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Tauc-Lorentz Dispersion Formula

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Tauc-Lorentz Dispersion Formula TN11 Tauc-Lorentz Dispersion Formula Spectroscopic ellipsometry (SE) is a technique based on the measurement of the relative phase change of re- flected and polarized light in order to characterize thin film optical functions and other properties. The meas- ured SE data are used to describe a model where layers refer to given materials. The model uses mathematical relations called dispersion formulae that help to evaluate the material’s optical properties by adjusting specific fit parameters. This technical note deals with Tauc-Lorentz dispersion formula. Theoretical model The real part εr,TL of the dielectric function is derived from the expression of εi using the Kramers-Kronig integration. Jellison and Modine developed this model (1996) using Then, it comes the following expression for εi: the Tauc joint density of states and the Lorentz oscillator. The complex dielectric function is : 2 ∞ ξ ⋅ε ()ξ ε ()E = ε ()∞ + ⋅ P ⋅ i dξ ()5 ~ε =ε + i ⋅ε =ε + i ⋅(ε × ε ) (1) r r π ∫ ξ 2 − E 2 TL r,TL i,TL r,TL i,T i, L Eg Here the imaginary part εi,TL of the dielectric function is where P is the Cauchy principal value containing the resi- given by the product of imaginary part of Tauc’s (1966) dues of the integral at poles located on lower half of the dielectric εi,T function with Lorentz one εi,L. In the approx- complex plane and along the real axis. imation of parabolic bands, Tauc’s dielectric function de- According to Jellison and Modine (Ref. 1), the derivation scribes inter-band transitions above the band edge as : of the previous integral yields : E − E 2 ⎛ g ⎞ 2 2 εi,T ()E > Eg = AT ⋅⎜ ⎟ ()2 A⋅C ⋅a ⎡ E + E + α ⋅ E ⎤ ⎜ E ⎟ ln 0 g g ⎝ ⎠ εr,TL ()E = ε∞ + 4 ⋅ln⎢ 2 2 ⎥ where : 2⋅π ⋅ζ ⋅α ⋅ E0 ⎣⎢ E0 + Eg − α ⋅ Eg ⎦⎥ -A is the Tauc coefficient T A a ⎡ ⎛ 2⋅ E + α ⎞ - E is the photon energy − ⋅ a tan ⋅ π − arctan⎜ g ⎟ + 4 ⎢ ⎜ ⎟ K -Eg is the optical band gap π ζ ⋅ E0 ⎣ ⎝ C ⎠ The imaginary part of Tauc’s dielectric function gives the ⎛ α − 2⋅ E ⎞⎤ response of the material caused by inter-band mecha- g + arctan⎜ ⎟⎥ nisms only : thus εi, T (E ≤ Eg) = 0. ⎝ C ⎠⎦ The imaginary part of the Lorentzian oscillator model is 2 2 given by : 4⋅ A⋅ E0 ⋅ Eg ⋅()E − γ + 4 ×K AL ⋅ E0 ⋅C ⋅ E π ⋅ζ ⋅α ε i, L ()E = 2 ()3 ()E 2 − E 2 + C 2 ⋅ E 2 ⎡ ⎛ α + 2⋅ E ⎞ ⎛ α − 2⋅ E ⎞⎤ 0 × arctan⎜ g ⎟ + arctan⎜ g ⎟ where : ⎢ ⎜ ⎟ ⎜ ⎟⎥ ⎣ ⎝ C ⎠ ⎝ C ⎠⎦ -AL is the strength of the ε2, TL(E) peak - C is the broadening term of the peak 2 2 ⎛ ⎞ A⋅ E ⋅C ⋅()E + E E − Eg -E is the peak central energy − 0 g ⋅ln⎜ ⎟ + 0 π ⋅ζ 4 ⋅ E ⎜ E + E ⎟ K