## Models of dielectric response in disordered solids

Optics Express, Vol. 15, Issue 24, pp. 16230-16244 (2007)

http://dx.doi.org/10.1364/OE.15.016230

Acrobat PDF (330 KB)

### 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 SiO_{2}-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 SiO_{2}-like.

© 2007 Optical Society of America

## 1. Introduction

*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]

*E*,

*e*,

*h*,

*m*,

*ε*

_{0}and B

_{0}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

*𝒩*(

_{j}*S*) and

*𝒩*(

_{k}*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

*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.

*E*. The complete complex dielectric function can be obtained by incorporating antisymmetry of the imaginary part and by Kramers–Kronig relations [8]

*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

_{2}S

_{3}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]

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]

*δ*, inside the band gap and only the transitions ξ→ξ* and

*δ*→

*δ** were taken into account in the case of amorphous SiO

*C*

_{x}*H*

_{y}*films [11*

_{z}11. L. Zajíčková, V. Buršíková, D. Franta, A. Bousquet, A. Granier, A. Goullet, and J. Buršík, “Comparative Study of Films Deposited from HMDSO/O_{2} in Continuous Wave and Pulsed rf Discharges,” Plasma Process. Polym. **4**, S287–S293 (2007). [CrossRef]

*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:SiO

*films [12, 13*

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

^{3}/sp

^{2}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:SiO

*films were clearly observed by an evaluation of the optical measurements using the PDOS model [14*

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

_{2}Sb

_{2}Te

_{5}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].

### 2.1. Application to DLC

*𝒩*(

_{j}*S*),

*N*(

_{j}*S*) is not normalized to the concentration of

*j*electrons N

*. Therefore,*

_{j}*N*(

_{j}*S*) was called unnormalized DOS in [12, 14

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

*Q*, proportional to the concentration of

_{j}*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}.

*N*(

_{j}*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,

*N*(

_{j}*S*) for valence electrons is expressed as

*N*

_{j*}(

*S*) for conduction electrons is

*e*(

_{j}*E*) is an elliptic integral which has to be calculated numerically (see Appendix B).

*Q*

_{j},

*E*

_{gj}and

*E*

_{hj}, that was suggested first in [17], a slightly different set of parameters employing the parameter

*A*instead of

_{j}*Q*was used in [12–16

_{j}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]

*A*and the quantity

_{j}*Q*, interesting from the viewpoint of material structure, can be calculated afterwards. However, it is less convenient than the parameterization with

_{j}*Q*.

_{j}*N*(

_{σ}*S*) and

*N*(

_{π}*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]

*Q*,

_{σ}*E*

_{gσ},

*E*

_{hσ},

*Q*,

_{π}*E*

_{gπ}and

*E*

_{hπ}, are summarized in Table 1.

*π*-to-

*σ*electrons and quantity proportional to the concentration of valence electrons if the probability ratio

*κ*=|

*p*

_{π→π}*|/|

*p*

_{σ→σ}*| is known.

*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

_{H}, n

_{C}and n

_{a}are the number of hydrogen, carbon and all atoms in the film, respectively. The symbol

*X*denotes the atomic fractions.

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]

*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.

*α*and considering the fact that each sp

^{3}-bonded carbon corresponds to four

*σ*electrons, each sp

^{2}-bonded carbon to three

*σ*and one

*π*electrons and each hydrogen atom to one s electron, the sp

^{3}-to-sp

^{2}ratio

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. 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. 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]

### 2.2. Application to a-Si

_{2}S

_{3}, etc.) the absorption shows one broad band which absorption edge is given by the well known Tauc equation [5]

*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]

*Q*,

*E*

_{g},

*E*

_{h}in wide spectral range. It has

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]

*C*(

*S*) and

*N*(

_{λ}*S*).

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]

### 2.3. SiO_{2}-like materials

_{2}-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 SiO

_{2}[5, 23

23. L. Pajasová, “Optical properties of GeO_{2} in the ultraviolet region,” Czech. J. Phys. **19**, 1265–1270 (1969). [CrossRef]

*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

## 3. PJDOS model

*ε*

_{r}. An example of such parameterization is shown below.

*N*is defined by Eq. (7) the unnormalized JDOS corresponding to

_{j}*j*→

*j** transitions can be defined as

*J*(

*E*), is a sum of the above introduced

*J*

_{j→j}* (

*E*) and have a straight forward relation to the

*ε*

_{i}(

*E*):

*N*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→j}*(

*E*). In case of proposed parameterization (9) and (10) the elliptic integral

*e*(

_{j}*E*) in Eq. (11) should be replaced by a similar function. It leads to the following parameterization of

*J*

_{j→j}*(

*E*)

*j*→

*j** transitions to the real part of dielectric function can be analytical expressed as

*E*=0,

*E*

_{gj},

*E*

_{hj}and ∞, it is necessary to take into account following limits

*π*→

*π** and

*σ*→

*σ** transitions had to be taken into account:

_{2}-like materials supposed the ξ→ξ*,

*δ*→

*δ** and phonon contributions are all described by Eqs. (30) and (31).

