franta_2007
===========

D\. Franta, D\. Nečas and L\. Zajíčková, "Models of dielectric response in disordered solids", Opt\. Express 15, 16230 \(2007\)

`Source record <https://github.com/LupoLab/optmat/blob/main/src/optmat/data/sources/franta_2007.toml>`__ · :doc:`Catalogue </the-data/sources>` · :doc:`Review register </register>`

DOI: `10\.1364/oe\.15\.016230 <https://doi.org/10.1364/oe.15.016230>`__

**Source kind:** theory

**Recorded technique:** Two dispersion models of disordered \(amorphous and polycrystalline\) solids built on the electronic structure\: PDOS, which parameterises the densities of states of the initial and final bands and convolves them numerically \(Eqs\. \(1\)\-\(25\)\), and PJDOS, which parameterises the joint density of states directly so that the real part has a closed form \(Eqs\. \(26\)\-\(39\)\); Kramers\-Kronig integration by an adaptive quadrature \(Appendix A\), the elliptic integrals of the square\-root DOS \(Appendix B\)\. Applied \(PDOS\) to films fitted to ellipsometry and reflectance\: diamond\-like carbon films of ref\. \[15\] \(as deposited and annealed\), a typical a\-Si\:H film \(with a Rayleigh\-Rice rough surface and a native oxide\), and an SiO2\-like film \(1\.13 um on c\-Si, like those of ref\. \[11\], measured by ellipsometry and reflectometry at 1\.2\-6\.5 eV, with surface roughness, a transition layer and the monochromator's spectral width taken into account\) \(pp\. 16236\-16241\)\.

**Materials in structured entries:** SiO2\-like\_film\_HMDSO, a\-C\_H, a\-Si\_H

**Dataset quantities:** band\_gap

**Contents:** 3 datasets; 8 formulas.

Curation notes
--------------

Journal pages 16230\-16244 = pdf pages 1\-15 \(the reference list is on pp\. 16230\-16231\)\. Identity \(three authors\) checked against the first page\. A model paper\: no optical constants are tabulated \(Figs\. 4, 6 and 7 show eps of the DLC, a\-Si and SiO2\-like films, Figs\. 3 and 5 the fitted DOS; not digitised\)\. The three parameter tables are the formulas below, checked cell by cell against their images; the band\-gap parameters are also band\_gap datasets \(kind 'effective'\)\.