By multiplying equation (2) by equation (3), Jellison sets ⎝ g ⎠ up a new expression for ε (E): i,L ⎡ ⎤ 2⋅ A⋅ E ⋅C E − Eg ⋅()E + Eg 2 0 ⎢ ⎥ ⎧ 1 A E C (E − E ) + 4 × Eg ⋅ln ()6 0 g π ⋅ζ ⎢ 2 2 2 2 2 ⎥ ⎪ 2 for E > Eg E − E + E ⋅C ε E = ε × ε = E 2 2 2 2 4 ⎢ ()0 g g ⎥ i, TL () i,L i,T ⎨ ()E − E0 + C E () ⎣ ⎦ ⎪ ⎩0 for E ≤ Eg where A=AT x AL . TN11 where: The parameters of the equations 2 2 2 2 2 2 2 2 ⎧aln = ()Eg − E0 ⋅ E + Eg ⋅C − E0 ⋅ (E0 + 3⋅ Eg ) The name of the different Tauc-Lorentz formulae present ⎪ 2 2 2 2 2 2 in the DeltaPsi2 software depending on the number of os- ⎪aa tan = ()()E − E0 ⋅ E0 + Eg + Eg ⋅C cillators are given below. It indicates the number of pa- ⎪ rameters too. ⎪ 2 2 ⎨α = 4⋅ E0 − C ()7 ⎪ Number of Number of 2 2 Formula ⎪γ = E0 − C / 2 oscillators parameters ⎪ 4 2 2 2 2 2 ⎩⎪ζ = ()E −γ +α ⋅C / 4 Tauc-Lorentz N=1 5 Extension to multiple oscillators Tauc-Lorentz 2 N=2 8 In the case of multiple oscillators (N>1) the expression of Tauc-Lorentz 3 N=3 11 εi contains now a series over all the oscillators. Increasing the number of oscillators leads to a shift of the peaks of absorption toward the ultraviolet region. N 2 ⎧ 1 Ai ⋅ Ei ⋅Ci ⋅()E − Eg ⎪ × for E > E 1 parameter is linked to the real part of the dielectric ε = ∑ E 2 2 2 2 2 g 8 i ⎨ i=1 ()E − Ei + Ci ⋅ E () function ⎪ ε ∞ ε ⎩0 for E ≤ Eg • r ( ) = ∞ is the high frequency dielectric constant. This is an additional fitting parameter that prevents ε1 from and εr is re-written as the following sum : converging to zero for energies below the band gap. Generally, ε∞>1. N 2 ∞ ξ ⋅ε ()ξ ε E = ε ∞ + ⋅ P ⋅ i dξ 9 r ()r ()∑ ∫ 2 2 () At least 4 parameters describe the imaginary part of the i=1 π ξ − E Eg dielectric function th Deriving this integral yields the analytical expression of •Ai (in eV) is related to the strength of the i absorption the real part of the dielectric function : peak. The subscript «i» refers to the number (i=1, 2 or 3) of oscillators. As Ai increases, the amplitude of the N ⎧ ⎡ 2 2 ⎤ peak increases and the Full Width At Half Maximum ⎪ Ai ⋅Ci ⋅aln Ei + Eg + α ⋅ Eg εr ()E = εr ()∞ + ⎨ ⋅ln⎢ ⎥ (FWHM) of that peak gets slightly larger. Generally, ∑ 2⋅π ⋅ζ 4 ⋅α ⋅ E E 2 + E 2 − α ⋅ E i=1 ⎩⎪ i ⎣⎢ i g g ⎦⎥ 10<Ai<200. •C (in eV) is the broadening term ; it is a damping coef- A a ⎡ ⎛ 2⋅ E + α ⎞ i th − i ⋅ a tan ⋅ π − arctan⎜ g ⎟ + ficient linked to FWHM of the i peak of absorption. The 4 ⎢ ⎜ ⎟ K higher it is the larger that peak becomes and at the π ζ ⋅ Ei ⎝ Ci ⎠ ⎣ same time the smaller its amplitude is. Generally ⎛ α − 2⋅ Eg ⎞⎤ 0<Ci<10. + arctan⎜ ⎟⎥ ⎜ ⎟ •Eg (in eV) is the optical band gap energy. ⎝ Ci ⎠⎦ •E (in eV) is the energy of maximum transition probability 2 2 i 4⋅ Ai ⋅ Ei ⋅ Eg ⋅()E − γ ⎡ ⎛ α + 2⋅ Eg ⎞ or the energy position of the peak of absorption. The + ⋅ ⎢arctan⎜ ⎟ π ⋅ζ 4 ⋅α ⎜ C ⎟ subscript «i» refers to the number (i=1, 2 or 3) of oscil- ⎣ ⎝ i ⎠ lators. Always, Eg<Ei. ⎛ α − 2⋅ Eg ⎞⎤ + arctan⎜ ⎟⎥ Limitation of the model ⎝ Ci ⎠⎦ The Tauc-Lorentz model requires ε to be zero for energies 2 2 ⎛ ⎞ i A ⋅ E ⋅C ⋅()E + E E − Eg − i i i g ⋅ln⎜ ⎟ less than the band gap. Consequently, Tauc-Lorentz mod- 4 ⎜ ⎟ el does not take into account intra-band absorption: any π ⋅ζ ⋅ E E + Eg ⎝ ⎠ defect or intra-band absorption increases εi below the ⎫ band gap and generates bad fits in that region. ⎡ E − E ()E + E ⎤ 2 Ai Ei Ci ⎢ g g ⎥⎪ + Eg ln ⎬ ()10 4 ⎢ 2 ⎥ π ⋅ζ E 2 − E 2 + E 2 C 2 ⎪ Valid spectral range ⎣⎢ ()0 g g i ⎦⎥⎭ The Tauc-Lorentz model well describes the behaviour of materials for energies E ≤ Ei where Ei is the transition en- ergy of the oscillator of highest order. TN11 Parameter setup Materials Eg ε?A∞ E0 C S. R. (eV) The Tauc-Lorentz model works particularly well for amor- a-C 1.305 3.774 38.835 6.335 6.227 1.5 - 5 phous materials exhibiting an absorption in the visible AlGaN 3.300 3.000 100.000 4.500 1.000 1.5 - 2.5 and/or FUV range (absorbing dielectrics, semiconduc- As2S3 2.349 1.240 133.819 4.574 5.935 0.75 - 4.75 tors, polymers). AsSSe 1.000 1.600 50.000 3.500 2.000 0.75 - 4.5 Note that : DLC 0.374 2.960 11.558 3.533 3.346 0.7 - 2 - The graph below shows the different contributions (in GaN 3.200 3.500 100.000 4.500 1.000 1,5 - 3 red dashed lines) to the imaginary part of the Tauc GeSbSe 1.851 2.062 54.703 3.412 2.045 0.65 - 3 Lorentz dielectric function (in red bold line). InGaN 3.000 3.500 100.000 4.500 1.000 1,5 - 3 - The sign «»∞ before a given parameter means that ei- Polymer 3.027 1.701 32.921 5.759 2.416 0.7 - 4 ther the amplitude or the broadening of the peak is poly-Si 0.882 3.937 49.597 3.711 0.321 1.5 - 3.25 linked to that parameter. a-Si 2.030 1.692 142.599 3.840 1.908 0.75 - 3.65 Tauc-Lorentz function a-Si:B 1.267 2.148 143.380 3.617 1.994 0.75 - 4.75 a-Si:H 1.393 0.626 171.105 3.582 2.201 0.75 - 4.75 SixOyNz 3.864 1.760 77.340 8.363 3.550 1.5 - 6.5 Ta-C 1.395 2.558 67.312 8.713 9.760 1.5 - 5 ZnS 2.976 4.500 57.167 3.954 1.200 1.5 - 6.5 GeSbTe 0,450 2,024 158,795 1,688 2,397 0,65 - 3 Materials following the Double Tauc-Lorentz model Materials Eg ε∞? A 1 E 1 Blue 1.451 2.346 6.236 1.617 Filter Oxide 1.168 0.894 41.060 6.464 coating Polymer 3.420 2.771 113.260 4.234 Dielectric function of a-Si represented by a single Tauc-Lorentz SiC 1.521 -1.657* 4.026 5.678 oscillator Ta2O5 4.261 1.827 188.580 4.188 Double Tauc-Lorentz function Materials C 1 A 2 E 2 C 2 S.
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