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