EQUATION MAP \(E photon energy in eV; S the electron energy, E\_F = 0\)\: \- Eq\. \(1\)/\(4\), p\. 16230\-16232\: eps\_i from the sum over band pairs of \|p\_jk\|^2 times the convolution of the occupied DOS N\_j\(S\) and the empty DOS N\_k\(S \+ E\); Eq\. \(3\) the JDOS; Eq\. \(5\) the Kramers\-Kronig real part; Appendix A, Eqs\. \(40\)\-\(45\), its numerical form \(checked 2026\-09\-27\: Eq\. \(43\) with the PJDOS eps\_i of Eq\. \(30\) reproduces Eq\. \(31\) to 1e\-11\)\. \- PDOS, DLC \(Section 2\.1, pp\. 16233\-16235\)\: Eq\. \(6\) \(stepwise Fermi\-Dirac, sigma and pi bands\), Eq\. \(7\) the 'unnormalized DOS' N\_j, Eq\. \(8\) its integral Q\_j \(proportional to the concentration of j electrons, times the unknown \|p\|\), Eqs\. \(9\)\-\(10\) the symmetric square\-root DOS of the valence and conduction bands between E\_g/2 and E\_h/2, Eq\. \(11\) eps\_i = \(1/E^2\) sum\_j \(32 Q\_j / \(pi \(E\_hj \- E\_gj\)^2\)\)^2 e\_j\(E\), e\_j the elliptic integral of Appendix B \(Eqs\. \(46\)\-\(53\)\); Eq\. \(12\) the older parameter A\_j = 32 Q\_j / \(pi \(E\_hj \- E\_gj\)^2\) of refs\. \[12\]\-\[16\]\. Eqs\. \(13\)\-\(17\)\: the pi\-to\-sigma ratio alpha = Q\_pi / \(kappa Q\_sigma\), with kappa = \|p\_pi\| / \|p\_sigma\| = 0\.5736 found by matching Eqs\. \(15\) and \(16\) for the annealed films \(p\. 16237\), and the sp3/sp2 ratio\. \- PDOS, a\-Si \(Section 2\.2, pp\. 16237\-16238\)\: Eq\. \(18\) the Tauc edge; Eq\. \(19\) eps\_i with the extended\-extended and the \(doubled\) localized\-extended terms; Eqs\. \(20\)\-\(22\) the extended DOS, A\_xi C\(S\) sqrt\(\.\.\.\)sqrt\(\.\.\.\), with C\(S\) = 1 \+ two Gaussians in \|S\| \(A\_1, B\_1, E\_1/2; A\_2, B\_2, E\_2/2\); Eq\. \(23\) the localized \(Urbach\) DOS A\_lambda exp\(\-\|\|S\| \- E\_g/2\| / E\_lambda\) inside the gap\. \- PDOS, SiO2\-like \(Section 2\.3, pp\. 16239\-16241\)\: Eqs\. \(9\)\-\(10\) for the extended states \(j = xi\), Eq\. \(24\) Gaussian two\-level defect terms G\(A, B, E\_delta\) and one Gaussian for the infrared phonons, all inside the bracket of Eq\. \(25\), which divides them by E^2\. MISPRINT\: Eq\. \(24\) prints both exponents without the minus sign, exp\[\(E \- E\_delta\)^2 / \(2 B^2\)\] \- exp\[\(E \+ E\_delta\)^2 / \(2 B^2\)\] \(page image, p\. 16239\), which diverges; exp\[\-\(E \- E\_delta\)^2 / \(2 B^2\)\] \- exp\[\-\(E \+ E\_delta\)^2 / \(2 B^2\)\] is meant\: with it the model reproduces Fig\. 7 \(formula pdos\_sio2\_like, check\)\. \- PJDOS \(Section 3, pp\. 16241\-16242\)\: Eq\. \(26\) the unnormalised JDOS, Eq\. \(27\) its integral Q\_j^2, Eq\. \(28\) eps\_i = J\(E\)/E^2, Eq\. \(29\) J proportional to \(E \- E\_g\)^2 \(E \- E\_h\)^2, Eq\. \(30\) eps\_i = 30 Q^2 \(E \- E\_g\)^2 \(E \- E\_h\)^2 / \(\(E\_h \- E\_g\)^5 E^2\) on \(E\_g, E\_h\), Eqs\. \(31\)\-\(35\) its closed\-form real part, Eqs\. \(36\)\-\(38\) limits\. MISPRINT\: Eq\. \(37\), the E \-\> 0 limit, prints '\- 3\(E\_h^2 \- E\_g^2\)/2' in the bracket; the limit of Eq\. \(31\) is 60 Q^2 / \(pi \(E\_h \- E\_g\)^5\) \[\(E\_h^2 \+ E\_g^2 \+ 4 E\_g E\_h\) ln\(E\_h/E\_g\) \- 3\(E\_h^2 \- E\_g^2\)\] \(series expansion; numerically, with Q = 74\.8, E\_g = 7\.05, E\_h = 25\.9\: Eq\. \(31\) at 1e\-3 eV gives 1\.1046, the corrected limit 1\.1046, the printed one 42\.94\)\. Eq\. \(33\) \(D, with the /2\) is right\. No PJDOS parameters are given here \(the DLC and UNCD fits are in refs\. \[17\] and \[26\]\)\.

