Two dispersion models of disordered solids, one parameterizing density of states (PDOS) and the other parameterizing joint density of states (PJDOS), are presented. Using these models, not only the complex dielectric function of the materials, but also some information about their electronic structure can be obtained. The numerical integration is necessary in the PDOS model. If analytical expressions are required the presented PJDOS model is, for some materials, a suitable option still providing information about the electronic structure of the material. It is demonstrated that the PDOS model can be successfully applied to a wide variety of materials. In this paper, its application to diamond-like carbon (DLC), a-Si and SiO2-like materials are discussed in detail. Unlike the PDOS model, the presented PJDOS model represents a special case of parameterization that can be applied to limited types of materials, for example DLC, ultrananocrystalline diamond (UNCD) and SiO2-like.
© 2007 Optical Society of America
The evaluation of optical measurements on amorphous or polycrystalline material is usually performed by parameterization of their dielectric functions using dispersion models based on the Lorentz oscillator, e.g. Forouhi–Bloomer , Tauc-Lorentz  and Urbach–Cody–Lorentz . These models are not derived on the basis of electronic structure and, therefore, they do not allow the density of states (DOS) or joint density of states (JDOS) to be determined correctly. In this paper, the models based on the parameterization of DOS (PDOS) or JDOS (PJDOS) will be summarized. Their main advantage, as compared to the Lorentz-oscillator-based models, is the ability to obtain parameters closely related to the electronic structure of materials, i. e. the band gap energy, quantity proportional to the density of electrons etc..
Light absorption in materials with disordered structures, i. e. amorphous and polycrystalline, is caused by electron transitions from the initial state in j band with the energy S to final state in k band with the energy S+E (see Fig. 1). The imaginary part of dielectric function is calculated as the sum of terms corresponding to all the possible transitions [4–7]
where E, e, h, m, ε 0 and B0 are photon energy, electron charge, Planck’s constant, electron mass, permittivity of vacuum and certain part of Brillouin zone of corresponding crystalline material, respectively. The functions 𝒩j(S) and 𝒩k(S) represent energy distributions of the DOS in j and k bands, respectively. The concentration of j electrons, i. e. number of j electrons per unit volume, is given by
The transitions considered above are possible only if the j states are occupied whereas the k states are not. This fact is taken into account by the insertion of the Fermi–Dirac statistics for electrons, f e, and holes, f h, into the integral. The probability of transitions is given by the squared momentum-matrix element |p j→k|2. It was factored out of the integral supposing constant for all the transitions between particular j and k bands.
Equation (1) describes both the interband and intraband transitions as schematically depicted in Fig. 1. The interband transitions between valence and conduction band cause broad absorption peak in the visible or UV spectral region. The intraband transitions are related to free carrier absorption steeply increasing towards the infrared region. Note the DOS functions, 𝒩j(S) or 𝒩k(S), in Eq. (1) can represent both extended and localized states. If j valence band is fully occupied and k conduction band fully unoccupied, then the Fermi–Dirac statistics can by omitted from Eq. (1) and then the integral defines the JDOS function 𝒥 j→k(E) corresponding to the j→k transitions
The expression in Eq. (1) defines the imaginary part of dielectric function only for positive values of E. The complete complex dielectric function can be obtained by incorporating antisymmetry of the imaginary part and by Kramers–Kronig relations 
Equations (4) and (5) form the basis of PDOS and JPDOS models presented in the following sections. The aim of this article is to find suitable parameterizations of 𝒩 or 𝒥 for particular materials. Thanks to the power of today’s computers, it is not necessary to limit this search to the class of functions permitting an analytical expression of the integrals. Most of the models presented below employ numerical methods, for details see Appendix.
2. PDOS model
The model based on the parameterization of DOS (PDOS) can be used for a wide variety of materials if suitable DOS functions are used (see Fig. 2). It was applied first to a-As2S3 chalcogenide thin films . Two types of electronic states, one forming bands of extended states (ξ and ξ*) and the other creating localized states closed to the valence and conduction bands (λ and λ*), were considered in this case. Only ξ→ξ* interband transitions causing a broad absorption peak and λ→ξ*, ξ→λ* transitions causing the Urbach tail were taken into account. One year later, it was used for polymorphous silicon (pm-Si) thin films  considering the same types of transitions. The extended states, ξ, and the localized defect states, δ, inside the band gap and only the transitions ξ→ξ* and δ→δ* were taken into account in the case of amorphous SiOxCyHz films . The contributions of δ→ξ* and ξ→δ* transitions were neglected. Contrary to the previously discussed materials, the contribution of the Urbach tail to the dielectric function did not reveal itself because the band gap was outside the measured spectral range. The PDOS model was successfully used also for diamond-like carbon (DLC) and DLC:SiOx films [12, 13] which mixed sp3/sp2 bonding structure gives rise to two types of electrons, σ and π. The transitions σ→σ* and π→π* contribute to the dielectric function whereas σ→π* and π→σ* transitions can be ignored. The power of this approach was further demonstrated when structural changes in the DLC and DLC:SiOx films were clearly observed by an evaluation of the optical measurements using the PDOS model [14–17]. The structural changes in Ge2Sb2Te5 chalcogenide films undergoing the phase transitions were recently studied by the optical method employing the PDOS model containing both the interband and intraband transitions .
