|
|
| Line 1: |
Line 1: |
| '''Free carrier absorption''' occurs when a material absorbs a photon and a carrier is excited from a filled state to an unoccupied state (in the same band). This is different from interband absorption in semiconductors because the excited electron is a conduction electron (i.e. it can move freely). In interband absorption the electron in question would be raised from a valence (nonconducting) band to a conducting one.
| | I’m Aurelio from Gouda studying Integrated International Studies. I did my schooling, secured 91% and hope to find someone with same interests in Stamp collecting.<br><br>Take a look at my blog post - [http://www.slideshare.net/davnec samsung galaxy note 4 manual t-mobile] |
| | |
| It is well known that the optical transition of electrons and [[Electron hole|hole]]s in the solid state is a useful clue to understand the physical properties of the material. However, the dynamics of the [[Charge carrier|carrier]] is affected by other carriers, not only by the periodic lattice potential. Moreover, the thermal fluctuation of each electron should be taken into account. Therefore a statistical approach is needed. To predict the optical transition in an appropriate precession, one should choose an approximation, called assumption of quasi-thermal distributions, of the electrons in the conduction band and of the holes in the valence band. In this case, the diagonal components of the [[density matrix]] become negligible after introducing thermal distribution function,
| |
| | |
| <math>\rho _{\lambda \lambda }^0 = \frac{1}{{e^{(\varepsilon _{\lambda ,k} - \mu )\beta } + 1}} = f_{\lambda ,k}</math>
| |
| | |
| This is the famous [[Fermi dirac statistics|Fermi-Dirac distribution]] for the distribution of electron energies. Thus, summing over possible l and k yields the total number of carriers N.
| |
| | |
| <math>N_\lambda = \sum\limits_\lambda {f_{\lambda ,k}}</math>
| |
| | |
| ==The optical susceptibility==
| |
| | |
| Using the above distribution function, the time evolution of density matrix does not have to be solved and the complexity is simplified.
| |
| | |
| <math> \rho _{cv}^{{\mathop{\rm int}} } (k,t) = \int {\frac{{d\omega }}{{2\pi }}\frac{{d_{cv} \varepsilon (\omega )e^{i(\varepsilon _{c,k} - \varepsilon _{v,k} - \omega )t} }}{{\hbar (\varepsilon _{c,k} - \varepsilon _{v,k} - \omega - i\gamma )}}(f_{v,k} - f_{c,k} )}</math>
| |
| | |
| The optical polarization is,
| |
| | |
| <math>\displaystyle P(t) = tr[\rho (t)d]</math>
| |
| | |
| With this relation and after adjusting the Fourier transformation, the optical susceptibility is
| |
| <math>\chi (\omega ) = - \sum\limits_k {\frac{{\left| {d_{cv} } \right|^{_2 } }}{{L^3 }}} (f_{v,k} - f_{c,k})\left( {\frac{1}{{\hbar (\varepsilon _{v,k} - \varepsilon _{c,k}+ \omega + i\gamma )}} - \frac{1}{{\hbar (\varepsilon _{c,k} - \varepsilon _{v,k} + \omega + i\gamma )}}} \right)</math>
| |
| | |
| ==Absorption coefficient==
| |
| | |
| The transition amplitude corresponds to the absorption of energy and the absorbed energy is proportional to the optical conductivity which is the imaginary part of the optical susceptibility after frequency is multiplied. Therefore, in order to obtain the absorption coefficient that is crucial quantity for investigation of electronic structure, we can use the optical susceptibility.
| |
|
| |
| <math> \alpha (\omega ) = \frac{{4\pi \omega }}{{n_b c}}\chi ''(\omega )</math> | |
| | |
| <math>{\rm{ }} = \frac{{4\pi \omega }}{{n_b c}}\sum\limits_k {\left| {d_{cv} } \right|^2 (f_{v,k} - f_{c,k} )\delta (\hbar (\varepsilon _{v,k} - \varepsilon _{c,k} + \omega ))}</math>
| |
| | |
| Considering the gap energy Eg, energy dispersion relation of free carrier proportional to the square of momentum and the relation of electron-hole distribution function, we can obtain the absorption coefficient with some kind of mathematical calculation. The final result is
| |
| <math>\alpha (\omega ) = \alpha _0^d \frac{{\hbar \omega }}{{E_0 }}\left( {\frac{{\hbar \omega - E_g - E_0^{(d)} }}{{E_0 }}} \right)^{(d - 2)/2} \sum\limits_k {\Theta (\hbar \omega - E_g - E_0^{(d)} )A(\omega )} </math>
| |
| | |
| The application of this result to semiconductor is important to understand the optical measurement data and the electronic properties. Some example shows the negative absorption coefficient that is fundamental presentation of [[Laser diode|semiconductor laser]].
| |
| | |
| ==References==
| |
| 1. H. Haug and S. W. Koch, "[http://books.google.co.kr/books?id=-UoG0Hx0w04C&dq=Quantum+Theory+of+the+Optical+and+Electronic+Properties+of+Semiconductors&printsec=frontcover&source=bn&hl=ko&ei=DoneSeT7Ipne7APqv6AJ&sa=X&oi=book_result&ct=result&resnum=5 Quantum Theory of the Optical and Electronic Properties of Semiconductors]
| |
| ", World Scientific (1994). sec.5.4 a
| |
| | |
| [[Category:Quantum mechanics]]
| |
I’m Aurelio from Gouda studying Integrated International Studies. I did my schooling, secured 91% and hope to find someone with same interests in Stamp collecting.
Take a look at my blog post - samsung galaxy note 4 manual t-mobile