COMPARISON WITH THE LATER FRANTA MODELS \(franta\_2013, franta\_2013b, franta\_2015b, franta\_2016 and optmat\.solid\.udm\)\: \- The PJDOS term, Eqs\. \(30\)\-\(35\), is exactly the symmetric valence\-to\-conduction interband term of the universal dispersion model \(franta\_2015b Eqs\. \(4\)\-\(12\); udm\.\_interband\)\: the same \(E \- E\_g\)^2 \(E \- E\_h\)^2 / E^2 shape and the same X\(E\), Y\(E\) in the real part\. Only the normalisation differs\: here the integral of E^2 eps\_i is Q^2 \(Eq\. \(27\)\); in the UDM the Thomas\-Reiche\-Kuhn integral of E eps\_i is the strength N \(eV^2\), with C\_N = \(E\_h^2 \- E\_g^2\)\(E\_h^2 \+ E\_g^2 \- 8 E\_g E\_h\)/12 \+ E\_g^2 E\_h^2 ln\(E\_h/E\_g\)\. So N\_vc = 30 Q^2 C\_N / \(E\_h \- E\_g\)^5 \(checked 2026\-09\-27\: udm\.\_interband scaled so reproduces Eqs\. \(30\)\-\(31\) to 1e\-14 at 1\-30 eV; for the SiO2\-like film's Q, E\_g, E\_h this is N = 357 eV^2\)\. \- The PDOS square\-root DOS convolution \(Eqs\. \(9\)\-\(11\)\) is not in the UDM; the PJDOS form replaced it as its analytic stand\-in \(Section 3\)\. franta\_2013b's 'PDOS' model column cites this paper \(its ref\. \[24\]\)\. \- The 2007 Gaussians \(Eq\. \(24\)\) sit inside the JDOS bracket and are divided by E^2 \(Eq\. \(25\)\), with an amplitude A/\(sqrt\(2 pi\) B\); the UDM's Gaussian peaks \(franta\_2015b Eqs\. \(41\)\-\(43\), udm\.\_gaussian\) are in eps\_i itself, normalised to unit sum\-rule strength \(1/\(sqrt\(pi\) s E\_p\)\)\. Not the same line shape\. \- The Urbach tail here is a localized DOS \(Eq\. \(23\)\) convolved with the conduction band; the UDM's Urbach term \(franta\_2013b Eq\. \(12\), franta\_2015b Eqs\. \(32\)\-\(40\)\) is an exponential JDOS joined to the interband edge\. Different constructions\. \- No sum rule\: Q\_j carries the unknown \|p\_jk\| \(Eq\. \(8\)\), so the strengths are not electron densities; the later papers \(franta\_2013 on\) normalise every term to the Thomas\-Reiche\-Kuhn sum rule, which makes N a density of electrons\.

TABLE 1'S OTHER ROWS \(notes only; no quantity\)\: the thickness ratio d\_f\(t\)/d\_f\(0\) 1\.021\-1\.114, alpha \(pi\-to\-sigma\) 0\.043\-0\.133, beta\* \(Eq\. \(15\)\) and beta† \(Eq\. \(16\)\), the ERDA hydrogen fraction X\_H 0\.38, 0\.36, 0\.37, 0\.34, 0\.30, 0\.25, and the sp3/sp2 ratio 4\.23\-0\.96, for the six films \(the formulas' checks reproduce them\)\. The films are those of ref\. \[15\] \(Franta et al\., Diamond and Related Materials 14, 1795 \(2005\)\), which used the A\_j parameterisation \(p\. 16235\); whether Table 1 is a refit in Q\_j or a conversion of \[15\]'s values is not stated\. Text values\: SiO2\-like films of ref\. \[11\], band gap 7\.4\-8\.2 eV and defect transition energy 4\.6\-5\.4 eV depending on the deposition; EELS of amorphous SiO2 \(ref\. \[25\]\) has defect states at 5\.5 and 7\.2 eV \(p\. 16240\)\.

