Lorentz Dispersion Model - Horiba

1 Lorentz Dispersion ModelSpectroscopic ellipsometry (SE) is a technique based on the measurement of the relative phasechange of reflected and polarized light in order to characterize thin film optical functions and otherproperties. The measured data are used to describe a Model where each layer refers to a given ma-terial. The Model uses mathematical relations called Dispersion formulae that help to evaluate thethickness and optical properties of the material by adjusting specific fit parameters. This application note deals with the Lorentzian Dispersion that the technical notes Classical Dispersion Model and Drude Dispersion Model are com-plementary to this modelThe Lorentz classical theory (1878) is based on theclassical theory of interaction between light and matterand is used to describe frequency dependent polariza-tion due to bound charge.

2 The bindings between elec-trons and nucleus are supposed to be similar to thethat of a mass-spring system. Electrons react to an electromagnetic field by vibratinglike damped harmonic oscillators. The way the dipolereplies to a submitted electric field is given by the fol-lowing equation of motion of a bound electron:where: m d2r/dt2 is the acceleration force; m 0 dr/dt is the viscous force; 0 is the dampingfactor; m t2r is the Hooke s force; m is the electronic massand t is the resonant frequency of the oscillator; -e Eloc is the local electric field driving force; e is themagnitude of the electronic charge and Eloc is the lo-cal electric field acting on the electron. The assump-tion is made that the macroscopic and local electricfields are equal and vary in time as ei solution to the previous equation yields the expres-sion for the amplitude of oscillation r depending on thephoton energy :At low frequencies << t, the amplitude r has a me-dium finite value and is in phase with the resonance frequency t the amplitude isimaginary and maximum when denominator is mini-mum.

3 More, at t there is a 90 phase shift be-tween E and r. At high frequencies >> t, the amplitude r 1 Restoring force between the orbiting electronand the atomic centre (Ref. 4).E()12022 E er mdtrd mdtrdmloctrrrr = + + ()()()2 1022 + =iEemrtlocrrFig. 2 The oscillator amplitude as a function offrequency (Ref. 5).PhaseAmplitude << t >> tOSCILLATOR DISPLACEMENTTN08 The induced dipole moment is related to r throughthis relation:The polarizability ( ) is given by ( )= ( )E( )where:Taking the sum of the single atom dipole moment overall atoms in a volume, it comes that the polarizationper unit volume is given by:The susceptibility ( ) is deduced from the previousequation:where the prefactor (Ne2/ 0m) is the plasma frequencysquared dielectric function is then given through this rela-tionThe limits s and of the dielectric function respec-tively at low and high frequencies are given by: The complex dielectric function can also be expressedin terms of the constants s and by substitutingequations (8) into (7) which yields the following equa-tion.

4 Where s is defined as: Lorentz Model describes radiation absorption due to in-ter-band transitions (quantum-mechanical interpreta-tion). Interband transitions are transitions for which theelectron moves to a final state corresponding to a dif-ferent band without changing its k-vector in Brillouin sfirst to multiple oscillatorsIf there is more than one oscillator, the dielectric func-tion is assumed to be equal to the sum of contributionsfrom individual oscillators. This situation fits better tothe case of real materials. In DeltaPsi2 software, the mathematical formulationused for a collection of:- three Lorentz oscillators is:- N (N 1) Lorentz oscillators is:Increasing the number of oscillators leads to a shift ofthe peaks of absorption toward the ultraviolet 3 The response of an oscillator amplitude to a periodic drivingforce depends on the resonance frequency (Ref 5).

5 Driving ForceFrequency >> t << t tTime()() ()()[]()3 0222 + = =imEeretlocrrrr()()()4 10222 + =imet()() ()() ()()5 0 EENP = =AtomAtom in E-field rEFig. 4 Polarization of the electronic cloud due to external E-field.(Hecht, Ref. 3)()()6 102202 + =imeNt() ()()711~0222 + +=+=i tp()()()81~10~22 = =+= = tps ()()()9 ~0222 + += itts22tps += ()() = + + + +=2122200222~jjojjjttsifi () = + +=Njjojjj i f 12220~TN08 The parameters of the equations4 parameters may be used in the expressions of thesingle Lorentz oscillator but it may be possible to char-acterize the function with fewer describing the real part of the dielectric function The constant is the high frequency dielectric con-stant; it takes into account the contribution of highenergy inter-band transition.

