New Amorphous Dispersion Formula - HORIBA

1 New Amorphous theoretical modelThe New Amorphous Dispersion Formula was derivedby HORIBA Jobin Yvon on the basis of Forouhi-Bloomerformulation. This new formulation was established inorder to give a Lorentzian shape to the expressions ofthe extinction coefficient and refractive index. The ab-sorption coefficient is given by :The refractive index is written through this Formula : where :Extension to multiple oscillators: N - New Amorphous The New Amorphous formulation can be extended tothe case where many oscillators are present in the ma-terial. Such dispersions are called N NewAmorphous and are given in the array equations for k( ) and n( ) for N oscillators aregiven below :- for the extinction coefficient :- for the refractive index :Increasing the number of oscillators leads to a shift ofthe peaks of absorption toward the ultraviolet parameters of the equationParameter describing the refractive indexThe term n( ) is an additional parameter, at leastgreater than one and equal to the value of the refrac-tive index when.

2 ()()()()1 for; 0 for ; f kgg2j2j2gj >+ =()()()()2 C Bn n2j2jj+ + += ()()()()3 f2 CfBgjjjj2gj2jjjj = =FormulaNumber of oscillatorsNumber of parametersNew AmorphousN=15 Double New AmorphousN=28 Triple New AmorphousN=311()()()()4 for ; 0 for ;g1g222 > + = =Njjjgjfk ()()()()5 CBnnN1j2j2jjjj = + + += New Amorphous Dispersion FormulaSpectroscopic 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 data are used to describe a model where each layer refers to a given material.

3 The model usesmathematical relations called Dispersion formulae that help to evaluate the material s optical properties byadjusting specific fit application note deals with the new Amorphous Dispersion Formula . For further information about theoriginal theory derived from Forouhi-Bloomer, please refer to the technical note Forouhi Bloomer aliasAmorphous Dispersion Formula .TN124 parameters describe the extinction coefficient. fj (j=1, 2, 3) (in eV) is related to the strength (ampli-tude) of the extinction coefficient peak. As the value offj increases both values of the refractive index and ex-tinction coefficient increases. Generally, 0<fj<1. j (j=1, 2, 3) (in eV) is the broadening term of thepeak of absorption.

4 The larger j is the larger the ab-sorption peak but the smaller its amplitude. General-ly, < j<8. j (in eV) is approximately the energy at which the ex-tinction coefficient is maximum (peak of absorption).As the value of j increases the absorption peak isshifted towards the UV region. Generally, < j<10. g (in eV) is the energy band gap. It is equal to theminimum of energy required for a transition from thevalence band to the conduction band. It is the energyfrom which the absorption starts to be non-zero:k(E Eg) 0. Always, g< between Amorphous and New Amor-phous parametersThe array below gives the relations between the pa-rameters of Forouhi-Bloomer and New Amorphous dis-persion : In DeltaPsi2 software, the user is advised to per-form fitting using the New Amorphous Formula (andits extensions) instead of the Amorphous one becauseB and C parameters are often setupNote that : - The graphs below show the different contributions (inred dashed lines) to the imaginary part of the DoubleNew Amorphous dielectric function (in red bold line).

5 - The sign before a given parameter means thateither the amplitude or the broadening of the peak islinked to that parameter. New Amorphous functionDouble New Amorphous functionApplications to materialsThe new Amorphous model works particularly well foramorphous materials exhibiting an absorption in thevisible and/or FUV range (absorbing dielectrics, semi-conductors, polymers).We advise the user to compare the results obtained be-tween the new Amorphous and Tauc-Lorentz dispersionformula. The Tauc-Lorentz model may fit better the ab-sorption part of the experimental AmorphousAfjB2. jC 2j+ 2jn Eg g Optical properties of Alq3 given by the Double New Amorphous functionOptical properties of Amorphous silicon given by the New Amorphous functionTN12 Materials following the New Amorphous modelThe asterisk * refers to parameters that are negativeand thus do not have any physical meaning but repre-sent good starting values to perform the fit on the following the Double New Amorphous modelMaterialsn?

6 Gfj jГjS. R. (eV) - - - - 5 - 4 Amino- - - - - - - - Fe - - doped * - 6 - - - * - 4Fe - - - - 5G l a ss - - * - - * - * - - - 4n Materialsn? gf1 1Г1S. R. (eV)e-PEDOT-PSS - - - - 4 - 6e - PPV - - si - - - - - - - - - - - - - - - 4n Materialsn? gf1 * * *n TN12 Materials following the Triple New Amorphous modelReferences1.

7 E. D. Palik, Handbook of Optical Constants of Solids II, Chap. A. R. Forouhi, I. Bloomer, Phys. Rev. B 34, 7018-7026 (1986).3. A. R. Forouhi, I. Bloomer, Phys. Rev. B 38, 1865 (1988).4. Jellison, F. A. Modine, Phys. Rev. , 69(3), 371-374 (1996).MaterialsГ1f2 2Г2S. R. (eV) - - - * * - 4 FeCo * - 4 Blue - - 2 - - - - - - n? gf1 1Г rbona coating Materials 2Г2f3 3Г3S. R. (eV) - - rbona - coa - ting - - document is not contractually binding under any circumstances - Printed in France - 11/2006