Application of Inverse Scattering Method to Investigate Electronic Structure of Solids

# Introduction

Since early 1960s, there has been enormous growth in the number of studies of solid structure; however, few have focused on their electronic structure. The number of techniques available for studying solid structure has also tremendously increased over the past two to three decades. The most significant reason for this trend is the growing awareness of the significance of proper understanding of properties of solid structure. Furthermore, the fact that various works of solid surfaces have had great impacts on their applications in the real world has also furthered this interest among various upcoming scientists. Over the past years, scientists have come up with various methods of studying property of different solid structure. Understanding the structure of solid has drawn great interest because, at a fundamental level, it represents a special kind of defect in solid state, which has not fully comprehended by scientists.

Scientists’ understanding of solid structure is more or less based on the fact that they are three dimensional and perfectly periodic within that state. Due to this nature, it is apparently possible to describe their electronic and vibrational properties with great detail by simply using methods that are dependent on periodicity. However, the major problem of this approach is that the introduction of surface breaks to this periodicity has greater potential of causing structural changes to the structure as well as the potential of introducing localized vibrational and electronic states. At a fundamental level, understanding of the electronic structure of solid surfaces would therefore be a greater and an essential advancement in science. Gaining proper understanding of the electronic structure of solid is not only of academic interest but also essential in development of electronic devices for everyday use.

Perhaps, the most important motivation for studying electronic structure of solid is the primary goal of understanding the principle and application of heterogeneous analysis. Scientists understand that there is increased rate of chemic interactions occurring in the solid structure in the presence of solid catalyst. According to Harrison (67) analysis, the understanding is that the increased rate of chemical interactions results from the modification of certain constituent chemicals which are adsorbed on the solid structure thus the need of understanding electronic structure of solid structure. Furthermore, it is not yet clear what is endowing such chemicals with enhanced ability to interact with other constituents on solid structure. It is therefore essential to understand the electronic structure of solid surfaces, what is causing the modification, what kind of site of the solid are active, as well as the rate limiting steps and activation energy.

Scientists have come up with different techniques, procedures and technologies of studying electronic structure of solids; however, one method which has not been extensively explored is the use of inverse scattering technique. Although there are very few pieces of literature about the application of inverse scattering method in the study of solid surface, the method has been widely used on other fields of academic study. The inverse scattering method was discovered by Gardner, Green, Miura and Kruskal in 1967 and since then it has become an essential technique in the modern scientific and mathematic study. Since 1967, numerous attempts have been made to extend the application of inverse scattering method especially in mechanics and theoretical physics.

In the study of the electronic structure of solids, the inverse scattering method has the advantage of being the most elementary as compared to other techniques. In essence, the method is contained within the calculation of conditions that are necessary for conserving eigenvalues within spectral problems. The simplicity of this method is based on the idea of linear operators containing variable coefficients which are easy to be obtained simply by means of transformation of operators using constant coefficients. With such transformation of operators, the condition for compatibility will assume none linear form thus leading to the establishment of the sought-after integral equation. Another advantage of using inverse scattering method in investigating electronic structure of solids is that possibility of being extended in many variables. Because we are considering the application of inverse scattering method in physics, it provides simplest exact solutions which are extremely informative and interesting (Harrison 67).

## Research problems

For many years, scientists have been studying electronic properties associated with solid matter for different aims and objectives. The recent technological evolution has subsequently accelerated theoretical and experimental development in this area of research. As a result, the development of new techniques and methods of studying electronic structure of solids is necessary to add to the currently available methods. Although majority of the methods currently being used by scientists enable them to detect the electronic properties of solids, they do not extend to theoretically explain the electronic structure of solids.

In solid state, matter is known to display a wide range of challenging phenomena and properties. Electrical properties of solids are one of the areas which have not been extensively studied and there are few available pieces of literature in this field of study. In order to ease and enhance our understanding, there is need to develop new approaches and methods of studying the electrical properties of solid surfaces. Furthermore, despite is wide application in the field of physics and mathematics little has been done to extend the application of inverse scattering method in the study of electronic properties of solid surfaces. In essence, there are few available pieces of literature the extensively discuss the use of inverse scattering method in studying the electronic structure of solids.

In order to ease our understanding, most solid surfaces are classified without deeper consideration of their electronic properties, an incidence which is limiting our understanding of their possible applicability. For example, solids are crudely classified with respect to their electrical conductivity as metals (conductors), semiconductors, and insulators without considering other aspects of their electronic properties. Little has been done to classified solids in terms of the electronic properties of their surfaces thus limiting our understanding of their surfaces. In addition, every solid has a unique surface with unique electronic properties.

