Software engineering

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Software_engineering

Software engineering is an engineering-based approach to software development. A software engineer is a person who applies the engineering design process to design, develop, maintain, test, and evaluate computer software. The term programmer is sometimes used as a synonym, but may also refer more to implementation rather than design and can also lack connotations of engineering education or skills.

Engineering techniques are used to inform the software development process, which involves the definition, implementation, assessment, measurement, management, change, and improvement of the software life cycle process itself. It heavily uses software configuration management, which is about systematically controlling changes to the configuration, and maintaining the integrity and traceability of the configuration and code throughout the system life cycle. Modern processes use software versioning.

History

Main article: History of software engineering

Beginning in the 1960s, software engineering was seen as its own type of engineering. Additionally, the development of software engineering was seen as a struggle. It was difficult to keep up with the hardware which caused many problems for software engineers. Problems included software that was over budget, exceeded deadlines, required extensive de-bugging and maintenance, and unsuccessfully met the needs of consumers or was never even completed. In 1968 NATO held the first Software Engineering conference where issues related to software were addressed: guidelines and best practices for the development of software were established.

The origins of the term "software engineering" have been attributed to various sources. The term "software engineering" appeared in a list of services offered by companies in the June 1965 issue of COMPUTERS and AUTOMATION and was used more formally in the August 1966 issue of Communications of the ACM (Volume 9, number 8) "letter to the ACM membership" by the ACM President Anthony A. Oettinger, it is also associated with the title of a NATO conference in 1968 by Professor Friedrich L. Bauer, the first conference on software engineering. Margaret Hamilton described the discipline "software engineering" during the Apollo missions to give what they were doing legitimacy. At the time there was perceived to be a "software crisis". The 40th International Conference on Software Engineering (ICSE 2018) celebrates 50 years of "Software Engineering" with the Plenary Sessions' keynotes of Frederick Brooks and Margaret Hamilton.

In 1984, the Software Engineering Institute (SEI) was established as a federally funded research and development center headquartered on the campus of Carnegie Mellon University in Pittsburgh, Pennsylvania, United States. Watts Humphrey founded the SEI Software Process Program, aimed at understanding and managing the software engineering process. The Process Maturity Levels introduced would become the Capability Maturity Model Integration for Development(CMMI-DEV), which has defined how the US Government evaluates the abilities of a software development team.

Modern, generally accepted best-practices for software engineering have been collected by the ISO/IEC JTC 1/SC 7 subcommittee and published as the Software Engineering Body of Knowledge (SWEBOK). Software engineering is considered one of major computing disciplines.

Definitions and terminology

Notable definitions of software engineering include:

• "The systematic application of scientific and technological knowledge, methods, and experience to the design, implementation, testing, and documentation of software"—The Bureau of Labor Statistics—IEEE Systems and software engineering – Vocabulary
• "The application of a systematic, disciplined, quantifiable approach to the development, operation, and maintenance of software"—IEEE Standard Glossary of Software Engineering Terminology
• "an engineering discipline that is concerned with all aspects of software production"—Ian Sommerville
• "the establishment and use of sound engineering principles in order to economically obtain software that is reliable and works efficiently on real machines"—Fritz Bauer
• "a branch of computer science that deals with the design, implementation, and maintenance of complex computer programs"—Merriam-Webster
• "'software engineering' encompasses not just the act of writing code, but all of the tools and processes an organization uses to build and maintain that code over time. [...] Software engineering can be thought of as 'programming integrated over time.'"—Software Engineering at Google

The term has also been used less formally:

• as the informal contemporary term for the broad range of activities that were formerly called computer programming and systems analysis;
• as the broad term for all aspects of the practice of computer programming, as opposed to the theory of computer programming, which is formally studied as a sub-discipline of computer science;
• as the term embodying the advocacy of a specific approach to computer programming, one that urges that it be treated as an engineering discipline rather than an art or a craft, and advocates the codification of recommended practices.

Etymology of "software engineer"

Margaret Hamilton promoted the term "software engineering" during her work on the Apollo program. The term "engineering" was used to acknowledge that the work should be taken just as seriously as other contributions toward the advancement of technology. Hamilton details her use of the term:

When I first came up with the term, no one had heard of it before, at least in our world. It was an ongoing joke for a long time. They liked to kid me about my radical ideas. It was a memorable day when one of the most respected hardware gurus explained to everyone in a meeting that he agreed with me that the process of building software should also be considered an engineering discipline, just like with hardware. Not because of his acceptance of the new "term" per se, but because we had earned his and the acceptance of the others in the room as being in an engineering field in its own right.

Suitability of the term

Individual commentators have disagreed sharply on how to define software engineering or its legitimacy as an engineering discipline. David Parnas has said that software engineering is, in fact, a form of engineering. Steve McConnell has said that it is not, but that it should be. Donald Knuth has said that programming is an art and a science. Edsger W. Dijkstra claimed that the terms software engineering and software engineer have been misused and should be considered harmful, particularly in the United States.

