Friedel oscillations

Friedel oscillations arise from localized perturbations in a metallic or semiconductor system caused by a defect in the Fermi gas or Fermi liquid. Friedel oscillations are a quantum mechanical analog to electric charge screening of charged species in a pool of ions. Whereas electrical charge screening utilizes a point entity treatment to describe the make-up of the ion pool, Friedel oscillations describing fermions in a Fermi fluid or Fermi gas require a quasi-particle or a scattering treatment. Such oscillations depict a characteristic exponential decay in the fermionic density near the perturbation followed by an ongoing sinusoidal decay resembling sin(x)/x.

The electrons that move through a metal or semiconductor behave like free electrons of a Fermi gas with wave function

Electrons in a metal behave differently than particles in a normal gas because electrons are Fermions and they obey Fermi–Dirac statistics. This means that every k-state in the gas can only be occupied by two electrons with opposite spin. The occupied states fill a sphere in the band structure k-space up to a fixed energy level, the so-called Fermi level E_F. The radius of the sphere in k-space, k_F, is called the Fermi wave vector.

If there is a foreign atom embedded in the metal or semiconductor, a so-called impurity, the electrons that move freely through the solid are scattered by the deviating potential of the impurity. During the scattering process the initial state wave vector k_i of electron wave function is scattered to a final state wave vector k_f. Because the electron gas is a Fermi gas only electrons with energies near the Fermi level can participate in the scattering process because there must be empty final states for the scattered states to jump to. Electrons that are too far below the Fermi level E_F can't jump to unoccupied states. The states around the Fermi level that can be scattered occupy a limited range of k-values or wavelengths. So only electrons within a limited wavelength range near the Fermi level are scattered resulting in a density modulation around the impurity like

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