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Organic/Metal Interface Properties
Published in Sam-Shajing Sun, Larry R. Dalton, Introduction to Organic Electronic and Optoelectronic Materials and Devices, 2016
where z is the direction normal to the surface. While it is uniformly parallel to the surface, the electron density perpendicular to the surface is not. The electrons “spill out” into the vacuum with respect to the background positive charge, creating an electrostatic dipole at the surface, and the electron density oscillates along z in the near surface region. The charge density oscillations arise from the screening behavior of the electrons and are called Friedel oscillations. The modification of the electrons’ spatial distribution is to screen out the effects of the change in background charge density at the surface, as shown in Figure 20.10. The spatial variation of the electron distribution is a direct consequence of the presence of the sharp step in the background charge density.
Concluding Remarks and Perspectives
Published in Chiun-Yan Lin, Jhao-Ying Wu, Chih-Wei Chiu, Ming-Fa Lin, Coulomb Excitations and Decays in Graphene-Related Systems, 2019
Chiun-Yan Lin, Jhao-Ying Wu, Chih-Wei Chiu, Ming-Fa Lin
The Friedel oscillations can be studied with the use of static dielectric screening in the generalized tight-binding model. In a monolayer graphene, the static dielectric function ϵ(q, 0) is calculated by specifying the general results of the full charge screening due to the massless-Dirac quasiparticle, which is incorporated by the development of the modified RPA, summarized in Chapter 2. As shown in Figure 15.1, the induced charge distribution and the screened potential are presented for the static case of a charge impurity e at r = 0 in the doped graphene under various Fermi levels. The most outstanding feature of Friedel oscillations is the strong oscillation behavior in real space, a phenomenon being deduced from a discontinuity of the second derivative of the static dielectric function. At higher dopings, the screening charges become rather extended in r-space, with a 1 /r long-wavelength decay of oscillations at a distance from the impurity [497]. It should be noticed that there is no interband contribution from the long-wavelength behavior of polarization, since the corresponding polarization approaches zero at q ~ 0. This indicates that the nonoscillating part at small-r distances comes from the intraband polarization with a characteristic decaying length that can be evaluated within the Thomas-Fermi approximation. The previous work has indicated that as q increases the interband transitions greatly enhance the large-q screening when compared with the intraband transitions, and consequently, make the small-r screening effects rather effective from an impurity [498]. However, and very importantly, at a long distance from the impurity, the decaying oscillation behavior with a unique period given by π/2kF comes from the discontinuity at ħυFq = 2EF as a result of singularities in the second derivative of the static dielectric function. The aforementioned features are expected to have important influences on, for example, the ordering of impurities and the resistivity in graphene-related systems [498].
Pseudo pair potential between protons in dense hydrogen from first principles
Published in Molecular Physics, 2022
Robbie S. Robinson, Praveer Tiwari, Jeffrey M. McMahon
Continuing under the static assumption for ε, ε takes the form . Showing that ε is diagonal in -space is straightforward, which means with extending over all values. It is expected that Additionally, because the system is isotropic, it is independent of direction. Therefore In the second limit (of ε where ), and T = 0 K, becomes the full Lindhard function where is the scalar form of the Fermi wave vector. Using Equation (16) in Equation (10), becomes The screened model implemented by using the full Lindhard function contains singularities for . This means that the potential is very sensitive to the charge susceptibility (response ) where the Fermi surface develops sharp potential [43] areas due to the presence of impurities in a metal, or such as in this screened pair potential model, each neighbouring proton in the perfect crystal acting as a localised perturbation. The Fourier transform of Equations (10) and (17) gives the long-range (LR) form of the potential where and Z = 1 for the atomic number. Equation (18) exhibits behaviour known as Friedel oscillations, which account for the charge density around an impurity [44].
Density embedding with constrained chemical potential
Published in Molecular Physics, 2019
K. Niffenegger, Y. Oueis, J. Nafziger, A. Wasserman
The effect of the finite-distance interactions on the energy of the atom can be seen in Figure 5. The energy of the atom is defined as the sum of occupied orbitals minus the energy contribution from the reactivity potential: In Figure 5, the dashed line shows the energy at large separation, R=15. It consists of straight line segments [2–4]. At short distances (e.g. R=3, solid red in Figure 5) the line segments have a slight curvature. As shown in the inset plot of Figure 5, the curvature is more noticeable for in the range of 2 to 3, where values are more evenly spaced. This curvature is the consequence of the inter-fragment interactions, but it does not smoothen the cusps at integer occupations. The atomic fragment density at large values of R jumps abruptly when going through integer occupations, as can been seen in the top () and middle () panels of Figure 6. For each value of , increasing the Fermi energy of the system changes almost exclusively . As these changes occur, we observe an increase in the value of the metal density accompanied by a decrease in the period of density (Friedel) oscillations. The bottom panel of Figure 6 shows the representative behaviour of fragment densities at small separations. Densities corresponding to non-integer values of begin to appear. We note that, in this regime, the density of the metal fragment appears unchanged for different values of . The density response of the system to infinitesimal changes of μ is thus largely localised to either atom or metal fragments.