films and Films on Powders

and Rough Surfaces Surface Enhancement / 219

3.9.1 Manifestation of Particle Shape in IR

Spectra / 2203.9.2 Coated Particles / 223

3.9.3 Composite, Porous, and Discontinuous

3.9.4 Interpretation of IR Surface-Enhanced

Spectra / 2323.9.5 Rough Surfaces / 241

3.10 Determination of Optical Constants of Isotropic Ultrathin

Films: Experimental Errors in Reflectivity

Measurements / 243

3.11 Determination of Molecular Packing and Orientation in

Ultrathin Films: Anisotropic Optical Constants of Ultrathin

3.11.1 Order–Disorder Transition / 253

3.11.2 Packing and Symmetry of Ultrathin Films / 257

3.11.3 Orientation / 266

3.11.4 Surface Selection Rule for Dielectrics / 280

3.11.5 Optimum Conditions for MO Studies / 282

References / 284

4 Equipment and Techniques 307

4.1 Techniques for Recording IR Spectra of Ultrathin Films on

4.4 Mapping, Imaging, and Photon Scanning Tunneling

Microscopy / 352

4.5 Temperature-and-Environment Programmed Chambers for In

Situ Studies of Ultrathin Films on Bulk and Powdered

Supports / 356

4.6 Technical Aspects of In Situ IR Spectroscopy of Ultrathin

Films at Solid–Liquid and Solid–Solid Interfaces / 360

4.8 IRRAS of Air–Water Interface / 381

4.9 Dynamic IR Spectroscopy / 383

4.9.1 Time Domain / 383

4.9.2 Frequency Domain: Potential-Modulation

Spectroscopy / 3874.10 Preparation of Substrates / 389

4.10.1 Cleaning of IREs / 389

5.1 Thermal SiO2 Layers / 416

5.2 Low-Temperature SiO2 Layers / 421

5.3 Ultrathin SiO2 Layers / 427

5.4 Silicon Nitride, Oxynitride, and Carbon Nitride

5.6.2 Boron Nitride and Carbide Films / 448

5.7 Porous Silicon Layers / 450

5.8 Other Dielectric Layers Used in Microelectronics / 454

5.8.1 CaF2, BaF2, and SrF2 Layers / 454

Al– PSG Interface / 484

6.2.3 Determination of Metal Film and Oxide Layer

Thicknesses in MOS Devices / 4866.3 Modification of Oxides in Metal–Same-Metal Oxide–InP

Devices / 488

6.4 Dielectric Layers in Sandwiched Semiconductor

Structures / 492

6.4.1 Silicon-on-Insulator / 492

6.4.2 Polycrystalline Silicon–c-Si Interface / 493

6.4.3 SiO2 Films in Bonded Si Wafers / 494

6.6.2 Cleaning and Etching of Si Surfaces / 504

6.6.3 Initial Stages of Oxidation of H-Terminated Si

Surface / 506References / 508

7 Ultrathin Films at Gas– Solid, Gas– Liquid, and Solid– Liquid

7.1 IR Spectroscopic Study of Adsorption from Gaseous Phase:

Catalysis / 514

7.1.1 Adsorption on Powders / 515

7.1.2 Adsorption on Bulk Metals / 527

7.2 Native Oxides: Atmospheric Corrosion and Corrosion

Inhibition / 532

7.3 Adsorption on Flat Surfaces of Dielectrics and

Semiconductors / 542

7.4 Adsorption on Minerals: Comparison of Data Obtained

In Situ and Ex Situ / 547

7.4.1 Characterization of Mineral Surface after Grinding:

Adsorption of Inorganic Species / 5477.4.2 Adsorption of Oleate on Calcium

Minerals / 5517.4.3 Structure of Adsorbed Films of Long-Chain

Amines on Silicates / 5547.4.4 Interaction of Xanthate with Sulfides / 561

CONTENTS xi

7.5 Electrochemical Reactions at Semiconducting Electrodes:

Comparison of Different In Situ Techniques / 570

7.5.1 Anodic Oxidation of Semiconductors / 571

7.5.2 Anodic Reactions at Sulfide Electrodes in Presence ofXanthate / 583

7.6 Static and Dynamic Studies of Metal Electrode–Electrolyte

Interface: Structure of Double Layer / 595

7.7 Thin Polymer Films, Polymer Surfaces, and

Polymer–Substrate Interface / 600

7.8 Interfacial Behavior of Biomolecules and Bacteria / 613

7.8.1 Adsorption of Proteins and Model Molecules at

Different Interfaces / 6147.8.2 Membranes / 624

In this book, we will designate ultrathin films, or, as they are also called inthe literature, nanolayers, to mean layers ranging from submonolayers to severalmonolayers; these may be formed from a wide range of organic and inorganicsubstances or present adsorbed atoms, molecules, biological species, on a sub-strate or at the interface of two media These films play an important role in manycurrent areas of research in science and technology, such as submicroelectronics,optoelectronics, optics, bioscience, flotation, materials science of catalysts, sor-bents, pigments, protective and passivating coatings, and sensors It could even beargued that the rapid advances in thin-film technology has necessitated the devel-opment of special approaches in the synthesis and investigation of nanolayers andsuperlattices Nowadays, these approaches are generally applicable in so-callednanotechnology, which includes the synthesis–deposition and characterization

