Physics:Quantum Band structure - Revision history

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	The '''electronic band structure''' (or simply '''band structure''') of a [[solid]] describes the range of [[energy level]]s that electrons may have within it, as well as the ranges of energy that they may not have, called ''[[band gap]]s'' or forbidden bands. Band theory explains these bands and gaps by examining the allowed quantum-mechanical [[wave function]]s for an electron in a large, periodic lattice of atoms or molecules. It provides the foundation for understanding the electrical and optical properties of solids and underlies the operation of ~~[[Solid-state (electronics)\|~~solid-state devices]] such as transistors and solar cells.<ref>{{Cite book \|last=Simon \|first=Steven H. \|url=http://archive.org/details/oxfordsolidstate0000simo \|title=The Oxford Solid State Basics \|date=2013 \|publisher=Oxford University Press \|location=Oxford \|isbn=978-0-19-150210-1}}</ref><ref name="girvin">{{Cite book \|last1=Girvin \|first1=Steven M. \|last2=Yang \|first2=Kun \|title=Modern Condensed Matter Physics \|date=2019 \|publisher=Cambridge University Press \|isbn=978-1-107-13739-4 \|location=Cambridge}}</ref>		The '''electronic band structure''' (or simply '''band structure''') of a [[solid]] describes the range of [[energy level]]s that electrons may have within it, as well as the ranges of energy that they may not have, called ''[[band gap]]s'' or forbidden bands. Band theory explains these bands and gaps by examining the allowed quantum-mechanical [[wave function]]s for an electron in a large, periodic lattice of atoms or molecules. It provides the foundation for understanding the electrical and optical properties of solids and underlies the operation of solid-state devices such as transistors and solar cells.<ref>{{Cite book \|last=Simon \|first=Steven H. \|url=http://archive.org/details/oxfordsolidstate0000simo \|title=The Oxford Solid State Basics \|date=2013 \|publisher=Oxford University Press \|location=Oxford \|isbn=978-0-19-150210-1}}</ref><ref name="girvin">{{Cite book \|last1=Girvin \|first1=Steven M. \|last2=Yang \|first2=Kun \|title=Modern Condensed Matter Physics \|date=2019 \|publisher=Cambridge University Press \|isbn=978-1-107-13739-4 \|location=Cambridge}}</ref>

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	* electrons moving in an average static potential, without fully treating interactions with other electrons, phonons, or photons.		* electrons moving in an average static potential, without fully treating interactions with other electrons, phonons, or photons.

	These assumptions can break down in important cases. Near surfaces, interfaces, and junctions, the bulk band structure is altered and effects such as [[surface states]], [[dopant]] states, and [[band bending]] become important. In very small systems such as molecules or [[quantum dot]]s, the concept of a continuous band structure no longer applies. Strongly correlated materials such as [[Mott ~~insulator]]s~~ also fall outside the normal single-electron band picture.<ref name="girvin" />		These assumptions can break down in important cases. Near surfaces, interfaces, and junctions, the bulk band structure is altered and effects such as [[surface states]], [[dopant]] states, and [[band bending]] become important. In very small systems such as molecules or [[quantum dot]]s, the concept of a continuous band structure no longer applies. Strongly correlated materials such as Mott insulatorss also fall outside the normal single-electron band picture.<ref name="girvin" />

	=== Crystalline symmetry and wavevectors ===		=== Crystalline symmetry and wavevectors ===
	For a crystal, band structure calculations take advantage of lattice periodicity. The single-electron [[Schrödinger equation]] in a periodic potential has solutions of the Bloch form		For a crystal, band structure calculations take advantage of lattice periodicity. The single-electron Schrödinger equation in a periodic potential has solutions of the Bloch form
	<math display="block">\psi_{n\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}} u_{n\mathbf{k}}(\mathbf{r}),</math>		<math display="block">\psi_{n\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}} u_{n\mathbf{k}}(\mathbf{r}),</math>
	where {{math\|'''k'''}} is the wavevector and {{math\|''n''}} labels the band. For each value of {{math\|'''k'''}}, there are multiple allowed energies, and these vary smoothly with {{math\|'''k'''}} to produce the dispersion relation {{math\|''E''<sub>''n''</sub>('''k''')}}.<ref name=Kittel>{{cite book		where {{math\|'''k'''}} is the wavevector and {{math\|''n''}} labels the band. For each value of {{math\|'''k'''}}, there are multiple allowed energies, and these vary smoothly with {{math\|'''k'''}} to produce the dispersion relation {{math\|''E''<sub>''n''</sub>('''k''')}}.<ref name=Kittel>{{cite book
	\|author=Charles Kittel		\|author=Charles Kittel
	\|title=[[Introduction to Solid State Physics ~~(Kittel book)\|Introduction to Solid State Physics]]~~		\|title=Introduction to Solid State Physics
	\|year=1996		\|year=1996
	\|edition=Seventh		\|edition=Seventh
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	}}</ref>		}}</ref>

