Why n-type has many conduction electrons

In an n-type semiconductor the conduction electrons do not come from the valence band. They come from donor atoms, whose energy levels lie just below the conduction band, so very little energy is needed to free them.

Explanation

1. A pentavalent impurity (P, As, Sb) replaces a Si atom. Four of its five valence electrons form covalent bonds; the fifth electron is loosely bound to the donor ion, like the electron of a hydrogen atom placed in a medium of dielectric constant $\varepsilon_r$.
2. Its binding (ionisation) energy is very small because it is reduced by $\varepsilon_r^2$ and by the effective mass:

$$E_b = 13.6\,\frac{m_e^*/m_e}{\varepsilon_r^2}\ \text{eV} \approx 0.03\text{–}0.05\ \text{eV}\ (\text{Si})$$

3. On the band diagram this is a donor level $E_d$ only about 0.05 eV below $E_c$ (e.g. P: 0.045 eV, As: 0.049 eV in Si).

 Ec ---------------------------
      ^   ^   ^    ~0.05 eV (easy)
 Ed  -*- -*- -*-   donor level
 
      band gap 1.1 eV (hard)
 
 Ev ---------------------------

4. Average thermal energy at room temperature is $kT \approx 0.026$ eV. This is far too small to excite many electrons across the 1.1 eV gap (the chance goes as $e^{-E_g/2kT} \approx e^{-21}$), but it is comparable to $E_c - E_d$. With many lattice vibrations, almost every donor gets enough energy ($\sim e^{-2}$ chance per attempt, attempted very frequently) and is ionised.
5. So, at 300 K, essentially all donors give their electron to the conduction band: $n \approx N_d$, e.g. $10^{16}$ cm⁻³, compared with only $1.45\times10^{10}$ cm⁻³ produced by band-to-band excitation.
6. Ionising a donor leaves a fixed positive ion, not a mobile hole, so electrons greatly outnumber holes.

Thus the large electron population in the conduction band of an n-type semiconductor is due to the small ionisation energy of donor levels, not to excitation across the band gap.