of ultrathin films with a prescribed composition, morphology–architecture andthicknesses on the order of 1 nm

Common features in all studies in the field of nanotechnology arise fromproblems connected with the physicochemical investigation of ultrathin films,which originate in general from their extremely small thickness To solve theseproblems, a number of technically complicated physical methods that oper-ate under UHV conditions, such as AES, XPS, LEED, HREELS are used.Infrared (IR) spectroscopy and in particular Fourier transform IR (FTIR) spec-troscopy — a method that enables the determination of molecular composition andstructure — offers important advantages in that the measurements can be carriedout for nanolayers located not only on a solid substrate but also at solid–gaseous,solid–liquid, liquid–gaseous, and solid–solid interfaces, including semiconduc-tor–semiconductor, semiconductor–dielectric, or semiconductor–metal, with nodestruction of either medium Thus IR spectroscopy is one of a few physicalmethods that can be used for both in situ studies of various processes on surfaceand at interfaces and technological monitoring of thin-film structures in fieldssuch as microelectronics or optoelectronics under serial production conditions.The versatility of modern FTIR spectroscopy provides means to characterizeultrathin coatings on both oversized objects (e.g., works of art) and small (10–20-

µm) single particles, substrates with unusual shapes (e.g., electronic boards), andrecessed areas (e.g., internal surfaces of tubes) It should be emphasized that IR

xiii

spectroscopy can be highly sensitive to ultrathin films: Depending on the system,the sensitivity is 10−5–10% monolayer.

However, the various IR spectroscopy techniques must be adapted to measurespectra of very small amounts of substance in the form of ultrathin films Whilefor analyses of bulk materials it is possible to select the optimum mass of sub-stance to record its spectrum, in the case of ultrathin films it is only possible tovary the conditions under which the spectra are recorded (measurement technique,polarization, angle of incidence, immersion media, number of radiation passagesthrough the sample) For this purpose, it is necessary first to theoretically assessthe effect of the recording conditions on the intensity of the absorption bands

By understanding optical theory for stratified media, it is also possible todistinguish optical effects (artifacts), which present in each IR spectrum of anultrathin film, and, hence, to avoid misinterpretation of the experimental data.Although the optimum conditions for a number of simple systems are known(e.g., for ultrathin films on metals reflection–absorption (IRRAS) at grazingangles of incidence is commonly used, and for ultrathin films on transparentsubstrates, multiple internal reflection (MIR) is most suitable in many cases),the spectral contrast can be further enhanced by employing additional specialtechnical approaches

The material in this handbook is presented in such a way as to address theseissues Thus, the theoretical concepts associated with the interaction of IR radia-tion with matter and with thin films are considered and, to evaluate the optimumconditions routinely, simple algorithms for programming are given in Chapter 1

In Chapter 2, a theoretical evaluation of the optimum conditions for measuringnanolayer spectra is presented In Chapter 3, a more detailed interpretation of the

IR spectra of ultrathin films on flat and powdered substrates is discussed fromthe viewpoint of optical theory, including the authors’ methods of determination

of the optical constants of ultrathin films [1, 2] and molecular orientation [3] InChapter 4, technical approaches to measure good-quality IR spectra of ultrathinfilms are considered Along with the techniques considered routine for studies

of ultrathin films in many laboratories and original techniques described in theliterature, techniques developed by the authors are included here, namely IRspectroscopy of single and multiple transmission inp-polarized radiation [4, 5];

IRRAS of the surface of semiconductors and dielectrics [6]; IRRAS of metals,semiconductors, and dielectrics in immersion media [7, 8]; diffuse transmission

of disperse materials [9, 10]; and the special attenuated total reflection (ATR)technique for studying the semiconductor–solution interface [11] An impor-tant feature of the optical accessories described is that they are placed in thesample compartment of a conventional continuous-scan or step-scan FTIR spec-trometer, without any change in the spectrometer optical scheme In addition,different attachments for specialized measurements are considered, including

IR microscope objectives, in situ chambers, and spectroelectrochemical cells.Time-resolved spectroscopy and enhanced surface and structure sensitivity in IR