Tasks in large scale projects

Main article: Outline of software engineering

Software requirements

Main article: Software requirements

Requirements engineering is about the elicitation, analysis, specification, and validation of requirements for software. Software requirements can be of three different types. There are functional requirements, non-functional requirements, and domain requirements. The operation of the software should be performed and the proper output should be expected for the user to use. Non-functional requirements deal with issues like portability, security, maintainability, reliability, scalability, performance, reusability, and flexibility. They are classified into the following types: interface constraints, performance constraints (such as response time, security, storage space, etc.), operating constraints, life cycle constraints (maintainability, portability, etc.), and economic constraints. Knowledge of how the system or software works is needed when it comes to specifying non-functional requirements. Domain requirements have to do with the characteristic of a certain category or domain of projects.

Software design

Main article: Software design

Software design is about the process of defining the architecture, components, interfaces, and other characteristics of a system or component. This is also called software architecture. Software design is divided into three different levels of design. The three levels are interface design, architectural design, and detailed design. Interface design is the interaction between a system and its environment. This happens at a high level of abstraction along with the inner workings of the system. Architectural design has to do with the major components of a system and their responsibilities, properties, interfaces, and their relationships and interactions that occur between them. Detailed design is the internal elements of all the major system components, their properties, relationships, processing, and usually their algorithms and the data structures.

Software construction

Main article: Software construction

Software construction, the main activity of software development, is the combination of programming, unit testing, integration testing, and debugging so as to implement the design. Testing during this phase is generally performed by the programmer while the software is under construction, to verify what was just written and decide when the code is ready to be sent to the next step.

Software testing

Main article: Software testing

Software testing is an empirical, technical investigation conducted to provide stakeholders with information about the quality of the product or service under test, with different approaches such as unit testing and integration testing. It is one aspect of software quality. As a separate phase in software development, it is typically performed by quality assurance staff or a developer other than the one who wrote the code.

Software analysis

Main article: Software analysis

Software analysis is the process of analyzing the behavior of computer programs regarding a property such as performance, robustness, and security It can be performed without executing the program (static program analysis), during runtime (dynamic program analysis) or in a combination of both.

Software maintenance

Main article: Software maintenance

Software maintenance refers to the activities required to provide cost-effective support after shipping the software product. Software maintenance is modifying and updating software applications after distribution to correct faults and to improve its performance. Software has a lot to do with the real world and when the real world changes, software maintenance is required. Software maintenance includes: error correction, optimization, deletion of unused and discarded features, and enhancement of features that already exist. Usually, maintenance takes up about 40% to 80% of the project cost therefore, focusing on maintenance keeps the costs down.

Education

Knowledge of computer programming is a prerequisite for becoming a software engineer. In 2004 the IEEE Computer Society produced the SWEBOK, which has been published as ISO/IEC Technical Report 1979:2005, describing the body of knowledge that they recommend to be mastered by a graduate software engineer with four years of experience. Many software engineers enter the profession by obtaining a university degree or training at a vocational school. One standard international curriculum for undergraduate software engineering degrees was defined by the Joint Task Force on Computing Curricula of the IEEE Computer Society and the Association for Computing Machinery, and updated in 2014. A number of universities have Software Engineering degree programs; as of 2010, there were 244 Campus Bachelor of Software Engineering programs, 70 Online programs, 230 Masters-level programs, 41 Doctorate-level programs, and 69 Certificate-level programs in the United States.

In addition to university education, many companies sponsor internships for students wishing to pursue careers in information technology. These internships can introduce the student to interesting real-world tasks that typical software engineers encounter every day. Similar experience can be gained through military service in software engineering.

Software engineering degree programs

Half of all practitioners today have degrees in computer science, information systems, or information technology. A small, but growing, number of practitioners have software engineering degrees. In 1987, the Department of Computing at Imperial College London introduced the first three-year software engineering Bachelor's degree in the UK and the world; in the following year, the University of Sheffield established a similar program. In 1996, the Rochester Institute of Technology established the first software engineering bachelor's degree program in the United States, however, it did not obtain ABET accreditation until 2003, the same time as Rice University, Clarkson University, Milwaukee School of Engineering and Mississippi State University obtained theirs. In 1997, PSG College of Technology in Coimbatore, India was the first to start a five-year integrated Master of Science degree in Software Engineering.

Since then, software engineering undergraduate degrees have been established at many universities. A standard international curriculum for undergraduate software engineering degrees, SE2004, was defined by a steering committee between 2001 and 2004 with funding from the Association for Computing Machinery and the IEEE Computer Society. As of 2004, in the U.S., about 50 universities offer software engineering degrees, which teach both computer science and engineering principles and practices. The first software engineering Master's degree was established at Seattle University in 1979. Since then graduate software engineering degrees have been made available from many more universities. Likewise in Canada, the Canadian Engineering Accreditation Board (CEAB) of the Canadian Council of Professional Engineers has recognized several software engineering programs.

In 1998, the US Naval Postgraduate School (NPS) established the first doctorate program in Software Engineering in the world. Additionally, many online advanced degrees in Software Engineering have appeared such as the Master of Science in Software Engineering (MSE) degree offered through the Computer Science and Engineering Department at California State University, Fullerton. Steve McConnell opines that because most universities teach computer science rather than software engineering, there is a shortage of true software engineers. ETS (École de technologie supérieure) University and UQAM (Université du Québec à Montréal) were mandated by IEEE to develop the Software Engineering Body of Knowledge (SWEBOK), which has become an ISO standard describing the body of knowledge covered by a software engineer.

