Inverse dielectric matrix

This is the so-called random stopping power, which i.s also obtained as the average over impact parameters of the position-dependent stopping power of equation (44). For simple metals like Al the diagonal elements of the inverse dielectric matrix A g,g(Q) rather isotropic, in which case there is little... 

The quantity needed to calculate the total induced potential, as seen by a test charge, is the inverse dielectric matrix ... 

The inverse dielectric matrix by definition satisfies the relations ... 

The density response matrix x Eq- (1)) is related to the inverse dielectric matrix e through... 

Microscopic theory of lattice dynamics studies the response of electron-ion systems to displacements of nuclei the "direct" method generally means an approach in which the undistorted crystal and the crystal with displacements are treated from the very beginning as two distinct and unrelated systems. The "direct" treatment of phonons is an alternative to the classical perturbation approach, in which phonons are viewed as a small perturbation of the ground state to be treated by linear response theory. This method, based on the inverse dielectric matrix, is explained in this Volume in the lecture-notes by J. T. Devreese, R. Resta and A. Baldereschl. 

So far we have been dealing with various forms of the "response" to displacements of atoms. In Section 6 also certain electric fields have been studied, we benefited from the fact that displacing atoms in GaAs generates dipoles and therefore electric fields all the reasonings of Section 6 were, however, limited to polar crystals and e.g. determination of static dielectric constant e as in Section 6.2 would be impossible in Ge or other homopolar substances. From the point of view of studying dielectric properties, the main drawback of Section 6 was our dependence upon the various displacement patterns the electric fields could not be varied at will, as an independent variable. The present Section summarizes the most recent applications of the DF which tend to fill this blank and to open the way to "direct" treatment of dielectric properties of semiconductors, within the framework of the Density Functional. They are the treatment of constant macroscopic electric field imposed from outside (Section 8.1) and "direct" evaluation. of the individual elements of the inverse dielectric matrix s ("q + +"g ) (Section 8.2). 

All local variations have their origin in the microscopic inhomogeneity of solid they are traditionally studied by the linear response theory, which relates them to the off-diagonal elements of the inverse dielectric matrix this is explained in the articles by A. Baldereschi and R. Resta in the present volume. A quantitative compai sc of the local variations in charge density with the predictions using the RPA dielectric matrix showed excellent agreement between both approaches, both in magnitude and detailed shape. 

The linea response of a crystal to a small external perturbation V is given by the inverse dielectric matrix e"... 

Table 8.1 Some elements t e syTtime r zed inverse dielectric matrix g /1g 1 e (G,G ) for calculated by the present "direct" gthod and through the Adler-Wiser formula (RPA). 

A number of new results obtained with the "direct" ab initio methods include phonon frequencies, anharmonicities, predictions of displacement patterns, soft-mode phase transitions, effective charges, dielectric constant, local field variations, elements of inverse dielectric matrix, etc. they were all obtained from the same fundamental equations. The Density Functional method opens a way to unified description of ground state properties of solids static, dynamic and dielectric ones. Though all the partial results above are interesting by themselves, they are even more important by providing further tests of and support for the validity of the Density Functional theory. 

In the first approach, the dynamical matrix is expressed in terms of the inverse dielectric matrix describing the response of the valence electron-density to a periodic lattice perturbation. For a number of systems the linear-response approach is difficult, since the dielectric matrix must be calculated in terms of the electronic eigenfunctions and eigenvalues of the perfect crystal. 

We study the dielectric and energy loss properties of diamond via first-principles calculation of the (0,0)-element ( head element) of the frequency and wave-vector-dependent dielectric matrix eg.g CQ, The calculation uses all-electron Kohn-Sham states in the integral of the irreducihle polarizahility in the random phase approximation. We approximate the head element of the inverse matrix hy the inverse of the calculated head element, and integrate over frequencies and momenta to obtain the electronic energy loss of protons at low velocities. Numerical evaluation for diamond targets predicts that the band gap causes a strong nonlinear reduction of the electronic stopping power at ion velocities below 0.2 a.u. 

