franta_2015

D. Franta, D. Nečas and I. Ohlídal, “Universal dispersion model for characterization of optical thin films over a wide spectral range: application to hafnia”, Appl. Opt. 54, 9108 (2015)

Source record · Catalogue · Review register

DOI: 10.1364/ao.54.009108

Source kind: fit

Recorded technique: Universal dispersion model (UDM) fitted, with newAD2’s automatic weighting, to ellipsometry (Horiba UVISEL2 VUV 0.6-8.7 eV at 70 deg in vacuum; UVISEL 0.6-6.5 eV, 55-75 deg; Woollam IR-VASE 300-6500 cm-1, reflection and transmission) and spectrophotometry (McPherson VUVAS 2000 3.3-10.8 eV; Perkin Elmer Lambda 1050 0.67-6.7 eV; Bruker Vertex 80v 80-7500 cm-1, reflectance from both sides and transmittance) of a HfO2 film (Leybold SYRUSpro DUV vacuum evaporation at 150 C, Meopta) on double-side polished, slightly P-doped Czochralski silicon; structural model with Rayleigh-Rice roughness, wedge non-uniformity, a SiOx transition layer and a back-side native oxide (pp. 9108-9112).

Materials in structured entries: HfO2

Dataset quantities: band_gap

Contents: 1 dataset; 1 formula.

Curation notes

Journal pages 9108-9119 = PDF pages 1-12. Identity checked against the first page (the extraction renders ‘HfO2’ as ‘HCO2’ in places). The film: HfO2 (hafnia), evaporated at 150 C, thickness 112.7 +- 0.1 nm (IR data; Table 1, p. 9115), roughness 2.19 +- 0.06 nm rms (AFM 0.56 nm); amorphous (‘In the case of amorphous HfO2, the four Gaussian contributions were sufficient’, p. 9118). The substrate c-Si uses the ADM of franta_2014b and Franta et al., Thin Solid Films 571, 490 (2014); the SiOx layers use UDM parameters ‘fixed in values determined for typical SiO2 layers’ (p. 9112, not printed). n and k are figures only (Figs. 10-12; uncertainties ‘of the order of 1e-3-1e-4’ inside the measured range, p. 9116; at the phonon peaks 2.8e-3 in n and 3.8e-3 in k, p. 9117).