Profession

Main articles: Software engineering professionalism and Regulation and licensure in engineering

Legal requirements for the licensing or certification of professional software engineers vary around the world. In the UK, there is no licensing or legal requirement to assume or use the job title Software Engineer. In some areas of Canada, such as Alberta, British Columbia, Ontario, and Quebec, software engineers can hold the Professional Engineer (P.Eng) designation and/or the Information Systems Professional (I.S.P.) designation. In Europe, Software Engineers can obtain the European Engineer (EUR ING) professional title.

In the United States, the NCEES began offering a Professional Engineer exam for Software Engineering in 2013, thereby allowing Software Engineers to be licensed and recognized. NCEES ended the exam after April 2019 due to lack of participation. Mandatory licensing is currently still largely debated, and perceived as controversial.

The IEEE Computer Society and the ACM, the two main US-based professional organizations of software engineering, publish guides to the profession of software engineering. The IEEE's Guide to the Software Engineering Body of Knowledge – 2004 Version, or SWEBOK, defines the field and describes the knowledge the IEEE expects a practicing software engineer to have. The most current SWEBOK v3 is an updated version and was released in 2014. The IEEE also promulgates a "Software Engineering Code of Ethics".

Employment

The examples and perspective in this United States may not represent a worldwide view of the subject. You may improve this United States, discuss the issue on the talk page, or create a new United States, as appropriate. (September 2021) (Learn how and when to remove this template message)

See also: Software engineering demographics

There are an estimated 26.9 million professional software engineers in the world as of 2022, up from 21 million in 2016.

Many software engineers work as employees or contractors. Software engineers work with businesses, government agencies (civilian or military), and non-profit organizations. Some software engineers work for themselves as freelancers. Some organizations have specialists to perform each of the tasks in the software development process. Other organizations require software engineers to do many or all of them. In large projects, people may specialize in only one role. In small projects, people may fill several or all roles at the same time. Many companies hire interns, often university or college students during a summer break, or externships. Specializations include analysts, architects, developers, testers, technical support, middleware analysts, project managers, educators, and researchers.

Most software engineers and programmers work 40 hours a week, but about 15 percent of software engineers and 11 percent of programmers worked more than 50 hours a week in 2008. Potential injuries in these occupations are possible because like other workers who spend long periods sitting in front of a computer terminal typing at a keyboard, engineers and programmers are susceptible to eyestrain, back discomfort, and hand and wrist problems such as carpal tunnel syndrome.

United States

The U. S. Bureau of Labor Statistics (BLS) counted 1,365,500 software developers holding jobs in the U.S. in 2018. Due to its relative newness as a field of study, formal education in software engineering is often taught as part of a computer science curriculum, and many software engineers hold computer science degrees. The BLS estimates from 2014 to 2024 that computer software engineering would increase by 17% . This is down from the 2012 to 2022 BLS estimate of 22% for software engineering. And, is further down from their 30% 2010 to 2020 BLS estimate. Due to this trend, job growth may not be as fast as during the last decade, as jobs that would have gone to computer software engineers in the United States would instead be outsourced to computer software engineers in countries such as India and other foreign countries. In addition, the BLS Job Outlook for Computer Programmers, 2014–24 predicts an −8% (a decline, in their words), then a decline in the Job Outlook, 2019-29 of -9%, and a 10% decline for 2021-2031 for those who program computers. Furthermore, women in many software fields has also been declining over the years as compared to other engineering fields. Then there is the additional concern that recent advances in Artificial Intelligence might impact the demand for future generations of Software Engineers. However, this trend may change or slow in the future as many current software engineers in the U.S. market leave the profession or age out of the market in the next few decades.

Certification

The Software Engineering Institute offers certifications on specific topics like security, process improvement and software architecture. IBM, Microsoft and other companies also sponsor their own certification examinations. Many IT certification programs are oriented toward specific technologies, and managed by the vendors of these technologies. These certification programs are tailored to the institutions that would employ people who use these technologies.

Broader certification of general software engineering skills is available through various professional societies. As of 2006, the IEEE had certified over 575 software professionals as a Certified Software Development Professional (CSDP). In 2008 they added an entry-level certification known as the Certified Software Development Associate (CSDA). The ACM had a professional certification program in the early 1980s, which was discontinued due to lack of interest. The ACM examined the possibility of professional certification of software engineers in the late 1990s, but eventually decided that such certification was inappropriate for the professional industrial practice of software engineering.

In the U.K. the British Computer Society has developed a legally recognized professional certification called Chartered IT Professional (CITP), available to fully qualified members (MBCS). Software engineers may be eligible for membership of the Institution of Engineering and Technology and so qualify for Chartered Engineer status. In Canada the Canadian Information Processing Society has developed a legally recognized professional certification called Information Systems Professional (ISP). In Ontario, Canada, Software Engineers who graduate from a Canadian Engineering Accreditation Board (CEAB) accredited program, successfully complete PEO's (Professional Engineers Ontario) Professional Practice Examination (PPE) and have at least 48 months of acceptable engineering experience are eligible to be licensed through the Professional Engineers Ontario and can become Professional Engineers P.Eng. The PEO does not recognize any online or distance education however; and does not consider Computer Science programs to be equivalent to software engineering programs despite the tremendous overlap between the two. This has sparked controversy and a certification war. It has also held the number of P.Eng holders for the profession exceptionally low. The vast majority of working professionals in the field hold a degree in CS, not SE. Given the difficult certification path for holders of non-SE degrees, most never bother to pursue the license.

