Dipartimento di Scienze Fisiche e Astronomiche Dottorato di Ricerca in Fisica XX Ciclo

Transcription

1 Ministero dell Universitá e Dipartimento di Scienze Universitá degli Studi della Ricerca Fisiche ed Astronomiche di Palermo Dipartimento di Scienze Fisiche e Astronomiche Dottorato di Ricerca in Fisica XX Ciclo Vacuum UV transparency of a-sio 2 : the interplay of intrinsic absorption, structural disorder and silanol groups Eleonora Vella S.S.D. FIS/01 Palermo, Febbraio 2009 Supervisore: prof. Roberto Boscaino Coordinatore: prof. Antonio Cupane

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3 Acknowledgments The realization of this Thesis has represented a significant occasion of scientific and personal growth and, like all the important proofs in one person s life, it would have been impossible without the contribution and support of many people. It is a real pleasure for me to express my gratitude to Prof. R. Boscaino, for having closely followed my research activity while allowing the development of my autonomy and for having shared with me his wide experience and knowledge. Moreover, I am indebted with him because he has showed me how a solid enthusiasm for science and a careful capability to organize people and work can coexist. I sincerely acknowledge Dr. S. Agnello, Prof. M. Cannas, Prof. F. M. Gelardi and Prof. Leone because of their many suggestions and advices. Aside from their valuable contribution to this Thesis, I am grateful to all of them because they helped to build my idea of scientific research. I would like to thank also the colleagues whom I shared my PhD experience with. They had a fundamental role both for their precious scientific collaboration and for their friendship and support, which made these three years an important human experience. Dr. G. Buscarino and Dr. F. Messina deserve my most sincere gratefulness because of the time they spent discussing with me of my work and because their passionate scientific spirit has been a guidance for me. It has been a real pleasure to work among these people. I genuinely acknowledge all the scientists I had the luck to meet and talk with during the last three years. Among them, it a pleasure for me to thank Prof. A. Shluger and Prof. A. Thrukhin. I express a special gratitude to Prof. L. Skuja, who carefully read this Thesis, offering me many useful comments and suggestions. I would like to express my appreciation for the work of Prof. A. Emanuele who thoroughly proofread this work providing many helpful advices and corrections. Finally, I gratefully acknowledge the technical assistance that Mr. G. Lapis, Mr. G. Napoli and Mr. G. Tricomi offered during my experiments. iii

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5 Contents Introduction 1 I Background 3 1 Structural disorder in amorphous materials Intrinsic absorption in crystalline systems Microscopic theories for the Urbach rule Intrinsic absorption in amorphous materials Amorphous silicon dioxide The fictive temperature Silanol groups in amorphous silicon dioxide Infrared absorption of silanol groups in a-sio Vacuum UV absorption of silanol groups in a-sio II Experiments and Results 45 3 Experimental methods: a theoretical background Optical absorption spectroscopy Vibrational spectroscopy: infrared absorption Experimental procedures and samples Amorphous silicon dioxide samples Vacuum UV absorption measurements The VUV spectrophotometer The cryogenic equipment Measurements and correction procedure Infrared absorption measurements v

6 Contents 5 Intrinsic absorption and interplay with silanol groups Introduction Infrared absorption spectra of as-grown samples The IR absorption band of silanol groups The band at 2260 cm 1 and the fictive temperature Vacuum UV absorption spectra of as-grown samples Discussion Conclusions Structural disorder and silanol groups content Introduction The Urbach edge of a-sio IR absorption spectra at room temperature VUV absorption spectra as a function of temperature Discussion and conclusions Temperature dependence of the IR and VUV absorption of silanol groups Introduction The IR absorption band of silanol groups at room temperature IR absorption spectra as a function of temperature VUV absorption of silanol groups as a function of temperature Discussion Temperature induced changes of IR absorption spectra Temperature induced changes of VUV absorption spectra Conclusions Conclusions 119 Bibliography 131 List of related papers 133 vi

7 Introduction The intrinsic absorption of a material, either crystalline or amorphous, is strictly related to the properties of its microscopic structure. The study of the characteristics of the intrinsic absorption is thus interesting on one side from the viewpoint of the comprehension of the basic structural features of materials and on the other because of the many applications requiring the knowledge of the factors controlling their transparency. In the present work we examine the features of the intrinsic optical absorption of amorphous silicon dioxide, or silica (a-sio 2 ), a material of great scientific and technological importance. Silica has always been regarded as a simple model system for the understanding of the general properties of amorphous insulators. Moreover, it has received great attention and thorough study for many years because of its extensive employment in applications such as optical and microelectronics devices. Notwithstanding the many theoretical and experimental studies dealing with different structural features of this material, many of its fundamental physical properties are still poorly understood and lively debated. One of the main still open and unclear questions concerning a-sio 2 is related with the description of its overall microscopic structure. Being an amorphous system, silica is characterised by a disordered structure, that is by a structure lacking of long-range order. This topological disorder is a key factor in determining many fundamental properties of amorphous materials and in particular their intrinsic absorption. Although several models have been proposed for the description of the amorphous silica matrix, none of them can account for the many experimental evidences related to the macroscopic effects of the structural disorder. In this context the study of the intrinsic absorption in a-sio 2 is particularly relevant for at least two reasons: it can provide useful information concerning the overall structure of the material and it can, besides, clarify the role of topological disorder in affecting the transparency properties in the near-edge spectral region. The study of the intrinsic absorption of a-sio 2 cannot disregard the examination of the properties of silanol groups (Si-OH) in the material, because they were suggested contributing to the absorption in the near-edge spectral region. Silanol groups are one of the most common impurities embedded in the a-sio 2 network and they have been extensively studied because they are easily incorporated into the amor- 1