## Research objective and question

The primary objective of this research study is to identify how inverse scattering technique can be used to study the electronic property of solid surfaces. This would be a great milestone in our understanding of solid surfaces and their applicability in physics. The research question that guided this study is, “Can inverse scattering method be used to investigate the electronic structure of solids?”

# Inverse Scattering Method

Inverse scattering is a unique method that is highly useful and effective tool in the study of electronic structure – most scientists have successfully used this method to analyze electronic structure of various matter. As illustrated by Pike and Pierre (76), the discovery and correct interpretation of inverse scattering technique represented a huge move towards scientists’’ understanding of physics. This technique was used to prove that x-rays can act bot as particles (photons) and waves. This method has also been used by other scientists to prove that the behavior of electron in metals is in accordance to Fermi-Dirac statistics. These studies played major roles in the birth and development of quantum mechanics in physicals. It is essential to note this given that the studies of electron structure of solids utilize quantum mechanics.

Today, inverse scattering method is utilized in many different research fields including the study of plasmons, phonons, valence-electron excitations, as well as resonance phenomena among other areas of application. Our study of electronic structure of solids using inverse scattering method is based on valence-electron excitations, quantum mechanics, as well as polarization of nuclear. Furthermore, the inverse scattering method can be used uniquely in the probing of spin polarization of magnetic properties. The most recent development in the field of inverse scattering technique is the discovery of sources of high-brilliance synchrotron radiation which have significantly allowed accuracy and enhanced experimentation. In addition, the use of synchrotron radiation sources has made inverse scattering analysis possible. This technique is able to provide highly intense polarized radiation which is essential in discovering the electronic structure of solids (Kaxiras 56).

Most of the experimental tools used in inverse scattering method are largely based on scattering of electrons, photons, neutrons, positrons as well as other identified particles. In this case, the interaction between the probe and the system being studied will give more information about the electronic nature of solids; however, this requires an advanced knowledge of the theory of the interaction. When a particle is impinged on the system, it would create an excitation thus subsequently leaving the system with momentum. In addition, it will leave the system when its polarization state and energy has been significantly changed allowing the probing of electronic structure of solids to be undertaken. The scattering process takes place and it is considered inelastic if total energy of the problem changes. The final state of the system also changes in the process. In this experimentation, it is essential to keep in mind that the type of excitation is determined by the overall amount of momentum which is transferred to the system (Antonov, Bruce and Alexander 212).

## Determination of Electronic Structure of Solids

The electronic structure of a solid material is basically determined by the quantum numbers of the available electrons. This varies significantly depending on the nature and type of the material. Crystalline materials, for example, are associated with momentum, energy, spin as well as point group symmetry. Plotting the energy band dispersion of the material with appropriate labels for the spin and point group symmetry as able to generate this information about the electronic structure of solid materials. However, when it comes to disordered materials, they can be categorized using average values of energy, spin, momentum and point group symmetry. The disorder introduces the material to characteristic broadenings. Therefore, the most ideal approach of investigating an electronic structure of solids is to determine their respective quantum numbers (Dreysse 87). This is the technique that is used ion inverse scattering technique in investigating the electronic structure of solids. Using the inverse scattering technique to probe quantum numbers, it is essential to first determine variety of particles then draw an outcome in the table as shown below.

Table 1: Techniques of probing electronic states

OUTGOING PARTICLE | ||||

PROTON | ELECTRON POSITRON | ION ATOM | ||

INCOMING PARTICLE | PHOTON | Optical Luminescence Raman Cyclotron resonance De Haas V. Alphen | Photoemission Auger | Photon stimulated Desorption |

ELECTRON POSITION | Appearance potential Positron annihilation Inverse photoemission | Appearance potential Electron energy loss Auger Tunneling | Electron stimulated Desorption | |

ION ATOM | Chemo-luminescence | Penning ionization Ion neutralization | Secondary ion |

This technique concentrates in the spectroscopy involving only electronic transitions without considering the motion of atoms. This is done to ensure the outstanding validity of Frank-Condon principle. When ions or atoms are present in either outgoing or incoming channels, it would be very difficult to achieve the validity of Frank-Condon principle. Among other available techniques, the inverse scattering approach distinguishes itself through its ability to simultaneously probe both momentum electrons and energy. In addition, the technique is distinguished by its ability to measure the absolute electron energies. However, it is not able to measure the energy differences. This technique is designed to impinge an electron onto the solid surface so that a photon is emitted. This is typical reverse method of photoemission (Canadell, Marie-Liesse and Christophe 203).