	The relevant wavevectors lie inside the [[Brillouin zone]], the fundamental region of reciprocal space. Band structure plots typically show energy versus wavevector along lines connecting high-symmetry points such as Γ, X, L, or K. These diagrams make it possible to distinguish, for example, between [[direct band gap]] and [[indirect band gap]] materials.<ref>{{Cite web \| url=http://www.ioffe.ru/SVA/NSM/Semicond/AlGaAs/bandstr.html \| title=NSM Archive - Aluminium Gallium Arsenide (AlGaAs) - Band structure and carrier concentration \| website=www.ioffe.ru }}</ref><ref name="SpringerBandStructure">{{cite web\|title=Electronic Band Structure\|url=https://web.archive.org/web/20180117001810/https://www.springer.com/cda/content/document/cda_downloaddocument/9783642007095-c1.pdf?SGWID=0-0-45-898341-p173918216\|website=www.springer.com\|publisher=Springer\|page=24}}</ref>		The relevant wavevectors lie inside the Brillouin zone, the fundamental region of reciprocal space. Band structure plots typically show energy versus wavevector along lines connecting high-symmetry points such as Γ, X, L, or K. These diagrams make it possible to distinguish, for example, between [[direct band gap]] and [[indirect band gap]] materials.<ref>{{Cite web \| url=http://www.ioffe.ru/SVA/NSM/Semicond/AlGaAs/bandstr.html \| title=NSM Archive - Aluminium Gallium Arsenide (AlGaAs) - Band structure and carrier concentration \| website=www.ioffe.ru }}</ref><ref name="SpringerBandStructure">{{cite web\|title=Electronic Band Structure\|url=https://web.archive.org/web/20180117001810/https://www.springer.com/cda/content/document/cda_downloaddocument/9783642007095-c1.pdf?SGWID=0-0-45-898341-p173918216\|website=www.springer.com\|publisher=Springer\|page=24}}</ref>

	[[File:Brillouin Zone (1st, FCC).svg\|thumb\|300px\|[[Brillouin zone]] of a [[face-centered cubic lattice]], showing several special symmetry points.]]		[[File:Brillouin Zone (1st, FCC).svg\|thumb\|300px\|Brillouin zone of a [[face-centered cubic lattice]], showing several special symmetry points.]]

	=== Density of states ===		=== Density of states ===
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	=== Filling of bands ===		=== Filling of bands ===
	At thermal equilibrium, the probability that a state of energy {{math\|''E''}} is occupied is given by the ~~[[Fermi–Dirac statistics\|~~Fermi–Dirac distribution]]		At thermal equilibrium, the probability that a state of energy {{math\|''E''}} is occupied is given by the Fermi–Dirac distribution
	<math display="block">f(E) = \frac{1}{1 + e^{(E-\mu)/(k_\text{B} T)}}</math>		<math display="block">f(E) = \frac{1}{1 + e^{(E-\mu)/(k_\text{B} T)}}</math>
	where {{math\|''\mu''}} is the chemical potential, usually called the [[Fermi level]], and {{math\|''k''<sub>B</sub>''T''}} is the thermal energy. The number of electrons per unit volume is obtained by integrating the occupied density of states:		where {{math\|''\mu''}} is the chemical potential, usually called the Fermi level, and {{math\|''k''<sub>B</sub>''T''}} is the thermal energy. The number of electrons per unit volume is obtained by integrating the occupied density of states:
	<math display="block">N/V = \int_{-\infty}^{\infty} g(E) f(E)\, dE</math>		<math display="block">N/V = \int_{-\infty}^{\infty} g(E) f(E)\, dE</math>

