Editing Nonmetal (section)

=== Structure, quantum mechanics and band structure ===
The early terminologies were empirical categorizations based upon observables. As the 20th century started there were significant changes in understanding. The first was due to methods, mainly [[x-ray crystallography]], for determining how atoms are arranged in the various materials. As early as 1784 [[René Just Haüy]] discovered that every face of a crystal could be described by simple stacking patterns of blocks of the same shape and size ([[Law of rational indices|law of decrements]]).<ref>{{Cite book |last=Authier |first=André |title=Early days of X-ray crystallography |date=2013 |publisher=Oxford university press |isbn=978-0-19-965984-5 |location=Oxford}}</ref>  Haüy's study led to the idea that crystals are a regular three-dimensional array (a [[Bravais lattice]]) of atoms and [[molecule]]s, with a single [[unit cell]] repeated indefinitely, along with other developments in the [[Physical crystallography before X-rays|early days of physical crystallography]]. After [[Max von Laue]] demonstrated in 1912 that x-rays diffract,<ref>{{cite journal |vauthors=von Laue M |date=1914 |title=Concerning the detection of x-ray interferences |url=http://nobelprize.org/nobel_prizes/physics/laureates/1914/laue-lecture.pdf |url-status=live |journal=Nobel Lectures, Physics |volume=1901–1921 |archive-url=https://web.archive.org/web/20101207113911/http://nobelprize.org/nobel_prizes/physics/laureates/1914/laue-lecture.pdf |archive-date=2010-12-07 |access-date=2009-02-18}}</ref> fairly quickly [[William Lawrence Bragg]] and his father [[William Henry Bragg]] were able to solve previously unknown structures.<ref>{{cite journal |vauthors=Bragg WL |date=1913 |title=The Structure of Some Crystals as Indicated by their Diffraction of X-rays |journal=Proc. R. Soc. Lond. |volume=A89 |issue=610 |pages=248–277 |bibcode=1913RSPSA..89..248B |doi=10.1098/rspa.1913.0083 |jstor=93488 |doi-access=free}}</ref><ref>{{cite journal |vauthors=Bragg WL, James RW, Bosanquet CH |date=1921 |title=The Intensity of Reflexion of X-rays by Rock-Salt |url=https://zenodo.org/record/1430965 |url-status=live |journal=Phil. Mag. |volume=41 |issue=243 |page=309 |doi=10.1080/14786442108636225 |archive-url=https://web.archive.org/web/20200329132638/https://zenodo.org/record/1430965 |archive-date=2020-03-29 |access-date=2019-09-10}}</ref><ref>{{cite journal |vauthors=Bragg WL, James RW, Bosanquet CH |date=1921 |title=The Intensity of Reflexion of X-rays by Rock-Salt. Part II |url=https://zenodo.org/record/1430951 |url-status=live |journal=Phil. Mag. |volume=42 |issue=247 |page=1 |doi=10.1080/14786442108633730 |archive-url=https://web.archive.org/web/20200329132627/https://zenodo.org/record/1430951 |archive-date=2020-03-29 |access-date=2019-09-10}}</ref> Building on this, it became clear that most of the simple elemental metals had [[close packed]] structures. With this determined the concept of [[Dislocation|dislocations]] originally developed by [[Vito Volterra]] in 1907<ref>[[Vito Volterra]] (1907) [https://eudml.org/doc/81250 "Sur l'équilibre des corps élastiques multiplement connexes"], ''Annales Scientifiques de l'École Normale Supérieure'', Vol. 24, pp. 401–517</ref> became accepted, for instance being used to explain the ductility of metals by [[G. I. Taylor]] in 1934.<ref>{{cite journal |author=G. I. Taylor |author-link=G. I. Taylor |year=1934 |title=The Mechanism of Plastic Deformation of Crystals. Part I. Theoretical |journal=Proceedings of the Royal Society of London. Series A |volume=145 |issue=855 |pages=362–87 |bibcode=1934RSPSA.145..362T |doi=10.1098/rspa.1934.0106 |jstor=2935509 |doi-access=free}}</ref>