Impact of globalization

The initial impact of outsourcing, and the relatively lower cost of international human resources in developing third world countries led to a massive migration of software development activities from corporations in North America and Europe to India and later: China, Russia, and other developing countries. This approach had some flaws, mainly the distance / time zone difference that prevented human interaction between clients and developers and the massive job transfer. This had a negative impact on many aspects of the software engineering profession. For example, some students in the developed world avoid education related to software engineering because of the fear of offshore outsourcing (importing software products or services from other countries) and of being displaced by foreign visa workers. Although statistics do not currently show a threat to software engineering itself; a related career, computer programming does appear to have been affected. Nevertheless, the ability to smartly leverage offshore and near-shore resources via the follow-the-sun workflow has improved the overall operational capability of many organizations. When North Americans are leaving work, Asians are just arriving to work. When Asians are leaving work, Europeans are arriving to work. This provides a continuous ability to have human oversight on business-critical processes 24 hours per day, without paying overtime compensation or disrupting a key human resource, sleep patterns.

While global outsourcing has several advantages, global – and generally distributed – development can run into serious difficulties resulting from the distance between developers. This is due to the key elements of this type of distance that have been identified as geographical, temporal, cultural and communication (that includes the use of different languages and dialects of English in different locations). Research has been carried out in the area of global software development over the last 15 years and an extensive body of relevant work published that highlights the benefits and problems associated with the complex activity. As with other aspects of software engineering research is ongoing in this and related areas.

Prizes

There are several prizes in the field of software engineering:

• The Codie awards is a yearly award issued by the Software and Information Industry Association for excellence in software development within the software industry.
• Jolt Awards are awards in the software industry.
• Stevens Award is a software engineering award given in memory of Wayne Stevens.
• Harlan Mills Award for "contributions to the theory and practice of the information sciences, focused on software engineering".

Criticism

Software engineering sees its practitioners as individuals who follow well-defined engineering approaches to problem-solving. These approaches are specified in various software engineering books and research papers, always with the connotations of predictability, precision, mitigated risk and professionalism. This perspective has led to calls for licensing, certification and codified bodies of knowledge as mechanisms for spreading the engineering knowledge and maturing the field.

Software engineering extends engineering and draws on the engineering model, i.e. engineering process, engineering project management, engineering requirements, engineering design, engineering construction, and engineering validation. The concept is so new that it is rarely understood, and it is widely misinterpreted, including in software engineering textbooks, papers, and among the communities of programmers and crafters.

One of the core issues in software engineering is that its approaches are not empirical enough because a real-world validation of approaches is usually absent, or very limited and hence software engineering is often misinterpreted as feasible only in a "theoretical environment."

Edsger Dijkstra, the founder of many of the concepts used within software development today, rejected the idea of "software engineering" up until his death in 2002, arguing that those terms were poor analogies for what he called the "radical novelty" of computer science:

A number of these phenomena have been bundled under the name "Software Engineering". As economics is known as "The Miserable Science", software engineering should be known as "The Doomed Discipline", doomed because it cannot even approach its goal since its goal is self-contradictory. Software engineering, of course, presents itself as another worthy cause, but that is eyewash: if you carefully read its literature and analyse what its devotees actually do, you will discover that software engineering has accepted as its charter "How to program if you cannot."

The future of software engineering

New and emerging technologies combined with changing industry dynamics are set to bring about a transformative era in the realm of software engineering.

A recent report by Gartner shows that a significant shift towards low code and no code development is set to bring in more non-technical individuals into software development.

Additionally, the advent of artificial intelligence (AI) and machine learning (ML) will continue to play a crucial role in automating software development, all while improving the quality of software built.

The rise of cloud computing and DevOps will further accelerate software development processes and improve the pace of software delivery and deployment.

These modern technologies work with both structured and unstructured data and encourage the implementation of secure coding practices to ensure that the software built is reliable and secure. 

Electronic band structure

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Electronic_band_structure

In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called band gaps or forbidden bands).

Band theory derives these bands and band gaps by examining the allowed quantum mechanical wave functions for an electron in a large, periodic lattice of atoms or molecules. Band theory has been successfully used to explain many physical properties of solids, such as electrical resistivity and optical absorption, and forms the foundation of the understanding of all solid-state devices (transistors, solar cells, etc.).

Why bands and band gaps occur

A hypothetical example of a large number of carbon atoms being brought together to form a diamond crystal, demonstrating formation of the electronic band structure. The right graph shows the energy levels as a function of the spacing between atoms. When far apart (right side of graph) all the atoms have discrete valence orbitals p and s with the same energies. However, when the atoms come closer (left side), their electron orbitals begin to spatially overlap. The orbitals hybridize into N molecular orbitals each with a different energy, where N is the number of atoms in the crystal. Since N is such a large number, adjacent orbitals are extremely close together in energy so the orbitals can be considered a continuous energy band. At the actual diamond crystal cell size (denoted by a), two bands are formed, called the valence and conduction bands, separated by a 5.5 eV band gap. Decreasing the inter-atomic spacing even more (e.g., under a high pressure) further modifies the band structure.