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	=== Tight binding model ===		=== Tight binding model ===
	The [[tight binding]] approach assumes that electrons remain largely localized around atoms and that crystal wavefunctions can be approximated as linear combinations of atomic orbitals. This model works especially well in systems where orbital overlap is limited, such as [[silicon]], [[gallium arsenide]], [[diamond]], and many insulators. A related and more refined formulation uses [[Wannier ~~function]]s~~.<ref name=Kittel /><ref name=Mattis>{{cite book \| author=Daniel Charles Mattis \| title=The Many-Body Problem: Encyclopaedia of Exactly Solved Models in One Dimension \| year = 1994 \| publisher=World Scientific \| page=340 \| isbn=978-981-02-1476-0 \| url=https://books.google.com/books?id=BGdHpCAMiLgC&pg=PA332}}</ref><ref name=Harrison>{{cite book		The [[tight binding]] approach assumes that electrons remain largely localized around atoms and that crystal wavefunctions can be approximated as linear combinations of atomic orbitals. This model works especially well in systems where orbital overlap is limited, such as [[silicon]], [[gallium arsenide]], [[diamond]], and many insulators. A related and more refined formulation uses Wannier functionss.<ref name=Kittel /><ref name=Mattis>{{cite book \| author=Daniel Charles Mattis \| title=The Many-Body Problem: Encyclopaedia of Exactly Solved Models in One Dimension \| year = 1994 \| publisher=World Scientific \| page=340 \| isbn=978-981-02-1476-0 \| url=https://books.google.com/books?id=BGdHpCAMiLgC&pg=PA332}}</ref><ref name=Harrison>{{cite book
	\|author=Walter Ashley Harrison		\|author=Walter Ashley Harrison
	\|title=Electronic Structure and the Properties of Solids		\|title=Electronic Structure and the Properties of Solids
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	=== KKR model ===		=== KKR model ===
	The [[Korringa–Kohn–Rostoker method]] (KKR), also known as multiple scattering theory, reformulates the problem using scattering matrices and Green's functions. It is particularly useful for alloys and disordered systems, and often uses the ''muffin-tin'' approximation for the crystal potential.<ref name=Galsin>{{cite book \|title=Impurity Scattering in Metal Alloys \|author=Joginder Singh Galsin \|page=Appendix C \|url=https://books.google.com/books?id=kmcLT63iX_EC&pg=PA498 \|isbn=978-0-306-46574-1 \|year=2001 \|publisher=Springer}}</ref><ref name=Ohtaka>{{cite book \|title=Photonic Crystals \|author=Kuon Inoue, Kazuo Ohtaka \|page=66 \|url=https://books.google.com/books?id=GIa3HRgPYhAC&pg=PA66 \|isbn=978-3-540-20559-3 \|year=2004 \|publisher=Springer}}</ref>		The Korringa–Kohn–Rostoker method (KKR), also known as multiple scattering theory, reformulates the problem using scattering matrices and Green's functions. It is particularly useful for alloys and disordered systems, and often uses the ''muffin-tin'' approximation for the crystal potential.<ref name=Galsin>{{cite book \|title=Impurity Scattering in Metal Alloys \|author=Joginder Singh Galsin \|page=Appendix C \|url=https://books.google.com/books?id=kmcLT63iX_EC&pg=PA498 \|isbn=978-0-306-46574-1 \|year=2001 \|publisher=Springer}}</ref><ref name=Ohtaka>{{cite book \|title=Photonic Crystals \|author=Kuon Inoue, Kazuo Ohtaka \|page=66 \|url=https://books.google.com/books?id=GIa3HRgPYhAC&pg=PA66 \|isbn=978-3-540-20559-3 \|year=2004 \|publisher=Springer}}</ref>

	=== Density-functional theory ===		=== Density-functional theory ===
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	=== Green's function and GW methods ===		=== Green's function and GW methods ===
	To include many-body effects more accurately, one can use ~~[[Green's function (many-body theory)\|~~Green's function]] methods. In these approaches, the poles of the Green's function correspond to the [[quasiparticle]] energies of the solid. The [[GW approximation]] is especially important because it often corrects the systematic band-gap underestimation of standard DFT and gives results in closer agreement with experiment.		To include many-body effects more accurately, one can use Green's function methods. In these approaches, the poles of the Green's function correspond to the [[quasiparticle]] energies of the solid. The GW approximation is especially important because it often corrects the systematic band-gap underestimation of standard DFT and gives results in closer agreement with experiment.

	=== Dynamical mean-field theory ===		=== Dynamical mean-field theory ===
	Some materials, such as [[Mott ~~insulator]]s~~, cannot be understood in terms of ordinary single-electron bands. In these cases, strong electron-electron interactions dominate. Methods such as the [[Hubbard model]] and [[dynamical mean-field theory]] are used to describe the resulting behaviour.		Some materials, such as Mott insulatorss, cannot be understood in terms of ordinary single-electron bands. In these cases, strong electron-electron interactions dominate. Methods such as the Hubbard model and [[dynamical mean-field theory]] are used to describe the resulting behaviour.

