Electrical – Generation Rate and Recombination Coefficient for Intrinsic Semiconductors

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?