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Jona Grümbel, Rüdiger Goldhahn, Martin Feneberg, [Yuichi Oshima](https://orcid.org/0000-0001-8293-4891), Hazem Abu-Farsakh, Abdallah Qteish

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[Dielectric function and electronic structure of nondegenerate rocksalt ScN: Spectroscopic ellipsometry and                    <math>                      <mrow>                        <mi>GW</mi>                      </mrow>                    </math>                    calculations](https://mdr.nims.go.jp/datasets/c7dc1ac7-4723-41a2-a35f-ab2b0cd0e6b3)

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Dielectric function and electronic structure of non-degenerate rocksalt ScN:Spectroscopic ellipsometry and GW -calculationsJona Grümbel,∗ Rüdiger Goldhahn, and Martin FenebergInstitut für Physik, Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, 39106, Magdeburg, GermanyYuichi OshimaResearch Center for Electronic and Optical Materials,National Institute for Materials Science , 1-1 Namiki, Tsukuba, Ibaraki 305-0044, JapanHazem Abu-FarsakhDepartment of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi-ArabiaAbdallah QteishDepartment of Physics, Yarmouk University, Irbid 21163, JordanIn this work we determine the dielectric function of ScN in a spectral range from 0.9 eV to 6.4 eVby spectroscopic ellipsometry from non-degenerately doped, bulk-like samples. Several models areapplied to the obtained dielectric functions yielding main critical point transitions energies. Theseresults are compared with state-of-the-art computations, where the band structure of ScN is cal-culated using Heyd-Scuseria-Ernzerhof (HSE06) hybrid functionals and quasi-particle correctionsin the G0W0 approach. Furthermore, the dielectric function of ScN is derived from the computedband structure by solving the Bethe-Salpeter-equation to account for electron-hole-pair interactions.We �nd exceptional agreement between computed and experimentally determined dielectric func-tions, where discrete excitons are not observed experimentally. We extrapolate an intrinsic directbandgap of (2.182±0.004)eV and an intrinsic indirect bandgap of (1.08±0.02eV) by taking into ac-count many-body-e�ects, while higher energy critical point transition energies of Γ-point transitionsare averaged over all samples, yielding EΓ = (3.853± 0.006)eV and EΓ′ = (5.21± 0.02)eV. Criticalpoint transitions in the computed band structure, though, deviate from the experimental ones dueto the omission of electron-hole-interaction, where the deviation increases with increasing energy.The dielectric limit of the electronic system is determined as ε∞ = (8.38 ± 0.04) from experiment,where the computed dielectric function reveals almost the same value (ε∞ = 8.33). Along with otherprevious publications we conclude that solving the Bethe-Salpeter-equation is indispensable for thecomputation of the dielectric function of semiconductors even in the absence of discrete excitons.I. INTRODUCTIONAlthough the vast majority of device applications consistof thin, strained, and doped �lms as active layers, thereis a fundamental need to understand the bulk propertiesof a given material to control the desired properties of athin �lm properly. This goal has not yet been achievedsu�ciently on ScN for mainly two reasons: (i) there areonly few bulk-like crystals available[1�6] and (ii) mostof them still exhibit either carrier densities beyond thedegeneracy limit[5, 6], high structural disorder[1, 4], ormacroscopically rough surfaces[1, 4, 5]. The bandgap of asemiconductor is particularly sensitive to the free carrierconcentration by means of bandgap-renormalization (BGR)and the absorption edge by means of band-�lling, knownas Burstein-Moss-shift (BMS). Corrections regarding thelatter have been discussed only for degenerately doped ScNsamples[7, 8], while the bandgap renormalization, to ourknowledge, was not accounted for in ScN. Knowledge of theintrinsic indirect and direct bandgaps of ScN is desirableas it exhibits promising properties for several applications,e.g. as (transparent) conductive bu�er layer for latticematched GaN growth[9] or for thermoelectric conversiondevices[10, 11], not speaking of the exploding research inter-est in the ternary ScAlN alloy system due to e.g. enhancedpiezoelectricity[12] and ferroelectricity[13]. In this work,∗ jona.gruembel@ovgu.dewe conduct a precise evaluation of electronic transitionenergies in ScN in the spectral range from 0.9 eV-6.4 eVby employing spectroscopic ellipsometry on high qualitybulk-like and non-degenerately doped ScN �lms. For thedetermination of direct bandgaps we use methods beyondthe widely misused Tauc-plot method, whose applicabilityon the optical response of semiconductors underlies severallimitations[14]. The experimental results are backed upby state-of-the-art calculations: a quasiparticle G0W0approach for the electronic structure, and by solving theBethe-Salpeter equation, starting from the G0W0 results,for the dielectric function. The latter approach takes intoaccount the electron-hole interactions in the optical response.II. METHODS AND SAMPLESA. ComputationThe �rst-principles calculations were performed em-ploying the projector-augmented-wave (PAW) method asimplemented in the Vienna Ab initio Simulation Package(vasp) [15�17]. Soft PAW pseudopotentials were used torepresent the electron-ion interactions, in which the 3s,3p, and 3d states of Sc and the 2s and 2p states of N areexplicitly treated as valence states. The adopted pseudopo-tentials are optimized for accurately describing unoccupiedstates far above the Fermi level. The quasiparticle (QP)mailto:jona.gruembel@ovgu.de2Table I: List of substrate and ScN layer orientation, ScN �lm thickness, selected ellipsometry layer model parameters(bold), x-ray rocking curve full width at half maximum (XRC-FWHMs), and free carrier concentration determined fromHall measurements. The roughness displays the thickness of the Bruggeman EMA layer (1:1 layer/void ratio). The XRCtwist FWHMs correspond to (200) and (220) re�ections and the XRC tilt FWHMs correspond to (131) and (2̄40)re�ections for m- and r-sapphire substrates, respectivelysubstrate ScN layer thickness thickness roughness point-by-point XRC FWHM strain nHallorientation non-uniformity �t cuto� twist tilt out-of-plane(µm) (%) (nm) eV (◦) (◦) (10−4) (1018 cm−3)A1r-sapphire (100)38.4 2.2 4.1 1.44 0.07 0.34 -2.9 7.7B1 19.2 2.0 5.1 1.55 0.09 0.45 -0.22 1.3C1 9.6 3.9 4.4 1.8 0.12 0.55 -2.4 15D 1.36 0 4.1 0.93 0.28 0.80 -2.9 27A2m-sapphire (110)44.8 0 5.3 1.44 0.03 0.05 -0.95 1.5B2 22.4 0 6.9 1.67 0.04 0.08 -1.6 3.2C2 11.2 0 7.1 1.8 0.05 0.14 -1.6 5.2E 3.7 0 7.5 1.94 0.09 0.35 3.1 12electronic band structure was obtained employing a single-shot GW approach starting from HSE06 hybrid functional[18, 19] data (hereafter denoted as HSE06-G0W0). TheHSE06 calculations, in turn, were performed starting fromDFT wavefunctions, employing the generalized gradientexchange-correlation functional of Perdew, Burke, andErnzerhof (GGA-PBE) [20]. The electronic wavefunctionswere expanded in a plane-wave basis set with a cuto�energy of 350 eV, and the Brillouin zone was sampled usinga regular 20× 20× 20 Monkhorst-Pack k-mesh [21]. A totalof 256 bands were used to obtain the QP energies. Conver-gence with respect to the plane-wave cuto� energy, k-pointsampling, and number of bands was carefully veri�ed. TheQP band structure along the high-symmetry directions wasinterpolated using maximally-localized Wannier functions(MLWFs) [22]. The HSE06-G0W0 approach provides areliable description of the underlying electronic structurefor the calculation of the dielectric function [23�25].The frequency-dependent dielectric function was then cal-culated by solving the Bethe-Salpeter equation (BSE), whichaccounts for the excitonic and local-�eld e�ects (LFEs) [26].To obtain the dielectric function over a wide spectral range(up to 60 eV) at a�ordable computational cost, we adopteda multi-grid strategy similar to that used by Schleife et al.[24]. In our approach, the Brillouin zone was sampled withsuccessively coarser k-point meshes for progressively widerphoton-energy regions. An adequate number of conductionbands (CBs) was used for each case. As the k-point samplingwas reduced, the number of CBs was increased, and conver-gence with respect to the included CBs was veri�ed. We usedregular 20×20×20, 16×16×16, 12×12×12, and 8×8×8 Γ-centered k-meshes for transitions up to 12.3 eV, 21.7 eV, 38.0eV, and 60.0 eV, respectively. The resulting imaginary partsof the dielectric function ε2 were found to merge smoothlyat 8.3 eV, 20.7 eV, and 36.3 eV. The real part ε1 was thencalculated using the Kramers-Kronig relation. A Lorentzianbroadening of 0.15 eV was used for the �rst two meshes and0.20 eV for the coarser meshes. As high-energy transitionsare explicitly included in our multi-grid BSE scheme, the ob-tained dielectric constant (ε∞) does not require a rigid shiftof ε1. For comparison, the dielectric function was also cal-culated using the independent-particle approximation (IPA)starting from the HSE06-G0W0 electronic structure, with aregular 20×20×20 Γ-centered k-mesh across the whole spec-trum up to 60 eV.B. Spectroscopic ellipsometry and transmissionSpectroscopic ellipsometry is capable of determining bothreal and imaginary parts of the dielectric function by mea-suring the angle Ψ and retardance ∆ of polarization upon re-�ection of polarized light. For an isotropic sample, the mea-sured quantities Ψ and ∆ are linked to the so-called pseudodielectric function by⟨ε⟩ = sin2 Φ[1 + tan2 Φ(1− ρ1 + ρ)2](1)where Φ is the angle of incidence (AOI) and ρ is the ratio ofthe diagonal Fresnel coe�cients rpp and rss:ρ =rpprss= tanΨei∆ (2)We use a Woollam VASE rotating compensator laboratoryellipsometer for up to 6.4 eV photon energy, where thespectral resolution is set to 4meV, 12meV, and 23meV at2.2 eV, 3.8 eV, and 5.2 eV by keeping the grating monochro-mator slit width constant at 400 nm, respectively. Weacquire data with an energy step width of 5meV at 45◦,60◦, and 75◦ AOI. Data acquisition and analysis is done bythe Woollam WVASE32 software. To obtain the dielectricfunction of the material, a layer model is applied, consistingof an anisotropic sapphire substrate model and a ScNlayer with a Bruggeman e�ective