Structural, Electronic, Optical And Transport Properties .
Structural, Electronic, Optical and TransportProperties of Pristine and Alloyed UltrathinNanowires of Noble MetalsSynopsis for Ph.D. in PhysicsSubmitted byArun KumarinFaculty of Physical SciencesDepartment of PhysicsHimachal Pradesh UniversitySummerhill, Shimla-171005Himachal Pradesh (India)Presented on 5th January 2013inResearch Degree Committee meetingofPhysics DepartmentSupervisorDr. P. K. AhluwaliaProfessorDepartment of PhysicsHimachal Pradesh UniversitySummerhillShimla-1710055th January 2013
Structural, Electronic, Optical and TransportProperties of Pristine and Alloyed UltrathinNanowires of Noble Metals1IntroductionIn the last two decades, nanoscience and nanotechnology have initiated muchinterest in fundamental research of the properties of nanomaterials and theirindustrial applications. As a result nanostructured materials as a foundationof nanoscience and nanotechnology, have become the hottest topics of research[2, 3, 49, 147]. Normally, nanostructures are defined as the structures with atleast one dimension less than 100 nm. In this dimension, the number of atomsare countable, making the properties of nanostructures different from those oftheir bulk counterparts due to distinct density of states (DOS) and increasedsurface to volume ratio. According to the number of dimensions less than 100nm, nanostructures can be classified into two-dimensional (2D, nanofilm), onedimensional (1D, nanotube or nanowire), and zero-dimensional (0D, quantumdot) structures.1.1One-dimensional nanostructuresIn a 1D nanostructure there is a confinement in two dimensions perpendicular to the longitudinal extent of the structure. This confinement is quantummechanical in nature. Due to the combination of both quantum confinementin the nanoscaled dimensions and the bulk properties in the third dimension,many interesting properties and applications can be expected based on a widevariety of 1D nanostructures. Also it has been observed [1] that 1D nanostructures represent the smallest dimension structure that can efficiently transportelectrical carriers and can be exploited as both the wiring and device elementsin future architectures for functional nanosystems.Depending on the topology and morphology, the 1D nanostructures can beclassified into following main groups: Nanotubes [3]. Nanowires [4, 5]. Coaxial cable structures [6]. Side-by-side biaxial nanowires [7]. Nanobelts (or nanoribbons) [8].1
The first three nanostructures listed above have a common characteristic ofcylindrical cross section, biaxial nanowires have the stacking of two parallelnanowires of different materials and nanobelts have a rectangular cross section(belt like morphology). The distinctive geometrical shapes of these 1D nanostuctures are important as their mechanical, electrical, optical, and thermal transport properties are geometry dependent. To investigate the uniqueness offeredby these shapes, new techniques have been developed to measure the propertiesof individual wire-like structures quantitatively and their structures are wellcharacterized by electron microscopy techniques [2].1.2Review of LiteratureThe quantum confined nanowires have a wide range of applications in electronics [60–69], optoelectronics [10–13], thermoelectrics [70, 71], optics [62, 72, 73],chemo and bio-sensing [18, 21, 25–27, 74, 75, 77], magnetic media [78–85], photocatalysis [30–32] and piezoelectronics [33–39] etc. As a result of the rapidprogression of modern nanoelectronics, nanowires (NWs) have begun to drawthe attention of researchers in cross disciplinary areas of physics, chemistry andengineering. In nanoelectronics, nanowires can function as interconnects in thefabrication of integrated circuits [64, 65], resonators [66–69], diodes [9–13, 60],light emitting diodes (LED) [61], multifunctional devices [63], logic gates [18–23]using nanowire field-effect transistors (NW-FETs) [62] and single electron transistors [24]. Their small size and their high electrical conductivity makes themvery attractive for applications in nanoelectronics [76].Sensing is another area in which the application of nanowires is expected tohave a great impact. Nanowire sensors have been reported, that can detect thepresence of many gases like O2 , N O2 and N H3 [74] at very low concentrations.Nanowires also have been used for the detection of ultraviolet light [75] as wellas highly sensitive biological and chemical species [77]. These sensors oftenfunction on the basis of changes in the electrical or physical properties of thenanowires when they come in contact of targeted chemical/biological [74–76]species. The sensing capabilities of nanowires can be controlled by selectivedoping that raises their affinities to certain substances. Also the nanowiresof noble metals particularly have been used as barcode tags for optical readout [86, 87]. Single-crystalline NW have also been used in batteries [47], solarcells [48] and photoelectrochemcal cells [31] for effective charge separation andcollection.In these areas, NW structures exhibit unique and superior properties compared to their bulk counterparts, resulting from 1D confined transport of electrons or photons, large surface area, quantum confinement, and excellent mechanical properties [40–46].In recent years, long metallic nanowires with well defined structures anda diameter of several nanometers have been fabricated using different methods [88–91]. For example, stable gold nanobridge with 0.8-3 nm in thicknessand 5-10 nm in length has been produced by electron beam irradiation of gold(001) oriented thin film [88]. Also suspended gold nanowire with 6 nm in length2
