Forouhi-Bloomer alias Amorphous Dispersion …

1 Forouhi-Bloomer alias Amorphous Dispersion FormulaTN13 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 data are used to describe a model where each layer refers to a given material. The model usesmathematical relations called Dispersion formulae that help to evaluate the thickness and optical propertiesof the material by adjusting specific fit application note deals with the original forouhi & Bloomer Dispersion formula also known as Amor-phous (name used in the DeltaPsi2 software).

2 We recommend the use of the updated New Amorphous for-mulation, please refer to the corresponding technical alias Amorphous theoretical modelThe Forouhi-Bloomer formulation (1986, 1988) is ap-plicable to Amorphous semiconductors and dielectricsand is based on the quantum-mechanical theory of ab-sorption. It takes into account the optical band gap inthe inter-band region. It is supposed that the valenceand conduction bands are parabolic and are separat-ed by a forbidden band whose width is Eg. Peaks thatcan be seen in the optical spectrum correspond totransitions of electrons between two states.

3 For anamorphous material, a single peak is present in the op-tical spectrum which refers to the transition between thebonding state in the valence band and the anti-bond-ing state in the conduction band. For crystalline semiconductors dielectrics and metals,several peaks can be observed, indicating the transi-tions occurring between the critical-point in the valenceand conduction the interband region, metals, semiconductors anddielectrics have similar structure in their optical func-tions. Thus, the same physical processes are involvedfor these three types of properties depend on inter-band transitions ofelectrons that are related to photon absorption.

4 The equation of ( ) below derives from the quantumexpression of the absorption. Equation of ( )where: - ( ) is the total number of ways a photon, of energy is removed from the incident intensity in a unit vol-ume and a layer of infinitesimal thickness x; ( )varies with the type ( Amorphous , dielectric, metallic)of the material ( ) is the transition probability rate at which a pho-ton is absorbed from the incident intensity in the fre-quency range [ ; + d ].-I0 is the incident photon ( ) defined as:The equation of ( ) is deduced from first-order time-dependant perturbation theory for direct and finiteelectron transitions.

5 ( ) depends on the dipole posi-tion matrix element squared between initial final statesand also on the lifetime of the excited state through thefollowing equation. For an Amorphous material, theexpression of ( ) is given by:()() ()()1 ck2 IxII1lim00x = = = ()()() 4 34222202 + = EE x I ec **hhhCONDUCTION BANDVALENCE BANDANTIBONDINGBONDINGEFig. 1 Orbital states of the energiesof the energies E and E * (Ref. 2) (E)E E E = E Egwhere:- e is the electron charge- c is the light speed in vacuum- is the Planck constant-E=h is the photon energy-E is the energy of the initial state-E * is the energy of the final excited state-In the case of crystalline semiconductors , dielectricsand metals, ( ) is the sum over the number of typesof :- the sum over N refers to the sum over the number ofobserved peaks associated with the ( ) defined as: ( ) is proportional to the number of possible transi-tions for a given photon of energy to be removed.

6 ( ) depends on the product of the number of occu-pied electron states in the valence band times the num-ber of unoccupied electron states in the conductionband. The densities of states in the valence and con-duction bands are assumed to have a parabolic shape. Therefore, ( ) is written aswhere:--- the band gap energy is For Amorphous and crystalline materials the final ex-pression of a( ) is For metals Eg=0, m( ) has the following expression:Equation of k( )Knowing the equations of ( ), ( ) and ( ) it isthen possible to determine k( ):By neglecting all the second order processes and re-placing all a( ) and a( ) by their respective expres-sions the corresponding extinction coefficient is derivedthrough these equations.

7 The final equation for the extinction coefficient ofamorphous materials is written as:withThe final expression for the extinction coefficient ofcrystalline materials is written this way:his the dipole matrix element squared be-tween the bonding and anti-bonding states. x r ()() ()[]{}() 4 kEkE kxkI ec3 4 2j2jcritvcritcjN1j2vcritccrit02 + = =hhrrKrrhj is the dipole matrix element squaredbetween the ith critical-point state in the valenceband and the ith critical-point state in the ()( )()( ) + E E dE vcvvvhis the density of states inthe valence band.

8 ()()() 2/1 EEcstEtopvv = is the density ofstates in the conduction band.()()() 2/1bottomccEEcstE = (). topvbottomcgEEE =()()()() 2gcaE cst = =h ()()() 2 = hcstm()()()()( ) 1220 ==Icck()()() EEfor;0 EEfor;gg22 >+ =CEBEEEAEkg()()() 422222 + = = = EECEEB x cstA h()()() EEfor0 EEforg1g22 >+ = =;;CEBEEEAEkNjjjgjTN13where Aj, Bj and Cj are given by: The final expression for the extinction coefficient ofmetals is written as:where Aj, Bj and Cj are given by: Crystalline semiconductors and metals may have sever-al peaks present in the structure of the extinction coeffi-cient which reveals the presence of several number of oscillators determines the name of thedispersion formula as shown below: Equation of n( )Then Kramers-Kronig (K-K) relation is used to deter-mine the refractive index through a Hilbert transform ofk(E).

9 Where P is the Cauchy principal value containing theresiduals of the integrand at poles located on the lowerhalf of the complex plane and along the real derivation of K-K yields the final expression for therefractive index:In the case of an Amorphous material:where In the case of a crystalline semiconductors and metals:where:FormulaNumber of oscillatorsNumber of parametersAmorphousN=15 Double AmorphousN=28 Triple AmorphousN=311 Quatre AmorphousN=414() ()[]() ()[]() 422222 + = = =jjcritvcritcjjcritvcritcjjvcritccritjj kEkECkEkEBkxkcst Ahrrrrrrr()() EEfor;0 EEfor.

10 G1g22 >+ = =NjjjjCEBEEAEk() ()[]() ()[]() 422222 + = = =jjcritvcritcjjcritvcritcjjvcritccritjj kEkECkEkEBkxkcst Ahrrrrrrr() ()() ()() 1 EdEEkEkPnEn = + ()() 200 CEBECEB En+ + += ()() 421222220220 = + = + + =BCQCEBCEQACCEEBBQAB gggg()() 1200 = + + +=NjjjjjCEBECEB En()() 421222220220 = += + + =jjjjgjjgjjjjgjgjjjjBCQCEBCEQACCEBEBQABT N13 The parameters of the equationParameter describing the refractive indexThe term ( ) is an additional parameter correspond-ing to the high-frequency dielectric constant.