franta_2007#
D. Franta, D. Nečas and L. Zajíčková, “Models of dielectric response in disordered solids”, Opt. Express 15, 16230 (2007)
Source record · Catalogue · Review register
DOI: 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#
{
"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#
{
"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#
{
"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#
{
"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#
{
"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#
{
"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#
{
"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#
{
"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#
{
"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#
{
"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#
{
"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 review register for the checks and decisions.