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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 24 — Nov. 26, 2007
  • pp: 16230–16244
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Models of dielectric response in disordered solids

Daniel Franta, David Nečas, and Lenka Zajíčková  »View Author Affiliations


Optics Express, Vol. 15, Issue 24, pp. 16230-16244 (2007)
http://dx.doi.org/10.1364/OE.15.016230


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Abstract

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

1. Introduction

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

4. N. F. Mott and E. A. Davis, Electronic Processes in Non-Crystalline Materials (Clarendon Press, Oxford, 1971).

7

7. P. Y. Yu and M. Cardona, Fundamentals of Semiconductors (Springer, Berlin, 2001).

]

εi(E)=(ehmE)214πε0B0j,kpjk2fe(S)𝒩j(S)fh(S+E)𝒩k(S+E)dS,
(1)

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

Nj=𝒩j(S)dS.
(2)

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 jk|2. It was factored out of the integral supposing constant for all the transitions between particular j and k bands.

Fig. 1. Schematic diagram of interband and intraband electronic transitions. Shaded area depicts occupied electronic states according to the Fermi-Dirac statistics. Symbol E F denotes the Fermi energy.

𝒥jk(E)=𝒩j(S)𝒩k(S+E)dS.
(3)

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 [8

8. F. Wooten, Optical Properties of Solids (Academic Press, New York, 1972).

]

εi(E)=(ehmE)2sgn(E)4πε0B0j,kpjk2fe(S)𝒩j(S)fh(S+E)𝒩k(S+E)dS,
(4)
εr(E)=1+1πXεi(X)X2E2dX=1+2π0Xεi(X)X2E2dX.
(5)

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

Fig. 2. Schematic diagram of electronic structure of three types of amorphous solids. Shaded area depicts occupied electronic states according to the Fermi–Dirac statistics. Symbol E F denotes the Fermi energy.

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 [9

9. D. Franta, I. Ohlídal, M. Frumar, and J. Jedelský, “Expression of the Optical Constants of Chalcogenide Thin Films Using the New Parameterization Dispersion Model,” Appl. Surf. Sci. 212–213, 116–121 (2003). [CrossRef]

]. 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 [10

10. D. Franta, I. Ohlídal, P. Klapetek, and P. Roca i Cabarrocas, “Complete Characterization of Rough Polymorphous Silicon Films by Atomic Force Microscopy and the Combined Method of Spectroscopic Ellipsometry and Spectroscopic Reflectometry,” Thin Solid Films 455–456, 399–403 (2004). [CrossRef]

] 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 δδ* were taken into account in the case of amorphous SiOxCyHz films [11

11. L. Zajíčková, V. Buršíková, D. Franta, A. Bousquet, A. Granier, A. Goullet, and J. Buršík, “Comparative Study of Films Deposited from HMDSO/O2 in Continuous Wave and Pulsed rf Discharges,” Plasma Process. Polym. 4, S287–S293 (2007). [CrossRef]

]. 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

12. D. Franta, L. Zajíčková, V. Buršíková, and I. Ohlídal, “New Dispersion Model of the Optical Constants of the DLC Films,” Acta Phys. Slov. 53, 373–384 (2003).

, 13

13. D. Franta, I. Ohlídal, V. Buršíková, and L. Zajíčková, “Optical properties of diamond-like carbon films containing SiOx,” Diamond Relat. Mater. 12, 1532–1538 (2003). [CrossRef]

] which mixed sp3/sp2 bonding structure gives rise to two types of electrons, σ and π. The transitions σσ* and ππ* contribute to the dielectric function whereas σπ* and πσ* 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

14. D. Franta, I. Ohlídal, V. Buršíková, and L. Zajíčková, “Optical Properties of Diamond-Like Carbon Films Containing SiOx Studied by the Combined Method of Spectroscopic Ellipsometry and Spectroscopic Reflectometry,” Thin Solid Films 455–456, 393–398 (2004). [CrossRef]

17

17. D. Franta, V. Buršíková, D. Nečas, and L. Zajíčková, “Modeling of optical constants of diamond-like carbon,” Diamond Relat. Mater. (submitted for publication).