**Recorded bibliographic identity status:** VALIDATED\_WITH\_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents.

Datasets
--------

franta\_2007/dlc\_band\_gap\_pi
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "dlc_band_gap_pi",
     "material": "a-C_H",
     "file": "a-C_H/franta_2007__dlc_band_gap_pi.csv",
     "table": "tbl_363cc8a41358",
     "quantity": "band_gap",
     "unit": "eV",
     "abscissa": "label",
     "use": "check",
     "parameters": {
       "kind": "effective",
       "form": "film",
       "technique": "ellipsometry and reflectance (PDOS model fit)"
     }
   }

Table 1, E\_g\_pi \(the lower limit of the pi \-\> pi\* transitions, the optical gap of the model\) of the six DLC films of ref\. \[15\], rows labelled by the annealing\: as deposited, 280, 325, 406, 460, 510 C\. A fitted model parameter\. The sigma band's E\_g\_sigma \(1\.546\-1\.674 eV\) is in the formulas\. 'check'\: possibly a re\-expression of ref\. \[15\]'s fits \(see the record's notes\)\.

franta\_2007/a\_si\_h\_band\_gap
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "a_si_h_band_gap",
     "material": "a-Si_H",
     "file": "a-Si_H/franta_2007__a_si_h_band_gap.csv",
     "table": "tbl_54c954601fc2",
     "quantity": "band_gap",
     "unit": "eV",
     "abscissa": "none",
     "use": "primary",
     "parameters": {
       "kind": "effective",
       "form": "film",
       "technique": "ellipsometry and reflectance (PDOS model fit)"
     }
   }

Table 2, E\_g of 'the typical a\-Si\:H film'\: the lower limit of the extended\-state DOS \(Eqs\. \(20\)\-\(21\)\), below which the localized states of Eq\. \(23\) \(Urbach energy E\_lambda = 0\.093 eV\) absorb\. A fitted model parameter; no uncertainty\. Deposition and thickness not stated\.

franta\_2007/sio2\_like\_band\_gap
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "sio2_like_band_gap",
     "material": "SiO2-like_film_HMDSO",
     "file": "SiO2-like_film_HMDSO/franta_2007__sio2_like_band_gap.csv",
     "table": "tbl_e67d945ff5b3",
     "quantity": "band_gap",
     "unit": "eV",
     "abscissa": "none",
     "use": "primary",
     "parameters": {
       "kind": "effective",
       "form": "film",
       "technique": "ellipsometry and reflectance (PDOS model fit)"
     }
   }

Table 3, E\_g of the SiO2\-like film \(1\.13 um on c\-Si, 'prepared similarly as in \[11\]', i\.e\. plasma\-deposited from HMDSO/O2\)\: the lower limit of the extended\-state DOS, fitted from data at 1\.2\-6\.5 eV, so it lies outside the measured range and is constrained only through the dispersion below the gap\. A fitted model parameter; no uncertainty\.

Formulas
--------

franta\_2007/pdos\_dlc\_as\_deposited
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "pdos_dlc_as_deposited",
     "material": "a-C_H",
     "equation": "Eqs. (5), (6), (9)-(11), (46)-(53), pp. 16231-16235 and 16244; Table 1 (p. 16236), column 'as deposited'",
     "printed": "eps_i(E) = (1/E^2) sum_{j = pi, sigma} (32 Q_j / (pi (E_hj - E_gj)^2))^2 e_j(E) (Eq. 11), e_j(E) = (2/3) M^3 [(k^2 - 1) K(k) + (k^2 + 1) E(k)], k = m/M, m = min(E/2 - E_g/2, E_h/2 - E/2), M = max(...), on E_g < E < E_h (Eqs. 50, 52); eps_r by Kramers-Kronig (Eq. 5)",
     "variable": "E: photon energy (eV); Q_j in eV^(3/2); energies in eV",
     "range": "the measured range of ref. [15]'s DLC films is not stated here",
     "coefficients": {
       "Q_pi": "3.37",
       "E_gpi": "0.949",
       "E_hpi": "6.71",
       "Q_sigma": "136",
       "E_gsigma": "1.584",
       "E_hsigma": "51.7"
     },
     "parameters": {
       "form": "film"
     },
     "check": "Table 1's derived rows reproduced (2026-09-27): alpha = Q_pi / (kappa Q_sigma), Eq. (13), kappa = 0.5736: 0.0432 (printed 0.043); sp3/sp2, Eq. (17), with X_H = 0.38: 4.235 (printed 4.23). Appendix B: Eq. (52) equals the direct quadrature of Eq. (46) to 1e-12 (E_g 1.5, E_h 51.7 eV, E = 2-40 eV); Eq. (9) integrates to Q (Eq. (8)). No eps value is printed (Fig. 4 only)."
   }

