ELECTRONIC PROPERTIES OF SOME BCC TRANSITION METALS

Abstract

A number of important physical properties, such as heat capacity, thermal conductivity, electrical conductivity, magnetic susceptibility and electrodynamics, of a metal can be understood from the electronic structure. In this thesis, an attempt has been made to study the electronic structure of some body-centered-cubic transition metals. For the last fifteen years the use and developments in linear methods for solving the band structure problem have totally erased the limitations that are faced in other techniques in use for such problems. Among these linear methods, linear augmented plane wave (LAPW) method and linear muffin tin orbital (LMTO) method are the most commonly used methods. In this work we use the LMTO formalism within the atomic sphere approximation (ASA) to perform self-consistent band structure calculations. The LMTO method combines the desirable features of the classical methods using fixed basis functions with those of the partial wave methods and has none of their drawbacks. Owing to small size of the basis sets LMTO is very well suited for self-consistent calculations. The variational principle for the one-electron Hamiltonian is used and the trial lunctions are taken to be linear combination of energy-independent muffin-tin orbitals (MTOs). This makes the secular equations linear in energy, i.e. they reduce to simple eigenvalue equations, from which all eigenvalues and eigen vectors of a given Bloch vector can be found by a single diagonalization. The Wigner Seitz cell is replaced by the muffin tin (MT) sphere of the same size to fill the volume of the first Brillouin zone. In the interstitial region i.e. outside the MT sphere, the tails of the MTO have a more restricted form; they have zero kinetic energy i.e. a solution of Laplace's equation. The Bloch sum of these tails is expanded in terms of structure constants. In contrast to the KKR structure constants, those of the LMTO are canonical in the sense that they do not depend on the energy and remain invariant under a uniform scaling of the crystal. The LMTO-ASA makes possible a complete separation (i) between the potential and energy dependency, expressed in terms of the logarithmic derivative functions and the dependence on the structure and wave vector, contained in the canonical structure constants. The LMTO method is best suited for closely packed structures and low energies and is traditionally used to perform density functional calculations. In this thesis we are interested in the FS of bcc transition metals, which can be calculated within the density functional formalism. The density functional theory (DFT) is based on the quantum statistical approach. The formal mathematical foundation of the DFT is provided by two theorems of Hohenberg and Kohn. In DFT, the density is utilized as a fundamental quantity and is supposed to contain all the relevant information about the ground state of a many-body system. During the past a two and half decades, several formal developments and applications of this theory have proved its enormous power and success. In DFT, the exchange and correlation (XC) energy is unknown, this requires approximations which can limit the accuracy of the calculated ground state properties. In this thesis, we have used the local density approximation (LDA) which is a reasonable choice for exchange and correlation. The LDA is valid when the density is nearly uniform. To study the effect of XC potential on our problem, we have used different XC potentials such as von Barth-Hedin (BH), Barth-Hedin modified by Janak (BHJ), Vosko-Wilk and Nussair (VWN), and Slater Xa, approximated by different workers. We have also included a nonlocal-XC potential in LDA given by Langreth and Mehl. In case of Slater Xa XC potential, a has been treated as an adjustable parameter and the values, calculated separately for each element, are for the best agreement with the experimental data. The electronic properties to be studied here, are band structure, density of states,Fermi surface topology, and enhancement factor etc. There exist a number of powerful experimental methods for measuring the FS, such as de Haas van Alphen (dHvA), cyclotron resonance, magneto-resistance etc. Among these dHvA has proved to be the most accurate and reliable in probing the electronic structure near the Fermi energy. In the first chapter of the thesis, we briefly review the LMTO theory, DFT and some of the experimental methods for measuring FS. In chapter 2, the FS of group VB transition metals vanadium (V), niobium (Nb) and tantalum (Ta) is represented in detail. We have chosen these metals as they play a special role among the transition metals due to their large heat capacity and high superconducting temperature arising from large electron-phonon effects. These metals are also basic building blocks in the host of transition metal compounds. The FS of these metals is similar and consists of three sheets, i.e. hole octahedron centered at r, hole ellipsoidal centered at N and a multiply connected hole sheet which extends from r to H in <100> direction and possesses branches at r and H. The extremal cross-sectional areas are calculated for fourteen different FS orbits. The agreement between computed extremal areas and experimentally measured areas is obtained by calculating the shift in the Fermi energy, AEF' required to bring the calculated FS areas in agreement with the experiment. We have performed calculations to study the effect of (i) various XC potentials (ii) increasing the number of TZ4 points (iii) including angular momentum expansion of wave functions up to 1=3 and (iv) inclusion of relativistic effects. In our study, it is found that the choice of XC potential plays an important role while rest are not that much significant. For tantalum, incorporating relativistic effects does improve the extreme AEF. We have also calculated the enhancement factor X (arising because of many-body interactions such as electron-electron, electron-phonon and electron-paramagnon) for these metals and compare this with other theoretical results. The compilation of similar studies for molybdenum is made in chapter 3. Molybdenum a group VIB transition metal is very interesting for such studies as its electronic structure, as that of the VB matals, is characterized by the overlap and hybridization of a wide nearly-free-electron s-p band with a relatively narrower d-band complex leading to a complex FS of several sheets and a low density of states of the d-electrons at the Fermi energy. The FS of molybdenum consists of four sheets i.e. electron jack centered at F, hole octahedron centered at H, hole ellipsoidal centered at N and electron lenses centered on the mi line. The work on molybdenum involves the study of the same effects as that for vanadium, niobium and tantalum. The relativistic effects play a much more important part as compared to that in case of VB transition metals. But the increase in number of k points and inclusion of the angular momentum expansion up to 1=3 does not influence the FS much as found earlier. The enhancement factor (X) is also reported for the XC potential which gives the best agreement with the experimental data. The chapter 4 of the thesis is devoted to iron, a transition metal which shows ferromagnetism and hencee the effect of exchange interaction and spin orbit interaction has to be included. Even though the FS of iron is studied in great details, its geometry has not been established unambiguously. This and the fact that all other transition metals studied in the present thesis are paramagnetic, prompted us to make an attempt with iron. Extremal areas of a large number of FS orbits are calculated for magnetic field along [001], [110] and [111] directions using different XC potentials. Our results for iron give 'fair agreement with experiments and other theoretical results. We discuss our results in the light of recent calculations which seem to suggest that LDA may not give the correct ground state of ferromagnetic iron. Studies of influence of pressure on the FS provide a stringent test of the electronic structure. Even when a satisfactory FS has been obtained for the normal lattice spacing, the calculated changes with lattice spacing may not agree with experiments. Thus the effect of hydrostatic pressure on the FS of all these bcc transition metals is studied and reported in chapter 5. The pressure derivatives of extremal areas, [d(lnA)/dP], of different (iv) FS orbits are calculated for different XC potentials and compared with the available experimental and theoretical results. Even though such studies are important, thereis a dearth of experimental as well as theoretical work. We hope that our work will enthuse others to work in this direction