Is charge carrier density an intrinsic property of a material and is thus constant?

Question

I was studying the equation $$I = nAvq$$ where $n$ = the charge carrier density, $A$ = cross-sectional area of the conductor, $v$ = mean drift velocity of the charge carriers, and $q$ = the charge on each charge carrier. I was confused about whether the charge carrier density is an intrinsic property of a material and has a constant value for a certain material? If not, then what factors affect it?

Answer 1

No, the carrier density will depend on the material, and for metals you might argue that it is a property of the material, but it will still depend on the details. If you look at it from a condensed matter physics point of view, it will depend on the details of the band structure which also depends on the arrangement and spacing of atoms (the crystal structure).

In a metal you end up with electrons in the conduction band and up to a point that will give you about the number of carriers per volume even if the metal isn't perfect, polycrystaline, or has impurities or even if it is alloyed, and in the drift velocity term in you equation will be related to the amount of scattering, meant time between collisions for the electrons (this relates to mobility of the electrons).

In a semiconductor, you still use your equation, but the carrier density will change tremendously on how you dope the material. If the material is very pure (intrinsic) with no dopants, the carrier density will depend strongly on the temperature as carriers get excited by temperature from the valence band to the conduction band. If you put dopants into the material, you can control the carrier density by the number of dopants. You will still have the behavior of the excitation from the valence band to the conduction band, but the dopant ionize at a much lower temperature and so one dopant atom gives one carrier. This allows you to control the carrier concentration and is really ultimately one of the reasons we can make diodes, transistors etc. in a controlable way.

If you like this kind of stuff I really recommend 'The modular series on solid state devices' volumes 1 and 2 are really well written and have good examples and usually there is one book per device type. if you want to get into how semiconductor devices work. I think the books are pretty cheap on e-bay only 4 or 5 dollars each.

so the short answer to our question is that the carrier density will depend on the band structure of your material which depends on the crystal structure and atom, but the number of carriers the conduction band will also depend on the details: temperature, number of defects, number of dopants are especially important in semiconductors.