## 4. Conclusion

_{2}-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 SiO

_{2}-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.

## A. Kramers-Kronig integral

*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

*E*) and (

*E*,∞) in which the following substitutions

*(*

**I***x*) is a finite function with the following limits at the integration boundaries:

## B. Elliptic integrals

*e*(

_{j}*E*)=0 otherwise. This is an elliptic integral that can be easily transformed to the Legendre normal form using substitution

**and**

*m***denote**

*M**x*=

**sint and making use of integrand symmetry about zero a complete elliptic integral is obtained:**

*m***(**

*K**k*) and

**(**

*E**k*) are the complete elliptic integrals of the first and second kind, respectively [28]:

## Acknowledgments

## References and links

1. | A. R. Forouhi and I. Bloomer, “Optical dispersion relations for amorphous semiconductors and amorphous dielectrics,” Phys. Rev. B |

2. | G. E. Jellison and F. A. Modine, “Parameterization of the optical functions of amorphous materials in the interband region,” Appl. Phys. Lett. |

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. |

4. | N. F. Mott and E. A. Davis, |

5. | J. Tauc, “Optical Properties of Non-Crystaline Solids,” in |

6. | S. Adachi, |

7. | P. Y. Yu and M. Cardona, |

8. | F. Wooten, |

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. |

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 |

11. | L. Zajíčková, V. Buršíková, D. Franta, A. Bousquet, A. Granier, A. Goullet, and J. Buršík, “Comparative Study of Films Deposited from HMDSO/O |

12. | D. Franta, L. Zajíčková, V. Buršíková, and I. Ohlídal, “New Dispersion Model of the Optical Constants of the DLC Films,” Acta Phys. Slov. |

13. | D. Franta, I. Ohlídal, V. Buršíková, and L. Zajíčková, “Optical properties of diamond-like carbon films containing SiO |

14. | D. Franta, I. Ohlídal, V. Buršíková, and L. Zajíčková, “Optical Properties of Diamond-Like Carbon Films Containing SiO |

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. |

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. |

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 Ge |

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 |

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 |

21. | D. Wood and J. Tauc, “Weak Absorption Tails in Amorphous Semiconductors,” Phys. Rev. B |

22. | D. Franta and I. Ohlídal, “Ellipsometric Parameters and Reflectances of Thin Films with Slightly Rough Boundaries,” J. Mod. Opt. |

23. | L. Pajasová, “Optical properties of GeO |

24. | H. R. Philipp, “Silicon Dioxide (SiO |

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. |

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, |

28. | M. Abramowitz and I. A. Stegun, |

**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

- A. R. Forouhi and I. Bloomer, "Optical dispersion relations for amorphous semiconductors and amorphous dielectrics," Phys. Rev. B 34, 7018-7026 (1986). [CrossRef]
- 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]
- 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]
- N. F. Mott and E. A. Davis, Electronic Processes in Non-Crystalline Materials (Clarendon Press, Oxford, 1971).
- J. Tauc, "Optical Properties of Non-Crystaline Solids," in Optical Properties of Solids, F. Abel`es, ed., pp. 277-313 (North-Holland, Amsterdam, 1972).
- S. Adachi, Optical Properties of Crystaline and Amorphous Semiconductors: Matrials and Fundamental Principles (Kluwer, Boston, 1999). [CrossRef] [PubMed]
- P. Y. Yu and M. Cardona, Fundamentals of Semiconductors (Springer, Berlin, 2001).
- F. Wooten, Optical Properties of Solids (Academic Press, New York, 1972).
- 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]
- 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]
- 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]
- 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).
- 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]
- 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]
- 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]
- 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]
- 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).
- 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).
- 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]
- 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]
- D. Wood and J. Tauc, "Weak Absorption Tails in Amorphous Semiconductors," Phys. Rev. B 5, 3144-3151 (1972). [CrossRef]
- D. Franta and I. Ohlýdal, "Ellipsometric Parameters and Reflectances of Thin Films with Slightly Rough Boundaries," J. Mod. Opt. 45, 903-934 (1998). [CrossRef]
- L. Pajasov’a, "Optical properties of GeO2 in the ultraviolet region," Czech. J. Phys. 19, 1265-1270 (1969). [CrossRef]
- 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]
- 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]
- 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).
- 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).
- 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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