UDM EQUATION MAP (this paper; journal pages). Eq. (1), p. 9109: sum rule N = int eps_i E dE = n_e U N_a (N_a = script-N_a/M, franta_2013); for HfO2 n_e = 88/3, U = 1.00023098, and with the assumed mass density 9680 kg/m3 (ref. 23) N = 5280 eV^2 and N_a = 180 eV^2. Eq. (2), p. 9110: eps = 1 + N_vc eps0_vc + N_vx eps0_vx + N_ut eps0_ut + sum_p N_loc,p eps0_loc,p + sum_p N_ph,p eps0_ph,p; Eq. (3): J_j = N_j eps0_i,j E^2; Eq. (4): int_0^inf eps0_i,j E dE = 1; Eq. (5): N_ve = N_vc + N_vx + N_ut + sum N_loc = n_ve N_a (core electrons omitted). - IBT (‘vc’, interband): Eq. (6), J_vc = (N_vc/C_N)(E - E_g)^2(E_h - E)^2 Pi_{Eg,Eh}(E); C_N Eq. (7); Pi Eq. (8); eps0_i,vc Eq. (9); eps0_r,vc Eqs. (10)-(14) (identical to franta_2015b Eqs. (7)-(12)). Lorentz-modified shape Eq. (15), p. 9111 (C_N and a, b, c in ‘Appendix A of Ref. [10]’, Thin Solid Films 539, 233 (2013), not in the library; printed in full in franta_2015b Eqs. (16)-(25)). - EXCITONS: Eq. (16), p. 9111: J_vc = (N_vc/C_N)(E - E_g)^2(E_h - E)^2 [A_0 + sum_k A_ex,k/((E - E_ex,k)^2 + B_ex,k^2)] Pi_{Eg,Eh}, ‘C_N is again the normalization factor of J_vc(E) given by Eq. (4)’; one A must be fixed (‘for HfO2 ... n_ex = 2 and fix A_ex,1 = 1’). NOTE (TC-387): franta_2015b (SPIE 9628, same group, same year) writes the same term differently, eps0_vc = (1/sum A)[A_0 eps0_vc0 + sum_p A_p eps0_ex,p] with each term normalised to unit strength (its Eqs. (26)-(27)). The two definitions give different spectra for the same A’s (the relative weights differ by each term’s C_N). Which one Table 2’s A values belong to is not stated beyond Eq. (16). - HET (‘vx’): Eq. (17), p. 9111: J_vx = N_vx 3 E_xi (E - E_xi)^2/E^3 Pi_{E_xi,inf}(|E|), E_xi = E_x + E_g/2 (E_x measured from the mid-gap Fermi level, p. 9111); real part ‘presented in Ref. [10]’ (printed in franta_2015b Eqs. (29)-(31)). It replaces eps_inf. - URBACH TAIL (‘ut’): Eq. (18), p. 9111, four pieces (exponential rise from E_g/2 to E_g, parabola from E_g to E_m, exponential fall from E_m to E_m + E_g/2, minus a constant for continuity), E_m = E_g/2 + E_h/2 (Eq. (19), p. 9112); real part and C_N ‘in Appendix B of Ref. [10]’ (printed in franta_2015b Eqs. (34)-(40)). The paper reserves ‘Urbach tail’ for disorder tails (E_u ~ 50 meV) and calls HfO2’s 0.55-0.65 eV tail an ‘exponential tail’ (p. 9112). - LOCALIZED STATES and PHONONS: Gaussian-broadened discrete lines, ‘detailed mathematical expressions ... in Refs. [10] and [31]’ (p. 9112); printed in franta_2015b Eqs. (41)-(47). Four Gaussian phonon peaks for HfO2 (Table 4), which ‘do not correspond to individual vibration modes’ (p. 9118). - No free-carrier term for HfO2 (p. 9110); no core-electron term in the UDM.

DERIVED QUANTITIES (text, not extracted): N_ve = 2064 +- 280 eV^2 (Eq. (30), p. 9116), n_ve = 34.41/3 = 11.47 (Eq. (31)), above the 4 real valence electrons per atom because the vx term absorbs Hf 4f, O 2s and Hf 5p excitations; N_lambda = N_loc + (N_ve-bar/N_vc) N_ut/2 = 0.067 + 5.587 = 5.654 (Eq. (36), p. 9117), n_lambda = 0.03141, stoichiometry x = (4 - n_lambda)/(2 + n_lambda) = 1.954 (Eq. (37)). CHECK (2026-09-26): N = (88/3) x 1.00023098 x 180 = 5281 eV^2 (printed 5280); N_ve = 421 + 1637 + 7 + 0.067 = 2065 (printed 2064 +- 280); n_lambda = 5.654/180 = 0.03141 and x = 1.9536, as printed; but (720/421) x 7/2 = 5.99, not 5.587: the printed 5.587 implies an unrounded N_ut of 6.53 (Table 3 prints 7 +- 2), so the chain was evaluated with unrounded parameters.

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_2015/band_gap

{
  "id": "band_gap",
  "material": "HfO2",
  "file": "HfO2/franta_2015__band_gap.csv",
  "table": "tbl_ad5b7faa65e7",
  "quantity": "band_gap",
  "unit": "eV",
  "abscissa": "none",
  "use": "primary",
  "uncertainty": {
    "column": true,
    "level": "1 sigma"
  },
  "parameters": {
    "kind": "tauc",
    "form": "film"
  }
}

Table 2 (p. 9117), E_g = 5.311 +- 0.004 eV: the UDM’s minimum energy of interband transitions, the onset of the (E - E_g)^2 JDOS (Eq. (6)). Uncertainty: the least-squares standard deviation. Film evaporated at 150 C (amorphous). Checked against the table image.