6 Generally, =1 butcan be greater than 1 if oscillators in higher energiesexist and are not taken into account. The constant s ( s> ) gives the value of the staticdielectric function at a zero frequency. The difference s- represents the strength of the single Lorentz os-cillator. The larger it is then the smaller the width 0of the peak of the single Lorentz describing the imaginary part of the sin-gle Lorentz oscillator dielectric function t (in eV) is the resonant frequency of the oscillatorwhose energy corresponds to the absorption t increases then the peak is shifted to higherphoton energies. Generally, 1 t 20. 0 (in eV) is the broadening of each oscillator alsoknown as the damping factor. The damping effect isdue to the absorption process involving transitionsbetween two states.

7 On a graphic representing i( ), 0 is generally equal to the Full Width At Half Maxi-mum (FWHM) of the peak. As 0 increases the widthof the peak increases, but its amplitude , 0 0 describing the imaginary part of the mul-tiple Lorentz oscillators dielectric function. fj (j = 1, 2 .. N) term is the oscillator strengthpresent in the expression of the multiple Lorentz os-cillator. As fj increases then the peak amplitude in-creases, but the width of the peak j , 0 fj 10. 0j (in eV) (j = 1, 2 .. N) is the resonant (peak) en-ergy of an oscillator for a collection of severalLorentzian oscillators. It is similar to t. Generally,1 0j 8. j (in eV) (j = 1, 2 .. N) parameter is the broadeningparameter corresponding to the peak energy of eachoscillator. It behaves like 0.

8 Generally, 0 j of the modelThe Lorentz oscillator is not suitable for describing theproperties (presence of gap energy and quantum ef-fects) of real absorbing (amorphous, semiconductors) set upNote that: 1. The Lorentzian dielectric function is available in theClassical Dispersion formula in the DeltaPsi2 The sign before a given parameter means thateither the amplitude or the broadening of the peakis linked to that parameter. 3. For each multiple oscillator the graphs show the dif-ferent contributions (in red dashed lines) of the NLorentzian oscillators to the imaginary part of theLorentz dielectric function (in red bold line).> Transparent Lorentz function - This function exhibits no absorption: 0=0. - This case corresponds to normal Dispersion where r( ) increases with photon energy.

9 > Absorbing Lorentz function- This function exhibits absorption: 0 0. - The real part of the dielectric function increases withincreasing frequency (normal Dispersion ) except for aregion between [3eV - 4eV] where the Dispersion be-comes anomalous. The absorption peak is given bythe imaginary part of the dielectric function i ( ) andis always positive. Dielectric function of SiO2TN08 > Multiple oscillator Lorentz functionApplications to materialsThe Lorentz oscillator Model is applicable to describes well for example the behaviour of a trans-parent or weakly absorbent material (insulators, semi-conductors). The spectral range of validity of theLorentz formula depends on the material but usuallythe fit is performed over the region < t for the singleLorentz oscillator and < i in case of multiple oscilla-tors where i is the transition energy of the oscillator ofhighest of materials following single Lorentz oscillator modelRepresentation of a Lorentz absorbing functionDielectric function of CuPc described by 2 oscillatorsDielectric function of a green colored filter described by 4 oscillatorsMaterials ?

10 S tГ0S. R. (eV) - 3 AlGa - - - - 6 Aminoa cid - 5Au - 5Ca - d Color Filter - 2 GaAs - - - - - - 6o - - - - :Eu3+ ,0 - - - t sTN08 List of materials following single Lorentz oscillator modelList of materials following multiple oscillator Model References1) H. M. Rosenberg, The Solid State, Oxford University Press2) F. Wooten, Optical Properties of Solids, Academic Press (1972)3) Eugene Hecht, Optics, Chap. 3, Hardcover (2001)4) ~boser/ ) ? s tГ0S. R. (eV) - - - - - - - 5 Spincoated - - - si - - : - - - - - - 5 SiO2 doped - - - - - 3 YAG:Tb(10%) ,0 - - - 3 s t CuPcGreen Color FilterPentacene/Sie 0, * 0, 0, 0, document is not contractually binding under any circumstances - Printed in France - 09/2006 MaterialsParameters TN08