medium approximation(EMA)[27] roughness layer on top, where we use a 1:1layer/void ratio. Additionally, for samples A1, B1, and C1,we account for a non-ideal model by means of thicknessnon-uniformity as displayed in Tab. I. The thickness of theScN layer is �xed to the values obtained from initial SEMmeasurements, only for sample D it is a free parameterwithin the layer model (see Tab. I). As discussed below,the surface of (110) layers is quite di�erent to that of (100)3(a) (b)Figure 1: (a) Quasi-particle band structure of ScN and (b) squared transition dipole moment for optical transitions fromthe three uppermost valence bands (6-8) to the �ve lowest conduction bands (9-13) along high symmetry directions.layers, where the former exhibit a higher surface roughness.Because an AOI of 75◦ is particularly surface sensitive, wewere not able to model the data of (110) ScN layers when75◦ AOI is included. Consequently, we exclude this AOIfrom our analysis for those samples. The dielectric functionof the ScN layer then is obtained by a point-by-point �t ofits optical constants.Normal incidence transmission is recorded by a Shi-madzu spectrophotometer (UV-3101PC + MPC-3100 UV-VIS-NIR) in the spectral range from 830 nm to 1800 nm(0.69 eV to 1.49 eV), where a wave length step width of 1 nmwas used, which corresponds to a energy step width of 1meVat 1.1 eV photon energy, similar to the selected spectral res-olution.C. SamplesSamples were grown by halide-vapor-phase epitaxy(HVPE) either on r- or m-plane sapphire substrates, usingScCl3 and NH3 gases in a reactor with speci�c parametersto avoid oxygen incorporation, yielding single crystallineScN layers as reported in ref. [3]. Sample parameters arelisted in Tab. I, where shared letters (e.g. A1 and A2)denote same-run samples. Among others, the samples wereinitially characterized by x-ray di�raction (XRD), Hallmeasurements, and scanning electron microscopy (SEM),which reveals smooth surfaces for (100) ScN �lms but lesssmooth surfaces for (110) ScN �lms. The surfaces of (100)ScN �lms exhibit large plateaus while the (110) �lms showa hill-valley-type structure[3]. From XRD rocking curves(XRCs), full widths of half maxima (FWHMs) between0.03◦ and 0.28◦ are obtained from the symmetric (200) and(220) re�exes for (100) and (110) ScN layers, respectively,indicating excellent structural quality for all 8 samples.They exhibit thicknesses from 1.3 µm to 45 µm and freecarrier densities between 1.3×1018 cm−3 and 2.7×1019 cm−3(see Tab. I). The degeneracy limit in ScN is reported tobe around 3 × 1019 cm−3 [28, 29], hence all samples havefree carrier concentrations below or near the degeneracypoint. Taking into account the literature value for strainfree ScN (a0 = 4.5047Å[30]) the strain in our samples is|ϵ| < 4×10−4 (see Tab. I), which is negligible in the contextof our study. For our analysis we conduct new Hall mea-surements in van-der-Pauw geometry (see Tab. I) to ensurethat we use the correct free carrier concentrations, whichmay have changed due to aging processes. Nevertheless, weexpect the aging, particularly a possible oxidation, to beweak due to the high structural quality of both the bulk andthe sample surfaces. Samples were cleaned ultrasonically inethanol or acetone to remove possible surface contaminantsprior to ellipsometry measurements. Results on sample A1are already published[31], but the cleaning procedure andslightly di�erent measurement settings reveal changes in theoptical constants, which are discussed below.III. RESULTSA. Band structureThe band structure of ScN was determined in numerousworks before[8, 28, 32�43], where only refs. [8, 35, 36, 42, 43]use the full quasi-particle approach. Qualitatively, though,they all agree that ScN is an indirect band gap semiconduc-tor with its band gap located between the Γ-point valenceband maximum (VBM) and the X-point conduction bandminimum (CBM), while the lowest direct gap occurs at theX-point. The present calculation shows the same behavioras displayed in Fig. 1a. The optical response, i.e. the dielec-tric function, is particularly sensitive to critical points in theband structure, which are characterized by |∇kECV| = 0.In the absence of discrete exciton states (see Sec. III C), weassume the qualitative band shape around each critical pointto be preserved upon electron-hole-interaction renormaliza-tion. Four critical point transitions, labeledX, Γ,X ′, and Γ′,are of interest in this work as they appear at energies below6.4 eV. The present band structure calculation yields transi-4Figure 2: Measured ellipsometric parameters Ψ (upper panel) and ∆ (middle panel) for incidence angles of 45◦ (blacklines), 60◦ (dark gray lines), and 75◦ (light gray lines) with corresponding point-by-point �t results (red lines) for samplesA1, C1, and D. The lower panel displays the measured depolarization factors for these samples. The reference lines indicatethe point-by-point �t cuto� energies as displayed in Tab. I.tion energies of EX = 2.21 eV, EΓ = 4.05 eV, EX′ = 4.65 eV,and EΓ′ = 5.45 eV. The large energy di�erence between theX ′ and Γ′ transitions questions a previously assumed com-bined Γ′X ′ transition[31]. For closer