Tauc–Lorentz model

The general form of the model is given by

:$\backslash varepsilon(E)\; =\; \backslash varepsilon\_\{\backslash infty\}\; +\; \backslash chi^\{TL\}(E)$

where

• $\backslash varepsilon$ is the relative permittivity,
• $E$ is the photon energy (related to the angular frequency by $E=\backslash hbar\backslash omega$),
• $\backslash varepsilon\_\{\backslash infty\}$ is the value of the relative permittivity at infinite energy,
• $\backslash chi^\{TL\}$ is related to the electric susceptibility.

The imaginary component of $\backslash chi^\{TL\}(E)$ is formed as the product of the imaginary component of the Lorentz oscillator model and a model developed by Jan Tauc for the imaginary component of the relative permittivity near the bandgap of a material. The real component of $\backslash chi^\{TL\}(E)$ is obtained via the Kramers-Kronig transform of its imaginary component. Mathematically, they are given by

:

: $\backslash Re\backslash left(\; \backslash chi^\{TL\}(E)\; \backslash right)\; =\; \backslash frac\{2\}\{\backslash pi\}\; \backslash int\_\{E\_\{g\}\}^\{\backslash infty\}\; \backslash frac\{\backslash xi\; \backslash Im\backslash left(\; \backslash chi^\{TL\}(\backslash xi)\; \backslash right)\}\{\backslash xi^\{2\}\; -\; E^\{2\}\}\; d\backslash xi$

where

• $A$ is a fitting parameter related to the strength of the Lorentzian oscillator,
• $C$ is a fitting parameter related to the broadening of the Lorentzian oscillator,
• $E\_\{0\}$ is a fitting parameter related to the resonant frequency of the Lorentzian oscillator,
• $E\_\{g\}$ is a fitting parameter related to the bandgap of the material.

Computing the Kramers-Kronig transform,

: $\backslash Re\backslash left(\; \backslash chi^\{TL\}(E)\; \backslash right)\; \backslash ,\backslash !$

$=\; \backslash frac\{A\; C\}\{\backslash pi\; \backslash zeta^\{4\}\}\; \backslash frac\{a\_\{\backslash mathrm\{ln\}\}\}\{2\; \backslash alpha\; E\_\{0\}\}\; \backslash ln\{\backslash left(\; \backslash frac\{E\_\{0\}^\{2\}\; +\; E\_\{g\}^\{2\}$

+ \alpha E_{g}}{E_{0}^{2} + E_{g}^{2} - \alpha E_{g}} \right)} \,\!

$-\; \backslash frac\{A\}\{\backslash pi\; \backslash zeta^\{4\}\}\; \backslash frac\{a\_\{\backslash mathrm\{atan\}\}\}\{E\_\{0\}\}\; \backslash left[\; \backslash pi\; -\; \backslash arctan\{\backslash left(\; \backslash frac\{\backslash alpha\; +\; 2\; E\_\{g\}\}\{C\}\; \backslash right)\}\; +\; \backslash arctan\{\backslash left(\; \backslash frac\{\backslash alpha\; -\; 2\; E\_\{g\}\}\{C\}\; \backslash right)\}\; \backslash right]\; \backslash ,\backslash !$

$+\; 2\; \backslash frac\{A\; E\_\{0\}\}\{\backslash pi\; \backslash zeta^\{4\}\; \backslash alpha\}\; E\_\{g\}\; \backslash left(\; E^\{2\}\; -\; \backslash gamma^\{2\}\; \backslash right)\; \backslash left[\; \backslash pi\; +\; 2\; \backslash arctan\{\backslash left(\; 2\; \backslash frac\{\backslash gamma^\{2\}\; -\; E\_\{g\}^\{2\}\}\{\backslash alpha\; C\}\; \backslash right)\}\; \backslash right]\; \backslash ,\backslash !$

$-\; \backslash frac\{A\; E\_\{0\}\; C\}\{\backslash pi\; \backslash zeta^\{4\}\}\; \backslash frac\{E^\{2\}\; +\; E\_\{g\}^\{2\}\}\{E\}\; \backslash ln\{\backslash left(\; \backslash frac\{\backslash left|\; E\; -\; E\_\{g\}\; \backslash right|\}\{E+E\_\{g\}\}\; \backslash right)\}\; \backslash ,\backslash !$

$+\; 2\; \backslash frac\{A\; E\_\{0\}\; C\}\{\backslash pi\; \backslash zeta^\{4\}\}\; E\_\{g\}\; \backslash ln\{\backslash left[\; \backslash frac\{\backslash left|\; E\; -\; E\_\{g\}\backslash right|\; \backslash left(\; E\; +\; E\_\{g\}\; \backslash right)\}\{\backslash sqrt\{\backslash left(\; E\_\{0\}^\{2\}\; -\; E\_\{g\}^\{2\}\; \backslash right)^\{2\}\; +\; E\_\{g\}^\{2\}\; C^\{2\}\; \}\}\; \backslash right]\}$

where

• $a\_\{\backslash mathrm\{ln\}\}\; =\; \backslash left(\; E\_\{g\}^\{2\}\; -\; E\_\{0\}^\{2\}\; \backslash right)\; E^\{2\}\; +\; E\_\{g\}^\{2\}\; C^\{2\}\; -\; E\_\{0\}^\{2\}\; \backslash left(\; E\_\{0\}^\{2\}\; +\; 3\; E\_\{g\}^\{2\}\; \backslash right)$,
• $a\_\{\backslash mathrm\{atan\}\}\; =\; \backslash left(\; E^\{2\}\; -\; E\_\{0\}^\{2\}\; \backslash right)\backslash left(\; E\_\{0\}^\{2\}\; +\; E\_\{g\}^\{2\}\; \backslash right)\; +\; E\_\{g\}^\{2\}\; C^\{2\}$,
• $\backslash alpha\; =\; \backslash sqrt\{4\; E\_\{0\}^\{2\}\; -\; C^\{2\}\}$,
• $\backslash gamma\; =\; \backslash sqrt\{E\_\{0\}^\{2\}\; -\; C^\{2\}/2\}$,
• $\backslash zeta^\{4\}\; =\; \backslash left(\; E^\{2\}\; -\; \backslash gamma^\{2\}\; \backslash right)^\{2\}\; +\; \backslash frac\{\backslash alpha^\{2\}\; C^\{2\}\}\{4\}$.

• Cauchy equation
• Sellmeier equation
• Lorentz oscillator model
• Forouhi–Bloomer model
• Brendel–Bormann oscillator model

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{{DEFAULTSORT:Tauc-Lorentz model}}

Category:Condensed matter physics

Category:Electric and magnetic fields in matter

Category:Optics