2.1. Application to DLC
The ideas of the PDOS model will be demonstrated first using the example of the DLC films. In this case the subscripts j distinguish the quantities related to s and p electrons. Since only the σ→σ* and π→π* transitions are significant for the absorption, Eq. (4) can be simplified using an assumption of stepwise Fermi-Dirac statistic (it is well fulfilled when the bands lay far from the Fermi energy) and by an introduction of Nj(S) that includes the constants in front of the integral
where the quantity Qj, proportional to the concentration of j electrons, can be one of the parameters of the PDOS model. The other two parameters required for the description of j→j* transitions are the minimum energy limit (band gap) E gj and the maximum energy limit E hj.
From a similarity with the DOS distribution in crystals in the vicinity of band energy extrema, Nj(S) and N j*(S) can be modeled as square root functions. Since the dielectric function is proportional to their convolution it is reasonable to assume their symmetry with respect to the Fermi level. Then, Nj(S) for valence electrons is expressed as
and N j*(S) for conduction electrons is
where ej(E) is an elliptic integral which has to be calculated numerically (see Appendix B).
Eqs. (9)–(11) can be simplified using the parameter Aj and the quantity Qj, interesting from the viewpoint of material structure, can be calculated afterwards. However, it is less convenient than the parameterization with Qj.
The results of unnormalized DOS, Nσ(S) and Nπ(S), for as deposited and annealed (510 °C) DLC films studied in  are shown in Fig. 3. The corresponding fitting parameters, Qσ, E gσ,E hσ, Qπ, E gπ and E hπ, are summarized in Table 1.
The advantage of the model is the possibility to evaluate the ratio of π-to-σ electrons
In  the method for the determination of κ experimentally was suggested using the as deposited and annealed DLC films. When the DLC films are annealed the hydrogen atoms desorb  and the films undergo structural changes. These changes are related to the metastability of diamond leading to its graphitization. The effect of hydrogen desorption is reflected in a decrease in the concentration of valence electrons Ne because one electron is lost per one leaving hydrogen atom. It is important to realize that the total number of valence electrons is equal to Ne multiplied by the film volume. Although the film surface does not change due to the annealing the original film thickness generally differs from the thickness after annealing. Then the ratio of the number of valence electrons ne in the film after and before annealing is
where superscripts (t) and (0) denote the quantities corresponding to the annealed and as deposited films and d f is the film thickness. In order to calculate κ from Eq. 15 the ratio β has to be evaluated from hydrogen fraction in the DLC films before and after annealing. Assuming the number of carbon atoms remains constant, β can be derived in following way
where nH, nC and na are the number of hydrogen, carbon and all atoms in the film, respectively. The symbol X denotes the atomic fractions.
The DLC films studied in  were analyzed by Elastic Recoil Detection Analysis (ERDA) for the hydrogen fraction and the results are summarized in Table 1. Annealing up to 325 °C did not change the hydrogen atomic fraction significantly and, therefore, the films annealed at higher temperatures are much more important for the calculation of κ. In order to determine κ with the best precision the differences of β calculated from Eqs. (15) and (16) for each couple of films before and after annealing to the particular temperature were minimized. The resulting κ is 0.5736. The ratio α of π-to-σ electrons in all the films calculated from Eq. (13) using the already known κ and the ratio β for all the annealed films calculated from Eqs. (15) and (16) using κ and hydrogen atomic fractions, respectively, are given in Table 1.
Using the calculated ratio α and considering the fact that each sp3-bonded carbon corresponds to four σ electrons, each sp2-bonded carbon to three σ and one π electrons and each hydrogen atom to one s electron, the sp3-to-sp2 ratio can be calculated as [9, 15, 20]:
The obtained values for all the DLC films are also given in Table 1.
