Method to Reveal and Investigate Almost 2D Fermi Surfaces in Layered Conductors: Universal Resistivity in a Parallel Magnetic Field

We suggested an original method to investigate the Fermi surfaces (FSs) in the quasi-two-dimensional conductors some time ago [A.G. Lebed and N.N. Bagmet, Phys. Rev. B 55, R8654 (1997)]. It was based on a consideration of a perpendicular conductivity in quasi-two-dimensional metals in parallel magnetic fields in the framework of the Boltzmann kinetic equation, where it was shown that the conductivity was independent on impurities. In this paper, we demonstrate that the above mentioned result is much more general than the kinetic equation and can be obtained even in a fully quantum mechanical case. We suggest to investigate this possible phenomenon in the quasi-two-dimensional organic, high-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{T}_{c}}$$\end{document}, and some others superconductors in a metallic phase to judge if the Fermi liquid picture is valid for them or not. If the Fermi liquid picture is valid, then study of the perpendicular resistivity in the rotated parallel magnetic field allows to extract important information about the two-dimensional Fermi surfaces.

One of the most important questions in area of the quasi-two-dimensional (Q2D) conductors and superconductors is the existence or not of the Fermi surfaces (FSs) in their non-superconducting phases. This question is still very controversial, where some experiments show the existence of the so-called pseudogap non-Fermi-liquid (n-FL) phase (for reviews, see [1,2]), whereas some others demonstrate the FL quantum oscillations of resistivity [3].
In [4,5], we suggested a new original method to investigate the FSs in the quasi-two-dimensional (Q2D) conductors in parallel magnetic fields. In the framework of the Boltzmann kinetic equation in the so-called -approximation [6,7], it was shown that in a clean limit perpendicular resistivity did not depend on impurities concentration and was a linear function of the parallel magnetic field, (1) in a broad region of the fields. On the other hand, it was also shown that, in the presence of inflection points on the 2D FS, there existed such in-plane directions of a magnetic field where the perpendicular resistivity changed into (2) where depends on impurities. These theoretical results were confirmed later in some publications (see, for example, [8]) and were experimentally observed [4,8,9]. The important point is that coefficient depends on some characteristics of a 2D cross-section of the Q2D electron FS, which can be measured in in-plane rotated magnetic fields. Nevertheless, the suggested method has not received still a broad application mainly due to the fact that a validity of the Boltzmann kinetic equation in metallic phases of organic and high-superconductors, in particular in the τ-approximation, is not generally justified.
The goal of our paper is a three-fold. First, we show that Eq. (1) has to be valid for perpendicular resistivity of a clean Q2D conductors also in a quantum case. We demonstrate that it survives even in the quantum picture, where electrons in a strong parallel magnetic field tunnel from one conducting layer to another. This indicates that the universal resistivity is a very general phenomenon. The second our goal is that, using the quantum approach, we show that in the case, where a 2D cross-section of the Q2D FS has some inflection points, there are such directions of magnetic fields, where goes to zero in Eq. (1), The third our goal is to suggest possible observations of the above discussed phenomena as the good tests of the existence of the 2D FSs, which is a central ques-