investigation, the tran-sition dipole moments (TDMs) are calculated along the samehigh symmetry directions. For the X, Γ, X ′, and Γ′ criti-cal point transitions there are 3, 6, 2, and 5 bands involvedof which 1, 3, 1, and 2 are conduction bands, respectively.Tab. II shows that the contributions from each valence bandto the total TDM of a certain transition are equal indepen-dently on the number of conduction bands contributing toeither transition. The total squared TDMs of X, Γ, X ′, andΓ′ transitions, in units of e2Å2, amount to 14.0, 8.46, 0, and11.6, respectively. We want to emphasize that the X ′ tran-sition is symmetry forbidden (TDM ≡ 0), thus, what waslabeled as combined Γ′X ′ transition previously[31] is a Γ′transition only.Table II: Total and valence band resolved squaredtransition dipole moments of rocksalt ScN for selectedcritical point transitions in units of e2Å2. Bands arelabeled according to Fig. 1atransition transition dipole momenttotal VB 6 VB 7 VB 8X 14.0 - 6.99 6.99Γ 8.46 2.82 2.82 2.82X ′ 0 0 - -Γ′ 11.6 3.86 3.86 3.86B. Spectroscopic ellipsometryThe data obtained from spectroscopic ellipsometry andcorresponding point-by-point �t results are shown exemplaryfor samples A1, C1, and D in Fig. 2. The point-by-point �tmatches the data nearly perfect for all AOIs above a cer-tain energy, depending on the sample. This cuto� is theenergy at which the measured depolarization factors (Fig.2 lower panel) deviate for di�erent AOIs, which coincideswith the goodness of the point-by-point �t. The depolariza-tion factors at di�erent AOIs are mainly due to the appear-ance of incoherent interface re�ections arising from the high�lm thicknesses[31]. Apart from sample D, the cuto� energy(Tab. I) increases with decreasing �lm thickness due to thelow absorption by the indirect transitions in this spectral re-gion. For sample D Fabry-Pérot fringes are observed, whichare well modeled by our layer model and the point-by-point�t (see Fig. 2, right column). Here, for E < 0.93 eV thepoint-by-point �t does not describe the data adequately (seeFig. 2, right column), so for this sample the point-by-pointcuto� is 0.93 eV. The depolarization factors of sample D de-viate for E < 2 eV, but they exhibit moderate values andoscillate consistently with the Fabry-Perot oscillations. Thisallows us to obtain the dielectric function of sample D downto 0.93 eV and hence direct evidence for the indirect bandgapin ScN from the imaginary part of the dielectric function (ε2),although the sensitivity of ellipsometry to weak ε2 for ε1 > 0is rather low. For sample A1 the noise increases signi�cantlyat higher energies (see Fig. 2, left panel) due to the nar-row slit width, which causes low signal-to-noise-ratio at thedetector, and light di�raction at the surface roughness. Forall samples the applied point-by-point �t results in excellentagreement with the data at photon energies larger than theabove introduced cuto� energy.5(a) (b)Figure 3: (a) Real (blue) and imaginary (orange) parts of the dielectric functions of all non-degenerate ScN samplesdetermined from spectroscopic ellipsometry. The reference lines represent the averaged critical point transition energies.The black lines represent the dielectric function of sample A1. (b) Comparison of experimental (sample A2, blue/orangesolid) and computed (G0W0: black dashed, G0W0 + BSE: black solid) real and imaginary parts of the dielectric function.C. Dielectric function of ScNThe point-by-point �tted dielectric functions of allsamples are displayed in Fig. 3a and show an excellentqualitative agreement. The quantitative deviation, which isstronger at higher energies, mainly stems from the treatmentof the surface roughness, which is not su�cient for thepurpose of quantitatively determining the absorption withhigh accuracy for the given surface structures. Below theabsorption onset at ≈ 2.16 eV, the real parts of the point-by-point �tted dielectric functions show a good agreementwhich strengthens the assumption of residual roughnesse�ects at higher energies due to the low penetration depthof the incident light and increased di�raction and scatteringe�ects for decreasing wave length. The imaginary partfor sample A1 (see Fig. 3a, black solid line) exhibits alower amplitude above the absorption edge than previouslypublished (e.g. 6.4 here and 7.3 in ref. [31] at 3 eV), whichindicates, e.g., organic contamination on the surface whichis now removed by the cleaning procedure in this work. Thisexplains also the decrease of the roughness layer thicknessfrom 8 nm[31] to 4.1 nm (see Tab. I).Fig. 3b shows real and imaginary parts of the dielec-tric function of sample A2, which exhibits the lowestfree electron density, along with results from presenttheoretical calculations. The computational approach,including the solution of the Bethe-Salpeter-equation toaccount for electron-hole-interactions, shows an excellenteven quantitative agreement with the experimental data,while the dielectric function calculated without solving theBSE deviates strongly in both amplitudes and transitionenergies. This behavior, which was observed before ine.g. AlN and CaO[44] or SnO2[45], once again emphasizesthe