Formulas

franta_2015/udm_HfO2

{
  "id": "udm_HfO2",
  "material": "HfO2",
  "equation": "Eqs. (2)-(19), pp. 9110-9112; Tables 2-4, pp. 9117-9118",
  "printed": "\\varepsilon(E) = 1 + N_{vc}\\varepsilon^0_{vc} + N_{vx}\\varepsilon^0_{vx} + N_{ut}\\varepsilon^0_{ut} + N_{loc}\\varepsilon^0_{loc} + \\sum_{p=1}^{4} N_{ph,p}\\varepsilon^0_{ph,p}",
  "variable": "E: photon energy (eV); phonon positions and widths (FWHM) in cm^-1; strengths N in eV^2",
  "coefficients": {
    "N_vc": "421",
    "N_vc_sigma": "32",
    "E_g": "5.311",
    "E_g_sigma": "0.004",
    "A_0": "0",
    "A_0_sigma": "2",
    "A_ex1": "1",
    "E_ex1": "6.98",
    "E_ex1_sigma": "0.01",
    "B_ex1": "0.86",
    "B_ex1_sigma": "0.01",
    "A_ex2": "4",
    "A_ex2_sigma": "2",
    "E_ex2": "11",
    "E_ex2_sigma": "1",
    "B_ex2": "4",
    "B_ex2_sigma": "1",
    "E_h": "20",
    "E_h_sigma": "3",
    "N_vx": "1637",
    "N_vx_sigma": "314",
    "E_x": "12",
    "E_x_sigma": "2",
    "N_ut": "7",
    "N_ut_sigma": "2",
    "E_u": "0.65",
    "E_u_sigma": "0.02",
    "N_loc": "0.067",
    "N_loc_sigma": "0.003",
    "E_loc": "4.547",
    "E_loc_sigma": "0.004",
    "B_loc": "0.212",
    "B_loc_sigma": "0.005",
    "nu_ph1": "302",
    "nu_ph1_sigma": "1",
    "beta_ph1": "103",
    "beta_ph1_sigma": "1",
    "N_ph1": "0.0047",
    "N_ph1_sigma": "0.0003",
    "nu_ph2": "322",
    "nu_ph2_sigma": "1",
    "beta_ph2": "369",
    "beta_ph2_sigma": "2",
    "N_ph2": "0.0220",
    "N_ph2_sigma": "0.0001",
    "nu_ph3": "366",
    "nu_ph3_sigma": "5",
    "beta_ph3": "129",
    "beta_ph3_sigma": "4",
    "N_ph3": "0.0039",
    "N_ph3_sigma": "0.0004",
    "nu_ph4": "750",
    "nu_ph4_sigma": "1",
    "beta_ph4": "144",
    "beta_ph4_sigma": "2",
    "N_ph4": "0.00089",
    "N_ph4_sigma": "0.00002"
  },
  "check": "Tables 2-4 checked cell by cell against their images (2026-09-26); Table 3's extraction splits '7 +- 2' into two columns, the image is unambiguous. Not evaluated: the exciton convention (Eq. (16) against franta_2015b Eqs. (26)-(27)) is ambiguous (TC-387), and no n, k table is printed to compare with."
}

Units: N in eV^2; E_g, E_ex, B_ex, E_h, E_x, E_u, E_loc, B_loc in eV; nu_ph and beta_ph (FWHM) in cm^-1 (converted as franta_2015b Eq. (46)). The *_sigma keys are the printed least-squares standard deviations. Table 2 marks A_0 (0 +- 2), A_ex1 (1), E_h (20 +- 3) and E_x (12 +- 2) ‘(fixed)’; the text (p. 9111) says only A_ex,1 was fixed and (p. 9116) that E_h and E_x were fixed from first-principles estimates. A fixed parameter printed with a standard deviation is contradictory (TC-387); the values are recorded as printed. No loc peak count other than one; four phonon peaks.

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.