]. 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 [18

18. D. Franta, M. Hrdlička, D. Nečas, M. Frumar, I. Ohlídal, and M. Pavlišta, “Optical characterization of phase changing Ge2Sb2Te5 chalcogenide films,” Phys. Status Solidi A-Appl. Mat. (to be published).

].

2.1. Application to DLC

εi(E)=sgn(E)E2j=π,σEF=0Nj(S)Nj*(S+E)dS,
(6)

where

Nj(S)=ehpjj*2mπε0B0𝒩j(S).
(7)

Unlike 𝒩j(S), Nj(S) is not normalized to the concentration of j electrons Nj. Therefore, Nj(S) was called unnormalized DOS in [12

12. D. Franta, L. Zajíčková, V. Buršíková, and I. Ohlídal, “New Dispersion Model of the Optical Constants of the DLC Films,” Acta Phys. Slov. 53, 373–384 (2003).

, 14

14. D. Franta, I. Ohlídal, V. Buršíková, and L. Zajíčková, “Optical Properties of Diamond-Like Carbon Films Containing SiOx Studied by the Combined Method of Spectroscopic Ellipsometry and Spectroscopic Reflectometry,” Thin Solid Films 455–456, 393–398 (2004). [CrossRef]

] although it is normalized obviously by

Nj(S)dS=Njehpjj*2mπε0B0Qj,
(8)

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 jj* 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

Nj(S)={32QjSEgj2Ehj2+Sπ(EhjEgj)2forEhj2<S<Egj20otherwise
(9)

and N j*(S) for conduction electrons is

Nj*(S)={32QjSEgj2Ehj2Sπ(EhjEgj)2forEgj2<S<Ehj20otherwise.
(10)

Substituting Eqs. (9) and (10) into Eq. (6) one obtains the expression for imaginary part of dielectric function of DLC

εi(E)=1E2j=π,σ(32Qjπ(EhjEgj)2)2ej(E),
(11)

where ej(E) is an elliptic integral which has to be calculated numerically (see Appendix B).

Besides the parameterization with Q j, E gj and E hj, that was suggested first in [17

17. D. Franta, V. Buršíková, D. Nečas, and L. Zajíčková, “Modeling of optical constants of diamond-like carbon,” Diamond Relat. Mater. (submitted for publication).

], a slightly different set of parameters employing the parameter Aj instead of Qj was used in [12

12. D. Franta, L. Zajíčková, V. Buršíková, and I. Ohlídal, “New Dispersion Model of the Optical Constants of the DLC Films,” Acta Phys. Slov. 53, 373–384 (2003).

16

16. D. Franta, V. Buršíková, I. Ohlídal, P. St’ahel, M. Ohlídal, and D. Nečas, “Correlation of thermal stability of the mechanical and optical properties of diamond-like carbon films,” Diamond Relat. Mater. 16, 1331–1335 (2007). [CrossRef]

]:

Aj32Qjπ(EhjEgj)2.
(12)

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.

Fig. 3. Unnormalized DOS of two DLC films calculated from the fitted parameters summarized in Table 1. Solid and dashed lines correspond to as deposited and annealed (510 °C) DLC films.
Fig. 4. Real (top) and imaginary (bottom) part of dielectric function for all the studied DLC films.

The results of unnormalized DOS, Nσ(S) and Nπ(S), for as deposited and annealed (510 °C) DLC films studied in [15

15. D. Franta, V. Buršíková, I. Ohlídal, L. Zajíčková, and P. St’ahel, “Thermal stability of the optical properties of plasma deposited diamond-like carbon thin films,” Diamond Relat. Mater. 14, 1795–1798 (2005). [CrossRef]

] 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.

Table 1. Fitting parameters of PDOS model applied to the DLC films from [15] together with hydrogen atomic fractions XH determined by ERDA, the π-to-σ ratio α, the ratio β of the number of valence electrons in the film after and before annealing and sp3-to-sp2 ratio XC(sp3)XC(sp2).

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The advantage of the model is the possibility to evaluate the ratio of π-to-σ electrons
α=NπNσ=1κQπQσ
(13)
and quantity proportional to the concentration of valence electrons
Ne=Nπ+NσQπ+κQσ
(14)
if the probability ratio κ=|p ππ*|/|p σσ*| is known.