importance of electron-hole-interactions to be includedin the computation of dielectric functions. The doublepeak structure at the X-point transition, i.e. the lowestdirect band gap, is due to the overestimation of the excitonbinding energy (EbX ≈ 100meV). An estimate of (6-7)meVis obtained from the present calculations by taking intoaccount the known convergence behavior of EbX with themesh size. As discussed below, this explains the absence ofdiscrete excitons in ScN at room temperature. Additionally,the dielectric function determined from BSE shows someresidual oscillations, where it was demonstrated in ref. [44]that such oscillations are due to a coarse k-point sampling.Calculations with coarser k-point sampling [46] revealthat, e.g., the Γ-point transition of ScN shows up as aclearly separable peak only if the mesh size is larger than14 × 14 × 14. Thus, a comparison of distinct features inthe dielectric functions from theory and experiment is onlyvalid if the mesh size is su�ciently large. The rather strongresidual oscillations above 4.5 eV indicate that the presentmesh size is not su�ciently large in this particular region.This is reasonable since the number of bands contributingto the dielectric response as well as o�-symmetry pointcontributions increase with increasing photon energy.To obtain the transition energies of critical point transi-tions from the experimental data we use the critical-point-parabolic-band (CPPB) model, where the nth derivative ofE2ε is described by[47, 48]dndEn(E2ε) =∑jCjeiΦj(E − Ej + iΓj)n+1. (3)Here, Cj , Φj , Ej , and Γj are amplitude, phase, energyposition, and broadening of the jth CP transition where wedo not account for the dimensionality of each critical point.Although the model neglects excitons, the experimentaldata is described reasonably well and hence the energypositions of those transitions can be determined accurately,as discussed before[48]. Note, that eq. (3) is only applicablefor n ≥ 2 [48]. We use n = 3 for this work. The thirdderivatives of the dielectric functions are calculated usingthird order polynomial interpolation at each data point6Figure 4: Exemplary results of the critical-point-parabolic-band (CPPB) model �tted to the third derivatives (real andimaginary part �tted simultaneously) of E2ε for samples A2 (left panel), A1 (middle panel), and D (right panel).with dynamical interpolation widths [minimum 29 datapoints (=̂0.14 eV) around 2.16 eV and maximum 81 datapoints (=̂0.4 eV) above 5.6 eV]. Due to the better signal-to-noise-ratio, d3dE3 (E2ε) = E2 d3εdE3 + 6E d2εdE2 + 6 dεdE was used toobtain d3dE3 (E2ε) from the �rst three derivatives of ε.Eq. (3) was �tted simultaneously to the real and imag-inary parts of d3dE3 (E2ε), which is exemplarily shown inFig. 4 for samples A1, A2, and D, respectively. Onlysmall deviations are visible around 2 eV in the spectraof all samples. We observe three distinct critical pointtransitions referred to EX , EΓ, and EΓ′ , located at around2.16 eV, 3.85 eV, and 5.21 eV, respectively. The results looksimilar for all samples and the obtained �t parameters (seeSupplement[46]) show high conformity across all samples.Because the residual strain is weak in our samples (seeTab. I), we choose to average the results of all samples toobtain a highly reliable value for the EΓ and EΓ′ transitionenergies. We determine EΓ = (3.853 ± 0.006)eV andEΓ′ = (5.21 ± 0.02)eV where the error is given by theStudent factor corrected (95% con�dence interval) standarddeviation. Theoretical calculations indicate that the energyshift due to residual strain is ≤ 2meV (band gap defor-mation potential ≤ 2 eV [35]). The value for EΓ is slightlyhigher than previously experimentally obtained 3.8 eV [8],where the deviation arises possibly from high strain in thethin �lms[8] and a di�erent derivation technique. The lowstandard deviation of 6meV indicates also that the bandgaprenormalization does not a�ect the Γ-point, which is dis-cussed in detail below. In contrast to the Γ-point transition,the X-point transition is expected to be in�uenced stronglyby free electrons, which are located at the X-point. Forcomparison later on, we determine an averaged transitionenergy of E(CPPB)X = (2.13 ± 0.02)eV. The third derivativeof (E2ε) is very sensitive to detect barely separated or lowmagnitude transitions, so we have experimental evidencethat the third critical point transition is not a combinedX ′/Γ′ transition as we concluded before[31], unless theyappear at the exact same energy (±0.05 eV). Because thepresent theoretical calculation predicts zero transition dipolematrix element for the X ′ transition, the third critical pointtransition can be assigned to EΓ′ only.For any M0 type CP transition, the imaginary part of thedielectric function can be described by Elliott's model[49],which includes coulomb interactions between electron-hole-pairs. Assuming Gaussian broadening, the imaginary partof the dielectric function is given byε2 =CE1 + erf(E−EXγX)1− exp(−2π√∣∣∣ EbXE−EX∣∣∣) + εi (4)where εi denotes the contribution of the indirect bandgapabsorption, EbX the exciton binding energy, and EX thedirect bandgap to be distinguished from the fundamentalindirect