8 Introduction phous matrix during the manufacture procedure and because they influence several features of the material that are exploited by applications, such as its transparency in the infrared spectral region or its radiation hardness in the ultraviolet spectral range. Many of the properties of these impurities in a-sio 2 are up to now not clear. In particular, only few data are available concerning their electronic transition, which, as mentioned, was suggested being located in the energy region close to that of intrinsic absorption of silica. This Thesis reports an experimental study on the intrinsic absorption of a-sio 2. In order to understand the way disorder affects this fundamental feature, the properties of the intrinsic absorption in different a-sio 2 materials, i.e. silica produced via different manufacture procedures, were compared and characterised. This comparison, indeed, can provide useful information about the microscopic mechanisms determining the observed macroscopic differences. Researches previously reported in literature mainly focused on the analysis of a-sio 2 materials having very low concentrations of silanol groups, principally because of the difficulties related with the closeness of their absorption to the intrinsic one. Here we examined materials produced by several different techniques and encompassing the whole range of typical concentrations of silanol groups in a-sio 2. Our experimental approach is based on the combined use of two spectroscopic methods: infrared and vacuum ultraviolet absorption spectroscopy. The Thesis is organized into two parts. Part I comprises Chapters 1 and 2, while Part II includes Chapters from 3 to 7. Part I is devoted to the introduction of the background of the specialized theoretical and experimental studies dealing with the features of the intrinsic absorption in crystalline and amorphous materials. An overview of the general properties of silanol groups in silica is given as well. In Part II the adopted experimental techniques and the main results are reported. After a brief theoretical background on the spectroscopic methods we used (Chapter 3), a description of the experimental setup, of the instruments and of the materials employed is provided (Chapter 4). In Chapters from 5 to 7 the experiments and the main achieved results are discussed. Finally, the most relevant conclusions are summarized. Most of the results presented in this work have been published as papers on scientific Journals. Bibliographic references to these papers and to a few others on closely related topics are reported in a List of related papers at the end of Part II. 2

9 Part I Background 3

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11 Chapter 1 Structural disorder in amorphous materials In this chapter we start the Background Part of the Thesis with a brief review of the commonly adopted description of the effects of disorder on the electronic structure of crystalline and amorphous materials, focusing our attention on the disorder induced effects on the intrinsic absorption edge. These topics are particularly relevant in the context of the present work, whose main purpose is the study of the features of the intrinsic absorption edge of an amorphous insulator, amorphous silicon dioxide. Finally, several structural properties of this material will be introduced. 1.1 Intrinsic absorption in crystalline systems An ideal or perfect crystal is an arrangement of atoms or molecules which can be obtained by the infinite and regular spatial repetition of the same structural unit. The fundamental feature of perfect crystals is their spatial periodicity: it implies perfect ordering of the atoms, as regards position and composition both on an atomic, or short, distance scale and on a macroscopic, or long, scale. Unlike crystalline solids, experiments of X rays and neutron scattering showed [1 5] that the structure of amorphous systems is characterised by a short range order analogous to that of the correspondent crystalline system, if it exists, while it lacks of the long range order. The description of the electronic structure of crystals is based on the hypothesis that their conduction electrons are subjected to a periodic potential caused by the regular placement of nuclei in their lattice positions. An immediate consequence of this periodicity is the Bloch theorem [6] stating that the one-electron wave functions are of the form ψ( r ) = ψ nk ( r ) = exp(i k r )u nk ( r ) (1.1) where the wave vector k is in the first Brillouin zone, u nk ( r ) has the periodicity of the crystal structure and n (=1,2,...) is the so called band index. The wave functions 5