and diameter down to 0.6 nm have been made and the novel multishell structurewere observed [91]. In 1998, Ohnishi et. al. [53] used scanning tunneling microscope and Yanson et. al. [54] through mechanically break-junction experimentsproduced atomically thin bridge of gold atoms and calculated the conductanceequal to Go 2e2 /h and the interatomic distance was reported as 2.6 Å [92].The break-junction experiments have also been performed for Ag [55], Cu [52]2and Pt [52] chains, with conductance nearly 1Go (Go 2eh ) for Cu, Ag and Auand 1.5Go to 2.5Go for Pt chains.Noble metal nanowires, which is the subject matter of the proposed work,have been drawing a great deal of attention due to their significantly differentstructural, electronic, magnetic, optical and transport properties compared totheir bulk manifestation [49, 50]. The increased surface to volume ratio andincreased density of states (DOS) makes them different from their correspondingbulk materials. Also the DOS does not vanish at subband edges and remainfinite. This makes these materials an interesting and exciting subject to exploreoptical properties. Noble metal nanowires, particularly, can also be used tocreate materials that exhibit negative index of refraction in the near-infraredregion [51]. The finite monoatomic chains of noble metals can be produced bymechanical break junction experiments [52–55] for Cu, Ag, Au and Pt chains.Many innovative experimental studies on one dimensional systems [53,54,56,57],revealing their fascinating properties, have boosted related theoretical research.Depending upon the type of structure, the electronic, magnetic, optical andtransport properties of these systems show interesting variations [58, 59].Very few studies have been made of the properties of metal alloys at the atomicscale (alloyed metal nanowires) [93–98]. In 2002 and 2003, point contact studieswere made of random alloys of a transition metal and a noble metal, namelygold and palladium [93], copper and nickel [94] and gold and platinum [96] fordifferent concentration ratios. In these experiments peak has been found at1Go , that is characteristic of the noble metals, survives for transition metalconcentrations well over 50%. The interpretation for this observation requiresfurther study. There is an evidence for segregation of the noble metals awayfrom the contact under the application of a high bias current [98].2Theoretical BackgroundThe universe around us is made of condensed matter i.e. matter whose energy is low enough that it gets condensed to form stable system of atoms andmolecules usually in solid or liquid phases. These atoms and molecules are further made up of electrons and nuclei. The quantum mechanics has proven to bethe best formulation to describe interacting system of electrons and nuclei. TheSchrödinger equation is the fundamental quantum mechanical equation that describes a system of electrons and nuclei in terms of wave function ψ, which isfundamental entity in quantum mechanics.3
The systems under study are indeed many electron systems and for a multielectron system the Schrödinger wave equation can be written as: 2 X2 XX ZA ZB e2X e2X ZA e2 ψ Eψ 2i 2A 2me i2MA4π o RAB4π r4π roijoAii jAA BAiHere i and j are indices used for electrons and A and B are the indices used fornuclei. On the left hand side of above equation first two terms represent the kinetic energy of electrons and nuclei respectively and the following terms describethe inter-nuclear, electron-electron and electron-nuclear Coulomb interaction energies respectively. If we use atomic units i.e. (e me 4π o 1), theSchrödinger equation becomes XXXXX1ZA 1ZA ZB 1 2i 2A ψ Eψ (1)2 i2MARABrrijAii jAA BAiThe ultimate aim of any physicist or a chemist for a typical system is to solvethis equation.However, as Paul Dirac at the dawn of theoretical quantum mechanics has saidthat all the answers of chemistry could be calculated from schrödinger equation[103], but it is the most challenging task to solve this equation analytically.Unfortunately the schrödinger equation can be solved exactly for only a fewsystems such as hydrogen atom and even numerically to systems containingsmall number of electrons [102].2.1ApproximationsTo solve the equation (1) one uses various approximations, which do not significantly