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tion for the majority of organic, high-, and some other layered superconductors in a metallic phase.
Below, we consider a layered superconductor in the so-called tight-binding model [6] with the following Q2D electron spectrum, which is an anisotropic one within the conducting plane: , is the integral of the overlapping of electron wavefunctions in a perpendicular to the conducting planes direction; is the Fermi energy; .] As to the parallel magnetic field, it is assumed to be inclined by angle β with respect to x axis, whereas alternative current (ac) electric field with small frequency is applied perpendicular to the conducting layer, (5) and (6) (see Fig. 1).
Let us rewrite Hamiltonian (4) in the absence of the fields in the form (7) which can be used before the tight binding procedure, performed in Eq. (4) for electron motion along z axis; m is free electron mass, is potential energy from atomic plane, located at the position along z axis. Let us introduce the magnetic field (5) into the Hamiltonian (7) by using the so-called Peierls substitution method [6,7]: As a result, we obtain: (9) Note that the Peierls substitution method is exact one for an in-plane isotropic case. In our case, since the characteristic energies of in-plane electron motion are of the order of Fermi energy, , then, according to the general theory [6,7], the Peierls procedure for considered here anisotropic case disregards only energies of the order of , with where is a perpendicular component of electron velocity to 2D cross-section of the Q2D FS (4), α is the polar angle counted from x axis (see Fig. 1). From equation (9), it follows that an arbitrary solution of the Hamiltonian can be written in the form (11) which corresponds to a free electron in-plane motion: (12) Let us expand the in-plane electron energy in Eq. (12) up to the first order of the magnetic field (5): Since we use the approximation , the in-plane electron motion is not affected by the magnetic field and, therefore, (14) where and are the corresponding components of in-plane electron velocity. Now, we x y cos .
x y x y   13) and (14) can be rewritten as (15) where the cyclotron frequency of electron motion along open trajectories in perpendicular to the conducting layers direction z, , is defined by the following expression: (16) [Note that the omitted quadratic term with respect to magnetic field, , in Eq. (13) is of the order of , .] The Schrödinger equation for the Hamiltonian, corresponding Eqs. (12) and (15), can be written as (17) where the electron wavefunction is given by Eq. (11).
Let us first consider the following one-dimensional problem: one layer, parallel to conducting plane, in the absence of a magnetic field. The corresponding Schrödinger equation is (18) where is the electron valence state wavefunction. In contrast to [10], we doesn't consider to be the so-called Dirac δ-function. Nevertheless, in the same way as in [10] we suggest that the potential energy satisfies the so-called tight binding approximation [6] [see also Eq. (4)], which is a typical case for Q2D compounds in perpendicular to the conducting plane direction. Then, we can represent many-body electron wavefunction (11) as Performing the tight binding procedure for Hamiltonian (17) and wavefunction (19) in the presence of magnetic field in the same way as in [10], we find the following infinite set of equations, where is the Bessel function of order. In this article, we use the quantum Kubo equation [12] for conductivity (22) where is the Fermi-Dirac distribution function, is a matrix element of the momentum operator along z axis. Since we know the electron wavefunctions and electron spectrum (21) in the magnetic field (5), we can calculate the matrix elements for the momentum operator, as it is done by us in [10]. As a result, the perpendicular conductivity (22) can be written as a contour integral over a 2D cross-section of the Q2D FS (4): where . Using the following mathematical formulas (24) and (25) where p.v. stands for the principle value of the corresponding integral, we can express the real part of conductivity in Eq. (23) as: Note that the main difference of our results from the existing (see [10]) is that we are interested in this paper in low frequencies [i.e., almost direct current (dc)] conductivity and resistivity, contrary to the so-called optical conductivity considered in [10]. Therefore, we can rewrite Eq. (26) in the form (27) As seen from Eq. (27), the dc conductivity is defined by zeros of the cyclotron frequency, of the Q2D FS (4). Near zeros of the cyclotron frequency (28) (i.e., when ), we can write in expression for dc conductivity (27) Now, let us recall that in the Q2D superconductors (see, for example, [4,5,8,9]), the following inequalities are valid: Taking account of them, we obtain for the perpendicular resistivity component the following linear magneto-resistance: (34) which justifies our main Eq. (1) and generalizes the kinetic equation results [4,5,8] to the considered in this paper quantum case.
Note that in the Letter we have used the Landau FL theory [6] and quantum mechanical description for electron motion in a Q2D conductor as well as quantum mechanical description of conductivitythe Kubo formula [12]. So, if the FL theory is valid in some Q2D conductor, experimentalists can conduct experiments in parallel magnetic fields and can discover unusual linear magnetoresistance (1) and (34). Then, they can rotate the field and extract angular dependence of the important 2D FS parameter (see Fig. 1) which allows to make some conclusions about the shape of the 2D FS. We stress that, if there are the inflection points on the 2D cross-section (29) of the Q2D FS (4), then for some directions of the magnetic field (5), which are parallel to the Fermi velocities at the inflection points, JETP LETTERS Vol. 118 No. 2 2023 LEBED author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.