	== Band diagrams ==		== Band diagrams ==

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	The '''electronic band structure''' (or simply '''band structure''') of a [[solid]] describes the range of [[energy level]]s that electrons may have within it, as well as the ranges of energy that they may not have, called ''[[band gap]]s'' or forbidden bands. Band theory explains these bands and gaps by examining the allowed quantum-mechanical [[wave function]]s for an electron in a large, periodic lattice of atoms or molecules. It provides the foundation for understanding the electrical and optical properties of solids and underlies the operation of [[Solid-state (electronics)\|solid-state devices]] such as transistors and solar cells.<ref>{{Cite book \|last=Simon \|first=Steven H. \|url=http://archive.org/details/oxfordsolidstate0000simo \|title=The Oxford Solid State Basics \|date=2013 \|publisher=Oxford University Press \|location=Oxford \|isbn=978-0-19-150210-1}}</ref><ref name="girvin">{{Cite book \|last1=Girvin \|first1=Steven M. \|last2=Yang \|first2=Kun \|title=Modern Condensed Matter Physics \|date=2019 \|publisher=Cambridge University Press \|isbn=978-1-107-13739-4 \|location=Cambridge}}</ref>		The '''electronic band structure''' (or simply '''band structure''') of a [[solid]] describes the range of [[energy level]]s that electrons may have within it, as well as the ranges of energy that they may not have, called ''[[band gap]]s'' or forbidden bands. Band theory explains these bands and gaps by examining the allowed quantum-mechanical [[wave function]]s for an electron in a large, periodic lattice of atoms or molecules. It provides the foundation for understanding the electrical and optical properties of solids and underlies the operation of [[Solid-state (electronics)\|solid-state devices]] such as transistors and solar cells.<ref>{{Cite book \|last=Simon \|first=Steven H. \|url=http://archive.org/details/oxfordsolidstate0000simo \|title=The Oxford Solid State Basics \|date=2013 \|publisher=Oxford University Press \|location=Oxford \|isbn=978-0-19-150210-1}}</ref><ref name="girvin">{{Cite book \|last1=Girvin \|first1=Steven M. \|last2=Yang \|first2=Kun \|title=Modern Condensed Matter Physics \|date=2019 \|publisher=Cambridge University Press \|isbn=978-1-107-13739-4 \|location=Cambridge}}</ref>

	~~<div style="float:right; background:#fff8cc; border:1px solid #e0d890; padding:8px; margin:0 0 1em 1em; width:420px; text-align:center;">~~
	<div style="font-size:90%; line-height:1.4; margin-top:6px;">A hypothetical example of band formation when a large number of carbon atoms is brought together to form a diamond crystal. As the atoms approach one another, atomic orbitals split into many closely spaced levels that merge into bands. At the diamond lattice spacing, two bands are separated by a band gap of about 5.5 eV.</div>
	~~</div>~~
	</div>		</div>

	<div style="width:300px;">		<div style="width:300px;">
	[[File:Solid state electronic band structure.svg\|thumb\|280px\|~~Quantum Band structure~~.]]		[[File:Solid state electronic band structure.svg\|thumb\|280px\|A hypothetical example of band formation when a large number of carbon atoms is brought together to form a diamond crystal. As the atoms approach one another, atomic orbitals split into many closely spaced levels that merge into bands. At the diamond lattice spacing, two bands are separated by a band gap of about 5.5 eV.]]
	</div>		</div>

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			{{Short description\|Quantum Collection topic on Quantum Band structure}}

	{{Quantum book backlink\|Condensed matter and solid-state physics}}		{{Quantum book backlink\|Condensed matter and solid-state physics}}

