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20 FREE ELECTRON IN METALS II - Coggle Diagram
20 FREE ELECTRON IN METALS II
Fermi–Dirac distribution function
According to quantum theory, at absolute zero of temperature, the free electrons occupy different energy levels continuously without any vacancy in between filled states
dropping the free electrons of a metal one by one into the potential well
first electron dropped would occupy the lowest available energy
next electron dropped also occupy the same energy level
the third electron dropped would occupy the energy level E1 (> E0) and so on because of Pauli’s exclusion principle
Fermi energy can also be defined as the highest energy possessed by an electron in the material at 0 K
At 0 K, the Fermi energy EF is represented as EF0
The Fermi function f(E) specifies how many of the existing states at the energy E will be filled with electrons
Fermi-Dirac distribution: Consider T = 0 K
Fermi-Dirac distribution: Consider T > 0 K
As T increases, the distribution rounds off slightly
Because of thermal excitation, states near and below EF lose population and states near and above EF gain population.
The Fermi energy EF also depends on temperature, but the dependence is weak in metals.
E = EF ,f(EF) = ½
kT (at 300 K) = 0.025eV, Eg(Si) = 1.1eV, so 3kT is very small in comparison.
Temperature dependence of Fermi-Dirac distribution
Quantum-based free-electron theory of metals
the outer-shell electrons are free to move through the metal but are trapped within a three-dimensional box formed by the metal surfaces
each electron is represented as a particle in a box
Particles in a box are restricted to quantized energy levels
One-dimensional region of space
if a particle is bouncing elastically back and forth along the x axis between two impenetrable walls separated by a distance L.
Because the walls are impenetrable, there is zero probability of finding the particle outside the box, so the wave function 𝜑(x) must be zero for x < 0 and x > L
Energy of particle
Three-dimensional region of space
the energies are quantized, and each allowed value of the energy is characterized by this set of three quantum numbers (one for each degree of freedom).
Density of states function
g(E) is called the density of states function
Number of electrons per unit volume