2.2. Application to a-Si
In the case of many amorphous materials such as amorphous silicon (a-Si), germanium (a-Ge) and various chalcogenides (a-As2S3, etc.) the absorption shows one broad band which absorption edge is given by the well known Tauc equation 
This absorption is caused by interband electronic transitions between valence and conduction extended states, i. e. ξ→ξ*. However, at the energies below E g a weak absorption occurs due to an existence of localized states inside the band gap. This absorption is called Urbach tail and was explained by transitions from occupied localized states to conduction extended states λ→ξ* and from valence extended states to unoccupied localized states ξ→λ . Then, similar as for DLC, Eq. (4) can be simplified as
where the second term represents both ξ→λ and λ→ξ* contributions because of the DOS symmetry. However, unlike for DLC, the DOS of extended states of the a-Si and similar materials cannot be parameterized only by three parameters Q, E g, E h in wide spectral range. It has
to be expressed for example as
According to Urbach tail expression the DOS of localized states is assumed in an exponential form for the energies inside the band gap
Note that in our original work  dealing with characterization of chalcogenide films in shorter spectral range a slightly different parameterization was used for C(S) and Nλ(S).
Results of the application of the above parameterization to fitting of ellipsometric and reflectance measurements on a typical a-Si film are shown in Table 2 and Figs. 5 and 6. In order to achieve agreement between the measurement and the fit a surface roughness described by Rayleigh-Rice theory  and a native oxide layer had to be taken into account.
2.3. SiO2-like materials
Silicon oxides (SiO2-like materials) exhibit interband absorption for the energies above 7 eV, i. e. in vacuum UV range. Therefore, the multipeak structure of the density of extended states known for SiO2 [5, 23] does not need to be taken into account in commonly used measurement range. Then, N ξ and N ξ* can be parameterized by Eqs. (9) and (10), respectively, in which the subscript j is ξ. The absorption inside the band gap is caused by sharp energy transitions between a couple of defect states, i. e. between a ground and excited states. The localized defect states corresponding to i-th type of defect compose two-level systems with a mean transition energy E δ,i. Therefore, their contributions to ε i(E) can be modeled by Gaussian broadened peaks which represent energy distribution of these transitions
where A δ,i is proportional to the density of defects and Bδ,i is the broadening factor. Besides ξ→ξ* and δi→δ*i contributions, the phonon absorption in infrared region (IR) has to be taken into account for a precise expression of the real part of the dielectric function. Although the absorption in IR had a much more complicated structure consisting of several absorption peaks related to Si-O-Si network vibrations as well as silanol, water and hydrocarbon groups the measurement range down to 1.2 eV (104 cm-1) allowed taking only an average contribution of these effects. Therefore, it can be simply modeled by a single Gaussian broadened peak. The final expression of ε i(E) is
where the last term represents the phonon absorption.
This model has already been applied to SiO2-like films in  taking into account just one type of defect states. The results as concern the band gap (7.4–8.2 eV depending on the deposition conditions) and transition energy corresponding to the defect states (4.6–5.4 eV) were in relatively sound agreement with the results obtained by EELS for amorphous SiO2 . The EELS measurements revealed that beside the defect states at 5.5 eV there is second at 7.2 eV. In our previous paper  the higher energy peak was not taken into account because the measurement range of ellipsometer was restricted to 1.5–5.1 eV and was too weak to make a significant contribution to the dielectric function when outside the measurement range. Here, a selected similar SiO2-like film was measured by ellipsometer and reflectometer, both in the wider range 1.2–6.5 eV. Therefore, these measurements were attempted to fit by the model in Eq. (25) taking into account both types of defect states. The resulting dielectric function is compared with tabulated data for fused silica  in Fig. 7. It can be seen from the imaginary part of dielectric function that the defect states contribute only by factors in the order of 10-3. In order to obtain reliable results with such precision in real and imaginary part of dielectric function the surface roughness, the transition layer between the c-Si substrate and SiO2-like film and the final spectral width of incident monochromatized light beam were taken into account. The last factor causes depolarization of reflected light that changes the formalism of ellipsometry because three instead of two conventional measured quantities are necessary for the description. Twelve parameters obtained from the fit are summarized in Table 3. The last two parameters related to the broadening factor and position of phonon absorption peak in IR, i. e. B p and E p, respectively, could not be determined because the absorption lay too far outside the measurement range and, therefore, these two parameters were fixed.
3. PJDOS model
A certain disadvantage of the PDOS model consists in the necessity of numerical calculations. Therefore, we developed the PJDOS model that represents a compromise between the requirement of an analytical expression and the parameterization reflecting band structure of amorphous materials. Using this approach, the JDOS is parameterized directly and the integration required in Eq. (4) is avoided. However, the real part of dielectric function has to be calculated, in general, by a numerical integration given by Eq. (5). Therefore, it is useful to find a parameterization of the JDOS that allows analytical expression of ε r. An example of such parameterization is shown below.