bandgap Eg. In contrast to our recent work[31] wechoose a more physically motivated model for the indirectbandgap absorption, following ref. 50:εi =CiE[E − Eg]2ΘH(E − Eg). (5)Here, Ci is an arbitrary amplitude of the indirect bandgapabsorption and ΘH denotes the heaviside step-function.The C/E term in eqs. (4) and (5) is used instead of thenominal C/E2 term to account for band non-parabolicity atenergies above the respective transition. Discrete excitonsare not included, as they are not observed[31]. The above-mentioned estimate for the exciton binding from the presentcalculation is EbX ≈ (6−7)meV, while e�ective-mass-theoryyields an even lower value (EbX = µ∗DOS/ε2S × Ry = 4meV)using e�ective masses and static dielectric constant fromrefs. [28] and [31], respectively. Thus, excitons are mostlikely thermally dissociated at room temperature andtherefore not observed. Furthermore, a free electron density> 1018 cm−3 gives rise to signi�cant screening of discreteexcitons.The dispersion of ε1 can be described by Shokhovets'model[51], which can be further simpli�ed for ScN. The di-electric function of the present theoretical calculation showsno prominent high energy contribution to ε2 as it is the case7(a) (b) (c)Figure 5: Exemplary results of modi�ed Shokovets' and Elliott's model �tted to the real and imaginary parts of thepoint-by-point �tted dielectric function, respectively, of (a) sample A1 and (b) sample D. The inset in (b) shows a close-upof the indirect absorption region. (c) Corrected normal incidence transmission (black dotted) and derived absorptioncoe�cient α (black solid) for sample A2. The model �t applied to the absorption coe�cient is shown as well (red solid).for e.g. AlN, GaN, and InN. So we choose to model theimaginary part of the dielectric function of ScN byε2 = AX [ΘH(E − EX)−ΘH(E − Eh)] (6)which, after a Kramers-Kronig-transformation, yieldsε1 = 1 +AXπln∣∣∣∣ E2h − E2E2X − E2∣∣∣∣ . (7)If we treat the absorption edge EX as a known value,this simpli�ed version of Shokhovets' model has two freeparameters. The dielectric limit of the electronic system,referred to as ε∞, is given by ε∞ = ε1(E → 0).Eqs. (4) and (7) are �tted to the imaginary and realparts of the point-by-point dielectric functions, respectively.For this purpose, the indirect bandgap Eg in eq. (5) is�xed at 1.08 eV + ∆EBMS (see supplement for the equationof ∆EBMS) and the upper limit of the �t range is set to2.7 eV to exclude residual absorption contributions fromhigher energy transitions within the Elliott model �t. Whenapplying eq. (7) to the real part of the dielectric function,EX was �xed to the value obtained by Elliotts' model andthe upper limit of the �t range is set to EX −γX . Fit resultsare displayed in Figs. 5a and 5b exemplarily for samples A1and D, respectively. We �nd an excellent agreement betweendata and �t for the real part of the dielectric function forall samples except of sample D. Because sample D exhibitsthe highest free electron concentration, we add a Drudecontribution with �xed plasma-frequency, as derived fromthe given carrier density and m∗e,DOS = 0.35me, yieldingan ε∞ very close to the other samples. Note, that theline shape is still not described adequately and that theresulting plasma broadening is γP = 5meV (25 cm−1). Thedetermined values for ε∞ exhibit high conformity, whereonly sample C1 is an outlier (see Tab I. of supplement [46]).Nevertheless, the �t describes the data excellently for lowdoped samples. Because ε1 is very similar for all samplesbelow the direct bandgap (see Fig. 3a), results of sampleA2 (second lowest free carrier density and lowest pbp cuto�energy) are taken as an estimate for intrinsic ScN. Wehave ε∞ = 8.38 ± 0.04, which is slightly higher than in ourprevious work[31], but well within the given uncertainty.The dielectric limit derived from the BSE dielectric functionis ε∞ = 8.33 which perfectly matches the experimental value.Applying eqs. (4) and (5) to the imaginary parts of thedielectric functions results in a good agreement between dataand model (see Figs. 5a and 5b for samples A1 and D exem-plarily). The inset of Fig. 5b demonstrates that the indirectband gap model with the assumed Eg = 1.08 eV + ∆EBMSdescribes the weak absorption reasonably for sample D,which, in contrast to all other samples, has a pbp cuto�energy below its bandgap. The lowest direct absorptionedge energies EX , as determined from Elliott's model, arein a narrow range from 2.150 eV to 2.160 eV. Given theexperimental spectral resolution of 4meV at 2.2 eV, we canconclude that the absorption edges do not shift signi�cantlyover the entire carrier density range (1.3 × 1018 cm−3 to2.7 × 1019 cm−3). In average we have (2.156±0.007)eV,which is a slightly higher value than determined from theCPPB model: E(CPPB)X = (2.13 ± 0.02)eV. This is notsurprising, since the CPPB model does not include excitonsand hence one determines the sharp onset of absorptionrather than the true absorption edge which is located nearthe in�ection point of ε2 (see vertical lines in Figs. 5aand 5b). The broadenings range from 72meV to 91meV,which is well above room temperature kBT , indicatingadditional broadening