The electrons of a single, isolated atom occupy atomic orbitals each of which has a discrete energy level. When two or more atoms join to form a molecule, their atomic orbitals overlap and hybridize.

Similarly, if a large number N of identical atoms come together to form a solid, such as a crystal lattice, the atoms' atomic orbitals overlap with the nearby orbitals. Each discrete energy level splits into N levels, each with a different energy. Since the number of atoms in a macroscopic piece of solid is a very large number (N~10²²) the number of orbitals is very large and thus they are very closely spaced in energy (of the order of 10⁻²² eV). The energy of the adjacent levels is so close together that they can be considered as a continuum, an energy band.

This formation of bands is mostly a feature of the outermost electrons (valence electrons) in the atom, which are the ones involved in chemical bonding and electrical conductivity. The inner electron orbitals do not overlap to a significant degree, so their bands are very narrow.

Band gaps are essentially leftover ranges of energy not covered by any band, a result of the finite widths of the energy bands. The bands have different widths, with the widths depending upon the degree of overlap in the atomic orbitals from which they arise. Two adjacent bands may simply not be wide enough to fully cover the range of energy. For example, the bands associated with core orbitals (such as 1s electrons) are extremely narrow due to the small overlap between adjacent atoms. As a result, there tend to be large band gaps between the core bands. Higher bands involve comparatively larger orbitals with more overlap, becoming progressively wider at higher energies so that there are no band gaps at higher energies.

Basic concepts

Assumptions and limits of band structure theory

Band theory is only an approximation to the quantum state of a solid, which applies to solids consisting of many identical atoms or molecules bonded together. These are the assumptions necessary for band theory to be valid:

• Infinite-size system: For the bands to be continuous, the piece of material must consist of a large number of atoms. Since a macroscopic piece of material contains on the order of 10²² atoms, this is not a serious restriction; band theory even applies to microscopic-sized transistors in integrated circuits. With modifications, the concept of band structure can also be extended to systems which are only "large" along some dimensions, such as two-dimensional electron systems.
• Homogeneous system: Band structure is an intrinsic property of a material, which assumes that the material is homogeneous. Practically, this means that the chemical makeup of the material must be uniform throughout the piece.
• Non-interactivity: The band structure describes "single electron states". The existence of these states assumes that the electrons travel in a static potential without dynamically interacting with lattice vibrations, other electrons, photons, etc.

The above assumptions are broken in a number of important practical situations, and the use of band structure requires one to keep a close check on the limitations of band theory:

• Inhomogeneities and interfaces: Near surfaces, junctions, and other inhomogeneities, the bulk band structure is disrupted. Not only are there local small-scale disruptions (e.g., surface states or dopant states inside the band gap), but also local charge imbalances. These charge imbalances have electrostatic effects that extend deeply into semiconductors, insulators, and the vacuum (see doping, band bending).
• Along the same lines, most electronic effects (capacitance, electrical conductance, electric-field screening) involve the physics of electrons passing through surfaces and/or near interfaces. The full description of these effects, in a band structure picture, requires at least a rudimentary model of electron-electron interactions (see space charge, band bending).
• Small systems: For systems which are small along every dimension (e.g., a small molecule or a quantum dot), there is no continuous band structure. The crossover between small and large dimensions is the realm of mesoscopic physics.
• Strongly correlated materials (for example, Mott insulators) simply cannot be understood in terms of single-electron states. The electronic band structures of these materials are poorly defined (or at least, not uniquely defined) and may not provide useful information about their physical state.

Crystalline symmetry and wavevectors

Fig 1. Brillouin zone of a face-centered cubic lattice showing labels for special symmetry points.

 

Fig 2. Band structure plot for Si, Ge, GaAs and InAs generated with tight binding model. Note that Si and Ge are indirect band gap materials, while GaAs and InAs are direct.

 

Main articles: Bloch's theorem and Brillouin zone

See also: Symmetry in physics, Crystallographic point group, and Space group

Band structure calculations take advantage of the periodic nature of a crystal lattice, exploiting its symmetry. The single-electron Schrödinger equation is solved for an electron in a lattice-periodic potential, giving Bloch electrons as solutions

${\displaystyle {\psi }_{n\mathbf{k}}(\mathbf{r})=e^{i\mathbf{k}\cdot \mathbf{r}}{u}_{n\mathbf{k}}(\mathbf{r})}$,

where k is called the wavevector. For each value of k, there are multiple solutions to the Schrödinger equation labelled by n, the band index, which simply numbers the energy bands. Each of these energy levels evolves smoothly with changes in k, forming a smooth band of states. For each band we can define a function E_n(k), which is the dispersion relation for electrons in that band.

The wavevector takes on any value inside the Brillouin zone, which is a polyhedron in wavevector (reciprocal lattice) space that is related to the crystal's lattice. Wavevectors outside the Brillouin zone simply correspond to states that are physically identical to those states within the Brillouin zone. Special high symmetry points/lines in the Brillouin zone are assigned labels like Γ, Δ, Λ, Σ (see Fig 1).

It is difficult to visualize the shape of a band as a function of wavevector, as it would require a plot in four-dimensional space, E vs. k_x, k_y, k_z. In scientific literature it is common to see band structure plots which show the values of E_n(k) for values of k along straight lines connecting symmetry points, often labelled Δ, Λ, Σ, or [100], [111], and [110], respectively. Another method for visualizing band structure is to plot a constant-energy isosurface in wavevector space, showing all of the states with energy equal to a particular value. The isosurface of states with energy equal to the Fermi level is known as the Fermi surface.