The second was the advent of quantum mechanics. In 1924 [[Louis de Broglie]] in his PhD thesis ''Recherches sur la théorie des quanta''<ref name="Broglie">{{cite web |last1=de Broglie |first1=Louis Victor |title=On the Theory of Quanta |url=https://fondationlouisdebroglie.org/LDB-oeuvres/De_Broglie_Kracklauer.pdf |access-date=25 February 2023 |website=Foundation of Louis de Broglie |edition=English translation by A.F. Kracklauer, 2004.}}</ref> introduced his theory of [[electron]] waves. This rapidly became part of what was called by [[Erwin Schrödinger]] ''undulatory mechanics'',<ref name="Schroedinger">{{Cite journal |last=Schrödinger |first=E. |date=1926 |title=An Undulatory Theory of the Mechanics of Atoms and Molecules |url=https://link.aps.org/doi/10.1103/PhysRev.28.1049 |journal=Physical Review |language=en |volume=28 |issue=6 |pages=1049–1070 |bibcode=1926PhRv...28.1049S |doi=10.1103/PhysRev.28.1049 |issn=0031-899X}}</ref> now called the [[Schrödinger equation]], wave mechanics or more commonly in contemporary usage [[quantum mechanics]]. While it was not so easy to solve the mathematics in the early days, fairly rapidly ideas such as the [[chemical bond]] terminology of [[Linus Pauling]]<ref>{{Cite book |last=Pauling |first=Linus |title=The nature of the chemical bond and the structure of molecules and crystals: an introduction to modern structural chemistry |date=2010 |publisher=Cornell Univ. Press |isbn=978-0-8014-0333-0 |edition=3. ed., 17. print |location=Ithaca, NY}}</ref> as well as [[electronic band structure]] concepts were developed.<ref name=":0">{{Cite book |last1=Ashcroft |first1=Neil W. |title=Solid state physics |last2=Mermin |first2=N. David |date=1976 |publisher=Saunders college publ |isbn=978-0-03-083993-1 |location=Fort Worth Philadelphia San Diego [etc.]}}</ref>{{Band structure filling diagram}}From this the concept of nonmetals as "not-a-metal" originates. The original approach to describe metals and nonmetals was a band-structure with [[Delocalized electron|delocalized electrons]] (i.e. spread out in space). A nonmetal has a [[Band gap|gap]] in the [[Band structure|energy levels]] of the electrons at the [[Fermi level]].<ref name=":0" />{{Rp|location=Chpt 8 & 19}} In contrast, a metal would have at least one  partially occupied band at the Fermi level;<ref name=":0" /> in a semiconductor or insulator there are no delocalized states at the Fermi level, see for instance [[Ashcroft and Mermin]].<ref name=":0" /> (A [[semimetal]] is similar to a metal, with a slightly more complex band structure.) These definitions are equivalent to stating that metals conduct electricity at [[absolute zero]], as suggested by [[Nevill Francis Mott]],<ref name=":3">{{Cite book |last=Yonezawa |first=Fumiko |title=Physics of metal-nonmetal transitions |date=2017 |publisher=IOS Press |isbn=978-1-61499-786-3 |location=Washington, DC |pages= |quote=}}</ref>{{Rp|page=257}} and the equivalent definition at other temperatures is also commonly used as in textbooks such as ''Chemistry of the Non-Metals'' by [[Ralf Steudel]]<ref>{{Cite book |last=Steudel |first=Ralf |url=https://www.degruyter.com/document/doi/10.1515/9783110578065/html |title=Chemistry of the Non-Metals: Syntheses - Structures - Bonding - Applications |date=2020 |publisher=De Gruyter |isbn=978-3-11-057806-5 |pages=154 |doi=10.1515/9783110578065}}</ref> and work on [[Metal–insulator transition|metal–insulator transitions]].<ref>{{Cite journal |last=MOTT |first=N. F. |date=1968-10-01 |title=Metal-Insulator Transition |url=https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.40.677 |journal=Reviews of Modern Physics |volume=40 |issue=4 |pages=677–683 |doi=10.1103/RevModPhys.40.677|bibcode=1968RvMP...40..677M }}</ref><ref>{{Cite journal |last1=Imada |first1=Masatoshi |last2=Fujimori |first2=Atsushi |last3=Tokura |first3=Yoshinori |date=1998-10-01 |title=Metal-insulator transitions |url=https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.70.1039 |journal=Reviews of Modern Physics |volume=70 |issue=4 |pages=1039–1263 |doi=10.1103/RevModPhys.70.1039|bibcode=1998RvMP...70.1039I }}</ref>