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			__TOC__
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	The '''electronic band structure''' (or simply '''band structure''') of a [[solid]] describes the range of [[energy level]]s that electrons may have within it, as well as the ranges of energy that they may not have, called ''[[band gap]]s'' or forbidden bands. Band theory explains these bands and gaps by examining the allowed quantum-mechanical [[wave function]]s for an electron in a large, periodic lattice of atoms or molecules. It provides the foundation for understanding the electrical and optical properties of solids and underlies the operation of [[Solid-state (electronics)\|solid-state devices]] such as transistors and solar cells.<ref>{{Cite book \|last=Simon \|first=Steven H. \|url=http://archive.org/details/oxfordsolidstate0000simo \|title=The Oxford Solid State Basics \|date=2013 \|publisher=Oxford University Press \|location=Oxford \|isbn=978-0-19-150210-1}}</ref><ref name="girvin">{{Cite book \|last1=Girvin \|first1=Steven M. \|last2=Yang \|first2=Kun \|title=Modern Condensed Matter Physics \|date=2019 \|publisher=Cambridge University Press \|isbn=978-1-107-13739-4 \|location=Cambridge}}</ref>		The '''electronic band structure''' (or simply '''band structure''') of a [[solid]] describes the range of [[energy level]]s that electrons may have within it, as well as the ranges of energy that they may not have, called ''[[band gap]]s'' or forbidden bands. Band theory explains these bands and gaps by examining the allowed quantum-mechanical [[wave function]]s for an electron in a large, periodic lattice of atoms or molecules. It provides the foundation for understanding the electrical and optical properties of solids and underlies the operation of [[Solid-state (electronics)\|solid-state devices]] such as transistors and solar cells.<ref>{{Cite book \|last=Simon \|first=Steven H. \|url=http://archive.org/details/oxfordsolidstate0000simo \|title=The Oxford Solid State Basics \|date=2013 \|publisher=Oxford University Press \|location=Oxford \|isbn=978-0-19-150210-1}}</ref><ref name="girvin">{{Cite book \|last1=Girvin \|first1=Steven M. \|last2=Yang \|first2=Kun \|title=Modern Condensed Matter Physics \|date=2019 \|publisher=Cambridge University Press \|isbn=978-1-107-13739-4 \|location=Cambridge}}</ref>

	<div style="float:right; background:#fff8cc; border:1px solid #e0d890; padding:8px; margin:0 0 1em 1em; width:420px; text-align:center;">		<div style="float:right; background:#fff8cc; border:1px solid #e0d890; padding:8px; margin:0 0 1em 1em; width:420px; text-align:center;">
	[[File:Solid state electronic band structure.svg\|400px\|A hypothetical example of band formation when a large number of carbon atoms is brought together to form a diamond crystal. As the atoms approach one another, atomic orbitals split into many closely spaced levels that merge into bands. At the diamond lattice spacing, two bands are separated by a band gap of about 5.5 eV.]]
	<div style="font-size:90%; line-height:1.4; margin-top:6px;">A hypothetical example of band formation when a large number of carbon atoms is brought together to form a diamond crystal. As the atoms approach one another, atomic orbitals split into many closely spaced levels that merge into bands. At the diamond lattice spacing, two bands are separated by a band gap of about 5.5 eV.</div>		<div style="font-size:90%; line-height:1.4; margin-top:6px;">A hypothetical example of band formation when a large number of carbon atoms is brought together to form a diamond crystal. As the atoms approach one another, atomic orbitals split into many closely spaced levels that merge into bands. At the diamond lattice spacing, two bands are separated by a band gap of about 5.5 eV.</div>
	</div>		</div>
			</div>

			<div style="width:300px;">
			[[File:Solid state electronic band structure.svg\|thumb\|280px\|Quantum Band structure.]]
			</div>

			</div>

	== Introduction ==		== Introduction ==
	In an isolated atom, electrons occupy discrete [[atomic orbital]]s with specific energies. When many identical atoms are brought together in a crystal, these orbitals overlap and split into a very large number of closely spaced levels. Since a macroscopic solid contains an enormous number of atoms, the levels become so densely packed that they can be treated as continuous energy bands.<ref name="Holgate">{{cite book		In an isolated atom, electrons occupy discrete [[atomic orbital]]s with specific energies. When many identical atoms are brought together in a crystal, these orbitals overlap and split into a very large number of closely spaced levels. Since a macroscopic solid contains an enormous number of atoms, the levels become so densely packed that they can be treated as continuous energy bands.<ref name="Holgate">{{cite book

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	{{Quantum book backlink\|Condensed matter and solid-state physics}}		{{Quantum book backlink\|Condensed matter and solid-state physics}}
	{{Quantum article nav\|previous=Physics:Quantum Thermodynamics\|previous label=Thermodynamics\|next=Physics:Quantum Fermi surfaces\|next label=Fermi surfaces}}		{{Quantum article nav\|previous=Physics:Quantum Thermodynamics\|previous label=Thermodynamics\|next=Physics:Quantum Fermi surfaces\|next label=Fermi surfaces}}