Estimate temperature for silicon (p-n junction)
Some terms are missing in \$n_i\$...
$$
n_i = \sqrt{N_cN_v}e^{-E_g/2kT} = 2\left (\frac{2\pi k}{h^2}\right )^{3/2}(m_e^*m_h^*)^{3/4}T^{3/2}e^{-E_g/2kT}
$$
Note that, in non-degenerate semiconductors, effective density of states in conduction band \$N_c\$ and that in valence band \$N_v\$
are expressed as (see Principle of Semiconductor Devices)
$$
Nc = 2\left(\frac{2\pi m_e^* kT}{h^2}\right)^{\frac{3}{2}}
$$
$$
Nv = 2\left(\frac{2\pi m_h^* kT}{h^2}\right)^{\frac{3}{2}}
$$
Now, using effective mass \$m_e=1.08m_0\$ and \$m_h=0.81m_0\$ given in the reference, and at 650 K, it will yield,
$$
n_i=3.29\times 10^{15}\ (cm^{-3})
$$
which is about \$10^{15}\ (cm^{-3})\$.