βne(t)ne(0)=df(t)Ne(t)df(0)Ne(0)=df(t)(Qπ(t)+κQσ(t))df(0)(Qπ(0)+κQσ(0)),
(15)

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

β=nH(t)+4nC(t)nH(0)+4nC(0)=4na(t)3nH(t)4na(0)3nH(0)=na(t)(43XH(t))na(0)(43XH(0))=
na(t)nC(0)(43XH(t))na(0)nC(t)(43XH(0))=na(t)na(0)XC(0)(43XH(t))na(0)na(t)XC(t)(43XH(0))=(1XH(0))(43XH(t))(1XH(t))(43XH(0))
(16)

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 [15

15. D. Franta, V. Buršíková, I. Ohlídal, L. Zajíčková, and P. St’ahel, “Thermal stability of the optical properties of plasma deposited diamond-like carbon thin films,” Diamond Relat. Mater. 14, 1795–1798 (2005). [CrossRef]

] 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 XC(sp3)XC(sp2) can be calculated as [9

9. D. Franta, I. Ohlídal, M. Frumar, and J. Jedelský, “Expression of the Optical Constants of Chalcogenide Thin Films Using the New Parameterization Dispersion Model,” Appl. Surf. Sci. 212–213, 116–121 (2003). [CrossRef]

, 15

15. D. Franta, V. Buršíková, I. Ohlídal, L. Zajíčková, and P. St’ahel, “Thermal stability of the optical properties of plasma deposited diamond-like carbon thin films,” Diamond Relat. Mater. 14, 1795–1798 (2005). [CrossRef]

, 20

20. F. Demichelis, C. F. Pirri, and A. Tagliaferro, “Evaluation of the [C(sp3)]/[C(sp2)] ratio in diamondlike films through the use of a complex dielectric constant,” Phys. Rev. B 45, 14,364–14,370 (1992). [CrossRef]

]:

XC(sp3)XC(sp2)=(13α)XH(12α)α(43XH).
(17)

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 [5

5. J. Tauc, “Optical Properties of Non-Crystaline Solids,” in Optical Properties of Solids, F. Abelès, ed., pp. 277–313 (North-Holland, Amsterdam, 1972).

]

εi(E)(EEg)2E2.
(18)

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 [21

21. D. Wood and J. Tauc, “Weak Absorption Tails in Amorphous Semiconductors,” Phys. Rev. B 5, 3144–3151 (1972). [CrossRef]

]. Then, similar as for DLC, Eq. (4) can be simplified as

εi(E)=sgn(E)E2EF=0[Nξ(S)Nξ*(S+E)+2Nλ(S)Nξ*(S+E)]dS,
(19)

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

Table 2. Fitting parameters of model applied to the typical a-Si:H film.

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to be expressed for example as

Nξ(S)={AξC(S)SEg2Eh2+SforEh2<S<Eg20otherwise,
(20)
Nξ*(S)={AξC(S)SEg2Eh2SforEg2<S<Eh20otherwise,
(21)

where

C(S)=1+A1exp[(SE12)22B12]+A2exp[(SE22)22B22].
(22)

According to Urbach tail expression the DOS of localized states is assumed in an exponential form for the energies inside the band gap

Nλ(S)=Aλexp(SEg2Eλ).
(23)

Note that in our original work [9

9. D. Franta, I. Ohlídal, M. Frumar, and J. Jedelský, “Expression of the Optical Constants of Chalcogenide Thin Films Using the New Parameterization Dispersion Model,” Appl. Surf. Sci. 212–213, 116–121 (2003). [CrossRef]

] dealing with characterization of chalcogenide films in shorter spectral range a slightly different parameterization was used for C(S) and Nλ(S).

Fig. 5. Unnormalized DOS of a-Si film calculated from the fitted parameters summarized in Table 2.

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 [22

22. D. Franta and I. Ohlídal, “Ellipsometric Parameters and Reflectances of Thin Films with Slightly Rough Boundaries,” J. Mod. Opt. 45, 903–934 (1998). [CrossRef]

] and a native oxide layer had to be taken into account.

