Friedel oscillation

A first step towards a systematic improvement over DFT in a local region is the method of Aberenkov et al [189]. who calculated a correlated wavefiinction embedded in a DFT host. However, this is achieved using an analytic embedding potential fiinction fitted to DFT results on an indented crystal. One must be cautious using a bare indented crystal to represent the surroundings, since the density at the surface of the indented crystal will have inappropriate Friedel oscillations inside and decay behaviour at the indented surface not present in the real crystal. 

If V is localized, say, near the origin, then for locations far from the origin, this behaves like j 2kFr)/r2, which means as cos(2kFr)/ r3. These damped oscillations of frequency 2kF are the Friedel oscillations, which always arise when an electron gas is perturbed the frequency of oscillation comes from the kink in the dielectric function at 2kF. We see the Friedel oscillations (in planar rather than in spherical geometry) for the electron gas at a hard wall [Eq. (12) et seq.] and for the electron density at the surface of a metal. 

One knows, however, that the simple density-functional theories cannot produce an oscillatory density profile. The energy obtained by Schmickler and Henderson55 is, of course, lower than that of Smith54 because of the extra parameters, but the oscillations in the profile found are smaller than the true Friedel oscillations. Further, the density-functional theories often give seriously inexact results. The problem is in the incorrect treatment of the electronic kinetic energy, which is, of course, a major contributor to the total electronic energy. The electronic kinetic energy is not a simple functional of the electron density like e(n) + c Vn 2/n, but a... 

Table II clearly indicates that none of the previously mentioned OF-KEDF s has the eorreet LR behavior at the FEG limit. Even more interestingly, the TF funetional is supposed to be exact at the FEG limit, but its LR funetion has no momentum dependence. At first glance, one would think that there is some ineonsistency involved. In fact, there is no confliet beeause the TF functional is only the zeroth-order perturbation result, while the Lindhard function is the first-order result. A similar paradox exists for the asymptotic Friedel oscillations in Eq. (87).

The physical origin of these asymptotic Friedel oscillations of wave vector, 2/cf, can be traced back to eqn (6.35) for the response function, x0(q). We see from the numerator that there are only contributions to the sum for the states, k, that are occupied and the states + q that are unoccupied, or vice versa. This is to be expected considering Pauli s exclusion principle in that an electron in state, k, can only scatter into state, + q, if it is empty. Moreover, we see from the denominator in eqn (6.35) that the individual contributions will be largest for the case of scattering between states that are very close to the Fermi surface, since then k2 — (k + q)2 0. We deduce from Fig. 6.5 that the maximum number of such scattering events will occur... 

Thus, whereas the asymptotic Friedel oscillations in eqn (6.84) have their phase fixed with respect to the underlying lattice for a given valence Z, the oscillations of the long-range potential are electron density or atomic volume sensitive through the phase shift a3 which from eqs (6.97) and (6.99) is given by... 

Fig. 1. Density profiles averaged over the period of Friedel oscillations for a potential with Ui < U2- The averaged densities show drops at the impurity positions. The amplitudes of the density drops depend on the direction of the incident wave.

We do not take Friedel oscillations into account since they do not change our estimate qualitatively [15]. 

Stabilization of Superlattices by Friedel Oscillations in Surface States. 249... 

An impurity atom in a solid induces a variation in the potential acting on the host conduction electrons, which they screen by oscillations in their density. Friedel introduced such oscillations with wave vector 2kp to calculate the conductivity of dilute metallic alloys [10]. In addition to the pronounced effect on the relaxation time of conduction electrons, Friedel oscillations may also be a source of mutual interactions between impurity atoms through the fact that the binding energy of one such atom in the solid depends on the electron density into which it is embedded, and this quantity oscillates around another impurity atom. Lau and Kohn predicted such interactions to depend on distance as cos(2A pr)/r5 [11]. We note that for isotropic Fermi surfaces there is a single kp-value, whereas in the general case one has to insert the Fermi vector pointing into the direction of the interaction [12,13]. The electronic interactions are oscillatory, and their 1 /r5-decay is steeper than the monotonic 1 /r3-decay of elastic interactions [14]. Therefore elastic interactions between bulk impurities dominate the electronic ones from relatively short distances on. 

The required 2D nearly free electron gas is realized in Shockley type surface states of close-packed surfaces of noble metals. These states are located in narrow band gaps in the center of the first Brillouin zone of the (lll)-projected bulk band structure. The fact that their occupied bands are entirely in bulk band gaps separates the electrons in the 2D surface state from those in the underlying bulk. Only at structural defects, such as steps or adsorbates, is there an overlap of the wave functions, opening a finite transmission between the 2D and the 3D system. The fact that the surface state band is narrow implies extremely small Fermi wave vectors and consequently the Friedel oscillations of the surface state have a significantly larger wave length than those of bulk states. 

Figure 1(c) shows Friedel oscillations around Cu atoms adsorbed onto a Cu(lll) surface, which equally has a surface state (kp = 0.21 A [21]). The STM image is taken out of a sequence of images recorded at 13.5 K where Cu adatoms readily diffuse (for videos see the author s website under gallery). Despite the fact that the atoms quite often come close to each other, they do not form islands but remain isolated during the observation time of several hours. This is remarkable for a metallic system and can only be reconciled by a significant short-range repulsion. For the present... 

In Fig. 4 we plot the correlation-kinetic field z[ (z) and observe that it too is concentrated about the surface. It is long-ranged in the vacuum, decaying asymptotically as a frO/z2. In the metal, it exhibits the requisite Bardeen-Friedel oscillations. c The field z[1)(z), however, is an order of magnitude smaller than the Pauli component field (z). 

In Fig. 5 the correlation-kinetic potential component Wt (z) is plotted. For these densities, the potential is entirely positive, possesses the correct asymptotic structure of Eq. (45) in the vacuum, and exhibits the Bardeen-Friedel oscillations. Once again, thepotential w[ z) is an order of magnitude smaller than the Pauli component Wx (z). For higher density metals (rs < 2), the correlation-kinetic contribution to vx(z) will be less significant. It will vanish entirely for the very slowly varying density case for which33,34 vx(z) = Wx (z). 

Friedel oscillations — Oscillations of the electronic density caused by a disturbance such as a surface or an excess charge. At surfaces, they decay asymptotically with 1/z2, where z is the distance from the surface. Within electrochemistry they play a role in - double-layer theories that represent the metal as -+ jellium. 

A treatment of transport properties in terms of this surface is no more complicated in principle than that in the polyvalent metals, but there is not the simple free-clectron extended-zone scheme that made that case tractable. Friedel oscillations arise from the discontinuity in state occupation at each of these surfaces, just as they did from the Fermi sphere. When in fact there arc rather flat surfaces, as on the octahedra in Fig. 20-6, these oscillations become quite strong and directional. A related effect can occur when two rather flat surfaces are parallel, as in the electron and hole octahedra, in which the system spontaneously develops an oscillatory spin density with a wave number determined by the difference in wave number between the two surfaces, the vector q indicated in Fig. 20-5. This generally accepted explanation of the antiferromagnetism of chromium, based upon nesting of the Fermi surfaces, was first proposed by Lomer (1962). 

---