12 1. Structural disorder in amorphous materials represented by Equation 1.1 are extended: the wave function of an electron described by Equation 1.1 extends over every cell of the crystal just as in the free electron case. The structural order is reflected in the wave function in Equation 1.1: on one side its amplitude shows long-range order because of its extended nature, on the other side it shows a perfect phase coherence, i.e. long-range order in the phase. Given the phase at one point, one can determine the phase at any other point in the crystal provided that one knows the wave vector k. Periodicity has a profound and characteristic influence on the energy spectrum of the electrons in a crystal: it consists of continuous bands of allowed levels separated by forbidden gaps. The eigenvalues of energy E n ( k ) are continuous functions of k within each band. Perfect crystals are thus characterised by bands having well defined edges, which correspond to the absolute minimum or maximum of E as a function of k for each band. The density of states n(e) approaches zero at the band edges as E E min, where E min is the minimum value of energy as a function of k for the considered band. All these features, and particularly the extended nature of the states, their perfect phase coherence and the sharp band edges, are universal properties of the electronic structures of perfect crystals stemming from the periodicity of the lattice. On the base of what it was just mentioned, perfect crystals cannot absorb photons having energy lower than the minimum energy difference between the energy levels of the conduction band and those of the valence band, meaning with valence and conduction band the last populated and the first non-populated band respectively in the scheme of energy levels of a solid crystalline insulator. Perfect crystals are totally transparent to photons having energy in the forbidden band gap and their absorption coefficient in this spectral region is thus zero. In correspondence to the value of energy of the minimum difference between the conduction and the valence band the absorption coefficient shows a sharp and sudden rise. Its absorption spectrum should have a step-wise profile. The experimentally observed profile of the absorption coefficient in crystalline solids is different from the one we have just described. In real crystalline solids the boundary between the transparency and the opacity spectral ranges is actually characterised by a sudden and intense rise of the absorption coefficient, which, however, cannot be represented as a step-wise profile. In the following the profile of the absorption coefficient, α, in this transition region will be referred to as absorption edge. It is now useful to consider some of the experimental evidences reported in literature concerning the absorption edge in crystalline materials. In particular we focus our attention on the absorption edge of α-quartz, one of the crystalline forms of The absorption coefficient is defined as α(e) = d 1 ln(i 0 (E)/I(E)), where d is the sample thickness expressed in cm, I 0 (E) and I(E) are the incident and transmitted intensity as a function of energy E. For a more detailed discussion of the absorption coefficient see Section

13 1.1. Intrinsic absorption in crystalline systems silicon dioxide. Figure 1.1: Absorption spectra of α-quartz at different temperatures: symbols are experimental data, whereas lines were obtained using the Urbach law (Equation 1.2). Temperature ranges from 100 to 800 K. The coordinates of the crossing point in a semi-logarithmic scale are α0 0=106.8±0.1 cm 1 and E0 0 =(9.1 ± 0.05) ev. Figure taken from Godmanis et al. [7]. In Figure 1.1 the spectra of the absorption edge of a sample of α-quartz at different temperatures are shown [7]. Temperature ranges from 100 to 800 K. Spectra in Figure 1.1 are spectra of the absorption coefficient α, in units of cm 1, as a function of energy, in units of ev. As it can be seen, spectra in Figure 1.1 are linear over at least three decades of α in a semi-logarithmic scale. The slope of the absorption spectra depends on temperature and it exists, moreover, a crossing point of the absorption spectra at different temperatures in a semi-logarithmic scale. Similar behaviours of the temperature dependence of the spectra in the absorption edge region were observed in many crystalline systems, including ionic crystals, semiconductors and organic crystals [8 21]. The absorption spectra reported in Figure 1.1 can be described using a phenomenological law known as Urbach law [8, 9]: ( ) E E α(e) = α0 0 0 exp 0 (1.2) E u (T ) α 0 0 and E 0 0 are temperature independent parameters representing the coordinates of the crossing point of the absorption spectra at different temperatures in a semilogarithmic scale. In the case of α-quartz the values of α 0 0 and E 0 0 as experimentally determined are ±0.1 cm 1 and (9.10 ± 0.05) ev respectively [7]. E u, which is 7

14 1. Structural disorder in amorphous materials generally referred to as Urbach energy, is a temperature dependent parameter representing the inverse of the slope of the absorption edge in a semi-logarithmic scale. For crystalline materials the Urbach energy shows the following temperature dependence [8 21]: E u (T ) = ω ( ) 0 ω0 coth (1.3) 2σ 0 2kT ω 0 is the energy of an Einstein oscillator in the Einstein model of the vibrational modes of a crystal and σ 0 is a temperature independent parameter characteristic of the material. The values of the Urbach energy at room temperature reported in literature for α-quartz are between 45 and 51 mev [7, 22 24]. Figure 1.2: Density of states n(e) as a function of energy E for a) a perfect crystal and b) a crystal containing only one localized imperfection. Figure adapted from Tauc [18]. In order to interpret the profile of the absorption coefficient we have just shown, it is useful to introduce a simple model for the description of the main features of the electronic structure of real crystals. We will start with a general qualitative discussion of the effects of disorder on the electronic structure of an otherwise ideal crystal [18]. Let us consider a crystal containing a single localized imperfection such as an impurity. We call imperfection any modification of the lattice parameters produced either by the presence of an impurity in the lattice, meaning with impurity an atom different from the constituent elements of the crystal, or by the thermal vibrations of the lattice itself. The problem can be dealt with as one of scattering by the imperfection, where the Bloch states of the perfect crystal (Equation 1.1) play the role of the incident waves. The localized change V introduced by the imperfection in the crystal potential can be both positive and negative. If V is weak, one can use the Born approximation to obtain the solution of the scattering problem. The result is then that the states remain extended and retain their phase coherence. There are 8

