# Introduction to the Physics of Semiconductor Devices

## Defects and Doping - Charge Carrier Capture and Emission Rates

• An electronic defect state occupied by an electron can emit the electron into the conduction band or capture a hole from the valence band.
• An empty electronic state (which can also be considered to be occupied by a hole) can capture an electron from the conduction band or emit the hole into the valence.
• The emission processes can either be optical (capture of a photon) and/or thermally activated (capture of phonons). The emission is always a processes in which the charge carrier gains energy.
• During the capture of charge carriers energy from the charge carrier is emitted in form of either a photon or phonons or both.
• The time evolution of the probability to find an electronic defect state occupied by an electron is given by:
$$\frac{\mathsf{d}q}{\mathsf{d}t}= \frac{c_{\mathsf{n}}+e_{\mathsf{p}}^{\mathsf{o}}+e_{\mathsf{p}}^{\mathsf{th}}} {c_{\mathsf{n}}+e_{\mathsf{p}}^{\mathsf{o}}+e_{\mathsf{p}}^{\mathsf{th}}+ c_{\mathsf{p}}+e_{\mathsf{n}}^{\mathsf{o}}+e_{\mathsf{n}}^{\mathsf{th}}}$$

• Electronic defect states - charge carrier capture and emission rates. (Taken from M. Schmidt, PhD thesis (2012))

• The capture rates are proportional to the charge carrier concentration, the capture cross-section and the thermal velocity of the charge carriers:
$$c_{\mathsf{n,p}}=(n,p)\sigma_{\mathsf{n,p}}(T)v_{\mathsf{n,p}}(T)$$.
• The thermal emission rates are temperature-dependent:
$$e_{\mathsf{n,p}}^{\mathsf{th}}\propto T^2\sigma_{\mathsf{n,p}}(T) \exp\left(-\frac{E_{\mathsf{A}}}{kT}\right)$$,
where Ea is the activation enthalpy of the defect state.
• The optical emission rates are given by the photon flux of the incident monochromatic light and the cross-section for the transition:
$$e_{\mathsf{n,p}}^{\mathsf{o}}=\sigma_{\mathsf{n,p}}^{\mathsf{o}}(h\nu)\Phi(h\nu)$$,