mechanisms. The exciton bindingenergies derived from Elliott's model scatter from 14.2meVto 21.5meV, which demonstrates that this �t parameteris very sensitive to the underlying indirect bandgap ab-sorption strength, although no clear trend of the indirectabsorption strength with respect to the exciton bindingenergy is observed (see Tab. I. of supplement[46]). This is8reasonable, since in the absence of discrete exciton statesthe exciton binding energy only determines the strengthof the Sommerfeld enhancement factor [denominator ineq. (4)], which of course overlaps with any other residualabsorption above the absorption edge. The average excitonbinding energy is EbX = (17 ± 5)meV in good agreementwith our previous work[31], but about three times thevalue estimated from e�ective-mass-theory and the presenttheoretical calculations. Thus, the exciton binding energycan not be determined unambiguously in this work.To determine the indirect band gap, normal incidencetransmission is measured on sample B2. The absorption-coe�cient α is obtained byT = (1−N∑j=1Rj) exp {−αd} (8)where d denotes the sample thickness and Rj the re�ectionlosses at the jth interface. Losses due to multiple interfacere�ections are omitted in eq (8), which is a reasonableapproximation given the large thicknesses of ScN �lmand sapphire substrate. Re�ection losses are included forair/ScN, ScN/sapphire, and sapphire/air interfaces, i.e.N = 3. Therefore, the model dispersion curves of ε1, i.e.the Shokhovets' model �t results for ScN and an averageof ordinary and extraordinary component from ref. [52]for sapphire, are used to calculate the respective interfacere�ections via Rj = (√ε1,j−1 −√ε1,j)2/(√ε1,j−1 +√ε1,j)2,where j = 0, 3 correspond to the front and backside ambientair, respectively. The resulting absorption coe�cient isslightly negative, which is likely due to a slight misalignmentof probe and reference light hole sizes. Thus, the measuredtransmission is multiplied by a factor of 0.93 to ensurethat α is positive at all wave lengths. The resulting dataand thereof calculated absorption coe�cient are displayedin Fig. 5c. The onset of absorption due to the indirectband gap around 1.1 eV is clearly visible. To derive anestimate for the indirect band gap, the derived α is �ttedusing α = ωε2/c√ε and eqs. (7) and (5) for ε1 and ε2,respectively, where the parameters for ε1 are set to thevalues obtained from �tting the pbp ε1 of sample B2. Fitresults are also shown in Fig. 5c (red line), where a goodagreement between �t and data is found for E > 1.2 eV.Because the model does not account for broadening, thelower limit of the �t range is set to 1.22 eV, which isobtained by a maximization of the adjusted coe�cient ofdetermination r̃ upon varying lower �t range limit. Theobtained band gap is Eg = (1.09 ± 0.02)eV, where theuncertainty is estimated from the variation of Eg uponvarying lower �t range limit.To derive estimates for true intrinsic direct and indi-rect bandgap, we have to account for many-body e�ects onthe absorption edge. Free carriers shift both, the bandgapand the absorption edge by means of bandgap renormaliza-tion (BGR) and band-�lling, known as Burstein-Moss-shift(BMS), respectively. The parabolic band approximation isnear-exact in the non-degenerate regime and the functionaldependence of a transition energy ECV on the free carrierdensity reads qualitativelyECV(n) = ECV,0 +∆EBGR(n, m∗e/h,DOS, εS, ν) +∆EBMS(n, m∗e,DOS, ν). (9)Here, ν is the manifold of the conduction band minimumand εS the low frequency dielectric constant. Note that weassume the valence band contribution to the BMS to benegligible in the non-degenerate regime. Using the e�ectivevalence and conduction band masses from Ref. 28 andεS = 29.5 from Ref. 31, ECV,0 is the only free parameterremaining in eq. 9. The explicit equations for the BGR, asderived in ref. [53], and the BMS as well as values for thee�ective masses are given in the supplement[46].Figure 6: Direct absorption edge EX as determined fromElliotts' model (black squares) plotted against the freecarrier concentration n along with model �t results (redsolid) and respective contributions (bandgaprenormalization: blue dashed, Burstein-Moss-Shift: greendashed).Fig. 6 displays the applied �t of eq. (9) to E(Elliott)X overn, where n is the free electron concentration determinedfrom Hall measurements. We obtain a slight decrease ofEX with increasing n due to the BGR before the BMSshifts the absorption edge to higher energies again. Theassumed e�ective masses and εS re�ect the near constantabsorption edges of our samples from 1.3 × 1018 cm−3 to2.7 × 1019 cm−3 reasonably, where the scattering aroundthe �t curve may arises from e.g. small residual straine�ect and �tting errors. The �t yields an intrinsic directbandgap of EX,0 = (2.182 ± 0.004)eV, which is close toan earlier reported absorption edge of non-degenerateScN (EX = 2.15 eV[54]), but deviates from previouslypublished values extrapolated from degenerately doped�lms (2.1 eV[7] and (2.07±0.05)eV[8]). This is becausethe BGR was not taken into account in these works. Thepresent band structure calculations yield EX = 2.21 eV,hence less than 1.5% deviation