On Screening In Some Low-Dimensional Crystals

On the basis of the proposed diagram technique in quantizing magnetic eld, two problems of interelecteren screening are investigated. In the rst one in the RPA framework the Fourier-transformation of Coulomb interactions is analysed. It is obtained to have an oscillating character with respect magnetic eld and band lling. In ultraquan-tum magnetic eld the polarization loop which is diagonal over Landau numbers formally coincides with that in the one-dimensional case. In the second problem in RPA the renormalization of the interelectron potential in a one-dimensional crystal with the narrow band is studied. It is shown, that at some conditions such potential is an oscillating sign-variable function on an interelectron distance.


Introduction
One-dimensional (1D) model is not only simple and convenient for calculations. There is a number of objects described by such a model: a class of anisotropic conductors whose structures consist of parallel linear chains, along which conduction electrons propagate essentially in one dimension (such as TTF-TCNQ, inorganic salts such as the \mixed-valence" platinocyanides).
Recently this model was widely used for the description of stage ordering phenomenon i.e. a periodic sequence of layers intercalated in the host crystal matrix 1]. In the frame of 1D model the average layer occupancies of the intercalant sites were analysed, ignoring the electron subsystem. Particularly, for better understanding of the electron subsystem in uence on, for instance, the stage ordering in a layer crystal the studying of screening e ects is necessary. However, actuality of such problems is much wider.
Some problems of screening are solved below i) in crystal placed into external magnetic eld ii) renormalization of the interelectron interaction potential.

Crystal in the external magnetic eld
Before studying of 1D problem let's consider 3D-crystal placed in the magnetic eld. Results of such problem, as it will be shown below, can be used in 1D-case. For better understanding of physical processes in solid state it is necessary to analyse many-body e ects. In spite of the essential improvements of such type analysis for an ideal crystal, they are more modest when external magnetic eld is applied. It means impossibility of the straight usage of many-body investigation methods, suited for the ideal crystal without eld, to the case for the ideal crystal in the eld. Really for the ideal crystal without eld diagram technique is usually based on the Fourier transformations of Green's functions taking into account the space homogeneity, that occurs in this case 2]. For the crystal placed in the eld, studied system becomes unhomogeneous because of vector potential dependence on coordinate, that signi cantly complicates this problem from the mathematical point of view.
2. Each wave line corresponds to the interaction potential with 3D momentum q, frequency ! and U(q). 3. For each top, where Green functions with quantum numbers n and n 0 and a wave line with momentum q come to, the factor corresponds to * n! p n 1 !n! exp ? 2  withq ? = (q x ; q y ), n = min(n; n 1 ). 4. Frequencies and z-components of momentum of internal lines must satisfy conservation laws in each top, and for diagrams of any order higher than the rst one the conservation law of 3D-momentum occurs too.
5. Diagram of m-th order has m integrations over independent momentum (3D -momentum and z-component), as well as summation on quantum numbers of internal lines and integration over independent frequencies are carried out.
6. Before the obtained expression one should write coe cient i m (?1) f , where f is a number of closed loops.
For instance, diagram of the rst order that describes exchange interaction corresponds to the expression (2.14) Using (2.14) and considering connection between V (n; n 0 ; q z ) and W nn 0 (q) (2.11) one can show that singularity of V (n; n 0 ; q z ) in the long wave approximation (i.e. at q z ! 0) disappears when polarization e ects are taken into account. By the way, the terms diagonal over Landau index play the main role here.
Two more conclusions, which come from the requirement =md n = 0, are: 1. Renormalized V (n; n 0 ; q z ) has an oscillating character at changing of the eld or band occupancy ( ). 2. In the ultraquantum case all the polarization e ects are determined only by electron state n = 0, when an electron moves in nitely along k z . That's why such behaviour can be considered formally as 1D case.

