Generation Rate and Recombination Coefficient for Intrinsic Semiconductors

Generation Rate

In an intrinsic semiconductor, the generation rate due to thermal energy is often given as $$G_{th} \propto \exp \left( - \frac{E_G}{k \cdot T} \right), \tag{1}\label{1}$$ with the band gap $E_G$, the temperature $T$ and the Boltzmann Constant $k$.

Question 1: Why is this formula shown with an proportional-to-sign "$\propto$" instead of the equal-sign "$=$"? Is there a more exact formula for $G_{th}$?

Recombination Coefficient

The recombination rate in a semiconductor is $$R = r(T) \cdot n \cdot p, \tag{2}$$ with the concentration of electrons $n$, the concentration of holes $p$ and the recombination coefficient $r(T)$.
For thermodynamic equilibrium, one can write $$G_{th} = R = r(T) \cdot n_0 \cdot p_0, \tag{3}$$ where $G_{th}$ is the generation rate from above, see equation \ref{1}. With the Maxwell-Boltzmann-Approximation, it is now possible to use the approximate intrinsic carrier concentration $$n_0 \cdot p_0 = n_i^2 \approx \left( \frac{4 \sqrt{2}}{h^3} \cdot \left(\pi \cdot k \cdot\sqrt{m_n \cdot m_p} \right)^{3/2} \cdot T^{3/2} \cdot \exp \left( - \frac{E_G}{2 \cdot k \cdot T} \right) \right)^2 \tag{4}$$ $$\hookrightarrow \quad n_0 \cdot p_0 = n_i^2 \approx n_{i,0}^2 \cdot T^3 \cdot \exp \left( - \frac{E_G}{k \cdot T} \right) \tag{5}$$ $h$ is Planck's Constant, and $m_n$ and $m_p$ are the effective masses of the electrons and holes, respectively.

Solving for the recombination coefficient, one obtains $$r(T) = \frac{G_{th}}{n_0 \cdot p_0} \tag{6}$$ $$\hookrightarrow \quad r(T) \propto \frac{\exp \left( - \frac{E_G}{k \cdot T} \right)}{n_{i,0}^2 \cdot T^3 \cdot \exp \left( - \frac{E_G}{k \cdot T} \right)} \tag{7}$$ $$\hookrightarrow \quad r(T) \propto \frac{1}{T^3} \tag{8}$$

Question 2: Does the recombination coefficient (theoretically and in the simplest physical model possible) drop with increasing temperature as suggested by the above formula?

• You have several problems with this question that should be cleaned up. Your second equation language has the recombination rate on both sides of the equation, capital R and r(T) should be more clearly defined. Gth is not defined ... etc. Also, can you put in equation #'s to assist in any subsequent write ups? – placeholder May 16 '16 at 13:21
• Question 1: There is some unspecified constant. Question 2: That may be right. The density of states (for a Fermi gas in 3-D) goes as T^3/2. (Sze.) The rate will go as (one over) the product of the electron and hole density. – George Herold May 16 '16 at 14:26