affect the involved physics of the system under study and facilitatemeaningfully the study of variety of many body problems.2.1.1Born-Oppenheimer approximationOne of the most simplifying approximation is based upon the idea that mass ofelectron is much smaller than that of the nucleus and thus electrons move muchmore rapidly than nucleus. Thus for a given set of positions of nuclei, electronsadjust almost immediately to movement of nuclei. This is known as BornOppenheimer approximation [99]. In other words we can say that forces onboth electrons and nuclei due to their charge are of same order of magnitude, sochanges which occur in their momenta as a result of these forces must also beof the same magnitude. But since nuclei are much more massive than electronsso accordingly they have much smaller velocities. While solving the schrödingerequation given by equation (1), one can assume that nuclei are stationary andsolve it for electronic ground state first and then calculate the energy of thesystem in that configuration and then later solve for nuclear motion. This helpsto separate the electronic and nuclear motion. Furthermore, this allows us toseparate the wavefunction as a product of nuclear and electronic terms. Theelectronic wave function φe (r, R) is solved for a given set of nuclear coordinates4
Ĥe φe (r, R) { X 11 X 2 X ZA i }φe (r, R)2 iRAi i j rijA,i Ee (R)φe (r, R)(2)and the electronic energy obtained contributes a potential term to the motionof nuclei described by the nuclear wave function φN (R).X ZA ZBX 1ĤN φN (R) { 2A Ee (R) }φN (R)2MARABA BA 2.1.2EφN (R)(3)Independent Electron ApproximationAnother approximation called independent electron approximation [120] whichassumes electrons to be non-interacting with each other has quantum manifestation because electrons obey Pauli’s exclusion principle. This manifestationof Pauli’s exclusion principle resulted in Hartree-Fock method and allows oneto express total Hamiltonian for N-electronsystem (H) as summation of singlePelectron Hamiltonian (Hi ) i.e. H i Hi and total wave function as Slater determinant of single electron wave functions. The slater determinant approximation does not take into account Coulomb Correlation leading to a total electronicenergy different from the exact solution of non-relativistic Schrödinger equationwithin Born-Oppenheimer Approximation. Therefore, Hartree-Fock Energy isalways above the exact energy. This difference is called the Correlation Energy,a term coined by Löwdin [100].2.2Density Functional Theory: An Ab initio ApproachTo overcome the difficulty of correlation energy and problem of 3N variables,a new approach ‘Density Functional Theory (DFT)’ was adopted for electronic structure calculations. In 1964 Hohenberg and Kohn, showed in a conference paper [101] that schrödinger equation (for N electron system containingwave function of 3N variables) could be reformulated as an equation of electrondensity with only three variables. This theory gives approximate solutions toboth Exchange and Correlation Energies. The main objective of DFT is to replace the many-particle electronic wavefunction with the electronic density asthe basic quantity. Our interest is in solving Schrödinger’s equation by meansof ab initio Density Functional Theory (DFT) as described below.The term Ab initio used here originates from Latin word which means ‘fromthe beginning’. A method is said to be Ab initio or from first principles if itrelies on basic and established laws of nature without additional assumptionsor special models based upon particular material. Density functional theory(DFT) is an extremely successful ab initio approach to compute properties ofmatter at microscopic scales. DFT is a quantum mechanical modelling methodused in physics to investigate the electronic structure (principally the groundstate) of many-body systems, in particular atoms, molecules, and the condensedphases (bulk, surfaces, chains).5
The fundamental pillars of density functional theory are two physical theorems proved by Kohn and Hohenberg [101,118, 119]. The first Hohenberg Kohn(HK) theorem states that: The ground-state energy from Schrödingers equation is a unique functional of the electron density. This theorem provides oneto one mapping between ground state wave function and ground state chargedensity. The first HK theorem is stated as: the ground state charge density canuniquely describe all the ground state properties of system. The fundamentalconcept behind density functional theory is that charge density (3-Dimensional)can correctly describe the ground state of N-particle instead of explicit usage ofwave function (3N-Dimensional) [117]. Thus by using charge density instead ofwave functions a 3N dimensional problem reduces to just a three dimensionalproblem.The second HK theorem states that: The electron density that minimizes theenergy of the overall functional is the true electron density corresponding to thefull solution of the Schrödinger