### Transcription of Tauc-Lorentz Dispersion Formula - HORIBA

1 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 r,TL of the dielectric function is derived from the expression of i using the Kramers-Kronig integration. Jellison and Modine developed this model (1996) using Then, it comes the following expression for i: the Tauc joint density of states and the **lorentz** oscillator.

2 The complex dielectric function is : ( ).. 2. r (E ) = r ( ) + P 2 i 2 d (5). TL = r, TL + i i, TL = r, TL + i ( i, T i, L ) (1). ~ Eg E. Here the imaginary part i,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 i,T function with **lorentz** one i,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 : 2. E Eg . i, T (E > E g ) = AT (2) A C aln E02 + E g2 + E g . E r,TL (E ) = + ln 2 . where : 2 4 E0 E0 + E g E g . 2. - AT is the Tauc coefficient A a 2 Eg +.

3 - E is the photon energy 4a tan arctan + K. - Eg is the optical band gap E0 C . The imaginary part of Tauc's dielectric function gives the 2 Eg . response of the material caused by inter-band mecha- + arctan . nisms only : thus i, T (E Eg) = 0. C . The imaginary part of the Lorentzian oscillator model is given by : +. (. 4 A E0 E g E 2 2 ) K. AL E0 C E 4. i , L (E ) = (3). (E 2. E 0). 2 2. + C2 E2 + 2 Eg 2 Eg . where : arctan + arctan . C C . - AL is the strength of the 2, TL(E) peak - C is the broadening term of the peak - E0 is the peak central energy . (. A E0 C E 2 + E g2 ) ln E E . +K. g By multiplying equation (2) by equation (3), Jellison sets 4 E E + Eg .. up a new expression for i,L(E): 2 A E0 C.. E E g (E + E g ) . 1 A E0 C (E E g ) 2 + E ln (6).. ( ) 2 . g for E > E g 4. 2 2. E0 E g + E g C.

4 2 2. i, TL (E ) = i,L i,T ( 2. ). = E E 2 E02 + C 2 E 2 (4) .. 0 for E E g where A=AT x AL . TN11. where: The parameters of the equations ( 2. g 2. 0 ). aln = E E E + E C E E + 3 E. 2 2. g 2 2. 0 ( 2. 0. 2. g ) The name of the different **Tauc-Lorentz** formulae present . ( 2 2. )(. aa tan = E E0 E0 + E g + E g C. 2 2 2 2. ) in the DeltaPsi2 software depending on the number of os- cillators are given below. It indicates the number of pa- rameters too. = 4 E0 C. 2 2. (7 ). Number of Number of = E0 C / 2. 2 2. **Formula** oscillators parameters 4. ( 2. ). = E 2 2 + 2 C 2 / 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. i 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.

5 N 1 Ai Ei Ci (E E g )2. for E > E g 1 parameter is linked to the real part of the dielectric ( 2. i = i =1 E E 2 Ei2 + Ci2 E 2 ) (8) function . 0 for E E g r ( ) = is the high frequency dielectric constant. This is an additional fitting parameter that prevents 1 from and r is re-written as the following sum : converging to zero for energies below the band gap. Generally, >1. N. 2.. ( ). r (E ) = r ( ) + P 2 i 2 d (9) At least 4 parameters describe the imaginary part of the i =1 E. E g dielectric function Deriving this integral yields the analytical expression of Ai (in eV) is related to the strength of the ith **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 . A C a Ei2 + E g2 + E g peak increases and the Full Width At Half Maximum r (E ) = r ( ) + i 4i ln ln 2 (FWHM) of that peak gets slightly larger.

6 Generally, i =1 . 2 Ei . Ei + E g 2. E g 10<Ai<200. Ci (in eV) is the broadening term ; it is a damping coef- Ai aa tan 2 Eg + ficient linked to FWHM of the ith peak of **absorption** . The arctan + K. 4 Ei Ci higher it is the larger that peak becomes and at the same time the smaller its amplitude is. Generally 2 Eg 0<Ci<10. + arctan . Eg (in eV) is the optical band gap energy. Ci . Ei (in eV) is the energy of maximum transition probability +. (. 4 Ai Ei E g E 2 2 .. ). arctan + 2 Eg .. or the energy position of the peak of **absorption** . The subscript i refers to the number (i=1, 2 or 3) of oscil- 4 Ci lators. Always, Eg<Ei. 2 Eg . + arctan Limitation of the model Ci . The **Tauc-Lorentz** model requires i to be zero for energies . (. Ai Ei Ci E 2 + E g2 ) ln E E.. g less than the band gap. Consequently, **Tauc-Lorentz** mod- E4.

7 E + Eg el does not take into account intra-band **absorption** : any defect or intra-band **absorption** increases i below the E E (E + E ) band gap and generates bad fits in that region. +. 2 Ai Ei Ci E g ln . g g (10 ).. ( ). 2 Valid spectral range 4 2. E 2. E 2. + E 2. C . 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 ? A E0 C S. R. (eV). The **Tauc-Lorentz** model works particularly well for amor- a-C - 5. phous materials exhibiting an **absorption** in the visible AlGaN - and/or FUV range (absorbing dielectrics, semiconduc- As2S3 - tors, polymers). AsSSe - Note that : DLC - 2. - The graph below shows the different contributions (in GaN 1,5 - 3. red dashed lines) to the imaginary part of the Tauc GeSbSe - 3.

8 **lorentz** dielectric function (in red bold line). InGaN 1,5 - 3. - The sign before a given parameter means that ei- Polymer - 4. ther the amplitude or the broadening of the peak is poly-Si - linked to that parameter. a-Si - **Tauc-Lorentz** function a-Si:B - a-Si:H - Si xOyNz - Ta-C - 5. ZnS - GeSbTe 0,450 2,024 158,795 1,688 2,397 0,65 - 3. Materials following the Double **Tauc-Lorentz** model M aterials Eg ? A1 E1. Blue Filte r Ox ide coa ting Polym e r Dielectric function of a-Si represented by a single **Tauc-Lorentz** SiC * oscillator Ta 2 O 5 Double **Tauc-Lorentz** function M a t e ria ls C1 A2 E2 C2 S . R . (e V ). B lu e - F i l te r O x id e 4 . 1 4 0 -6 . 2 6 9 * - 5. c o a ti n g P o lym e r - 5. S iC -5 . 1 2 9 * 5 1 . 2 6 4 - T a 2O 5 - This document is not contractually binding under any circumstances - Printed in France - 11/2006.

9 Materials following the Triple **Tauc-Lorentz** model Materials Eg ? A1 E1 C1 A2. a - Si * HfO 2 Dielectric function of a blue filter represented by a double Tauc- **lorentz** oscillator Materials E2 C2 A3 E3 C3 S. R. (eV). a - Si - 6. HfO 2. Applications to materials - Materials following the **Tauc-Lorentz** model References The asterisk * refers to parameters that are negative and 1) Jellison and Modine, Appl. Phys. Lett. 69 (3), 371-374 (1996). thus do not have any physical meaning but represent 2) Erratum, G. E. Jellison & F. A. Modine, Appl. Phys. Lett. 69 (14), 2137 (1996). good starting values to perform the fit on the material. 3) H. Chen, W. Z. Shen, Eur. Phys. J. B. 43, 503-507 (2005).