## The origin of electronic structure of atoms

The electric phenomena have been known for centuries but little was known about electronic structure until the late 1890s following the discovery of electron as a fundamental constituent of matter. Later on, Maxwell’s theory of electromagnetism was modified by Hendrik to allow the interpretation of the electric properties in terms of motion of particles which are charged. Korepin and Bogoliubov (324) made an observation and concluded that the stability and observed spectra of atoms could be explained by quantum mechanics. Here, the quantum mechanics accounts for atomic spectra and stability in terms of discrete sets of allowed levels of electron energies. This set the stage for the establishment of the law of quantum mechanics in analyzing electronic structure of atoms. Further progress in the study of atomic structure led to improved understanding of electrons in solids and molecules.

Understanding the nature of electron structure in solids has been a many-body problem that requires the use of statistical concepts that can effectively describe the intrinsic properties of solids. Many developments took place within the last decades of 1900s which have led to new understanding of electronic structure of solid matter. The most important one was the discovery of new materials such as high temperature superconductors and fullerenes as well as discovery of phenomena such as quantum hall effect and superconductivity among others. The theory of superconductivity was the most influential discovery in physics because it influenced all other fields by providing basis for emergence of new phenomena (Kaxiras 65).

## Energy bands

Analysis using inverse scattering method indicates that similar to the formation of diatomic molecules, the number of electron states is conserved when many atoms are brought together to form a solid. The only difference from diatomic molecules is that in solids the number of basis states tends to be relatively greater. The simplicity of crystalline solids allows for deeper and more accurate investigation of their structure using the inverse scattering technique. Analysis of crystalline solid using this method indicates that atomic energy levels are split into bands when atoms are brought together in the solid structure. However, it is essential to note that rather than a single antibonding or bonding state, the atomic level split into an entire band of states that are distributed between antibonding and bonding limits.

Within each band in solid are so many different states. The presence of so many states allows both spins in each spatial state to accommodate electrons. In solids, all states are fully filled. Analysis of electronic structure of solid exposes one important feature of insulators, that is, the state of the system can only be changed if an electron with several electron volts of energy is excited and thus transfer to an empty band of higher energy. Without this, it is practically impossible to change state of the system in insulators. For this reason, the material cannot absorb light with lesser frequency and the crystal takes a transparent form. This makes it very difficult to induce currents using small voltages (Korepin and Bogoliubov 324).

In solids, electrical conductivity results from full bands, thus, insulators cannot conduct electricity because they do not have full bands. In addition, presence of conductivity does not result from localization of electrons at bonds or atoms in solids. Analysis of electronic structure of solids using inverse scattering method indicates that bands is freely present in crystals and that their electrons are in states of crystals. In this case, rather than atomic states, electrons free form bonding and antibonding molecular states at the individual atoms. One major characteristic of metal that distinguishes it from insulators is that its bands are only partly filled. The electrical conductivity of metal is given as a proportion of time between scattering event expressed by the following formula:

And,

The figure below gives optical absorption spectra of metals at low frequency.

Figure 1: optical absorption spectra of metals

## Dynamics of electrons

An analysis of solid structure using inverse scattering method shows an interesting behavior of individual electrons in the band in circumstances when the electron energy bands are neither completely empty nor completely full. The electron dynamics are shown to provide important link between the electronic properties of solids ad the band properties. Considering the Brillouin Zone of a solid matter, the magnitude of wave function does not change with time. In a similar manner, the probability density of the wave function at any point does not change. The electron dynamics of solids is determined by the choice of wave packets. This implies that the electron dynamic of solid is influenced by the slow varying of magnetic fields relative to interaction spacing (Novikov 99).

The use of inverse scattering method allowed us to note some qualitative aspects of the electron dynamics in solids. The analysis established that electron velocities will tend to be smaller if bands are narrow in energy, therefore, they start behaving like very heavy particles. This quality of electron dynamic is clearly observed in transition metal bands as well as in insulator valence bands. On the other hand, the electrons are more mobile in semiconductors and simple metals simply because their bands tend to be broader thus not behaving like some heavy particles. Furthermore, the electrons in metals are behaving as free particles whose masses near that of a true electron mass. Inverse scattering analysis shows that when an electron is accelerated into the Brillouin Zone, it immediately jumps across the zone and enters the opposite face (Cakoni and David 92).

