franta_2018

D. Franta, P. Franta, J. Vohánka, M. Čermák and I. Ohlídal, “Determination of thicknesses and temperatures of crystalline silicon wafers from optical measurements in the far infrared region”, J. Appl. Phys. 123, 185707 (2018)

Source record · Catalogue · Review register

DOI: 10.1063/1.5026195

Source kind: fit

Recorded technique: Revision of franta_2017’s temperature-dependent advanced dispersion model (ADM) of float-zone c-Si, fitted in newAD2 by least squares to 85 078 points at 0.0086-40 eV and 5-700 K (Table III, p. 185707-7): the data of franta_2017 plus a 0.25 mm wafer and a 14.33 mm slab, Bruker Vertex 80v transmittance at 0 deg in the FIR (70-680 cm-1, 1 cm-1 resolution) and MIR (450-2000 cm-1, 4 cm-1; the slab to 3000 cm-1) at 300-450 K (heated holder, INSTEC mK1000 controller, Pt100 sensor ‘with an accuracy of 0.01 K’), Lambda 1050 and UVISEL on the new samples, and n at 2 um for 5-700 K in 1 K steps (696 points) from the two-term Sellmeier formula of Table II, which was fitted to Frey et al.’s published formula values. The partial coherence of the FIR transmittance is modelled by convolving the coherent transmittance with the instrument’s line profile (Eq. (13)); the samples’ temperatures at 300-450 K are fitted (offsets dT from the holder, Fig. 12).

Materials in structured entries: Si

Dataset quantities: None recorded.

Contents: 0 datasets; 2 formulas.

Curation notes

Journal article pages 185707-1 to 185707-11 = PDF pages 2-12 (PDF page 1 is the publisher’s cover page). Identity checked against the first article page. The paper prints no n or k: the model’s n and k are in Figs. 6-7 (FIR and MIR at three temperatures), their LSM uncertainties in Fig. 9, ADM minus the Sellmeier formula at 2 um in Fig. 11. The model has ‘more than one thousand parameters’ (p. 185707-6), none printed. The refractiveindex.info pages main/Si/nk/Franta-*.yml cite this paper for their ‘Numerical procedures’: their numbers are curated in franta_2017 (datasets rii_*), whose notes show that they follow this paper’s two-term Sellmeier formula at 2 um within 1e-5 over 10-500 K.

CHANGES TO THE MODEL (pp. 185707-1, -3, -7, -9). New samples and data (above); the 2 um pseudo-data from the two-term Sellmeier formula with the ABESF (Eqs. (9)-(10), Table II) in place of franta_2017’s Eq. (3); the linear thermal expansion from the three-phonon ABESF formula (Eq. (3), the 4th fit of Table I) in place of Watanabe et al.’s polynomial (‘a negligible effect on the dispersion model’); N_a(T) = N_a^300K [1/(1 + e(T))]^3 (Eq. (5)) and d(T) = d^300K [1 + e(T)] (Eq. (6)), reference 300 K; Gaussian broadening by cubic-spline approximation and analytic integration in place of Romberg integration, which removed the numerical noise of franta_2017’s comparison at 2 um (Fig. 11 and p. 185707-9).

TEXT VALUES (pp. 185707-5 to -9). Frey et al.’s index (Proc. SPIE 6273, 62732J (2006), not in the library): uncertainty ‘estimated to be +-1 x 10^-4’, ‘published ... together with values calculated using formula (7) on the mesh consisting of 12 x 13 temperature-spectral points’ at 30-295 K and 1.1-5.5 um; ‘the original Frey et al. data determined by the MDM were not published’; the authors ‘suspect that the Frey et al. model exhibits a systematic error because one of the strength parameters S_j(T) is negative’. The one-term formula deviates from Frey’s mesh values by -6e-4 to +14e-4, the two-term formula systematically by about the stated uncertainty (Fig. 5). ADM minus the two-term formula at 2 um: an undulation 5-10 times smaller than 1e-4 (Fig. 11). Statistical (LSM) uncertainty of n ‘approximately 2 x 10^-5’ at room temperature in the infrared (Fig. 9). Wafer thicknesses at 300 K: 0.252260 +- 0.000002 mm (LSM standard error; about 7.4 nm from the 1e-4 index uncertainty, Eq. (14)) and 0.577568 +- 0.000006 mm. Effective numbers of electrons of the Sellmeier poles (Eq. (11)) ‘approximately 2.6 and 2.8’ (281.961/108.2 = 2.61; (273.563 + 23.447)/108.2 = 2.74, with franta_2014c’s N_a(300 K) = 108.2 eV^2), against 4.12 for the valence electrons of the ADM. Lattice spacings of Lu et al. 0.078390 nm (300 K) and 0.078372 nm (near 0 K), whence e_0 ~ -2.30e-4.