Fig. 6. Real (top) and imaginary (bottom) part of dielectric function for a-Si film and c-Si substrate.

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

5. J. Tauc, “Optical Properties of Non-Crystaline Solids,” in Optical Properties of Solids, F. Abelès, ed., pp. 277–313 (North-Holland, Amsterdam, 1972).

, 23

23. L. Pajasová, “Optical properties of GeO2 in the ultraviolet region,” Czech. J. Phys. 19, 1265–1270 (1969). [CrossRef]

] 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

G(Aδ,i,Bδ,i,Eδ,i)=Aδ,i2πBδ,i{exp[(EEδ,i)22Bδ,i2]exp[(E+Eδ,i)22Bδ,i2]},
(24)

εi(E)=sgn(E)E2[EF=0Nξ(S)Nξ*(S+E)dS+iG(Aδ,i,Bδ,i,Eδ,i)+G(Ap,Bp,Ep)],
(25)

Table 3. Fitting parameters of model applied to a SiO2-like film prepared similarly as in [11].

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where the last term represents the phonon absorption.

Fig. 7. Real (top) and imaginary (bottom) parts of dielectric function of SiO2-like film with the thickness of 1.13 µm deposited on c-Si substrate. Data tabulated for fused silica [24] are shown for comparison.

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 jj* transitions can be defined as

(ehm)2pjj*24πε0B0𝒥jj*(E)=Jjj*(E).
(26)

In the analogy to Eq. (8) it is, evidently, normalized to the parameter Q 2 j

0Jjj*(E)dE=Qj2.
(27)

The unnormalized JDOS corresponding to all possible transitions, J(E), is a sum of the above introduced J jj* (E) and have a straight forward relation to the ε i(E):

εi(E)=J(E)E2=1E2jJjj*(E).
(28)

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 jj*(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 jj*(E)

Jjj*(E)(EEgj)2(EEhj)2
(29)

for the energies of interband transitions. Then

εi,jj*(E)={30Qj2(EEgj)2(EEhj)2(EhjEgj)5E2forEgj<E<Ehj0otherwise.
(30)

Integrating the Kramers-Kronig relation (5), contribution of jj* transitions to the real part of dielectric function can be analytical expressed as

εi,jj*(E)=60Qj2π(EhjEgj)5[B(E)lnE+EhjE+Egj+C(E)lnEEhjEEgjD(E)]
(31)

with following substitutions:

B(E)=Y(E)+X(E)2E2,C(E)=Y(E)X(E)2E2,
(32)
D(E)=Egj2Ehj2E2lnEhjEgj+3(Ehj2Egj2)2,
(33)
X(E)=2E[Ehj(Egj2+E2)+Egj(Ehj2+E2)],
(34)

and

Y(E)=E2(Ehj2+Egj2+4EgjEhj+E2)+Egj2Ehj2.
(35)

For some energy values, i. e. E=0, E gj, E hj and ∞, it is necessary to take into account following limits

limEEhj,EgjC(E)lnEEhjEEgj=0,
(36)
limE0εr,jj*(E)=60Qj2π(EhjEgj)5[(Ehj2+Egj2+4EgjEhj)lnEhjEgj3(Ehj2Egj2)2]
(37)

and

limEεr,jj*(E)=0.
(38)

The PJDOS model has been first applied to the DLC films [17

17. D. Franta, V. Buršíková, D. Nečas, and L. Zajíčková, “Modeling of optical constants of diamond-like carbon,” Diamond Relat. Mater. (submitted for publication).

]. As in the PDOS model of the DLC described in section 2.1 two contributions to the dielectric function corresponding to ππ* and σσ* transitions had to be taken into account:

ε̂(E)=1+ε̂ππ*(E)+ε̂σσ*(E).
(39)

The results obtained by both, PDOS and PJDOS, models are compared in [17

17. D. Franta, V. Buršíková, D. Nečas, and L. Zajíčková, “Modeling of optical constants of diamond-like carbon,” Diamond Relat. Mater. (submitted for publication).

] in case of DLC films. Recently, the PJDOS model was applied also to the ultrananocrystalline diamond (UNCD) films [26

26. D. Franta, L. Zajíčková, M. Karásková, O. Jašek, D. Nečas, P. Klapetek, and M. Valtr, “Optical Characterization of Ultrananocrystalline Diamond Films,” Diamond Relat. Mater. (submitted for publication).