DLC film of ref\. \[15\], as deposited\. The films' deposition and thickness are in ref\. \[15\] \(not in the library\)\.

franta\_2007/pdos\_dlc\_280C
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "pdos_dlc_280C",
     "material": "a-C_H",
     "equation": "Eqs. (5), (6), (9)-(11), (46)-(53); Table 1 (p. 16236), column '280 C'",
     "printed": "As pdos_dlc_as_deposited (Eq. 11).",
     "variable": "E: photon energy (eV); Q_j in eV^(3/2); energies in eV",
     "coefficients": {
       "Q_pi": "4.67",
       "E_gpi": "1.109",
       "E_hpi": "7.33",
       "Q_sigma": "131",
       "E_gsigma": "1.611",
       "E_hsigma": "51.7"
     },
     "parameters": {
       "form": "film"
     },
     "check": "alpha = 0.0621 (printed 0.062); beta*, Eq. (15), with d_f(t)/d_f(0) = 1.021: 1.0013 (printed 1.001); beta†, Eq. (16), X_H 0.38 -> 0.36: 0.9891 (0.989); sp3/sp2 2.746 (2.75). Checked 2026-09-27."
   }

DLC film of ref\. \[15\], annealed at 280 C\.

franta\_2007/pdos\_dlc\_325C
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "pdos_dlc_325C",
     "material": "a-C_H",
     "equation": "Eqs. (5), (6), (9)-(11), (46)-(53); Table 1 (p. 16236), column '325 C'",
     "printed": "As pdos_dlc_as_deposited (Eq. 11).",
     "variable": "E: photon energy (eV); Q_j in eV^(3/2); energies in eV",
     "coefficients": {
       "Q_pi": "5.41",
       "E_gpi": "1.104",
       "E_hpi": "7.65",
       "Q_sigma": "127",
       "E_gsigma": "1.633",
       "E_hsigma": "51.7"
     },
     "parameters": {
       "form": "film"
     },
     "check": "alpha = 0.0743 (printed 0.074); beta* 0.9982 (0.998); beta† 0.9944 (0.994); sp3/sp2 2.153 (2.15). Checked 2026-09-27."
   }

DLC film of ref\. \[15\], annealed at 325 C\.

franta\_2007/pdos\_dlc\_406C
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "pdos_dlc_406C",
     "material": "a-C_H",
     "equation": "Eqs. (5), (6), (9)-(11), (46)-(53); Table 1 (p. 16236), column '406 C'",
     "printed": "As pdos_dlc_as_deposited (Eq. 11).",
     "variable": "E: photon energy (eV); Q_j in eV^(3/2); energies in eV",
     "coefficients": {
       "Q_pi": "6.75",
       "E_gpi": "1.104",
       "E_hpi": "8.07",
       "Q_sigma": "120",
       "E_gsigma": "1.674",
       "E_hsigma": "51.7"
     },
     "parameters": {
       "form": "film"
     },
     "check": "alpha = 0.0981 (printed 0.098); beta* 0.9919 (0.992); beta† 0.9788 (0.979). sp3/sp2 by Eq. (17) with X_H = 0.34 is 1.480 (1.481 with the printed alpha), NOT the printed 1.45: the other five films reproduce to the printed digits, so this entry of Table 1 is off by 0.03 (a misprint or an X_H of about 0.36; not resolved). Checked 2026-09-27."
   }

DLC film of ref\. \[15\], annealed at 406 C\.