PREFACE xv

spectroscopic techniques such as modulation spectroscopy and two-dimensionalcorrelation analysis are described Subsequent chapters illustrate some appli-cations of these techniques in the study of thin-layer structures and semicon-ductor–electrolyte interfaces, which are currently of great practical significance

in electronics, solar energy storage, sensors, catalysis, bioscience, flotation, andcorrosion inhibition Recommendations are given regarding application of thesetechniques for automated and on-line analysis of thin-film structures PAS [12,13] are not included here, because its sensitivity is insufficient for the measure-ments in question Infrared emission studies of ultrathin films have recently beenreviewed [14, 15] Since this method, although rather surface sensitive under spe-cific conditions, has not yet experienced extensive application, it is not consideredhere as well Instead, fundamentals and techniques of transmission, diffuse trans-mission, IRRAS, ATR, and DRIFTS methods will be presented, with emphasis

on their application to ultrathin films Complementary information on how to usethese methods and their history may be found in other monographs [16–21]

It should be noted that because the material is presented in such a manner,this monograph may serve as a handbook It includes the theoretical founda-tions for the interaction of IR radiation with thin films, as well as the optimumconditions of measuring spectra of various systems, which are analyzed by com-puter experiments and illustrated by specific examples Complementary to this,the basic literature devoted to the application of IR spectroscopy in the inves-tigation of nanolayers of solids and interfaces is presented, and the necessaryreference material for the interpretation of spectra is tabulated Thus this bookwill be extremely useful for any laboratory employing IR spectroscopy, and foreach industrial firm involved in the production of thin-film structures, as well

as by final-year and postgraduate students specializing in the fields of optics,spectroscopy, or semiconductor technology

Dr Valeri Tolstoy (St Petersburg State University, Russia) authored Chapter 2and Sections 3.5, 3.10, 4.1.1, 4.1.2, 4.2.1, 4.2.2, 4.4, 4.5, 7.1–7.3 Dr IrinaChernyshova (St Petersburg State Polytechnical University, Russia) wroteChapters 1, 3, 4, and 7 (except for the sections mentioned above) andcoauthored Sections 2.3, 2.5, and 2.7 Prof Valeri Skryshevsky (Kyiv NationalTaras Shevchenko University, Ukraine) presents Chapters 5 and 6 Tables

in the Appendix were collected by Valeri Tolstoy and Irina Chernyshova.The language and style were edited by Dr Roberta Silerova (University ofSaskatchewan, Canada) Dr Nadezhda Reutova (St Petersburg State University,Russia) translated into English Chapters 2, 5, and 6 and Sections 3.3–3.5, 3.10,4.1, and 4.2, and helped with translation of Chapter 1 and Sections 3.1 and 3.2

Acknowledgements

We thank Dr Silerova for editing the English language, improving the handbookstyle, and her great patience during the joint work several years long; Dr Reutova

for translating part of the text into English; and all the authors who have mitted citation of their results VPT acknowledges a partial financial supportgranted by Interfanional Scientific Foundation (USA), Russian Foundation forBasic Research (RFBR), and Civilian Research and Development Foundation(CRDF) (USA) IVC thanks RFBR, the Swedish Institute, and Elena Chernet-skaya VAS thanks Ministry of Ukraine for Education and Science We all thankfriends and relatives who helped us with this work.

per-REFERENCES

1 V P Tolstoy, G N Kusnetsova, and I I Shaganov, J Appl Spectrosc 40, 978 (1984).

2 V P Tolstoy and S N Gruzinov, Opt Spektrosc 71, 129 (1991).

3 I V Chernyshova and K Hanumantha Rao, J Phys Chem B 105, 810 (2001).

4 V P Tolstoy, L P Bogdanova, and V B Aleskovski, Doklady AN USSR, 291, 913

(1986) (in Russian).

5 V P Tolstoy, A I Somsikov, and V B Aleskovski, USSR Inventor Certificate

No 1099255, Byul Izobr 23, 145 (1984).

6 S N Gruzinov and V P Tolstoy, J Appl Spectrosc 46, 480 (1987).

7 V P Tolstoy and V N Krylov, Opt Spectrosc 55, 647 (1983).

8 V P Tolstoy and S N Gruzinov, Opt Spectrosc 63, 489 (1987).

9 S P Shcherbakov, E D Kriveleva, V P Tolstoy, and A I Somsikov, Russ J.

Equipment Tech Exper 1, 159 (1992).