Energy band gaps can be classified using the wavevectors of the states surrounding the band gap:

• Direct band gap: the lowest-energy state above the band gap has the same k as the highest-energy state beneath the band gap.
• Indirect band gap: the closest states above and beneath the band gap do not have the same k value.

Asymmetry: Band structures in non-crystalline solids

Although electronic band structures are usually associated with crystalline materials, quasi-crystalline and amorphous solids may also exhibit band gaps. These are somewhat more difficult to study theoretically since they lack the simple symmetry of a crystal, and it is not usually possible to determine a precise dispersion relation. As a result, virtually all of the existing theoretical work on the electronic band structure of solids has focused on crystalline materials.

Density of states

Main article: Density of states

The density of states function g(E) is defined as the number of electronic states per unit volume, per unit energy, for electron energies near E.

The density of states function is important for calculations of effects based on band theory. In Fermi's Golden Rule, a calculation for the rate of optical absorption, it provides both the number of excitable electrons and the number of final states for an electron. It appears in calculations of electrical conductivity where it provides the number of mobile states, and in computing electron scattering rates where it provides the number of final states after scattering.

For energies inside a band gap, g(E) = 0.

Filling of bands

Main articles: Fermi level and Fermi–Dirac statistics

 

Filling of the electronic states in various types of materials at equilibrium. Here, height is energy while width is the density of available states for a certain energy in the material listed. The shade follows the Fermi–Dirac distribution (black: all states filled, white: no state filled). In metals and semimetals the Fermi level E_F lies inside at least one band.

In insulators and semiconductors the Fermi level is inside a band gap; however, in semiconductors the bands are near enough to the Fermi level to be thermally populated with electrons or holes.

At thermodynamic equilibrium, the likelihood of a state of energy E being filled with an electron is given by the Fermi–Dirac distribution, a thermodynamic distribution that takes into account the Pauli exclusion principle:

${\displaystyle f(E)=\frac{1}{1+e^{(E-\mu )/k_{\mathrm{B}}T}}}$

where:

• k_BT is the product of Boltzmann's constant and temperature, and
• µ is the total chemical potential of electrons, or Fermi level (in semiconductor physics, this quantity is more often denoted E_F). The Fermi level of a solid is directly related to the voltage on that solid, as measured with a voltmeter. Conventionally, in band structure plots the Fermi level is taken to be the zero of energy (an arbitrary choice).

The density of electrons in the material is simply the integral of the Fermi–Dirac distribution times the density of states:

${\displaystyle N/V={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}g(E)f(E)dE}$

Although there are an infinite number of bands and thus an infinite number of states, there are only a finite number of electrons to place in these bands. The preferred value for the number of electrons is a consequence of electrostatics: even though the surface of a material can be charged, the internal bulk of a material prefers to be charge neutral. The condition of charge neutrality means that N/V must match the density of protons in the material. For this to occur, the material electrostatically adjusts itself, shifting its band structure up or down in energy (thereby shifting g(E)), until it is at the correct equilibrium with respect to the Fermi level.

Names of bands near the Fermi level (conduction band, valence band)

A solid has an infinite number of allowed bands, just as an atom has infinitely many energy levels. However, most of the bands simply have too high energy, and are usually disregarded under ordinary circumstances. Conversely, there are very low energy bands associated with the core orbitals (such as 1s electrons). These low-energy core bands are also usually disregarded since they remain filled with electrons at all times, and are therefore inert. Likewise, materials have several band gaps throughout their band structure.

The most important bands and band gaps—those relevant for electronics and optoelectronics—are those with energies near the Fermi level. The bands and band gaps near the Fermi level are given special names, depending on the material:

• In a semiconductor or band insulator, the Fermi level is surrounded by a band gap, referred to as the band gap (to distinguish it from the other band gaps in the band structure). The closest band above the band gap is called the conduction band, and the closest band beneath the band gap is called the valence band. The name "valence band" was coined by analogy to chemistry, since in semiconductors (and insulators) the valence band is built out of the valence orbitals.
• In a metal or semimetal, the Fermi level is inside of one or more allowed bands. In semimetals the bands are usually referred to as "conduction band" or "valence band" depending on whether the charge transport is more electron-like or hole-like, by analogy to semiconductors. In many metals, however, the bands are neither electron-like nor hole-like, and often just called "valence band" as they are made of valence orbitals. The band gaps in a metal's band structure are not important for low energy physics, since they are too far from the Fermi level.

Theory in crystals

The ansatz is the special case of electron waves in a periodic crystal lattice using Bloch's theorem as treated generally in the dynamical theory of diffraction. Every crystal is a periodic structure which can be characterized by a Bravais lattice, and for each Bravais lattice we can determine the reciprocal lattice, which encapsulates the periodicity in a set of three reciprocal lattice vectors (b₁, b₂, b₃). Now, any periodic potential V(r) which shares the same periodicity as the direct lattice can be expanded out as a Fourier series whose only non-vanishing components are those associated with the reciprocal lattice vectors. So the expansion can be written as:

${\displaystyle V(\mathbf{r})=\sum _{\mathbf{K}}{V}_{\mathbf{K}}{e}^{i\mathbf{K}\cdot \mathbf{r}}}$

where K = m₁b₁ + m₂b₂ + m₃b₃ for any set of integers (m₁, m₂, m₃).