Originally<ref>{{Cite journal |last1=Wilson |first1=A. H. |date=1931 |title=The theory of electronic semi-conductors |journal=Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character |language=en |volume=133 |issue=822 |pages=458–491 |bibcode=1931RSPSA.133..458W |doi=10.1098/rspa.1931.0162 |issn=0950-1207 |doi-access=free}}</ref><ref>{{Cite journal |last1=Wilson |first1=A. H. |date=1931 |title=The theory of electronic semi-conductors. - II |journal=Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character |language=en |volume=134 |issue=823 |pages=277–287 |bibcode=1931RSPSA.134..277W |doi=10.1098/rspa.1931.0196 |issn=0950-1207 |doi-access=free}}</ref> this band structure interpretation was based upon a single-electron approach with the Fermi level in the band gap as illustrated in the Figure, not including a complete picture of the [[many-body problem]] where both [[Exchange interaction|exchange]] and [[Electronic correlation|correlation]] terms matter, as well as [[Relativistic effect|relativistic effects]] such as [[spin-orbit coupling]]. For instance, the passivity of gold is typically associated with relativistic terms.<ref>{{Cite journal |last=Pyykkö |first=Pekka |date=2012-05-05 |title=Relativistic Effects in Chemistry: More Common Than You Thought |url=https://www.annualreviews.org/content/journals/10.1146/annurev-physchem-032511-143755 |journal=Annual Review of Physical Chemistry |language=en |volume=63 |issue=1 |pages=45–64 |doi=10.1146/annurev-physchem-032511-143755 |pmid=22404585 |bibcode=2012ARPC...63...45P |issn=0066-426X}}</ref> A key addition by Mott and [[Rudolf Peierls]] was that these could not be ignored.<ref>{{Cite journal |last1=Mott |first1=N F |last2=Peierls |first2=R |date=1937 |title=Discussion of the paper by de Boer and Verwey |url=https://iopscience.iop.org/article/10.1088/0959-5309/49/4S/308 |journal=Proceedings of the Physical Society |volume=49 |issue=4S |pages=72–73 |bibcode=1937PPS....49...72M |doi=10.1088/0959-5309/49/4S/308 |issn=0959-5309}}</ref> For instance, [[Nickel(II) oxide|nickel oxide]] would be a metal if a single-electron approach was used, but in fact has quite a large band gap.<ref>{{Cite journal |last1=Boer |first1=J H de |last2=Verwey |first2=E J W |date=1937 |title=Semi-conductors with partially and with completely filled 3 d -lattice bands |url=https://iopscience.iop.org/article/10.1088/0959-5309/49/4S/307 |journal=Proceedings of the Physical Society |volume=49 |issue=4S |pages=59–71 |bibcode=1937PPS....49...59B |doi=10.1088/0959-5309/49/4S/307 |issn=0959-5309}}</ref> As of 2024 it is more common to use an approach based upon [[density functional theory]] where the many-body terms are included.<ref>{{Cite web |last=Burke |first=Kieron |date=2007 |title=The ABC of DFT |url=https://dft.uci.edu/doc/g1.pdf}}</ref><ref>{{Cite book |last1=Gross |first1=Eberhard K. U. |url=https://books.google.com/books?id=aG4ECAAAQBAJ&q=density+functional+theory |title=Density Functional Theory |last2=Dreizler |first2=Reiner M. |date=2013 |publisher=Springer Science & Business Media |isbn=978-1-4757-9975-0 |language=en}}</ref>  Although accurate calculations remain a challenge, reasonable results are now available in many cases.<ref>{{Cite journal |last1=Ferreira |first1=Luiz G. |last2=Marques |first2=Marcelo |last3=Teles |first3=Lara K. |date=2008 |title=Approximation to density functional theory for the calculation of band gaps of semiconductors |url=https://link.aps.org/doi/10.1103/PhysRevB.78.125116 |journal=Physical Review B |language=en |volume=78 |issue=12 |page=125116 |arxiv=0808.0729 |bibcode=2008PhRvB..78l5116F |doi=10.1103/PhysRevB.78.125116 |issn=1098-0121}}</ref><ref>{{Cite journal |last1=Tran |first1=Fabien |last2=Blaha |first2=Peter |date=2017 |title=Importance of the Kinetic Energy Density for Band Gap Calculations in Solids with Density Functional Theory |journal=The Journal of Physical Chemistry A |language=en |volume=121 |issue=17 |pages=3318–3325 |bibcode=2017JPCA..121.3318T |doi=10.1021/acs.jpca.7b02882 |issn=1089-5639 |pmc=5423078 |pmid=28402113}}</ref>

