Carrier Concentration in Metals

Carrier concentration in metals is the number of electrons N(E) per unit volume in the energy interval E and E + dE.

Each electron energy level can accommodate two electrons as per Pauli’s exclusion principle.

We have,

\begin{aligned} & \mathrm{N}(\mathrm{E}) \mathrm{dE}=\mathrm{D}(\mathrm{E}) \mathrm{dE} \cdot \mathrm{F}(\mathrm{E}) \\ & \mathrm{N}(\mathrm{E}) \mathrm{dE}=2 \times \frac{\pi}{4 \mathrm{~h}^3}(8 \mathrm{~m})^{\frac{3}{2}} \mathrm{E}^{\frac{1}{2}} \mathrm{dEF}(\mathrm{E}) \end{aligned}

The actual number of electrons in dE,

\mathrm{N}(\mathrm{E}) \mathrm{dE}=\frac{\pi}{2 \mathrm{~h}^3}(8 \mathrm{~m})^{\frac{3}{2}} \mathrm{E}^{\frac{1}{2}} \mathrm{dE} \frac{1}{1+\exp \left(\frac{\mathrm{E}-\mathrm{E}_{\mathrm{F}}}{\mathrm{KT}}\right)}

Calculation of density of electrons and Fermi energy at 0K

At T = 0K,

F(E) = 1

\begin{aligned} \int_{0}^{\mathrm{N}} \mathrm{dN} & =\int_{0}^{\mathrm{E}_{F}} \frac{\pi}{2 \mathrm{~h}^{3}}(8 \mathrm{~m})^{\frac{3}{2}} \mathrm{E}^{\frac{1}{2}} \mathrm{dE} \\ \mathrm{N} & =\frac{\pi}{3 \mathrm{~h}^{3}}(8 \mathrm{~m})^{\frac{3}{2}}\left(\mathrm{E}_{\mathrm{F}}\right)^{\frac{3}{2}} \end{aligned}

Normally,

\mathrm{N}=\frac{\left(\begin{array}{c} \text { Number of free electrons per atom } \times \\ \text { Density } \times \text { Avagadro Number } \end{array}\right)}{\text { Atomic Weight }}

\mathrm{E}_{\mathrm{F}}=\left(\frac{3 \mathrm{Nh}^{3}}{\pi(8 \mathrm{~m})^{\frac{3}{2}}}\right)^{\frac{2}{3}}

\mathrm{E}_{\mathrm{F}}=\frac{\mathrm{h}^{2}}{8 \mathrm{~m}}\left(\frac{3 \mathrm{~N}}{\pi}\right)^{\frac{2}{3}}

Hence the Fermi energy of a metal depends only on the density of electrons of that metal.

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Hard and soft magnetic material