10 V P Tolstoy and S P Shcherbakov, J Appl Spectrosc 57, 577 (1992).

11 I V Chernyshova and V P Tolstoy, Appl Spectrosc 49, 665 (1995).

12 J F McClelland, R W Jones, S Luo, and L M Seaverson, in P B Coleman (Ed.),

Practical Sampling Techniques for Infrared Analysis, CRC Press, Boca Raton, FL,

1993, Chapter 5.

13 J F McClelland, S J Bajic, R W Jones, and L M Seaverson, in F M Mirabella

(Ed.), Modern Techniques in Applied Molecular Spectroscopy, Wiley, New York,

1998, Chapter 6.

14 W Suetaka, Surface Infrared and Raman Spectroscopy Methods and Applications,

Plenum, New York, 1995, Chapter 4.

15 S Zhang, F S Franke, and T M Niemczyk, in F M Mirabella (Ed.), Modern

Techniques in Applied Molecular Spectroscopy, Wiley, New York, 1998, Chapter 9.

16 F M Mirabella (Ed.), Modern Techniques in Applied Molecular Spectroscopy, Wiley,

New York, 1998.

17 W Suetaka, Surface Infrared and Raman Spectroscopy Methods and Applications,

Plenum, New York, 1995.

18 F Mirabella (Ed.), Internal Reflection Spectroscopy, Marcel Dekker, New York, 1993.

19 P B Coleman, Practical Sampling Techniques for Infrared Analysis, CRC Press,

Boca Raton, FL, 1993.

PREFACE xvii

20 J Workman, Jr and A W Springsteen (Eds.), Applied Spectroscopy: A Compact

Reference for Practitioners, Academic, San Diego, 1998.

21 J Chalmers and P Griffiths (Eds.), Handbook of Vibrational Spectroscopy, Vol 1,

Wiley, New York, 2002, Chapter 1.

VALERITOLSTOY

IRINACHERNYSHOVA

VALERISKRYSHEVSKY

ACRONYMS AND SYMBOLS

ARUPS angle-resolved UV photoelectron spectroscopy

DTGS deuterated triglicine sulfate (detector)

DTIFTS diffuse transmittance infrared Fourier transform

spectroscopy

xix

EDS energy dispersive X-ray spectroscopy

EMIRS electrochemically modulated infrared spectroscopy

EXAFS extended X-ray absorption fine-structure

HREELS high resolution electron energy loss spectroscopy

IRRAS infrared reflection absorption spectroscopy

L monolayer Langmuir monolayer

ACRONYMS AND SYMBOLS xxi

MIRE multireflection internal reflection element

OTTLE optically transparent thin-layer electrochemical (cell)

PDIR potential-difference infrared (spectroscopy)

RBS Rutherford backscattering

SIPOS semi-insulating polycrystalline silicon

SNIFTIRS subtractively normalized interfacial FTIR spectroscopy

SPAIRS single potential alternation IR spectroscopy

STIRS surface titration by internal reflectance spectroscopy

TIRF total internal reflection fluorescence

ACRONYMS AND SYMBOLS xxiii

X, EX, BX, AX xanthate (ethyl-,n-butyl-, amyl-)

XANES X-ray absorption near-edge structure

2D IR two dimensional correlation analysis of IR dynamic

ε Permittivity≡dielectric constant≡dielectric function

ε∞ High-frequency dielectric constant≡screening factor

εst Static (low-frequency) dielectric constant

In the infrared (IR) spectroscopic range (200–4000 cm−1), radiation is generallycharacterized by its wavenumber ν (cm−1), related to the wavelength λ (µm),frequency ˜ν (s−1), and angular frequencyω (s−1) as

When IR radiation containing a broad range of frequencies passes through

a sample, which can be represented as a system of oscillators with resonancefrequenciesν0,i, then according to the Bohr rule,

(whereE is the difference between the energy of the oscillator in the excited

and ground states,ν is the frequency of photons, and h = 6.626069 × 10−34 J·s

is Planck’s constant), photons with frequenciesν = ν0,i will be absorbed Thesephotons will be eliminated from the initial composition of the radiation Since

all the elementary excitations have unique energy levels (fingerprints),

mea-surements of the disappearing energy as a function of ν (absorption spectrum

of the sample) enable these excitations to be identified, and microscopic mation about the sample (e.g., molecular identity and conformation, intra- andintermolecular interactions, or crystal-field effects, etc.) may be obtained.†

infor-†See M Hollas, Fundamental Aspects of Vibrational Spectroscopy Modern Spectroscopy , 3rd ed.,

Wiley, Chichester, 1996.