Similarly as unnormalized DOS Nj is defined by Eq. (7) the unnormalized JDOS corresponding to j→j* transitions can be defined as
In the analogy to Eq. (8) it is, evidently, normalized to the parameter Q 2 j
The unnormalized JDOS corresponding to all possible transitions, J(E), is a sum of the above introduced J j→j* (E) and have a straight forward relation to the ε i(E):
Since the parameterization of Nj with a good physical meaning that would lead to an analytical integration of Eqs. (4) and (5) was not yet found it is advantageous, as concern the efficiency of the fitting algorithm, to parameterize directly J j→j*(E). In case of proposed parameterization (9) and (10) the elliptic integral ej(E) in Eq. (11) should be replaced by a similar function. It leads to the following parameterization of J j→j*(E)
for the energies of interband transitions. Then
Integrating the Kramers-Kronig relation (5), contribution of j→j* transitions to the real part of dielectric function can be analytical expressed as
with following substitutions:
For some energy values, i. e. E=0, E gj, E hj and ∞, it is necessary to take into account following limits
The PJDOS model has been first applied to the DLC films . As in the PDOS model of the DLC described in section 2.1 two contributions to the dielectric function corresponding to π→π* and σ→σ* transitions had to be taken into account:
The results obtained by both, PDOS and PJDOS, models are compared in  in case of DLC films. Recently, the PJDOS model was applied also to the ultrananocrystalline diamond (UNCD) films  where three different type of transitions contribute to the dielectric function. It is apparent that the presented PJDOS model can be used also in case of SiO2-like materials supposed the ξ→ξ*, δ→δ* and phonon contributions are all described by Eqs. (30) and (31).
Two dispersion models of disordered solids, the first parameterizing density of states (PDOS) and the second parameterizing joint density of states (PJDOS), were presented. Using these models not only the optical constants, i. e. the complex dielectric function, of the materials but also some information about their electronic structure can be obtained. In the frame of the PDOS model, it is possible to include the contributions of both, interband and intraband transitions, whereas the presented PJDOS model is applicable only to the selected interband transitions. The numerical integration required in the PDOS model could be seen as a certain disadvantage. However, it is important to notice that today PCs are able to calculate the value of complex dielectric function for one photon energy within 1–10 ms using optimized numerical procedures. If even faster calculation time is required the presented PJDOS model is a suitable option for some materials still providing information about the electronic structure of the material.
It was demonstrated that the PDOS model can be successfully applied to a wide variety of materials. In this paper, its application to DLC, a-Si and SiO2-like materials was discussed in details. Unlike the PDOS model, the presented PJDOS model allowing analytical expressions of the complex dielectric function represents a special case of parameterization that can be applied to a limited types of materials, for example DLC, UNCD and SiO2-like. The application of the PDOS or PJDOS models to the DLC was especially advantageous because it provided information about the ratio of π-to-σ electrons.
Although the PDOS and PJDOS models reached excellent agreement with the optical measurement one can see, from the theoretical point of view, their two weaknesses (i) DOS symmetry of valence and conduction bands was assumed, (ii) high excitation states were neglected. Since the optical methods, i. e. ellipsometry and spectrophotometry, measure the convolution of DOS of valence and conductive bands they have to be combined with other methods, such as photoelectron spectroscopy or X-ray absorption spectroscopy, providing independent information about the DOS in order to avoid the first assumption. The second weakness of the PDOS and PJDOS models could be eliminated simply by adding more conductive bands corresponding to higher excitations. The models are still under development and this problem will be confronted in future.
A. Kramers-Kronig integral
Let us consider a Kramers-Kronig integral (5) for the real part of dielectric function ε r(E) calculated from the imaginary part ε i(E). The integrand can have singularities of 1/X type at -E, 0 and E that should be integrated in the sense of principal value. By decomposition to partial fractions and making use of the time-reversal antisymmetry of the imaginary part, i. e. ε i(X)=-ε i(-X), the integral can be transformed
Therefore, Eq. (5) can be rewritten in the following form:
To calculate the principal value by a numerical integration, the integral can be split into three integrals over intervals (-∞,0), (0,E) and (E,∞) in which the following substitutions
are applied, respectively. All the substitutions transform the integration ranges to the same interval (0,1). Thus, we obtain
where the integrand I(x) is a finite function with the following limits at the integration boundaries:
where the function J is defined in Eq. (28) and primed symbols denote derivatives. The integral (43) can be calculated by suitable adaptive method, for example, by adaptive Gaussian method available in GSL .
B. Elliptic integrals
Function e(E) introduced in Eq. (11) is expressed for E g<E<E h as
with integration limits
and ej(E)=0 otherwise. This is an elliptic integral that can be easily transformed to the Legendre normal form using substitution
which leads to
The symbols m and M denote
Substituting further x=m sint and making use of integrand symmetry about zero a complete elliptic integral is obtained:
It can be expressed as follows:
where K(k) and E(k) are the complete elliptic integrals of the first and second kind, respectively :
They can be calculated, for example, by functions available in GSL .
This work was supported by the Grant Agency of Czech Republic under the contract 202/05/0607 and by the Ministry of Education of the Czech Republic under the contracts MSM0021622411 and 1K05025.
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