Free electron concentration range between semiconductors and metals

[cryptist]

A structure with free electron density around 10^26 m^-3 is considered as a highly doped semiconductor or a metal?

Or in other words, what is the lowest possible free electron concentration for a metal and what is the highest possible free electron concentration for a doped semiconductor?

 

[Andrea M.]

The electron densities in a metal are typically of order $10^{28}/m^3$.
I'm not an expert but i think that the distinction is more profound: in the ground state of a metal at least one band is partially filled; in the ground state of an insulator all bands are either completely filled or completely empty. The insulators can be characterized by the energy gap $E_g$ between the top of the highest filled band and the bottom of the lowest filled band. A solid with an energy gap will be non conducing at $T=0$ but when the temperature is not zero there is a non vanishing probability that some electron can be thermally excited across the energy gap into the lowest unoccupied bands (conduction bands), leaving an unoccupied level in the highest occupied bands (valence bands). The thermally exited electron and the hole are capable of conducing.
The probability of such excitations depend on the size of the gap and is roughly of order $e^{-E_g/2k_bT}$. Solids that are insulator at $T=0$, but whose energy gap are of such size that thermal excitation can lead to observable conductivity at temperatures below the melting point are called semiconductors.

 

[cryptist]

Thank you for the detailed answer but I know the band gap theory. (Perhaps I've got misunderstood in this question, since I superficially got stamped as "starter")

I can calculate electron concentration for undoped semiconductors, and theoretically it is possible to obtain ANY electron concentration when you keep increase the temperature in the following formula:
$$n_i=2[\frac{2\pi kT}{h^2}]^{3/2}(m_n m_p)^{3/4}exp(\frac{-E_g}{2kT})$$

However, in reality there is no semiconductor with band gap of 10^-22 J, so actually you cannot obtain any carrier concentration.

I can also calculate free electron density in a metal with this formula:
$$\frac{\pi}{3}(\frac{8m_e E_F}{h^2})^{3/2}$$

By adding one condition (assuming low temperatures (T around 10K)) what I'm asking are these:

- Is there any metal with free electron density around 10^26 m^-3?
- Is there any metal with Fermi energy say around 0.05 ev?
- Is there any doped semiconductor with free electron density around 10^26 m^-3?

I'll be glad if someone can answer any of these three questions. (No need for answering them all, just one of them is enough)

Thank you in advance.