] 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*, δδ* and phonon contributions are all described by Eqs. (30) and (31).

4. Conclusion

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

Xεi(X)X2E2dX=12[εi(X)X+E+εi(X)XE]dX=
12εi(X)X+EdX+12εi(X)XEdX=εi(X)XEdX.
(40)

Therefore, Eq. (5) can be rewritten in the following form:

εr(E)=1+1πεi(X)XEdX.
(41)

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

X=xE1x,X=ExandX=Ex,
(42)

are applied, respectively. All the substitutions transform the integration ranges to the same interval (0,1). Thus, we obtain

εr(E)=1+1π01εi(Ex)xεi(xE)xεi(xE1x)x(1x)dx=1+1π01I(x)dx,
(43)

where the integrand I(x) is a finite function with the following limits at the integration boundaries:

limx0+I(x)=J(0)E,
(44)
limx1I(x)=2Eεi(E)+εi(E),
(45)

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 [27

27. M. Galassi, J. Davies, J. Theiler, B. Gough, G. Jungman, M. Booth, and F. Rossi, GNU Scientific Library Reference Manual, 2nd ed. (Network Theory Limited, Bristol, 2005).

].

B. Elliptic integrals

Function e(E) introduced in Eq. (11) is expressed for E g<E<E h as

e(E)=SminSmax(SEg2)(Eh2+S)(E+SEg2)(Eh2ES)dS,
(46)

with integration limits

Smin=max(Eh2,Eg2E)andSmax=min(Eg2,Eh2E)
(47)

and ej(E)=0 otherwise. This is an elliptic integral that can be easily transformed to the Legendre normal form using substitution

S=xE2,
(48)

which leads to

e(E)=mm(M2x2)(m2x2)dx.
(49)

The symbols m and M denote

m=min(E2Eg2,Eh2E2)andM=max(E2Eg2,Eh2E2).
(50)

Substituting further x=m sint and making use of integrand symmetry about zero a complete elliptic integral is obtained:

e(E)=2m2M0π21k2sin2tcos2tdt,wherek=mM.
(51)

It can be expressed as follows:

e(E)=23M3[(k21)K(k)+(k2+1)E(k)],
(52)

where K(k) and E(k) are the complete elliptic integrals of the first and second kind, respectively [28

28. M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, 1964).

]:

K(k)=0π211k2sin2tdt,E(k)=0π21k2sin2tdt.
(53)

They can be calculated, for example, by functions available in GSL [27

27. M. Galassi, J. Davies, J. Theiler, B. Gough, G. Jungman, M. Booth, and F. Rossi, GNU Scientific Library Reference Manual, 2nd ed. (Network Theory Limited, Bristol, 2005).

].

Acknowledgments

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.

References and links

1.

A. R. Forouhi and I. Bloomer, “Optical dispersion relations for amorphous semiconductors and amorphous dielectrics,” Phys. Rev. B 34, 7018–7026 (1986). [CrossRef]

2.

G. E. Jellison and F. A. Modine, “Parameterization of the optical functions of amorphous materials in the interband region,” Appl. Phys. Lett. 69, 371–373 (1996). [CrossRef]

3.

A. S. Ferlauto, G. M. Ferreira, J. M. Pearce, C. R. Wronski, R. W. Collins, X. M. Deng, and G. Ganguly, “Analytical model for the optical functions of amorphous semiconductors from the near-infrared to ultraviolet: Applications in thin film photovoltaics,” J. Appl. Phys. 92, 2424–2436 (2002). [CrossRef]

4.

N. F. Mott and E. A. Davis, Electronic Processes in Non-Crystalline Materials (Clarendon Press, Oxford, 1971).

5.

J. Tauc, “Optical Properties of Non-Crystaline Solids,” in Optical Properties of Solids, F. Abelès, ed., pp. 277–313 (North-Holland, Amsterdam, 1972).

6.

S. Adachi, Optical Properties of Crystaline and Amorphous Semiconductors: Matrials and Fundamental Principles (Kluwer, Boston, 1999). [CrossRef] [PubMed]

7.