equation. If the true functional form of energyin terms of density gets known, then one could vary the electron density untilthe energy from the functional is minimized, giving us required ground statedensity. This is essentially a variational principle and is used in practice withapproximate forms of the functional designed by quantum chemists/physiciststo study different types of systems [120]. The simplest possible choice of afunctional can be a constant electron density all over the space.The total charge density can be written in terms of single particle wave functions as:Xn(r) ψµ (r)ψµ (r)(4)µAn important step towards applying DFT to real systems was taken by Kohnand Sham in 1965 in the form of Kohn-Sham (KS) [119] equations. The KohnSham equation reformulate the Schrödinger equation of interacting electronsmoving in an external ion potential into a problem of non interacting electrons moving in an effective potential. The KS equations are defined by alocal effective external potential (called Kohn-Sham potential) in which thenon-interacting particles move. The Kohn-Sham equations have the form 2 2 Veff (r) ψµ (r) εµ ψµ (r)(5)2mwhereVeff (r) V (r) VH (r) VXC (r)(6)The contribution to the total energy here gets divided into two parts. The 2 2 ), the hartree potential enfirst part contains terms: the kinetic energy ( 2mergy (VH (r)) and classical Coulomb energy (V (r)) and second part contains theexchange correlation energy (VXC (r)) which includes many body and quantumeffects. It is customary to divide the exchange correlation into exchange part(for which there exists an exact expression although computationally expensiveto calculate) and correlation part (which is unknown).6
Kohn and Sham introduced a set of orbitals from which electron density canbe calculated. These Kohn -Sham orbitals do not in general correspond to actualelectron density. The only connection the Kohn-Sham orbitals have to the realelectronic wave function is that they both give rise to the same charge density.To calculate the kinetic energy term, the Kohn-Sham orbitals are used as shownbelow:Ts [ρ] N ZXi 1 2 2 ψµ (r)dr ψµ (r) 2m(7)On the right-hand side of equation (6) there are three potentials, V , VH , andVXC . The first potential defines the interaction of an electron with differentatomic nuclei present which is basically Coulomb potential. One takes care ofthis term with the help of a trick which replaces Coulomb potential by a Pseudopotential [120]. It is well known that since core electrons do not participatein bond formation, it was natural to assume that they can be replaced with apseudo core. That means we have to deal with fewer number of electrons (valence electrons). Thus a pseudopotential is an approximation for the full corepotential. How good a pseudopotential is, infact successful for the generatedpseudopotential decided by how well it reproduce the results from all electroncalculations. The effort in generating a pseudopotential lies in the fact that,all electron wave function must match with the pseudo wave function after acertain cut off radii.The second potential is called the Hartree potential. This potential describesthe Coulomb repulsion between the electron being considered in one of theKohn-Sham equations and the total charge density defined by all electrons in theproblem. The Hartree potential includes a so called self-interaction contributionbecause the electron we are describing in the Kohn-Sham equation is also part ofthe total electron density, so part of VH involves a Coulomb interaction betweenthe electron and itself. The self interaction is unphysical, and the correction forit is one of several effects that are lumped together into the final potential inthe KS equations, VXC , which defines exchange and correlation contributionsto the single electron equations.To solve KS equation we need Hartree potential which depends upon thecharge density of the system and to know the charge density we need the singleparticle wave functions, which can be obtained only after solving KS equations.Thus the problem reduces to solving a set of self consistent equations. Theyare solved in an iterative way by starting with a trial set of single particle wavefunctions from which Hartree potential is obtained. The solution obtained inthis way is called self consistent solution. The algorithm used to solve it is asgiven in figure 1.2.3Local Density Approximation(LDA)The most important potential term in KS equations is VXC [103] which is usedto describe exchange and correlation effects. Unfortunately, exact form of the7
Figure 1: Algorithm to solve Kohn Sham equations in DFT Codesexchange correlation functional whose existence is ensured by the HohenbergKohn theorem is not known. There is only one case of uniform electron gaswhere this functional can be derived exactly as the electron density is constantat all points in space. Therefore, the exchange-correlation po
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