15 1.1. Intrinsic absorption in crystalline systems only minor changes in the density of states and in particular the sharp band edges remain. When the strength of the potential change exceeds a certain critical value, electronic states develop in the density of states near the bottom of the band for V attractive or near the top of the band for V repulsive. These states are δ-functions in the band gap and it can be shown that they are bound and consequently localized around the imperfection. In Figure 1.2 an example of the densities of states as a function of energy for a perfect crystal and for a crystal containing a single localized imperfection are reported. As it was mentioned, both attractive and repulsive localized potential changes can give rise to electronic localized states in a crystal, in contrast to the ordinary scattering problem in which one cannot produce bound states out of repulsive potentials. The reason is that the energy spectrum in a crystal is comprised of continua of levels having both lower and upper limits. A repulsive potential, if strong enough, can create a localized state above the upper boundary of the continuum in the same way that an attractive potential can produce a localized state below the lower boundary of the continuum. The next step is to consider a real crystal as resulting from a low but finite concentration of randomly distributed individual imperfections. For weak scattering potentials and for energies well inside the band, the wave functions remain extended, however, they lose their perfect phase coherence. Because of the multiple scattering by many randomly distributed scatterers, the phase at a certain point r becomes uncorrelated with the phase at another point r if the distance r r is much larger than a characteristic length ξ, called the phase coherence length. For an ideal crystal the phase coherence length is infinite. The mean free path is another measure of the linear dimension over which phase coherence is retained and it differs from the phase coherence length by a factor of order unity. The one-electron wave function can be expanded in the original Bloch functions (Equation 1.1) where ψ = k a k ψ k (1.4) a k = (ψ k, ψ) (1.5) The integral in Equation 1.5 can be split into a sum of integrals each over regions of linear dimension of order ξ. The phase of ψ in different regions is uncorrelated. Thus the contributions of each region to the integral have random phases. From another point of view, it is possible to show that the random multiple scattering introduces a factor exp( R/ξ) in the average autocorrelation function c( R ) = ψ ( R + r ), ψ( r ) (1.6) 9

16 1. Structural disorder in amorphous materials when the average is carried out over an ensemble of imperfect crystals with a random distribution of imperfections. The factor exp( R/ξ) simply means that phase coherence has been lost over distances greater than ξ. There is no long-range order in the phase, but short-range order only. Figure 1.3: Density of states n(e) as a function of energy E for a crystal containing a random distribution of localized imperfections. Figure adapted from Tauc [18]. Since the concentration of imperfections is much smaller than that of regular atoms, changes in the density of states are limited. When the strength of the imperfection scattering potential of each imperfection is large enough to produce localized states at each imperfection, there will be a non-zero density of states inside the gap with a peak around the energy value of the bound state corresponding to a single impurity. These features are summarized in the density of states as a function of energy shown in Figure 1.3. According to the model which was just described, both structural and thermal disorder can give rise to localized states within the band gap [18, 25]. Due to these localized states in the immediate neighbourhood of the band limits, the absorption edge looses its step-wise shape. The two main theories proposed in literature to interpret the exponential character of the absorption edge in crystals are briefly exposed in the next section. 1.2 Microscopic theories for the Urbach rule The theories of the Urbach edge are based on the idea that a sharp absorption edge is broadened by some mechanism. In crystals there is little doubt that thermal vibrations are responsible for the Urbach edges. Equation 1.3 can be deduced from the assumption that the broadening of the edge, and thus the Urbach energy E u, is proportional to the mean square displacement of atoms, < U 2 >, from their equilibrium positions. < U 2 > is proportional to the mean potential energy of their oscillations and therefore to their total energy. In the Einstein model for the vibra- 10

17 1.2. Microscopic theories for the Urbach rule tional modes of a crystal it results ( ) 1 < U 2 > = ω exp( ω 0 /k B T ) 1 = ω ( ) (1.7) 0 2 coth ω0 2k B T Within this interpretation of the Urbach tail at low temperatures the broadening is determined by the zero-point vibrations, whereas at higher temperatures the contribution of thermal vibrations becomes more and more relevant. There have been many attempts to theoretically explain the exact nature of interactions leading to the Urbach rule in crystals [26 47]. Although a detailed description of the several models proposed in literature is beyond the purpose of the present work, it is useful to mention and qualitatively describe the two main fundamental approaches to this issue, both presenting the Urbach rule as a result of the exciton-phonon interactions. In general an exciton in a crystal is given by an excited bounded state arising from the Coulomb interaction of a valence band hole with a conduction band electron. Excitons can be considered as a complex manifestations of the electronic band structure of crystals. In order to clarify this point, let us consider a way to form an excited state in a crystal different from that of removing one electron from the highest level of the valence band and placing it into the lowest level of the conduction band. Let us suppose we form an electronic level by superimposing enough levels near the minimum of the conduction band to form a well localized wave packet. Because levels in the neighbourood of the minimum are needed in order to produce the wave packet, the energy E C of the wave packet will be somewhat greater than E C, the minimum of the conduction band. Let us suppose in addition that the valence band level that was depopulated is also a wave packet, formed of levels in the neighbourhood of the valence band maximum, so that its energy E V is somewhat less than E V, and chosen so that the center of the wave packet is spatially very near the center of the conduction band wave packet. Neglecting electron-electron interaction, the energy required to move an electron from valence to conduction band wave packets would be E C E V >E C E V, but because the levels are localized, there will be, in addition, a non-negligible amount of negative Coulomb energy due to the electrostatic interaction of the conduction band electron and valence band hole reducing the excitation energy of this more complicate type of excited state. From this point of view, excitons can be considered as the true lowest excited state of crystals. In general, an exciton can be more or less localized depending on whether the localized electron and hole levels extend over many lattice constant. The two principal models proposed for the interpretation of the Urbach rule in crystals present it as the result of the broadening of the exciton absorption line caused by the exciton-phonon interactions. The first is the model of the momentarily trapped exciton, developed by Toyozawa and coworkers [26, 27, 41, 42, 47], and the second 11