AS PRINTED. The Frey range is given as 1.1-5.6 um on p. 185707-4 and 1.1-5.5 um elsewhere, and 20-300 K against the mesh’s 30-295 K. Table I’s polynomial column is Watanabe et al.’s formula with their reference 293 K (e(293 K) = -8e-7, e(300 K) = 1.80e-5), although the text says the results use 300 K; its e_0 = -4.2668e-4 is the value quoted in the text. Table I’s ABESF columns (the 1st to 4th fits) print e_1-e_3 without their factor 10^-4: corrected in the formula below.

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_2018/sellmeier_abesf

{
  "id": "sellmeier_abesf",
  "material": "Si",
  "equation": "Eqs. (9)-(10), p. 185707-5; Table II, p. 185707-5",
  "printed": "n^2(E, T) - 1 = \\frac{2}{\\pi} \\sum_{j=1}^{1 or 2} \\frac{N_j}{E_j^2(T) - E^2}, E_j(T) = E_j^{0K} + (E_j^{300K} - E_j^{0K}) [\\exp(\\Theta_j/300 K) - 1]/[\\exp(\\Theta_j/T) - 1]",
  "variable": "E: photon energy (eV); T: temperature (K); N_j in eV^2, E_j in eV, Theta_j in K",
  "range": "fitted to Frey et al.'s formula values at 30-295 K and 1.1-5.5 um; used for n at 2 um, 5-700 K",
  "coefficients": {
    "one_N1": "281.961",
    "one_E1_0K": "4.13650",
    "one_E1_300K": "4.09926",
    "one_Theta1": "321.00",
    "two_N1": "273.563",
    "two_E1_0K": "4.38129",
    "two_E1_300K": "4.36617",
    "two_Theta1": "592.56",
    "two_N2": "23.447",
    "two_E2_0K": "3.24469",
    "two_E2_300K": "3.10798",
    "two_Theta2": "208.97"
  },
  "check": "Table II cells agree with the image (2026-10-14). Computed (2026-10-14; hc = 1.239841984 eV um): two-term n(2 um) = 3.424838 (10 K), 3.453226 (293.15 K), 3.453541 (295 K), 3.454397 (300 K), 3.492349 (500 K); one-term 3.453498 (295 K); franta_2017's Eq. (3) gives 3.45351 at 295 K. The refractiveindex.info Franta pages' n at 2 um equals the two-term values within 1e-5 at 10-500 K (franta_2017 notes). The Theta_j correspond to 6.69, 12.35 and 4.35 THz, as the text says."
}

Prefixes one_ and two_: the one-term and two-term formulas. The N_j are temperature independent (the expansion factor neglected, p. 185707-5). The two-term formula’s parameters were fixed in the ADM fit, which used its n at 2 um for 5-700 K as pseudo-data (p. 185707-9); franta_2019’s Table 3 reprints the two-term values. A fit of n data, not an independent measurement: it carries Frey et al.’s values (and their suspected systematic error) and must not be fitted beside Frey et al.’s silicon data.