HANDBOOK OF

INFRARED SPECTROSCOPY

OF ULTRATHIN FILMS

Valeri P Tolstoy Irina V Chernyshova Valeri A Skryshevsky

A JOHN WILEY & SONS, INC., PUBLICATION

ABSORPTION AND REFLECTION OF INFRARED RADIATION BY

ULTRATHIN FILMS

The primary characteristics that one identifies from infrared (IR) spectra ofultrathin films for further analysis are the resonance frequencies, oscillatorstrengths (extinction coefficients), and damping (bandwidths), related to differentkinds of vibrational, translational, and frustrated rotational motion inside thethin-film material [1–7] However, the microscopic processes inside or at thesurface of a film (motion of atoms and electrons) give rise to the frequency

dependence (the dispersion) not only of the extinction coefficient but also of

the refractive index of the film As a result, a real IR spectrum of an ultrathin

film is, as a rule, distorted by so-called optical effects Specifically, the spectrum

strongly depends upon the conditions of the measurement, the film thickness,and the optical parameters of the surroundings and substrate impeding extraction

of physically meaningful information from the spectrum Thus after introduction

of the nomenclature accepted in optical spectroscopy and a brief discussion ofthe physical mechanisms responsible for absorption by solids on a qualitative

level, this introductory chapter will concentrate on the basic macroscopic or phenomenological theory of the optical response of an ultrathin film immobilized

on a surface or at an interface

The theoretical analysis of the IR spectra of ultrathin films on various strates and at interfaces will involve two assumptions: (1) the problem is linearand (2) the system under investigation is macroscopic; that is, one can usethe macroscopic Maxwell formulas containing the local permittivity The firstassumption is valid only for weak fields The second assumption means that thevolume considered for averaging, a (the volume in which the local permittiv-

sub-ity is formed), is lower than the parameter of inhomogenesub-ity of the medium,

d (e.g., the effective thickness of the film, the size of islands, or an effective

dimension of polariton),a < d In this case, the response of the medium to the

external electromagnetic field is essentially the response of a continuum The

1

description of the medium properties using the macroscopic permittivity is calledthe dielectric continuum theory (DCT) One issue discussed repeatedly in theliterature is the correctness of the DCT approximation for spectra of ultrathinfilms In this context, semiphenomenological and microscopic models establish-ing the correlation between local field effects, molecular and atomic dynamics,and the IR spectrum of the film on the surface were proposed (for reviews, seeRefs [1, 6, 7]) These models greatly enhance the understanding of ultrathinfilms of submonolayer coverage However, the corresponding relationships arecumbersome, deal with specific (as a rule, vacuum–metal) interfaces, require fur-ther refinements for most of the systems studied, and, hence, are unfeasible forroutine analysis of the spectra On the other hand, the macroscopic relationshipscan be used without substantial difficulties to solve many problems arising inthe spectroscopy of ultrathin films First, they allow one to choose the optimumconditions for recording IR spectra by comparing band intensities (Chapter 2),which is of prime technical importance Second, by using spectral simulationsbased on the macroscopic formulas, it is possible to distinguish optical effectsfrom physicochemical effects in the film, such as film inhomogeneity (porosity)and changes in the orientation of the film species (Chapter 3).

1.1 MACROSCOPIC THEORY OF PROPAGATION

OF ELECTROMAGNETIC WAVES IN INFINITE MEDIUM

Optical thin-film theory is essentially based on the Maxwell theory (1864) [8],

which summarizes all the empirical knowledge on electromagnetic phenomena.Light propagation, absorption, reflection, and emission by a film can be explained

based on the concept of the macroscopic dielectric function of the film

mate-rial In this section, we will present the results of the Maxwell theory relating

to an infinite medium and introduce the nomenclature used in the followingsections dealing with absorption and reflection phenomena in layered media Thebasic assertions of macroscopic electrodynamic theory can be found in numeroustextbooks (see, e.g., Refs [9–16])

charge other than that due to polarization are described by Maxwell’s equations[in International System (SI) of units]:

Here, E and H are, respectively, the vectors of the macroscopic electric and the

magnetic field; ε0= 8.854 × 10−12C·N−1·m−2 and µ0= 4π × 10−7N·A−2 are

1.1 MACROSCOPIC THEORY OF PROPAGATION OF ELECTROMAGNETIC WAVES 3

the permittivity and permeability of free space, respectively; ˆσ is the electrical

conductivity (it appears as a coefficient in Ohm’s law, j = σE, where j is the

current density), andt is time The quantities D and B are the electric

displace-ment and the magnetic induction, respectively They characterize the action of

the field on matter and are related to the field vectors by the so-called localmaterial equations

where P and M are the vectors of the electric and magnetic polarization,

respec-tively; ˆε and ˆµ represent relative local permittivity (dielectric constant) and

magnetic permeability, respectively The electric polarization of the medium, P,

is defined as the sum of the dipole moments in a unit volume of the medium:

of accuracy, the magnetic permeability ˆµ can be taken to be equal to unity in the

optical range [11] There is a straightforward correlation between the ity ˆσ of the material and the imaginary part of the permittivity ˆε [see Eq (1.16)

conductiv-below] Thus, the parameter that most completely describes the interaction ofthe IR radiation with a medium is the permittivity ˆε of the medium For an

isotropic medium (or a cubic crystal), uniaxial crystals, and the biaxial crystals

of an orthorhombic system, the permittivityˆε is a symmetric tensor of the second

rank with six distinct, complex (1.1.9◦) elements, which can be transformed to adiagonal form by the proper choice of coordinate system [9, 14–16] Depending

on the symmetry of the film, the number of different principal values of the sor ˆε can be 1, 2, or 3 (Section 3.11.2) In particular, for many materials (e.g.,

ten-quartz, tourmaline, calcite) in the visible spectral region, where their absorption

is small, two principal values of the refractive index [Eq (1.9)] may be equal,

n x = n y = n z As a result, along an arbitrary direction of propagation different

from z (z being the optical axis) there exist two independent linearly

polar-ized plane waves (1.1.5◦) traveling with different phase velocities (1.1.4◦) This

phenomenon is named double refraction, or (for the real refractive index) fringence If an anisotropic material is absorbing [e.g., in the case of the minerals

bire-mentioned above, but in the region of their lattice vibrations (1.2.4◦)], the tion coefficient (1.1.10◦) is also dependent on polarization This phenomenon is

extinc-referred to as pleochroism A special case of pleochroism is dichroism, or the

imaginary refractive index birefringence For the biaxial crystals belonging tomonoclinic and triclinic systems, the principal axes of the real and imaginaryparts of the permittivity do not coincide and the tensor ˆε cannot be put in the

diagonal form In inhomogeneous or discontinuous media, the permittivity ˆε

varies from region to region

Eqs (1.1a) and (1.1b) can be transformed into the standard wave equations,

which represents the transverse monochromatic plane wave propagating in the

direction k Here,ω is the angular frequency of the wave, r is the location vector

that depends on the coordinate system being used, E0 and H0 are the amplitudes

of the electric and magnetic field, respectively, and k is the wave vector, which

describes the propagation of the wave in the given medium The terms transverse

and longitudinal waves refer to the direction of the electric field vector E with

respect to the direction of wave propagation, k Transversality (E ⊥ H ⊥ k) of the

electromagnetic wave can be obtained by substitution of Eqs (1.5a) and (1.5b)into (1.1a) with allowance made for (1.1b) and (1.1d) (see also 1.3.7◦)

vector k This plane is characterized by a constant phase = ωt − const and is called the surface of constant phase or the wavefront The wavefront travels in a

medium in the direction k with the velocity

This is called the phase velocity of the electromagnetic wave in this medium.

phase , and the direction of propagation k, an electromagnetic wave is

char-acterized by the direction of oscillation of the electric field vector E This is

known as the polarization state of the electromagnetic wave If during wave propagation the electric vector lies in a plane, the wave is said to be lin- early polarized The plane that contains the electric vectors and the direction

of the wave propagation is called the plane of polarization Superposition of

two linearly polarized waves that are in phase results in a third linearly ized wave

rela-tionship for the wave vector:

1.1 MACROSCOPIC THEORY OF PROPAGATION OF ELECTROMAGNETIC WAVES 5

Here,c = 1/(ε0µ0)1/2is the velocity of electromagnetic radiation in vacuum The

dependence ofω on k (the energy–momentum relation) for an electromagnetic

wave propagating through a crystal is called the dispersion law Hence, Eq (1.7)

represents the dispersion law of a transverse electromagnetic wave in an infinitecrystal [17]

expres-sion for the absolute value of the phase velocity of a wave v= |v| in the medium

and, thus, the parameters ˆε and ˆn to be constant and σ = 0 However, if the

medium absorbs electromagnetic radiation, these quantities become dependent

on the frequency of incident radiation, the function ˆε(ω) termed the dielectric function Below we will neglect the so-called spatial dispersion effects [18] con-

nected with the dependence of the dielectric function on the wave vector ε(k).

