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 `__ · :doc:`Catalogue ` · :doc:`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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 :doc:`review register ` for the checks and decisions.