18 1. Structural disorder in amorphous materials is the model of the internal electric micro-fields, worked out by Dow and Redfield [28, 31, 39, 40, 43]. Toyozawa and coworkers [26, 27, 41, 42, 47] showed that the broadening of the exciton line caused by the momentary trapping of an exciton by the local lattice deformation due to the thermal lattice vibrations can give rise to exponential absorption edges, having the temperature dependence experimentally observed. According to this model, the trapped exciton states appear from place to place and from time to time during vibrations. Depending on the intensity of exciton-phonon interaction, the states of the trapped exciton may be more or less stable relative to the free exciton ones. A criterion for a strong exciton-phonon interaction can be given in terms of the value of the exciton-phonon-coupling constant g. This constant is defined in the following way. After the creation of an exciton, the lattice can gain energy by relaxing to the position of minimal energy. The ratio of this lattice relaxation energy to the maximum energy gain due to the transfer defines g. In the Toyozawa s theory the parameter σ 0 in Equation 1.3 is proposed being inversely proportional to the exciton-phonon-coupling constant and the criterion for a strong exciton-phonon interaction causing the momentary self-trapping is σ 0 <1. The Toyozawa model [27, 42, 47] provides furthermore a detailed interpretation of the other parameters of the Urbach rule. The energy ω 0 in Equation 1.3 is the average energy of the phonons interacting most strongly with excitons while forming the Urbach edge. The energy parameter E0 0 is approximately equal, but slightly higher than, the energy E max of the peak of the exciton absorption band at 0 K. For a large variety of crystals the difference between E0 0 and E max is smaller than 0.1 ev. The second class of studies [28, 31, 39, 40, 43] explains the exponential Urbach edges as originating by the ionization of excitons by internal electric fields. For the electric fields responsible for this effect the term micro-fields was proposed. The spatial variation of micro-fields occurs over distances large compared with atomic sizes but small compared with macroscopic dimensions (i.e. distances >1 Å and 10 4 Å). There are several possible sources of internal electric fields, one of them is the oscillating potential fluctuations associated with longitudinal acoustical (LO) phonons. According to this model the parameter ω 0 in Equation 1.3 has to be interpreted as the energy ω LO of LO phonons, associated with the corresponding electric fields. Unlike the Toyozawa s model the theory by Dow and Redfield cannot account quantitatively for the temperature dependence of the Urbach energy as given in Equation Intrinsic absorption in amorphous materials Passing from a crystalline system to an amorphous one, a shift of the absorption edge is observed either towards lower or higher energies. For example in germanium and silicon going from the crystal to the amorphous material this shift is towards 12

19 1.3. Intrinsic absorption in amorphous materials lower energies, whereas it is in the opposite direction in selenium and tellurium [18]. No simple general rule governing these changes has been suggested. Notwithstanding these differences, the shape of the absorption curve appears to be similar for many amorphous materials. Figure 1.4: Absorption edge typical of amorphous semiconductors and insulators: three different regions are distinguishable, named as A, B and C. Figure adapted from Tauc [18]. In many amorphous insulators and semiconductors the absorption edge has the profile shown in Figure 1.4. Three different regions of the absorption edge can be distinguished. The first one, named as A in Figure 1.4, typically starts from values of the absorption coefficient α between 10 3 and 10 4 cm 1, whereas the exponential part of the spectrum, indicated in figure as B, extends over four order of magnitude of α. Finally, in the lower energy range of the absorption edge a weak absorption tail can be seen, which is referred to as C in figure. We will first discuss on the features of the absorption coefficient in the regions A and C and we will then focus our attention on the characteristics of the part B of the spectrum. In the part A of the spectrum, which is generally referred to as Tauc region, the absorption coefficient has the following energy dependence [18] α(e) (E E g) r E (1.8) where r is a constant proper of the material and of the order 1. For amorphous silicon dioxide (a-sio 2 ) r = 2. Equation 1.8 is generally used to define the optical gap E g as 13