Figure 2: dispersion of electron wave functions

# Summary of Electronic Structure of Solids

The structure of solids is such that the atomic valence levels are broadened into bands that consist of as many states as the number of atoms present in the solid. These band states are discovered to contain mobile electrons with each state being characterized by the wave number and momentum all of which are restricted within the Brillouin Zone in the solid. In case solid contains only four neighboring atoms, then the atomic valence orbitals would combine to form bond orbitals to stabilize such arrangement – the atomic valence orbitals are formed between two electrons per bond and each set of neighbors. This creates covalent structures in solids consisting of bands of state that are based upon bond orbital. In such case some band orbitals are fully occupied by electrons while others are completely empty. It is essential to note that the covalent structure in solids is only stable if every bond does not contain exactly two electrons. Other factors that would make the covalent structure in solids unstable include availability of too little bond energy and too polar bond.

When the covalent structure in solid is not stable, there is a general tendency of the lattice to collapse to a denser structure. This would determine the structure of the solid. For instance, the solid would be an ionic crystal if the redistribution of electrons leaves every atomic shell either empty or full – this is the most stable arrangement. On the other hand, it would be a metallic structure if the redistribution of electrons leaves the bands of state partially occupied. Analysis of electronic structure of solid by inverse scattering method indicates that electron energy bands are dependent on interatomic matrix elements and a set or orbital energies (Pike and Pierre 76). This is clearly seen by representing electron states in linear combinations of atomic orbitals. This observation also implies that in solids all bands are given by free-electron approximation. In covalent solids, the kinetic energy can be as a small correction and that pseudo-potential is a dominant effect of a minor problem.

Inverse scattering method was used to analyze the variation of electronic structure beginning from an elemental semiconductor to more polar solids. For this reason, the starting point was germanium moving across the series. The analysis indicated that across this series, these metals contain the same total number of electrons as well as the same structure. That is to say, they are all isoelectronic in nature. The only difference comes in the number of nuclear charge that each metal contains – the nuclear charge actually increases on anion while it decreases on cation (Novikov 87). The analysis points to a qualitative variation in terms of electronic structure of each solid within this series. This qualitative variation in electronic structure is also exhibited in nonpolar solids whereby there are basically two types of atomic sites on the two opposite sides of the horizontal bonds.

Polar solids exhibit a peculiar quality whereby the nuclear charge on the atom to the right of horizontal bond increases across the series. This behavior tends to have a displacing effect on the bond charge; the displacing effect is higher for atoms that have higher nuclear charge. In most cases, the corresponding transfer of charge when the bond charge is displaced is larger than the overall change in nuclear charge. This behavior makes an atom with greater nuclear charge more negative than atoms with lesser nuclear charge. In most cases, electronic charge seems to reside on the nonmetallic atoms at high polarities. The most noticeable change is that as polarity of the atom increases, the gap between the valence bands opens up widely; in addition, the gap between the conduction band and valence bands also widens as polarity of the atom increases (Cakoni and David 209).

Works Cited

Antonov, Victor, Bruce Harmon, and Alexander Yaresko. *Electronic Structure and Magneto-Optical Properties of Solids*. Dordrecht: Kluwer Academic Publishers, 2004. Print.

Cakoni, Fioralba, and David Colton. *Qualitative Methods in Inverse Scattering Theory: An Introduction*. Berlin: Springer, 2006. Print.

Cakoni, Fioralba, and David L. Colton. *A Qualitative Approach to Inverse Scattering Theory*. , 2014. Internet resource.

Canadell, Enric, Marie-Liesse Doublet, and Christophe Iung. *Orbital Approach to the Electronic Structure of Solids*. Oxford: Oxford University Press, 2012. Print.

Dreysse, Hugues. *Electronic Structure and Physical Properties of Solids: The Uses of the Lmto Method*. Berlin [etc.: Springer, 2000. Print.

Harrison, Walter A. *Electronic Structure and the Properties of Solids: The Physics of the Chemical Bond*. Dover Publications, 2012. Print.

Kaxiras, Efthimios. *Atomic and Electronic Structure of Solids*. Beijing: Shi jie tu shu chu ban gong si Beijing gong si, 2007. Print.

Korepin, Vladimir E, N M. Bogoliubov, and A G. Izergin. *Quantum Inverse Scattering Method and Correlation Functions*. Cambridge, [England: Cambridge University Press, 1997. Print.

Novikov, S P. *Theory of Solitons: The Inverse Scattering Method*. New York: Consultants Bureau, 1984. Print.

Pike, E R, and Pierre C. Sabatier. *Scattering: Scattering and Inverse Scattering in Pure and Applied Science*. San Diego, California: Academic Press, 2002. Print.