ELECTRONIC PROPERTIES OF NOBLE METALS AND SOME FCC TRANSITION METALS

Abstract

In this thesis, an attempt has been made to study the electronic properties of noble metals and some fcc transition metals. During the past two decades the techniques for solving the band structure problems have reached a point where, with the aid of large computers, an accurate solution may be obtained. In this work we use the linear-muffin-tin-orbital (LMTO) method in the atomic sphere approximation (ASA) to solve the band structure problem. The variational principle for a one-electron Hamiltonian is used and the trial function is a linear combination of energy independent muffin-tin orbitals (MTO). The secular equation reduces to an eigen value equation . The triah function is defined with respect to a muffin-tin (MT) potential and the energy bands depend on the potential inside the spheres through potential parameters which describe the energy dependence of the logarithmic derivatives. The energy independent MTO is the linear combination which matches on the solution of the Laplace equation in the interstitial region and is regular at infinity. The LMTO method is particularly suited for closely packed structure and it combines the desirable features of the Korringa Kohn Rostoker (KKR), linear combination of atomic orbitals (LCAO) and Cellular methods. The secular matrix is linear in energy, the overlap integrals factorise as potential parameters and structure constants, the later are canonical in the sense that they neither depend on energy nor on the cell volume and they specify the boundary conditions on a single MT or atomic sphere in the most convenient way. This method is very well suited for self-consistent calculations. In this thesis we are interested in the Fermi surface (FS - a ground state property) of metals which can be calculated within the density functional formalism. This requires that we perform self-consistent electronic struture calculations. The electron density can be utilised as a central quantity and the formulation of a many-particle pr'oblem into a single-particle like frame work, is the essence of the density functional theory (DFT). Starting from the Thomas-Fermi method and several modifications, the DFT has been rejuvenated by the pioneering works of Hohenberg, Kohn and Sham who have laid its strict mathematical foundation and thus provided a formal justification for the use of density as a basic quantity. In the DFT, the problem faced is that exchange and correlation energy function can only be approximated. Hence to overcome this problem, we have used the local density approximation (LDA) for exchange-correlation (XC) which is valid when the density varies slowly in space. To check the effect of exchange and correlation 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. There are a number of powerful experimental methods for measuring the FS. These methods include de Haas-van Alphen (dHvA), cyclotron resonance, magneto-resistance etc. The dHvA effect has proved to be most accurate and reliable tool for probing the electronic struture near the fermi energy. There exists voluminous data on FS and cyclotron masses. We have compared our results with these data. Hence in the first chapter we review briefly the LMTO theory, OFT as well as some of the experimental methods for measuring FS. Chapter 2 is devoted to the FS of noble metals. The main reason for choosing the noble metals is that their FS is easy to study as it consists of single sheet and has already been studied experimentally and theoretically in great detail so that a study of these metals can be used to debug the programs. Another reason for choosing these metals is that spin-orbit effects vary from negligible to predominent as we move from copper (Cu) to gold (Au). We have calculated extremal areas for four orbits in the noble metals and studied the effect of (i) various XC potentials (ii) increasing the number of k points in the Brillouin-zone (BZ) summations (iii) including angular momentum expansion up to i=3 and (iv) inclusion of relativistic effects. We observe t.I.,:.7,17' the case of noble metals the choice of XC potential plays an important role while the other effects are not significant. Here we have adopted a different criterion to determine the agreement between the computed FS extremal areas and the experimentally measured areas. This is done by calculating the shift in the Fermi energy AEF required to bring the calculated FS area in agreement with the experiment. Our results show that in case of copper and silver the extreme AEF is 4.1 and 0.9 mRyd, respectively, with the Slater Xa (a=0.77) XC potential while for gold AEF is around 3.5 mRyd with Slater Xa (a=0.693) XC potential. Here a has been treated as an adjustable parameter and the values reported are for the best agreement with the experimental data. The success with which the LMTO method gives the FS topology of noble metals prompted us to perform similar calculations for the transition metals palladium and platinum. These metals possess a complicated FS and we believe that the accuracy with which the LMTO method can give FS topologies will be born out by our results. Interest in palladium and platinum has kept alive because of their fascinating electronic properties such as high density of states, large paramagnetic susceptibilities with unique temperature dependence, alloying, catalysis etc. The FS of palladium consists of four sheets i.e. a closed electronic surface centred at r, a sheet of hole centred at X, an open hole surface and a sheet of L pocket hole. However platinum has only the first three sheets and has no L pocket hole. The study of palladium and platinum also involves the study of same effects as we have studied for noble metals. We observe that in case of palladium and platinum the relativistic effects play important role while these effects are found to be negligible in noble metals. Similar to the case of the noble metals, the increase of number of k points and including angular momentum expansion up to 1=3 does not affect FS areas for palladium but in case of platinum these effects are significant. The results of this phase of investigation are compiled in chapter 3. We found that the inclusion of relativistic effects, in case of palladium and platinum, brings forth a dramatic improvement in results. The AEF reduces to 4.0 mRyd with Slater Xa (a=0.75) for palladium and 2.7 mRyd for platinum with Slater Xa la=0.815/ XC potential from AEF of 16.0 and 40.0 mRyd, respectively. Civ) The many-body interactions (such as electron-electron, electron-phonon and electron-paramagnon ) renormalise the dynamic properties of bare electrons. We have calculated the enhancement factor X (X=0 in absence of many-body interactions) and Fermi velocities for these metals. These are compared with other theoretical results. All the metals discussed so far are paramagnetic. As an example of a ferromagnetic metal, we have chosen nickel because here the effect of exchange interaction and spin-orbit interaction has to be included. In chapter 4 , we have discussed FS of nickel. We have calculated various FS orbit areas for magnetic field along C001], [1107 and (111l directions. We have calculated .EF with different XC potentials,. Our results for nickel are in agreement with the experiment and other theoretical results. The interpretation of pressure effects on both, electron transport and crystallographic properties of metals, usually requires some knowledge of the way in which the FS is affected by pressure. We have studied the effect of hydrostatic pressure on the FS of noble metals, palladium, platinum and nickel. The effect of pressure on the FS provides a valuable check on the reliability of band-structure calculations. We have calculated pressure derivatives of extremal areas (d(lnAl/dF] by performing self- consistent band-structure calculations at two different radii. Our results are compared with the experimental results. Thus chapter 5 is devoted to the effect of hydrostatic pressure on the FS.