20 1. Structural disorder in amorphous materials distinguished from the electrical gap, determined from the temperature dependence of electrical conductivity. Region C of the absorption edge is a weak absorption tail, whose intensity and shape depend on the preparation, purity and thermal history of the material. In this region the absorption coefficient is generally lower than 0.5 cm 1. On the contrary if defects related absorption bands are present in the same region, α(e) can reach much higher values. The attribution of this absorption tail has been controversial. Several hypothesis were put forward with regard to the optical transitions responsible for the absorption in this region [18], however, being of only marginal interest in the present work, we will not deal with these topics here. The exponential region of the absorption edge (part B in Figure 1.4) extends over values of the absorption coefficient from 1 to 10 4 cm 1. The characteristics of the exponential region of the absorption edge are not common to all amorphous materials. The main differences between one material and the other concern in particular the temperature dependence of the parameters of this exponential profile. There exists a first class of amorphous materials, like for example amorphous silicon and a-sio 2, characterised by an exponential part of the absorption edge which can be described using the Urbach law, analogously to the case of crystalline solids. For convenience we report here again the equation for the Urbach law ( ) E E α(e) = α0 0 0 exp 0 (1.9) E u (T ) For this first class of materials the Urbach energy E u is almost temperature independent al low temperatures, usually below room temperature, and it increases for higher temperatures. We recall that an increase of the Urbach energy corresponds to a decrease of the slope of the absorption coefficient profile in a semi-logarithmic scale. Moreover, in many amorphous materials parts A and B move as a whole as a function of temperature. Just like in the case of crystalline systems, for this family of amorphous solids there exists a crossing point in a semi-logarithmic scale of the absorption spectra acquired at different temperatures. In Equation 1.9 α0 0 and E0 0 are the coordinates of this crossing point and are thus temperature independent parameters characteristic of the material. It is worth noting that Equation 1.9 is not the only expression reported in literature for the Urbach law; another commonly used equation is for example ( ) E Eg (T ) α(e) = α g (T ) exp (1.10) E u (T ) where E g is the optical gap as independently determined by the absorption coefficient spectrum in the Tauc region and α g (T ) is a temperature dependent parameter. For a-sio 2 the values reported in literature for E g and α g at room temperature are 8.5 ev and cm 1 respectively [22, 48]. 14

21 1.3. Intrinsic absorption in amorphous materials As mentioned before, there exists a second class of amorphous materials, like for example vitreous As 2 Se 3 [49] and Na 2 O 3 GeO 2 [50], showing a different temperature dependence of the exponential part of the absorption edge. For these systems the semi-logarithmic slope of the exponential edge is temperature independent. In such a case in a semi-logarithmic scale the absorption spectra acquired at different temperatures are parallel to each other. For these materials the law describing the energy and temperature dependence of the absorption spectra can be written in the form [21] ) α(e) = I 0 exp (AE + TT1 (1.11) where A is a temperature independent slope parameter and T 1 is a characteristic temperature. Although this issue has been analysed in several studies [51 53], there is no agreement on the mechanisms causing a rule of the form of Equation Expression 1.11 is generally referred to as glass-like Urbach law, so as to distinguish it by the ordinary crystal-like Urbach law, as reported in Equations 1.9 and In the following we will indicate Equations 1.9 and 1.10 simply as Urbach law, whereas we will refer to Equation 1.11 as glass-like Urbach law. Figure 1.5: Absorption spectra of a-sio 2 at different temperatures: symbols are experimental data, whereas lines were obtained using the Urbach law (Equation 1.9). Temperature ranges from 100 to 800 K. The coordinates of the crossing point in a semi-logarithmic scale are α0 0=105.5±0.1 cm 1 and E0 0 =(8.70 ± 0.05) ev. Figure taken from Godmanis et al. [7]. Before focusing our attention on the first category of materials, it is useful to show two examples representative of the different behaviours just discussed. In Figure 1.5 the spectra of a sample of a-sio 2 at different temperatures are reported as observed 15

22 1. Structural disorder in amorphous materials Figure 1.6: Absorption spectra of Na 2 O 3 GeO 2 at different temperatures: symbols are experimental data, whereas lines were obtained using the glass-like Urbach law (Equation 1.11). Temperature ranges from 300 to 700 K. Figure taken from Trukhin et al. [21]. by Godmanis et al. [7]: the symbols represent the experimental data, whereas the lines were obtained by fitting the experimental curves using Equation 1.9. Spectra were acquired in the temperatures range from 100 and 800 K. The exponential profile extends over three orders of magnitude of α and the semi-logarithmic slope of the curves increases with increasing temperature. Moreover, the spectra were fitted using Equation 1.9 and the best fitting curves are reported in Figure 1.5 as well. As it can be seen, the extrapolations of the best fitting curves cross in a point, whose coordinates are estimated as α 0 0=10 5.5±0.1 and E 0 0=(8.70 ± 0.05) ev. In Figure 1.6 the exponential parts of the absorption edge of vitreous Na 2 O 3 GeO 2 are shown at different temperatures [21]. Temperature ranges from 300 to 700 K. The symbols correspond to the experimental data and the lines are the extrapolations of the experimental profiles obtained using Equation As it is possible to see, the semi-logarithmic slope of the spectra does not change as a function of temperature. Up to now we have traced a phenomenological picture of the features of the absorption edge in amorphous materials. We will proceed with the discussion of a simple model for the description of the properties of the electronic structure of disordered systems. As it has been said, in crystalline systems the exponential tail of the absorption edge is generally considered as caused by thermal disorder. This interpretation suggests that in amorphous materials, where a non-thermal static component to disorder is present, there should exist a temperature-independent contribution to the Urbach edge. It is useful to start considering an experimental study where 16

