Free electron concentration range between semiconductors and metals

In summary, there is no metal with free electron density around 10^26 m^-3, but it is theoretically possible to obtain any electron concentration when you keep increasing 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})$$. Fermi energy around 0.05 ev is also required for a metal to be a doped semiconductor.
  • #1
cryptist
121
1
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?
 
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  • #2
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.
 
  • #3
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.
 

1. What is the difference between the free electron concentration in semiconductors and metals?

The main difference between the free electron concentration in semiconductors and metals is the number of available electrons. In semiconductors, the number of free electrons is relatively low, typically on the order of 10^16 to 10^18 electrons per cubic centimeter. In metals, the free electron concentration is much higher, ranging from 10^22 to 10^29 electrons per cubic centimeter.

2. How does the free electron concentration affect the electrical conductivity of a material?

The free electron concentration directly affects the electrical conductivity of a material. In metals, the high concentration of free electrons allows electric current to flow easily, resulting in high electrical conductivity. In semiconductors, the lower concentration of free electrons means that electric current can only flow under certain conditions, resulting in lower electrical conductivity compared to metals.

3. Can the free electron concentration in a material be controlled?

Yes, the free electron concentration in a material can be controlled through various methods such as doping or applying an external electric field. Doping involves introducing impurities into the material to increase or decrease the number of free electrons. Applying an external electric field can also alter the free electron concentration by influencing the movement of electrons within the material.

4. How does temperature affect the free electron concentration in a material?

Temperature has a significant impact on the free electron concentration in a material. In semiconductors, an increase in temperature can cause more electrons to break free from their bonds, resulting in an increase in free electron concentration. In metals, higher temperatures can cause the free electrons to move more rapidly, leading to a decrease in concentration.

5. What are the factors that determine the free electron concentration in a material?

The free electron concentration in a material is determined by factors such as the type of material, its atomic structure, and external influences such as temperature and electric fields. In semiconductors, the concentration can also be affected by the presence of impurities or defects in the crystal structure. In metals, the number of available valence electrons also plays a significant role in determining the free electron concentration.

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