franta\_2007/pdos\_dlc\_460C
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "pdos_dlc_460C",
     "material": "a-C_H",
     "equation": "Eqs. (5), (6), (9)-(11), (46)-(53); Table 1 (p. 16236), column '460 C'",
     "printed": "As pdos_dlc_as_deposited (Eq. 11).",
     "variable": "E: photon energy (eV); Q_j in eV^(3/2); energies in eV",
     "coefficients": {
       "Q_pi": "7.80",
       "E_gpi": "0.957",
       "E_hpi": "8.33",
       "Q_sigma": "110",
       "E_gsigma": "1.604",
       "E_hsigma": "51.7"
     },
     "parameters": {
       "form": "film"
     },
     "check": "alpha = 0.1236 (printed 0.124); beta* 0.9609 (0.960); beta† 0.9600 (0.960); sp3/sp2 1.052 (1.05). Checked 2026-09-27."
   }

DLC film of ref\. \[15\], annealed at 460 C\.

franta\_2007/pdos\_dlc\_510C
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "pdos_dlc_510C",
     "material": "a-C_H",
     "equation": "Eqs. (5), (6), (9)-(11), (46)-(53); Table 1 (p. 16236), column '510 C'",
     "printed": "As pdos_dlc_as_deposited (Eq. 11).",
     "variable": "E: photon energy (eV); Q_j in eV^(3/2); energies in eV",
     "coefficients": {
       "Q_pi": "7.94",
       "E_gpi": "0.791",
       "E_hpi": "8.04",
       "Q_sigma": "104",
       "E_gsigma": "1.546",
       "E_hsigma": "51.7"
     },
     "parameters": {
       "form": "film"
     },
     "check": "alpha = 0.1331 (printed 0.133); beta* 0.9253 (0.925); beta† 0.9394 (0.939); sp3/sp2 0.965 (0.96). Checked 2026-09-27."
   }

DLC film of ref\. \[15\], annealed at 510 C \(the solid lines of Fig\. 3 are this film and the as\-deposited one\)\.

franta\_2007/pdos\_a\_si\_h
~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "pdos_a_si_h",
     "material": "a-Si_H",
     "equation": "Eqs. (5), (19)-(23), pp. 16237-16238; Table 2 (p. 16238)",
     "printed": "eps_i(E) = (sgn(E)/E^2) int_{-inf}^{0} [N_xi(S) N_xi*(S + |E|) + 2 N_lambda(S) N_xi*(S + |E|)] dS (Eq. 19); N_xi(S) = A_xi C(S) sqrt(-S - E_g/2) sqrt(E_h/2 + S) on -E_h/2 < S < -E_g/2, N_xi*(S) = A_xi C(S) sqrt(S - E_g/2) sqrt(E_h/2 - S) on E_g/2 < S < E_h/2 (Eqs. 20-21); C(S) = 1 + A_1 exp[-(|S| - E_1/2)^2 / (2 B_1^2)] + A_2 exp[-(|S| - E_2/2)^2 / (2 B_2^2)] (Eq. 22); N_lambda(S) = A_lambda exp(-||S| - E_g/2| / E_lambda) (Eq. 23, inside the gap)",
     "variable": "E, S: energies (eV); A_xi in eV^-1/2, A_lambda in eV^1/2, A_1 and A_2 dimensionless; the other parameters in eV",
     "range": "not stated (the ellipsometric and reflectance data; Fig. 6 shows 1.2-6.5 eV)",
     "coefficients": {
       "A_xi": "0.255",
       "E_g": "1.822",
       "E_h": "27.8",
       "A_1": "11.7",
       "B_1": "0.429",
       "E_1": "3.648",
       "A_2": "11.2",
       "B_2": "1.945",
       "E_2": "0.812",
       "A_lambda": "1.28",
       "E_lambda": "0.093"
     },
     "parameters": {
       "form": "film"
     },
     "check": "No value is printed to reproduce. Evaluated numerically (2026-09-27; Eq. (23) taken inside the gap, |S| < E_g/2): eps_i peaks at 23.3 at 3.63 eV (0.91 at 2 eV, 11.4 at 5 eV, 4.5 at 6.5 eV); eps_r is 12.5 at 1.2 eV and 21.7 at 2.7 eV. The shape and the peak positions follow Fig. 6, but the amplitude is about 10 % below the figure's a-Si curve as read by eye (peak near 26). Not resolved: a factor of 1.1 in A_xi^2 is not a plausible misprint, and the figure cannot be read more precisely."
   }