23 1.3. Intrinsic absorption in amorphous materials the problem of the effects of thermal and structural disorder on the Urbach edge of amorphous materials was dealt with. Figure 1.7: Absorption spectra of hydrogenated amorphous silicon: the solid symbols refer to data obtained at different measurement temperatures, i.e. 12.7, 151 and 293 K, whereas the open symbols refer to samples subjected to under vacuum isochronally heating in the range of temperatures ( ) K. Lines represent extrapolations of the experimental curves obtained using the Urbach law (Equation 1.9). Figure taken from Cody et al. [54]. In order to determine the relative sizes of the two components of disorder, Cody et al. [54] measured the optical absorption edge on films of hydrogenated amorphous silicon (a-sih) subjected to two different treatments. The effects of thermal disorder were estimated by measuring the absorption spectra of samples maintained at different temperatures, while the effects of static disorder were evaluated by intentionally inducing structural disorder in the samples via the introduction of dangling bonds through the thermal evolution of hydrogen. This was done isochronally heating the films in vacuum in the range of temperatures from 300 to 900 K. In Figure 1.7 the absorption spectra of the films subjected to the two treatments are reported: the solid symbols refer to data obtained at different temperature measurements, the open symbols refer to films isochronally heated at different temperatures, whereas lines are the extrapolations of the experimental curves determined using the Urbach law (Equation 1.9). In the spectra of Figure 1.7 the Urbach and the Tauc regions are clearly distinguishable. As it can be seen, the effects of the increase of any of the two kinds of disorder are comparable: both produce a shift of the absorption edge to lower energies and an increase of the Urbach energy. This result is also confirmed by the trend of the optical energy gap E g (see Equation 1.8) as a function of the Urbach energy in the samples subjected to the two treatments. In Figure 1.8 the op- 17

24 1. Structural disorder in amorphous materials Figure 1.8: Optical energy gap (see Equation 1.8) as a function of the Urbach energy (see Equation 1.9): filled and open circles refer to a-sih films maintained at different temperatures, whereas filled and open triangles represent measurements on the isochronally heated samples. Figure taken from Cody et al. [54]. tical energy gap is plotted against the Urbach energy: filled and open circles refer to a-sih films maintained at different temperatures, whereas filled and open triangles represent measurements on the isochronally heated samples. As it can be seen, the effects of both treatments determine a linear relationship between E g and E u. In order to interpret the experimental evidences just discussed, we will present qualitatively a simple model for the electronic structure containing the essential features common to all disordered materials. This model, known as Mott-FCO model (Fritzsche-Cohen-Ovshinsky) [55, 56], can be considered as a natural extrapolation of the results discussed in Section 1.1 for ideal and real crystals. Let us start considering a single isolated band of a crystal and hypothesizing that the ordered structure of this crystal has been made random through some disordering process. As the randomness increases, the band becomes broader and the nature of the wave functions changes. We will use as an example the scheme of the density of states displayed in Figure 1.9. For E c <E<E c the states remain extended with a finite phase coherence length. At the energies E c and E c the character of the states abruptly changes from extended to localized so that for E<E c and E>E c there are tails of localized states [18]. A few words are necessary in order to clarify what we mean by extended states for a disordered structure. In the case of an electron in a periodic potential the electronic wave function extends over every unit cell with equal 18

25 1.3. Intrinsic absorption in amorphous materials Figure 1.9: Density of states of a single isolated energy band in a disordered material. States having energy E c <E<E c are extended. Shaded areas represent tails of localized states. Figure adapted from Tauc [18]. intensity. In the more general case of disordered systems wave functions are called extended if there are paths extending to infinity in both directions lying entirely in regions where the wave function is not negligible. As it is possible to see from the scheme of Figure 1.9, the loss of long range order in the potential has a twofold effect. For the states in the middle of the band, the long range phase order is lost, but the states remain extended. On the other hand, for the states in the tails of the band the effect of disorder is more drastic. The amplitude of the wave function is different from zero only in a finite region and as consequence the character of the states changes from extended to localized. There should therefore be two characteristic energies E c and E c separating the regions of localized states from that of extended states. Anderson [57] has actually shown that electrons in localized states cannot diffuse at T = 0 K. At finite temperatures they presumably can contribute to the conductivity only by phonon assisted processes. This is the reason why the energy values E c and E c are called mobility edges. In disordered materials localized states are not in general associated with single imperfections and they are instead due to the distribution of the atomic configurations, and consequently in the potential, caused by disorder. When the fluctuations in the positions are strong enough or large enough in spatial extent, the modifications they induce in the potential can bind or localize one or more states. The associated energy levels depend on the details of such fluctuations and the density of states can be determined from the probability distribution of fluctuations. For example the localized states furthest from the centre of the band are associated with the widest or deepest, and thus less likely, fluctuations, explaining the existence of band tails. As disorder increases, more and more localized states are created and the mobility edges move inward into the band. At the same time, the mean free paths of the extended states are reduced. When disorder reaches a certain critical value, the two mobility edges merge and all the states in the band are localized. This is the so-called Anderson transition [57]. The Mott-FCO model explains several macroscopic properties of amorphous materials. In particular in the case of amorphous semiconductors the activated tempera- 19