franta_2018/thermal_expansion

{
  "id": "thermal_expansion",
  "material": "Si",
  "equation": "Eqs. (1)-(3), pp. 185707-2 to -3; Table I, p. 185707-2",
  "printed": "e(T) = \\sum_{k=0}^{4} e_k T^k (2); e(T) = e_0 + \\sum_j e_j/[\\exp(\\Theta_j/T) - 1] (3)",
  "variable": "T: temperature (K); e: linear thermal expansion (L(T) - L_0)/L_0 (dimensionless); Theta_j in K; poly_ek in K^-k",
  "range": "Watanabe et al. 293-1000 K (e and alpha); Ibach below room temperature (alpha); the 4th fit 20-1000 K (conclusion)",
  "coefficients": {
    "poly_e0": "-4.2668e-4",
    "poly_e1": "-9.0926e-8",
    "poly_e2": "6.5359e-9",
    "poly_e3": "-4.7219e-12",
    "poly_e4": "1.3822e-15",
    "fit1_e0": "-2.7131e-4",
    "fit1_e1": "35.188",
    "fit1_Theta1": "790.4",
    "fit2_e0": "-3.4529e-4",
    "fit2_e1": "18.739",
    "fit2_e2": "20.391",
    "fit2_Theta1": "566.4",
    "fit2_Theta2": "1556",
    "fit3_e0": "-2.2278e-4",
    "fit3_e1": "-1.036",
    "fit3_e2": "36.241",
    "fit3_Theta1": "165.6",
    "fit3_Theta2": "718.7",
    "fit4_e0": "-2.3741e-4",
    "fit4_e1": "-1.748",
    "fit4_e2": "33.153",
    "fit4_e3": "18.635",
    "fit4_Theta1": "197.6",
    "fit4_Theta2": "652.2",
    "fit4_Theta3": "3033",
    "chi_e_Watanabe_poly": "0.201",
    "chi_e_Watanabe_fit1": "0.899",
    "chi_e_Watanabe_fit2": "0.131",
    "chi_e_Watanabe_fit3": "1.106",
    "chi_e_Watanabe_fit4": "0.275",
    "chi_alpha_Watanabe_poly": "0.806",
    "chi_alpha_Watanabe_fit3": "1.211",
    "chi_alpha_Watanabe_fit4": "0.115",
    "chi_alpha_Ibach_fit3": "0.469",
    "chi_alpha_Ibach_fit4": "0.257"
  },
  "check": "Table I cells compared with the image (2026-10-14); the CSV loses the minus signs and the powers of ten of the exponent column, so the values were read from the image ('a × 10^-b' written ae-b). With the corrections below, e(300 K) = 6.5e-7, 4.3e-7, -1.0e-7 and 5.5e-7 for the 1st-4th fits, zero to the rounding of e_0 (the reference temperature is 300 K, Eq. (6) and p. 185707-3); as printed it is 2.7, 3.5, 2.2 and 2.4. The polynomial gives e(293 K) = -8e-7 (Watanabe et al.'s reference). The Theta_j of the text check: 790.4 K = 16.5 THz, 566.4 K = 11.8 THz, 1556 K = 32.4 THz, 197.6 K = 4.1 THz, 652.2 K = 13.6 THz.",
  "corrections": [
    {
      "term": "fit1_e1",
      "printed": "35.188",
      "corrected": "35.188e-4",
      "reason": "Table I prints the ABESF coefficients e_j of Eq. (3) without their factor 10^-4: as printed, e(300 K) = 2.7, where the reference temperature 300 K requires e(300 K) = 0.",
      "evidence": "With e_j x 10^-4, e(300 K) = 6.5e-7, zero to the rounding of the printed e_0 = -2.7131e-4 (computed 2026-10-14). franta_2019 Table 1 reprints the 4th fit as e_1 = -0.00017(2), e_2 = 0.00332(3), e_3 = 0.00186(12) (here -1.748, 33.153, 18.635)."
    },
    {
      "term": "fit2_e1",
      "printed": "18.739",
      "corrected": "18.739e-4",
      "reason": "As fit1_e1: Table I prints e_j without the factor 10^-4.",
      "evidence": "With e_j x 10^-4, the 2nd fit's e(300 K) = 4.3e-7 (computed 2026-10-14); see fit1_e1."
    },
    {
      "term": "fit2_e2",
      "printed": "20.391",
      "corrected": "20.391e-4",
      "reason": "As fit1_e1: Table I prints e_j without the factor 10^-4.",
      "evidence": "As fit2_e1."
    },
    {
      "term": "fit3_e1",
      "printed": "-1.036",
      "corrected": "-1.036e-4",
      "reason": "As fit1_e1: Table I prints e_j without the factor 10^-4.",
      "evidence": "With e_j x 10^-4, the 3rd fit's e(300 K) = -1.0e-7 (computed 2026-10-14); see fit1_e1."
    },
    {
      "term": "fit3_e2",
      "printed": "36.241",
      "corrected": "36.241e-4",
      "reason": "As fit1_e1: Table I prints e_j without the factor 10^-4.",
      "evidence": "As fit3_e1."
    },
    {
      "term": "fit4_e1",
      "printed": "-1.748",
      "corrected": "-1.748e-4",
      "reason": "As fit1_e1: Table I prints e_j without the factor 10^-4.",
      "evidence": "With e_j x 10^-4, the 4th fit's e(300 K) = 5.5e-7 (computed 2026-10-14); franta_2019 Table 1, '4-th fit in [12]': e_1 = -0.00017(2)."
    },
    {
      "term": "fit4_e2",
      "printed": "33.153",
      "corrected": "33.153e-4",
      "reason": "As fit1_e1: Table I prints e_j without the factor 10^-4.",
      "evidence": "As fit4_e1; franta_2019 Table 1: e_2 = 0.00332(3)."
    },
    {
      "term": "fit4_e3",
      "printed": "18.635",
      "corrected": "18.635e-4",
      "reason": "As fit1_e1: Table I prints e_j without the factor 10^-4.",
      "evidence": "As fit4_e1; franta_2019 Table 1: e_3 = 0.00186(12)."
    }
  ]
}

The linear thermal expansion of c-Si, no optmat quantity: fitted to Watanabe et al. (Int. J. Thermophys. 25, 221 (2004); e and alpha, 293-1000 K) and, for the 3rd and 4th fits, Ibach (Phys. Status Solidi 31, 625 (1969); alpha below room temperature, chi with an assumed uncertainty 1e-7 K^-1). The 4th fit is the one used in the revised ADM (p. 185707-3). chi: the rms deviation over the data’s uncertainty. franta_2019 Table 1 reprints the 4th fit with uncertainties (‘4-th fit in [12]’) and gives newer fits.

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.