from the experimentalvalue. Note, that in ref. [42] almost the same transitionenergy was obtained from an extended QSGW scheme,where the electron-phonon interaction renormalization wasalso considered[42]. They obtained a corrected X-pointgap of 2.23 eV, which deviates less than 2.5% from thepresent experimental value. Thus, the renormalization of9Table III: Comparison of intrinsic indirect and direct band gaps, higher energy critical point transition energies, and thedielectric constant determined in this work and from selected theoretical works..Eg (eV) EX (eV) EΓ (eV) EΓ′ (eV) ε∞experiment (this work) 1.08(2) 2.182(4) 3.853(6) 5.21(2) 8.38(4)theory (this work, G0W0) 1.09 2.21 4.05 5.45 8.33 [BSE]theory (OEPx[cLDA]-G0W0) [35] 0.84 1.98 3.51 4.72 -theory (LDA-G0W0) [35] 1.14 2.06 3.71 - -theory (HSE25 with0.79 1.91 3.58 4.74 9.825% quasiparticle mix) [28]theory (HSE06 with hybrid0.92 2.02 3.72 4.9 7.6exchange correlation potential) [8]the exciton continuum transition energy at the X-pointdue to electron-hole-pair interactions is of low magnitude.The remaining deviation between theory and experimentis reasonable since measurements are performed at roomtemperature while the present computations yield the bandstructure and dielectric function at T = 0K.We now include experimental results by angular-resolvedphotoelectron-spectroscopy (ARPES) of degeneratelydoped, bulk ScN into our discussion[6]. We can conduct fur-ther conclusions regarding the bandgap renormalization inScN from the following assessments: The expected bandgapshrinkage due to BGR amounts to 28meV at 1× 1018 cm−3and 74meV at 3× 1019 cm−3, while no shrinkage of the EΓand EΓ′ transition energies between those carrier densitiesis found. Additionally, this transition is almost perfectlyresembled by the present theory. Hence, as Γ−point transi-tions do not underlie the BMS to compensate the BGR, wemay conclude that the BGR does not a�ect any transitionslocated near the Γ-point. Ref. 6 consistently reports a BMSof ≈ 370meV and a BGR of ≈ −340meV at the X-pointwhile the indirect bandgap (ΓX) is 1.2 eV, so we furtherconclude that the bandgap renormalization in ScN a�ectsonly the local environment in reciprocal space of the valenceband at the X-point and does not a�ect the conductionband at all. This is clearly visible by the formation ofa local valence band maximum in their experimentallydetermined electronic band structure (see Fig. 2c in Ref.6). This implies that the indirect bandgap, as determinedfrom transmission measurements on sample B2, is a�ectedby the BMS only, yielding an intrinsic indirect bandgap ofEg,0 = (1.08 ± 0.02)eV, which is only slightly lower thanthat in Ref. 6 (1.2 eV).In summary, we derived highly reliable electronic transi-tion energies from non-degenerately doped quasi-bulk ScNsingle crystals as listed in Tab. III and presented a detailedcomparison with state-of-the-art computations. Intrinsicdirect and indirect bandgaps are extrapolated with anapproximate accuracy ≤ 20meV from experiment by takinginto account free carrier e�ects. The EΓ and EΓ′ criticalpoint transition energies are derived in a statistical mannerwith uncertainties of 6meV and 20meV, respectively, fromexperiment. We �nd further evidence that the bandgaprenormalization does not a�ect the band structure nearthe Γ-point in ScN. The present theory shows exceptionalagreement of the direct X and Γ-point transition energiesas well as the dielectric limit of the electronic system ifelectron-hole-interaction, i.e. the Bethe-Salpeter-equation,are considered. Additionally we demonstrated that theX ′-transition is symmetry forbidden and therefore notobserved in experiment, while the Γ′-transition energiesderived from theory and experiment align. The criticalpoint transition energies obtained from the G0W0 bandstructure, i.e. in the omission of electron-hole-interactions,show almost linearly increasing deviation from the ex-perimental values for increasing energies (see Tab. III).Hence, this work once again demonstrates the importanceof electron-hole-interactions to be included to derive theoptical response of semiconductors properly. 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B 70, 045303 (2004).https://doi.org/10.1103/PhysRevB.109.075142https://doi.org/10.1103/PhysRevB.109.075142https://doi.org/10.1103/PhysRevB.108.165104https://arxiv.org/abs/2508.05247https://arxiv.org/abs/2508.05247https://arxiv.org/abs/2508.05247https://doi.org/10.1103/PhysRevB.84.075218https://doi.org/10.1063/1.4882237https://doi.org/10.1063/1.4882237https://doi.org/10.1016/0039-6028(73)90337-3https://doi.org/10.1103/PhysRevB.45.11749https://doi.org/10.1103/PhysRev.108.1384https://doi.org/10.1103/PhysRevB.93.045203https://doi.org/10.1063/1.1582369https://doi.org/10.1103/PhysRevB.24.1971https://doi.org/10.1103/PhysRevB.24.1971https://doi.org/10.1103/PhysRevB.70.045303 Dielectric function and electronic structure of non-degenerate rocksalt ScN:  Spectroscopic ellipsometry and GW-calculations Abstract Introduction Methods and samples Computation Spectroscopic ellipsometry and transmission Samples results Band structure Spectroscopic ellipsometry Dielectric function of ScN References