It is common to nuance the early definition of [[Alan Herries Wilson]] and Mott which was for a  zero temperature. As discussed by [[Peter Edwards (chemist)|Peter Edwards]] and colleagues,<ref>{{Cite journal |last1=Edwards |first1=P. P. |last2=Lodge |first2=M. T. J. |last3=Hensel |first3=F. |last4=Redmer |first4=R. |date=2010 |title='… a metal conducts and a non-metal doesn't' |journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |language=en |volume=368 |issue=1914 |pages=941–965 |bibcode=2010RSPTA.368..941E |doi=10.1098/rsta.2009.0282 |issn=1364-503X |pmc=3263814 |pmid=20123742}}</ref> as well as [[Fumiko Yonezawa]],<ref name=":3" />{{Rp|pages=257–261}}it is important to consider the temperatures at which both metals and nonmetals are used. Yonezawa provides a general definition for both general temperatures and conditions (such as standard temperature and pressure):<ref name=":3" />{{Rp|page=260}}

{{block quote|text=When a material conducts and at the same time the temperature coefficient of the electric conductivity of that material is not positive under a certain environmental condition, the material is metallic under that environmental condition. A material which does not satisfy these requirements is not metallic under that environmental condition.}}

The precise meaning of semiconductor needs a little care. In terms of the temperature dependence of their conductivity they are all classified as insulators; the pure forms are [[Intrinsic semiconductor|intrinsic semiconductors]]. When they are doped their conductivity continues to increase with temperature,<ref name=":0" /> and can become substantial; hence the ambiguities with an empirical categorisation using conductivity described earlier. Indeed, some elements that are normally considered as insulators have been exploited as semiconductors. For instance diamond, which has the largest band gap of the elements that are solids under normal conditions,<ref>{{Cite journal |last1=Strehlow |first1=W. H. |last2=Cook |first2=E. L. |date=1973-01-01 |title=Compilation of Energy Band Gaps in Elemental and Binary Compound Semiconductors and Insulators |url=https://pubs.aip.org/aip/jpr/article-abstract/2/1/163/241551/Compilation-of-Energy-Band-Gaps-in-Elemental-and?redirectedFrom=fulltext |journal=Journal of Physical and Chemical Reference Data |volume=2 |issue=1 |pages=163–200 |doi=10.1063/1.3253115 |bibcode=1973JPCRD...2..163S |issn=0047-2689}}</ref> has a number of semiconductor applications.<ref>{{Cite journal |last=Collins |first=Alan T. |date=1997 |title=The optical and electronic properties of semiconducting diamond |url=https://royalsocietypublishing.org/doi/10.1098/rsta.1993.0017 |journal=Philosophical Transactions of the Royal Society of London, Series A |volume=342 |issue=1664 |pages=233–244 |doi=10.1098/rsta.1993.0017}}</ref><ref>{{Cite journal |last=Umezawa |first=Hitoshi |date=2018-05-01 |title=Recent advances in diamond power semiconductor devices |url=https://linkinghub.elsevier.com/retrieve/pii/S1369800117322217 |journal=Materials Science in Semiconductor Processing |series=Wide band gap semiconductors technology for next generation of energy efficient power electronics |volume=78 |pages=147–156 |doi=10.1016/j.mssp.2018.01.007 |issn=1369-8001}}</ref>

Band structure definitions of metals and nonmetals are widely used in current research into materials, and apply both to single elements such as insulating boron<ref>{{Cite journal |last1=Ogitsu |first1=Tadashi |last2=Schwegler |first2=Eric |last3=Galli |first3=Giulia |date=2013 |title=β-Rhombohedral Boron: At the Crossroads of the Chemistry of Boron and the Physics of Frustration |url=https://pubs.acs.org/doi/10.1021/cr300356t |journal=Chemical Reviews |language=en |volume=113 |issue=5 |pages=3425–3449 |doi=10.1021/cr300356t |issn=0009-2665 |osti=1227014 |pmid=23472640}}</ref> as well as compounds such as [[strontium titanate]].<ref>{{Cite journal |last1=Reihl |first1=B. |last2=Bednorz |first2=J. G. |last3=Müller |first3=K. A. |last4=Jugnet |first4=Y. |last5=Landgren |first5=G. |last6=Morar |first6=J. F. |date=1984 |title=Electronic structure of strontium titanate |url=https://link.aps.org/doi/10.1103/PhysRevB.30.803 |journal=Physical Review B |language=en |volume=30 |issue=2 |pages=803–806 |bibcode=1984PhRvB..30..803R |doi=10.1103/PhysRevB.30.803 |issn=0163-1829}}</ref> The characteristics associated with metals and nonmetals in early work such as their appearance and mechanical properties are now understood to be consequences of how the atoms and electrons are arranged.