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In solid-state physics, the free electron model is a quantum mechanical model for the behaviour of charge carriers in a metallic solid. It was developed in 1927, [1] principally by Arnold Sommerfeld, who combined the classical Drude model with quantum mechanical Fermi–Dirac statistics and hence it is also known as the Drude–Sommerfeld model.
Under the free electron model, the electrons in a metal can be considered to form a uniform Fermi gas. The number density N / V {\displaystyle N/V} of conduction electrons in metals ranges between approximately 10 28 and 10 29 electrons per m 3 , which is also the typical density of atoms in ordinary solid matter.
Friedel oscillations of the electron density in 1D electron gas occupying the half-space >.Here, = /, and is the Fermi wave vector. As a simple model, consider one-dimensional electron gas in a half-space >.
Luttinger liquid theory describes low energy excitations in a 1D electron gas as bosons. Starting with the free electron Hamiltonian: = † is separated into left and right moving electrons and undergoes linearization with the approximation () over the range :
Jellium, also known as the uniform electron gas (UEG) or homogeneous electron gas (HEG), is a quantum mechanical model of interacting electrons in a solid where the positive charges (i.e. atomic nuclei) are assumed to be uniformly distributed in space; the electron density is a uniform quantity as well in space.
When talking about solid materials, the discussion is mainly around crystals – periodic lattices. Here we will discuss a 1D lattice of positive ions. Assuming the spacing between two ions is a, the potential in the lattice will look something like this: The mathematical representation of the potential is a periodic function with a period a.
In this case, the carrier density (in this context, also called the free electron density) can be estimated by: [5] n = N A Z ρ m m a {\displaystyle n={\frac {N_{\text{A}}Z\rho _{m}}{m_{a}}}} Where N A {\displaystyle N_{\text{A}}} is the Avogadro constant , Z is the number of valence electrons , ρ m {\displaystyle \rho _{m}} is the density of ...
Although the Drude model was fairly successful in describing the electron motion within metals, it has some erroneous aspects: it predicts the Hall coefficient with the wrong sign compared to experimental measurements, the assumed additional electronic heat capacity to the lattice heat capacity, namely per electron at elevated temperatures, is also inconsistent with experimental values, since ...