P. Y. Yu and M. Cardona, Fundamentals of Semiconductors (Springer, Berlin, 2001).

8.

F. Wooten, Optical Properties of Solids (Academic Press, New York, 1972).

9.

D. Franta, I. Ohlídal, M. Frumar, and J. Jedelský, “Expression of the Optical Constants of Chalcogenide Thin Films Using the New Parameterization Dispersion Model,” Appl. Surf. Sci. 212–213, 116–121 (2003). [CrossRef]

10.

D. Franta, I. Ohlídal, P. Klapetek, and P. Roca i Cabarrocas, “Complete Characterization of Rough Polymorphous Silicon Films by Atomic Force Microscopy and the Combined Method of Spectroscopic Ellipsometry and Spectroscopic Reflectometry,” Thin Solid Films 455–456, 399–403 (2004). [CrossRef]

11.

L. Zajíčková, V. Buršíková, D. Franta, A. Bousquet, A. Granier, A. Goullet, and J. Buršík, “Comparative Study of Films Deposited from HMDSO/O2 in Continuous Wave and Pulsed rf Discharges,” Plasma Process. Polym. 4, S287–S293 (2007). [CrossRef]

12.

D. Franta, L. Zajíčková, V. Buršíková, and I. Ohlídal, “New Dispersion Model of the Optical Constants of the DLC Films,” Acta Phys. Slov. 53, 373–384 (2003).

13.

D. Franta, I. Ohlídal, V. Buršíková, and L. Zajíčková, “Optical properties of diamond-like carbon films containing SiOx,” Diamond Relat. Mater. 12, 1532–1538 (2003). [CrossRef]

14.

D. Franta, I. Ohlídal, V. Buršíková, and L. Zajíčková, “Optical Properties of Diamond-Like Carbon Films Containing SiOx Studied by the Combined Method of Spectroscopic Ellipsometry and Spectroscopic Reflectometry,” Thin Solid Films 455–456, 393–398 (2004). [CrossRef]

15.

D. Franta, V. Buršíková, I. Ohlídal, L. Zajíčková, and P. St’ahel, “Thermal stability of the optical properties of plasma deposited diamond-like carbon thin films,” Diamond Relat. Mater. 14, 1795–1798 (2005). [CrossRef]

16.

D. Franta, V. Buršíková, I. Ohlídal, P. St’ahel, M. Ohlídal, and D. Nečas, “Correlation of thermal stability of the mechanical and optical properties of diamond-like carbon films,” Diamond Relat. Mater. 16, 1331–1335 (2007). [CrossRef]

17.

D. Franta, V. Buršíková, D. Nečas, and L. Zajíčková, “Modeling of optical constants of diamond-like carbon,” Diamond Relat. Mater. (submitted for publication).

18.

D. Franta, M. Hrdlička, D. Nečas, M. Frumar, I. Ohlídal, and M. Pavlišta, “Optical characterization of phase changing Ge2Sb2Te5 chalcogenide films,” Phys. Status Solidi A-Appl. Mat. (to be published).

19.

D. C. Ingram, J. A. Woollam, and G. Bu-Abbud, “Mass density and hydrogen concentration in diamond-like carbon films: proton recoil, rutherford backscattering and ellipsometric analysis,” Thin Solid Films 137, 225–230 (1986). [CrossRef]

20.

F. Demichelis, C. F. Pirri, and A. Tagliaferro, “Evaluation of the [C(sp3)]/[C(sp2)] ratio in diamondlike films through the use of a complex dielectric constant,” Phys. Rev. B 45, 14,364–14,370 (1992). [CrossRef]

21.

D. Wood and J. Tauc, “Weak Absorption Tails in Amorphous Semiconductors,” Phys. Rev. B 5, 3144–3151 (1972). [CrossRef]

22.

D. Franta and I. Ohlídal, “Ellipsometric Parameters and Reflectances of Thin Films with Slightly Rough Boundaries,” J. Mod. Opt. 45, 903–934 (1998). [CrossRef]

23.

L. Pajasová, “Optical properties of GeO2 in the ultraviolet region,” Czech. J. Phys. 19, 1265–1270 (1969). [CrossRef]

24.