From this theory, an attempt can be made to predict the band structure of a particular material, however most ab initio methods for electronic structure calculations fail to predict the observed band gap.

Nearly free electron approximation

Main articles: Nearly free electron model, Free electron model, and pseudopotential

In the nearly free electron approximation, interactions between electrons are completely ignored. This approximation allows use of Bloch's Theorem which states that electrons in a periodic potential have wavefunctions and energies which are periodic in wavevector up to a constant phase shift between neighboring reciprocal lattice vectors. The consequences of periodicity are described mathematically by the Bloch's theorem, which states that the eigenstate wavefunctions have the form

${\displaystyle {\mathrm{\Psi }}_{n,\mathbf{k}}(\mathbf{r})=e^{i\mathbf{k}\cdot \mathbf{r}}{u}_n(\mathbf{r})}$

where the Bloch function ${\displaystyle u_n(\mathbf{r})}$ is periodic over the crystal lattice, that is,

${\displaystyle u_n(\mathbf{r})=u_n(\mathbf{r}-\mathbf{R})}$.

Here index n refers to the n-th energy band, wavevector k is related to the direction of motion of the electron, r is the position in the crystal, and R is the location of an atomic site.

The NFE model works particularly well in materials like metals where distances between neighbouring atoms are small. In such materials the overlap of atomic orbitals and potentials on neighbouring atoms is relatively large. In that case the wave function of the electron can be approximated by a (modified) plane wave. The band structure of a metal like aluminium even gets close to the empty lattice approximation.

Tight binding model

Main article: Tight binding

The opposite extreme to the nearly free electron approximation assumes the electrons in the crystal behave much like an assembly of constituent atoms. This tight binding model assumes the solution to the time-independent single electron Schrödinger equation ${\displaystyle \mathrm{\Psi }}$ is well approximated by a linear combination of atomic orbitals ${\displaystyle {\psi }_n(\mathbf{r})}$.

${\displaystyle \mathrm{\Psi }(\mathbf{r})=\sum _{n,\mathbf{R}}{b}_{n,\mathbf{R}}{\psi }_n(\mathbf{r}-\mathbf{R})}$,

where the coefficients ${\displaystyle b_{n,\mathbf{R}}}$ are selected to give the best approximate solution of this form. Index n refers to an atomic energy level and R refers to an atomic site. A more accurate approach using this idea employs Wannier functions, defined by:

${\displaystyle a_n(\mathbf{r}-\mathbf{R})=\frac{V_C}{(2\pi {)}^3}{\int }_{\text{BZ}}d\mathbf{k}{e}^{-i\mathbf{k}\cdot (\mathbf{R}-\mathbf{r})}{u}_{n\mathbf{k}}}$;

in which ${\displaystyle u_{n\mathbf{k}}}$ is the periodic part of the Bloch's theorem and the integral is over the Brillouin zone. Here index n refers to the n-th energy band in the crystal. The Wannier functions are localized near atomic sites, like atomic orbitals, but being defined in terms of Bloch functions they are accurately related to solutions based upon the crystal potential. Wannier functions on different atomic sites R are orthogonal. The Wannier functions can be used to form the Schrödinger solution for the n-th energy band as:

${\displaystyle {\mathrm{\Psi }}_{n,\mathbf{k}}(\mathbf{r})=\sum _{\mathbf{R}}{e}^{-i\mathbf{k}\cdot (\mathbf{R}-\mathbf{r})}{a}_n(\mathbf{r}-\mathbf{R})}$.

The TB model works well in materials with limited overlap between atomic orbitals and potentials on neighbouring atoms. Band structures of materials like Si, GaAs, SiO₂ and diamond for instance are well described by TB-Hamiltonians on the basis of atomic sp³ orbitals. In transition metals a mixed TB-NFE model is used to describe the broad NFE conduction band and the narrow embedded TB d-bands. The radial functions of the atomic orbital part of the Wannier functions are most easily calculated by the use of pseudopotential methods. NFE, TB or combined NFE-TB band structure calculations, sometimes extended with wave function approximations based on pseudopotential methods, are often used as an economic starting point for further calculations.

KKR model

Main article: Multiple scattering theory

The KKR method, also called "multiple scattering theory" or Green's function method, finds the stationary values of the inverse transition matrix T rather than the Hamiltonian. A variational implementation was suggested by Korringa, Kohn and Rostocker, and is often referred to as the Korringa–Kohn–Rostoker method. The most important features of the KKR or Green's function formulation are (1) it separates the two aspects of the problem: structure (positions of the atoms) from the scattering (chemical identity of the atoms); and (2) Green's functions provide a natural approach to a localized description of electronic properties that can be adapted to alloys and other disordered system. The simplest form of this approximation centers non-overlapping spheres (referred to as muffin tins) on the atomic positions. Within these regions, the potential experienced by an electron is approximated to be spherically symmetric about the given nucleus. In the remaining interstitial region, the screened potential is approximated as a constant. Continuity of the potential between the atom-centered spheres and interstitial region is enforced.