26 1. Structural disorder in amorphous materials Figure 1.10: Band electronic structure for amorphous materials within the Mott-FCO model. Localized states (shaded areas) appear as tails of the conduction and valence bands, leaving a well defined mobility gap. States having energy E<E V and E>E C are extended. Figure adapted from Tauc [18]. ture dependence of the electric conductivity and the optical absorption data suggest a model for the electronic structure with valence and conduction bands separated by a forbidden gap. An example of the simplest model that can be used for the discussion of amorphous semiconductors and which incorporates the basic properties of disordered materials is shown in Figure In this band model there exist tails of localized states above E V and below E C. Within this scheme the electrons and holes in the extended states within kt of the mobility edges are the charge carriers of the system. Figure 1.11: Density of states g(e) as a function of energy E in amorphous materials according to the Mott-FCO model. Shaded areas represent localized states, whereas open areas extended states. Figure adapted from Tauc [18]. We can now briefly describe the hypothesis proposed within the Mott-FCO model 20

27 1.3. Intrinsic absorption in amorphous materials with regard to the attribution of the absorption in the two characteristic regions of the absorption edge of amorphous materials which we are interested in (A and B in Figure 1.4) to specific electronic transitions. In Figure 1.11 the density of states as a function of energy according to the Mott-FCO model is displayed. The energy values E V and E C represent the mobility edges. Several theoretical models [58, 59] for the calculation of the transition probability ascribed the Tauc region of the absorption edge to transitions between the extended valence band and the conduction band. As to the Urbach region, it was hypothesized that this is associated to transitions between the localized band tails above the valence band and extended states of the conduction band or between extended states of the valence band and localized band tails below the conduction band [60]. In this context it is worth noting that in recent years the development of models for the electronic structure based on the density functional theory allowed accurate calculations of the density of states of several amorphous systems. In the computational studies where the problem of the electronic transitions responsible for the Urbach and Tauc regions was addressed descriptions similar to that we have just mentioned were proposed [2, 61 69]. In the last part of this section we will focus again our attention on the exponential part of the absorption edge of amorphous materials. The approach generally adopted in literature is a natural extension of the interpretation of the Urbach edge of crystalline systems discussed in Section 1.2. The description of the Urbach edge in amorphous materials is based, indeed, on the hypothesis of the substantial equivalence of the thermal and structural frozen-in disorder. To include the effect of structural disorder on the Urbach energy the following generalization is generally made [54] E u (T, X) = K ( U 2 T + U 2 X ) (1.12) where U 2 X is the contribution of topological disorder to the mean-square deviation of the atomic positions from a perfectly ordered configuration. As a justification of this central hypothesis the dynamic phonon disorder and static structural disorder should have similar effects on the electronic energy levels. For convenience the contribution of structural disorder is expressed in units of the zero-point energy ω 0 /2, estimated using the Einstein model for the vibrational modes of a crystal. Thus, for an amorphous material Equation 1.3 becomes E u (T, X) = 1 ( E T (T ) + X ω ) 0 (1.13) σ 0 2 where E T (T ) = ω ( ) 0 2 coth ω0 2k B T (1.14) 21

28 1. Structural disorder in amorphous materials and X = U 2 X U 2 0 (1.15) is a measure of the structural disorder normalized to U 2 0, the zero-point uncertainty in the atomic positions. As to the optical energy gap E g, its explicit temperature dependence in crystalline materials can be written as [70] E g (T ) = E g (0) D ( U 2 T U 2 0 ) (1.16) where E g (0) is the zero-temperature optical gap and D is a second-order deformation potential. If Equation 1.16 is generalized analogously to Equation 1.3 to include the effects of structural disorder, then the mean-square lattice displacements in Equation 1.16 can be expressed in terms of the experimentally measured quantity E u (T, X), and Equation 1.16 can be rewritten as follows: ( ) E g (T, X) = E g (0, 0) U 2 Eu (T, X) 0 D E u (0, 0) 1 (1.17) From Equation 1.17 it results that when the alterations of the lattice parameters are due to structural and thermal disorder, E u and E g are linearly related. It was verified that this linear relation holds true for different amorphous materials [22, 54, 71]. In the context of the present work it is useful to mention the results of a study where Equations 1.13 and 1.17 were used to describe the temperature dependence of the Urbach energy and the optical energy gap of a-sio 2. Saito et al. [22] measured the optical absorption edge on a sample of a-sio 2 in the range of temperatures from 4 to 1900 K. They found that for temperatures lower than the glass transition temperature (T g 1200 K) the temperature dependencies of the Urbach energy and of the optical energy gap can be well described respectively using Equations 1.13 and In Figure 1.12 the Urbach energy and the optical energy gap in a-sio 2 as a function of temperature are reported [22]: solid symbols represent the experimental data, whereas solid lines were obtained by fitting data below 1200 K with Equations 1.13 and Moreover, from the fitting the following values of the parameters ω 0, σ 0, X and D were obtained: ω 0 =(0.079 ± 0.008) ev, σ 0 =(0.66 ± 0.05), X=(0.33 ± 0.03) and D=(10.3 ± 0.06). These results require a few remarks, especially that concerning the value of X. The value of this parameter in a-sio 2 is 0.3 for temperatures below T g : this is much smaller than the value ( 8) which has been reported for amorphous silicon [71]. This actually means that in a-sio 2 the contributions of thermal vibrations to E u and E g are larger than that of the frozen in structural disorder. This fact is contrary to the common sense that the absorption edge in an amorphous structure is mainly determined by the static disorder. Thus, from the viewpoint of 22