H. R. Philipp, “Silicon Dioxide (SiO2) (Glass),” in Handbook of Optical Constants of Solids , E. Palik, ed., vol. I, pp. 749–763 (Academic Press, New York, 1985). [CrossRef]

25.

L. A. J. Garvie, P. Rez, J. R. Alvarez, and P. R. Buseck, “Interband transitions of crystalline and amorphous SiO2: An electron energy-loss spectroscopy (EELS) study of the low-loss region,” Solid State Commun. 106(5), 303–307 (1998). [CrossRef]

26.

D. Franta, L. Zajíčková, M. Karásková, O. Jašek, D. Nečas, P. Klapetek, and M. Valtr, “Optical Characterization of Ultrananocrystalline Diamond Films,” Diamond Relat. Mater. (submitted for publication).

27.

M. Galassi, J. Davies, J. Theiler, B. Gough, G. Jungman, M. Booth, and F. Rossi, GNU Scientific Library Reference Manual, 2nd ed. (Network Theory Limited, Bristol, 2005).

28.

M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, 1964).

OCIS Codes
(120.4530) Instrumentation, measurement, and metrology : Optical constants
(160.2750) Materials : Glass and other amorphous materials
(260.2030) Physical optics : Dispersion

ToC Category:
Physical Optics

History
Original Manuscript: October 17, 2007
Revised Manuscript: November 20, 2007
Manuscript Accepted: November 20, 2007
Published: November 21, 2007

Citation
Daniel Franta, David Nečas, and Lenka Zajíčková, "Models of dielectric response in disordered solids," Opt. Express 15, 16230-16244 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-24-16230