This is permissible for the IR range (the limit k→ 0)

In the case of absorption of radiation, the wave is damped and the wave

equation (1.4a) transforms into

∇2E− ε0ˆεµ0ˆµ ∂2E

∂t2 − µ0µσ ∂E

wave vector, the permittivity, and the refractive index are allowed to be complexquantities,

k = k− ik= ˆks = (k− ik)s, (1.12a)

respectively Here, s is the unit vector along the direction of propagation of the

wave Notice that Eq (1.12a) is legitimate only for homogeneous plane waves,

for which k  ||k [9, 10, 14–16] The quantity ˆk = k− ik is confusingly called

the wavenumber, like the quantity ν = 1/λ Equations (1.7), (1.10), and (1.12a)

give the following relationship between the wavenumber and the refractive index:

of the optical constants:

E = E0e i(ωt−ksr) e −ksr= E0e i[ωt−(2πn/λ)sr ]e −(2πk/λ)sr (1.14)

From Eq (1.14), we can see that the real parts of the wavenumber and therefractive index,k andn, respectively, determine the phase velocity of the wave

in the medium, whereas the imaginary parts,k andk, determine the attenuation

of the electromagnetic field along the direction of propagation of the wave Byvirtue of this, the imaginary part of the refractive index,k, is called the extinction coefficient or absorption index Note that the symbol k is used to designate the

wavenumber and the extinction coefficient To distinguish between them, we labelthe wavenumber with a hat, as ˆk.

Eq (1.14) into Eq (1.11a) gives



(ε2+ ε2)1/2 + ε1/2

,

k =1 2

1.1 MACROSCOPIC THEORY OF PROPAGATION OF ELECTROMAGNETIC WAVES 7

Wavenumber, cm–1α

Figure 1.1 Frequency dependences of n, k, ε,ε, and Im(1/ˆε) for β-SiC characterized by

Fig 1.1 shows these functions forβ-SiC, which is a strong oscillator with weak

damping (1.3.20◦)

the optical constants, which do not assume a model for calculatingˆε (Section 1.3), are rigorously described by the Kramers–Kr¨onig (KK) relations, first introduced

in 1926 [19, 20],

ε(ω) − ε∞= 2

π P

 ∞0

ε(ω) − ε∞

ω2− ω2 d ω, (1.19)where P refers to the principal-value integral and ε∞ is the offset value (alsocalled the anchor point) that accounts for the background of the ε(ω) func-

tion (1.3.6◦).

The theoretical aspects of the KK relations are discussed in detail elsewhere[10, 21–23] Using the KK relations, the optical parameters of a substance can becalculated from the measured spectra [24, 25] or the absorption spectrum can be

calculated from the measured reflection spectrum [26] The latter mathematical

operation is called the KK transform(ation) The efficient algorithms for this

procedure are available for both normal and oblique incidence spectra [26–32]and are provided in most software packages of present-day Fourier transforminfrared (FTIR) spectrometers

electro-magnetic field energy normal to a unit surface containing the vectors of theelectric and magnetic field is equal to the vector product

= E × H,

called the Poynting vector or the “ray vector” Note that in the general case

of an anisotropic crystal, the direction of the Poynting vector differs from that

of the wave vector k The velocity of the energy propagation in the medium

is named the ray (or energy) velocity It can be shown [10, 14] that the phasevelocity (1.1.4◦) is the projection of the ray velocity onto the direction of thewave normal

period of oscillation, transferred by the wave across a unit area perpendicular tothe direction of the Poynting vector per unit time For a damped homogeneouswave of the type (1.14) propagating in thez-direction [6–13],

I = |
| = 1

2cnε0|E|2= 1

2cnε0|E(0)|2e −4πkz/λ , (1.20)

where k = Im(ˆn) and n = Re(ˆn) are the extinction coefficient and the refractive

index of the medium, respectively, the angular brackets denote a time average

over one period, and E(0) is the electric field at z = 0 Averaging the square

of the vector of the electric field, we have E2 =1

2|E|2, which allows one torewrite Eq (1.20) as

Thus the quantityI is proportional to the mean square of the electric field, E2 ,and the refractive index of the medium (the speed of light in the medium)

k(1.1.10◦), absorption of IR radiation is characterized by the decay constant(the absorption coefficient), the imaginary component of the permittivity, Imˆε,

and the imaginary component of the reciprocal of the permittivity, Im(1/ˆε) The

conductivity σ is reserved to describe conductors in the frequency range where

0ε [33].

α ≡ I1 d

Here, I is the intensity of electromagnetic radiation and z is the distance that

it has passed through the absorbing medium The distance 1/α measured in a

1.1 MACROSCOPIC THEORY OF PROPAGATION OF ELECTROMAGNETIC WAVES 9

direction normal to the surface plane is defined as the penetration depth of theradiation in the medium When the electromagnetic wave has propagated throughthe penetration depth, its intensity decreases by a factor ofe.

