franta_2014c#
D. Franta, D. Nečas, L. Zajíčková and I. Ohlídal, “Utilization of the sum rule for construction of advanced dispersion model of crystalline silicon containing interstitial oxygen”, Thin Solid Films 571, 490 (2014)
Source record · Catalogue · Review register
DOI: 10.1016/j.tsf.2014.03.059
Source kind: fit
Recorded technique: Distribution of the total transition strength of c-Si (the Thomas-Reiche-Kuhn sum rule) among its absorption processes, from a dispersion model fitted to ‘ellipsometric and spectrophotometric characterizations of a set of double side polished wafers and slabs produced using float zone and Czochralski processes’, 8.7 meV (70 cm-1) to 8.7 eV, 300-600 K; ‘Experimental details are out of scope of this paper and will be published elsewhere’ (pp. 490-491).
Materials in structured entries: Si
Dataset quantities: None recorded.
Contents: 0 datasets; 1 formula.
Curation notes#
Journal pages 490-495 = PDF pages 1-6. Identity checked against the first page. A theory paper on the construction of the advanced dispersion model of c-Si (the model of franta_2017): no n, k, epsilon or absorption value is printed, and the dispersion formulas of the individual processes are ‘out of this paper’ (p. 492). Figs. 1-6 plot N_a(T), the transition strength function F(E) above the gap at 300 and 600 K, n_eff(E), F(E) below the gap at 300-600 K, the transition strengths below the gap against T, and F(E) of pure c-Si against Czochralski Si with 20 ppm interstitial oxygen. Nothing printed has an optmat quantity: the effective numbers of electrons of Table 1 and the relative strengths in the text are recorded as the formula below.
MODEL MAP. Eq. (8): int_0^inf F(E) dE = N, F(E) = E eps_i(E) (Eq. (9)), the total transition strength. Eq. (10): F as a sum of damped harmonic oscillators, sum_j N_j = N, N_j in eV^2. Eq. (14): N = N_e + N_Si + N_O; Eqs. (15)-(17): their dependence on T and the atomic fraction f_Oi of interstitial oxygen through the density parameter N_a(T), Eq. (18): N_a(T) = N_a(300 K) [(1 + e(300 K))/(1 + e(T))]^3, e(T) the linear thermal expansion (Watanabe et al., Int. J. Thermophys. 25, 221 (2004); ‘linear thermal expansion coefficient’ in the text), N_a(300 K) = 108.2 eV^2 (from franta_2013). Eq. (20): N ~ N_ee ~ N_e ~ Z_Si N_a, Z_Si = 14; N = N_e U with U = 1.0002735 for silicon (p. 492). Eq. (21): N_t(T) = n_t N_a(T), n_t the effective number of electrons of process t (K, L, vx, vc), temperature independent; Eq. (22): n_ve = n_vc + n_vx; Eq. (26): n_idt(T) and n_dt(T) depend on T, n_idt ~ 1 + 2 f_BE(E_p, T) (Eq. (27)), n_idt + n_dt = n_vc (Eq. (28)); Eq. (29): N_fc ~ T^(3/2) exp(-E_g/(2 k_B T)). Interstitial oxygen: Eq. (30), N_A2u = alpha_A2u N_O, alpha_A2u = 0.1914 (‘corresponds to the effective charge of 3.5e determined by ab initio calculations’; a value from the ASTM F1188 calibration was ‘approximately 5% larger’), A2u mode at 1107 cm-1; Eqs. (31)-(33), the A2u + A1g two-phonon peak at 1721 cm-1, alpha_A2u+A1g(300 K) = 0.0025, temperature dependent through Bose-Einstein factors; Eq. (34), the one-phonon Si band induced by O_i (0-520 cm-1), N_1ph = alpha_1ph [2 f_Oi/(1 - f_Oi)] N_Si, alpha_1ph = 0.0172 (pp. 493-494).