Polarization properties in the narrow band crystal
Using above results, let's consider polarization properties of 1D narrow band crystal with the dispersion law " (k) = a + (1 ? cos k) ; (3.1) where is the integral of the electron mixing neighbouring atoms and it equals to the half bandwidth; here the lattice constant along c-axis is chosen to be equal to unity. In layered crystal the system of stage ordered intercalant atoms can be described by just such a dispersion law, because these atoms are bound by weak van der Waals interaction. That's why in this case, at the electron system description, one can use tight-binding approximation, that gives (3.1).
Comparing (3.1) with (2.3), it can be seen, that a = ! c (n + 1=2) when magnetic eld is applied. Thus form for a in "(k) (3.1) describes law dispersion of the chain crystal with eld, coinciding with its axis or without it. Applying of such a nonparabolic dispersion law for the screen e ect study distinguishes this work from a numerous ones, where similar problems were solved for the electrons with a square or linear (at the Fermi level) dispersion law (see for instance 5]).
A polarization loop with electron dispersion law (3.1) has been obtained in 6,7] in extreme magnetic eld. After integration over frequency in (2.12) we'll have 6] Let us present <e in the static limit. It has a form <e (q; 0) <e (q) = ? 1 2 sin (q=2) ln tan (k 1 =2) tan (k 0 =2) ; (3.6) where k 0 , k 1 are lower and upper limits of integration. They are determined by Q. As it follows from mentioned above situation <e (q) = 0 at jQj > 1, i.e. for entirely full or empty band; <e (q) as function of q has two singularities: at q = 2k F and q = 2( ? k F ) (k F is Fermi's wave vector). The rst one (at q = 2k F ) coincides with the same for electrons with k 2 dispersion law and shows itself as Peierls transition in 1D lattice or Kohn's anomaly in the phonon spectrum which appears in a softening of the phonon mode 8].
In the given case such an additional e ect can occur at q = 2( ?k F ).
It should be noted that such an complemental point was overlooked in 7]. Behaviour of =m (q; !) is analysed in 9].
Let's research renormalization of interelectron interaction in the narrow band crystal. Let a bare potential of interelectron interaction has a form of the Lorentz-type curve V 0 (r) = D r 2 + D 2 ; (3.7) where r = jr i ?r j j, and D is its halfwidth. The choice of such a potential will not change the qualitative character of the conclusions about the screening. From another point of view, this potential doesn't have any singularities (as for instance Coulomb potential at r = 0) and thus one will have no doubts on received below conclusions, as the result of such singularities. Fourier transformation of (3.7) is V 0 (q) = q ( =2) exp (?qD) : (3.8) The screened potential in the RPA is 10] V (q) = V 0 (q) 1 + (q; ) V 0 (q) : (3.9) Considering the connection of real and imaginary parts of dielectric function with (q), we can write 10] " 1 (q) = 1 + <e (q) V 0 (q) ; (3.10) " 2 (q) = =m (q) V 0 (q) : (3.11) Using (3.9) screened potential V (r) = ZṼ (q) cos (qr) dq (3.12) has been found in the static case 11]. All energy values below are written in eV.
As it was mentioned above the value and character of (q) essentially depends on the value Q, i.e. on the degree of band lling ( g.1); with band width increasing, at xed , the area of q, where <e (q) and =m (q) di ers from zero, is signi cantly restricted ( g.1; curve 4). Thus, in the limit ! 1,Ṽ (r) ! V 0 (r). " 1 (q) and " 2 (q) sharp extrema ( g.2, curves 1,2) correspond to singularities in <e (q) at q = 2k F and q = 2( ? k F ).

Conclusions
Interaction between electrons in 1D crystal with a narrow nonparabolic band leads to the oscillating sign-variable character of potentialṼ (r) depending on the interelectron distance. Physically it means, that depending on a distance such an interaction has di erent nature -attractive or repulsive. SimilarṼ (r) dependence on r (at large r) takes place for the ion-ion interaction at the considering of the indirect exchange \ion-electron-ion"   on r at = 0:1 and 0:9 < Q < 1.