Local Field Effects (LFEs) are illustrated in Fig. 4. Im Eo o reproduces some values from Fig. 3. For comparison, Im[l/ATq ol includes an estimate of the local field obtained by calculating Kq qi as the inverse of a 9 X 9 dielectric matrix which contains G = 0 and the eight vectors of the closest shell in the bcc reciprocal lattice. The reduction of the values without LFE (lim eo,oL open symbols) compared to those with LFE (llm(l/ATqo)I, filled symbols) is of the order reported by Van Vechten and Martin [24] (without their dynamical correlations ). The different sign of the effect for frequencies above and below the peak has been noticed before [25]. The differences are even smaller for the energy loss function. Hence the energy loss reported in the next paragraph was calculated from so o(q, w) alone. 

In terms of the inverse of this matrix, the test-charge-test-charge dielectric matrix can then easily be calculated ... 

It should be emphasized that these derivations can be fully extended for time dependent perturbations. For that purpose it is sufficient to introduce a frequency w, apart from the wave vectors, in the densities and potentials, in the polarizability matrices and dielectric matrices, and in the exchange and correlation function which then becomes G[q,u). This latter point will be discussed in more detail in the next chapter. Note also that, strictly speaking, causality and the related dispersion relations only hold for the inverse dielectric matrices, since they describe the response. For the dielectric matrix itself, to the best of our knowledge no proof of the dispersion relations has been given. 

The inversion of the dielectric matrix e to second order is easily done by expanding the dielectric matrix and its inverse in the definition of the inverse. 

The evaluation of the inverse test-charge—test-charge dielectric matrix is then straightforward ... 

The most recent of the "dire pj " methods concerng the evaluation of the elements of the e dielectric matrix it is based on tl fact that the modification in the electronic charge dmslty An(r) is related to the modification in total potential V (r) by the polarizability matrix x(q+G,q+G ). The method does ng fe u j.r achieving self-consistency and provides the elements of e j (q+G,q+G ) within the RPA approximation. After inversion of jthe e the combination of the two direct methods (for e and e ) seems to offer a possibility of switching on and off, at will, the exchange-correlation its effects on various physical properties could then be studied in detail. Beside tl original work Ref. 77,... 

In semiconductor nanostructures, local field effects are mostly due to a surface charge polarization contribution, which is essentially a macroscopic classical term, that is very important in optical properties calculations. For instance, surface polarization is responsible for a strong optical anisotropy of elongated nanocrystals [36]. We can say that the one-particle contribution represents the optical properties of an isolated, stand-alone nanocrystal, the intrinsic properties due to the delocalized states and the quantum confinement effects. One-particle contributions do not take into account the influence of the external environment into the optical properties, such as the macroscopic polarization of the surface bonds. On the contrary, the methods beyond one-particle calculations, based on the inversion of the dielectric matrix or, as we will see below, a time-dependent tight-binding formulation, take into account more properly the influence of the external environment, in particular the charge transfer within the nanocrystals and at the surface. 

Figure 5.2 reports the absorption cross section of a small silicon nanocrystal. It is clear that the tight-binding approach with inclusion of local field effects (calculated by inversion of the dielectric matrix) compares very well to the formulation with a classical model of the surface polarization, based on the Clausius Mossotti equa-... 

In contrast to the SHG coefficient, the electro-optic coefficient r,y/t is symmetric in its first two indices for a lossless, not optically active, material. This follows from Eq. (11), which relates the electro-optical coefficient to the impermeability tensor Tiy (the matrix inverse of the dielectric permeability tensor y), a real and symmetric tensor under the stated conditions [3],... 

Figure 7.3 Dielectric constant ep versus (inverse) temperature for Stockmayer fluids at fixed fluid density p = 0.7. Curves aie labeled according to values of the matrix density.

Two common properties which can be calculated from the minimum-energy structure are the elastic and dielectric constants. The elastic constant matrix is used to relate the strains of a material to the internal forces, or stresses It is defined as the second derivative of the energy with respect to the strain, normalised by the cell volume. The inverse of the elastic... 

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