AS PRINTED (page images checked 2026-10-14): Eq. (17) prints N_O with Z_Si^2 m_e/m_O, where Eq. (8) gives Z_O^2 m_e/m_O; Eq. (23) prints n_ve + n_L + n_L ~ 14, where Table 1 and the text (n_K = 1.55) read n_ve + n_L + n_K. Neither enters a value recorded here. n_ve = 4.12, n_L = 8.33, n_K = 1.55 are Shiles et al.’s values for aluminium (Phys. Rev. B 22, 1612 (1980)), taken for a-Si:H in franta_2013b and ‘suitable also for crystalline silicon’ (p. 493): not determined for c-Si here. Their sum, 14.00, checks Eq. (23).
TABLE 1 CHECKS (2026-10-14, image): the CSV agrees with the image; n_ve = n_vx + n_vc (0.24 + 3.88 = 4.12) and n_vc = n_idt + n_dt (0.50 + 3.38 = 3.88 at 300 K; 3.88 at 400-600 K) as the caption says; n_mph = n_2ph + n_3ph + n_4ph holds to rounding (1.00e-7 + 1.51e-8 + 1.09e-9 = 1.162e-7, printed 1.16e-7). Later values: franta_2017 gives n_ve = 4.51 or 4.37 for its float-zone model (p. 415) and franta_2018 cites 4.12 for the ‘real strength of valence electrons determined using the advanced dispersion model’.
Recorded bibliographic identity status: VALIDATED. This concerns identification of the source; it does not establish the scientific correctness of its contents.
Datasets#
No datasets are recorded.
Formulas#
franta_2014c/effective_electrons#
{
"id": "effective_electrons",
"material": "Si",
"equation": "Table 1, p. 493; Eqs. (18), (21)-(28), (30)-(34), pp. 492-494",
"printed": "N_t(T) = n_t N_a(T)",
"variable": "T: temperature (K); n_t: effective number of electrons of process t (dimensionless); N_a: density parameter (eV^2); alpha: relative transition strengths (dimensionless)",
"range": "300-600 K (Table 1)",
"coefficients": {
"N_a_300K": "108.2",
"U": "1.0002735",
"n_K": "1.55",
"n_L": "8.33",
"n_ve": "4.12",
"n_vx": "0.24",
"n_vc": "3.88",
"n_idt_300K": "0.50",
"n_idt_400K": "0.61",
"n_idt_500K": "0.72",
"n_idt_600K": "0.84",
"n_dt_300K": "3.38",
"n_dt_400K": "3.27",
"n_dt_500K": "3.16",
"n_dt_600K": "3.04",
"n_fc_300K": "3.23e-12",
"n_fc_400K": "1.01e-9",
"n_fc_500K": "3.42e-8",
"n_fc_600K": "3.76e-7",
"n_mph_300K": "1.16e-7",
"n_mph_400K": "1.48e-7",
"n_mph_500K": "1.84e-7",
"n_mph_600K": "2.22e-7",
"n_2ph_300K": "1.00e-7",
"n_2ph_400K": "1.26e-7",
"n_2ph_500K": "1.52e-7",
"n_2ph_600K": "1.79e-7",
"n_3ph_300K": "1.51e-8",
"n_3ph_400K": "2.08e-8",
"n_3ph_500K": "2.84e-8",
"n_3ph_600K": "3.78e-8",
"n_4ph_300K": "1.09e-9",
"n_4ph_400K": "1.91e-9",
"n_4ph_500K": "3.17e-9",
"n_4ph_600K": "5.01e-9",
"alpha_A2u": "0.1914",
"alpha_A2u_A1g_300K": "0.0025",
"alpha_1ph": "0.0172",
"nu_A2u": "1107",
"nu_A2u_A1g": "1721"
},
"check": "Table 1 cells compared with the image (2026-10-14); 'a × 10^-b' printed, written here as ae-b. The sums of the caption hold (see notes). N_a_300K, U and the alphas are from the text (pp. 492-494); nu in cm-1."
}
Table 1 rows above E_g (n_K, n_L, n_ve, n_vx, n_vc) are printed once, ‘Const.’ in T; n_idt, n_dt, n_mph and n_2ph are marked ‘Lin.’, n_3ph ‘Quad.’, n_4ph ‘Cubic’, n_fc ‘Exp.’ (Eq. (29)). Pure (oxygen-free) c-Si for the phonon rows (p. 493). The model’s phonon, free-carrier and electronic formulas are not given; the values cannot be turned into n or k.
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.