Friedel oscillations

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Solution to the Poisson–Boltzmann equation of the "screening by charged rods" problem. Areas of negative density can be translated to regions with a net positive charge.

Screening of negatively charged particle in a pool of positive ions.

Friedel Oscillations arise from localized perturbations in a metallic or semiconductor system caused by a defect in the 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 treats the make-up of the ion pool as point entities, Friedel Oscillations describing fermions in a Fermi fluid require a quasi-particle 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.

“Hand-Waving” Qualitative Description

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Local density of states image taken by an scanning tunneling microscope of an epitaxially grown InAs pillar on a clean surface . The image on the left depicts the dynamics of low energy electrons

In the classic scenario of electric charge screening, a dampening in the electric field is observed in a mobile charge carrying fluid upon the presence of a charged object. Since electric charge screening considers the mobile charges in the fluid as point entities, the concentration of these charges with respect to distance away from the point decreases exponentially. This phenomenon is governed by Poisson–Boltzmann equation .
The quantum mechanical description of a perturbation in a fermionic fluid is modelled by the Tomanaga-Luttinger liquid. The fermions in the fluid that take part in the screening cannot be considered as a point entity but a wave-vector is required to describe them. Charge density away from the perturbation is not a continuum but fermions arrange themselves at discrete spaces away from the perturbation. This is the cause of the circular ripples around the impurity.

N.B. Where classically near the charged perturbation an overwhelming number of oppositely charged particles can be observed, in the quantum mechanical scenario of Friedel Oscillations periodic arrangements of oppositely charged fermions followed by spaces with same charged regions.

In the figure to the right, a 2-dimenisional Friedel Oscillations has been illustrated with an STM image of a clean semiconductor surface containing an InAs pillar with a net positive charge. The electron density around pillar in the center of the images can be visualized by the dark regions depicting high electronic density and light regions low density. As the image is taken on a surface, the regions of low electron density leave the atomic nuclei ‘exposed’ which result in a net positive charge. The figure also shows the change in the wavelength of the Friedel Oscillations as a function of electron energy, an increase in wavelength with decreasing electron energy.