By integrating Eq (1.22), we find that the intensity of light in an absorbingmedium is attenuated according to

the wavenumber,k, the dielectric functionε, and the extinction coefficientk:

betweenα and k is demonstrated in Fig 1.1.

of the Poynting vector equals the energy dissipated from the electromagnetic fieldper unit volume per second This quantity is related to the imaginary part of thepermittivity of the medium,ε(ω), the frequency of the field, ω, and the complex

electric field vector E by

Re(∇·
) = 1

2[ε(ω)ε0ω]E · E∗= ε(ω)ε0ωE2 , (1.25)

where E2 is the mean square of the electric field Comparing Eqs (1.25)and (1.21), we see that, as expected, the rate of absorption is proportional tothe intensity of radiation and the imaginary part of the permittivity,ε(ω) The

frequency of the maximum of the functionε(ω) is, by definition, the frequency

of the transverse optical (TO) vibrations (1.3.7◦) of the medium, ωTO, and thefunction Imˆε = ε(ω) is called the TO energy loss function For vibrations with

weak damping ( 0), the frequency of the TO vibrations is close to theresonance frequency,ωTO≈ ω0 (Section 1.3) [35]

direct proportion to the quantity (Sections 1.5 and 1.6)

ˆε(ω) =

ε(ω)

known as the longitudinal optical (LO) energy loss function By definition, the

LO energy loss function has its maxima at the frequencies of the LO vibrations

of the medium,ωLO (see also 1.3.7◦)

The energy loss functions for β-SiC are shown in Fig 1.1 with the other

optical parameters

1.2 MODELING OPTICAL PROPERTIES OF A MATERIAL

In this section, the physical mechanisms and selection rules of IR absorption bybulk material due to vibrations, electronic excitations, and free carriers (elec-trons and holes) are briefly discussed on the qualitative level from the viewpoint

of quantum mechanics In general, this problem is highly specialized, and forfuller details several standard textbooks [21, 34, 36–50] are recommended All

of these mechanisms also apply to ultrathin films, but their appearance, whichwill be discussed in Chapters 3 and 5–7, is quite specific Note that although

we will discuss absorption by solids, the mechanisms to be considered are alsoapplicable to liquids

(metals and semiconductors) differ considerably from the optical properties ofdielectrics These differences can be explained in the following manner In asolid, there is an abundance of both electrons and energy levels (states) theycan occupy However, because of the periodicity of the crystal lattice, the energystates of electrons are confined to energy bands The highest fully occupied energy

band is called the valence band The lowest partially filled or completely empty energy band is called the conduction band The distance between the bottom of the conduction band and the top of the valence band is called the band gap.

The energy of the band gap is symbolized by E g (Fig 1.2) The conductivity

is determined not only by the amount of free electrons (holes) in the material,but also by the number of vacant places for their transfer Therefore, the lowestconductivity will be observed when the valence band is completely filled and

thus when the conduction band is empty Such a material is called dielectric In

Figure 1.2 Schematic representation of band structure of (a, b) metals (overlapping bands

width of band gap and dash line indicates position of Fermi level.

1.2 MODELING OPTICAL PROPERTIES OF A MATERIAL 11

dielectrics the valence electrons are “strongly” bound to the constituent atoms butcapable of displacement by an electric field, that is, they can be polarized (1.3.2◦)

energy space with the conduction band, the solids have the maximal conductivity

In such a material under the influence of an electric field, electrons can move toneighboring vacant states causing an electric current to flow This phenomenon

is characteristic for metals

has a small concentration of either empty or filled states and the temperature ficient of the electrical resistance is negative at high temperatures, the substance

coef-is classified as a semiconductor

using as an example the transmission and reflection spectra of galena ral PbS, semiconductor withE g ∼ 0.4 eV [46]) (Fig 1.3) In these spectra, the

(natu-regions corresponding to different excited states in solids are clearly able In region I, falling in the range of 1800–200 cm−1 for the majority ofsubstances (ν < 500 cm−1 for PbS), the interaction of light with TO vibrations

discern-of the crystal lattice — phonons — takes place Here, the extinction coefficient

k (1.1.10◦) reaches values on the order of 10−1–101 [25, 51, 52] The secondregion, II, dependent on the width of the band gapE g is called the transparency region of the crystal For dielectrics and semiconductors this region lies in the IR

range For many oxides its short-wavelength boundary (ν ∼ 3000 cm−1 for PbS)extends to vacuum ultraviolet (UV) However, because of the presence of defectsand impurities in crystals and mutliphonon processes, the extinction coefficientdoes not reduce to zero everywhere over region II At the maxima, the quantity

k can reach values on the order of 10−3–10−1 [25, 48, 51] depending on the

III

0

10

T, %

Figure 1.3 Absorption according to different mechanisms in transmission spectrum of a

2-mm-thick PbS plate I — phonon absorption, II — transparency region, III — fundamental absorption Asterisks indicate artifact due to atmosphere absorption.