Density-functional theory

Main article: Density functional theory

See also: Kohn–Sham equations

In recent physics literature, a large majority of the electronic structures and band plots are calculated using density-functional theory (DFT), which is not a model but rather a theory, i.e., a microscopic first-principles theory of condensed matter physics that tries to cope with the electron-electron many-body problem via the introduction of an exchange-correlation term in the functional of the electronic density. DFT-calculated bands are in many cases found to be in agreement with experimentally measured bands, for example by angle-resolved photoemission spectroscopy (ARPES). In particular, the band shape is typically well reproduced by DFT. But there are also systematic errors in DFT bands when compared to experiment results. In particular, DFT seems to systematically underestimate by about 30-40% the band gap in insulators and semiconductors.

It is commonly believed that DFT is a theory to predict ground state properties of a system only (e.g. the total energy, the atomic structure, etc.), and that excited state properties cannot be determined by DFT. This is a misconception. In principle, DFT can determine any property (ground state or excited state) of a system given a functional that maps the ground state density to that property. This is the essence of the Hohenberg–Kohn theorem. In practice, however, no known functional exists that maps the ground state density to excitation energies of electrons within a material. Thus, what in the literature is quoted as a DFT band plot is a representation of the DFT Kohn–Sham energies, i.e., the energies of a fictive non-interacting system, the Kohn–Sham system, which has no physical interpretation at all. The Kohn–Sham electronic structure must not be confused with the real, quasiparticle electronic structure of a system, and there is no Koopmans' theorem holding for Kohn–Sham energies, as there is for Hartree–Fock energies, which can be truly considered as an approximation for quasiparticle energies. Hence, in principle, Kohn–Sham based DFT is not a band theory, i.e., not a theory suitable for calculating bands and band-plots. In principle time-dependent DFT can be used to calculate the true band structure although in practice this is often difficult. A popular approach is the use of hybrid functionals, which incorporate a portion of Hartree–Fock exact exchange; this produces a substantial improvement in predicted bandgaps of semiconductors, but is less reliable for metals and wide-bandgap materials.

Green's function methods and the ab initio GW approximation

Main articles: Green's function (many-body theory) and Green–Kubo relations

To calculate the bands including electron-electron interaction many-body effects, one can resort to so-called Green's function methods. Indeed, knowledge of the Green's function of a system provides both ground (the total energy) and also excited state observables of the system. The poles of the Green's function are the quasiparticle energies, the bands of a solid. The Green's function can be calculated by solving the Dyson equation once the self-energy of the system is known. For real systems like solids, the self-energy is a very complex quantity and usually approximations are needed to solve the problem. One such approximation is the GW approximation, so called from the mathematical form the self-energy takes as the product Σ = GW of the Green's function G and the dynamically screened interaction W. This approach is more pertinent when addressing the calculation of band plots (and also quantities beyond, such as the spectral function) and can also be formulated in a completely ab initio way. The GW approximation seems to provide band gaps of insulators and semiconductors in agreement with experiment, and hence to correct the systematic DFT underestimation.

Dynamical mean-field theory

Main article: Dynamical mean-field theory

Although the nearly free electron approximation is able to describe many properties of electron band structures, one consequence of this theory is that it predicts the same number of electrons in each unit cell. If the number of electrons is odd, we would then expect that there is an unpaired electron in each unit cell, and thus that the valence band is not fully occupied, making the material a conductor. However, materials such as CoO that have an odd number of electrons per unit cell are insulators, in direct conflict with this result. This kind of material is known as a Mott insulator, and requires inclusion of detailed electron-electron interactions (treated only as an averaged effect on the crystal potential in band theory) to explain the discrepancy. The Hubbard model is an approximate theory that can include these interactions. It can be treated non-perturbatively within the so-called dynamical mean-field theory, which attempts to bridge the gap between the nearly free electron approximation and the atomic limit. Formally, however, the states are not non-interacting in this case and the concept of a band structure is not adequate to describe these cases.

Others

Calculating band structures is an important topic in theoretical solid state physics. In addition to the models mentioned above, other models include the following:

• Empty lattice approximation: the "band structure" of a region of free space that has been divided into a lattice.
• k·p perturbation theory is a technique that allows a band structure to be approximately described in terms of just a few parameters. The technique is commonly used for semiconductors, and the parameters in the model are often determined by experiment.
• The Kronig–Penney model, a one-dimensional rectangular well model useful for illustration of band formation. While simple, it predicts many important phenomena, but is not quantitative.
• Hubbard model

The band structure has been generalised to wavevectors that are complex numbers, resulting in what is called a complex band structure, which is of interest at surfaces and interfaces.

Each model describes some types of solids very well, and others poorly. The nearly free electron model works well for metals, but poorly for non-metals. The tight binding model is extremely accurate for ionic insulators, such as metal halide salts (e.g. NaCl).

Band diagrams

To understand how band structure changes relative to the Fermi level in real space, a band structure plot is often first simplified in the form of a band diagram. In a band diagram the vertical axis is energy while the horizontal axis represents real space. Horizontal lines represent energy levels, while blocks represent energy bands. When the horizontal lines in these diagram are slanted then the energy of the level or band changes with distance. Diagrammatically, this depicts the presence of an electric field within the crystal system. Band diagrams are useful in relating the general band structure properties of different materials to one another when placed in contact with each other.