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References

  1. A. R. Forouhi and I. Bloomer, "Optical dispersion relations for amorphous semiconductors and amorphous dielectrics," Phys. Rev. B 34, 7018-7026 (1986). [CrossRef]
  2. G. E. Jellison and F. A. Modine, "Parameterization of the optical functions of amorphous materials in the interband region," Appl. Phys. Lett. 69, 371-373 (1996). [CrossRef]
  3. A. S. Ferlauto, G. M. Ferreira, J. M. Pearce, C. R. Wronski, R. W. Collins, X. M. Deng, and G. Ganguly, "Analytical model for the optical functions of amorphous semiconductors from the near-infrared to ultraviolet: Applications in thin film photovoltaics," J. Appl. Phys. 92, 2424-2436 (2002). [CrossRef]
  4. N. F. Mott and E. A. Davis, Electronic Processes in Non-Crystalline Materials (Clarendon Press, Oxford, 1971).
  5. J. Tauc, "Optical Properties of Non-Crystaline Solids," in Optical Properties of Solids, F. Abel`es, ed., pp. 277-313 (North-Holland, Amsterdam, 1972).
  6. S. Adachi, Optical Properties of Crystaline and Amorphous Semiconductors: Matrials and Fundamental Principles (Kluwer, Boston, 1999). [CrossRef] [PubMed]
  7. P. Y. Yu and M. Cardona, Fundamentals of Semiconductors (Springer, Berlin, 2001).
  8. F. Wooten, Optical Properties of Solids (Academic Press, New York, 1972).
  9. D. Franta, I. Ohlýdal, M. Frumar, and J. Jedelský, "Expression of the Optical Constants of Chalcogenide Thin Films Using the New Parameterization Dispersion Model," Appl. Surf. Sci. 212-213, 116-121 (2003). [CrossRef]
  10. D. Franta, I. Ohlýdal, P. Klapetek, and P. Roca i Cabarrocas, "Complete Characterization of Rough Polymorphous Silicon Films by Atomic Force Microscopy and the Combined Method of Spectroscopic Ellipsometry and Spectroscopic Reflectometry," Thin Solid Films 455-456, 399-403 (2004). [CrossRef]
  11. L. Zajý¡cková, V . Bur¡sýková, D . Franta, A.  Bousquet, A.  Granier, A.  Goullet, and J. Bur¡sýk, "Comparative Study of Films Deposited from HMDSO/O2 in Continuous Wave and Pulsed rf Discharges," Plasma Process. Polym. 4, S287-S293 (2007). [CrossRef]
  12. D. Franta, L. Zajý¡cková, V. Bur¡sýková, and I. Ohlýdal, "New Dispersion Model of the Optical Constants of the DLC Films," Acta Phys. Slov.  53, 373-384 (2003).
  13. D. Franta, I. Ohlýdal, V. Burýková, and L. Zajý¡cková, "Optical properties of diamond-like carbon films containing SiOx," Diamond Relat. Mater. 12, 1532-1538 (2003). [CrossRef]
  14. D. Franta, I. Ohlýdal, V. Burýková, and L.  Zajý¡cková, "Optical Properties of Diamond-Like Carbon Films Containing SiOx Studied by the Combined Method of Spectroscopic Ellipsometry and Spectroscopic Reflectometry," Thin Solid Films 455-456, 393-398 (2004). [CrossRef]
  15. D. Franta, V. Burýková, I. Ohlýdal, L. Zajý¡cková, and P. Stáhel, "Thermal stability of the optical properties of plasma deposited diamond-like carbon thin films," Diamond Relat. Mater. 14, 1795-1798 (2005). [CrossRef]
  16. D. Franta, V. Burýková, I. Ohlýdal, P. Stáhel, M. Ohlýdal, and D. Ne¡cas, "Correlation of thermal stability of the mechanical and optical properties of diamond-like carbon films," Diamond Relat. Mater. 16, 1331-1335 (2007). [CrossRef]
  17. D. Franta, V. Burýková, D. Ne¡cas, and L. Zajý¡cková, "Modeling of optical constants of diamond-like carbon," Diamond Relat. Mater. (submitted for publication).
  18. D. Franta, M. Hrdli¡cka, D. Ne¡cas, M. Frumar, I. Ohlýdal, and M. Pavli¡sta, "Optical characterization of phase changing Ge2Sb2Te5 chalcogenide films," Phys. Status Solidi A-Appl. Mat. (to be published).
  19. D. C. Ingram, J. A. Woollam, and G. Bu-Abbud, "Mass density and hydrogen concentration in diamond-like carbon films: proton recoil, rutherford backscattering and ellipsometric analysis," Thin Solid Films 137, 225-230 (1986). [CrossRef]
  20. F. Demichelis, C. F. Pirri, and A. Tagliaferro, "Evaluation of the [C(sp3)]/[C(sp2)] ratio in diamondlike films through the use of a complex dielectric constant," Phys. Rev. B 45, 14,364-14,370 (1992). [CrossRef]
  21. D. Wood and J. Tauc, "Weak Absorption Tails in Amorphous Semiconductors," Phys. Rev. B 5, 3144-3151 (1972). [CrossRef]
  22. D. Franta and I. Ohlýdal, "Ellipsometric Parameters and Reflectances of Thin Films with Slightly Rough Boundaries," J. Mod. Opt. 45, 903-934 (1998). [CrossRef]
  23. L. Pajasov’a, "Optical properties of GeO2 in the ultraviolet region," Czech. J. Phys. 19, 1265-1270 (1969). [CrossRef]
  24. H. R. Philipp, "Silicon Dioxide (SiO2) (Glass)," in Handbook of Optical Constants of Solids, E. Palik, ed., vol. I, pp. 749-763 (Academic Press, New York, 1985). [CrossRef]
  25. L. A. J. Garvie, P. Rez, J. R. Alvarez, and P. R. Buseck, "Interband transitions of crystalline and amorphous SiO2: An electron energy-loss spectroscopy (EELS) study of the low-loss region," Solid State Commun. 106(5), 303-307 (1998). [CrossRef]
  26. D. Franta, L. Zaj ¡cková, M. Karásková, O. Ja¡sek, D. Ne¡cas, P. Klapetek, and M. Valtr, "Optical Characterization of Ultrananocrystalline Diamond Films," Diamond Relat. Mater. (submitted for publication).
  27. M. Galassi, J. Davies, J. Theiler, B. Gough, G. Jungman, M. Booth, and F. Rossi, GNU Scientific Library Reference Manual, 2nd ed. (Network Theory Limited, Bristol, 2005).
  28. M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, 1964).

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