'a typical a\-Si film', a\-Si\:H per the caption; its deposition, substrate \(c\-Si\) and thickness are not given; the fit included a Rayleigh\-Rice rough surface and a native oxide \(p\. 16238\)\.

franta\_2007/pdos\_sio2\_like
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "pdos_sio2_like",
     "material": "SiO2-like_film_HMDSO",
     "equation": "Eqs. (5), (9), (10), (24), (25), pp. 16239-16241; Table 3 (p. 16240)",
     "printed": "eps_i(E) = (sgn(E)/E^2) [int_{-inf}^{0} N_xi(S) N_xi*(S + |E|) dS + sum_i G(A_delta_i, B_delta_i, E_delta_i) + G(A_p, B_p, E_p)] (Eq. 25), N_xi and N_xi* from Eqs. (9)-(10) with Q_xi, E_g, E_h; G(A, B, E_delta) = A/(sqrt(2 pi) B) {exp[(E - E_delta)^2/(2 B^2)] - exp[(E + E_delta)^2/(2 B^2)]} (Eq. 24, printed without the minus signs in the exponents: see notes)",
     "variable": "E: photon energy (eV); Q_xi in eV^(3/2); A_delta, B_delta, A_p, B_p, E_p printed in meV, E_g, E_h, E_delta in eV",
     "range": "1.2-6.5 eV (the ellipsometric and reflectance data)",
     "coefficients": {
       "Q_xi": "74.8",
       "E_g": "7.05",
       "E_h": "25.9",
       "A_delta1": "29.0",
       "B_delta1": "440",
       "E_delta1": "5.20",
       "A_delta2": "190",
       "B_delta2": "414",
       "E_delta2": "6.70",
       "A_p": "4.72",
       "B_p": "100",
       "E_p": "300"
     },
     "fixed": [
       "B_p",
       "E_p"
     ],
     "parameters": {
       "form": "film"
     },
     "check": "No value is printed to reproduce. Evaluated numerically (2026-09-27) with the corrected Eq. (24) (exp[-...]) and every meV parameter converted to eV (A read as 1e-3 of its printed value, the unit that keeps eps_i dimensionless with E in eV): eps_r = 2.070 at 1.5 eV, 2.084 at 2 eV, 2.174 at 4 eV, 2.443 at 6.4 eV; the defect terms give eps_i = 0.99e-3 at 5.1 eV and the total 3.45e-3 at 6.4 eV. Fig. 7 shows eps_r rising from about 2.07 at 1.2 eV to 2.45 at 6.5 eV and a defect peak near 1.0e-3 at about 5.1 eV (by eye): consistent. With the printed Eq. (24) the defect term at 5 eV is of order -1e100. Eq. (9) integrates to Q (Eq. (8)). The interband part equals the UDM's symmetric interband term with N = 30 Q^2 C_N / (E_h - E_g)^5 = 30 x 74.8^2 x 5069 / 18.85^5 = 357 eV^2 in the UDM's units (see the record's notes)."
   }

SiO2\-like film, 1\.13 um on c\-Si, 'prepared similarly as in \[11\]' \(Zajickova et al\., Plasma Process\. Polym\. 4, S287 \(2007\)\: HMDSO/O2 plasma\), measured by ellipsometry and reflectometry at 1\.2\-6\.5 eV\. B\_p and E\_p 'could not be determined because the absorption lay too far outside the measurement range and, therefore, these two parameters were fixed' \(p\. 16241\); the phonon Gaussian is 'an average contribution' of the Si\-O\-Si, silanol, water and hydrocarbon bands\. Fig\. 7 compares the result with Palik's fused silica \(philipp\_1985\)\: the film's eps\_r lies slightly below it in the visible and meets it near 6\.4 eV\.

Corrections
-----------

No dataset errata are recorded in this source record.

Formula corrections, where recorded, appear with their formulas above. Curation notes may also discuss corrections or unresolved interpretation. Consult the :doc:`review register </register>` for the checks and decisions.